Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Novel surface energy sensor for detecting protein adsorption and subsequent conformation change Clark, Alison Jane 2004

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2005-994546.pdf [ 17.33MB ]
Metadata
JSON: 831-1.0085812.json
JSON-LD: 831-1.0085812-ld.json
RDF/XML (Pretty): 831-1.0085812-rdf.xml
RDF/JSON: 831-1.0085812-rdf.json
Turtle: 831-1.0085812-turtle.txt
N-Triples: 831-1.0085812-rdf-ntriples.txt
Original Record: 831-1.0085812-source.json
Full Text
831-1.0085812-fulltext.txt
Citation
831-1.0085812.ris

Full Text

N O V E L SURFACE E N E R G Y SENSOR FOR DETECTING PROTEIN ADSORPTION AND SUBSEQUENT CONFORMATION C H A N G E by ALISON JANE C L A R K B.Sc, University of Waterloo, 1993 M.Sc., University of British Columbia, 1997 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Physics.and Astronomy We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH C OLU M B IA OCTOBER, 2004 © Alison Jane Clark, 2004 ABSTRACT This thesis describes the development of a novel experimental technique to measure the change in surface energy of a sensing membrane as molecules adsorb to its surface. The sensor is constructed from a thin elastomeric membrane mounted on an annular support that is immersed in an aqueous solution. The sensor is acoustically actuated to resonate in a selected mode of oscillation and the change of resonant frequency of the sensor as a function of time is monitored. These sensing membranes have a low inherent tension such that the surface energy on its two interfaces dominate the membrane tension and the adsorption^of molecules decreases the surface energy and thus reduces the resonant frequency of the membrane. The adsorption behaviour of two types of molecules were investigated; surfactants and proteins. While the sensor responds quickly (~l-2 minutes) to the adsorption of the small surfactant molecules, it exhibits a long time response over many hours to the adsorption of protein molecules. This long time response is attributed to the slow conformation change of the protein molecules once they have adsorbed. An auxiliary method to measure the amount of protein molecules on the surface of the membrane was devised to run simultaneously with the observations of resonant frequency. This technique employed fluorescent excitation of tagged protein molecules in an optical evanescent field. This measurement confirmed that the population of protein molecules on the surface did indeed reach a steady-state value within 30 minutes, in turn confirming the sensitivity of the sensing membrane to molecular conformation change. A step-wise kinetic protein adsorption model was developed and compared to the experimental data generated in the simultaneous measurement described above. This model was able to successfully describe the puzzling kinetics of protein adsorption to, and desorption from, the sensing membrane. This required a key, non-obvious term in the rate equations - the molecules in the bulk solution are found to make an important contribution to both the desorption of bound molecules and also their slow conformation change. This observation provides an effective and self-consistent explanation for the previously conflicting notions of adsorption isotherms and irreversibility. TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS Hi LIST OF TABLES vi LIST OF FIGURES vii ACKNOWLEDGEMENTS xi 1 INTRODUCTION 1 1.1 DESCRIPTION OF ADSORPTION SENSOR 3 1.1.1 Resonant frequency sensor 3 1.1.2 Mass effect calculation 4 1.1.3 Surface energy 5 1.1.4 Surface fluorescence sensor 6 1.2 THESIS OVERVIEW 8 2 BACKGROUND 9 2.1 R E S O N A N T FREQUENCY SENSORS 9 2.1.1 Quartz crystal microbalance 10 2.1.2 Surface acoustic wave sensor 12 2.1.3 Flexural plate wave sensor 13 2.2 ADSORPTION A T INTERFACES 18 2.2.1 Interfacial thermodynamics 18 2.2.2 Solute adsorption at a liquid/solid interface 25 2.2.3 Properties of adsorbate molecules 29 2.2.4 Adsorption of surfactants at air/liquid and liquid/solid interfaces 36 2.2.5 Adsorption of globular proteins at solid/liquid interfaces 39 2.2.6 Experimental methods to measure and describe adsorption ; 47 3 EXPERIMENTAL INVESTIGATIONS 54 3.1 CONFIGURATION FOR RESONANT FREQUENCY M E A S U R E M E N T . : 55 3.1.1 Sensing membrane 56 3.1.2 Sensing membrane support 56 3.1.3 Acoustic transducer 58 3.1.4 Vibration sensor 59 3.1.5 Aqueous solution 61 3.2 CONFIGURATION FOR SURFACE FLUORESCENCE M E A S U R E M E N T 62 iii 3.2.1 Fluorescent tag 63 3.2.2 Laser 66 3.2.3 Waveguide construction 69 3.2.4 Waveguide holder , 71 3.2.5 Fluorescent radiation collection..:....!.... 73 3.3 FLUID FLOW A N D CONTROL 77 3.3.1 Circulation 77 3.3.2 Flushing 78 3.3.3 Temperature control 80 3.4 INSTRUMENTATION A N D CONTROL 82 3.4.1 Resonant frequency measurement 82 3.4.2 Fluorescent waveguide experiment 92 3.4.3 Simultaneous experiments 95 4 M O D E L I N G O F A D S O R P T I O N R E L A T E D T E N S I O N C H A N G E 97 4.1 M E M B R A N E TENSION CALCULATION 98 4.1.1 Experimental sensing membrane 99 4.1.2 Water loading of membrane : 99 4.2 FLUID MIXING M O D E L 103 4.2.1 Initial mixing 105 4.2.2 Flushing 105 4.2.3 Re-circulation 108 4.3 ADSORPTION M O D E L 110 4.3.1 Bulk concentration I l l 4.3.2 Diffusion across boundary layer 112 4.3.3 Initial adsorption 113 4.3.4 Surfactant adsorption model 114 4.3.5 Protein adsorption model 117 4.4 D A T A PROCESSING A N D M O D E L IMPLEMENTATION 122 4.4.1 Raw data processing 122 4.4.2 Discrete adsorption model 126 4.4.3 Model implementation 128 5 R E S U L T S A N D DISCUSSION 131 5.1 E X P E R I M E N T A L SETUP 131 5.1.1 Membrane tension measurement 131 5.1.2 Membrane leaching 135 5.1.3 Bulk concentration measurement 13 7 5.1.4 Pre-adsorbing experimental surfaces 138 5.1.5 Artifacts in surface fluorescence experiment 140 5.2 S U R F A C T A N T ADSORPTION 148 5.2.1 Typical sensor response 148 5.2.2 Comparison with surfactant adsorption model 151 5.3 PROTEIN ADSORPTION 154 iv 5.3.1 Resonant frequency measurements 154 5.3.2 Simultaneous measurement of surface fluorescence and resonant frequency 162 5.3.3 Comparison with protein adsorption model 168 6 CONCLUSIONS 177 R E F E R E N C E S 181 APPENDIX A IDEAL MEMBRANE 186 APPENDIX B TOTAL INTERNAL REFLECTION AND WAVEGUIDES... 188 APPENDIX C GOTTLIEB/AEBISCHER PROCEDURE 192 APPENDIX D "READ_DATA.C" PROGRAM 196 APPENDIX E "RATEFCN_SDS.F" PROGRAM 203 APPENDIX F " R A T E F C N L Y Z . F " PROGRAM 208 APPENDIX G MINUIT OUTPUT FOR MODEL FIT ON FIGURE 5.18 213 APPENDIX H MINUIT OUTPUT FROM MODEL FIT IN FIGURE 5.19 214 APPENDIX I MINUIT OUTPUT FROM MODEL FIT IN FIGURE 5.36 215 LIST OF TABLES T A B L E 2 - 1 P R O P E R T I E S O F A C O U ST IC W A V E S E N S O R S 1 6 T A B L E 2 - 2 P R O P E R T I E S O F S U R F A C T A N T S U S E D IN THIS S T U D Y 3 1 T A B L E 2 -3 T H E 2 0 A M I N O A C I D S A N D T H E I R A B B R E V I A T I O N S 3 2 T A B L E 2 - 4 P H Y S I O C H E M I C A L P R O P E R T I E S O F L Y S O Z Y M E 3 4 T A B L E 4 - 1 P A R A M E T E R S F O R S U R F A C T A N T A D S O R P T I O N M O D E L 1 2 9 T A B L E 4 - 2 P A R A M E T E R S F O R P R O T E I N A D S O R P T I O N M O D E L 1 3 0 T A B L E 5 -1 C A L C U L A T E D T E N S I O N P E R S Y M M E T R I C M O D E N U M B E R F R O M G O T T L I E B -A E B I S C H E R M O D E L 1 3 2 T A B L E 5 -2 C O M P A R I S O N O F M E M B R A N E T E N S I O N S F O R A I R A N D W A T E R L O A D E D C A S E S F O R 3 D I F F E R E N T M E M B R A N E T Y P E S 135 T A B L E 5-3 M O D E L F IT T O D A T A S H O W N I N F I G U R E 5 . 3 6 173 vi LIST OF FIGURES F I G U R E 1.1 T Y P I C A L P R O T E I N A D S O R P T I O N I S O T H E R M I L L U S T R A T I N G I R R E V E R S I B L E P R O T E I N A D S O R P T I O N . 2 F I G U R E l .2 R E S O N A N T F R E Q U E N C Y A D S O R P T I O N S E N S O R . . .4 F I G U R E 1.3 T E N S I O N I N A N E L A S T O M E R I C T H I N M E M B R A N E S H O W I N G C O N T R I B U T I O N F R O M S U R F A C E E N E R G Y 6 F I G U R E l .4 S U R F A C E F L U O R E S C E N C E A D S O R P T I O N S E N S O R 7 F I G U R E 2.1 Q U A R T Z C R Y S T A L M I C R O B A L A N C E G E O M E T R Y A ) I S O T R O P I C V I E W , B ) C R O S S - S E C T I O N I N X Z P L A N E S H O W I N G D I R E C T I O N O F P A R T I C L E M O T I O N 1 0 F I G U R E 2 .2 S A W S E N S O R A ) S I D E V I E W A N D B ) T O P V I E W 12 F I G U R E 2 .3 A N T I - S Y M M E T R I C A N D S Y M M E T R I C L A M B W A V E 13 F I G U R E 2 .4 M I C R O - F A B R I C A T E D F P W D E V I C E 1 4 F I G U R E 2.5 I N T E R F A C I A L R E G I O N B E T W E E N T W O P H A S E S A N D T H E G I B B S D I V I D I N G P L A N E 1 9 F I G U R E 2 .6 I N T E R F A C E B E T W E E N B U L K L I Q U I D A N D V A P O U R 2 0 F I G U R E 2 .7 D E F I N I T I O N O F C O N T A C T A N G L E F O R L I Q U I D D R O P A T S O L I D I N T E R F A C E 2 3 F I G U R E 2 .8 P O L Y D I M E T H Y L S I L O X A N E M O L E C U L E 2 4 F I G U R E 2 .9 C O M P A R I S O N O F F O U R C O M M O N I S O T H E R M T Y P E S A N D T H E I R E Q U A T I O N S 2 6 F I G U R E 2 . 1 0 S C H E M A T I C F O R T H E K I N E T I C S O F L A N G M U I R A D S O R P T I O N 2 8 F I G U R E 2 . 1 1 S O D I U M D O D E C Y L S U L F A T E S U R F A C T A N T M O L E C U L E 2 9 F I G U R E 2 . 1 2 T R I T O N X - 1 0 0 S U R F A C T A N T M O L E C U L E 3 0 F I G U R E 2 . 1 3 S U R F A C T A N T M I C E L L E F O R M A T I O N 3 0 F I G U R E 2 . 1 4 D S C T H E R M O G R A P H O F U N F O L D I N G F O R H E N E G G - W H I T E L Y S O Z Y M E 3 3 F I G U R E 2 . 1 5 L Y S O Z Y M E P R I M A R Y S T R U C T U R E S H O W I N G A M I N O A C I D S E Q U E N C E 3 5 F I G U R E 2 . 1 6 L Y S O Z Y M E A ) S E C O N D A R Y S T R U C T U R E A N D B ) T E R T I A R Y S T R U C T U R E 3 6 F I G U R E 2 . 1 7 L A N G M U I R A D S O P R T I O N I S O T H E R M F O R T R I T O N X - 1 0 0 A T T H E A I R / W A T E R I N T E R F A C E 3 7 F I G U R E 2 . 1 8 S C H E M A T I C S F O R T H E K I N E T I C S O F I R R E V E R S I B L E A D S O R P T I O N F R O M M O D I F I E D L A N G M U I R M O D E L 4 3 F I G U R E 2 . 1 9 F R A C T I O N A L S U R F A C E C O V E R A G E A S A F U N C T I O N O F T I M E F O R M O D I F I E D L A N G M U I R M O D E L S H O W N m F I G U R E 2 . 1 8 4 4 F I G U R E 2 . 2 0 M I C R O C A N T I L E V E R E D S T R E S S S E N S O R 5 2 F I G U R E 3.1 B A S I C E X P E R I M E N T A L C O N F I G U R A T I O N T O M E A S U R E C H A N G E I N R E S O N A N T F R E Q U E N C Y 5 5 F I G U R E 3 .2 A N N U L A R S U P P O R T S F O R M O U N T I N G S E N S I N G M E M B R A N E 5 7 F I G U R E 3.3 A C O U S T I C A U D I O T R A N S D U C E R 5 8 F I G U R E 3 .4 C R O S S - S E C T I O N O F V I B R A T I O N D E T E C T I O N P R O B E : A ) S I D E V I E W , B ) E N D -O N V I E W 5 9 F I G U R E 3.5 • D I S P L A C E M E N T S E N S I T I V I T Y O F V I B R A T I O N P R O B E 6 0 F I G U R E 3 .6 S E N S I N G M E M B R A N E U S E D A S A W A V E G U I D E T O E X C I T E F L U O R E S C E N T L Y T A G G E D P R O T E I N M O L E C U L E S O N S U R F A C E 6 2 F I G U R E 3 .7 A L E X A F L U O R 5 3 2 C A R O B X Y L I C A C I D , S U C C I N I M I D Y L E S T E R 6 4 F I G U R E 3.8 E X C I T A T I O N A N D E M I S S I O N C U R V E S F O R A L E X A 5 3 2 F L U O R E S C E N T T A G 6 4 F I G U R E 3 .9 M E T H O D O F C O U P L I N G L A S E R P O I N T E R T O O P T I C A L W A V E G U I D E 6 6 F I G U R E 3 . 1 0 M E T H O D O F C O U P L I N G P I G - T A I L E D L A S E R T O O P T I C A L W A V E G U I D E 6 7 F I G U R E 3 . 1 1 P O W E R O U T P U T F R O M E N D O F 5 3 2 P I G - T A I L E D L A S E R U S E D I N E X P E R I M E N T 6 8 F I G U R E 3 . 1 2 C O N S T R U C T I O N O F S E N S I N G M E M B R A N E A S A W A V E G U I D E 6 8 F I G U R E 3 . 1 3 P R O C E D U R E T O M O U N T E L A S T O M E R I C M E M B R A N E S A S O P T I C A L W A V E G U I D E S 7 0 F I G U R E 3.14 To S C A L E D I A G R A M O F W A V E G U I D E H O L D E R ; A ) F R O M A B O V E , B ) C R O S S -S E C T I O N A , C ) C R O S S - S E C T I O N B 71 F I G U R E 3.15 D I F F E R E N T I A L S C R E W B E T W E E N H O L D E R L E G S A N D A N N U L A R M E M B R A N E S U P P O R T 72 F I G U R E 3.16 C R O S S - S E C T I O N S H O W I N G A N N U L A R M E M B R A N E S U P P O R T I N S E R T E D I N ' • H O L D E R A N D I M M E R S E D I N E X P E R I M E N T A L C E L L : A ) A L O N G C R O S S -S E C T I O N A , B ) A L O N G C R O S S - S E C T I O N B.'...! 73 F I G U R E 3.17 L A R G E C O R E O P T I C A L F I B E R A R R A Y : A ) O B L I Q U E V I E W , B ) S I D E - O N V I E W S H O W N I M M E R S E D I N E X P E R I M E N T A L C E L L 74 F I G U R E 3.18 S P E C T R A O F F I L T E R S U S E D T O I S O L A T E F L U O R E S C E N T R A D I A T I O N 74 F I G U R E 3.19 S C H E M A T I C O F F L U O R E S C E N T L I G H T C O L L E C T I O N A P P A R A T U S 75 F I G U R E 3.20 S C H E M A T I C O F S O L U T I O N F L O W I N E X P E R I M E N T 77 F I G U R E 3.21 A D A P T A T I O N O F B A S I C F L U I D F L O W T O A L L O W F O R F L U S H I N G , A ) N O R M A L C I R C U L A T I O N B ) F L U S H I N G 79 F I G U R E 3.22 T E M P E R A T U R E C O N T R O L S Y S T E M 81 F I G U R E 3.23 E V O L U T I O N O F T E M P E R A T U R E I N E X P E R I M E N T A L C E L L F O R A P P R O A C H T O T H E R M A L E Q U I L I B R I U M 81 F I G U R E 3.24 S C H E M A T I C O F I N S T R U M E N T S U S E D F O R R E S O N A N T F R E Q U E N C Y M E A S U R E M E N T 83 F I G U R E 3.25 M O S F E T C I R C U I T U S E D T O C O N T R O L M O T O R M I X E R 84 F I G U R E 3.26 E X A M P L E R E S O N A N T F R E Q U E N C Y S P E C T R U M W I T H P R O B E P O S I T I O N E D A T C E N T R E O F M E M B R A N E 85 F I G U R E 3.27 F I R S T T H R E E S Y M M E T R I C R E S O N A N T M E M B R A N E M O D E S 86 F I G U R E 3.28 E X A M P L E O F R E S P O N S E O F S E N S O R T O F R E Q U E N C Y S W E E P I N A ) A M P L I T U D E A N D B ) P H A S E F O R (0,2) R E S O N A N C E 87 F I G U R E 3.29 F L O W C H A R T O F L A B V I E W P R O G R A M T O M O N I T O R R E S O N A N T F R E Q U E N C Y 89 F I G U R E 3.30 E X A M P L E O F U N D E R D A M P E D S Y S T E M , G = 50, o-= 0.3 Hz 90 F I G U R E 3.31 E X A M P L E O F A N O V E R D A M P E D S Y S T E M , G = 400, cr= 0.005 Hz 91 F I G U R E 3.32 E X A M P L E O F W E L L T U N E D S Y S T E M , G = 100, a= 0.01 Hz 92 F I G U R E 3.33 S C H E M A T I C F O R I N S T R U M E N T A T I O N U S E D I N F L U O R E S C E N C E M E A S U R E M E N T 92 F I G U R E 3.34 F I L T E R C I R C U I T U S E D T O T I M E A V E R A G E T H E P H O T O M U L T I P L I E R S I G N A L 93 F I G U R E 3.35 F L O W C H A R T O F L A B V I E W P R O G R A M T O D O S U R F A C E F L U O R E S C E N C E M E A S U R E M E N T 94 F I G U R E 3.36 F L O W C H A R T O F L A B V I E W P R O G R A M T O D O S I M U L T A N E O U S E X P E R I M E N T 96 F I G U R E 4. l V A R I A T I O N O F K0N P A R A M E T E R V A L U E A C C O R D I N G T O M E M B R A N E T H I C K N E S S F O R ,4 = 0.0095 M 101 F I G U R E 4.2 V A R I A T I O N O F K0N P A R A M E T E R V A L U E A C C O R D I N G T O M E M B R A N E R A D I U S VOKH= 25 U M 102 F I G U R E 4.3 B A S I C M O D E L T O D E S C R I B E E X P E R I M E N T A L M I X I N G 104 F I G U R E 4.4 C O N C E N T R A T I O N I N E X P E R I M E N T A N D M I X I N G T A N K A S P R O T E I N IS A D D E D A T T = 0 C A L C U L A T E D F R O M M O D E L 105 F I G U R E 4.5 M I X I N G S C H E M A T I C F O R F L U S H I N G S C E N A R I O 106 F I G U R E 4.6 C H A N G E O F C O N C E N T R A T I O N I N E X P E R I M E N T A N D M I X I N G T A N K W H E N F L U S H IS S T A R T E D A T T= 0, C A L C U L A T E D F R O M M O D E L 107 F I G U R E 4.7 C O N C E N T R A T I O N O F P R O T E I N PN E X P E R I M E N T A L V E S S E L A S F L U S H P R O C E E D S F O R D I F F E R E N T F L U S H A M O U N T S C A L C U L A T E D F R O M M O D E L 109 F I G U R E 4.8 E X P A N S I O N O F F I G U R E 4.7 T O S H O W B E H A V I O U R A T L O N G E R F L U S H T I M E S . ... 109 F I G U R E 4 . 9 E X A M P L E O F E V O L U T I O N O F B U L K C O N C E N T R A T I O N I N E X P E R I M E N T A L C E L L , C A L C U L A T E D F R O M M O D E L 1 1 2 F I G U R E 4 . 1 0 S C H E M A T I C O F K I N E T I C S O F S U R F A C T A N T A D S O R P T I O N . I . . , . . . . - 1 1 4 F I G U R E 4 . 1 1 S C H E M A T I C O F K I N E T I C S O F P R O T E I N A D S O R P T I O N '. 1 1 9 F I G U R E 4 . 1 2 E X A M P L E R U N D A T A F I L E U S E D T O G E N E R A T E M O D E L I N P U T 123 F I G U R E 5.1 C A L C U L A T E D T E N S I O N P E R M O D E N U M B E R F O R A W A T E R L O A D E D M E M B R A N E 1 3 2 F I G U R E 5 .2 C O M P A R I S O N O F F I R S T T H R E E C I R C U L A R L Y S Y M M E T R I C M O D E S F O R W A T E R A N D A I R L O A D I N G 133 F I G U R E 5.3 C O M P A R I S O N O F A I R L O A D E D R E S O N A N T S P E C T R A F O R 8 1 4 1 A N D 8 1 6 1 M E M B R A N E S 1 3 4 F I G U R E 5 .4 E X A M P L E O F C H A N G E I N R E S O N A N T F R E Q U E N C Y O F F R E S H 8 1 4 1 S A M P L E I M M E R S E D I N B U F F E R S O L U T I O N 1 3 6 F I G U R E 5.5 M E A S U R E M E N T O F B U L K F L U O R E S C E N C E C O M P A R E D W I T H P R E D I C T E D B U L K C O N C E N T R A T I O N 1 3 7 F I G U R E 5 .6 C O M P A R I S O N O F B U L K F L U O R E S C E N T S I G N A L F R O M C L E A N A N D P R E -A D S O R B E D G L A S S B E A K E R 1 3 9 F I G U R E 5 .7 C O M P A R I S O N O F S I G N A L S F R O M P H O T O M U L T I P L I E R A N D P H O T O D I O D E A S L A S E R IS W A R M I N G U P 1 4 1 F I G U R E 5 .8 F L U O R E S C E N C E S I G N A L , F C A L C U L A T E D F R O M D A T A S H O W N I N F I G U R E 5 . 7 . . 1 4 2 F I G U R E 5 .9 E X A M P L E O F P H O T O - B L E A C H I N G O B S E R V E D D U R I N G S U R F A C E F L U O R E S C E N T E X P E R I M E N T 143 F I G U R E 5 . 1 0 E X A M P L E O F A D S O R P T I O N E X P E R I M E N T W I T H S H U T T E R I N G A N D F I L T E R I N G T O E L I M I N A T E P H O T O - B L E A C H I N G 1 4 4 F I G U R E 5 . 1 1 E X A M P L E O F S U R F A C E F L U O R E S C E N C E E X P E R I M E N T P E R F O R M E D U S I N G F I B E R P I G T A I L E D L A S E R 145 F I G U R E 5 . 1 2 C O R R E C T I O N T O F L U O R E S C E N C E D A T A T O A C O U N T F O R P H O T O - B L E A C H I N G . . 1 4 6 F I G U R E 5.13 E X A M P L E O F A S U R F A C E F L U O R E S C E N T E X P E R I M E N T W H E R E T H E B U L K S O L U T I O N IS C O N T R I B U T I N G T O F L U O R E S C E N T S I G N A L 1 4 7 F I G U R E 5 . 1 4 D A T A F R O M F I G U R E 5 . 1 3 C O R R E C T E D T O A C C O U N T F O R C O N T R I B U T I O N F R O M B U L K F L U O R E S C E N C E 1 4 7 F I G U R E 5 . 1 5 E X A M P L E O F C H A N G E I N F R E Q U E N C Y A S S D S IS A D D E D T O S E N S O R M E M B R A N E 1 4 9 F I G U R E 5 . 1 6 C H A N G E I N T E N S I O N A S S D S is A D D E D F O R E X A M P L E S H O W N I N F I G U R E 5 . 1 6 1 4 9 F I G U R E 5 . 1 7 E X A M P L E O F S E N S O R R E S P O N S E T O I N C R E M E N T A L L Y A D D I N G S D S T O M I X I N G T A N K , C O M P A R E D W I T H B U L K C O N C E N T R A T I O N 1 5 0 F I G U R E 5 . 1 8 R E S P O N S E O F S E N S O R T O CB= 0 . 0 2 M G / M L T R I T O N X - 1 0 0 1 5 1 F I G U R E 5 . 1 9 D A T A F R O M E X P E R I M E N T S H O W N I N F I G U R E 5 . 1 7 C O M P A R E D W I T H O U T P U T F R O M S U R F A C T A N T A D S O R P T I O N M O D E L 1 5 2 F I G U R E 5 . 2 0 S D S A D S O R P T I O N I S O T H E R M G E N E R A T E D F R O M M O D E L F I T A N D C O M P A R E D T O L A N G M U I R I S O T H E R M E Q U A T I O N 153 F I G U R E 5 . 2 1 C H A N G E I N T E N S I O N O F T H E S E N S O R F O R L Y S O Z Y M E A D S O R P T I O N , CB = 0 . 0 0 0 2 2 M G / M L A N D CB = 0 . 0 0 1 M G / M L 155 F I G U R E 5 . 2 2 E X A M P L E O F T H E L O N G T I M E R E S P O N S E O F T H E S E N S O R F O R L Y S O Z Y M E A D S O R P T I O N , CB = 0 . 0 0 1 M G / M L A N D CB = 0 . 0 0 0 5 M G / M L 1 5 6 F I G U R E 5 . 2 3 E X A M P L E O F T W O E X P E R I M E N T A L R U N S S H O W I N G R E P E A T A B I L I T Y F O R L Y S O Z Y M E A D S O R P T I O N , CB = 0 . 0 0 1 M G / M L 1 5 7 F I G U R E 5 . 2 4 E X A M P L E S H O W I N G T W O D I F F E R E N T S E N S O R R E S P O N S E S F O R L Y S O Z Y M E A D S O R P T I O N F O R CB = 0 . 0 0 1 M G / M L 1 5 7 F I G U R E 5 . 2 5 R E S P O N S E O F S E N S O R T O L Y S O Z Y M E A D S O R P T I O N F O R CB = 0 . 0 0 1 5 M G / M L W I T H F L U S H I N G 1 5 8 F I G U R E 5 . 2 6 C O M P A R I S O N O F F L U S H I N G A F T E R D I F F E R E N T A M O U N T S O F T I M E F O R T W O S I M I L A R B U L K S O L U T I O N C O N C E N T R A T I O N S O F L Y S O Z Y M E 1 6 0 F I G U R E 5 . 2 7 C O M P A R I S O N O F T W O S E N S O R R E S P O N S E S T O L Y S O Z Y M E A D S O R P T I O N A T CB = 0 . 0 0 0 2 M G / M L A N D F L U S H I N G A F T E R 3 0 M I N U T E S 1 6 1 F I G U R E 5 . 2 8 C O M P A R I S O N O F T W O S E N S O R R E S P O N S E S T O L Y S O Z Y M E A D S O R P T I O N A T CB = 0 . 0 0 0 2 M G / M L A N D F L U S H I N G A F T E R 6 0 M I N U T E S 1 6 1 F I G U R E 5 . 2 9 R E S P O N S E O F S E N S O R T O 0 . 0 0 0 3 6 M G / M L L Y S O Z Y M E C O N C E N T R A T I O N S H O W I N G B O T H C H A N G E I N T E N S I O N A N D S U R F A C E F L U O R E S C E N C E 163 F I G U R E 5 . 3 0 C O M P A R I S O N O F F L U O R E S C E N C E S I G N A L W I T H C H A N G E I N B U L K C O N E N T R A T I O N F O R E X P E R I M E N T S H O W N I N F I G U R E 5 . 2 9 1 6 4 F I G U R E 5 . 3 1 S I M U L T A N E O U S M E A S U R E M E N T O F C H A N G E I N T E N S I O N A N D S U R F A C E F L U O R E S C E N C E F O R L Y S O Z Y M E A D S O R P T I O N A T CB = 0 . 0 0 0 7 2 M G / M L 165 F I G U R E 5 . 3 2 C O M P A R I S O N O F T H E C H A N G E I N T E N S I O N F O R T H E S I M U L T A N E O U S M E A S U R E M E N T S S H O W N I N F I G U R E 5 . 2 9 A N D F I G U R E 5 . 3 1 165 F I G U R E 5 . 3 3 . C O M P A R I S O N O F T H E F L U O R E S C E N C E S I G N A L F O R T H E S I M U L T A N E O U S M E A S U R E M E N T S S H O W N I N F I G U R E 5 . 2 9 A N D F I G U R E 5 . 3 1 1 6 6 F I G U R E 5 . 3 4 R E S P O N S E O F S E N S O R T O T H E I N C R E M E N T A L A D D I N G O F L Y S O Z Y M E F O R CB = 0 . 0 0 0 3 7 M G / M L A N D CB= 0 . 0 0 0 5 5 M G / M L 1 6 7 F I G U R E 5 . 3 5 C O M P A R I S O N O F D A T A S H O W N I N F I G U R E 5 . 3 4 C O M P A R E D W I T H T H E C H A N G E I N B U L K C O N C E N T R A T I O N F O R E X P E R I M E N T A L R U N 1 6 8 F I G U R E 5 . 3 6 M O D E L F I T F O R T H E S I M U L T A N E O U S E X P E R I M E N T A L D A T A S H O W N I N F I G U R E 5 . 3 4 1 7 1 F I G U R E 5 . 3 7 T H E E V O L U T I O N O F E A C H S U R F A C E S T A T E P O P U L A T I O N A S A F U N C T I O N O F T I M E F O R T H E M O D E L F I T P R E S E N T E D I N F I G U R E 5 . 3 6 1 7 2 F I G U R E 5 . 3 8 C O M P A R I S O N O F T H E E X P E R I M E N T A L D A T A S H O W N I N F I G U R E 5 . 2 9 W I T H T H E M O D E L F I T W I T H P A R A M E T E R S L I S T E D I N T A B L E 5-3 1 7 4 F I G U R E 5 . 3 9 C O M P A R I S O N O F T H E E X P E R I M E N T A L D A T A S H O W N I N F I G U R E 5 . 3 1 W I T H T H E M O D E L F I T W I T H P A R A M E T E R S L I S T E D I N T A B L E 5-3 175 F I G U R E 5 . 4 0 I S O T H E R M G E N E R A T E D F R O M P R O T E I N A D S O R P T I O N M O D E L U S I N G T H E M O D E L F I T P A R A M E T E R S L I S T E D I N T A B L E 5-3 1 7 6 F I G U R E 5 . 4 1 C O M P A R I S O N O F M O D E L FIT I S O T H E R M S H O W N I N F I G U R E 5 . 4 0 W I T H E X P E R I M E N T A L R E S U L T S F O R L Y S O Z Y M E O N N E G A T I V E L Y C H A R G E D P O L Y S T Y R E N E L A T E X 1 7 6 F I G U R E A . 1 T H E O R E T I C A L M O D E S H A P E S F O R F I R S T 9 I D E A L M E M B R A N E M O D E S 1 8 7 F I G U R E B . 1 L I G H T R A Y I N C I D E N T O N A B O U N D A R Y B E T W E E N T W O M E D I A 1 8 8 F I G U R E B . 2 L I G H T R A Y I N C I D E N T A T B O U N D A R Y B E T W E E N T W O M E D I A A T T H E C R I T I C A L A N G L E 1 8 9 F I G U R E B .3 P L A N A R W A V E G U I D E S T R U C T U R E , N 2 < N I 1 9 0 F I G U R E B .4 F A L L O F F O F I N T E N S I T Y O F E V A N E S C E N T W A V E I N T O T H E S E C O N D M E D I A 1 9 1 x ACKNOWLEDGEMENTS I would like to thank the following people who have contributed to my endeavors over the course of the six years I have worked on my Ph.D. Firstly, sincere thanks go to my supervisor, Lome Whitehead, for creating an environment, which fosters scientific innovation, and supporting this project. Secondly, thanks to my co-supervisor, Charles Haynes, for providing the motivation and background in protein adsorption and for always being excited about my results. Thanks also to the rest of my supervisory committee, Andrzej Kotlicki, Boye Ahlborn and Ranjith Divigalpitiya for making my committee meetings informative and collaborative discussions. Thanks to my fellow researchers and graduate students in the Structured Surface Physics Laboratory who have helped me overcome various experimental and theoretical obstacles: Michele Mossman, Vincent Kwong, Januk Aggarwal, Anne Liptak, Peter Kan and Robin Coope. Thanks to Helge Seetzen for always being able to solve any computer problems I had, and to Andrzej Kotlicki for helping in the design and construction of all the electronic circuitry that was needed in this project. In the Biotechnology Laboratory I would like to thank Sue Liu for helping me prepare and measure my protein solutions, Louise Creagh for preparing.the lysozyme samples for my use and Brad Maclean for assisting in the fluorescent labeling of the lysozyme. A sincere thanks to everyone who has helped look after Amelie so that I was able to return after my maternity leave: Nancy Brideau, Stacie Coughlan, Therese Coughlan, Jennifer Clark and Keith Coughlan. Thanks to my parents, Peter and Celia Clark for supporting me in everything I do and to my Dad for instilling in me a love of science at an early age. Finally, to sincere thanks to my husband, Kevin, for supporting me completely and for sorting out all the technical glitches at my final defence, and to Amelie for her help " a l'Universite"! 1 INTRODUCTION The idea for the adsorption sensor described in this thesis came from a prior project undertaken by the candidate. That project investigated the propagation of planar surface waves in a thin elastomeric membrane supported on one side with water and their use as a variable spacing diffraction grating. ' These waves had a characteristically low wave speed (~ 1 m/s) and as such could generate small wavelengths (~ 0.1 mm) at audio frequencies (~1 kHz). At these wavelengths a ~ 1° deflection in the first order diffraction beam was achieved for a laser beam reflecting at glancing angle from the surface wave. Because the elastomeric membranes were thin with only a small amount of inherent tension, the surface energies on either side of the membrane influenced the wave speed, by contributing to the overall tension. It was postulated that the adsorption of biomaterials to the membrane surface could affect the wave speed of the surface waves. Preliminary results indicated that this indeed was the case and the experimental endeavour described in this thesis to develop an adsorption sensor based on elastomeric surface waves was undertaken. Presently, there is a considerable amount of research being conducted on the adsorption of proteins to solid surfaces. The irreversible adsorption of blood proteins to the surfaces of biomedical implant materials is the first step in a cascading process that leads to blood clotting and device rejection. However, the phenomenon of irreversible protein adsorption is very poorly understood. For example, Figure 1.1 shows a typical protein adsorption isotherm that plots the adsorbed surface population as a function of bulk solution concentration. On the ascending slope of the isotherm, steady-state surface populations are achieved which are well below a monolayer. This implies that the amount of molecules adsorbing at a particular bulk concentration is balanced by an equal amount desorbing. However, upon dilution of the bulk solution concentration, the majority of the adsorbed molecules remain bound to the surface, which is indicated by the arrows pointing back towards the y-axis. This indicates that there is no rate of desorption from the surface upon bulk solution dilution. This discrepancy is usually termed the paradox of irreversible protein adsorption. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 bulk concentration xlO (mg/mL) Figure 1.1 Typical protein adsorption isotherm illustrating irreversible protein adsorption. Irreversible protein adsorption, as will be described in section 2.2.5.1, is thought to be due to protein molecules changing their conformation upon adsorption such that they become progressively bound to the surface as a function of time. Measuring and understanding the kinetics of protein adsorption and conformation change on solid surfaces is therefore the first stage in the engineering of surfaces that are resistant to either protein adsorption or subsequent conformation change. However, as will be described in section 2.2.6, there remains a need for new experimental techniques to improve our understanding of the kinetics of protein conformation change. This thesis describes the development of a novel adsorption sensor that is sensitive to both the adsorption and conformation change of molecules at its surfaces. It is unique in that it is capable of measuring the kinetics of both these processes simultaneously. Therefore the development of this device could be an important breakthrough in the field of protein adsorption. Furthermore, a new protein adsorption model has been developed which describes the behaviour of the adsorption sensor. It possesses several key ideas that resolve the previously described paradox of irreversible protein adsorption. Thus, this thesis 2 therefore describes two major contributions to the study of protein adsorption at solid/liquid interfaces. 1.1 Description of adsorption sensor The adsorption sensor consists of an elastomeric sensing membrane which is mounted on an annular holder with an inner diameter of ~ 2 cm. There were two types of measurements that were performed on this sensing membrane. The first type involved measuring the change in resonant frequency of an oscillation mode as molecules adsorbed to its surface. As will be explained in the following sections, this measurement was primarily sensitive to the conformation change of the adsorbed molecules. The second type involved measuring the surface fluorescence emitted from the sensing membrane as fluorescently tagged protein molecules adsorbed to its surface. This method was sensitive to the total amount of adsorbed proteins on the surface of the membrane. Thus by simultaneously performing both of these measurements as a function of time, the kinetics of both adsorption and conformation change could be determined. 1.1.1 Resonant frequency sensor In order to use the elastomeric surface waves described above in a sensor application, the original configuration was redesigned so that it could operate in a resonant frequency mode. The basic configuration of this resonant frequency adsorption sensor is illustrated in Figure 1.2. The elastomeric membrane was mounted on an annular support and the round membrane modes were acoustically actuated from underneath. The motion of the membranes was detected using a reflection-based vibration detection probe positioned above the center of the membrane. The sensing membranes were immersed in an aqueous solution containing the adsorbing molecule of interest. This configuration produced well-defined mode shapes with a fundamental mode frequency of ~ 2 Hz at a membrane radius of ~ 1 cm. The sensor operated by monitoring the drop in resonant frequency of a particular mode as a function of time as the molecules adsorbed to its surface. 3 vibration detection probe measuring membrane oscillation attached molecules that change the resonant frequency of oscillating membrane elastomeric membrane oscillating in resonant mode aqueous solution containing acoustic transducer molecules of interest Figure 1.2 Resonant frequency adsorption sensor 1.1.2 Mass effect calculation When protein molecules and other molecules adsorb to the sensor surface, the resonant frequency of a particular vibration mode drops significantly. For example, drops in frequency of up to 10% were observed. Since, the mass of a monolayer of the molecule is significantly less than the mass of the membrane itself, this effect is not due to changes in the mass of the membrane. For example, the mass of the membrane is given in Equation (1-1 ), where r is the radius, h is the thickness and p is the density. For a typical membrane used in this study, mmembrane - 6.95 mg. The mass of a monolayer of molecules adhering to both sides of the membrane is given in Equation ( 1-2 ), where /"«, is the maximum possible surface population of a given molecule in mg/m2 and is calculated from the inverse of the expected area the molecule will take up on the surface. Note however, that this is only an estimate as the actual area of the membrane could be greater than A = nr2 due to surface topography. For two of the molecules used in this study, lysozyme and sodium dodecyl sulfate, mmonoiayer = 0.0013 mg and 0.0005 mg respectively. For a typical membrane used in this study, this change in mass would result in a percentage drop in frequency of A/7/~ 0.03%, much smaller than that observed due to the adsorption of molecules. Furthermore, as will be described in section 4.1.2, the membranes were already heavily mass loaded by the surrounding fluid medium, therefore any additional adsorbed mass could only affect the 4 resonant frequency i f its surface density was significantly greater than the fluid density. Therefore, in practice the observed change in frequency due to mass loading of a monolayer would be much less than the percentage drop quoted above. mmembrane = n r h P '> ( 1 - 1 ) mmonolayer = 2/~oo &r ( 1-2 ) 1.1.3 Surface energy As will be fully described in section 2.2.1.1, at an interface between two phases, there exists an excess energy or surface free energy, y due to imbalances in the molecular forces between molecules in the bulk versus those at the surface. Surface energy has units of N/m and is also commonly referred to as surface tension. For a thin membrane of elastomeric material freely mounted on an annular holder, the total tension, Ttotai, of the membrane wil l have a contribution from the surface free energy of the membrane, y according to Equation ( 1 -3 ), where Tinherent is the tension due to added mechanical stresses on the membrane. This is demonstrated in Figure 1.2. Providing there is little inherent tension, Tinnerenh i.e. the film has not been stretched by thermal, curing or mounting stresses, the surface tension could dominate the total tension of the membrane. 2"total = Tinherent ( 1-3 ) 5 Figure 1.3 Tension in an elastomeric thin membrane showing contribution from surface energy. As will be described in section 2.2.1, molecular adsorption to an interface reduces the surface energy at the interface by changing the molecular forces experienced by molecules at the surface. Furthermore, as will be described in section 5.1.1, the inherent tension of the thin elastomeric films used as sensor membranes is low enough that the surface energy of the membranes is a major contributor to the overall membrane tension. From the theory of ideal membranes described in Appendix A, the resonant frequencies of the membrane vibrational modes are dependent on the membrane tension. Therefore, the adsorption of molecules to the surface of the membrane can result in a reduction of the surface energy at the interface that in turn reduces the resonant frequency of the membrane mode. In testing the sensitivity of the sensor to adsorption of different types of molecules, some surprising results were achieved. While the sensor responded within minutes to the adsorption of a simple surfactant molecule, as wil l be described in section 5.2, it had a long-time response, over many hours, to the adsorption of protein molecules, as will be described in section 5.3. It was for this reason that the sensor was thought to be sensitive to not only the adsorption of proteins, but also to their slow conformation change. 1.1.4 Surface fluorescence sensor The surprising result described above prompted the need to independently measure the amount of protein molecules adsorbed on the surface of the sensor as a function of time. I f it could be determined that the amount of molecules on the surface reached a steady-state value within ~30 minutes (as expected from other protein adsorption studies), but the change in resonant frequency continued for well past this time, it would provide direct evidence of the sensitivity of this sensor to adsorbed protein conformation change in the resonant frequency signal. Therefore, a means was devised whereby the evolution of the number density of protein molecules as a function of time could be measured simultaneously with the change in frequency of the sensor membrane. Figure 1.4 shows a schematic of how this measurement was accomplished. The protein molecules were tagged with a fluorescent label that absorbed laser radiation and fluoresced at a different wavelength. Since the sensor membrane material was optically clear, it could be used as a waveguide for the exciting laser radiation. Therefore, protein molecules adsorbing to the sensor surface would absorb laser radiation in the evanescent field of the waveguide and only those molecules adsorbed to the sensor surface would be selectively excited to emit fluorescent radiation. A light collection system was devised to collect the fluorescent radiation emitted from the sensor surface. This enabled the time evolution of the population of adsorbed protein molecules to be measured simultaneously with the change in tension of the sensor membrane. fluorescent radiation collection array solution containing fluorescently labeled protein molecules pigtailed laser injected into edge of sensing membrane" adsorbed protein molecules absorbing laser radiation and emitting fluorescent radiation sensing membrane waveguide evanescent field of laser radiation Figure 1.4 Surface fluorescence adsorption sensor The adsorption sensor described herein probes the non-specific adsorption of molecules to the surface of the sensing membrane. However, this is the first step toward the development of an inexpensive biosensor platform. For example, specific ligands can be attached to the membrane to monitor the adsorption and detection of analytes of interest. 7 1.2 Thesis overview This thesis concerns the development of a surface energy sensor capable of measuring the kinetics of protein adsorption and protein conformation change at the solid/liquid interface. Chapter 2 contains some background material necessary to understand the device operation and the experimental results it generates. Section 2.1 describes a few examples of other resonant frequency sensors that are used in the study of molecular adsorption. Section 2.2 outlines the phenomenon of molecular adsorption at interfaces, the thermodynamics that govern its behaviour and current methods to measure and describe it. Chapter 3 outlines the construction and operation of the sensor described in the introduction. Both the configurations for the resonant frequency measurement and the surface fluorescence measurement wil l be fully explained. The fluid flow system that supplies the solution containing the adsorbing molecules to the sensor surface is outlined. Finally, the instrumentation and control mechanisms for operating the sensor and generating experimental data is described. Chapter 4 describes how the sensor behaviour is modeled. The method used to extract the membrane tension from the resonant frequency of a particular resonant mode is outlined. A method to calculate the evolution of the bulk concentration in the experimental cell is then described. Finally, a step-wise kinetic model to describe the kinetics of molecular adsorption and its affect on the membrane tension is derived. Chapter 5 presents the experimental results obtained and describes the response of the sensor to different types of molecules. Certain aspects of sensor behaviour and the conditions necessary for the successful operation of the device are described. The response of the sensor to the adsorption of surfactant molecules is then presented. Finally, the response of the sensor to the adsorption of protein molecules is presented, along with a discussion of how these results may shed some light on the mechanism of protein adsorption. 8 2 BACKGROUND This thesis describes the development of a new resonant frequency sensor and its application as a molecular adsorption sensor. Therefore, the first part of this chapter wil l describe other resonant frequency sensors commonly used, their principles of operation and application in the field of bio-sensing. The second part of the chapter wil l provide the necessary background in the field of interfacial adsorption. 2.1 Resonant frequency sensors A resonant frequency sensor is a mechanical vibrator whose resonant frequency is sensitive to changes in its environment, such as temperature, pressure or adsorption to its surfaces. For it to be a useful sensor, there must exist a well-defined relationship between its resonant frequency and the parameter of interest. The most widely developed and commercialized resonant frequency sensors are those based on the piezoelectric material quartz because of its high stiffness and sharp resonances. A piezoelectric material is an anisotropic crystal that undergoes mechanical stress under the action of an electric field. By applying an alternating electric field, acoustic waves in the ultrasonic frequency range can be propagated in the crystal. The type of waves generated depends on the shape of the crystal, its orientation and the geometry of the electrodes. For example, waves can be bulk waves propagating through the volume of the crystal, or surface waves whose energy is confined to within a wavelength of the crystal surface. Waves are either shear, where the particles move in a direction perpendicular to the direction of the wave, or longitudinal, where the particle displacement is parallel to the direction of propagation of the wave. By attaching two electrodes to the crystal, standing waves can be generated whose amplitude will be maximized at certain resonant frequencies. The type of sensor relevant to the work presented in this thesis measures the adsorption of molecules to its surface from an aqueous solution, therefore, the device must be able to operate effectively with the sensing surface immersed in a liquid medium. Three types of acoustic wave devices that have been applied as an adsorption sensor are the quartz crystal 9 microbalance (QCM), the surface acoustic wave (SAW) sensor, and the flexural plate wave (FPW) sensor. 2.1.1 Q u a r t z c rys ta l m ic roba lance -The QCM device is the most widely used of all the acoustic wave sensors. It is also known as a thickness-shear mode (TSM) device and operates with shear or tranverse waves propagating through the bulk of a quartz disc with the particle motion parallel to the sensor surface. This is shown in Figure 2.1, where there is a corresponding thin film metal electrode on the bottom surface of the device as well. 3 metal electrode a) b) CD-Figure 2.1 Quartz crystal microbalance geometry a) isotropic view, b) cross-section in xz plane showing direction of particle motion. The velocity of the bulk wave, v, is given by Equation (2-1 ), where Y is the shear modulus of the quartz in the direction of wave propagation and p is its density (Y- 29.47 GPa and p -10 2648 kg/m3 for AT-cut quartz4). Equation ( 2-2 ) gives the resonant frequencies,/,, of the device where (XQ is the thickness of the quartz disc. v = J - (2-1) V P f " = ^ (» = ^ - 5 - ) (2-2) As the name "microbalance" implies, it is sensitive to the adsorption of mass at its surface. The added mass at the surface is simply considered an extension of the thickness of the quartz crystal. The change in frequency due the added mass, Am, is given by Sauerbrey's equation, Equation ( 2-3.), where A is the area of the crystal. Equation (2-3 ) is only valid for frequency shifts of less than 2% of the unloaded resonant frequency. . . r 2 Am 1 Immersing the sensor in a fluid medium also causes a shift in frequency due to the energy lost to viscous coupling of the bulk wave with the fluid. The decay length of the resulting transverse shear wave that propagates into the liquid is given by Equation ( 2-4 ), where, pi is the fluid density and TJL is the fluid viscosity. Equation (2-5 ) gives the shift in frequency due to the mass loading of the fluid. (2-4) \G>PL \ nip The frequency shifts due to the viscous damping and the added mass can be decoupled by modeling the system as an electric equivalent circuit. In this model, the added mass acts as an inductor and the viscous damping acts as both a resistor and an inductor. 11 Cavic and Thompson measured the adsorption of seven different proteins using a QCM sensor. When a solution of 0.003 mg/mL fibrinogen was exposed to the surface of the sensor resonating a t / ~ 9 MHz, they recorded a drop in frequency of 400 Hz. 2.1.2 Sur face acoust ic w a v e sensor Acoustic resonators that can produce surface waves are inherently much more sensitive to changes at their surfaces than devices based on bulk oscillations. This is because the majority of the energy of the wave is confined to within a wavelength's distance from the surface of the device. One example of this is the SAW sensor based on propagating Rayleigh waves along the top surface of a quartz crystal as shown in Figure 2.2. The spacing of the inter-digitized electrodes defines the wavelength of the propagating wave. These sensors have shown very high mass sensitivity as gas sensors (see Table 2-1), but have too much damping when immersed in a liquid to have an application as a biosensor.6 X a) b) Figure 2.2 SAW sensor a) side view and b) top view. 12 2.1.3 Flexural plate wave sensor Flexural plate wave (FPW) sensors are based on the propagation of Lamb waves, surface waves that propagate along both faces of a plate that is of the order of the wavelength thickness. There are two modes of propagation, symmetric and anti-symmetric, as shown in Figure 2.3. The phase velocities, vp, of the propagating wave for the symmetric and anti-symmetric cases are given in Equations ( 2-6 ) and ( 2-7 ) respectively, where E is the Young's modulus, pthe density, v the Poisson's ratio, A, the wavelength and t the thickness. Note that in the anti-symmetric case, the phase velocity is proportional to the plate thickness, so by reducing the plate thickness the phase velocity of the wave can be reduced. It is therefore possible to manufacture a plate whose phase velocity is less than the speed of sound in a liquid, meaning that it may propagate immersed in a liquid medium without significant energy loss. This makes Lamb wave sensors feasible for use as biosensors. Symmetric Lamb wave Anti-symmetric Lamb wave Figure 2.3 Anti-symmetric and symmetric lamb wave. VP = P P5) (2-6) VP = v A J 1 EV P\2 P5) (2-7) 13 Since quartz plates cannot be micro-fabricated thin enough for this application, the wave is generated in a non-piezoelectric layer such as silicon nitride, SiaN4. There have been a few examples of FPW devices reported in the literature. For example, White and Wenzel7 describe the fabrication of a FPW device by depositing a silicon nitride film on a silicon wafer, depositing a thin film metal ground plane, and then a thin layer of piezoelectric material, ZnO. Interdigitated electrodes are deposited on top of the ZnO layer, and then the silicon wafer is etched from underneath, leaving just the thin composite layer, as shown in Figure 2.4. White and Wenzel found the velocity of the lowest order Lamb mode to be 474 m/s in air and 304 m/s in water. Since the speed of sound in water is co=1420 m/s, the phase velocity of the wave is low enough to propagate while in contact with the water. However, there is a mass-loading effect causing the phase velocity to be lowered in water with respect to air. The decay length of the evanescent acoustic field traveling in the fluid medium is given by Equation (2-8 ), where c is the speed of sound in the fluid medium.8 By assuming that the fluid in this evanescent field effectively adds to the mass per unit area of the plate, M, according to Equation ( 2-9 ) where / is the thickness of the plate and pf is the density of the fluid medium, the phase velocity of a fluid loaded plate can be calculated according to Equation ( 2-10 ), where T denotes the inherent tension in the silicon nitride layer created in the micro-fabrication process. 14 X v, M - pt + Pf$E ( 2 - 8 ) ( 2 - 9 ) M T + Et' 1 2 ( l - v 2 ) ( 2 - 1 0 ) Costello et al. report using a similar sensor to the one described above to measure the adsorption of Bovine Serum Albumin (BSA). With a bulk concentration of protein of 3 mg/mL exposed to the sensor surface oscillating at 5.8 MHz, they found a frequency drop of 2 kHz, which they calculated to correspond to an adsorbed layer of approximately 2.8 ug/cm2. The mass sensitivity, Sm, of a sensor is defined by Equation (2-11 ) 9 , where vp is the phase velocity, and Avp is the change in phase velocity for the addition of mass, Am. Table 2-1 shows a comparison between the three resonant frequency sensors discussed in this section, listing their frequency range of operation, theoretical formulas for mass sensitivity derived from Equation (2-11 ), and typical values of mass sensitivity. Sm = lim 1 dv, Am^OVj, Am ( 2 - 1 1 ) 15 sensor type frequency rangea (MHz) theoretical mass sensitivity Sm (cm2/g) * Mass sensitivity typical value (cm2/g) Quartz crystal microbalance 5-20 1 pt -201" Surface acoustic wave device 30-500 f0K1% pv -(100-200)8 Flexural plate wave sensor 2-7 - 1 II 2(pt + dE) -HSU * reference i | calculated from theoretical value for a 9MHz crystal (/ = 185 urn, p = 2648 kg/m3) % Kt constant depending on properties of SAW device § taken from reference 9 || reference 8 Table 2-1 Properties of acoustic wave sensors. Clearly the FPW device has the capability to be more sensitive than the QCM. However, due to the difficulty in micro-fabricating such devices, it has not been as widely developed and applied as the QCM. Furthermore, there exist only a few examples of its application as a protein adsorption sensor. QCM devices, in comparison, due to their ease of manufacturing are a mature technology that has found widespread use in many fields. Furthermore, considerable effort has been expended to develop sophisticated electric equivalent circuits models to fully account for the effect the viscous loading has on the mass sensitivity when immersed in an aqueous environment.1 0'1 1'1 2 This section has given an overview of the resonant frequency sensors .that have been successfully applied to measure the adsorption of protein molecules. These sensors operate at a high frequency in the MHz range and are principally sensitive to the addition of mass to their surfaces. In contrast, the resonant frequency sensor described in this thesis operates at low frequencies in the Hz range and is sensitive to changes in the surface energy at its 16 surfaces. However, issues such as the mass loading of the sensor by the surrounding fluid medium, also affect the operation of the resonant frequency sensor described in this thesis. 17 2 .2 Adsorption at interfaces As described in the Introduction, the sensor described in this thesis is sensitive to surface energy changes as molecules adsorb to its surface, this section will give a base introduction to the concept of surface energy at interfaces. It will describe two surface active molecules, surfactants and proteins, that are used as model adsorbates, and will discuss the adsorption of these molecules to interfaces. Furthermore, the current methods used to measure molecular adsorption to interfaces will be outlined, with a discussion of some of the models used to describe adsorption. 2.2.1 Interfacial thermodynamics An interface is defined here as the boundary between two phases, of which at least one is a fluid, i.e. liquid/vapour, solid/vapour, liquid/liquid or liquid/solid. Solutes dissolved in fluid phases wil l have a tendency to adsorb at the interfaces of the fluid. The basic thermodynamic concepts and definitions that govern this behaviour wil l be described in this section. 2.2.1.1 Basic concepts and definitions An important thermodynamic potential is the Gibbs energy, G, which denotes the system energy that is available to do useful work. It is a fundamental potential because for a given process at constant temperature and pressure, a decrease in G indicates a favourable process while an increase in G indicates an unfavourable reaction. For a system at constant temperature and pressure, thermodynamic equilibrium occurs when the system is in a state of minimum Gibbs energy. The Gibbs energy, G, is given by Equation ( 2-12 ), where H is the enthalpy, Tis the absolute temperature in K and S is the entropy. It is an extensive system variable that in a bulk phase (i.e. liquid or gas) depends on the volume or mass of the phase. Note that the Gibbs energy is reduced by both a decrease in enthalpy and an increase in entropy. 18 G = H-TS (2-12 ) Figure 2.5 shows the density profile across the interface between phase a and phase {3. There is not a sharp transition, rather an interfacial region over which the density changes between the two values. However, to simplify the thermodynamic description of the interface, it is useful to define a plane in the interfacial region with properties that reflect the volume averaged properties of the interfacial region. This plane is called the Gibbs dividing surface. c D 4— interfacial region—^. phase a \ ! Gibbs phase p \ p4- dividing \| surface ! Figure 2.5 Interfacial region between two phases and the Gibbs dividing plane. At an interface, the two dimensional analog of the Gibbs energy is the interfacial free energy. Equation (2-13 ) gives the interfacial free energy for a one component, two phase system, where P is the pressure, Tthe temperature and n the number of moles. It is an intensive variable that does not depend on the area of the interface. dG_ 8A (2-13) P.T.n 19 From a conceptual point of view, the interfacial energy can by explained as an excess energy at the interface due to density differences between molecules in the two phases. For example, consider the interface between a liquid and its vapour, for which Figure 2.5 can be considered a plot of the density profile where a is the liquid phase and J3 is its vapour phase. In the bulk fluid, the forces acting on the molecules are balanced, however, at the surface there is a net inward force, perpendicular to the surface, that acts on the molecules as shown in Figure 2.6. It therefore requires work to move molecules to the surface, resulting in an excess energy at the interface. gas < • • < — • — • - - < • - — > y 1 I I bulk liquid 4 • • < • • 4 • • I I I Figure 2.6 Interface between bulk liquid and vapour. Since the units of the interfacial free energy are [J/m 2], which is equivalent to [N/m], the interfacial free energy is often considered analogous to interfacial tension. Fundamentally, y, the interfacial free energy is "the work required to create a unit area of surface". 1 3 The interfacial tension is "the work necessary to stretch or compress an existing surface".13 At a liquid interface these can be considered equivalent, as the expansion of a liquid surface involves bringing more atoms to the surface. However, in a solid surface that is extensible, the interface can be stretched without adding more atoms to the surface. Interfacial tension and free energy cannot be considered equivalent, though the interfacial free energy does contribute to the overall tension in the solid. In systems comprised of more than one component, for instance a solute dissolved in a fluid, the solute has a tendency to adsorb to the interface as this wil l lower the interfacial free 20 energy. The Gibbs adsorption equation, ( 2-14 ), gives the thermodynamic relationship between the interfacial free energy, adsorbed amount and concentration in solution, where S is the surface excess entropy per unit area, is the. surface concentration of component i and jUj is the chemical potential of component i. This is an ideal surface equation of state analogous to the ideal gas law and can only be applied under equilibrium conditions. It states that any adsorption at the interface will reduce the interfacial surface tension, y. The following sections address the measurement of y at fluid/fluid, solid/fluid and elastomer/fluid interfaces. 2.2.1.2 Fluid/fluid interfaces The measurement of the surface tension of a liquid with its vapour, yLV, can be done by slowly and reversibly expanding the surface and measuring the work required, W, to increase the interfacial area, A, according to Equation ( 2-15 ). There are a variety of methods for accomplishing such a measurement.14 The Wilhelmy plate method measures the force acting on a vertical plate suspended and in contact with a liquid interface, such that the lower edge of the plate is in line with the interface far away from the plate. The surface tension of the interface causes a net downward force on the plate from which can be calculated the surface tension. In the pendant drop method, a drop of fluid is suspended from a capillary tube and the shape it forms is a balance between the gravity acting on it and the surface tension of the liquid interface. In the capillary rise method, the height that a fluid rises in a small diameter capillary tube is related to the radius of the tube, the densities of two fluids and the surface tension of the interface. dy = -SdT-Y,ridMi ( 2-14) dW = yLV dA (2-15) 21 Solutes dissolved in the bulk fluid either have a tendency to conglomerate or to adsorb to interfaces, as described above. For ideal dilute solutions, Equation (2-14 ) reduces to Equation ( 2-16 ) which shows the relationship between adsorption and interfacial tension at a fluid interface.15 In this equation, R is the gas constant, T the absolute temperature, /"the surface concentration in moles per unit area and ct, is the bulk solution concentration. dy = -RTrd\ncb (2-16) At a liquid interface, for instance pure water in equilibrium with its vapour in air, the surface tension of a clean water interface at 20°C is yo = 72.8 mN/m. However, even a minute amount of contamination wil l cause adsorption at the interface and have the effect of lowering this value according to Equation ( 2-16 ). Since the concept of surface tension is well defined and easy to measure at fluid interfaces there has been considerable work done to measure the changes in surface tension as adsorption to the interface occurs. ' Equation ( 2-16 ) can then be applied to these data to calculate the surface population, r. 2.2.1.3 Fluid/solid interfaces The thermodynamics at solid interfaces is considerably more complicated than at fluid interfaces.18 The surface tension of a solid interface with its own vapour is denoted as ys°. It is more difficult to accurately measure the surface tension of a solid interface. The methods used to measure that of a liquid described in section 2.2.1.2 wil l not work, as solids are generally not expansible. Furthermore, i f the solid is extensible, these methods would generate the interfacial tension, which for solids cannot be considered equivalent to the interfacial free energy. An estimation of yi can be made by measuring the surface tension of the liquid phase, assuming that the surface tension is continuous across the solid-liquid phase transition and extrapolating the value to 20°C. Measurement of the interfacial free energy at the solid/liquid interface, y$L, is generally done by contact angle measurements. A small drop of the liquid is placed on the solid and the 22 angle the tangent to the edge of the drop makes with the solid surface, 6, is determined. This is illustrated in Figure 2.7. Then the forces acting on the 3 phase boundary between the solid, liquid and vapour are balanced according to Equation ( 2-17 ), the Young-Dupre equation.19 This equation assumes that the surface is ideally smooth and not deformable due to the forces acting upon it. Figure 2.7 Definition of contact angle for liquid drop at solid interface. YSV = YSL + YLV c o s 0 ( 2-17 ) While the determination of the contact angle, 9, is straight forward, the measurement of ysv, as discussed above, is problematic. This is further complicated by the fact that ysv is the interfacial tension of the solid with the liquid vapour and is different from y$° as adsorption of the liquid vapour at the solid interface reduces the interfacial energy. The differences between the two values is denoted by the film pressure, ne, according to Equation ( 2-18 ). However, for low energy solids (y£ < 100 mN/m), the adsorption from the vapour phase is generally considered negligible and it is valid to assume that ne - 0. YSV=YS°-*e <2"18) Solid/liquid interfaces can broadly be separated into two classes, hydrophilic and hydrophobic. A hydrophilic (meaning "water-loving") surface has a higher surface energy 23 than that of water so that a water droplet wil l spread out over the surface. In this case, 0=0. Metals and glass surfaces have high surface energies14 (-500 mN/m), and are consequently hydrophilic. Hydrophobic (meaning "water-hating") surfaces have lower surface energies than that of water and are not wetted. In general the lower the surface energy of the solid, the higher the contact angle of a water droplet on that solid. For example, Teflon™ has a Ys = 18.9 mN/m 2 1 and G= 108°.1 4 2.2.1.4 Fluid/elastomer interfaces It has been suggested that the mobile interface of an elastomer with a liquid is more akin to that of a hydrophobic oil/liquid interface than that of a rigid solid/liquid interface.13 Synthetic elastomers are made from long chain polymeric molecules that when "cured" or cross-linked behave as a flexible and extensible solid. They have extremely low Young's moduli in their rubbery regime (E ~ 106 Pa), compared to conventional solid materials (E ~ 109 Pa). Because of their flexibility and apparent inertness, silicone elastomers in particular are considered a candidate for compatible biomaterials. Applications include cardiac pacemakers, cochlear implants, contact lenses and catheters. The polymer molecules comprising silicone elastomers consist of a long, flexible, inorganic backbone with pendant organic groups. Figure 2.8 shows the formula of a polydimethylsiloxane molecule, PDMS, a silicone elastomer, where n can range up to 2500. CH CH CH CHo — Si — O CH O CH Si — CHo I 3 CHo -Jn Figure 2.8 Polydimethylsiloxane molecule. PDMS polymers have unique surface properties because of the flexibility of the backbone and the pendant groups.23 They are low surface energy solids which are hydrophobic with an 24 interfacial surface tension with water of 23 mN/m. 2 4 Because of the mobility of the pendant groups, the polymer molecule may reorient itself in the presence of water to lower its interfacial free energy. The time dependency of this relaxation will depend on the rigidity of the molecule. • 2.2.2 Solute adsorption at a liquid/sojid interface Adsorption, the binding of an adsorbate molecule to ah interface, can occur at gaseous/liquid, liquid/liquid, or solid/liquid interfaces. This investigation focuses on the adsorption of chain molecules (both surfactants and proteins) to the solid/liquid interface of the sensing elastomeric membrane. 2.2.2.1 Adsorption isotherms and models describing them The amount of adsorbate at an interface, r, is usually dependent on the concentration in the bulk, c/j. A plot of /"versus is often called an adsorption isotherm and can be used to describe the thermodynamics of an adsorption process i f equilibrium is reached at all c. Figure 2.9 shows four typical isotherm shapes and the equations that are used to describe them. Figure 2.9a) shows the Freundlich isotherm, which has a concave shape and slowly increasing surface population; p and q are parameters depending on the adsorbate and surface. Figure 2.9b) shows the Langmuir isotherm, which shows the behaviour of surface saturation above a certain bulk concentration. The parameters for this model are K a and rm the latter of which represents the maximum sorbate loading on the sorbent, and is often estimated as Ma, where a is the expected projected area each adsorbate molecule will take up on the surface. Figure 2.9c) shows a high affinity isotherm, which can also be described mathematically with the Langmuir isotherm equation. Finally, Figure 2.9d) shows the linear isotherm, which is valid only at small surface populations since it assumes that there is no limitation on surface sites available for adsorption. A perturbation solution of the Langmuir isotherm for small c, the linear isotherm has one parameter, KH, a Henry's type equilibrium constant that is considered to be the thickness of the layer of solution that contains the same population as is adsorbed on the surface. 25 c) High affinity, monolayer isotherm d) Linear isotherm Figure 2.9 Comparison of four common isotherm types and their equations. 2.2.2.2 Adsorption thermodynamics: the Langmuir and Von Szyckowski equations Equation (2-19) gives the form of the Langmuir adsorption isotherm as it is applied to macromolecule adsorption, is the maximum possible population on the surface and K a is the equilibrium adsorption constant. The fraction of sorbent area occupied by sorbate, 6, is given by Equation ( 2-20 ) and is simply the surface population, r, multiplied by the area per molecule, a. In the Langmuir model, the available sorbent fraction function, <f>, is given by Equation ( 2-21 ) and is simply 1 minus the occupied fraction. The Langmuir model assumes that there are a discrete number of identical sites available for adsorption, that there is no lateral interaction between adsorbed molecules and that there is no conformation change 26 upon adsorption. It was originally devised to describe the adsorption of gases at a metal surface. However, despite its simplicity it has been applied to a wide variety of adsorption processes including protein adsorption. ( r = r„ Ka cb J + Ka cb e = ra </> = \-e (2-19) ( 2-20 ) (2-21) By combining the Langmuir adsorption isotherm with the Gibbs equation, Equation ( 2-16 ), the Von Szyckowski surface equation of state is generated, given in Equation ( 2-22 ), where n is the number of moles of sorbate. This equation relates the equilibrium change in surface free energy, yeq-yo, to the equilibrium surface population, req. At low surface populations, the surface equation of state becomes linear in req as given by Equation ( 2-23 ). Yeq=Yo -nRTr^ In eq 1 v r<*> J Yeq=Yo-nRTrt eq ( 2-22 ) ( 2-23 ) 2.2.2.3 The Langmuir model of adsorption kinetics In many adsorption systems, particularly those involving adsorption of chain molecules, equilibrium is not reached. Measuring the adsorption kinetics is then more relevant. This involves determining the surface population as a function of time, r(t). Adsorption kinetics generally involves three subprocesses: transport to the interface, adsorption to the interface and, in the case of chain molecules such as proteins, conformational rearrangement at the interface. Transport to the interface depends upon the hydrodynamics of the system, but is often limited by diffusion within the surface boundary layer. Adsorption depends on the affinity for the macromolecule to the surface, and the likelihood of both spontaneous and 27 cooperative desorption from the surface. In the case of proteins, the rate and extent of conformation change on the surface is also important and depends on many factors, including the conformational freedom of the polymer chain and the amount of available surface area onto which to spread. The kinetics of surfactant adsorption will be fully discussed in section 2.2.4 along with the models that have been used to describe experimental results. The kinetics of protein adsorption will be discussed in section 2.2.5, along with several different models that have been proposed to describe this more complicated process. Figure 2.10 shows the schematic for Langmuir adsorption. Molecules adsorb to the surface with a rate constant, ka, and desorb with a rate constant, kj. The system will be at equilibrium, i.e. a steady state population on the surface when kaCb kdT. The kinetic equation for the rate of change of the population on the surface, dr/dt, is given by Equation (2-24). Note that for this kinetic model, i f the bulk population is reduced to zero, all the molecules will desorb from the surface back into the bulk. bulk population k. a surface population i Figure 2.10 Schematic for the kinetics of Langmuir adsorption. dr = kacb</>-kdr (2-24) dt 28 2.2.3 Properties of adsorbate molecules In order to demonstrate the capability of the surface energy sensor described in this thesis, two different types of molecules were used as model adsorbates; surfactants and globular proteins. The adsorption of globular proteins to the sensor was of particular interest due to the apparent sensitivity of the surface energy change to sorbate conformation change. The adsorption of surfactants to the sensor surface was also studied to show how adsorption of a simpler molecule compared to the adsorption of globular proteins. This section will describe the structure and solution behaviour of these two types of molecules. 2.2.3.1 Structure and solution behaviour of surfactants The word surfactant is derived from "surface active agents". They are molecules that are amphiphilic, meaning that they have both a hydrophobic and a hydrophilic part. They are most commonly used as detergents and stabilizers for colloidal suspensions. Surfactants typically consist of a hydrophobic tail bonded to a hydrophilic head group. Figure 2.11 and Figure 2.12 show the structures of two surfactants used in this study: Sodium Dodecyl Sulfate, (SDS), an anionic surfactant and Triton X-100, a non-ionic surfactant. O C H 3 C H 2 O U CH 2 GH 2 ChU 0 _ s _ _ 0 " N + ^ n o L / i i o v>no v^rio r\. w V/ \J V/ 1/ \./ hydrophilic C H 2 C H 2 Q hydrophobic tail head Figure 2.11 Sodium dodecyl sulfate surfactant molecule. 29 H 3 C - C - CH 0(CH 2CH 20) - H hydrophilic head hydrophobic tail Figure 2.12 Triton X-100 surfactant molecule. The amphiphilic structure of surfactants makes them highly surface active in aqueous solutions, such that the hydrophobic tail wil l preferentially organize itself to minimize contact with water. At a certain bulk concentration of surfactant in solution, called the critical micelle concentration, cmc, the surfactant molecules spontaneously aggregate to form micelles of very well defined size and stoichiometry. Figure 2.13 shows an example of a micelle. It is a thermodynamically favourable structure as the hydrophobic tails are all hidden from the aqueous environment. At concentrations above the critical micelle concentration, the monomer concentration remains at the cmc while the micelle concentration increases as more surfactant molecules are added to the solution. hydrophobic tail hydrophilic head surfactant molecule micelle Figure 2.13 Surfactant micelle formation. 30 Table 2-2 shows the properties of the two surfactants used in this study. Surfactant SDS Triton X-100 type anionic non ionic M w [g/mol] 288 625 cmc 8x 10"3 mol/L 2 x 10"4 mol/L (pure water) 2.3 mg/mL 0.125 mg/mL Table 2-2 Properties of surfactants used in this study. 2.2.3.2 Globular proteins Proteins are a class of linear polymers of amino acids linked together by peptide bonds. Globular proteins are proteins that fold into compact forms with a volume packing fraction between 0.70 and 0.80. They comprise enzymes and antibodies and many of the proteins important to cellular function. 2.2.3.2.1 Structures The twenty different amino acids found in proteins are shown in Table 2-3 along with their common abbreviations. The different amino acids can either be classed as non-polar and hydrophobic, polar and hydrophilic, positively charged and hydrophobic or negatively charged and hydrophilic. Proteins are therefore both amphiphilic like surfactants, as well as being polyelectrolytes. 31 Amino acid 3 letter abbreviation 1 letter abbreviation Behaviour Glycine Gly G Non-polar Alanine Ala A (hydrophobic) Valine Val V Leucine Leu L Isoleucine He I Methionine Met M Phenylalanine Phe F Tryptophan Trp W Proline Pro P Serein Ser S Polar Threonine Thr T (hydrophilic) Cysteine Cys C Tyrosine Tyr Y Asparagine Asn N Glutamine Gin Q Aspartic acid Asp D Electrically Charged Glutamic acid Glu E (negative, hydrophilic) Lysine Lys K Electrically charged Arginine Arg R (positive, hydrophobic) Histidine His H Table 2-3 The 20 amino acids and their abbreviations. Protein structure can be described in levels of increasing complexity. Primary structure describes the basic sequence of amino acid monomers and the location of any disulfide bridges linking pairs of cysteine amino acids in the chain. Secondary structure describes the local arrangement of the polypeptide backbone and any areas of local structure such as a-helix or /?-sheet formations. Tertiary structure describes the overall three dimensional topology of the protein. 2.2.3.2.2 Conformations and the forces that determine them The linear chain of the protein can be folded into an almost infinite number of shapes or conformations, however many of these conformations are energetically unfavourable. The native protein structure is that which it will form in an aqueous environment to provide its biological function. There are four main forces that act between the amino acid residues in 32 the protein chain and the solvent that determine the native protein structure it wil l form in an aqueous solution; hydrogen bonding, hydrophobic interaction, ionic pairing and the formation of disulfide bridges. Competing with the above forces that determine protein structure is the loss of conformational entropy due to the formation of an ordered, compact structure. This entropy loss is usually balanced by the increase in entropy caused by the dehydration of hydrophobic residues as they situate themselves in the interior of the folded molecule.27 As such, the folded structure is thermodynamically stable but changes in the environment of the molecule can easily disrupt the protein structure. A globular protein can be denatured, i.e. lose its compact globular form, by heating, chemical interaction or adsorption to a surface. In each of these cases the thermodynamic environment of the protein changes, such that the protein may now change its conformation to access a lower energy state. For example, Figure 2.14 shows the heat capacity of lysozyme as a function of temperature measured by differential scanning calorimetry. As can be seen, the heat capacity has a prominent endothermic peak around 70°C, corresponding to the denaturing of the molecule. - 1 U H 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20 30 40 50 60 70 80 90 Temperature (°C) Figure 2.14 DSC thermograph of unfolding for hen egg-white lysozyme. 33 2.2.3.2.3 Composition, native-state structure and solution properties of hen egg-white lysozyme (HEWL) In this experiment, a globular protein called Hen Egg White Lysozyme (HEWL) was used. For the remainder of this thesis it shall be referred to as simply "lysozyme". It is considered a model protein because its structure is well understood and documented. It is relatively small with a molecular weight of 14,600 atomic mass units. Table 2-4 gives some of the physiochemical properties of lysozyme in aqueous solutions. 2 8 The isoelectric point is the pH at which the positive and negative charges on the protein are balanced. The Gibbs energy of denaturation is the change in Gibbs energy from the native state to the denatured state and is a measure of the thermodynamic stability of the native state. property symbol value Molecular weight Dimensions Specific density Isoelectric point Eisenberg scale hydrophobicity index Diffusivity in water Gibbs energy of denatuaration (25°C, pH7) (thermal) (denaturant) P Pi D AN-DG 14600 g/mol 45A x 30Ax30A -1.4 x 10 3kg/m 3 -11.1 -7.6 J/g (total) 1.04 x lO - 1 0 m2/s 4.1 J/g 4.0 J/g Table 2-4 Physiochemical properties of lysozyme. Lysozyme is an enzyme found in egg whites whose function is to break down the polysaccharide walls of many bacteria, and as such provides protection against infection. Figure 2.15 shows the primary structure of lysozyme with the different amino acids labeled 34 according to Table 2-3 and the hydrophobic residues shaded grey. Since lysozyme has a relatively small molecular weight, it cannot form a conformation whereby all hydrophobic residues are shielded on the inside of the molecule. The active site where the polysaccharide bond is attacked is inside the cleft of the protein. Figure 2.16a) shows a ribbon diagram of the lysozyme backbone and its /?-sheet and cr-helix regions. Figure 2.16b) shows a space-filling model of the tertiary structure where the polar/charged residues are shown in dark grey and the hydrophobic residues are shown in light grey.29 Both representations of the lysozyme molecule are shown in the same orientation. 35 a) b) Figure 2.16 Lysozyme a) secondary structure and b) tertiary structure. 2.2.4 Adsorption of surfactants at air/liquid and liquid/solid interfaces The adsorption of surfactants at the air/water interface has been extensively studied due to the ease of measuring the change in the surface tension of the air/water interface.3 0'3 1'3 2 Surfactant adsorption at this interface is usually reversible and may therefore be treated as an equilibrium process. Langmuir adsorption isotherms generally provide a good description of the data. Lin et al. found good agreement fitting the Langmuir isotherm to data generated from the adsorption of Triton X-100 at the air/water interface, with rx = 2.91xl0"6 mol/m2 and Ka=l.5xl0 m /mol. Figure 2.17 shows the shape of the isotherm generated using these parameters as an example of the adsorption behaviour at the air/water interface. 36 2.0 0 0.002 0.004 0.006 0.008 0.01 Cb [mg/mL] Figure 2.17 Langmuir adsoprtion isotherm for Triton X-100 at the air/water interface. At the solid/liquid interface, equilibrium studies have been performed by solution depletion methods as will be described in section 2.2.6.1. For example, S.-Y. Lin et al.M performed a study to measure the adsorption of SDS onto polystyrene latex particle suspensions by measuring the change of surface tension of the solution interface and calculating the amount adsorbed from mass balance considerations. The isotherms generated were in good agreement with the Langmuir isotherm model. Kinetic data have been difficult to obtain because of the short time constants involved, and the fact that the adsorption is fundamentally limited by the hydrodynamics of the experimental system. This means that in order to solve for the kinetic rate constants, the hydrodynamics of the system must be well understood. For example, Brinck et al.35 used ellipsometry to measure the adsorption of a non-ionic surfactant at a silica surface. They found that once surfactant was added to the system, equilibrium surface concentrations were achieved quickly on the order of 5 minutes. They found their experimental system to be completely reversible with respect to dilution of the bulk solution concentration. Similarly, Geffroy et al. measured the adsorption of nonionic surfactants to hydrophobic polystyrene using reflectometry. Again, they found the surface concentration to equilibrate quickly, within a few minutes, but found incomplete desorption upon dilution of the bulk solution 37 concentration. However, they attributed the residual adsorbed amount to be within the error for an adsorbed amount corresponding to a low bulk surface concentration. However, surfactant adsorption to solid surfaces is not always as simple as the above examples would indicate. Firstly, it is not always reversible. Romero-Cano et al31 measured the adsorption and desorption of Triton X-100 to polystyrene particles by solution depletion methods. In all cases they found a residual amount of surfactant adsorbed despite multiple washing steps. Secondly, the formation of surface micelles or hemi-micelles is possible.38 Furthermore, in the case of SDS adsorption, it may be complicated by the fact that SDS will hydrolyze to form dodecanol that also adsorbs at the surface. For example Turner et al.39 measured the adsorption of SDS to polystyrene using neutron reflection and attenuated total reflection infrared spectroscopy, they found that the hydrolysis of SDS affected the adsorption kinetics below the cmc, and only purified solutions of SDS did not exhibit this behaviour. 2.2.4.1 Current models for adsorption thermodynamics and kinetics Adsorption of surfactants at the air/water interface is most commonly measured by determining the change in surface tension of the interface. Therefore, equilibrium isotherm models are combined with the Gibbs adsorption law, Equation ( 2-14 ), to produce a surface equation of state for the equilibrium surface tension, y. Note that this only applies below the critical micelle concentration, as above this concentration very few monomers exist in solution that may contribute to adsorption. The most commonly used surface equations of state are the von Szyckowski equation, Equation (2-22 ), and the linear form, Equation ( 2-23 ), generated from the Langmuir and linear isotherms respectively. For non-ionic surfactants on non-ideal surfaces the Frumkin isotherm, Equation ( 2-25 ), has also been used , where KF is the Frumkin equilibrium adsorption constant and A is a parameter relating to the ideality of the surface (A = 0 for an ideal surface). The corresponding surface equation of state for the Frumkin isotherm is given in Equation ( 2-26 ). The Frumkin isotherm has also been adapted for use with ionic surfactants by taking into account the electric double layer.40 38 (2-25) ro-y = -nRTrao\n\i-f eq CO J — nRTArc 2 CO ( 2-26 ) Models for surfactant adsorption kinetics are generated by assuming that the dynamic surface tension, y(t), of the interface is related to the surface population, r(t), in the same way the equilibrium values are related in the surface equation of states, although the mass transport of the molecules to the interface has to be solved in order to fully interpret the kinetic parameters. At the solid/water interface the kinetics of surfactant adsorption appears to be more complicated as several investigators have reported the formation of micelles at the interface even for concentrations below the cmc. 4 1 , 4 2 This effect appears to be more pronounced at hydrophilic surfaces. Several investigators have formulated models to describe the kinetics of surfactant adsorption for their particular combination of surfactant/surface/experimental method. For example, Brinck et al. devised a model that described surfactant adsorption from a micellar solution through a stagnant boundary layer to a silica surface. 2.2.5 Adsorption of globular proteins at solid/liquid interfaces Due to the complexity of protein molecules, there are many different driving forces for protein adsorption at interfaces. These include, hydrophobic, electrostatic and conformational entropy gain. Therefore, protein adsorption behaviour varies with different proteins and different surfaces. The following section will give a broad overview of protein adsorption behaviour to solid surfaces. 2.2.5.1 Experimental studies: protein adsorption behaviour The most notable characteristic of protein adsorption on strongly attractive surfaces is that it is generally irreversible with respect to dilution of the bulk solution concentration. 39 Ascending isotherms which are measured as bulk concentration is increased are well described by the Langmuir isotherm equation, Equation ( 2-19 ). However, descending isotherms that are measured as the bulk solution concentration is diluted show hysteresis, indicating that a fraction of the adsorbed protein molecules cannot be desorbed under thermodynamically favourable conditions. An extreme example of this is given by Norde4 3 who showed the ascending and descending isotherms for Bovine Serum Albumin (BSA) on borosilicate powder. The descending isotherm indicated that the BSA has a high affinity for the surface and no desorption was seen to occur. The apparent irreversibility of protein adsorption on solid surfaces is unusual in that the finite initial slope of the ascending adsorption isotherm suggests apparent equilibrium surface concentrations are achieved. The Langmuir kinetic model, Equation ( 2-24 ), predicts that the equilibrium surface concentration, req, will be reached at bulk concentration, Q,, when dr/dt = 0 and kaCb (/> = kd req, i.e. the on-rate and off-rate of the protein adsorption balance each other out. However, i f protein adsorption is irreversible, then kd = 0 and the surface should become saturated at any bulk concentration. The fact that this does not happen is the fundamental paradox of irreversible protein adsorption, and much of the protein adsorption research community is working to understand this phenomena. Complicating the understanding of protein adsorption is the fact that while in general protein molecules appear to adsorb irreversibly, exchange reactions occur between protein molecules in the bulk and on the sorbent. This has been experimentally observed with use of radio-labeled protein molecules. For example, Ball et al44 studied the exchange of radio-labeled Human Immunoglobulin G (IgG) with unlabeled IgG adsorbed on the surface of latex particles. In this way they were able to measure an exchange rate constant and determine that the exchange process was first order with respect to the concentration of molecules in the bulk solution. This means that while protein molecules are unlikely to desorb spontaneously when protein molecules are removed from the solution phase, there is some sort of cooperative reaction that allows protein molecules at sufficient concentration in the bulk to contribute to desorption. A possible explanation for this effect has been given by Nadarajah et a/. 4 5 who suggest that protein molecules in the bulk can cluster around an adsorbed molecule changing the hydrophobic forces in its immediate vicinity, making it favourable to ' ' 40 desorb. Without these protein molecules in the bulk solution, a kinetic barrier is present that favours the protein to shield its hydrophobic areas from the aqueous solution by staying adsorbed to the surface. The irreversibility of protein adsorption is thought to be connected to the conformation change of protein molecules as they adsorb and form more contacts with the surface. There have been numerous studies that show that protein conformation changes upon adsorption to solid surfaces.46 For example, results from Su etal41 indicate that the thickness of the adsorbed layer of BSA on the.silica/water interface was significantly less than that expected from a BSA molecule in its native state, indicating that conformational changes had occurred. As described in section 2.2.3.2.2, the balance of forces that stabilize the native protein structure in solution is disrupted when the protein adsorbs to an interface. The hydrophobic effect is a strong driving force for both adsorption and subsequent conformation change as the protein shields its hydrophobic residues from the aqueous environment. This is supported by experimental evidence of a greater degree of protein conformation change at hydrophobic surfaces. Furthermore, when comparing adsorption of the same protein on surfaces of increasing hydrophobicity, the amount of adsorbed protein tends to increase as the surface becomes more hydrophobic. For example, Kondo et al. adsorbed ^-amylase to ultrafine silica particles whose surfaces had been modified to change their surface hydrophobicity. Not only did they find an increased adsorbed amount with increasing hydrophobicity but also an increased degree of conformation change. Proteins and their adsorption behaviour can be broadly classified based on their conformational stability. Conformationally stable protein molecules such as lysozyme at pH 7 maintain much of their globular native-state conformation upon adsorption. Protein molecules with lower native-state stabilities such as Human Serum Albumin (HSA) have more conformational freedom and are much more likely to radically change their conformation upon adsorption. For example, Norde and Favier49 measured the loss in the a-helix secondary structure of lysozyme and HSA as they adsorbed to silica particles. They found that adsorption of lysozyme at pH 4.0 resulted in a change of 32% to 25% a-helix 41 content within the protein molecule at the plateau surface concentration, while adsorption of HSA at pH 7.0 resulted in a much larger change of 74% to 38% or-helix content at the plateau surface concentration. Evidence that a slow conformation change during protein adsorption is linked to the irreversibility of the adsorption process is given by experiments that measure adsorbed amounts as a function of time and as a response to bulk concentration dilution. Wahlgren et al.50 performed a study that measured the amount of lysozyme adsorbed to a silicon oxide surface in the presence of a pH 7, phosphate buffer. For a range of bulk concentrations (0.001 mg/mL to 1 mg/mL) the plateau surface population was reached within 30 minutes. At a bulk concentration of 1 mg/mL, adsorption irreversibility was then assayed by rinsing the sorbent following specific contact times with the bulk protein solution. For a rinse performed after 15 min., -30% of the adsorbed amount was removed, while after 60 min., only -15% of the adsorbed protein molecules rinsed off. The authors concluded that the protein molecules were slowly becoming irreversibly bound due to a slow conformation reaction. 2.2.5.2 Current models for adsorption thermodynamics and kinetics Protein adsorption models are devised to predict the amount of protein that adsorbs under different experimental conditions. There are numerous examples of different protein adsorption models that have been proposed to predict the kinetics of protein adsorption and its apparent irreversibility. While individual researchers often devise their own models to describe highly specific results garnered from their experimental method, the following section outlines some current models that have a broader applicability and how these models attempt to capture the progressively irreversible nature of the protein adsorption process. 2.2.5.2.1 Modified Langmuir kinetics A common modification to Langmuir adsorption kinetics to account for protein irreversibility is to add a second type of surface population, which is irreversibly bound to the surface. This second type of surface population is usually assumed to have undergone conformation changes and have a larger area of interaction on the surface than those molecules in the 42 reversibly bound population, {0:2 > a{). The rate constant for the transition from reversibly bound molecules to irreversibly bound is given by kirr. The schematic for this kinetic model is given in Figure 2.18 and the kinetic equations are given in Equation ( 2-27 ). Note that for this case the fraction of available sorbent surface, <f>, is now defined according to Equation (2-28). bulk population reversibly bound surface population irreversibly bound surface population Figure 2.18 Schematics for the kinetics of irreversible adsorption from modified Langmuir model. d£\ dt = ka cb<j)-kdrx -kirrrx dr2 dt - kirr / " j ( 2-27 ) (t> = \-r1al-r2a2 (2-28) Note that this model, although it accounts for the irreversibility of protein adsorption, does not have the behaviour that the adsorbed amount wil l be dependent on the bulk concentration. This is demonstrated in Figure 2.19, where the total fractional surface concentration as a function of time is shown for a range of bulk concentrations. It can be seen that at any bulk concentration eventually all molecules wil l become irreversibly bound and the surface will proceed to saturation. 43 time Figure 2.19 Fractional surface coverage as a function of time for modified Langmuir model shown in Figure 2.18. 2.2.5.2.2 Random sequential adsorption with post adsorption transition Random sequential adsorption (RSA) treats the protein molecules as hard particles that are adsorbed sequentially on the surface. The adsorption position is selected randomly and the particle adsorbs only i f the selected adsorption site does not overlap with an already adsorbed particle. Van Tassel51 devised a model whereby protein molecules are modeled as disks with diameter da that adsorb with rate ka according to RSA. Upon adsorption they may either desorb with rate kd or spread instantaneously to a disk with a larger diameter, dp, with rate ks. This post-adsorption transition may only occur i f there is sufficient space for that particular particle to spread to a larger diameter. The kinetic rate equations for this model are given in Equations ( 2-29 ) and ( 2-30 ), where <f>a is the probability that an adsorbing particle lands in an unoccupied space, y/ap is the probability that the spreading of the molecule is not blocked by adjacent adsorbed molecules and ra and fp are the surface populations of the two types of molecules. 44 ^ - = kacb<l>a-ksra^ap-kdra ( 2 - 2 9 ) — = ksray/ap ( 2 - 3 0 ) A method to calculate the probability functions <j)a and y/ap, is given by Brutasori and VanTassel52 using scaled particle theory. Calonder and Van Tassel53 later published a paper whereby they coupled this model with mass transport considerations to test the model under real experimental conditions. However, model fits were only made during the adsorption phase and no attempt was made to model the data upon bulk concentration dilution. Note that although this model is a more sophisticated approach to irreversible adsorption, it has the same basic form as the modified Langmuir kinetics above, and will therefore also have the characteristic that the surface will eventually saturate even at low bulk solution concentrations. 2.2.5.2.3 Wahlgren model Wahlgren et al.50 devised a model that attempted to account for the following protein adsorption behaviours. Firstly, it allowed for two types of adsorbed molecules: reversibly bound and irreversibly bound. Secondly, it allowed for an adsorbed amount dependent on the bulk concentration at infinite times. Thirdly, it allowed for multilayer adsorption, since they concluded from their data that dimer formation was likely occurring. They accomplish this by having a four state surface model, whereby, 0] is the reversibly bound state, 62 the irreversibly bound, 63 a reversibly bound state formed in a second layer and 64, a irreversibly bound state formed in the second layer. The rate of adsorption into the first reversibly bound state is given by k\ and the rate of desorption of this state is given by r/. The rate of transformation from reversibly bound to irreversibly bound in the first layer is given by s. An exchange reaction is allowed between irreversibly bound molecules and bulk molecules at rate, e. Finally, the rate of adsorption and desorption into the second layer for both reversibly and irreversibly bound molecules is given by k^ and rs respectively. The kinetic rate equations for this model are given by Equations ( 2-31 ) to (2-34 ). 45 1 = kjC(l -0X- 02)-(n + s)9i + eC02 (2-31) 802 s9x-eC02 dt (2-32) d03 = k3C(Ox dt (2-33) d04 = k3C(02 04)-r303 -eC04 dt (2-34 ) With this model they found it difficult to reproduce the fast adsorption rates at low bulk concentrations, so they incorporated adsorption and desorption rate constants that were dependent on the adsorbed amount according to Equations ( 2-35 ) and ( 2-36 ). With this model they were able to get good agreement with their experimental results for both adsorption and desorption regimes. However, they don't account for an increase in area due to protein spreading, or give a physical explanation of why the rates may be dependent on the adsorbed amounts as shown in Equations ( 2-35 ) and ( 2-36 ). 2.2.5.3 Protein adsorption to polymer surfaces Silicone elastomers have long been thought of as good candidates for biomaterials due to their low surface energy and apparent inertness. However, there have been recent studies investigating the conformational stability of proteins adsorbed onto silicones. These studies have found that not only do proteins adsorb at the silicone/aqueous interface, but the nature of the surface itself induces a greater degree of denaturation than observed at other solid/water interaces.54 k(0)=ke ad (2-35) r(0)=reP* (2-36) 46 Green et al. investigated the adsorption of lysozyme using FTIR to a variety of different polymeric surfaces including a hydrophobic elastomeric polymer.55 They found a greater degree of unfolding on the elastomeric surface and attributed this to the fact that the elastomer can be considered more of a hydrophobic liquid than a polymer that is below its glass transition temperature. 2.2.6 Experimental methods to measure and describe adsorption Traditionally, protein adsorption has been characterized by measuring isotherms. In section 2.2.6.1, the most common method to measure equilibrium isotherms, solution depletion, will be discussed. When adsorption is irreversible, a potentially more useful measurement is adsorption kinetics, and techniques capable of making kinetic measurements will be discussed in section 2.2.6.2. Many of the techniques for kinetic measurements are also used to produce isotherms. Most of these methods are primarily sensitive to the amount of protein in the adsorbed layer; however, a few can also offer some clues to adsorption induced conformation change. In section 2.2.6.3, methods that provide only information about the protein's average conformational state will be discussed. Finally, in section 2.2.6.4, two methods will be discussed that measure changes in surface energy at the solid/liquid interface. 2.2.6.1 Equilibrium methods Solution depletion is often used to characterize the adsorption of protein molecules (and other easily assayed analytes) at the solid/liquid interface.56 A protein solution of known total concentration, c0, is allowed to equilibrate with a suspension of particles presenting a large surface area, A, to volume, V, ratio. After a fixed amount of time, a sample of the solution is taken and its bulk concentration, c&, is measured. The assumption is that adsorption onto the large surface area causes a significant amount of protein to be depleted from the solution, i.e. enough that is measurable. A mass balance will allow for the calculation of how much protein is adsorbed on the particles, r, according to Equation (2-37). 47 r = ( c o - ^ ) -A ( 2 - 3 7 ) The bulk concentration, ct, can be determined through a variety of methods. For relatively high bulk concentrations, the intrinsic absorbance or fluorescence of the macromolecule can be measured with a spectrometer. For lower bulk concentrations, the macromolecule may be labeled with either a radioactive or fluorescent marker to increase the sensitivity of detection. 2.2.6.2 Kinetic methods Some commonly used kinetic methods that are sensitive only to adsorbed amounts include surface plasmon resonance (SPR)5 7, quartz crystal microbalance (QCM) 5 8 and total internal reflection fluorescence (TIRF) 5 9. Some kinetic methods that can extract information about adsorbed amounts and protein conformation include ellipsometry50 and neutron reflection47. In ellipsometry, the refractive index and thickness of an adsorbed layer are probed by means of measuring changes in the state of polarized light upon reflection. Giacomelli et al.60 measured the adsorption of BSA onto hydrophobic TiC>2 using ellipsometry. They reported two stages to the observed adsorption. The first was fast and caused the surface to be saturated with protein, the second was much slower and caused the thickness of the layer to increase with time, but the adsorbed amount remained constant. Neutron reflection is similar to ellipsometry, with the exception that the wavelength of the neutron beam is much smaller than that of visible light and can therefore investigate the structure and composition of films of molecular dimensions with a higher resolution. Specular neutron reflection results from Su et al.41 indicate that the thickness of the adsorbed layer of BSA on the hydrophilic silica/water interface was significantly less than that expected from a BSA molecule in its native state, indicating that conformational changes had occurred and the protein had most likely spread on the surface. The difference between these two results, thickness increase vs. thickness decrease for the same molecule, BSA, is possibly due to the fact that in the first case the surface was hydrophobic and in the second it was hydrophilic. 48 2.2.6.3 Monitoring changes in sorbate conformation A few techniques exist which can extract some information about the conformation of the adsorbed layer such as fourier transform infrared spectroscopy (FTIR) 6 1, circular dichroism (CD) spectroscopy49 and differential scanning calorimetry (DSC) 2 8. FTIR measures the absorbance spectrum in the infrared region when molecules are adsorbed on a flat crystal and light is reflected off the interface. Information about the a-helix and P-sheet structure of the molecule can be garnered from the FTIR spectrum. CD spectroscopy is an ultraviolet spectroscopicmethod that shows the difference between the adsorption of left and right circularly polarized light. It compares the features on the CD spectrum due to specific structural features of the native protein to the CD spectrum of the adsorbed state. In this way it can probe the type and amount of protein secondary structure. DSC is based on the fact that protein unfolding is an endothermic process. A thermogram of unfolding is measured which gives two parameters that relate to the stability of the protein: Tm, the temperature at which the heat capacity reaches a maximum and AH, the enthalpy change. The evidence that there can be a conformation change upon adsorption to a solid has been garnered from the methods described above. Yokoyama et al.62 measured the FTIR spectrum when lysozyme adsorbed on to a hydrophobic polymer. They found that adsorbed molecules had a higher /9-sheet content and lower a-helix content than the molecules in solution. Kondo and Fukada63 measured adsorbed amounts of human hemoglobin on ultra fine silica particles by both circular dichroism and solution depletion methods as a function of time. While the adsorbed amount derived from the solution depletion experiments equilibrated within 20 minutes, the observed conformation changes from the CD spectra continued for many hours. Billsten et al.64 performed DSC and CD experiments of the adsorption of various mutants of varying native-state stability of human carbonic anhydrase I I (HC Al l ) on silica particles. The mutants exhibited thermal transitions after adsorption, indicating them to be much less stable and more prone to post adsorption conformation changes. 2.2.6.4 Surface energy methods As described in section 2.2.1.3, the measurement of the surface energy at the solid/liquid interface is technically challenging. However, there are a few experimental techniques that 4 9 are capable of measuring changes in the surface energy at the solid/liquid interface as adsorption occurs. Since, the experimental technique described in this thesis is based on the change in surface energy at the solid/liquid interface, these methods will be described in more detail than other experimental methods described in sections 2.2.6.1, 2.2.6.2 and 2.2.6.3. These methods are particularly relevant, as the results garnered from them seem to suggest that they are sensitive to protein conformation change due to protein adsorption. 2.2.6.4.1 Axisymmetric drop shape analysis by profile The first method, axisymmetric drop shape analysis by profile, ADSA-P 6 5, is based on the contact angle method described in section 2.2.1.3. An image of a liquid droplet on a well defined flat surface is filmed and digitized. Therefore, the drop shape and contact angle, 9, can be calculated as a function of time. I f there are protein molecules dissolved in the liquid, they will adsorb at both the liquid/vapour interface and the solid/liquid interface. The droplet is enclosed in a chamber with 100% humidity so that there is no evaporation. Adsorbing protein molecules at the liquid/vapour interface will change the shape of the drop according to Equation ( 2-38 ), which is the Laplace equation of capillarity, where yv is the surface energy of the liquid/vapour interface, AP is the pressure change across the interface and Ri and R2 are the principal radii of curvature of the drop. Adsorbing protein molecules on the solid/liquid interface will change the contact angle, 9, that the drop makes with the solid interface. The interfacial free energy of the solid/liquid interface, ysi(t), can then be obtained from Young's Equation as given in Equation (2-39), where ysv(t) the interfacial free energy of the solid/vapour interface is assumed to be constant during the course of the experiment. In this way, the surface energy of the solid liquid interface can be monitored as a function of time as adsorption proceeds. rsi{t)=rSv{t)-riv(thos9(t) ( 2-38) (2-39) 50 Van der Vegt et al. measured the interfacial free energy changes due to the adsorption of human immunoglobulin G, IgG, onto fluoroethylenepropylene-Teflon using ADSA-P 6 5. The amount of adsorbed IgG for a specified adsorption time was measured using drops of solution made with radiolabeled IgG, and counting the surface activity after rinsing. They found that ys\ and yv continued to change for at least an hour, while the radiolabeling experiment confirmed that steady state surface concentrations are achieved within 30 min. This implies that the slow changes were due to conformation changes in the adsorbed protein molecules rather than continued protein adsorption. When applying the Gibbs adsorption equation, Equation ( 2-16 ), to their data to solve for the adsorbed amounts, it did not match the adsorbed amounts they had measured in the radiolabeling experiment. They therefore concluded that it was not a relevant treatment due to the irreversible nature of protein adsorption. Noordmans et al. describe a method whereby they can simultaneously perform ADSA-P and ellipsometry on a sessile solution drop.6 6 This is a true simultaneous experiment, whereas for the method described above, the radiolabeling experiment was performed separately. They find that interfacial tension at the solid/liquid interface continues to change for at least 2 hours, while the adsorbed amounts reach a stationary state within 20 min, again indicating that the method is sensitive to protein conformation change. While this method shows promise, there are several problems with it that make it difficult to get accurate results for the change in interfacial surface energy. Firstly, the droplet often changes its contact area during the course of the experiment, meaning that certain assumptions made in the calculation of yv with Equation ( 2-38 ) are invalid. Secondly, the hydrodynamics of mass transport to both interfaces are quite complicated and unaccounted for in the analysis. Lastly, due to the finite amount of time (seconds) it takes to initially form the droplet, t = 0 is ambiguous and could cause a fairly large error in calculated values of 2.2.6.4.2 Micro-cantilevered biosensor The second method involves measuring the radius of curvature of an atomic force microscopy micro-fabricated silicon nitride cantilever. This method is based on Stoney's 51 equation given in Equation (2-40 ), where R is the radius curvature, vis the Poisson's ratio, E the elasticity of the cantilever, / is the thickness and Acxi and Ao~2 are the changes in the surface stresses on the top and bottom sides of the cantilever. Figure 2.20 shows a schematic of a cantilever with molecules adsorbed on the bottom surface. In this figure, the adsorbed molecules cause a compressive stress on the bottom surface compared to the top, causing the cantilever to bend. This method requires functionalizing the top surface in such a way as to prevent adsorption on this surface, as only a differential in the change in stress between the two sides wil l cause the cantilever to bend. By enclosing the cantilever in a flow cell, protein molecules can be added to the environment immediately surrounding the cantilever and the adsorption of protein molecules as a function of time can be studied. (2-40) Figure 2.20 Microcantilevered stress sensor. Moulin et al.6S used a silicon nitride microcantilever that was sputter coated with a thin layer of hydrophilic gold on the top surface and functionalized on the bottom surface with an inert thiol, (HS(CH2)io(C02C)2COCH3) to prevent protein adsorption. They measured the cantilever deflection for the addition of both IgG and BSA into the flow cell surrounding the 5 2 cantilever. In the case of IgG, a slow decrease in the surface stress was observed for 11 hours, with a total change of Aa=- 0.2 N/m over that time. On the other hand, BSA caused a slow increase in the surface stress over the same time frame, with a total change of A<J= + 0.08 N/m over 11 hours. While this method is very promising for the study of conformation change due to protein adsorption, there seems to be no recent developments in the literature beyond the paper cited above. Since, the microcantilever could be treated with a wide variety of surface treatments the adsorption of different proteins could be studied on different surfaces. Both of these methods show that it is possible to be sensitive to the slow conformation change of adsorbed protein molecules when measuring the changes in surface energy as adsorption proceeds. This supports the fundamental claim of this experimental work, namely that the adsorption sensor described herein is capable of detecting conformation changes in the adsorbed molecules at its surface through changes in surface energy at its interfaces. 53 3 EXPERIMENTAL INVESTIGATIONS The goal of this experimental program was to measure the interaction of molecules with the surface of a thin, elastomeric membrane as they adsorbed to its surface to learn more about the kinetics of adsorption and conformation change. There are two types of measurements that could be done simultaneously to extract information about molecular adsorption to the surface of the sensor. The primary measurement involved monitoring the change in surface vibrational resonant frequency of the membrane as molecules adsorbed. As described in the introduction, this change in resonant frequency is due to changes in surface energy upon adsorption. Mass loading on the surface contributes a negligible effect. Results obtained during the adsorption of protein molecules to the sensor showed that the sensor continued to have a substantial signal change well after the population on the surface should have equilibrated. It seemed likely that the conformation change of adsorbing protein molecules was contributing to a slow change in the resonant frequency signal, therefore a second measurement was devised to independently monitor the amount of protein molecules on the surface of the membrane. This was accomplished by tagging the protein molecules with a fluorescent molecule that is excited by laser radiation. When the sensor membrane was used as a waveguide for this excitation radiation, any protein molecules adsorbing to the membrane surface absorbed laser radiation from the evanescent wave of the waveguide and emitted light at a higher wavelength. The amount of light that was emitted at this wavelength was proportional to the amount of protein molecules adsorbed to the surface. By simultaneously performing both of these measurements, it was possible to verify that the resonant frequency measurement was indeed sensitive to the surface energy, which itself was a function of the conformation change in the protein molecules. This section wil l describe the implementation of both of these kinds of measurements. The construction of the sensor membrane and how it was actuated and monitored to measure the change in resonant frequency will be outlined. Furthermore, it wil l be shown how the sensor membrane was modified so that it could be used as a waveguide for the surface fluorescence measurement. The fluid delivery system that pumped solution into the basic experimental cell under controlled, well-defined hydrodynamics, and allowed for flushing of the 54 experiment with clean protein-free solution will be described. Finally, all the peripheral instrumentation and the control protocols that were used to probe the behaviour of the system will be outlined. 3 .1 Configuration for resonant frequency measurement Figure 3.1 shows the basic experimental configuration for the resonant frequency measurement; including the sensor membrane, the means to actuate it and a means to measure its response. A thin membrane of elastomeric material, the sensor membrane, was mounted on an annular support and immersed in an aqueous solution. The annular support was positioned over an audio acoustic transducer that excited the membrane to resonate. The vibration of the membrane was detected with an optical vibration detection probe; the vibration sensor. This entire configuration was immersed in an aqueous solution in the experimental cell, a -250 mL glass beaker. vibration sensor experimental cell aqueous solution support ring and sensing membrane acoustic transducer 1.5 mm 0.25 mm u 2 mm! -v- vibration sensor \ sensing membrane -/acoustic transducer 4 mm Figure 3.1 Basic experimental configuration to measure change in resonant frequency. 55 3.1.1 Sensing membrane The sensing membrane must be a thin, uniformly thick sheet of elastomeric material. Results have been achieved using a variety of different elastomeric materials during the course of this investigation including silicone rubber, polyurethane rubber and acrylic elastomer. However, the two former elastomers produced films that weren't reproducible in terms of thickness or tension and hence were not persued as serious candidates for the sensing membranes needed in this study. Initially, a small amount of a two-part silicone rubber69 was sandwiched between two sheets of mylar with spacers used to define the thickness. However, since these sheets were rolled by hand and cured under a variety of different conditions, there was an unacceptably large variability in thickness and inherent tension. Furthermore, uncured silicone oils remaining in the membrane had a tendency to leach out while immersed in aqueous solutions. Polyurathane rubber films were created by first dissolving pellets of the rubber in isopropyl alchohol and then allowing a small drop of this solution to spread on the surface of water. However, the difficulties in mounting this delicate film on an annular holder produced an unacceptable variation from sample to sample. A source of uniformly thick elastomeric films was found by using a commercially available, optically clear, elastomeric transfer adhesive made from 2000MP Acrylic. The product is supplied sandwiched between two sheets of plastic release liner material and is available in two thicknesses, 25 um 7 0 and 50 urn 7 1. Since it is adhesive, it was easy to mount on the membrane holders without the need to use excess glue material. 3.1.2 Sensing membrane support In order for the resonant modes of the membrane to be well defined, the sensing membrane had to be mounted on an annular support with clean, sharp edges on the inner diameter. Figure 3.2 shows a diagram of the annular supports used to mount the adhesive films. The film was sandwiched between two similar annular supports, the bottom with screw holes for 56 attaching the legs and the top with clear holes. Since the sensing membrane was an adhesive elastomeric material, it was held securely between the top and bottom support. The annular supports were machined from stainless steel and were approximately 2mm thick. The legs were also made from stainless steel and screwed into the holders such that the top of the cone of the acoustic transducer was within a fraction of a millimeter from the bottom surface of the sensing membrane. This ensured that there was good acoustic coupling between the two and there was sufficient amplitude at the resonant frequency to get a good signal to noise ratio. Top holder r Bottom holder screw holes Figure 3.2 Annular supports for mounting sensing membrane The adhesive film material was mounted on the annular supports using the following procedure: A piece of the transfer adhesive film was cut of sufficient size to cover the annular supports shown in Figure 3.2. One side of the release liner was removed and the bottom annular support was firmly pressed onto the exposed adhesive film. Because the film was an elastomeric material, it was somewhat challenging to then remove the backing side of the release liner. However, i f the indentation formed by the ring of the holder was filled with water and then frozen, the remaining release layer could be easily peeled off. When the ice melted, this left the sensing membrane freely mounted on the annular holder. The release liner could also be easily removed i f the holder and film were cooled down to a temperature that was below the glass transition temperature of the elastomeric film. This was done by using liquid nitrogen to cool a slab of metal, such as brass, and placing the holder and 57 sensing membrane onto the slab. The release liner could then be peeled off as long as care was taken not to shatter the brittle, cold membrane. The top holder was then placed carefully over the bottom holder to ensure that the screw holes and inner diameters lined up and pressed firmly together. The legs were attached and the entire holder and sensing membrane was then ready to be placed into the experimental cell. 3.1.3 A c o u s t i c t ransducer An acoustic transducer suitable for immersing in an aqueous solution was constructed as shown in Figure 3.3. A small audio speaker was disassembled from a portable headphone set, inserted in a short tube and sealed underneath and mounted with silicone elastomer. It was then sealed on top with another thin smooth layer of silicone elastomer. This layer had to be thick enough to seal the speaker from the aqueous solution, but thin enough so as not to impede the vibration of the speaker, (~Vi mm). A plastic cone, cut from the tip of a plastic syringe, was mounted on top of the transducer. This cone focused the acoustic energy onto the most sensitive region near the center of the sensing membrane. This arrangement ensured good coupling with the circularly symmetric membrane modes. A thin plastic tube was inserted into the smooth side of the cone and attached to a syringe. This allowed for the removal of any trapped air in the cone when the transducer was immersed in the aqueous solution in the experimental cell. I.D = 2 mm cone for focusing acoustic energy — headphone speaker — mounting silicone elastomer sealing silicone elastomer v 15 mm V f -0.5 Figure 3.3 Acoustic audio transducer. 58 The audio acoustic transducer consisted of a magnetically driven, small speaker that is designed to operate up to a frequency of 3kHz while operating in air. By immersing the speaker in an aqueous medium one would expect to reduce this range by a factor of 30, meaning that reasonable amplitudes could be generated up to approximately 100 Hz. Since this was still well above the frequency (~7 Hz) that the sensor was typically operated at, good acoustic coupling between the acoustic transducer and the vibrating membrane was always achieved. 3.1.4 V i b r a t i o n sensor In order to monitor the vibration of the sensing membrane, a surface motion transducer system, based on measuring the change in reflection from a vibrating surface, was used.72 The probe consists of a bundle of optical fibers with an illuminating fiber, /, at the core, surrounded by return fibers, R, as shown in Figure 3.4. When this probe is positioned at standoff distance, y, and the vibrating surface has an amplitude of motion, dy, the amount of light reflected back through the return fibers is modulated with the same frequency as the vibrating surface. The light collected by the return fibers is sent to a photo-sensor to produce a representative output voltage signal. Figure 3.4 Cross-section of vibration detection probe: a) side view, b) end-on view. The calibration curve for the detector used in this experiment is shown in Figure 3.5. It shows the relationship between voltage measured by the photo-sensor and standoff distance 59 between the vibration probe and a stationary surface. As the probe was lowered towards the surface, the measured light level went through a maximum when the probe was approximately 0.6 mm away from the surface. There are two approximately linear regions: a front slope and a back slope. When the probe was positioned in the middle of either of these linear regions the amplitude of the voltage signal was linearly proportional to the amplitude of motion of the vibrating surface. The optimal probe standoff position on the front slope linear region was approximately 0.25 mm from the surface and on the back slope it was approximately 0.9 mm from the surface. The front slope region was chosen because the length of its linear region was significantly longer than that of the back slope. To position the probe in the middle of the front slope linear range it was carefully lowered while monitoring the DC voltage light level shown on the device. As it was lowered it went through a maximum value and then was carefully lowered until the DC level was approximately half of its maximum value. It was then centered in the linear region of the curve such that any fluctuation in the light level measured by the probe would be directly proportional to the fluctuation in the surface. This light level was converted to a voltage signal by the photo-sensor and by phase sensitive detection the amplitude and phase of the vibration of the surface directly below where the probe was positioned could be determined. To convert the amplitude of the voltage signal to the amplitude of motion of the surface, the slope from the calibration curve in Figure 3.5 was used. a 0 0.5 1 1.5 2 probe standoff (mm) Figure 3.5 Displacement sensitivity of vibration probe. 60 3.1.5 Aqueous solution As described in section 2.2.3.2.2, the thermodynamic stability of protein molecules is highly dependent upon solution conditions, therefore it is common procedure to perform adsorption studies at constant pH. A 50 mM potassium phosphate monobasic-sodium hydroxide pH 7.00 buffer solution was prepared with a commercially available buffer solution concentrate7 and de-ionized, distilled water. This solution was then de^gassed by mixing under a vacuum pump aspirator. De-gassing of the solution was essential in order to minimize the formation of air bubbles on the sensor surface over the time span of the experiment. 61 3.2 Configuration for surface fluorescence measurement As previously discussed, a second experiment, capable of running simultaneously with the one already described, was designed to probe for information about how much protein was adsorbing to the surface of the sensor. This was done by labeling the protein with a fluorescent marker, which, when exposed to a specific wavelength of light, will emit light at a higher wavelength. A method was devised to selectively excite only those protein molecules adsorbed on the surface of the sensor. The signal from the fluorescent light emitted from the surface of the membrane should then be proportional to the amount of protein on the surface. This was accomplished by using the sensing membrane as an optical waveguide as described in Appendix B. An evanescent light radiation field surrounds optical waveguides. This field can excite fluorescent molecules with which the protein molecules are tagged. Thus, by measuring the amount of fluorescent light emitted, the concentration of tagged protein molecules in the very close vicinity of the optical waveguide can be measured. Figure 3.6 illustrates this concept. l y sozyme molecules evanescent field membrane fluorescent tags Figure 3.6 Sensing membrane used as a waveguide to excite fluorescently tagged protein molecules on surface. The transfer adhesive films described in section 3.1.1 were ideally suited to this purpose as they were optically clear with optically smooth surfaces and an index of refraction of n = 62 1.47. Since the evanescent field from the guided light extends only a short distance past the outside surface of the membrane, only those fluorophores that were adsorbed to the surface were selectively excited. The penetration depth of the evanescent field is given in Equation (B-4 ). For this experiment A = 532nm, and 6 = 1 AT (from output of fiber of laser), n\ = 1.47 and ri2 - 1.31, so 8= 645 nm. The penetration depth of the evanescent field was much larger than the size of a lysozyme molecule (~40A). This was not a problem because the protein molecules have sufficient affinity for the surface that the relative signal from the protein molecules in the bulk solution within the evanescent wave was negligible relative to the signal due to the molecules in contact with the surface. 3.2.1 Fluorescent tag Fluorophores absorb photons of light at the exciting wavelength and may therefore reach an excited state. As the molecule relaxes, a photon of a longer wavelength is emitted. This process is usually reversible. However, occasionally the fluorophore is destroyed in the excited state, resulting in irreversible photo-bleaching. Photo-bleaching can be minimized by reducing the intensity of the exciting radiation, while still generating a good signal to noise ratio. The fluorescent dye used to tag the protein molecules in this experiment was Alexa Fluor 532. It is a carboxylic acid, succinimidyl ester with a molecular weight of approximately 724 a.m.u.. Figure 3.7 shows the chemical formula of this fluorescent dye. Alexa Fluor forms photostable conjugates with relatively high fluorescent yields.7 4 The succinimidyl ester forms a covalent bond with primary amine groups on proteins. The excitation and emission spectra of the Alexa 532 dye conjugated to the goat anti-mouse IgG antibody are shown in Figure 3.8. It was designed to be excited by the frequency-doubled Nd-Yag laser at 532nm. Its emission maximum occurs at 555nm. 63 Figure 3.7 Alexa Fluor 532 carobxylic acid, succinimidyl ester. 450 500 550 600 650 wavelength (nm) Figure 3.8 Excitation and emission curves for Alexa 532 fluorescent tag. An Alexa Fluor 532nm protein labeling kit was ordered from Molecular Probes and was used to label the lysozyme according to the labeling protocol provided with the kit. The following will describe an example of a particular protein labeling experiment: A lysozyme solution of approximately 6.15 mg/mL was prepared by dissolving 62.3 mg lysozyme in 10 mL of buffer 64 solution. A portion of this solution was diluted 1 in 3 to produce ~1.5 mL of ~2mg/mL lysozyme solution for use in the labeling reaction. Since succinimidyl esters react most efficiently at pH 7.5-8.5, 150 uL of 1 M bicarbonate solution was added to raise the pH of the protein solution. The solution was then added to the vial of reactive dye and mixed for an hour at room temperature. The next step involved separating the un-reacted dye molecules from the reacted dye conjugates. Rather than using the separation columns provided with the kit that were optimized for use with higher molecular weight proteins, a size exclusion column pre-packed with P6 Biogel was used. This gel retains all molecules smaller than -6000 a.m.u while allowing larger molecules to pass through the column void. Thus, the unreacted dye molecules of 724 a.m.u are eluted substantially later than the reacted dye-protein conjugates. After separation ~2.5mL of concentrated pure dye-conjugate solution was recovered. A portion of this was diluted 1 in 20 with buffer solution in order to measure the degree of labeling. A spectrometer was used to measure the absorbance at 280 nm, A280, and 530 nm, A530, in a cuvette with a 1 cm pathlength. The protein concentration, M, and degree of labeling, n = moles dye per mole protein, could be determined according to Equations ( 3-1 ) and ( 3-2 ); where 37600 crn^M"1 is the molar extinction coefficient for lysozyme at 280 nm 7 5 and 81000 cm^M" 1 is the molar extinction coefficient of the Alexa 532 dye at 530 nm 7 6 . The factor of 20 represents the degree to which the solution was diluted in order to make the measurement with the spectrometer. To convert from molar concentration (mol/L) to mg/mL multiply the result of Equation (3-1 ) by the molecular weight of the protein, Mw. Following the above procedure, the lysozyme concentration was measured to be 0.828 mg/mL and the degree of labeling was determined to be 0.715. This meant that on average, approximately 7 out of every 10 lysozyme molecules had a dye molecule attached to it. In order to maximize the usefulness of this solution it was diluted 1 in 10 with unlabelled M = 20[A280 - (0.09A530)] 37600 . (3-1) n = 20^530 81000 M (3-2) 6 5 protein. The resulting solution was re-measured and was determined to have a protein concentration of 1.82 mg/mL lysozyme and a degree of labeling of 0.0697. This solution was divided into 10 lmL vials and refrigerated for later use. 3.2.2 Laser Two methods of injecting green 532nm laser light into the waveguide membrane were used. The first method involved using a 532nm laser pointer with a lens to focus the light onto the edge of the membrane. The second method involved the use of a solid-state 532nm green laser pigtailed to a single mode fiber that was butt-coupled to the edge of the membrane. Figure 3.9 shows a diagram of how the laser pointer was coupled to the optical waveguide. Although, using the laser pointer allowed more light to enter the waveguide, the laser beam had to travel through the bulk solution before hitting the edge of the membrane. Furthermore, a significant amount of light was scattered as the focused laser beam hit the edge of the membrane. Both of these effects contributed to a portion of the fluorescent signal to be due to bulk solution fluorescence and not due to surface fluorescence. And the apparent advantage of having more light enter the membrane was mitigated by the fact that this enhanced the photo-bleaching of the immobilized protein molecules on the surface of the waveguide. laser beam Figure 3.9 Method of coupling laser pointer to optical waveguide. 6 6 Figure 3.10 shows a diagram of how the pig-tailed laser was coupled to the optical waveguide. The details of the construction of the waveguide holder and how the waveguide was precisely aligned with the output from the end of the fiber are discussed in section 3.2.4. Figure 3.10 Method of coupling pig-tailed laser to optical waveguide. The fiber pigtailed laser proved to be a more efficient method of injecting light into the edge of the membrane. The fiber terminator used was a standard ceramic ferrule. The output from the fiber had a mode field diameter of 3.3 urn with a Gaussian angular distribution with a 1/e half angle of 7.47 degrees. It was rated to have 0.3 mW of laser power from the end of the fiber with a stability of +/- 5%. However, solid-state green 532nm lasers at this time are relatively unstable compared to the more established technologies of red solid-state lasers. Figure 3.11 shows a typical plot of the power output from the end of the fiber as the laser was warming up. As can be seen, after about an hour the output stabilized, but was still considerably lower than its rated value. For this reason, the laser power was monitored 67 during the course of the experiment, so that any fluctuations in laser power could be divided out of the fluorescent signal. 2 3 time (hours) Figure 3.11 Power output from end of 532 pig-tailed laser used in experiment. stainless steel annular holder stainless steel annular holder + teflon thin film stainless steel annular holder + teflon thin film + sensing membrane sensing membrane - — 1 « mm mm stainless steel annular holder teflon thin film Figure 3.12 Construction of sensing membrane as a waveguide. 68 3.2.3 Waveguide construction In order to use the sensing membrane as a waveguide it had to be carefully mounted so that laser radiation could be efficiently injected into the membrane. The membrane was mounted between two stainless steel annular supports as described in section 3.1.2 with a modification shown in Figure 3.12. A thin film of Teflon A F ™ was painted on to the surface of the annular support where laser light was to be injected into the edge of the sensing membrane. Since, the Teflon™ has a lower index of refraction, (n = 1.31) than the 8141 membrane, this allowed for the injected light to be guided efficiently until it reached the edge of the annular support. To maximize the amount of laser light that was injected into the membrane, it was imperative that the edge of the membrane where the laser light was to be injected was a smooth, clean edge. This was achieved by the following procedure which is depicted in Figure 3.13: When the adhesive fi lm was attached to the first side of the annular support, a small amount of silicone rubber was dabbed at the edge where the laser light was to be injected. This was cured in the oven, and the annular support with the attached film was cooled by placing the support on a piece of brass cooled down with liquid nitrogen. With the annular support and membrane side down on the brass, the well created by the inner diameter of the support was filled with enough water to cover the surface of the membrane. When the water was frozen, the support was flipped over and the mylar backing of the membrane could be peeled off. The portion of support where the Teflon™ had been painted had very poor adhesion to the adhesive film. However, the fact that the edges were sealed by the silicone on one side and the ice on the other, ensured that as the mylar backing was peeled off, the membrane didn't lift off the support above the Teflon™, creating wrinkles. The edge of the membrane where the laser was to be injected was then trimmed off with a sharp razor blade. Whilst still cool, the other support was carefully positioned to iine up with the first and clamped together. This support/membrane composite was then allowed to return to room temperature. I f the membrane support was held in a vertical position while the ice melted, it was unlikely that the edges of the adhesive film would be distorted by the melting ice and become stuck to the inner diameter of the support. In order to preserve and seal the edges, another small dab of silicone was put on the edge and covered with a small piece of mylar and cured in the oven. When the mylar was peeled off, the edge was sealed with a clean, smooth interface to allow for the laser ferrule to be butted up against the edge of the membrane. 1) Peel off top mylar release liner. 2) Attach annular holder to exposed film L Dab small amount of 3) silicone elastomer to outside edge. r 4) Add water to indent and freeze I 5) Peel off back side of mylar release liner 6) Cut along dotted lines with sharp razor blade. mylar release liner 8141/8142 elastomeric film annular holder silicone elastomer 7) 9) 10) Attach bottom side of annular holder Melt ice, leaving 8) elastomeric membrane freely mounted. 1 • a Dab small amount of optically clear optically clear silicone JB| WEL^^ silicone elastomer elastomer, and cover ISii • « * — - — mylar with mylar for smooth edge. Peel off mylar, leaving smooth edge. Figure 3.13 Procedure to mount elastomeric membranes as optical waveguides. 70 3.2.4 Waveguide holder The laser emitting fiber was coupled to the sensing membrane by butting it up against the edge of the membrane. However, since the output of the fiber was ~3 um in diameter and the membrane itself was ~25 or 50 um it required very precise positioning in order to align the fiber with the edge of the membrane. A special holder was designed so that the position of the membrane support sandwich could be adjusted precisely using differential screws. Figure 3.14 shows a diagram of this holder; a) from above, b) along cross-section A, and c) along cross-section B. The end of the fiber pigtail was inserted through a hole shown in cross-section A. There was a symmetrical hole on the other side of the holder in order to view the opposite edge of the membrane and determine how well the laser light was being guided through the membrane. P holes for legs a) ^—>\ b) b = d . b=^* end of fiber pigtail inserted here c) B 3.65 cm < > Figure 3.14 To scale diagram of waveguide holder; a) from above, b) cross-section A, c) cross-section B. 71 The membrane support sandwich was inserted into this holder and secured in position with the use of the differential screws and the supporting legs. One end of the differential screw was screwed into the holder legs, while the other end of the screw was screwed into the threaded side of the annular membrane support as shown in Figure 3.15. The spacing between the top of the holder legs and the bottom of the annular support, dy, was what varied when the differential screw was adjusted. annular support differential screw holder leg elastomeric membrane thread for nut used to hold legs in waveguide holder Figure 3.15 Differential screw between holder legs and annular membrane support. Figure 3.16 shows the construction shown in Figure 3.15 inserted in the waveguide holder shown in Figure 3.14 and immersed in the experimental cell. A nut was used to secure the legs in position against the holder. The thread differential on the screw was 0.05766 mm per turn, meaning that one turn in the screw resulted in a total vertical displacement between the two sides of the screw of approximately 60p.m. In order to align the membrane with the laser, it was imperative that the initial distance, dy, was close to the desired alignment, otherwise adjusting the differential screw over its entire range was not sufficient to align the laser with edge of the membrane. In order to align the membrane, the opposite edge of the membrane was viewed as the differential screw was adjusted. Optimal alignment was achieved when the maximum amount of light was guided along the waveguide. 72 vibration detection Figure 3.16 Cross-section showing annular membrane support inserted in holder and immersed in experimental cell: a) along cross-section A, b) along cross-section B. 3.2.5 Fluorescent radiation collection The light emitted by the excited fluorophore was collected by a series of large core optical fibers positioned above the membrane surface. The large core optical fibers were made from a clear, flexible plastic, surrounded by a lower index cladding. Therefore, according to the principles of waveguides described in section Appendix B, they efficiently transmitted radiation along their length. In order to have reproducibility in the positioning of the fibers, five fibers of approximate diameter 9mm were bunched around the tip of the vibration detection probe as shown in Figure 3.17a. As the vibration detection probe was lowered to the surface, the ends of the fibers were immersed in the aqueous solution and positioned directly above the surface of the sensing membrane as shown in Figure 3.17b. A large portion of the fluorescent radiation that was emitted upwards therefore entered the fibers and could be transmitted along their length. 73 a) b) Figure 3.17 Large core optical fiber array: a) oblique view, b) side-on view shown immersed in experimental cell. In order to isolate the signal due to light emitted by the fluorophores, two filters were used. The first filter, XF3021 7 7 , was used to cut out any scattered light from to the 532 nm laser. The second filter, a cyan subtractive filter78, was used to cut out any scattered light from the near infrared LED of the vibration detection probe. Figure 3.18 shows the transmission spectra of both of these filters. 100 i 500 525 550 575 600 625 wavelength (nm) Figure 3.18 Spectra of filters used to isolate fluorescent radiation. 74 The light collected from the fiber array was fed into a photomultiplier tube to measure and amplify the signal. Figure 3.19 shows the schematic of how the fluorescent radiation was collected, filtered, spatially concentrated and fed in the photomultiplier tube. To improve coupling, the radiation was spatially concentrated into the side of the photomultiplier tube. A cone of specular multilayer mirror f i lm 7 9 was used to concentrate the light down to a smaller diameter. At the end of the fiber array one of the filters was bonded with clear silicone rubber, such that the end of each individual fiber was completely covered with the filter. The large end of the concentration cone was installed at the end of the fiber array. The second filter was placed at the small end of the concentration cone. Another large core optical fiber was used to transmit the fluorescent light from the end of the concentration cone to the opening of the photomultiplier tube. The entire light collection cone was enclosed in a light tight box, so that no background light radiation could leak into the photomultiplier tube. photomultiplier tube LrSJ Figure 3.19 Schematic of fluorescent light collection apparatus. In order to shield the experiment from ambient light, a black box with two curtained sides was constructed. The curtained sides were made from black fabric and multilayered so all necessary tubing and wiring could enter into the darkened box without leaking in too much light. For added precaution, the room lights were turned off during the course of the experiment. With the experiment shielded by the box, the room lights off and the laser and vibration detection probe turned off, the signal measured by the photomultiplier tube was comparable to the dark noise current of the photomultiplier tube itself. 75 However, the filters shown in Figure 3.18 that are used to block the radiation from the laser and vibration detection probe's LED are not 100% efficient at their respective wavelengths. Furthermore, even with the laser carefully aligned with the edge of the membrane using the holder described in section 3.2.4 and illustrated in Figure 3.16, there was always some scattered laser light that leaked from the inside edge of the annular support. Therefore, with the laser and the vibration detection probe turned on for the simultaneous experiment, there would always be a background light level detected by the photomultiplier tube before any fluorescent protein molecules were added to the system. How these two types of contributions to the background signal can be subtracted out, wil l be discussed in section 5.1.5.1. The surface fluorescence measurement described in section 3.2 was designed so that it may be run either independently or concurrently with the resonant frequency measurement described in section 3.1. The signal collected in the surface fluorescence measurement should be proportional to the amount of fluorescently labeled protein on the surface. The signal collected in the resonant frequency measurement is related to the change in surface energy as protein molecules adsorb and unfold on the surface. By comparing the two responses, it was hoped that the effects on the tension due to the adsorption could be decoupled from those due to conformation change. 76 3.3 F l u i d f l o w a n d c o n t r o l It was possible to perform experiments using only the basic configuration as described sections 3.1 and 3.2, however, there were several reasons why a means to pump fluid into and out of the experimental cell was advantageous. Firstly, the surface of the sensing membrane needed to be exposed to a new solution of known protein concentration in a controlled manner with well defined hydrodynamic properties. Secondly, it was important to have a method to easily flush out the protein solution from the experimental cell and replace it with clean solution to examine the reversibility of the protein adsorption reaction. Lastly, a means to control and maintain a constant temperature was required. Al l of these goals were achieved with the fluid delivery system that is described in the following sections. vibration detector sensing membran transducer-experimental cell open ended test tube peristaltic pump mixing tank Figure 3.20 Schematic of solution flow in experiment. 3.3.1 C i r c u l a t i o n Figure 3.20 shows the basic schematic for pumping fluid into and out of the experimental cell. There was a storage volume, called the mixing tank, from which the fluid was peristaltically pumped at approximately 140 mL/min to the experimental cell, whereupon it drained back to the mixing tank by means of gravity. The flow in the return pipe, was 77 initiated by filling the tubing with fluid using a syringe, and then inserting into the mixing tank. The vertical position of the mixing tank could be precisely adjusted manually with a jack so that the rate of flow of the solution returning into the mixing tank due to gravity matched the rate of flow pumping into the experimental cell. In this way, the level of liquid in the experimental cell remained constant. An open-ended test tube was used to remove the pulses in the flow stream due to the peristaltic pump. Also included in the circulation path between the pump and the open-ended test tube was an inline filter with a 5um pore size filter paper. This was to remove any small contaminants such as dust particles from the circulating fluid. A 12V motor with a small paddle attached was used to thoroughly mix the solution in the experimental cell. Generally, the experiment was run with a total fluid volume of 500mL, which divided itself as follows: -200 mL in the experimental cell, -200 mL in the mixing tank and -100 mL in the tubing and test tube. The combination of the mixing motor and the flow into the experimental cell ensured that turbulent mixing occurred in the experimental cell. I f a concentrated protein solution was added to the mixing tank and then pumped into the experimental cell, the bulk concentration of the solution surrounding the sensing membrane reached a new equilibrium value quickly. As will be described in section 4.2, it was possible to accurately calculate the bulk concentration as a function of time in the experimental cell i f turbulent mixing was assumed. 3.3.2 Flushing The basic flow schematic described in the previous section was adapted to allow for the ability to flush out the solution in the experimental cell with clean, buffered aqueous solution. This allowed for the study of the reversibility of the adsorption process. Also, flushing with clean solution allowed for the experiment to be continued for many consecutive runs simply by inserting a new sensor membrane, once the system had been thoroughly flushed out. To carry out this flushing, a second larger storage tank called the flush tank was added surrounding the mixing tank. The flush tank capacity was almost 2L. The mixing tank was fully immersed inside the flush tank so that they would be in thermal equilibrium with each other. A 3-way stopcock was connected to the tubing such that the in-flowing experimental 78 fluid could be switched from the mixing tank to the flush tank. A second 3-way stopcock was inserted in the tubing up-stream from the first stopcock to allow for the fluid in the mixing tank to be pumped out as waste. The waste tubing line was attached to a second head on the peristaltic pump such that the pump rate out of the mixing tank would exactly match the rate of flow of the fluid out of the flush tank. In this way, the level of fluid in both the experimental cell and the mixing tank remained constant during the flushing operation. This adaptation to the basic fluid flow is shown in Figure 3.21. Figure 3.21a) shows the fluid flow path for fluid circulation from the mixing tank. Figure 3.21b) shows the fluid flow path for the flushing operation when the fluid from the flush tank was pumped into the experimental cell. When the fluid in the flush tank was depleted, the 3-way stopcocks were switched back to the initial positions to resume the normal circulation path of fluid from the mixing tank. 3-way stopcock circulation positions a) mixing tank flush tank 3-way stopcock flushing positions b) IS 1 waste mixing tank flush tank Figure 3.21 Adaptation of basic fluid flow to allow for flushing, a) normal circulation b) flushing. 79 The evolution of the bulk concentration in the mixing tank as the flushing operation proceeded could also be calculated. As described in section 4.2.2, it was determined that flushing with 1.5L of clean buffer solution should reduce the residual solution concentration in the experimental cell to around 0.2% of its original value. 3.3.3 Temperature control Since temperature fluctuations could affect the resonant frequency of the sensor, it was important to maintain a constant temperature in the experimental cell. Isothermal conditions were achieved using a re-circulating temperature control bath of approximately 30L maintained at 27 °C. Figure 3.22 shows how a third pump head on the peristaltic pump was used to circulate the temperature control fluid, or thermal fluid, around to various parts of the experiment so that a thermal equilibrium could be established. Firstly, the thermal fluid was pumped through tubing that was wrapped around the flush tank, the temperature regulation coil. The flush tank with the temperature regulation coil was also inserted into a larger, insulated plastic container and filled with some water to minimize heat loss to the environment. It was then pumped through an insulated aluminum manifold that was in thermal contact with the experimental cell with some silicone heat sink paste. The thermal fluid was then pumped back into the temperature control bath. Figure 3.23 shows an example of the evolution of the temperature in the experimental cell as the system approached thermal equilibrium. A long-term temperature stability of ± 0.1 °C RMS was achieved with this system. 80 Figure 3.22 Temperature control system. 25 i 20 4 - — r - — r - — r - , , 1 0 2 4 6 8 10 12 time (hours) Figure 3.23 Evolution of temperature in experimental cell for approach to thermal equilibrium. 81 3.4 I n s t r u m e n t a t i o n a n d c o n t r o l In order to use the sensor in real time it must be monitored and controlled with a computer. LabVIEW computer software was used to automate the instrumentation using the general purpose interface bus (GPIB). Al l of the instruments used were programmable with the GPIB language and connected to the computer using GPIB cables. The following sections wil l describe the instrumentation and control programs that were used to operate the sensor for both the resonant frequency measurement and the surface fluorescence measurement. In each case there were several different aspects of the sensor behaviour or experimental conditions that could be monitored by using different LabVIEW programs. 3.4.1 Resonant frequency measurement Figure 3.24 shows a schematic the instruments that were used to monitor and control the vibrating membrane sensor experiment. The LabVIEW program controlled the driving frequency of the acoustic transducer, monitored the sensor response with a lock-in amplifier, measured the experimental temperature with a thermocouple and used a GPIB controller to turn a motor mixer on and off. 82 ice water thermos TypeK thermocouple vibration detection sensor MOSFET circuit multimeter T, temperature lock-in amplifier (f>, phase . R, amplitude GPIB controller i on I off signal function generator / , frequency Figure 3.24 Schematic of instruments used for resonant frequency measurement. A function generator80 was used to drive the acoustic transducer with a sinusoidal wave. While the driving frequency was continually changing as the experiment progressed in order to keep the membrane vibrating at resonance, the amplitude was, in general, kept at its maximum of 10V peak-peak to maximize the amplitude response of the sensor. A lock-in amplifier81 was used to measure the signal from the vibration detection probe with the driving signal of the function generator used as a reference. The output could be chosen to either be the magnitude, R, and phase, <f>, the phase difference between the signal and driving frequency, or the x and y values of the signal where x = R cos (0)andy = R sin(<f>). The time constant of the lock-in amplifier, r, gave an indication of how long the signal was time averaged. A larger, r, meant less random noise in the sampled data points, but limited how quickly data could be sampled. In general, it was necessary to wait 3 r between data measurements to ensure the lock-in amplifier had responded to changes in experimental conditions since the previous data set was sampled. The experimental temperature was determined by measuring the voltage of a type K thermocouple with a multimeter . The thermocouple created a voltage differential due to a temperature differential between its two ends. With one end immersed in the experimental 83 cell and the other in a thermos of ice water, Equation ( 3-3 ) gives temperature in the experimental cell, T, based on the voltage, V, of the thermocouple. T = -0.0087 V4 +0.131 V3 -0.5401 V2 +24.922 V + 0.169 (3-3) As described in section 3.3, it was necessary to use a motor mixer to ensure the solution in the experimental cell was thoroughly mixed. However, continually running this motor mixer produced too much disturbance of the surface of the sensor membrane. A GPIB stepping motor controller with a programmable bit set output was used to enable the motor mixer to be turned off for a set amount of time before data was read from the lock-in amplifier. Since the output from the controller was an open collector TTL output, a MOSFET circuit connected to a 12V dc power supply was necessary to trigger the motor mixer on and off. Figure 3.25 shows a diagram of the circuit used. GPIB controller 2K< + V D Mixing motor I I R F 5 1 0 D C ( J ^ ) 1 2 V Figure 3.25 MOSFET circuit used to control motor mixer. 3.4.1.1 Resonant frequency spectra By scanning through the driving frequency of the acoustic transducer and measuring the magnitude of the membrane's motion with the vibration detection probe, the resonance frequencies of the membrane could be measured. The (R, <p) output setting of the lock-in amplifier was used. Depending on the location of the vibration probe, different resonances 84 could be determined. With the probe positioned in the center of the sensor membrane, the circularly symmetric modes, (0,ri) n >1, could be scanned. Figure 3.26 shows the amplitude of a sensor membrane immersed in a pH 7.0, buffered solution in response to sweeping frequency from 0 to 30 Hz. The first four circularly symmetric resonances were clearly defined and labeled on the figure. l i 0.9 -0.8 -0 5 10 15 20 25 30 frequency (Hz) Figure 3.26 Example resonant frequency spectrum with probe positioned at centre of membrane. 3.4.1.2 Mode shape mapping I f the vibration probe was physically moved across the surface of the membrane while it was vibrating at a resonant frequency, the shape of the resonance could be mapped. In this case, the (x, y) output setting of the lock-in amplifier was used and the sign and amplitude of the x output was graphed as a function of position. Figure 3.27 shows the mode-shapes and frequencies of the first three circularly symmetric modes of a typical sensor membrane. Note that was a different membrane than the one used for the example in Figure 3.26. As can be seen, the mode shapes were clearly defined and agree with the expected theoretical mode shapes as described in Appendix A. 85 (0,1) mode 1.68 Hz (0,2) mode 6.65 Hz (0,3) mode 14.05 Hz Figure 3.27 First three symmetric resonant membrane modes. 3.4.1.3 Resonant frequency tracking The resonant frequency of the sensor could be determined as a function of time by continually adjusting the frequency of the drive signal to keep it at the center frequency of a selected mode. This was achieved by using phase information from the displacement signal. As the driving frequency was scanned through resonance, the amplitude of the response went through a maximum and the phase went through zero, as shown in Figure 3.28. 86 6 7 8 9 10 frequency (Hz) / , frequency (Hz) Figure 3.28 Example of response of sensor to frequency sweep in a) amplitude and b) phase for (0,2) resonance. As can be seen, the phase response was fairly linear in the frequency range near the resonance. Thus, by monitoring the phase shift of the signal detected by the vibration detection probe, it could be determined whether or not the resonant frequency had shifted. A negative shift indicated that the resonant frequency had dropped, while a positive shift indicated an increase in the resonant frequency. Thus, a new driving frequency could be calculated to shift the system back towards resonance according to Equation ( 3-4 ) where a new driving frequency,^, is calculated from a previous frequency,^./ and phase shift, $ . 7 . By choosing an appropriate relationship between shift in phase and shift in frequency, called gain, G, the driving frequency calculated would be held near the center frequency of the 87 resonant response. Note that, in theory, the gain was actually the slope of the linear region of the phase shift versus frequency graph, shown in Figure 3.28b. f i = f i - i + ^ r f (3 -4) The procedure to track the resonant frequency as a function of time was as follows. The vibration probe was positioned at the center of the sensor membrane and lowered to the appropriate distance from the surface as described in section 3.1.4. The function generator was then manually adjusted to find the (0,2) mode by finding the frequency with the highest amplitude in the appropriate range, (7-8 Hz usually). Since there was an adjustable offset for the phase on the lock-in amplifier, this had to be manually adjusted to zero at the resonant frequency. Once this was done, the LabVIEW program could be started to monitor the resonant frequency over time. The inputs to the program were: starting frequency,//, function generator amplitude, gain, G, and output filename. A flow chart of the program used to monitor the resonant frequency of the sensor is shown in Figure 3.29. The phase, <j>, and amplitude, R, are read from the lock-in amplifier. The program then calculated an appropriate shift in frequency based on Equation ( 3-4 ) and outputted the modified drive frequency to the function generator. The motor mixer was then turned on for x seconds. The motor mixer was turned off for y seconds before the loop started again and the data was sampled from the lock-in amplifier. A file was created that output time, t, phase shift, <f>, frequency,/ amplitude, R and temperature, T. In this way the resonant frequency of the sensor could be monitored as it responded to various different changes in its environment, for example temperature and adhesion of macromolecules. 88 record initial time, t, initialize driving frequency and amplitude on function generator read phase shift, if), from lock-in read amplitude, R, from lock-in 1 set bit to turn on motor mixer calculate frequency shift Af=}/G output new frequency to function generator record time, t 1 measure voltage, V, from multimeter and calculate temperature, T wait x seconds I clear bit to turn off motor mixer wait y seconds : 1 increment counter, j i output: counter '^, elapsed time, t-t„ phase, <j>, frequency,/^  amplitude, R and temperature, T to file Figure 3.29 Flow chart of LabVIEW program to monitor resonant frequency. The noise in the resonant frequency signal was primarily due to disturbance of the sensor membrane by the movement of the liquid in the experimental cell. As mentioned earlier, to minimize the noise due to the motor mixer, it had to be turned off for a sufficient amount of time so that the membrane had a chance to settle down and the time averaged signal on the lock-in amplifier had a chance to respond. This had to be at least 3 time constants on the lock-in amplifier. Since, r = 3 s was used, the motor mixer was turned on for 10s and off for 10s, making approximately 20s between sampled data points. 89 With the proper tuning of the iteration parameters, namely, gain, G, time constant on the lock-in amplifier, r, and time between data measurements, At, noise in the signal could be minimized. The time interval was selected based on the time constant of the lock-in amplifier as previously discussed. The gain, G, was selected to ensure that frequency change would bring the system back to resonance. Theoretically, it is equal to the slope of the phase versus frequency graph as shown in the example of Figure 3.28b. Ideally, the noise in the resonant frequency data would be close to the inherent noise in the actual resonant frequency. For example, for the resonant mode shown in Figure 3.28, the slope of the phase versus frequency graph is -87 degrees per Hz. I f the selected value of G were too small, the resonant frequency would oscillate around the true resonance. This is shown in Figure 3.30, where G was chosen to be 50. The driving frequency oscillated wildly with a <r= 0.3 Hz. I f too large a value were chosen for G, extra structure appeared in the signal. This is shown in Figure 3.31, where for the same mode G was chosen to be 400. 7.8 n 6.8 0 5 10 15 20 25 time (min.) Figure 3.30 Example of under damped system, G = 50, a- 0.3 Hz 90 7.4 ~ 7.3 N PC S 7.2 4> 0> .1-7.1 ^ 10 20 time (min.) 30 40 Figure 3.31 Example of an over damped system, G = 400, cr= 0.005 Hz In between, excellent results were obtained. For example, for the resonance depicted in Figure 3.28 this system could stay tuned in the center of a 2 Hz FWHM resonance with a frequency noise of about 0.01 Hz RMS when the time constant of the lock-in amplifier was set to 3 seconds and data was sampled every 20 seconds with a gain of 100. This is shown in Figure 3.32, where the size of the error bars was the standard deviation calculated from the data itself, a= 0.01 Hz. 91 N fi >> U s 1) 3 B" 9 7.4 7.35 7.3 7.25 7.2 J 7.15 7.1 7.05 80 85 90 95 100 time (min.) 105 110 Figure 3.32 Example of well tuned system, G = 100, ar= 0.01 Hz 3.4.2 Fluorescent waveguide experiment Figure 3.33 shows the schematic of the instruments used to measure the fluorescent radiation from the sensor during the experiment described in section 3.2. optical fiber array light -tight box photomultiplier tube V J -600 V dc fiber pigtailed laser multi-meter 1 fluorescent signal, p filter circuits multi-meter 2 laser J power, I J PC mini integrating sphere with photo-diode Figure 3.33 Schematic for instrumentation used in fluorescence measurement. 92 The photomultiplier tube 8 4 was powered by a high voltage, DC power supply set to -600V. The dark current reading from this particular photomultiplier tube corresponded to ~ 0.035 mV, as read by a multimeter. A low pass filter circuit was used in order to minimize the noise in the signal from the photomultiplier tube. Figure 3.34 shows the circuit diagram for the filter that has a time constant of 5s. Since the data was sampled at most once every 20 seconds, the signal was not impaired by the low pass filter. 470 kn A A A r in 10 uF out Figure 3.34 Filter circuit used to time average the photomultiplier signal. Since the pigtailed green laser used to illuminate the waveguide sensor was relatively unstable, it was necessary to measure the laser output power as a function of time. A small o r integrating structure with a linear photodiode was used to measure the light leaking from the fiber immediately adjacent to the laser head. This integrating structure simply consisted of a small cylinder lined with a highly reflective, diffuse white plastic material with a baffled photodiode inserted inside. Another low pass filter circuit and multimeter82 were used to monitor the output from this photodiode. Although, this method did not measure the total laser power, any fluctuations in the total power were reflected in the fluctuations in the leaking light from the fiber. Since the signal measured by the photodiode was divided out of the signal from the photomultiplier tube, it was important that this signal, too be filtered with the same circuit as shown in Figure 3.34. It is important to emphasize that this measurement of the fluorescent signal was not absolute. For instance, the optical fiber array only captured a percentage of the emitted fluorescent radiation and the losses inherent in the filters and light concentration cone described in 93 section 3.2.5 were uncharacterized. Furthermore, the relationship between quantity of adsorbed protein and amount of fluorescent radiation was not calibrated. However, the aim was to measure a fluorescent signal that was proportional to the total amount of fluorescently labeled protein on the surface. It was the evolution of this signal, as a function of time in comparison to the evolution of the resonant frequency signal, which was of interest. 3.4.2.1 Surface fluorescence measurement ; ' In order to measure the fluorescent signal as a function of time, the voltage of multi-meters connected to the photodiode and photomultiplier were read with a LabVIEW program. A flow chart showing the steps of this program is given in Figure 3.35. record initial time, t, > measure photomultiplier voltage, p i set bit to turn on motor mixer . z measure photodiode voltage, / 1 + , record time, t A wait x seconds ^ r clear bit to turn off motor mixer * r increment counter, j i output: counter, j, elapsed time, t-t,, photomultiplier voltage, p, laser power, / to file Figure 3.35 Flowchart of LabVIEW program to do surface fluorescence measurement. 94 3.4.2.2 Bulk fluorescence measurement Using the fluorescent collection apparatus described in section 3.2.5, it was also possible to do an experiment to measure the bulk fluorescence as a function of time. The bulk fluorescence signal was proportional to the bulk concentration in the experimental cell. This was accomplished by simply using the waveguide holder described in section 3.2.4 without the waveguide. When the pigtailed laser was inserted into the holder, the laser light shone in a narrow beam into the aqueous solution. With the large core optical fiber array positioned above this beam of laser light, the fluorescence from bulk solution could be monitored as labeled protein molecules were added to the experiment. The same LabVIEW program was used as described in the previous section. This measurement allowed for the study of the evolution of the bulk concentration in the experimental cell as a function of time. 3.4.3 Simultaneous experiments Both the vibrating membrane experiment and the surface fluorescent experiment could be performed simultaneously. A flow chart showing the steps of the LabVIEW program to do this is given in Figure 3.36. 95 record initial time, initialize driving frequency and amplitude on function generator * read phase shift, <f>, from lock-in 4 read amplitude, R, from lock-in set bit to turn on motor mixer I measure photomultiplier voltage, p i measure photodiode voltage, / • i ; calculate frequency shift Af=}/G 1 output new frequency to function generator i record time, t 1 wait x seconds I clear bit to turn off motor mixer waity seconds increment counter, y' i output: counter j, elapsed time, t-U, phase, <j>, frequency,/, amplitude, R , photomultiplier voltage, p and photodiode voltage, /, to file Figure 3.36 Flow chart of LabVIEW program to do simultaneous experiment. 96 4 MODELING OF ADSORPTION RELATED TENSION CHANGE An adsorption model was devised to account for the effect macromolecular adsorption has on the resonant frequency of the sensor described in section 3.1.1. The model was based on physically reasonable adsorption processes and was able to predict and decouple the sensor response to both adsorption and conformation change. This model is critical to the interpretation of the experimental results obtained with the adsorption sensor described in Chapter 3. In order to completely describe the experimental conditions outlined in Chapter 3, three mathematical models were used. The first step was to derive the change in tension of the sensor from the change in resonant frequency that was measured in the experiment. A method based on the fluid loading of a baffled membrane was used and is outlined in section 4.1. The next step involved a mathematical description of the fluid flow described in section 3.3 so that the change in bulk concentration in the experimental cell as a function of time was known. This was done by treating the fluid flow scheme as a three tank mixing model with a time delay due to transport through the connecting pipes. The model is outlined in section 4.2. The adsorption model described in section 4.3 is novel. It tracks the populations of molecules in various bulk and adsorbed states as a function of time, where the key model parameters are the rates of transfer between molecular states. Finally, section 4.4 describes how all three of these steps were combined together and implemented so that data from a particular experimental run could be compared with the adsorption model and meaningful model parameters could be generated. 97 4.1 Membrane tension calculation This experiment monitored the resonant frequency o f a round membrane immersed in an aqueous solution as molecules adsorbed to its surface. A s previously discussed, the mass o f the adsorbed layer was insignificant compared to the mass o f the membrane itself. Hence, the adsorption was ; affecting the resonant frequency by changing its surface energy or tension. It was therefore necessary to understand how the resonant frequency, fmn, depended on the tension in the membrane, T. A n ideal membrane is defined as a thin stretched film c lamped at the perimeter whose restoring force is solely due to the film tension and not its stiffness. T h e theory for the resonant modes o f an ideal membrane is reviewed in A p p e n d i x A . T h e sensing membranes in this study were oscil lating in an aqueous solution, therefore, the general equation, ( 4-1 ) needed to be solved, where w(r, is the out-of-plane displacement o f the membrane, p, the density and h the thickness. T h e interaction o f the membrane with the surrounding fluid is brought in through the pressure term, P. A solution to this problem is described in section 4.1.2. dt2 Note that this analysis ignores the energy loss due to radiative and viscous dampl ing which would require a term proportional to dwldt in Equat ion (4-1 ). Radiative damping can be ignored i f the wavelength o f the radiation in the f luid is m u c h greater than the wavelength in the vibrating membrane. T h e speed o f sound i n water is cv=1420 m/s and at a typical operating frequency o f ~ 7 H z this would yie ld a wavelength o f - 2 0 0 m in the f luid, m u c h greater than the associated wavelength in the membrane, - 0.01 m. V i s c o u s damping is expected to also be an insignificant contribution due to the l ow viscosity o f the aqueous solution. T h e validity these claims are supported by the fact that wel l defined harmonic modes are generated in the vibrating membranes, as shown in Figure 3.26. L i k e l y , the dominant loss mechanism is mechanical hysteresis in the rubber. 9 8 4.1.1 Experimental sensing membrane In order to use Equation (4-1 ) as the basis for the model, the restoring force due the membrane stiffness could not contribute to the mechanical behaviour of the membrane. Tong et al. define the conditions for when this is true.86 Equation ( 4-2 ) gives the equation for the flexural rigidity of a membrane; where E is defined as the Young's modulus in [N/m2], h is the thickness of the membrane in [m] and vis the Poisson's ratio of the membrane. Equation (4-3 ) gives the conditions when the stiffness effect can be neglected; where cr= T/h is the stress in the membrane in [N/m ] and a is the radius of the membrane in [m]. For the membranes used in this device, E ~ 10 N/m ,h = 25 um, v ~ 0.5, a ~ 0.01m and 7~ 0.05 N/m. According to Equation (4-2 ), this yielded a C value of 2xl0" 9, which is significantly less than one. Therefore, the stiffness was not contributing to the restoring force of the membrane and the model to describe the sensing membrane could be based on Equation (4-1 ). D = Eh3 2~ ( 4 - 2 ) 1 2 ( l - v 2 ) v ' c"Vs<<l («) 4.1.2 Water loading of membrane Recall from Appendix A that Equation (4-4 ) gives the solution to Equation ( 4-1 ) with P = 0, and therefore does not account for the interaction of the membrane with a surrounding fluid. The effect of this mass loading is to lower the resonant frequencies of the membrane modes with respect to those of an ideal membrane. With a thin membrane, even a low density surrounding fluid such as air will modify the resonant frequencies, as is well known in the analysis of the acoustics of drums.87 Clearly, the mass loading of the membrane by a much denser, aqueous solution would have a greater effect on the resonant frequencies and must be accounted for in order to extract the true tension of the membrane. 99 r _ Mmn T m n ~ 2naiph (4-4> A procedure to solve for the mass loading of a round membrane by a surrounding fluid is outlined by Tong et al.S6 based on an analysis of the problem by Gottlieb and Aebischer.88 They solve, for the circularly symmetric modes, the radiation mass impedance terms that act on a membrane when immersed in a fluid such as air or water. It is an iterative method to solve for the resonant frequencies in a vacuum, fondeal, given the experimental resonant frequencies measured in a fluid, fon"'d- In turn, solving Equation ( 4-4 ) would then yield a value for membrane tension. A main parameter of their model, /?, is given by Equation (4-5 ); where po is the density of the fluid and p is the density of the membrane material and the 2 accounts for the fact that the fluid is acting on both sides of the membrane.89 It is a dimensionless parameter that is used in the calculation of the ratio between ideal eigenfrequency and fluid loaded frequency as shown in Equation (C-3 ). A second parameter used in the model is c0, the speed of sound in the fluid. The properties of buffer solution, i.e. po = 1009 kg/m and c0 = 1420 m/s were used for these calculations. a 2 A) a P = —7- (4-5) ph v ' For a given set of membrane and fluid properties, the Gottlieb-Aebischer procedure gave a ratio of the experimental resonant frequency to that of a circularly symmetric ideal membrane mode k0„, according to Equation (4-6) . r fluid ko^%r <4"6> J0n A detailed explanation of the procedure of how fo„'ea is generated given fon'd is given in Appendix C. Figure 4.1 shows how the ko„ parameter of the first four circularly symmetric modes depends on the membrane thickness for a given membrane radius, (a = 0.0095m). It can be seen that 1 0 0 the thinner the membrane, the smaller the value of the ko„ parameter, and greater the extent the resonant frequency is lowered for a given mode. This effect is more pronounced as the mode number increases. Figure 4.2 shows how the ko„ parameter depended on the membrane radius for a given membrane thickness, (h = 25 um). The smaller the radius, the larger the value of ko„, therefore the surrounding fluid had a lesser effect on lowering the resonant frequencies. 101 0.18 i 0 0 0.01 0.02 0.03 0.04 0.05 0.06 membrane radius (m) Figure 4.2 Variation of k0n parameter value according to membrane radius for h = 25um. The second circularly symmetric mode of the membrane was usually monitored in this experiment, which required determining ko2. According to the Gottlieb-Aebischer model, the fa value for the typical membrane values used in this experiment is ko2 = 0.07457. To extract the tension of the membrane one must simply rearrange Equation (4-6 ) to solve for fo„,deal and insert this value into Equation (4-4 ) and solve for T, resulting in the relation depicted in Equation (4-7). T = 2n a ph J (4-7) 102 4.2 Fluid mixing model The response of the sensor membrane depended on the bulk concentration of the aqueous solution in which it was immersed. Therefore, it was necessary to understand how the bulk concentration changed when protein was added to the experiment and also when the experiment was flushed with clean buffer solution. Recall the general circulation schematic outlined for the experiment in Figure 3.20. There were two main tanks, with a pump controlling the flow from the mixing tank to the experimental tank, and an open-ended test tube was used to eliminate pulses in the fluid stream from the peristaltic pump. Although gravity controlled the drainage back into the mixing tank, the height of the mixing tank was carefully adjusted so that the drainage rate equaled the pumping rate and fluid level in the experimental tank remained constant. This system was represented as a three-tank mixing problem with connecting pipes. This model is a macro-mixing model; the micro-mixing is assumed to be essentially perfect. The rate of fluid flow between the tanks was denoted as R. Tank A, with volume Va was used to represent the mixing tank. Tank B with volume V\, was used to represent the experimental cell. Finally, tank C with volume, Vc, was used to represent the open-ended test tube. The volumes of the 3 connecting pipes were denoted as Vpi, Vp2, and VP3. This is shown in Figure 4.3 with the volumes of the 3 tanks and 3 pipes during a typical experimental run denoted. 103 B Experimental tank Vb Vp3 C test-tube Vc Vpl Vp2 A Mixing tank Va Va = Volume of tank A = 182 mL Vb = Volume of tank B = 217 mL Vc = Volume of tank C = 20 mL Vpl = Volume of pipe 1 = 50 mL Vp2 = Volume of pipe 2 = 26 mL Vp3 = Volume of pipe 3 = 5 mL Figure 4.3 Basic model to describe experimental mixing. To represent the change in amount of protein in the tanks as a function of time, a series of coupled differential equations were written. The amount of protein (in mg) in tanks A, B and C were defined as x(t), y(t) and z(t), respectively. The rate of change of protein in tank A was simply the amount of protein that entered from tank B minus the amount that left from tank A. However, since it took VP2/R seconds for the protein solution to travel along the connecting pipe between tanks B and A, the amount of protein that entered tank A at time t was based on the concentration of protein in tank B, at time (t-VP2/R) seconds. Equation (4-8 ) gives the differential equation describing the rate of change of protein in tank A, dx/dt. Similarly, Equations (4-9 ) and (4-10 ) give the rate of change of protein in tank B and tank C, respectively. The assumption was made that homogeneous mixing occurs in all tanks. dx{t) _ dt t-R R - ^ R ( 4 - 8 ) dyif) dt R R - ^ R Vb ( 4 - 9 ) 104 dz(t) dt 4.2.1 Initial mixing Figure 4.4 shows an example of how the concentrations in both the experimental and mixing tank changed as protein was added at time t = 0 to the mixing tank, tank A. For this particular case, R = 2.0 mL/s, and 10 mg of protein was added. It can be seen that it took approximately 3 minutes for the bulk concentration in the experimental cell to reach its equilibrium value. t- hi1 R R - ^ R (4-10) 0.07 n Mixing tank Experimental tank Figure 4.4 Concentration in experiment and mixing tank as protein is added at t = 0 calculated from model. 4.2.2 Flushing When flushing of the system with clean buffer solution occurred according to Figure 3.21, the mixing model was modified according to Figure 4.5. Fresh buffer solution was pumped 105 into the test tube and on into the experimental tank, and the waste solution was pumped out of the mixing tank. Al l of theses processes occurred at the same pump rate, R. C test-tube Flush tank Vp3 Vb B Experimental tank Vc Vp2 Va A Mixing tank -V waste Figure 4.5 Mixing schematic for flushing scenario. The new differential equations that describe the rate of change of protein in tank A, tank B and tank C are given by Equations (4-11 ), ( 4-12 ) and, respectively. For the case of the amount of protein in the test tube in Equation (4-13 ), the flushing scenario only applied Vpi/R seconds after the flushing had commenced and any remaining protein in pipe 1 had been pumped out. dx(t) dt R •R x(t) R (4-11) dy{f) dt t-R R-^R ( 4-12) 1 0 6 dz{t) x\ R dt dt if if t-R < flush start (4-13) R > flush start Figure 4.6 shows how the bulk concentration in the experiment and mixing tank changed when the flush was started at t = 0 according to these equations. Notice that the lowering of the concentration in the mixing tank lagged behind that of the experimental tank. This is because the clean buffer solution was pumped directly into the experimental tank, while the mixing tank has the waste from the experimental tank draining into it. It can be seen, however, that it took approximately 10 minutes for the concentrations of both tanks to be reduced to close to zero levels. 0 . 0 3 Mixing tank Experimental tank 4 6 time (min.) 1 0 1 2 Figure 4.6 Change of concentration in experiment and mixing tank when flush is started at t = 0, calculated from model. 1 0 7 4.2.3 Re-circulation In order to completely reduce the bulk concentrations in the mixing and experimental tanks to zero, the flush would have to occur for an infinite amount of time. In practice, however, the degree to which the experiment could be flushed was limited by the volume of the flushing tank. To model the bulk concentrations in the mixing and experimental tanks once flushing was stopped and fluid was once again circulated according to the schematic in Figure 4.3, Equations ( 4-8 ) and ( 4-9 ) were once again applied. For the case of amount of protein in tank C, Equation (4-14 ) was applied, which again took into account the time it took for the fluid in pipe 1 to be pumped through into tank C. dz(t) 4) dt vc y f t-R J R dt R — <0 R if if f y . \ t- p l V R > flush end (4-14) t p l V R > flush end Figure 4.7 shows the concentration of protein in the experimental tank once the flush was started for 3 different flush amounts. It can be seen that 0.5 L was clearly inadequate, as the bulk concentration returned to almost a third of its original concentration. Similarly, 1.0 L also returned to a small significant concentration. Figure 4.8 shows the same graph with the y-scale expanded to show the behaviour at longer flush times. Clearly, a flush amount of 1.5L was sufficient to reduce the bulk concentration down to close to zero levels (0.6% of its original concentration). 108 0 . 0 2 5 - i 0 2 4 6 8 1 0 1 2 time (min.) Figure 4.7 Concentration of protein in experimental vessel as flush proceeds for different flush amounts calculated from model. time (min.) Figure 4.8 Expansion of Figure 4.7 to show behaviour at longer flush times. As will be described in section 5.1.3, an experiment was conducted as described in section 3.4.2.2 to measure the bulk solution concentration as a function of time. In comparing this data to the bulk concentration predicted by the equations of this section, excellent agreement was achieved. 109 4.3 Adsorption model The adsorption model that is described in the following section was an attempt to describe the data generated by this experiment. Recall that the sensor measured the change in tension of the sensing surface as molecules adsorbed to the surface. The tension change was caused by changes in surface energy due to the initial adsorption of the molecules, and in the case of proteins, the slow change in conformation as unfolding occurred. The adsorption of both a simple surfactant molecule and a protein molecule were described by means of two separate versions of the model. The model consisted of a series of discrete particle states, whereby particles in adjacent states moved from one state to another. There are two types of populations in the model, bulk populations that were expressed in [molecules/m3], and surface populations that were expressed in units of [molecules/m ] . For the sake of clarity, all variables denoting particle populations will be shown in bold-type. The bulk populations included the particles in the bulk fluid in the experimental cell, B and a boundary layer population in the bulk fluid immediately above the sensor surface, L. The surface populations were a series of surface states, each with an effective area of interaction with the surface {a*} with units of [m2/molecule]. The flux of particles, 0, between adjacent states was the number of molecules per unit surface area, moving between states per second in units of [molecules/m s]. For the simplest case, it was defined as the rate of transfer from the first state to the second multiplied by the population of the first state. For example, consider adjacent surface states, ,4 and B, the flux from A to B is given by Equation (4-15 ), where RA->B is the rate, which in this case has units of [1/s] . The main parameters of this adsorption model were the rates of transfer between adjacent states. 0 A ^ B = R A ^ B A (4-15) Since the sensor measured the energy of interaction of the molecules with the surface, the change in tension predicted by the model was simply the number of molecules in each 110 surface state multiplied by an energy factor, -e*, in units of [Nm] to predict the change in tension of the sensor, AT(t) according to Equation (4-16 ), where the {e*} are model parameters. The index k indicates the adsorbed surface state, where ; > and m indicates the total number of adsorbed surface states allowed in the model. m AT(t)=-^rk(t)ek (4-16) k = \ The following sections wil l describe in more detail how the bulk concentration, the boundary layer, and the initial adsorption of the molecules are defined in this model. Furthermore, two specific cases of surfactant adsorption and protein adsorption are presented. 4.3.1 Bulk concentration The amount and rate of adsorption that occurs on a surface depends on the amount of molecules present in the bulk fluid solution. To model the bulk fluid concentration, c&, as a function of time in the experimental cell, the equations of section 4.2 were used. For a given experimental run, the experimental times when adsorbate amounts were added and when the flush operation was started and stopped were recorded. In this way the evolution of molecules in the experimental cell could be predicted. Figure 4.9 shows an example of how the bulk concentration in the experimental cell was calculated from the equations of section 4.2 for a given experimental run. For this particular case, 5 mg of SDS was incrementally added to the mixing tank at the times indicated by the dashed lines. The flush was started and stopped at the times indicated by the solid lines. I l l 20 40 60 time (min) 80 100 120 Figure 4.9 Example of evolution of bulk concentration in experimental cell, calculated from model. To convert from bulk concentration, in [mg/mL] to bulk concentration, B in [molecules/m3], Equation (4-17 ) was used where MW is the molecular weight of the molecule in [g/mol] and NA is Avogadro's Number which is 6.022x1023. In this way, Equation (4-17 ) could be used to generate the bulk concentration as a function of time B(t) which was then used as input to the adsorption model so that it could be compared with the experimental results to predict model parameters. B = -f-—cb (4-17) 4.3.2 Diffusion across boundary layer In order to model the adsorption of molecules to the surface, the transport of molecules to the surface had to be well understood. For a moving fluid, the molecules arrive at the surface by convective diffusion. Transport cannot be purely convective because although the bulk fluid is moving, there is zero fluid velocity at the solid/liquid interface due to the no-slip condition and molecules have to diffuse across a stagnant boundary layer. The diffusion boundary layer thickness, d, is defined as the distance at which transport by convection and diffusion is 112 the same. Equation (4-18) gives the Nernstian expression for the net flux of particles across the boundary layer with concentration L, where D is the diffusion coefficient in units of [m /s]. The boundary layer concentration, L, is considered the concentration of molecules immediately above, but not adsorbed to, the surface. < * W = ( * " * ) f ( 4 - l 8 ) The rate of change of molecules in the boundary layer was modeled by assuming diffusion across a stagnant layer according to (4-18 ). This should be a valid assumption as long as turbulent mixing was occurring in the experimental cell. The value of d depended upon the velocity of the fluid moving above the surface and the geometry of the flow cell. It can be calculated directly for simple, linear flow cells with well-defined flow rates. However, for this system, d remained a model parameter that was determined by fitting the adsorption model to the experimental data. In all cases, the required value of d had physically sensible values. 4.3.3 Initial adsorption To model the initial adsorption of molecules from the boundary layer to the sensor surface, the assumption was made that the rate was limited by the amount of surface area available for adsorption. In this way, the sorbate could approach a monolayer as adsorption progressed at high bulk concentrations. The flux of particles from the boundary layer, L, to the surface state, r, is given by Equation (4-19). It states that the flux was proportional to the population of the boundary layer, L and to the fraction of the surface area available for adsorption (1-6), where 6 is the excluded surface area and is given by Equation (4-20 ). It is simply the sum over all surface states of the population of each state multiplied by the effective surface area of each state. 0 L _ ^ r = R L ^ r L(l-6) (4-19) 113 m (4-20 ) 4.3.4 Surfactant adsorption model The first case of the adsorption model was used to describe the adsorption of a surfactant such as SDS. It was a simple case involving just one state of molecules on the surface. Surfactant molecules diffused from the bulk fluid across the boundary layer and adsorbed to the surface and back again. In this case, the area of a surfactant molecule was assumed to not change upon adsorption and was therefore not a model parameter. Figure 4.10 shows the schematic of the different molecular populations and fluxes in the surfactant adsorption model. Since the surfactant molecules are assumed to not change conformation upon adsorption, there is only one surface state, r, with effective surface area, a (~ 50A2 for SDS and Triton X-100) 9 2. bulk population 0, 0, L-*B boundary layer thickness d boundary layer population 0, L-*r surface population Figure 4.10 Schematic of kinetics of surfactant adsorption. 114 The diffusion across the boundary layer depicted as @B->L and 0L-+B are modeled as described in the previous section, where OB-^L is considered the net flux across the boundary layer. Equation (4-21 ) gives the flux, 0L-+r, from the boundary layer, L, into the surface bound state, r. It states that the flux was proportional to the rate, RL->H multiplied by the average population of the boundary layer, L, multiplied by the fraction of unoccupied surface available for adsorption as described in Equation (4-19 ). The backward flux, @r->u is given by Equation ( 4-22 ) and is simply proportional to the rate, Rr-+L, and the population of the surface bound state, R The rate of change in population of molecules in the boundary layer, L, is therefore given by Equation (4-23 ) and includes the fluxes into and out of the bulk population, B, and the fluxes into and out of the surface bound state, r. The rate of change of populations of molecules in the surface bound state is given by Equation ( 4-24 ), it includes the rates into and out of the surface bound state from the boundary layer. By substituting the already defined expressions for the fluxes into Equations (4-23 ) and (4-24), Equations ( 4-25 ) and ( 4-26 ) are generated which show the rate of change of the populations in the boundary layer and adsorbed state respectively. 0 L ^ r = R L _ > r L(l-ra) ®r->L=Rr->Lr ( 4 - 2 1 ) ( 4 - 2 2 ) dt d ( 4 - 2 3 ) dr dt = &L->r-®r->L (4-24 ) D dt (B-L)+Rr^Lr-RL^r L(l-ra) ( 4 - 2 5 ) 115 ~jfT = RL^>r L(l-ra)-Rr^.Lr (4-26) Finally, the expected tension change generated by the adsorption of molecules in ris given by Equation ( 4-27 ), where e is the energy factor in units of [N/m]. The Tcf shown in Equation ( 4-27 ) is a correction factor that is added in to account for a phenomenon that was observed during the adsorption of SDS. It was observed that during the course of surfactant adsorption, there would always be a slight drift in tension signal with time that seemed to leave a residual positive AT ones the SDS was rinsed from the surface. The magnitude of this effect seemed to be proportional to the amount of surfactant adsorbed on the surface. A possible explanation for this effect is the formation of small air bubbles on the surface during adsorption of the surfactant that did not dissipate once the surfactant is desorbed. How this correction factor is implemented in the adsorption model is discussed further in section 4.4.2. AT = -era + Tcf (4-27) In order to represent the rate constant parameters in a meaningful way, it is useful to convert them to time constants in units of [s]. Equation (4-28 ) gives the formula for the time constant for the boundary layer to surface state rate parameter, r^r- Equation ( 4-29 ) gives the formula for the time constant for the surface state to boundary layer rate parameter, rr-*L-d TL^r=~R~Z (4-28) zr^L=J?bl (4-29> The parameters in the surfactant adsorption model are: d, Rr-+L, R^r, e and a, where a is estimated based on knowledge of the geometry of the molecule and the remaining parameters are allowed to vary. 116 4.3.5 P r o t e i n a d s o r p t i o n m o d e l A new adsorption model was constructed to describe the adsorption of protein molecules at a solid-liquid interface. This model includes diffusion across the boundary layer, initial adsorption to the surface and spreading of the molecule on the surface. As protein adsorption is much more complicated and not as well understood as surfactant adsorption, this model is empirical in nature, and seeks to represent the observed behaviour of the adsorption process. Currently, protein molecules are thought to adsorb to surfaces according to the following steps: Initially, the protein molecules are brought in close proximity to a surface by either convective or diffusive transport. Then the protein molecules adsorb because of a variety of different reasons listed in section 2.2.5. Once adsorbed, the protein wil l rearrange from its compact globular form and "unfold" on the surface, due to the various reasons outlined in section 2.2.3.2.2. This unfolding wil l cause a larger area of interaction with the surface as more amino acid residues come into contact with the sorbent. Therefore, as unfolding occurs, there are more interactions with the surface that need to be disrupted in order for the protein to desorb from the surface. This would then make spontaneous desorption more and more unlikely as unfolding proceeds. However, it is known that protein molecules in the bulk wil l exchange with adsorbed molecules on the surface, and that this process ceases when the bulk concentration goes to zero. Therefore, there is a mechanism by which the molecules in the bulk can assist the adsorbed molecule to desorb and leave bare sorbent available for a new molecule to adsorb. In order to capture some of the characteristic protein adsorption behaviour listed above, the protein adsorption model derived here has the following features: To account for the protein conformation change, it is a multi-state model with an increasing area of interaction, ak, for each surface state i * . To account for the slow conformation change, the forward rate of transfer between states decreases as the area of the state increases. Furthermore, to account for the decreasing likelihood of desorption as conformation change proceeds, the backward rates also decrease as the area increases. Finally, in order to account for the interaction of the bulk molecules with the adsorbed molecules (i.e. exchange reactions and conformation change), a series of forward and backward rates are made proportional to the bulk solution 117 concentration {R*}. The dependence of the backward rates of transfer between adsorbed molecules is what differentiates this model from others. Notice that this feature allows for an off-rate from the surface while there are molecules in the bulk solution, exchange reactions between molecules on the surface and those in the bulk and for the off-rate to go to zero when the bulk solution is diluted. Models by Wahlgren et al.50 and Daly et alP allow direct exchange between molecules in the bulk and an adsorbed molecule that is otherwise irreversibly bound. Whereas the model described in this section only allows a molecule in a higher adsorbed state to desorb once it has devolved back through the intermediary states into a native-state conformation. This suggests that the bulk solution molecules, instead of directly exchanging with the adsorbed molecules, supply kinetic energy through collisions such that they achieve a conformation that is more loosely bound to the surface. The dependence of the forward rates of transfer on the bulk solution concentration implies that the bulk molecules also supply kinetic energy to the adsorbed molecules to allow them to change their conformation and become more tightly bound to the surface. This feature was allowed in the model by Daly et al. who found that the reorientation rate of lysozyme adsorption on slica was first order with respect to the bulk solution concentration. As described above, as a protein molecule changes its conformation on the surface, it transforms through a continuum of intermediary states. The challenge was to derive a model with a minimum number of discrete adsorbed states that successfully predicted the response of the sensor to conformation change. In the following section a three state surface model will be described. Figure 4.11 shows the basic schematic for this model with the different states and fluxes defined. Note that the fluxes listed here are the total fluxes and include contributions that are proportional to the surface populations as well as those proportional to the bulk concentration described above. 118 ~7N~ boundary layei( thickness B ) bulk population 0, L -»5 boundary layer population 0 0; ->z. 0 2 - W 0 3->2 0 2->3 surface state 1 surface state 2 surface state 3 Figure 4.11 Schematic of kinetics of protein adsorption. Equation ( 4 - 3 0 ) defines the flux of molecules from the boundary layer, L, into the first surface state, / } , as described in Equation ( 4 - 1 9 ) . This state is adsorbed in the native conformation and has a known effective surface area of interaction with the surface, ai, based on information on the geometry of the native state of the protein (a/ ~ 40A 2 for lysozyme). Equation ( 4 - 3 1 ) gives the flux of molecules from state 1, JT}, back into the boundary layer, L. 0L^1 =RL^1 L \ l - r 1 a l - r 2 a 2 - r 3 a 3 ) (4-30) 01^L={Rl^L+RULL)r1 (4-31) Equation ( 4 - 3 2 ) gives the flux of molecules from state 1, / } , into state 2 , / } , with effective surface area of interaction, a2, (a2>ai)- The back flux from state 2 , r2, to state 1, /"}, is given 119 by Equation ( 4-33 ). Note that in this particular case this caused protein molecules to revert back to state 1 where they could more spontaneously desorb. The model could also be modified to allow for particles to desorb directly say from state 2 directly into the boundary layer. 0I^2=(Rj^2+RU2L)r1 (4-32) 02^1 = (*2-W + RUl l) r2 (4-33 ) In the same way, Equations (4-34) and (4-35 ) give the forward and backward fluxes between state 2, and state 3, / } . 02- >3 = [R2^3 + *2->5 L)r2 ( 4-34 ) 03- >2 =(R3^2 + R3->2 L)r3 (4-35) Therefore, Equations (4-36 ), ( 4-37 ), (4-38 ) and (4-39 ) give the rate of change of the populations in each state as a function of time by adding and subtracting the relevant fluxes into and out of each state. d T d <4"36> -^- = {0L^l-®l->L+<?>2^1-®1^2) (4-37) = (®1^2-®2^1+&3->2-02-*3) (4-38) dt dr3 = 02->3-&3->2 dt ^ J ^ Z (4-39) 1 2 0 Equations ( 4-40 ), ( 4-41 ), (4-42 ), and (4-43 ) give the full expressions for the rate of change of the populations in each state by substituting the definitions of the fluxes into the above equations. dL f < * _ £ ) ~ R l ^ i L & " a i F * " a^r2 ~ asF^+ (R]^L + R*-*L L)Fl (4 -40 ) _ _ _ _ -~- = RL-^l L^-TjOY-r2a2-r3a3)-{R1^L +R*_>LL)rJ -(ff/->2+*/->2^7 +(*2-W+*2->l L)r2 ( 4 - 4 1 ) h -(R2->i+RUi L)r2 {R3->2 + R*3->2 L)r3 - ( R 2 ^ 3 + R * 2 ^ 3 L)T2 dt + 1 ( 4 - 4 2 ) d^f- = { R 2 ^ 3 + R*2_>3 L]r2 - (R3^2 + R * M L)r3 ( 4.43 ) Finally, Equation ( 4-44 ) gives the change in surface tension expected from the model and Equation (4-45 ) gives the surface fluorescence, AF, expected from the model where/is a scaling factor. The parameters for this particular version of the protein adsorption model are: d, ei, e2, es, aj, 02, a$, RI->L, RL->I, R1-+2, R i->2, R2->u R 2->u R2->i, R 2^3, R3->2, R 3-^2 and / ATm=-(r1e1+r2e2+r3e3) ( 4 _ 4 4 ) AFm=f(r1 + r2+r3) (4_45) 121 4.4 Data processing and model implementation The models for both the evolution of the bulk concentration described in section 4.2 and the molecular adsorption described in section 4,3 are given as a series of coupled differential equations. In both cases explicit solutions to these equations were not possible. However, it was relatively simple to construct a numerical implementation with a discrete time step, At. Populations of molecules in each state were calculated at each time step based on the model parameters. For the case of the equations of section 4.2, model parameters were all known and all that remained was to input the times that molecules were added and the flush was started and stopped. However, for the equations of section 4.3, the model parameters, namely the rates of transfer between molecular states, were unknown and a functional minimization program had to be used to compare the experimental results to the model and to therefore determine the model parameters. The following sections describe in detail how both the models for the bulk concentration evolution in the experimental cell and the adsorption and spreading of molecules on the sensor surface were implemented. 4.4.1 Raw data processing The adsorption model described in section 4.3 had to be compared to data from a real experiment in order to determine the model parameters. The following section describes how information from a particular experimental run was used to create a file with all necessary information to input into the adsorption model program. This information included the bulk concentration as a function of time, B(t), and the experimental change in the sensor tension as a function of time, ATe(t). Firstly, a RUN text file was manually created which contained all relevant information from an experimental run. Figure 4.12 shows the format of the RUN file. This included the filename of the raw data file generated by the LABVIEW program, the pump rate, the volumes of all tanks and pipes, the counts at which each amount of protein was added to the mixing tank, plus the counts at which the flush was started and stopped. Firstly the name of the raw data file was identified, then the molecular weight, Mw, of the adsorbate was listed, 122 followed by the pump rate, R, and the volumes of the mixing tank, Va, pipe 1, Vpi, experimental tank, Vb, pipe 2, Vp2, the test-tube, Vc, and finally pipe 3, Vp3. The next lines contained for each count, j the amount of adsorbate, Pj, which was added. Then, the counts at which the flush was started and stopped were indicated by a -1 and -2 respectively in the amount position. Finally, a zero was used to indicate end of file. "raw data file name " M , •-w R Va Vpl Vb Vp2 Vc Vp3 j P J J k Pk I -1 m -2 0 0 } count at which amount P. in (mg) added count at which flush started YYi -  < count at which flush ended Figure 4.12 Example RUN data file used to generate model input The RUN data file was used as input to a program called readdata.c that in turn was used to create the files necessary to run the adsorption model. The C program, readdata.c is included in Appendix D. It created two output files; one containing the bulk concentration as a function of time, B(t), and the other containing the experimental change in sensor tension as a function of time, ATe(f). Initially, the program matched the counts, j, listed in the run file with the corresponding experimental time step, te, from the raw data file. It then reset zero time as 30 minutes prior to the first addition of protein molecules to the experiment. In this way, all experimental runs could be compared as starting at the same time. The bulk concentration in the experimental cell as a function of time was calculated using the coupled differential equations of section 4.2. The continuous equations for the amount of adsorbate in the mixing tank, x(t), the experimental cell, y(t) and the test tube, z(t) were rewritten in a discrete time step form as shown in Equations (4-46 ), (4-48 ) and (4-49 ) . 123 For example, Equation ( 4-8 ) is rewritten as shown in Equation (4-46 ), where x, the amount of protein in the mixing tank, is calculated at each time, th based on its value at the previous time step, trAt. At the times, corresponding to the addition of an amount of adsorbate, Pj, Equation (4-47 ) applies. The discrete forms of the rest of the equations of section 4.2 are constructed in an analogous manner. x(ti) = x(ti-At)+ y ti-At- R x(tj-At) Vu RAt (4-46 ) x(ti) = x(ti-At)+Pj (ti=tj) (4-47) 7 y(ti)=y(ti-At)+ tt - At R y (ti-At) Vu \R At ( 4-48) 4i)=z(ti-At)+ ti-At V R Ah-At) v„ RAt (4-49 ) The bulk concentration of the experimental cell as a function of time, B(t,), in molecules/m3, is given by Equation ( 4-50 ). It was important that the size of the time step, At, chosen for the evaluation of the bulk concentration values was the same as was used in the subsequent adsorption model evaluation, so that the adsorption model and the bulk concentration, B(t) were calculated at the same times. B { t i ) J ^ l A M ( 4 . 5 0 ) 124 The readdata.c program also calculated the experimental change in tension of the membrane, AT(te), from the raw data file. Recall that the raw data file contained the resonant frequency, fof^ite), of the sensor as a function of time, where, te indicated the times at which the experimental data was collected. The corresponding experimental sensor tension, Te(te), was calculated according to Equation (4-7 ). Next, the start of the data file was adjusted so that te = 0 is redefined as 30 minutes before the first adsorbate amount was added. This allowed for a consistent baseline of data between all data sets. From this baseline, the average sensor tension, Te, prior to adding adsorbate was calculated according to Equation (4-51 ), where te is defined in seconds and n is the total number of data points in the baseline. Once the average had been calculated, the standard deviation of the baseline, cr, could be calculated according to (4-52 ). This standard deviation was considered representative of the noise in the rest of the experimental tension data. The change in tension of the sensor as a function of time, ATe(te), could then be calculated according to (4-53 ). Te=-^ / e <1800 ( 4-51) . #i-7 te < 1800 ( 4-52 ) ATe(te)=Te(te)-Te (4-53 ) Finally, the readata.c program output a file containing the total number of data points in the experimental change in tension list, the noise, cr, and a list of {te, ATe(te)} and also a file containing the total number of data points in the bulk concentration data and a list of {th B(ti)}. These two files were used as input to the adsorption model that is described in the next section. 125 4.4.2 Discrete adsorption model In the same way that the continuous equations of the fluid mixing model were converted to discrete forms, as described in the previous section, the adsorption model equations given in section 4.3 could be rewritten so that they could be solved numerically. Both the adsorption model and the bulk concentration were calculated at the same time step interval, At, so that the bulk concentration B(tf) from Equation ( 4-50 ) could be used as the bulk concentration, B, in the adsorption model. Equations (4-54) and (4-55 ) show the discrete time step solutions to the coupled differential equations for the surfactant adsorption model described in section 4.3.4. L(ti)=L(ti-At) + j{B{ti - At)-L(tt - At)}+ R r ^ L r(tt - At) -RL^rLiti-At^l-ariti-At)} At_ d (4-54) r(tj) = r(ti-At)+[Rr^L r{ti-At)-RL^r L^-^{1-0^-At))] At (4-55) Equation (4-56 ) shows how the correction factor, Tcf, described in section 4.3.4 was implemented. It had the behaviour that the magnitude of the effect was proportional to the adsorbed surface population and also increased monotonically with time. A new model parameter was introduced, cf, which was 0(10" 2 2) since it was multiplying a surface population which is 0(10 2 2 ) . Equation (4-57) gives the equation for the predicted change in tension of the sensor surface from the surfactant adsorption model. Tcf(ti)=Tcf(ti-At)+cfr(ti) (4-56) (4-57) Equations (4-58 ) through to (4-61 ) give the discrete time step solutions to the coupled differential equations of the protein adsorption model described in section 4.3.5. Equation 126 (4-62 ) gives the predicted change in tension of the sensor from the protein adsorption model and Equation (4-63 ) gives the predicted change in fluorescence from the protein adsorption model, where/is the fluorescent scaling factor. L(tt) = L(ti -At)+ D (B(tj-At)-L(tj -At)) + a (R^L + RUL L ^  - At))r} (tt - At) -Rj^LLiti-Atj f 3 N i-Takrk(u-At) V k=l At_ d (4-58) ri{ti)=r1{ti-At)+ RL->1 L(tt-At) 1 - J^akrk(ti-At) V k = l - (RJ^L + RUL L(tt - At"j)rj (tt - At) -(R,^2+ RU2 L(ti - At))rj (ti - At) + (R2^I + R*2^i L(ti - At))r2 (tt - At) At (4-59) (R1^2+RU2L(ti-At))r1(ti-At) ~ (R2^3 + R2-+3 L(ti - At))r2(tt - At) - (R2^J + RUJ L(tt - At))r2 (ti - At) + (R3^2 + R]^2 L(<i - At))r3 (ti - At) At (4-60) r3{ti)=r3(ti-At)+ (R2^3 + R2->3 L(tt - At))r2 (^ - At) ~{R3->2 +RU2 L(ti-At))r3(ti-At) At (4-61) ATm(ti)=-^ek rk(ti) k=l (4-62) AFm(ti)=fYjrk(ti) k=l (4-63) 127 To compare the model data, ATm(tl) and AFm(t,), to the experimental data generated from a particular experimental run, ATe(te) and AFe(te), the chi-squared parameter given by ( 4-64 ) was minimized, where the sum is over the total number of data points in the experimental data set,{r e}. Note that the model and the experimental data set must be compared at the same time, therefore, tt = te. Since the model time interval, Atm, is much smaller than the experimental time interval, Ate, the chi-squared parameter had to be calculated at the time value of the experimental data set, te. Note that the second term for the chi-squared fit of the fluorescent data only applies to experiments where the surface fluorescence experiment was run simultaneously. ATe{te)-ATm{ti) c j + 1 Up (?i=te) (4-64) 4.4.3 Model implementation The model could be implemented either in a spreadsheet format or a computer program. For the spreadsheet implementation, columns corresponded to populations in each state and rows to successive time steps. To minimize (4-64 ), the solver capabilities of the spreadsheet were used, which varied selected model parameters until the chi-squared value was minimized. However, this was not a rigorous parameter fitting method. A second more rigorous way of fitting the experimental data to the model was to use MINUIT a function minimization and error analysis package from CERN. 9 4 The model was generated using a FORTRAN program that was linked to the MINUIT code to minimize the chi-squared parameter given in Equation (4-64). I f MINUIT converged, it output parameter values and error bars, plus a correlation matrix. For input into the MINUIT program, the parameters were placed into an array called XVAL. Inside the FORTRAN program the actual model parameters were modified according to a scaling factor. Appendix E includes the FORTRAN program that was used for input to MINUIT when modeling surfactant adsorption as described in section 4.3.4. Table 4-1 summarizes all the 128 parameters involved in the surfactant adsorption rate model, the name used in the FORTRAN program, and how its value was scaled from the MINUIT XVAL array. model parameter units FORTRAN parameter FORTRAN representation 1/s RS1L XVAL( l ) d m D 10"4 XVAL(2) RL-+r m/s RLS1 10"5 XVAL(3) e N/m El * a x XVAL(4) cf N/m CF 10"22 XVAL(5) Table 4-1 Parameters for surfactant adsorption model Appendix F includes the FORTRAN program that was used for input to MINUIT when modeling protein adsorption. Table 4-2 shows all the parameters and their units involved in the three-surface-state model described in section 4.3.5. 129 model parameter units FORTRAN FORTRAN parameter representation d m D I f / 4 XVAL( l ) ei N/m E l AlxXVAL(2) RL^>1 m/s RLS1 I f /6 XVAL(3) R1->L 1/s RS1L I f / 3 XVAL(4) * R1->L m3/(molecules-s) RSI LB 10"21 XVAL(5) Rl^>2 1/s RS1S2 10"4 XVAL(6) Rl^>2 m /(molecules-s) RS1S2B 10"22 XVAL(7) R2->1 1/s RS2S1 10"5 XVAL(8) R2^>1 m3/(molecules-s) RS2S1B I f / 2 3 XVAL(9) R2^>3 1/s RS2S3 10'5 XVAL(IO) R2^>3 m /(molecules-s) RS2S3B 10"23 X V A L ( l l ) R3^>2 1/s RS3S2 lO"2 4 XVAL(12) R3^>2 m3/(molecules-s) RS3S2B 10"24 XVAL(13) e2 N/m E2 AlxXVAL(14) N/m E3 E2xXVAL(15) f - F XVAL(16) Table 4-2 Parameters for protein adsorption model. 5 RESULTS AND DISCUSSION This chapter wil l present data generated from the LabVIEW programs described in section 3.4, that monitor and control the adsorption sensor described in chapter 3. Section 5.1 discusses the measurement of various different sensor and system properties necessary to understand the response of the sensor. Section 5.2 presents results from surfactant adsorption experiments and compares these results to the surfactant adsorption model described in section 4.3.4. Section 5.3 presents some results from protein adsorption experiments, both from the resonant frequency measurement and the surface fluorescent experiment, and compares these results to the protein adsorption model described in section 4.3.5. 5 .1 Experimental setup There are various sensor properties that had to be well characterized in order that meaningful results could be generated from performing adsorption experiments. Firstly, the calculation of the membrane tension from the measured resonant frequency, described in section 4.1, will be discussed in reference to some experimental results. Secondly, the leaching behaviour of the sensor membranes once immersed in the aqueous solutions will be described. Thirdly, measurements of the bulk concentration will be compared with the 3-tank mixing model described in section 4.2. Next, the need to pre-adsorb the system tanks and tubing with a high protein concentration will be discussed. Finally, some of the artifacts present in data generated in the surface fluorescent experiment wil l be outlined. 5.1.1 Membrane tension measurement Changes in the surface energy due to the adsorption of molecules to the sensor surface are determined through changes in the overall membrane tension. This is possible because the surface energy contributes to the overall membrane tension from Equation (1-3 ). Therefore, it is important that the method described in section 4.1, that calculates the membrane tension from the resonant frequency, be validated and the results discussed. 131 As an example, the resonant frequencies of the first four circularly symmetric membrane modes for the spectrum shown in Figure 3.26 were used to determine the membrane tension using this method. Table 5-1 summarizes the measured experimental frequencies and the estimated tension for each of the four modes. Figure 5.1 shows the tension calculated in each of these modes according to the Gottlieb-Aebischer model. As can be seen, the resultant tension was fairly consistent between all four modes with a RMS fractional error of less than 5%, an amount that was consistent with the intrinsic noise observed in the frequency. The average tension in this particular example was 77 = 0.0314 ± 0.0006 N/m, where the quoted error value is the standard error of the average. Therefore, this method produces consistent results when comparing different resonant modes from the same membrane. Table 5-1 Membrane mode fwater (Hz) fvacuum (Hz) Tension (N/m) (0,1) 1.9 46.2 0.0322 (0,2) 7.6 101.9 0.0297 (0,3) 16.0 163.9 0.0313 (0,4) 26.4 226.8 0.0323 Calculated tension per symmetric mode number from Gottlieb-Aebischer model. a c 'So c 0) 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 0 1 2 3 4 5 symmetric mode number Figure 5.1 Calculated tension per mode number for a water loaded membrane. 132 As a further check for the consistency of this model, the resonant frequencies of a given membrane were measured while immersed in both air and water. Figure 5.2 shows the calculated tensions per symmetric mode number for both the air loaded and water loaded cases. While the calculated tensions for each case were consistent between mode numbers, the average calculated tension for the air loaded case was higher on average by 0.018 N/m than the water loaded case. This was consistent with the fact that the water molecules will interact with the polymer molecules to lower the surface tension much in the same way that the protein molecules lower the surface tension upon adsorption. 0.08 -j 0.07 -0.06 -s 0.05 -a 0.04 -mo "5 a 0.03 -<u H 0.02 -0.01 -0 -50 Hz • 2.2 Hz 130 Hz • 8.95 Hz • air loaded • water loaded 212 Hz • 18.6 Hz 1 2 3 symmetric mode number Figure 5.2 Comparison of first three circularly symmetric modes for water and air loading. It should be noted that the magnitude of the membrane tensions calculated for these membranes is remarkably low. Recall from section 2.2.1 that the surface tension of a clean air/water interface is 0.072 N/m, and the surface energy of a typical hydrophobic surface such as Teflon™ is 0.0189 N/m. For the air loaded case, a typical average tension was -0.06 N/m and for the water loaded case was -0.04 N/m. This membrane material is hydrophobic and hence would have a ys° value of the order of 0.020 N/m. According to Equation ( 1-3 ), this would mean the intrinsic membrane tension was - 0.020 N/m for the air loaded case. Therefore, the surface energy is a major contributor to the measured tension of the 133 membrane. This conclusion is further reinforced by the observation that when the membranes were immersed in an aqueous solution, they appeared to be highly fluid under the influence of the moving liquid. Therefore, it is quite plausible that changes in the surface energy at the membrane surfaces due to adsorption can have a significant and measurable effect on the overall membrane tension, as the surface energy at these surfaces is a major contributor to the overall membrane tension. Using the program described in section 3.4.1.1, the resonant spectra of 3 different membrane types could be measured both in air and water when the membranes were both actuated and measured at their center. Figure 5.3 shows a comparison of the air-loaded spectra for the 8161 and the 8141 membranes. Clearly, these two membrane types, although the same thickness, are quite different. The 8161 membrane appears to have a higher inherent tension than the 8141 membrane. 1.8 n f 1-6 t 1.4 ° 12 C 1 0 1 1 0.8 1 0.6 1 0-4 £ 0.2 > — 8161 8141 50 100 150 200 250 300 350 400 driving frequency (Hz) Figure 5.3 Comparison of air loaded resonant spectra for 8141 and 8161 membranes. From the resonant spectra shown above, the resonant frequencies of the first four circularly symmetric modes, {fo„} can be determined. The Gottlieb-Aebischer model described in section 4.1.2 was used to determine the membrane tension. Table 5-2 shows both the air loaded and water loaded average tensions for the three different types of 3M transfer 134 adhesive membranes, where the average is of all the measured modes and the given error bar is the standard deviation of the average. Clearly, from the limited data set available, there is some variation between samples produced from the same membrane material type. However, the measured tension calculated from different modes of the same sample is consistent. Since it was extremely difficult to mount the adhesive membranes without defect, i.e., small dust particles or minute puckers in the membrane material, it is quite probable that there could be some added tension due to the mounting procedure and this would account for the sample to sample variation. A more refined fabrication procedure would therefore be useful, but was not the focus of these initial studies. membrane Air loaded Tension Water loaded Tension A T material (N/m) (N/m) (N/m) 8141 0.065 ± 0.002 0.073 ±0.001 0.036 ±0.001 0.037 ± 0.002 8142 0.076 ± 0.004 0.042 ± 0.002 0.034 ± 0.006 8161 0.183 ±0.006 0.107 ±0.005 0.08 ±0.01 0.155 ±0.009 0.091 ±0.005 0.06 ± 0.01 Table 5-2 Comparison of membrane tensions for air and water loaded cases for 3 different membrane types. 5.1.2 Membrane leaching As was mentioned in section 3.1.1 when a fresh membrane was initially immersed in the clean buffer solution, and the resonant frequency tracking program described in section 3.4.1.3 was run, there would be a gradual lowering of the resonant frequency as a function of time. An example of this effect is shown in Figure 5.4. As can be seen, this is an approximately exponentially decaying function that levels off after approximately 20 hours. 135 8 6 -I • 1 1 — i • 1 0 10 20 30 40 time (hours) Figure 5.4 Example of change in resonant frequency of fresh 8141 sample immersed in buffer solution. There are several possibilities for the cause of this effect. The first is the leaching of chemicals from the membrane material. Since the chemistry of the 8141 transfer adhesive is proprietary, it is difficult to assess the feasibility of this explanation on the basis of chemistry. However, for the case of membranes made from a 2 part curing silicone elastomer, it is very likely that this effect would be caused by uncured silicone oils leaving the membrane. There is also the possibility that it is the slow change in polymer structure at the elastomer/water interface as discussed in section 2.2.1.4. However, since the membrane is elastomeric, it would be expected that this effect would be relatively quick compared to a polymer that is below its glass transition temperature. I f indeed this were the cause, this would call into question the slow response of the sensor due to protein adsorption. For example, the long time response of the sensor that was being attributed to protein conformation change could be due to conformation changes at the polymer surface in response to protein adsorption. However, as will be shown in section 5.1.5, the sensor does not exhibit this slow response to surfactant adsorption, which is a highly surface active molecule. In order to ensure that this leaching effect did not contribute to any signal during protein adsorption experiments, all samples were immersed in the experiment at least a day prior to 136 running an adsorption experiment. The change in resonant frequency of the membrane was monitored over this time to ensure that the sample had in fact reached a steady state resonant frequency prior to the addition of any adsorbents. 5.1.3 Bulk concentration measurement In order to validate the fluid mixing model described in section 4.2, an experiment was performed as described in section 3.4.2.2 to measure the bulk concentration using the fluorescently labeled protein molecules. Figure 5.5 shows a comparison between the experimentally measured bulk fluorescence and the predicted bulk concentration from the model, where the bulk fluorescence was calculated according to Equation ( 5-1) and scaled to match the experimental data. As can be seen, there is excellent agreement between the two data sets, meaning that the fluid mixing model is sufficient to describe the evolution of the bulk concentration in the experimental cell as protein or surfactant was both added into the mixing tank and also when the flush was initiated. Furthermore, as can be seen from Figure 5.5, the flush operation was efficient at removing any residual adsorbents from the experimental cell. This is important because the fluid mixing model described in section 4.2, is used as an input to the adsorption model described in section 4.3. 0.0007 E "Bb 0.0005 a _© 2 0.0003 e a u a w 0.0001 "a m -0.0001 bulk fluorescence fluid mixing model 40 45 50 time (min.) 65 Figure 5.5 Measurement of bulk fluorescence compared with predicted bulk concentration. 137 5.1.4 Pre-adsorbing experimental surfaces Protein will adsorb to all surfaces including the optical vibration sensor, the transducer that actuates the sensor, and the experimental vessels and tubing. Protein adsorption on the emitting and collecting fibers of the vibration sensor wil l effect only the amplitude of the signal and not its frequency response. This is also true of the transducer. Furthermore, the mass of a monolayer of protein is such a small fraction of the mass of the transducer that it is unlikely to affect its operation. The protein fouling of the experimental vessels and tubing is a more serious potential complication and will be addressed in the following section. One potential weakness of this system is the relatively low surface area of the sensor (~6 cm2) compared to the large surface area of the fluid delivery system (-900 cm2). As the protein in solution wil l adsorb to all available surfaces it is likely to significantly deplete the bulk solution concentration. However, this effect can be mitigated by pre-coating the experimental vessels and tubing with a high bulk concentration of the protein solution, one sufficient to form a saturated monolayer on all surfaces. The inherent irreversibility of this adsorption means that the large experimental surface area should have minimal effect on subsequent experimental runs at lower bulk concentrations. The effect of pre-coating the experimental vessels with protein can be illustrated by the following quick demonstration. A sample of fluorescently labeled lysozyme was prepared as described in section 3.2.1. Laser light at 532 nm was shone through the experimental cell to excite the fluorescently labeled protein molecules in the bulk solution. To prevent bleaching of the fluorescent tag by the laser beam, the laser was both shuttered and filtered with a neutral density filter. A partial reflection of the laser beam was deflected to a small integrating sphere with a photodiode detector to monitor the laser power. The emitted fluorescent light was collected as described in section 3.2.5. A small paddle attached to a motor was used as a mechanical mixer to quickly equilibrate the bulk solution concentration. To illustrate the irreversibility of binding to the experimental vessels, a clean glass beaker was inserted into the experiment as the experimental cell. The pump was turned off and 250 ml of pH 7.0 buffered solution prepared with de-ionized water was added. Firstly, a 138 small amount of the concentrated, labeled protein solution was added to the experimental cell and the fluorescence of the bulk concentration was monitored as a function of time as described in section 3.4.2.2. Secondly the experimental beaker was drained, rinsed and refilled with clean, buffered solution. Then the experiment was repeated. Figure 5.6 shows the results of this experiment. The signal received by the photomultiplier was divided by the signal received by the photodiode monitoring the laser power. This ensured that any variation in the laser power was not reflected in the presented data. It can be seen that in the first case of the clean glass beaker there was a small but easily measured depletion of the bulk concentration by adsorption onto the glass surfaces of the beaker. However, in the subsequent experiment, the bulk concentration remained constant, indicating that there was very little further adsorption onto the surface of the glass. This simple demonstration establishes that i f the surfaces of the experiment, including glass vessels and tubing, are pre-coated with protein, there wi l l be minimal depletion of the bulk concentration by these surfaces during experimentation. 0.0050 0.0045 0.0040 0.0035 0.0030 0.0025 g 0.0020 -I ~ 0.0015 0.0010 0.0005 ^ 0.0000 preadsorbed glass beaker clean glass beaker 0 0.5 1 1.5 2 time (hours) 2.5 Figure 5.6 Comparison of bulk fluorescent signal from clean and pre-adsorbed glass beaker Another implication of a large apparatus surface area is the possibility of exchange between native-state protein in the bulk and pre-adsorbed protein on the apparatus, leading to a 139 population of non native-state proteins in the bulk. Although the apparatus surface area is large, (-900 cm2) there is also a large bulk solution volume (-500 mL). I f a typical saturation surface population of - 1 mg/m2 is assumed, this would give a total amount of adsorbed protein of - 0.0009 mg. In the solution phase, a typical bulk solution concentration of 0.001 mg/mL would yield a total amount of solution phase protein to be 0.5 mg. Therefore, there is significantly more solution phase protein than adsorbed protein and any contribution to the solution phase from exchanged proteins would be insignificant. 5.1.5 Artifacts in surface fluorescence experiment When performing the surface fluorescence measurement described in section 3.2, there were several artifacts in the signal measured from the photomultiplier tube. This section will describe each of them and explain how they were removed from the fluorescent signal, F. 5.1.5.1 Calculation of fluorescence signal In order to generate a fluorescent signal, F, that was independent of any fluctuations in the power of the laser, it was important to divide the signal measured from the photomultiplier tube, p, by the signal measured from the integrating cylinder, /. This was done with the use of Equation (5-1 ), where Pback and hack are the background levels of the photomultiplier tube and the photodiode respectively measured before the laser was turned on. F = -JP-Pback) ( 5 . ! ) I'hack This is illustrated in Figure 5.7, where both the signals from the photomultiplier tube and photodiode are shown for a period before the laser is turned on and afterwards. The background signals hack and pback were calculated from the average of the signals from the photodiode and photomultiplier for the period of time before the laser was turned on. The background signal measured by the photomultiplier tube, hack, was due to scattered light from the red LED used in the vibration detection probe described in section 3.1.4. The background signal measured by the photodiode, Pback, was due to an offset existing in the amplifier measuring the output from the photodiode. Clearly, even though a filter was being used to cut-out any scattered light from the 532 nm laser, the photomultiplier tube measured a response due to this light. Figure 5.8 shows the fluorescence signal, F, calculated according to Equation (5-1 ), for the data shown in Figure 5.7. As can be seen by dividing the signal from the photodiode out of the photomultiplier signal, a flat, non-zero, response was seen in the fluorescence data. Shown on the figure is the average fluorescence, F , and the standard deviation of the data, crF, calculated over the time period indicated by the arrow. Once the fluorescently tagged protein molecules were added to the experiment, the actual fluorescent signal of interest was the change in fluorescence from this baseline, AF = F - F. The standard deviation, a/r, was used as a value representative of the noise in the fluorescent signal when calculating the %2 value for the fitting program described in section 4.4.3. time (min.) Figure 5.7 Comparison of signals from photomultiplier and photodiode as laser is warming up. 141 0.1 0.08 £ u .uu o £ 0.04 o 0.02 20 F = 0.0276 OF = 0 . 0 0 0 9 70 120 170 time (min.) 220 Figure 5.8 Fluorescence signal, F calculated from data shown in Figure 5.7. 5.1.5.2 Photo-bleaching of fluorophores As described in section 3.2.1, photo-bleaching is an irreversible process that destroys the fluorescent behaviour of the fluorophores, and the more laser radiation they are exposed to, the more bleaching wil l occur. Figure 5.9 shows an example of an adsorption experiment performed using the laser pointer described in section 3.2.2 as a method for injecting light into the membrane. Recall that for these adsorption experiments, the amount of protein molecules on the surface were expected to level off within about 30 minutes. While there is an excellent response observed in the fluorescent signal, photo-bleaching resulted in a steep decline in the fluorescent signal after it reached a maximum value at around 15 minutes. Note that the fluorescent signal dropped linearly with time, which would be expected, as the photo-bleaching effect should be proportional to the amount of laser radiation the fluorophores were exposed to. 142 0.029 0.024 -^ 0.019 -u § 0.014 -u o E 0.009 -0.004 --0.001 -r 1( Figure 5.9 Example of photo-bleaching observed during surface fluorescent experiment. One of the reasons for the pronounced photo-bleaching effect observed in the above example was the large amount of light emitted into the sample by the laser pointer. Although the signal from the photo-multiplier tube was only sampled periodically (every ~5 s), the laser light was continually bombarding the immobilized protein molecules with radiation. An experiment was therefore devised to verify that photo-bleaching was indeed occurring. The time between sampled data points was reduced to every ~2.5 minutes, the laser beam was manually shuttered so that laser radiation was only allowed to enter the membrane for a period of a few seconds and a neutral density filter was used to reduce the intensity of the laser radiation. The amount of laser radiation that the protein molecules were exposed to was therefore dramatically reduced. Figure 5.10 shows the results of this crude experiment. Clearly this data does not exhibit the same steep decline as in Figure 5.9 and the fluorescent signal appears to level off after 30 minutes. Therefore, this demonstrates that the linear drop in the fluorescent signal was due to photo-bleaching of the fluorophores by the exciting laser radiation. 143 0.016 0.014 0.012 fe, 0.01 a 0.008 o u £ 0.006 o 3 S 0.004 0.002 0 -0.002 time (min.) Figure 5.10 Example of adsorption experiment with shuttering and filtering to eliminate photo-bleaching. However, the laser pointer was used only for preliminary surface fluorescent experiments and the fiber pigtailed laser was determined to be a more desirable method to inject the laser radiation into the waveguide membranes. This method produced less scattered radiation and was also a less powerful laser. However, it was not physically possible to shutter this laser as the fiber was pigtailed directly to the laser and the end of the fiber was butt-coupled up to the edge of the membrane. Furthermore, due to the instability of the laser upon warm-up, turning the laser on and off was not an option. It should be possible to use a shutter at the input to the fiber, however this was not an option for the pigtailed laser used in this experiment. Figure 5.11 shows an example of an adsorption experiment performed using the fiber pigtailed laser. As can be seen there was a small decline observed in the fluorescent signal over a 2 hour period once it reached a maximum after around 30 minutes. Note that in the example shown in Figure 5.9, the fluorescent signal dropped 50% over 40 minutes. While in the example shown in Figure 5.11, the signal dropped 15% over 2 hours. So while still present, the photo-bleaching effect was reduced with the use of the fiber pigtailed laser. • • • • ~ i 1 9 10 20 30 40 50 60 70 80 144 0.018 < u C <a o o E 0.013 H 0.008 0.003 ^ ^ • ^ f — , , ,— -0.002 ^ 20 40 60 80 100 time (min.) 120 140 160 Figure 5.11 Example of surface fluorescence experiment performed using fiber pigtailed laser. As proven above, the cause of the linear decline in the surface fluorescent data was due to the photo-bleaching effect. Therefore, it was quite simple to remove this artifact from the fluorescent signal by fitting a straight line to the declining portion of the data, which was defined to be from the times, t\ to ti, according to Equation ( 5-2 ). The fluorescent signal could then be corrected to account for the effect of the photo-bleaching according to Equation ( 5-3 ). This is illustrated in Figure 5.12. AF(t)=mt + b (t = tj...t2) ( 5 _ 2 ) AFcorr(t)=AF(t) {t = 0...t,) ( 5 . 3 ) AFcorr {t) = AF{t)-(mt + ^+AFitj) (t = t]...t2) 145 Figure 5.12 Correction to fluorescence data to acount for photo-bleaching. 5.1.5.3 Bulk contribution to measured fluorescence As described in section 5.1.5.1, there was a significant amount of laser light scattered into the experimental cell. This scattered radiation could interact with the bulk solution and produced a small amount of fluorescent radiation. Depending on the magnitude of this effect, it is plausible that it could contribute to the fluorescent signal measured by the photomultiplier tube. Figure 5.13 shows an example of a surface fluorescent experiment where this was indeed the case. To illustrate this, the bulk concentration as a function of time, B(t), calculated from the equations of section 4.2, has been scaled appropriately and added to the graph. Clearly, the very sharp rise in fluorescent signal as the protein was added mimics exactly the change in the bulk solution concentration. To correct for this artifact, it was simply necessary to scale the bulk concentration to match the data and then subtract the scaled data,/B(f), from the fluorescent signal AF. This is shown in Figure 5.14, where the fluorescent signal now purely represents the fluorescent signal due the immobilized protein molecules on the waveguide surface. 146 0.006 < 0.005 -0.004 -0.003 0.002 0.001 0.000 -0.001 / 100 150 time (min.) 200 250 Figure 5.13 Example of a surface fluorescent experiment where the bulk solution is contributing to fluorescent signal. 0.004 0.003 0.002 ST o.ooi < -0.001 o.ooo {•: '*< 50 100 150 time (min.) 200 250 Figure 5.14 Data from Figure 5.13 corrected to account for contribution from bulk fluorescence. 147 5.2 Surfactant adsorption This section wil l discuss the response of the sensor membrane to the adsorption of two surfactants, SDS and Triton X-100, the properties of which are listed in section 2.2.3.1. Since it was not possible to label these surfactant molecules with the fluorescent tag described in section 3.2.1, only the resonant frequency measurements described in section 3.1 and 3.4.1 were performed. Section 5.2.1 will show some examples of the typical sensor response to surfactant adsorption. Section 5.2.2 will show that the surfactant adsorption model described in section 4.3.4 well describes the experimental data. These results are mainly presented to emphasize the differences in the response of the sensor between surfactant and protein adsorption. A comprehensive study on the phenomenon of surfactant adsorption is outside the scope of this thesis. 5.2.1 Typical sensor response Figure 5.15 shows an example of the response of the sensor to the addition of SDS at a concentration of 0.0057 mg/mL. The lines indicate the points in time at which the SDS was added to the mixing tank, and also when the flush was started and stopped. Clearly, the response of the sensor was flat prior to the addition of the surfactant, but showed a fast response such that within a few minutes a new steady-state resonant frequency was reached. This corresponds to a change in resonant frequency of Af= 7.35 - 7.20 = 0.15 Hz, a 2% change. When the flush operation was started, and the bulk solution concentration started to dilute, the resonant frequency started to climb as the surfactant molecules desorbed back into the bulk solution. After the flush was finished, the steady state resonant frequency returned to its value prior to the adsorption of SDS. Therefore, it is clear that the adsorption of SDS to the sensor membrane is a completely reversible process. 148 7.4 N X 7.35 7.3 o § 7.25 3 CT <D <h 7.2 7.15 7.1 add SDS start -flush end flush 20 30 40 50 60 time (min.) 70 80 Figure 5.15 Example of change in frequency as SDS is added to sensor membrane. The tension of the membrane corresponding to the resonant frequency was calculated from Equation (4-7 ). Figure 5.16 shows the change in tension of the sensor membrane for the data represented in Figure 5.15. This data was generated using the readdata.c program listed in in Appendix D. c H < I 0.0014 0.001 H 35 0.0006 0.0002 -0.0002 20 30 40 50 60 time (min) 70 80 Figure 5.16 Change in tension as SDS is added for example shown in Figure 5.16. 149 Figure 5.17 shows another example of SDS adsorption in which incrernental amounts of SDS were added every -20 minutes to the mixing tank. The graph shows both the change in tension of the sensor membrane and the evolution of the bulk solution concentration for this experimental run. Notice that the change in tension did not scale linearly with the change in the bulk concentration, as would be expected for surface adsorption limited to a monolayer. Notice also that this example exhibited a slight drift in tension upwards when the surfactant molecules were adsorbed on the surface. This resulted in the sensor membrane tension to return to a slightly higher value after the SDS was flushed from the system, than before it was added. The origin of this phenomenon is unclear; it may have been due to a slow build up of tiny air bubbles that were trapped on the surface, or perhaps to the hydrolysis of SDS as it formed dodecanol. It appeared only to happen during the adsorption of SDS and not Triton X-100. Recall that a correction factor to account for this behaviour in the surfactant model was discussed in section 4.3.4. 0.007 n 0.006 0.005 >~? 0.004 e & 0.003 0.002 0.001 r-«--10 -0.001 reduction in tension bulk concentration r 10 30 50 70 time (min.) 0.035 0.025 e 0.015 c + 0.005 -0.005 Figure 5.17 Example of sensor response to incrementally adding SDS to mixing tank, compared with bulk concentration. 150 5.2.2 Comparison with surfactant adsorption model As described in section 4.4.3, data from a given experimental run could be fed into the functional minimization program, MINUIT, to determine the parameters of the surfactant adsorption model described in section 4.3.4. Since this model is relatively simple compared to the protein adsorption model, MINUIT was always able to converge successfully and present accurate values for the given parameters. For example, Figure 5.18 shows the results of fitting the model to a data set for the addition of 10 mg of Triton X-100 to the mixing tank. The experimental values for change in tension of the sensor are shown as filled black diamonds, and the resulting model fit is shown as a solid line. The parameter values for this model fit are also shown. The output from the MINUIT program including the correlation matrix is shown in Appendix G. Figure 5.18 Response of sensor to Cb = 0.02 mg/mL TritonX-100. Clearly, there is an excellent agreement with the data. When an attempt was made to fit the experimental data with 4 parameters (Rr->L, d, R L->r and e), Rr->L, d and R i-+r were highly correlated as shown by the first correlation matrix in Appendix G. However, when the fit was repeated with d, the boundary layer, fixed and 3 model parameters (Rr-^L, R L->r and e), 151 Rr->L and R L->r were no longer correlated as shown by the second correlation matrix in Appendix G. This makes their values more physically meaningful and the adsorption of Triton X-100 molecules from the boundary layer to the surface happened about twice as fast as desorption from the surface back into the boundary layer. Figure 5.19 shows the result of fitting the adsorption model to the data presented in Figure 5.17 for the incremental adsorption of SDS. The output from MINUIT for this model fit is shown in Appendix H, for both a 5 parameter model {Rr->L, d, R i-+r, e and cf) with d variable, and a 4 parameter model with d fixed (Rr->L, R L->r > e and cf). Figure 5.19 Data from experiment shown in Figure 5.17 compared with output from surfactant adsorption model. Again, excellent agreement is achieved between the model and the data, although, the parameters appear to be quite different than for the adsorption of Triton X-100. In this case, the parameters Tr->i and r^r remained highly correlated even when the boundary layer thickness, d, was fixed. For this example it appears that adsorption from the boundary layer to the surface happens thirteen times faster than desorption from the surface back to the boundary layer. This could be explained by the fact that the hydrophobic tail on the SDS molecule appears to be significantly longer than that of the Triton X-100 molecule, as shown 152 in Figure 2.11 and Figure 2.12. This would indicate that an SDS molecule would have a greater affinity for a surface in an aqueous solution than the Triton X-100 molecule. Note that the value of the boundary layer parameter, d, turned out to be slightly different in the two examples of surfactant adsorption discussed. While the pump rate was the same in both cases, the fluid inlet pipes or the motor mixer paddle could have been orientated differently causing a difference in the thickness of the boundary layer. The model fit parameters for SDS adsorption shown in Figure 5.19 were used to generate the expected surface concentration, r, for a given bulk solution concentration, Q,, and an adsorption isotherm was generated. This is shown in Figure 5.20 where the solid diamonds represent data points generated from the model fit and the solid line corresponds to a Langmuir isotherm generated from Equation (2-19 ) and fitted to the data points with the given parameters. This predicts that the surface should saturate with a surface concentration of ~ 1.0 mg/m2. This compares well with results from Turner et al29 who measured the adsorption of SDS to polystyrene using neutron reflection and found a plateau surface concentration of rsat~ 1.1 mg/m 2. 0.0 4 , , , , , , , 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 cb (mg/mL) Figure 5.20 SDS adsorption isotherm generated from model fit and compared to Langmuir isotherm equation. 153 5.3 Protein adsorption This section will discuss the response of the sensor membrane to the adsorption of lysozyme, the properties of which are listed in section 2.2.3.2.3. Results from both the resonant frequency and the surface fluorescence measurements will be presented. However, the majority of data was collected using the resonant frequency measurement, prior to the completion of the apparatus necessary to perform the surface fluorescence measurement. As will be described, the preliminary results of the response of the sensor membrane to the adsorption of lysozyme prompted the invention of an auxiliary method to quantify the amount of protein on the surface of the sensor: the surface fluorescence measurement. Section 5.3.1 wil l show some examples of the typical sensor response during the resonant frequency measurement to the adsorption of lysozyme. Section 5.3.2 will show some examples of simultaneously performing the resonant frequency measurement and the surface fluorescence measurement. Section 5.3.3 will present examples of fitting the protein adsorption model described in section 4.3.5 to the data collected during the simultaneous measurements. 5.3.1 Resonant frequency measurements One of the surprising initial results achieved by measuring the response of the sensor to the adsorption of protein molecules, was its long time response. Recall that other methods that measure the mass effect due to protein adsorption found that the amount of protein at the surface, for a given bulk solution concentration, equilibrated quickly (usually within 30 minutes depending on the hydrodynamics of mass transport to the surface). However, by measuring the change in tension of the sensor membranes described in section 3.1.1, as protein molecules adsorbed to its surface, a comparatively long time response was observed. In fact, for many of the long time experiments conducted, the tension of the sensor membranes never appeared to reach an equilibrium value. For example, Figure 5.21 shows the change in tension of a sensor membrane for a bulk solution concentration of 0.00022 mg/mL and 0.001 mg/mL. As can be seen, even after 1000 minutes (16 hours), the tension in the sensor membrane was still noticeably declining for both cases. 154 0.008 0.007 0.006 0.005 0.004 0.003 <• 1 0.002 0.001 0 -0.001 400 600 time (min.) 1200 Figure 5.21 Change in tension of the sensor for lysozyme adsorption, Q, = 0.00022 mg/mL and = 0.001 mg/mL. As previously discussed (sections 2.2.1.4 and 5.1.1), the surface energy of the two sides of these sensor membranes is a major contributor to the overall membrane tension. The change in the membrane tension, AT, due to adsorption is actually due a change in the surface energy at the elastomer/fluid interface. As described in section 2.2.6.3, protein molecules adsorbing at the solid/water interface are known to undergo slow conformation changes that render them irreversibly bound to the surface over time. This irreversibility is thought to be due to more and more contacts being made with the surface as the protein molecules unfold. At the air/water interface, a slow decline in the surface tension of protein solutions has been shown to be due to protein denaturation at the interface.95 Therefore, it seems very plausible that the long time response of the sensor membranes was due to slow protein conformation change. Figure 5.22 compares two experimental runs, one at bulk solution concentration, c* = 0.001 mg/mL and the other at Cb = 0.0005 mg/mL. The response of the sensor to the higher bulk concentration appeared to level off after 10 hours, while the lower concentration run appeared to be still changing even after 30 hours. The higher bulk solution concentration should correspond to a higher concentration of protein molecules adsorbed on the surface. This would therefore leave less space available for these protein molecules to unfold and 155 spread out on the surface. It is therefore not surprising that the higher bulk concentration should reach a steady state tension faster than the lower bulk solution concentration. 1000 time (min.) 2000 Figure 5.22 Example of the long time response of the sensor for lysozyme adsorption, cb = 0.001 mg/mL and cb = 0.0005 mg/mL. Adsorption experiments are always highly dependent on the nature of the interface at which the molecules are adsorbing. A small amount of contamination can affect both the surface energy at the interface and the adsorbed amount. For these experiments, the sensor membranes were immersed in the circulating fluid for at least 24 hours in order to pre-leach them as described in section 5.1.2. Although every attempt was made to eliminate contaminants from the fluid, i.e. the use of distilled, de-ionized water and a 5 um in-line filter, it is not surprising that some contamination occurred. However, repeated runs at the same bulk concentration often had remarkably similar responses. Figure 5.23 shows an example where two different sensor membranes were exposed to a bulk solution concentration of cb - 0.001 mg/mL. As can be seen, the two data sets are barely distinguishable. In fact such agreement persisted for up to 16 hours. However, they would be indistinguishable i f the graph were displayed over that time scale. 156 s < I 160 210 260 time (min.) Figure 5.23 Example of two experimental runs showing repeatability for lysozyme adsorption, Cb = 0.001 mg/mL. However, there are examples of repeated runs at the same concentrations that did not produce the same response in the sensor membrane. Figure 5.24 shows two runs for the bulk solution concentration, Cb =0.0011 mg/mL. As can be seen, these two responses diverged initially, although appear to have a similar functional response at longer times. 0.006 n Figure 5.24 Example showing two different sensor responses for lysozyme adsorption for Cb = 0.001 mg/mL. 157 As already discussed, the irreversibility of protein adsorption is thought to be connected to the conformation change of protein molecules as they adsorb. As described in section 3.3.2, the fluid circulation system was adapted to allow the capability to flush out the fluid in the experimental cell and replace it with fresh, buffer solution. By diluting the bulk solution concentration down to near zero values, the irreversibility of protein adsorption could be studied. Figure 5.25 shows an example where the sensor membrane was exposed to a bulk solution concentration of Cb = 0.0015 mg/mL for ~3.5 hours and then flushed with clean buffer solution. The times at which protein was added and the flush was started and stopped are indicated on the graph by solid lines. 0 50 100 150 200 250 300 time (min.) Figure 5.25 Response of sensor to lysozyme adsorption for Cb = 0.0015 mg/mL with flushing. As can be seen, in contrast to the surfactant adsorption examples discussed in section 5.2.1, the sensor frequency did not return to its pre-adsorbed protein value. Instead, after the protein flush was initiated, it reached a new steady-state resonant frequency, well below that of the original value. This implies that protein adsorption to the sensor membrane was partially irreversible and that only a portion of the protein molecules were able to desorb back into the bulk solution. Note also that after the flush operation was finished, a new steady state resonant frequency was achieved that had a flat response. Therefore, even 158 though protein molecules were still adsorbed on the surface, they had ceased the slow conformation change process that was evidently occurring prior to the flush. There are two possible explanations for this behaviour. The first is simply that all the protein molecules that were not finished changing conformation at the surface desorbed, and all those remaining had become irreversibly bound and static. This implies an instantaneous change of state between reversibly and irreversibly bound, perhaps with a small probability of occurring in a given time period. This seems unlikely because the conformation change is thought to be a progression through a continuum of intermediate states each becoming progressively bound to the surface. A second explanation is that the protein molecules in the bulk solution help to cause the conformation change of the protein molecules on the surface. For example, they could supply kinetic energy to the bound molecules to allow them to overcome small energy barriers such that they may change conformation into a lower energy state. Clues to the link between protein irreversibility and conformation change can be found by examining the response of the sensor at different flush times. For example, Figure 5.26 shows two experimental runs conducted at similar bulk solution concentrations, Cb = 0.0002 mg/mL and Cb = 0.00022 mg/mL. For the first run, the system was flushed after - 1 hour, while the second was flushed after -20 hours. In the first example, it appears that a large proportion of the molecules desorbed from the surface, while in the second example an almost insignificant amount appears to have desorbed. This data supports the postulation that the protein molecules do indeed become more progressively bound to the surface the longer they are adsorbed. Again, a flat response in the tension was observed in the second example after flushing, but there is not sufficient data in the first example to determine i f the conformation change reaction had ceased for the short flush time. Notice also these two runs show a marked difference in their responses even though their bulk concentrations were very close. 159 a < 0.005 0.004 0.003 0.002 0.001 0 -0.001 0.00022 mg/mL 0.0002 mg/mL 200 400 600 800 1000 1200 1400 time (min.) Figure 5.26 Comparison of flushing after different amounts of time for two similar bulk solution concentrations of lysozyme. However, experiments conducted to examine the repeatability of the desorption step at short flush times showed that there was also a large variation in this behaviour at short surface exposure times. For example, Figure 5.27 shows the response of the sensor for two experimental runs conducted at a bulk solution concentration of Cb = 0.0002 mg/mL and flushing after 30 minutes of exposure to the surface. As can be seen, in one case a large proportion of the protein molecules on the surface appeared to desorb corresponding to an 85% decrease in the tension value, while in the other case, only a 23% decrease in the tension was observed. A smaller variation in desorption behaviour was also observed for the two examples shown in Figure 5.28 for a solution concentration of Cb = 0.0002 mg/mL and flushing after 60 minutes of exposure to the surface. In one case the tension decreased by about 25% upon flushing, while in the other, no apparent desorption appeared to occur. These two examples underline the difficulty of drawing conclusions about the kinetics of protein adsorption by comparing different adsorption experiments, particularly for short exposure times. Whether this variability was due to differences between the two surfaces to which the protein molecules were adsorbing, to subtle differences in the hydrodynamics for the two cases, or just to unpredictable desorption behaviour of the protein molecules themselves, is unclear. 160 Figure 5.27 Comparison of two sensor responses to lysozyme adsorption at cb 0.0002 mg/mL and flushing after 30 minutes. 0.0019 -, -0.0001 J 90 140 190 240 290 time (min.) Figure 5.28 Comparison of two sensor responses to lysozyme adsorption at cb 0.0002 mg/mL and flushing after 60 minutes. 5.3.2 Simultaneous measurement of surface fluorescence' and resonant frequency The change in tension of these sensor membranes as a function of time provides some interesting clues to the kinetics of protein adsorption and conformation change on the sorbent surfaces. Their long time response certainly indicates that they were sensitive to the slow conformation changes of the protein molecules. However, it became clear that independent conformation of the amount of adsorbed protein molecules on the surface was needed. This would not only confirm conclusively that the sensor membranes were measuring the conformation change, but also provide extra information to allow the kinetics of both the adsorption and conformation change to be well understood. It was for these reasons that the surface fluorescence measurement described in section 3.2 was devised. By simultaneously measuring the surface fluorescence and the change in resonant frequency, the time evolution of both signals could be compared. Recall that the surface fluorescence should be proportional to the amount of protein molecules on the surface, which is expected to reach a steady state value within about 30 minutes. Unfortunately, the implementation of the surface fluorescence measurement proved to be difficult. Constructing membrane waveguides that efficiently coupled the injected laser light was problematic. In fact, in order to get a sufficient amount of radiation into the membranes at all, the 50 um thick membranes had to be used. Furthermore, often the membrane appeared to guide the laser light efficiently in air, but the handling required to insert the membranes into the aqueous solution could easily hamper this capability. Often, after the waveguide membranes had been immersed in the aqueous solution for 24 hours to pre-leach them, the amount of efficiently guided laser radiation had significantly deteriorated. Nonetheless, several successful experiments were conducted that generated some very encouraging results. Figure 5.29 shows an example of a simultaneous measurement of both the change in membrane tension and the surface fluorescence for a bulk solution concentration of Cb = 0.00036 mg/mL and flushing after 2 hours. For this example, the surface fluorescence, F, was corrected to account for the photo-bleaching effect described in section 5.1.5.2, and 162 shown in Figure 5.12 for this particular example. Clearly, the surface fluorescence shows that the amount of protein molecules on the surface reached a plateau value after approximately 40 minutes, while the change in membrane tension was clearly still decreasing up until the flush time. Upon flushing it appears that 25% of the protein molecules desorbed from the surface according to the surface fluorescence. However, there was a much smaller proportional change in the tension signal due to flushing. The fact that these two data sets were measured simultaneously, allows for the kinetics of protein adsorption and conformation change for this experimental run to be better understood. Furthermore, it gives some insight into how the adsorbing protein molecules were changing the surface energy of the sensor membranes. 0.0025 0.0020 ^ 0.0015 H & 0.0010 I 0.0005 0.0000 -0.0005 J II if n n * i ( W l 0 add protein 50 start flush end flush -100 time (min.) T 0.020 0.016 0.012 0.008 + 0.004 0.000 < o c <u a xn CD s-< O 3 ->- -0.004 Figure 5.29 Response of sensor to 0.00036 mg/mL lysozyme concentration showing both change in tension and surface fluorescence. Clearly, the above example confirms that the signal due to the change in tension of the membrane continued to change well after the population of protein molecules on the surface had reached a steady state value. This strongly supports the hypothesis that the slow change in the tension of the membrane was due to the conformation change of the adsorbed protein molecules. The alternate possibility that the slow change was due to the rearrangement of the polymer molecules at the elastomer interface, as discussed in section 2.2.1.4, is dispelled by 163 the fact that the slow change ceased when the bulk solution was diluted and there were still adsorbed protein molecules on the surface. One would not expect that the rearrangement of the polymer molecules in the membrane itself to be affected by the dilution of the bulk solution concentration. Futhermore, this effect was not evident in the surfactant adsorption data discussed in section 5.2. To confirm that the fluorescence signal shown in Figure 5.29 had no contribution due to the bulk fluorescence, the fluorescence signal, &Fcorr was compared with the change in the bulk concentration, cj, and shown in Figure 5.30. Clearly, the time evolution of the bulk concentration in the experimental cell as the protein molecules were added, was much faster than the increase in the surface fluorescence. Therefore, for this particular example the surface fluorescence signal, AFcorr, had no contribution from the bulk fluorescence. 0.018 l | 0.014 < o 0.010 c • <D O g 0.006 o 3 0.002 -0.002 lb -add protein start flus end flush fluorescence • bulk concentration 60 . . . ,110 time (mm.) 160 0.00045 0.00035 0.00025 <^  0.00015 § + 0.00005 -1- -0.00005 Figure 5.30 Comparison of fluorescence signal with change in bulk conentration for experiment shown in Figure 5.29 Figure 5.31 shows another example of a simultaneous measurement conducted at a bulk solution concentration of Cb = 0.00072 mg/mL, twice the concentration of the previous example. For this example, the fluorescent signal, F, was corrected for both a photo-bleaching effect as described in section 5.1.5.2 and also a bulk fluorescent contribution as described in section 5.1.5.3. 0.0035 ^ 0.0025 -] < 0.0015 0.0005 4 -0.0005 J 0 add protein if start flush>| end flush • 50 100 150 time (min.) 0.07 0.06 + 0.05 0.04 200 < 0.03 g 0.02 g o 3 o . o i E o.oo - o . o i Figure 5.31 Simultaneous measurement of change in tension and surface fluorescence for lysozyme adsorption at cb = 0.00072 mg/mL. Figure 5.32 shows a graph comparing the change in tension response for these two examples. 0.004 -0.001 ® 100 150 time (min.) 200 Figure 5.32 Comparison of the change in tension for the simultaneous measurements shown in Figure 5.29 and Figure 5.31 165 Figure 5.33 shows a graph comparing the fluorescence signals obtained for these two examples. Note that there is no physical meaning to the scaling of these two curves. In each case, the strength of the fluorescent signal from the surface depended on the intensity of the laser radiation that was efficiently coupled into the waveguide of the sensor membrane. 0.06 0.05 8 0.04 0.03 o 8 0.02 <U o 0.00072 m g / m L 0.01 0.00 0 -0.01 1 50 0.0003^ mg /mL 100 time (min.) 150 200 Figure 5.33 Comparison of the fluorescence signal for the simultaneous measurements shown in Figure 5.29 and Figure 5.31. Because of the questionable repeatability of the resonant frequency measurements and the scaling mismatch issue on the fluorescent data, it was difficult to make meaningful comparisons between different experimental runs conducted using different bulk concentrations. Therefore an experiment was conducted whereby protein was incrementally added to the experiment such that the sensor membrane would be exposed to two bulk solution concentrations sequentially. Figure 5.34 shows the results from conducting a simultaneous experiment in this manner. The sensor membrane was initially exposed to a bulk solution concentration of c* = 0.00037 mg/mL for a time of 70 minutes, and then = 0.00055 mg/mL for a time of-120 minutes. The experiment was then flushed after - 3 hours of total exposure. Note that the fluorescent data generated for this experiment had a good signal-to-noise ratio and did not need to be corrected for either photo-bleaching effects or any bulk fluorescent contribution. The fluorescent data clearly responded to both the initial 166 addition of protein and the subsequent addition by reaching a new steady state value. The resonant frequency data, however, showed only a small kink at the time of the second addition of protein, indicating a slight increase in the rate of conformation change due to the higher bulk solution concentration. 0.050 0.040 0.030 O c O <L> 0.010 J 0.020 0.000 -0.010 Figure 5.34 Response of sensor to the incremental adding of lysozyme for Cb = 0.00037 mg/mL and = 0.00055 mg/mL. Figure 5.35 shows a comparison of the fluorescent signal, AF, for the experiment described above compared with the evolution of the bulk solution concentration, Cb, in the experimental cell. Note that despite the fact that the second addition of protein brought the bulk solution concentration to one and a half times its original value, the fluorescent signal only increased by about 25% in response. This indicated that, as expected, the first exposure to the protein solution caused a significant portion of the surface to become covered in adsorbed protein, such that increasing the bulk concentration did not result in a linear increase in the amount adsorbed. 167 0.06 -0.05 I*, <I 0.04 8 0.03 -<u M 0.02 -<u o M °- 01 -0.00 --0.01 _ , , - u . w u i time (min.) Figure 5.35 Comparison of data shown in Figure 5.34 compared with the change in bulk concentration for experimental run. 5 . 3 . 3 Comparison with protein adsorption model The 3-state protein adsorption model described in section 4.3.5 is a complex model with 18 parameters. Many of the forward and backward rate calculation algorithms have the option of being proportional to the population of molecules in the boundary layer, or not, and it cannot be determined which case is more realistic. It was the hope, in fitting this model to the experimental data, that many of the associated rate parameters could be set to zero, and those remaining would be physically meaningful. The data generated from the simultaneous measurement gives both information regarding the rate of conformation change of the protein molecules in the change of tension of the membrane and the total amount of protein molecules adsorbed to the surface in the surface fluorescence measurement. By combining these two data sets it was the hope that a physically meaningful fit of the model to the experimental data could be generated. Recall from Equation (4-44 ), it was stated that the change in tension of the sensor membrane is predicted by summing over the product of each surface state population, / with a weighting factor, eu. Furthermore, from Equation (4-45 ), that the change in surface fluorescence is predicted by multiplying the total surface population, r,0, = / }+ by a fluorescent scaling factor,/ Initially it was thought that the weighting factors in Equation (4-44 ) should be proportional to the area of each state, { ^ } . However, by examining the data generated in the simultaneous experiments, it became clear that there was no way that this would be able to simultaneously fit both sets of data unless a? and a3 were much greater than aj. However, as described in section 2.2.3.2, lysozyme is a relatively stable globular protein and not expected to radically change its conformation or projected area of interaction with the surface significantly. Therefore, the model parameters a2 and a3 are used only to predict the fractional excluded surface area as molecules adsorb and not the extent of change that the adsorbed molecules have on the surface tension. Recall from section 2.2.5.1 that even though protein molecules are considered to adsorb irreversibly, there is an exchange reaction with molecules in the bulk. Therefore it is speculated that bulk molecules may interact with adsorbed molecules making them more likely to desorb. It was for this reason that the protein adsorption model included a series of backward rates that were proportional to the population of molecules in the boundary layer. This feature of the model allowed for an off-rate to exist while there were protein molecules in the bulk solution, therefore allowing for steady state surface populations to be established that were concentration dependent. Once the protein molecules were removed from the bulk solution, these bulk dependent back rates dropped to zero and the protein molecules which had changed their conformation became irreversibly bound to the surface. Once the bulk solution was depleted, and the molecules were irreversibly bound, they did not desorb even over long time periods. Therefore, it seems reasonable that the backward rates for particles in state 2 and state 3 should be solely proportional to the bulk, and not have any contribution from a spontaneous rate. Note that this explanation accounts for the paradox of irreversible protein adsorption that was described in section 2.2.5.1. Recall from the experimental results of section 5.3.1 that upon flushing the slow change in the tension of the membrane ceased, indicating that the adsorbed protein molecules had ceased their conformation change reaction. This implies that the conformation change reaction was also influenced by protein molecules in the bulk. Therefore the forward rates of transfer from state 1 to 2 and 2 to 3 should also be solely proportional to the bulk. Note that 169 while there may be a spontaneous rate of conformation change independent of the bulk, on the time scale of these experiments for lysozyme it appears to be essentially zero. In order to rationalize the response of the sensor to the initial adsorption and subsequent conformation change with the above discussion, the following sequence of events was proposed: The molecule diffused from the bulk to the boundary layer at rate RB->L and adsorbed in its native state at rate RL->I. Since it made minimal contact with the surface it could also spontaneously desorb at-rate Ri-n. or transfer to state 2 at rate R*i^L. Once in state 2, it could either revert back to state 1 at rate R*2->i L, or transfer to state 3 at rate R*2->3L. In state 3 it could revert back to state 2 at rate R* 1-+2L. Therefore, in order to fit the protein model to the experimental data the following assumptions were made: Firstly, the areas of state 2 and state 3 were fixed constant at 1.1 x ai and 1.2 x a/, respectively. Secondly, all forward and backward rates (except for state 1 into and out of the boundary layer) were made solely proportional the population of molecules in the boundary layer. This reduced the full, 18 parameter model to an 11 parameter model. The experimental data set shown in Figure 5.34 was considered a good candidate to fit the model to, as the fluorescent data had good signal-to-noise and no artifacts due to photo-bleaching or bulk fluorescence had to be removed from the data. An initial attempt to guess some meaningful model parameters was done on the spreadsheet version of the model. When a reasonable fit was achieved, these model parameters were input into the MINUIT version of the model and allowed to run. MINUIT was able to converge to a successful solution and output its predicted model parameters. Figure 5.36 shows the result of this operation, where the experimental data for both the change in tension of the membrane and the surface fluorescence are compared with the model. The output from MINUIT for this model fit is given in Appendix I. As can be seen, an excellent agreement was achieved, and the model was able to successfully predict the response of the sensor to the incremental increase in bulk concentration and also to flushing. 170 time (min.) Figure 5.36 Model fit for the simultaneous experimental data shown in Figure 5.34. Figure 5.37 shows the predicted populations of the molecules in each surface state and the total population as a function of time for the model fit shown in Figure 5.36. The model predicted that the molecules transferred into state 2 relatively quickly and then more slowly into state 3. Upon flushing, it was mostly the molecules in state 1 that desorbed, while a small proportion of the molecules in state 2 seemed to revert back to state 1 and then desorbed as the bulk concentration was diluted. Al l molecules in state 3 remained irreversibly bound to the surface. 171 1 1 0 50 100 150 200 250 time (min). Figure 5.37 The evolution of each surface state population as a function of time for the model fit presented in Figure 5.36. Table 5-3 shows the model parameters that were used to produce the fit that is shown in Figure 5.36, along with an approximate time constant, r, for the listed rates. The time constants for RL-+i and RI-+L are given by Equations (4-28 ) and ( 4-29 ) respectively, while those listed for the rates proportional to the bulk are estimated by inverting the rate and multiplying by the bulk concentration in molecules/m3. Note that this model fit predicted that ei was essentially zero, and therefore molecules in that first state had a neglible affect on the surface energy of the sensor membrane as they adsorbed. This phenomenon has actually been observed for the adsorption of lysozyme at the air/water interface96 where an induction time was observed in the reduction of surface tension as the molecules adsorbed, and it was only after they had changed their conformation that they reduced the surface tension. It may seem surprising that the surface energy of the sensor was not affected by the initial adsorption of lysozyme molecules to its surface, yet exhibited a dramatic response to the adsorption of surfactant molecules. However, since the surfactant molecules are much smaller (-288 a.m.u) than the lysozyme molecules (14000 a.m.u), the number density of adsorbed surfactant molecules was much greater than for lysozyme. This implies that there were more physical contacts with the sensor surface that were able to affect the surface energy. For example, the adsorption model predicted-a typical surface population for 172 surfactant adsorption to be r= 1 x 10 1 8 molecules/m2, while for lysozyme it was r= 1 x 10 molecules/m , which is two orders of magnitude less. model parameter value units time constant T d 0.014 mm ei 0 N/m RL->1 2.18 x 10"5 m/s 0.9 s R1->L 0.0151 1/s 66.2 s * R1^>L - m3/(molecules-s) Rl->2 - 1/s * Rl-*2 1.61.x io; 2 2 m /(molecules-s). 6.6 min. R2->1 - 1/s R2->1 v 9.96 x.'lO"23 m3/(molecules;s) 10.6 min. R2^>3 - 1/s R2^>3 3.43 x 10 - 2 4 m3/(molecules-s) 5.1 hours R3^>2 - 1/s * R3^>2 1.03 x 10"23 m /(molecules-s) 1.7 hours e2 -0.00305 ai N/m e3 -0.0228 ai N/m f 0.0108 -Table 5-3 Model fit to data shown in Figure 5.36. Due to the great challenge of generating data with the simultaneous measurement method, it was not possible to assess the predictive power of this model under a wide avariety of 173 different conditions. However, the fact that the model was fit to an experimental data set with two different bulk solution concentrations, means that it should be reasonably good at predicting the sensor response to bulk solution concentrations close to the ones for which it was applied. To test this, the model parameters shown in Table 5-3 were used to predict the sensor response for the experimental data shown in Figure 5.29 and Figure 5.31. Note that the only model parameter that was varied was/ the scaling parameter for the fluorescent data as this is fully expected to be different for different experimental runs. As can be seen, while not in exact agreement with the experimental data, the model does do a good job at representing the data. 0.0025 0.0020 'g 0.0015 ^0.0010 < I -0.0005 -model fit experimental data 100 time (min) 150 0.016 0.011 . 0.006 0.001 200 -0.004 Figure 5.38 Comparison of the experimental data shown in Figure 5.29 with the model fit with parameters listed in Table 5-3. 174 0.005 0.004 ^0.003 &0.002 ^0.001 0.000 -0.001 o 50 m o d e l fit e x p e r i m e n t a l d a t a 100 time (min) 150 u 0.02 | o 93 u 0.01 § 3 0.00 -0.01 Figure 5.39 Comparison of the experimental data shown in Figure 5.31 with the model fit with parameters listed in Table 5-3. While the model parameters listed in Table 5-3 may not be able to predict the sensor over a wide range of different conditions, the fact that the model was able to predict the sensor response for both protein addition and flushing for a particular experimental run, means the form of the model as presented is a good candidate for describing and understanding the kinetics of protein adsorption. In order to check that the protein adsorption model predicts the expected behaviour, an ascending isotherm was generated using the model fit parameters listed in Table 5-3. The steady-state total surface population, rto, in mg/m2, was plotted against the corresponding bulk solution concentration, Cb in mg/mL. This is shown in Figure 5.40. As can be seen, the isotherm has the expected shape with a plateau surface population, rsat ~ 1.2 mg/m2, reached at a bulk solution concentration of csat ~ 0.003 mg/mL. Figure 5.41 shows a comparison of the model fit isotherm shown in Figure 5.40 with some experimental results from Haynes et al. who measured the adsorption of lysozyme to negatively charged polystyrene latex using the solution depletion method. In order to compare the shapes of the two isotherms, they are plotted on a graph with rtotirsat vs. Cblcsat, where rsat is an estimate of the plateau surface concentration corresponding to ~ 90 % of a monolayer, and csat is the corresponding bulk concentration at which this is achieved. As the shapes of the two isotherms are similar, this 175 is further validation that the protein adsorption model presented here exhibits characteristic protein adsorption behaviour. a 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.001 0.002 0.003 0.004 Cb (mg/mL) 0.005 0.006 Figure 5.40 Isotherm generated from protein adsorption model using the model fit parameters listed in Table 5-3. I 0.8 - • • _ 5 0.6 - • negative polystyrene latex ^ 0.4 - ' • model ft 0.2 0 0.5 1 1.5 2 2.5 Cb/cSttt Figure 5.41 Comparison of model fit isotherm shown in Figure 5.40 with experimental results for lysozyme on negatively charged polystyrene latex. 176 6 CONCLUSIONS The central motivation of this thesis was the investigation of the apparent paradox of irreversible protein adsorption, where despite the irreversibility of adsorption, sub-monolayer coatings can exist in apparent equilibrium with protein in the bulk solution. In part this problem has been difficult to tackle because of a dearth of experimental techniques capable of measuring the kinetics of protein conformation change at the solid/water interface. The construction and operation of a resonant frequency sensor membrane sensitive to molecular conformation change, as is described here, has made it possible to provide information that sheds some light on this problem. The resonant frequency of the sensor membranes is dependent on the membrane tension, so in general the mass of the membranes is not critical, however the fact that the membrane must operate underwater created important issues which needed to be taken into account. Specifically, the fluidic mass loading significantly reduced the resonant frequencies compared to those of a similar membrane, oscillating in a vacuum. A method to account for the fluid mass loading was successfully applied to the elastomeric sensor membranes and this enabled the membrane tension as a function of time to be determined from the resonant frequency signal. Typically, the sensor membranes operated in the (0,2) membrane mode at a frequency of 7-8 Hz. This corresponded to a net membrane tension of -0.04 N/m, of the same order as the surface tension often associated with air/fluid interfaces. To evaluate the behaviour of the surface energy sensor, its response to two different types of molecules was investigated; surfactants and proteins. Surfactants are highly surface-active molecules that tend to adsorb reversibly at most solid interfaces. As would be expected in response to adding surfactant molecules to the bulk solution surrounding the sensor membrane, it responded quickly, within minutes, and a new steady-state resonant frequency was achieved. Upon flushing the surfactant molecules out of the bulk solution, the resonant frequency returned to its initial value. In contrast, protein molecules are large complex molecules whose native state conformation is a delicate balance between competing thermodynamic driving forces. Correspondingly, the sensor response was much more complex. In response to adding protein molecules to the bulk solution surrounding the 177 sensor membrane, the resonant frequency dropped slowly over time for a period of many hours. Upon flushing, the resonant frequency rose slightly and achieved a new steady-state well below that of its initial value. Therefore, it was concluded that the surface energy sensor described herein was also sensitive to the slow conformation change of adsorbed protein molecules and was successfully able to observe the partial irreversibility of protein adsorption. Returning to the problem of the permanence of such protein adhesion, this has usually been attributed to the conformation change of the protein molecules as they adsorb and change their shape such that more and more physical bonds are made with the surface. The molecule therefore becomes increasingly bound to the surface over time, making spontaneous desorption extremely unlikely. However, this interpretation seems to be inconsistent with the observation that at low bulk solution concentrations, steady state sub-monolayer surface populations are achieved. Such a sub-monolayer equilibrium would normally be explained by the existence of a rate of desorption from the surface that counterbalances the adsorption rate so that a steady state surface concentration is maintained. But the permanence of protein adhesion would appear to be inconsistent with the existence of a non-zero desorption rate. It is this apparent paradox of protein irreversibility that we hoped to tackle with using the new information at hand. It was therefore important to confirm that the long time response of the sensor membrane was due to molecular conformation change. To do so, a means of independently measuring the amount of protein molecules on the surface of the membrane was devised. The protein molecules were tagged with a fluorescent label that absorbed laser radiation at 532 nm and emitted fluorescent radiation with a peak at 555 nm. Since the sensor membrane material was optically clear, it could be used as a waveguide for the exciting laser radiation. Therefore, protein molecules adsorbing to the sensor surface would absorb laser radiation in the evanescent field of the waveguide and only those molecules adsorbed to the sensor surface would be selectively excited to emit fluorescent radiation. A light collection system was devised to transmit the fluorescent radiation emitted from the sensor surface to a photomultiplier tube such that it could be measured as a function of time. This enabled the 178 time evolution of the population of adsorbed protein molecules to be measured simultaneously with the change in tension of the sensor membrane. While the realization of this experimental technique proved to be technically challenging, several successful experimental runs were conducted using this simultaneous measurement technique. As shown in section 5.3.2, it was found that in response to adding protein molecules to the bulk solution surrounding the sensor membrane, the surface fluorescent signal reached a steady-state value within 30 minutes, while the membrane tension continued to drop well after this time. This strongly supports the interpretation that the membrane sensor is responding to molecular conformation change. Furthermore, since the fluorescence measurement signal was proportional to the population of protein molecules on the surface of the membrane, this provided additional information about the kinetics of protein adsorption to the sensor surface. The challenge, then, was to devise a credible model for the time evolution of the protein molecules on the surface, which could match the experimental observations. To this end, a step-wise kinetic rate model was devised in which protein molecules transferred between various bulk and surface states. The surface states consisted of a series of states with increasing projected area on the surface. The main model parameters comprised of the forward and backward rates of transfer between surface states and bulk states. In order to account for the desorption of molecules while exposed to the bulk solution concentration, the backward rates for adsorbed molecules were assigned a component that was proportional to the bulk solution concentration. Similarly, to account for the fact that the slow change in the membrane tension attributed to conformation change ceases upon bulk solution dilution, the forward rates of transfer between adsorbed surface states were also assigned a component that was proportional to the bulk concentration. These are novel features that differentiate this model from others that attempt to capture exchange reactions and irreversibility. Such a model was successfully applied to an experimental data set taken using the simultaneous measurement. A surprising result of this fit was that it was able to accurately model the sensor response by having, except for molecules in state 1, all forward and backward rates of transfer be solely proportional to the bulk solution concentration. Since 179 molecules in state 1 are considered to be in their native state, they are able to spontaneously adsorb and desorb without interaction from bulk solution molecules. However, once adsorbed it seems that substantially all conformation change results from interaction with bulk solution molecules. It seems plausible that this interaction could involve the transfer of kinetic energy such that an activation energy barrier could be surmounted, thus allowing the molecule to access a lower energy conformation, or revert back to a previous surface state. While the work described in this thesis is in no means a comprehensive study of the phenomenon of irreversible protein adsorption, these new modeling and measurement techniques show potential as useful tools for extended investigations. There are many possibilities for the future application of this sensor to the study of the kinetics of protein adsorption. The response of the sensor to different protein molecules under a wide range of bulk solution concentrations and flush times could be examined. Furthermore, the sensor surface could be adapted with various surface treatments or self-assembled monolayers devised to inhibit protein adsorption. In this way the effectiveness of these treatments could be evaluated. Perhaps more importantly, the protein adsorption model suggested in this thesis seems to be a viable step forward in tackling the paradox of the irreversibility of protein adsorption. It was successfully able to predict the sensor response to the adsorption of protein molecules and its response to bulk solution dilution. Furthermore, it was able to predict an ascending slope isotherm in which surface populations were dependent on bulk solution concentrations. While it may be premature to say that the problem has been entirely solved, it does seem highly probable that the key to the answer is the idea that transitions between surface conformation states are driven by the interactions with protein molecules in the bulk solution. 180 REFERENCES Alison Jane Clark, A variable spacing diffraction grating created with elastomeric surface waves. (University of British Columbia, [Vancouver], 1997). L. A. Whitehead and A. J. Clark, Applied Optics 37 (22), 5063 (1998). B. A. Cavic, G. L. Hayward, and M. Thompson, Analyst 124 (10), 1405 (1999). R.C. and Ward Ebersole, M.D., Journal of American Chemical Society 110, 8623 (1988) . B. A. Cavic and M. Thompson, Analyst 123 (10), 2191 (1998). B. Drafts, IEEE Transactions on Microwave Theory and Techniques 49 (4), 795 (2001). R. M. White and S. W. Wenzel, Applied Physics Letters 52 (20), 1653 (1988). B. J. Costello, B. A. Martin, and R. M. White, 1989 Ultrasonics symposium, 977 (1989) . M. J. Vellekoop, Ultrasonics 36 (1-5), 7 (1998). 0 C. E. Reed, K. K. Kanazawa, and J. H. Kaufman, Journal of Applied Physics 68 (5), 1993 (1990). 1 F. Ferrante, A. L. Kipling, and M. Thompson, Journal of Applied Physics 76 (6), 3448 (1994). 1 2 H. L. Bandey, S. J. Martin, R. W. Cernosek et al, Analytical Chemistry 71 (11), 2205 (1999). 1 3 Joseph D. Andrade, Surface and interfacial aspects of biomedical polymers. (Plenum Press, New York, 1985). 1 4 Clarence A. Miller and P. Neogi, Interfacial phenomena : equilibrium and dynamic effects. (M. Dekker, New York, 1985). 1 5 Hans Lyklema, H. P. van Leeuwen, M. van Vliet et al., Fundamentals of interface and colloid science. (Academic Press, London ; San Diego, 1991). 1 6 C. J. Beverung, C. J. Radke, and H. W. Blanch, Biophysical Chemistry 81 (1), 59 (1999). 1 7 Q. Jiang and Y. C. Chiew, Colloids and Surfaces B-Biointerfaces 20 (4), 303 (2001). 1 8 A. I. Rusanov, Surface Science Reports 23 (6-8), 173 (1996). 1 9 Finlay MacRitchie, Chemistry at interfaces. (Academic Press, San Diego, 1990). 2 0 Paul C. Hiemenz, Principles of colloid and surface chemistry, 2nd , rev. and expand ed. (M. Dekker, New York, 1986). 181 D. K. Chattoraj and K. S. Birdi, Adsorption and the Gibbs surface excess. (Plenum Press, New York, 1984). D. Fallahi, H. Mirzadeh, and M. T. Khorasani, Journal of Applied Polymer Science 88 (10), 2522 (2003). M. J. Owen, Industrial & Engineering Chemistry Product Research and Development 19 (1), 97 (1980). A. W. Feinberg, A. L. Gibson, W. R. Wilkerson et al., Synthesis and Properties of Silicones and Silicone-Modified Materials 838, 196 (2003). Milton J. Rosen, Surfactants and interfacial phenomena, 2nd ed. (Wiley, New York, 1989). Charles R. Cantor and Paul Reinhard Schimmel, The conformation of biological macromolecules. (W. H. Freeman, San Francisco, 1980). W. Norde, Macromolecular Symposia 103, 5 (1996). C. A. Haynes and W. Norde, Journal of Colloid and Interface Science 169 (2), 313 (1995). 193L from Protein Database, in www, proteinexplorer. org. J. K. Ferri and K. J. Stebe, Advances in Colloid and Interface Science 85 (1), 61 (2000). A. J. Prosser and E. I. Franses, Colloids and Surfaces A-Physicochemical and Engineering Aspects 178 (1-3), 1 (2001). C. H. Chang and E. I. Franses, Colloids and Surfaces A-Physicochemical and Engineering Aspects 100, 1 (1995). S. Y. Lin, K. Mckeigue, and C. Maldarelli, Aiche Journal 36 (12), 1785 (1990). S. Y. Lin, C. D. Dong, T. J. Hsu et al., Colloids and Surfaces A-Physicochemical and Engineering Aspects 196 (2-3), 189 (2002). J. Brinck, B. Jonsson, and F. Tiberg, Langmuir 14 (5), 1058 (1998). C. Geffroy, M. A. C. Stuart, K. Wong et al, Langmuir 16 (16), 6422 (2000). M. S. Romero-Cano, A. Martin-Rodriguez, and F. J. de las Nieves, Journal of Colloid and Interface Science 227 (2), 329 (2000). A. A. Levchenko, B. P. Argo, R. Vidu et al, Langmuir 18 (22), 8464 (2002). S. F. Turner, S. M. Clarke, A. R. Rennie et al, Langmuir 15 (4), 1017 (1999). R. P. Borwankar and D. T. Wasan, Chemical Engineering Science 41 (1), 199 (1986). P. Levitz, Langmuir 7 (8), 1595 (1991). R. Atkin, V. S. J. Craig, E. J. Wanless et al, Advances in Colloid and Interface Science 103 (3), 219 (2003). W. Norde, Journal of Dispersion Science and Technology 13 (4), 363 (1992). 182 V. Ball, P. Huetz, A. Elaissari et al, Proceedings of the National Academy of Sciences of the United States of America 91 (15), 7330 (1994). . A. Nadarajah, C. F. Lu, and K. K. Chittur, Proteins at Interfaces II602, 181 (1995). C. A. Haynes and W. Norde, Colloids and Surfaces B-Biointerfaces 2, 517 (1994). T. J. Su, J. R. Lu, R. K. Thomas et al, Journal of Physical Chemistry B 102 (41), 8100(1998). A. Kondo, T. Urabe, and K. Yoshinaga, Colloids and Surfaces A-Physicochemical and Engineering Aspects 109, 129 (1996). W. Norde and J. P. Favier, Colloids and Surfaces 64 (1), 87 (1992). M. Wahlgren, T. Arnebrant, and I. Lundstrom, Journal of Colloid and Interface Science 175 (2), 506(1995). P. R. Van Tassel, L. Guemouri, J. J. Ramsden et al, Journal of Colloid and Interface Science 207 (2), 317(1998). M. A. Brusatori and P. R. Van Tassel, Journal of Colloid and Interface Science 219 (2) , 333 (1999). C. Calonder and P. R. Van Tassel, Langmuir 17 (14), 4392 (2001). M. Prokopowicz, B. Banecki, J. Lukasiak et al, Journal of Biomaterials Science-Polymer Edition 14 (2), 103 (2003). R. J. Green, I. Hopkinson, and R. A. L. Jones, Langmuir 15 (15), 5102 (1999). L. Shi and K. D. Caldwell, Journal of Colloid and Interface Science 224 (2), 372 (2000). P. Schuck, Annual Review of Biophysics and Biomolecular Structure 26, 541 (1997). D. Z. Shen, M. H. Huang, L. M. Chow et al., Sensors and Actuators B-Chemical 77 (3) , 664 (2001). B. K. Lok, Y. L. Cheng, and C. R. Robertson, Journal of Colloid and Interface Science 91 (1), 87 (1983). C. E. Giacomelli, M. J. Esplandiu, P. I. Ortiz et al, Journal of Colloid and Interface Science 218 (2), 404(1999). A. Ball and R. A. L. Jones, Langmuir 11 (9), 3542 (1995). Y. Yokoyama, R. Ishiguro, H. Maeda et al., Journal of Colloid and Interface Science 268 (1), 23 (2003). A. Kondo and H. Fukuda, Journal of Colloid and Interface Science 198 (1), 34 (1998). P. Billsten, U. Carlsson, B. H. Jonsson etal, Langmuir 15 (19), 6395 (1999). W. Vandervegt, H. C. Vandermei, and H. J. Busscher, Journal of Colloid and Interface Science 156 (1), 129 (1993). 183 J. Noordmans, H. Wormeester, and H. J. Busscher, Colloids and Surfaces B-Biointerfaces 15 (3-4), 227 (1999). H. J. Butt, Journal of Colloid and Interface Science 180 (1), 251 (1996). A. M. Moulin, S. J. O'Shea, R. A. Badley et al, Langmuir 15 (26), 8776 (1999). GE RTV615 polydimethylsiloxane elastomer. 3 M ™ Optically Clear Adhesive 8141. 3 M ™ Optically Clear Adhesive 8142: Opto Acoustic Sensors Angstrom Resolver1* Series Dual Channel Models 201, Inc. Fisher SD109B-500. N. Panchuk-Voloshina, R. P. Haugland, J. Bishop-Stewart et al, Journal of Histochemistry & Cytochemistry 47 (9), 1179 (1999). Y. Ito, A. Yoshikawa, T. Hotani et al., European Journal of Biochemistry 259 (1-2), 456(1999). Product Information Sheet for Alexa Fluor® 532 Protein Labeling Kit (A10236), (www.probes.com). XF3021 (OG550) available from Omega Optical Inc. F52-536 available from Edmund Optics Inc. 3 M ™ Radiant Mirror Film. Hewlet Packard 33120 Function Generator. Stanford Research Systems SR530 Lock-in Amplifier. Hewlett Packard 34401A Multimeter. UBC - PHYSICS Stepping Motor Controller 88-028. Burle 931A photomultiplier tube. TSL250. Q. K. Tong, M. A. Maden, A. Jagota et a l , Journal of the American Ceramic Society 77 (3), 636 (1994). R.S. Christian, R.E. Davis, A Tubis et al, Journal of Acoustical Society of America 76(5), 1336(1984). H. P. W. Gottlieb and H. A. Aebischer, Acustica 61 (4), 223 (1986). H. P. W. Gottlieb and H. A. Aebischer, Acustica 84 (4), 779 (1998). Martin Malmsten, Biopolymers at interfaces edited by Martin Malmsten. (Marcel Dekkcr, New York, 1998). V .. : ' \ , J. J. Ramsden, Quarterly Reviews of Biophysics 27 (1), 41 (1994). R. Porcely A. B. Jodar, M. A. Cabrerizo et al, Journal of Colloid and Interface Science 239' (2), 568 (2001). . . ' ;• : 184 S. M. Daly, T. M. Przybycien, and R. D. Tilton, Langmuir 19 (9), 3848 (2003). F James, MINUIT Function Minization and Errror Analysis Reference Manual, Version 94.1, Computing and Networks Division, CERN, Geneva, Switzerland. B. C. Tripp, J. J. Magda, and J. D. Andrade, Journal of Colloid and Interface Science 173(1), 16(1995). J. S. Erickson, S. Sundaram, and K. J. Stebe, Langmuir 16 (11), 5072 (2000). C. A. Haynes, E. Sliwinsky, and W. Norde, Journal of Colloid and Interface Science 164 (2), 394 (1994). Eugene Hecht, Optics, 3rd ed. (Addison-Wesley, Reading, Mass., 1998). Miles V. Klein and Thomas E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986). 185 APPENDIX A IDEAL MEMBRANE Equation ( A - l ) gives the equation of motion of an ideal membrane, where, w(r, <j>,f) is the out of plane displacement, Tis the uniform tension in [N/m] in the membrane, p is the density of the membrane in [kg/m 3], h is the thickness of membrane in [m] and P is the external =0 and Equation ( A - l ) is solved by separating the variables according to Equation (A-2 ). Equation (A-3 ) gives the solution for the normal mode shapes of the membrane where pmn is the «th root of J m , the Mh-order Bessel function of the first kind and Anm is the amplitude. Equation (A-4 ) gives the solution for the time dependence of the normal modes, where comn are the angular frequencies of the normal modes and is given by Equation (A-5 ). The eigen-frequencies are related to the angular frequencies according tofmn=comn/27r. pressures in [N/m ] acting on the membrane. To find the normal modes of the membrane P TV2w(r,(p,t) + P = ph (A-l) w(r,(p,t) = W(r,(p)Y(t) (A-2) W(r,q>) = AnmJm \COS (A-3) Y(t) = cos(comnt) (A-4) (A -5 ) 186 The mode shapes for the first few modes according to Equation (A-3 ) are shown in Figure A.L (0,1) u Q 1 = 2.4048 (1,1) u „ = 3.8317 (2,1) u 2 =5.1356 (0,2) u Q 2= 5.5201 (3,1) p.3 = 6.3802 (1,2) n 1 2 = 7.0516 (4,1) u 4 = 7.5883 (2,2) p 2 2 = 8.4172 (0,3) u 0 3 = 8.6537 Figure A . l Theoretical mode shapes for first 9 ideal membrane modes. 187 APPENDIX B TOTAL INTERNAL REFLECTION AND WAVEGUIDES The second part of the sensor described in section 3.2, is based on the concept of using the sensing membrane as an optical waveguide. An optical waveguide is any structure in which light is trapped and travels along its length by means of total internal reflection. These concepts will be defined in the following section. B.l Snell's Law Figure B.l shows a light ray originating in a medium with index of refraction, «/, incident at a boundary with another medium, index of refraction, n2. It wil l refract and the angle of refraction, 92, wil l be related to the angle of incidence, 9j, according to Equation ( B-l ) which is called Snell's Law. n\Sin6\ = n%Sin02 ( B - l ) When the angle of incidence, 9i, is such that the angle of refraction, 92, is 90° as shown in Figure B.2, it is called the critical angle, 0C. Equation (B-2) gives the relationship necessary 188 to solve for the critical angle. In order for this relationship to be true, n2<nh For angles greater than the critical angle, 9C, the light ray will totally internally reflect from the surface.98 Figure B.2 Light ray incident at boundary between two media at the critical angle. @crit ~ Sin \ n \ J (B -2 ) B.2 Waveguide concept A waveguide is an optical structure in which light radiation can travel by means of total internal reflection. Total internal reflection can occur at the boundary between two media of different indices of refraction when the index in the outer layer (n2) is lower than that of the inner layer (ni). For a planar structure of index, «/, clad on top and bottom with a material of lower index, n2, shown in Figure B.3, as long as the angle of incidence is greater than the critical angle, the light rays will totally internally reflect at both boundaries and be confined to the waveguide as they travel along its length. 189 n 2 Figure B.3 Planar waveguide structure, n2<ni Although, the light ray does reflect without loss of energy from the boundary, there is a portion of the light ray that wil l penetrate into the second medium. This is called the evanescent wave. The intensity of the evanescent wave in the outer medium will drop off according to Equation (B -3 ) where, z, is the distance into the second medium and X is the wavelength of the incident light ray." The distance that the evanescent wave penetrates into the second medium is described by the penetration depth, S, as given by Equation ( B -4) . " Figure B.4 shows the way in which the intensity of the evanescent wave will fall off as a function of distance into the second medium according to Equation (B -4 ) . It can be seen that the intensity of the evanescent wave is virtually diminished over the distance of 3S. z l(z) = l(0)e 5 (B-3) 5 = (B-4) 190 APPENDIX C GOTTLIEB/AEBISCHER PROCEDURE This Appendix wil l outline the procedure to solve for the ko„ parameters used to extract the membrane tension, T, from the experimentally measured resonant frequency,/;,,. The full derivation of the method can be found in the paper by Gottlieb and Aebischer88. First, two dimensionless frequency parameters are defined,/, which is derived from the eigen-frequency of an ideal membrane, a>onldeal, as shown in Equation ( G-l ) and FN which is derived from the experimentally measured resonant frequency,/)/7""', as shown in Equation ( C-2 ). Note that the FN is known while the corresponding ideal membrane frequency for that particular mode is unknown and is what the analysis will solve for. fn , ideal c0 ( C - l ) Fi 2nf, fluid N On a c0 (C-2) Equation (C-3 ) is used to generate t h e / from the FN with the coefficients on the sum, bmN, being defined by Equation ( C-4 ). Equations (C-3 ) and ( C-4 ) are coupled and have to be solved iteratively. ( f \ 2 J n VFNJ l + /1mNN(FN)+B Yj>mmNm(FN) (C-3) b£=fi mmN{FN)+ mmsiFN) m*N,s*m J m FN J [l + Bmmm{FN)l (C-4) The dimensionless radiation mass impedance matrix,-wm„ is defined by the, integral in Equation (C-5 ). Equation (C-5 ) can be solved numerically, however Gottlieb and Aebischer88 tabulate in their paper values for mmn(FN) in intervals of FN = 0.1. They suggest using a 192 linear interpolation for intermediary values. However, a typical value of FN under the experimental values of this experiment is much lower than what they encountered. For example for a sensor membrane vibrating in the (0,2) mode, a typical resonant frequency is fofluid ~7 Hz. Using Equation ( C-2 ), this generates a FN value of FN = 0.00029, with a radius of a = 0.0095 m and c0 = 1420 m/s. Therefore, in practice it was acceptable to use the mmn(0) values from the paper. (C -5) The procedure used to iteratively solve Equations ( C 3^ ) and ( C-4 ) is as follows. First, an initial calculation of the/, value is made by ignoring the terms in the sum of Equation ( C-3 ), as shown in Equation ( C-6 ). Then all other ideal frequencies are generated from the expected ratios of the Bessel function zeroes as shown in Equation ( C-7 ). fn =FN41+ Bmnn(FN) (n = N) (C -6 ) fm fn M0m M0n (n * m) (C -7 ) In the same way, an initial calculation of the bmN values are made from Equation ( C-4 ), by ignoring the terms in the summation, as shown in Equation ( C-8 ). mmNiFN) '7 ^ 2 J m WFN J \l+Bmmm{FN)] ( C - 8 ) Once initial "guesses" of the fn and bmN values are made, the full expressions of Equations (C-3 ) and (C-4) can be solved iteratively until the generated values for / , converge. Once 193 the ideal membrane frequencies f o n d e a l from/, according to Equation ( C-9 ), the ko„ value can be determined using Equation (4-6 ). rideal _ cofn JOn -~Z 2na (C-9) For clarity, Equations ( C-3 ) and ( C-4) are written out with all their indices defined for «<=4. r£i2 \F1J = 7 + /?m7(F7)+/j{z>J m 7 2(F ;)+£j m13(Fj)+b4 m14(Fj)} KF2J 1 + Pm22(F2)+ «{bj m2](F2)+b23 m23(F2) + b24 m24{F2)} \F3J = l + Pm33(F3)+n{tf m31{F3)+b32 m32(F3) + b34 m34 \F4J = l+Rm44{F4)+p{b41m41(F4)+b42m4^ b'2=P m21 {Fj)+blm23 (Fj)+b\m24 (Fj) \F1J \l + pm22{F1)} b'3=P m31(Fj )+b]2m32 (Fj)+bjm24 (Fj) <f?2 [l + Pms^Fj)] bi=P m41 (Fj)+b\ m32 (Fj)+bl3m23 (Fj ) \FlJ [l + Pm44(Fj)) bj = B m12 (F2 )+b23m23 (F2 )+bjm24 (F2 ) \F2J \l + pmjj(F2)] 194 bj = B m32{F2)+bjm2](F2)+bjm24(F2) \F2) [l+Bm33{F2)] b24=B m42{F2)+bfm21{F2)+b23m23 {F2 ) \F2 -[l + Pm44(F2)] bf=P ml3 (F3)+b \m32 (F3)+b34m34 (F3) -[l+Bmu{F3)} b32=P m23{F3)+ bjm31(F3)+ b\m34 (F3 ) (A)2 -[l + Pm22{F3)] bhP m43 (F3 ) + b] m3 j(F3)+b32 m32 JF3 ) r J ± 2 KF3j [l + Pm44{F3)] bUP m14 (F4 )+b42m42 [F4 )+b43m43 (F4 ) \F4J \l + Pmn{F4)\ b42=P m24 {F4 )+bfm4]{F4 )+bjm43 (F4 ) \F4J -[l + Bm22{F4)] b3=P m34 (F4 )+bjm41 (F4 )+b42 m42 JF4 ) (A)2 KF4) [j + Pm33(F4)] 195 APPENDIX D "READ_DATA.C" PROGRAM # i n c l u d e < s t d l i b . h > # i n c l u d e <math.h> i i n c l u d e < s t dio.h> • i n c l u d e " s t r i n g . h " f l o a t P a ddVol, PaddConc, time a d d , t i m e r l u s h , F l u s h V o l , PumpRate; f l o a t t i m e , t e n s i o n , f r e q u e n c y ; f l o a t t i m e s [ 5 0 0 0 ] , f r e q u e n c i e s [ 5 0 0 0 ] , t e n s i o n s [ 5 0 0 0 ] , b u l k [ 5 0 0 0 0 0 0 ] , x p [ 5 0 0 0 0 0 0 ] , y p [ 5 0 0 0 0 0 0 ] , z p [ 5 0 0 0 0 0 0 ] , p h o t o m u l t i p l i e r s [ 5 0 0 0 ] , p h o t o d i o d e s [ 5 0 0 0 ] , f l u o r e s c e n c e [ 5 0 0 0 ] ; f l o a t a v e r a g e T , a v e r a g e f ; FILE * f p l , * f p 2 , * f p 3 , * f p 4 ; c h a r c h; ch a r f i l e n a m e [ 2 5 6 ] , o u t f i l e n a m e l [ 2 5 6 ] , d a t a f i l e [ 2 5 6 ] , o u t f i l e n a m e 2 [ 2 5 6 ] ; c h a r c o m p r e s s [ 2 5 6 ] , s d e l t t [ 1 0 ] ; i n t f l a g ; i n t n d a t a , x , i , j , n b d a t a , n n b d a t a ; i n t r e a d c o u n t e r ; f l o a t readamount; i n t c o u n t e r [ 2 0 ] ; d o u b l e d e l t t , t ; i n t n e w d a t a c o u n t , n t d a t a ; i n t i c o u n t ; i n t t i m e t r a c k ; f l o a t addtime,newtime; f l o a t c o u n t ; f l o a t phase, a m p l i t u d e , p h o t o m u l t i p l i e r , p h o t o d i o d e ; f l o a t s u m s q , s i g , s u m s q f , s i g f ; i n t a d d p r o t e i n s ; i n t c o u n t e r [ 2 0 ] ; f l o a t amount[20]; f l o a t f c o u n t e r ; i n t c o u n t e r T ; f l o a t t i m e e n d ; i n t t i ; f l o a t b i g t i m e ; f l o a t p b a c k , l b a c k ; f l o a t r a d i u s , t h i c k n e s s , m d e n s i t y ; f l o a t t _ a d d _ p r o t e i n [ 2 0 ] , t _ f l u s h _ s t a r t , t _ f l u s h _ e n d ; i n t c _ a d d _ p r o t e i n , c _ f l u s h _ s t a r t , c _ f l u s h _ e n d ; f l o a t P_add_Vol, P_add_conc; f l o a t t s t a r t , t i m e ^ s t a r t ; . ... f l o a t y p s t a r t ; ' ' f l o a t Vb,Va,Vc,R,Vpl,Vp2,Vp3; i n t t i x l a g , t i y l a g , t i z l a g ; f l o a t Mw; .. f l o a t xend, yend, P, xe'ndj, yendj;. .-•, i . . . : main () 196 { /** membrane p r o p e r t i e s **/ radius=0.0095, t h i c k n e s s = 0 . 0 0 0 0 5 0 , mdensity=980; f l a g = 0; p r i n t f ( " E n t e r t h e name o f t h e e x p e r i m e n t a l d a t a r u n f i l e : " ) ; s c a n f ( " % s " , & f i l e n a m e ) ; p r i n t f ( " % s \ n " , f i l e n a m e ) ; p r i n t f ( " W h a t i s t h e t i m e i n t e r v a l i n seconds f o r t h e m o d e l ? " ) ; s c a n f ( " I s " , S s d e l t t ) ; d e l t t = a t o f ( s d e l t t ) ; p r i n t f ( " T h e t i m e i n t e r v a l i n seconds f o r t h e model i s % f s . \ n " , d e l t t ) p r i n t f ( " W h a t i s t h e s m a l l e s t t i m e i n t e r v a l i n seconds f o r t h e d a t a f i l e ? " ) ; s c a n f ( " % s " , & c o m p r e s s ) ; a d d t i m e = a t o f ( c o m p r e s s ) ; p r i n t f ( " T h e t i m e i n t e r v a l i s % f s . \ n " , a d d t i m e ) ; i f ( ( f p l = f o p e n ( f i l e n a m e , " r " ) ) = = N U L L ) { p r i n t f ( " c a n n o t open f i l e % s \ n " , f i l e n a m e ) ; f l a g = l ; - ' } . - ' -i f ( f l a g ! = l ) { i = l ; f s c a n f ( f p l , " % s \ n " , S d a t a f i l e ) ; p r i n t f ( " % s \ n " , d a t a f i l e ) ; f s c a n f ( f p l , " % f \ n " , &Mw); p r i n t f ( " T h e m o l e c u l a r w i e g h t i s :%f\n",Mw); f s c a n f ( f p l , " % f % f % f % f % f % f %f\n",&R, &Va,&Vpl,&Vb,&Vp2,&Vc,&Vp3) p r i n t f ( " T h e pump r a t e i s : % f \ n " , R ) ; p r i n t f ( " T h e Va v a l u e i s : % f \ n " , V a ) ; p r i n t f ( " T h e V p l v a l u e i s : % f \ n " , V p l ) ; p r i n t f ( " T h e Vb v a l u e i s : % f \ n " , V b ) ; p r i n t f ( " T h e Vp2 v a l u e i s : % f \ n " , V p 2 ) ; p r i n t f ( " T h e Vc v a l u e i s : % f \ n " , V c ) ; p r i n t f ( " T h e Vp3 v a l u e i s : % f \ n " , V p 3 ) ; do { f s c a n f ( f p l , " % i \ t % f \ n " , S r e a d c o u n t e r , & r e a d a m o u n t ) ; p r i n t f ( " % i \ t % f \ n " , r e a d c o u n t e r , r e a d a m o u n t ) ; c o u n t e r [ i ] = r e a d c o u n t e r ; amount[i]=readamount; i++; } w h i l e ( r e a d c o u n t e r ! = 0 ) ; f c l o s e ( f p l ) ; f o r (j=l;j<20;j++) p r i n t f ( " \ t % d \ t % d \ t % f \ n " , j , c o u n t e r [ j ] ,amount[j] ) } p r i n t f ( " E n t e r t h e b a c k g r o u n d s i g n a l f o r t h e p h o t o d i o d e : " ) ; s c a n f ( " % f " , S l b a c k ) ; p r i n t f ( " E n t e r t h e b a c k g r o u n d s i g n a l f o r t h e p h o t o m u l t i p l i e r : " ) ; s c a n f ( " % f " , & p b a c k ) ; p r i n t f ( " E n t e r t h e name o f t h e t e n s i o n d a t a o u t p u t d a t a f i l e : " ) ; s c a n f ( " % s " , & o u t f i l e n a m e l ) ; p r i n t f ( " E n t e r t h e name o f t h e b u l k c o n c e n t r a t i o n d a t a o u t p u t d a t a f i l e : ") ; s c a n f ( " % s " , S o u t f i l e n a m e 2 ) ; i f ( ( f p 2 = f o p e n ( d a t a f i l e , " r " ) ) = = N U L L ) { p r i n t f ( " c a n n o t open f i l e % s \ n " , d a t a f i l e ) ; f l a g = l ; } i f ( f l a g ! = l ) { i = l ; a d d p r o t e i n s = 0 ; w h i l e ( f s c a n f ( f p 2 , " % f " , & c o u n t ) !=EOF){ f s c a n f ( f p 2 , " \ t % f \ t % f \ t % f \ t % f \ t % f \ t % f \ n " , S t i m e , S p h a s e , & f r e q u e n c y , S a m p l i t u d e , S p h o t o m u l t i p l i e r , & p h o t o d i o d e ) ; i c o u n t = ( i n t ) c o u n t ; i f ( i c o u n t = = c o u n t e r [ i ] ) { i f (amount[i]==-1) t _ f l u s h _ s t a r t = t i m e ; e l s e i f (amount[i]==-2) t _ f l u s h _ e n d = t i m e ; e l s e { t _ a d d _ p r o t e i n [ i ] = t i m e ; p r i n t f ( " t h e %d t h p r o t e i n addtime i s % f \ n " , i , t _ a d d _ p r o t e i n [ i ] ) ; a d d p r o t e i n s + + ; }' i++; 1 t i m e s [ i c o u n t ] = t i m e ; f r e q u e n c i e s [ i c o u n t ] = f r e q u e n c y ; p h o t o m u l t i p l i e r s [ i c o u n t ] = p h o t o m u l t i p l i e r ; p h o t o d i o d e s [ i c o u n t ] = p h o t o d i o d e ; } n d a t a = i c o u n t ; f o r (j=l;j<20;j++) p r i n t f ( " \ t % d \ t % d \ t % f \ t % f \ n " , j , c o u n t e r [ j ] , a m o u n t [ j ] , t _ a d d _ p r o t e i n [ j ] ) ; p r i n t f ( " t _ f l u s h _ s t a r t i s % f \ n " , t _ f l u s h _ s t a r t ) ; p r i n t f ( " t _ f l u s t _ e n d i s % f \ n " , t _ f l u s h _ e n d ) ; f c l o s e ( f p 2 ) ; } averageT=0; counterT=0; newdatacount=0; j = i ; f o r ( i = l ; i < = n d a t a ; i + + ) { i f ( t i m e s [ c o u n t e r [ 1 ] ] > = 1 8 0 0 ) { t i m e _ s t a r t = 1 8 0 0 ; i f ( t i m e s [ i ] > t _ a d d _ p r o t e i n [ 1 ] - 1 8 0 0 ) newdatacount++; t e n s i o n s [ i ] = p o w ( ( f r e q u e n c i e s [ i ] * 9.5091*2*3.14159*radius/5.5201), 2 ) * ( m d e n s i t y * t h i c k n e s s ) ; i f ( ( t i m e s [ i ] > t _ a d d _ p r o t e i n [ 1 ] - 1 8 0 0 ) & & ( t i m e s [ i ] < = t _ a d d _ p r o t e i n [ 1 ] ) ) { a v e r a g e T = a v e r a g e T + t e n s i o n s [ i ] ; 198 counterT++; } } e l s e { t i m e _ s t a r t = 0 ; newdatacount++; t e n s i o n s [ i ] = p o w ( ( f r e q u e n c i e s [ i ] * 1 3 . 4 0 9 * 2 * 3 . 1 4 1 5 9 * r a d i u s / 5 . 5 2 0 1 ) , 2 ) * ( m d e n s i t y * t h i c k n e s s ) ; ' i f ( t i m e s [ i ] < = t _ a d d _ p r o t e i n [ 1 ] ) { a v e r a g e T = a v e r a g e T + t e n s i o n s [ i ] ; counterT++; ) ) / * * p r i n t f ( " % f \ n " , t e n s i o n s [ i ] ) ; * * / } t i m e e n d = t i m e s [ n d a t a ] ; bigtime=10;. i f ( t i m e _ s t a r t = = 1 8 0 0 ) { n e w t i m e = t i m e e n d - t _ a d d _ p r o t e i n [ 1 ] + 1 8 0 0 ; n n b d a t a = ( i n t ) ( ( n e w t i m e - 1 8 0 0 ) / d e l t t ) + ( i n t ) ( 1 8 0 0 / b i g t i m e ) ; ) e l s e { newtime=timeend; n n b d a t a = ( i n t ) ( n e w t i m e -t _ a d d _ p r o t e i n [ 1 ] ) / d e l t t + ( i n t ) t _ a d d _ p r o t e i n [ 1 ] / b i g t i m e ; } p r i n t f ( " t h e n d a t a v a l u e i s % d \ n " , n d a t a ) ; p r i n t f ( " t h e t i m e e n d v a l u e i s % f \ n " , t i m e e n d ) ; p r i n t f ( " t h e newtime v a l u e i s % f \ n " , n e w t i m e ) ; p r i n t f ( " t h e newdatacount v a l u e i s %d\n",newdatacount); p r i n t f ( " t h e addtime v a l u e i s % f \ n " , a d d t i m e ) ; n t d a t a = ( i n t ) n e w t i m e / a d d t i m e ; p r i n t f ( " t h e n t d a t a v a l u e i s % d \ n " , n t d a t a ) ; p r i n t f ( " t h e addtime v a l u e i s % f \ n " , a d d t i m e ) ; /** c a l c u l a t e t h e b u l k c o n c e n t r a t i o n p r o f i l e a c c o r d i n g t o t h e i n p u t t e d e x e r i m e n t a l r u n i n f o **/ t i = 0 ; xp[0]=0; yp[0]=0; zp[0]=0; j = i ; t i y l a g = ( i n t ) V p 2 / ( R * d e l t t ) ; t i x l a g = ( i n t ) V p l / ( R * d e l t t ) ; t i z l a g = ( i n t ) V p 3 / ( R * d e l t t ) ; f o r ( t = 0 ; t < = t i m e e n d ; t = t + d e l t t ) { ,-./ t i + + ; i f ( t < t _ a d d _ p r o t e i n [ 1 ] ) { x p [ t i ] = 0 ; y p [ t i ] = 0 ; '• • ' - -z p [ t i j = 0 ; b u l k [ t i ] = 0 ; ' 199 i f (t<=10) p r i n t f ( " % d % f % f % f \ n " , t i , t , x p [ t i ] , y p [ t i ] ) ; } i f ( ( t < = t _ a d d _ p r o t e i n [ j ] ) && ( ( t + d e l t t ) > t _ a d d _ p r o t e i n [ j ] ) ) { x p [ t i ] = x p [ t i - 1 ] + a m o u n t [ j ] ; y p [ t i ] = y p [ t i - 1 ] ; z p [ t i ] = z p [ t i - 1 ] ; b u l k [ t i ] = ( y p [ t i ] * l e 6 ) / ( V b * M w * l . 6 6 e - 2 1 ) ; p r i n t f ( " \ t % d \ t % f \ t % d \ t % f \ t % f \ n " , j , a m o u n t [ j ] , t i , t , t _ a d d _ p r o t e i n [ j ] ) ; j++; p r i n t f ( " h i i'm i n t h e add p r o t e i n t i = %d t= % f and xp= % f and yp= % f \ n " , t i , t , x p [ t i ] , y p [ t i ] ) ; } i f ( ( j > l ) && ( t > = t _ a d d _ p r o t e i n [ j - 1 ] ) && ( t < t _ f l u s h _ s t a r t ) ) { x p [ t i ] = x p [ t i - l ] + ( y p [ t i - t i y l a g - l ] / V b - x p [ t i - l ] / V a ) * R * d e l t t ; y p [ t i ] = y p [ t i - l ] + ( z p [ t i - t i z l a g - 1 ] / V c - y p [ t i - l ] / V b ) * R * d e l t t ; z p [ t i ] = z p [ t i - 1 ] + ( x p [ t i - t i x l a g - 1 ] / V a - z p [ t i - l ] / V c ) * R * d e l t t ; i f ( t < = t _ a d d _ p r o t e i n [ j - 1 ] + 5 0 ) p r i n t f ( " ! %d % f % f % f \ n " , t i , t , x p [ t i ] , y p [ t i ] ) ; b u l k [ t i ] = ( y p [ t i ] * l e 6 ) / ( V b * M w * 1 . 6 6 e - 2 1 ) ; }• i f ( ( t > = t _ f l u s h _ s t a r t ) & & ( t < t _ f l u s h _ e n d ) ) { x p [ t i ] = x p [ t i - l ] + ( y p [ t i - t i y l a g - l ] / V b - x p [ t i - l ] / V a ) * R * d e l t t ; y p [ t i ] = y p [ t i - 1 ] + ( z p [ t i - t i z l a g - 1 ] / V c - y p [ t i - l ] / V b ) * R * d e l t t ; i f ( ( t - V p l / R ) < t _ f l u s h _ s t a r t ) z p [ t i ] = z p [ t i - 1 ] + ( x p [ t i - t i x l a g - 1 ] / V a -z p [ t i - l ] / V c ) * R * d e l t t ; e l s e z p [ t i ] = z p [ t i - l ] - z p [ t i - l ] * R * d e l t t / V c ; b u l k [ t i ] = ( y p [ t i ] * l e 6 ) / ( V b * M w * l . 6 6 e - 2 1 ) ; i f ( t < t _ f l u s h _ s t a r t + 1 0 ) p r i n t f ( " h i i'm f l u s h i n g t = % f \ n " , t ) ; i f ( ( t > = t _ f l u s h _ e n d ) ) { x p [ t i ] = x p [ t i - l ] + ( y p [ t i - t i y l a g - l ] / V b - x p [ t i - l ] / V a ) * R * d e l t t ; y p [ t i ] = y p [ t i - 1 ] + ( z p [ t i - t i z l a g - 1 ] / V c - y p [ t i - l ] / V b ) * R * d e l t t ; i f ( ( t - V p l / R ) < t _ f l u s h _ e n d ) z p [ t i ] = z p [ t i - 1 ] + ( - z p [ t i - l ] / V c ) * R * d e l t t ; e l s e z p [ t i ] = z p [ t i - 1 ] + ( x p [ t i - t i x l a g - 1 ] / V a - z p [ t i - 1 ] / V c ) * R * d e l t t ; b u l k [ t i ] = ( y p [ t i ] * l e 6 ) / ( V b * M w * l . 6 6 e - 2 1 ) ; i f ( t < t _ f l u s h _ e n d + 1 0 ) p r i n t f ( " h i i'm a f t e r t h e f l u s h i n g t = % f \ n " , t ) ; } } n b d a t a = ( i n t ) ( t i m e e n d / d e l t t ) - ; -p r i n t f ("the n b d a t a v a l u e i s %d", nbdata) ;' averageT=averageT/counterT; p r i n t f ("The counter,T v a l u e is'%d\n",covinterT) ; "' .• p r i n t f ("The averageT v a l u e i s : % f \ n " , averageT ;) ; ••- ' for ( i = l ; i < = n d a t a ; i + + ) { f l u o r e s c e n c e [ i ] = - ( p h o t o m u l t i p l i e r s [ i ] - p b a c k ) / ( p h o t o d i o d e s [ i ] - l b a c k ) ; } 200 f o r ( i = l ; i < = n d a t a ; i + + ) { i f ( t i m e _ s t a r t = = 1 8 0 0 ) { i f ( ( t i m e s [ i ] > t _ a d d _ p r o t e i n [ 1 ] - 1 8 0 0 ) & & ( t i m e s [ i ] < = t _ a d d _ p r o t e i n [ 1 ] ) ) { a v e r a g e f = a v e r a g e f + f l u o r e s c e n c e [ i ] ; } } e l s e { i f ( t i m e s [ i ] < = t _ a d d _ p r o t e i n [ 1 ] ) { a v e r a g e f = a v e r a g e f + f l u o r e s c e n c e [ i ] ; } } } a v e r a g e f = a v e r a g e f / c o u n t e r T ; f o r ( i = l ; i < = n d a t a ; i + + ) { t e n s i o n s [ i ] = t e n s i o n s [ i ] - a v e r a g e T ; f l u o r e s c e n c e [ i ] = f l u o r e s c e n c e [ i ] - a v e r a g e f ; i f ( t i m e _ s t a r t = = 1 8 0 0 ) { i f ( ( t i m e s [ i ] > t _ a d d _ p r o t e i n [ 1 ] - 1 8 0 0 ) & & ( t i m e s [ i ] < = t _ a d d _ p r o t e i n [ 1 ] ) ) { sumsq=sumsq+pow((tensions[ i ] ) , 2) ; s u m s q f = s u m s q f + p o w ( ( f l u o r e s c e n c e [ i ] ) , 2) ; } } e l s e '{ " -i f ( t i m e s [ i ] < = t _ a d d _ p r o t e i n [ 1 ] ) { sumsq=sumsq+pow((tensions[i]) , 2) ; s u m s q f = s u m s q f + p o w ( ( f l u o r e s c e n c e [ i ] ) , 2) ; } } } s i g = s q r t ( s u m s q / ( c o u n t e r T - 1 ) ) ; s i g f = s q r t ( s u m s q f / ( c o u n t e r T - 1 ) ) ; /** w r i t e o u t p u t f i l e c o n t a i n i n g t e n s i o n d a t a w i t h t h e t i m e i n t e r v a l chosen **/ a d d t i m e = a t o f ( c o m p r e s s ) ; p r i n t f ( " t h e n d a t a v a l u e i s %d\n",ndata) ; p r i n t f ( " t h e t i m e e n d v a l u e i s % f \ n " , t i m e e n d ) ; p r i n t f ( " t h e newtime v a l u e i s % f \ n " , n e w t i m e ) ; p r i n t f ( " t h e newdatacount v a l u e i s % d\n",newdatacount); p r i n t f ( " t h e addtime v a l u e i s % f \ n " , a d d t i m e ) ; n t d a t a = ( i n t ) n e w t i m e / a d d t i m e ; p r i n t f ( " t h e new f l u s h s t a r t t i m e i s % f \ n " , t _ f l u s h _ s t a r t -t _ a d d _ p r o t e i n [ 1 ] + 1 8 0 0 ) ; p r i n t f ( " t h e new f l u s h end t i m e i s % f \ n " , t _ f l u s h _ e n d -t _ a d d _ p r o t e i n [ 1 ] + 1 8 0 0 ) ; p r i n t f ( " t h e n t d a t a v a l u e i s % d \ n " , n t d a t a ) ; f p 3 = f o p e n ( o u t f i l e n a m e l , " w " ) ; f p r i n t f ( f p 3 , " % d %1.4e % 1 . 4 e \ n " , n e w d a t a c o u n t , s i g , s i g f ) ; t i m e t r a c k = a d d t i m e ; f o r (x=l;x<=ndata;x++){ i f ( t i m e s [ x ] > = t i m e t r a c k ) { 201 i f ( t i m e _ s t a r t = = 0 ) f p r i n t f ( f p 3 , " % f % 1 . 4 e \ n " , t i m e s [ x ] , t e n s i o n s [ x ] ) ; e l s e { i f ( t i m e s [ x ] > t _ a d d _ p r o t e i n [ 1 ] - t i m e _ s t a r t ) f p r i n t f ( f p 3 , " % f %1.4e % 1 . 4 e \ n " , t i m e s [ x ] - t _ a d d _ p r o t e i n [ 1 ] + t i m e _ s t a r t , t e n s i o n s [ x ] , f l u o r e s c e n c e [ x ] ) ; } t i m e t r a c k = t i m e t r a c k + a d d t i m e ; } } f c l o s e ( f p 3 ) ; /** w r i t e t h e b u l k c o n c e n t r a t i o n o u t p u t f i l e w i t h t h e t i m e i n t e r v a l chosen f o r t h e model **/ f p 4 = f o p e n ( o u t f i l e n a m e 2 , " w " ) ; f p r i n t f ( f p 4 , "%d % f \ n " , n n b d a t a - 5 , d e l t t ) ; t=0; f o r (x=l;x<=nbdata;x++){ i f ( t i m e _ s t a r t = = 0 ) { i f ( ( t < t _ a d d _ p r o t e i n [ 1 ] ) & & ( t > = b i g t i m e ) ) { f p r i n t f ( f p 4 , " % f % 1 . 4 e \ n " , t , b u l k [ x ] ) ; b i g t i m e = b i g t i m e + 1 0 ; } i f ( t > = t _ a d d _ p r o t e i n [ l ] ) f p r i n t f ( f p 4 , " % f % 1 . 4 e \ n " , t , b u l k [ x ] ) ; } e l s e { i f ( ( t > t _ a d d _ p r o t e i n [ 1 ] - t i m e _ s t a r t ) & & ( t < t _ a d d _ p r o t e i n [ 1 ] ) & & ( ( t -t _ a d d _ p r o t e i n [ 1 ] + t i m e _ s t a r t ) > = b i g t i m e ) ) { f p r i n t f ( f p 4 , " % f % 1 . 4 e \ n " , ( t - t _ a d d _ p r o t e i n [ 1 ] + t i m e _ s t a r t ) , b u l k [ x ] ) ; b i g t i m e = b i g t i m e + 1 0 ; } i f ( t > t _ a d d _ p r o t e i n [ l ] ) f p r i n t f ( f p 4 , " % . 2 f % 1 . 4 e \ n " , ( t -t _ a d d _ p r o t e i n [ 1 ] + t i m e _ s t a r t ) , b u l k [ x ] ) ; } t = t + d e l t t ; } f c l o s e ( f p 4 ) ; } 202 APPENDIX E "RATEFCNSDS.F" PROGRAM SUBROUTINE FCN(NPAR,GRAD,FVAL,XVAL,IFLAG) COMMON I F I L E S COMMON TFILENAMES DIMENSION TFILENAMES(10) CHARACTER TFILENAMES*15 CHARACTER TFILENAME*15 COMMON BFILENAMES DIMENSION BFILENAMES(10) CHARACTER BFILENAMES*15 CHARACTER BFILENAME*15 CHARACTER PARFILE*15 DIMENSION OUTFILES(IO) CHARACTER 0UTFILES*15 IMPLICIT DOUBLE PRECISION(A-H,0-Z) DIMENSION NPTT(IO),NPTB(10),SIG(10) DIMENSION TIMESB(100000,10),TENSIONS(100000, 10), +TIMEST(100000,10),BULK(100000,10),PMODEL(100000,10) DIMENSION GRAD(*), XVAL(*) C- INPUT arguments:NPAR = number o f c u r r e n t l y v a r i a b l e p a r a m e t e r s . C- XVAL = t h e v e c t o r o f ( c o n s t a n t & v a r i a b l e ) p a r a m e t e r s . C- IFLAG = i n d i c a t e s what i s t o be c a l c u l a t e d . C- OUTPUT arguments:FVAL = t h e f u n c t i o n v a l u e . C- GRAD = t h e ( o p t i o n a l ) v e c t o r o f f i r s t d e r i v a t i v e s . ***************************************************************** * READ IN DATA C PRINT*, I F I L E S I F ( I F L A G .EQ. 1 )THEN DO 20 ICOUNT=l,IFILES TFILENAME=TFILENAMES(ICOUNT) OPEN(UNIT=8,FILE=TFILENAME, STATUS='OLD') READ(8,*) NPTT(ICOUNT),SIG(ICOUNT) PRINT*, NPTT(ICOUNT),SIG(ICOUNT) DO 10 1=1,NPTT(ICOUNT) READ(8,*) TIMEST(I,ICOUNT),TENSIONS(I,ICOUNT) C PRINT*,TIMEST(I,ICOUNT),TENSIONS(I,ICOUNT) 10 CONTINUE CLOSE(8) 20 CONTINUE ENDIF I F ( I F L A G .EQ. 1 )THEN DO 40 ICOUNT=l,IFILES BFILENAME=BFILENAMES(ICOUNT) OPEN(UNIT=8,FILE=BFILENAME,STATUS='OLD') READ(8,*) NPTB(ICOUNT),DELTT PRINT*, NPTB(ICOUNT),DELTT DO 30 I=l,NPTB(ICOUNT) READ(8,*) TIMESB(I,ICOUNT),BULK(I,ICOUNT) C PRINT*,TIMESB(I,ICOUNT),BULK(I,ICOUNT) 30 CONTINUE CLOSE(8) 4 0 CONTINUE ENDIF C Compute c h i - s q u a r e d Al=4.81E-19 DCOEFF=5E-10 E1=XVAL(4)*A1 D=XVAL(2)*lE-4 RBL=DCOEFF/D RS1L=XVAL(1) RLS1=XVAL(3)*lE-5 CF=XVAL(5)*lE-22 C PRINT*,RS1L,D,RLS1,El,CF, DELTT CHISQ=0.0 DO 150 ICOUNT=l,IFILES 1=1 C DELTT i s t i m e i n t e r v a l f o r model TIME=0 C i n i t i a l i z e t h e p o p u l a t i o n s t o z e r o B=0 BL1=0 BL2=0 BL3=0 BL4=0 BL5=0 S1=0 TENSION=0 ADDTEN=0.. • , ' ." \' DO 50 J=0,NPTB(ICOUNT),1 DELTT=TIMESB(J+l,ICOUNT)-TIMESB(J,ICOUNT) TENSIONO=TENSION • • - . BO=B BL50=BL5 S10=S1 204 c B=BULK(J,ICOUNT) PRINT*,B BL5=BL50+(RBL*(BO-BL50)-RLS1*BL50*(1-S10*A1) ++RS1L*S10)*DELTT/D S1=S10+(RLS1*BL50*(1-S10*A1)-RS1L*S10)*DELTT I F (TIME.GT.1800)THEN ADDTEN=ADDTEN+CF*S1 ENDIF TENSI0N=S1*E1+ADDTEN I F ( (TIME.GT.1800) .AND. (TIME.LT.1900))THEN C PRINT*,SI,ADDTEN,TENSION ENDIF I F (TIME.GT.TIMEST(I,ICOUNT)) THEN PMODEL(I,ICOUNT)=TENSIONO+((TENSION-TENSIONO)/DELTT)* + (TIMEST ( I , ICOUNT)-TIME+DELTT)- • ' C PRINT*,I,ICOUNT,PMODEL,TENSION CHISQ=CHISQ+((TENSIONS(I,ICOUNT)-PMODEL(I,ICOUNT)) +/SIG(ICOUNT))**2 1 = 1 + 1 ENDIF TIME=TIME+DELTT 50 CONTINUE 150 CONTINUE FVAL=CHISQ PRINT 1000,XVAL(i; ,XVAL(2),XVAL(3) , XVAL(4),XVAL(5),CHISQ 60 I F (IFLAG .EQ. 3) THEN PRINT*,'Enter t h e name o f t h e o u t p u t p a r a m e t e r f i l e (' READ*, PARFILE OPEN(UNIT=8,FILE=PARFILE,STATUS='NEW') DO 60 I=1,IFILES WRITE(8,*) TFILENAMES(I) CONTINUE , PAR) WRITE(8 WRITE(8 WRITE(8 WRITE(8 WRITE(8 WRITE(8 WRITE(8 DO 355 *) 'FINISH' 1001) RS1L 1001) D 1001) RLS1 1001) E l 1001) CF *) CHISQ ICOUNT=l,IFILES 1=1 DELTT i s t i m e i n t e r v a l f o r model 205 TIME=0 C i n i t i a l i z e t h e p o p u l a t i o n s t o z e r o B=0 BL1=0 BL2=0 BL3=0 BL4=0 BL5=0 S1=0 TENSION=0 ADDTEN=0 DO 350 J=0,NPTB(ICOUNT),1 DELTT=TIMESB(J+l,ICOUNT)-TIMESB(J,ICOUNT) TENSIONO=TENSION BO=B BL40=BL4 BL50=BL5 S10=S1 B=BULK(J,ICOUNT) C PRINT*,B BL5=BL50+(RBL*(BO-BL50)-RLS1*BL50*(1-S10*A1) ++RS1L*S10)*DELTT/D S1=S10+(RLS1*BL50*(1-S10*A1)-RS1L*S10)*DELTT I F (TIME.GT.1800)THEN ADDTEN=ADDTEN+CF*S1 ENDIF TENSI0N=S1*E1+ADDTEN I F ((TIME.GT.1800).AND.(TIME.LT.1900))THEN C PRINT*,SI,ADDTEN,TENSION ENDIF I F (TIME.GT.TIMEST(I,ICOUNT)) THEN PMODEL(I,ICOUNT)=TENSIONO+((TENSION-TENSIONO)/DELTT)* +(TIMEST(I,ICOUNT)-TIME+DELTT) WRITE(8,1005) TIME,TENSIONS(I,ICOUNT),PMODEL(I,ICOUNT) 1 = 1 + 1 ENDIF TIME=TIME+DELTT 350 CONTINUE 355 CONTINUE CLOSE (8) ENDIF RETURN 1000 FORMAT (F10. 6, ' ',F10.6,' \ F 1 0 . 6 , ' \ F 1 0 . 6 , 1 ',F10.6' ',E16.8) 1001 FORMAT(E14.8) 1002 FORMAT(F10.6) 1005 FORMAT(F14.6,' ',F10.6,' ',F10.6) END 207 APPENDIX F "RATEFCNLYZ.F" PROGRAM SUBROUTINE FCN(NPAR,GRAD,FVAL,XVAL,IFLAG) COMMON I F I L E S COMMON TFILENAMES DIMENSION TFILENAMES(10) CHARACTER TFILENAMES*15 CHARACTER TFILENAME*15 COMMON BFILENAMES DIMENSION BFILENAMES(10) CHARACTER BFILENAMES*15 CHARACTER BFILENAME*15 CHARACTER PARFILE*15 IMPLICIT DOUBLE PRECISION(A-H,O-Z) DIMENSION NPTT(IO),NPTB(10),SIGT(10),SIGF(10) DIMENSION TIMESB(100000,10),TENSIONS(100000, 10) , +TIMEST(100000,10),BULK(100000, 10),PMODELT(100000, 10) , +PMODELF(100000,10),FLUORESCENCE(100000,10) DIMENSION GRAD ( *•) , XVAL ( * ) C- INPUT arguments:NPAR = number o f c u r r e n t l y v a r i a b l e p a r a m e t e r s . C- XVAL = t h e v e c t o r o f ( c o n s t a n t & v a r i a b l e ) p a r a m e t e r s . C- IFLAG = i n d i c a t e s what i s t o be c a l c u l a t e d . C- OUTPUT arguments:FVAL = t h e f u n c t i o n v a l u e . C- GRAD = t h e ( o p t i o n a l ) v e c t o r o f f i r s t d e r i v a t i v e s . ********************************************** * READ IN DATA ***************************************************************** C PRINT*, I F I L E S I F ( I F L A G .EQ. 1 )THEN DO 20 ICOUNT=l,IFILES TFILENAME=TFILENAMES(ICOUNT) OPEN(UNIT=8,FILE=TFILENAME,STATUS='OLD') READ ( 8 , * ) NPTT(ICOUNT),SIGT(ICOUNT),SIGF(ICOUNT) PRINT*, NPTT(ICOUNT),SIGT(ICOUNT),SIGF(ICOUNT) DO 10 1=1,NPTT(ICOUNT) READ-'(8, * j . TIMEST'd,,ICOUNT) , TENSIONS ( I , ICOUNT) ,• +FLUORESCENCE(I,ICOUNT) • PRINT*,TIMEST(I,ICOUNT),TENSIONS(I,ICOUNT), +FLUORESCENCE(I,ICOUNT) 10 .- CONTINUE > " " ' "' CLOSE ( 8 l ' "•' 20 CONTINUE ENDIF I F ( I F L A G .EQ. 1 )THEN DO 40 ICOUNT=l,IFILES BFILENAME=BFILENAMES(ICOUNT) OPEN(UNIT=8,FILE=BFILENAME,STATUS='OLD') READ(8,*) NPTB(ICOUNT),DELTT PRINT*, NPTB(ICOUNT),DELTT DO 30 1=1,NPTB(ICOUNT) READ(8,*) TIMESB(I,ICOUNT),BULK(I,ICOUNT) C PRINT*,TIMESB(I,ICOUNT),BULK(I,ICOUNT) 30 CONTINUE CLOSE(8) 4 0 CONTINUE ENDIF-************************************************ C Compute c h i - s q u a r e d DCOEFF=l.04E-10 • A l = l . 6 e - 1 7 A2=l.1*A1 A3=1.2*A1 D=XVAL(1)*lE-4 C PRINT*,D RBL=DCOEFF/D E1=XVAL(2)*A1 RLS1=XVAL(3)*lE-6 RS1L=XVAL(4)*lE-3 RS1LB=XVAL(5)*1E-21 RS1S2=XVAL(6)*lE-4 RS1S2B=XVAL(7)*lE-22 RS2S1=XVAL(8)*lE-5 RS2S1B=XVAL(9)*lE-23 RS2S3=XVAL(10)*lE-5 RS2S3B=XVAL(11)*lE-23 RS3S2=XVAL(12)*lE-5 RS3S2B=XVAL(13)*lE-24 E2=XVAL(14)*A1 E3=XVAL(15)*E2 F=XVAL(16)*1E-16 CHISQ=0.0 DO 150 ICOUNT=l,IFILES 1=1 C DELTT i s t i m e i n t e r v a l f o r model TIME=0 C i n i t i a l i z e t h e p o p u l a t i o n s t o z e r o B=0 209 BL5=0 S1=0 S2=0 S3=0 TENSION=0 FLUO=0 DO 50 J=0,NPTB(ICOUNT),1 DELTT=TIMESB(J+l,ICOUNT)-TIMESB(J,ICOUNT) TENSIONO=TENSION FLUOO=FLUO BO=B BL50=BL5 S10=S1 S20=S2 S30=S3 B=BULK(J,ICOUNT) C PRINT*,B BL5=BL50+(RBL*(BO-BL50)-RLS1*BL50*(1-S10*A1-S20*A2 +-S30*A3)+(RS1L+RS1LB*BL50)*S10)*DELTT/D S1=S10+(RLS1*BL50*(1-S10*A1-S20*A2-S30*A3) +-(RS1L+RS1LB*BL50)*S10-(RS1S2+RS1S2B*BL50)*S10 ++(RS2S1+RS2S1B*BL50)*S20)*DELTT S2=S20+((RS1S2+RS1S2B*BL50)*S10 +-(RS2S3+RS2S3B*BL50)*S20-(RS2S1+RS2S1B*BL50)*S20 ++(RS3S2+RS3S2B*BL50)*S30)*DELTT S3=S30+((RS2S3+RS2S3B*BL50)*S20 +-(RS3S2+RS3S2B*BL50)*S30)*DELTT TENSI0N=S1*E1+S2*E2+S3*E3 FLUO=(S1+S2+S3)*F I F (TIME.GT.TIMEST(I,ICOUNT)) THEN PMODELF(I,ICOUNT)=FLUOO+((FLUO-FLUOO)/DELTT)* +(TIMEST(I,ICOUNT)-TIME+DELTT) PMODELT(I,ICOUNT)=TENSIONO+((TENSION-TENSIONO)/DELTT)* +(TIMEST(I,ICOUNT)-TIME+DELTT) C PRINT*,I,ICOUNT,PMODELF,FLUO CHISQ=CHISQ+((FLUORESCENCE(I,ICOUNT)-PMODELF(I,ICOUNT)) + /SIGF(ICOUNT))**2+((TENSIONS(I,ICOUNT)-PMODELT(I, ICOUNT)) +/SIGT(ICOUNT))**2 1=1 + 1 ENDIF TIME=TIME+DELTT 210 50 CONTINUE 150 CONTINUE FVAL=CHISQ PRINT 1000, X V A L ( 1 ) , X V A L ( 2 ) , X V A L ( 3 ) , X V A L ( 4 ) , X V A L ( 5 ) , +XVAL(6) ,XVAL(7),XVAL(8),XVAL(9),XVAL(10) , XVAL(11),XVAL(12) + ,XVAL(13),XVAL(14),XVAL(15),XVAL(16) ,CHISQ I F (IFLAG .EQ. 3) THEN PRINT*,'Enter t h e name o f t h e o u t p u t p a r a m e t e r f i l e (' READ*, PARFILE OPEN(UNIT=8,FILE=PARFILE,STATUS='NEW') DO 60 1=1,IFILES WRITE(8,*) TFILENAMES(I) 60 CONTINUE WRITE(8,*) 'FINISH' PAR) WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 WRITE(8,1001 XVAL(1) XVAL(2) XVAL(3) XVAL(4) XVAL(5) XVAL(6) XVAL(7) XVAL(8) XVAL(9) XVAL(10) XVAL(11) XVAL(12) XVAL(13) XVAL(14) XVAL(15) XVAL(16) WRITE(8,*) CHISQ C C DO 355 ICOUNT=l,IFILES 1=1 TIME=0 B=0 DELTT i s ti m e i n t e r v a l f o r model i n i t i a l i z e t h e p o p u l a t i o n s t o z e r o BL5=0 S1=0 S2=0 S3=0 TENSION=0 FLUO=0 . DO 350 J=0,NPTB(ICOUNT),1 DELT.T=TIMESB(J+l,ICOUNT)-TIMESB(J,ICOUNT) TENSIONO=TENSION FLUOO=FLUO BO=B BL50=BL5 S10=S1 S20=S2 S30=S3 B=BULK(J,ICOUNT) BL5=BL50+(RBL*(BO-BL50)-RLS1*BL50*(1-S10*A1-S20*A2 +-S30*A3)+(RS1L+RS1LB*BL50)*S10)*DELTT/D S1=S10+(RLS1*BL50*(1-S10*A1-S20*A2-S30*A3) +-(RS1L+RS1LB*BL50)*S10-(RS1S2+RS1S2B*BL50)*S10 ++(RS2S1+RS2S1B*BL50)*S20)*DELTT S2=S20+((RS1S2+RS1S2B*BL50)*S10 +-(RS2S3+RS2S3B*BL50)*S20-(RS2S1+RS2S1B*BL50)*S20 ++(RS3S2+RS3S2B*BL50)*S30)*DELTT S3=S30+((RS2S3+RS2S3B*BL50)*S20 +-(RS3S2+RS3S2B*BL50)*S30)*DELTT TENSI0N=S1*E1+S2*E2+S3*E3 FLU0=(S1+S2+S3)*F I F (TIME.GT.TIMEST(I,ICOUNT)) THEN PMODELT(I,ICOUNT)=TENSIONO+((TENSION-TENSIONO) +/DELTT)*(TIMEST(I,ICOUNT)-TIME+DELTT) PMODELF(I,ICOUNT)=FLUOO+((FLUO-FLUOO) +/DELTT)*(TIMEST(I,ICOUNT)-TIME+DELTT) WRITE(8,1005) TIME,TENSIONS(I,ICOUNT), + PMODELT(I,ICOUNT) ,FLUORESCENCE(I,ICOUNT) , PMODELF(I, ICOUNT) , +S1,S2,S3 1=1 + 1 ENDIF TIME=TIME+DELTT 350 CONTINUE 355 CONTINUE CLOSE(8) ENDIF RETURN 1000 FORMAT (F10. 6, ' V F 1 0 . 6 , ' \ F 1 0 . 6 , ' \ F 1 0 . 6 , ' »,F10.6' ',E16.8) 1001 FORMAT(E14 .8) . 1002 FORMAT(F10.6) 1005 FORMAT (F14 . 6, ' ',F10.6,' ',F10.6,' ',F10.6,' ',F10.6,' \ E 1 6 . 8 , + ' ',E16.'8,' '-,.E16.-8.) .. END . •-. • -APPENDIX G MINUIT OUTPUT FOR MODEL FIT ON FIGURE 5.18 4 Parameter model - d variable FCN= 464.4692 FROM MIGRAD STATUS=CONVERGED 266 CALLS 268 TOTAL EDM= 0.57E-05 STRATEGY= 1 ERROR MATRIX ACCURATE EXT PARAMETER NO. NAME 1 2 3 4 5 RS1L D RLS1 E CF VALUE 0.65911E-02 1.4878 0.49888E-01 -0.74326E-02 0. ERROR 0.36692E-03 0.86241 E-01 0.46747E-02 0.26956E-03 constant STEP SIZE 0.74127E-06 0.20030E-04 0.91314E-06 0.21498E-05 FIRST DERIVATIVE -32.030 0.49086 9.7761 -3.3707 EXTERNAL ERROR MATRIX. NDIM= 50 NPAR= 4 ERR DEF= 1.0 0.135E-06 0.304E-04 0.137E-05 0.327E-07 0.304E-04 0.744E-02 0.330E-03 0.926E-05 0.137E-05 0.330E-03 0.219E-04 0.105E-05 0.327E-07 0.926E-05 0.105E-05 0.727E-07 PARAMETER CORRELATION COEFFICIENTS NO. GLOBAL 1 2 3 4 1 0.99760 1.000 0.960 0.799 0.330 2 0.96373 0.960 1.000 0.819 0.398 3 0.99914 0.799 0.819 1.000 0.830 4 0.99788 0.330 0.398 0.830 1.000 3 Parameter model - d fixed FCN= 464.4889 FROM MIGRAD STATUS=CONVERGED 13 CALLS 343 TOTAL EDM= 0.17E-11 STRATEGY=1 ERROR MATRIX UNCERTAINTY= 1.6% EXT PARAMETER NO. NAME 1 2 3 4 5 RS1L D RLS1 E CF VALUE 0.66415E-02 1.5000 0.50426E-01 -0.74182E-02 0. ERROR 0.10474E-03 constant 0.27160E-02 0.24670E-03 constant STEP SIZE -0.68822E-08 -0.49208E-07 -0.14664E-07 FIRST DERIVATIVE 0.85177E-01 -0.11923E-01 0.11105E-01 EXTERNAL ERROR MATRIX. NDIM= 50 NPAR= 3 ERR DEF= 1.0 0.110E-07 0.233E-07 -0.518E-08 0.233E-07 0.738E-05 0.641 E-06 -0.518E-08 0.641 E-06 0.609E-07 PARAMETER CORRELATION COEFFICIENTS NO. GLOBAL 1 3 4 1 0.96954 1.000 0.082 -0.200 3 0.99739 0.082 1.000 0.957 4 0.99748 -0.200 0.957 1.000 213 APPENDIX H MINUIT O U T P U T F R O M M O D E L FIT IN F I G U R E 5.19 5 parameter model - d variable FCN= 465.0720 FROM MIGRAD STATUS=CONVERGED 494 CALLS 496 TOTAL EDM= 0.48E-05 STRATEGY= 1 ERROR MATRIX ACCURATE EXT PARAMETER STEP FIRST NO. NAME VALUE ERROR SIZE DERIVATIVE 1 RS1L 2.3570 1.0189 0.32681 E-05 8.6423 2 D 2.0277 0.15668E-01 0.10560E-04 -0.58238 3 RLS1 3.6715 1.5950 0.41060E-05 -6.8792 4 E -0.13009E-01 0.23924E-03 0.17409E-05 16.016 5 CF 0.44906E-03 0.20160E-04 0.13875E-06 165.58 EXTERNAL ERROR MATRIX. NDIM= 50 NPAR= 5 ERR DEF= 1.0 0.104E+01 0.159E-02 0.163E+01 0.159E-04 •0.577E-06 0.159E-02 0.245E-03 0.222E-02 -0.781 E-06 0.131 E-06 0.163E+01 0.222E-02 0.255E+01 0.465E-04 -0.185E-05 0.159E-04 - 0.781 E-06 0.465E-04 0.572E-07 -0.274E-08 -0.577E-06 0.131E-06 -0.185E-05 -0.274E-08 0.406E-09 PARAMETER CORRELATION COEFFICIENTS NO. GLOBAL 1 2 3 4 5 1 0.99999 1.000 0.100 0.998 0.065 -0.028 2 0.43354 0.100 1.000 0.089 -0.208 0.415 3 0.99999 0.998 0.089 1.000 0.122 -0.058 4 0.99694 0.065 -0.208 0.122 1.000 -0.568 5 0.75742 -0.028 0.415 -0.058 -0.568 1.000 4 parameter model - d fixed FCN= 465.8982 FROM MIGRAD STATUS-CONVERGED 514 CALLS 516 TOTAL EDM= 0.77E-05 STRATEGY= 1 ERROR MATRIX ACCURATE EXT PARAMETER NO. NAME 1 2 3 4 5 RS1L D RLS1 E CF VALUE 0.39830 2.0000 0.62324 -0.12960E-01 0.44272E-03 ERROR 0.11756 constant 0.18583 0.23241 E-03 0.18477E-04 STEP SIZE 0.13488E-05 0.13317E-04 0.17396E-05 0.13839E-06 FIRST DERIVATIVE 20.571 -2.0787 15.379 223.98 EXTERNAL ERROR MATRIX. NDIM= 50 NPAR= 4 ERR DEF= 1.0 0.138E-01 0.218E-01 0.249E-05 -0.386E-06 0.218E-01 0.345E-01 0.741 E-05 -0.743E-06 0.249E-05 0.741 E-05 0.540E-07 -0.232E-08 -0.386E-06 -0.743E-06 -0.232E-08 0.341 E-09 PARAMETER CORRELATION COEFFICIENTS NO. GLOBAL 1 3 4 5 1 0.99998 1.000 0.997 0.091 -0.178 3 0.99998 0.997 1.000 0.172 -0.216 4 0.99678 0.091 0.172 1.000 -0.540 5 0.70425 -0.178 -0.216 -0.540 1.000 \ APPENDIX I MINUIT OUTPUT FROM MODEL FIT IN FIGURE 5.36 COVARIANCE MATRIX CALCULATED SUCCESSFULLY FCN= 5849.663 FROM MIGRAD STATUS=CONVERGED 794 CALLS 2724 TOTAL EDM= 0.27E-04 STRATEGY= 1 ERROR MATRIX ACCURATE EXT PARAMETER STEP FIRST NO. NAME VALUE ERROR SIZE DERIVATIVE 1 D 0.13782 0.97674E-03 0.23717E-05 18.345 2 E1 -0.11280E-12 0.71910E-06 0.63345E-04 ** at limit ** 3 RLS1 21.793 0.25515 0.41561E-05 -37.537 4 RS1L 15.063 0.89808 0.34089E-05 44.710 5 RS1LB 0. constant 6 RS1S2 0. constant 7 RS1S2B 1.6093 0.12749 0.31198E-05 -39.500 8 RS2S1 0. constant 9 RS2S1B 9.9574 0.24782 0.24741 E-05 48.629 10 RS2S3 0. constant 11 RS2S3B 0.34253 0.53847E-01 0.16852E-05 -44.902 12 RS3S2 0. constant 13 RS3S2B 10.292 0.27355 0.15238E-04 4.0785 14 E2 -0.30462E-02 0.77420E-04 0.90634E-06 77.910 15 E3 7.4950 1.2272 0.76624E-05 -8.7403 16 F 0.10831 E-01 0.16541E-03 0.73909E-06 -881.79 EXTERNAL ERROR MATRIX. NDIM= 50 NPAR= 11 ELEMENTS ABOVE DIAGONAL ARE NOT PRINTED. 0.954E-06 ERR DEF= 1.0 0.955E-16 0.240E-03 -0.306E-03 -0.575E-04 0.324E-18 0.255E-13 -0.317E-12 •0.502E-13 -0.550E-04 -0.106E-12 0.263E-04 0.142E-13 -0.137E-03 -0.340E-07 -0.614E-03 -0.826E-07 •0.114E-12 •0.185E-16 •0.352E-12 •0.558E-16 0.651 E-01 -0.680E-01 -0.131 E-01 -0.890E-02 0.645E-02 -0.341 E-01 -0.799E-05 0.807E+00 0.110E+00 0.130E+00 •0.459E-01 0.220E+00 0.620E-04 -0.150E+00 0.107E+01 -0.200E-04 0.145E-03 0.163E-01 0.232E-01 0.614E-01 •0.623E-02 -0.515E-02 0.290E-02 0.317E-01 0.386E-01 -0.128E-01 0.946E-05 0.140E-04 -0.351 E-05 0.149E+00 0.149E+00 -0.654E-01 0.200E-04 0.220E-04 -0.857E-05 0.748E-01 0.197E-04 0.599E-08 0.311E+00 0.863E-04 0.413E-04 0.111E-07 0.151E+01 0.198E-03 0.274E-07 PARAMETER CORRELATION COEFFICIENTS NO. GLOBAL 1 2 3 4 7 9 11 0.97048 0.00412 0.99797 0.99992 0.99997 0.99965 0.99999 13 0.99897 14 0.99985 15 0.99999 16 0.99967 1 1.000 0.000 0.963 -0.349 -0.462 -0.227 0.500 -0.514 -0.450 -0.512 -0.511 2 0.000 1.000 0.000 -0.001 -0.001 -0.001 0.000 -0.001 0.000 -0.001 -0.001 3 0.963 0.000 1.000 -0.297 -0.402 -0.141 0.469 -0.488 -0.405 -0.478 -0.473 4 -0.349 -0.001 -0.297 1.000 0.964 0.585 -0.949 0.896 0.892 0.967 0.975 7 -0.462 -0.001 -0.402 0.964 1.000 0.733 -0.908 0.909 0.958 0.950 0.947 9 11 13 14 15 16 -0.227 0.500 -0.514 -0.450 -0.512 -0.511 -0.001 0.000 -0.001 0.000 -0.001 -0.001 -0.141 0.469 -0.488 -0.405 -0.478 -0.473 0.585 -0.949 0.896 0.892 0.967 0.975 0.733 -0.908 0.909 0.958 0.950 0.947 1.000 -0.386 0.570 0.730 0.491 0.536 -0.386 1.000 -0.870 -0.841 -0.989 -0.962 0.570 -0.870 1.000 0.932 0.927 0.913 0.730 -0.841 0.932 1.000 0.908 0.865 0.491 -0.989 0.927 0.908 1.000 0.975 0.536 -0.962 0.913 0.865 0.975 1.000 215 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0085812/manifest

Comment

Related Items