{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","GraduationDate":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","RightsURI":"https:\/\/open.library.ubc.ca\/terms#rightsURI","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Applied Science, Faculty of","@language":"en"},{"@value":"Electrical and Computer Engineering, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Duan, Yan","@language":"en"}],"DateAvailable":[{"@value":"2015-10-24T05:39:21","@language":"en"}],"DateIssued":[{"@value":"2015","@language":"en"}],"Degree":[{"@value":"Master of Applied Science - MASc","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"This work presents a study of cholesterol's effects on archaebacterial cell membranes using coarse grain molecular dynamic simulations. As a major component in biological membranes, cholesterol is closely related to the dynamics of lipids and biomechanical properties of the membrane. A coarse grained molecular dynamics (CGMD) model is constructed to study the membrane properties. The CMGD model provides insights into the diffusion dynamics of lipids, membrane thickness, line tension, and surface tension as a function of cholesterol content. The CGMD simulation results are validated using experimental measurements from a tethered archaebacterial bilayer lipid membrane. The membrane is tethered to an inert gold bioelectronic interface, which allows the experimental measurements to be performed using standard laboratory equipments. A fractional order macroscopic model is introduced to link microscopic simulation results with macroscopic experimental measurements. To ensure the bioelectronic interface does not affect the membrane dynamics and biomechanics, it is shown that variations in the position dependent water density are negligible near the surface of the membrane. Furthermore, the Percus-Yevick equation is used to confirm that harsh repulsive forces play a negligible role in the long range dynamics of the water density profile.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/54885?expand=metadata","@language":"en"}],"FullText":[{"@value":"Study of Cholesterol in TetheredMembrane Using Coarse-GrainedMolecular Dynamics SimulationsbyYan DuanB.Sc in Engineering Physics, The University of Alberta, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2015c Yan Duan 2015AbstractThis thesis presents a study of cholesterol's e\u000bects on archaebacterial cellmembranes using coarse grain molecular dynamic simulations. As a majorcomponent in biological membranes, cholesterol is closely related to the dy-namics of lipids and biomechanical properties of the membrane. A coarsegrained molecular dynamics (CGMD) model is constructed to study themembrane properties. The CMGD model provides insights into the dif-fusion dynamics of lipids, membrane thickness, line tension, and surfacetension corresponding to di\u000berent cholesterol levels. The CGMD simulationresults are validated using experimental measurements from a tethered ar-chaebacterial bilayer lipid membrane. The membrane is tethered onto aninert gold bioelctronic interface, which allows the experimental measure-ments to be performed using standard laboratory equipments. A fractionalorder macroscopic model is introduced to link microscopic simulation resultswith macroscopic experimental measurements. To ensure the bioelectronicinterface does not a\u000bect the membrane dynamics and biomechanics, it isshown that variations in the position dependent water density are negligiblenear the surface of the membrane. Furthermore, the Percus-Yevick equationis used to con\frm that harsh repulsive forces play a negligible role in thelong range dynamics of the water density pro\fle.iiPrefaceThis thesis consists of three main chapters. The \frst chapter gives the mo-tivation of the thesis and \fve levels of abstraction for cell membrane model-ing. The second chapter provides a comprehensive introduction on necessaryconcepts and theories to understand molecular dynamics, and its simulationschemes and tools. The third chapter is largely based on work conduct-ed in statistical signal processing lab at University of British Columbia, byDr.William Hoiles, Prof.Vikram Krishnamurthy and Yan Duan. I was re-sponsible for all the coarse-grained simulation, derivation and applicationof Percus-Yevick equation for the water density pro\fle, numerical solutionof Volterra di\u000berential-integral equation, and parameter estimations for thefractional order macroscopic model.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation: the Biosensor . . . . . . . . . . . . . . . . . . . . 11.2 Cell Membrane System . . . . . . . . . . . . . . . . . . . . . 21.3 Bioelectronic Interface . . . . . . . . . . . . . . . . . . . . . . 31.4 Abstractions in Bio-System Modeling . . . . . . . . . . . . . 41.4.1 Ab-initio Molecular Dynamics . . . . . . . . . . . . . 51.4.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . 51.4.3 Coarse-Grained Molecular Dynamics . . . . . . . . . 61.4.4 Continuum Theory . . . . . . . . . . . . . . . . . . . 61.4.5 Macroscopic Model . . . . . . . . . . . . . . . . . . . 61.5 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . 72 Fundamental Molecular Dynamics . . . . . . . . . . . . . . . 92.1 Basic Algorithm and Force Fields . . . . . . . . . . . . . . . 92.1.1 Global MD Algorithm . . . . . . . . . . . . . . . . . . 92.1.2 Force Fields . . . . . . . . . . . . . . . . . . . . . . . 112.2 GROMACS: the MD Simulation Engine . . . . . . . . . . . . 132.2.1 Initialize GROMACS Files and Programs . . . . . . . 132.2.2 Energy Minimization . . . . . . . . . . . . . . . . . . 15ivTable of Contents2.2.3 Running the Production Simulation . . . . . . . . . . 162.2.4 Analysis of the Simulation Results . . . . . . . . . . . 182.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Study of Cholesterol in Tethered Membranes . . . . . . . . 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 CGMD Simulation Setup and Membrane Formation . . . . . 223.2.1 CGMD Model . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Coarse-Grained Molecular Dynamics Simulation Pro-tocol . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Formation of Membrane in Experiment . . . . . . . . 253.3 Laterial Di\u000busion Dynamics of Lipids and Cholesterol . . . . 263.3.1 Numerical Solution of Volterra Di\u000berential-Integral E-quation . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 CGMD Simulation Results on Di\u000busion Dynamics . . 313.4 Water Density Pro\fle at Bioelectronic Interface . . . . . . . 333.4.1 Derivation of Percus-Yevick Equation . . . . . . . . . 343.4.2 Match the Percus-Yevick Equation with CGMD Re-sults . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 Archaebacterial Membrane Biomechanics . . . . . . . . . . . 383.5.1 Archaebacterial Membrane Thickness . . . . . . . . . 383.5.2 Archaebacterial Membrane Line Tension and SurfaceTension . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6 Fractional Order Macroscopic Model . . . . . . . . . . . . . . 423.6.1 Fractional Order Macroscopic Model and ParameterEstimation . . . . . . . . . . . . . . . . . . . . . . . . 433.6.2 Experimental Results Utilizing the Macroscopic Mod-el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Summary and Future Work . . . . . . . . . . . . . . . . . . . 51Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A Matlab Code to Calculate the Percus Yevick Pair Distribu-tion Function for Spherical Particles . . . . . . . . . . . . . . 60vList of Tables3.1 Lipid and Cholesterol Di\u000busion (nm2\/\u0016s) . . . . . . . . . . . 323.2 Biomechanic Parameters of Membrane . . . . . . . . . . . . . 393.3 Macroscopic Model Parameters for Archaebacterial Membrane 48viList of Figures1.1 Overview of biosensor . . . . . . . . . . . . . . . . . . . . . . 21.2 Schematic of natural cell membrane . . . . . . . . . . . . . . 31.3 Schematic of \fve levels of abstraction for membrane models . 52.1 Some MARTINI mapping examples . . . . . . . . . . . . . . . 123.1 A brief summary of recent related works on study of choles-terol in membrane using MD\/CGMD simulation methods . . 203.2 Ball structures of Dphpc, GDPE lpids, and Cholesterol . . . . 213.3 Coarse graiend molecular dynamics structure . . . . . . . . . 243.4 Computed mean-square displacement . . . . . . . . . . . . . . 313.5 The computed water density computed from the CGMD sim-ulation results, and analytical results from the Percus-Yevickequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Ribbon structure of archaebacterial membrane . . . . . . . . 413.7 Fractional order macroscopic model of the tethered archae-bacterial membrane . . . . . . . . . . . . . . . . . . . . . . . 453.8 Parameter estimation using least squares method . . . . . . . 473.9 Experimentally measured (gray dots) and numerically pre-dicted current response for tethered membranes containingcholesterol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49viiAcknowledgementsI would like to \frst thank my supervisor Prof.Vikram Krishnamurthy atUBC. Not only he o\u000bers me valuable directions and support for my graduatestudy, but also his great passion and insights on science and engineeringencourage me to make more contributions to our community.I am truly honored to work with Dr.William Hoiles, as a beginner inthis \feld, I got valuable training from him. His keen insights, comprehensiveknowledge and kind patience showed me the characteristics that an excellentscholar would possess.I am very grateful for my family members. Their encouragements andloving supports are so important, that I could have been a totally di\u000berentperson without them.Finally I would like to express my appreciation to my dear friends andcolleagues, it's always good time to talk to them and learn from them.viiiDedicationTo my family: Shaoan, Defang, Bing, my cousins and my grandparents.ixChapter 1Introduction1.1 Motivation: the BiosensorIn 1956, Dr.Leland C. Clark conducted his famous experiment on oxygen de-tection using platinum, where he placed the enzyme Glucose Oxidase (GOD)close to the surface of platinum, thus to trap oxygen against the electrodesusing a piece of dialysis membrane [13]. It was the \frst time that elemen-t recognition was achieved by using bio-materials. Since then, substantialresearch has emerged to investigate the element recognition process involv-ing bio-materials, which has further led to the developments of biologicalsensors, or the simply-called \"biosensors\".Commercialized biosensors are already in use in areas such as healthcaremonitoring, industrial processing and environmental pollution control [50].Even though this is the case, further improvements on biosensors to achievehigher sensitivity of element detection, wider scope of applications, and low-er production costs for biosensors are strongly demanded by markets [51].This necessitates a thorough understanding of element recognition processinvolved in biosensors.A standard biosensor consists of a recognition component called bio-receptor, and a corresponding transducer component. A bioreceptor inter-acts with the desired chemical analytes and the interaction is detected by thetransducer. The transducer produces a measurable electrical signal, which issent to an electronic system composed of an ampli\fer and a display circuit.The biosensor structure is summarized in Fig. 1.1. The class of biorecep-tor includes a variety of biomolecules such as membrane proteins, enzymes,antibodies and nucleic acids. An electrochemical transducer is normally ametal electrode, while other types such as optical and thermal transducersare also in use [20]. More details on choices of transducers are introducedin Sec. 1.3.Since many of the element recognition processes occur at the cell mem-brane, investigation of cell membrane properties has played an importantrole in biosensor research. Furthermore, due to the di\u000eculty of using livinganimal cell membranes, arti\fcial lipid membranes have become the main11.2. Cell Membrane Systemsource to obtain insights into the recognition processes occurring in theproximity of the cell membrane. In this thesis, a study on synthetic archae-bacterial membrane will be presented.Figure 1.1: Overview of a biosensor, consisting of a bioreceptor (membraneproteins) and a transducer (metal electrode\/bioelectronic interface), whichare connected to an ampli\fer and a display circuit1.2 Cell Membrane SystemThe cell membrane is composed of three primary components: membranelipids, macromolecules, and the cytoskeletal \flaments [44]. Fig. 1.2 summa-rizes the membrane structure.Membrane lipids are a group of biological compounds that possess am-phiphilic property, which means that each lipid molecule has one water-soluble end and one non-water-soluble end. The amphiphilic property forceslipid molecules to group together to form a double-layer structure in waterenvironment, so a cell membrane is often referred as \"lipid bilayer\" in manycontexts. Di\u000berent components of membrane lipids have di\u000berent e\u000bectson cell shape, permeability and organization of macromolecules [44]. Mem-brane lipids contain a wide variety of biological compounds, the most com-mon ones include phospholipids and sterols such as cholesterol. For arti\fcialmembranes, various synthetic phospholipid derivatives can be used [16]. Inthis work, a synthetic archaebacterial membrane composed of zwittrion-ic C20 diphytanylether-glycero-phosphatidylcholine lipid (DphPC) and C20diphytanyl-diglyceride ether lipid (GDPE) is investigated.Macromolecules refer to a group of large (more than 1000 atoms) moleculesincluding nucleic acids, carbonhydrates and proteins. In the case of cellmembrane, macromolecules usually refer to proteins. The membrane pro-21.3. Bioelectronic Interfaceteins can be classi\fed into two functional classes: transport protein andmembrane receptor [44]. As one type of bioreceptors, membrane recep-tor proteins transmit electrical signals between intracecullar and extracel-lular environments. Transport proteins, on the other hand, aid in trans-membrane movements of ions and large molecules by forming channels\/-pores. Furthermore, cytoskeletal \flaments provide the physical structuralsupport for the membrane.This thesis studies the properties of lipids, such as di\u000busion dynamics,membrane thickness and surface\/line tension. Therefore, a simple membranemodel (archaebacterial membrane) that only includes membrane lipids isconstructed.Figure 1.2: Schematic of a natural biological membrane. The extracellularuid represents the contents outside the cell, and the cytosol is the interi-or of the cell with the membrane separating the two domains. Note thatbiological membranes are composed of thousands of di\u000berent components(macromolecules, lipids, chemical species)1.3 Bioelectronic InterfaceElectrical instrumentation is required to measure the electrical signals gener-ated from arti\fcial membrane. Thus to perform measurements for a biosen-sor, we need to introduce a transducer, that is, a bioelectronic interface toconnect the biological system to the electrical instrumentation.We consider a tethered membrane system connected to a bioelectronic31.4. Abstractions in Bio-System Modelinginterface, which is the standard set up for an electrochemical biosensor. Aparticularly useful bioelectronic interface comprises of inert gold electrodes.They have advantages over redox active electrodes such as biopolymers,because redox active electrodes will tend to force the tethers to dissociatefrom the electrode surface, thus to destroy the membrane. Also, redoxactive electrodes will release metal ions into solution, which can interferewith the electrophysiological response of proteins and peptides [14]. Theinert gold electrode, however, capacitively couples the electronic domain tothe physiological domain without the issues associated with redox electrodes.However, with the presence of gold interface, the so-called \"di\u000busionlimited e\u000bect\" need to be considered in modeling of tethered membranesystem. The di\u000busion limited e\u000bect refers to the fact that a fast reaction willbe triggered if membrane components di\u000buse and contact the gold surface,such e\u000bect at the interface can cause the charge transfer of bio-molecules,which are highly undesirable [53]. The problem of di\u000busion limited e\u000bect,and its solution will be discussed with more details in Sec. 3.6 of this thesis.1.4 Abstractions in Bio-System ModelingAs mentioned above, a bioelectronic interface should connect the tetheredmembrane system and electric instrumentation. However, the problem is,how to make interpretations of electrical measurements thus to explain mem-brane characteristics? The investigation on membrane system is to studymicroscopic properties such as di\u000busion dynamics and membrane thickness,but the data that can be measured is macroscopic, such as voltage and cur-rent. Thus a key to the development of novel membrane biosensor is anaccurate mathematical model of the cell membrane, which is able to inter-pret the macroscopic measurements and reects corresponding microscopicmembrane properties correctly. Such a model must link the microscop-ic dynamics of water, lipids, peptides to experimental measurements at amacroscopic time and length scale.There are \fve levels of modeling abstraction of mathematical models thathave been used for modeling dynamics of cell membrane. For these abstrac-tions, more the details considered for the system, more is the computationalpower and time required for the simulation, and vice versa. A schematicdiagram for the \fve levels of abstraction for bio-systems is summarized inFig. 1.3.41.4. Abstractions in Bio-System ModelingFigure 1.3: Schematic diagram illustrating the length and timescale achiev-able by the atomistic to macroscopic simulation methods1.4.1 Ab-initio Molecular DynamicsThe \"Ab-initio\" molecular dynamics, or the \"from the beginning\" moleculardynamics model, is the model that takes into account the most comprehen-sive details for a system. Ab-initio includes both classical Newton's physicsand quantum mechanical Schrodinger's equations for the dynamics of tar-get system, which might include particles such as water, ions, membranelipids, proteins, and peptides. This is the model that provides the mostdetailed description of a real system. However, due to these (sometimesunnecessarily) detailed equations, an excessive computational power is re-quired. Thus Ab-initio molecular dynamics could only attain membranelength scale of nanometer and simulation time in an order of femtosecond,which is too small compared to the data that can be obtained from any ex-perimental measurement [39]. Therefore such method is not quite popularin bio-system modeling so far.1.4.2 Molecular DynamicsMolecular dynamics (MD) is a simpli\fcation of the above \"Ab-initio\" molec-ular dynamics model. The quantum mechanical Schrodinger's equationsconsidered in \"Ab-initio\" molecular dynamics are ignored in this model,that is, the matrix representation of semi-empirical potential from quan-tum mechanics is not used in MD, and only empirical potentials associatedwith chemical bonds, bond angles and non-bonded forces are considered intoMD [42]. More details of molecular dynamics would be discussed in the nextchapter. It's also worth noting that, although only Newtonian equations areevaluated for each time step of MD simulation, still merely a length orderof nanometer and time order of nanosecond can be achieved by using MD.51.4. Abstractions in Bio-System Modeling1.4.3 Coarse-Grained Molecular DynamicsCoarse-Grained Molecular Dynamics (CGMD) is a further simpli\fcationbased on molecular dynamics. By grouping certain atoms together intocoarse-grained beads, with the bead-to-bead interactions empirically pa-rameterized, membrane dynamics can still be evaluated using Newton's e-quations of motion [47]. Such simpli\fcation allows CGMD simulations toachieve simulation time scale of microseconds with a system size of tens ofnanometers, which are good enough to match most real membrane dynamicsthat can be measured by experiments. The main results presented in thisthesis would be based on CGMD simulations.1.4.4 Continuum TheoryDespite their di\u000berent levels of abstraction, the above three molecular dy-namics models are considering discrete particles of a system. Thus if thenumber of particles in target system is large, the e\u000eciency of computationcan be limited. Therefore, a more simpli\fed class of models is to treat thediscrete entities as continuous densities, which represent the space-time aver-age of the microscopic motion of particles. The most well-known continuummodel should be the Poisson-Nernst-Planck system of equations for di\u000busionprocess of ion transport, which combines the Poisson equation from electro-statics, and the Nernst-Planck equation for di\u000busion [60]. Such a signi\fcantsimpli\fcation allows continuum models to achieve simulation time scale ofthe the order of microseconds, with a system size of micrometers.1.4.5 Macroscopic ModelAs its name suggests, a macroscopic model describes a bio-system systemby using macroscopic parameters, which can be obtained from experimentalmeasurements. Normally these parameters can be de\fned to be any entity,and there's even no speci\fc physical interpretation required for them: aslong as these estimated parameters can \ft the experimental measurements,the macroscopic model can be regarded to be successful. In this thesis, afractional macroscopic model is introduced to estimate biological param-eters, such as tethered membrane conductance and capacitance based onexperimental measurements. Due to the free choice of parameters for themacroscopic models, there is no limit to the attainable scales of length andtime.61.5. Thesis Contributions1.5 Thesis ContributionsThis thesis is composed of four chapters, which include an introductorychapter for fundamental molecular dynamics, and a chapter on the study ofcholesterol in tethered membrane using CGMD simulations.Chapter 2 provides an introduction to the fundamental newtonian physic-s involved in molecular dynamics algorithms, along with common force \felds(potential functions) including MARTINI force \feld used in CGMD simula-tions. Then a four-step MD simulation scheme is introduced along with thesimulation engine GROMACS. Details on con\fguration of GROMACS \flesand programs are provided, with an introduction to common terminologiesand algorithms used in MD simulations.Chapter 3 presents the core research work in this thesis. The e\u000bectsof cholesterol on tethered archaebacterial membrane system are studied byusing Coarse-Grained molecular dynamics simulations. The archaebacterialmembrane consists of 70% DphPc and 30% GDPE lipids, which are teth-ered onto a gold bioelectronic interface. By following the simulation schemeintroduced in Chapter 2, the CGMD simulation based on MARTINI force\feld is used to study properties of membrane lipids, as cholesterol level variesfrom 0% to 50%.The following results are presented regarding membrane properties af-fected by cholesterol:\u000f In order to match the simulation model with real membrane used inexperiments, a CGMD model for a tethered archaebacterial membraneis introduced. The CGMD simulation protocol is provided followingthe four-step scheme introduced in Chapter 2.\u000f To make reliable measurements in experiments, it is crucial to \fgureout the region that is not a\u000bected by the bioelectronic interface, whichcan change the di\u000busion dynamics of membrane and water. Thus theposition-dependent water density pro\fle at the bioelectronic interfaceis studied. An analytical solution of Percus-Yevick equation is ob-tained, to show that the water density pro\fle doesn't vary much as faras 4nm from the interface, and short-range interactions in Lennard-Jones potential contribute little to a\u000bect water density pro\fle.\u000f Lipids undergo three regimes of di\u000busion: ballistic, subdifusion andFickian di\u000busion, which are predicted by mode-coupling theory. Al-though the three di\u000busion dynamics can be obtained by Volterra integral-di\u000berential equation, however, due to the excessive computational pow-71.5. Thesis Contributionser required by its numerical solution, CGMD simulation results areused to analyze di\u000busion dynamics of membrane instead. It is shownthat cholesterol concentration does not a\u000bect the transition time be-tween the subdi\u000busion and Fickian di\u000busion, and cholesterol concen-tration only a\u000bects the di\u000busion coe\u000ecient of lipids in Fickian di\u000busionregime. As cholesterol concentration increases, the lipid di\u000busion co-e\u000ecients decrease.\u000f The e\u000bects that cholesterol have on biomechanics properties of mem-brane are also presented by using CGMD simulation results. It isshown that as the cholesterol concentration increases from 0% to 50%,the membrane thickness increases, the membrane line tension increas-es, while the membrane surface tension increases up to 30% cholesterolin membrane then decreases.\u000f To link the macroscopic experimental measurements with microscop-ic simulation results, and to account for the di\u000busion limited e\u000bectsdue to the bioelectronic interface presented in the tethered membranesystem, a fractional order macroscopic model is introduced. The teth-ered membrane is modeled by a fractional order RC circuit, whoseparameters are obtained by a least-square estimator.Finally, Chapter 4 briey summarizes the key results, and commentson the suggestions for future research related to the work presented in thisthesis.8Chapter 2Fundamental MolecularDynamics2.1 Basic Algorithm and Force FieldsCurrently there are two classes of simulation techniques in computationalchemistry and biology: molecular dynamics (MD) and Monte Carlo simu-lation [22]. Monte Carlo molecular modeling method starts from an initialmicrostate, then keeps moving to the desired state according to the desiredensemble's Boltzman probability distribution [22]. In compasiron, MD is amore universal technique, especially in non-equilibrium ensemble and anal-ysis of dynamic scenarios. Ever since it was \frst proposed by Alder andWainwright in the late 1950's [1] to study the interactions of hard spheres,MD has been used extensively for studying the structural and dynamicalproperties of molecules.2.1.1 Global MD AlgorithmAn MD algorithm starts by assigning initial positions and velocities to allparticles within the system being considered. The algorithm looks for aglobal optimal value of energy potential function for the initial system. Astep called \"energy minimization\" guarantees that the initial positions andvelocities of all particles can be accepted (i.e states of atoms are within therange pre-de\fned by the simulation engine). Finally, for each time step, aset of classical Newtonian equations of motion is solved for ithparticle of asystem, which includes N interacting particles [28]:mi@2ri@t2= Fidridt= vividt=dvidt(2.1)92.1. Basic Algorithm and Force Fieldswhere miis the mass of the ithparticle, t the time, rithe vector of parti-cle relative to the origin, and Fithe interactive forces applied on the par-ticle. The interactive forces are the derivatives of the potential functionE(r1; r2; :::rN):Fi=\u0000@E@ri(2.2)where the potential function is given as a combination of bonded and non-bonded interactions between the particles. A potential function can beshown as follows:E(r1; :::; rN) =Xbondska(l \u0000 l0)2+Xbondskb(\u0012 \u0000 \u00120)2+Xtorsionkc[1 + cos(n! \u0000 )]+Xatompairs4\u000fij[(ffijrij)12\u0000 (ffijrij)6]+Xatompairskqiqjrij(2.3)The \frst and second term are the covalently bonding energies, which areinduced by deformations of bond length l and bond angle \u0012, respectively;the third term describes the energy due to rotation around the chemicalbond, the \frst three terms represent the bonded potential. The last twoterms describe the non-bonded interactions between all atom pairs, whichinclude Lennard-Jones potential and the Coulomb electrostatic potential,respectively. ka, kb, kcare all constants related to speci\fc atom interactions,\u000f is the permittivity constant and k is the Coulomb constant. r is thedistance between atoms, while ff is the distance at which inter-atom potentialis zero.By solving the newtonian equations 2.1, the particle trajectories whichare represented by coordinates ri=1;2;:::;Ncan be obtained. Based on theinitial conditions: potential interaction E, initial positions r and velocitiesv , with temperature and pressure con\fgured to be desired values, the sim-ulation keeps updating the coordinates of the atoms by solving Newtonianequations, thus the coordinates of particles are generated at speci\fc timesteps.102.1. Basic Algorithm and Force FieldsTo summarize, a molecular dynamics simulation involves the followingthree steps:1. Input initial conditions, which include pre-de\fned potential functions(force \felds), initial positions and velocities of particles2. Compute forces by solving equation 2.23. Update the system states (positions, velocities and energies of all par-ticles) by solving equation 2.12.1.2 Force FieldsAs described in equation 2.3, force \feld (or equivalently, potential func-tion) can be fully described by bonded and non-bonded interactions. Beforestarting any molecular dynamics simulation, we need to pre-de\fne the pa-rameters that describe the potential function properly, thus to compute theforces and update the system trajectories. Di\u000berent force \felds are madeand used for di\u000berent purposes. The most comprehensive force \feld is the so-called \"all-atom\" force \feld{that is, all atoms are considered to be assignedinteraction parameters [69]. However, limited by computational power andsimulation time, mainstream force \felds usually ignore some unimportantatoms such as non-polar hydrogens, which contribute insigni\fcant interac-tions to a system. In this section, one common widely used force \feld isintroduced, which will be used in the simulations in Chapter 3 of this thesis.Coarse Grained Force Fields: MARTINIThe signi\fcance of molecular dynamics simulations is based on the fact that,mechanical and chemical processes at atomistic level are able to reect realstates of system at macroscopic level. Therefore, an MD simulation needsto match a real experiment at a reasonable scale of space and time. Thismeans that, an ideal MD simulation should not only take into account max-imum details for topology (nano-meter scale) of the target system, but alsoproceed with long enough time that is comparable to the real experiment,which is at time scale of at least micro-second. However, a bio-system forstudy can contain thousands of particles, thus to keep essential topology ofthe system, simulation time is limited to nano-seconds due to restriction ofcomputational power.A compromising solution for such problem is to use coarse-graining{thatis, to select atoms that share common chemical\/biological characteristics,112.1. Basic Algorithm and Force Fieldsthen group them together and treat each group as a single bead. Clearly,such approximation will reduce degrees of freedoms of the original system,and ignore many inside interactions between the atoms. It is expected thatsuch approximation might induce errors of simulation results. However, itturns out that, with clever choices of grouped atoms, coarse-grained modelsare able to generate simulation results that are in excellent agreements withexperiments, while the time scale of simulation can be signi\fcantly extendedto micro-seconds.The MARTINI force \feld [47] is a popular coarse-grained molecular dy-namics force \feld, which is suitable for biomolecular systems. It follows asimple four-to-one mapping, that is, four selected atoms are grouped andtreated as one coarse-grained bead, and only polar, non-polar, apolar andcharged interactions are de\fned in the force \feld. Some MARTINI mappingexamples are visualized in Fig 2.1 [47].Figure 2.1: Martini mapping examples of selected molecules: (A) Stan-dard water particle representing four water molecules, (B) Polarizable wa-ter molecule with embedded charges, (C) DMPC lipid, (D) Polysaccharidefragment, (E) Peptide, (F) DNA fragment, (G) Polystyrene fragment, (H)Fullerene molecule.122.2. GROMACS: the MD Simulation Engine2.2 GROMACS: the MD Simulation EngineAmong currently available simulation packages of molecular dynamics, GRO-MACS (GROningen MAchine for Chemical Simulations)[6] is known for be-ing lightweight, fast, free and open source. As an e\u000ecient engine to performmolecular dynamics simulations and energy minimization, GROMACS hasvarious applications in computational biology and molecular modeling[6].To accomplish a molecular dynamics simulation, GROMACS follows a four-step scheme to obtain the simulation results. In the following subsections,each of the four steps will be introduced in details.2.2.1 Initialize GROMACS Files and ProgramsGROMACS programs perform MD simulation steps by reading and writing\fles in several speci\fc formats [69], which are listed below:\u000f xxx.mdp: the main parameter input \fle that records parameters andconditions for energy minimization, position restraints, and main MDsimulation step\u000f xxx.pdb: short for the Protein DataBank \fle format,this input \flecontains molecular structure and coordinate information of the parti-cles\u000f xxx.gro: similar to .pdb \fle, this input \fle contains molecular structurein Gromos 86 format, which indexes particles according to their types\u000f xxx.itp: an input topology \fle that de\fnes the system particles' topolo-gies, such as charge, mass, and radius of particles\u000f xxx.top: the input topology \fle, which de\fnes the force \felds, particletypes and number of particles, its particles speci\fcations are based onthe .itp \fle\u000f xxx.tpr: the input \fle for simulation steps, it contains starting coor-dinates and velocities of the system particles\u000f xxx.xtc: the output \fle generated by production run, it includes tra-jectory information of all system particles for each time step\u000f xxx.edr: the output \fle generated by production run, it contains en-ergy information at the end of simulation time132.2. GROMACS: the MD Simulation EngineFurthermore, being a Unix-based program package, GROMACS com-mand follows standard Unix\/Linux command protocol, the most commonfunctional routines are:\u000f mdrun: the main computational chemistry engine to perform molecu-lar dynamics simulation and energy minimization\u000f editconf: a program that convert one \fle format to another\u000f grompp: before any production run, this program could take .top and.mdp \fle as inputs and generate the input \fle .tpr for the productionrun\u000f genbox: add solvants (water in most cases) into the coordinates \flelike .gro \fles\u000f g msd: it takes .xtc \fle as input, and compute mean square displace-ment from the trajectory \fle, thus to compute di\u000busion constant ofspeci\fc particlesInitial ConditionsBefore any MD simulation, the topology \fle (.itp, .top) has to be properlyedited, thus to load the force \feld, which de\fnes the interactions betweenparticles of the system. Then three parameters need to be initialized: coor-dinates and velocities of the particles, along with the pre-de\fned box size.These parameters will be included in .gro \fle. Setting initial velocities isoptional, if velocities are not initialized manually, then GROMACS wouldassign initial velocities to the particles following MaxwellBoltzmann distri-bution in statistical mechanics [69], given temperature T :p(vi) =rmi2\u0019BTexp(\u0000miv2i2BT) (2.4)where B is the Boltzman constant.Periodic Boundary ConditionsOne possible problem in molecular dynamics simulation is the \"edge e\u000bec-t\": the simulated particles in a system might move out of the pre-de\fnedsimulation box. To solve such problem, periodic boundary conditions areintroduced into GROMACS, that is, the original single box is replaced bya box-array, which contains multiple translated copy of the same unit. For142.2. GROMACS: the MD Simulation Enginethe convenience of de\fning various systems, GROMACS allows di\u000berent s-tandard shapes for cell units (while for e\u000eciency, rectangular cells are themost popular).It is also worth noting that, to solve the problem that there are multi-ple particle images inside translated cells, GROMACS follows the so-called\"minimum image\" convention, that is, only the nearest particle image is con-sidered for short-range non-bonded interactions. Speci\fcally, a cut-o\u000b radiusRcutis applied to truncate short range non-bonded interactions, where Rcutis less than half the minimum box vector [69]:Rcut<12min(jjajj; jjbjj; jjcjj) (2.5)where a; b; c are the vectors de\fning the simulation box. The cut-o\u000b radiusis set in the .mdp \fle.Solvating the SystemFor the simulations of bio-systems, such as cell membrane, the whole systemmust be immersed into water to imitate a real bio-system. The program toadd solvent water is \"genbox\". After solvating the system, topology \fles(.top) need to be edited to record the added water molecules. The numberof added water molecules can be obtained by checking the output .gro \flegenerated by genbox.2.2.2 Energy MinimizationIf initial state of the system (i.e initial velocities and positions of particles)is out of acceptable range, the force induced by the interactions will betoo large for a simulation to start, due to a large value of potential energy.Thus a step of potential energy minimization is required to make sure thesimulation can be started properly, as the system state will be close toequilibrium if its potential energy is approximately minimized.This is a typical optimization problem that searches for the global min-imum point of the potential function given by (2.3). There are numerousoptimization algorithms (conjugate gradient, L-BFGS) available for energyminimization. In GROMACS, the most common one is the steepest descentmethod [69]. It chooses a step in the direction of the negative gradient,which guarantees the step to be descending, thus if the step size is proper,the global minimum point would be reached after enough steps of simulation.To make sure all simulation steps start with a close-to-equilibrium state,energy minimization must be done whenever a system is changed (i.e after152.2. GROMACS: the MD Simulation Enginea new system \fle is created or after solvents are added to a system). Theparameters involved in energy minimization are de\fned in a speci\fc .mdp\fle, which is basically same as the .mdp \fle used in production run, exceptthat the \"integrator\" needs to be set to be \"steep\".2.2.3 Running the Production SimulationOnce the system is well-equilibrated via energy minimization, it is ready forthe production run of MD simulation. The program \"grompp\" records all theparameters in .mdp \fle to set the simulation in the desired environment, anexample production run .mdp \fle containing common parameters is shownas follows:integrator =mdnsteps =500dt =0:002nstlist =10rlist =0:9coulombtype =pmercoulomb =0:9vdw \u0000 type =cut\u0000 offrvdw =0:9tcoupl =Berendsentc\u0000 grps =protein non\u0000 proteintau\u0000 t =0:1 0:1ref \u0000 t =298 298Pcoupl =Berendsentau\u0000 p =1:0 1:0compressibility =5e\u0000 5 5e\u0000 5ref \u0000 p =1:0(2.6)This .mdp \fle initializes the simulation environment, such that equationsof motion are integrated using the so-called leap-frog integrator (\"integrator =md\"), which is the default integrator in GROMACS, in 500 time steps with162.2. GROMACS: the MD Simulation Engine0.002 ps for each step. \"nstlist = 10\" means the system is updated for ev-ery 10 steps, rlist is the cut-o\u000b distance (in nm), beyond which short-rangenon-bonded interactions for a certain particle would be ignored. Similarly,\"rcoulomb\" and \"rvdw\" set the cut-o\u000b distance for coulomb force and vander waals interactions, with units in nm. \"coulomtype\" de\fnes the summa-tion method used to calculate total energy of long range electrostatics, formore details, reader can refer to [69]. The rest of above parameters involvea concept of temperature and pressure coupling in GROMACS, which isintroduced as follows.Groups: Temperature and Pressure CouplingGenerally there are four types of ensembles for simulation of bio-systems:\u000f NVE: Constant number of particles (N), system volume (V) and energy(E)\u000f NVT: Constant number of particles (N), system volume (V) and tem-perature (T)\u000f NPT: Constant number of particles (N), system pressure (P) and tem-perature (T)\u000f NAPxyT: Constant number of particles (N), system area (A), systempressure (P) along x-y direction, and temperature (T)Since isothermal and isobaric simulations (NPT) are the most relevantto experimentl data, the temperature and pressure should be controlledproperly in simulations. However, due to a result of integration errors andheating e\u000bects from interactive forces, temperature and pressure tend to driftslightly inside the simulated system. To solve such problems, GROMACSallows user to de\fne groups to control the temperature and pressure.For example, in the .mdp \fle presented in 2.6, two particle groups \"pro-tein\" and \"non-protein\" are pre-de\fned, thus temperature-coupling andpressure-coupling algorithms can be applied to control the desired tempera-ture\/pressure. The most common coupling algorithm is the Berendsen weakcoupling [69], it corrects the deviation of temperature T and pressure P withrates according to:dTdt=T0\u0000 TfitdPd=P0\u0000 Tfip(2.7)where T0, P0are the reference temperature and pressure, fitand fipare pre-de\fned constants for correction, with units in ps.172.3. Summary2.2.4 Analysis of the Simulation ResultsThe production run will output the trajectory \fle (.xtc) to record all thesystem state in each time step, and energy \fle (.edr) that contains all the en-ergy terms at the end of simulation. With these information as a function oftime, various system properties such as di\u000busion coe\u000ecient, thickness, sur-face tension and line tension of membrane can be obtained. Visualizationtools such as pymol and VMD [31] can be used to visualize simulation result-s. Although output \fles can be processed and analyzed by any user-de\fnedprogram, GROMACS o\u000bers various programs to analyze the simulation re-sults. For example, g msd can obtain mean square displacement of particles,which is used to calculate di\u000busion coe\u000ecients. g energy can extract poten-tial energy information, thus to calculate line tension and surface tension ofa membrane system.2.3 SummaryIn this chapter, the global MD algorithm is explained using classical Newto-nian physics equations. A coarse grained MARTINI force \feld is introducedin details. The MD simulation engine GROMACS is discussed by introduc-ing its commonly used \fles and programs, along with details of con\fguration.Then a four-step scheme of MD simulation is introduced:\u000f Set up GROMACS \fles and programs with proper initializations\u000f Minimize potential energy to ensure the system start at a near-equilibriumstate\u000f The production run simulation with con\fgured .mdp \fle\u000f Analysis of the simulation results using various toolsAs shown in next chapter, although CGMD simulation applies quite sim-pli\fed physics to particles of a bio-system, by following the above four stepsin GROMACS, the obtained simulation results are in excellent agreementswith experimental measurements.18Chapter 3Study of Cholesterol inTethered Membranes3.1 IntroductionAs a major sterol component in most eukaryotic membranes, Cholesterol(C27H45OH) is an important membrane components, which can regulatemembrane properties, such as lipid di\u000busion and membrane stability [34, 49].Although signi\fcant research work has focused on the e\u000bects that choles-terol has on eukaryotic membranes, little attention has been paid to howcholesterol a\u000bects the properties of archaebacterial membranes. The aim ofthis chapter is to present a study of cholesterol on a multicomponent syn-thetic archaebacterial membrane, where CGMD simulation makes crucialcontribution to justify the experimental results.From coarse-grained molecular dynamics simulation, it can be shownthat archaebacterial membrane properties including electroporation, di\u000bu-sion dynamics, and biomechanics, are all a\u000bected by the concentration ofcholesterol present in cell membrane. The experimental data also con\frmsthe simulation results. Given the unique structure of archaebacterial lipid-s (i.e. the hydrocarbon chains are \fctionalized with methyl groups), thecholesterol content has a noticeably di\u000berent e\u000bect on the membrane prop-erties compared to that of eukaryotic and prokaryotic membranes. Theresults provide novel insights into how cholesterol a\u000bects archaebacterialmembrane properties, which are of use for the design of synthetic biomimet-ic membranes [57].Related WorksStudies of cholesterol span over the past several decades, and have involvedboth experimental measurements and molecular dynamics techniques. Fora review the reader can be referred to [34, 49] and citations therein. Dueto the microscopic scale of length and time involved in membrane research,atomistic-level simulations are necessary for predicting the dynamic proper-193.1. Introductionties of membranes, which are di\u000ecult to be measured experimentally. Sig-ni\fcant insights have been gained based on the use of molecular dynamicssimulations. In [5, 10, 17, 52] it is shown that increasing the cholesterolcontent in POPC, DOPC, DSPC, and DPPC membranes will result in anincrease in membrane stability, that is, an increasing physical resistance tomembrane defects. While increasing the cholesterol in DOPC membranesreduces the membrane line tension [7]. In DPPC membranes, increasing theconcentration of cholesterol will cause an increase in membrane thickness,and a decrease in the lateral di\u000busion of lipids [5, 25].Recently coarse-grained molecular dynamics (CGMD) has been appliedto study the atomistic e\u000bects cholesterol has on POPC, DOPC, and DPPCmembranes [15]. The CGMD results show that for cholesterol concentra-tions between 0% to 40% the membrane thickness decreases, however for50% the membrane thickness increases:possibly as a result of the interdigi-tation between lipid tails resulting from the free space under cholesterol [15].So far a substantial amount of work has focused on lipids containing PO,DO, DS, and DP lipid tails. However, an important question is, what e\u000bectsdoes cholesterol have on lipids with phytanyl tails, which are typically foundin acrhaebacterial membranes? By using experimental measurements fromblack lipid membranes, Uitert et. al. [68] show that low concentrations ofcholesterol increase the membrane stability, however above 20% the mem-brane stability begins to decrease. Several questions remain of how choles-terol a\u000bects the dynamics and biomechanics of archaebacterial membranescomposed of phytanyl lipids.Recent related works are summarized in Fig. 3.1.Figure 3.1: A brief summary of recent related works on study of cholesterolin membrane using MD\/CGMD simulation methods203.1. IntroductionSimulation Setup and Key ResultsIn this thesis a CGMD model based on the MARTINI force \feld [45, 46] isconstructed to study the di\u000busion dynamics and biomechanics of archaebac-terial membranes containing cholesterol, whose concentrations range from0% to 50%. The archaebacterial membrane is composed of 70% DphPC and30% GDPE lipids (both of which contain phytanyl tails). Furthermore, tosimulate a tethered archaebacterial membrane, two bioelectronic gold sur-face attaching membrane are included in the CGMD model. In Fig. 3.2, thebrief ball structures of Dphpc, GDPE and cholesterol are provided, we canobserve that compared with Dphpc and GDPE lipids, cholesterol is smallerin size.Figure 3.2: Ball structures of Dphpc, GDPE lpids, and CholesterolFirst, the CGMD simulation results are used to compute the membraneproperties such as: di\u000busion dynamics of lipids, membrane thickness, surfacetension, and line tension of the archaebacterial membrane. In [19], it wasshown that lipids undergo three primary regimes of di\u000busion: the ballistic,subdi\u000busion, and Fickian di\u000busion. By using the results from mode-couplingtheory and CGMD simulations, it will be shown that the transition time be-tween the subdi\u000busion and Fickian di\u000busion regimes is not dependent onthe concentration of cholesterol in the membrane. Therefore, the choles-terol only a\u000bects the di\u000busion coe\u000ecient of lipids in the Fickian di\u000busionregime. To validate the CGMD simulation results, we use experimentalmeasurements from tethered archaebacterial membranes containing di\u000ber-ent concentrations of cholesterol.Second, to ensure that the tethers and bioelectronic interface would nota\u000bect the membrane response, we must be able to compute the positiondependent density of water at the bioelectronic interface. The fact thatdensity has negligible variations at the membrane surface suggests that thebioelctronic interface does not impact the dynamics: this result is validatedusing di\u000busion measurements of the lipids in the distal and proximal layer213.2. CGMD Simulation Setup and Membrane Formationof the archaebacterial membrane. Additionally, by using approximationsto the Yvan-Born-Green integral equation [72], we will show that harshrepulsive forces play a negligible role in the long range dynamics of theposition dependent density pro\fle of water at the bioelectronic interface.Finally, in order to link the microscopic molecular dynamics simulationresults and macroscopic experimental results, a fractional order macroscopicmodel is introduced. Experimental measurements are performed by mea-suring the current response of the tethered archaebacterial membranes to agiven voltage excitation.3.2 CGMD Simulation Setup and MembraneFormationTo investigate the e\u000bects of Cholesterol on archaebacterial membranes, aCGMD model of an archaebacterial membrane containing cholesterol is con-structed. By using the MARTINI force \feld [45, 46] in simulation, thedynamics of lipid di\u000busion, membrane thickness, surface tension, and linetension will be studied. The CGMDmodel is validated by experimental mea-surements from a tethered archaebacterial lipid membrane, while a fractionalorder macroscopic model is to be introduced, thus to allow experimental re-sults to be compared with simulation results. The CGMD model, along withthe experimental measurements, provides key insights into how cholesteroland the bioelectronic interface impact the dynamics and biomechanics ofarchaebacterial membranes.3.2.1 CGMD ModelTo model the tethered archaebacterial membrane, we use the MARTINIforce \feld [45, 46], which is a popular CGMD force \feld designed for biomolec-ular systems. The main set-up of the MARTINI force-\feld is to map ap-proximately four heavy non-hydrogen atoms into one coarse-grained bead.Normally, each bead has a mass of 72 amu. For example, in the CGMDmodel four water atoms are represented by a single coarse-grained bead. Tokeep the model as simple as possible, only four main interactions are de-\fned: polar (P), non-polar (N), apolar (C) and charged (Q), each of themhas several subtypes. Q and N types have four subtypes, Qda, Qd, Qa, Q0and Nda, Nd, Na, N0, which mean they have di\u000berent hydrogen-bondingcapabilities of the atom group: da = donor or acceptor, d = donor, a =acceptor, 0 = no hydrogen bonding. On the other hand, P and C types223.2. CGMD Simulation Setup and Membrane Formationhave \fve subtypes, P1, P2, P3, P4, P5and C1, C2, C3, C4, C5, where thesubscripts 1-5 denote their increasing polar a\u000enity [45, 46].The CGMD model is constructed to imitate the essential dynamics ofthe tethered archaebacterial membrane system, which is composed of lipids,cholesterol, a gold bioelectronic interface, tethers and spacers. Given the factthat tether density of the tethered archaebacterial membrane is only aboutless than 1% (i.e. for every 268 lipid in the proximal layer one is tethered),the contribution of the tethers and spacers is regarded to be negligible, andtherefore the tethering is not included in the CGMD model. The mappingof the lipids and gold surface into the MARTINI force \feld is provided asfollows:Lipids: The molecular components include the zwittrionic C20 diphytanyl-ether-glycero-phosphatidylcholine lipid (DphPC), C20 diphytanyl-diglycerideether lipid (GDPE), cholesterol, and the gold surface. The tethered archae-bacterial membrane we consider is composed of a 30% GDPE to 70% DphPCratio of lipids. The lipid ratio is identical to that used for the experimentalmeasurements. The phosphatidylcholine headgroup of the DphPC lipid isrepresented by two beads: the positive choline by the Q0bead, and thenegative phosphate by the Qabead. The ether-glycol is represented by aSNabead, and each of the phytanyl tails by four C1beads.The phytanyland either glycerol moieties of GDPE are represented by the same mappingas for the DphPC, however the hydroxyl headgroup of GDPE is represent-ed by a P4bead. In total the DphPC lipid is composed of 12 beads, andthe GDPE lipid by 11 beads. The cholesterol is represented by 8 beads asde\fned in [15].Gold Surface: The gold surface is composed of a square lattice withcustom Pfbeads. The distance between adjacent beads is 0.3 nm. Theinteraction of the Pfbead is designed to reduce the e\u000bects of excess adsorp-tion to the surface. The interaction between Pfand P4is 1\/3 the valuebetween P4and P4, and the interaction between Pfand other bead typesis \u0018 12% of the MARTINI value between P4and respective bead types.The following interactions are excluded: interaction between Pfbeads, andbetween the C5beads of the tethers and spacers, and P4and Qobeads ofthe lipids. Note that a similar interaction is used in [43] to represent thegold surface in the MARTINI force-\feld.The complete CGMD simulation structure is provided in Fig.3.3 for ref-erence. The tethering reservoir is selected to have a height of 4 nm to matchthe experimentally measured tethering reservoir thickness from [23].233.2. CGMD Simulation Setup and Membrane FormationBioelectronic InterfaceBulk ElectrolyteTethering Reservoirhm4 nmFigure 3.3: Coarse grained molecular dynamics structure of 0% tetheredDphPC membrane with hmdenoting the membrane thickness. Lipid tailsare represented by the green beads, Qabead is displayed in blue, the Qobead in orange, P4hydroxyl headgroup of GDPE by the red bead, the SNabead as pink, and the water beads as a translucent blue. The gold surface isindicated by the gold planes. Cholesterol is represented by the black beads.The coloring scheme of the axis is red for x, green for y, and blue for z.3.2.2 Coarse-Grained Molecular Dynamics SimulationProtocolThe molecular dynamics simulations were performed using GROMACS [24]version 4.6.2 with the MARTINI force \feld [45, 46]. The interactions of theCGMD beads are de\fned by the Lennard-Jones (LJ) potential, and harmon-ic potentials (constraints=none) are utilized for bond and angle interactions.A shift function is added to the Coulombic force (coulombtype=shift) to s-moothly and continuously decay to zero from 0 nm (rcoulomb-switch) to1.2 nm (rcoulomb). The LJ interactions were treated likewise except thatthe shift function was turned on between 0.9 nm (rvdw-switch) and 1.4 nm243.2. CGMD Simulation Setup and Membrane Formation(rvdw). The grid-type neighbour searching algorithm is utilized for the sim-ulation, that is, atoms in the neighbouring grid were updated every 10 timesteps. The equations of motion are integrated using the leapfrog algorithmwith a time step of 0.02 ps. Periodic boundary conditions are implementedin xy-dimensions (Fig. 3.3). Simulations are performed in the NAPzT en-semble using a temperature of 350 K to match that used in [43] for similarmembrane structures. The temperature is held constant using a velocityrescaling algorithm [9] with a time constant of 0.5 ps. Furthermore, Berend-sen pressure coupling was applied with semi-isotropic type. The lipid andwater molecules are coupled separately for temperature and pressure con-trol. The gold surface is modeled using the walls option in GROMACS.Note that CGMD simulation times are reported as e\u000bective time, that is,four times the actual simulation time. The e\u000bective time is introduced toaccount for the speed-up in the CGMD model [45, 46].All the systems studied here were \frst energy minimized using the steep-est descent method in GROMACS. A 50 ns equilibration run is performedprior to the production run. Production runs are performed for a simulationtime of up to 1 \u0016s. Visualization of the CGMD results are reported usingPyMOL.3.2.3 Formation of Membrane in ExperimentThe tethered archaebacterial membrane is constructed using the solvent-exchange formation process presented in [28,29]. The tethered membrane issupported on a polycarbonate slide containing a 100 nm thick sputtered goldelectrodes each with dimensions 0:7\u00023 mm. The formation of the tetheredmembrane proceeds in two steps.First, benzyl disul\fde tetra-ethyleneglycol and benzyl disul\fde tetra-ethyleneglycol are \fxed to the surface of the gold electrode. Speci\fcal-ly, an ethanolic solution containing 370 \u0016M of 1% benzyl disul\fde tetra-ethyleneglycol and 99% benzyl disul\fde tetra-ethyleneglycol is exposed tothe gold surface for 30 min, then the surface is ushed with ethanol and airdried for approximately 2 min.The second stage involves the formation of the tethered archaebacterialmembrane. 8 \u0016L of 3 mM ethanolic solution containing a mixture of 70%DphPC (zwitterionic C20 diphytanylether-glycerophosphatidylcholine) and30% GDPE (C20 diphytanyl-diglyceride ether) lipids, and the rest choles-terol is brought into contact with the gold surface from the \frst step. Thissolution is incubated for 2 min at 20\u000eC in allowing the formation of thetethered archaebacterial membrane. Proceeding the 2 min incubation, 300253.3. Laterial Di\u000busion Dynamics of Lipids and Cholesterol\u0016L of phosphate bu\u000bered saline solution at a pH of 7.2 is ushed throughthe chamber.The tethered archaebacterial membrane is equilibrated for 30 min priorto performing any experimental measurements. The quality of the tetheredarchaebacterial membrane is measured continuously using an SDx tetheredmembranes tethaPodTM swept frequency impedance reader operating atfrequencies of 1000, 500, 200,100,40,20,10,5,2,1,0.5,0.1 Hz and an excitationpotential of 20 mV (SDx Tethered Membranes, Roseville, Sydney).3.3 Laterial Di\u000busion Dynamics of Lipids andCholesterolThe di\u000busion dynamics of a homogeneous medium like water can be de-scribed using standard Fickian di\u000busion, where the mean-square displace-ment (MSD) is proportional to time:MSD = h(x(t)\u0000 x0)2i = 4Dt (3.1)Here xois the initial position, x(t) is the current position at time t, D is thedi\u000busion coe\u000ecient and h\u0001i is the ensemble average, the ensemble is takenover all particles.However, as a result of the polymeric structure of lipids, the dynamic-s of lipids are more complex than simple liquids like water. Recently theso-called Mode-coupling theory (MCT), which is originally used for investi-gating dynamics of glass-forming liquids, has been applied to describe thedi\u000busion dynamics of lipids in membranes [19]. In this section, we provide amethod to model the di\u000busion dynamics of lipids, to see if the lipids are inthe ballistic, subdi\u000busion, or Fickian di\u000busion regimes, by using the resultsfrom CGMD simulationsAs predicted by MCT theory, the time evolution of the mean-squaredisplacement of lipids is given by a generalized form of Fickian di\u000busion:h(x(t)\u0000 xo)2i \/ t\f(3.2)where \f is the power-law exponent. For standard Fickian di\u000busion \f = 1,in which case the proportionality constant in Eq. 3.2 is related to the lipiddi\u000busion coe\u000ecient D by ht Einstein relation D = h(x(t)\u0000 xo)2i=4t.The MCT theory [12] predicts that at the femtosecond timescale, thelipids are in a ballistic regime where \f = 2. As time evolves, the lipids enterthe subdi\u000busive region, with \f < 1, as a result of local caging e\u000bects from263.3. Laterial Di\u000busion Dynamics of Lipids and Cholesterolneighboring aggregated lipids. Then at the nanosecond timescale, the lipidsenter the Fickian di\u000busion regime with \f = 1, allowing for the di\u000busioncoe\u000ecient to be computed as in Eq. 3.1. An estimate of the power-law coef-\fcient \f in Eq. 3.2 can be computed from the CGMD simulation trajectoriesusing the following relation:\f(t) =@ ln(h(x(t)\u0000 xo)2i)@ ln(t): (3.3)To model the di\u000busion dynamics of the lipids, we consider the dynamicsto satisfy the following linear Volterra integro-di\u000berential equation, whichdescribes the time dependence of MSD in lipid bilayers [19]:@h(x(t))2i@t+tZ0M(t\u0000 s)h(x(s))2ids = 4(kBTmL)t;M(t) =\u000e(t)fi3+Be\u0000t=fi1(1 + (t=fi2)\f); (3.4)for xo= 0.where M(t) is the memory kernel, kBis Bolztmann's constant, T istemperature, mLis the mass of the lipid, and fi1; fi2; fi3; \f and B are modelparameters. fi3is the transition time at which the MSD transitions fromthe ballistic region to subdi\u000busion region, fi2the onset of the subdi\u000busionregion, and fi1the transition from subdi\u000busion to Fickian di\u000busion region.The di\u000busion coe\u000ecient D can be computed from the results of Eq. 3.4 inthe Fickian di\u000busion region, where for t ! 1 the di\u000busion coe\u000ecient isrelated to the MSD by h(x(t))2i \u0019 4Dt, and can be evaluated using:D = (kBTmL)h1ZoM(t)dti\u00001: (3.5)Notice that Eq. 3.5 provides another method to compute the di\u000busion coef-\fcient D compared to the standard Eienstein relation given below Eq. 3.2.3.3.1 Numerical Solution of Volterra Di\u000berential-IntegralEquationAlthough to our best knowledge, there is not any e\u000ecient method to solveEqn. 3.4 analytically, it is still possible to obtain an approximate numerical273.3. Laterial Di\u000busion Dynamics of Lipids and Cholesterolsolution, where numerical di\u000berentiation and numerical integration can becalculated by using \fnite di\u000berence method:f0(t) \u0019f(t+ h)\u0000 f(t)h(3.6)where the subdivided interval of integration h is represented by:h =tN\u0000 aN(3.7)here N is the number of subdivided intervals chosen and tNis the last timestep. Similarly, tiis the ithtime step. Then by using the trapezoidal rule:Ztn0f(x) \u0019 h(12(f(x1) + 2[f(x2) + f(x3) + :::f(xn\u00001)] +12fn) (3.8)and simpson's rule:Zt2n0f(x)dx \u0019h3[f0+ 4(f1+ f3+ :::+ f2n\u00001)+2(f2+ f4+ f2n\u00002) + f2n](3.9)the integration term in volterra integral-di\u000berential equation 3.4 can beapproximated by plugging in trapezoidal rule and simpson's rule for oddand even time steps, respectively, then clearly we have the approximatedintegrals for each time step:Zt10M(t\u0000 s)h(x(s))2ids \u0019h[12M(t1\u0000 s1)h(x(s1))2i]Zt20M(t\u0000 s)h(x(s))2ids \u0019h3[M(t2\u0000 s0)h(x(s0))2ids+4M(t2\u0000 s1)h(x(s1))2ids+M(t2\u0000 s2)h(x(s2))2ids]283.3. Laterial Di\u000busion Dynamics of Lipids and CholesterolZt30M(t\u0000 s)h(x(s))2ids \u0019h[12M(t3\u0000 s0)h(x(s0))2ids+M(t3\u0000 s1)h(x(s1))2ids+M(t2\u0000 s2)h(x(s2))2ids+12M(t3\u0000 s3)h(x(s3))2ids]\u0001 \u0001 \u0001 \u0001 \u0001 \u0001ZtN\u000010M(t\u0000 s)h(x(s))2ids \u0019h[12M(tN\u00001\u0000 s0)h(x(s0))2]+M(tN\u00001\u0000 s1)h(x(s1))2]ids+ :::M(tN\u00001\u0000 sN\u00002)h(x(sN\u00002))2]ids+12M(tN\u00001\u0000 sN\u00001)h(x(sN\u00001))2]idsZtN0M(t\u0000 s)h(x(s))2ids \u0019h3[M(tN\u0000 s0)h(x(s0))2ids+4[M(tN\u0000 s1)h(x(sN))2ids+ :::M(tN\u0000 sN\u00001)h(x(sN\u00001))2ids]2[M(tN\u0000 s1)h(x(sN))2ids+ :::M(tN\u0000 sN\u00002)h(x(sN\u00002))2ids]+M(t2\u0000 s2)h(x(s2))2]ids(3.10)So if we re-write the above approximated representations in a more com-pact form, that is, if we renameM(ti\u0000sj) to beMij, and rename h(x(sj))2]ito be xj, then the numerical volterra integral-di\u000berential equation 3.4 canbe written as:x2\u0000 x02h=4kBTmLt1\u0000h2[M10x0+M11x1]x3\u0000 x12h=4kBTmLt2\u0000h3[M20x0+ 4M21x1+M22x2]\u0001 \u0001 \u0001 \u0001 \u0001 \u0001xN\u0000 xN\u000022h=4kBTmLtN\u00001\u0000h2[MN\u00001;0x0+2[MN\u00001;1x1+ :::MN\u00001;N\u00002xN\u00002] +MN\u00001;N\u00001xN\u00001]293.3. Laterial Di\u000busion Dynamics of Lipids and CholesterolxN\u0000 xN\u000012h=4kBTmLtN\u0000h3[MN0x0+ 4[MN1x1+ :::MN;N\u00001xN\u00001]+2[MN2x2+ :::MN;N\u00002xN\u00002] +MN;NxN](3.11)Therefore the Eqn 3.11 can be re-written in a matrix form AX = B, whereX = [x1; x2; :::xN]0, andA =266664h2k111 0 : : : 008h23k21+2h23k22\u0000 1 1 : : : 0: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :0 : : : AN\u00001;N\u000021 00 0 : : : AN;N\u000011377775(3.12)withAN\u00001;N\u00002= 2[h2MN\u00001;1+ :::h2MN\u00001;N\u00002] + h2MN\u00001;N\u00001andAN;N\u00001=8h23[MN1+MN3+:::MN;N\u00001]+4h23[MN2+MN4+:::MN;N\u00002]+2h23MNNFurthermore,B =266666666666648hkBTmLt1\u0000 h2M10x1+ 18hkBTmLt2\u00002h23M20x28hkBTmLt3\u0000 h2M10x3\u0001\u0001\u00018hkBTmLtN\u00001\u0000 h2MN\u00001;0xN\u000018hkBTmLtN\u00002h23MN0xN37777777777775(3.13)Now we are able to solve the matrix equation AX = B and obtain the mean-square displacements as a function of time. Thus the ballistic, subdi\u000busiveand Fickian region can be observed. However, as mentioned above, the bal-listic region and subdi\u000busive region will occur at timescale of femtosecond,or 10\u00006nanosecond. This means that if we require a 1 ms computation, thedimension of matrix A is at least 109\u0002 109. Although A is a sparse matrix,still the required computational power and time for solving this equation isconsidered to be excessive. Further work of matrix sparsi\fcation is expectedto be done regarding this part. Since in this thesis, we are only consideringthe e\u000bects of cholesterol on Fickian di\u000busion dynamics of membrane, CGMDsimulation results are used to analyze the di\u000busion dynamics of lipids.303.3. Laterial Di\u000busion Dynamics of Lipids and Cholesterol3.3.2 CGMD Simulation Results on Di\u000busion DynamicsIn this subsection, the results of CGMD simulations are utilized to gaininsights into the concentration e\u000bects, that cholesterol has on the di\u000busiondynamics of lipids in the archaebacterial membrane.The key question addressed in this section is to con\frm that, if the lipiddynamics in CGMD model indeed shows a short time ballistic regime, anextended subdi\u000busion regime, and a Fickian di\u000busion regime, as predictedby the MCT theory. Furthermore, if these regimes are present, then whatis the transition time between each regime. Knowledge of these transitiontimes is crucial, as based on Eq. 3.1, the di\u000busion coe\u000ecients can only becomputed using CGMD trajectories in the Fickian di\u000busion regime.10\u22121 100 101 102 10310\u22121100101102103Time [ns]MSD[nm2]t [ns]\u3008(x(t))2\u3009[nm2]Subdiffusion Fickian\u03b2 \u2248 0.5 \u03b2 \u2248 1Figure 3.4: Computed mean-square displacement for the DphPC, GDPE,and cholesterol for the 0% and 50% cholesterol membranes with \f de\fne inEq. 3.3. Notice that the di\u000busion dynamics are in the subdi\u000busion regime(\f \u0019 0:5) for t \u0014 3 ns, and for t \u0015 20 ns the di\u000busion dynamics are inthe Fickian regime (\f \u0019 1) . This is in agreement with the mode-couplingtheory for exible macromolecules [11, 19].Fig. 3.4 presents the numerically computed mean-square displacementof DphPC, GDPE, and cholesterol from the CGMD simulation results forcholesterol concentrations of 0% and 50%, which are generated from the313.3. Laterial Di\u000busion Dynamics of Lipids and Cholesteroltrajectory .xtc \fle of simulation. From Fig.3.4, we see that for t \u0014 3 nsthe molecules di\u000buse in the subdi\u000busion regime, and for t \u0015 20 ns thedi\u000busion of the molecules is in the Fickian di\u000busion regime. This is inagreement with the results predicted using mode-coupling theory for exiblemacromolecules [11, 19]. Given the time step of the CGMD simulation is 20fs, the ballistic region \f > 1 is not observed in Fig. 3.4 for any of the lipidsor cholesterol. This is expected as the ballistic region is typically observedfor t \u0014 10 fs [19].Surprisingly, the transition times between the subdi\u000busion and Fickiandi\u000busion regime is not dependent on the cholesterol content. This suggeststhat the concentration of cholesterol presented contributed negligibly to thecaging e\u000bect, that is, only the Fickian di\u000busion dynamics are strongly de-pendent on the concentration of cholesterol present. The results in [18, 25]for SOPC, SLPC, SAPC, SDPC, and DPPC suggest that as the cholesterolcontent increases there should be a decrease in the di\u000busion coe\u000ecient D.Table 3.1: Lipid and Cholesterol Di\u000busion (nm2\/\u0016s)DphPC GDPE Cholesterol0% 69.6\u00063.2 57.4\u00062.7 -10% 69.0\u00060.4 56.3\u00062.4 106.5\u000640.920% 49.3\u00061.3 44.4\u00062.6 92.5\u000633.730% 50.0\u00063.8 48.5\u00061.2 86.8\u000629.140% 54.5\u000610 67.5\u00067.3 55.1\u00068.650% 39.5\u00061.6 31.3\u00061.8 46.7\u00068.2To gain insight into how the concentration of cholesterol a\u000bects the d-i\u000busion dynamics in the Fickian regime, we compute the di\u000busion coe\u000e-cient D for DphPC, GDPE, and cholesterol for archaebacterial membranescontaining 0% to 50% cholesterol. The results are provided in Table 3.1.The di\u000busion coe\u000ecient of DphPC for 0% cholesterol is in excellent agree-ment with the experimentally measured di\u000busion coe\u000ecient of 18:1 \u0006 5:6nm2\/\u0016s [4]. Furthermore, the numerically computed di\u000busion coe\u000ecientof cholesterol are in excellent agreement with the experimentally measureddi\u000busion coe\u000ecient of cholesterol which are in the range of 10 nm2\/\u0016s to100 nm2\/\u0016s [56, 63]. The cholesterol has a higher di\u000busion coe\u000ecient thanDphPC and GPDE, and the di\u000busion of cholesterol monotonically decreasesas the concentration of cholesterol increases. If we consider only the massand size of cholesterol, it is expected that the lower mass and size of choles-323.4. Water Density Pro\fle at Bioelectronic Interfaceterol compared to DphPC and GDPE will allow cholesterol to have a largerdi\u000busion coe\u000ecient. Another contributing factor is that the headgroup ofDphPC and GDPE both have a larger dipole moment than that of choles-terol which will also reduce the di\u000busion coe\u000ecient of the lipids comparedto the cholesterol [25].An interesting question is, why a \"slow\" lipid system could be evenslowed down by faster-moving cholesterol, which has higher di\u000busion co-e\u000ecients than GDPE and Dphpc molecules? Such phenomenon can bewell explained by the free volume theory [67], which is used to predict aliquid-ordered-liquid-disordered (lo\u0000 ld) coexistence region in phosphatidyl-choline\/cholesterol mixture [55]. If the environment temperature is abovethe phase transition temperature, the membrane lipids are in the so-calleddisordered phase (ld), where the acyl chains of the phospholipid moleculesare in a disordered state that contains a high fraction of gauche conformer-s [2]. When cholesterol are added into the membrane, cholesterol \frst spanover the hydrocarbon cores of the bilayer, if concentration of cholesterol in-creases, free-volum models [54, 55] predict that excessive cholesterol wouldpack tightly into the lipids, thus the cavities and deects from the membraneare \flled by cholesterols, and more regions of membrane are converted in-to the ordered state (lo), which has much less free volume, and therefore,smaller di\u000busion coe\u000ecients.3.4 Water Density Pro\fle at BioelectronicInterfaceBefore conducting actual experimental measurements for the tethered mem-brane, a crucial problem is to \fgure out the region where the properties oftethered membrane are not a\u000bected by bioelectronic interface. Thus it be-comes necessary that water density pro\fle nearby the gold surface should beinvestigated: the membrane should be set in the region where water densitypro\fle is not a\u000bected much by the bioelectronic interface.As the dynamics of water in proximity to the bioelectronic interface havevery di\u000berent characteristics from that in the bulk region [3, 21, 76], the ther-modynamic and structural properties of water molecules at the interface area\u000bected by two primary factors: a smaller number of neighboring moleculeinteractions, and a change in the potential energy of the uid as a result ofinteractions with the surface. The density pro\fle of water near the interfacetypically consists of oscillations that are similar to sinusoidal waves , withperiod close to the mean thickness of each water layer in proximity to the333.4. Water Density Pro\fle at Bioelectronic Interfaceinterface [21, 76]. This observation suggests that the oscillations occur at asimilar length scale to the molecular diameter of the water molecules.In our simulation the interface is modeled using a lattice of coarse-grainedbeads which interact with the water beads via a Lennard-Jones potential.Then a key question we would like to answer is: does the density pro\flefrom the CGMD model match that from a hard-sphere uid at a hard-wall interface? In this section we will show that, approximations to theYvan-Born-Green integral equation [72], which lead to an analytical resultrepresented by the so-called \"Percus-Yevick Equation\", will be able to e-valuate the density pro\fle of a hard-sphere uid at a hard-wall interface.This density pro\fle can then be compared with the density pro\fle from theCGMD simulation, thus to evaluate the e\u000bects of harsh repulsive forces thatCGMD water beads have on the water density.Now let's provide a derivation of the Percus-Yevick equation \frst.3.4.1 Derivation of Percus-Yevick EquationFirst we consider a monoatomic uid with density \u001a, which is interactingwith pair potential u, and an external potential ffi that represents the con-tribution of interactions from the interface. Then the mean force acting onparticle 1 by a particle 2 can be represented by the gradient of potential\u0000r1u(r1; r2). The total mean force on particle 1 by the other particles andinterface is given by the Yvan-Born-Green integral equation [72]:kBTr1ln(\u001a(r1)) =\u0000r1ffi(r1)\u0000Zr1u(r1; r2)\u001a(r2jr1)dr2(3.14)where kBis the Boltzmann's constant, T is the temperature, and \u001a(r2jr1)the conditional singlet density (i.e. the conditional density at r2given aparticle is \fxed at r1).In (3.14) the so-called correlation function is represented as function ofthe external \feld ffi. Note that the right hand side of (3.14) is the averageforce acting on a particle \fxed at r1. To estimate \u001a(r), approximations to(3.14) are to be made to only account for long-range interactions. The mainidea is to construct an external reference potential ffiR, which only accountsfor long range interactions of the particles and the interface, and pair inter-action that only accounts for short range interactions, which is denoted asthe reduced system. The only requirement is that the density pro\fle of thefull system and reduced system must be equal. Using this method, Weekset. al. [72] propose the approximation that the singlet density for both the343.4. Water Density Pro\fle at Bioelectronic Interfacefull system and reduced system must be similar at short range. The externalpotential ffi and reference potential ffiRare then related by [21, 72]:ffiR(r1) = ffi(r1) +Z[\u001aR(r2)\u0000 \u001aB]u1(r1; r2)dr2(3.15)with \u001aBthe bulk density, u1(r1; r2) the attractive part of the pair potential,and \u001aRthe density in the external reference potential. Note that for agiven \u001a(r), which can be computed from the CGMD simulation, an e\u000bectivereference potential ffiRcan be evaluated by solving (3.15) self-consistently.To solve for \u001a(r) analytically we must determine an expression for ffiRin(3.15). It is reasonable (we will see why very soon) to approximate ffiRby ffiR0where we have neglected the harsh repulsive forces at the interfaceleaving only the attractive forces in the full system. Then, for a hard-wall,ffi =1 for r \u0014 1 and 0 otherwise, the density \u001a(r) is given by [71]:\u001a(r1) = \u001aB+ \u001aBZ[\u001a(r1)\u0000 \u001aB]c(r12)dr2;\u001a(r1) = 0 for r1\u0014 1 (3.16)with c(\u0001) the direct correlation function of the uid, r12= r1\u0000r2. For a hard-sphere, the direct correlation function is given by a cubic polynomial whichis dependent on the radius of the hard-sphere, denoted by R, and the bulkwater density \u001aB[73]. For a hard-sphere and hard-wall interface Eq. 3.16 isgiven by the Percus-Yevick equation which can be solved analytically usingthe method in [70], which is shown as follows.The close form of Percus-Yevick equation is derived directly from Ornstein-Zernike equation, which simply states that total inuence h of particle i onparticle j is a superposition of both direct and indirect inuence:h(rij) = c(rij) + \u001aBZdrkc(rik)[g(rkj)\u0000 1] = g(r)\u0000 1 (3.17)where g(r) is the \"scaled\" density of particles, or the pair distribution func-tion; the direct correlation function c(rij) describes the direct inuence ofparticle i on particle j; and second term of the equation, which is the indi-rect correlation function, suggests that the indirect inuence of particle i onparticle j is an integrated result of particle i acting on a reference particlek, which in turn has inuence on j.By assuming the case of hard sphere potential:u(r) =(1 if r < b0 if r \u0015 b:(3.18)353.4. Water Density Pro\fle at Bioelectronic Interfacewhere b is the radius of particle, after de\fning function y(r) to be:h(r)\u0000 c(r) = y(r)\u0000 1;withy(r) =(\u0000c(r) if r < bg(r) if r \u0015 b:(3.19)it's straightforward to conclude thatc(r) =(\u0000y(r) if r < b0 if r \u0015 b:(3.20)Then by setting particle j to be the original point, rename ri= r, rj= r0,we can express Ornstein-Zernike equation in terms of y(r) [65]:y(r) = 1 + \u001aBZr0 1+2\u0003 f31 break ;32 end3334 alpha=f \/denom ;35 beta=1\u0000lamd\u0003 f+3\u0003alpha ;36 gamma=3\u0000lamd\u0003denom ;37 % Open the f i l e to wr i t e c ( r )38 fpk=fopen ( ' pys f . dat ' , 'w+' ) ;3940 pk=0;41 pp1=alpha\u0003(4\u0000 lamd+3\u0003alpha )+1;42 pp2=0;43 pyhk=(1\/(pp1^2+pp2^2)\u00001)\/n0 ;44 pys f=1+pyhk\u0003n0 ;45 f p r i n t f ( fpk , '%6u %14.9 f nn ' , pk , pys f ) ;4647 % ca l c u l a t e h( r ) in terms o f c ( r )48 f o r i k =1:nk49 pk=ik \u0003dk ;50 x=pk\u0003 dia \/2 ;51 snx=s i n (x ) ;52 csx=cos (x ) ;53 ps ix=snx\/x ;54 phix=3\u0003(snx\u0000x\u0003 csx ) \/(x^3) ;55 pp1=alpha \u0003( beta \u0003phix+gamma\u0003 ps ix )+csx ;56 pp2=alpha \u0003x\u0003phix+snx ;57 pyhk=(1\/(pp1^2+pp2^2)\u00001)\/n0 ;58 pys f=1+pyhk\u0003n0 ;59 f p r i n t f ( fpk , '%6u %14.9 f nn ' , pk , pys f ) ;60 hk ( ik+1)=pk\u0003pyhk ;61Matlab Code61 end62 f c l o s e ( fpk ) ;6364 % f o u r i e r trans form65 hw=\u00002\u0003 f f t ( hk ) ;66 hr=imag (hw( 2 : nk+1) ) ;6768 f p r=fopen ( ' pypdf . dat ' , 'w+' ) ;69 % reve r s e f o u r i e r trans form and wr i t e h( r )70 f o r i r =1:nk71 r=i r \u0003dr\/ dia ;72 i f r >= 173 g=1+hr ( i r ) \/(rm\u00034\u0003 pi \u0003 r \u0003 dia ^2) ;74 f p r i n t f ( fpr , ' %14.9 f %14.9 f nn ' , r , g ) ;75 end76 end77 f c l o s e ( f p r ) ;78 % plo t pa i r d i s t r i b u t i o n func t i on79 load pypdf . dat \u0000a s c i i ;80 load pys f . dat \u0000a s c i i ;81 % sc a l e the d i s t r i b u t i o n p l o t82 f o r i =[1 : l ength ( pypdf ( : , 2 ) ) ]83 i f \u0000pypdf ( i , 1 ) +41.55>39 && \u0000pypdf ( i , 1 ) +41.55<4184 pypdf ( i , 2 )=pypdf ( i , 2 ) \/1 .5+0 .31 ;85 end86 end87 % Plot the a n a l y t i c a l s o l u t i o n o f Percus\u0000Yevickequat ion88 f i g u r e (1 ) ;89 x = 19.7\u0003 ones (1 ,2300) ;90 y = 1 : 2300 ;91 p lo t (x , y , ' l i n ew id th ' , 2 ) ;92 hold on ;93 x an a l y t i c a l=(\u0000pypdf ( 8 : 8 00 , 1 ) +41.55) \u00030 . 4755 ;94 y an a l y t i c a l=(pypdf ( 8 : 8 00 , 2 ) \u00000.45) \u00031800 ;95 p lo t ( x ana l y t i c a l , y ana l y t i c a l , ' r ' , ' l i n ew id th ' , 2 ) ;96 ax i s ( [ 1 2 , 2 0 , 0 . 0 , 2 7 0 0 ] ) ;9798 x l ab e l ( ' r (nm) ' ) ;99 y l ab e l ( ' n rho ( r ) ' ) ;62Matlab Code100 hold on ;101 % Import the CGMD s imula t i on r e s u l t s102 A=importdata ( ' dens i ty . xvg ' ) ;103104 x CGMD=A( : , 1 ) ;105 y CGMD=A( : , 2 ) ;106 p lo t (A( : , 1 ) ,A( : , 2 ) , '\u0000\u0000 ' , ' l i n ew id th ' , 2 ) ;107108 l egend ( ' Ana ly t i c a l ' , 'CGMD' ) ;109 f i l l P a g e ( gcf , ' pape r s i z e ' , [ 5 3 ] , ' margins ' , [ 0 0 0 0 ] ) ;110 x l ab e l ( ' p o s i t i o n z [nm] ' ) ;111 y l ab e l ( ' water dens i ty ' ) ;112 pr in t \u0000deps epsFig63","@language":"en"}],"Genre":[{"@value":"Thesis\/Dissertation","@language":"en"}],"GraduationDate":[{"@value":"2015-11","@language":"en"}],"IsShownAt":[{"@value":"10.14288\/1.0165816","@language":"en"}],"Language":[{"@value":"eng","@language":"en"}],"Program":[{"@value":"Electrical and Computer Engineering","@language":"en"}],"Provider":[{"@value":"Vancouver : University of British Columbia Library","@language":"en"}],"Publisher":[{"@value":"University of British Columbia","@language":"en"}],"Rights":[{"@value":"Attribution-NonCommercial-NoDerivs 2.5 Canada","@language":"*"}],"RightsURI":[{"@value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/2.5\/ca\/","@language":"*"}],"ScholarlyLevel":[{"@value":"Graduate","@language":"en"}],"Title":[{"@value":"Study of cholesterol in tethered membrane using coarse grained molecular dynamics simulations","@language":"en"}],"Type":[{"@value":"Text","@language":"en"}],"URI":[{"@value":"http:\/\/hdl.handle.net\/2429\/54885","@language":"en"}],"SortDate":[{"@value":"2015-12-31 AD","@language":"en"}],"@id":"doi:10.14288\/1.0165816"}