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Enzyme design for the steady-state of metabolism Ballantyne, James Stuart 1981

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ENZYME  DESIGN  FOB THE  STEADY-STATS Of HETABOLISH  BY JAMES  SXOABT  EALLA8TXNE  B . S c  University  o f  Guelph,  Gueljh,  Canada,1973  M.Sc.  University  of  Guelph,  Guelph,  Canada,1976  A  THESIS THE  SUBMISTID  IN PABTIAI  BEQUIBEHENTS CGCTOB  FOIfIL1HE8T CP  P C I T B I BEGB1E OF  OF PHILOSOPHY ID  THE  We  FACULTY  accept  CP GBADOAIE  (Department  of  the thesis  as  required  The  University  of  STUDIES  Zoology)  conforming  standard  B r i t i s h  Columbia  J a n u a r y 1981 (c)  t o  James S t u a r t B a l l a n t y n e 1981  the  In  presenting  requirements of  B r i t i s h  i t  freely  agree for  this for  an  available  that  I  understood  that  financial  by h i s  or  her or  shall  Date  DE-6  (2/79)  the  Library  copying  publication  the  not  be  allowed  Columbia  of  the  shall  and study.  by  of  University  I of  make  further this  head  representatives.  of  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook Place Vancouver, Canada V 6 T 1W5  at  granted  permission.  Department  fulfilment  the  extensive  may be  copying  gain  that  reference  for  purposes  or  degree  agree  for  permission  scholarly  i n p a r t i a l  advanced  Columbia,  department  for  thesis  It  this  without  thesis  of  my  i s thesis my  written  ii  The  steady-state  of  metabolism  equilibrium  thermodynamic  been  to  related  Simplified  free  reaction  have  variety  of  the  their  change  reverse  d i r e c t i o n  energy  substrate It  i s  of  be  suggested  __G£.  _ f G £ _ a y  The  importance  metabolism.  continuous  ^Gp,i s  two  range  of  the  i s  the  Km f o r  adjusted. of the  the  The  the  these the  of ^ G ^ .  to  a  to  in  to  where  i s  G g  v e l o c i t i e s . conditions  shared  a  that  two  constraints of  to  that  equal  the  The  i n  Gibbs of  event.  fumarase-  a f f i n i t i e s  have  to  achieve  the  imposed  on  a to t h i s  on The  same  concentration  steady-state  equal  imposed  of are  selected  substrate. be  of  steady-state  given  are  the  determination  order  must  of  c a t a l y t i c  numerically In  free  binding  be  enzyme  the  Gibbs  must  substrate  i s  magnitude  the  different systems  sharing  of  the  B i o l o g i c a l  of  interpretation  extent  i n  reverse a  the  the  l i e  the  under  products. of  have  metabolism.  fumarase  compared not  giving  ^  of  i n  considerable  suggested  enzymes  second  When  enzymes  size  two  found  The  enzymes  include  enzyme  important  to  important  design  the  forward  steady-states,  a  that  independently  perturbed  metabolic  difference  been  is  steady-state  steady-state  enzymes.  this  of  investigated-  thermodynamic  reactants  possible.  over  the  The  has  l&Qfg)  of  for  activation  that  Many  theoretically  that  using  been  steady-state  p r o f i l e s  perturbed  be  the  constructed  change  may  a f f i n i t y  i n  r e l a t i o n s h i p .  energy  free  role  conditions  Haldane  properties  energy  been  has  the f i r s t  i n must  both be  the  design  a f f i n i t y  equals  iii  T A B L E OF CONTENTS  ABSTRACT  i  L I S T OF T A B L E S  i v  L I S T OF F I G U R E S  v i  ACKNOWLEDGEMENTS  vli  CHAPTER 1. GENERAL INTRODUCTION  1  C H A P T E R 2. E Q U I L I B R I U M THERMODYNAMICS OF ENZYMES  6  CHAPTER  3. THE R E L A T I O N S H I P BETWEEN B I N D I N G AND C A T A L Y S I S  INTRODUCTION  11  M A T E R I A L S AND METHODS  12  Enzyme a s s a y s Kinetic  12  and thermodynamic  calculations  12  R E S U L T S AND D I S C U S S I O N  14  Perturbation  o f *>G  Relationship  b e t w e e n ^G  Free  energy  and G  14  £  and  G  14  profiles  C H A P T E R 4. P e r t u r b a t i o n  15 of ^ G R  INTRODUCTION  21  M A T E R I A L S AND METHODS  23  R E S U L T S AND D I S C U S S I O N  23  CHAPTER  5. ENZYME D E S I G N AND M E T A B O L I S M  INTRODUCTION  30  M A T E R I A L S AND METHODS  30  R E S U L T S AND D I S C U S S I O N  31  Three  dimensional  representation  o f an enzyme  catalyzed  reaction  31  The  32  general.steady-state  iv  The  role  of a f f i n i t y  The  steady-state  Constancy The  of metabolism  of a f f i n i t y  stability  33  during  35  steady-state  of the s t e a d y - s t a t e  transitions  of metabolism  39 41  CONCLUSIONS  42  SUMMARY  43  REFERENCES  CITED  ABBREVIATIONS APPENDIX  1  USED  '....52 55 56  V  L I S T OF  Table  2-1 H a l d a n e  of Table  several 3-1  for Table  Table  3-2  Gibbs various  4-1  4-2  product Table for  enzyme  4-3  reaction  interpretation  mechanisms  The e f f e c t s o f p e r t u r b i n g  10  conditions  on  and^G  reaction  free  17  energy changes  f o r the fumarase  reaction  conditions  Comparison  catalyzed Table  r e l a t i o n s h i p s and thermodynamic  the fumarase  under  TABLES  18  o f ^Gjj.and/G°  for several  enzyme  reactions  Independent  22 modulation  i n the fumarase Independent  the fumarase  of binding  of s u b s t r a t e  and  reaction  perturbation  reaction  26 o f ^Qf  with  respect  to^Gy 27  vi  L I S T OF  Figure for  3-la the  Example  of  fumarase  Figure  Km  and  3-2  Figure  4-la  fumarase  Figure  4-2  •of  AG3.  Figure  5-1  uni-uni  by  Effect under  Hoff  The  role  of  5-5 the  energy  profile  f o r the  reaction 20  on  In  (Vmaxf/Vmaxr)  for  conditions  plot  of  28  fumarase  reaction  under  enthalpy  and  entropy  in  perturbations 29  Enzyme  reaction surface  for a  simple  mechanisms  ^ Gj^ v e r s u s  Figure  19  28  of  5-3  the  plot  conditions  Figure  on  Cornish-Bowden/Eisenthal  temperature  various  The  5-4  19  modified  free  of  5-2  Figure  concentration  fumarase  Figure  several  substrate  Vmax d e t e r m i n a t i o n  4- l b v a n ' t various  of  Simplified  catalyzed  versus  reaction  3- l b E x a m p l e for  velocity  FIGURES  45  general  steady-state affinity  46  f o r enzymes and  metabolites  from  tissues The  47  effect  of  changing  Km  f o r the  shared  substrate  steady-state The  effect  continuous  Figure  5-6  The  Figure  5-7  Three  demonstrate  -  48 of  enzyme c o n c e n t r a t i o n  on  the  achievement  steady-state  intersection dimensional  stability  of  49  E l and  plot  of  E2 the  50 steady-state  to 51  vii  ACKNOWLEDGEMENTS  I  should  like  t o thank  Dr. Peter  to  pursue  these  investigations  in  which  they  could  stimulating Dr.  John  discussions Gosline,  Shoubridge Gosline  and  and  contributing helped  with  especially  be  Dr.  Lionel to some  like  John  Mary  Suarez.  Harrison  the final  of the assays t o thank  Mary  Dr.  Tom  many  Chamberlin,  from  Dr.Chris  many French,  Mommsen, Dave  Eric  Randall,  useful  of the thesis.  i n Chapter  freedom  the.environment  benefited  Professors  made  form  I  Chamberlin,  Hanrahan,  Raul  f o rthe  and f o rcreating  pursued. with  Hochachka  John  suggestions  Dr. Chris  French  3. F i n a l l y ,  I  primus  pares.  inter  should  1  CHAPTER  GENERAL The in  steady-state  of  studies the  case  of  the  concentration  this  definition  enzyme  Why  present  the  lower  invoked  (Atkinson,1977). example  may  other  of  availability to explain The  of the laws  simply  be  a  i n a pathway.  i s to describe  metabolism  produce  the  steady-state  and what  i s  stemming  from  achieved. in a  enzymes, enzymes?  Mass  inhibition metabolic  crucial action  have  presents  a c t i o n upon w h i c h much o f  feature  As s u c h ,  of  the  metabolism.  further definition.  the  Is  our  steady-  to postulate  The p u r p o s e  The  an  non-regulatory  n o t be n e c e s s a r y  o f enzyme d e s i g n  been  regulation  metabolism  rest.  are  pathway  are  some o f t h e p r o p e r t i e s o f t h e  needs  How  regulatory  the steady-state.  of  wherein of  parameters  of of  of physics  features  In  rates  steady-state.  aspects  I f s o i t may  steady-state itself  question  and feedback  design  itself.  non-regulatory  kinetic  of l e a s t  few  their  the  than  but  state  s e q u e n t i a l enzymes  steady-state  effectors to explain  thesis  two  of the metabolic  of the p r i n c i p l e  understanding  enzymes  of  i t i s n o t known what  cofactor  variously  basic  homeostasis  activity  achievement  effects,  state  this  One  and  a r e some e n z y m e s , n o t a b l y  in  Similarly,  of t h e s t e a d y - s t a t e i s that  forms  examples of  (Roberts,1977)  metabolites  constant.  i s how  a r e many  the steady-state  of  concentrations  related?  to  properties  are  There  transitions  metabolism  interconversion  i s one o f t h e s i m p l e s t  be d i s c u s s e d .  steady-state  of t h e b a s i c  INTRODUCTION  of metabolism  w h i c h m e t a b o l i s m may  studies  1  of  this  steady-state  are required to nature i t  of the  stable  or  2  unstable? occurs  When  transition  i s i t along  Chemical affect are  a  most  the  subject  physical  another  the  work  on  and  To of  date a  I  between  force  It  relationship and  the  It other  i s widely  of  is a  to  the  affect  the  major  reaction  of  He  the  that by  organism class  has  of  over  the  a  of  the  the  enzymes  has  previous  effects  of  While  i t is  the  force  for  on  the  role  of  of  the  enzyme  reaction  living the  the  and  its  rate  of  reaction  range  of  more t h a n  to  investigate reaction  mus,t be  the  van  force  function.  chemical  it.This  and  investigated  the a  also  affinity  Donder  of  on  and  enzyme  to 2 the  in  vivo  done w i t h i n  the  steady-state. behave  known c a t a l y s t s  A l l enzymes of  (exoenzymes or of  . Most  work  pressure  the  this  has  shown  of  concept that  De  design  therefore,  catalyze  assumed A l l  an  affinity  the  the  affinity  equilibrium.  large  on  no  which  systems are  that  driving  of  of  and  b i o l o g i c a l systems  (1973)  rate  the  shown  another  forces  called  ratio/Keq).  do  important  between  catalysts.  outside there  is  imposed  approach  been  concentrated  aware  the  Chemical  enzyme d e s i g n  affinity.  enzymes  constraints  the  or  has  same  temperature  function  Rottenberg  linear function  kcal/mol.  state  am  chemical  relationship  a  has  to  path?  the  these,  action  on  pressure  i t .  be  a  pressure  catalyzing  driving  of  which provides  systems. affinity  which  thermodynamics  and  temperature affinity  by  (1936) have a d e q u a t e l y  enzyme  temperature  Of  steady-state  stable  investigated.  ln(mass  attributes  one  and  affected  systems.  force  = RT  Rysselberghe  are  frequently  (affinity=A3  all  predictable  systems  most to  a  from  function  course,  others  enzymes w h i c h are  essentially  i_n  do  to  speed  this  when  vitro  designed  to  as  )  but  function  3  inside  of  approach  the  cell  ; they  to e q u i l i b r i u m but  maintain  the  compounds  constant.  capable work  of  on  state  metabolic  are  termed  subject  to  'signals'  enzymes  activation  or  a much more  non-regulatory proposed by  later  is critical  regulatory this  detail  but  because  the  problem,  of  functional  properties  development Implicit  is  a  flux  and  enzyme t h e  net  difference  between  the  flux  flux  produced  of  At  through the  by  flux  the  this  and  this  study  to  pace  of  any of  than  steady-state,  Atkinson,  1977)  is  availability, As  role  I shall  show  for regulatory  explain  how  non-  an  enzymes  For  initial on  requisite  the for  steady-state.  s t e a d y - s t a t e .is t h a t i t  point by  considered  regulation  given  i n a pathway  produced  are  concentrate  given  the  metabolic  r e g u l a t o r y ones.  mainly  point  reverse  being  former  represents  non-regulatory  that  the  factors.  5),  I will  every  which  substrate  such  idea  categories  etc.)  any  the  a r i s e n the  various  1973;  other  understanding  situation.  by  Start,  instead  has  general  insufficient  maintenance  i n an  this  particular  other  each  and  exploration  the  emphasized  in metabolic  (Chapter  is  enzymes pace  reason,  enzymes  useful  previous  between  c o n c e n t r a t i o n , and  greater  enzymes  balance  and  s t e a d y - s t a t e s , most  two  (Newsholme and  flux  clearly  inhibition  In  the  are  From  role  of  speed  metabolically  non-regulatory;  important  simply  rate  cations, metabolites  enzymes.  a  a l l  has  into  and  do  in  fall  regulatory  play  modulator  different  to  the  metabolic  regulation  (hormones,  maintained  of  (Atkinson,1977).  to  is  at  not  to c o n t r o l  Although  function  that  designed  concentrations  metabolic  transient  it  are  is the  reaction.  c a t a l y z e d by  described forward  This  by  reaction  simplest  an the and  concept  4  has  been  design.  almost  To u n d e r s t a n d  therefore enzymes be  i t . Both  account  of  and  features general  model  the study  of  o f enzyme  metabolism  i t i s  the f u n c t i o n a l design  forward  and r e v e r s e  of the  reactions  must  related. have  systems.  of metabolic context,  in  steady-state  relationships  simulations  ignored  t o understand  govern  into  These  this  the  necessary  which  taken  design  universally  been  analyzed  Using  this  enzymes  then,  have  using  approach been  computer  some o f t h e  determined.  I s e t o u t t o answer  In  the following  questions. 1) What of  relationships exist  between  sequential non-regulatory  allowing 2)  enzymes  function at different  What  relationships  sequential  between  enzymes  allowing  function at different  3)  are the concentrations  How  allow  function at different  4) D o e s  the a f f i n i t y  in  a  properties  metabolic  pathway  steady-states?  exist  non-regulatory  t h e thermodynamic  the kinetic  in  a  p r o p e r t i e s of  metabolic  pathway  steady-states? o f s e q u e n t i a l enzymes  related  to  steady-states?  f o r any g i v e n  r e a c t i o n change  at  different  steady-states? 5)  Is  the  concentration 6) Why vary in  steady-state of pathway  d o t h e maximum  by up t o f o u r  the steady-state I  found  approaching place,  I used  intermediates  activities  orders they  remarkably them  perturbed  with  simple the  fumarase as a  fluctuations  or i s i t a stable  o f enzymes  of magnitude a l l must  by  i n any g i v e n  (Guppy  answers t o a l l these use of model  'model'  state? pathway  e t a_l.,1979)  f u n c t i o n a t t h e same  systems.  non-regulatory  in  when  rates?  questions In the  by  first  enzyme.  A  5  thermodynamic since  i t  approach  provides  processes  i n terms  study  found  I  was  a  employed  way  of  for this  comparing  o f t h e same u n i t s  that  i n the forward  direction  direction  (AG^)  necessarily  observation value  of  had metabolic ^Gfj_ with  metabolic  equal  To e x p l o r e , I used  regulatory imposed  that  second  vivo  part  how  this  a computer  enzymes  wherein the  i s that  appropriately  Km  affinity  found of  i s equal  that  the concentrations adjusted  with  respect  reverse this  by c o m p a r i n g  the  that  twelve  —G^  i s  may  be a c h i e v e d  f o r two s e q u e n t i a l constraints  to  allow  must two  t o each  in non-  must  be  steady-state  to a f f i n i t y .  substrate of the  of  f o r the reaction.  numerically  f o r the shared  energy  That  showed  two  enzymes  of  the  to^G*.  of s t e a d y - s t a t e scheme  free  of  part  f o r a s e r i e s of  affinity  modeling  and  equal  of the study  sort  to  I established  t o t h e in v i v o  on s u c h a s e q u e n c e  function is  impact  i_n  enzymes. T h i s  numerically  vivo  not  compared  types  In t h i s  i n the Gibbs  activation  of the t h e s i s  different  (calories).  the d i f f e r e n c e  i s  part  The  first  be t h e same; t h e enzymes  other.  must  be  6  CHAPTER 2  E Q U I L I B R I U M THERMODYNAMICS OF ENZYMES  INTRODUCTION For  the simplest  fully  reversible  enzyme c a t a l y z e d  reaction  i s: k, A + E  k  net  flux  z  (2-1)  k-3.  = E B k , K a = k |/k_| a n d K b = k s / k ^ .  The  of A t o B i s given by:  Equation presence  absent  the  rate  r e a c t a n t s and product.  equation  At equilibrium  equation  following  Vmaxf([A]/Ka)-Vmaxr([B]/Kb) 1+([A]/Ka) + ([Bj/Kb)  (2-2) d e s c r i b e s  of both  equation. of  E + B  k._  = E A k , Vmaxr  v=  totally  3  EA T=* EB k.,  Where Vmaxf  k  r  (2-2)  equation  i s  (2-2) reduces the flux set  reaction  i n the  When o n e o r t h e o t h e r i s to  i s zero.  equal  of  (2-2)  to  the  Michaelis-Menten  I f the left  zero  hand  side  and rearranged t h e  i s obtained: [B]/[A]  = Keq = VmaxfKb  (2-3)  VmaxrKa Equation for types has  this  (2-3) i s t h e Haldane  type  of reaction.  relationship  Haldane  o f enzyme c a t a l y z e d r e a c t i o n s . previously only  been used  (Haldane,  relationships The  to verify  exist  Haldane the v a l i d i t y  1930) for a l l  relationship of  kinetic  7  determinations Keq  of  al.,  1973).  be  i t sconstituent constants  or to determine  reactions i n which i t i s otherwise  Since may  of  equation  simply  (2-3) r e p r e s e n t s  expressed  RT  I n Keq  In  ( P u r i c h e_t  the e q u i l i b r i u m  i n thermodynamic  = RT  difficult  the  state,  i t  terms:  (VmaxfKb)  (2-4)  (VmaxrKa) If RT  equation  In Keq  (2-4) i s expanded  = RT  I n Vmaxf  + RT  and  I n Kb  the terms  -RT  separated  I n Vmaxr  -RT  In  Ka) (2-5)  rearranging RT  (2-5) g i v e s ,  In Keq  Fersht  = RT  In  (1977)  (Vmaxf/Ka)  described  - RT  In  (Vmaxf/Kb)  the r e l a t i o n s h i p  between  (2-6) the  free  b energy in  of b i n d i n g  terms of  kcat  (*G  * ) and  a n d Km.  the f r e e energy of a c t i v a t i o n  (^G  Since,  Vmax = k c a t / ( e n z y m e c o n c e n t r a t i o n ) RT  In  The either by  left  part  the minus  also  be  adequately  described In  Substituting  of  hand  worked  however,  -RT  side  In  (kT/h) -  (^Gf-^G )  equation  (2-8) c l o s e l y  In Keq  a  s i d e of e q u a t i o n  are  with  irreversible  describes  those  reversible  (2-8)  (2-9)  (kcatr/Kb) (2-6),  = RT  In  and  = -RT  (2-8) and (kT/h) -  proteases  enzymes.  Most  the reverse  by a F e r s h t i a n e q u a t i o n  resembles separated  with  opposite  r  (2-9)  ( AGf+  this  type  metabolic  reaction  TT b I n ( k T / h ) + (^G +^Gy)  *  RT  (2-7)  sign. Fersht  equation  enzymes  hand  = RT  of the r i g h t  Since of  (kcatf/Ka)  )  sign: (2-9)  i n (2-6) g i v e s :  b *G ) a  - RT  In  (kT/h)  (2-10)  may  8  *  + Simplifying  b  ( ^G +  <*G_  r  )  (2-10) RT I n K e q -*G .+*G^-( ^ G f + ^ G * )  (2-11)  v  The  terms  complex Eyring 11)  kT/h r e p r e s e n t s and  i s  the  and Stover,  since  reaction.  the rate of  decay  same f o r a l l c h e m i c a l  1974).  This  i ti s identical  term  of  activated  reactions  disappears  but of opposite  the  in  (Johnson,  equation  sign f o rthe reverse  Since, -RT I n K e q = ^ G ° equation  (2-12)  ( 2 - 1 1 ) may b e r e a r r a n g e d *  b  *  and w r i t t e n ,  •  k  4 Q f + ^Gj,= ^ G + ^Gfc+ ^ G  (2-13)  r  Equation  (2-13) d e s c r i b e s t h e f r e e  enzyme c a t a l y z e d r e a c t i o n The  Gibbs  free  product  are negative  in  the forward  both In  derived  a  from  describing  the Table  these  bi-bi,  energy  the  way  2-1  i s  that  free  the  of C l e l a n d  important  energy of  a n d one i n t h e r e v e r s e  on).  In the  case be  of  bi-reactions  random  or  activation  profiles more  in  the  equations  order .  The  tothe  forward  i n the forward  the  may b e complex  (1970) and r e f e r s  i sa bi-uni  ordered  of  The t e r m i n o l o g y u n i -  a u n i - u n i mechanism;  forward  s u b s t r a t e and  thermodynamic  reactants  ( i e . One r e a c t a n t  i s designated  f o r an  and p o s i t i v e .  reactions at equilibrium.  etc.  may  energies  relationships  the  reactants  binding of both  Gibbs  presents  profile  i n F i g u r e 3-1.  . The f r e e  the  Haldane  directions  reverse  of  energy  and reverse d i r e c t i o n s  number o f k i n e t i c a l l y reverse  as d e p i c t e d  as i s ^ G  similar  reactions.  uni,  (2-  and  and one i n  two r e a c t a n t s  in  mechanism and so of  binding  of  Theorrell-Chance  9  mechanism  i s similar  concentration substrates termed  equal  complex  any p r o d u c t s  bound  constants  the  1/2  in  before  The s t r e n g t h  i s  complex.  not  binding,  the thus  coefficients The approach  the  the  same  average  t h e sum o f  relationship  t o understanding  the  enzymes.  A study  essential  t o an u n d e r s t a n d i n g from  understanding , metabolism.  of  The  assumed  inhibitory in  energies  of  the that  to i t s  "proper"  i n the unproductive  ternary  the  of the proper free  a l l  particular  binding energies  ligand  and the of  i s  inhibitory  binding  with  o f 1/2.  Haldane  displaced  of  are  results  of the s u b s t r a t e  binding  before  ping-pong.  The o c c u r e n c e  as b i n d i n g  the  i s termed  free  the  low. I f a l l  released  mechanisms  that  the reaction i s  and products  binding  of binding  The e f f e c t i v e  therefore  is  mechanism  these  except  are released  for a l l substrates  complexes  type  i s vanishingly  a n d i f some p r o d u c t  have  coefficients ligand.  bi-bi  the dissociation constants.  ternary  site  before  sequential  Michaelis to  of the t e r n a r y  bind  substrates  to the ordered  equilibrium  of the p r o p e r t i e s  equilibrium of  therefore  their  function  a  useful  c h a r a c t e r s i t i c s of  o f enzymes a t e q u i l i b r i u m i s  of t h e i r and  provides  will in  function  in  ultimately the  situations lead  steady-state  t o an of  10  Table  2-1 H a l d a n e r e l a t i o n s h i p s a n d t h e r m o d y n a m i c i n t e r p r e t a t i o n s o f s e v e r a l enzyme r e a c t i o n  Mechanism  Haldane  Theorell-Chance  *  VIKpKipKqKiq Keq=  Non-sequential  V2KaKiaKbKib  Ping-pong  Thermodynamic  interpretation  AG°=^JG.f + l / 2 4 G + 1/2&G ^ + , £ 1/2 G^ +1/2 G ^ - ( A G + %  a  r  l/24Gp  +l/2AGj  Random b i - b i  + 1/2AG<^ +  1/24G!J, )  * Ordered  mechanisms  uni-bi  VIKpKipKqKiq Keq= V2KaKia  AG°=AG.f  VIKpKip Keq= V2KaKiaKbKib  il -  i b ^ (AGJT + l / 2 A G p + l / 2 A G j > + 1/2AG* + 1 / M G ^ )  •* Bi-uni  b + 1 / 2 A G * +l/2&Gg.  b  it  dG°= A G f + 1 / 2 A G + l/2AGft. + {, ,'{> 1/2AG^•+l/2AGfc A  (AG,. + l / 2 A G p +1/2AG*^ ) * see  l i s t of abbreviations of p a r a m e t e r s .  a t end of t h e s i s  for descriptions  11  CHAPTER  THE  3  R E L A T I O N S H I P BETWEEN B I N D I N G AND  CATALYSIS  Introduction An  important  lower  the  Gibbs  that  enzymes  c h a r a c t e r i s t i c of  free (and  of  other  catalysts)  p o s i t i o n of  accelerate  the  enzymes  bind  it  seem l i k e l y  would  substrate  and  several  termed  kinetics)  have  reactions.  (1975)  envisaged  maximize  reaction  relationship proteases reaction  of  considered. that  maximizing  theoretical  the  rates.  one  for  both  the Since  of  binding  of  kcat/Km  Km,  i n the  two  (the  binding  of  was  is  shown t h a t  Km  to  on  the over  Crowley  i n order a  catalysis  to  the  to  direct by  reverse  direction  were  1975)  concluded  have  a d v a n t a g e o u s and,  and  and  and  turn  i s high  consequences  necessarily  forward  (1977) and  based  forward  enzyme  Michaelis  implying  (Cornish-Bowden,  i s not  the  the  function  d i r e c t i o n ; the  have  as  affinities  displaying  conclusion,  and  rates  between  q u i c k l y , Fersht  ratio  kcat  kcat/Km  shown.  differing  true  enzymes  was  perturbation  enzymes  Their  workers  grounds,  with  that extent  this  stated  the  same  respectively  clarified  substrate  maximizing Other  kcat  enzymes whose  that  alter  to  occur.  the  been  between  in only  in which  (1930)  stated  the  products  and  for  For  q u a n t i t i e s of  also  relationships  Km  not  cannot  to  independent  could  the  constants  large  examples  that  product  catalytic  reverse  reaction  and  is its ability  a c t i v a t i o n . Haldane  r e a c t i o n . He  substrates  Surprisingly, properties,  a  reverse  giving  catalyst  energy  equilibrium  forward  a  kcat  can  be  not  on  treated  12  as  independent  variables.  However,  the  possible  b e t w e e n Km a n d k c a t w e r e n o t i n v e s t i g a t e d As  a  first  relationships enzyme,  approach  reversible  experimentally.  t o improving our understanding of the  b e t w e e n Km a n d k c a t ,  fumarase  relationships  a relatively  ( E . C . 4 . 2 . 1 . 2 ) was s e l e c t e d  simple metabolic  which  catalyzes the  hydration of fumarate. forward fumarate  + H20  ^=---___'  malate  reverse A variety kcat,  or  of conditions  both,  in  were used  order  to perturb either  Km  to explore the relationship  or  between  them.  M A T E R I A L S AND  Enzyme a s s a y s . P u r i f i e d obtained assayed  from  2.44  Sigma  heart  fumarase  Chemical  Co.,  i n a 50 mM p o t a s s i u m  otherwise malate  the  pig  stated.  direction  The  using  phosphate  change the  in  coefficient  recording constant  (+0.1  0  C) w i t h  a t pH  with  unless  a t 2 4 0 nm  with  was  Mo. I t w a s 7.3  extinction  direction  Cuvette  a Lauda  St.Louis,  absorbance  0.26 w a s f o l l o w e d  spectrophotometer.  (E.C.  buffer  millimolar  a n d a t 2 8 0 nm i n t h e f u m a r a t e  extinction  METHODS  i n the  coefficient  the millimolar  a Unicam  temperature  SP 1 8 0 0  was  constant temperature  kept  bath and  c irculator. Kinetic  and  determined C using the  thermodynamic in triplicate modified  calculations at five  .  Km  temperatures  Cornish-Bowden/Eisenthal  and from  Vmax  were  20° C t o 40°  plot  (Cornish-  13  Bowden 3-1.  and  Eisenthal,  1978).  An e x a m p l e  S u b s t r a t e c o n c e n t r a t i o n s ranged  two  or three times  the  relationship  pKa  of  activity  and  reaction  then  was  directions.  ln(}(VVf)  fumarate  coefficients  relationship  from  one t e n t h  t h e Km. ^ G ° was d e t e r m i n e d = -{RT  malate  i s presented  Vm  are  and  Vf  simplifies  determined  The G i b b s  + RT I n similar  o f t h e Km t o t h e Keq u s i n g  Keq}.  Since  (Sober,  were assumed  the  1970) t h e  t o be e q u a l .  The  t o ^ G ° = - RT I n K e q . K e q f o r  in  free  from  in Figure  both  energy  the  forward  of binding  and  the  reverse  was d e t e r m i n e d  from  b the Km  Km  values using  i s expressed Fumarase  t h e r e l a t i o n s h i p ^G  i n molar obeys  Michaelis-Menten  below  and  dialysis  indicate  that  t h e Km  kcat  was d e t e r m i n e d  The  Gibbs  method  free  five  i s equal from  Vmax  energy  using  the least As  where the  = 4.576 kcat  enzyme  sec. AS  from  kcat/(enzyme  and H i l l ,  1968)  constant.  The  concentration).  was d e t e r m i n e d  using the  of l o g  kcat  versus  1/T  analysis. - logT  = Vmax/mg  substituted  + E a / -4.756 T )  enzyme x m o l e c u l a r w e i g h t o f  x 10  mmol/ m o l .  into  the following  obtain AG . AG  (Teipel  1971)  = E a - RT  f o r fumarase)  were  substrate  where:  ( l o g k c a t - 10.753  (194,000  and'AH  using  at  and T e i p e l ,  to the dissociation  regression  ( i n 1/sec.)  (Hill  studies  Arrhenius plots  squares  kinetics  t h e Km  of a c t i v a t i o n  o f Low et. aJL. ( 1 9 7 3 )  was d e t e r m i n e d  times  binding  AH  Ea  . The  units.  concentrations equilibrium  = - RT I n (1/Km)  = A H - TdS  X  lmin./60  equation to  14  R E S U L T S AND  Perturbation  of  catalyzed  by  fumarase  possible  t o a l t e r ^G  ^G^is  fixed  not  independently  AG  by  and  and  four  Increasing G  2.  No  3.  I n c r e a s i n g ^G  i n *> G  By  salt,  necessity  with  then  I n c r e a s i n g ^G  The above  five  plus  Since  no  the  no  four  i t  in  conceptually  For  fumarase, 3-1).  reverse  respectively; forward The  with  the  possible:  /G  simply  change  that  the  opposites  under  a l l nine  second  salt  and  and  the  r e l a t i o n s h i p between  to  and  third  reaction during  are  any  combinations  ( T a b l e 3 - 1 ) can site  contains  substrate agrees the  best two  binding  with  other  fumarase  be  these  but  a l l  four  only  of  temperature  the  AG^  changes  studies  active site  second  the  transitions  r e l a t i o n s h i p s hold  .  The  fact  in  the  in  by  ^G  and  assuming  with  the  that  a l l  # ^G  were  that  first  catalysis.  (Hill  and  to  specialized into  be  in  transitions.  and  explained  patterns  r e l a t i o n s h i p s occur  fourth  same  _-G^  obtain  f u n c t i o n a l domains,  and  the  occurs.  of  possible  b possible  of  i n e i t h e r parameter  observed  possible  first  reaction  change  important.  i t was  The  can  ^G  are  been  assumed  are  (Table  increasing  has  is  two  and  ^G  change  w h e r e no  hysteresis  conditions,  the  is  i n c r e a s i n g *• G  possiblities  case  i t  A with  other  reaction  temperature  r e l a t i o n s h i p s are  ir 4.  or  the  r e l a t i o n s h i p b e t w e e n ^G  decreasing  with  perturbing  pH  . I f the  primary  with  a  .  ^G  chemical  1.  change  AG  with and  DISCUSSION  T e i p e l , 1971)  the  obtained active  concerned  with  This  concept  which  consider  binding  regions  15  (interacting catalytic fumarate of  with  the carboxyl  regions  (involved  or malate  changes  groups in  of  hydration  r e s p e c t i v e l y ) . In such  i n *> G i n d e p e n d e n t  the  of changes  substrate)  or  and  dehydration  event,  of  the p o s s i b i l i t y  i n *>G  can  easily  be  envi saged. Although  t h e model  relationships  observed  widely  applicable  enzymes  i s the spatial  catalysis important without  for  a  m^G  the  and v i c e  of  versa,  and  flexibility  in  design  it  may  strong  environments,  facilitate binding  catalytic Free the  the  fumarase,  o f many,  of  since  of binding  always that  lead  would and  should  enzymes  adjustments  design  of  interactions  and  may b e a n for  to similar not  allow  activation.  allow  greater  suited to function i n may b e made  necessarily a f f e c t i n g the other. the  i f not a l l  o f ' enzymes  binding  . catalysis of  site  i t i s  metabolic  a r e important  in  In addition  enzymes without  either  in  which  sacrificing  efficiency. energy  profiles  reaction  constructed  using  *G  the  f u m +*G  reverse  by  fumarase  energy (Figure  profiles 3-2)  can  for be  equation:  b  *  f u m =^G  b Because  Simplified free  catalyzed  *  the  should  a situation  modulation  binding'  without  ^G  the specific  Such a s e p a r a t i o n  c o n s t r a i n t on t h e a c t i v e  independent  parameter  characteristic  e_t a _ l . , 1 9 7 6 ) .  Independent  different  for  and f u n c t i o n a l s e p a r a t i o n  i t ,perturbation  changes  i n understanding  b e t w e e n ^ G a n d *>G  since  (Holbrook design  i s useful  b m a l + ^G  mal + ^G°  (3-1)  t and ^G  reactions,  were d e t e r m i n e d  f o r both  the forward  i t i s important  to point  out that  and  t h e ^G°  16  calculated value ^G.  from  determined This  validity itself  o f ^G  a n d ^G  (Table  3-2).  under  several  points  negative  activation positive.  a  of  estimates An  The  the  the  the  products  (^G°),  a  reaction.  Chapter  4 will  is  achieved.  catalyzed most where  the The  function freedom  data  The  the  The  of  the  with  to  3-2)  reverse free  reveals proceeds  energy  of  direction  is  energies  of  reverse  directions  reactants  to  binding  of  ligands in  the  mechanism of  ^G^  escaped in  by  which  and^G  general  0  steady-state.Other  of  notice  thermodynamic  b i n d i n g and  terms,  catalysis  is a  enzymes which  must  features required i n the  since  *  reversible  dealt with  this  i n enzyme  obvious. of  for  of  expressed  modulation  profiles  free  the  profile  substrates  the  of  drop  the  t o have  independent  be  of  the and  enrgy  (Table  Gibbs  energy  non-equivalence  will  free  free energy  binding  compared  i s most  design  the  Gibbs  p r e r e q u i s i t e for design  of  of  the  relationship  in a  check  in  deal  not  independent  and  result  are  either  forward  free  r e a c t i o n s appears  kinetic  crucial  This  and  with  of  conditions  forward  equal  an  energy.  difference  of  (<^Gg) d o e s n o t  of  free  both  independently  a n a l y s i s of  interest.  Gibbs  i s i n good agreement  supplies  variety  in  activation  (3-1)  experimentally,  equivalence  fumarase  with  equation  next  for  complete  chapter.  17  Table  3-1. The e f f e c t s for  o f p e r t u r b i n g c o n d i t i o n s on M G  the fumarase  transitions vs.  high  were  Two  w i t h and without  temperature  obtained  parameters  from  obtained  were  kinds  i n the i n t i a l  are given  a r e s u b t r a c t e d means  conditions  andA&G^  s t a t e and under  calculations  Transition  and low  the d i f f e r e n c e i n thermodynamic  and  error  step-wise  u t i l i z e d . A A G^  p e r t u r b i n g c o n d i t i o n s . A l l assay  standard  of  0.25 M s a l t ,  the  Values  malate,  reaction.  andMG  conditions  i n the text. f o r which  t h e maximal  i s + 2.7%.  G (cal/mol)  4A G (cal/mol)  pH 7 . 3 , 30°C 945  -645  0-0.25 M K C l  malate,  pH  8.0 220  20-40°C  fumarate,  pH 7 . 3 , 30°C 210  -10  270  220  0-0.25 M K C l  fumarate, 20-40°C  pH  8.0  18  Table  3-2. G i b b s under  free energy various  calculations  of b i n d i n g and  conditions.  are given  in  Assay  f o r AG° were o b t a i n e d  Values  given  using  a r e means + s t a n d a r d  AGf  AG  activation  conditions  the text.  values  £T  Conditions  changes  The  and  calculated  equation  (3-1)  error. AG'  AG  1  AG,  ' cal/mol  cal/mol  cal/mol  calculated observed cal/mol cal/mol cal/mol cal/mol  -4010 + 42  13100 + 76  -3235 + 151  13425 + 95  -1100 + 81  -930 + 10  -325 + 110  30°C  -3980 + 36  13245 + 51  -3375 + 72  13510 + 74  -865 + 55  -790 ± 16  -265 + 46  40°C  -3915 i 35  13420 + 51  -3275 + 57  13560 + 61  -780 + 52  -680 + 20  -140 + 34  pH 7.3,+ 0.25 M K C l -4025 20°C + 78  13420 + 132  -4150 + 44  14545 + 76  -1000 + 85  -975 + 20  -1125 + 159  30°C  -3985 + 54  13455 + 93  -4020 + 91  14455 + 101  -965 + 99  -835 + 23  -1000 + 139  40°C  -3565 + 212  13465 + 115  -3980 + 39  14410 + 122  -525 + 330  -725 + 21  -945 + 219  •4135 + 61  13405 + 56  -3050 + 22  13165 + 23  845 + 29  -875 + 18  240 + 34  30°C  -4075 + 53  13600 + 14  -2835 + 90  13150 + 38  •795 + 77  -785 + 24  450 + 24  40°C  •3865 + 73  13625 + 115  -2775 + 74  13170 + 111  •640 + 23  -700 + 25  455 + 54  pH 7 . 3 , 20°C  pH  8.0, 20°C  19  Figure  3-1  Example a) for  and Km  of  velocity  modified and  versus  substrate  concentration  Cornish-Bowden/Eisenthal  Vmax d e t e r m i n a t i o n s  with  plots  fumarase.  b)  300  20  Figure  3-2.  Simplified catalyzed KP04.  free by  Symbols  energy p r o f i l e  fumarase  f o r the  a t 4 0 ° C , pH  are explained  i n the  7.3,  reaction and  text.  50  mM  ibbs Free Energy (kcal mol' ) 1  21  CHAPTER  INDEPENDENT  4  MODULATION OF FORWARD AND R E V E R S E  BINDING  Introduction The worthy  observation  of f u r t h e r  phenomenon implies in  made  consideration.  of metabolic  that  such  i n the last  chapter  Firstly,  enzymes as T a b l e  a design  feature  i t  that ^G^ f i s  a  widespread  4-1 d e m o n s t r a t e s .  must have i m p o r t a n t  This  functions  metabolism. The t h e o r e m  that: kl/k-l-Keq  has but and for  been w i d e l y also  1950) even  an enzyme c a t a l y z e d and  respectively. catalysis (1957) more  assumed t o h o l d  f o r enzyme c a t a l y z e d  Meyerhof,  forward  reverse  suggested  general  reactions  later that  the  expressed  case  not only  this  reaction  Gadsby e t a l .  and  (4-1)  reactions.  used  Early  reactions  workers  (Oesper  method t o c a l c u l a t e t h e Keq  using  (1946)  equation  f o rchemical  Vmax  as the rate  Manes, H o f e r  values  constants  dealing and Weller  with  f o r the k l a n d k-1  heterogenous  (1950) and H o r i u t i  (4-1) i s a s p e c i a l  case  of  the  equation (kl/k-1)  In  ^G° i s  = Keq  o f enzyme c a t a l y z e d  (4-2) reactions  the factor  z may b e  as ln(Kb)(k-1) Z = (Ka)(kl)  +1  (4-3)  22  Table  4-1.  of A G ^ *  Comparison  catalyzed  and A c  for several  reactions.  Enzyme  Ref .  AG  (cal/mol)  phosphoglucomutase pyruvate  kinase  phosphoglycerate lactate  *  AGJJ  was  1.  3.  -1746  1  -2410  2  -5263  3  -613  4  -4767  4  -850  5  -4955  5  at  In  the  published  Vmaxf/Vmaxr  (Vmaxf/Vmaxr),  J.V.Passonneau.1969.  the  of  phosphates  and  reversibility  of  assessment  i t s capacity  skeletal  250:3316-3321.  McQuate,J.T.  and  of  the  M.F.Utter.  pyruvate  enzyme.  mannose,  ribose,  R.J.Barsotti.1975.  muscle  J.Biol.Chem.  studies  Vmaxr  with, and  244:910-916.  Dyson,R.D.,J.M.Cardenas,  of  of  and  the  Phosphoglucomutase  fructose, glucose,  J.Biol.Chem.  using  where Vmaxf  same c o n c e n t r a t i o n s  Lowry,O.H. and  galactose. 2.  = RT  AGJJ  determined  from  (cal/mol)  1  kinase  determined  Ref .  0  -618  dehydrogenase  equation are  enzyme  to  pyruvate  support  1959.  kinase  kinase  The and  as  glyconeogenesis.  E q u i l i b r i u m and  kinetic  r e a c t i o n . J.Biol.Chem.234:2151-  2157. 4.  Krietsch,W.K.G. from  and  rabbit skeletal  T.Bucher. muscle  1970.  and  yeast.  580 5.  Ballantyne,J.S.  unpublished  3-phosphoglycerate  data.  Eur.J.Biochem.  kinase 17:568-  23  When e q u a t i o n expressed the  (4-2)  is  substituted  i n thermodynamic  Haldane  utilization  terms  relationship  into  the  equation  thermodynamic  results.  Equation  of  methods worked  out  et  a l . (1946) and  Horiuti  (4-1)  expression  (4-2)  for chemical  and  permits  systems  of the  i n enzyme  systems. Gadsby possibility  of  catalysis  factor  not  equal  Z was  been done  in this  enzyme c a t a l y z e d  The  use  This  Little  chapter  hinted at  situations  further  work  where  seems t o  p r o b e s AGfc , u s i n g a  . the the have  simple  reaction.  same m e t h o d o l o g y  additional  i n heterogenous  t o one.  area.  (1957) both  of  M A T E R I A L S AND  METHODS  was  as  employed-  in  Chapter  v a r i o u s c o n c e n t r a t i o n s of  KC1  3  with  i n the  the  reaction  medium.  R E S U L T S AND  Table  4-2  fumarase binding  shows t h a t wherein  of  adaptational effect  the  i t is possible  i t  is  s u b s t r a t e and  "artificial"  expediency  possible product.  conditions  same  DISCUSSION  to  may  to  select  to  independently  Presumably  perturb alter  the  the  conditions  i f  we  ratio  amino  acid  modulate  can of  for  Ks  select to  Kp,  sequence  to  response.  b  with  Since,  as  we  respect  to  ^G^  have  seen  in  any  i n Chapter way  and  3, ^G  may  since  the  be  perturbed  binding  of  24  substrate  and  product  w h e t h e r ^G^  ask  Table  4-3  shows  selecting The last the  the  the  characteristic conditions.  a  given  and  at  the  from  van't  Hoff  absolute  versus  enthalpic  By  with  we  may  respect  next  to  ^G . r  p o s s i b l e to achieve  nor  and  the  he  the was  reaction.  a  This  by  the of  event  versus  under  plots.  of In  the the  result. of  the  does  suggests  of  Figure  the  is a  reaction  plots.  Although  d i f f e r e n c e s of  the  r e a c t i o n s would as  and  determined  catalysis  demonstrates  slope on  the  AG . 0  one this  the  inverse  of  In  Keq  ordinate i s  determined  4-2  in  that  the  is  to  i t i s assumed  4-1  position  e n t r o p i c d i f f e r e n c e s b e t w e e n AG^  of  enthalpy  biological  the  Figure  set  the  reaction  equal  entropic effects.  inverse  given  reverse  not  fl  of  (enthalpy)  In(Vmaxf/Vmaxr) versus  t h a t ^G  substrate  heat  a  these  enthalpy  case  of  graphical technique  from  overall  nature  the  rate limiting  and  the  to determine  studied this  forward  of  understanding  change  similar  Arrhenius  change  result  on  a l s o p o s s i b l e to c a l c u l a t e  the  slope  our  able  part  ratios  In Keq  reaction  used  later  present  to  r e a c t i o n s the  in  i n the  equilibrium  plotting  Additionally,  This  of  the  activation  temperature  1/T.  changed.  and  The  Hoff  fundamental  was  a different  difference. of  Hoff  enthalpy  expect  perturbed  way  is  r e q u i r e d f o r the  the  event  the  on  a chemical  for chemical  equal  may  of  of  van't  chemical  energy  van't  activation  this  temperature  direction.lt  least  i n any  beginning  Arrhenius  entropy  neither  of  now  catalysis.  a given  determine  the  are  absolute of  that  temperature  products  change  again  and  of  biochemical  independently  perturbed  investigations  effect  of  be  be  appropriate conditions.  century  and  can  may  by  both  shows t h e e n t h a l p i c Overall  there  is a  25  trend  towards  compensating  corresponding  change  an  of  TflS .  enthalpy  change  As  4-3  Table  with  has  a  already dp  demonstrated respect value, later  ^G  to  . This  r  to  independently  means  in effect  positive,negative small this  directly the  i t is possible  is  an  affects  important the  enzyme c a t a l y z e s The  next  modulation required  of  nature in  chapter binding  for biological  steady-state transitions.  and  or  the  and  feature  metabolic  demonstrate catalysis  catalysis  also provide  will  are  and  how the  necessary  for s t a b i l i t y  ^ G.f  may  R  with  assume  become of  steady-state  i t s specific will  that ^G  l a r g e . As  design of  perturb  apparent  enzymes  i n the  any  and  reaction  pathway. the  independent  freedom to  during  of  permit  design the  steady-state  26  Table  4-2  Independent product Values  of binding  i n the fumarase are subtracted  standard  Transition  modulation  error  means  f o r which  7.3-pH  pH 7 . 3 , 0.25M  8.0 KC1,30-40°C  3 0 ° C , pH 7 . 3 , 0-0.25M pH 8 . 0 ,  20-40°C  KC1  the maximal  i s 5.9%.  (cal/mol)  20°C,pH  and  reaction.  AAGjf.  conditions  of substrate  b m (cal/mol)  •125  185  420  40  -5  -645  270  275  Table  4-3  Independent for  t h e fumarase  Values  error  conditions  means  f o r which  2 0 ° C , pH 7 . 3 - p H pH 8 . 0 ,  20-40°C  8.0  the maximal  — A G^  20-40°C  pH 7 . 3 , 4 0 ° C , 0 - 0 . 2 5 M  to AG  i s 0.85%.  (cal/mol)  pH 7 . 3 ,  respect  reaction.  are subtracted  standard  Transition  perturbation of A w i t h  KC1  (cal/mol)  320  135  45  850  305  •260  220  5  28  Figure  4-la Effect  of  fumarase 4-lb  t e m p e r a t u r e on under  van't Hoff plot various  various of  conditions  In(Vmaxf/Vmaxr)  for  conditions.  fumarase  reaction  under  slope - 1690  3.403  3.45  K3 0 0 03  29  Figure  4-2  The r o l e  of enthalpy  o f ZiGj^.A  plot  entropy  change  AH =AH -AH° e  and entropy  of enthalpy (As  a n d AS  change  inperturbations (AH  ) under v a r i o u s =AS -AS°. 0  ) versus  conditions.  30  CHAPTER  ENZYME D E S I G N  AND  5  METABOLISM  Introduct ion The  steady-state  of a metabolic  p a t h w a y must  encompass  the  following: 1) . F l u x  of metabolic  catalyzed any  reaction)  other  except  point  in  compounds  a t any p o i n t  i n t h e pathway must the  same  pathway  ( i . e . a t any  enzyme  be t h e same a s t h a t a t  (branch  points  provide  ions).  2) .  The  concentrations  any  given  velocity.  transitions Ideally  but they  strict  of a l l i n t e r m e d i a t e s  Changes  may  must a g a i n  control  occur  be c o n s t a n t  should  be  must  not change a t  during  steady-state  a t t h e new  maintained  velocity.  during  the  transition. In first  order  necessary  steady-states examined system  the and  sequential state  to understand to look  the steady-state  a t a model system  can e x i s t .  Once  biological any  design  enzymes  in  features  t o see what  general  constraints  can  (those  a pathway must  within the b i o l o g i c a l  be  have t o permit  AND  f l u x e s f o r t h e two u n i - u n i  of  has been  imposed  features  i ti s  range  steady-state  c o n s t r a i n t s ) c a n be  MATERIALS The  this  of metabolism  upon t h e  which a  two  steady-  determined.  METHODS mechanisms c a t a l y z i n g :  31  E2  El A v were  c a l c u l a t e d using  for  E l a n d E2  maximal  for Kc  Vbl([A]/Kb2)~Vc([B]/Kb2) 1+([A]/Kb2) + ([B]/Kc)  for  and  while  the shared for  specific  the  visualization  Centre  representation concentration substrate  5-1  enzyme.  On  Y-axis Z  for  parameters were  used  to  reactions  substrates  were  used  velocity  A  and  and C  constants  except  tested.  produce  a  A  files  of the  UBC  dimensional  varying  concentration  where surface  three  5-la,b  substrate  the  respectively. Arbitrary  being  equations  at constant  over  product a  range  concentrations.  representat ion  shows  the  the X a x i s  represents  (above  reverse  to  r e s p e c t i v e l y o f E 2 . Ka  B f o r E l a n d E2  from  dimensional  Figure  and  refer  r e f e r to the M i c h a e l i s  R E S U L T S AND  Three  Vbl  a v a i l a b l e i n the public  was  (5-lb)  r e f e r to the maximal  constants  kinetic  routine  and  forward  reactions  K b l a n d Kb2  substrate  Va  a n d Vc  reverse  relationships  Computer  where the  refer to the Michaelis  values  of  v=  respectively  the forward  equations:  Va([A 3/Ka)-Vbl([B]/Kbl) 1+([A]/Ka) + (IB]/Kbl)  f o r E l a n d Vb2  respectively, for  the f o l l o w i n g  C  v=  velocity  respectively  *  t h e X-Y  DISCUSSION  o f an enzyme c a t a l y z e d  surfaces  produced  the concentration  the concentration plane' represents  of A  o f B. T h e  by a  react ion  representative  i s represented. region  positive flux  The  of p o s i t i v e  i n the  direction  32  B production.  of  Regions  in  the  direction  the  i n t e r s e c t i o n of  surface The  equation  the  of  to  the  law  The  General  of  For  the  this  X-Y  if state  line  will  is for  parameters  or  of  be  El  and  surface  E2  One  (net  reaction  flux  the  is zero). (Keq)  reaction  in a path  for would  described  by  the  the  values^ of  a  large  number of  B  of  values  v  for  B which  demonstrates  of  each  v  pair  increases  Changing  the  and  (the  shared  kinetic  permit  steady-  such a  'series to  the  other  the  parameters  of  where  this  steady-state.  such- r e l a t i o n s h i p s e a c h d e p e n d i n g  assigned  t o E l and  biochemical  c o n s t a i n t s are  E2.  However,  systems cannot placed  on  the  the  the  lines,  valid  steady-  refers  kinetic  of  the  for E l equals  5-2  equally  are  constant  velocity  another  defines  Important  B as C.  held  such p a i r  lines  section  are  another  of  parameters  >  concentration  line  B and  generates  next  C  calculated. Figure  pair  designed.  system  arbitrary  between  E2  A and  Using  relationship  the  the  and  between A and  in  isolated  the  constant  calculations.  kinetic  enzyme  by  mass.  of  whenever  E2.  of  may  and  the  produced  equilibrium constant  El E2 A ^ = = = = =* B ^=====* C  relationship  El  and  flux  sequence:  exist  substrate)  states  straight line  (Z=0)  i s the  represent  Steady-State  concentrations  velocity  The  Z-axis  equilibrium situation  conservation  the  the  negative  plane  e q u i l i b r i u m along  --->  of  the  r e a c t i o n . In a c l o s e d  procede  the  A production.  of  represents  of  as be  new  There on  the  outlined so  biological  loosely systems  33  that  severely  actually  observed  apparently The  of  the  the  may  relate  be  of  effectively a  to  control  enzymes w i t h  an  opposite  enzymes  p a t h w a y s and  like  to  reverse  so-called they  affinities  With of  the  the  the by  by-pass  This  difference  other  catalyze  reactions  are  i n low  with  activity  enzymes t h a t  will  with  steady-states  (as  measured  catalyze  far by  reactions  from Vmax) close  are  higher  in order from  enzymes  to  in  which  equilibrium the  to the  i s evident  and  in  reactions  far  q u a l i t a t i v e observation  steady-states  invested  require  those  regulatory  These  usually  The  e q u i l i b r i u m constant  the  other  reactions  given  between  that  reaction  are  control  a  largest  with  pass  equilibrium  value.  of  pathway.  to  as  most  enzymes,  close  same f l u x  which To  with  equilibrium values).  for a  of  replaced  a  pathway  flux  must  be  the  the  to  activities  regulatory  the  invested  on  steady-  1955).  (close  steady-states  oft cited  therefore  (Atkinson,1977).  small  found  two  direction  some  flux,  steps  typically  the  are  imposed  of  pathway,  "regulatory"  affinity  enzyme  affinity  These  the  enzymes,  equilibrium  of  the  permitted  (Prigogine,  enzymes a r e  the  regulatory  the  and  constitute  regulated  maintain  i n the  negative  pathway.  direction  of  c o n t r o l . The  affinity  number of  the  enzyme  that  consider  constraints  number  affinities  the  determine  with  shall  determines  obvious the  others  i n a l l the  closely  I  solutions  below.  a pathway  most  affinity  a l l the  small  of  limiting  affinities  latter  pathways.  constraints  tha  p o s i t i v e and  pathway  steady-state  affinity  pathway,  sum  of  in metabolic  perhaps  metabolic  is  kind  overall affinity  flux,  states  the  most c r i t i c a l  r o l e of  Since  limit  and non-  equilibrium  34  and  a r e found  most  easily  surface is  i n greater  understood  (eg. Figure  farther  from  where  the flux  case  when m e a s u r i n g observed than  the  even  varying enzyme enzyme  An  the  steady-states biological  is  why  for turns vivo  if ^Gj^  will (with  which  (discussed on  enzyme  out i d e n t i f i e s  observed  case futher  Vmax  values  function at  t h e Vmaxf  and and  the  specific  f o r one next  of p o s s i b l e  below). for  (of  f o r the  the range  important  be  organization  Thus  one  steady-state  affinities.  i_n v i v o . T h i s  one o f t h e most  will  (Newsholme  the determinants  reaction  Therefore,  of r e g u l a t o r y  of t h i s  conditions  to explore  flux  equilibrium  t o t h e Vmaxf  function at relatively  given  similar  levels  i s always  respect  the  i t  have a lower  the pathway) i s that  in  response  the f a r from e q u i l i b r i u m  b y t h e Vmax m e t h o d  consequence  large with  i t i s important  of  That  affinity  exploration,i t  c o n s t r a i n t s on i n  c o n d i t i o n s f o r the steady-state.  metabolic  out i n the previous enzymes w h i c h  not a l l other refers  activation may  along  constraint  As p o i n t e d of  indeed  This i s  enzyme  t o i t s Vmax v a l u e .  enzyme  important  narrows  an  in v i v o a t s t e a d y - s t a t e b o t h  pathway,  enzyme  any  than  d i f f e r e n c e i n Vmax  enzymes  to  to the e q u i l i b r i u m value  activities  though  be v e r y  in  requires  Close  i s much c l o s e r  affinities may  reference  i t s Vmax v a l u e  This  non-regulatory 1979).  ( a s m e a s u r e d by V m a x ) .  r e g u l a t o r y enzymes  same r a t e .  Start,  5-1)  non-regulatory  constant), the  with  enzyme  that  activity  expect  to  kinds the  difference  a  to distinguish  of c a t a l y s t s  of the forward i t  seems  chapters,  in  characteristic them  from  i s the parameter 4G^. the  Gibbs  compared t o the r e v e r s e  t o be c o r r e l a t e d i n some way  with  most Since  f r e e energy reaction, substrate  of one and  35  product the  concentrations  substrate  and  product  reactions  in v i v o are  affinity  or  seen of  to  be  enzymes  obtained.  well  as  there  is a  =  These  the  reasonably  the  constraint and  catalyzing AGj^and  Control  2.  Equivalence  of  steady-  for  terms  i s shown  on  various  (i.e.  in metabolic occurs  of  ^GR  and  steady-state,  r e l a t i o n s h i p between and  the  the  vivo  could as  i_n v i v o ,  yet  coefficient 5-3).  Thus  second  important  vivo  substrate  in_  design  be  function  (Figure a  As  series  levels  relationship (correlation _i_n v i v o  may  for a  substrate  If  of  the  enzymes  equivalence  between  constraints:  of A G R w i t h steady-state  the  metabolic  of  the  state  to,be of  a  may  be  to  developed  reexamined and  allow . a and  the  to  determine  design  biologically  the  features meaningful  maintained.  Metabolism  restricted sequence  affinity  steady-state  enzymes  i n v e s t i g a t e the  more  simulate  two  affinity  Steady-state  To this  over  general  required  The  observed.  ratios  catalysis  r e a c t i o n , a l l o w i n g _in  these  1.  natire  This  data  which  concentrations the  is  affinity.  Using  the  affinity  is this  product  at  good  maintenance  5-3).  enzymes v a r y  affinities  this  equilibrium), a correlation  i_n v i v o  twelve  fact,  i n thermodynamic  (Figure  for which  0.9371) between  for  from  with AGg  In  concentration  expressed  distance  exist  12  _in v i v o .  of  enzyme context, two  requirements the  enzymes El  for  steady-state  f o l l o w i n g s c h e m e was in E2  metabolism:  used  in to  36  where E l  represents  considered E2  which  c a t a l y z e s the  from  the  important not  ways.  parameters  on  constraints  always  vivo  closely  the  Km  important  above  vales  A  and  that  the  ^ G R  and  the  were  to  steady-state  differs  A G ^  affinity by  in  two  affinity kinetic  E2  is also  of  imposing, case  the  in ofthe  additional  i n the  biologically obtained.  important  are  the  i n the  changes  concentrations  adjusted,  Similarly  i s i s assumed  the  steady-state,  B  B.  concentrations  accomplished  . Iterative  and  Michaelis acceptable  Through  this  of  both  and  apparently  f u n c t i o n . Each  enzymes  of  in  these  below.  shared  of  C  c o n s t r a i n t s used  C  to  c a l c u l a t e what  critically  outcome  A  being  steady-state  imposing  was  sequence  treatment  case,  be  f o r the  of  This  made • u n t i l  separately  the  general  to  Km  One  By  general  between  considered  same.  the  such  steady-state  found  the  C.  second  This  Vmax's were  the  were  at  the  f o r E2  listed  and  to  place,  affinity.  be  of  relationships  1.The  In  B of  first  in  conversion  E l , i t i s p o s s i b l e to  the  determination  is  the  t o AG-g f o r E2. to  enzyme  of  treatment  the  must  equivalent  procedure  In  to  relationship  constants  conversion  constant.  equivalent  first  c a t a l y z e s the  previous  held  addition  the  substrate  this  f o r both  analysis i s that  enzymes must  f o r the  be  sequence  steady-state, El >  the  Km  substrate  for  B  as  f o r enzyme  A  _  =  =  product 2.  In  E2 =  A  B  _  =  =  =  i .  >  c  f o r enzyme 1 e q u a l s  biological  terms  the  this  Km  for B  as  relationship  37  may  need  Figure  only  5-4a  this  the  the  Km  procedure  ratio  Kbl/Ka  substrate  and  calculated  f o r E2  concentration shared less  than  lowering  Km  The  slopes  of  the  these  from  Kbl  increasing  Kbl  time  ratio  the  conclude  that  Kbl=Kb2.  At  tolerances  results  the  Kb2/Kbl ideal  present  t o Km  5-4b  the  Km  linear  similar. ideal  in  i t  i s not  non-equivalence  Km  Kbl  same  for  the the  the  over  the  effect  of  close  to  region  increases.  demonstrates =  ratio  of  1  results  Kb2/Kbl  similar  way  as  this  Kbl  by  slope. was  intsections  i_n v i v o . Km  less  intermediate  5-4  From  the  is  but  Kb2, this  analysis I  and  p o s s i b l e to determine are  the  continuous  extends  the  between  then  substrate  Kb2/Kbl  same  is  (Kbl=Kb2)  to  Figure  lowered.  relationship  C  concentration  the  been  of  i s not  the  intersection  to  increasing  E l at  shows  Lowering  fold.  eventually  (low  discontinuous has  as  when  situation  5  of  equal  negative  drops  as  by  longer  Figure  changed in  no  and  ratio  ratio  of  curve  As  a  and  the  acceptable  is  ideal  ratio  4,  value  steady-state  intersection.  produces  however,  of  steady-state.  more t h e  concentrations  are the  The  region  the  becomes  lines  discontinuous Kb2  E2.  at  The  becomes  steady-state part  3,  calculated  the  the  words  2,  same v e l o c i t y  from  substrate  i n c r e a s i n g Kb2  changing If  other  but  1,  is fixed  substrate.  onto  f o r B of  curves  a  the  concentrations). the  2 by  B/A  steady-state  important  origin  in  give  deviates  In  the  that  of  the  product/substrate  velocity  shared  Moreover,  substrate  the  achieve  concentrations.  which the  on  to  f o r Enzyme  the  large  metabolically  B  and  passes  to  Kb2.  of  ratio  the  concentration. zero  effect  tThe  substrate  intersection  approximated  product  of  linear  from  be  shows the  increasing In  to  Kb2  is  what  the  equivalence  is  38  an  ideal  situation  which  evolution  produce.  However, other  factors  actually  being  in  2.The adjusted That to  achieved  enzyme  concentration  the concentration of  analysis. enzymes  enzyme  this  ideal  o f one o r b o t h  in  from  the forward  enzyme  one  E1V1  o f enzyme one i n t h e s e q u e n c e is  another  describing this  important  is  related  outcome of  relationship is  this  for  the  of changing  a n d E1V2  reverse  5-2).  the  Figure  o f one enzyme away As  the  one  activity  of  maximal  direction.  t h e amount  (equation  i s  o f enzyme  amount  ( 5 - 5 ) shows t h e from  o f E2  the  ideal  increases  the and  intersection  describing the steady-state  becomes l e s s l i n e a r  again  into  substrate  passes  implying  a  decreasing will  of  slopes. find  in  While  f o r other an  Again,  steady-state. i n continuous  increasing concentration Thus,  i s done  mechanisms presumably combinations of  be l i m i t e d  other  equation  E l in a 5-2  biologically  to this  system  similar relationships  o f e n z y m e s . An e q u a t i o n the observed  to  of E l r e s u l t s  perturbing  ( 5 - 2 ) may  the  intersections with  i n t e r s e c t i o n s which are  equation  explanation  concentration  Increasing  i f t h e same e x p e r i m e n t  discontinuous  two u n i - u n i  permits  of negative  results  intersection.  unacceptable.  exist  E2  that  discontinuous  of  the region  discontinuous  concentration  results  two  (5-2)  r e f e r s t o t h e Vmax a c t i v i t y  direction  in  situation  we  be  i s (when Kb2 = K b l ) :  designation  effect  to  enzymes must  E1V1-E2V1 = E1V2-E2V2  The  tend  steady-state.  two  The e q u a t i o n  prevent  would  vivo.  to the continuous  that  may  presumably  constant  like  (5-2)  relationship  39  between groups  enzyme a c t i v i t i e s of  Pette,  Once  the  sequential obtained.  enzymes  empirical values  (the  constant-proportion  (1965)). above  This  i n a pathway  two  the  result  data  in  design  f e a t u r e s are  relationship which  is a  figure  5-3,  imposed  between AGp  satisfactory may  be  and  on  in_  the  two  vivo  is  explanation  proven  by  of  the  substituting  of: [A]  =[  B]  x  affl  [B]  =  [C]  x  aff2  and  Where a f f l reactions E2  and set  a f f 2 are equal  to  the the  respectively.(This  e x p r e s s i o n AGJJ = a f f i n i t y The  equations  point  synthesis  due  to  and  Kc/Kb2  equivalent the  Haldane  steady-state - Vbl(  B  for E l  statement  of  the and the  relationship).  between  E l and  E2  are:  /Kbl)  B xaffl)/Ka  of  B /Kb2)  B /Kb2)  along B by  +  +  /Kbl)  - Vc(  ( B  the  ( B  /aff2)/Kc  /aff2)/Kc  line  E l equals  B  of  intersection  e x a c t l y the  r a t e of  the its  rate  of  utilization  E2.  Constancy  It of  an  Kbl/Ka  of  (5-4)  l+(  by  is  B xaffl)/Ka  Vbl(  every  ratio  ratios  = l+(  At  Km  d e s c r i b i n g the Vat  flux  product/substrate  of  may  affinity  be  kinetic  relationships  seen  during  from  Figures  constraints are  Km  steady-state  5-4  and  satisfying  equivalence  and  transitions  5-5 the the  that  the  observed Vmax  only ir\  set vivo  relationships  40  contained  in  steady-states 5-4) 5)  equation are either  or are s t r a i g h t  which  will  affinity this  never  simple  placed  to  5-6a.  In  so  the  dimensional side  i s figure  (Figure occurs,  authors  scales  and  glycolysis  are  increased  flux  pathway  increase  communication,  when  the  when  of  from  close  from  to  substantially  one r a t e even  product  has  i n the equilibrium  the three  though  of  many  equilibrium  of  the This  steps  even  (Lowry,O.H.  situation  to  several in  during  of intermediates  most  from  of f l u x  change.  that  This  The  and viewed  Hintz,C.S.,Lowry,C.V., 1980).  steady-  the origin  t h e s i d e when  1964)  are  increases.  t h e same  the concentration  And Lowry,O.H.,  axis  the observations  Passonneau,  in  substrate i s  the metabolic  from  and  of enzymes  other  velocity  remains  substrate  1980,  that  5AG^  are superimposed  the intersection  R.,Chi,M.M.-Y. an a n a l o g y  on  As a t r a n s i t i o n  maintained  the  ranges  surfaces  t o be l i n e a r  as  e x p l a i n s and confirms  (Lowry  broad  the shared  describes  continuous  5-6b).  of  this  c a n be s e e n  the a f f i n i t y  concentrations relationship  to achieve  i s cut along  (Figure  product.  intersection  also  (Figure  desribes  over  of metabolism  concentrations  give  on two s e q u e n t i a l e n z y m e s a  dimensional  . The  The s t e a d y - s t a t e  infinite  another  order  o n t h e same a x i s  intersection  the  three  solution  I t follows that  maintained and  which  the o r i g i n  which  i_n v i v o .  f o r the steady-state  adjusted state.  two  from  line  i s imposed  of substrate  Accordingly,  Figure  the  i s  solutions  to the ideal  radiating  observed  affinity  concentrations  designed  lines  relationship  of  A l l other  parallel  cross  equivalence  constancy  5-2.  i n the  Personal Ing,J.,Fell,  important most  result chemical  41  systems  where  Stabi1ity A  of  kj/k-j = the  new  state  which  of  substrate  adding  X-Y  will  the  result  the  arrested  for  and  X-Y  the  regardless  the  of  is  steady-state  observations, above  E2  flux  treatment  of  activity  has (Lowry  for  feature been and  of  this  of  flux of  the  of  flux  with  and  velocity . The  enzymes.  In  1964;  Guppy  to  the  will  be  previously  In  under the  but  substrate  steady-state  be  f u n c t i o n of  reported  and  thermodynamic  seems t o  affinity  towards  perturbation.  i f i t explains  metabolic  Passoneau,  surface  return  steady-state.  of  The  shared  the  by  5-5).  negative  resembles  indeed  is a  frequently  the  is  direction  the  reached  (Figure  will  the  metabolism  is positive  the  is useful this  steady-  be  It  i t predicts a constancy  i s a design  explained  and  the  may  concentrations  self-stabilizing.  in  perturbations  which  stable  contruct  constancy  the  stable  to  E l and  plane  a  respect  velocity  flux  to  conclusion  plane  be  in  of  intersection  X-Y  theoretical  of  by  the  respect.  instance, states  as  in this  the  This  small  result  i t may  back  case  steady-state  i n a new  at  perplexing  The  A perturbation  therefore  A  Below  steady-state  equilibrium  1977).  stable with  the  in which  i t will  lead  steady-state.  steady-state Since  will  generated  Above  steady-state.  toward  being  appears  plane.  unstable  Alternatively,  concentration.  now  be  variables determining  DiPrima, of  metabolism  s o r t may  reached.  surfaces  steady-state  of  perturbations  property  the  any  the  (Boyce and  the  the  of  small  has  the  of  being  case  state  is  steady-state  steady-state  perturbations  Keq.  never e_t  the  case  the  first  different  steady-state fact,  such  adequately al. ,  1979)  42  Whereas  feedback  magnitude  of the f l u x  flux  and s u b s t r a t e  feedback  control  1968)),  i t  sufficient and  to zero,  steady-state inaccurate  grouped  that  fumarase  1980,  enzymes  term  treatment  (Sel'kov,  they  are not  steady-state  into  two b r o a d  flux  whose  helps  Since  i n the  sense  the system  is  i s an  (steady-state  (Prigogine,1955).  understand  why  t h e Km  values  for  malate  t h e same  ( e g . T h e Km  malate  dehydrogenase  and numerous  affinity  affinity  production state  classes:  an  "equilibrium" in this  entropy  are often  data  of  enzymes  misleading.  and m i t o c h o n d r i a l  unpublished  invoking  step, that  the  glycoltyic by  reactions with  e q u i l i b r i u m as a s t a b l e  substrates  in  explained  treatment  catalyze  completely  this  influence  concentration.  i s l a r g e . The  replaces  shared  this  and n o n - e q u i l i b r i u m  and  Finally  of  from  one, the s t a t e of l e a s t  flux)  are best  phosphofructokinase  substrate  enzymes  can  oscillations  f o r the properties  e a r l y workers  equilibrium  open  the  evident  steady-state  close  concentration  at  is  undoubtedly  (eg. stable  to account  These  of  mechanisms  examples  in  (Ballantyne  glycolysis).  CONCLUSIONS  The 1. T h e  s i x major  2. F o r t w o e n z y m e s  3. T h e  during  chapter  are:  t o t h e i_n v i v o  a substrate  t h e Km's  affinity.  f o r that  same.  of a r e a c t i o n  changes  non-regulatory 4. E n z y m e s  sharing  are the  affinity  of t h i s  of A G ^ i s e q u i v a l e n t  magnitude  substrate  conclusions  in flux  i n metabolism  (a c o n s t r a i n t t h a t  remains may  constant  apply  only  enzymes).  catalyzing reactions with  affinities  close  to  to  43  the  e q u i l i b r i u m value  measured  by  reactions 5.  with  Vmax m e t h o d affinities  Enzyme c o n c e n t r a t i o n s reactions  6.  the  should  The to  substrate these,  (5)  may  catalyzing  consecutive  E1V1-E2V1  be  four  supra).  =  uni-uni  E1V2-E2V2.  i s stable with  have  respect  that  mimic  the  possible.  The  proof  of  concentration  some s u p p o r t i n g  More c o n c l u s i v e  obtained  conditions  concentrations  when  from e q u i l i b r i u m .  metabolism  first  (vide  only  substrate  far  those  activities  fluctuations. the  literature  of  than  two  r e l a t e d by  steady-state  Of the  are  of  have h i g h e r  by  measuring  intracellular (6)  will  while  proof  Km's  of  and  milieu  require  evidence  a method  monitoring  flux  (3)  Vmax's  as  from and under  closely  as  of  perturbing  and  substrate  in_ v i . v o . SUMMARY  Metabolic magnitude  of  activation of  that  from  the  reaction  zero  to  steady-state. enzyme not  i n the  compared  i n the  to  This  stability  and  this  a l l metabolic  reverse  parameters,  the  enzyme  over  the  of  notably  concentration  affinity  of  the  i s equal  and  is  displacement  has  been  a  Km  must  for be  reaction.  of  affinity function  e q u i l i b r i u m of  the  for  a  simple  i t applies  sharing. shared  modulated The  the  linear  steady-state  the  change  the  suggest  substrate the  to  The  i s e s s e n t i a l to  from  proven  The  steady-state.  free energy  enzyme p a r a m e t e r  sequences.  consequence  f o r the  Gibbs  steady-state  infinity.  While  designed  s e q u e n c e _in v i v o m e a s u r e m e n t s  inevitable design  are  difference  forward  attainment,  if  enzymes  Km  is  Several  three to not  for  the  an  enzyme  substrate  to permit  many  and  control shared  44  substrate  must  be a d j u s t e d  be t h e s a m e .  presumably  substrate  (due  of  synthesis  protein  With as  by  the  enzyme  or  by  prerequisite  steady-state  i s achieved.  relationship  between  "plasticity"  o f enzyme  A G R design  as  control and  an  we of  also  accumulated  equivalent  affinity  outlined  must  on t h e m a c h i n e r y  enzymes t h a t  metabolism The  of  behaviour)  metabolically  of metabolic  for  concentration  interaction  to non-steady-state  the c h a r a c t e r i s t i c a  The  know  AGJJ  =  affinity  i t , the stable  metabolism depends  i n previous  signals.  via upon  chapters.  the the  45  Figure  5-1  Enzyme  reaction surface  for simple  uni-uni  mechanism.  46 /  Figure  5-2 T h e g e n e r a l Five  steady-state.  p a i r s o f enzymes  simulated. Va Vbl  1-5E1 a n d 1-5E2  The k i n e t i c  = 20,30,40,50,60  parameters  f o r sets  were  f o r E l were  1-5 r e s p e c t i v e l y  = 10  Ka = 3 0 Kbl  = 50  The  kinetic  Vb  = 10  Vc  = 90  Kb2 Kc The  parameters  f o r E2 w e r e  = 40 = 30 concentrations  incremented ratio  of s u b s t r a t e s  C/A m a i n t a i n e d  constant  ratio  1-5  r e s p e c t i v e l y . The v e l o c i t y  catalyzed  of  A f o r E l was  by 1,2,3,4 a n d 5 a n d t h e c o n c e n t r a t i o n  C/A  and  for a l l sets,  f o r each p a i r .  was 0 . 5 , 0 . 6 , 0 . 7 , 0 . 8 ,  b y E l a n d E2 was  5 f o r sets  1-5  a n d 0.9  (Va  incremented  r e s p e c t i v e l y . The  - v) + (Vc +v)C Ka  Kc  A  B = (Vbl  + v ) + (Vb2 - v ) Kb2/Kbl  f o r sets  of the r e a c t i o n by  1,2,3,4  concentration  B f o r E l a n d E2 w a s c a l c u l a t e d u s i n g  KblA  The  the  equation  47  Figure  5-3 A G g , v e r s u s A G several  f o r enzymes a n d m e t a b o l i t e s from  tissues.Data  A G R was d e t e r m i n e d  sources a r e appended.  from Vmaxf/Vmaxr  i n the  same way a s f o r T a b l e 4 - 1 . Unless  otherwise designated a l l data are  muscle  enzymes and m e t a b o l i t e s .  PGI= PGM=  phosphoglucose  isomerase  phosphoglucomutase  AK=  adenylate  ald=  aldolase  FDPase=  kinase  fructose  CPK= c r e a t i n e LDH=  diphosphatase  phosphokinase  lactate  dehydrogenase  G3PDH= g l y c e r a l d e h y d e - 3 - p h o s p h a t e TPI=  triosephosphate isomerase  PK= p y r u v a t e HK=  for  kinase  hexokinase  dehydrogenase  48  Figure  5-4 T h e e f f e c t on  Va Vbl Ka  5-l»was u s e d  the kinetic velocity  ratio The  f o r the shared  t o compute  t h e same  E 2 a s f o r E l a t t h e same c o n c e n t r a t i o n  with The  t h e Km  substrate  the steady-state.  Equation of  of changing  parameters  was c o m p u t e d  of B t o A equal  value = 200 = 100 = 1000  Kbl  = 100  The  solid  to  velocity of B  below.  f o r E l using  a  constant  Kbl/Ka.  o f Kb2 i s i n d i c a t e d  beside  each  line.  Vb2 = 200 Vc = 100 Kb2 =  varied  Kc, = v a r i e d lines  concentration  indicate  to satisfy the region  of substrates  Haldane  relationshi  where t h e  i s positive.  49  Figure  5-5 T h e e f f e c t  o f enzyme c o n c e n t r a t i o n  achievement  of the continuous  steady-state.  Equation  5-lb  velocity  o f E2 a s f o r E l a t t h e same  of  B with  The  the kinetic  velocity  ratio The  was u s e d  on t h e  Vb2=varied  Vbl=225  Vc=varied  Ka=400  Kb2=100  Kbl=100  of  solid  to  t h e same concentration  below. •  f o r E l using  a  constant  Kbl/Ka.  Vb2/Vc i s i n d i c a t e d  Va=300  The  parameters  was c o m p u t e d  of B t o A equal  ratio  t o compute  beside  each  line.  Kc=200 lines  substrates  indicate  where  i s positive.  the concentration  SUBSFRRFE B 0.0  SD.D  1DD.Q  15D.D  2QQ.D  25D.0  300.0  50  Figure5-6  a)  Three  dimensional  plot  of  two  simple  uni-uni  m e c h a n i sms. b)  Two  dimensional  plot  of  a)  viewed  from  x-axis.  50a  gure  5-7  Stability  of  the  steady-state.  h •  52  REFERENCES  Atkinson,D.E.  1977.  regulation.  Cellular  Academic  equations  and  energy  Press.  Boyce,W.E. a n d D i Pr ima,R.C.  CITED  San  value  and  i t s  Francisco.  1977 .  boundary  metabolism  Elementary  differential  problems. John Wiley  and  Sons,  Toronto. Cleland,W.W.  1970.  Boyer,P.D.  (ed.),  Cornish-Bowden, enzymic  A.  Michaelis  1976.  Crowley,P.H.  and  1975.  Donder,Th. theory  Fersht,  and  Natural  and  1977.  effect  Eisenthal,  enzymes  of n a t u r a l  I I .  selection  on  101:1-10. 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J.Biol.Chem.  and  243:5679.  55  ABBREVIATIONS net Vmaxf,Vmaxr  V I , V2  kcatf,kcatr  USED  rate of production  maximal  velocity  forward  or reverse  maximal  velocity  forward  (1) o r r e v e r s e  catalytic  of substrate or  product  o f enzyme c a t a l y z e d r e a c t i o n i n direction.  o f enzyme c a t a l y z e d r e a c t i o n i n (2) d i r e c t i o n ,  rate constant'for forward  or reverse  reaction kl,k-l  rate or  constant  reverse  Ka,Kb,Kp,Rq  Michaelis  Kia,Kib,Kip,  Inhibition  f o r chemical  (-1)  reaction  i n forward  direction  constant  f o r s u b s t r a t e A,B,P o r Q  constant  f o r A,B,P o r Q  Kiq Keq  Equilibrium  R  Gas  T  Temperature  i n °K  k  Boltzmann's  constant  h  Planck's  A  *  constant  Gibbs  Gibbs  constant  free  forward  Ik p  A  ,&G  %  Gibbs  enrgy  change  ( f )or reverse  free  substrate  * G  constant  free  substrate  energy A,B,P, energy A,B,P,Q  of a c t i v a t i o n i n (r) direction  change  of binding of  o r Q. change  of binding of  in inhibitory  mode.  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