Introduction to enzyme kinetics

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Introduction to enzyme kinetics
frank.bruggeman_at_falw.vu.nl or frans.bruggeman_at_manc
hester.ac.uk

• Frank Bruggeman
• Systems Biology Group
• MCISB, MIB
• University of Manchester
• June 16th - Biomodeling network

http//home.tiscali.nl/frankbruggeman
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contents

• Proteins structural and functional aspects
• Mass-action kinetics (no catalysis)
• Enzyme kinetics (catalysis)
• Quasi-steady state enzyme kinetics
• Rapid-equilibrium enzyme kinetics
• Multimeric enzyme kinetics
• Networks of enzymes

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Proteins structural aspects
Secondary structure ?-helix, ?-sheet, turns
Primary structure sequence of amino acids
connected through peptide bonds
Quaternary structure proteins composed out of
subunits (subunitsingle amino acid chain not
all proteins have subunits)
Tertiary structure single amino acid chain
composed out of secondary structures
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Proteins functional aspects

• Most important function of proteins is to
  function as reaction catalysts, i.e. as enzymes.

metabolism
signal transduction
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Reactions kinetics (not catalysed)mass action
kinetics

• Consider the following isomerization reaction

P
S
This reaction is reversible P is formed out of S
and S is formed out of P. This reaction has a
net rate v defined as, in the direction of
product formation, the number of moles product
formed per unit time, e.g. mol/min. This net
rate v is the difference between the elementary
rates of production and consumption of P. The
net rate v depends on the concentration of
substrate S and product P as
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Reactions kinetics (not catalysed)mass action
kinetics
Per unit time v mol of S is consumed and v mol of
P is produced. Therefore, at time t, the rate of
change in the concentration of S and P are given
by
kinetic model in the form of set of ordinary
differential equations (ODES)
(For notational convenience, most of the time we
will omit (t).)
Using these equations, given an initial condition
(e.g. S(0)100, P(0)0), and parameterisation of
elementary rate constants (k100,k-10) the
concentrationof S and P as function of time can
be obtained by way of integration. In this
case,the integration can be done analytically,
to obtain
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Reactions kinetics (not catalysed)mass action
kinetics
Kinetic model
Eventually the net rate is zero, then the system
has attained thermodynamicequilibrium. No net
changes occur any more in the concentrations of S
and P.
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Reactions kinetics (not catalysed)mass action
kinetics
The rate can now be written as
P/S is called the mass action ratio, often
denoted by ?.
Measure for the distance from thermodynamic
equilibrium 1-?/Keq
Molar gibbs free energy potential ?? (unit
Joule/mol) stored in reaction ratio P/S
The equilibrium constant is a thermodynamic
property of the reactants of the reaction. It
does not derive from kinetics. But it does
constrain the kinetic properties as is evident
from the Haldane relationship. Thus hexokinase
in yeastwill have the same equilibrium constant
as hexokinase in E. coli even though thekinetic
properties of the two enzymes will differ
equilibrium constant are not subjectto evolution.
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Reactions kinetics (catalysed)enzyme kinetics

• An enzyme catalyses a reaction in its catalytic
  site. The substrates and products bind at this
  location on the enzyme. Effectors may bind in
  the catalytic as well. If they bind elsewhere on
  the enzyme then the effectors are allosteric
  effectors. Effectors may be activating or
  inhibiting the rate of catalysis.
• Consider the following enzyme catalyzed reaction
• In the simplest case, this reaction may involve
  the following elementary reactions

v
S
P
v1
v2
ES
ES
EP
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Reactions kinetics (catalysed)enzyme kinetics

• This set of reactions was first analysed by
  Michaelis and Menten (1913)
• Note that consideration of complex EP would have
  been better because the isomerization takes place
  on the enzyme.
• The following ODES constitute the kinetic model
• The elementary rate equations are

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Reactions kinetics (catalysed)enzyme kinetics
On a very short time scale we observe, saturation
of e with s and small production of p (negligible
with respect to total S100).
Conditions S100, e1, p0, es0 k1100,
k-110, k2100, k-210,
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Reactions kinetics (catalysed)enzyme kinetics
After we observe significant changes in S and P
at constant concentrations fores and e.
Finally, at the longest time scale p and s
become constant.
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Reactions kinetics (catalysed)enzyme kinetics
Actually, finally the system reaches
thermodynamic equilibrium.
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Reactions kinetics (catalysed)enzyme kinetics
We observed that for a long period of time, es
and e have been constant. During this period the
rates were not zero so the system was not in
thermodynamicequilibrium. This state is called
a quasi-steady state (QSS) quasi because not
allvariables were stationary. During this
period we have
These allow us to remove the variables e and es,
because they have become constant. Then we can
simplify the model from 4 variables to the 2
variables s and p. But how?
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QSS enzyme kinetics
Because QSS we have
We define the Michaelis-Menten constant for S and
for P as
The total concentration of enzyme is constant
Because vv2, we obtain for the rate, after some
rearrangements
Now we define the maximal rate in the forward and
backward reaction as
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QSS enzyme kinetics
Does the simplified QSS description indeed
describe the dynamics of the originalsystem?
Yes, it does! This approximation of mass-action
description of enzyme mechanisms by QSS rate
equations depends on the assumption spgtgteT. In
fact, in signal transduction and gene expression
networks this assumption is questionable.
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QSS enzyme kinetics

• Finally we obtain the reversible,
  product-dependent Michaelis-Menten rate equation
  
• At thermodynamic equilibrium v0 and then we have

We can now write the MM rate equation differently
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QSS enzyme kinetics
In the absence of product, we obtain the
irreversible, product-independent rate equation
(this could also have been obtained for cases
with large Keq and KM,P)
If sltltKM,S then vVMAXs/KM,S If sKM,S then
v1/2VMAX If sgtgtKM,S then vVmax
VMAX2 and KM,S1
If s0 similar plot for p but with vlt0. In the
presence of s and p more complicated plots and
V-MAXltvltVMAX (try for yourself).
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QSS enzyme kinetics multiple substrates and
products

• Most enzyme-catalyzed reactions in cells are not
  isomerisations and therefore have multiple
  substrates and/or products. Consider the
  following two substrate and two product reaction.
  (By the way, this would be called a bi-bi
  reaction.)
• S1S2?P1P2

Now multiple mechanisms are possible, e.g.
1. Ordered binding
2. Binding in random order
EP1
ES1S2 EP1P2
ES1
ES2
EP2
E
E
3. S could bind randomly but P does not or vice
versa.
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QSS enzyme kinetics multiple substrates and
products

• Again QSS rate equations can be derived. For
  instance by using the King-Altman technique or
  methods from linear algebra (see syllabus, papers
  or textbooks).
• The papers by Cleland gives rate equations for
  most mechanisms encountered in cells using QSS
  kinetics. Note that those equations can become
  very complicated. He also discusses methods to
  derive kinetic properties from experimental data.
  Cornish-Bowdens and Segels book are also good
  references for those subjects.
• The disadvantage of the use of QSS rate equations
  is their large number of parameters. But they
  are exact if based on experiment! In those
  cases where not all parameters are known or for
  exploratory modeling rapid-equilibrium (REQ) rate
  equations are very useful.
• This is what we will discuss next.

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REQ enzyme kinetics multiple substrates and
products
Consider the following mechanism

• If
• the binding of reactants to the enzyme does not
  depend on which other reactants have already
  bound to the enzyme, and
• those binding reactants can be considered to be
  in thermodynamic equilibrium
• Then
• the states of the protein can be
  straightforwardly expressed in terms of free
  reactant concentrations and dissociation
  constants.

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REQ enzyme kinetics multiple substrates and
products
Then
The rate of the enzyme is given by
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REQ enzyme kinetics multiple substrates and
products
Very often we also consider the mixed complexes
es1p2 and es2p1 then
This rate equation can be very straightforwardly
expanded to more substrates and products
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Action of effectors
The rate of enzymes can be inhibited or activated
by effectors. Those may bind to any of the
states of the enzyme. In each case the effect on
the enzyme rates is different.
Simple case
REQ rate equation
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Action of effectors
More complicated case
REQ rate equation
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Action of effectors
Terminology of types of effectors
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Experimental determination of enzyme mechanism,
effectors, and parameters

• Various plotting techniques have been developed
  to determine the mechanism by which an enzyme
  catalyses its reaction, how do effectors
  influence the rate (competitively,
  uncompetitively, noncompetitively, or mixed), and
  what are kinetic parameters. Those methods rely
  on parameter estimation methods applied to
  datasets of enzyme rates as function of the
  concentrations of reactants and effectors.

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Multimeric (multi-subunit) enzymes
The most influential models of multimeric enzyme
kinetics have been by Monod, Wyman Changeux and
by Koshland, Nemethy Filmer. Here we will
consider the simplest of these two, the model by
Monod et al. This is called the
concerted-symmetry model. It consider a
multimeric enzyme composed out n subunits that
can each by in a taut T state or in a relaxed
R state. The model considers only a
monosubstrate reaction and consider products only
as an inhibitor it can not describe reversible
reactions (this was done later by Popova Selkov
and by Hofmeyr Cornish-Bowden). The R state is
the state with the highest affinity for the
substrate. This model is called the concerted
symmetry model because all subunits change at the
same time from R to T and vice versa. This
assumption was relaxed by Koshland et al.
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Concerted-symmetry model by Monod, Wyman
Changeux
The models uses the REQ assumption. It considers
the following reactions.
With
as the equilibrium constant for the isomerization
of the enzyme between its T and R state,
as the ratio of the substrate concentration over
the dissociation constant of the substrate from
the enzyme in the R state
as the ratio of the dissociation constants of
the R state over the T state.
After tedious algebra (see Syllabus)
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Concerted-symmetry model by Monod, Wyman
Changeux
Monod-Wyman-Changeux model. v/VMAX as function of
? for different values of c (Figure A), L (Figure
B), and n (Figure C). If constant the values for
c, L, and n were 0, 1000, and 4, respectively.
Adapted from (Monod et al., 1965).
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Reversible concerted-symmetry model by Popova
Selkov
The upper figures portray a K-system and the
lower figures a V-system. For the figure in the
top left corner the value of L was varied. For
the top figure in the right corner the value of ?
was varied. In the left lower figure L was
varied. The right bottom figure varied ? . These
plots were recalculated from Popova and Selkov
(1975).
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Networks of enzymes
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Networks of enzymes
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References

• 1. Syllabus at biomodelling network site
• 2. Classical enzyme kinetics papers
• Cleland, W.W., The kinetics of enzyme-catalyzed
  reactions with two or more substrates or
  products. I. Nomenclature and rate equations.
  Biochim Biophys Acta, 1963. 67 p. 104-37.
• Cleland, W.W., The kinetics of enzyme-catalyzed
  reactions with two or more substrates or
  products. II. Inhibition nomenclature and
  theory. Biochim Biophys Acta, 1963. 67 p.
  173-87.
• Cleland, W.W., The kinetics of enzyme-catalyzed
  reactions with two or more substrates or
  products. III. Prediction of initial velocity and
  inhibition patterns by inspection. Biochim
  Biophys Acta, 1963. 67 p. 188-96.
• Monod, J., J. Wyman, and J.P. Changeux, On the
  Nature of Allosteric Transitions A Plausible
  Model. J Mol Biol, 1965. 12 p. 88-118.
• Koshland, D.E., Jr., G. Nemethy, and D. Filmer,
  Comparison of experimental binding data and
  theoretical models in proteins containing
  subunits. Biochemistry, 1966. 5(1) p. 365-85.
• 3. Books
• Cornish-Bowden, A., Fundamentals of enzyme
  kinetics. 1995, London Portland Press.
• Segel, I.H., Enzyme kinetics behavior and
  analysis of rapid equilibrium and steady-state
  enzyme systems. Wiley classics library edition.
  1993, New York John Wiley Sons, Inc.