Enzyme Kinetics
Callie Martins
Spring 2012
Contents
1 Introduction
1.1 Chemical Reactions . . . .
1.2 Catalysts/Enzymes . . . .
1.3 Kinetics, Rate Constants,
Reaction Rates . . . . . .
. . . . . . . . . .
. . . . . . . . . .
Equilibrium, and
. . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
Factors that Effect
. . . . . . . . . . . .
1
1
1
1
2 Assumptions
2
3 Rate Equations
3.1 Definition of Variables . . . . . . . . . . . . . . . . . . . . . . . .
3.2 General Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Rates for Specific Compounds in Reaction . . . . . . . . . . . . .
2
2
3
3
4 Solving The System of Differential Equations
4
5 The
5.1
5.2
5.3
5
6
7
7
Michaelis-Menten Model
From Four Rate Equations to Two Rate Equations . . . . . . . .
The Quasi-Steady-State Assumption . . . . . . . . . . . . . . . .
The Michaelis-Menten Equation . . . . . . . . . . . . . . . . . . .
6 Conclusion
10
6.1 Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
A MATLAB Coding
11
1
Abstract
Enzyme kinetics is the study of how biological catalysts increase the reaction
rate in biochemical reactions. Without such catalysts, the biological processes
necessary for organisms would not proceed at a rate that would sustain life. The
document herein will introduce how the kinetics of enzyme catalyzed reactions
can be modeled qualitatively. It will also describe the Michaelis-Menten model,
which allows experimental data to be studied more easily.
1
Introduction
Enzyme kinetics is the study of how biological catalysts increase the reaction
rate in biochemical reactions. Without such catalysts, the biological processes
necessary for organisms would not proceed at a rate that would sustain life.
Reaction rates can be represented mathematically, illustrating how the different
chemical species involved in the reaction are affected throughout the reaction.
This report will show the derivation of a system of differential equations describing a general enzyme catalyzed biochemical reaction. Solutions to the system
can then be qualitatively discerned using a numerical solver such as MATLAB’s
ode45. The graphed solutions show how the concentrations of the various species
change throughout the reaction. A simpler way to model enzyme kinetics is the
Michealis-Menten model. After performing qualitative analysis to obtain a clear
image of how the species behave, we will derive the Michaelis-Menton model and
discuss its applicability.
Before deriving the mathematical model for enzyme kinetics there are several
basic definitions that need to be explained.
1.1
Chemical Reactions
A chemical reaction is the process by which two or more chemical compounds,
called reactants, combine to form one or more different compounds, called products. A reactant can be comprised of a single element or it may be a molecule
containing multiple elements. The same holds true for the products of the reaction. In biochemistry, a reactant is referred to as a substrate.
1.2
Catalysts/Enzymes
A certain amount of energy must be available for a reaction to occur. This
is referred to as the activation energy of the reaction. A catalyst lowers the
activation energy of a given chemical reaction, increasing the rate of the reaction.
Enzymes, usually proteins, are catalysts in biochemical reactions.
1.3
Kinetics, Rate Constants, Equilibrium, and Factors
that Effect Reaction Rates
Chemical kinetics is the study of the rates of chemical reactions. A rate law is
an equation that describes the rate at which that particular reaction proceeds.
Rate constants, conventionally denoted as k, are determined experimentally.
The Arrhenius equation demonstrates the variables that determine the value of
k.
k = Ae−EA /RT
A, called the frequency factor, accounts for factors such as the frequency of
collisions between molecules and the orientation of the molecules. The components of the exponent are the activation energy (EA ), the gas constant (R),
1
and the temperature at which the reaction is taking place (T) in Kelvins. From
this equation we can also see that the presence of a catalyst increases the rate
constant by decreasing the activation energy.
Reactants will be converted to products until the reaction reaches chemical
equilibrium at which point in time the concentrations of the reactants and the
products no longer change, assuming a closed system.
2
Assumptions
• Constant temperature, pH, and similar stable conditions
An increase in temperature increases the molecular motion resulting in
more collisions and, therefore, an increase in the overall reaction rate.
However, too much heat can denature an enzyme and render it ineffective. Likewise, cold temperatures decrease molecular motion, leading to
fewer collisions and a slower reaction rate. Each enzyme has an optimal
pH at which it is able to maintain it’s tertiary structure (3-dimensional
conformation). A change in the pH can denature the enzyme as well.
• Each enzyme can catalyze only one substrate molecule at a time
This allows for a more consistent rate for our model.
• Enzyme is unchanged throughout reaction
In order to easily model the reaction, we will assume that the enzymes
remain unchanged. Should the enzymes become denatured, this would
alter the concentration of effective enzymes.
• Substrate concentration is much greater than enzyme concentration This
is necessary for the quasi-steady-state assumption which will be addressed
later in this document. In short, it assumes that all of the enzymes will
bind to substrate almost immediately in solution. With a high proportion
of substrate available, the enzyme will be able to bind to substrate and
release product at a fairly steady rate. The significance of this will be
explained in due time.
3
Rate Equations
3.1
Definition of Variables
• S=Concentration of substrate
• E=Concentration of unoccupied enzyme
• C=Concentration of Enzyme-Substrate Complex
• P=Concentration of product
2
• R1 =Reaction rate for S+E→ C
• R2 =Reaction rate for C→ S+E
• R3 =Reaction rate for C→ P
• k1 =Rate constant for S+E→ C
• k2 =Rate constant for C→ S+E
• k3 =Rate constant for C→ P
3.2
General Rates
The following equations describe the reaction rate for each step in the overall
reaction. R1 is the rate at which the substrate binds to the enzyme to form
an enzyme-substrate complex. This rate is determined by the concentration of
the substrate, the concentration of the enzyme, and the rate constant for that
particular reaction. Sometimes the substrate dissociates from the enzyme’s
binding site before the reaction proceeds to the final product. R2 represents
the rate at which this reverse reaction occurs and it is dependent upon the
concentration of the enzyme-substrate complex and the specific rate constant
for the reaction. R3 is the rate at which the enzyme-substrate complex is able
to produce the product of the reaction. This is dependent on the concentration
of the enzyme-substrate complex and rate constant for this process.
• R1 =k1 ES
• R2 =k2 C
• R3 =k3 C
3.3
Rates for Specific Compounds in Reaction
Using the rate equations for each step in the reaction, a differential equation
can be derived for each species in the reaction to model how it changes as the
reaction proceeds. For example, the concentration of the substrate will decrease
with the formation of the enzyme-substrate complex, R1 , and increase with
collapse of the enzyme-substrate complex, R2 . This provides the reasoning for
the sign associated with each component.
S’=−k1 E·S+k2 C
E’=−k1 E·S+k2 C+k3 C
C’=k1 E·S-k2 C-k3 C
P’=k3 C
3
4
Solving The System of Differential Equations
In order to solve the system of equations, it will be beneficial to express the
equations through a common variable. Let’s designate this variable as x. In
doing so, we express the entire system of differential equations as x0 = f (x),
where f (x) is a matrix of the rate laws in terms of x1 , x2 , x3 , x4 , (all of which
are time dependent), and the rate constants.
 
x1
x2 

x=
x3  .
x4
We will do this by making the following definitions:
x1 = S(t), x2 = E(t), x3 = C(t), x4 = P (t)
We are now able to describe the concentration rates of each species in terms
of the variable x.
x1 ’=−k1 ·x2 ·x1 +k2 ·x3
x2 ’=−k1 ·x2 ·x1 +k2 x3 +k3 x3
x3 ’=k1 ·x2 ·x1 -k2 x3 -k3 x3
x4 ’=k3 ·x3
Putting all the pieces together we obtain:
 0 

x1
k1 x2 x1 + k2 x3
x02  −k1 x2 x1 + k2 x3 + k3 x3 
 

x0 = f (x) = 
x03  =  k1 x2 x1 − k2 x3 − k3 x3 
x04
k3 x3
Having placed the system in this form, we can now use the MATLAB numerical solver, ode45. This is done by creating an m-file which can be found in
the appendix at the end of this document. This produces the graph seen below.
4
As you can see in this graph, the concentration of the substrate declines as
the concentration of the product increases. The concentrations of both species
eventually begin to approach equilibrium concentrations. The concentration
of the unoccupied enzyme initially decreases very rapidly as it combines with
the substrate to form the enzyme-substrate complex. The concentration of the
enzyme eventually begins to return to initial levels as there is less substrate
with which to bind with. The concentration of the enzyme-substrate complex
increases at the start of the reaction as the enzyme and substrate bind together,
however this concentration begins to decrease as the substrate is converted to
product and, consequently, the concentration of unoccupied enzyme increases
towards initial levels.
5
The Michaelis-Menten Model
The preceding qualitative analysis is nice for modeling the general behavior of
the reaction species, however, it forces us to assign arbitrary rate constants for
reaction. Actually finding such values experimentally would be nearly impossible. A better method for dealing with experimental data is required. Such a
method exists in the Michealis-Menten model, named for German biochemist
5
Leonor Michaelis and Canadian physician Maud Menten. The Michaelis-Menten
equation is so useful because it involves only two parameters, both of which can
be determined experimentally.
5.1
From Four Rate Equations to Two Rate Equations
The derivation of the Michaelis-Menten equation will require some algebraic
manipulation of the reaction differentials. The initial objective is to reduce the
system from four equations to two.
S 0 = −k1 ES + k2 C
(1)
E = −k1 ES + k2 C + k3 C
(2)
0
0
C = k1 ES − k2 C − k3 C
0
P = k3 C
(3)
(4)
Adding equation (2) and equation (3) we discover the following:
E 0 + C 0 = (−k1 ES + k2 C + k3 C) + (k1 ES − k2 C − k3 C) = 0
(5)
Since the change in unoccupied enzyme concentration plus the change in the
concentration of enzymes bound to a substrate equals zero, we can see that the
total enzyme concentration is constant. This implies the equation
E + C = ET
(6)
where is the concentration of the total amount of enzymes. Rearranging
equation (6) yields
E = ET − C
(7)
This expression for the enzyme concentration can be substituted into the
substrate concentration differential equation (2).
S 0 = −k1 ES + k2 C
= −k1 (ET − C)S + k2 C
= −k1 SET + k1 SC + k2 C
= −k1 SET + (k1 S + k2 )C
(8)
The same substitution can be made in the differential equation describing
the change in the concentration of the enzyme-substrate complex (3).
C 0 = k1 ES − k2 C − k3 C
= k1 (ET − C)S − k2 C − k3 C
= k1 SET − (k1 SC + k2 C + k3 C)
= k1 SET − (k1 S + k2 + k3 )C
6
(9)
5.2
The Quasi-Steady-State Assumption
It is now time to introduce a very important assumption. As stated prior, we
assume that the concentration of substrate far exceeds the concentration of the
enzyme. Under these conditions, we assume that when the substrate and enzyme
are combined in solution they will bind together almost immediately. From
this point the enzyme-substrate complex will produce product at a relatively
constant rate for the length of the reaction. This assumption allows us to say
C0 = 0
This is of course an approximation to facilitate our needs, but it is not
unreasonable for our purposes. We now have the following equations to deal
with:
5.3
S 0 = −k1 SET + (k1 S + k2 )C
(10)
0 = k1 SET − (k1 S + k2 + k3 )
(11)
The Michaelis-Menten Equation
Solving (11) for ET :
ET =
k1 SET
(k2 + k3 + k1 S)
Substitute (12) into (10) and simplify:
S 0 = −k1 SET + (k2 + k1 S)
= −k1 SET +
=
=
k1 SET
(k2 + k3 + k1 S)
k1 k2 SET + k12 S 2 ET
k2 + k3 + k1 S
−k1 SET (k2 + k3 + k1 S) + (k1 k2 SET + k12 S 2 ET )
(k2 + k3 + k1 S)
−k1 k2 SET − k1 k3 SET − k12 S 2 ET + k1 k2 SET + k12 S 2 ET
(k2 + k3 + k1 S)
All but one term in the numerator cancels and we are left with
S0 =
−k1 k3 SET
(k2 + k3 + k1 S)
We divide top and bottom by k1
−k1 k3 SET
·
(k2 + k3 + k1 S)
1
k1
1
k1
7
k3 SET
= − k1 +k3
+S
k1
(12)
After simplifying, we finally arrive at the Michaelis-Menten equation
S0 =
Vmax S
KM + S
3
. These two parameters are the maxwhere Vmax = k3 ET and KM = k2k+k
1
imum velocity of the reaction and the Michaelis-Menten constant, respectively.
Graphing this differential equation using Matlab’s odesolve, we obtain the
following graph.
This is reasonably good approximation of the originally derived rate equation for the change in substrate concentration. Since the rate at which the
concentration of the substrate changes is a measure of the rate of the reaction,
we will redefine S’ as V, for velocity.
Vmax S
KM + S
Recall that Vmax is dependent on ET by definition. Therefore, an increase
in the total enzyme concentration will result in an increase in the maximum
velocity of the reaction. This makes sense when we consider that additional
enzymes in the solution will increase the probability of a collision with a substrate. Similarly, additional substrate will increase the velocity of the reaction.
If we take the derivitate of V
S0 = V =
V0 =
(KM + S)Vmax − Vmax S
(KM + S)2
=
KM Vmax
(KM + S)2
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There comes a point at which additional substrate does not speed up the
reaction. When the total enzyme concentration is held constant, the enzymes
will reach a point at which they are binding with substrate at a maximum rate,
so a higher concentration of substrate will not increase the reaction rate. This
is reflected in the term Vmax , the velocity that the reaction will never surpass
under the assumptions made herein. The graph below shows how the velocity
of the reaction changes with increasing concentrations of substrate.
The reaction velocity and substrate concentrations can be found through
experimentation and graphed in such a manner. The values of Vmax and KM
can be roughly calculated just from the graph. Vmax is the horizontal asymptote
that you can see the line approaching. A little algebra shed light on how to
discern the value of KM .
1
Vmax S
Vmax =
2
KM + S
KM + S =
Vmax S
1
2 Vmax
KM = S
Therefore, KM is equal to the concentration of the substrate when the velocity is at half of the maximum velocity. Examining the graph above, it appears
that the velocity is approaching 1.2. Therefore, the Michaelis-Menten constant
for this reaction is 0.6.
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6
Conclusion
Enzymes are a vital aspect of life processes. Enzymes are the most powerful
catalysts known. In fact, an enzyme can increase the rate of a reactoin by a
factor of up to 1020 over the uncatalyzed reaction. Nonenzymatic catalysts can
increase the reaction rate by factors of 102 to 104 .
6.1
Inhibition
There are two main types of enzyme inhibition: competitive and non-competitive.
A competitive inhibitor is a molecule other than the substrate that is able to
bind to the active site of the enzyme. The enzyme is thus rendered inactive
and unable to carry out the reaction. In the presence of a competitive inhibitor,
the maximum velocity of the reaction doesn’t change, but the rate constants do.
This means that the reaction will still approach the same Vmax , but V at a given
substrate concentration will be less than when the inhibitor is not present. A
non-competitive inhibitor binds to an area that is not the active site; however,
its binding causes a conformational change in the enzyme which makes it inactive and unable to bind substrate. In the presence of a non-competitive inhibitor
the rate constants remain the same, but the maximum velocity of the reaction
will be lower. Put in terms of our earlier defined variables: KM doesn’t change,
but V approaches a lesser value than when the inhibitor is not present. Enzyme
inhibition takes place naturally as part of maintaining homeostasis. However,
through scientific inquiry and mathematical modelling (i.e. enzyme kinetics)
humans have learned to manipulate enzymatic activity and better understand
topics such as pharmacology.
For example, aspirin inhibits COX-1 and COX-2, enzymes responsible for
the production of the inflammation messenger prostaglandin. Other substances
inhibit reactions that are necessary to sustain life such as cyanide which inhibits
cytochrome c oxidase, halting cellular respiration in the mitochondria.
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Enzyme kinetics plays a crucial role in our daily lives, whether it be medication or cellular respiration.
A
MATLAB Coding
This following m-file was created to show the solution curves calculated by
ode45. The m-file must be saved as the name of the function (G, in this example).
function xprime=G(t,x,a,b,c)
xprime=zeros(4,1);
xprime(1)=-a*x(2)*x(1)+b*x(3);
xprime(2)=-a*x(2)*x(1)+b*x(3)+c*x(3);
xprime(3)=a*x(2)*x(1)-b*x(3)-c*x(3);
xprime(4)=c*x(3)
After saving the m-file as the name of the function (in this case, G), the
following lines of commands will produce a graph of the solutions in MATLAB.
We will assign initial conditions of S(0)=10,E(0)=7, and P(0)=0 for this example. The rate constants used in this case will be k1 =3, k2 =1, and k3 =3 (Since
R1 and R3 proceed at a greater rate than R2 –the reaction wouldn’t get very far
otherwise!).
[t,x]=ode45(@G,[0,6],[1;.1;0;0],[],10,1,5);
plot(t,x)
title(’General Solution to System of Differential Equations Derived from
Rate Laws to Model Enzyme Kinetics’)
xlabel(’Time’),ylabel(’Concentration of Substrate, Enzyme, Enzyme-Substrate
Complex, and Product’)
legend(’[Substrate]’,’[Enzyme]’,’[Enzyme-Substrate Complex]’,’[Product]’),grid
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References
[1] Chem Guide,http://www.chemguide.co.uk/physical/basicrates/arrhenius.html
[2] Worcester Polytechnic Institute,
http://www.math.wpi.edu/Course_Materials/Calc3/Labs/node2.html
[3] CmePy, http://fcostin.github.com/cmepy/example_mm.html
[4] Li,Shen,Li,
http://www.bioinfoamss.org/module/7publications/workingpaper
/steady_state_laws_in_enzyme_kinetics.pdf
[5] Inhibition Graph,http://faculty.une.edu/com/courses/bionut/distbio
/obj-512/Chap9-enzymeinhibition.html
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