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Computational Biology, Part 17 Biochemical Kinetics I Robert F. Murphy Copyright 1996, 1999-2006. All rights reserved.

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Biochemical Kinetics The recursion relations we have used before could be expressed as difference equations. The recursion relations we have used before could be expressed as difference equations. This is because an equation of the form x i+1 =f(x i ) can always be rewritten as x i =f(x i )-x i This is because an equation of the form x i+1 =f(x i ) can always be rewritten as x i =f(x i )-x i Analysis of the kinetics of biochemical reactions requires the use of differential equations. Analysis of the kinetics of biochemical reactions requires the use of differential equations.

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Differential equations vs. difference equations A difference equation expresses the change in some variable as a result of a finite change in another variable. A difference equation expresses the change in some variable as a result of a finite change in another variable. A differential equation expresses the change in some variable as a result of an infinitesimal change in another variable. A differential equation expresses the change in some variable as a result of an infinitesimal change in another variable.

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Difference equations Difference equations allow direct, exact integration to calculate the values of dependent variables at all values of the independent variable (such as generation number) Difference equations allow direct, exact integration to calculate the values of dependent variables at all values of the independent variable (such as generation number) Difference equations imply a “synchronicity” to changes in variables Difference equations imply a “synchronicity” to changes in variables

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Differential equations Differential equations can sometimes be solved analytically to yield an equation for the dependent variable as a function of the independent variable(s) that does not involve derivatives Differential equations can sometimes be solved analytically to yield an equation for the dependent variable as a function of the independent variable(s) that does not involve derivatives An alternative is to approximate the solution by numerical integration An alternative is to approximate the solution by numerical integration

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Numerical integration Numerical integration of differential equations only yields an approximation because we cannot calculate infinitesimal changes Numerical integration of differential equations only yields an approximation because we cannot calculate infinitesimal changes We must use a finite integration interval or step size and thereby convert a differential equation into a difference equation We must use a finite integration interval or step size and thereby convert a differential equation into a difference equation

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Numerical integration The simplest numerical integration method is Euler’s method. It simply converts each differential to a difference and calculates the value of the dependent variables by multiplying the right hand side of each differential equation by the step size. The simplest numerical integration method is Euler’s method. It simply converts each differential to a difference and calculates the value of the dependent variables by multiplying the right hand side of each differential equation by the step size.

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Numerical integration The smaller the step size is, the greater the accuracy obtained but the greater the number of calculations that must be done to get to a specific value of the independent variable The smaller the step size is, the greater the accuracy obtained but the greater the number of calculations that must be done to get to a specific value of the independent variable To increase efficiency, the step size can be changed from one step to another To increase efficiency, the step size can be changed from one step to another If the change in the dependent variable from the previous step to the current one is “small,” the step size can be increased (and vice versa)

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Goal As with the example from population dynamics, our goal is to describe how the behavior of a system depends on parameters (e.g., rate constants) and boundary conditions (e.g., initial concentrations) As with the example from population dynamics, our goal is to describe how the behavior of a system depends on parameters (e.g., rate constants) and boundary conditions (e.g., initial concentrations)

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Boundary conditions Boundary conditions can be divided into two categories Boundary conditions can be divided into two categories Initial value problems occur when all dependent variables are known at some starting value of the independent variable Two-point boundary problems occur when some dependent variables are known only at one value of the independent variable and the rest are known only at some other value of the independent variable

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Initial value problems We will consider only initial value problems, where we wish to calculate the values of the dependent variables at some point or set of points different from the initial point We will consider only initial value problems, where we wish to calculate the values of the dependent variables at some point or set of points different from the initial point

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Example biochemical system For illustration, we will consider a simple, well-studied biochemical reaction, the enzyme-catalyzed conversion of a substrate into a product For illustration, we will consider a simple, well-studied biochemical reaction, the enzyme-catalyzed conversion of a substrate into a product

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Enzyme-substrate kinetics We can write four differential equations describing this system. We will use E as shorthand for E(t), S for S(t), C for C(t), and P for P(t). We can write four differential equations describing this system. We will use E as shorthand for E(t), S for S(t), C for C(t), and P for P(t). What is an expression for dE/dt? What is an expression for dE/dt?

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Enzyme-substrate kinetics

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Boundary conditions Boundary conditions Normally, enzyme and substrate are mixed at time 0, so product and complex concentrations are initially 0: C 0 =P 0 =0. Normally, enzyme and substrate are mixed at time 0, so product and complex concentrations are initially 0: C 0 =P 0 =0.

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What now? We have a set of four coupled differential equations that cannot be solved analytically. We have a set of four coupled differential equations that cannot be solved analytically. We can We can Try to simplify them using various assumptions so that they can be solved analytically, or Integrate them numerically

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First simplification: Assumption of substrate excess To simplify system, we first assume that the substrate is present in such a high concentration that it is always in vast excess over the enzyme concentration. In this case, the substrate concentration may be viewed as remaining constant: To simplify system, we first assume that the substrate is present in such a high concentration that it is always in vast excess over the enzyme concentration. In this case, the substrate concentration may be viewed as remaining constant:

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Assumption of substrate excess Enzyme is either free or in complex. Mass balance gives an expression for E Enzyme is either free or in complex. Mass balance gives an expression for E Substituting this for E and S 0 for S in the original differential equation for C gives Substituting this for E and S 0 for S in the original differential equation for C gives

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Assumption of substrate excess This can be integrated directly to give This can be integrated directly to give

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Assumption of substrate excess Conclusion: Complex concentration asymptotically approaches the steady-state concentration, Conclusion: Complex concentration asymptotically approaches the steady-state concentration,

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Timescale How long does it take to reach the steady state? It must depend on the k’s since they are in the term in front of t in the exponential. How long does it take to reach the steady state? It must depend on the k’s since they are in the term in front of t in the exponential. One characterization of the timescale of a process: How long does it take the function describing the process to go from its minimum value to its maximum value if going at its maximum rate. One characterization of the timescale of a process: How long does it take the function describing the process to go from its minimum value to its maximum value if going at its maximum rate.

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Timescale Definition Definition

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Timescale In our case, C(t) follows an exponential so we consider f(t)=e -kt with k=k 1 S 0 +k -1 +k 2. In our case, C(t) follows an exponential so we consider f(t)=e -kt with k=k 1 S 0 +k -1 +k 2.

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Timescale and step size The timescale of a process is a useful guide to determining the step size for numerical integration. The timescale of a process is a useful guide to determining the step size for numerical integration. A rule of thumb if using a fixed step size is to set it to no more than one-tenth of the timescale. A rule of thumb if using a fixed step size is to set it to no more than one-tenth of the timescale.

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Numerical Integration using Excel

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Enzyme-substrate kinetics

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Interactive demonstration (Model enzyme-substrate kinetics using (Model enzyme-substrate kinetics using Euler’s method - consider timescale Use Named cells) (Explore effect of step size) (Explore effect of step size)

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Second simplification: Assumption of quasi-steady state The assumption of substrate excess enables an exact solution for the differentials. A less demanding assumption is that S can change but only “slowly” such that C “keeps up” or The assumption of substrate excess enables an exact solution for the differentials. A less demanding assumption is that S can change but only “slowly” such that C “keeps up” or

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Assumption of quasi-steady state In this case, we can substitute S(t) for S 0 in the definition of In this case, we can substitute S(t) for S 0 in the definition of This leads to the Michaelis-Menten formulation This leads to the Michaelis-Menten formulation

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Interactive demonstration (Compare full kinetic model with analytical solution for C(t) under assumption of substrate excess) (Compare full kinetic model with analytical solution for C(t) under assumption of substrate excess) [Compare with quasi-steady state formulation] [Compare with quasi-steady state formulation] (Modify model to allow product feedback) (Modify model to allow product feedback)

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