Biochemical Kinetics: Difference & Differential Equations in Computational Biology, Slides of Computational Biology

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Computational Biology, Part 15

Biochemical Kinetics I

Robert F. Murphy

Biochemical Kinetics

 The recursion relations we have used before could be expressed as difference equations.

 This is because an equation of the form xi+1=f(xi) can always be rewritten as

 xi=f(xi)-xi

 Analysis of the kinetics of biochemical reactions requires the use of differential equations.

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 imply a “synchronicity” to changes in variables

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

 An alternative is to approximate the solution by numerical integration

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.

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

 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)

Boundary conditions

 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

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

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 ).

 What is an expression for dE/dt?

E  S

k 1   k  1

C

k 2  E  P

Enzyme-substrate

kinetics E  S

k 1   k  1

C

k 2  E  P

dE dt

  k 1 ES   k  1  k 2  C

dS dt

?

dE dt

  k 1 ES   k  1  k 2  C

dS dt

  k 1 ES  k  1 C dC dt

 k 1 ES   k  1  k 2  C

dP dt

?

Enzyme-substrate

kinetics E  S

k 1   k  1

C

k 2  E  P

dE dt

  k 1 ES   k  1  k 2  C

dS dt

  k 1 ES  k  1 C dC dt

 k 1 ES   k  1  k 2  C

dP dt

 k 2 C

Enzyme-substrate

kinetics E  S

k 1   k  1

C

k 2  E  P

What now?

 We have a set of four coupled differential equations that cannot be solved analytically.

 We can

Try to simplify them using various assumptions so that they can be solved analytically, or Integrate them numerically

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: S ( t )  S 0