Biochemical Kinetics III: Numerical Methods for ODE Integration, Slides of Computational Biology

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Computational Biology, Part 17

Biochemical Kinetics III

Robert F. Murphy

General form of ordinary

differential equations

 For a set of n unknown functions yi (for i = to n ) we are given a set of n functions fi  that specify the derivatives of each yi with respect to some independent variable x

dyi ( x )

dx

 f i ( x , y 1 , ..., yn ) for i  1, ..., n

Euler’s method

 We can rewrite this as a recursion formula that allows us to calculate values of the functions yi at a series of x values. We introduce a second subscript j to indicate which x value we refer to (note that yi,j now refers to a value not a function). yi , j  1  yi , j   x  f (^) i ( x (^) j , y 1, j , ..., yn , j ) x (^) j  1  x (^) j   x

Euler’s method

 Note the asymmetry of this method: the derivative ( fi  ) that is used to span the  x is calculated only for the x value at the beginning of the interval. In regions where fi  is increasing with x, this leads to underestimation of  y, and, in regions where fi  is decreasing with x, to overestimation of  y.

Midpoint method

 A better estimate would come from evaluating the fi  at the midpoint of the  x interval. The problem: we know x at the midpoint but we don’t know the yi at the midpoint (yet). The solution is to use Euler’s method to estimate  y and then re- estimate  y using the derivatives evaluated halfway along the line segment encompassing the original  y.

Midpoint method (2nd order

Runge-Kutte)

 This is called the midpoint method or the second-order Runge-Kutte method.

 y (^) j   x  f (^) ( x (^) j , y (^) j )

 y (^) j  0.5   x  f ( x (^) j 

 x 2

, y (^) j 

 y (^) j 2

y (^) j  1  y (^) j   y (^) j  0.

Midpoint method (2nd order

Runge-Kutte)

 Now consider dy/dx =3 y. The analytical solution is y =e^3 x.  The graph shows the Euler, Midpoint and analytical solutions (along with derivatives). Three methods for dy/dx =3 y

(^100) 2030

4050

6070

8090

100

0 0.5 (^) x 1 1.

y

0

50

100

150

200

250 y (2°RK)^300 y (Analy t)y (Euler) dy /dx(x)dy /dx(x+Dx/2)

Note that the midpoint method is better than Euler’s method but it does not give results identical to the analytical solution since dy/dx now depends on y.

Fourth-order Runge-Kutta

 The midpoint method can be extended by considering other intermediate estimates. The most frequently used variation is the fourth-order Runge-Kutta method which considers one estimate at the initial point, two estimates at the midpoint, and one estimate at a trial endpoint.