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LecturesPart18

LecturesPart18 - Computational Biology Part 18 Biochemical...

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Computational Biology, Part 18 Biochemical Kinetics II Robert F. Murphy Robert F. Murphy Copyright Copyright 1996, 1999-2006. 1996, 1999-2006. All rights reserved. All rights reserved.
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General form of ordinary differential equations For a set of For a set of n n unknown functions unknown functions y y i (for (for i i =1 =1 to to n n ) we are given a set of ) we are given a set of n n functions functions f f i that specify the derivatives of each that specify the derivatives of each y y i with with respect to some independent variable respect to some independent variable x x
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Euler’s method The simplest numerical integration method The simplest numerical integration method is is Euler’s method Euler’s method . It simply converts each . It simply converts each differential to a difference differential to a difference and then calculates the value of and then calculates the value of y y i by by multiplying the right hand side of each multiplying the right hand side of each differential equation by the step size differential equation by the step size x x
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Euler’s method We can rewrite this as a recursion formula We can rewrite this as a recursion formula that allows us to calculate values of the that allows us to calculate values of the functions functions y y i at a series of at a series of x x values. We values. We introduce a second subscript introduce a second subscript j j to indicate to indicate which which x x value we refer to (note that value we refer to (note that y y i,j i,j now now refers to a refers to a value value not a not a function function ). ).
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Euler’s method Note the asymmetry of this method: the Note the asymmetry of this method: the derivative ( derivative ( f f i ) that is used to span the ) that is used to span the x x is is calculated calculated only only for the for the x x value at the value at the beginning beginning of the interval. In regions where of the interval. In regions where f f i is increasing with is increasing with x, x, this leads to this leads to under under estimation of estimation of y, y, and, in regions and, in regions where where f f i is decreasing with is decreasing with x, x, to to over over estimation of estimation of y.
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