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LecturesPart17

# LecturesPart17 - Computational Biology Part 17 Biochemical...

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Biochemical Kinetics The recursion relations we have used before The recursion relations we have used before could be expressed as could be expressed as differen differen ce ce equations equations . . This is because an equation of the form This is because an equation of the form x x i+1 i+1 =f(x =f(x i ) ) can always be rewritten as can always be rewritten as x x i =f(x =f(x i )-x )-x i Analysis of the Analysis of the kinetics kinetics of biochemical of biochemical reactions requires the use of reactions requires the use of differen differen tial tial equations equations . .
Differential equations vs. difference equations A differen A differen ce ce equation expresses the change equation expresses the change in some variable as a result of a in some variable as a result of a finite finite change in another variable. change in another variable. A differen A differen tial tial equation expresses the change equation expresses the change in some variable as a result of an in some variable as a result of an infinitesimal infinitesimal change in another variable. change in another variable.

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Difference equations Difference equations allow direct, Difference equations allow direct, exact exact integration integration to calculate the values of to calculate the values of dependent variables at all values of the dependent variables at all values of the independent variable (such as generation independent variable (such as generation number) number) Difference equations imply a Difference equations imply a “synchronicity” to changes in variables “synchronicity” to changes in variables
Differential equations Differential equations can sometimes be Differential equations can sometimes be solved solved analytically analytically to yield an equation for to yield an equation for the dependent variable as a function of the the dependent variable as a function of the independent variable(s) that does not independent variable(s) that does not involve derivatives involve derivatives An alternative is to An alternative is to approximate approximate the the solution by solution by numerical integration numerical integration

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Numerical integration Numerical integration of differential Numerical integration of differential equations only yields an approximation equations only yields an approximation because we cannot calculate infinitesimal because we cannot calculate infinitesimal changes changes We must use a finite We must use a finite integration interval integration interval or or step size step size and thereby convert a differential and thereby convert a differential equation into a difference equation equation into a difference equation
Numerical integration The simplest numerical integration method The simplest numerical integration method is is Euler’s method

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LecturesPart17 - Computational Biology Part 17 Biochemical...

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