Computational Biology, Part 15 Biochemical Kinetics IBiochemical KineticsDifferential equations vs. difference equationsDifference equationsDifferential equationsNumerical integrationSlide 7Slide 8GoalBoundary conditionsInitial value problemsExample biochemical systemEnzyme-substrate kineticsSlide 14Slide 15Slide 16Slide 17Slide 18What now?First simplification: Assumption of substrate excessAssumption of substrate excessSlide 22Slide 23TimescaleSlide 25Slide 26Timescale and step sizeNumerical Integration using ExcelSlide 29Interactive demonstrationSlide 31Second simplification: Assumption of quasi-steady stateAssumption of quasi-steady stateSlide 34Slide 35Slide 36Slide 37Computational Biology, Part 15Biochemical Kinetics IComputational Biology, Part 15Biochemical Kinetics IRobert F. MurphyRobert F. MurphyCopyright Copyright  1996, 1999-2009. 1996, 1999-2009.All rights reserved.All rights reserved.Biochemical KineticsBiochemical KineticsThe recursion relations we have used before The recursion relations we have used before could be expressed as could be expressed as differendifferencece equations equations..This is because an equation of the form This is because an equation of the form xxi+1i+1=f(x=f(xii)) can always be rewritten as can always be rewritten as xxii=f(x=f(xii)-x)-xiiAnalysis of the Analysis of the kineticskinetics of biochemical of biochemical reactions requires the use of reactions requires the use of differendifferentialtial equationsequations..Differential equations vs. difference equationsDifferential equations vs. difference equationsA differenA differencece equation expresses the change equation expresses the change in some variable as a result of a in some variable as a result of a finitefinite change in another variable.change in another variable.A differenA differentialtial equation expresses the change equation expresses the change in some variable as a result of an in some variable as a result of an infinitesimalinfinitesimal change in another variable. change in another variable.Difference equationsDifference equationsDifference equations allow direct, Difference equations allow direct, exact exact integrationintegration 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 variablesDifferential equationsDifferential equationsDifferential equations can sometimes be Differential equations can sometimes be solved solved analyticallyanalytically 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 derivativesinvolve derivativesAn alternative is to An alternative is to approximateapproximate the solution the solution by by numerical integrationnumerical integrationNumerical integrationNumerical 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 changeschangesWe must use a finite We must use a finite integration interval integration interval or or step sizestep size and thereby convert a differential and thereby convert a differential equation into a difference equationequation into a difference equationNumerical integrationNumerical integrationThe simplest numerical integration method The simplest numerical integration method is is Euler’s methodEuler’s method. It simply converts each . It simply converts each differential to a difference and calculates the differential to a difference and calculates the value of the dependent variables by value of the dependent variables 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.Numerical integrationNumerical integrationThe smaller the step size is, the greater the The smaller the step size is, the greater the accuracy obtained but the greater the number of accuracy obtained but the greater the number of calculations that must be done to get to a specific calculations that must be done to get to a specific value of the independent variablevalue of the independent variableTo increase efficiency, the step size can be To increase efficiency, the step size can be changed from one step to anotherchanged from one step to anotherIf the change in the dependent variable from the If the change in the dependent variable from the previous step to the current one is “small,” the step previous step to the current one is “small,” the step size can be increased (and vice versa)size can be increased (and vice versa)GoalGoalAs with the example from population As with the example from population dynamics, our goal is to describe how the dynamics, our goal is to describe how the behavior of a system depends on behavior of a system depends on parameters parameters (e.g., rate constants) and (e.g., rate constants) and boundary conditions boundary conditions (e.g., initial (e.g., initial concentrations)concentrations)Boundary conditionsBoundary conditionsBoundary conditions can be divided into two Boundary conditions can be divided into two categoriescategoriesInitial value problems Initial value problems occur when all dependent occur when all dependent variables are known at some starting value of the variables are known at some starting value of the independent variableindependent variableTwo-point boundary problems Two-point boundary problems occur when some occur when some dependent variables are known only at one value of dependent variables are known only at one value of the independent variable and the rest are known only the independent variable and the rest are known only at some other value of the independent variableat some other value of the independent variableInitial value problemsInitial value problemsWe will consider only initial value We will consider only initial value problems, where we wish to calculate the problems, where we wish to