Computational Biology, Part 17 Biochemical Kinetics I Robert F. Murphy Copyright 1996, 1999-2006. All rights reserved

Computational Biology, Part 17

Biochemical Kinetics I

Computational Biology, Part 17

Biochemical Kinetics I

Robert F. MurphyRobert F. Murphy

Copyright Copyright 1996, 1999- 1996, 1999-2006.2006.

All rights reserved.All rights reserved.

Biochemical KineticsBiochemical Kinetics

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

This is because an equation of the This is because an equation of the form form xxi+1i+1=f(x=f(xii)) can always be can always be rewritten as rewritten as xxii=f(x=f(xii)-x)-xii

Analysis of the Analysis of the kineticskinetics of of biochemical reactions requires the biochemical reactions requires the use of use of differendifferentialtial equations equations..

Differential equations vs. difference equations

Differential equations vs. difference equations A differenA differencece equation expresses equation expresses the change in some variable as a the change in some variable as a result of a result of a finitefinite change in change in another variable.another variable.

A differenA differentialtial equation equation expresses the change in some expresses the change in some variable as a result of an variable as a result of an infinitesimalinfinitesimal change in another change in another variable.variable.

Difference equationsDifference equations

Difference equations allow Difference equations allow direct, direct, exact exact integrationintegration to to calculate the values of dependent calculate the values of dependent variables at all values of the variables at all values of the independent variable (such as independent variable (such as generation number)generation number)

Difference equations imply a Difference equations imply a “synchronicity” to changes in “synchronicity” to changes in variablesvariables

Differential equationsDifferential equations

Differential equations can Differential equations can sometimes be solved sometimes be solved analyticallyanalytically to yield an equation for the to yield an equation for the dependent variable as a function dependent variable as a function of the independent variable(s) of the independent variable(s) that does not involve derivativesthat does not involve derivatives

An alternative is to An alternative is to approximateapproximate the solution by the solution by numerical numerical integrationintegration

Numerical integrationNumerical integration

Numerical integration of Numerical integration of differential equations only differential equations only yields an approximation because yields an approximation because we cannot calculate infinitesimal we cannot calculate infinitesimal changeschanges

We must use a finite We must use a finite integration integration interval interval or or step sizestep size and thereby and thereby convert a differential equation convert a differential equation into a difference equationinto a difference equation


The simplest numerical The simplest numerical integration method is integration method is Euler’s Euler’s methodmethod. It simply converts each . It simply converts each differential to a difference and differential to a difference and calculates the value of the calculates the value of the dependent variables by dependent variables by multiplying the right hand side multiplying the right hand side of each differential equation by of each differential equation by the step size.the step size.


The smaller the step size is, the The smaller the step size is, the greater the accuracy obtained but the greater the accuracy obtained but the greater the number of calculations that greater the number of calculations that must be done to get to a specific value must be done to get to a specific value of the independent variableof the independent variable

To increase efficiency, the step size To increase efficiency, the step size can be changed from one step to anothercan be changed from one step to another If the change in the dependent variable If the change in the dependent variable from the previous step to the current one from the previous step to the current one is “small,” the step size can be increased is “small,” the step size can be increased (and vice versa)(and vice versa)

GoalGoal

As with the example from As with the example from population dynamics, our goal population dynamics, our goal is to describe how the is to describe how the behavior of a system depends behavior of a system depends on on parameters parameters (e.g., rate (e.g., rate constants) and constants) and boundary boundary conditions conditions (e.g., initial (e.g., initial concentrations)concentrations)

Boundary conditionsBoundary conditions

Boundary conditions can be divided Boundary conditions can be divided into two categoriesinto two categories Initial value problems Initial value problems occur when all occur when all dependent variables are known at some dependent variables are known at some starting value of the independent starting value of the independent variablevariable

Two-point boundary problems Two-point boundary problems occur when occur when some dependent variables are known only some dependent variables are known only at one value of the independent variable at one value of the independent variable and the rest are known only at some and the rest are known only at some other value of the independent variableother value of the independent variable

Initial value problemsInitial value problems

We will consider only initial We will consider only initial value problems, where we wish value problems, where we wish to calculate the values of to calculate the values of the dependent variables at the dependent variables at some point or set of points some point or set of points different from the initial different from the initial pointpoint

Example biochemical systemExample biochemical system For illustration, we will For illustration, we will consider a simple, well-consider a simple, well-studied biochemical reaction, studied biochemical reaction, the enzyme-catalyzed the enzyme-catalyzed conversion of a substrate conversion of a substrate into a productinto a productE + S

k1→

k−1← C

k2→ E + P

Enzyme-substrate kineticsEnzyme-substrate kinetics

We can write four differential We can write four differential equations describing this equations describing this system. We will use system. We will use EE as as shorthand for shorthand for EE((tt), ), SS for for SS((tt), ), CC for for C C((tt), and ), and PP for for PP((tt).).

What is an expression for What is an expression for dE/dtdE/dt??

E + Sk1→

k−1← C

k2→ E + P

Enzyme-substrate kinetics

Enzyme-substrate kinetics E + S

k1→

k−1← C

k2→ E + P

dE

dt=−k1ES+ k−1 + k2( )C

dSdt

=?

dE

dt=−k1ES+ k−1 + k2( )CdSdt

=−k1ES+ k−1CdCdt

=?



k1→

k−1← C

k2→ E + P

dE



=k1ES− k−1 + k2( )CdPdt

=?



k1→

k−1← C

k2→ E + P

dE




=k2C



k1→

k−1← C

k2→ E + P

Enzyme-substrate kineticsEnzyme-substrate kinetics Boundary conditionsBoundary conditions

Normally, enzyme and substrate are Normally, enzyme and substrate are mixed at time 0, so product and complex mixed at time 0, so product and complex concentrations are initially 0: concentrations are initially 0: CC00==PP00=0.=0.

E (0 ) = E0

S(0 ) = S0C(0 ) = C0

P(0 ) = P0

What now?What now?

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

We canWe can Try to simplify them using Try to simplify them using various assumptions so that they various assumptions so that they can be solved analytically, orcan be solved analytically, or

Integrate them numericallyIntegrate them numerically

First simplification: Assumption of substrate excess

First simplification: Assumption of substrate excess To simplify system, we first To simplify system, we first assume that the substrate is assume that the substrate is present in such a high present in such a high concentration that it is always concentration that it is always in vast excess over the enzyme in vast excess over the enzyme concentration. In this case, concentration. In this case, the substrate concentration may the substrate concentration may be viewed as remaining be viewed as remaining constant:constant:

S( t ) ≡ S0

Assumption of substrate excessAssumption of substrate excess Enzyme is either free or in Enzyme is either free or in complex. Mass balance gives an complex. Mass balance gives an expression for expression for EE

Substituting this for Substituting this for EE and and SS00 for for SS in the original differential in the original differential equation for equation for CC gives gives

E + C = E0 + C0

E = E0 + C0 −C

Assumption of substrate excessAssumption of substrate excess This can be integrated This can be integrated directly to givedirectly to giveC ( t ) = C0 −C( )e− k1S0 +k−1 +k2( ) t + C

where

C =E0 + C0( )S0Km + S0and

Km =k−1 + k2k1

Assumption of substrate excessAssumption of substrate excess Conclusion: Complex Conclusion: Complex concentration asymptotically concentration asymptotically approaches the steady-state approaches the steady-state concentration, concentration,

C

TimescaleTimescale

How long does it take to reach the How long does it take to reach the steady state? It must depend on the steady state? It must depend on the kk’s since they are in the term in ’s since they are in the term in front of front of tt in the exponential. in the exponential.

One characterization of the One characterization of the timescaletimescale of a process: How long does it take of a process: How long does it take the function describing the process the function describing the process to go from its minimum value to its to go from its minimum value to its maximum value if going at its maximum maximum value if going at its maximum rate.rate.

TimescaleTimescale

DefinitionDefinition

timescale ( f ) =fmax − fmindf

dt max

TimescaleTimescale

In our case, In our case, CC((tt) follows an ) follows an exponential so we consider exponential so we consider ff((tt)=)=ee--ktkt with with kk==kk11SS00++kk-1-1++kk22..for 0 ≤ t < ∞

fmax = e−k0 =1 andfmin = e−k∞ = 0

dfdt max

= −ke−ktmax

= k

∴ timescale(e−kt ) =1 −0

k= k−1

Timescale and step sizeTimescale and step size

The timescale of a process is The timescale of a process is a useful guide to determining a useful guide to determining the step size for numerical the step size for numerical integration.integration.

A rule of thumb if using a A rule of thumb if using a fixed step size is to set it fixed step size is to set it to no more than one-tenth of to no more than one-tenth of the timescale.the timescale.

Numerical Integration using ExcelNumerical Integration using Excel

dE




=k2C



k1→

k−1← C

k2→ E + P

Interactive demonstrationInteractive demonstration (Model enzyme-substrate (Model enzyme-substrate kinetics usingkinetics using Euler’s method - consider Euler’s method - consider timescaletimescale

Use Named cells)Use Named cells) (Explore effect of step size)(Explore effect of step size)

k1= 0.1 deltat= 0.1km1= 0.005 timescale= =1/(_k1*C6+_km1+_k2)k2= 0.1

t E S C P0 1 10 0 0=A6+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A7+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A8+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A9+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A10+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A11+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A12+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A13+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A14+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A15+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A16+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A17+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A18+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A19+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A20+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A21+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A22+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A23+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A24+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A25+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A26+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A27+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A28+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A29+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A30+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A31+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A32+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A33+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A34+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A35+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A36+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A37+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A38+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A39+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat

Second simplification: Assumption of quasi-steady state

Second simplification: Assumption of quasi-steady state The assumption of substrate The assumption of substrate excess enables an exact excess enables an exact solution for the solution for the differentials. A less differentials. A less demanding assumption is that demanding assumption is that SS can change but only can change but only “slowly” such that “slowly” such that CC “keeps “keeps up” orup” or

dC

dt≈ 0

Assumption of quasi-steady stateAssumption of quasi-steady state In this case, we can In this case, we can substitute substitute SS((tt) for ) for SS00 in the in the definition of definition of

This leads to the Michaelis-This leads to the Michaelis-Menten formulationMenten formulation

C

dS

dt≈−k2C ≈−k2 C

dSdt

=−k2 E0 + C0( )SKm + S( )

Interactive demonstrationInteractive demonstration (Compare full kinetic model (Compare full kinetic model with analytical solution for with analytical solution for CC((tt) under assumption of ) under assumption of substrate excess)substrate excess)

[Compare with quasi-steady [Compare with quasi-steady state formulation]state formulation]

(Modify model to allow (Modify model to allow product feedback)product feedback)

k1= 0.01 deltat= =E2/10km1= 0.005 timescale= =1/(_k1*C6+_km1+_k2)k2= 0.01 Km= =(_km1+_k2)/_k1

Cbar= =(B6+D6)*C6/(E3+C6)t E S C P0 1 10 0 0=A6+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A7+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A8+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A9+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A10+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A11+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A12+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A13+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A14+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A15+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A16+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A17+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A18+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A19+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A20+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A21+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A22+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A23+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A24+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A25+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A26+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A27+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A28+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A29+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A30+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A31+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A32+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A33+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A34+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A35+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A36+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A37+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A38+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat=A39+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C)*deltat=P+(_k2*_C)*deltat

k1= 0.01 deltat= =E2/10km1= 0.005 timescale= =1/(_k1*C6+_km1+_k2)k2= 0.01 Km= =(_km1+_k2)/_k1km2= 0.01 Cbar= =(B6+D6)*C6/(E3+C6)t E S C P0 1 10 0 0=A6+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A7+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A8+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A9+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A10+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A11+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A12+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A13+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A14+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A15+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A16+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A17+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A18+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A19+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A20+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A21+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A22+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A23+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A24+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A25+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A26+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A27+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A28+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A29+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A30+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A31+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A32+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A33+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A34+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A35+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A36+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A37+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A38+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat=A39+deltat =E+(-_k1*E*S+(_km1+_k2)*_C)*deltat=S+(-_k1*E*S+_km1*_C)*deltat=_C+(_k1*E*S-(_km1+_k2)*_C+_km2*P)*deltat =P+(_k2*_C-_km2*P)*deltat

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Computational Biology, Part 17 Biochemical Kinetics I Robert F. Murphy Copyright 1996, 1999-2006. All rights reserved