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Thread Subject:
Modeling Stochasticity of a ODE model consisting of Hill Kinetics and Michaelis-Menten equation.

Subject: Modeling Stochasticity of a ODE model consisting of Hill Kinetics and Michaelis-Menten equation.

From: James Ooi

Date: 9 Jun, 2011 18:01:04

Message: 1 of 4

Hi there,

I am working on a model where its equations are of Michaelis-Menten and Hill Kinetics type of equation. I want to study the stochastic nature of my model since I have pretty much explored its deterministic behavior.

Does anyone has a suggestion on what algorithm can be used or any method to simulate stochasticity?

I have explored Gillespie and Monte Carlo, both of which require me to break down my equations to simple mass action kinetics. I would like to avoid it if possible.

Example of equations that I m using is as below:
x1=k2*x2/(x2+1)*x3*(x5^2/(1/(ka*kb)+1/kc*x5+x5^2));
x2=k1*ct/(1+ks*ct);
x5=k2*x5/(x5+1)*x2*(x5^2/(1/(kk*kc)+1/kl*x5+x5^2));

Appreciate your comment and time looking into my problem. Thank you.


James

Subject: Modeling Stochasticity of a ODE model consisting of Hill Kinetics and Michaelis-Menten equation.

From: Roger Stafford

Date: 9 Jun, 2011 18:50:21

Message: 2 of 4

"James Ooi" wrote in message <isr1p0$fth$1@newscl01ah.mathworks.com>...
> Hi there,
>
> I am working on a model where its equations are of Michaelis-Menten and Hill Kinetics type of equation. I want to study the stochastic nature of my model since I have pretty much explored its deterministic behavior.
>
> Does anyone has a suggestion on what algorithm can be used or any method to simulate stochasticity?
>
> I have explored Gillespie and Monte Carlo, both of which require me to break down my equations to simple mass action kinetics. I would like to avoid it if possible.
>
> Example of equations that I m using is as below:
> x1=k2*x2/(x2+1)*x3*(x5^2/(1/(ka*kb)+1/kc*x5+x5^2));
> x2=k1*ct/(1+ks*ct);
> x5=k2*x5/(x5+1)*x2*(x5^2/(1/(kk*kc)+1/kl*x5+x5^2));
>
> Appreciate your comment and time looking into my problem. Thank you.
> James
- - - - - - - - - -
  James, I seriously doubt if many in this newsgroup are sufficiently familiar with this area of enzyme kinetics to be able to intelligently answer your question as it stands. Few, including myself, would ever have heard of the Michaelis-Menten and Hill type equations.

  I would advise you to rephrase your request in such a manner that it can be understood by a much wider audience. In particular, how does stochasticity enter into the equations you describe? What quantities involve stochastic variation? If you do not wish to use a Monte Carlo method of simulation, what manner of algorithm do you envision carrying out the desired analysis? Can you give any concrete examples of such algorithms as you have in mind?

Roger Stafford

Subject: Modeling Stochasticity of a ODE model consisting of Hill Kinetics and Michaelis-Menten equation.

From: James Ooi

Date: 9 Jun, 2011 19:34:05

Message: 3 of 4

"Roger Stafford" wrote in message <isr4ld$otb$1@newscl01ah.mathworks.com>...
> "James Ooi" wrote in message <isr1p0$fth$1@newscl01ah.mathworks.com>...
> > Hi there,
> >
> > I am working on a model where its equations are of Michaelis-Menten and Hill Kinetics type of equation. I want to study the stochastic nature of my model since I have pretty much explored its deterministic behavior.
> >
> > Does anyone has a suggestion on what algorithm can be used or any method to simulate stochasticity?
> >
> > I have explored Gillespie and Monte Carlo, both of which require me to break down my equations to simple mass action kinetics. I would like to avoid it if possible.
> >
> > Example of equations that I m using is as below:
> > x1=k2*x2/(x2+1)*x3*(x5^2/(1/(ka*kb)+1/kc*x5+x5^2));
> > x2=k1*ct/(1+ks*ct);
> > x5=k2*x5/(x5+1)*x2*(x5^2/(1/(kk*kc)+1/kl*x5+x5^2));
> >
> > Appreciate your comment and time looking into my problem. Thank you.
> > James
> - - - - - - - - - -
> James, I seriously doubt if many in this newsgroup are sufficiently familiar with this area of enzyme kinetics to be able to intelligently answer your question as it stands. Few, including myself, would ever have heard of the Michaelis-Menten and Hill type equations.
>
> I would advise you to rephrase your request in such a manner that it can be understood by a much wider audience. In particular, how does stochasticity enter into the equations you describe? What quantities involve stochastic variation? If you do not wish to use a Monte Carlo method of simulation, what manner of algorithm do you envision carrying out the desired analysis? Can you give any concrete examples of such algorithms as you have in mind?
>
> Roger Stafford

Hi Roger,
Thanks for response.

For an equation that count each molecule as a discreet event, Gillespie is a good approach, taking a random number then selecting which step (reaction) it will take place.
However, in my ODE model, it is a continuous event (concentration of a species) rather than a discreet event. Hence the stochastic event is the random selection of its concentration.

What I have in mind is that for every integration step, instead of taking a fixed time step(ie. 1s), it will take a random time step(ie. 0.5 - 1.5). The exact numerical solution should also be randomized so that each random time step, the solution falls randomly between a range.

Does my description of an algorithm seems plausible?

Thanks again.

James
 

Subject: Modeling Stochasticity of a ODE model consisting of Hill Kinetics and Michaelis-Menten equation.

From: Roger Stafford

Date: 9 Jun, 2011 19:59:04

Message: 4 of 4

"James Ooi" wrote in message <isr77c$3d5$1@newscl01ah.mathworks.com>...
> Hi Roger,
> Thanks for response.
>
> For an equation that count each molecule as a discreet event, Gillespie is a good approach, taking a random number then selecting which step (reaction) it will take place.
> However, in my ODE model, it is a continuous event (concentration of a species) rather than a discreet event. Hence the stochastic event is the random selection of its concentration.
>
> What I have in mind is that for every integration step, instead of taking a fixed time step(ie. 1s), it will take a random time step(ie. 0.5 - 1.5). The exact numerical solution should also be randomized so that each random time step, the solution falls randomly between a range.
>
> Does my description of an algorithm seems plausible?
>
> Thanks again.
>
> James
- - - - - - - - -
  Well that helps, but I still don't understand how this relates to the equations you quoted. How does time enter into these equations? What is the meaning of the variables x1, x2, x3, x5, ka, kb, etc., and where do the time steps enter in? Which are concentrations? You need to tell us a lot more.

  Remember I said that I, and probably many others, have never before been exposed to these equations of Michaelis-Menten and Hill.

Roger Stafford

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