Team:Aberdeen Scotland/internal/stochastic

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= Stochastic Simulations =
= Stochastic Simulations =
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Due to the low levels of proteins involved when the input signal is activated (lacI repression being lifted and the subsequent lifting of the repression of tetR that induces lysis) we decided to create a stochastic model to create a more realistic simulation of the process. The model we chose was the “tau leap” model is it is quite computationally efficient and easy to integrate when a deterministic model is already in place. The method can be implemented in the following simple steps...
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Due to the low levels of proteins involved when the input signal is activated (LacI repression being lifted and the subsequent lifting of the repression of TetR that induces lysis) we decided to perform a stochastic simulation of the model to take into consideration stochastic effects not reproduced by the deterministic model. The method we chose was the “Tau Leap” model - as is it is quite computationally efficient and easy to integrate when a deterministic model has already been established. The method can be implemented by the following simple steps:
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1. Create a Poisson random number generating script ( we did this in C++) that accepts a number λ and outputs  an integer distributed with a Poisson distribution around this number.
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1. Choose a time, τ, which is large enough so that all reactions have a possibility of taking place, but small enough so that not too many reactions take place. This is the leap condition from [1].
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2. Chose a time, tau, which is large enough that all reactions have a possibility of taking place, but small enough that not too many reactions will take place.
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2. Generate a random number froma Poisson distribution with mean λ=ζ.τ, where τ is the chosen time interval for the stochastic simulation, and ζ represents a term on the right hand side of the ordinary differential equation from the deterministic model.[2]
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3. Take the deterministic equations, and multiply each term by tau.
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3. Multiply each term in the deterministic equations by tau.
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4. Take the new value of each term and call it λ.
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4. Assign the label λ to the new value of each term.
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5. Input this value of λ into the Poisson random number generator
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5. Input the value of λ into the Poisson random number generator
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6. Now have now replaced each term in the deterministic equation with an integer distributed around the value of the original term, so we simply add or subtract the integers from the old value of [X_t] to create [X_(t+tau)]
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6. We have now replaced each term in the deterministic equations with an integer distributed around the value of the original term, so we simply add or subtract the integers from the old value of [X<sub>t</sub>] to create [X<sub>(t+tau)</sub>]
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This process can be very fast computationally depending on the choice of tau. We ran several simulations to determine the dependence of tau on how the model worked and found that the choice was fairly arbitary when taken between ~0.01 seconds and ~1.5 seconds. However, the larger tau is, the quicker the simulation runs. We see the comparison between tau = 0.01 seconds, tau = 0.1 seconds and tau =1.5 seconds below.
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This process can be computationally very fast depending on the choice of tau. We ran several simulations to determine the dependence of the results on tau and found that the results were robust for tau between ~0.01 seconds and ~1.5 seconds. As expected, the larger tau is, the quicker the simulation runs.  
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[[Image:Stochastic 3.jpg|center|700px]]
 
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[[Image:Stochastic 1.jpg|center|700px]]
 
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[[Image:Stochastic 2.jpg|center|700px]]
 
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== References ==
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So the three simulations with different tau timesteps all work almost identically. We note however that increasing tau above 2 seconds can lead to some very strange behaviour which we will not show here.
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[1] Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems. Journal of Chemical Physics, 115:1716-1733
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[2] T. Tian, K.Burrage, P. M. Burrage and M. Carletti (2006): Stochastic Delay Differential Equations for Genetic Regulatory Networks, Special Issue of J. Comp and Applied Maths, doi:10.1016/j.cam.2006.02.063
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= Stochastic VS deterministic =
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= Stochastic vs. deterministic =
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here are some sample graphs showing how the stochastic simulation compares to the deterministic simulation. Not all proteins and mRNAs are plotted as it can become very confusing
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[[Image:Stochastic 4.jpg|center|700px]]
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[[Image:Mrna2.png|center|700px]]
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<a href="https://2009.igem.org/Team:Aberdeen_Scotland/internal/deterministic"><img src="https://static.igem.org/mediawiki/2009/e/ed/Aberdeen_Left_arrow.png">&nbsp;&nbsp;Back to Deterministic Model</a>
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<a href="https://2009.igem.org/Team:Aberdeen_Scotland/internal/SimBiology">Continue to SimBiology&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2009/4/4c/Aberdeen_Right_arrow.png"></a>
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Latest revision as of 09:40, 18 August 2009

University of Aberdeen iGEM 2009

Stochastic Simulations

Due to the low levels of proteins involved when the input signal is activated (LacI repression being lifted and the subsequent lifting of the repression of TetR that induces lysis) we decided to perform a stochastic simulation of the model to take into consideration stochastic effects not reproduced by the deterministic model. The method we chose was the “Tau Leap” model - as is it is quite computationally efficient and easy to integrate when a deterministic model has already been established. The method can be implemented by the following simple steps:


1. Choose a time, τ, which is large enough so that all reactions have a possibility of taking place, but small enough so that not too many reactions take place. This is the leap condition from [1].

2. Generate a random number froma Poisson distribution with mean λ=ζ.τ, where τ is the chosen time interval for the stochastic simulation, and ζ represents a term on the right hand side of the ordinary differential equation from the deterministic model.[2]

3. Multiply each term in the deterministic equations by tau.

4. Assign the label λ to the new value of each term.

5. Input the value of λ into the Poisson random number generator

6. We have now replaced each term in the deterministic equations with an integer distributed around the value of the original term, so we simply add or subtract the integers from the old value of [Xt] to create [X(t+tau)]

This process can be computationally very fast depending on the choice of tau. We ran several simulations to determine the dependence of the results on tau and found that the results were robust for tau between ~0.01 seconds and ~1.5 seconds. As expected, the larger tau is, the quicker the simulation runs.



References

[1] Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems. Journal of Chemical Physics, 115:1716-1733

[2] T. Tian, K.Burrage, P. M. Burrage and M. Carletti (2006): Stochastic Delay Differential Equations for Genetic Regulatory Networks, Special Issue of J. Comp and Applied Maths, doi:10.1016/j.cam.2006.02.063

Stochastic vs. deterministic

here are some sample graphs showing how the stochastic simulation compares to the deterministic simulation. Not all proteins and mRNAs are plotted as it can become very confusing

Stochastic 4.jpg
Mrna2.png


  Back to Deterministic Model Continue to SimBiology  



 

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