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| | + | <a href="https://2012.igem.org/wiki/index.php?title=Team:Grenoble/Modeling/Amplification/ODE"><img src="https://static.igem.org/mediawiki/2012/e/ef/ODE.png" alt="" /></a> |
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| | + | <a href="https://2012.igem.org/wiki/index.php?title=Team:Grenoble/Modeling/Amplification/Stochastic"><img src="https://static.igem.org/mediawiki/2012/a/ad/Stochastic_analysis.png" alt="" /></a> |
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| | <section> | | <section> |
| - | <h1> Goal </h1>
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| - | </br>
| + | <a href="https://2012.igem.org/ wiki/ index. php?title= Team: Grenoble/ Modeling/ Amplification/ Stochastic/ what"><img src="https://static.igem.org/mediawiki/2012/ d/ d9/ What.png" alt="" /></a> |
| - | Statistic modeling is a technique of presenting data or predicting outcomes that takes into account a certain degree of randomness or unpredictability. The stochastic process is often used to represent the evolution of some random value, or system, over time.
| + | <a href="https://2012.igem.org/ Team: Grenoble/ Modeling/ Amplification/ Stochastic/ results"><img src="https://static.igem.org/mediawiki/2012/ 1/ 17/ Results.png" alt="" /></ a> |
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| - | <center><img src="https://static.igem.org/mediawiki/2012/e/e7/Stochastic_def.png" alt="" /></center> | + | |
| - | </br> | + | |
| - | It is the probabilistic counterpart to a deterministic process.
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| - | <center><img src="https://static.igem.org/mediawiki/2012/3/38/Diagram_stoch.png" alt="" /></center>
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| - | <h1> Why </h1>
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| - | </br>
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| - | We have used a stochastic process because in one bacteria we don't have enough molecules to consider that one miological element has a continuous value. The behavior of those biological elements is regulated by probability laws.
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| - | </br>
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| - | In biology systems, introducing stochastic noise has been found to help improve the signal strength of the internal feedback loops for balance and other vestibular communication.
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| - | </br>
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| - | <h1> How </h1>
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| - | </br>
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| - | Rather than using fixed variables such as in other mathematical modeling, a stochastic model incorporates random variations to predict future conditions and to see what they might be like.
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| - | </br>
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| - | To introduce that randomness we use a new function : propensities.
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| - | </br>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/c/ca/Propensity.png" alt="" /></center>
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| - | </br>
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| - | For example we take four possible reactions. Each reaction has a probability to happen in the next amount of time.
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| - | </br>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/8/8c/Reactions.png" alt="" /></center>
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| - | </br>
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| - | We randomly chose the next reaction regarding the propensities.
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| - | </br>
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| - | </br>
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| - | When we run the script once, we get a graph. This graph represent the rando evolution of an element. Because of this randomness, if we run the script an other time we will get a different graph. That is why to be able to interpret the results we have to run the cripts hundreds or thousands of times.
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| - | </br>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/ 9/ 9c/ Curves.png" alt="" /></ center> | + | |
| - | </br>
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| - | Instead of describing a process which can only evolve in one way, in a stochastic or random process there is some indeterminacy : even if the initial condition is known, there are several directions in which the process may evolve.
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| - | </br>
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| - | </br> | + | |
| - | To model that randomness we use a Gillepsie algorithm or Stoachastic Simulation Algorithm (SSA).
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| - | </br>
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| - | </br>
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| - | <h1> Gillespie _ Stochastic Simulation Algorithm </h1>
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| - | </br>
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| - | The Gillespie algorithm generates a statistically correct trajectory of a stochastic equation.
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| - | </br>
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| - | </br>
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| - | Traditional continuous and deterministic biochemical rate equations do not accurately predict cellular reactions since they rely on bulk reactions that require the interactions of millions of molecules.
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| - | In contrast, the Gillespie algorithm allows a discrete and stochastic simulation of a system with few reactants because every molecule is explicitly simulated. When simulated, a Gillespie realization represents a random walk of the entire system.
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| - | </br>
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| - | </br>
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| - | We assert :
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| - | </br>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/3/37/Assert.png" alt="" /></center>
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| - | </br>
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| - | Each reaction Rj is characterized mathematically by two quantities :
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| - | </br>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/5/50/State_change_vector.png" alt="" /></center>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/9/9a/Change.png" alt="" /></center>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/ b/ bc/ Plot.png" alt="" /></ center> | + | |
| - | <center><img src="https://static.igem.org/mediawiki/2012/d/d0/Prop.png" alt="" /></center>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/6/61/Prop2.png" alt="" /></center>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/0/09/Prop3.png" alt="" /></center>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/f/f9/Prop4.png" alt="" /></center>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/3/32/Prop5.png" alt="" /></center>
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| - | </br>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/e/e8/More_precisely.png" alt="" /></center>
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| - | </br>
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| - | Once we have put the theory of the stochastic approach, we can determine the algorithm.
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| - | </br>
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| - | </br>
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| - | <h1> One iteration of Gillespie algorithm </h1>
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| - | </br>
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| - | <center><img src="https://static.igem.org/mediawiki/2012/f/f3/Iteration1.png" alt="" /></center>
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| | </section> | | </section> |
| | </div> | | </div> |
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