Team:Grenoble/Modeling/Amplification/Stochastic/what

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<h1> Goal </h1>
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Stochastic modeling is a technique of predicting outcomes that takes into account a certain degree of randomness or unpredictability. In a stochastic modeling, a small amount of randomness is added at each time step of the simulation.
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It is the probabilistic counterpart to a deterministic process.
<center><img src="https://static.igem.org/mediawiki/2012/3/38/Diagram_stoch.png" alt="" /></center>
<h1> Why </h1>
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The deterministic model, who studies a population of bacteria, considers continuous concentrations of molecules. However, in a single bacteria, the quantity of the different proteins is of the order of 100, and the concentrations take thus discrete values. These values depend on events (production, degradation) which are hard to predict, and must therefore be approached in terms of probability of occurence (or preopensity).
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<h1> How </h1>
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A stochastic model incorporates random variations to predict future conditions and to see what they might be like.
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To introduce that randomness we use a new function : propensities.
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<center><img src="https://static.igem.org/mediawiki/2012/c/ca/Propensity.png" alt="" /></center>
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As an example, to illustrate this we consider 4 possible reactions. Each reaction has a probability to happen in the next time step.
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<center><img src="https://static.igem.org/mediawiki/2012/8/8c/Reactions.png" alt="" /></center>
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We randomly chose the next reaction regarding the propensities.
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When we run a stochastic simulation once, we get a trajectory. This graph represents the random evolution of the variables. Because of the randomness, if we run the script another time we will get a different trajectory. 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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<center><img src="https://static.igem.org/mediawiki/2012/9/9c/Curves.png" alt="" /></center>
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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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To model that randomness we use a Gillepsie algorithm or Stoachastic Simulation Algorithm (SSA).
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Traditional continuous and deterministic biochemical rate equations do not accurately predict cellular reactions since they rely on random reactions that require the interactions of millions of molecules.
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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We will use the following notations :
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<center><img src="https://static.igem.org/mediawiki/2012/3/37/Assert.png" alt="" /></center>
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Each reaction Rj is characterized mathematically by two quantities :
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<center><img src="https://static.igem.org/mediawiki/2012/5/50/State_change_vector.png" alt="" /></center>
<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>
<center><img src="https://static.igem.org/mediawiki/2012/6/61/Prop2.png" alt="" /></center>
<center><img src="https://static.igem.org/mediawiki/2012/f/f9/Prop4.png" alt="" /></center>
<center><img src="https://static.igem.org/mediawiki/2012/0/09/Prop3.png" alt="" /></center>
<center><img src="https://static.igem.org/mediawiki/2012/3/32/Prop5.png" alt="" /></center>
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<center><img src="https://static.igem.org/mediawiki/2012/e/e8/More_precisely.png" alt="" /></center>
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Once we have presented the theory behind the stochastic approach, let us have a look at the algorithm.
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<center><img src="https://static.igem.org/mediawiki/2012/f/f3/Iteration1.png" alt="" /></center>
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<center><img src="https://static.igem.org/mediawiki/2012/4/47/Iteration2.png" alt="" /></center>
<center><img src="https://static.igem.org/mediawiki/2012/8/8c/Equations.png" alt="" /></center>
<center><img src="https://static.igem.org/mediawiki/2012/8/86/Iteration3.png" alt="" /></center>
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The algorithm comprises 5 steps.
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<center><img src="https://static.igem.org/mediawiki/2012/7/79/Scheme.png" alt="" /></center>
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Conerning the parameters, we use the same as in the ODE modeling but we have to make a conversion to remove the dependence in volume and mol. Thus, in most cases, we need to multiply those parameters by the Avogadro number and by the cell's volume.
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You can find our <b>script</b> for the stochastic modeling <a href="https://2012.igem.org/wiki/index.php?title=Team:Grenoble/Modeling/Amplification/Stochastic/Scripts">here</a>.
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