Team:Colombia/Stochastic

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I could not help laughing at the ease with which he explained his process of deduction. ‘When I hear you give your reason,’ I remarked, ‘the thing always appears to me to be so ridiculously simple that I could easily do it myself, though at each successive instance of your reasoning I am baffled, until you explain your process. And yet I believe that my eyes are as good as yours.
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‘Quite so,’ he answered, lighting a cigarette, and throwing himself down into an armchair. ‘You see, but you do not observe. The distinction is clear. For example, you have frequently seen the steps which lead up from the hall to this room.
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‘Frequently.
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<h1 class="curs1">Stochastic Model.</h1>
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‘How often?’
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‘Well, some hundreds of times.
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‘Then how many are there?’
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‘How many! I don’t know.
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‘Quite so. You have not observed. And yet you have seen. That is just my point. Now, I know that there are seventeen steps, because I have both seen and observed. By the way, since you are interested in these little problems, and since you are good enough to chronicle one or two of my trifling experiences, you may be interested in this.’ He threw over a sheet of thick, pink-tinted note-paper which had been lying open upon the table. ‘It came by the last post,’ said he. ‘Read it aloud.
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The note was undated, and without either signature or address.
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‘There will call upon you tonight, at a quarter to eight o’clock,’ it said, ‘a gentleman who desires to consult you upon a matter of the very deepest moment. Your recent services to one of the Royal Houses of Europe have shown that you are one who may safely be trusted with matters which are of an importance which can hardly be exaggerated. This account of you we have from all quarters received. Be in your chamber then at that hour, and do not take it amiss if your visitor wear a mask.
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Until now, all we have is a deterministic model of our system of detection of Cholerae. This model takes into account exact concentrations of molecules as the variables in the differential equations. This means that we make calculations with the mean values of the amount of molecules.
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‘This is indeed a mystery,’ I remarked. ‘What do you imagine that it means?’
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It is well known that we can simulate this kind of molecular dynamics as stochastic processes. In particular, such events will be simulated as Markovian random walk in N dimension, where N represents the number of substances involved.
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“…For “ordinary” chemical systems in which fluctuations and correlations play no significant role, the method stands as an alternative to the traditional procedure of numerically solving the deterministic reaction rate equations. For nonlinear systems near chemical instabilities, where fluctuations and correlations may invalidate the deterministic equations, the method constitutes an efficient way of numerically examining the predictions of the stochastic master equation. Although fully equivalent to the spatially homogeneous master equation, the numerical…” (Gillespie 1976)
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Dealing with such a processes, the first possible approach will be to solve the stochastic master equation. The function that satisfies that equation will measure the probability of finding different numbers of molecules of the N different substances at each instant of time. But as you should probably know, this kind of numerical computation will be hard to develop and won’t be efficient. One of the reason is, for example, that, due to the very low probability of occurrence of the events (chemical reactions, ARN transcription, proteins production, …), and the very short time intervals that we should use, most of the time won’t be happening nothing.  
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<br><br>
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For that reason, we decided to use a nice very efficient algorithm proposed by Daniel Gillespie, which is based in a different probability function density: The reaction probability density function. These functions will tell us how much time will pass until the next event occurrence and how this time is distributed. After that we decided which one of the events will occur based on the number of molecules of each species and how they interact with each other.  
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<br><br>
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To give you an idea of how does our system works and how will be the events related, we present a Stochastic Petri Net. In Figure 1 we can appreciate the different molecules species interacting between them or undergoing process such as degradation or production.<br>
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[[File:Colombia_PetriNet.png|center|580px]]
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<center> <b>Figure 1</b> </center>
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In this case, we have 24 different possible events to take into account. (Table 1)
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[[File:Colombia_TableEst1.jpg|left|600px]]
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<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
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<b>Table 1.</b> Events involved in our detection system.
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In Table 1 every variable represents the number of molecules of this kind. n_events is then a real number.
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<br><br>
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The time τ between events is calculated by:
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[[File:Col_Ecu1.jpg|left|600px]]
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<br><br><br><br><br>
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Which event will occur is calculated by using another random number R between 0 and 1 in the following way:
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Let’s define
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[[File:Col_Ecu2.png|left|600px]]
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Now divide [0,1] in 25 interval such that (Figure 2)
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[[File:Col_Ecu3.png|left|600px]]
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[[File:Col_Img4.png|left|600px]]
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<center> <b>Figure 2</b> </center>
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The size of the ith interval depends on the value of n_i. <br><br><br>
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The event that will occur is the one that correspond to in which interval is located the random number R.
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<font color="#8A0808" size="5" ><b>Results -></b></font>
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Here we present a graphic for each molecule species involved in our system and how it responds to the presence of CAI-1 in the environment. The red line in the Figure 3 represents the amount of CAI-1 molecules in the environment.
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<br>
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The blue lines represent the behavior of each cell and the green line presents the mean behavior of the total population.  
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[[File:Col_Gra0Est.png|center|600px]]
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<center><b> Figure 3 </b></center>
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[[File:Col_Gra555Est.png|center|600px]]
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<center><b> Figure 4 </b></center>
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[[File:Col_Gra5Est.png|center|600px]]
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<center><b> Figure 5 </b></center>
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[[File:Col_Gra8Est.png|center|600px]]
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<center><b> Figure 6 </b></center>
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[[File:Col_Gra10Est.png|center|600px]]
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<center><b> Figure 7 </b></center>
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[[File:Col_Gra310Est.png|center|600px]]
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<center><b> Figure 8</b></center>
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<div class="button-fill orange" ><div class="button-text"> Back to modeling</div><div class="button-inside"><div class="inside-text"><a style="text-decoration: none; background-color: none; color: red;" href="https://2014.igem.org/Team:Colombia/Modeling">Go! </a></div></div></div>
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Latest revision as of 03:59, 18 October 2014

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Stochastic Model.


Until now, all we have is a deterministic model of our system of detection of Cholerae. This model takes into account exact concentrations of molecules as the variables in the differential equations. This means that we make calculations with the mean values of the amount of molecules. It is well known that we can simulate this kind of molecular dynamics as stochastic processes. In particular, such events will be simulated as Markovian random walk in N dimension, where N represents the number of substances involved.

“…For “ordinary” chemical systems in which fluctuations and correlations play no significant role, the method stands as an alternative to the traditional procedure of numerically solving the deterministic reaction rate equations. For nonlinear systems near chemical instabilities, where fluctuations and correlations may invalidate the deterministic equations, the method constitutes an efficient way of numerically examining the predictions of the stochastic master equation. Although fully equivalent to the spatially homogeneous master equation, the numerical…” (Gillespie 1976)


Dealing with such a processes, the first possible approach will be to solve the stochastic master equation. The function that satisfies that equation will measure the probability of finding different numbers of molecules of the N different substances at each instant of time. But as you should probably know, this kind of numerical computation will be hard to develop and won’t be efficient. One of the reason is, for example, that, due to the very low probability of occurrence of the events (chemical reactions, ARN transcription, proteins production, …), and the very short time intervals that we should use, most of the time won’t be happening nothing.

For that reason, we decided to use a nice very efficient algorithm proposed by Daniel Gillespie, which is based in a different probability function density: The reaction probability density function. These functions will tell us how much time will pass until the next event occurrence and how this time is distributed. After that we decided which one of the events will occur based on the number of molecules of each species and how they interact with each other.

To give you an idea of how does our system works and how will be the events related, we present a Stochastic Petri Net. In Figure 1 we can appreciate the different molecules species interacting between them or undergoing process such as degradation or production.
<

Colombia PetriNet.png

Figure 1

In this case, we have 24 different possible events to take into account. (Table 1)

Colombia TableEst1.jpg



































Table 1. Events involved in our detection system.
In Table 1 every variable represents the number of molecules of this kind. n_events is then a real number.



The time τ between events is calculated by:

Col Ecu1.jpg






Which event will occur is calculated by using another random number R between 0 and 1 in the following way:

Let’s define

Col Ecu2.png





Now divide [0,1] in 25 interval such that (Figure 2)

Col Ecu3.png





Col Img4.png







Figure 2


The size of the ith interval depends on the value of n_i.


The event that will occur is the one that correspond to in which interval is located the random number R. Results -> Here we present a graphic for each molecule species involved in our system and how it responds to the presence of CAI-1 in the environment. The red line in the Figure 3 represents the amount of CAI-1 molecules in the environment.
The blue lines represent the behavior of each cell and the green line presents the mean behavior of the total population.

Col Gra0Est.png

Figure 3

Col Gra555Est.png

Figure 4


Col Gra5Est.png

Figure 5

Col Gra8Est.png

Figure 6


Col Gra10Est.png

Figure 7


Col Gra310Est.png

Figure 8


Back to modeling