Team:Colombia/Modeling/Stochastic

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Stochastic Model

The previous sections showed how to know the mean behavior of the system for one cell, but this is just an average of the total proteins within the cell. All the biological systems are controlled by probability events. The cell is a huge space where there are a lot of small molecules, if we want a biological process to happen, two of this little molecules have to find themselves between million of other molecules in a huge pool. Also there is a possibility that a cell may need more concentration of a promoter’s activator while other may need less if the day is sunny but if is cloudy the second cell may need more. The differential equations don’t take into account these uncontrollable events that can change the response dramatically.

If we look only one cell, with all the uncontrollable events this may not behave like we want and the system may not respond or even worse the probability of dying exist and our cell may die. But dealing with one cell is not real, we always work with hundreds of cells. Within this population some cells may no produce the expected response but the others will and the average of cells would be able to respond to the presence of the pest.

The stochastic algorithms are a way to model these probability events within a population. This simulation are made in order to confirm that the system dynamics are robust and consistent and show us if the response is still behaving like we want after taking probabilities considerations. We use the Gillispie method to develop our model. Here is a brief explanation of how it works:

The complete method consists of eight steps.

1. Define the number of cells.
2. Define the time of the simulation
3. Define and name all the constants involved.
4. Define creation and destruction expression for each substance involved: The differential equations in this part have to be divided in two, the creation and the destruction expression.
5. Apply Gilliespie algorithm:
   1. Calculate the sample space of the analysed system: This is the sum of all the changes presented at specific time t.
      
   2. Calculate time distribution that depends on a random number between 0 and 1.
      
   3. Generate ranges for the space created in the step 5.1, taking into account the random number from step 5.1. After that, assign an answer to each range.
6. Take the outputs from the simulation and convert them into regular interval vectors.
7. Obtain the Gilliespie function mean values.
8. Plot the obtained functions.