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Team:ETHZ Basel/Modeling/Stochastic Simulations
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| + | ==Existing Stochastic Models== | ||
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Molecules are discretely represented, as number of particles and reactions happen probabilistically. At each time step, two molecules are randomly selected and, after this, another random number is generated (used to see if a reaction between the two molecules will occur). The random number is compared to the probability of that particular reaction occurring (value retrieved from a look – up table). If the probability exceeds the random number, the particles don’t react, otherwise, they do and the system is updated accordingly. | Molecules are discretely represented, as number of particles and reactions happen probabilistically. At each time step, two molecules are randomly selected and, after this, another random number is generated (used to see if a reaction between the two molecules will occur). The random number is compared to the probability of that particular reaction occurring (value retrieved from a look – up table). If the probability exceeds the random number, the particles don’t react, otherwise, they do and the system is updated accordingly. | ||
Both unimolecular and bimolecular reactions are permitted. In the case of the former, a dummy variable, pseudo – molecule, is created. The probabilities for unimolecular and bimolecular reactions happening have fixed formulae, dependent on the rate of unimolecular/bimolecular reactions, the total number of molecules in the system, the time – slice duration, the volume of the system. | Both unimolecular and bimolecular reactions are permitted. In the case of the former, a dummy variable, pseudo – molecule, is created. The probabilities for unimolecular and bimolecular reactions happening have fixed formulae, dependent on the rate of unimolecular/bimolecular reactions, the total number of molecules in the system, the time – slice duration, the volume of the system. | ||
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| + | ====Conclusions==== | ||
| + | StochSim has been applied on the study of CheYp level in the chemotaxis network. The findings of Morton-Firth & Bray (1998) are that CheYp concentration undergoes fluctuations around the deterministic value. The amplitude of fluctuations and their duration decreases as the number of molecules in the system is increased. | ||
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| + | ===StochSim vs Gillespie=== | ||
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| + | Both the Gillespie algorithm and the Stochsim algorithm are based on identical assumptions (discrete time steps and molecule numbers, reactions happening probabilistically, with probability depending on the deterministic rate constants). | ||
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| + | The Gillespie algorithm makes time steps of variable length, based on the rate constants and population size of each chemical species. Two random numbers determine which reaction will take place next (based on each reaction's individual probability; the most likely reaction is chosen) and how long the time step will last (the most likely time step to optimize the chosen reaction is chosen). Both random numbers are used to calculate the (theoretical - based) density functions for the next reaction and the time step. The gives a handle on the stochasticity of the system and makes Gillespie less restrictive than StochSim, which uses a fixed time step for each iteration. | ||
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| + | On the other hand, StochSim is more robust than Gillespie to computational explosion, especially if the system contains multi - state molecules. In this case, the total possible configurations of a multi - state molecule is an exponential number, with the exponent being the number of possible states of the molecules (e.g. different binding sites for a protein). Since the Gillespie algorithm scales with the number of reaction channels, therefore also with the total possible number of chemical species involved, this will lead to computational explosion and render the simulation not feasible. In StochSim, the different states of the molecule can be associated to the molecule itself, without the need for creating additional states. | ||
| + | In conclusion, when the number of reactions is small and the number of molecules is large, the Gillespie algorithm is more efficient than Stochsim. However, if the system has multi - states species, StochSim is likely to be faster. | ||
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| + | ==Stochastic Model of the chemotaxis network (Gillespie Algorithm) == | ||
Latest revision as of 01:28, 19 September 2010
Open Issues
Debating Questions
Why would stochastic simulation (on the molecular level) be good:
• It sounds more cool than ODE – based models;
• It generally gives (or at least it should give) a more realistic view of the biological process studied;
• Individual complexes could be traced all through the simulation (not sure if we’d ever need that; probably not).
Why would stochastic simulation (on the molecular level) not be of interest:
• In terms of the results, the predicted values of the CheY concentration levels fluctuate around the deterministically obtained ones, with amplitude of fluctuations indirectly proportional to population size (number of molecules).
Existing Stochastic Models
StochSim
General
StochSim (Morton-Firth & Bray 1998) is a stochastic program for modeling the intracellular signaling pathway responsible for chemotaxis behavior. Molecules are represented as individual software objects and their interactions are probability - governed (based on empirically - derived distributions of concentrations and kinetic rates).
Algorithm
Molecules are discretely represented, as number of particles and reactions happen probabilistically. At each time step, two molecules are randomly selected and, after this, another random number is generated (used to see if a reaction between the two molecules will occur). The random number is compared to the probability of that particular reaction occurring (value retrieved from a look – up table). If the probability exceeds the random number, the particles don’t react, otherwise, they do and the system is updated accordingly. Both unimolecular and bimolecular reactions are permitted. In the case of the former, a dummy variable, pseudo – molecule, is created. The probabilities for unimolecular and bimolecular reactions happening have fixed formulae, dependent on the rate of unimolecular/bimolecular reactions, the total number of molecules in the system, the time – slice duration, the volume of the system.
Conclusions
StochSim has been applied on the study of CheYp level in the chemotaxis network. The findings of Morton-Firth & Bray (1998) are that CheYp concentration undergoes fluctuations around the deterministic value. The amplitude of fluctuations and their duration decreases as the number of molecules in the system is increased.
StochSim vs Gillespie
Both the Gillespie algorithm and the Stochsim algorithm are based on identical assumptions (discrete time steps and molecule numbers, reactions happening probabilistically, with probability depending on the deterministic rate constants).
The Gillespie algorithm makes time steps of variable length, based on the rate constants and population size of each chemical species. Two random numbers determine which reaction will take place next (based on each reaction's individual probability; the most likely reaction is chosen) and how long the time step will last (the most likely time step to optimize the chosen reaction is chosen). Both random numbers are used to calculate the (theoretical - based) density functions for the next reaction and the time step. The gives a handle on the stochasticity of the system and makes Gillespie less restrictive than StochSim, which uses a fixed time step for each iteration.
On the other hand, StochSim is more robust than Gillespie to computational explosion, especially if the system contains multi - state molecules. In this case, the total possible configurations of a multi - state molecule is an exponential number, with the exponent being the number of possible states of the molecules (e.g. different binding sites for a protein). Since the Gillespie algorithm scales with the number of reaction channels, therefore also with the total possible number of chemical species involved, this will lead to computational explosion and render the simulation not feasible. In StochSim, the different states of the molecule can be associated to the molecule itself, without the need for creating additional states. In conclusion, when the number of reactions is small and the number of molecules is large, the Gillespie algorithm is more efficient than Stochsim. However, if the system has multi - states species, StochSim is likely to be faster.
