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Hybrid approaches for multiple-species stochastic reaction-diffusion models.

Spill F, Guerrero P, Alarcon T, Maini PK, Byrne H - J Comput Phys (2015)

Bottom Line: In this way errors due to the flux between the domains are small.Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles.The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.

View Article: PubMed Central - PubMed

Affiliation: Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA ; Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.

ABSTRACT

Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. In this way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.

No MeSH data available.


The domain is decomposed into a deterministic region where the system is described by the mean field equations, and a stochastic region in which it is described by the stochastic equations. At the interface, which is a single compartment, between these domains the flux into the mean field domain is deterministic (equation (13)), whereas the reactions and flux into the stochastic domain are calculated in a stochastic way (equations (14) and (16)).
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fg0010: The domain is decomposed into a deterministic region where the system is described by the mean field equations, and a stochastic region in which it is described by the stochastic equations. At the interface, which is a single compartment, between these domains the flux into the mean field domain is deterministic (equation (13)), whereas the reactions and flux into the stochastic domain are calculated in a stochastic way (equations (14) and (16)).

Mentions: We now explain how the stochastic and deterministic regimes are coupled at the interface. We identify at all times(12)ns(kI)=Ns(kI)h, to emphasise that the interface compartment will exhibit both deterministic and stochastic behaviour. Three processes contribute to changes in particle numbers in compartment : fluxes into and from compartment , which is part of the mean field domain, fluxes into and from compartment , which is part of the stochastic domain, and local reactions (see Fig. 1). Hence, we model the flux between compartments and deterministically, and the flux between compartments and in a stochastic manner. If τ denotes the current Gillespie time step, we calculate(13)ns(kI,t+τ)=ns(kI,t)+τDsh2(ns(kI,t)−ns(kI−1,t)). The flux between boxes and is accounted for by the transition rates(14)TNs(kI)−1,Ns(kI+1)+1/N(kI),N(kI+1)=Dh2Ns(kI),TNs(kI)+1,Ns(kI+1)−1/Ns(kI),Ns(kI+1)=Dh2Ns(kI+1). We also have(15)TNs(kI)±1,Ns(kI−1)∓1/Ns(kI),Ns(kI−1)=0, as the corresponding flux is already accounted for by equation (13). Finally, we specify the local reactions in a stochastic way via transition rates(16)TN1(kI)+ρ1,r,…,Nsmax(kI)+ρsmax,r/N1(kI),…,Nsmax(kI)=Rr(N1(kI),…,Nsmax(kI)). In Appendix A.3 we discuss an alternative formulation for which local reactions are calculated in the mean field framework.


Hybrid approaches for multiple-species stochastic reaction-diffusion models.

Spill F, Guerrero P, Alarcon T, Maini PK, Byrne H - J Comput Phys (2015)

The domain is decomposed into a deterministic region where the system is described by the mean field equations, and a stochastic region in which it is described by the stochastic equations. At the interface, which is a single compartment, between these domains the flux into the mean field domain is deterministic (equation (13)), whereas the reactions and flux into the stochastic domain are calculated in a stochastic way (equations (14) and (16)).
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4554296&req=5

fg0010: The domain is decomposed into a deterministic region where the system is described by the mean field equations, and a stochastic region in which it is described by the stochastic equations. At the interface, which is a single compartment, between these domains the flux into the mean field domain is deterministic (equation (13)), whereas the reactions and flux into the stochastic domain are calculated in a stochastic way (equations (14) and (16)).
Mentions: We now explain how the stochastic and deterministic regimes are coupled at the interface. We identify at all times(12)ns(kI)=Ns(kI)h, to emphasise that the interface compartment will exhibit both deterministic and stochastic behaviour. Three processes contribute to changes in particle numbers in compartment : fluxes into and from compartment , which is part of the mean field domain, fluxes into and from compartment , which is part of the stochastic domain, and local reactions (see Fig. 1). Hence, we model the flux between compartments and deterministically, and the flux between compartments and in a stochastic manner. If τ denotes the current Gillespie time step, we calculate(13)ns(kI,t+τ)=ns(kI,t)+τDsh2(ns(kI,t)−ns(kI−1,t)). The flux between boxes and is accounted for by the transition rates(14)TNs(kI)−1,Ns(kI+1)+1/N(kI),N(kI+1)=Dh2Ns(kI),TNs(kI)+1,Ns(kI+1)−1/Ns(kI),Ns(kI+1)=Dh2Ns(kI+1). We also have(15)TNs(kI)±1,Ns(kI−1)∓1/Ns(kI),Ns(kI−1)=0, as the corresponding flux is already accounted for by equation (13). Finally, we specify the local reactions in a stochastic way via transition rates(16)TN1(kI)+ρ1,r,…,Nsmax(kI)+ρsmax,r/N1(kI),…,Nsmax(kI)=Rr(N1(kI),…,Nsmax(kI)). In Appendix A.3 we discuss an alternative formulation for which local reactions are calculated in the mean field framework.

Bottom Line: In this way errors due to the flux between the domains are small.Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles.The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.

View Article: PubMed Central - PubMed

Affiliation: Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA ; Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.

ABSTRACT

Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. In this way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.

No MeSH data available.