Software development for multiscalereaction-diffusion modelling
Martin Robinson, Mark Flegg, Radek ErbanOxford Centre for Collaborative Applied Mathematics (OCCAM)
http://people.maths.ox.ac.uk/robinsonm/
Multiscale Reaction-DiffusionModelling
• Partition the domain amongst different modelling methods.
– Trade-off between accuracy and computational error
– Vary assumptions (e.g. stochastic effects important in subset of domain)
– Combine strength of different modelling techniques
(left) Growth of actin filaments in filopodia (elongated organelle that growout of cells to probe the environment). The tip is modelled with an accuratemolecular-based method, and the remainder of the filament using a spatiallyaveraged 1D compartment-based method.(right) Diffusion of calcium in intracellular signalling. The ion channel (thecalcium source) is modelled using the molecular-based method, and thelarge surrounding volume uses a more efficient, but less spatially accurate,compartment-based method.
Simulation Methods
Molecular-based
•Diffusion of particles by Brownian random motion
• Bimolecular reactions may occur when particle pairsare closer than a set binding radius (Smoluchowskimodel)
• Errors converge as δt→ 0
• Accurate surface geometry
• Ideal for low molecule numbers or high spatial varia-tion (e.g. cell membrane)
Compartment-based
•Domain partitioned into subvolumes with width h.Each subvolume assumed well-mixed.
• Number of molecules within each subvolume istracked instead of individual particles.
• Stochastic, event-based algorithm (Next SubvolumeMethod)
•Diffusion error O(h2). Reaction error does not con-verge with small h
• Ideal for homogeneous and/or high molecule concen-trations
PDE-based
•Mean-field PDE approximation: dpidt = Di∇2pi +
R(p1, p2, . . . , pN ) where pi is the concentration forspecies i.
•Mixed-mode FEM discritisation for diffusion
• Free diffusion across interface boundaries, no-fluxotherwise
• Ideal for high concentration areas
Software DevelopmentC++ libraries
•Kairos - Compartment-based library
https://github.com/martinjrobins/Kairos
– Diffusion & Reactions (any order). Per compartment or global
– Compartment geometry currently regular grid. Arbitrary 3D Mesh geome-try planned
– Real-time output via concentration plots (VTK) or output to file.
•Tyche - Molecular-based library
https://github.com/martinjrobins/Tyche
– Brownian Diffusion
– Zeroth order, Unimolecular and Bimolecular Reactions
– Axis-aligned plane and rectangle surface geometries.
– Surfaces can transmit, reflect or generate molecules.
– Real-time 3D viz & concentration plots (VTK) or output to file.
•Moirai - Pde-based library
https://github.com/martinjrobins/Moirai
– Finite Element Discritisation on 3D mesh
– Brownian Diffusion of N species
– Reactions are work in progress
– Output timesteps to VTK-format file
(left) Kairos coupled to Smoldyn (right) Moirai coupled to Tyche
Integration with Smoldyn
• Smoldyn is a molecular-based spatial stochastic simulator
http://www.smoldyn.org/ [Andrews, 2012]
– Easy to use - system defined via simple text file input
– Isotropic/Anisotropic diffusion with drift and on surfaces. Excluded volumesupported.
– Zeroth order, Unimolecular and Bimolecular Reactions (reversible & irre-versible). Surface or compartment-specific reactions.
– Surfaces can be constructed from rectangles, triangles, spheres, hemi-spheres, cylinders, or disks
– Many options for user-defined output or system commands
– and much more . . .
• Kairos is currently integrated within Smoldyn, adding compartment-basedmodelling and coupling with particles [Robinson et al., 2013a]
• Integration of Moirai planned.
Two Regime Method (TRM)
BrownianMolecule
• Preserves correct diffusion flux across interface [Flegg et al., 2012, 2013]
•Molecules moving from Particle (ΩP ) to Compartment (ΩC) domain loseposition information. This results in unphysical diffusion flux in this direction.Balanced via increased diffusion flux from ΩC to ΩP
• New particles in ΩP placed near interface with a randomly sampled distanceaccording to molecular-based timestep δt
TRM with Reactions
3D simulation domain. Periodic boundary condition atx = 0 and reflective boundaries otherwise. Moleculesgenerated at x = L with constant rate λ
•D = 0.1
• L = 1.0
• k1 = 1.0
• k2 = 10−4
• λ = 105
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.2
0.4
0.6
0.8
1.0
ni/nmax
t = 4.97
Pure diffusion
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.2
0.4
0.6
0.8
1.0
ni/nmax
t = 4.97
Ak1→ ∅
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.2
0.4
0.6
0.8
1.0
ni/nmax
t = 4.97
A + Ak2→ ∅
• Accurate results for unimolecular and bimolecular reactions
• Slight flattening of concentration profile to left of interface
• Error scales as O(h) and is minimised when h =√πDδt
TRM with Moving Interface
Infinite domain as x→ −∞. All other bound-aries reflective
•Molecular species A generated at x = L anddiffuses through domain. One unimolecular
destruction reaction: Ak→ ∅
Interface moves in x-direction with constantstep-size h
• Goal to keep max concentration on particle side ΩP less than cmax
• Interface moves left if A(x, t) > cmax and right if A(x, t) < cmax −∆r
• Threshold separation ∆r must be sufficiently large to prevent spurious move-ment due to stochastic fluctuations
•More details in [Robinson et al., 2013b]
0.0
0.2
0.4
0.6
0.8
1.0
ni/nmax
t = 0.1 t = 0.501
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.2
0.4
0.6
0.8
1.0
ni/nmax
t = 1.002
0.0 0.2 0.4 0.6 0.8 1.0x
t = 1.507
Concentration histogram (Blue = ΩP ,Red = ΩC)
0 2 4 6 8 10t
0
2000
4000
6000
8000
RM
Ser
ror
molecularcompartmentTRM - static interfaceTRM - moving interface
RMS Error versus time
Outlook
•Moving Interface
– Travelling wave problem (Fisher equation) - Interface movement dependenton wave speed
– Requires reversible reactions
• C++ libraries and Smoldyn integration
– Encourage outside use
– Implement arbitrary compartment geometry via unstructured mesh
– Integrate Moirai (PDE-library) in Smoldyn
– New features as needed/requested. . .
• PDE-based coupling
– 3D generalisation of [Franz et al., 2012] (Diffusion)
– Add reactions to mean-field model
• Applications
– Intracellular calcium signalling (Marcin Paczkowski)
– Min system in E. Coli (Robert Ross)
References
Steven S Andrews. Spatial and stochastic cellular modeling with the smoldyn simulator. In Bacterial Molecular Networks, pages 519–542. Springer, 2012.
Mark B Flegg, S Jonathan Chapman, and Radek Erban. The two-regime method for optimizing stochastic reaction–diffusion simulations. J. R. Soc. Interface, 9(70):859–868, 2012.
Mark B Flegg, S Jonathan Chapman, Likun Zheng, and Radek Erban. Analysis of the two-regime method on square meshes. arXiv preprint arXiv:1304.5487, 2013.
Benjamin Franz, Mark B Flegg, S Jonathan Chapman, and Radek Erban. Multiscale reaction-diffusion algorithms: Pde-assisted brownian dynamics. arXiv preprint arXiv:1206.5860, 2012.
Martin Robinson, Steven S. Andrews, and Radek Erban. Multiscale stochastic simulations with smoldyn. In Preparation, 2013a.
Martin Robinson, Mark Flegg, and Radek Erban. Multiscale reaction-diffusion simulation: Two-regime method with moving interface. In Preparation, 2013b.
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