Software development for multiscalereaction-diffusion modelling

Martin Robinson, Mark Flegg, Radek ErbanOxford Centre for Collaborative Applied Mathematics (OCCAM)

martin.robinson@maths.ox.ac.uk

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.