Spontaneous separation of bi-stable biochemical systemsinto spatial domains of opposite phases

Johan Elf and Måns Ehrenberg

Presented by Jonathan Blakes

Computational Foundations of Nanoscience Journal Club

2008-05-16

Outline

Introduction: Bi-stable chemical systems Well-mixed assumptions violated even in bacteria The Next Subvolume Method

Implementation Ideas for Multi-Compartmental Gillespie

Influence of Diffusion Specifying Geometries Influence of Geometry Conclusions

Bi-stable chemical systems Biochemical systems can be in different, self-perpetuating states depending

on previous stimuli: loss of ‘potency’ when stem cells differentiate, switching on and off of genes in quorum sensing, etc.

Bi-stable chemical systems have two reasonably steady states and switch between these unpredictably due to the underlying stochasticity of the system.

Stochasticity arises from small numbers of a particular molecular species in the system and slow reactions.

Stochastic simulation algorithms based on Gillespie Direct Method allow us to sample trajectories of the Markov process corresponding to the Chemical Master Equation (CME).

Assumptions violated “Bistability can vanish due to spatial localised fluctuations for inorganic

catalysts, and thus invalidate any macroscopic description of the kinetics.” ‘Macroscopic’ in this case could mean a bacterial cell. Our model of quorum sensing in P. aeruginosa treats each bacteria as an

single volume because they are defined by a single membrane and very small (rod shaped ~0.3-0.8μm wide 1.0-1.2μm long ≈ 0.311μm3).

We use a version of the Gillespie algorithm to determine which reactions in our system will happen next.

Because Gillespie samples CME it assumes a homogenous (well-mixed) system, where diffusion of reactants in the system occurs on a much faster timescale than the reactions; i.e. no patches of higher or lower reactant concentrations (domain separation).

However, diffusion of molecules in vivo is much slower than in vitro, due to intracellular organisation like the actin cytoskeleton and genome (next slide)

Cells often have non-uniform shapes: macrophages, budding yeast; domain separation should be expected, and is in fact crucial for functioning.

Macrophage and Bacterium 2,000,000X

2002

Watercolor by David S. Goodsell

Next Subvolume Method (NSM)

Partition large volumes (cell) into many smaller volumes, where each subvolume small enough relative to rate of diffusion that it can be considered well-mixed.

As well a rate constant for each reaction,

each reactant has diffusion constant D, which

summarises the intracellular congestion.

The authors tool MesoRD has been used to model the stochastic contribution to different mutant phenotypes in the Min-system in E. coli.

Visualisation of a stochastic simulation of a wild type E. coli cell: MinD on the cell membrane and MinE in complex with MinD.

Implementation

• Connectivity matrix defines

neighbours and therefore geometry

(boundaries is connection to self)

• Configuration is a multiset

• Q is order in event queue

• Heap structure• Scales logarithmically

with number of

subvolumes

• Could equally be used for storing next compartment in multi-compartmental Gillespie algorithm...

Implementation

Multi-Compartmental Gillespie

Influence of Diffusion

• A and B inhibit the production

of each other at identical rates• Slower diffusion (upper) leads to domain separation, while

faster diffusion (lower) does not, ascribed to faster transitioning between attractors, however this is not so at boundary (corners).

Specifying Geometries

Shape achieved in MesoRD using Constructive Solid Geometry (CSG)

Describe shape by extending

SBML:

Influence of Geometry• Domain separation in tube and plane, but not for cube, as mixing

time shorter in cube.• Shape determines domains as much as diffusion rate.

Conclusions

• Localisation of molecules within even small volumes can affect behaviour of system, dependent on diffusion rates of species and geometry.

• Next Subvolume Method is a scalable algorithm for modelling these affects.

• NSM could be used in our simulator as next level down of multiscale approach for 3D (cytoplasm) and 2D (membrane) volumes, not having this facility could mean our models cannot reproduce observed phenomena.

• Constructive Solid Geometry of cytoplasm would define membrane shape.

• Heap may be better way of finding next reaction in multiple compartments.

• That’s it, thanks for listening.

References

1. Elf, J. and Ehrenberg, M. Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases Syst. Biol. 2004 1(2): 230-236

2. Hattne, J., Fange, D. and Elf, J. Stochastic reaction-diffusion simulation with MesoRD Bioinformatics 2005 21(12) 2923—2924

3. Fange, D. and Elf, J. Noise induced Min phenotypes in E. coli. PLoS Comp. Biol. 2006 2(6): 637-648

4. M. Ander et al. SmartCell, a framework to simulate cellular processes that combines stochastic approximation with diffusion and localisation: analysis of simple networks Syst. Biol. 2004 1: 129-138

5. Lemerle, C., Di Ventura, B. and Serrano, L.Space as the final frontier in stochastic simulations of biological systems FEBS Letters 2005 579: 1789-1794