Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases Johan Elf and M

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Spontaneous separation of bi-stable biochemical
systemsinto spatial domains of opposite phases
Johan Elf and Måns EhrenbergPresented by
Jonathan BlakesComputational Foundations of
Nanoscience Journal Club2008-05-16
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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

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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).

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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.

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Macrophage and Bacterium 2,000,000X 2002 Waterco
lor by David S. Goodsell
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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.
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Implementation

• Connectivity matrix defines
• neighbours and therefore geometry
• (boundaries is connection to self)
• Configuration is a multiset
• Q is order in event queue

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Implementation

• Heap structure
• Scales logarithmically
• with number of
• subvolumes
• Could equally be used for storing next
  compartment in multi-compartmental Gillespie
  algorithm...

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Multi-Compartmental Gillespie
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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).

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Specifying Geometries

• Shape achieved in MesoRD using Constructive Solid
  Geometry (CSG)
• Describe shape by extending
• SBML

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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.

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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.
• Thats it, thanks for listening.

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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) 29232924
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

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