From Signal Transduction to Spatial Pattern Formation in E. Coli: A Paradigm for Multiscale Modeling

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From Signal Transduction to Spatial Pattern
Formation in E. Coli A Paradigm for Multiscale
Modeling in BiologyR. Erban and H.G. Othmer,
Multiscale Model. Simul. (2005) 3362-394

• Presented by
• Scott Christley and Yilin Wu

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Motivation

• Collective behavior of bacteria from cell-level
  decision making to population-level behavior.
  Examples spatial pattern formation in E. Coli
  and Myxobacteria, formation of biofilms
• Cell-level respond to external signal, induce
  internal signals, and change individual behaviors
• Model issue how to incorporate micro-behavior
  into macro-models?

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E. Coli movement

• Good paradigm relevant physical processes occur
  over large range of time scales

Swimming Velocity-jump process through the CCW
and CW rotating of helical flagella. (CCW
bundle, run CW dissociation,
tumble) Signal transduction simplified
model
Upper right picture from Physics Today by Howard
Berg, lower right figure from PNAS by Edward
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Without Internal Dynamics

• p(x,v,t) -density of bacteria at x, with velocity
  v, at time t
• ?-turning rate 1/? is mean run time between
  velocity jumps
• T(v,v) turning kernel the prob. of a velocity
  jump from v to v

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With Internal Dynamics

• For the simplified model of signal transduction
• After reference 47

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Lift to Macroscopic Level

• Define moments as integrals across the internal
  state space (z1, z2)

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Lift to Macroscopic Level

• Plug in the transport equation and integrate with
  respect to z (internal state).
• Assume signal gradients are small, so the
  higher-order moments can be neglected.

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Solve Macroscopic

• Either integrate the equations with respect to
  velocity, v (hyperbolic). This introduces
  higher-order velocity moments.
• Or, use a scaling argument and asymptotic
  analysis (parabolic).

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Asymptotic Analysis

• Time, length, velocity scaling
• T 1sec, L 1mm, s0 10?/sec
• Dimensionless parameters

• Scaling parameter ?

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Asymptotic Analysis

• Plug in dimensionless and scaling parameters.

• Regular perturbation expansion.

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Asymptotic Analysis

• Plug into equations and collect terms by orders
  of ?

• Couple pages of mathematical arguments, and you
  get the chemotaxis equation for macroscopic
  density

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Results

• 106 particles
• Their macroscopic PDE gives similar results to
  stochastic simulation of the particles.