Numerical effects in modeling and simulating chemotaxis in biological reaction-diffusion systems. Anand Pardhanani, David Pinto, Amanda Staelens, and Graham Carey Institute for Computational Engineering & Sciences The University of Texas-Austin. Supported in part by NSF grant 791AT-51067A.
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Anand Pardhanani, David Pinto, Amanda Staelens, and Graham Carey
Institute for Computational Engineering & Sciences
The University of Texas-Austin
Supported in part by NSF grant 791AT-51067A
[e.g., Murray, 1989; Woodward, Tyson et al., 1995; Ross, 2001; Luca & Ross, 2001]
Movement of cell or organism
in response to chemical stimulus
can migrate up (attractant) or down (repellent)
n = cell density, c = chemoattractant density
(many possibilities, depending on form)
* Simple Keller-Segel model admits travelling waves
* Interplay of diffusion+reaction+chemo. produces wide range of behavior, patterns, nonlinear dynamics
* models typically strongly nonlinear (derived from microscopic or macroscopic approaches)
Experimental results (Budrene and Berg, 1995):
E. coli: biological reaction-diffusion systemsPDE model
3-species: [Woodward et al., 1995; Murray et al., 1998]
Chemoattractant produced by bacteria themselves.
Discrete formulation based on:
- Finite difference or finite element spatial approx.
- Self-adjoint FD treatment of chemotaxis terms
- Explicit or implicit integration in time [upto O(∆t4)]
- Fully-coupled space-time formulation
- Parallel scheme: nonoverlapped domain decomp.
Approximation parameters are key:
Usual issues: (1) Accuracy, (2) Stability
Many “real” applications convection-dominated
stability & accuracy are key challenges
many techniques developed to address this
Fictitious solutions pervasive if numerics inadequate
Illustrative example [Pearson, 1993: Gray-Scott model]
- Numerical studies focus on new/challenging regimes
- Pick some reasonable scheme & parameters
- Obtain results that look plausible
* Discrete (nonlinear) model often admits different solutions from those of the PDE system
* In particular, adequate resolution is critical
* Requires mesh refinement & adaptive formulations
All results for same parameter values, & plotted at the same time-instant. Only difference is in grid resolution.
m = 800x800
m = 200x200
m = 400x400
m = 400x400
m = 200x200