Numerical effects in modeling and simulating chemotaxis in biological reaction diffusion systems
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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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Numerical effects in modeling and simulating chemotaxis in biological reaction-diffusion systems

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Numerical effects in modeling and simulating chemotaxis in biological reaction diffusion systems

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


Outline

Outline

  • Introduction

  • Chemotaxis: mechanism & models

  • Objectives of study

  • Numerical approach & issues

  • Sample results

  • Summary


Introduction

Introduction

  • Chemotaxis important in many bio-systems:

    • Aggregation of glial cells in Alzheimer's disease.

    • Proliferation & migration of micro-organisms.

    • Tumor growth processes, via angiogenesis.

    • Atherogenesis & cardiovascular disease.

      [e.g., Murray, 1989; Woodward, Tyson et al., 1995; Ross, 2001; Luca & Ross, 2001]

Chemotaxis

Movement of cell or organism

in response to chemical stimulus


Chemotaxis mechanism models

Chemotaxis: mechanism & models

  • Cells respond to chemical gradient

    can migrate up (attractant) or down (repellent)

  • Simple 2-eqn model (Keller-Segel Theory) :

    n = cell density, c = chemoattractant density


Chemotaxis mechanism models1

Chemotaxis: mechanism &models

OR

Diffusion alone

w/ chemotaxis

(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)


Overall goals of our study

Overall goals of our study

  • Focus on bacteria PDE models

  • Mathematical modeling issues:

    • Realistic chemotaxis & reaction terms

    • Parameter space study: pattern & behavior types

    • Stability analysis around steady-states

  • Numerical model & algorithms

    • Efficient, robust discrete approximations

    • Implement on parallel cluster platforms

    • Investigate accuracy, efficiency, reliability


Bacteria aggregation patterns

Bacteria aggregation patterns

Experimental results (Budrene and Berg, 1995):

Numerical results:


Numerical effects in modeling and simulating chemotaxis in biological reaction diffusion systems

E. coli: PDE model

3-species: [Woodward et al., 1995; Murray et al., 1998]

Chemoattractant produced by bacteria themselves.


Numerical approximation issues

Numerical approximation & issues

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


New numerical issues

“New” numerical issues

  • Strongly nonlinear operators

    Fictitious solutions pervasive if numerics inadequate

  • Situation compounded by sensitivity to parameters and/or initial conditions

    Illustrative example [Pearson, 1993: Gray-Scott model]


Numerical issues

Numerical issues

  • Reaction-diffusion-chemotaxis typical scenario:

    - Numerical studies focus on new/challenging regimes

    - Pick some reasonable scheme & parameters

    - Obtain results that look plausible

  • Our experience: results are often spurious!

    * Discrete (nonlinear) model often admits different solutions from those of the PDE system

    * In particular, adequate resolution is critical

    * Requires mesh refinement & adaptive formulations


Bacteria sample results

Bacteria: Sample results

  • Spatial resolution effects

    All results for same parameter values, & plotted at the same time-instant. Only difference is in grid resolution.

  • All calculations on parallel cluster using 16 processors.

m = 800x800

m = 200x200

m = 400x400


Bacteria sample results1

Bacteria: Sample results

  • Spatial resolution study for another chemotaxis model (Salmonella)

m = 400x400

m = 200x200


Summary

Summary

  • Chemotaxis-based models growing in importance in many areas

  • Often used in conjunction with strongly nonlinear reaction terms

  • Numerical models prone to spurious solutions & fictitious bifurcations

  • Mesh refinement studies critical for investigating nonlinear dynamics and pattern formation.

  • Many open questions: Existence/uniqueness; analytical techniques for validation; multigrid solution strategies; convection-dominated cases?


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