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

Introduction Chemotaxis: mechanism & models Objectives of study Numerical approach & issues Sample results Summary

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]

Introduction

Chemotaxis Movement of cell or organism in response to chemical stimulus

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

∂n∂t

=∇⋅(Dn∇n)−∇[χ(n,c)∇c]+f(n,c)

∂c∂t

=g(n,c)

Chemotaxis: mechanism &models

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)

OR

χ(n,c)

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 patternsExperimental results (Budrene and Berg, 1995):

Numerical results:

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

Chemoattractant produced by bacteria themselves.

E. coli: PDE model

Numerical approximation & issuesDiscrete 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 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 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 Spatial resolution effects

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

m = 200x200 m = 400x400 m = 800x800

All calculations on parallel cluster using 16 processors.

Bacteria: Sample results Spatial resolution study for another chemotaxis model (Salmonella)

m = 200x200 m = 400x400

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?