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

1. 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

2. Outline • Introduction • Chemotaxis: mechanism & models • Objectives of study • Numerical approach & issues • Sample results • Summary

3. 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

4. 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

5. 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)

6. 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

7. Bacteria aggregation patterns Experimental results (Budrene and Berg, 1995): Numerical results:

8. E. coli: PDE model 3-species: [Woodward et al., 1995; Murray et al., 1998] Chemoattractant produced by bacteria themselves.

9. 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

10. “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]

11. 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

12. 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

13. Bacteria: Sample results • Spatial resolution study for another chemotaxis model (Salmonella) m = 400x400 m = 200x200

14. 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?