Journal of Mathematical Biology

, Volume 55, Issue 3, pp 331–352

# An equation-free approach to analyzing heterogeneous cell population dynamics

• Katherine A. Bold
• Yu Zou
• Ioannis G. Kevrekidis
• Michael A. Henson
Article

DOI: 10.1007/s00285-007-0086-6

Bold, K.A., Zou, Y., Kevrekidis, I.G. et al. J. Math. Biol. (2007) 55: 331. doi:10.1007/s00285-007-0086-6

## Abstract

We propose a computational approach to modeling the collective dynamics of populations of coupled, heterogeneous biological oscillators. We consider the synchronization of yeast glycolytic oscillators coupled by the membrane exchange of an intracellular metabolite; the heterogeneity consists of a single random parameter, which accounts for glucose influx into each cell. In contrast to Monte Carlo simulations, distributions of intracellular species of these yeast cells are represented by a few leading order generalized Polynomial Chaos (gPC) coefficients, thus reducing the dynamics of an ensemble of oscillators to dynamics of their (typically significantly fewer) representative gPC coefficients. Equation-free (EF) methods are employed to efficiently evolve this coarse description in time and compute the coarse-grained stationary state and/or limit cycle solutions, circumventing the derivation of explicit, closed-form evolution equations. Coarse projective integration and fixed-point algorithms are used to compute collective oscillatory solutions for the cell population and quantify their stability. These techniques are extended to the special case of a “rogue” oscillator; a cell sufficiently different from the rest “escapes” the bulk synchronized behavior and oscillates with a markedly different amplitude. The approach holds promise for accelerating the computer-assisted analysis of detailed models of coupled heterogeneous cell or agent populations.

### Mathematics Subject Classification (2000)

37N25 39C15 70G60

## Authors and Affiliations

• Katherine A. Bold
• 1
• Yu Zou
• 1
• Ioannis G. Kevrekidis
• 1
• Michael A. Henson
• 2
1. 1.Department of Chemical Engineering and Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
2. 2.Department of Chemical EngineeringUniversity of MassachusettsAmherstUSA