Drivers of individual and collective movement in biochemical systems
Space weather phenomena, disease spread, and cancer cells’ metastasis are a few of the myriad examples of systems in which multiple ‘players’ interact and move through a medium. Mathematics provides useful tools with which to characterise and quantify such systems for greater understanding, predictive ability and, potentially, system control. With the support of the Marie Skłodowska-Curie Actions (MSCA) programme, the STOPATT project advanced the theory and maths behind spatial pattern formation in such systems, integrating stochastic input. To do so, MSCA fellow Erika Hausenblas of the University of Leoben focused on slime mould, eukaryotic organisms whose life cycle includes both free-living single cells and spores.
Reaction-advection-diffusion, chemotaxis and stochastic perturbation
The slime mould is an interesting reaction-advection-diffusion system described by equations accounting for three underlying ‘drivers’ of pattern formation in space and time: chemical reactions, bulk motion in the system (advection), and the random motion of individual units (diffusion). It also exhibits chemotaxis (the movement of cells or organisms in response to concentration gradients of chemicals, including nutrients, toxins, or signalling molecules). “If there is insufficient food, the slime mould produces a chemical that attracts other slime mould cells. The cells aggregate, building a ‘tower’ that enables cells at the peak to be blown by the wind to a potentially more bountiful place. So chemotaxis is a strategy for survival,” Hausenblas explains. “The mathematical equations model the average behaviour of a population, neglecting the inherent random noise associated with the noisy environments in which biological systems live. One can model this intrinsic randomness by perturbing the system with stochastic input. Ours is a stochastic process in the form of a function describing food resources availability.”
Defining and testing the mathematical models
A ‘solution’ to any real-life ‘problem’ exists de facto since the system exists. However, creating a mathematical model that adequately represents the system and its solution(s) is quite complicated, particularly for complex biological systems. Hausenblas successfully derived equations modelling the slime mould’s reaction-advection-diffusion with chemotaxis and stochastic perturbation and proved the existence of a solution for the equations in both 1D and 2D. The next step was evaluating the solution(s) under various conditions, asking questions such as is the solution ‘regular’, what are its features (for example, is there more than one region of aggregation) and what happens to the spatial patterns as time goes to infinity. “We are still working on this goal. To better answer these questions, we have developed code to model bifurcation – when a slight change in certain parameters results in a sudden qualitative change in the system’s behaviour. Generally, a bifurcation point is associated with changes in local stability or equilibria. In our case, this can model slime mould aggregation in more than one region because of the stochastic food function,” notes Hausenblas. Randomness makes determining bifurcation much more difficult than it is with deterministic variables, but Hausenblas is up to the challenge. In such complex systems, it is rare to find analytical (exact) solutions, and many system properties cannot be determined through experimentation since the measurement apparatus modifies the natural system’s behaviour. Hausenblas will develop the numerical solutions of her equations, enabling numerical simulation to approach the exact solution to a reasonable and sufficient degree for practical insight into stochastic pattern formation in biochemical systems.
Keywords
STOPATT, stochastic, slime mould, chemotaxis, reaction advection diffusion, bifurcation, biological systems, mathematical models, aggregation, spatial pattern
