Homeostasis and dynamic phase transition in a simple model of cells with chemical signaling: Can renormalization group teach us something nontrivial about biology?

Anatolij Gelimson (left) and Ramin Golestanian

Authors: Anatolij Gelimson, Ramin Golestanian

Affiliation: Rudolf Peierls Centre for Theoretical Physics, University of Oxford, United Kingdom.

The motility of bacteria or cells in response to chemicals (chemotaxis) has attracted a lot of interest in biological and medical research [2]. It plays a crucial role in cancer metastasis [3], the early stages of bacterial colony formation, wound healing and development of embryos [2]. However, the underlying mechanisms of these important processes are not fully understood due to the high complexity of these living many-body systems.

Figure 1: The interplay between the two general processes growth and chemotaxis (figure 1) leads to a variety of collective phenomena.

In our recent publication [1] in Physics Review Letters we have developed a simple model to shed some light on these interacting cells, also taking into account cellular growth and death. To study it, we have applied the method of so-called Dynamical Renormalization Groups common for the theory of phase transitions [4]. Similar to physical systems, it turns out that details of the microscopic behavior of cells do not impact the collective behavior on a large scale, whereas the interplay between the two general processes growth and chemotaxis (figure 1) leads to a variety of collective phenomena, which includes a sharp transition from a phase that has moderate controlled growth and death, and regulated chemical interactions, to a phase with strong uncontrolled growth/death and no chemical interactions [1]. Remarkably, for a range of parameters, the transition point shows nontrivial collective motion, which can even be described analytically. [1]

Bacteria such as E. coli have developed an elaborate run-and-tumble search strategy for the needed chemicals by coupling sensing of the chemicals to their motility machinery [5]. In eukaryotic cells, the chemotaxis mechanism is even more complex, often involving thousands of molecular motors or actin polymerization [6].

However, if one regards the effects of these microscopic mechanisms on a more macroscopic level, the resulting motion of bacteria and cells can effectively be modeled as a directed motion towards (or away from) increasing concentrations of chemicals [7]. On this coarse-grained level of description, the motion of bacteria in a field of chemicals is therefore somewhat analogous to the motion of particles in a gravitational or electrical field [8, 9].

But other than in non-active matter, distinctive features of a living system are also growth and death, which we need to take into account in a generic model for the formation of cellular or bacterial aggregations [10]. Interestingly, it turns out that the interplay between chemotactic interactions and growth-death processes leads to a range of different collective behaviors of cells.

We have studied our cellular model with the method of Dynamical Renormalization groups [4]. The basic idea behind it is simple: while microscopically a large number of particles, cells or bacteria might show very complicated behavior with a variety of different interactions, on a more macroscopic level only very few of these interactions will actually determine the collective effects. The so-called renormalization is basically a systematic way of observing a many-particle system from a coarser and coarser level. Coarsening the system will result in make some interactions disappear, whereas others will become stronger. In Physics, this development is called a flow in parameter space. [4]

Figure 2

In our model we have found a threshold in growth and chemotactic strength at which the flow in parameter space changes, which corresponds to a critical change of the macroscopic behavior of cells (figure 2). Below the threshold, the bacteria show jamming and aggregation due to chemotaxis. But above the threshold, chemotaxis becomes irrelevant and the behavior of cells is dominated by uncontrolled growth and death [1].

This threshold could potentially be tested experimentally and also contribute towards answering of fundamentally challenging questions in metastatic growth or bacterial colony formation. The hope is that our research will help understand what controls the communication between strongly dividing cells that are far apart and their collective behavior. The method of Dynamical Renormalization groups we have applied is very generic and could be powerful to shed light on more complex scenarios, like for example adhesive metastatic cells or chemical-dependent cell growth.

References:
[1] Anatolij Gelimson, Ramin Golestanian, "Collective Dynamics of Dividing Chemotactic Cells", Physical Review Letters, 114, 028101 (2015). Abstract.
[2] S.J. Singer, Abraham Kupfer, "The Directed Migration of Eukaryotic Cells", Annual Review of Cell Biology, 2, 337 (1986).
 Abstract.
[3] Douglas Hanahan, Robert A. Weinberg, Cell, 144, 646 (2011). Full Article.
[4] Ernesto Medina, Terence Hwa, Mehran Kardar, Yi-Cheng Zhang, "Burgers equation with correlated noise: Renormalization-group analysis and applications to directed polymers and interface growth", Physical Review A, 39, 3053 (1989). Abstract.

[5] Howard C. Berg, "E. coli in Motion" (Springer-Verlag, New York, 2004).
[6] Herbert Levine, Wouter-Jan Rappel, "The physics of eukaryotic chemotaxis", Physics Today, 66 (issue 2), 24 (2013). Abstract.

[7] Evelyn F. Keller, Lee A. Segel, "Traveling bands of chemotactic bacteria: A theoretical analysis", Journal of Theoretical Biology, 30, 235 (1971). 
Abstract.
[8] Pierre-Henri Chavanis, Carole Rosier, Clément Sire, "Thermodynamics of self-gravitating systems", Physical Review E, 66, 
036105 (2002). Abstract. 

[9] Pierre-Henri Chavanis, Clément Sire, "Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions", Physical Review E, 69, 016116 (2004). 
Abstract.
[10] Martin Nowak, "Evolutionary Dynamics", Harvard University Press (2006).