Proliferating active matter

Abstract

The fascinating patterns of collective motion created by autonomously driven particles have fuelled active-matter research for over two decades. So far, theoretical active-matter research has often focused on systems with a fixed number of particles. This constraint imposes strict limitations on what behaviours can and cannot emerge. However, a hallmark of life is the breaking of local cell number conservation by replication and death. Birth and death processes must be taken into account, for example, to predict the growth and evolution of a microbial biofilm, the expansion of a tumour, or the development from a fertilized egg into an embryo and beyond. In this Perspective, we argue that unique features emerge in these systems because proliferation represents a distinct form of activity: not only do the proliferating entities consume and dissipate energy, they also inject biomass and degrees of freedom capable of further self-proliferation, leading to myriad dynamic scenarios. Despite this complexity, a growing number of studies document common collective phenomena in various proliferating soft-matter systems. This generality leads us to propose proliferation as another direction of active-matter physics, worthy of a dedicated search for new dynamical universality classes. Conceptual challenges abound, from identifying control parameters and understanding large fluctuations and nonlinear feedback mechanisms to exploring the dynamics and limits of information flow in self-replicating systems. We believe that, by extending the rich conceptual framework developed for conventional active matter to proliferating active matter, researchers can have a profound impact on quantitative biology and reveal fascinating emergent physics along the way.

Keywords: Biological physics; Statistical physics, thermodynamics and nonlinear dynamics.

Fig. 1. Proliferation generates tree structures.

Charles Darwin’s 1837 sketch, his first diagram of an evolutionary tree (1837). (Source: https://commons.wikimedia.org/wiki/File:Darwin_tree.png).

Fig. 2. Self-organization driven by the feedback between growth and form.

Stresses induced by differential growth in layered materials induce buckling instabilities, as shown here for different systems. a, Bacillus subtilis pellicles floating on liquid culture media. b, Vibrio cholerae biofilms. c, ±1/2 defects of dense nematics as seen in human fingerprints have been hypothesized to play key roles in directing layer formation. Part a adapted with permission from ref. , National Academy of Sciences. Part b adapted with permission from ref. , National Academy of Sciences. Part c adapted with permission from ref. , IOP.

Fig. 3. Natural selection.

Combining two different types of proliferating systems generally leads to competition for space and resources. a, In confined space, competition often leads to moving interfaces, here simulated for two tissue types (red and blue-green): the blue-green tissue, having a higher homeostatic pressure, invades the red tissue, which also has a lower apoptosis rate with a constant velocity. As the difference in homeostatic pressure increases, the blue-green tissue invades the red ever faster (arrows), and the interface becomes unstable. b–d, With open boundaries, species compete to invade unoccupied territory, as shown here for colonies grown from a mixture of two differently labelled strains of budding yeast (S. cerevisiae) (b) and E. coli (strain Dh5α) grown at two different temperatures 21 °C (c) and 37 °C (d). The strains that expand faster (yellow) tend to increase in fractional abundance. The initial mixture of each colony was 0.5% yellow and 99.5% blue. The yellow strains grow faster by 15%, yet take over only in discrete sectoring events, the number of which is controlled by fluctuations early in the expansion process (jackpot events). Part a adapted with permission from ref. under a Creative Commons licence CC BY 4.0. Parts b–d adapted with permission from ref. , Wiley.

Fig. 4. Proliferating particles phase separate due to crowding-induced slowdown of passive diffusion.

a, Bacteria (Acetobacter indonesiensis) colonizing cavities (numbered 1…5) of different length. The lower parts of the longer cavities 1 and 2 exhibit a dark phase where bacteria are densely packed (‘jammed’ phases); the population in cavities 3, 4 and 5 are far more dilute (‘gaseous phase’). b, c, A model of proliferating hard spheres reproduces the length-dependent transition from gaseous to jammed. The maximum fraction Φ(0) at the floor of the cavities is shown in b as a function of vertical length L of the colonized region. Colonization is only possible if L is larger than a critical length L_est, the ‘establishment’ length. The computed density profiles Φ(y) are shown in c for a few select points in b. Figure adapted with permission from ref. , National Academy of Sciences.

Fig. 5. Proliferating motile matter.

Chemotactic range expansions are guided by self-produced attractant gradients (top). The resulting propagating fronts are faster than unguided range expansions, which are described by Fisher–Kolmogorov wave equations. Figure adapted with permission from ref. , Springer Nature Ltd.

Fig. 6. Four aspects of proliferation.

a–d, Proliferation injects: biomass (part a); sources of proliferation (part b); degrees of freedom (part c); and, by making heritable errors, it also injects information (part d).