Synchrony in reaction-diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

Diversity and survival of artificial lifeforms under sedimentation and random motion

Cellular automata are often used to explore the numerous possible scenarios of what could have occurred at the origins of life and before, during the prebiotic ages, when very simple molecules started to assemble and organise into larger catalytic or informative structures, or to simulate ecosystems. Artificial self-maintained spatial structures emerge in cellular automata and are often used to represent molecules or living organisms. They converge generally towards homogeneous stationary soups of still-life creatures. It is hard for an observer to believe they are similar to living systems, in particular because nothing is moving anymore within such simulated environments after few computation steps, because they present isotropic spatial organisation, because the diversity of self-maintained morphologies is poor, and because when stationary states are reached the creatures are immortal. Natural living systems, on the contrary, are composed of a high diversity of creatures in interaction having limited lifetimes and generally present a certain anisotropy of their spatial organisation, in particular frontiers and interfaces. In the present work, we propose that the presence of directional weak fields such as gravity may counter-balance the excess of mixing and disorder caused by Brownian motion and favour the appearance of specific regions, i.e. different strata or environmental layers, in which physical-chemical conditions favour the emergence and the survival of self-maintained spatial structures including living systems. We test this hypothesis by way of numerical simulations of a very simplified ecosystem model. We use the well-known Game of Life to which we add rules simulating both sedimentation forces and thermal agitation. We show that this leads to more active (vitality and biodiversity) and robust (survival) dynamics. This effectively suggests that coupling such physical processes to reactive systems allows the separation of environments into different milieux and could constitute a simple mechanism to form ecosystem frontiers or elementary interfaces that would protect and favour the development of fragile auto-poietic systems.

Observability coefficients for predicting the class of synchronizability from the algebraic structure of the local oscillators

Understanding the conditions under which a collective dynamics emerges in a complex network is still an open problem. A useful approach is the master stability function-and its related classes of synchronization-which offers a necessary condition to assess when a network successfully synchronizes. Observability coefficients, on the other hand, quantify how well the original state space of a system can be observed given only the access to a measured variable. The question is therefore pertinent: Given a generic dynamical system (represented by a state variable x) and given a generic measure on it h(x) (which may be either an observation of an external agent, or an output function through which the units of a network interact), are classes of synchronization and observability actually related to each other? We explicitly address this issue, and show a series of nontrivial relationships for networks of different popular chaotic systems (Rössler, Lorenz, and Hindmarsh-Rose oscillators). Our results suggest that specific dynamical properties can be evoked for explaining the classes of synchronizability.

Cells as strain-cued automata

We argue in favor of representing living cells as automata and review demonstrations that autonomous cells can form patterns by responding to local variations in the strain fields that arise from their individual or collective motions. An autonomous cell's response to strain stimuli is assumed to be effected by internally-generated, internally-powered forces, which generally move the cell in directions other than those implied by external energy gradients. Evidence of cells acting as strain-cued automata have been inferred from patterns observed in nature and from experiments conducted in vitro. Simulations that mimic particular cases of pattern forming share the idealization that cells are assumed to pass information among themselves solely via mechanical boundary conditions, i.e., the tractions and displacements present at their membranes. This assumption opens three mechanisms for pattern formation in large cell populations: wavelike behavior, kinematic feedback in cell motility that can lead to sliding and rotational patterns, and directed migration during invasions. Wavelike behavior among ameloblast cells during amelogenesis (the formation of dental enamel) has been inferred from enamel microstructure, while strain waves in populations of epithelial cells have been observed in vitro. One hypothesized kinematic feedback mechanism, "enhanced shear motility", accounts successfully for the spontaneous formation of layered patterns during amelogenesis in the mouse incisor. Directed migration is exemplified by a theory of invader cells that sense and respond to the strains they themselves create in the host population as they invade it: analysis shows that the strain fields contain positional information that could aid the formation of cell network structures, stabilizing the slender geometry of branches and helping govern the frequency of branch bifurcation and branch coalescence (the formation of closed networks). In simulations of pattern formation in homogeneous populations and network formation by invaders, morphological outcomes are governed by the ratio of the rates of two competing time dependent processes, one a migration velocity and the other a relaxation velocity related to the propagation of strain information. Relaxation velocities are approximately constant for different species and organs, whereas cell migration rates vary by three orders of magnitude. We conjecture that developmental processes use rapid cell migration to achieve certain outcomes, and slow migration to achieve others. We infer from analysis of host relaxation during network formation that a transition exists in the mechanical response of a host cell from animate to inanimate behavior when its strain changes at a rate that exceeds 10-4-10-3s-1. The transition has previously been observed in experiments conducted in vitro.

Stability, complexity and robustness in population dynamics

The problem of stability in population dynamics concerns many domains of application in demography, biology, mechanics and mathematics. The problem is highly generic and independent of the population considered (human, animals, molecules,…). We give in this paper some examples of population dynamics concerning nucleic acids interacting through direct nucleic binding with small or cyclic RNAs acting on mRNAs or tRNAs as translation factors or through protein complexes expressed by genes and linked to DNA as transcription factors. The networks made of these interactions between nucleic acids (considered respectively as edges and nodes of their interaction graph) are complex, but exhibit simple emergent asymptotic behaviours, when time tends to infinity, called attractors. We show that the quantity called attractor entropy plays a crucial role in the study of the stability and robustness of such genetic networks.

Predictive power of "a minima" models in biology

Many apparently complex mechanisms in biology, especially in embryology and molecular biology, can be explained easily by reasoning at the level of the "efficient cause" of the observed phenomenology: the mechanism can then be explained by a simple geometrical argument or a variational principle, leading to the solution of an optimization problem, for example, via the co-existence of a minimization and a maximization problem (a min-max principle). Passing from a microscopic (or cellular) level (optimal min-max solution of the simple mechanistic system) to the macroscopic level often involves an averaging effect (linked to the repetition of a large number of such microscopic systems with possible random choice of the parameters of each of them) that gives birth to a global functional feature (e.g. at the tissue level). We will illustrate these general principles by building in four different domains of application "a minima" models and showing the main properties of their solutions: (1) extraction of a minimal RNA structure functioning as the first "peptidic machine," a kind of ancestral ribosome; (2) study of a genetic regulatory network of Drosophila centred on Engrailed gene and expressing successively two genes inside a limit cycle; (3) study of a genetic network regulating neural activity and proliferation in mammals; and (4) study of a simple geometric model of epiboly in zebrafish.

Demography and diffusion in epidemics: malaria and black death spread

The classical models of epidemics dynamics by Ross and McKendrick have to be revisited in order to incorporate elements coming from the demography (fecundity, mortality and migration) both of host and vector populations and from the diffusion and mutation of infectious agents. The classical approach is indeed dealing with populations supposed to be constant during the epidemic wave, but the presently observed pandemics show duration of their spread during years imposing to take into account the host and vector population changes as well as the transient or permanent migration and diffusion of hosts (susceptible or infected), as well as vectors and infectious agents. Two examples are presented, one concerning the malaria in Mali and the other the plague at the middle-age.

Biological boundaries and biological age

The chronologic age classically used in demography is often unable to give useful information about which exact stage in development or aging processes has reached an organism. Hence, we propose here to explain in some applications for what reason the chronologic age fails in explaining totally the observed state of an organism, which leads to propose a new notion, the biological age. This biological age is essentially determined by the number of divisions before the Hayflick's limit the tissue or mitochondrion in a critical organ (in the sense where its loss causes the death of the whole organism) has already used for its development and adult phases. We give a precise definition of the biological age of an organ based on the Hayflick's limit of its cells and we introduce a desynchronization index (the cell entropy) for some critical tissues or membranes, which are mainly skin, intestinal endothelium, alveoli epithelium and mitochondrial inner membrane. In these actively metabolising interface tissues or membranes, there is a rapid turnover of cells, of their cytoplasmic constituents such as proteins, and of membrane lipids. The boundaries corresponding to these tissues, cells or membranes have vital functions of interface with the environment (protection, homeothermy, nutrition and respiration) and have a rapid turnover (the total cell renewal time is in mice equal to 3 weeks for the skin, 1.5 day for the intestine, 4 months for the alveolae and 11 days for mitochondrial inner membrane) conditioning their biological age. The biological age of a tissue is made of two major components: (1) first, its embryonic age based on the distance (in number of divisions) between the birth date of its first differentiated cell and the time until it reaches its final boundary at the end of its development and (2) second, its adult age whose complement until its death is just the lapse of time made of the sum of remaining cell cycle durations authorized by its Hayflick's limit. From this definition, we calculate the global biological lifespan of an organism and revisit notions like demographic survival curves, duration and synchrony of cell cycles, living boundaries from proto-cells to organs, and embryonic and adult phases duration.