Predictive Power of “A Minima” Models in Biology

Abstract

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.

Gap geometry dictates epithelial closure efficiency

“…Similar force patterns were observed in such conditions including force dipoles at negatively curved regions and backward-pulling forces at positively curved ones ( Supplementary Movie 6 ). Overall, these force distributions confirmed that a predominant crawling mechanism with perpendicular forces was observed in regions of positive curvature, whereas a purse-string mechanism with tangential forces with respect to the cell boundary was favored in negative regions (sustaining theoretical conjectures 31 41 42 ).…”

“…To better understand the interaction and regulation occurring between the two mechanisms, we use our mathematical model. Indeed, the decomposed contribution of the two mechanisms suggests that cells have evolved to select the mechanism that drives the closure in an efficient way 31 41 42 . This representation explains the prevalence of one or the other closure mechanism depending on the local curvature and, more generally, on the size of the gap.…”

Orientational Coupling Locally Orchestrates a Cell Migration Pattern for Re‐Epithelialization

“…This implies that the orientation dependent cell migratory behavior involves a “zipper‐like” mechanism: the full occupancy over nonadhesive upright triangle regions constitutes the “zip on” function for cells migrating along converging paths; whereas the unoccupied “hole” above inverted triangle region is a manifestation of the “zip off” function. The activation of “zip on” function could enhance the efficiency for re‐epithelialization and vice versa …”

Mean curvature flow with obstacles

“…We point out that the study of the positive curvature flow in Section 6 is related to some biological models which originally motivated our work: in several recent studies of actomyosin cable contraction in morphogenesis and tissue repair there is increasing evidence that the contractile structure forms only in the positive curvature part of the boundary curve (see [4,3] and references therein). Since the contraction of such actomyosin structures can be associated with curvature terms (see [22,1,2]), this leads very naturally to consider the positive curvature flow problem.…”