Attraction Basins as Gauges of Robustness against Boundary Conditions in Biological Complex Systems

Abstract

One fundamental concept in the context of biological systems on which researches have flourished in the past decade is that of the apparent robustness of these systems, i.e., their ability to resist to perturbations or constraints induced by external or boundary elements such as electromagnetic fields acting on neural networks, micro-RNAs acting on genetic networks and even hormone flows acting both on neural and genetic networks. Recent studies have shown the importance of addressing the question of the environmental robustness of biological networks such as neural and genetic networks. In some cases, external regulatory elements can be given a relevant formal representation by assimilating them to or modeling them by boundary conditions. This article presents a generic mathematical approach to understand the influence of boundary elements on the dynamics of regulation networks, considering their attraction basins as gauges of their robustness. The application of this method on a real genetic regulation network will point out a mathematical explanation of a biological phenomenon which has only been observed experimentally until now, namely the necessity of the presence of gibberellin for the flower of the plant Arabidopsis thaliana to develop normally.

Discrete causal model view of biological networks

“…This regulatory behavior typically stresses the role of an initiator in a process which does not interact once launched. This behavior is confirmed in other theoretical [15,6] and experimental [28] studies, since LFY is known to be only transiently expressed in the developing flowers and reaches its highest level during the flower induction. Weak-Or regulators are dominated in the control of expression by other regulators when they are not available.…”

“…generators) to qualify the behavior of the interactions more precisely. We illustrate the model inference process using cabin in Arabidopsis thaliana morphogenesis, originally modeled in [14] and also in [6] using Neural networks model and in [15] with René Thomas' model. The ABC developmental model [4] postulates that the development of flower organs is the result of the combination of .…”

Synchronism versus asynchronism in monotonic Boolean automata networks

“…In particular, the additional complexity brought by the different possible ways of updating automata states has been dealt with in the past by imposing restrictions on the updating mode (e.g. restrictions to the nondeterministic asynchronous one [22,23,27,29], the parallel one, or some deterministic [1,2,8,14] or probabilistic ones [9,15,25]). …”

“Immunetworks”, intersecting circuits and dynamics

“…The set of configurations leading towards a given attractor is called the attraction basin of this attractor [12,13]. The notion of attraction basin will be of importance here because, as in [14], attraction basins will serve later as gauges of the robustness and plausibility of some attractors that will be considered. More precisely, we will abide by the idea that the greater the size of the attraction basin of an attractor A is, the most likely is this attractor.…”

“…Structurally composed of seven positive circuits (one of size 1 and six of size 2 that intersect pairwise) and two negative circuits (one of size 3 and one of size 4 that share two arcs, i.e., three nodes), this network has six fixed points amongst which four represent floral tissues (sepals, petals, stamens and carpels), one corresponds to a tissue which is in the plant but not in the flower (inflorescence) and one has never been observed neither in nature nor by experimentation (mutant). In [14], the authors have emphasised mathematical relationships between some structural features of the network, its attractors and the physiological functions of the plant that ensure its flowering. Other studies [81,82] about the floral morphogenesis of Arabidopsis thaliana, closer to the domain of statistical physics, have emphasised links between networks dynamics and cellular differentiation.…”