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.

Figure 1. Iteration graph.

Iteration graph representing the dynamics of an arbitrary discrete dynamical system having four attractors : two fixed points, and , and two limit cycles, and . The attraction basins of these attractors are respectively , , and .

Figure 2. Original Mendoza & Alvarez-Buylla network.

Original genetic regulation network modeling the flower morphogenesis of the plant Arabidopsis thaliana. Above is pictured the underlying interaction graph. Repressions (resp. activations) are represented by empty arrows (resp. full arrows). Below, the matrix of size contains the interaction weights between genes and is the thresholds vector.

Figure 3. Reduced Mendoza & Alvarez-Buylla network.

Genetic regulatory network with two non-trivial strongly connected symmetric components (in grey). The asymptotic dynamics of this network has the same attractors as the original network.

Figure 4. General iteration graph of the strongly connected symmetric component {ap3, bfu, pi}.

General iteration graph of the strongly connected symmetric component of the reduced Mendoza & Alvarez-Buylla network pictured in Figure 3. In this graph, for the sake of clarity, we have represented arcs with the same beginning and ending as one unique arc labelled by . The sub-graph in grey corresponds to a limit cycle of the connected component with the parallel iteration mode. It induces limit cycles 1, 3, 4, 6 and 7 of Table 1. Note that when the state of nodes lfy, ufo and sup is fixed to in (this always becomes true after a few steps according to the proof of Proposition 1), then the connected component is free to evolve as pictured by this general iteration graph.

Figure 5. General iteration graph of the strongly connected symmetric component {ap1, ag}.

General iteration graph of the strongly connected symmetric component of the reduced Mendoza & Alvarez-Buylla network pictured in Figure 3 (a) when the states of nodes emf1 and tfl1 are both fixed to and (b) when they are both fixed to . In this graph, for the sake of clarity, we have represented arcs with the same beginning and ending as one unique arc labelled by . The sub-graph in grey is a limit cycle of the connected component with the parallel iteration mode. It induces limit cycles 2, 3, 4 and 5 of Table 1. Note that when the state of nodes lfy and lug is fixed to in (this always becomes true after a few steps according to the proof of Proposition 1), then the connected component is free to evolve as pictured by one of these two general iteration graphs since no other nodes than emf1 and tfl1 whose states are either both or both have an influence on them.

Figure 6. Toy model.

Variation of the original Mendoza & Alvarez-Buylla network. This version of the network includes a supplementary node corresponding to gene rga (in dashed lines) to account for the gibberellin's influence on the rest of the network. Above is pictured the interaction graph of this network. Repressions (resp. activations) are represented by empty arrows (resp. full arrows). Nodes and interactions added to the original network are indicated in dashed grey. The matrix of size contains the interaction weights. is the activation thresholds vector.

Figure 7. Attraction basins sizes.

Histograms representing the absolute sizes (left panel) and the relative sizes (right panel) of the attraction basins of the genetic regulation network of the floral morphogenesis of the plant Arabidopsis thaliana, depending on the absence or presence of gibberellin.

Figure 8. Algorithm.

Algorithm that writes in explicit form the characteristic polynomials (with indeterminate the state perturbation rate ) of the probabilities of passage from any attraction basin to any other.

Figure 9. Characteristic polynomials.

Curves of the characteristic polynomials with indeterminate the state perturbation rate (in percents). The top panels (resp. the bottom panels) plot the curves of the characteristic polynomials of the probabilities of passage from the sepal (resp. inflorescence) attraction basin to another attraction basin when the state of rga is free (left panel) and when it is fixed to by the presence of gibberellin (right panel). Every panel plots six curves, one for each ordered couple of attraction basins. In the bottom panels, several curves are superimposed: the curves of the characteristic polynomials corresponding to the probabilities to become either a Pet configuration or a Sta configuration in the left panel as well as the curves of the characteristic polynomials corresponding to the probabilities to become either a Sep configuration or a Car configuration in the right panel.