Uncovering the role that specific genes play in reachability properties of complex biological Gene Regulatory Networks (GRNs): The Arabidopsis thaliana flower development case

The dynamic processes of multicellular organism development are regulated and coordinated by Gene Regulatory Networks (GRN’s). Therefore, a sustained effort to understand the dynamical properties of these modularly structured networks has shown the great utility of experimentally grounded and dynamically characterized discrete Boolean models, as an ideal formalism based on dynamical systems modeling tool for its qualitative description. Up to now, several low-dimensional Boolean GRNs have been proposed to recover gene activation configurations observed for specific cell types, and they have been validated via robustness and mutant analyses. Nevertheless, systematic studies that elucidate the role of individual genes implicated on transitions between given attractors in the context of morphogenetic patterns of development, are still very scarce. Such sort of studies is in fact quite relevant because genes belonging to a given GRN do not work in isolation. Indeed, they could interact with others GRN’s and/or with micro-environmental cues. Consequently, the structural specificities of the involved genes at the network level should be assessed in order to uncover the functional nature of the larger network involved. This is particularly meaningful when considering the role played by specific genes on transient dynamics related to cell fate specification. Following this idea, we propose here a computer-based analytical procedure intended to elucidate the role that specific genes play on the reachability properties of GRNs. As a structural property of a given dynamical network, reachability characterizes the attainability of specified attractors from given initial attractors as a consequence of the action of specific driving exogenous stimuli. Our proposal is based on algebraic systems approaches built around the Semi-Tensor Product (STP). We illustrate here our proposed procedure through the exploration of the reachability properties of the well-known Floral Organ Specification GRN of Arabidopsis thaliana (FOS-GRN), that recovers ten fixed-point attractors. Our findings suggest that there exist 79 inducible transitions among all possible pairs of attractors, with a suitable external Boolean control input over different well-characterized nodes of the network. Additionally, we found that such potentiality of these genes to produce attractor transitions is maintained by the continuous approximation model of the FOS-GRN, recovering not only qualitative but also useful quantitative information. Finally, we discussed the biological significance of our results and, even if we do not establish the specific molecular nature of the characterized exogenous control input, we concluded that reachability analysis can give us some important insights on the network level role that individual genes acquire by their collaboration with the GRN, becoming then targets in cell-fate decisions during development. Author summary Bringing to light the specific role that given genes play in gene regulatory networks is of particular importance, especially when it is necessary to quantify the influence of the environment on their dynamics. However, this becomes difficult by the fact that the genes in the network interact both nonlinearly and in the presence of feedback-based interactions. This requires then the development of methods of analysis that take into account such complex interdependencies. In this regard, Control Theory offers tools that allow characterizing the changes in the transition patterns between stable configurations (that is, cellular phenotypes) of the networks as a result of the presence of exogenous stimuli . In this work we propose a method based on the algebraic representation of small size gene regulatory networks, we first described in discrete Boolean terms, focused on delimiting the influence that specific nodes of the network play in the enhancement of transitions that define trajectories in the space of stable configurations of the state of expression of the genes involved. The proposed method uses the key control-oriented concept of reachability and is illustrated by the characterization of some induced morphogenetic trajectories that explain the development stages of Arabidopsis thaliana flower organs. Our proposal allows us to confirm that, in biological terms, the reachability analysis offers powerful tools to deepen the understanding of the interplay between the structure of specific gene regulatory networks involving both their own constitutive elements and including the networks with which they interact. This contributes to the understanding of biological development, which opens an access route to the exploration of the basic principles that associate the structure of gene regulatory processes not only with cell reprogramming and cell dedifferentiation, but also with dynamic processes underlying phenotypic plasticity and its evolutionary consequences. This is because the explanation of phenotypic change responses to environmental variability requires specifying how the constraints that govern developmental trajectories, potentially elucidated through reachability analysis, modulate the balance between phenotypic robustness and evolvability. ### Competing Interest Statement The authors have declared no competing interest.