The Impact of Self-Loops on Boolean Networks Attractor Landscape and Implications for Cell Differentiation Modelling

Abstract

Boolean networks are a notable model of gene regulatory networks and, particularly, prominent theories discuss how they can capture cellular differentiation processes. One frequent motif in gene regulatory networks, especially in those circuits involved in cell differentiation, is autoregulation. In spite of this, the impact of autoregulation on Boolean network attractor landscape has not yet been extensively discussed in literature. In this paper we propose to model autoregulation as self-loops, and analyse how the number of attractors and their robustness may change once they are introduced in a well-known and widely used Boolean networks model, namely random Boolean networks. Results show that self-loops provide an evolutionary advantage in dynamic mechanisms of cells, by increasing both number and maximal robustness of attractors. These results provide evidence to the hypothesis that autoregulation is a straightforward functional component to consolidate cell dynamics, mainly in differentiation processes.

Breaking reflection symmetry: evolving long dynamical cycles in Boolean systems

“…Unlike Hopfield networks, the lack of symmetry in the interaction matrix ( ) of Boolean networks implies the non-existence of a Lyapunov function, making them difficult to study analytically [ 27 , 28 ]. Further, the inclusion of self-loops has been shown to increase the number and robustness of attractor states, thereby increasing the complexity of our model’s dynamics [ 29 ]. For these reasons, such models are remarkably expressive and useful in explaining real biological observations such as cell differentiation [ 30 ].…”

Excitation and inhibition imbalance affects dynamical complexity through symmetries

“…Unlike Hopfield networks, the lack of symmetry in the interaction matrix (a ij = a ji ) of Boolean networks implies the non-existence of a Lyapunov function, making them difficult to study analytically [55,56]. Further, the inclusion of self-loops has been shown to increase the number and robustness of attractor states, thereby increasing the complexity of our model's dynamics [57]. For these reasons, such models are remarkably expressive and useful in explaining real biological observations such as cell differentiation [58].…”

diffeRenTES: An R package for computing cell differentiation trees from Boolean networks

“…Dynamical systems have been successfully employed to investigate and reproduce biological phenomena. Among these, Boolean networks (BNs) [1,2], thanks to their simplicity combined with the rich bouquet of complex behaviors they can exhibit, have proved capable of reproducing relevant properties and dynamics of cell differentiation [2][3][4][5][6][7][8]. In these works, the focus of the analysis is on the identification of the attractors -portions of the state space of a dynamical system that attract distinct trajectories -which can express gene expression patterns similar to those found in real cells.…”