Progress in cell reprogramming recently has revived Waddington's concept of an epigenetic landscapein the term of a quasi-potential function and cell attractors in a complex dynamical system embodied by the cell's gene regulatory network (GRN) . The quasi-potential of network states have biological significance because the relative stability  in a multi-stable dynamical system offers a measure of the effort for the transition between attractors . However, quasi-potential landscapes that have been developed for continuous systems by multiple groups are not suitable for discrete networks , which are generally used to study large networks or ensembles thereof. Here we introduce the relative stability of network states to Boolean networks by using the noise-perturbed Markov matrices. With an ensemble approach of a minimal gene network for pancreas development , we show that Boolean networks with canalized / sign-compatible Boolean functionscan capture essential features of cell fate dynamics and allow the estimation of relative stabilities of network states and, hence, of transition barriers. Our Boolean network framework for calculating the relative stabilities and transition rates of network states can be used to quantify the influence of different geneson cell transitions, estimate the time sequence of cell differentiation and facilitate the rational design of cell reprogramming protocols.
Progress in cell type reprogramming has revived the interest in Waddington's concept of the epigenetic landscape. Recently researchers developed the quasi-potential theory to represent the Waddington's landscape. The Quasi-potential U(x), derived from interactions in the gene regulatory network (GRN) of a cell, quantifies the relative stability of network states, which determine the effort required for state transitions in a multi-stable dynamical system . However, quasi-potential landscapes , originally developed for continuous systems , are not suitable for discrete-valued networks which are important tools to study complex systems. In this paper, we provide a framework to quantify the landscape for discrete Boolean networks (BNs). We apply our framework to study pancreas cell differentiation where an ensemble of BN models is considered based on the structure of a minimal GRN for pancreas development . We impose biologically motivated structural constraints (corresponding to specific type of Boolean functions) and dynamical constraints (corresponding to stable attractor states) to limit the space of BN models for pancreas development. In addition, we enforce a novel functional constraint corresponding to the relative ordering of attractor states in BN models to restrict the space of BN models to the biological relevant class. We find that BNs with canalyzing / sign-compatible Boolean functions best capture the dynamics of pancreas cell differentiation. This framework can also determine the genes' influence on cell state transitions, and thus can facilitate the rational design of cell reprogramming protocols.
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, we show that Boolean networks with canalized
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sign-compatible Boolean functionscan capture essential features of cell fate dynamics and allow the estimation of relative stabilities of network states and, hence, of transition barriers. Our Boolean network framework for calculating the relative stabilities and transition rates of network states can be used to quantify the influence of different geneson cell transitions, estimate the time sequence of cell differentiation and
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