  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.
