190 research outputs found
Collectively canalizing Boolean functions
This paper studies the mathematical properties of collectively canalizing
Boolean functions, a class of functions that has arisen from applications in
systems biology. Boolean networks are an increasingly popular modeling
framework for regulatory networks, and the class of functions studied here
captures a key feature of biological network dynamics, namely that a subset of
one or more variables, under certain conditions, can dominate the value of a
Boolean function, to the exclusion of all others. These functions have rich
mathematical properties to be explored. The paper shows how the number and type
of such sets influence a function's behavior and define a new measure for the
canalizing strength of any Boolean function. We further connect the concept of
collective canalization with the well-studied concept of the average
sensitivity of a Boolean function. The relationship between Boolean functions
and the dynamics of the networks they form is important in a wide range of
applications beyond biology, such as computer science, and has been studied
with statistical and simulation-based methods. But the rich relationship
between structure and dynamics remains largely unexplored, and this paper is
intended as a contribution to its mathematical foundation.Comment: 15 pages, 2 figure
Canalizing Kauffman networks: non-ergodicity and its effect on their critical behavior
Boolean Networks have been used to study numerous phenomena, including gene
regulation, neural networks, social interactions, and biological evolution.
Here, we propose a general method for determining the critical behavior of
Boolean systems built from arbitrary ensembles of Boolean functions. In
particular, we solve the critical condition for systems of units operating
according to canalizing functions and present strong numerical evidence that
our approach correctly predicts the phase transition from order to chaos in
such systems.Comment: to be published in PR
Evolution of Canalizing Boolean Networks
Boolean networks with canalizing functions are used to model gene regulatory
networks. In order to learn how such networks may behave under evolutionary
forces, we simulate the evolution of a single Boolean network by means of an
adaptive walk, which allows us to explore the fitness landscape. Mutations
change the connections and the functions of the nodes. Our fitness criterion is
the robustness of the dynamical attractors against small perturbations. We find
that with this fitness criterion the global maximum is always reached and that
there is a huge neutral space of 100% fitness. Furthermore, in spite of having
such a high degree of robustness, the evolved networks still share many
features with "chaotic" networks.Comment: 8 pages, 10 figures; revised and extended versio
Propagation of external regulation and asynchronous dynamics in random Boolean networks
Boolean Networks and their dynamics are of great interest as abstract
modeling schemes in various disciplines, ranging from biology to computer
science. Whereas parallel update schemes have been studied extensively in past
years, the level of understanding of asynchronous updates schemes is still very
poor. In this paper we study the propagation of external information given by
regulatory input variables into a random Boolean network. We compute both
analytically and numerically the time evolution and the asymptotic behavior of
this propagation of external regulation (PER). In particular, this allows us to
identify variables which are completely determined by this external
information. All those variables in the network which are not directly fixed by
PER form a core which contains in particular all non-trivial feedback loops. We
design a message-passing approach allowing to characterize the statistical
properties of these cores in dependence of the Boolean network and the external
condition. At the end we establish a link between PER dynamics and the full
random asynchronous dynamics of a Boolean network.Comment: 19 pages, 14 figures, to appear in Chao
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