2,225 research outputs found
Nested canalyzing depth and network stability
We introduce the nested canalyzing depth of a function, which measures the
extent to which it retains a nested canalyzing structure. We characterize the
structure of functions with a given depth and compute the expected activities
and sensitivities of the variables. This analysis quantifies how canalyzation
leads to higher stability in Boolean networks. It generalizes the notion of
nested canalyzing functions (NCFs), which are precisely the functions with
maximum depth. NCFs have been proposed as gene regulatory network models, but
their structure is frequently too restrictive and they are extremely sparse. We
find that functions become decreasingly sensitive to input perturbations as the
canalyzing depth increases, but exhibit rapidly diminishing returns in
stability. Additionally, we show that as depth increases, the dynamics of
networks using these functions quickly approach the critical regime, suggesting
that real networks exhibit some degree of canalyzing depth, and that NCFs are
not significantly better than functions of sufficient depth for many
applications of the modeling and reverse engineering of biological networks.Comment: 13 pages, 2 figure
Robustness of Transcriptional Regulation in Yeast-like Model Boolean Networks
We investigate the dynamical properties of the transcriptional regulation of
gene expression in the yeast Saccharomyces Cerevisiae within the framework of a
synchronously and deterministically updated Boolean network model. By means of
a dynamically determinant subnetwork, we explore the robustness of
transcriptional regulation as a function of the type of Boolean functions used
in the model that mimic the influence of regulating agents on the transcription
level of a gene. We compare the results obtained for the actual yeast network
with those from two different model networks, one with similar in-degree
distribution as the yeast and random otherwise, and another due to Balcan et
al., where the global topology of the yeast network is reproduced faithfully.
We, surprisingly, find that the first set of model networks better reproduce
the results found with the actual yeast network, even though the Balcan et al.
model networks are structurally more similar to that of yeast.Comment: 7 pages, 4 figures, To appear in Int. J. Bifurcation and Chaos, typos
were corrected and 2 references were adde
Inferring Biologically Relevant Models: Nested Canalyzing Functions
Inferring dynamic biochemical networks is one of the main challenges in
systems biology. Given experimental data, the objective is to identify the
rules of interaction among the different entities of the network. However, the
number of possible models fitting the available data is huge and identifying a
biologically relevant model is of great interest. Nested canalyzing functions,
where variables in a given order dominate the function, have recently been
proposed as a framework for modeling gene regulatory networks. Previously we
described this class of functions as an algebraic toric variety. In this paper,
we present an algorithm that identifies all nested canalyzing models that fit
the given data. We demonstrate our methods using a well-known Boolean model of
the cell cycle in budding yeast
Mutual information in random Boolean models of regulatory networks
The amount of mutual information contained in time series of two elements
gives a measure of how well their activities are coordinated. In a large,
complex network of interacting elements, such as a genetic regulatory network
within a cell, the average of the mutual information over all pairs is a
global measure of how well the system can coordinate its internal dynamics. We
study this average pairwise mutual information in random Boolean networks
(RBNs) as a function of the distribution of Boolean rules implemented at each
element, assuming that the links in the network are randomly placed. Efficient
numerical methods for calculating show that as the number of network nodes
N approaches infinity, the quantity N exhibits a discontinuity at parameter
values corresponding to critical RBNs. For finite systems it peaks near the
critical value, but slightly in the disordered regime for typical parameter
variations. The source of high values of N is the indirect correlations
between pairs of elements from different long chains with a common starting
point. The contribution from pairs that are directly linked approaches zero for
critical networks and peaks deep in the disordered regime.Comment: 11 pages, 6 figures; Minor revisions for clarity and figure format,
one reference adde
Random Boolean Network Models and the Yeast Transcriptional Network
The recently measured yeast transcriptional network is analyzed in terms of
simplified Boolean network models, with the aim of determining feasible rule
structures, given the requirement of stable solutions of the generated Boolean
networks. We find that for ensembles of generated models, those with canalyzing
Boolean rules are remarkably stable, whereas those with random Boolean rules
are only marginally stable. Furthermore, substantial parts of the generated
networks are frozen, in the sense that they reach the same state regardless of
initial state. Thus, our ensemble approach suggests that the yeast network
shows highly ordered dynamics.Comment: 23 pages, 5 figure
Relative Stability of Network States in Boolean Network Models of Gene Regulation in Development
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.Comment: 24 pages, 6 figures, 1 tabl
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