14,298 research outputs found
Dynamical complexity of discrete time regulatory networks
Genetic regulatory networks are usually modeled by systems of coupled
differential equations and by finite state models, better known as logical
networks, are also used. In this paper we consider a class of models of
regulatory networks which present both discrete and continuous aspects. Our
models consist of a network of units, whose states are quantified by a
continuous real variable. The state of each unit in the network evolves
according to a contractive transformation chosen from a finite collection of
possible transformations, according to a rule which depends on the state of the
neighboring units. As a first approximation to the complete description of the
dynamics of this networks we focus on a global characteristic, the dynamical
complexity, related to the proliferation of distinguishable temporal behaviors.
In this work we give explicit conditions under which explicit relations between
the topological structure of the regulatory network, and the growth rate of the
dynamical complexity can be established. We illustrate our results by means of
some biologically motivated examples.Comment: 28 pages, 4 figure
Discrete time piecewise affine models of genetic regulatory networks
We introduce simple models of genetic regulatory networks and we proceed to
the mathematical analysis of their dynamics. The models are discrete time
dynamical systems generated by piecewise affine contracting mappings whose
variables represent gene expression levels. When compared to other models of
regulatory networks, these models have an additional parameter which is
identified as quantifying interaction delays. In spite of their simplicity,
their dynamics presents a rich variety of behaviours. This phenomenology is not
limited to piecewise affine model but extends to smooth nonlinear discrete time
models of regulatory networks. In a first step, our analysis concerns general
properties of networks on arbitrary graphs (characterisation of the attractor,
symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc).
In a second step, focus is made on simple circuits for which the attractor and
its changes with parameters are described. In the negative circuit of 2 genes,
a thorough study is presented which concern stable (quasi-)periodic
oscillations governed by rotations on the unit circle -- with a rotation number
depending continuously and monotonically on threshold parameters. These regular
oscillations exist in negative circuits with arbitrary number of genes where
they are most likely to be observed in genetic systems with non-negligible
delay effects.Comment: 34 page
Ensembles, Dynamics, and Cell Types: Revisiting the Statistical Mechanics Perspective on Cellular Regulation
Genetic regulatory networks control ontogeny. For fifty years Boolean
networks have served as models of such systems, ranging from ensembles of
random Boolean networks as models for generic properties of gene regulation to
working dynamical models of a growing number of sub-networks of real cells. At
the same time, their statistical mechanics has been thoroughly studied. Here we
recapitulate their original motivation in the context of current theoretical
and empirical research. We discuss ensembles of random Boolean networks whose
dynamical attractors model cell types. A sub-ensemble is the critical ensemble.
There is now strong evidence that genetic regulatory networks are dynamically
critical, and that evolution is exploring the critical sub-ensemble. The
generic properties of this sub-ensemble predict essential features of cell
differentiation. In particular, the number of attractors in such networks
scales as the DNA content raised to the 0.63 power. Data on the number of cell
types as a function of the DNA content per cell shows a scaling relationship of
0.88. Thus, the theory correctly predicts a power law relationship between the
number of cell types and the DNA contents per cell, and a comparable slope. We
discuss these new scaling values and show prospects for new research lines for
Boolean networks as a base model for systems biology.Comment: 22 pages, article will be included in a special issue of J. Theor.
Biol. dedicated to the memory of Prof. Rene Thoma
Additive Functions in Boolean Models of Gene Regulatory Network Modules
Gene-on-gene regulations are key components of every living organism. Dynamical abstract models of genetic regulatory networks help explain the genome’s evolvability and robustness. These properties can be attributed to the structural topology of the graph formed by genes, as vertices, and regulatory interactions, as edges. Moreover, the actual gene interaction of each gene is believed to play a key role in the stability of the structure. With advances in biology, some effort was deployed to develop update functions in Boolean models that include recent knowledge. We combine real-life gene interaction networks with novel update functions in a Boolean model. We use two sub-networks of biological organisms, the yeast cell-cycle and the mouse embryonic stem cell, as topological support for our system. On these structures, we substitute the original random update functions by a novel threshold-based dynamic function in which the promoting and repressing effect of each interaction is considered. We use a third real-life regulatory network, along with its inferred Boolean update functions to validate the proposed update function. Results of this validation hint to increased biological plausibility of the threshold-based function. To investigate the dynamical behavior of this new model, we visualized the phase transition between order and chaos into the critical regime using Derrida plots. We complement the qualitative nature of Derrida plots with an alternative measure, the criticality distance, that also allows to discriminate between regimes in a quantitative way. Simulation on both real-life genetic regulatory networks show that there exists a set of parameters that allows the systems to operate in the critical region. This new model includes experimentally derived biological information and recent discoveries, which makes it potentially useful to guide experimental research. The update function confers additional realism to the model, while reducing the complexity and solution space, thus making it easier to investigate
The number and probability of canalizing functions
Canalizing functions have important applications in physics and biology. For
example, they represent a mechanism capable of stabilizing chaotic behavior in
Boolean network models of discrete dynamical systems. When comparing the class
of canalizing functions to other classes of functions with respect to their
evolutionary plausibility as emergent control rules in genetic regulatory
systems, it is informative to know the number of canalizing functions with a
given number of input variables. This is also important in the context of using
the class of canalizing functions as a constraint during the inference of
genetic networks from gene expression data. To this end, we derive an exact
formula for the number of canalizing Boolean functions of n variables. We also
derive a formula for the probability that a random Boolean function is
canalizing for any given bias p of taking the value 1. In addition, we consider
the number and probability of Boolean functions that are canalizing for exactly
k variables. Finally, we provide an algorithm for randomly generating
canalizing functions with a given bias p and any number of variables, which is
needed for Monte Carlo simulations of Boolean networks
Boolean network model predicts cell cycle sequence of fission yeast
A Boolean network model of the cell-cycle regulatory network of fission yeast
(Schizosaccharomyces Pombe) is constructed solely on the basis of the known
biochemical interaction topology. Simulating the model in the computer,
faithfully reproduces the known sequence of regulatory activity patterns along
the cell cycle of the living cell. Contrary to existing differential equation
models, no parameters enter the model except the structure of the regulatory
circuitry. The dynamical properties of the model indicate that the biological
dynamical sequence is robustly implemented in the regulatory network, with the
biological stationary state G1 corresponding to the dominant attractor in state
space, and with the biological regulatory sequence being a strongly attractive
trajectory. Comparing the fission yeast cell-cycle model to a similar model of
the corresponding network in S. cerevisiae, a remarkable difference in
circuitry, as well as dynamics is observed. While the latter operates in a
strongly damped mode, driven by external excitation, the S. pombe network
represents an auto-excited system with external damping.Comment: 10 pages, 3 figure
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