10,533 research outputs found
General Iteration graphs and Boolean automata circuits
This article is set in the field of regulation networks modeled by discrete
dynamical systems. It focuses on Boolean automata networks. In such networks,
there are many ways to update the states of every element. When this is done
deterministically, at each time step of a discretised time flow and according
to a predefined order, we say that the network is updated according to
block-sequential update schedule (blocks of elements are updated sequentially
while, within each block, the elements are updated synchronously). Many
studies, for the sake of simplicity and with some biologically motivated
reasons, have concentrated on networks updated with one particular
block-sequential update schedule (more often the synchronous/parallel update
schedule or the sequential update schedules). The aim of this paper is to give
an argument formally proven and inspired by biological considerations in favour
of the fact that the choice of a particular update schedule does not matter so
much in terms of the possible and likely dynamical behaviours that networks may
display
Negative circuits and sustained oscillations in asynchronous automata networks
The biologist Ren\'e Thomas conjectured, twenty years ago, that the presence
of a negative feedback circuit in the interaction graph of a dynamical system
is a necessary condition for this system to produce sustained oscillations. In
this paper, we state and prove this conjecture for asynchronous automata
networks, a class of discrete dynamical systems extensively used to model the
behaviors of gene networks. As a corollary, we obtain the following fixed point
theorem: given a product of finite intervals of integers, and a map
from to itself, if the interaction graph associated with has no
negative circuit, then has at least one fixed point
Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks
We are interested in fixed points in Boolean networks, {\em i.e.} functions
from to itself. We define the subnetworks of as the
restrictions of to the subcubes of , and we characterizes a
class of Boolean networks satisfying the following property:
Every subnetwork of has a unique fixed point if and only if has no
subnetwork in . This characterization generalizes the fixed point
theorem of Shih and Dong, which asserts that if for every in
there is no directed cycle in the directed graph whose the adjacency matrix is
the discrete Jacobian matrix of evaluated at point , then has a
unique fixed point. Then, denoting by (resp. )
the networks whose the interaction graph is a positive (resp. negative) cycle,
we show that the non-expansive networks of are exactly the
networks of ; and for the class of
non-expansive networks we get a "dichotomization" of the previous forbidden
subnetwork theorem: Every subnetwork of has at most (resp. at least) one
fixed point if and only if has no subnetworks in (resp.
) subnetwork. Finally, we prove that if is a conjunctive
network then every subnetwork of has at most one fixed point if and only if
has no subnetwork in .Comment: 40 page
Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems
A complete classification of the computational complexity of the fixed-point
existence problem for boolean dynamical systems, i.e., finite discrete
dynamical systems over the domain {0, 1}, is presented. For function classes F
and graph classes G, an (F, G)-system is a boolean dynamical system such that
all local transition functions lie in F and the underlying graph lies in G. Let
F be a class of boolean functions which is closed under composition and let G
be a class of graphs which is closed under taking minors. The following
dichotomy theorems are shown: (1) If F contains the self-dual functions and G
contains the planar graphs then the fixed-point existence problem for (F,
G)-systems with local transition function given by truth-tables is NP-complete;
otherwise, it is decidable in polynomial time. (2) If F contains the self-dual
functions and G contains the graphs having vertex covers of size one then the
fixed-point existence problem for (F, G)-systems with local transition function
given by formulas or circuits is NP-complete; otherwise, it is decidable in
polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24
versio
Homogeneous and Scalable Gene Expression Regulatory Networks with Random Layouts of Switching Parameters
We consider a model of large regulatory gene expression networks where the
thresholds activating the sigmoidal interactions between genes and the signs of
these interactions are shuffled randomly. Such an approach allows for a
qualitative understanding of network dynamics in a lack of empirical data
concerning the large genomes of living organisms. Local dynamics of network
nodes exhibits the multistationarity and oscillations and depends crucially
upon the global topology of a "maximal" graph (comprising of all possible
interactions between genes in the network). The long time behavior observed in
the network defined on the homogeneous "maximal" graphs is featured by the
fraction of positive interactions () allowed between genes.
There exists a critical value such that if , the
oscillations persist in the system, otherwise, when it tends to
a fixed point (which position in the phase space is determined by the initial
conditions and the certain layout of switching parameters). In networks defined
on the inhomogeneous directed graphs depleted in cycles, no oscillations arise
in the system even if the negative interactions in between genes present
therein in abundance (). For such networks, the bidirectional edges
(if occur) influence on the dynamics essentially. In particular, if a number of
edges in the "maximal" graph is bidirectional, oscillations can arise and
persist in the system at any low rate of negative interactions between genes
(). Local dynamics observed in the inhomogeneous scalable regulatory
networks is less sensitive to the choice of initial conditions. The scale free
networks demonstrate their high error tolerance.Comment: LaTeX, 30 pages, 20 picture
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
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