228 research outputs found
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
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
Mathematical approaches to differentiation and gene regulation
We consider some mathematical issues raised by the modelling of gene
networks. The expression of genes is governed by a complex set of regulations,
which is often described symbolically by interaction graphs. Once such a graph
has been established, there remains the difficult task to decide which
dynamical properties of the gene network can be inferred from it, in the
absence of precise quantitative data about their regulation. In this paper we
discuss a rule proposed by R.Thomas according to which the possibility for the
network to have several stationary states implies the existence of a positive
circuit in the corresponding interaction graph. We prove that, when properly
formulated in rigorous terms, this rule becomes a theorem valid for several
different types of formal models of gene networks. This result is already known
for models of differential or boolean type. We show here that a stronger
version of it holds in the differential setup when the decay of protein
concentrations is taken into account. This allows us to verify also the
validity of Thomas' rule in the context of piecewise-linear models and the
corresponding discrete models. We discuss open problems as well.Comment: To appear in Notes Comptes-Rendus Acad. Sc. Paris, Biologi
Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits
In this paper we consider a family of dynamical systems that we call
"arabesques", defined as closed chains of 2-element negative circuits. An
-dimensional arabesque system has 2-element circuits, but in addition,
it displays by construction, two -element circuits which are both positive
vs one positive and one negative, depending on the parity (even or odd) of the
dimension . In view of the absence of diagonal terms in their Jacobian
matrices, all these dynamical systems are conservative and consequently, they
can not possess any attractor. First, we analyze a linear variant of them which
we call "arabesque 0" or for short "A0". For increasing dimensions, the
trajectories are increasingly complex open tori. Next, we inserted a single
cubic nonlinearity that does not affect the signs of its circuits (that we call
"arabesque 1" or for short "A1"). These systems have three steady states,
whatever the dimension is, in agreement with the order of the nonlinearity. All
three are unstable, as there can not be any attractor in their state-space. The
3D variant (that we call for short "A1\_3D") has been analyzed in some detail
and found to display a complex mixed set of quasi-periodic and chaotic
trajectories. Inserting cubic nonlinearities (one per equation) in the same
way as above, we generate systems "A2\_D". A2\_3D behaves essentially as
A1\_3D, in agreement with the fact that the signs of the circuits remain
identical. A2\_4D, as well as other arabesque systems with even dimension, has
two positive -circuits and nine steady states. Finally, we investigate and
compare the complex dynamics of this family of systems in terms of their
symmetries.Comment: 22 pages, 12 figures, accepted for publication at Int. J. Bif. Chao
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
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