2,419 research outputs found
Dynamical Properties of a Two-gene Network with Hysteresis
A mathematical model for a two-gene regulatory network is derived and several
of their properties analyzed. Due to the presence of mixed continuous/discrete
dynamics and hysteresis, we employ a hybrid systems model to capture the
dynamics of the system. The proposed model incorporates binary hysteresis with
different thresholds capturing the interaction between the genes. We analyze
properties of the solutions and asymptotic stability of equilibria in the
system as a function of its parameters. Our analysis reveals the presence of
limit cycles for a certain range of parameters, behavior that is associated
with hysteresis. The set of points defining the limit cycle is characterized
and its asymptotic stability properties are studied. Furthermore, the stability
property of the limit cycle is robust to small perturbations. Numerical
simulations are presented to illustrate the results.Comment: 55 pages, 31 figures.Expanded version of paper in Special Issue on
Hybrid Systems and Biology, Elsevier Information and Computation, 201
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On Robust Stability of Limit Cycles for Hybrid Systems with Multiple Jumps
In this paper, we address stability and robustness properties of hybrid limit cycles for a class of hybrid systems with multiple jumps in one period. The main results entail equivalent characterizations of stability of hybrid limit cycles for hybrid systems. The hybrid limit cycles may have multiple jumps in one period and the jumps are allowed to occur on sets. Conditions guaranteeing robustness of hybrid limit cycles are also presented
An analysis of overall network architecture reveals an infinite-period bifurcation underlying oscillation arrest in the segmentation clock
Unveiling the mechanisms through which the somitogenesis regulatory network
exerts spatiotemporal control of the somitic patterning has required a
combination of experimental and mathematical modeling strategies. Significant
progress has been made for the zebrafish clockwork. However, due to its
complexity, the clockwork of the amniote segmentation regulatory network has
not been fully elucidated. Here, we address the question of how oscillations
are arrested in the amniote segmentation clock. We do this by constructing a
minimal model of the regulatory network, which privileges architectural
information over molecular details. With a suitable choice of parameters, our
model is able to reproduce the oscillatory behavior of the Wnt, Notch and FGF
signaling pathways in presomitic mesoderm (PSM) cells. By introducing
positional information via a single Wnt3a gradient, we show that oscillations
are arrested following an infinite-period bifurcation. Notably: the
oscillations increase their amplitude as cells approach the anterior PSM and
remain in an upregulated state when arrested; the transition from the
oscillatory regime to the upregulated state exhibits hysteresis; and an
opposing distribution of the Fgf8 and RA gradients in the PSM arises naturally
in our simulations. We hypothesize that the interaction between a limit cycle
(originated by the Notch delayed-negative feedback loop) and a bistable switch
(originated by the Wnt-Notch positive cross-regulation) is responsible for the
observed segmentation patterning. Our results agree with previously unexplained
experimental observations and suggest a simple plausible mechanism for
spatiotemporal control of somitogenesis in amniotes.Comment: 11 pages, 5 figures, added references, added figures, extended
supporting material, revised arguments in the discussion, corrected typo
Designer Gene Networks: Towards Fundamental Cellular Control
The engineered control of cellular function through the design of synthetic
genetic networks is becoming plausible. Here we show how a naturally occurring
network can be used as a parts list for artificial network design, and how
model formulation leads to computational and analytical approaches relevant to
nonlinear dynamics and statistical physics.Comment: 35 pages, 8 figure
Network Topology as a Driver of Bistability in the lac Operon
The lac operon in Escherichia coli has been studied extensively and is one of
the earliest gene systems found to undergo both positive and negative control.
The lac operon is known to exhibit bistability, in the sense that the operon is
either induced or uninduced. Many dynamical models have been proposed to
capture this phenomenon. While most are based on complex mathematical
formulations, it has been suggested that for other gene systems network
topology is sufficient to produce the desired dynamical behavior.
We present a Boolean network as a discrete model for the lac operon. We
include the two main glucose control mechanisms of catabolite repression and
inducer exclusion in the model and show that it exhibits bistability. Further
we present a reduced model which shows that lac mRNA and lactose form the core
of the lac operon, and that this reduced model also exhibits the same dynamics.
This work corroborates the claim that the key to dynamical properties is the
topology of the network and signs of interactions.Comment: 15 pages, 13 figures, supplemental information include
Emergence of switch-like behavior in a large family of simple biochemical networks
Bistability plays a central role in the gene regulatory networks (GRNs)
controlling many essential biological functions, including cellular
differentiation and cell cycle control. However, establishing the network
topologies that can exhibit bistability remains a challenge, in part due to the
exceedingly large variety of GRNs that exist for even a small number of
components. We begin to address this problem by employing chemical reaction
network theory in a comprehensive in silico survey to determine the capacity
for bistability of more than 40,000 simple networks that can be formed by two
transcription factor-coding genes and their associated proteins (assuming only
the most elementary biochemical processes). We find that there exist reaction
rate constants leading to bistability in ~90% of these GRN models, including
several circuits that do not contain any of the TF cooperativity commonly
associated with bistable systems, and the majority of which could only be
identified as bistable through an original subnetwork-based analysis. A
topological sorting of the two-gene family of networks based on the presence or
absence of biochemical reactions reveals eleven minimal bistable networks
(i.e., bistable networks that do not contain within them a smaller bistable
subnetwork). The large number of previously unknown bistable network topologies
suggests that the capacity for switch-like behavior in GRNs arises with
relative ease and is not easily lost through network evolution. To highlight
the relevance of the systematic application of CRNT to bistable network
identification in real biological systems, we integrated publicly available
protein-protein interaction, protein-DNA interaction, and gene expression data
from Saccharomyces cerevisiae, and identified several GRNs predicted to behave
in a bistable fashion.Comment: accepted to PLoS Computational Biolog
Controlling spatiotemporal pattern formation in a concentration gradient with a synthetic toggle switch
The formation of spatiotemporal patterns of gene expression is frequently guided by gradients of diffusible signaling molecules. The toggle switch subnetwork, composed of two cross-repressing transcription factors, is a common component of gene regulatory networks in charge of patterning, converting the continuous information provided by the gradient into discrete abutting stripes of gene expression. We present a synthetic biology framework to understand and characterize the spatiotemporal patterning properties of the toggle switch. To this end, we built a synthetic toggle switch controllable by diffusible molecules in Escherichia coli. We analyzed the patterning capabilities of the circuit by combining quantitative measurements with a mathematical reconstruction of the underlying dynamical system. The toggle switch can produce robust patterns with sharp boundaries, governed by bistability and hysteresis. We further demonstrate how the hysteresis, position, timing, and precision of the boundary can be controlled, highlighting the dynamical flexibility of the circuit
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