1,978 research outputs found
Learning and Designing Stochastic Processes from Logical Constraints
Stochastic processes offer a flexible mathematical formalism to model and
reason about systems. Most analysis tools, however, start from the premises
that models are fully specified, so that any parameters controlling the
system's dynamics must be known exactly. As this is seldom the case, many
methods have been devised over the last decade to infer (learn) such parameters
from observations of the state of the system. In this paper, we depart from
this approach by assuming that our observations are {\it qualitative}
properties encoded as satisfaction of linear temporal logic formulae, as
opposed to quantitative observations of the state of the system. An important
feature of this approach is that it unifies naturally the system identification
and the system design problems, where the properties, instead of observations,
represent requirements to be satisfied. We develop a principled statistical
estimation procedure based on maximising the likelihood of the system's
parameters, using recent ideas from statistical machine learning. We
demonstrate the efficacy and broad applicability of our method on a range of
simple but non-trivial examples, including rumour spreading in social networks
and hybrid models of gene regulation
Transcriptional delay stabilizes bistable gene networks
Transcriptional delay can significantly impact the dynamics of gene networks.
Here we examine how such delay affects bistable systems. We investigate several
stochastic models of bistable gene networks and find that increasing delay
dramatically increases the mean residence times near stable states. To explain
this, we introduce a non-Markovian, analytically tractable reduced model. The
model shows that stabilization is the consequence of an increased number of
failed transitions between stable states. Each of the bistable systems that we
simulate behaves in this manner
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
A stroboscopic averaging algorithm for highly oscillatory delay problems
We propose and analyze a heterogenous multiscale method for the efficient
integration of constant-delay differential equations subject to fast periodic
forcing. The stroboscopic averaging method (SAM) suggested here may provide
approximations with \(\mathcal{O}(H^2+1/\Omega^2)\) errors with a
computational effort that grows like \(H^{-1}\) (the inverse of the
stepsize), uniformly in the forcing frequency Omega
Optimal Path to Epigenetic Switching
We use large deviation methods to calculate rates of noise-induced
transitions between states in multistable genetic networks. We analyze a
synthetic biochemical circuit, the toggle switch, and compare the results to
those obtained from a numerical solution of the master equation.Comment: 5 pages. 2 figures, uses revtex 4. PR-E reviewed for publicatio
Mean-Field approximation and Quasi-Equilibrium reduction of Markov Population Models
Markov Population Model is a commonly used framework to describe stochastic
systems. Their exact analysis is unfeasible in most cases because of the state
space explosion. Approximations are usually sought, often with the goal of
reducing the number of variables. Among them, the mean field limit and the
quasi-equilibrium approximations stand out. We view them as techniques that are
rooted in independent basic principles. At the basis of the mean field limit is
the law of large numbers. The principle of the quasi-equilibrium reduction is
the separation of temporal scales. It is common practice to apply both limits
to an MPM yielding a fully reduced model. Although the two limits should be
viewed as completely independent options, they are applied almost invariably in
a fixed sequence: MF limit first, QE-reduction second. We present a framework
that makes explicit the distinction of the two reductions, and allows an
arbitrary order of their application. By inverting the sequence, we show that
the double limit does not commute in general: the mean field limit of a
time-scale reduced model is not the same as the time-scale reduced limit of a
mean field model. An example is provided to demonstrate this phenomenon.
Sufficient conditions for the two operations to be freely exchangeable are also
provided
Modeling delayed processes in biological systems
Delayed processes are ubiquitous in biological systems and are often
characterized by delay differential equations (DDEs) and their extension to
include stochastic effects. DDEs do not explicitly incorporate intermediate
states associated with a delayed process but instead use an estimated average
delay time. In an effort to examine the validity of this approach, we study
systems with significant delays by explicitly incorporating intermediate steps.
We show by that such explicit models often yield significantly different
equilibrium distributions and transition times as compared to DDEs with
deterministic delay values. Additionally, different explicit models with
qualitatively different dynamics can give rise to the same DDEs revealing
important ambiguities. We also show that DDE-based predictions of oscillatory
behavior may fail for the corresponding explicit model
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