2,210 research outputs found
Inference for stochastic chemical kinetics using moment equations and system size expansion
Quantitative mechanistic models are valuable tools for disentangling biochemical pathways and for achieving a comprehensive understanding of biological systems. However, to be quantitative the parameters of these models have to be estimated from experimental data. In the presence of significant stochastic fluctuations this is a challenging task as stochastic simulations are usually too time-consuming and a macroscopic description using reaction rate equations (RREs) is no longer accurate. In this manuscript, we therefore consider moment-closure approximation (MA) and the system size expansion (SSE), which approximate the statistical moments of stochastic processes and tend to be more precise than macroscopic descriptions. We introduce gradient-based parameter optimization methods and uncertainty analysis methods for MA and SSE. Efficiency and reliability of the methods are assessed using simulation examples as well as by an application to data for Epo-induced JAK/STAT signaling. The application revealed that even if merely population-average data are available, MA and SSE improve parameter identifiability in comparison to RRE. Furthermore, the simulation examples revealed that the resulting estimates are more reliable for an intermediate volume regime. In this regime the estimation error is reduced and we propose methods to determine the regime boundaries. These results illustrate that inference using MA and SSE is feasible and possesses a high sensitivity
Data-driven modelling of biological multi-scale processes
Biological processes involve a variety of spatial and temporal scales. A
holistic understanding of many biological processes therefore requires
multi-scale models which capture the relevant properties on all these scales.
In this manuscript we review mathematical modelling approaches used to describe
the individual spatial scales and how they are integrated into holistic models.
We discuss the relation between spatial and temporal scales and the implication
of that on multi-scale modelling. Based upon this overview over
state-of-the-art modelling approaches, we formulate key challenges in
mathematical and computational modelling of biological multi-scale and
multi-physics processes. In particular, we considered the availability of
analysis tools for multi-scale models and model-based multi-scale data
integration. We provide a compact review of methods for model-based data
integration and model-based hypothesis testing. Furthermore, novel approaches
and recent trends are discussed, including computation time reduction using
reduced order and surrogate models, which contribute to the solution of
inference problems. We conclude the manuscript by providing a few ideas for the
development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and
Multiscale Dynamics (American Scientific Publishers
A Minimal Model of Burst-Noise Induced Bistability
We investigate the influence of intrinsic noise on stable states of a
one-dimensional dynamical system that shows in its deterministic version a
saddle-node bifurcation between monostable and bistable behaviour. The system
is a modified version of the Schl\"ogl model, which is a chemical reaction
system with only one type of molecule. The strength of the intrinsic noise is
varied without changing the deterministic description by introducing bursts in
the autocatalytic production step. We study the transitions between monostable
and bistable behavior in this system by evaluating the number of maxima of the
stationary probability distribution. We find that changing the size of bursts
can destroy and even induce saddle-node bifurcations. This means that a bursty
production of molecules can qualitatively change the dynamics of a chemical
reaction system even when the deterministic description remains unchanged.Comment: 7 pages, 9 figure
Stochastic ordinary differential equations in applied and computational mathematics
Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation
Stochastic reaction networks with input processes: Analysis and applications to reporter gene systems
Stochastic reaction network models are widely utilized in biology and
chemistry to describe the probabilistic dynamics of biochemical systems in
general, and gene interaction networks in particular. Most often, statistical
analysis and inference of these systems is addressed by parametric approaches,
where the laws governing exogenous input processes, if present, are themselves
fixed in advance. Motivated by reporter gene systems, widely utilized in
biology to monitor gene activation at the individual cell level, we address the
analysis of reaction networks with state-affine reaction rates and arbitrary
input processes. We derive a generalization of the so-called moment equations
where the dynamics of the network statistics are expressed as a function of the
input process statistics. In stationary conditions, we provide a spectral
analysis of the system and elaborate on connections with linear filtering. We
then apply the theoretical results to develop a method for the reconstruction
of input process statistics, namely the gene activation autocovariance
function, from reporter gene population snapshot data, and demonstrate its
performance on a simulated case study
Approximation and inference methods for stochastic biochemical kinetics - a tutorial review
Stochastic fluctuations of molecule numbers are ubiquitous in biological
systems. Important examples include gene expression and enzymatic processes in
living cells. Such systems are typically modelled as chemical reaction networks
whose dynamics are governed by the Chemical Master Equation. Despite its simple
structure, no analytic solutions to the Chemical Master Equation are known for
most systems. Moreover, stochastic simulations are computationally expensive,
making systematic analysis and statistical inference a challenging task.
Consequently, significant effort has been spent in recent decades on the
development of efficient approximation and inference methods. This article
gives an introduction to basic modelling concepts as well as an overview of
state of the art methods. First, we motivate and introduce deterministic and
stochastic methods for modelling chemical networks, and give an overview of
simulation and exact solution methods. Next, we discuss several approximation
methods, including the chemical Langevin equation, the system size expansion,
moment closure approximations, time-scale separation approximations and hybrid
methods. We discuss their various properties and review recent advances and
remaining challenges for these methods. We present a comparison of several of
these methods by means of a numerical case study and highlight some of their
respective advantages and disadvantages. Finally, we discuss the problem of
inference from experimental data in the Bayesian framework and review recent
methods developed the literature. In summary, this review gives a
self-contained introduction to modelling, approximations and inference methods
for stochastic chemical kinetics.Comment: 73 pages, 12 figures in J. Phys. A: Math. Theor. (2016
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