5 research outputs found
Computation of biochemical pathway fluctuations beyond the linear noise approximation using iNA
The linear noise approximation is commonly used to obtain intrinsic noise
statistics for biochemical networks. These estimates are accurate for networks
with large numbers of molecules. However it is well known that many biochemical
networks are characterized by at least one species with a small number of
molecules. We here describe version 0.3 of the software intrinsic Noise
Analyzer (iNA) which allows for accurate computation of noise statistics over
wide ranges of molecule numbers. This is achieved by calculating the next order
corrections to the linear noise approximation's estimates of variance and
covariance of concentration fluctuations. The efficiency of the methods is
significantly improved by automated just-in-time compilation using the LLVM
framework leading to a fluctuation analysis which typically outperforms that
obtained by means of exact stochastic simulations. iNA is hence particularly
well suited for the needs of the computational biology community.Comment: 5 pages, 2 figures, conference proceeding IEEE International
Conference on Bioinformatics and Biomedicine (BIBM) 201
Improved estimations of stochastic chemical kinetics by finite state expansion
Stochastic reaction networks are a fundamental model to describe interactions
between species where random fluctuations are relevant. The master equation
provides the evolution of the probability distribution across the discrete
state space consisting of vectors of population counts for each species.
However, since its exact solution is often elusive, several analytical
approximations have been proposed. The deterministic rate equation (DRE) gives
a macroscopic approximation as a compact system of differential equations that
estimate the average populations for each species, but it may be inaccurate in
the case of nonlinear interaction dynamics. Here we propose finite state
expansion (FSE), an analytical method mediating between the microscopic and the
macroscopic interpretations of a stochastic reaction network by coupling the
master equation dynamics of a chosen subset of the discrete state space with
the mean population dynamics of the DRE. An algorithm translates a network into
an expanded one where each discrete state is represented as a further distinct
species. This translation exactly preserves the stochastic dynamics, but the
DRE of the expanded network can be interpreted as a correction to the original
one. The effectiveness of FSE is demonstrated in models that challenge
state-of-the-art techniques due to intrinsic noise, multi-scale populations,
and multi-stability.Comment: 33 pages, 9 figure
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
Model reconstruction for moment-based stochastic chemical kinetics
Based on the theory of stochastic chemical kinetics, the inherent randomness and stochasticity of biochemical reaction networks can be accurately described by discrete-state continuous-time Markov chains, where each chemical reaction corresponds to a state transition of the process. However, the analysis of such processes is computationally expensive and sophisticated numerical methods are required. The main complication comes due to the largeness problem of the state space, so that analysis techniques based on an exploration of the state space are often not feasible and the integration of the moments of the underlying probability distribution has become a very popular alternative. In this thesis we propose an analysis framework in which we integrate a number of moments of the process instead of the state probabilities. This results in a more timeefficient simulation of the time evolution of the process. In order to regain the state probabilities from the moment representation, we combine the moment-based simulation (MM) with a maximum entropy approach: the maximum entropy principle is applied to derive a distribution that fits best to a given sequence of moments. We further extend this approach by incorporating the conditional moments (MCM) which allows not only to reconstruct the distribution of the species present in high amount in the system, but also to approximate the probabilities of species with low molecular counts. For the given distribution reconstruction framework, we investigate the numerical accuracy and stability using case studies from systems biology, compare two different moment approximation methods (MM and MCM), examine if it can be used for the reaction rates estimation problem and describe the possible future applications.
In this thesis we propose an analysis framework in which we integrate a number of moments of the process instead of the state probabilities. This results in a more time-efficient simulation of the time evolution of the process. In order to regain the state probabilities from the moment representation, we combine the moment-based simulation (MM) with a maximum entropy approach: the maximum entropy principle is applied to
derive a distribution that fits best to a given sequence of moments.
We further extend this approach by incorporating the conditional moments (MCM) which
allows not only to reconstruct the distribution of the species present in high amount in the
system, but also to approximate the probabilities of species with low molecular counts.
For the given distribution reconstruction framework, we investigate the numerical accuracy
and stability using case studies from systems biology, compare two different moment
approximation methods (MM and MCM), examine if it can be used for the reaction rates
estimation problem and describe the possible future applications.Basierend auf der Theorie der stochastischen chemischen Kinetiken können die inhĂ€rente ZufĂ€lligkeit und StochastizitĂ€t von biochemischen Reaktionsnetzwerken durch diskrete zeitkontinuierliche Markow-Ketten genau beschrieben werden, wobei jede chemische Reaktion einem ZustandsĂŒbergang des Prozesses entspricht. Die Analyse solcher Prozesse ist jedoch rechenaufwendig und komplexe numerische Verfahren sind erforderlich. Analysetechniken, die auf dem Abtasten des Zustandsraums basieren, sind durch dessen GröĂe oft nicht anwendbar. Als populĂ€re Alternative wird heute hĂ€ufig die Integration der Momente der zugrundeliegenden Wahrscheinlichkeitsverteilung genutzt. In dieser Arbeit schlagen wir einen Analyserahmen vor, in dem wir, anstatt der Zustandswahrscheinlichkeiten, zugrundeliegende Momente des Prozesses integrieren. Dies fĂŒhrt zu einer zeiteffizienteren Simulation der zeitlichen Entwicklung des Prozesses. Um die Zustandswahrscheinlichkeiten aus der Momentreprsentation wiederzugewinnen, kombinieren wir die momentbasierte Simulation (MM) mit Entropiemaximierung: Die Maximum- Entropie-Methode wird angewendet, um eine Verteilung abzuleiten, die am besten zu einer bestimmten Sequenz von Momenten passt. Wir erweitern diesen Ansatz durch das Einbeziehen bedingter Momente (MCM), die es nicht nur erlauben, die Verteilung der in groĂer Menge im System enthaltenen Spezies zu rekonstruieren, sondern es ebenso ermöglicht, sich den Wahrscheinlichkeiten von Spezies mit niedrigen Molekulargewichten anzunĂ€hern. FĂŒr das gegebene System zur Verteilungsrekonstruktion untersuchen wir die numerische Genauigkeit und StabilitĂ€t anhand von Fallstudien aus der Systembiologie, vergleichen zwei unterschiedliche Verfahren der Momentapproximation (MM und MCM), untersuchen, ob es fĂŒr das Problem der AbschĂ€tzung von Reaktionsraten verwendet werden kann und beschreiben die mögliche zukĂŒnftige Anwendungen