296 research outputs found
Model Reduction for the Chemical Master Equation: an Information-Theoretic Approach
No full textThe complexity of mathematical models in biology has rendered model reduction
an essential tool in the quantitative biologist's toolkit. For stochastic
reaction networks described using the Chemical Master Equation, commonly used
methods include time-scale separation, the Linear Mapping Approximation and
state-space lumping. Despite the success of these techniques, they appear to be
rather disparate and at present no general-purpose approach to model reduction
for stochastic reaction networks is known. In this paper we show that most
common model reduction approaches for the Chemical Master Equation can be seen
as minimising a well-known information-theoretic quantity between the full
model and its reduction, the Kullback-Leibler divergence defined on the space
of trajectories. This allows us to recast the task of model reduction as a
variational problem that can be tackled using standard numerical optimisation
approaches. In addition we derive general expressions for the propensities of a
reduced system that generalise those found using classical methods. We show
that the Kullback-Leibler divergence is a useful metric to assess model
discrepancy and to compare different model reduction techniques using three
examples from the literature: an autoregulatory feedback loop, the
Michaelis-Menten enzyme system and a genetic oscillator
Approximation methods for stochastic systems biology
No full textBiochemical reactions involved in complex cellular mechanisms are driven by inherently
stochastic molecular interactions. Although the intrinsic noise is often negligible in the
macroscopic world, it has been established experimentally that intracellular processes
can be subject to substantial stochasticity due to a low number of molecules present.
Therefore, modelling the dynamics of such biological systems necessitates the use of
stochastic rather than deterministic methods.
The Chemical Master Equation (CME) gives an accurate mathematical description
of stochastic chemical reaction kinetics in well-mixed conditions. However, analytical
solutions to the CME are available only for a handful of biologically relevant systems
and its exact stochastic simulation with Monte Carlo methods can be prohibitively
computationally expensive. This in turn motivates the development of approximation
methods that provide more e cient ways of investigating the system behaviour. The
study and the development of novel analytical and computational approximations to
the CME is the focus of this thesis.
First, we develop an approximate time-dependent closed-form solution to the CME describing the Michaelis-Menten reaction mechanism of enzyme catalysis. The derivation
is based on a time scale separation technique called averaging, allowing us to treat the
Markovian dynamics on the slower time scale as a one-dimensional master equation
that can be solved exactly in time using methods from linear algebra and complex
analysis.
Second, we introduce MomentClosure.jl, a Julia package for automated derivation of the
moment equations applicable to any biochemical system. As the moment expansion of
the CME can lead to an in nite hierarchy of coupled moment equations, MomentClosure
implements a wide array of moment closure methods that truncate the moment hierarchy
and provide a closed set of equations describing approximate moment dynamics.
The package integrates seamlessly with other Julia libraries and makes moment closure
approximations more accessible to the broader scienti c community.
Lastly, we propose a surrogate modelling framework that allows us to approximate the
solution of the CME by training neural networks on stochastic simulation data. We
showcase our approach on several models of gene expression, nding that relatively
simple neural networks can learn to approximate highly complex distributions of molecule
numbers over time and parameter space, and hence greatly accelerate otherwise
computationally expensive parameter exploration and inference studies
Inference and Uncertainty Quantification of Stochastic Gene Expression via Synthetic Models
No full textEstimating uncertainty in model predictions is a central task in quantitative biology. Biological models at the single-cell level are intrinsically stochastic and nonlinear, creating formidable challenges for their statistical estimation which inevitably has to rely on approximations that trade accuracy for tractability. Despite intensive interest, a sweet spot in this trade off has not been found yet. We propose a flexible procedure for uncertainty quantification in a wide class of reaction networks describing stochastic gene expression including those with feedback. The method is based on creating a tractable coarse-graining of the model that is learned from simulations, a synthetic model, to approximate the likelihood function. We demonstrate that synthetic models can substantially outperform state-of-the-art approaches on a number of nontrivial systems and datasets, yielding an accurate and computationally viable solution to uncertainty quantification in stochastic models of gene expression
Stochastic modelling of regulation strategies in stem cell populations
No full textHealthy tissues exhibit remarkable resilience, maintaining functionality through continuous renewal of cells, and displaying an impressive regeneration capacity after injury. Achieving this demands the precise regulation of the tissue-resident stem cell numbers and their proliferation and differentiation rates. However, the mechanisms preventing the depletion or overgrowth of the stem cell pool and enabling regeneration remain elusive. Unravelling the regulation strategies of stem cell populations would shed light on tissue function and potential ways regulation can break down in disease.
Recent years witnessed technological advances that enabled the examination of tissues with an unprecedented level of detail. This revealed that there is a degree of randomness in the fate determination after stem cell division, and yet tissues manage to maintain a correct proportion of each cell type during homeostasis. Understanding how stem cell populations integrate environmental cues and communicate with other cell populations to maintain and regenerate tissues in the presence of randomness requires the development of mathematical models capturing homeostasis and regeneration as collective phenomena.
This thesis addresses various challenges of modelling stem cell populations in the presence of regulatory mechanisms. Can we propose stochastic models for stem cell populations with regulatory mechanisms in a mathematically tractable fashion? Stem cell populations can be composed of fewer than ten cells, how can we provide accurate descriptions of their behaviour in such cases? What are the main signatures of different regulation strategies at the population level? Can we use them to argue for or against their presence in different tissues?
First, we propose and investigate the dynamics of a stochastic model for stem cell populations in the presence of regulation through competition for niche access, prevalent in many tissues. Our model effectively captures the essential elements of competition for niche access while remaining mathematically tractable. We characterise the behaviours of the model analytically and numerically, addressing challenges such as non-Gaussian fluctuations, extinction dynamics or finite-size effects
Concentration oscillations in single cells : the roles of intracellular noise and intercellular coupling
No full textConcentration oscillations are a ubiquitous characteristic of intracellular dynamics. The period of these oscillations can vary from few seconds to many hours, well known examples being calcium oscillations (seconds to minutes), glycolytic oscillations (minutes) and circadian rhythms (1 day). Considerable advances into understanding the mechanisms and functionality of concentration oscillations have been made since glycolytic oscillations were observed in the late 1950s, and mathematical methods have been an essential part of this process. With increasing ability to experimentally measure oscillations in single cells as well as in cell ensembles, the gold standard of modelling is to provide tools that can elucidate how the system-wide dynamics in complex organisms emerge from a system of single cells. Both abstract and detailed mechanistic models are complementary in the insight they can bring, and for networks of coupled cells considerations such as intrinsic intracellular noise, cellular heterogeneity and coupling strength are all expected to play a part. Here, we investigate separately the potential roles played by intrinsic noise arising from finite numbers of interacting molecules and by coupling among cellular oscillators. Regarding the former, it is well known that internal or molecular noise induces concentration oscillations in chemical systems whose deterministic models exhibit damped oscillations. We show, using the linear-noise approximation of the chemical master equation, that noise can also induce oscillations in biochemical systems whose deterministic descriptions admit no damped oscillations, i.e., systems with a stable node. This non-intuitive phenomenon is remarkable since, unlike noise-induced oscillations in systems with damped deterministic oscillations, it cannot be explained by noise excitation of the deterministic resonant frequency of the system. We here prove the following general properties of stable-node noise-induced oscillations for systems with two species: (i) the upper bound of their frequency is given by the geometric mean of the real eigenvalues of the Jacobian of the system, (ii) the upper bound of the Q-factor of the oscillations is inversely proportional to the distance between the real eigenvalues of the Jacobian, and (iii) these oscillations are not necessarily exhibited by all interacting chemical species in the system. The existence and properties of stable-node oscillations are verified by stochastic simulations of the Brusselator, a cascade Brusselator reaction system, and two other simple chemical systems involving autocatalysis and trimerization. We also show that external noise induces stable node oscillations with different properties than those stimulated by internal noise. Having demonstrated and explored this non-intuitive effect of noise, we extend the work to investigate the phenomenon of noise induced oscillations in cellular reaction systems characterised by the ‘bursty’ production of one or more species. Experiments have shown that proteins are typically translated in sharp bursts and similar bursty phenomena have been observed for protein import into subcellular compartments. We investigate the effect of such burstiness on the stochastic properties of downstream pathways by considering two identical systems with equal mean input rates, except in one system molecules (e.g., proteins) are input one at a time and in the other molecules are input in bursts according to some probability distribution. We find that the stochastic behaviour falls in three categories: (i) both systems display or do not display noise-induced oscillations; (ii) the non-bursty input system displays noiseinduced oscillations whereas the bursty input system does not; (iii) the reverse of (ii). We derive necessary conditions for these three cases to classify pathways involving autocatalysis, trimerization and genetic feedback loops. Our results suggest that single cell rhythms can be controlled by regulation of burstiness in protein production. We go on to investigate roles played by intercellular coupling in whole organ-level oscillations with an experimental analysis of circadian rhythms in Arabidopsis thaliana †. Circadian clocks in animals are known to be tightly coupled among the cells of some tissues, and this coupling profoundly affects cellular rhythmicity. However, research on the clock in plant cells has largely ignored intercellular coupling. Our research group used luciferase reporter gene imaging to monitor circadian rhythms in leaves of Arabidopsis thaliana plants, with both a lower resolution, high throughput method and a high-resolution (cellular level), lower throughput method. Leaves were grown and imaged in a variety of light conditions to test the relative importance of intercellular coupling and cellular coupling to the environmental signal. We analysed the high throughput data and described the circadian phase by the timing of peak expression. Leaves grown for three weeks without entrainment reproducibly showed spatio-temporal waves of gene expression, consistent with intercellular coupling. A range of patterns was observed among the leaves, rather than a unique spatio-temporal pattern, although some patterns were more often observed. All of the measured leaves exposed to lightdark entrainment cycles had fully synchronised rhythms, which could desynchronise rather quickly when placed in a non-entraining environment (i.e., constant light conditions). After four days in constant light some of these leaves were as desynchronised as leaves grown without any rhythmic input, as described by the phase coherence across the leaf. The same fast transition was observed in the reverse experimental scenario, i.e., applying light-dark cycles to leaves grown in constant light resulted in full synchronisation within two to four days. From these results we conclude that single-cell circadian oscillators were coupled far more strongly to external light-dark cycles than to the other cellular oscillators. Leaves did not spontaneously completely desynchronise, which is consistent with a presence of intercellular coupling among heterogeneous clocks. We also developed a methodology, based on the notion of two functional spatial scales of expression across the leaf, to analyse the high-resolution microscope data and determine whether there is a difference in the phase of circadian expression between vein cells and mesophyll cells in the leaf. The result from a single leaf suggests that the global phase wave dominates the phase behaviour but that there are small delays in the veins compared to their nearby mesophyll cells. This result can be validated by applying the methodology developed here to new microscope leaf data which is currently being collected in the research group. † This work was performed as a collaboration between David Toner (DT) and Benedicte Wenden (BW). BW designed and carried out the experiments, DT performed the data analysis and led on data visualisation
Approximation methods and inference for stochastic biochemical kinetics
No full textRecent experiments have shown the fundamental role that random fluctuations play
in many chemical systems in living cells, such as gene regulatory networks. Mathematical
models are thus indispensable to describe such systems and to extract relevant
biological information from experimental data. Recent decades have seen a considerable
amount of modelling effort devoted to this task. However, current methodologies
still present outstanding mathematical and computational hurdles. In particular,
models which retain the discrete nature of particle numbers incur necessarily severe
computational overheads, greatly complicating the tasks of characterising statistically
the noise in cells and inferring parameters from data. In this thesis we study analytical
approximations and inference methods for stochastic reaction dynamics.
The chemical master equation is the accepted description of stochastic chemical
reaction networks whenever spatial effects can be ignored. Unfortunately, for most
systems no analytic solutions are known and stochastic simulations are computationally
expensive, making analytic approximations appealing alternatives. In the case
where spatial effects cannot be ignored, such systems are typically modelled by means
of stochastic reaction-diffusion processes. As in the non-spatial case an analytic treatment
is rarely possible and simulations quickly become infeasible. In particular, the
calibration of models to data constitutes a fundamental unsolved problem.
In the first part of this thesis we study two approximation methods of the chemical
master equation; the chemical Langevin equation and moment closure approximations.
The chemical Langevin equation approximates the discrete-valued process described
by the chemical master equation by a continuous diffusion process. Despite being frequently
used in the literature, it remains unclear how the boundary conditions behave
under this transition from discrete to continuous variables. We show that this boundary
problem results in the chemical Langevin equation being mathematically ill-defined if
defined in real space due to the occurrence of square roots of negative expressions.
We show that this problem can be avoided by extending the state space from real to
complex variables. We prove that this approach gives rise to real-valued moments and
thus admits a probabilistic interpretation. Numerical examples demonstrate better accuracy
of the developed complex chemical Langevin equation than various real-valued
implementations proposed in the literature.
Moment closure approximations aim at directly approximating the moments of a
process, rather then its distribution. The chemical master equation gives rise to an
infinite system of ordinary differential equations for the moments of a process. Moment
closure approximations close this infinite hierarchy of equations by expressing
moments above a certain order in terms of lower order moments. This is an ad hoc
approximation without any systematic justification, and the question arises if the resulting
equations always lead to physically meaningful results. We find that this is
indeed not always the case. Rather, moment closure approximations may give rise to
diverging time trajectories or otherwise unphysical behaviour, such as negative mean
values or unphysical oscillations. They thus fail to admit a probabilistic interpretation
in these cases, and care is needed when using them to not draw wrong conclusions.
In the second part of this work we consider systems where spatial effects have to be
taken into account. In general, such stochastic reaction-diffusion processes are only defined
in an algorithmic sense without any analytic description, and it is hence not even
conceptually clear how to define likelihoods for experimental data for such processes.
Calibration of such models to experimental data thus constitutes a highly non-trivial
task. We derive here a novel inference method by establishing a basic relationship
between stochastic reaction-diffusion processes and spatio-temporal Cox processes,
two classes of models that were considered to be distinct to each other to this date.
This novel connection naturally allows to compute approximate likelihoods and thus
to perform inference tasks for stochastic reaction-diffusion processes. The accuracy
and efficiency of this approach is demonstrated by means of several examples.
Overall, this thesis advances the state of the art of modelling methods for stochastic
reaction systems. It advances the understanding of several existing methods by elucidating
fundamental limitations of these methods, and several novel approximation and
inference methods are developed
Intrinsic noise analyzer:a software package for the exploration of stochastic biochemical kinetics using the system size expansion
No full textThe accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen's system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface. In particular, the present methods allow for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients, which typically outperforms stochastic simulations. These analytical tools are complemented by automated multi-core stochastic simulations with direct statistical evaluation and visualization. We showcase iNA's performance by using it to explore the stochastic properties of cooperative and non-cooperative enzyme kinetics and a gene network associated with circadian rhythms. The software iNA is freely available as executable binaries for Linux, MacOSX and Microsoft Windows, as well as the full source code under an open source license
Distributions of RNA polymerase and transcript numbers in models of gene expression describing the mRNA life-cycle
No full textTranscription, the production of RNA from a gene, is an inherently stochastic process, as recent
experiments have firmly established. This stochasticity makes the modelling of genetic networks
highly challenging. Recent decades have seen a rise in the development of new mathematical models
of gene regulatory networks that aim to extract relevant biological information from experimental
data. The telegraph model of gene expression, where the gene switches between active and inactive
states, is the most widely used in the literature. However, it has been shown that it cannot explain
several experimental observations, as it does not capture many biological details such as transcription
factor and polymerase binding to the gene, RNA nuclear retention, multi-step elongation, RNA
maturation, etc.
The chemical master equation (CME) describes stochastic chemical reaction networks and,
hence, is a commonly used tool in the mathematical modelling of such networks. Specifically, it
describes how the joint probability distribution of the copy number of different chemical species
evolves in time under spatially homogeneous conditions. Unfortunately, this equation can be solved
analytically only in a few cases, while on the other hand, stochastic simulations can be computationally
expensive and slow. For these reasons, various approximation techniques have been developed
lately to approximate solutions to hitherto unsolved complex master equations. For example, the
geometric singular perturbation theory serves as a very useful tool for finding approximate solutions
to CMEs of biological models which feature processes on different time scales.
In this thesis, we study the formulation and detailed analysis of three different analytically
tractable stochastic models that capture the main features of gene expression under various additional
assumptions and that can potentially provide means to infer parameter values from experimental
data. We quantify which and how different approximation methods can be applied to systems
of interest in order to obtain closed-form analytical solutions.
The first model presented in this thesis is a stochastic model of gene expression with polymerase
recruitment and pause release, two steps necessary for messenger RNA (mRNA) production. For
this model, which captures the bursty production of mRNA molecules, we derive the exact steadystate
distribution of mRNA numbers. Additionally, this model includes the translation process –
synthesis of protein from mRNA – and we apply perturbation techniques in order to obtain an
approximate steady-state distribution of protein numbers.
The second model that we are studying in this work is a stochastic model of RNA transcription,
which focuses on capturing the processes of transcriptional initiation, elongation, premature
detachment, pausing, and termination. In this model, the gene is divided into an arbitrary number
of segments. The results from our analysis uncover the explicit dependence of the statistics of
nascent (actively transcribed) and mature (cellular) RNA on transcriptional parameters. By performing
mathematical analysis, we derive exact closed-form expressions for the mean and variance
of nascent RNA fluctuations on each gene segment, as well as for the total nascent RNA on a gene.
Additionally, we obtain the exact expressions for the first two moments of mature RNA fluctuations
while we present an approximation approach for deriving distributions for the total numbers
of nascent and mature RNA in various parameter regimes.
The third model that we study in this thesis is a stochastic model that describes the dynamics of
signal-dependent gene expression and its propagation downstream of transcription. In this model,
the activation of the gene promoter is time-dependent due to the temporal variation in transcription
factor (protein) numbers; after transcription initiation, the produced mRNA undergoes an arbitrary
number of stages of its life cycle. For any time-dependent stimulus and in the case of bursty gene
expression, we developed a novel procedure that allows us to obtain approximate time-dependent
distributions of mRNA numbers at all stages of its life cycle. We derive an expression for the error
in the approximation and verify its accuracy via stochastic simulation. We show that, depending
on the frequency of oscillation and the time of measurement, a stimulus can lead to cytoplasmic
amplification or attenuation of transcriptional noise.
To summarize, this thesis presents a detailed explanation of the construction of three families
of stochastic models of gene expression and demonstrates how to perform mathematical analysis
of the complex CMEs that represent these models. A number of novel approximation methods that
address some difficulties in solving the CME are included in this study, while one of the main goals
of this work is to show that extracting biological information from mathematical models can provide
us with a better understanding of cells’ functions
The Order of St. John's galley squadron at sea
No full textIn the article, the name of the author is listed erroneously as Joseph A. Grima.In a scrupulously factual description of various aspects of life at sea on an Order's galley in the seventeenth century Mediterranean, Joseph F. Grima details the ship's movements in and out of harbour as well as in action, procedures relating to protocol and discipline, and also conditions of the crew on board, such as food provisions and medical services.peer-reviewe
Systematic approximation methods for stochastic biochemical kinetics
No full textExperimental studies have shown that the protein abundance in living cells varies from
few tens to several thousands molecules per species. Molecular fluctuations roughly
scale as the inverse square root of the number of molecules due to the random timing
of reactions. It is hence expected that intrinsic noise plays an important role in the
dynamics of biochemical networks.
The Chemical Master Equation is the accepted description of these systems under
well-mixed conditions. Because analytical solutions to this equation are available only
for simple systems, one often has to resort to approximation methods. A popular
technique is an expansion in the inverse volume to which the reactants are confined,
called van Kampen's system size expansion. Its leading order terms are given by the
phenomenological rate equations and the linear noise approximation that quantify the
mean concentrations and the Gaussian fluctuations about them, respectively. While
these approximations are valid in the limit of large molecule numbers, it is known that
physiological conditions often imply low molecule numbers.
We here develop systematic approximation methods based on higher terms in the
system size expansion for general biochemical networks. We present an asymptotic
series for the moments of the Chemical Master Equation that can be computed to
arbitrary precision in the system size expansion. We then derive an analytical approximation
of the corresponding time-dependent probability distribution. Finally, we devise
a diagrammatic technique based on the path-integral method that allows to compute
time-correlation functions. We show through the use of biological examples that the
first few terms of the expansion yield accurate approximations even for low number of
molecules. The theory is hence expected to closely resemble the outcomes of single cell
experiments
- …
