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Introduction

## Publications

Publications (48)

Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can
rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution
of the master equation using orthogonal polynomials. The solution is given in two alternative formulations:
a series with continuous...

Noise in gene expression can lead to reversible phenotypic switching. Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is impor...

Background
It is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effe...

The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework o...

Computing the stationary distributions of a continuous-time Markov chain (CTMC) involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too large, and the equations cannot be solved analytically or numerically. Several approximation schemes overcome this issue by truncating the state space to a...

We study a class of countably infinite linear programs (CILPs) whose feasible sets are bounded subsets of appropriately defined spaces of measures. The optimal value, optimal points, and minimal points of these CILPs can be approximated by solving finite-dimensional linear programs. We show how to construct finite-dimensional programs that lead to...

Metabolic heterogeneity is widely recognized as the next challenge in our understanding of non-genetic variation. A growing body of evidence suggests that metabolic heterogeneity may result from the inherent stochasticity of intracellular events. However, metabolism has been traditionally viewed as a purely deterministic process, on the basis that...

The chemical master equation and the stochastic simulation algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled for through effective dilution reactions and extrinsic noise sources. We here reexamine these paradigms through developing an analytical agent-ba...

Metabolic heterogeneity is widely recognised as the next challenge in our understanding of non-genetic variation. A growing body of evidence suggests that metabolic heterogeneity may result from the inherent stochasticity of intracellular events. However, metabolism has been traditionally viewed as a purely deterministic process, on the basis that...

Single-cell analyses are becoming increasingly important in cell biology and personalized approaches to medicine. Such analyses frequently reveal heterogeneity that exists within and between cells. We give a concise overview of stochastic methods used to analyze non-genetic heterogeneity in models of cell populations and examine several analytical...

Motivation:
Normalisation of single cell RNA sequencing (scRNA-seq) data is a prerequisite to their interpretation. The marked technical variability, high amounts of missing observations and batch effect typical of scRNA-seq datasets make this task particularly challenging. There is a need for an efficient and unified approach for normalisation, i...

Computing the stationary distributions of a continuous-time Markov chain involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too large, and cannot be solved analytically or numerically. Several approximation schemes overcome this issue by truncating the state space to a manageable size. In t...

The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these dis...

Phenotypic variation is a hallmark of cellular physiology. Metabolic heterogeneity, in particular, underpins single-cell phenomena such as microbial drug tolerance and growth variability. Much research has focussed on transcriptomic and proteomic heterogeneity, yet it remains unclear if such variation permeates to the metabolic state of a cell. Her...

Cell-to-cell heterogeneity is driven by stochasticity in intracellular reactions and the population dynamics. While these sources are usually studied separately, we develop an agent-based framework that accounts for both factors while tracking every single cell of a growing population. Apart from the common intrinsic variability, the framework also...

Phenotypic variation is a hallmark of cellular physiology. Metabolic heterogeneity, in particular, underpins single-cell phenomena such as microbial drug tolerance and growth variability. Much research has focussed on transcriptomic and proteomic heterogeneity, yet it remains unclear if such variation permeates to the metabolic state of a cell. Her...

We introduce the exit time finite state projection (ETFSP) scheme, a truncation-based method that yields approximations to the exit distribution and occupation measure associated with the time of exit from a domain (i.e., the time of first passage to the complement of the
domain) of time-homogeneous continuous-time Markov chains. We prove that: (i)...

How cells maintain their size has been extensively studied under constant conditions. In the wild, however, cells rarely experience constant environments. Here, we examine how the 24-h circadian clock and environmental cycles modulate cell size control and division timings in the cyanobacterium Synechococcus elongatus using single-cell time-lapse m...

Growth impacts a range of phenotypic responses. Identifying the sources of growth variation and their propagation across the cellular machinery can thus unravel mechanisms that underpin cell decisions. We present a stochastic cell model linking gene expression, metabolism and replication to predict growth dynamics in single bacterial cells. Alongsi...

We study a class of countably-infinite-dimensional linear programs (CILPs) whose feasible sets are bounded subsets of appropriately defined spaces of measures. The optimal value, optimal points, and minimal points of these CILPs can be approximated by solving finite-dimensional linear programs. We show how to construct finite-dimensional programs t...

Normalisation of single cell RNA sequencing (scRNA-seq) data is a prerequisite to their interpretation. The marked technical variability and high amounts of missing observations typical of scRNA-seq datasets make this task particularly challenging. Here, we introduce bayNorm, a novel Bayesian approach for scaling and inference of scRNA-seq counts....

Clonal cells of exponentially growing populations vary substantially from cell to cell. The main drivers of this heterogeneity are the population dynamics and stochasticity in the intracellular reactions, which are commonly studied separately. Here we develop an agent-based framework that allows tracking of the biochemical dynamics in every single...

Growth pervades all areas of life from single cells to cell populations to tissues. Cell size often fluctuates significantly from cell to cell and from generation to generation. Here we present a unified framework to predict the statistics of cell size variations within a lineage tree of a proliferating population. We analytically characterize (i)...

Cellular growth impacts a range of phenotypic responses. Identifying the sources of fluctuations in growth and how they propagate across the cellular machinery can unravel mechanisms that underpin cell decisions. We present a stochastic cell model linking gene expression, metabolism and replication to predict growth dynamics in single bacterial cel...

Population growth is often ignored when quantifying gene expression levels across clonal cell populations. We develop a framework for obtaining the molecule number distributions in an exponentially growing cell population taking into account its age structure. In the presence of generation time variability, the average acquired across a population...

How cells maintain their size has been extensively studied under constant conditions. In the wild, however, cells rarely experience constant environments. Here, we examine how the 24-hour circadian clock and environmental cycles modulate cell size control and division timings in the cyanobacterium Synechococcus elongatus using single-cell time-laps...

The decision to divide is the most important one that any cell must make. Recent single cell studies suggest that most bacteria follow an “adder” model of cell size control, incorporating a fixed amount of cell wall material before dividing. Mycobacteria, including the causative agent of tuberculosis Mycobacterium tuberculosis, are known to divide...

The stochastic nature of chemical reactions has resulted in an increasing research interest in discrete-state stochastic models and their analysis. A widely used approach is the description of the temporal evolution of such systems in terms of a chemical master equation (CME). In this paper we study two approaches for approximating the underlying p...

Cell size and individual growth rates vary substantially across genetically identical cell populations. This variation cannot entirely be explained by asynchronous cell division cycles, but also needs to take into account the differences in the histories that cells experience during their lifespan. We describe a stochastic framework to characterise...

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...

MATLAB code used for inference using SSE and MA.
This zip-file contains the MATLAB code for the simulation and application example presented in the paper. We provide implementations of all models, parameter estimation and uncertainty analysis to allow everybody to reproduce the results.
(ZIP)

Supplementary notes regarding modeling and computational analysis.
This document provides a detailed description of the biochemical reaction networks and their parameters, system size expansion and moment approximation, as well as the parameter estimation and the uncertainty analysis results.
(PDF)

Supporting Information.
Stochastic Simulation of Biomolecular Networks in Dynamic Environments.
(PDF)

An extensible, Mathematica implementation of the Extrande algorithm.
(NB)

Simulation of biomolecular networks is now indispensable for studying
biological systems, from small reaction networks to large ensembles of cells.
Here we present a novel approach for stochastic simulation of networks embedded
in the dynamic environment of the cell and its surroundings. We thus sample
trajectories of the stochastic process describ...

The stochastic nature of chemical reactions involving randomly fluctuating
population sizes has lead to a growing research interest in discrete-state
stochastic models and their analysis. A widely-used approach is the description
of the temporal evolution of the system in terms of a chemical master equation
(CME). In this paper we study two approac...

Few analytical methods exist for quantitative studies of large fluctuations
in stochastic systems. In this article, we develop a simple diagrammatic
approach to the Chemical Master Equation that allows us to calculate multi-time
correlation functions which are accurate to a any desired order in van Kampen's
system size expansion. Specifically, we p...

Background
The linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene reg...

The linear noise approximation (LNA) offers a simple means by which one can study intrinsic noise in monostable biochemical networks. Using simple physical arguments, we have recently introduced the slow-scale LNA (ssLNA), which is a reduced version of the LNA under conditions of timescale separation. In this paper we present the first rigorous der...

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 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...

It is commonly believed that, whenever timescale separation holds, the predictions of reduced chemical master equations obtained using the stochastic quasi-steady-state approximation are in very good agreement with the predictions of the full master equations. We use the linear noise approximation to obtain a simple formula for the relative error b...

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the s...

Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption...