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Timetable (SDBW03)

Advances in numerical and analytic approaches for the study of non-spatial stochastic dynamical systems in molecular biology

Monday 4th April 2016 to Friday 8th April 2016

Monday 4th April 2016
09:00 to 09:20 Registration INI 1
09:20 to 09:30 Welcome from Christie Marr (INI Deputy Director) INI 1
09:30 to 10:15 Thomas Kurtz
Approximations for Markov chain models
Co-author: David F. Anderson (Univ of Wisconsin - Madison)

The talk will begin by reviewing methods of specifying continuous-time Markov chains and classical limit theorems that arise naturally for chemical network models. Since models arising in molecular biology frequently exhibit multiple state and time scales, analogous limit theorems for these models will be illustrated through simple examples.

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10:15 to 11:00 James Faeder
Towards large scale models of biochemical networks
Co-authors: Jose Juan Tapia (University of Pittsburgh), John Sekar (University of Pittsburgh)

In this talk I will address some of the challenges faced in developing detailed models of biochemical networks, which encompass large numbers of interacting components. Although simpler coarse-grained models are often useful for gaining insight into biological mechanisms, such detailed models are necessary to understand how molecular components work in the network context and essential to developing the ability to manipulate such networks for practical benefits. The rule-based modeling (RBM) approach, in which biological molecules can be represented as structured objects whose interactions are governed by rules that describe their biochemical interactions, is the basis for addressing multiple scaling issues that arise in the development of large scale models. Currently available software tools for RBM, such as BioNetGen, Kappa, and Simmune, enable the specification and simulation of large scale models, and these tools are in widespread use by the modeling community. I will re view some of the developments that gave rise to those capabilities, and then I will describe our current efforts broaden the appeal of these tools as well as to better enable collaborative development of models through re-use of existing models and improving visual representations of models.

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  • - For more information about rule-based modeling tools described in this talk.
11:00 to 11:30 Morning Coffee
11:30 to 12:15 Simon Cotter
A constrained approach to the simulation and analysis of stochastic multiscale chemical kinetics
Co-authors: Radek Erban (University of Oxford), Ioannis Kevrekidis (Princeton), Konstantinos Zygalakis (University of Southampton)

In many applications in cell biology, the inherent underlying stochasticity and discrete nature of individual reactions can play a very important part in the dynamics. The Gillespie algorithm has been around since the 1970s, which allows us to simulate trajectories from these systems, by simulating in turn each reaction, giving us a Markov jump process. However, in multiscale systems, where there are some reactions which are occurring many times on a timescale for which others are unlikely to happen at all, this approach can be computationally intractable. Several approaches exist for the efficient approximation of the dynamics of the “slow” reactions, some of which rely on the “quasi-steady state assumption” (QSSA). In this talk, we will present the Constrained Multiscale Algorithm, a method based on the equation free approach, which was first used to construct diffusion approximations of the slowly changing quantities in the system. We will compare this method with other methods which rely on the QSSA to compute the effective drift and diffusion of the approximating SDE. We will then show how this method can be used, back in the discrete setting, to approximate an effective Markov jump generator for the slow variables in the system, and quantify the errors in that approximation. If time permits, we will show how these generators can then be used to sample approximate paths conditioned on the values of their endpoints.
12:15 to 13:30 Lunch at Wolfson Court
14:00 to 14:45 Raul Fidel Tempone
Efficient Simulation and Inference for Stochastic Reaction Networks

Stochastic Reaction Networks (SRNs), that are intended to describe the time evolution of interacting particle systems where one particle interacts with the others through a finite set of reaction channels. SRNs have been mainly developed to model biochemical reactions but they also have applications in neural networks, virus kinetics, and dynamics of social networks, among others. 

This talk is focused on novel fast simulation algorithms and statistical inference methods for SRNs. 

Regarding simulation, our novel Multi-level Monte Carlo (MLMC) hybrid methods provide accurate estimates of expected values of a given observable at a prescribed final time. They control the global approximation error up to a user-selected accuracy and up to a certain confidence level, with near optimal computational work. 

With respect to statistical inference, we first present a multi-scale approach, where we introduce a deterministic systematic way of using up-scaled likelihoods for parameter estimation. In a second approach, we derive a new forward-reverse representation for simulating stochastic bridges between consecutive observations. This allows us to use the well-known EM Algorithm to infer the reaction rates. 
14:45 to 15:30 Erkki Somersalo (Case Western Reserve University)
15:30 to 16:00 Afternoon Tea
16:00 to 16:45 Poster pitches
16:45 to 18:00 Welcome Wine Reception and Poster Session
Tuesday 5th April 2016
09:00 to 09:45 Rosalind Allen
Inherent variability in the kinetics of amyloid fibril formation
Co-authors: Juraj Szavits-Nossan, Kym Eden, Ryan Morris, Martin Evans and Cait MacPhee

In small volumes, the kinetics of filamentous protein self-assembly is expected to show significant variability, arising from intrinsic molecular noise. We introduce a simple stochastic model including nucleation and autocatalytic growth via elongation and fragmentation, which allows us to predict the effects of molecular noise on the kinetics of autocatalytic self-assembly. We derive an analytic expression for the lag-time distribution, which agrees well with experimental results for the fibrillation of bovine insulin. Our analysis shows that significant lag-time variability can arise from both primary nucleation and from autocatalytic growth and should provide a way to extract mechanistic information on early-stage aggregation from small-volume experiments.
09:45 to 10:30 Muruhan Rathinam
Analysis of Monte Carlo estimators for parametric sensitivities in stochastic chemical kinetics
Co-author: Ting Wang (University of Delaware)

We provide an overview of some of the major Monte Carlo approaches for parametric sensitivities in stochastic chemical systems. The efficiency of a Monte Carlo approach depends in part on the variance of the estimator. It has been numerically observed that in several examples, that the finite difference (FD) and the (regularized) pathwise differentiation (RPD) methods tend to have lower variance than the Girsanov Tranformation (GT) estimator while the latter has the advantage of being unbiased. We present a theoretical explanation in terms of system volume asymptotics for the larger variance of the GT approach when compared to the FD methods. We also present an analysis of efficiency of the FD and GT methods in terms of desired error and system volume.
10:30 to 11:00 Morning Coffee
11:00 to 11:45 David Doty
"No We Can't": Impossibility of efficient leader election by chemical reactions
Co-author: David Soloveichik (University of Texas, Austin)

Suppose a chemical system requires a single molecule of a certain species $L$. Preparing a solution with just a single copy of $L$ is a difficult task to achieve with imprecise pipettors. Could we engineer artificial reactions (a chemical election algorithm, so to speak) that whittle down an initially large count of $L$ to 1? Yes, with the reaction $L+L \to L+F$: whenever two candidate leaders encounter each other, one drops out of the race. In volume $v$ convergence to a single $L$ requires expected time proportional to $v$; the final reaction --- two lone $L$'s seeking each other in the vast expanse of volume $v$ --- dominates the whole expected time.

One might hope that more cleverly designed reactions could elect a leader more quickly. We dash this hope: $L+L \to L+F$, despite its sloth, is the fastest chemical algorithm for leader election there is (subject to some reasonable constraints on the reactions). The techniques generalize to establish lower bounds on the time required to do other computational tasks, such as computing which of two species $X$ or $Y$ holds an initial majority.

Democracy works... but it's painstakingly slow.

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11:45 to 12:30 Jay Newby
First-passage time to clear the way for receptor-ligand binding in a crowded environment
I will present theoretical support for a hypothesis about cell-cell contact, which plays a critical role in immune function. A fundamental question for all cell-cell interfaces is how receptors and ligands come into contact, despite being separated by large molecules, the extracellular fluid, and other structures in the glycocalyx. The cell membrane is a crowded domain filled with large glycoproteins that impair interactions between smaller pairs of molecules, such as the T cell receptor and its ligand, which is a key step in immunological information processing and decision-making. A first passage time problem allows us to gauge whether a reaction zone can be cleared of large molecules through passive diffusion on biologically relevant timescales. I combine numerical and asymptotic approaches to obtain a complete picture of the first passage time, which shows that passive diffusion alone would take far too long to account for experimentally observed cell-cell contact format ion times. The result suggests that cell-cell contact formation may involve previously unknown active mechanical processes.
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 14:45 John Albeck
Linking dynamic signaling events within the same cell
In intracellular signaling pathways, biochemical activation events are transmitted from one node within the signaling network to another.  Recent work examining the information capacity of signaling pathways has concluded that most signaling pathways have limited abilities to resolve different strengths of inputs.  However, these studies are based on data in which only a single signal is measured in each cell, in response to a given cell, with the limitation that transmission of a signal from one signaling node to another cannot be directly observed.  Other published data suggest that single cells may have a much higher capacity to transmit quantitative information, which is obscured by population heterogeneity.  To better understand the properties of information transmission through biochemical cascades in individual cells, we have developed a panel of live-cell reporters to monitor multiple signaling events in the cell proliferation and growth network (CPGN).  These reporters include activity biosensors for the kinases ERK, Akt, mTOR, and AMPK, and CRISPR-based reporters for ERK target gene expression.  Experimental analysis with these tools reveals the temporal and quantitative linkage properties between nodes of the CPGN.  I will discuss two studies currently underway in our lab.  The first examines the how the CPGN manages the interplay between ATP-producing and ATP-consuming processes during cell proliferation; we find that loss of Akt signaling results in unstable levels of ATP and NADH in proliferating cells.  The second project focuses on how variations in amplitude and duration of ERK activity control the expression of the target gene Fra-1, which is involved in metastasis; here, we show that cancer therapeutics directed at inhibiting this pathway create strikingly different kinetics of ERK activity at the single-cell level, with distinct effects on Fra-1 expression.

14:45 to 15:30 Aleksandra Walczak
15:30 to 16:00 Afternoon Tea
16:00 to 16:45 Vahid Shahrezaei
Inference of size dependence of transcription parameters from single cell data using multi-scale models of gene expression
Co-authors: Anthony Bowman (Imperial College London), Xi-Ming Sun (MRC CSC), Samuel Marguerat (MRC CSC)

Gene expression is affected by both random timing of reactions (intrinsic noise) and interaction with global stochastic systems in the cells (extrinsic noise). A challenge in inferring parameters of gene expression using models of stochastic gene expression is that these models usually only inlcude intrinsic noise. However, experimental distributions of transcripts are strongly influenced by extrinsic effects including cell cycle and cell division. Here, we present a multi-scale approach in stochastic gene expression to deal with this problem. We apply our methodology to data obtained using single molecule Fish technique in fission yeast. The data suggests cell size influences transcription parameters. We use Approximate Bayesian Computation (ABC) along with sequential Monte Carlo to infer the dependence of gene expression parameters on cell size. Our analysis reveals a linear increase of transcription burst size during the cell cycle.
Wednesday 6th April 2016
09:00 to 09:45 Omer Dushek
Cellular signalling in T cells is captured by a tractable modular phenotypic model
T cells initiate adaptive immune responses when their T cell antigen receptors (TCRs) recognise antigenic peptides bound to major histocompatibility complexes (pMHC). The binding of pMHC ligands to the TCR can trigger a large signal transduction cascade leading to T cell activation, as measured by the secretion effector cytokines/chemokines. Although the signalling proteins involved have been identified, it is still not understood how the cellular signalling network that they form converts the dose and affinity of pMHC into T cell activation. Here we use a holistic method to infer the signalling architecture from T cell activation data generated by stimulating T cells with a 100,000-fold variation in pMHC affinity/dose. We observe bell-shape dose-response curves and a different optimal pMHC affinity at different pMHC doses. We show that this can be explained by a unique, tractable, and modular phenotypic model of signalling that includes kinetic proofreading with limited sign alling coupled to incoherent feedforward but not negative feedback. The work provides a complementary approach for studying cellular signalling that does not require full details of biochemical pathways.

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09:45 to 10:30 Eric Deeds
10:30 to 11:00 Morning Coffee
11:00 to 11:45 Carlos Lopez
Intracellular signaling processes and cell decisions using stochastic algorithms
Cancer cells within a tumor environment exhibit a complex and adaptive nature whereby genetically and epigenetically distinct subpopulations compete for resources. The probabilistic nature of gene expression and intracellular molecular interactions confer a significant amount of stochasticity in cell fate decisions. This cellular heterogeneity is believed to underlie cases of cancer recurrence, acquired drug resistance, and so-called exceptional responders. From a population dynamics perspective, clonal heterogeneity and cell-fate stochasticity are distinct sources of noise, the former arising from genetic mutations and/or epigenetic transitions, extrinsic to the fate decision signaling pathways and the latter being intrinsic to biochemical reaction networks. Here, we present our results and ongoing work of a kinetic modeling study based on experimental time course data for EGFR-addicted non-small cell lung cancer (PC9) cells in both parental and isolated sublines. When PC9 c ells are treated with erlotinib, an EGFR inhibitor, a complex array of division and death cell decisions arise within a given population in response to treatment. Although deterministic (ODE) simulations capture the effects of clonal heterogeneity and describe the overall trends of experimentally treated tumor cell populations, these are not capable of explaining the observed variability of drug response trajectories, including response magnitude and time to rebound. Our stochastic simulations, instead, capture the effects of intrinsically noisy cell fate decisions that cause significant variability in cell population trajectories. These findings indicate that stochastic simulations are necessary to distinguish the contribution of extrinsic (clonal heterogeneity) and intrinsic (cell fate decisions) noise to understand the variability of cancer-cell response treatment. Furthermore, they suggest that, whereas tumors with distinct clon-al structures are expected to behave differently in response. 
11:45 to 12:30 Tomas Vejchodsky
Tensor methods for higher-dimensional Fokker-Planck equation
In order to analyse stochastic chemical systems, we solve the corresponding Fokker-Planck equation numerically. The dimension of this problem corresponds to the number of chemical species and the standard numerical methods fail for systems with already four or more chemical species due to the so called curse of dimensionality. Using tensor methods we succeeded to solve realistic problems in up to seven dimensions and an academic example of a reaction chain of 20 chemical species.

In the talk we will present the Fokker-Planck equation and discuss its well-posedness. We will describe its discretization based on the finite difference method and we will explain the curse of dimensionality. Then we provide the main idea of tensor methods. We will identify several types of errors of the presented numerical scheme, namely the modelling error, the domain truncation error, discretization error, tensor truncation error, and the algebraic error. We will present an idea that equilibration of these errors based on a posteriori error estimates yields considerable savings of the computational time.
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 17:00 Free Afternoon
Thursday 7th April 2016
09:00 to 09:45 Pieter Rein ten Wolde
Fundamental limits to transcriptional regulatory control
Gene expression is typically regulated by gene regulatory proteins that bind to the DNA. Experiments have shown that these proteins find their DNA target site via a combination of 3D diffusion in the cytoplasm and 1D diffusion along the DNA. This stochastic transport sets a fundamental limit on the precision of gene regulation. We derive this limit analytically and show by particle-based GFRD simulations that our expression is highly accurate under biologically relevant conditions.
09:45 to 10:30 Andrew Duncan
Hybrid modelling of stochastic chemical kinetics
Co-authors: Radek Erban (University of Oxford), Kostantinos Zygalakis (University of Edinburgh)

It is well known that stochasticity can play a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be adequately modelled by Markov processes and, for such systems, methods such as Gillespie's algorithm are typically employed. While such schemes are easy to implement and are exact, the computational cost of simulating such systems can become prohibitive as the frequency of the reaction events increases. This has motivated numerous coarse grained schemes, where the "fast" reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation for systems where all reactants are present in large concentrations, the approximation breaks down when the fast chemical species exist in small concentrations, giving rise to significant errors in the simulation. This is particularly problematic when using such methods to compute statistics of extinction times for chemical species, as well as computing observables of cell cycle models. In this talk, we present a hybrid scheme for simulating well-mixed stochastic kinetics, using Gillepsie-type dynamics to simulate the network in regions of low reactant concentration, and chemical Langevin dynamics when the concentrations of all species is large. These two regimes are coupled via an intermediate region in which a "blended"' jump-diffusion model is introduced. Examples of gene regulatory networks involving reactions occurring at multiple scales, as well as a cell-cycle model are simulated, using the exact and hybrid scheme, and compared, both in terms weak error, as well as computational cost.
10:30 to 11:00 Morning Coffee
11:00 to 11:45 Kevin Burrage
Sampling Methods for Exploring Between Subject Variability in Cardiac Electrophysiology Experiments
Co-authors: C. C. Drovandi (QUT), N. Cusimano (QUT), S. Psaltis (QUT), A. N. Pettitt (QUT), P. Burrage (QUT)

Between-subject and within-subject variability is ubiquitous in biology and physiology and understanding and dealing with this is one of the biggest challenges in medicine. At the same time it is difficult to investigate this variability by experiments alone. A recent modelling and simulation approach, known as population of models (POM), allows this exploration to take place by building a mathematical model consisting of multiple parameter sets calibrated against experimental data. However, finding such sets within a high-dimensional parameter space of complex electrophysiological models is computationally challenging. By placing the POM approach within a statistical framework, we develop a novel and efficient algorithm based on sequential Monte Carlo (SMC). We compare the SMC approach with Latin hypercube sampling (LHS), a method commonly adopted in the literature for obtaining the POM, in terms of efficiency and output variability in the presence of a drug block through an in-depth investigation via the Beeler-Reuter cardiac electrophysiological model. We show improved efficiency via SMC and that it produces similar responses to LHS when making out-of-sample predictions in the presence of a simulated drug block.
11:45 to 12:30 Vikram Sunkara
Insights into the dynamics of Hybrid Methods through a range of biological examples. A hands on approach
Biological systems can emerge complexity from simple yet multitude of interactions. Capturing such biological phenomenon mathematically for predictions and inference is being actively researched. Computing systems where the interacting components are inherently stochastic demands large amounts of computational power. Recently, splitting the dynamics of the system into deterministic and stochastic components has been a new strategy for computing biological networks. This hybrid strategy drastically reduces the number of equations to solve, however, the new equations are naturally stiff and nonlinear. Hybrid models are a strong candidate as a numerical method for probing large biological networks with intrinsic stochasticity. In this talk we will take on a new mathematical and numerical perspective of hybrid models. Through many biological examples, we will aim to gain insight into the benefits and stumbling blocks of the hybrid framework.

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12:30 to 13:30 Lunch at Wolfson Court
14:00 to 14:45 Carmen Molina-Paris
A stochastic story of two receptors and two ligands
In this talk, I will introduce the role of the co-receptors CD28 and CTLA-4 in the immune system. Both CD28 and CTLA-4 molecules are expressed on the membrane of T cells and can bind CD80 and CD86 ligand molecules, expressed on the membrane of antigen presenting cells. Classical immunology has identified CD28 co-receptor as enhancing the signal received by T cells from their T cell receptors (TCRs), and CTLA-4 as suppressing TCR signals. New experimental work is supporting a different role for the CTLA-4 molecule. In this talk, I will describe work in progress by our group, to model as  a multi-variate stochastic process the system of two receptors and two ligands. 
14:45 to 15:30 Ankit Gupta
Stability properties of stochastic biomolecular reaction networks: Analysis and Applications
Co-author: Mustafa Khammash (ETH Zurich)

The internal dynamics of a living cell is generally very noisy. An important source of this noise is the intermittency of reactions among various molecular species in the cell. The role of this noise is commonly studied using stochastic models for reaction networks, where the dynamics is described by a continuous-time Markov chain whose states represent the molecular counts of various species. In this talk we will discuss how the long-term behavior of such Markov chains can be assessed using a blend of ideas from probability theory, linear algebra and optimisation theory. In particular we will describe how many biomolecular networks can be viewed as generalised birth-death networks, which leads to a simple computational framework for determining their stability properties such as ergodicity and convergence of moments. We demonstrate the wide-applicability of our framework using many examples from Systems and Synthetic Biology. We also discuss how our results can hel p in analysing regulatory circuits within cells and in understanding the entrainment properties of noisy biomolecular oscillators.
15:30 to 16:00 Afternoon Tea
16:00 to 16:45 Mustafa Khammash
Subtle is the noise, but malicious it is not: dynamic exploits of intracellular noise
Co-authors: Ankit Gupta (ETH Zürich), Corentin Briat (ETH Zürich)

Using homeostasic regulation and oscillatory entrainment as examples, I demonstrate how novel and beneficial functional features can emerge from exquisite interactions between intracellular noise and network dynamics. While it is well appreciated that negative feedback can be used to achieve homeostasis when networks behave deterministically, the effect of noise on their regulatory function is not understood. Combining ideas from probability and control theory, we have developed a theoretical framework for biological regulation that explicitly takes into account intracellular noise. Using this framework, I will introduce a new regulatory motif that exploits stochastic noise, using it to achieve precise regulation and perfect adaptation in scenarios where similar deterministic regulation fails. Next I propose a novel role of intracellular noise in the entrainment of decoupled biological oscillators. I will show that while intrinsic noise may inhibit oscillatory activity in ind ividual oscillators, it can actually induce the entrainment of a population of such oscillators. Thus in both regulation and oscillatory entrainment, beneficial dynamic features exist not just in spite of the noise, but rather because of it.
19:30 to 22:00 Formal Dinner at Christ's College
Friday 8th April 2016
09:00 to 09:45 Yiannis Kaznessis
Closure Scheme for Chemical Master Equations - Is the Gibbs entropy maximum for stochastic reaction networks at steady state?
Stochasticity is a defining feature of biochemical reaction networks, with molecular fluctuations influencing cell physiology. In principle, master probability equations completely govern the dynamic and steady state behavior of stochastic reaction networks. In practice, a solution had been elusive for decades, when there are second or higher order reactions. A large community of scientists has then reverted to merely sampling the probability distribution of biological networks with stochastic simulation algorithms. Consequently, master equations, for all their promise, have not inspired biological discovery.

We recently presented a closure scheme that solves chemical master equations of nonlinear reaction networks [1]. The zero-information closure (ZI-closure) scheme is founded on the observation that although higher order probability moments are not numerically negligible, they contain little information to reconstruct the master probability [2]. Higher order moments are then related to lower order ones by maximizing the entropy of the network. Using several examples, we show that moment-closure techniques may afford the quick and accurate calculation of steady-state distributions of arbitrary reaction networks.

With the ZI-closure scheme, the stability of the systems around steady states can be quantitatively assessed computing eigenvalues of the moment Jacobian [3]. This is analogous to Lyapunov’s stability analysis of deterministic dynamics and it paves the way for a stability theory and the design of controllers of stochastic reacting systems [4, 5].

In this seminar, we will present the ZI-closure scheme, the calculation of steady state probability distributions, and discuss the stability of stochastic systems.

1. Smadbeck P, Kaznessis YN. A closure scheme for chemical master equations. Proc Natl Acad Sci U S A. 2013 Aug 27;110(35):14261-5.

2. Smadbeck P, Kaznessis YN. Efficient moment matrix generation for arbitrary chemical networks, Chem Eng Sci, 20
09:45 to 10:30 Darren Wilkinson
Scalable algorithms for Markov process parameter inference
Inferring the parameters of continuous-time Markov process models using partial discrete-time observations is an important practical problem in many fields of scientific research. Such models are very often "intractable", in the sense that the transition kernel of the process cannot be described in closed form, and is difficult to approximate well. Nevertheless, it is often possible to forward simulate realisations of trajectories of the process using stochastic simulation. There have been a number of recent developments in the literature relevant to the parameter estimation problem, involving a mixture of approximate, sequential and Markov chain Monte Carlo methods. This talk will compare some of the different "likelihood free" algorithms that have been proposed, including sequential ABC and particle marginal Metropolis Hastings, paying particular attention to how well they scale with model complexity. Emphasis will be placed on the problem of Bayesian pa rameter inference for the rate constants of stochastic biochemical network models, using noisy, partial high-resolution time course data.
10:30 to 11:00 Morning Coffee
11:00 to 11:45 Christian Ray
Lineage as a conception of space in compartmental stochastic processes across cellular populations
Co-author: Arnab Bandyopadhyay (University of Kansas)

Cytoplasmic regulatory networks often approximate well-mixed reaction kinetics in single cells, but with variability from cell to cell. As a result, inheritance dynamics and kin correlations have been implicated in effects on cell cycle, regulatory networks, and modulation of population growth rate. Based on an experimental result in our lab suggesting lineage correlations in bacterial growth arrest, we developed a cellular stochastic simulation framework to analyse the role of lineage in bacterial cells regulating growth rate by means of an intracellular molecular network. The simulation framework thus models both intrinsic and inherited noise sources while maintaining lineage data between cell agents assigned individual unique identifiers.

Our initial application of the framework demonstrates the role of lineage in the probability of bacterial growth arrest controlled by an endogenous toxin from a toxin-antitoxin system. These systems have tight binding between toxin and antitoxin, so that there is a discrete critical threshold in the toxin:antitoxin ratio below which a cell is essentially toxin-free and growth is unrestricted, and above which toxin rapidly slows the growth rate. The subset of high-toxin cells crossing into the growth arrested state are associated with antibiotic persistence. Our implementation of a simple toxin-antitoxin system in the simulation framework revealed the statistical dependence of growth arrest on cellular lineage: after several generations of growth, the probability of cellular growth arrest began to depend on lineage distance. Clusters of closely related cell agents had a high probability of transitioning into growth arrest, while the rest of the lineage continued to grow withou t restriction.

We consider various quantities of interest in multiscale lineage simulations, and conclude that growth transitions in a cellular colony cannot be fully understood without quantitative knowledge of its lineage.
11:45 to 12:30 Ramon Grima
The system-size expansion of the chemical master equation: developments in the past 5 years
Co-author: Philipp Thomas (Imperial College London)

The system-size expansion of the master equation, first developed by van Kampen, is a well known approximation technique for deterministically monostable systems. Its use has been mostly restricted to the lowest order terms of this expansion which lead to the deterministic rate equations and the linear-noise approximation (LNA). I will here describe recent developments concerning the system-size expansion, including (i) its use to obtain a general non-Gaussian expression for the probability distribution solution of the chemical master equation; (ii) clarification of the meaning of higher-order terms beyond the LNA and their use in stochastic models of intracellular biochemistry; (iii) the convergence of the expansion, at a fixed system-size, as one considers an increasing number of terms; (iv) extension of the expansion to describe gene-regulatory systems which exhibit noise-induced multimodality; (v) the conditions under which the LNA is exact up to second-order moments; (v i) the relationship between the system-size expansion, the chemical Fokker-Planck equation and moment-closure approximations.

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12:30 to 13:30 Lunch at Wolfson Court
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons