# Abstracts

#### Max Aldana (UNAM)

**Epigenetic inheritance of antibiotic resistance in bacteria: a stochastic formulation**

Adaptive resistance to antibiotics emerges when populations of bacteria are subjected to gradually increasing concentrations of antibiotics. It is characterized by a rapid appearance and a fast reversibility to the non-resistant phenotype when the antibiotic is removed from the media. Recent work shows that adaptive resistance requires noise-driven heterogeneity of gene expression patterns, particularly those associated to the production of membrane porins and efflux pumps. However, we will show in this presentation that stochastic heterogeneity is not enough to obtain adaptive resistance. Working with an efflux pump regulatory network model, we have obtained evidence showing that in addition to stochastic variability of gene expression patterns, epigenetic inheritance of this variability is also necessary to build up adaptive antibiotic resistance in bacterial populations. Furthermore, contrary to the common assumption, our model predicts that adaptive resistance cannot be accounted for increased mutation rates. We also show that adaptive resistance entails a delicate balance between being resistant to antibiotics on the one hand, and being able to reproduce on the other hand. This tradeoff is achieved by very few cells in the population, which are the ones that give rise to the resistant phenotype.

#### Tibor Antal (Edinburgh)

**Universal asymptotic of clone size distribution for arbitrary population growth**

Deterministically growing (wild-type) populations which seed stochastically developing mutant clones have found an expanding number of applications from microbial populations to cancer. The special case of exponential wild-type population growth, usually termed as Luria-Delbruck or Lea-Coulson model, is often assumed but seldom realistic. In this talk we generalize the model to different types of wild-type population growth, while the mutants evolve as a birth-death branching process. Our focus is on the size distribution of clones after some time, which can be mapped to the total number of mutants. Exact expressions are derived for exponential, power-law and logistic population growth. We prove that the large time limit of the clone size distribution has a general one-parameter form for any population growth. The clone size distribution always has a power law tail, and for subexponential wild-type growth the probability of a give clone size is inversely proportional to the clone size. We support our findings by analyzing a dataset on tumor metastasis sizes, and we find that a power-law tail is more likely than an exponential one. Joint work with Michael Nicholson.

#### Frank Ball (Nottingham)

**An epidemic in a dynamic population with importation of infectives**

We consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size n. A Markovian SIR (susceptible -> infective -> recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where n -> 1, keeping the basic reproduction number R_0 as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than 1/ log n. We show that, as n -> 1, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process S = {S(t); t >= 0} describing the limiting fraction of the population that are susceptible. The process S grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the start of the regenerative cycle. Properties of the process S, including the jump size and stationary distributions, are determined.

Based on work done jointly with Tom Britton (Stockholm University) and

Pieter Trapman (Stockholm University)

#### Golan Bel (Ben Gurion)

**Critical and gradual transitions in pattern-forming systems**

Critical transitions have attracted a great deal of attention due to their relevance to many natural and social systems. Much research has been devoted to the characterization and identification of imminent critical transitions. In spatially extended systems, the dynamics (close to and away from the critical point) is more complicated due to the expansion, shrinking and coalescence of alternative-state domains. Pattern-forming systems introduce additional complexity due to the patterned nature of one of the stable states. In this talk, I will present several works in which we used the context of drylands vegetation dynamics to study various aspects of this additional complexity: (i) Using a minimal model, we showed that in systems exhibiting a bistability of a patterned state with a uniform state, a multitude of intermediate stable localized states may appear, giving rise to step-like gradual shifts with extended pauses at these states. This result suggests that a combination of abrupt-shift indicators and gradual-shift indicators might be needed to unambiguously identify regime shifts. (ii) The existence of these localized states in models for the dynamics of drylands vegetation and the response of the systems described by these models to local perturbations will be discussed. (iii) We show how a simplified version of a model for drylands vegetation dynamics can explain the emergence and the observed dynamics of the spectacular phenomenon of “fairy circles” in southern Africa. If time permits, I will present recent results demonstrating the effects of heterogeneity on the pattern formation, survivability and resilience of water-limited vegetation.

#### Joseph Challenger (Imperial)

**A fresh look at within-host modelling of falciparum malaria**

In this work, we are taking a new look at within-host dynamics of the malaria parasite. Untreated falciparum malaria infections can persist for over a year. One of the motivations for developing within-host models of malaria is to attempt to explain and reproduce this behaviour. Typically, such models are informed by an unusual dataset of untreated malaria infections, known as the ‘malaria therapy dataset’. Our aim is to reproduce the behaviour observed, but with a model of lower complexity than found in existing models. As our model contains both asexual and sexual parasites, it will allow us to make predictions on clinical outcomes as well as the contribution to onward transmission. To this model, we are adding antimalarial drugs, using a full pharmacokinetic-pharmacodynamic (PKPD) approach. This modelling framework will allow us to examine the effects of dosing timings, as well as the consequences of non-adherence to treatment.

#### Christina Cobbold (Glasgow)

**The importance of stochasticity in antigenic variation models: an example from African Trypanosomes**

The antigenic archive of pathogens plays an important role in within-host infections. Antigenically varying pathogens escape host immunity by switching expression of different variants. The timing of this process and the number of antigenic variants expressed over the course of an infection are often stochastic and spontaneous, independent of host immune pressure. They result mainly from genetic properties of the antigenic archive of the pathogen. However, their subsequent interaction with dynamic host immune mechanisms determines the outcome of an infection: its peak, duration, and nature of chronicity.

African Trypanosomes are a prominent example of such pathogens exploiting antigenic variation to establish chronic infection in their hosts. With increasing availability of parasite genomic data, more realistic within- host models are now possible, through the integration of features of the antigenic archive. This integration has the potential to significantly improve our understanding of the fundamental mechanisms of parasite immune evasion. Here we study the within-host dynamics of trypanosomes. We use a mathematical model to investigate how antigenic archive properties modulate the balance between specific and general control of the pathogen. We show that introducing stochasticity has important implications for the comparisons between infections across hosts of different size and for understanding the architecture of the genetic archive of the parasite.

#### Rob Cross (Warwick)

**Single molecular mechanochemistry of kinesin**

Complex living cells contain networks of microtubules, linear protein polymers that serve as railways for the motor-driven transport of cellular components. Together with dyneins, kinesins are the molecular engines for this cellular railway. Most kinesins haul molecular cargo directionally along their microtubule rails, stepping repeatedly and directionally between binding sites spaced 8nm apart. But some kinesins are specialised to control the assembly dynamics of their microtubule tracks, and a few can do both. These two distinct activities, hauling cargo along microtubules and biasing subunit exchange at microtubule tips, are linked by a common thread, the generation and sensing of mechanical force via the chemistry of the kinesin active site. This connection, termed mechanochemical coupling, works reciprocally – it allows conformational changes (themselves relying on fluctuations in protein structure) in the kinesin active site to be amplified and coordinated and harnessed to drive larger scale motions that can do substantial work, and it also allows external forces to influence conformational events in the active site. Recently, the force-generating and force-sensing mechanisms of the kinesin active site have come much more clearly into focus. I will discuss mechanisms by which thermally-driven fluctuations in structure can be sensed, filtered and harnessed, via mechanochemical coupling, so as to allow kinesin motors to step productively along their microtubule track and to modulate microtubule dynamics. I will also describe new experiments, aimed at testing specific candidate models for the structural mechanisms by which thermal fluctuations are filtered and harnessed to produce sustained directional motion.

#### Jonathan Desponds (ENS Paris)

**Fluctuating fitness shapes the clone size distribution of immune repertoires**

The adaptive immune system relies on the diversity of receptors ex- pressed on the surface of B and T-cells to protect the organism from a vast amount of pathogenic threats. The proliferation and degrada- tion dynamics of different cell types (B cells, T cells, naive, memory) is governed by a variety of antigenic and environmental signals, yet the observed clone sizes follow a universal power law distribution. Guided by this reproducibility we propose effective models of so- matic evolution where cell fate depends on an effective fitness. This fitness is determined by growth factors acting either on clones of cells with the same receptor responding to specific antigens, or directly on single cells with no regard for clones. We identify fluctuations in the fitness acting specifically on clones as the essential ingredient leading to the observed distributions. Combining our models with experiments we characterize the scale of fluctuations in antigenic environments and we provide tools to identify the relevant growth signals in different tissues and organisms. Our results generalize to any evolving population in a fluctuating environment.

#### Maria D'Orsogna (UC Riverside)

**Stochastic Nucleation and Growth**

The binding of individual components to form composite structures is a ubiquitous phenomenon within the sciences. Within heterogeneous nucleation, particles may be attracted to an initial exogenous site, impurities or boundaries for example. Homogeneous nucleation instead describes identical particles spontaneously clustering upon contact. Given their ubiquity nucleation and growth have been extensively studied in the past decades, often assuming infinitely large numbers of building blocks and unbounded cluster sizes. These assumptions led to the use of mass-action, mean field descriptions such as the well known Becker Doering equations. In cellular biology, however, nucleation often take place in confined spaces, with a finite number of components, so that discrete and stochastic effects must be taken into account. In this talk we examine finite sized homogeneous nucleation by considering a fully stochastic master equation, solved via Monte-Carlo simulations and via analytical insight. We find striking differences between the mean cluster sizes obtained from our

discrete, stochastic treatment and those predicted by mean field treatments. We discuss first assembly times and consider both monomer attachment/detachment kinetics as well as general coagulation/fragmentation events.

#### Louise Dyson (Warwick), Thomas House (Manchester)

**Fluctuation-driven phenomena in epidemiology: a review of some past results and a prospectus for future work**

Models that account for stochasticity have a long history in mathematical epidemiology, but their popularity has varied over time. We will review some previous results, for example models for households that started with the work of Reed and Frost almost a century ago. We will also outline some future possibilities such as the analysis of diseases under eradication efforts and models that can interface with new forms of data that are becoming available.

#### Tobias Galla (Manchester)

**Fixation and extinction dynamics in individual-based models: from evolutionary games to the initiation of cancer**

Key fluctuation-driven phenomena in biological systems include the fixation and extinction of species in stochastic population dynamics. These are often not captured by traditional approaches based on deterministic rate equations, and only occur in finite populations. In this talk I will first give a brief general introduction to fixation problems. I will then discuss the fixation or extinction of single mutants in a birth-death process subject to fluctuating environments, and I will highlight some of the non-trivial phenomena that can occur in two-player and multi-player evolutionary games. In the final part of the talk I will briefly outline how the mathematics of fixation processes and ideas from statistical physics can be applied to analysing the acquisition of successive mutations in a model of cancer initiation.

#### Jeff Gore (MIT)

**Cooperation, cheating and collapse in biological populations**

Natural populations can suffer catastrophic collapse in response to small changes in the environment, and recovery after such a collapse can be difficult. We have used laboratory microcosms to directly measure theoretically proposed early warning signals of impending population collapse based on the phenomenon of critical slowing down. Our experimental yeast populations cooperatively break down the sugar sucrose, meaning that below a critical size the population cannot sustain itself. We find that catastrophic collapse is preceded by a change in the fluctuations of the population, suggesting that this and other indicators may provide advance warning of impending collapse. The cooperative nature of yeast growth on sucrose makes the population susceptible to the emergence of "cheater" cells, which do not contribute to the public good and reduce the resilience of the population.

#### Adam Kucharski (LSHTM)

**Modelling the stuttering dynamics of emerging infectious diseases**

Obtaining good estimates of disease transmission potential is crucial for effective surveillance and control of infectious diseases. However, when an infection transmits inefficiently between humans, estimates often have to be made using data from a limited number of observed transmission chains.

I will describe how we can use simple stochastic mathematical models to characterize the transmission dynamics of emerging infections from readily available data sources. Such techniques make it possible to identify the extent to which outbreaks are driven by inherent pathogen transmissibility or pre-existing population immunity. They can also be used to measure individual heterogeneity in transmission, and establish what control measures may be required to contain an outbreak.

As well as discussing the methodological challenges involved, I will explain how the work can give insights into the epidemiology and control of infections such as Ebola, avian influenza and MERS-CoV.

#### Yen Ting Lin (Manchester)

**Effects of bursting noise in gene regulation networks**

Including short-lived mRNA populations in models of gene regulation networks introduces both the transcriptional bursting (i.e., transcription occurs at random times) and translational bursting (i.e., random amounts of proteins are translated by each mRNA). In this talk, I will present a method to construct mesoscopic models for such type of dynamical systems, which fully accounts for the bursting noise. We systematically compare different levels of modeling, ranging from individual-molecule-based models including mRNA populations, over protein-only individual-based models to mesoscopic models such as diffusion-type models and our proposed model. We show that the proposed mesoscopic model outperforms conventional diffusion-type models. In a one-dimensional autoregulated network, we present closed-form analytic solutions for both the stationary distribution of protein expression as well as first-passage times of the dynamical system. We present numerical solutions for higher-dimensional gene regulation networks, in which case we also carry out analysis in the weak-noise limit.

The implications of the study are multi-faceted. Bursting noise is a ubiquitous feature of many biological systems. From a modeling perspective, our proposed method provides an alternative way to analyze dynamical systems with bursting noise. Biologically, the study presents quantitative evidences, the first to our knowledge, showing that bursting noise is the predominant form of intrinsic noise in gene regulation networks. Finally, from a mathematical point of view, the proposed model belongs to a class of stochastic processes named piecewise deterministic Markov processes, and our analysis on first-passage times and weak-noise limit of the process may inspire more rigorous analytic investigations in the future.

Time permitting, I will also briefly report recent generalisation of the method to investigate the mechanisms of noise-induced oscillations in different circuits of biological clock. To this end, we show that multiple mechanisms could excite stochastic oscillators in different parameter regimes. This part is a work in progress.

References: The talk is based on the following papers

• Physical Review E 93, 022409 (2016)

• Journal of the Royal Society Interface 13: 20150772 (2016)

#### Malwina Luczak (QM London)

**Extinction time for the weaker of two competing SIS epidemics**

We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. In our model, the spread of each strain is described by the stochastic logistic SIS epidemic process in the absence of the other strain, and we assume that there is perfect cross-immunity between the two virus strains, that is, individuals infected by one strain are temporarily immune to re-infections and infections by the other strain. For the case where each strain on its own is supercritical (that is, its basic reproductive ratio is larger than 1), and one strain has a strictly larger basic reproductive ratio than the other, we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further extend our results to the case where the difference between the two reproductive ratios may tend to 0.

#### Baruch Meerson (Hebrew University of Jerusalem)

**Noisy invasions: large fluctuations in stochastic invasion models**

Invasion fronts are recognized as important, and often fateful, phenomena in ecology, epidemiology and biological evolution. The position of an invasion front fluctuates because of the shot noise of elemental processes. What is the probability to observe, at a given time, a certain front position that considerably deviates from its deterministic counterpart? The answer strongly depends on whether the front propagates into a metastable or unstable state, and I will review recent theoretical progress in both cases [1-7]. The progress is mostly based on using a dissipative version of WKB theory which assumes many individuals in the front region. In this theory the most likely history of the system, conditioned on a given front displacement, is encoded in a special trajectory of the underlying effective Hamilton mechanics: a classical field theory. This special trajectory is described by a traveling front solution

of the field theory equations. For fronts, propagating into unstable states, unusually large front displacements are much more likely than unusually small ones. The leading contribution to the probability density of large displacements comes from a few fastest particles running ahead of the front. For such

invasion fronts the WKB theory breaks down. Here some useful results can be obtained by studying the large-displacement statistics of the branching Brownian motion.

1. B. Meerson, P.V. Sasorov and Y. Kaplan, Phys. Rev. E 84, 011147 (2011).

2. E. Khain and B. Meerson, J. Phys. A: Theor. Math. 46, 125002 (2013).

3. B. Meerson and P.V. Sasorov, Phys. Rev. E 84 030101 (R) (2011).

4. B. Meerson, A. Vilenkin and P.V. Sasorov, Phys. Rev. E 87, 012117 (2013).

5. B. Derrida, B. Meerson and P.V. Sasorov, arXiv:1601.08070.

7. B. Derrida, Z. Shi, arXiv:1601.04652.

#### Thierry Mora (ENS Paris)

**Physical limit to concentration sensing in a background of competing ligands**

Many biological objects, such as transmembrane receptors or gene promoters, sense concentrations of chemical entities by binding them specifically. This sensing is limited by noise stemming from the small number of sensed molecules. In addition, in many situations competing ligands may bind non-specifically to the target, potentially interfering with the signal of the correct ligand, creating cross-talk. What is the best accuracy of concentration sensing one can achieve in the face of this cross-talk? I will show how to derive the best achievable performance using the principle of maximum likelihood, and will discuss how the mathematical solution can inspire new network designs based on the principle of kinetic proof-reading.

#### Tim Rogers (Bath)

**Strength in numbers: how demographic noise can reverse the direction of selection**

Deterministic evolutionary theory robustly predicts that populations displaying altruistic behaviours will be driven to extinction by mutant cheats that absorb common benefits but do not themselves contribute. In this talk I will show how demographic stochasticity can in fact reverse the direction of selection in favour of the cooperative phenotype if their behaviour appreciably alters the carrying capacity of the population. I will present an analysis of a simple but general model of population dynamics in the specific context of public goods production, and derive explicit conditions for stochastic selection reversal.

#### Ludger Santen (Saarbruecken)

**Length regulation of microtubules by molecular motors: Exact solution, density profiles and confinement**

The length of microtubules (MT) can be regulated in different ways by MT associated proteins (MAPs). Here, a MT model is discussed, whose length is regulated by the action of processive kinesin motors. We treat first the case of infinite processivity, i.e. particle exchange in the bulk is neglected. The exact results can be obtained for model parameters which correspond to a finite length of the MT. In contrast to the model with particle exchange we find that the lengths of the MT are exponentially distributed in this parameter regime. The remaining parameter space of the model, which corresponds to diverging MT lengths, is analyzed by means of extensive Monte Carlo simulations and a macroscopic approach. For divergent MTs we find a complex structure of the phase diagram in terms of shapes of the density profile. We also discuss the case of kinesin-MT interactions in finite volumes.

Joint work with Chikashi Arita.

#### Alex Sigal (Durban)

**Quantitative study of transmission between cells in HIV and TB**

Infection dynamics may depend not only on the specific biology of host pathogen interactions, but also on the number of pathogens transmitted to a single target host cell. Here I will describe how pathogen numbers influence infection dynamics in two infectious diseases: HIV and TB. In HIV, I will discuss our recent results showing that multiplicity of infection per cell determines the rate of the viral cycle. For TB, I will describe a positive feedback loop which leads to expanding infection once multiple bacilli infect one human macrophage.

#### Orkun Soyer (Warwick)

**Design principles for multistability in cellular signaling networks**

#### Bartek Waclaw (Edinburgh)

**Phenotypic switching and the biological evolution of microbes**

Stochastic phenotype switching (SPS) is a biological phenomenon in which cells (usually microbes) randomly switch between two or more "states" (phenotypes) without altering their genetic code. SPS has been suggested to play a beneficial role in microbial populations by leading to the division of labour among cells, or ensuring that at least some of the population survives an unexpected change in environmental conditions. Here I will discuss an alternative possible function of stochastic phenotype switching - a way to adapt more quickly even in a static environment. Using computer modelling and simple mathematics I will show that when a genetic mutation causes a population to become less fit, switching to an alternative phenotype with higher fitness (growth rate) may give the population enough time to develop compensatory mutations that increase the fitness again. The possibility of switching phenotypes can reduce the time to adaptation by orders of magnitude if the "fitness valley" caused by the deleterious mutation is deep enough. The proposed mechanism may explain the observed rapid evolution of resistance to antibiotics.

#### Kit Yates (Bath)

**Hybrid frameworks for modeling cell migration**

Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, due to recent advances in computational power, the simulation, and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems in which regions of low copy numbers exist.

Individual-based stochastic models provide microscopic/mesoscopic accuracy but at the cost of significant computational resources. Models which have regions of both low and high concentrations often necessitate coupled macroscale and microscale modelling paradigms. This is because microscale models are not feasible to simulate at large concentrations and macroscale models are often inappropriate at small concentrations.

In this talk I will motivate the need for efficient hybrid modeling methodologies with biological examples from early embryo formation. I will then outline a few methodologies that we have developed to couple a range of different regimes together including PDEs, position-jump models, and fully and partially excluding volume exclusion processes.

#### Anton Zilman (Toronto)

**First passage processes in cellular transport, signaling and population dynamics**