Workshop 2: Stochastic Processes in Cell and Population Biology: Titles & Abstracts
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- Titles & Abstracts
In deterministic epidemic models, pathogen extinction in a population is determined by the magnitude of the basic reproduction number R0. In stochastic epidemic models, the probability of pathogen extinction depends on R0, the size of the population and the number of infectious individuals. For example, in the SIS Markov chain epidemic model, if the basic reproduction number R0>1, the population size is large and I(0)=a is small, then a classic result of Whittle (1955) gives an approximation to the probability of pathogen extinction: (1/R0)a. This classic result can be derived from branching process theory. We apply results from multitype Markov branching process theory to generalize this approximation for probability of pathogen extinction to more complex epidemic models with multiple stages, treatment , or multiple populations and to within host models of virus and cell dynamics.
Work done in collaboration with Yuan Yuan and Glenn Lahodny.
I will focus on computational methods for stochastically modeled biochemical reaction networks. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. I will show how different computational methods can be understood and analyzed by using different representations for the processes. Topics discussed will be a subset of: approximation techniques, variance reduction methods, parameter sensitivities.
I will start with an overview over various models for the dynamics of the genetic composition of populations evolving under recombination. For the deterministic treatment that applies in the infinite-population limit, one has large, nonlinear dynamical systems; for the stochastic treatment required for finite populations, the Moran, or Wright-Fisher model is appropriate.
I will focus on models involving only single crossovers in every generation and contrast the situations in continuous and in discrete time. In continuous time, the deterministic model has a simple closed solution, which is due to the independence of the individual recombination events. In contrast, discrete time introduces dependencies between the links and leads to a much more complex situation. Nevertheless, the situation becomes tractable by looking backwards in time, starting from single individuals at present in a Wright-Fisher population with recombination and tracing back the ancestry of the various gene segments that result from recombination. In the limit of population size to infinity, these segments become independent. We identify the process that describes their history, together with the tree structures they define, which we like to call ancestral recombination trees. It turns out that the corresponding tree _topologies_ play a special role: Surprisingly, explicit probabilities may be assigned to them, which then leads to an explicit solution of the recombination dynamics.
This is joint work with Ute von Wangenheim.
[1] E. Baake, Deterministic and stochastic aspects of single-crossover recombination, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, 3037-3053
Gene expression is the biological process that actualizes phenotypes (such as drug resistance) encoded in the genetic sequence. Correlating the gene expression mean with various phenotypes provided important insights into the genotype-phenotype connection. Yet, recently it became clear that cells with identical genomes exposed to the same environment can differ dramatically in their gene expression and phenotype. The importance of such random phenotypic variation (noise) for stress resistance is now well established. It still remains unknown, however, how the time-dependent aspects such as the duration (or "memory") of random cellular decisions affect sensitivity to drug treatment.
We engineered Saccharomyces cerevisiae cells to carry a synthetic gene circuit controlling the expression of a bifunctional fluorescent reporter, yEGFP::zeoR, which also counteracted the antibiotic Zeocin. Single cells randomly differentiated into drug-resistant and drug-sensitive phenotypes, which differed in their fitness both in the presence and absence of drug. We developed computational methods to predict the overall fitness of the cell population in arbitrary antibiotic concentrations. We found that only after incorporating nongenetic (cellular) memory of randomly established drug resistance states, the antibiotic response of cell populations exposed to drug became predictable.
A stochastic model for variable-length stepping of kinesins engineered with extended neck linkers is developed. This requires consideration of the separation in microtubule binding sites between the heads of the motor at the beginning of a step. It can be shown that the separation is a stationary Markov process and can be included in the calculation of standard experimental quantities, such as asymptotic velocity and effective diffusion, through the appropriate limits of a semi-Markov process. Using this framework, asymptotic results for randomly detached motors are also obtained and linked to the statistical analysis of velocity data from motor assays. In addition, we will discuss how the framework developed here could be used as one component of a larger scale model for motor-cargo systems of the type to be presented in Kramer's talk.
In a toy model of binary splitting branching processes with population size dependence (supercritical below and subcritical above a threshhold, the carrying capacity) the chance of a little population establishing itself in the sense of reaching a band around the carrying capacity is determined, and so is the persistence time of the population. Mutations and competition between morphs are introduced, and it turns out that the resulting processes exhibit evolutionary branchings which occur in a manner slightly different from that predicted by established deterministic theory. The validity of conclusions is discussed in terms of more general branching processes. (Joint work with Serik Sagitov, Fima Klebaner et al.)
The growing field of synthetic biology aims to forward engineer biology both for applications such as energy production, drug production, and bioremediation, as well as for the purpose of furthering the fundamental understanding of natural systems. However, engineering living cells is notoriously difficult due to issues such as mutation, epigenetic variation, fitness effects, and the interaction of synthetic components with host cell processes. Thus, simpler contexts such as cell-free expression systems offer great promise to engineering complex biological behavior in a quantitative fashion. Furthermore, the confinement of cell-free gene circuit reactions in nanofabricated reaction devices offers a flexible approach to investigating fundamental aspects of gene circuit function. Currently, we are using such devices to study noise in simple gene circuits. Cell-free reactions confined in different volume wells are imaged over time using fluorescent microscopy. The noise characteristics of the resulting gene expression trajectories are analyzed and compared for different gene circuits.
Recent years have seen increasing attention to the subtle effects on intracellular transport caused when multiple molecular motors bind to a common cargo. We develop and examine a coarse-grained model which resolves the spatial configuration as well as the thermal fluctuations of the molecular motors and the cargo. This intermediate model can accept as inputs either common experimental quantities or the effective single-motor transport characterizations obtained through the kind of systematic analysis of detailed molecular motor models described in Fricks' presentation. Through stochastic asymptotic reductions, we derive the effective transport properties of the multiple-motor-cargo complex, and provide analytical explanations for why a cargo bound to two molecular motors moves more slowly at low applied forces but more rapidly at high applied forces than a cargo bound to a single molecular motor.
Deterministic population dynamical models connect to reality through their interpretation as limits for systems size going to infinity of stochastic processes in which individuals are represented as discrete entities. In structured population models individuals may be born in different states (e.g. locations in space) after which they proceed through their h(eterogeneity)-state space, e.g. spanned by their i(dividual)-state and location. On such models one can graft evolutionary processes like random genetic drift or adaptive evolution by rare repeated substitutions of mutants in heritable traits affecting the state transition and reproduction processes of individuals. From this general perspective I will derive the so-called Canonical Equation of adaptive dynamics, a differential equation for evolutionary trait change derived under the additional assumption that mutations have small effect. In the CE approximation the rate of evolution is found to correspond to the product of a parameter $n_{e,A}$, equal to the population size times a dimensionless product of life history parameters (including spatial movements), times the gradient of the invasion fitness of potential mutants with respect to their trait vector. From a heuristic connection with the diffusion approximation for genetic drift it follows that $n_{e,A} = n_{e,D}$, the effective population size from population genetics.
We derive kinetic differential or integrodifference equations for the mean and variance or of a quantitative trait as a function of space and time, in some cases recovering known equations and in some cases obtaining new ones that capture effects, such as nonmonotonicity of traveling waves, that can be seen in stochastic simulations. We then reanalyze kinetic equations due to Kirkpatrick and Barton for population range limits, showing that they exhibit bistability and hysteresis. This suggests a possible mechanism for lag times between establishment and subsequent explosive growth and range expansion in the absence of an Allee effect.
Stochastic biochemical systems and population genetics models are described by similar mathematical equations, and hence similar phenomena should be observed in both systems. Here we focus on stochastic kinetics with time scale separation. We show how to integrate out the fast degrees of freedom, while rigorously preserving their effects on the fluctuations of slower variables. This procedure allows to speed up simulation of kinetic networks and reveals a number of interesting phenomena, previously unobserved in the context of classical stochastic kinetics. One of the most interesting is the emergence of geometric phases, which we show may have substantial effects on, in particular, the frequency of fixation of new mutations in slowly variable environments.
Pattern formation in a developing tissue frequently involves the proper spatial localization of the boundary between different cell types. A standard mechanism to accomplish this uses one or more diffusible chemical signals called morphogens that are produced at boundaries of the tissue, and whose concentration determines gene expression and phenotypic characteristics in the tissue. How such boundaries, which lead to distinguished fates between adjacent cells, are set is an important issue in developmental biology. Since the concentration of morphogens or downstream components may be small, stochastic fluctuations make the reliable determination of the boundary more difficult. Models of this process are frequently based on reaction-diffusion equations, and in this talk we will address several questions related to the simulation of such systems. Firstly, how does one chose a computational cell size for a complex reaction-diffusion network, secondly, how does does one eliminate fast reactions in a stochastic reaction network, and thirdly, how does the network structure affect the resilience of boundary location determination when stochastic effects are important.
Actin polymerization in vivo is regulated spatially and temporally by a web of signaling proteins. We developed a three-dimensional, physico-chemical, stochastic model of sheet-like lamellipodia, which are projected by eukaryotic cells during cell migration, and contain a dynamically remodeling three-dimensional actin mesh. A number of regulatory proteins and subtle mechano-chemical couplings determine the lamellipodial protrusion dynamics. Our work sheds light on how lamellipodial protrusion dynamics is affected by the concentrations of actin and actin-binding proteins. Overall, our work emphasizes that elongation and nucleation processes work highly cooperatively in determining the optimal protrusion speed for the actin mesh in lamellipodia. We also studied molecular mechanisms of growth retraction cycles in filopodia, finger-like protrusions based on bundles of actin filaments. In particular, we found that capping proteins and molecular motors may have a profound effect on filopodial dynamics. We also uncovered the rules of active transport in filopodia, mediated by molecular motors, allowing for highly efficient delivery of cytosolic proteins to the filopodial tip. We studies the concentration profile of motors and actin along the filopodial tube, and the way motor transport couples to filopodial growth dynamics.
In mammals, the suprachiasmatic nucleus (SCN), a brain region of about 20,000 neurons, serves as the master circadian clock, coordinating timing throughout the body and entraining the body to daily light cycles. Experiments in which cell-to-cell signaling between SCN neurons is disrupted by physical separation of the cells, or by blocking vasoactive intestinal polypeptide (VIP) mediated signaling, show that the remarkable precision of the circadian clock at the level of the organism relies on this intercellular signaling. In the absence of cell-to-cell signaling, each SCN neuron and the SCN as a whole exhibits a high degree of stochasticity, with significantly less stable oscillations. We describe several novel findings that were obtained via a combination of experiment and discrete stochastic models, explored through wavelet analysis.
Populations, whether they be viral particles, bio-chemicals, plants or animals, are subject to intrinsic and extrinsic sources of stochasticity. This stochasticity in conjunction with nonlinear interactions between individuals determines to what extinct populations are able to persist in the long-term. Understanding the precise nature of these interactive effects is a central issue in population biology from theoretical, empirical, and applied perspectives.
For the first part of this talk, I will discuss, briefly, the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable.
For the second part of the talk, I will discuss results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise can facilitate coexistence of competing species and how dispersal in stochastic environments can rescue locally extinction prone populations. Empirical demonstrations from Kansas prairies, acorn woodpecker populations, and microcosm experiments will be discussed.
How do cells execute decisions to cope with and survive under environmental conditions? My laboratory focuses on understanding how the dynamics of genetic circuits comprised of interactions between genes and proteins allow cells to govern decision-making. Interestingly, we find that stochastic fluctuations that are inherent to the biochemical reactions within genetic circuits can allow cells to cope with unpredictable environmental conditions. In addition, since cells have the ability to alter their own environment, the decisions at the single-cell level can depend on the context of the population. I will be presenting our attempts to understand these problems.
Evolution and co-evolution of ecological communities are stochastic processes often characterized by vastly different rates of reproduction and mutation and a coexistence of very large and very small sub-populations of competing species. This creates serious difficulties for accurate statistical modeling of evolutionary dynamics. In this talk, we review recent progress in this area and introduce a new exact algorithm for fast fully stochastic simulations of birth/death/mutation processes. It produces a significant speedup compared to the direct stochastic simulation algorithm in a typical case when the total population size is large and the mutation rates are much smaller than birth/death rates. We illustrate the performance of the algorithm on several representative examples: evolution on a smooth fitness landscape, Kauffmann's NK model, directed evolution of a regulatory gene network, and stochastic predator-prey system. (joint work with William Mather)
The SLFV process is a population model in which individuals live in a continuous space. Each of them also carries some heritable type or allele. We shall describe the long-term behaviour of this measure-valued process and that of the corresponding genealogical process of a sample of individuals in two cases : one that mimics the evolution of nearest-neighbour voter model (but in a spatial continuum), and one that allows some individuals to send offspring at very large distances. This is a joint work with Nathanaël Berestycki and Alison Etheridge.
The dynamics of a virus infection within its host is governed at its earliest stages by processes at the molecular and cellular scale. We are developing cell-culture measurements and computational models to better understand how these and other processes contribute to the early dynamics of virus growth and infection spread. As a model system we study vesicular stomatitis virus (VSV), a rabies-like RNA virus, growing on BHK cells. Established single-cycle measures of virus growth within infected cells provide population averages, which mask potential cell-to-cell variation. We used fluorescence-activated cell sorting to isolate single cells infected by single particles of a recombinant VSV expressing green fluorescent protein. Measured virus yields spanned a broad range from 8000 to below the detection limit of 10 infectious virus particles per cell. Viral genetic variation and host-cell cycle differences were unable to fully account for the observed yield differences. Computer simulations of the VSV dynamics within an infected cell suggest a potential role for stochastic gene expression to the observed yield variation. These studies are currently being extended to study the kinetics of virus production from individual infected cells.
Posters
Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. Two types of SSAs will be analysed. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. The connections between SSAs and the deterministic models (based on reaction-diffusion PDEs) will be presented. I will consider chemical reactions both at a surface and in the bulk. I will show how the "microscopic" parameters should be chosen to achieve the correct "macroscopic" reaction rate. This choice is found to depend on which SSA is used.
I will also present multiscale algorithms which use models with a different level of detail in different parts of the computational domain. Finally, I will discuss the derivation of macroscopic PDEs (collective behaviour) from individual based models of unicellular organisms (bacteria, amoeboid cells) and social insects (locusts).
We present a model of phototactic bacteria, that is, bacteria that move in the direction of light. These cyanobacteria, Synechocystis sp., form global patterns of fingers reaching towards the light. Our model is a novel multi-particle system that accounts for local social interactions between the cells, global forces due to the light and surface dependent movement. We also present results from a simulation of our model.
We consider a population in which each individual has a type in $\{1,\cdots,Q\}$ and a location in some metric space $E$. To each individual we assign a mass of $1/N$ and we call the total mass of individuals of type $i$ as the density of type $i$. The population evolves through type dependent reproduction, death and spatial migration. There are two time scales in our model separated by a factor of $N$. The spatial migration operates at the slower scale while the birth-death rates have fast density-dependent components that drive the density to an equilibrium. In such a setting we show that if we observe the process at the slow scale then in the limit $N \to \infty$, the overall population density is always at the equilibrium while the spatial variations are governed by an underlying Fleming-Viot process.
We then consider models in which there is immigration, dispersion and general spatial interaction at the slow scale. In these cases a Fleming-Viot process with mutation, recombination and selection will arise in the limit. To illustrate the usefulness of such a limit theorem, we use the results known for Fleming-Viot processes to understand the phenomenon of cell polarity.
The intrinsic stochasticity of gene expression can give rise to phenotypic heterogeneity in a population of genetically identical cells. Correspondingly, there is considerable interest in understanding how different molecular mechanisms impact the 'noise' in gene expression. Of particular interest are post-transcriptional regulatory mechanisms involving genes called small RNAs, which often act as central elements of regulatory pathways that control important processes such as development and cancer.
We propose and analyze general stochastic models of gene expression and, in certain limits, derive exact analytical expressions quantifying the noise in protein distributions. Focusing on specific regulatory mechanisms, we propose and analyze a general model for post-transcriptional regulation of stochastic gene expression. The results obtained provide new insights into the role of post-transcriptional mechanisms in controlling the noise in gene expression.
We present a new model for seed banks, where individuals may obtain their type from ancestors which have lived in the near as well as the very far past. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution by Kaj, Krone and Lascoux (2001) are boundary cases of our model. The mathematical methods are based on renewal theory as well as on a Gibbsian approach introduced by Hammond and Sheffield (2011) in a different context. This is joint work with Jochen Blath, Dario Spano and Adrian Gonzalez Casanova.
The ability to simulate molecular systems biology models enables new possibilities for exploring the implications of these models. I extend such systems biology models to compute candidate fitness correlates. These are properties of the system that emerge in simulations and that have some relevance for fitness as assessed by biological intuition. Computationally defined fitness correlates can be used for analyzing how many random changes in biochemical reaction rates might lead to an advantageous outcome for the system. Using the circadian clock of the green algae Ostreococcus as an example, I will show that surprisingly many random changes of biochemical reaction rates are advantageous from a mechanistic point of view.
Molecular noise restricts the ability of an individual cell to resolve input signals of different strengths and gather information about the external environment. Transmitting information through complex signaling networks with redundancies can overcome this limitation. We developed an integrative theoretical and experimental framework, based on the formalism of information theory, to quantitatively predict and measure the amount of information transduced by molecular and cellular networks. Analyzing tumor necrosis factor (TNF) signaling revealed that individual TNF signaling pathways transduce information sufficient for accurate binary decisions, and an upstream bottleneck limits the information gained via multiple pathways together. Negative feedback to this bottleneck could both alleviate and enhance its limiting effect, despite decreasing noise. Bottlenecks likewise constrain information attained by networks signaling through multiple genes or cells.
Work done in collaboration with Raymond Cheong, Alex Rhee, Chiaochun Joanne Wang, Ilya Nemenman, Andre Levchenko
Near critical catalyst-reactant branching processes with controlled immigration are considered. The reactant population evolves according to a branching process whose branching rate is proportional to the total mass of the catalyst. The bulk catalyst evolution is that of a classical continuous time branching process; in addition there is a specific form of immigration. Immigration takes place exactly when the catalyst population falls below a certain threshold, in which case the population is instantaneously replenished to the threshold. A diffusion limit theorem for the scaled processes is presented, in which the catalyst limit is described through a reflected diffusion, while the reactant limit is a diffusion with coefficients that are functions of both the reactant and the catalyst.
Stochastic averaging under fast catalyst dynamics is considered next. In the case where the catalyst evolves "much faster" than the reactant, a scaling limit, in which the reactant is described through a one dimensional SDE with coefficients depending on the invariant distribution of the reflected diffusion, is obtained.
Joint work with Amarjit Budhiraja.
A biotechnological application of artificial microRNAs (amiR) is the generation of plants resistant to virus infection. This resistance has proven to be highly effective and sequence-specific. However, before these transgenic plants can be deployed in the fields, it is important to evaluate the likelihood of emergence of resistance-breaking mutants. Two issues are of particular interest: (i) whether such mutants can arise in non-transgenic plants that may act as reservoirs; and (ii) whether suboptimal expression of the transgene, resulting in sub-inhibitory concentrations of the amiR would favor the emergence of escape mutants. To address the first issue, we experimentally evolved independent lineages of Turnip mosaic virus (TuMV, family Potyviridae) in fully susceptible wild-type Arabidopsis thaliana plants and then simulated the spill over of the evolving virus to the fully resistant A. thaliana transgenic plants. To address the second issue, the evolution phase took place in transgenic plants that expressed the amiR at sub-inhibitory concentrations. Our results show that TuMV populations replicating in susceptible hosts accumulated resistance-breaking alleles that resulted in overcoming the resistance of fully resistant plants. The rate at which resistance was broken was 7 times faster for TuMV populations that experienced sub-inhibitory concentrations of the antiviral amiR. To better understand the viral population dynamics taking place within each host we performed in silico stochastic simulations of the experiments. By using a Monte Carlo model simulating virus RNA evolution with digital genomes, we fitted the experimental data estimating relevant population genetics parameters for the two experimental setups. Together, our results contribute to the rational management of amiR-based antiviral resistance in plants.
Work done in collaboration with Lafforgue, G., Martínez, F., de la Iglesia, F., Niu, Q.W., Lin, S.S., Solé, R.V., Chua, N.H, Daròs, J.A., and Elena, S.F.
Branching actin network is the primary engine driving cell motility. At the front of a motile cell, inside the lamellipodium, is a dense mesh of actin filaments pushing the membrane forward. The structure of the mesh is believed to be largely regulated by three processes: branching of new filaments off of existing ones, capping of filament tips stopping filament growth, and filament growth. Filaments inside the lamellipodium have been observed to organize into a strict orientation pattern where filaments are angled approximately -35/35 degrees from the normal direction of the membrane. It has been previously hypothesized that the three processes above are sufficient to generate the unique orientation pattern.
We derive and analyze an integro-differential PDE for the angular density of branching actin network by incorporating the three constituent processes. Our analysis implies that there exist multiple equilibrium angular distributions, which strongly suggests additional process that regulates actin filament orientation.
Many of the biological networks inside cells can be thought of as transmitting information from the inputs (e.g., the concentrations of transcription factors or other signaling molecules) to their outputs (e.g., the expression levels of various genes). On the molecular level, the relatively small concentrations of the relevant molecules and the intrinsic randomness of chemical reactions provide sources of noise that set physical limits on this information transmission. Given these limits, not all networks perform equally well, and maximizing information transmission provides a optimization principle from which we might hope to derive the properties of real regulatory networks. Inspired by the precision of transmission of positional information in the early development of the fly embryo, I will discuss the properties of specific small networks that can transmit the maximum information. Concretely, I will show how the form of molecular noise drives predictions not just of the qualitative network topology but also the quantitative parameters for the input/output relations at the nodes of the network. I will show how the molecular details of regulation change the networks ability to transmit information.
To understand how existing organisms evolved to their present form, one can compare statistical features of genomes within a population to predictions of evolutionary models. Theoretical developments have produced a good understanding of how positive selection at a few sites affects genetic variation at linked neutral sites and of how strong selection at many sites affects variation at linked neutral sites. Recent sequence data from a variety of populations indicates that moderate selection acting on linked sites may be common and simulations show that it can have a significant impact on observed sequence variation. I will present a newly developed framework which allows us to understand the expected patterns of genetic variation when weak or moderate selection acts on many linked sites. I will show that in this limit the probability of allelic configurations cannot be described by any neutral model, indicating that it is possible to detect selection from patterns of sampled allelic diversity. I will then combine this analysis with the structured coalescence approach to trace the ancestry of individuals through the distribution of fitnesses within the population. I will show that selection alters the statistics of genealogies compared to neutral population with varying size, building a basis for a way to detect negative selection in sequence data.
The analysis of the dynamics of the biochemical regulatory interactions that govern the behavior of living organisms is a complicated task and an important challenge in understanding biological systems. Boolean networks have been increasingly used as a first step to model these regulatory systems, as they have been shown to display the essential aspects of the regulation dynamics without needing the kinetic details, just the logic of the interactions. One important question when using the Boolean framework concerns the introduction of noise and stochasticity in the dynamical behavior. Several different methods to take biological noise into account have been proposed, each of which usually involves different updating schemes. Currently it is unclear which of these methods, if any, is the optimal one.
In this work we use random Boolean networks to further characterize the advantages and disadvantages of the different updating schemes used for introducing noise in the dynamics. We focus on the effect that different levels of noise have in making two trajectories that start from the same initial condition differ from each other. To measure this we use the Hamming distance between the final states. We find that the noise-Hamming distance relationship varies with the connectivity of the network and between the different methods.
