Multitype Branching Processes Research Articles

Role of epistasis on the fixation probability of a non-mutator in an adapted asexual population

The mutation rate of a well adapted population is prone to reduction so as to have a lower mutational load. We aim to understand the role of epistatic interactions between the fitness affecting mutations in this process. Using a multitype branching process, the fixation probability of a single non-mutator emerging in a large asexual mutator population is analytically calculated here. The mutator population undergoes deleterious mutations at constant, but at a much higher rate than that of the non-mutator. We find that antagonistic epistasis lowers the chances of mutation rate reduction, while synergistic epistasis enhances it. Below a critical value of epistasis, the fixation probability behaves non-monotonically with variation in the mutation rate of the background population. Moreover, the variation of this critical value of the epistasis parameter with the strength of the mutator is discussed in the appendix. For synergistic epistasis, when selection is varied, the fixation probability reduces overall, with damped oscillations.

Individual‐based integral projection models: the role of size‐structure on extinction risk and establishment success

Summary Matrix models or integral projection models (IPMs) are commonly used to study the dynamics of structured populations, where discrete or continuous traits influence survival, growth or reproduction. When a population's size is small, as is often the case for threatened species or potentially invasive species arriving in novel habitats, extinction risk may be substantial due to demographic stochasticity. Branching processes, which are individual‐based counterparts to matrix models and IPMs, allow one to quantify these risks of extinction. For discretely structured populations, the theory of multitype branching processes provides analytic methods to compute how extinction risk changes over time and how it depends on the size and composition of the population. Building on prior work on continuous‐state branching processes, we extend these analytic methods to individual‐based models accounting for any mixture of discrete and continuous population structure. The individual‐based IPMs are defined by probabilistic update rules at the level of the individual which determine how each individual with a given trait value dies, changes trait values (e.g. grows in size) or produces individuals with the same or other trait values. We show that probabilities of extinction can be analytically determined by probability‐generating functionals associated with the individual‐based IPMs. In particular, we present analytical expressions for how extinction probabilities change over time and depend on the initial abundance and trait distribution of the population. We illustrate how to numerically implement these methods using data from the short‐lived desert shrub species Cryptantha flava and provide a more general discussion of how to implement these methods to other data sets including those involving fluctuating environmental conditions. As most IPM studies have the necessary data to parameterize individual‐based IPMs, these methods provide a computationally efficient means to explore how continuously structured populations differing in their evolutionary history and environmental context may differ in their vulnerability to extinction or ability to colonize new habitats.

Lyapunov Exponents for Branching Processes in a Random Environment: The Effect of Information

We consider multitype branching processes evolving in a Markovian random environment. To determine whether or not the branching process becomes extinct almost surely is akin to computing the maximal Lyapunov exponent of a sequence of random matrices, which is a notoriously difficult problem. We define Markov chains associated to the branching process, and we construct bounds for the Lyapunov exponent. The bounds are obtained by adding or by removing information: to add information results in a lower bound, to remove information results in an upper bound, and we show that adding less information improves the lower bound. We give a few illustrative examples and we observe that the upper bound is generally more accurate than the lower bounds.

Evolutionary escape on complex genotype–phenotype networks

We study the problem of evolutionary escape that is the process whereby a population under sudden changes in the selective pressures acting upon it try to evade extinction by evolving from previously well-adapted phenotypes to those that are favoured by the new selective pressure. We perform a comparative analysis between results obtained by modelling genotype space as a regular hypercube (H-graphs), which is the scenario considered in previous work on the subject, to those corresponding to a complex genotype–phenotype network (B-graphs). In order to analyse the properties of the escape process on both these graphs, we apply a general theory based on multi-type branching processes to compute the evolutionary dynamics and probability of escape. We show that the distribution of distances between phenotypes in B-graphs exhibits a much larger degree of heterogeneity than in H-graphs. This property, one of the main structural differences between both types of graphs, causes heterogeneous behaviour in all results associated to the escape problem. We further show that, due to the heterogeneity characterising escape on B-graphs, escape probability can be underestimated by assuming a regular hypercube genotype network, even if we compare phenotypes at the same distance in H-graphs. Similarly, it appears that the complex structure of B-graphs slows down the rate of escape.

Fixation probability of rare nonmutator and evolution of mutation rates.

Although mutations drive the evolutionary process, the rates at which the mutations occur are themselves subject to evolutionary forces. Our purpose here is to understand the role of selection and random genetic drift in the evolution of mutation rates, and we address this question in asexual populations at mutation-selection equilibrium neglecting selective sweeps. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large asexual population of mutators and find that a nonmutator is more likely to fix when the deleterious mutation rate of the mutator population is high. Compensatory mutations in the mutator population are found to decrease the fixation probability of a nonmutator when the selection coefficient is large. But, surprisingly, the fixation probability changes nonmonotonically with increasing compensatory mutation rate when the selection is mild. Using these results for the fixation probability and a drift-barrier argument, we find a novel relationship between the mutation rates and the population size. We also discuss the time to fix the nonmutator in an adapted population of asexual mutators, and compare our results with experiments.

Estimating the Propagation of Interdependent Cascading Outages with Multi-Type Branching Processes

In this paper, the multi-type branching process is applied to describe the statistics and interdependencies of line outages, the load shed, and isolated buses. The offspring mean matrix of the multi-type branching process is estimated by the Expectation Maximization (EM) algorithm and can quantify the extent of outage propagation. The joint distribution of two types of outages is estimated by the multi-type branching process via the Lagrange-Good inversion. The proposed model is tested with data generated by the AC OPA cascading simulations on the IEEE 118-bus system. The largest eigenvalues of the offspring mean matrix indicate that the system is closer to criticality when considering the interdependence of different types of outages. Compared with empirically estimating the joint distribution of the total outages, good estimate is obtained by using the multitype branching process with a much smaller number of cascades, thus greatly improving the efficiency. It is shown that the multitype branching process can effectively predict the distribution of the load shed and isolated buses and their conditional largest possible total outages even when there are no data of them.

Asymptotic behavior of critical indecomposable multi-type branching processes with immigration

In this paper the asymptotic behavior of a critical multi-type branching process with immigration is described when the offspring mean matrix is irreducible, in other words, when the process is indecomposable. It is proved that sequences of appropriately scaled random step functions formed from periodic subsequences of a critical indecomposable multi-type branching process with immigration converge weakly towards a process supported by a ray determined by the Perron vector of the offspring mean matrix. The types can be partitioned into nonempty mutually disjoint subsets (according to communication of types) such that the coordinate processes belonging to the same subset are multiples of the same squared Bessel process, and the coordinate processes belonging to different subsets are independent.

Beyond the Q-process: various ways of conditioning the multitype Galton-Watson process

Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated Q-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the Q-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained Q-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype branching process on having an infinite total progeny leads to a process presenting the features of a Q-process. It does not however coincide with the original associated Q-process, except in the critical regime.

Contagions in random networks with overlapping communities

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

Contagions in random networks with overlapping communities

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

Abstract B2-50: Stochastic population genetics modeling of the evolution of the myelodysplastic syndrome

Abstract We are developing new models to understand the evolution of Myelodysplastic Syndromes (MDS), a group of pre-leukemic blood diseases, which are increasing in incidence as the population is aging and which remain difficult to treat. In the early stage of MDS, exaggerated apoptosis of myeloid blood cells occurs, but as the disease evolves, survival of undifferentiated cells dominates. As a model we study progression to MDS/AML from severe congenital neutropenia (SCN), an inherited bone marrow failure with known germ-line mutations, e.g. ELANE which encodes the neutrophil elastase. Administration of granulocyte colony-stimulating factor (GCSF) improves the neutropenia but is associated with transformation to MDS and acute myeloid leukemia (AML), defined by greater than 20% blasts in the marrow. Of note, ~ 70% of SCN patients who progress to MDS acquire a truncating D715 mutation in the GCSF Receptor. We previously developed a model of the SCN -> MDS -> AML sequence [1], which however did not include the MDS stage. We hypothesize that the propagation of the D715 mutant in MDS secondary to SCN follows a newly developed Moran model with directional selection and co-localization. Selection is caused by alterations in the STAT/SOCS feedback loop with the truncated GCSF receptor, leading to changes in proliferation and differentiation [2]. Usually, mathematical models assume that bone marrow is a well-mixed environment and the normal and malignant precursors freely interact. However, these subpopulations occupy distinct niches [3] and their competition is limited by physical separation, which is a factor in our variant of the Moran model. We estimate the selective advantage of mutant cells based on competitive repopulation assay and flow cytometry cell-cycle distribution and find the time at onset of MDS consistent with the published median of 13 years [1]. We built an ODE model of the STAT/SOCS circuit, and investigated its sensitivity to reduced binding of JAK/STAT complexes to the D715 mutant receptor, even that the circuit exhibits extremely strong stabilizing properties. A link to cell cycle characteristics via distributions of cell-cycle-phase dependent activated STATs suggests connection with differentiation and proliferation. Further, to investigate a fuller model in which a number of driver mutations may occur at a different order, we developed a new version of the infinite-allele branching process of Griffiths and Pakes [4, 5]. Mathematical modeling of SCN->MDS -> AML will lead to a greater understanding of homeostatic and aberrant myelopoiesis as well as designing better predictive biomarkers for malignant clonal evolution.

How does the core sit inside the mantle?

The k-core, defined as the largest subgraph of minimum degree k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [Journal of Combinatorial Theory, Series B 67 (1996) 111–151] determined the threshold dk for the appearance of an extensive k-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the k-core is “embedded” into the random graph for any k≥3 and any fixed average degree d=np>dk. This generalises prior results on, e.g., the internal structure of the k-core.

Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata

A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multitype branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of termination probabilities, and computing these probabilities in turn boils down to computing the least fixed point (LFP) solution of a corresponding monotone polynomial system (MPS) of equations, denoted x = P ( x ). It was shown in Etessami and Yannakakis [2009] that a decomposed variant of Newton’s method converges monotonically to the LFP solution for any MPS that has a nonnegative solution. Subsequently, Esparza et al. [2010] obtained upper bounds on the convergence rate of Newton’s method for certain classes of MPSs. More recently, better upper bounds have been obtained for special classes of MPSs [Etessami et al. 2010, 2012]. However, prior to this article, for arbitrary (not necessarily strongly connected) MPSs, no upper bounds at all were known on the convergence rate of Newton’s method as a function of the encoding size |P| of the input MPS, x = P ( x ). In this article, we provide worst-case upper bounds, as a function of both the input encoding size |P|, and ε > 0, on the number of iterations required for decomposed Newton’s method (even with rounding) to converge to within additive error ε > 0 of q*, for an arbitrary MPS with LFP solution q*. Our upper bounds are essentially optimal in terms of several important parameters of the problem. Using our upper bounds, and building on prior work, we obtain the first P-time algorithm (in the standard Turing model of computation) for quantitative model checking, to within arbitrary desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) ω -regular or LTL property.

A multitype infinite-allele branching process with applications to cancer evolution

We extend the infinite-allele simple branching process of Griffiths and Pakes (1988) allowing the offspring to change types and labels. The model is developed and limit theorems are given for the growth of the number of labels of a specific type. We also discuss the asymptotics of the frequency spectrum. Finally, we present an application of the model's use in tumorigenesis.

A multitype infinite-allele branching process with applications to cancer evolution

We extend the infinite-allele simple branching process of Griffiths and Pakes (1988) allowing the offspring to change types and labels. The model is developed and limit theorems are given for the growth of the number of labels of a specific type. We also discuss the asymptotics of the frequency spectrum. Finally, we present an application of the model's use in tumorigenesis.

Likelihood-based inference for discretely observed birth-death-shift processes, with applications to evolution of mobile genetic elements.

Continuous-time birth-death-shift (BDS) processes are frequently used in stochastic modeling, with many applications in ecology and epidemiology. In particular, such processes can model evolutionary dynamics of transposable elements-important genetic markers in molecular epidemiology. Estimation of the effects of individual covariates on the birth, death, and shift rates of the process can be accomplished by analyzing patient data, but inferring these rates in a discretely and unevenly observed setting presents computational challenges. We propose a multi-type branching process approximation to BDS processes and develop a corresponding expectation maximization algorithm, where we use spectral techniques to reduce calculation of expected sufficient statistics to low-dimensional integration. These techniques yield an efficient and robust optimization routine for inferring the rates of the BDS process, and apply broadly to multi-type branching processes whose rates can depend on many covariates. After rigorously testing our methodology in simulation studies, we apply our method to study intrapatient time evolution of IS6110 transposable element, a genetic marker frequently used during estimation of epidemiological clusters of Mycobacterium tuberculosis infections.

A stochastic microscopic model for the dynamics of antigenic variation

We present a novel model that describes the within-host evolutionary dynamics of parasites undergoing antigenic variation. The approach uses a multi-type branching process with two types of entities defined according to their relationship with the immune system: clans of resistant parasitic cells (i.e. groups of cells sharing the same antigen not yet recognized by the immune system) that may become sensitive, and individual sensitive cells that can acquire a new resistance thus giving rise to the emergence of a new clan. The simplicity of the model allows analytical treatment to determine the subcritical and supercritical regimes in the space of parameters. By incorporating a density-dependent mechanism the model is able to capture additional relevant features observed in experimental data, such as the characteristic parasitemia waves. In summary our approach provides a new general framework to address the dynamics of antigenic variation which can be easily adapted to cope with broader and more complex situations.

Surviving evolutionary escape on complex genotype–phenotype networks

We study the problem of evolutionary escape and survival of cell populations with a genotype-phenotype structure. We refer to evolutionary escape as the process where a cell of a given ill-adapted population to reach a well-adapted phenotype. Similarly, survival refers to the dynamics of the population once the escape phenotype has been reached. The aim of this paper is to analyse the influence of topological properties associated to robustness and evolvability on the probability of escape and on the probability of survival. In order to explore these issues, we formulate a population dynamics model, consisting of a multi-type time-continuous branching process, where types are associated to genotypes and their birth and death probabilities depend on the associated phenotype (non-escape or escape). We exploit the separation of time scales introduced by the the difference in reproductive ratios between the ill-adapted phenotypes and the escape phenotype. Two dynamical regimes emerge: a fast-decaying regime associated to the escape process itself, and a slow regime which corresponds to the survival dynamics of the population once the escape phenotype has been reached. We exploit this separation of time scales to analyse the topological factors which determine escape and survival probabilities. We show that, while the escape probability depends on the degree of escape phenotype, the probability of survival is essentially determined by its robustness, measured in terms of a weighted clustering coefficient.