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PubMed2006

Bisexual branching processes in a genetic context: the extinction problem for Y-linked genes.

In this paper the inheritance of a Y-linked gene with alleles R and r in a population with both females and males is modelled using a two-type bisexual branching process. It is assumed that the reproductive distribution associated with the R allele can differ from the associated with the r allele and that females prefer to mate with a male having the R allele rather than with a male with the r allele. Under these assumptions, we provide some conditions for the extinction and/or survival of both alleles in the population. These conditions depend on the magnitudes of the average number of females and males per mating unit. Moreover, the almost sure extinction of the r allele is independent of the behaviour of the R allele. On the other hand, the survival of the R allele with positive probability may depend strongly on the reproductive behaviour of the other allele. Theoretical results are illustrated by means of simulated examples and some open problems are proposed to the reader as conjectures.

Alleles↗
PubMed1992

A branching process model of gene amplification following chromosome breakage.

We have devised a mathematical model of gene amplification utilizing recent experimental observations concerning dihydrofolate reductase (DHFR) gene amplification in CHO cells. The mathematical model, based on a biological model which proposes that acentric elements are the initial intermediates in gene amplification, includes the following features: (1) initiation of amplification by chromosomal breakage to produce an acentric structure; (2) replication of acentric DNA, once per cell cycle; (3) dissociation of replicated acentric DNA; (4) unequal segregation of acentric DNA fragments to daughter cells at mitosis; (5) subsequent reintegration of acentric fragments into chromosomes. These processes are assumed to be independent for each element present in a cell at a given time. Thus, processes of unequal segregation and integration may occur in parallel, not necessarily in a unique sequence, and may be reiterated in one or multiple cell cycles. These events are described mathematically as a Galton-Watson branching process with denumerable infinity of object types. This mathematical model qualitatively and quantitatively reproduces the major elements of the dynamical behavior of DHFR genes observed experimentally. The agreement between the mathematical model and the experimental data lends credence to the biological model proposed by Windle et al. (1991), including the importance of chromosome breakage and subsequent gene deletion resulting from resection of the broken chromosome ends as initial events in gene amplification.

Animals↗
PubMed2025

Bayesian inference of fitness landscapes via tree-structured branching processes.

MOTIVATION: The complex dynamics of cancer evolution, driven by mutation and selection, underlies the molecular heterogeneity observed in tumors. The evolutionary histories of tumors of different patients can be encoded as mutation trees and reconstructed in high resolution from single-cell sequencing data, offering crucial insights for studying fitness effects of and epistasis among mutations. Existing models, however, either fail to separate mutation and selection or neglect the evolutionary histories encoded by the tumor phylogenetic trees. RESULTS: We introduce FiTree, a tree-structured multi-type branching process model with epistatic fitness parameterization and a Bayesian inference scheme to learn fitness landscapes from single-cell tumor mutation trees. Through simulations, we demonstrate that FiTree outperforms state-of-the-art methods in inferring the fitness landscape underlying tumor evolution. Applying FiTree to a single-cell acute myeloid leukemia dataset, we identify epistatic fitness effects consistent with known biological findings and quantify uncertainty in predicting future mutational events. The new model unifies probabilistic graphical models of cancer progression with population genetics, offering a principled framework for understanding tumor evolution and informing therapeutic strategies. AVAILABILITY AND IMPLEMENTATION: The Python package FiTree and the analysis workflows are available at https://github.com/cbg-ethz/FiTree.

Bayes Theorem↗
PubMed1996

A discrete-time, multi-type generational inheritance branching process model of cell proliferation.

Mammalian cell populations, such as tumors, may contain subpopulations differing in parameters such as cell lifetimes, even if the populations are derived from single cells. The mode of inheritance of cell lifetimes has previously been the subject of experimental and mathematical investigation. To obtain data on cell lifetimes over more cell generations then previously available, Axelrod et al. [Cell Prolif. 26:235-249(1988)] measured the number of cells in primary colonies and secondary colonies derived form the primary colonies. The experimental results indicated large variance of cells per colony and highly significant correlations between the numbers of cells in primary and secondary colonies. To mathematically model these results we derive, for previously uninvestigated multi-type Galton-Watson branching process models, the covariance of the cell counts in the primary and secondary colonies. As a result, we are able to successfully model the data with two subpopulations having differing proliferation rates, in which the proliferation rate of a daughter cell is primarily determined by the proliferation rate of its mother. Interestingly, simulations display a trade-off between high values of variances and correlation coefficients. The values obtained from experiment are located on the boundary of the region attainable by simulation.

Animals↗
PubMed2006

Probability of fixation under weak selection: a branching process unifying approach.

We link two-allele population models by Haldane and Fisher with Kimura's diffusion approximations of the Wright-Fisher model, by considering continuous-state branching (CB) processes which are either independent (model I) or conditioned to have constant sum (model II). Recent works by the author allow us to further include logistic density-dependence (model III), which is ubiquitous in ecology. In all models, each allele (mutant or resident) is then characterized by a triple demographic trait: intrinsic growth rate r, reproduction variance sigma and competition sensitivity c. Generally, the fixation probability u of the mutant depends on its initial proportion p, the total initial population size z, and the six demographic traits. Under weak selection, we can linearize u in all models thanks to the same master formula u = p + p(1 - p)[g(r)s(r) + g(sigma)s(sigma) + g(c)s(c)] + o(s(r),s(sigma),s(c), where s(r) = r' - r, s(sigma) = sigma-sigma' and s(c) = c - c' are selection coefficients, and g(r), g(sigma), g(c) are invasibility coefficients (' refers to the mutant traits), which are positive and do not depend on p. In particular, increased reproduction variance is always deleterious. We prove that in all three models g(sigma) = 1/sigma and g(r) = z/sigma for small initial population sizes z. In model II, g(r) = z/sigma for all z, and we display invasion isoclines of the 'mean vs variance' type. A slight departure from the isocline is shown to be more beneficial to alleles with low sigma than with high r. In model III, g(c) increases with z like ln(z)/c, and g(r)(z) converges to a finite limit L > K/sigma, where K = r/c is the carrying capacity. For r > 0 the growth invasibility is above z/sigma when z < K, and below z/sigma when z > K, showing that classical models I and II underestimate the fixation probabilities in growing populations, and overestimate them in declining populations.

Alleles↗
PubMed2003

Functional renormalization group for anisotropic depinning and relation to branching processes.

Using the functional renormalization group, we study the depinning of elastic objects in presence of anisotropy. We explicitly demonstrate how the Kardar-Parisi-Zhang (KPZ) term is always generated, even in the limit of vanishing velocity, except where excluded by symmetry. This mechanism has two steps. First a nonanalytic disorder-distribution is generated under renormalization beyond the Larkin length. This nonanalyticity then generates the KPZ term. We compute the beta function to one loop taking properly into account the nonanalyticity. This gives rise to additional terms, missed in earlier studies. A crucial question is whether the nonrenormalization of the KPZ coupling found at 1-loop order extends beyond the leading one. Using a Cole-Hopf-transformed theory we argue that it is indeed uncorrected to all orders. The resulting flow equations describe a variety of physical situations: We study manifolds in periodic disorder, relevant for charge density waves, as well as in nonperiodic disorder. Further the elasticity of the manifold can either be short range (SR) or long range (LR). A careful analysis of the flow yields several nontrivial fixed points. All these fixed points are transient since they possess one unstable direction towards a runaway flow, which leaves open the question of the upper critical dimension. The runaway flow is dominated by a Landau-ghost mode. For LR elasticity, relevant for contact line depinning, we show that there are two phases depending on the strength of the KPZ coupling. For SR elasticity, using the Cole-Hopf transformed theory we identify a nontrivial 3-dimensional subspace which is invariant to all orders and contains all above fixed points as well as the Landau mode. It belongs to a class of theories which describe branching and reaction-diffusion processes, of which some have been mapped onto directed percolation.

Journal Article↗
PubMed2001

Model for studying branching processes, multiplicity distributions, and non-Poissonian fluctuations in heavy-ion collisions.

A change is made in a statistical framework by introducing a set of variables called ancestral or stochastic. This leads to an underlying dynamics based on branching laws, lines of descent in an hierarchical topology, period doublings, cascades, and clans. Above a certain branching probability, a percolative feature suddenly appears. Power laws emerge and cascade points arise and end at golden mean (square root (5-1)/2.

Journal Article↗
PubMed2005

Dopamine cell morphology and glial cell hypertrophy and process branching in the nigrostriatal system after striatal 6-OHDA analyzed by specific sterological tools.

Morphological changes in the dopamine neurons and glial cells of the rat mid-brain ascending dopamine pathways were investigated after a partial lesion induced by unilateral striatal injection of a small dose of 6-hydroxydopamine (6-OHDA). Fourteen days after lesion, animals showed contralateral rotation induced by apomorphine injection. After behavioral analysis, fats were killed and their brains processed for the immunohistochemistry tyrosine hydroxylase (TH), a marker for dopamine cells, as well as glial fibrillary acidic protein (GFAP) and OX-42, markers for astrocyte and microglia, respectively. Stereological tools were employed in the quantifications. The volumes of the regions of the striatal TH immunoreactive disappearance, as well as the astroglial and microglial activation were several folds increased compared to control saline-injected rats. The optical disector detected decreases in the estimated total number of dopamine cells in the entire ipsilateral pars compacta of the substantia nigra (SNc) and the ventral tegmental area (VTA) as well as in the estimated total number of varicosity profiles in the entire ipsilateral neostriatum. The stereological tool rotator showed no changes either in the mean or in the histogram distribution of the cytoplasmic volume of the nigral and VTA dopamine cells of 6-OHDA lesioned rats. Increases in the estimated total number of GFAP positive astrocytes were found in the entire neostriatum bilaterally as well as in the ipsilateral entire SNc and VTA of 6-OHDA lesioned rats. The estimated total number of OX-42 immunoreactive microglial profiles was elevated only in the ipsilateral entire neostriatum of the lesioned rats. The rotator detected cytoplasmic hypertrophy in the astrocytes, and also a shift to the fight of the gaussian curves of the normal distribution of the logarithmic plotted values of the astroglial cell body volumes of the neostriatum bilaterally as well as in the ipsilateral SNc and VTA of the striatal 6-OHDA injected rats. Cytoplasmic hypertrophy of microglia, and also a shift to the right of the gaussian curves of the values of microglia cell body volumes were seen only in the ipsilateral neostriatum; however, the point intercepts revealed an increased amount of microglial processes in the ipsilateral SNc and VTA of the lesioned rats. Specific stereological methods can be applied on detection of regionally different forms of cellular astroglial and microglial reaction after a partial lesion of dopamine pathway.

Animals↗
PubMed1991

A branching-process model for the evolution of transposable elements incorporating selection.

We have formulated a very general mathematical model to analyze the evolution of transposable genetic elements in prokaryotic populations. Transposable genetic elements are DNA sequences able to replicate and insert copies of themselves at new locations in the genome. This work characterizes the equilibrium distribution of copy number under the influence of copy number-dependent selection, transposition and deletion. Our principal results concern the equilibrium distribution of copy number in response to various selective regimes. For particular transposition patterns (e.g., unregulated transposition or copy number-dependent transposition), equilibrium distributions are calculated numerically for a variety of specific selection patterns. Selection is quantified through specification of the expected number of offspring for individuals of each type, which is generally a non-increasing function of copy number, in accord with the usual evolutionary speculations.

Animals↗
PubMed1987

Inference for an age-dependent, multitype branching-process model of mast cells.

We consider an age-dependent, multitype model for the growth of mast cells in culture. After a colony of cells is established by an initiator type, the two possible types of cells are resting and proliferative. Using novel inferential procedures, we estimate the generation-time distribution and the offspring distribution of proliferative cells, and the waiting-time distribution of resting cells.

Animals↗