Bulletin of Mathematical Biology最新文献

Bayesian phylogenetic inference is powerful but computationally intensive. Researchers may find themselves with two phylogenetic posteriors on overlapping data sets and may wish to approximate a combined result without having to re-run potentially expensive Markov chains on the combined data set. This raises the question: given overlapping subsets of a set of taxa (e.g. species or virus samples), and given posterior distributions on phylogenetic tree topologies for each of these taxon sets, how can we optimize a probability distribution on phylogenetic tree topologies for the entire taxon set? In this paper we develop a variational approach to this problem and demonstrate its effectiveness. Specifically, we develop an algorithm to find a suitable support of the variational tree topology distribution on the entire taxon set, as well as a gradient-descent algorithm to minimize the divergence from the restrictions of the variational distribution to each of the given per-subset probability distributions, in an effort to approximate the posterior distribution on the entire taxon set.

In this paper, we study the problem of cost optimisation of individual-based institutional incentives (reward, punishment, and hybrid) for guaranteeing a certain minimal level of cooperative behaviour in a well-mixed, finite population. In this scheme, the individuals in the population interact via cooperation dilemmas (Donation Game or Public Goods Game) in which institutional reward is carried out only if cooperation is not abundant enough (i.e., the number of cooperators is below a threshold $1\le t\le N-1$ , where N is the population size); and similarly, institutional punishment is carried out only when defection is too abundant. We study analytically the cases $t=1$ for the reward incentive under the small mutation limit assumption and two different initial states, showing that the cost function is always non-decreasing. We derive the neutral drift and strong selection limits when the intensity of selection tends to zero and infinity, respectively. We numerically investigate the problem for other values of t and for population dynamics with arbitrary mutation rates.

During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker–Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker–Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker–Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker–Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics.

Macrophages in atherosclerotic lesions exhibit a spectrum of behaviours or phenotypes. The phenotypic distribution of monocyte-derived macrophages (MDMs), its correlation with MDM lipid content, and relation to blood lipoprotein densities are not well understood. Of particular interest is the balance between low density lipoproteins (LDL) and high density lipoproteins (HDL), which carry bad and good cholesterol respectively. To address these issues, we have developed a mathematical model for early atherosclerosis in which the MDM population is structured by phenotype and lipid content. The model admits a simpler, closed subsystem whose analysis shows how lesion composition becomes more pathological as the blood density of LDL increases relative to the HDL capacity. We use asymptotic analysis to derive a power-law relationship between MDM phenotype and lipid content at steady-state. This relationship enables us to understand why, for example, lipid-laden MDMs have a more inflammatory phenotype than lipid-poor MDMs when blood LDL lipid density greatly exceeds HDL capacity. We show further that the MDM phenotype distribution always attains a local maximum, while the lipid content distribution may be unimodal, adopt a quasi-uniform profile or decrease monotonically. Pathological lesions exhibit a local maximum in both the phenotype and lipid content MDM distributions, with the maximum at an inflammatory phenotype and near the lipid content capacity respectively. These results illustrate how macrophage heterogeneity arises in early atherosclerosis and provide a framework for future model validation through comparison with single-cell RNA sequencing data.

While mean-field models of cellular operations have identified dominant processes at the macroscopic scale, stochastic models may provide further insight into mechanisms at the molecular scale. In order to identify plausible stochastic models, quantitative comparisons between the models and the experimental data are required. The data for these systems have small sample sizes and time-evolving distributions. The aim of this study is to identify appropriate distance metrics for the quantitative comparison of stochastic model outputs and time-evolving stochastic measurements of a system. We identify distance metrics with features suitable for driving parameter inference, model comparison, and model validation, constrained by data from multiple experimental protocols. In this study, stochastic model outputs are compared to synthetic data across three scales: that of the data at the points the system is sampled during the time course of each type of experiment; a combined distance across the time course of each experiment; and a combined distance across all the experiments. Two broad categories of comparators at each point were considered, based on the empirical cumulative distribution function (ECDF) of the data and of the model outputs: discrete based measures such as the Kolmogorov-Smirnov distance, and integrated measures such as the Wasserstein-1 distance between the ECDFs. It was found that the discrete based measures were highly sensitive to parameter changes near the synthetic data parameters, but were largely insensitive otherwise, whereas the integrated distances had smoother transitions as the parameters approached the true values. The integrated measures were also found to be robust to noise added to the synthetic data, replicating experimental error. The characteristics of the identified distances provides the basis for the design of an algorithm suitable for fitting stochastic models to real world stochastic data.

When hybridization or other forms of lateral gene transfer have occurred, evolutionary relationships of species are better represented by phylogenetic networks than by trees. While inference of such networks remains challenging, several recently proposed methods are based on quartet concordance factors-the probabilities that a tree relating a gene sampled from the species displays the possible 4-taxon relationships. Building on earlier results, we investigate what level-1 network features are identifiable from concordance factors under the network multispecies coalescent model. We obtain results on both topological features of the network, and numerical parameters, uncovering a number of failures of identifiability related to 3-cycles in the network. Addressing these identifiability issues is essential for designing statistically consistent inference methods.

Fred Brauer was an eminent mathematician who studied dynamical systems, especially differential equations. He made many contributions to mathematical epidemiology, a field that is strongly connected to data, but he always chose to avoid data analysis. Nevertheless, he recognized that fitting models to data is usually necessary when attempting to apply infectious disease transmission models to real public health problems. He was curious to know how one goes about fitting dynamical models to data, and why it can be hard. Initially in response to Fred's questions, we developed a user-friendly R package, fitode, that facilitates fitting ordinary differential equations to observed time series. Here, we use this package to provide a brief tutorial introduction to fitting compartmental epidemic models to a single observed time series. We assume that, like Fred, the reader is familiar with dynamical systems from a mathematical perspective, but has limited experience with statistical methodology or optimization techniques.

Fibrous dysplasia (FD) is a mosaic non-inheritable genetic disorder of the skeleton in which normal bone is replaced by structurally unsound fibro-osseous tissue. There is no curative treatment for FD, partly because its pathophysiology is not yet fully known. We present a simple mathematical model of the disease incorporating its basic known biology, to gain insight on the dynamics of the involved bone-cell populations, and shed light on its pathophysiology. We develop an analytical study of the model and study its basic properties. The existence and stability of steady states are studied, an analysis of sensitivity on the model parameters is done, and different numerical simulations provide findings in agreement with the analytical results. We discuss the model dynamics match with known facts on the disease, and how some open questions could be addressed using the model.

Human immunodeficiency virus (HIV) infects CD4+ cells and causes progressive immune function failure, and CD8+ cells lyse infected CD4+ cell via recognising peptide presented by human leukocyte antigens (HLA). Variations in HLA allele lead to observed different HIV infection outcomes. Within-host HIV dynamics involves virus replication within infected cells and lysing of infected cells by CD8+ cells, but how variations in HLA alleles determine different infection outcomes was far from clear. Here, we used mathematical modelling and parameter inference with a new analysis of published virus inhibition assay data to estimate CD8+ cell lysing efficiency, and found that lysing efficiency fall in the gap between low bound (0.1-0.2 day^-1 (Elemans et al. in PLoS Comput Biol 8(2):e1002381, 2012)) and upper boundary (6.5-8.4 day^-1 (Wick et al. in J Virol 79(21):13579-13586, 2005)). Our outcomes indicate that both lysing efficiency and viral inoculum size jointly determine observed different infection outcomes. Low lysing rate associated with non-protective HLA alleles leads to monostable viral kinetic to high viral titre and oscillatory viral kinetics. High lysing rate associated with protective HLA alleles leads monostable viral kinetic to low viral titre and bistable viral kinetics; at a specific interval of CD8+ cell counts, small viral inoculum sizes are inhibited but not large viral inoculum sizes remain infectious. Further, with CD8+ cell recruitment, HIV kinetics always exhibit oscillatory kinetics, but lysing rate is negatively correlated with range of CD8+ cell count. Our finding highlights role of HLA allele determining different infection outcomes, thereby providing a potential mechanistic explanation for observed good and bad HIV infection outcomes induced by protective HLA allele.

Maximum likelihood estimation is among the most widely-used methods for inferring phylogenetic trees from sequence data. This paper solves the problem of computing solutions to the maximum likelihood problem for 3-leaf trees under the 2-state symmetric mutation model (CFN model). Our main result is a closed-form solution to the maximum likelihood problem for unrooted 3-leaf trees, given generic data; this result characterizes all of the ways that a maximum likelihood estimate can fail to exist for generic data and provides theoretical validation for predictions made in Parks and Goldman (Syst Biol 63(5):798-811, 2014). Our proof makes use of both classical tools for studying group-based phylogenetic models such as Hadamard conjugation and reparameterization in terms of Fourier coordinates, as well as more recent results concerning the semi-algebraic constraints of the CFN model. To be able to put these into practice, we also give a complete characterization to test genericity.