Bayesian Finite Markov Mixture Model for Temporal Multi‐Tissue Polygenic Patterns

Finite mixture models can provide the insights about behavioral patterns as a source of heterogeneity of the various dynamics of time course gene expression data by reducing the high dimensionality and making clear the major components of the underlying structure of the data in terms of the unobservable latent variables. The latent structure of the dynamic transition process of gene expression changes over time can be represented by Markov processes. This paper addresses key problems in the analysis of large gene expression data sets that describe systemic temporal response cascades and dynamic changes to therapeutic doses in multiple tissues, such as liver, skeletal muscle, and kidney from the same animals. Bayesian Finite Markov Mixture Model with a Dirichlet Prior is developed for the identifications of differentially expressed time related genes and dynamic clusters. Deviance information criterion is applied to determine the number of components for model comparisons and selections. The proposed Bayesian models are applied to multiple tissue polygenetic temporal gene expression data and compared to a Bayesian model‐based clustering method, named CAGED. Results show that our proposed Bayesian Finite Markov Mixture model can well capture the dynamic changes and patterns for irregular complex temporal data (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolutionary dynamics of continuous strategy games on graphs and social networks under weak selection

Understanding the emergence of cooperation among selfish individuals has been a long-standing puzzle, which has been studied by a variety of game models. Most previous studies presumed that interactions between individuals are discrete, but it seems unrealistic in real systems. Recently, there are increasing interests in studying game models with a continuous strategy space. Existing research work on continuous strategy games mainly focuses on well-mixed populations. Especially, little theoretical work has been conducted on their evolutionary dynamics in a structured population. In the previous work (Zhong et al., BioSystems, 2012), we showed that under strong selection, continuous and discrete strategies have significantly different equilibrium and game dynamics in spatially structured populations. In this paper, we further study evolutionary dynamics of continuous strategy games under weak selection in structured populations. By using the fixation probability based stochastic dynamics, we derive exact conditions of natural selection favoring cooperation for the death–birth updating scheme. We also present a network gain decomposition of the game equilibrium, which might provide a new view of the network reciprocity in a quantitative way. Finally, we make a detailed comparison between games using discrete and continuous strategies. As compared to the former, we find that for the latter (i) the same selection conditions are derived for the general 2 × 2 game; especially, the rule b/c > k in a simplified Prisoner's Dilemma is valid as well; however, (ii) for a coordination game, interestingly, the risk-dominant strategy is disfavored. Numerical simulations have also been conducted to validate our results.     

A game theoretical model of deforestation in human-environment relationships

We studied a two-person game regarding deforestation in human-environment relationships. Each landowner manages a single land parcel where the state of land-use is forested, agricultural, or abandoned. The landowner has two strategies available: forest conservation and deforestation. The choice of deforestation provides a high return to the landowner, but it degrades the forest ecosystem services produced on a neighboring land parcel managed by a different landowner. Given spatial interactions between the two landowners, each landowner decides which strategy to choose by comparing the expected discounted utility of each strategy. Expected discounted utility is determined by taking into account the current and future utilities to be received, according to the state transition on the two land parcels. The state transition is described by a Markov chain that incorporates a landowner's choice about whether to deforest and the dynamics of agricultural abandonment and forest regeneration. By considering a stationary distribution of the Markov chain for land-use transitions, we derive explicit conditions for Nash equilibrium. We found that a slow regeneration of forests favors mutual cooperation (forest conservation). As the forest regenerates faster, mutual cooperation transforms to double Nash equilibria (mutual cooperation and mutual defection), and finally mutual defection (deforestation) leads to a unique Nash equilibrium. Two different types of social dilemma emerge in our deforestation game. The stag-hunt dilemma is most likely to occur under an unsustainable resource supply, where forest regenerates extremely slowly but agricultural abandonment happens quite rapidly. In contrast, the prisoner's dilemma is likely under a persistent or circulating supply of resources, where forest regenerates rapidly and agricultural abandonment occurs slowly or rapidly. These results show how humans and the environment mutually shape the dilemma structure in forest management, implying that solutions to dilemmas depend on environmental properties.     

Predicting trait values and measuring selection in complex life histories: reproductive allocation decisions in Soay sheep

Accurate prediction of life history phenomena and characterisation of selection in free-living animal populations are fundamental goals in evolutionary ecology. In density regulated, structured populations, where individual state influences fate, simple and widely used approaches based on individual lifetime measures of fitness are difficult to justify. We combine recently developed structured population modelling tools with ideas from modern evolutionary game theory (adaptive dynamics) to understand selection on allocation of female reproductive effort to singletons or twins in a size-structured population of feral sheep. In marked contrast to the classical selection analyses, our model-based approach predicts that the female allocation strategy is under negligible directional selection. These differences arise because classical selection analysis ignores components of offspring fitness and fails to consider selection over the complete life cycle.     

New Routes to Phylogeography: A Bayesian Structured Coalescent Approximation

Phylogeographic methods aim to infer migration trends and the history of sampled lineages from genetic data. Applications of phylogeography are broad, and in the context of pathogens include the reconstruction of transmission histories and the origin and emergence of outbreaks. Phylogeographic inference based on bottom-up population genetics models is computationally expensive, and as a result faster alternatives based on the evolution of discrete traits have become popular. In this paper, we show that inference of migration rates and root locations based on discrete trait models is extremely unreliable and sensitive to biased sampling. To address this problem, we introduce BASTA (BAyesian STructured coalescent Approximation), a new approach implemented in BEAST2 that combines the accuracy of methods based on the structured coalescent with the computational efficiency required to handle more than just few populations. We illustrate the potentially severe implications of poor model choice for phylogeographic analyses by investigating the zoonotic transmission of Ebola virus. Whereas the structured coalescent analysis correctly infers that successive human Ebola outbreaks have been seeded by a large unsampled non-human reservoir population, the discrete trait analysis implausibly concludes that undetected human-to-human transmission has allowed the virus to persist over the past four decades. As genomics takes on an increasingly prominent role informing the control and prevention of infectious diseases, it will be vital that phylogeographic inference provides robust insights into transmission history.     

Markov and Semi‐Markov Switching Linear Mixed Models Used to Identify Forest Tree Growth Components

Summary Tree growth is assumed to be mainly the result of three components: (i) an endogenous component assumed to be structured as a succession of roughly stationary phases separated by marked change points that are asynchronous among individuals, (ii) a time‐varying environmental component assumed to take the form of synchronous fluctuations among individuals, and (iii) an individual component corresponding mainly to the local environment of each tree. To identify and characterize these three components, we propose to use semi‐Markov switching linear mixed models, i.e., models that combine linear mixed models in a semi‐Markovian manner. The underlying semi‐Markov chain represents the succession of growth phases and their lengths (endogenous component) whereas the linear mixed models attached to each state of the underlying semi‐Markov chain represent—in the corresponding growth phase—both the influence of time‐varying climatic covariates (environmental component) as fixed effects, and interindividual heterogeneity (individual component) as random effects. In this article, we address the estimation of Markov and semi‐Markov switching linear mixed models in a general framework. We propose a Monte Carlo expectation–maximization like algorithm whose iterations decompose into three steps: (i) sampling of state sequences given random effects, (ii) prediction of random effects given state sequences, and (iii) maximization. The proposed statistical modeling approach is illustrated by the analysis of successive annual shoots along Corsican pine trunks influenced by climatic covariates.     

A new method for inferring hidden markov models from noisy time sequences

We present a new method for inferring hidden Markov models from noisy time sequences without the necessity of assuming a model architecture, thus allowing for the detection of degenerate states. This is based on the statistical prediction techniques developed by Crutchfield et al. and generates so called causal state models, equivalent in structure to hidden Markov models. The new method is applicable to any continuous data which clusters around discrete values and exhibits multiple transitions between these values such as tethered particle motion data or Fluorescence Resonance Energy Transfer (FRET) spectra. The algorithms developed have been shown to perform well on simulated data, demonstrating the ability to recover the model used to generate the data under high noise, sparse data conditions and the ability to infer the existence of degenerate states. They have also been applied to new experimental FRET data of Holliday Junction dynamics, extracting the expected two state model and providing values for the transition rates in good agreement with previous results and with results obtained using existing maximum likelihood based methods. The method differs markedly from previous Markov-model reconstructions in being able to uncover truly hidden states.     

Stochastic modelling of a single ion channel: an alternating renewal approach with application to limited time resolution

Stochastic models of ion channels have been based largely on Markov theory where individual states and transition rates must be specified, and sojourn-time densities for each state are constrained to be exponential. This study presents an approach based on random-sum methods and alternating-renewal theory, allowing individual states to be grouped into classes provided the successive sojourn times in a given class are independent and identically distributed. Under these conditions Markov models form a special case. The utility of the approach is illustrated by considering the effects of limited time resolution (modelled by using a discrete detection limit, xi) on the properties of observable events, with emphasis on the observed open-time (xi-open-time). The cumulants and Laplace transform for a xi-open-time are derived for a range of Markov and non-Markov models; several useful approximations to the xi-open-time density function are presented. Numerical studies show that the effects of limited time resolution can be extreme, and also highlight the relative importance of the various model parameters. The theory could form a basis for future inferential studies in which parameter estimation takes account of limited time resolution in single channel records. Appendixes include relevant results concerning random sums and a discussion of the role of exponential distributions in Markov models.     

Sequential patterns of sex allocation in simultaneous hermaphrodites: do we need models that specifically incorporate this complexity?

Theoretical and empirical studies of sex allocation usually treat sequential and simultaneous hermaphroditism as distinct and disparate forms of allocation. However, the sexual patterns of numerous species have both sequential (e.g., size-based) and simultaneous components. In most cases, we have drawn from sex allocation theory developed for sequential hermaphrodites to explain ontogenetic changes in allocation and from theory developed for simultaneous hermaphrodites to explain the remaining aspects of these sexual patterns rather than develop a more integrated theory. Here I present the evolutionary stable solution (ESS) to a dynamic statevariable model that explicitly combines the effects of size and simultaneous allocation to male and female function in a dynamic game. The model structure and initial parameter values are based on the sexual pattern of the blue-banded goby, Lythrypnus dalli, a simultaneous hermaphrodite. I then compare the natural patterns of sex allocation in L. dalli with the predictions of the model and with those of a dynamic version of the size advantage model. The integrated model predicted variation in allocation, sex-specific size distributions, and seasonal sex ratio better than the sequential hermaphroditism model did. Indeed, the sequential model, using L. dalli parameter values, predicts a dioecious rather than sequentially hermaphroditic allocation pattern. The comparison of these two models illustrates the disadvantage of drawing from two bodies of theory without a formal integrated framework. Furthermore, the comparison focuses attention on the role of costs of reallocation in the evolution of mixed (or intermediate) sexual patterns.     

Bringing consistency to simulation of population models--Poisson simulation as a bridge between micro and macro simulation

Population models concern collections of discrete entities such as atoms, cells, humans, animals, etc., where the focus is on the number of entities in a population. Because of the complexity of such models, simulation is usually needed to reproduce their complete dynamic and stochastic behaviour. Two main types of simulation models are used for different purposes, namely micro-simulation models, where each individual is described with its particular attributes and behaviour, and macro-simulation models based on stochastic differential equations, where the population is described in aggregated terms by the number of individuals in different states. Consistency between micro- and macro-models is a crucial but often neglected aspect. This paper demonstrates how the Poisson Simulation technique can be used to produce a population macro-model consistent with the corresponding micro-model. This is accomplished by defining Poisson Simulation in strictly mathematical terms as a series of Poisson processes that generate sequences of Poisson distributions with dynamically varying parameters. The method can be applied to any population model. It provides the unique stochastic and dynamic macro-model consistent with a correct micro-model. The paper also presents a general macro form for stochastic and dynamic population models. In an appendix Poisson Simulation is compared with Markov Simulation showing a number of advantages. Especially aggregation into state variables and aggregation of many events per time-step makes Poisson Simulation orders of magnitude faster than Markov Simulation. Furthermore, you can build and execute much larger and more complicated models with Poisson Simulation than is possible with the Markov approach.     

The dynamical attainability of ESS in evolutionary games

In this paper, the attainability of ESS of the evolutionary game among n players under the frequency-independent selection is studied by means of a mathematical model describing the dynamical development and a concept of stability (strongly determined stability). It is assumed that natural selection and small mutations cause the phenotype to change gradually in the direction of fitness increasing. It is shown that (1) the ESS solution is not always evolutionarily attainable in the evolutionary dynamics, (2) in the game where the interaction between two species is completely competitive, the Nash solution is always attainable, and (3) one of two species may attain the state of minimum fitness as a result of evolution. The attainability of ESS is also examined in two game models on the sex ratio of wasps and aphids in light of our criterion of the attainability of ESS.     

Stability in an evolutionary game

A behavior or strategy which is evolutionarily stable must be both optimal and stable. The strategy must be optimal in that it maximizes the expected fitness of all the individuals using it. In addition, the strategy must be resistant to invasion by a mutant. The difference between the Nash solution of game theory and the ESS used in ecology is that the Nash solution only satisfies an optimality criterion and not an evolutionary stability criterion. We extend the ESS definition of Maynard Smith and Price so that it can be applied directly to two-strategy evolutionary games. The concept of a balanced game is introduced, and necessary conditions are derived which are similar to the Nash necessary conditions. The balanced game necessary conditions may be used for direct calculation of ESS candidates. These results are used to examine the optimal flowering time of an annual plant experiencing competition from neighboring plants. The plant competition model is general, and the results may be applied to a wide range of interference competition problems.     

Cooperation should not be assumed

Evolutionary game theory provides a framework for explaining social interactions, including those between males and females. In a recent article, Roughgarden et al. discuss a new approach to sexual selection based on cooperative game theory and argue that cooperation rather than competition is fundamental in interactions between the sexes. However, compelling reasons for adopting this approach are not given and the authors do not adopt it consistently. We argue that non-cooperative game theory provides an adequate basis for understanding sexual selection, but that further work is needed to produce realistic models. We agree with Roughgarden and colleagues that bargaining is an important aspect of social interactions, but this is not a novel claim. Bargaining does not require the assumption of cooperation and does not necessarily lead to it.     

A comparison of strategies for Markov chain Monte Carlo computation in quantitative genetics

In quantitative genetics, Markov chain Monte Carlo (MCMC) methods are indispensable for statistical inference in non-standard models like generalized linear models with genetic random effects or models with genetically structured variance heterogeneity. A particular challenge for MCMC applications in quantitative genetics is to obtain efficient updates of the high-dimensional vectors of genetic random effects and the associated covariance parameters. We discuss various strategies to approach this problem including reparameterization, Langevin-Hastings updates, and updates based on normal approximations. The methods are compared in applications to Bayesian inference for three data sets using a model with genetically structured variance heterogeneity.     

Stabilization of Structured Populations via Vector Target-Oriented Control

In contrast with unstructured models, structured discrete population models have been able to fit and predict chaotic experimental data. However, most of the chaos control techniques in the literature have been designed and analyzed in a one-dimensional setting. Here, by introducing target-oriented control for discrete dynamical systems, we prove the possibility to stabilize a chosen state for a wide range of structured population models. The results are illustrated with introducing a control in the celebrated LPA model describing a flour beetle dynamics. Moreover, we show that the new control allows to stabilize periodic solutions for higher-order difference equations, such as the delayed Ricker model, for which previous target-oriented methods were not designed.     

A model of weak selection in the infinite alleles framework

Ewens (1972) proposed a model in the infinite allele framework for populations with neutrality of all alleles at a particular locus. This paper proposes a generalisation of Ewens' result for situations where there is a form of weak selection. The models considered here are continuous time, discrete state space Markov processes.     

A cellular automata model for simulating fed-batch penicillin fermentation process

A cellular automata model to simulate penicillin fed-batch fermentation process(CAPFM)was established in this study,based on a morphologically structured dynamic penicillin production model,that is in turn based on the growth mechanism of penicillin producing microorganisms and the characteristics of penicillin fed-batch fermentation.CAPFM uses the three-dimensional cellular automata as a growth space,and a Moore-type neighborhood as the cellular neighborhood.The transition roles of CAPFM are designed based on mechanical and structural kinetic models of penicillin batch-fed fermentation processes.Every cell of CAPFM represents a single or specific number of penicillin producing microorganisms,and has various state.The simulation experimental results show that CAPFM replicates the evolutionary behavior of penicillin batch-fed fermentation processes described by the structured penicillin production kinetic model accordingly.     

Sequential Decision Rules for Managing Nematodes with Crop Rotations

A dynamic model of nematode populations under a crop rotation that includes both host and nonhost crops is developed and used to conceptualize the problem of economic control. The steady state of the dynamic system is used to devise an approximately optimal decision policy, which is then applied to cyst nematode (Heterodera schachtii) control in a rotation of sugarbeet with nonhost crops. Long-run economic returns from this approximately optimal decision rule are compared with results from solution of the exact dynamic optimization model. The simple decision rule based on the steady state provides long-run average returns that are similar to the fully optimal solution. For sugarbeet and H. schachtii, the simplified rule can be calculated by maximizing a relatively simple algebraic expression with respect to the number of years in the sequence of nonhost crops. Maximization is easy because only integers are of interest and the number of years in nonhost crops is typically small. Solution of this problem indirectly yields an approximation to the optimal dynamic economic threshold density of nematodes in the soil. The decision rule requires knowledge of annual nematode population change under host and nonhost crops, and the relationship between crop yield and nematode population density.     

N-player quantum games in an EPR setting

The N-player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players' strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for N-qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the players' payoffs and mixed Nash equilibrium are determined. Players' N x N payoff matrices are then defined using linear functions so that common two-player games can be easily extended to the N-player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners' Dilemma game for general N ≥ 2. We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs. By dispensing with the standard unitary transformations on state vectors in Hilbert space and using instead rotors and multivectors, based on Clifford's geometric algebra (GA), it is shown how the N-player case becomes tractable. The new mathematical approach presented here has wide implications in the areas of quantum information and quantum complexity, as it opens up a powerful way to tractably analyze N-partite qubit interactions.     

Semi‐Markov Models with Phase‐Type Sojourn Distributions

Summary Continuous‐time multistate models are widely used for categorical response data, particularly in the modeling of chronic diseases. However, inference is difficult when the process is only observed at discrete time points, with no information about the times or types of events between observation times, unless a Markov assumption is made. This assumption can be limiting as rates of transition between disease states might instead depend on the time since entry into the current state. Such a formulation results in a semi‐Markov model. We show that the computational problems associated with fitting semi‐Markov models to panel‐observed data can be alleviated by considering a class of semi‐Markov models with phase‐type sojourn distributions. This allows methods for hidden Markov models to be applied. In addition, extensions to models where observed states are subject to classification error are given. The methodology is demonstrated on a dataset relating to development of bronchiolitis obliterans syndrome in post‐lung‐transplantation patients.