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PubMed2005

Markov chain-based numerical method for degree distributions of growing networks.

In this paper, we establish a relation between growing networks and Markov chains, and propose a computational approach for network degree distributions. Using the Barabási-Albert model as an example, we first show that the degree evolution of a node in a growing network follows a nonhomogeneous Markov chain. Exploring the special structure of these Markov chains, we develop an efficient algorithm to compute the degree distribution numerically with a computation complexity of O (t(2)), where t is the number of time steps. We use three examples to demonstrate the computation procedure and compare the results with those from existing methods.

Journal Article↗
PubMed2004

The relative importance of herbivory and carnivory on the distribution of energy in a stochastic tri-trophic food web.

A three-state, discrete-time Markov chain is used to model the dynamics of energy flow in a tri-trophic food web. The distribution of energy in the three trophic levels is related to the rates of flow between the trophic levels and calculated for the entire range of possible flow values. These distributions are then analysed for stability and used to test the idea that plants are resource-limited and herbivores are predation-limited. Low rates of death and decomposition, when coupled with low rates of herbivory and carnivory, tend to destabilize this food web. Food webs with higher rates of death and decomposition are relatively more stable regardless of rates of herbivory and carnivory. Plants are more prone to resource-limitation and herbivores are, in general, limited by their predators, which supports Hairston et al. (Am. Nat. 94 (1960) 421). The rate of decomposition often mediates the roles of top-down and bottom-up control of energy flow in the food web.

Animals↗
PubMed2006

GenRGenS: software for generating random genomic sequences and structures.

SUMMARY: GenRGenS is a software tool dedicated to randomly generating genomic sequences and structures. It handles several classes of models useful for sequence analysis, such as Markov chains, hidden Markov models, weighted context-free grammars, regular expressions and PROSITE expressions. GenRGenS is the only program that can handle weighted context-free grammars, thus allowing the user to model and to generate structured objects (such as RNA secondary structures) of any given desired size. GenRGenS also allows the user to combine several of these different models at the same time.

Computational Biology↗
PubMed2004

Information coding and oscillatory activity in synfire neural networks with and without inhibitory coupling.

When a population spike (pulse-packet) propagates through a feedforward network with random excitatory connections, it either evolves to a sustained stable level of synchronous activity or fades away (Diesmann et al. in Nature 402:529-533 1999; Cateau and Fukai Neur Netw 14:675-685 2001). Here I demonstrate that in the presence of noise, the probability of the survival of the pulse-packet (or, equivalently, the firing rate of output neurons) reflects the intensity of the input. Furthermore, inhibitory coupling between layers can result in quasi- periodic alternation between several levels of firing activity. These results are obtained by analyzing the evolution of pulse-packet activity as a Markov chain. For the Markov chain analysis, the output of the chain is a linear mapping of the input into a lower-dimensional space, and the eigenvalues and eigenvectors of the transition matrix determine the dynamics of the evolution. Synchronous propagation of firing activity in successive pools of neurons are simulated in networks of integrate-and-fire and compartmental model neurons, and, consistent with the discrete Markov process, the activation of each pool is observed to be predominantly dependent upon the number of cells that fired in the previous pool. Simulation results agree with the numerical solutions of the Markov model. When inhibitory coupling between layers are included in the Markov model, some eigenvalues become complex numbers, implying oscillatory dynamics. The quasiperiodic dynamics is validated with simulation with leaky integrate-and-fire neurons. The networks demonstrate different modes of quasiperiodic activity as the inhibition or excitation parameters of the network are varied.

Action Potentials↗
PubMed1995

Populations with constant immigration.

"We consider a Leslie-type model of a one-sex (female) population of natives with constant immigration. The fertility and mortality schedule of the natives may be below or above replacement level. Immigrants retain their fertility and mortality, their children adopt the fertility and mortality of the natives. It is shown how this model may be written in a homogeneous form (without additive term) with a Leslie-type matrix. Reproductive values of individuals in each age group are discussed in terms of a left eigenvector of this matrix. The homogeneous form of our projection model permits the transformation into a Markov chain with transient and recurrent states. The Markov chain is the basis for the definition of genealogies, which incorporate immigration. It is shown that genealogies describe the life histories of individuals in a population with immigration. We calculate absorption times of the Markov chain and relate them to genealogies. This extends the theory originally designed for closed populations to populations with immigration." (SUMMARY IN FRE)

Demography↗
PubMed2005

Metabolic flux distribution analysis by 13C-tracer experiments using the Markov chain-Monte Carlo method.

Metabolic flux analysis using 13C-tracer experiments is an important tool in metabolic engineering since intracellular fluxes are non-measurable quantities in vivo. Current metabolic flux analysis approaches are fully based on stoichiometric constraints and carbon atom balances, where the over-determined system is iteratively solved by a parameter estimation approach. However, the unavoidable measurement noises involved in the fractional enrichment data obtained by 13C-enrichment experiment and the possible existence of unknown pathways prevent a simple parameter estimation method for intracellular flux quantification. The MCMC (Markov chain-Monte Carlo) method, which obtains intracellular flux distributions through delicately constructed Markov chains, is shown to be an effective approach for deep understanding of the intracellular metabolic network. Its application is illustrated through the simulation of an example metabolic network.

Carbon Isotopes↗
PubMed2004

Coalescent process with fluctuating population size and its effective size.

We consider a Wright-Fisher model whose population size is a finite Markov chain. We introduce a sequence of two-dimensional discrete time Markov chains whose components describe the coalescent process and the fluctuation of population size. For the limiting process of the sequence of Markov chains, the relationship of the expectation of coalescence time to the harmonic and the arithmetic means of population sizes is shown, and the Laplace transform of the distribution of coalescence time is calculated. We define the coalescence effective population size (cEPS) by the expectation of coalescence time. We show that cEPS is strictly larger (resp. smaller) than the harmonic (resp. arithmetic) mean. As the population size fluctuates more quickly (resp. slowly), cEPS is closer to the harmonic (resp. arithmetic) mean. For the case of a two-valued Markov chain, we show the explicit expression of cEPS and its dependency on the sample size.

Genetics, Population↗
PubMed2005

Markov chain-based method for generating long-range dependence.

This paper describes a model for generating time series which exhibit the statistical phenomenon known as long-range dependence (LRD). A Markov modulated process based on an infinite Markov chain is described. The work described is motivated by applications in telecommunications where LRD is a known property of time series measured on the Internet. The process can generate a time series exhibiting LRD with known parameters and is particularly suitable for modeling Internet traffic because the time series is in terms of ones and zeros, which can be interpreted as data packets and interpacket gaps. The method is extremely simple, both computationally and analytically, and could prove more tractable than other methods described in the literature.

Journal Article↗
PubMed2005

Finite state automata resulting from temporal information maximization and a temporal learning rule.

We extend Linkser's Infomax principle for feedforward neural networks to a measure for stochastic interdependence that captures spatial and temporal signal properties in recurrent systems. This measure, stochastic interaction, quantifies the Kullback-Leibler divergence of a Markov chain from a product of split chains for the single unit processes. For unconstrained Markov chains, the maximization of stochastic interaction, also called Temporal Infomax, has been previously shown to result in almost deterministic dynamics. This letter considers Temporal Infomax on constrained Markov chains, where some of the units are clamped to prescribed stochastic processes providing input to the system. Temporal Infomax in that case leads to finite state automata, either completely deterministic or weakly nondeterministic. Transitions between internal states of these systems are almost perfectly predictable given the complete current state and the input, but the activity of each single unit alone is virtually random. The results are demonstrated by means of computer simulations and confirmed analytically. It is furthermore shown numerically that Temporal Infomax leads to a high information flow from the input to internal units and that a simple temporal learning rule can approximately achieve the optimization of temporal interaction. We relate these results to experimental data concerning the correlation dynamics and functional connectivities observed in multiple electrode recordings.

Action Potentials↗
PubMed2000

Performance of Markov chain-Monte Carlo approaches for mapping genes in oligogenic models with an unknown number of loci.

Markov chain-Monte Carlo (MCMC) techniques for multipoint mapping of quantitative trait loci have been developed on nuclear-family and extended-pedigree data. These methods are based on repeated sampling-peeling and gene dropping of genotype vectors and random sampling of each of the model parameters from their full conditional distributions, given phenotypes, markers, and other model parameters. We further refine such approaches by improving the efficiency of the marker haplotype-updating algorithm and by adopting a new proposal for adding loci. Incorporating these refinements, we have performed an extensive simulation study on simulated nuclear-family data, varying the number of trait loci, family size, displacement, and other segregation parameters. Our simulation studies show that our MCMC algorithm identifies the locations of the true trait loci and estimates their segregation parameters well-provided that the total number of sibship pairs in the pedigree data is reasonably large, heritability of each individual trait locus is not too low, and the loci are not too close together. Our MCMC algorithm was shown to be significantly more efficient than LOKI (Heath 1997) in our simulation study using nuclear-family data.

Algorithms↗
PubMed2004

A Bayesian approach to DNA sequence segmentation.

Many deoxyribonucleic acid (DNA) sequences display compositional heterogeneity in the form of segments of similar structure. This article describes a Bayesian method that identifies such segments by using a Markov chain governed by a hidden Markov model. Markov chain Monte Carlo (MCMC) techniques are employed to compute all posterior quantities of interest and, in particular, allow inferences to be made regarding the number of segment types and the order of Markov dependence in the DNA sequence. The method is applied to the segmentation of the bacteriophage lambda genome, a common benchmark sequence used for the comparison of statistical segmentation algorithms.

Algorithms↗
PubMed2003

A comparison of three different stochastic population models with regard to persistence time.

Results are summarized from the literature on three commonly used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.

Animals↗
PubMed2007

Quantitative analysis of the phenotypic variability of shoot architecture in two grapevine (Vitis vinifera) cultivars.

BACKGROUND AND AIMS: Plant architecture and its interaction with agronomic practices and environmental constraints are determinants of the structure of the canopy, which is involved in carbon acquisition and fruit quality development. A framework for the quantitative analysis of grapevine (Vitis vinifera) shoot architecture, based on a set of topological and geometrical parameters, was developed for the identification of differences between cultivars and the origins of phenotypic variability. METHODS: Two commercial cultivars ('Grenache N', 'Syrah') with different shoot architectures were grown in pots, in well-irrigated conditions. Shoot topology was analysed, using a hidden semi-Markov chain and variable-order Markov chains to identify deviations from the normal pattern of succession of phytomer types (P0-P1-P2), together with kinematic analysis of shoot axis development. Shoot geometry was characterized by final internode and individual leaf area measurements. KEY RESULTS: Shoot architecture differed significantly between cultivars. Secondary leaf area and axis length were greater for 'Syrah'. Secondary leaf area distribution along the main axis also differed between cultivars, with secondary leaves preferentially located towards the basal part of the shoot in 'Syrah'. The main factors leading to differences in leaf area between the cultivars were: (a) slight differences in main shoot structure, with the supplementary P0 phytomer on the lower part of the shoot in 'Grenache N', which bears a short branch; and (b) an higher rate and duration of development of branches bearing by P1-P2 phytomers related to P0 ones at the bottom of the shoot in 'Syrah'. Differences in axis length were accounted for principally by differences in individual internode morphology, with 'Syrah' having significantly longer internodes. This trait, together with a smaller shoot diameter, may account for the characteristic 'droopy' habit of 'Syrah' shoots. CONCLUSIONS: This study highlights the architectural parameters involved in the phenotypic variability of shoot architecture in two grapevine cultivars. Differences in primary shoot structure and in branch development potential accounted for the main differences in leaf area distribution between the two cultivars. By contrast, shoot shape seemed to be controlled by differences in axis length due principally to differences in internode length.

Kinetics↗
PubMed2002

Markovian models for the developmental study of social behavior.

Behavioral development involves changes in the probabilities of both social and nonsocial activities and the sequential pattern of activities over time. A number of methods have been offered for the analysis of these patterns of behavioral sequences. However, there continue to be problematic issues, including the analysis of nonstationary data; accommodation of changes in patterns within an observation period, or over repeated observations or age; and identification of differences in pattern changes between individuals or groups, and the factors responsible for these differences. In this work, we analyze data from 15 young monkeys (Macaca nemestrina) using classification and Markovian methods, including a new approach to nonstationary data called the double-chain Markov model (DCCM). These methods allowed us to identify differences in behavior patterns that differentiate between normal subjects and those presenting developmental anomalies.

Aging↗
PubMed1999

Overdominant alleles in a population of variable size.

An approximate method is developed to predict the number of strongly overdominant alleles in a population of which the size varies with time. The approximation relies on the strong-selection weak-mutation (SSWM) method introduced by J. H. Gillespie and leads to a Markov chain model that describes the number of common alleles in the population. The parameters of the transition matrix of the Markov chain depend in a simple way on the population size. For a population of constant size, the Markov chain leads to results that are nearly the same as those of N. Takahata. The Markov chain allows the prediction of the numbers of common alleles during and after a population bottleneck and the numbers of alleles surviving from before a bottleneck. This method is also adapted to modeling the case in which there are two classes of alleles, with one class causing a reduction in fitness relative to the other class. Very slight selection against one class can strongly affect the relative frequencies of the two classes and the relative ages of alleles in each class.

Alleles↗
PubMed2005

Phylogenetic MCMC algorithms are misleading on mixtures of trees.

Markov chain Monte Carlo (MCMC) algorithms play a critical role in the Bayesian approach to phylogenetic inference. We present a theoretical analysis of the rate of convergence of many of the widely used Markov chains. For N characters generated from a uniform mixture of two trees, we prove that the Markov chains take an exponentially long (in N) number of iterations to converge to the posterior distribution. Nevertheless, the likelihood plots for sample runs of the Markov chains deceivingly suggest that the chains converge rapidly to a unique tree. Our results rely on novel mathematical understanding of the log-likelihood function on the space of phylogenetic trees. The practical implications of our work are that Bayesian MCMC methods can be misleading when the data are generated from a mixture of trees. Thus, in cases of data containing potentially conflicting phylogenetic signals, phylogenetic reconstruction should be performed separately on each signal.

Algorithms↗
PubMed1992

DNA damage caused by ionizing radiation.

A survey is given of continuous-time Markov chain models for ionizing radiation damage to the genome of mammalian cells. In such models, immediate damage induced by the radiation is regarded as a batch-Poisson arrival process of DNA double-strand breaks (DSBs). Enzymatic modification of the immediate damage is modeled as a Markov process similar to those described by the master equation of stochastic chemical kinetics. An illustrative example is the restitution/complete-exchange model. The model postulates that, after being induced by radiation, DSBs subsequently either undergo enzymatically mediated restitution (repair) or participate pairwise in chromosome exchanges. Some of the exchanges make irremediable lesions such as dicentric chromosome aberrations. One may have rapid irradiation followed by enzymatic DSB processing or have prolonged irradiation with both DSB arrival and enzymatic DSB processing continuing throughout the irradiation period. Methods for analyzing the Markov chains include using an approximate model for expected values, the discrete-time Markov chain embedded at transitions, partial differential equations for generating functions, normal perturbation theory, singular perturbation theory with scaling, numerical computations, and certain matrix methods that combine Perron-Frobenius theory with variational estimates. Applications to experimental results on expected values, variances, and statistical distributions of DNA lesions are briefly outlined. Continuous-time Markov chains are the most systematic of those radiation damage models that treat DSB-DSB interactions within the cell nucleus as homogeneous (e.g., ignore diffusion limitations). They contain virtually all other relevant homogeneous models and semiempirical summaries as special cases, limiting cases, or approximations. However, the Markov models do not seem to be well suited for studying spatial dependence of DSB interactions, which is known to be important in some situations.

Animals↗
PubMed2003

The gene-orientation structure of eukaryotes.

It is shown that the sequence of gene orientations of four eukaryotes-for those that are presently known-are well modeled by a two-state, two-parameter (first order) Markov chain. These include the six chromosomes of nematodes (C. elegans), the 16 chromosomes of yeast (S. cerevisiae), the five chromosomes of Arabidopsis (A. thaliana), and the 19 scaffolds of fruit flies (D. melanogaster). Moreover, they are reasonably well modeled, more simply, by a one-parameter symmetric version of the Markov chain. Further, compelling statistical evidence is presented which suggests that the parameters particularizing the Markov chain are organism dependent rather than merely chromosome dependent. This surprising observation begs an appropriate biological explanation. Does there exist some kind of mechanism of "communication" among a eukaryote's chromosomes that serves to maintain common values for all of the chromosomal parameters? Or are the common parameter values merely a consequence of a common environment of origin for all of the chromosomes of an organism? (If so, why so?) A third possible explanation is ruled out: while there exists within the class of Markov chain models under consideration, a case that could be described as equivalent to "flipping a fair coin," a thorough-random-shuffling-of-the-genes explanation, via mutations, does not hold up to statistical scrutiny.

Animals↗