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PubMed1988

[Assessment of prognosis with Markov chains in chronic cardiovascular diseases in relation to various therapies].

On the basis of 5-years intervention results concerning chronic heart and vessel diseases of a check, which is representative for approx. 280.000 inhabitants of defined territories (EBMO-Cor Berlin), the prognose for the time of 5 and 10 years were estimated. It refers to ambulantories with the diagnosis hypertension and coronary heart disease, who were treated in a different way. The fundamental idea was to extract a prognostic index to make possible a choice of the treatment, adequate to the specific disease and its severe degrees, free of chance. The basic requirement of the application of the "Markoff-model" was the evidence of homogeneity concerning transition of severe degrees, used as reference, which was proved by means of the 2-k-chi 2-Felder-test. By using a starting vector of 1000 patients in each case inquiries about the distribution to the severe degrees had been made. The so achieved results illustrate a therapy-dependent susceptibility of these (morphologically defined) population suffering from a heart disease and allow a forecast, estimated also for a longer period, about the extent of the transition of a respecting and with the possibilities of the outpatient practice defined severe degrees of a heart and circulation disease and with that about chance and risk of a patient. By means of the epidemiological reference of the study the result gains special importance for outpatient practice.

Cardiovascular Diseases↗

PubMed2005

A stochastic automata network descriptor for Markov chain models of instantaneously coupled intracellular Ca2+ channels.

Although there is consensus that localized Ca(2+) elevations known as Ca(2+) puffs and sparks arise from the cooperative activity of intracellular Ca(2+) channels, the precise relationship between single-channel kinetics and the collective phenomena of stochastic Ca(2+) excitability is not well understood. Here we present a formalism by which mathematical models for Ca(2+)-regulated Ca(2+) release sites are derived from stochastic models of single-channel gating that include Ca(2+) activation, Ca(2+) inactivation, or both. Such models are stochastic automata networks (SANs) that involve a large number of functional transitions, that is, the transition probabilities of the infinitesimal generator matrix of one of the automata (i.e., an individual channel) may depend on the local [Ca(2+)] and thus the state of the other channels. Simulation and analysis of the SAN descriptors representing homogeneous clusters of intracellular Ca(2+) channels show that (1) release site density can modify both the steady-state open probability and stochastic excitability of Ca(2+) release sites, (2) Ca(2+) inactivation is not a requirement for Ca(2+) puffs or sparks, and (3) a single-channel model with a bell-shaped open probability curve does not lead to release site activity that is a biphasic function of release site density. These findings are obtained using iterative, memory-efficient methods (novel in this biophysical context and distinct from Monte Carlo simulation) that leverage the highly structured SAN descriptor to unambiguously calculate the steady-state probability of each release site configuration and puff statistics such as puff duration and inter-puff interval. The validity of a mean field approximation that neglects the spatial organization of Ca(2+) release sites is also discussed.

Algorithms↗

PubMed2001

Estimation of admixture proportions: a likelihood-based approach using Markov chain Monte Carlo.

When populations are separated for long periods and then brought into contact for a brief episode in part of their range, this can result in genetic admixture. To analyze this type of event we considered a simple model under which two parental populations (P1 and P2) mix and create a hybrid population (H). After that event, the three populations evolve under pure drift without exchange during T generations. We developed a new method, which allows the simultaneous estimation of the time since the admixture event (scaled by the population size t(i) = T/N(i), where N(i) is the effective population size of population i) and the contribution of one of two parental populations (which we call p1). This method takes into account drift since the admixture event, variation caused by sampling, and uncertainty in the estimation of the ancestral allele frequencies. The method is tested on simulated data sets and then applied to a human data set. We find that (i) for single-locus data, point estimates are poor indicators of the real admixture proportions even when there are many alleles; (ii) biallelic loci provide little information about the admixture proportion and the time since admixture, even for very small amounts of drift, but can be powerful when many loci are used; (iii) the precision of the parameters' estimates increases with sample size n = 50 vs. n = 200 but this effect is larger for the t(i)'s than for p1; and (iv) the increase in precision provided by multiple loci is quite large, even when there is substantial drift (we found, for instance, that it is preferable to use five loci than one locus, even when drift is 100 times larger for the five loci). Our analysis of a previously studied human data set illustrates that the joint estimation of drift and p1 can provide additional insights into the data.

Bayes Theorem↗

PubMed2006

Pattern statistics on Markov chains and sensitivity to parameter estimation.

BACKGROUND: In order to compute pattern statistics in computational biology a Markov model is commonly used to take into account the sequence composition. Usually its parameter must be estimated. The aim of this paper is to determine how sensitive these statistics are to parameter estimation, and what are the consequences of this variability on pattern studies (finding the most over-represented words in a genome, the most significant common words to a set of sequences,...). RESULTS: In the particular case where pattern statistics (overlap counting only) computed through binomial approximations we use the delta-method to give an explicit expression of sigma, the standard deviation of a pattern statistic. This result is validated using simulations and a simple pattern study is also considered. CONCLUSION: We establish that the use of high order Markov model could easily lead to major mistakes due to the high sensitivity of pattern statistics to parameter estimation.

Journal Article↗

PubMed2006

Graph transformation method for calculating waiting times in Markov chains.

We describe an exact approach for calculating transition probabilities and waiting times in finite-state discrete-time Markov processes. All the states and the rules for transitions between them must be known in advance. We can then calculate averages over a given ensemble of paths for both additive and multiplicative properties in a nonstochastic and noniterative fashion. In particular, we can calculate the mean first-passage time between arbitrary groups of stationary points for discrete path sampling databases, and hence extract phenomenological rate constants. We present a number of examples to demonstrate the efficiency and robustness of this approach.

Journal Article↗

PubMed1997

A Markov Chain Model of Coalescence with Recombination

Trees that describe the ancestry of DNA sequences sampled from a population may differ between loci because of genetic recombination. We seek to understand the relationship between such trees for loci that are linked with non-zero recombination rate. We consider a coalescent process model with recombination, as described by Hudson (1983; 1990). For two loci and a sample size of two sequences, a detailed analysis of this process yields the joint distribution of the two trees (one at each locus). A number of interesting results follow from this analysis, including the distribution of the number of recombination events in the history of the sample. For the general case of m loci and samples of size n, we describe an algorithm for simulating the tree building process. Because analytic results are difficult to obtain in this case, we use simulation to study properties of trees at multiple linked loci such as total tree time and number of recombination events. Copyright 1997 Academic Press

Journal Article↗

PubMed1965