Modeling haplotype block variation using Markov chains

Models of background variation in genomic regions form the basis of linkage disequilibrium mapping methods. In this work we analyze a background model that groups SNPs into haplotype blocks and represents the dependencies between blocks by a Markov chain. We develop an error measure to compare the performance of this model against the common model that assumes that blocks are independent. By examining data from the International Haplotype Mapping project, we show how the Markov model over haplotype blocks is most accurate when representing blocks in strong linkage disequilibrium. This contrasts with the independent model, which is rendered less accurate by linkage disequilibrium. We provide a theoretical explanation for this surprising property of the Markov model and relate its behavior to allele diversity.     

Integrals for continuous-time Markov chains

This paper presents a method of evaluating the expected value of a path integral for a general Markov chain on a countable state space. We illustrate the method with reference to several models, including birth-death processes and the birth, death and catastrophe process.     

Multilocus molecular mapping of the mouse X chromosome

Using restriction fragment length polymorphisms (RFLPs) and enzymatic variants between distantly related mouse species, we have assigned three genes to the mouse X chromosome and concurrently mapped a total of eight genes spanning an estimated 50 cM of the chromosome. Segregation of RFLPs in over 200 male progeny from interspecies backcrosses between the inbred strain C57BL/6JRos and either wild-derived Mus musculus or Mus spretus was followed for the murine genes Timp (tissue inhibitor of metalloproteinases), Cf-8 (coagulation factor VIII), and Rsvp (red-sensitive visual pigment) and the known X-linked markers Otc, Hprt, Cf-9, G6pd, and Ags. From the centromere, the gene order was defined as Otc, Timp, Hprt, Cf-9, (Cf-8/Rsvp/G6pd), Ags, by minimizing the number of multiple recombinational events. No significant differences in map order or frequency of recombination were observed between the two backcross series studied. The use of Southern analysis has allowed us to add new genes to the map in a cumulative manner, and as probes become available, additional markers can be mapped, using the same set of mice, by utilizing existing blots or resampling the DNAs. The use of probes for functional genes has allowed us to directly compare the X chromosomes of mouse and man and has provided insight into chromosomal rearrangements which have occurred during the evolutionary divergence of these species, as well as to define the extent of linkage homologies.     

Markov chains with infinite transition rates

We investigate finite state, continuous time Markov chains, whose transition rates have different orders of magnitude. Such a chain may be approximated by a simple one, which is obtained by grouping together states whose mutual communication is of the highest order. Using this reduced chain, the computation of both transition probability matrix and stationary distribution of the original chain is simplified significantly. The method is applicable in enzyme kenetics, and presumably in mathematical ecology or population genetics.     

Predicting protein subcellular locations using hierarchical ensemble of Bayesian classifiers based on Markov chains

Background  

The subcellular location of a protein is closely related to its function. It would be worthwhile to develop a method to predict the subcellular location for a given protein when only the amino acid sequence of the protein is known. Although many efforts have been made to predict subcellular location from sequence information only, there is the need for further research to improve the accuracy of prediction.     

Analysing grouping of nucleotides in DNA sequences using lumped processes constructed from Markov chains

The most commonly used models for analysing local dependencies in DNA sequences are (high-order) Markov chains. Incorporating knowledge relative to the possible grouping of the nucleotides enables to define dedicated sub-classes of Markov chains. The problem of formulating lumpability hypotheses for a Markov chain is therefore addressed. In the classical approach to lumpability, this problem can be formulated as the determination of an appropriate state space (smaller than the original state space) such that the lumped chain defined on this state space retains the Markov property. We propose a different perspective on lumpability where the state space is fixed and the partitioning of this state space is represented by a one-to-many probabilistic function within a two-level stochastic process. Three nested classes of lumped processes can be defined in this way as sub-classes of first-order Markov chains. These lumped processes enable parsimonious reparameterizations of Markov chains that help to reveal relevant partitions of the state space. Characterizations of the lumped processes on the original transition probability matrix are derived. Different model selection methods relying either on hypothesis testing or on penalized log-likelihood criteria are presented as well as extensions to lumped processes constructed from high-order Markov chains. The relevance of the proposed approach to lumpability is illustrated by the analysis of DNA sequences. In particular, the use of lumped processes enables to highlight differences between intronic sequences and gene untranslated region sequences.     

Continuous-time Markov chains as models for animal behaviour

A survey is given of the application of (functions of) continuous-time Markov chains in the statistical analysis of behavioural time series.     

Sojourn times for conditioned Markov chains in genetics

The use of diffusion methods to predict the distribution of the number of visits to a particular gene frequency in a class of finite population models is discussed. The models for which such approximations are accurate are found, and several results unified by this approach. Some difficulties in the application of diffusion methods to such sojourn times are highlighted.     

Direct products of genetic algebras and Markov chains

Summary The effect of the combination of breeding and migration on linkage in a subdivided population can be treated by means of a direct product structure. The eigenvalues of the combined operator are investigated and the theory placed in a general setting.     

Estimating statistics of neuronal dynamics via Markov chains

We present an efficient computational method for estimating the mean and variance of interspike intervals defined by the timing of spikes in typical orbits of one-dimensional neuronal maps. This is equivalent to finding the mean and variance of return times of orbits to particular regions of phase space. Rather than computing estimates directly from time series, the system is modelled as a finite state Markov chain to extract stationary behaviour in the form of invariant measures and average absorption times. Ergodic-theoretic formulae are then applied to produce the estimates without the need to generate orbits directly. The approach may be applied to both deterministic and randomly forced systems. Received: 26 January 2000 / Accepted in revised form: 9 June 2000