Estimation of inbreeding by random walks in pedigrees

Summary Wright and McPhee (1925) suggested a method of estimating the inbreeding coefficient of an individual based on the probability that a pair of lineages traced randomly, one through the maternal line and one through the paternal line, both contain a common ancestor. (One-half of this probability is an unbiased estimate of the inbreeding coefficient). In their procedure, maternal and paternal lines are chosen in pairs, and comparisons are made only between the lines in a pair. A more efficient procedure is to compare every maternal line with every paternal line, a procedure used by Robertson and Mason (1954). In this paper we provide estimates of the sampling variance of the inbreeding coefficient as estimated by the multiple comparison method, and we examine the relative efficiency of this method and the Wright-McPhee procedure. Formulae are also provided for ascertaining the optimal sampling method for estimating the average inbreeding coefficient of a group or herd.Work supported by NSF grant GB43209, NIH grant GM21732 and by Research Career Award GM0002301.     

Search along persistent random walks

Optimal search strategies and their implementations in biological systems are a subject of active research. Here we study a search problem which is motivated by the hunt of sperm cells for the egg. We ask for the probability for an active swimmer to find a target under the condition that the swimmer starts at a certain distance from the target. We find that success probability is maximal for a certain level of fluctuations characterized by the persistence length of the swimming path of the swimmer. We derive a scaling law for the optimal persistence length as a function of the initial target distance and search time by mapping the search on a polymer physics problem.     

From random walks to informed movement

The analysis of animal movement is a large and continuously growing field of research. Detailed knowledge about movement strategies is of crucial importance for understanding eco‐evolutionary dynamics at all scales – from individuals to (meta‐)populations. This and the availability of detailed movement and dispersal data motivated Nathan and colleagues to published their much appreciated call to base movement ecology on a more thorough mechanistic basis. So far, most movement models are based on random walks. However, even if a random walk might describe real movement patterns acceptably well, there is no reason to assume that animals move randomly. Therefore, mechanistic models of foraging strategies should be based on information use and memory in order to increase our understanding of the processes that lead to animal movement decisions. We present a mechanistic movement model of an animal with a limited perceptual range and basic information storage capacities. This ‘spatially informed forager’ constructs an internal map of its environment by using perception, memory and learned or evolutionarily acquired assumptions about landscape attributes. We analyse resulting movement patterns and search efficiencies and compare them to area restricted search strategies (ARS) and biased correlated random walks (BCRW) of omniscient individuals. We show that, in spite of their limited perceptual range, spatially informed individuals boost their foraging success and may perform much better than the best ARS. The construction of an internal map and the use of spatial information results in the emergence of a highly correlated walk between patches and a rather systematic search within resource clusters. Furthermore, the resulting movement patterns may include foray search behaviour. Our work highlights the strength of mechanistic modelling approaches and sets the stage for the development of more sophisticated models of memory use for movement decisions and dispersal.     

Distinct error rates for reference and nonreference genotypes estimated by pedigree analysis

Errors in genotype calling can have perverse effects on genetic analyses, confounding association studies, and obscuring rare variants. Analyses now routinely incorporate error rates to control for spurious findings. However, reliable estimates of the error rate can be difficult to obtain because of their variance between studies. Most studies also report only a single estimate of the error rate even though genotypes can be miscalled in more than one way. Here, we report a method for estimating the rates at which different types of genotyping errors occur at biallelic loci using pedigree information. Our method identifies potential genotyping errors by exploiting instances where the haplotypic phase has not been faithfully transmitted. The expected frequency of inconsistent phase depends on the combination of genotypes in a pedigree and the probability of miscalling each genotype. We develop a model that uses the differences in these frequencies to estimate rates for different types of genotype error. Simulations show that our method accurately estimates these error rates in a variety of scenarios. We apply this method to a dataset from the whole-genome sequencing of owl monkeys (Aotus nancymaae) in three-generation pedigrees. We find significant differences between estimates for different types of genotyping error, with the most common being homozygous reference sites miscalled as heterozygous and vice versa. The approach we describe is applicable to any set of genotypes where haplotypic phase can reliably be called and should prove useful in helping to control for false discoveries.     

Efficient inference of haplotypes from genotypes on a pedigree

We study haplotype reconstruction under the Mendelian law of inheritance and the minimum recombination principle on pedigree data. We prove that the problem of finding a minimum-recombinant haplotype configuration (MRHC) is in general NP-hard. This is the first complexity result concerning the problem to our knowledge. An iterative algorithm based on blocks of consecutive resolved marker loci (called block-extension) is proposed. It is very efficient and can be used for large pedigrees with a large number of markers, especially for those data sets requiring few recombinants (or recombination events). A polynomial-time exact algorithm for haplotype reconstruction without recombinants is also presented. This algorithm first identifies all the necessary constraints based on the Mendelian law and the zero recombinant assumption, and represents them using a system of linear equations over the cyclic group Z2. By using a simple method based on Gaussian elimination, we could obtain all possible feasible haplotype configurations. A C++ implementation of the block-extension algorithm, called PedPhase, has been tested on both simulated data and real data. The results show that the program performs very well on both types of data and will be useful for large scale haplotype inference projects.     

Stochasticity, invasions, and branching random walks

We link deterministic integrodifference equations to stochastic, individual-based simulations by means of branching random walks. Using standard methods, we determine speeds of invasion for both average densities and furthest-forward individuals. For density-independent branching random walks, demographic stochasticity can produce extinction. Demographic stochasticity does not, however, reduce the overall asymptotic speed of invasion or preclude continually accelerating invasions.     

Genetic relationships among wheat genotypes, as revealed by microsatellite markers and pedigree analysis

Genetic relationships among 20 elite wheat genotypes were studied using microsatellite markers and pedigree analysis. A total of 93 polymorphic bands were obtained with 25 microsatellite primer pairs. Coefficient of parentage (COP) values were calculated using parentage information at the expansion level of 5. The pedigree-based similarity (mean 0.115, range 0.00-0.53) was lower than the similarity assessed using microsatellite markers (mean 0.70, range 0.47-0.91). Similarity estimates were used to construct dendrograms by using the unweighted pair-group method with arithmetic averages (UPGMA). Clustering of genotypes in respect of marker-based similarity revealed two groups. Genotype PBW442 diverged and appeared as distinct from all other genotypes in both marker-based and pedigree-based analysis. The correlation of COP values with genetic similarity values based on microsatellite markers is low (r = 0.285, p < 0.05). The results indicate a need to develop wheat varieties with a diverse genetic background and to incorporate new variability into the existing wheat gene pool.     

GMCheck: Bayesian error checking for pedigree genotypes and phenotypes

GMCheck uses graphical modeling to find the posterior probabilities of data errors given genotypes or phenotypes in a specified pedigree structure.     

Predicting genetic interactions with random walks on biological networks

Background  

Several studies have demonstrated that synthetic lethal genetic interactions between gene mutations provide an indication of functional redundancy between molecular complexes and pathways. These observations help explain the finding that organisms are able to tolerate single gene deletions for a large majority of genes. For example, system-wide gene knockout/knockdown studies in S. cerevisiae and C. elegans revealed non-viable phenotypes for a mere 18% and 10% of the genome, respectively. It has been postulated that the low percentage of essential genes reflects the extensive amount of genetic buffering that occurs within genomes. Consistent with this hypothesis, systematic double-knockout screens in S. cerevisiae and C. elegans show that, on average, 0.5% of tested gene pairs are synthetic sick or synthetic lethal. While knowledge of synthetic lethal interactions provides valuable insight into molecular functionality, testing all combinations of gene pairs represents a daunting task for molecular biologists, as the combinatorial nature of these relationships imposes a large experimental burden. Still, the task of mapping pairwise interactions between genes is essential to discovering functional relationships between molecular complexes and pathways, as they form the basis of genetic robustness. Towards the goal of alleviating the experimental workload, computational techniques that accurately predict genetic interactions can potentially aid in targeting the most likely candidate interactions. Building on previous studies that analyzed properties of network topology to predict genetic interactions, we apply random walks on biological networks to accurately predict pairwise genetic interactions. Furthermore, we incorporate all published non-interactions into our algorithm for measuring the topological relatedness between two genes. We apply our method to S. cerevisiae and C. elegans datasets and, using a decision tree classifier, integrate diverse biological networks and show that our method outperforms established methods.     

Dwell time symmetry in random walks and molecular motors

The statistics of steps and dwell times in reversible molecular motors differ from those of cycle completion in enzyme kinetics. The reason is that a step is only one of several transitions in the mechanochemical cycle. As a result, theoretical results for cycle completion in enzyme kinetics do not apply to stepping data. To allow correct parameter estimation, and to guide data analysis and experiment design, a theoretical treatment is needed that takes this observation into account. In this article, we model the distribution of dwell times and number of forward and backward steps using first passage processes, based on the assumption that forward and backward steps correspond to different directions of the same transition. We extend recent results for systems with a single cycle and consider the full dwell time distributions as well as models with multiple pathways, detectable substeps, and detachments. Our main results are a symmetry relation for the dwell time distributions in reversible motors, and a relation between certain relative step frequencies and the free energy per cycle. We demonstrate our results by analyzing recent stepping data for a bacterial flagellar motor, and discuss the implications for the efficiency and reversibility of the force-generating subunits.     

Biological organization,Jarkievich measures and asymmetric random walks

In this paper we analyze the organization imposed by the energy input during the migration of enzymes on DNA. We attempt to measure that organization by means of a concept proposed by A.A. Jarkievich in 1961. We found relationships among a Jarkievich measure, the energy dissipation, and the fluctuations in the kinematic velocity of the enzyme on the DNA.     

Efficient inference of haplotypes from genotypes on a large animal pedigree

We present a simple algorithm for reconstruction of haplotypes from a sample of multilocus genotypes. The algorithm is aimed specifically for analysis of very large pedigrees for small chromosomal segments, where recombination frequency within the chromosomal segment can be assumed to be zero. The algorithm was tested both on simulated pedigrees of 155 individuals in a family structure of three generations and on real data of 1149 animals from the Israeli Holstein dairy cattle population, including 406 bulls with genotypes, but no females with genotypes. The rate of haplotype resolution for the simulated data was >91% with a standard deviation of 2%. With 20% missing data, the rate of haplotype resolution was 67.5% with a standard deviation of 1.3%. In both cases all recovered haplotypes were correct. In the real data, allele origin was resolved for 22% of the heterozygous genotypes, even though 70% of the genotypes were missing. Haplotypes were resolved for 36% of the males. Computing time was insignificant for both data sets. Despite the intricacy of large-scale real pedigree genotypes, the proposed algorithm provides a practical rule-based solution for resolving haplotypes for small chromosomal segments in commercial animal populations.     

Random walks, random sequences, and nonlinear dynamics in human optokinetic nystagmus.

Optokinetic nystagmus (OKN) is a reflexive eye movement with target-following slow phases (SP) alternating with oppositely directed fast phases (FP). We measured the following from OKN in three humans: FP beginning and ending positions, amplitudes, and intervals and SP amplitudes and velocities. We sought to predict future values of each parameter on the basis of past values, using state-space representation of the sequence (time-delay embedding) and local second-order approximation of trajectories. Predictability is an indication of determinism: this approach allows us to investigate the relative contributions of random and deterministic dynamics in OKN. FP beginning and ending positions showed good predictability, but SP velocity was less predictable. FP and SP amplitudes and FP intervals had little or no predictability. FP beginnings and endings were as predictable as randomized versions that retain linear autocorrelation; this is typical of random walks. Predictability of FP intervals did not change under random rearrangement, which is characteristic of a random process. Only linear determinism was demonstrated; nonlinear interactions may exist that would not be detected by our present approach.     

Lowering the barriers to random walks on the cell surface

We used fluorescence recovery after photobleaching (FRAP) and single particle tracking (SPT) techniques to compare diffusion of class I major histocompatibility complex molecules (MHC) on normal and alpha-spectrin-deficient murine erythroleukemia (MEL) cells. Because the cytoskeleton mesh acts as a barrier to lateral mobility of membrane proteins, we expected that diffusion of membrane proteins in alpha-spectrin-deficient MEL cells would differ greatly from that in normal MEL cells. In the event, diffusion coefficients derived from either FRAP or SPT analysis were similar for alpha-spectrin-deficient and normal MEL cells, differing by a factor of approximately 2, on three different timescales: tens of seconds, 1-10 s, and 100 ms. SPT analysis showed that the diffusion of most class I MHC molecules was confined on both cell types. On the normal MEL cells, the mean diagonal length of the confined area was 330 nm with a mean residency time of 40s. On the alpha-spectrin-deficient MEL cells, the mean diagonal length was 650 nm with a mean residency time of 45s. Thus there are fewer barriers to lateral diffusion on cytoskeleton mutant MEL cells than on normal MEL cells, but this difference does not strongly affect lateral diffusion on the scales measured here.     

Sampling rate effects on measurements of correlated and biased random walks

When observing the two-dimensional movement of animals or microorganisms, it is usually necessary to impose a fixed sampling rate, so that observations are made at certain fixed intervals of time and the trajectory is split into a set of discrete steps. A sampling rate that is too small will result in information about the original path and correlation being lost. If random walk models are to be used to predict movement patterns or to estimate parameters to be used in continuum models, then it is essential to be able to quantify and understand the effect of the sampling rate imposed by the observer on real trajectories. We use a velocity jump process with a realistic reorientation model to simulate correlated and biased random walks and investigate the effect of sampling rate on the observed angular deviation, apparent speed and mean turning angle. We discuss a method of estimating the values of the reorientation parameters used in the original random walk from the rediscretized data that assumes a linear relation between sampling time step and the parameter values.     

Adaptive walks by the fittest among finite random mutants on a Mt. Fuji-type fitness landscape II. Effect of small non-additivity

We examined properties of adaptive walks by the fittest on “rough Mt. Fuji-type” fitness landscapes, which are modeled by superposing small uncorrelated random component on an additive fitness landscape. A single adaptive walk is carried out by repetition of the evolution cycle composed of (1) mutagenesis process that produces random d-fold point mutants of population size N and (2) selection process that picks out the fittest mutant among them. To comprehend trajectories of the walkers, the fitness landscape is mapped into a (x, y, z)-space, where x, y and z represent, respectively, normalized Hamming distance from the peak on the additive fitness landscape, scaled additive fitness and scaled non-additive fitness. Thus a single adaptive walk is expressed as the dynamics of a particle in this space. We drew the “hill-climbing” vector field, where each vector represents the most probable step for a walker in a single step. Almost all of the walkers are expected to move along streams of vectors existing on a particular surface that overlies the (x, y)-plane, toward the neighborhood of a characteristic point at which a mutation-selection-random drift balance is reached. We could theoretically predict this reachable point in the case of random sampling search strategy. Received: 1 March 2000 / Published online: 3 August 2000     

Bounded random walks as a null model for evaluating population trends

Management of wildlife and protection of endangered species depend on determination of population trends. Because population changes are stochastic and autoregressive, there is reason to believe that population trends might not be properly determined by simple regression over short time periods. A bounded random walk (BRW) model is introduced as a null model for evaluating population trends. The BRW model shows long-term stability but rising and falling sequences of up to many decades. For a given variability and survey length, there will be an expected probability of finding a greater than X% slope simply by chance. This false positive probability needs to be considered when evaluating trends. Breeding Bird Survey data for 128 species over 46 years for two states were analyzed for trends for different series lengths. Trends estimated from short series were likely to not agree with the 46-year trends. Very short series (e.g., 5 years) tended to indicate no trend due to loss of statistical power. A 101-year series for sandwich term (Sterna sandvicensis) revealed that even for 40 year-long series, 33% of subset series had a negative trend compared to the strong 101 year full series positive trend. The BRW model simulations and both data sets pointed to 20 years as a minimum time period for estimating trends reliably, though this can be longer for species that tend to cycle. Proper inference should thus consider the implications of inherent time series variability.