Neurons communicate with each other via sequences of action potentials. The purpose of this study is to approximate the interval between action potentials which is also called the First Passage Time (FPT), the first time the membrane voltage passes a threshold. The subthreshold depolarization of a neuron receiving a multitude of random synaptic inputs has often been modelled as the Ornstein--Uhlenbeck (OU) process. This model provides an analytically tractable formalism of neuronal membrane voltage mean and variance in terms of a neuron's membrane time constant and the mean of input voltage. Some authors obtained an approximate mean and variance of the FPT for Stein's model with a constant threshold for firing by using Stein's method. They approximated the mean and variance of FPT by using the first term of the Taylor's series expansion. We expect this procedure works for the OU process, a diffusion process. This study finds that Stein's method works well for the OU process with the small Wiener process parameter. After adding a few other terms of the Taylor's series, the parameter range in which the approximation works well are almost the same as the range in which the first term does. The relationship between the approximation results and the confidence band of the mean and variance of the simulated FPT gives evidence that their parameter range is the same; but, the approximation by two terms of the Taylor's series gives less approximation error. The goodness--of--fit--test shows that the lognormal distribution is close to the distribution of FPT for all the Wiener parameters we used. We compared a lognormal distribution of the FPT, estimated from simulation of the OU process, with the probability density function (pdf) of the FPT, approximated from a transformation of the marginal distribution of membrane voltage at the time at which the mean of membrane voltage passes the threshold. We found that the approximation pdf and the lognormal pdf are almost equally close to the true and unknown pdf when the parameter of the Wiener process is small.

Cell populations typically exhibit binary or graded transcriptional responses to external stimuli. Transcription factor interactions with DNA have been hypothesized to account for both of these scenarios. To address this hypothesis, two stochastic models were constructed to describe transcription in simple, engineered eukaryotic systems. In the first system, three transcription factors bind independently to enhancer sites directing production of protein. This system has no regulation in the form of feedback loops, but the system nonetheless exhibits a clear binary response when transcription factor binding fluctuations are slow. The graded response occurs when transcription factor binding fluctuations are rapid. Thus, transcription factor binding fluctuation is an important mechanism underlying and reconciling the graded and binary transcriptional responses. In the second model, the influence of autoregulatory feedback loops on transcription was assessed. Autoregulated systems are capable of exhibiting bistability, a mechanism cited to explain the binary transcriptional response. In this autoregulated system, a dimeric protein acts as a transcription factor to increase its own production. Using biologically realistic parameter values, the system was determined not to be bistable. However, binary transcriptional responses were still observed in stochastic models due to discrete fluctuations in transcription factor binding. The results of both models suggest that transcription factor binding fluctuations play an important, and often overlooked role, in observed patterns of transcriptional activation.

One of the major goals of ecology is to understand the mechanisms which promote species coexistence. Much progress has been made in recent years in understanding how spatial processes influence diversity. Here we present a pair of models designed to investigate the role of one spatial process, localized dispersal, in promoting regional species coexistence when trophic structure is present. The model community considered is a two-predator, two-prey assemblage with cyclic endstates. Through analysis and simulation, we show that increasing the range of species dispersal can inhibit regional species coexistence. This result is at odds with the conclusions of some previous studies which considered locality within a single trophic level, and suggests that the relationship between dispersal distance and regional coexistence may be more complicated than previously realized.

The small intestine is prone to infections by bacteria that can adhere to the surface wall. We wish to better understand an infection of the small intestine caused by enteropathogenic Escherichia coli (EPEC) and to achieve this objective we present three models: a chemostat and two mechanistic spatial models in 1-D and 2-D. The chemostat model is a dynamic model where four biologically significant steady states of the infection were observed: washout, persistence, threshold, and blowup. These results concur with previous work done in this field; however in this instance our model is far less complex. The 2-D mechanistic spatial model suggests that bacteria that adhere to the intestinal wall cause the infection to persist. The 2-D model also suggests that the radial gradients of EPEC are less important than the longitudinal gradients, allowing us to proceed with a 1-D analysis. The 1-D model permits an in-depth realization of the infection process, including bacterial growth and microvilli growth kinetics. This paper will discuss how our 3 models merge EPEC pathogenesis mechanisms with current-day CSTR and PFR colonization models.

Multiple disease susceptibility alleles, genotype errors, or missing genotype data can create problems when testing for association between alleles or genotypes at a genetic marker and a dichotomous phenotype. I used likelihood methods to study the impact of each of these factors on detecting association. In the presence of multiple disease susceptibility alleles, I found that power of the likelihood ratio test (LRT) declines less when based on haplotypes made up of tightly linked single nucleotide polymorphisms (SNPs) than when based on individual SNPs. The result suggests that statistical methods based on haplotypes may be useful to identify and locate complex disease genes. Genotype errors can lead to excess type I error in nuclear family (case-parents) studies when errors resulting in Mendelian inconsistent families are corrected but other errors remain in the data. I developed a LRT for single SNPs or haplotypes that incorporates nuisance parameters for genotype errors and showed that type I error rate can be controlled at little cost to power. For nuclear family data in which missing parents and additional siblings create a diversity of family structures, I developed a unified approach to computing LRT power for a test of association. Comparison of LRT power with power of a family-based association test showed that LRT has greater power.

To increase the analytical tractability of lattice stochastic spatial population models, several approximations have been developed. The pair-edge approximation is a moment-closure method that is effective in predicting persistence criteria and invasion speeds on a homogeneous lattice. Here the effectiveness of the pair-edge approximation is evaluated on a spatially heterogeneous lattice in which some sites are unoccupiable, or ?dead?. This model has several possible interpretations, including a spatial SIS epidemic model, in which immobile host-species individuals occupy some sites while others are empty. As in the homogeneous model, the pair-edge approximation is found to be significantly more accurate than the ordinary pair approximation in determining conditions for persistence. However, habitat heterogeneity decreases invasion speed more than is predicted by the pair-edge approximation, and the discrepancy increases with greater clustering of dead sites. The accuracy of the approximation validates the underlying heuristic picture of population spread and therefore provides qualitative insight into the dynamics of lattice models. Conversely, the situations where the approximation is less accurate reveal limitations of pair approximation in the presence of spatial heterogeneity.

Genes homologous to a set of known, aligned, genes can be found by screening DNA libraries with PCR. PCR primers for such screens are commonly designed via a method described by Sells and Chernoff (1995). This standard design method does not make use of information about the evolutionary relationship between the known genes. The present study investigated the efficacy of using information about evolutionary relationships (inferred from the sequence data) in the design of PCR primers. This study compares the standard primer design method (represented herein by a modified multinomial distribution) with evolutionary model based primer design methods. The primer design method that, given an alignment of known sequences with one sequence left out, assigned a higher probability, on average, to the left-out sequence, was defined as the better method. By this measure of relative performance, an evolutionary model based primer design method sensitive to states correlated across sites of a sequence, outperformed the standard method, on the alignments studied.

This dissertation examines the problem of quantifying amino acid conservation in proteins molecular evolution. Ideally, this conservation is quantified by inferring the rate of evolution at each amino acid site of a multiple-alignment. However, current rate-inference methods have three problematic assumptions. The methods assume that (a) the rates of all sites are independent, (b) the rates are drawn from a known prior distribution, and (c) the mean rate across sites is approximately one. The problems are two-fold. First, the assumptions of site-rate independence and known mean rate are contradictory. To see the contradiction, consider a two-site alignment with known rate of ~0.5 at site one. The rate at site two is unknown under the independent-sites assumption, but is ~1.5 by the assumption of known mean rate. Second, if the rates are drawn from a known prior distribution, the assumption of known distribution implies the question ?which distribution??. Previous work has focused only on selecting better families of rate distributions, often at the expense of additionally parameterizing the evolutionary model. Herein, I develop a method of inferring rates requiring only the assumption of known mean rate, and not requiring additional parameterization. Thus a model of evolution based on our method is a more general framework for inferring rates than previous work. Since a known mean rate is required to distinguish evolutionary rate from time, our method is arguably the most general possible that allows rate and time to be fully and independently identified. The method is assessed by investigating conservation in the Myc, Max, and p53 transcription-factor families.

<p>Multi-site Leslie matrices for sub-divided populations are explored with respect to optimization of management goals and transient dynamics associated with implementing actions to achieve those goals. The management goals explored were minimizing the cost associated with controlling a pest species (Yellow Legged Herring gull, Larus cachinnans), and maximizing the yield from a commercially valuable species (Artco-Norwegian cod, Gadus morhua). Transient dynamics were evaluated for a representative r- and K-selected species, and time to convergence was compared between one-site versus multi-site models, and for different migration patterns, migration levels, and proportion of the population migrating.<br><br>In a density-independent model for the Yellow Legged Herring Gull, the most efficient control technique was to focus management actions on the better quality sites, because breeders at high quality sites had higher expected life-time reproductive values. The amount of harvest required to maintain equilibrium was a function of site quality and the balance between immigration and emigration-cost (and effort) increased as dispersal favored better quality sites. Given a choice between destroying eggs or culling adult breeders, culling required ten times less effort per-capita and would be the optimal strategy as long as per-capita culling cost is no more than ten times greater than the per-egg destruction cost.<br><br>A density-dependent model of Arcto-Norwegian cod revealed that the theoretical yield was maximized from harvesting age 6 individuals. If only the minimum age harvested could be controlled, then the constrained yield was maximized from harvesting ages five and older. Yields were compared between a reserve model with 25% of fishing area closed and a no-reserve model. Yields in the reserve model exceeded the non-reserve model when transfer rates out of the reserve were higher, when higher fecundity was realized in the reserve (which could result from improved habitat quality), and when fishing rates in the non-reserve model were 1.5 and 2.0 times the optimal level.<br><br>In both a density-independent and a density-dependent context, I showed that the optimal strategy could be determined from inspection of elements of the left eigenvector (i.e. reproductive value) divided by a vector of age specific harvest value (or cost of control action, in the case of a pest species). The maximum sustained yield was obtained when the age class with the smallest ratio was harvested; the minimum cost comes from removing individuals with the largest ratio. In one-site models, the optimal strategy involved the harvest of no more than two age classes, where the second (younger) age class had the second smallest ratio (for maximization) or largest ratio (for minimization). However, in multi-site models, the presence of migration permitted the replenishment of age classes beyond the one fully exploited in a harvested site, and thus the optimal strategy could involve the harvest of more than two age classes.<br><br>Optimal solutions to the above models corresponded to equilibrium conditions. However, the amount of time between the implementation of a management action and the attainment of equilibrium can be great. Analysis of transient dynamics revealed that the time to convergence is affected by many factors. I defined convergence as the time when a measured population growth rate (and the growth rate in all subsequent years) was within 1% of the asymptotic growth rate (corresponding to equilibrium for a given action). Comparing single-site versus multi-site models, the multi-site models converged more slowly. Multi-site models that incorporated low levels of migration and migration in only the first age class (as opposed to migration in all age classes) converged slowest. Models for the longer-lived K-selected species generally converged more slowly than the short-lived r-selected species, although for some migration patterns (particularly when emigration out of the site receiving a management action exceeded immigration into that site) models for both species converged quickly. <P>

<p><P> Ecologists and conservation biologists have had an increasing interest in landscapeecology, fragmentation and meta-population structures and dynamics for endangered or threatened species of wildlife (Nichols et al. 1992).They have realized the need for parameter estimates to use in the multi-state models;and have tried estimation of transitionprobabilities among stages using tag-return and capture-recapture models. Thesetransition probabilities are composed of survival and movement rates andcan only be estimated separately when an additional assumption is made(Brownie et al. 1993) that movement occurs at the end of theinterval between time + 1.We generalize this workto allow different movement patterns in the intervalfor multiple tag-recovery and capture-recapture experiments.<P><P> With methods of separating survival and movement rates in multi-state tag-return and capture-recapture models,we develop multi-state fishery tag return models with potential forfisheries that have multiple sites or patches with movement possible between sites. We build on models developed by Brownie et al. (1985), Pollock et al. (1991, 1995), Hoenig et al. (1998 a, b), and Hearn et al. (1998) on twice-a-year tagging for single state models.These methods allow the estimation of patch-specific natural and fishingmortality rates and movement rates between patches. <P><P> We then develop multi-state fishery tag-return and capture-recapture models with potential forfisheries that have multiple sites or patches withmovement possible between sites when taggingtakes place twice a year.These methods allow the estimation of patch-specific reporting, natural and fishing mortality rates, and movement rates between patcheswith movement time following a uniform distribution for two special cases: (1) a two-site tag-return fisheries model;(2) a two-site model with capture-recapture catch and releasesampling in a marine reserve closed to regular fishing and the more usual tag-return sampling in the fishery area, since marine reserves which provide a refuge from fishing are gaining in popularity as a tool of fisheries management, but their effectiveness needs to beevaluated. We present small simulation studies andcompare the Relative Bias and Relative Standard Errors (RES) of estimatesas well as the correlations between estimates.<P><P>