We investigate the properties of a Wright-Fisher diffusion process started from frequency x at time 0 and conditioned to be at frequency y at time T. Such a process is called a bridge. Bridges arise naturally in the analysis of selection acting on st
anding variation and in the inference of selection from allele frequency time series. We establish a number of results about the distribution of neutral Wright-Fisher bridges and develop a novel rejection sampling scheme for bridges under selection that we use to study their behavior.
We propose general principles for semantic networks allowing them to be implemented as dynamical neural networks. Major features of our scheme include: (a) the interpretation that each node in a network stands for a bound integration of the meanings
of all nodes and external events the node links with; (b) the systematic use of nodes that stand for categories or types, with separate nodes for instances of these types; (c) an implementation of relationships that does not use intrinsically typed links between nodes.
Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. To understand the interactive effects of environmental stochasticity, spatial heterogeneity
, and dispersal on population growth, we study the following model for population abundances in $n$ patches: the conditional law of $X_{t+dt}$ given $X_t=x$ is such that when $dt$ is small the conditional mean of $X_{t+dt}^i-X_t^i$ is approximately $[x^imu_i+sum_j(x^j D_{ji}-x^i D_{ij})]dt$, where $X_t^i$ and $mu_i$ are the abundance and per capita growth rate in the $i$-th patch respectivly, and $D_{ij}$ is the dispersal rate from the $i$-th to the $j$-th patch, and the conditional covariance of $X_{t+dt}^i-X_t^i$ and $X_{t+dt}^j-X_t^j$ is approximately $x^i x^j sigma_{ij}dt$. We show for such a spatially extended population that if $S_t=(X_t^1+...+X_t^n)$ is the total population abundance, then $Y_t=X_t/S_t$, the vector of patch proportions, converges in law to a random vector $Y_infty$ as $ttoinfty$, and the stochastic growth rate $lim_{ttoinfty}t^{-1}log S_t$ equals the space-time average per-capita growth rate $sum_imu_iE[Y_infty^i]$ experienced by the population minus half of the space-time average temporal variation $E[sum_{i,j}sigma_{ij}Y_infty^i Y_infty^j]$ experienced by the population. We derive analytic results for the law of $Y_infty$, find which choice of the dispersal mechanism $D$ produces an optimal stochastic growth rate for a freely dispersing population, and investigate the effect on the stochastic growth rate of constraints on dispersal rates. Our results provide fundamental insights into ideal free movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology.
Recent whole genome polymerase binding assays have shown that a large proportion of unexpressed genes have pre-assembled RNA pol II transcription initiation complex stably bound to their promoters. Some such promoter proximally paused genes are regul
ated at transcription elongation rather than at initiation; it has been proposed that this difference allows these genes to both express faster and achieve more synchronous expression across populations of cells, thus overcoming molecular noise arising from low copy number factors. It has been established experimentally that genes which are regulated at elongation tend to express faster and more synchronously; however, it has not been shown directly whether or not it is the change in the regulated step {em per se} that causes this increase in speed and synchrony. We investigate this question by proposing and analyzing a continuous-time Markov chain model of polymerase complex assembly regulated at one of two steps: initial polymerase association with DNA, or release from a paused, transcribing state. Our analysis demonstrates that, over a wide range of physical parameters, increased speed and synchrony are functional consequences of elongation control. Further, we make new predictions about the effect of elongation regulation on the consistent control of total transcript number between cells, and identify which elements in the transcription induction pathway are most sensitive to molecular noise and thus may be most evolutionarily constrained. Our methods produce symbolic expressions for quantities of interest with reasonable computational effort and can be used to explore the interplay between interaction topology and molecular noise in a broader class of biochemical networks. We provide general-purpose code implementing these methods.
The Chandra Source Catalog (CSC) is a general purpose virtual X-ray astrophysics facility that provides access to a carefully selected set of generally useful quantities for individual X-ray sources, and is designed to satisfy the needs of a broad-ba
sed group of scientists, including those who may be less familiar with astronomical data analysis in the X-ray regime. The first release of the CSC includes information about 94,676 distinct X-ray sources detected in a subset of public ACIS imaging observations from roughly the first eight years of the Chandra mission. This release of the catalog includes point and compact sources with observed spatial extents <~ 30. The catalog (1) provides access to the best estimates of the X-ray source properties for detected sources, with good scientific fidelity, and directly supports scientific analysis using the individual source data; (2) facilitates analysis of a wide range of statistical properties for classes of X-ray sources; and (3) provides efficient access to calibrated observational data and ancillary data products for individual X-ray sources, so that users can perform detailed further analysis using existing tools. The catalog includes real X-ray sources detected with flux estimates that are at least 3 times their estimated 1 sigma uncertainties in at least one energy band, while maintaining the number of spurious sources at a level of <~ 1 false source per field for a 100 ks observation. For each detected source, the CSC provides commonly tabulated quantities, including source position, extent, multi-band fluxes, hardness ratios, and variability statistics, derived from the observations in which the source is detected. In addition to these traditional catalog elements, for each X-ray source the CSC includes an extensive set of file-based data products that can be manipulated interactively.
We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the $m$-dimensional lattice and for which the $m$ matrices that record the transition probabi
lities in each of the lattice directions commute pairwise. One reason such processes are of interest is that the transition matrix is straightforward to diagonalize, and hence it is easy to compute $n$ step transition probabilities. The set of commuting birth-and-death processes decomposes as a union of toric varieties, with the main component being the closure of all processes whose nearest neighbor transition probabilities are positive. We exhibit an explicit monomial parametrization for this main component, and we explore the boundary components using primary decomposition.
If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting MRCA age process has be
en studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright--Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on $mathbb{R}_{++}timesmathbb{R}_{++}$ with intensity measures of the form $lambdaotimesmu$ where $lambda$ is Lebesgue measure, and $mu$ (the family lifetime measure) is an arbitrary, absolutely continuous measure satisfying $mu((0,infty))=infty$ and $mu((x,infty))<infty$ for all $x>0$. Special cases of this construction describe the time evolution of the MRCA age in $(1+beta)$-stable continuous state branching processes conditioned on nonextinction--a particular case of which, $beta=1$, is Fellers continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist.
New models for evolutionary processes of mutation accumulation allow hypotheses about the age-specificity of mutational effects to be translated into predictions of heterogeneous population hazard functions. We apply these models to questions in the
biodemography of longevity, including proposed explanations of Gompertz hazards and mortality plateaus, and use them to explore the possibility of melding evolutionary and functional models of aging.
We consider the asymptotic behavior as $ntoinfty$ of the spectra of random matrices of the form [frac{1}{sqrt{n-1}}sum_{k=1}^{n-1}Z_{nk}rho_n ((k,k+1)),] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $
rho_n((k,k+1))$ are obtained by applying an irreducible unitary representation $rho_n$ of the symmetric group on ${1,2,...,n}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $rho_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on ${1,2,...,n}$ are indexed by partitions $lambda_n$ of $n$. A consequence of the results we establish is that if $lambda_{n,1}gelambda_{n,2}ge...ge0$ is the partition of $n$ corresponding to $rho_n$, $mu_{n,1}gemu_{n,2}ge >...ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $mu_n$ is the transpose of the Young diagram of $lambda_n$), $lim_{ntoinfty}frac{lambda_{n,i}}{n}=p_i$ for each $ige1$, and $lim_{ntoinfty}frac{mu_{n,j}}{n}=q_j$ for each $jge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $theta Z$ and variance $1-theta^2$, where $theta$ is the constant $sum_ip_i^2-sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.
We study a continuous-time dynamical system that models the evolving distribution of genotypes in an infinite population where genomes may have infinitely many or even a continuum of loci, mutations accumulate along lineages without back-mutation, ad
ded mutations reduce fitness, and recombination occurs on a faster time scale than mutation and selection. Some features of the model, such as existence and uniqueness of solutions and convergence to the dynamical system of an approximating sequence of discrete time models, were presented in earlier work by Evans, Steinsaltz, and Wachter for quite general selective costs. Here we study a special case where the selective cost of a genotype with a given accumulation of ancestral mutations from a wild type ancestor is a sum of costs attributable to each individual mutation plus successive interaction contributions from each $k$-tuple of mutations for $k$ up to some finite ``degree. Using ideas from complex chemical reaction networks and a novel Lyapunov function, we establish that the phenomenon of mutation-selection balance occurs for such selection costs under mild conditions. That is, we show that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.
