Distributions of activation energy barriers that produce stretched exponential probability distributions for the time spent in each state of the two state reaction A⇌B

Many biochemical reactions consist of the spontaneous fluctuation between two states: A⇌B. For example these two states could be a ligand bound to an enzyme and the ligand and the enzyme separated from each other. A typical case would be the unbinding of CO from myoglobin (Mb), namely, MbCO⇌Mb+CO. Another example is the fluctuation in the ion channel protein in the cell membrane between conformations that are closed to the passage of ions and those that are open to the passage of ions, namely, closed⇌open. Such chemical reactions can be described as two energy levels corresponding to the two states, separated by a distribution of activation energy barriers. Since a kinetic rate can be associated with each energy barrier, this is also equivalent to a distribution of kinetic rate constants. We derive the distribution of the kinetic rates that produces the stretched exponential probability distribution, exp(−at ^b ) where 0<b≤1, which has been observed for such reactions. We also derive the form of the cumulative probability distribution when the pathways connecting the states have minimum or maximum rate constants.     

Exact sampling formulas for multi-type Galton-Watson processes

 Exact formulas for the mean and variance of the proportion of different types in a fixed generation of a multi-type Galton-Watson process are derived. The formulas are given in terms of iterates of the probability generating function of the offspring distribution. It is also shown that the sequence of types backwards from a randomly sampled particle in a fixed generation is a non-homogeneous Markov chain where the transition probabilities can be given explicitly, again in terms of probability generating functions. Two biological applications are considered: mutations in mitochondrial DNA and the polymerase chain reaction. Received: 10 June 2001 / Revised version: 21 November 2001 / Published online: 23 August 2002 Mathematics Subject Classification (2000): Primary 60J80, Secondary 92D10, 92D25 Key words or phrases: Multi-type Galton-Watson process – sampling formula – PCR – mitochondrial DNA     

Time development of probability distributions for interacting species

A general solution to the dynamical equation for the probability distribution associated withn interacting species is obtained by employing the author's generic canonical expression for the rate functions. Interacting species models with limit-cycle dynamics and no stable equilibrium points feature probability distributions that are asymptotic for large values oft to Dirac δ-distributions concentrated on the limit-cycles, as illustrated here for an analytically solvable two-species model. For ann-species Volterra model, a stationary or temporally-averaged probability distribution should generally be much more complicated than the specialized Poisson form studied by Kerner and others.     

A note on the dispersionless growth law for single cells

A population of initially synchronized cells is considered wherein each cell grows according to a dispersionless growth law and the probability of cell division is determined by cell age. The first and second moments of the distribution of birth volumes are considered as functions of time and it is shown that it is impossible for both moments to approach finite, nonzero limits ast→∞. This implies that the volume distribution of the population will not approach a limiting distribution on any finite, nonzero volume interval and that the population will not attain balanced exponential growth. An illustrative example is worked out in detail. The distribution of birth volumes is also analyzed as a function of generation number and it is found that the logarithm of the birth volume in thejth generation is normally distributed asj→∞, with an unbounded variance. Generalizations and implications of these results are briefly discussed. Work supported by the U.S. Atomic Energy Commission.     

Geographical invariance and the strong-migration limit in subdivided populations

Invariance under population subdivision and the strong-migration limit are investigated for digenic samples in neutral models. The monoecious, diploid population is subdivided into a finite number of panmictic colonies that exchange gametes. The backward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Results are derived for the distribution of the place and time of coalescence, for the probability of identity in the model of infinitely many alleles, and for the distribution of the number of nucleotide differences in the model of infinitely many sites without recombination. Received: 5 October 1999 / Revised version: 1 February 2000 / Published online: 4 July 2000     

Distribution of the oriental weatherloach, Misgurnus anguillicaudatus, in paddy fields and its implications for conservation in Sado Island, Japan

The distribution of the oriental weatherloach, Misgurnus anguillicaudatus, together with related environmental factors were surveyed at 185 paddy field locations on Sado Island in Japan. This was associated with the reintroduction of the Japanese crested ibis, Nipponia nippon, which prefers to feed on the loach. Loach were found to be present at 90 locations (49%). Analysis with GIS, GLM and AIC revealed that positive factors for the loach distribution included the presence of an earth ditch, the connections at outlets and the proportion of paddy field area within an 800 m radius. Conversely, pumping-up water irrigation and flow irrigation affected loach distribution negatively. In an interview survey that was conducted concurrently, older farmers in the area recalled that the loach had once been distributed over almost the entire island (89/96, 93%) about half a century ago. The farmers also suggested that the impact of agricultural chemicals was one of the main reasons for loach reduction or local extinction. A map of the current potential loach distribution was made using the best fit model from GIS variables. The loach was expected to be distributed in the Kuninaka region with high probability (60–100%) and in patches in the Osado and Kosado regions with low–medium probability (10–50%). As a feasible scenario for the conservation of the loach, another predictive map of the loach distributions was made using a model fit based on the GIS variables and outlet connection, in which the all of the outlets were presumed to be connected to the ditches without gaps. In this case, the loach were expected to be distributed almost throughout the paddy fields with medium–high probability (40–100%). In another scenario that presumed the complete land consolidation of all the paddy fields, the probability of loach occurrence was low (0–30%) throughout the island with the exception of the island’s center in the Kuninaka region.     

Bayesian image processing of data from constrained source distributions—I. non-valued,uncorrelated and correlated constraints

A series of Bayesian image processing algorithms which incorporate various classes ofa priori source information in treating data which obeys Poisson and Gaussian statistics is derived using maximum entropy considerations. The standard maximum likelihood equations are shown to be a special case of Bayesian image processing when thea priori information about a source distribution φ _j is solely that a non-vanishing probability for each element value φ _j exists only in some finite interval,a _j ≤φ _j ≤φ _j . Bayesian image processing equations for thea priori source information that all φ _j are finite -∞<φ _j <∞ and each φ _j distribution has a defined mean φ _j and a defined variance σ _j are derived. The Bayesian image processing equations are also derived when thea priori source information is that all φ _j ≥0 and that each φ _j distribution has a defined mean φ _j and a defined variance σ _j . The a priori source distribution constraint that a correlation exists among nearby elements is also considered. The results indicate improvement over standard methods.     

The spatial distribution of termites in shortgrass steppe: a geostatistical approach

The broad-scale distribution of subterranean termites (Reticulitermestibialis) was studied in a shortgrass-steppe ecosystem in northern Colorado, United States. Termite occurrence and abundance was measured over 4 months at 10-m intervals along a 900-m transect that spanned a topographic gradient. Geostatistics were used to model the probability of termite occurrence along the transect, and to identify the distributional extent and potential roles of termites in shortgrass steppe. Semivariance was calculated between sample pairs of differing distances and kriging was used to interpolate the probability of termite occurrence along the transect. The semivariogram showed spatial dependence in termite distribution between samples 10–330 m apart and converged on the population variance at distances >330 m, which suggested that spatial dependence explained much of the broad-scale variation in termite distribution. A relatively large nugget variance, however, indicated there was spatial dependence below the 10-m sampling resolution. Termites were most frequently found on a south-facing slope and in a lowland swale. Four-wing saltbush (Atriplexcanescens) was also common in these areas and is important in the production of woody litter. The distribution of termites was significantly associated with proximity to saltbush, which showed a strong spatial dependence at scales <500 m. Kriged probabilities of occurrence and cross-correlation between termites and shrubs showed that peak termite occurrence was shifted upslope 100 m from areas of closest shrub proximity. Other factors, such as soil temperature, texture, or organic matter, are therefore likely to influence termite distributions in shortgrass steppe. The geostatistical approach used here provides a basis for further study on termites in shortgrass steppe, where their roles in decomposition and nutrient cycling are unknown. Geostatistics could also be used to describe distribution patterns on other soil arthropods sampled from traps or soil cores along transects that span topographic or land-use changes. Received: 4 April 1997 / Accepted: 4 November 1997     

Limits of a multi-patch SIS epidemic model

 We start from a stochastic SIS model for the spread of epidemics among a population partitioned into M sites, each containing N individuals; epidemic spread occurs through within-site (`local') contacts and global contacts. We analyse the limit behaviour of the system as M and N increase to ∞. Two limit procedures are considered, according to the order in which M and N go to ∞; independently of the order, the limiting distribution of infected individuals across sites is a probability measure, whose evolution in time is governed by the weak form of a PDE. Existence and uniqueness of the solutions to this problem is shown. Finally, it is shown that the infected distribution converges, as time goes to infinity, to a Dirac measure at the value x ^* , the equilibrium of a single-patch SIS model with contact rate equal to the sum of local and global contacts. Received: 18 July 2001 / Revised version: 16 March 2002 / Published online: 26 September 2002 Mathematics Subject Classification (2000): 92D30, 60F99 Key words or phrases: SIS epidemic – Metapopulation – Markov population processes – Weak convergence of measures     

An age-dependent stochastic model of population growth

A stochastic model of population growth is treated using the Bellman-Harris theory of agedependent stochastic branching processes. The probability distribution for the population size at any time and the expectation are obtained when it is assumed that there is probability (1−σ), 0≤σ<1, of the organism dividing into two at the end of its lifetime, and probability σ that division will not take place.     

Evolutionary and dynamic stability in continuous population games

 Asymptotic stability under the replicator dynamics over a continuum of pure strategies is shown to crucially depend on the choice of topology over the space of mixed population strategies, namely probability measures over the real line. Thus, Strong Uninvadability, proved by Bomze (1990) to be a sufficient condition for asymptotic stability under the topology of variational distance between probability measures, implies convergence to fixation over a pure strategy x ^* only when starting from a population strategy which assigns to x ^* a probability sufficiently close to one. It does not imply convergence to x ^* when starting from a distribution of small deviations from x ^* , regardless of how small these deviations are. It is, therefore, suggested that when a metric space of pure strategies is involved, another topology, hence another stability condition, may prove more relevant to the process of natural selection. Concentrating on the case of a one dimensional continuous quantitative trait, we resort to the natural Maximum Shift Topology in which an ɛ-vicinity of the fixation on a pure strategy x ^* consists of all mixed population strategies with support which includes x ^* and is in the ɛ-neighborhood of x ^* . Under this topology, a relatively simple necessary and sufficient condition for replicator asymptotic stability, namely Continuous Replicator Stability (CRSS), is demonstrated. This condition is closely related to the static stability condition of Neighbor Invadability (Apaloo 1997), and slightly stronger than the condition of Continuous Stability (Eshel and Motro 1981). Received: 9 July 2002 / Revised version: 12 November 2002 / Published online: 19 March 2003 Key words or phrases: CSS – ESS – NIS – Strong uninvadability – Weak topology – Replicator dynamics – Long-term evolution – Continuous population games – Continuous replicator stability – Asymptotic stability     

Existence of steady-state probability distributions in multilocus models for genotype evolution

It is shown that a representative Fisher-Wright model withn(≥3) diallelic loci admits a necessary condition for existence of a time-independent steady-state probability distribution. This necessary condition states that a global integral depending on the phenotype fitness functions of natural selection must be larger than a certain quantity depending on the parameters associated with genetic drift.     

A Mathematical Model for Evolution and SETI

Darwinian evolution theory may be regarded as a part of SETI theory in that the factor f_l in the Drake equation represents the fraction of planets suitable for life on which life actually arose. In this paper we firstly provide a statistical generalization of the Drake equation where the factor f_l is shown to follow the lognormal probability distribution. This lognormal distribution is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution when the number of factors increased to infinity. In addition we show that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of lognormal distributions (b-lognormals) constrained between the time axis and the exponential growth curve. Finally, since each b-lognormal distribution in the family may in turn be regarded as the product of a large number (actually “an infinity”) of independent lognormal probability distributions, the mathematical way is paved to further cast Darwinian Evolution into a mathematical theory in agreement with both its typical exponential growth in the number of living species and the Statistical Drake Equation.     

Neutral Models with Generalised Speciation

Hubbell’s neutral theory claims that ecological patterns such as species abundance distributions can be explained by a stochastic model based on simple assumptions. One of these assumptions, the point mutation assumption, states that every individual has the same probability to speciate. Etienne et al. have argued that other assumptions on the speciation process could be more realistic, for example, that every species has the same probability to speciate (Etienne, et al. in Oikos 116:241–258, 2007). They introduced a number of neutral community models with a different speciation process, and conjectured formulas for their stationary species abundance distribution. Here we study a generalised neutral community model, encompassing these modified models, and derive its stationary distribution, thus proving the conjectured formulas.     

The effects of various spatial distributions of weak noise on rhythmic spiking

We consider the response of the classical Hodgkin–Huxley (HH) spatial system in the weak to intermediate noise regime near the bifurcation to repetitive spiking. The deterministic component of the input (signal) is restricted to a small segment near the origin whereas noise, with parameter σ, occurs either only in the signal region or throughout the whole neuron. In both cases small noise inhibits the spiking and there is a minimum in the spike counts at σ ≈ 0.15. At the same value of σ, the variance of the spike counts undergoes a pronounced maximum. For spatially restricted noise, the spike count continues to increase beyond the minimum until σ = 0.5, but in the case of spatially extended noise the spike count begins to decline around σ = 0.35 to give a local maximum. For both spatial distributions of noise, the variance of the spike count is found to also have a local minimum at about σ = 0.4. Examples are given of the probability distributions of the spike counts and the spatial distributions of spikes with varying noise level. The differences in behaviours of the spike counts as noise increases beyond 0.3 are attributable to noise-induced spiking outside the signal region, which has a larger probability of occurrence when the noise is over an extended region. This aspect is investigated by ascertaining the probability of noise-induced spiking as a function of noise level and examination of the corresponding latency distributions. These findings prompt a definition of weak noise in the standard HH model as that for which the probability of secondary phenomena is negligible, which occurs when σ is less than about 0.3. Finally, if signal and weak (σ < 0.3) noise are applied on disjoint intervals, then the noise has no effect on the instigation or propagation of spikes, no matter how large its region of application. These results are expected to apply to type 2 neurons in general, including the majority of cortical pyramidal cells.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

The probability of treatment induced drug resistance

We propose a discrete time branching process to model the appearance of drug resistance under treatment. Under our assumptions at every discrete time a pathogen may die with probability 1−p or divide in two with probability p. Each newborn pathogen is drug resistant with probability μ. We start with N drug sensitive pathogens and with no drug resistant pathogens. We declare the treatment successful if all pathogens are eradicated before drug resistance appears. The model predicts that success is possible only if p<1/2. Even in this case the probability of success decreases exponentially with the parameter m=μN. In particular, even with a very potent drug (i.e. p very small) drug resistance is likely if m is large.     

When and where was the most recent common ancestor?

 The structured coalescent is investigated for single-locus, digenic samples in the diffusion limit of the unidimensional stepping-stone model for homogeneous, isotropic migration and random genetic drift. Let T denote the scaled time to the most recent common ancestor (MRCA) of the two genes, and let Z designate the scaled deviation of the position of the MRCA from the average position of the two genes. The joint probability density of T and Z is evaluated explicitly. Both the marginal and conditional distributions of T have infinite expectation, as does the marginal distribution of Z. Conditioned on T = τ, the distribution of Z is Gaussian with mean zero and variance 2τ. The main results are extended to anisotropic migration. The results establish the existence of and define in the diffusion limit a retrospective stochastic process for digenic samples in one spatial dimension. Received: 1 May 2001 / Revised version: 2 September 2001 / Published online: 8 February 2002     

A partial digest approach to restriction site mapping

We present a new, practical algorithm to resolve the experimental data in restriction site analysis, which is a common technique for mapping DNA. Specifically, we assert that multiple digestions with a single restriction enzyme can provide sufficient information to identify the positions of the restriction sites with high probability. The motivation for the new approach comes from combinatorial results on the number of mutually homeometric sets in one dimension, where two sets ofn points are homeometric if the multiset ofn(n−1)/2 distances they determine are the same. Since experimental data contain errors, we propose algorithms for reconstructing sets from noisy interpoint distances, including the possibility of missing fragments. We analyse the performance of these algorithms under a reasonable probability distribution, establishing a relative error limit ofr=Θ(1/n ²) beyond which our technique becomes infeasible. Through simulations, we establish that our technique is robust enough to reconstruct data with relative errors of up to 7.0% in the measured fragment lengths for typical problems, which appears sufficient for certain biological applications.     

Survival and population growth of a long-lived threatened snake species, Drymarchon couperi (Eastern Indigo Snake)

Demographic data provide a basis for understanding the life history and ecology of species, factors which are vital for informing conservation efforts; however, little is known regarding the population ecology of most snake species, including the threatened Eastern Indigo Snake (Drymarchon couperi). We used 11 years (1999–2009) of capture-mark-recapture (CMR) and 2.5 years (2003–2005) of radiotelemetry data from southeastern Georgia, USA, in a CMR modeling framework to estimate apparent survival, capture and transition probabilities, and evaluate factors influencing these parameters. The model-averaged estimate of overall apparent annual survival probability was 0.700 (±0.030 SE) and is comparable to that obtained from known fate analysis (radiotelemetry) at the same site. Body size positively influenced survival, regardless of sex. Capture probability differed seasonally by sex, suggesting lower capture probability for females in fall and males in winter. There was no evidence for effect of precipitation or site-specific differences in survival. Model averaged estimate of annual adult survival estimated using multistate CMR models was 0.738 ± 0.030 and 0.515 ± 0.189 for subadults. We estimated population growth rate (λ) and elasticity (proportional sensitivity) of λ to vital rates using a stage-structured matrix population model. Population growth rate ranged from 0.96 to 1.03 depending on the value of the probability of transitioning from subadult to adult stage. The λ was proportionally most sensitive to changes in adult survival rate, followed by subadult survival. Our results suggest that protecting adult snakes and their habitats would result in the highest likelihood of long-term population stability and growth.