Stochastic prey-predator relationships: A random differential equation approach

A stochastic model describing two interacting populations is considered. The model involves a random differential equation of the form dX/dt=A(t)X+Y(t) where the random matrixA and vectorY represent the interactions and growth rates respectively andX is a (random) vector the components of which are the logarithms of the population's sizes. An expression for the solution of the above equation is obtained whence its statistical properties can be determined. Alternatively, a method based on Liouville's theorem is used to obtain the probability distribution of the solution. Application of both methods to simple cases indicates that the random solution is asymptotically stable in the mean even when the solution to the associated deterministic equation is not, viz. in the absence of self interactions.     

Analyzing stochastic events in multi-channel patch clamp data

In an effort to understand the gating properties of ionic channels in biological membranes, an efficient method was developed to estimate the kinetic constants from opening-closing events of the channels. Our method is suitable to single channel patch-clamp recordings that contain several ionic channels functioning simultaneously. It is different from the maximum likelihood method previous developed by Horn and Lange, in that our method is a continuum approach and makes uses of analytic expressions of the probability density functions of the event times. Combinatorial analysis was necessary to correctly include more typical multi-channel recordings. This yields computationally quicker results than the method of Horn and Lange, which uses a discretized time series. Model-dependent portions of the code are minimal and easily modified. To illustrate the goodness of our method, we have generated the open-close processes of the patch-clamp records on a digital computer using the exponential random number generators. For multi-channel patches, we have introduced a few plausible approximations to make our algorithm more efficient. The soundness of the our approximations were tested with such measures as the fraction of the open state at time t, p _open(t), and frequencies of the number of openings per run. A copy of the computer code implementing this algorithm can be obtained from the authors.This work is supported by NIH R01 HL33905-01     

Rate Constants and the Variation of Random Concentration Curves in a 2-Compartment System

In biology and medicine many substances and drugs enter the system not at regular time intervals but rather according to a random process. In the present article a situation is investigated where input enters a 2-compartment system according to a Poisson process. The arising two random concentration curves y(t), one for the central and one for the peripheral compartment are discussed (shot noise). The equations for E[y(t)] and Var [y(t)] are derived. The dependence of E[y(t)] and Var [y(t)] and of the index of dispersion ID[y(t)] on the rate parameters is analysed and discussed in both compartments. The arising calculations were considerably simplified by means of “Mathematica”, a computer program which allows to perform symbolic calculations.     

EVIDENCE FOR AN EXTREME BOTTLENECK IN A RARE MEXICAN PINYON: GENETIC DIVERSITY,DISEQUILIBRIUM, AND THE MATING SYSTEM IN PINUS MAXIMARTINEZII

Maxipiñon (Pinus maximartinezii Rzedowski), which is confined to a single population of approximately 2000 to 2500 mature trees, covers about 400 ha in southern Zacatecas, Mexico. Genetic diversity measured by expected heterozygosity was 0.122, which is moderate for pines. However, percentage polymorphic loci was low, 30.3%. The fixation index (F) of 0.081 indicated only slight heterozygote deficiency. Mating system analysis indicated a significant but low level of selling; the multilocus outcrossing rate, t_m, was 0.816. The mean of single locus estimates, t_s, was smaller (0.761), perhaps suggesting mating among relatives, although the difference between t_m and t_s was not statistically significant. The most striking features of maxipiñon's genetic structure were that no polymorphic locus had more than two alleles and most alleles at polymorphic loci were at intermediate frequencies. This is in contrast to other pines, which often have three to five or more alleles at some loci and in which the distribution of allele frequencies is U-shaped, most alleles being present at frequencies less than 10% or greater than 90%. A population with only two alleles per locus and at intermediate frequencies could occur if the population had been reduced to an extreme bottleneck and then expanded rapidly before random drift modified allele frequencies. A novel origin from a hybridization event would also explain the results. Significant gametic disequilibrium was detected at several pairs of loci in both maternal and paternal gametes. The presence of disequilibrium is in agreement with an origin from an extreme bottleneck, perhaps even a single seed. Furthermore, it demands that the event be relatively recent. The number of generations, as calculated from the observed mean disequilibrium, suggested that maxipiñon derived from an extreme bottleneck four to five generations ago, which is less than 1000 years in this species.     

The “Window <Emphasis Type="Italic">t</Emphasis> test”: a simple and powerful approach to detect differentially expressed genes in microarray datasets

This work focuses on differential expression analysis of microarray datasets. One way to improve such statistical analyses is to integrate biological information in the design of these analyses. In this paper, we will use the relationship between the level of gene expression and variability. Using this biological information, we propose to integrate the information from multiple genes to get a better estimate of individual gene variance, when a small number of replicates are available, to increase the power of the statistical analysis. We describe a strategy named the “Window t test” that uses multiple genes which share a similar expression level to compute the variance which is then incorporated a classic t test. The performances of this new method are evaluated by comparison with classic and widely-used methods for differential expression analysis (the classic Student t test, the Regularized t test (reg t test), SAM, Limma, LPE and Shrinkage t). In each case tested, the results obtained were at least equivalent to the best performing method and, in most cases, outperformed it. Moreover, the Window t test relies on a very simple procedure requiring small computing power compared with other methods designed for microarray differential expression analysis. Electronic Supplementary Material  Supplementary material is available for this article at and is accessible for authorized users.     

An urn model study of variability within a compartment

Formulas are derived for the mean and variance of the number of radioactive atoms present in a compartment (or urn). Initally,n ₁ radioactive atoms andb stable atoms are placed in the urn; and subsequently,r stable atoms are added and an equal number,r, of a random mixture of stable and radioactive atoms is removed per unit time. The expected number of radioactive atoms,E(t), present at timet is, as expected,n ₁ e^−λt where λ=(r/Δt)/(b+r+n ₁). The relative variance, σ²(t)/n ₁ ² , vanishes to zero forr=1, atoms per unit time and for a large number ofn ₁ radioactive atoms; but for a large number of bothr andn ₁ atoms the relative variance is ∼e ^−λt , equal to the fractional retention, fort>1/λ. Thus in studies where radionuclides are injected into animals and a single compartment represents the data, if a large variance is observed it might be due to the fact that large numbers of atoms are transferred out in unit time. When a small variance is observed, this is probably due to the fact that few atoms are transferred in smaller units of time (such that λ is the same in both cases). Research sponsored by the Energy Research and Development Administration under contract with Union Carbide Corporation.     

The Discrete Distribution Used for the Log-rank Test Can Be Inaccurate

When the number of tumors is small, a significance level for the Cox-Mantel (log-rank) test Z is often computed using a discrete approximation to the permutation distribution. For j = 0,…, J let N_j(t) be the number of animals in group j alive and tumor-free at the start of time t. Make a 2 × (1+J) table for each time t of the number of animals R_j(t) with newly palpated tumor out of the total N_j(t) at risk. There are a total of say K tables, one for each distinct time t with observed death or newly palpated tumor. The usual discrete approximation to the permutation distribution of Z is defined by taking tables to be independent with fixed margins N_j(t) and ΣR_j(t) for all t. However, the N_j(t) are random variables for the actual permutation distribution of Z, resulting in dependence among the tables. Calculations for the exact permutation distribution are explained, and examples are given where the exact significance level differs substantially from the usual discrete approximation. The discrepancy arisis primarily because permutations with different Z-scores under the exact distribution can be equal for the discrete approximation, inflating the approximate P-value.     

The convergence in distribution of some simple epidemics

A group of n susceptible individuals exposed to a contagious disease isconsidered. It is assumed that at each point in time one or more susceptible individuals can contract the disease. The progress of this simple batch epidemic is modeled by a stochastic process X_n(t), t[0, ∞), representing the number of infectiveindividuals at time t. In this paper our analysis is restricted to simple batch epidemics with transition rates given by [α²X_n(t){n −X_n(t) +X_n(0)}]^1/2, t[0, ∞), α(0, ∞). This class of simple batch epidemics generalizes a model used and motivated by McNeil (1972) to describe simple epidemic situations. It is shown for this class of simple batch epidemics, that X_n(t), with suitable standardization, converges in distribution as n→∞ to a normal random variable for all t(0, t₀), and t₀ is evaluated.     

Probability functions for components of theDendroctonus frontalis—Host tree population system and their potential use with population models

Detailed investigation of the within-tree population system of Dendroctonus frontalisZimmermann has resulted in a large data base consisting of abundance estimates for various life stages. This data base was used to construct histograms for transformed estimates and several life stage indices. Histograms were also constructed for transformed values of adult residence time, brood development time, and several host-tree characteristics. Probability and cumulative density functions of the Weibull distribution were fitted, in tandem, to the scaled frequencies and interval means for each histogram. The inverse cumulative function is known, and with a uniform random number generator, allows the selection of random deviates from each distribution. This technique can be used for generating initial (starting) values in D. frontalis population models.     

The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

Drift variances of FSTand GST statistics obtained from a finite number of isolated populations

Approximate formulas for the mean and variance of the F_STor G_ST statistic in a finite number of isolated populations are developed under the effect of random genetic drift. Computer simulation has shown that the approximate formulas give a fairly accurate result unless the initial frequency of one of the alleles involved is close to 1 and t/2N is large, where N is the effective size of a subpopulation and t is the number of generations. It is shown that when the number of subpopulations (s) is small, the mean of F_STor G_ST depends on the initial gene frequencies as well as on s. When the initial frequencies of all alleles are nearly equal to each other and the number of subpopulations is large, the distribution of F_ST in the early generations is bell-shaped. In this case Lewontin and Krakauer's k parameter is approximately 2 or less. However, if one of the initial allele frequencies is close to 1, the distribution is skewed and leptokurtic, and the k parameter often becomes larger than 2 in later generations. Thus, even under pure random genetic drift, Lewontin and Krakauer's test of selective neutrality of polymorphic genes in terms of F_ST is not always valid. It is also shown that Jacquard's approximate formula for k generally gives an overestimate.     

Relationship among acidophilic bacteria from diverse environments as determined by randomly amplified polymorphic DNA analysis (RAPD)

Summary Random amplification of polymorphic DNA (RAPD), a PCR-based technique was applied to evaluate genomic diversity among three strains of Acidithiobacillus thiooxidans, five strains of Acidithiobacillus ferrooxidans and one acidophilic moderate thermophile strain, using 45 random primers of five different series. More than 2200 bands were observed, with an average of 45 bands per primer. Primer OPC-3 produced the maximum number of fragments whereas minimum numbers of fragments were produced with primer OPA-5. A dendrogram was generated using cluster analysis by the unweighted pair group method of arithmetic means (UPGMA). The dendrogram showed three groups with similarity ranging from 29 to 85%. The maximum similarity (85%) was observed between the strains T.t₁ and T.t₂ of Acidithiobacillus thiooxidans.     

Mate choice for genetic compatibility in the house mouse

In house mice, genetic compatibility is influenced by the t haplotype, a driving selfish genetic element with a recessive lethal allele, imposing fundamental costs on mate choice decisions. Here, we evaluate the cost of genetic incompatibility and its implication for mate choice in a wild house mice population. In laboratory reared mice, we detected no fertility (number of embryos) or fecundity (ability to conceive) costs of the t, and yet we found a high cost of genetic incompatibility: heterozygote crosses produced 40% smaller birth litter sizes because of prenatal mortality. Surprisingly, transmission of t in crosses using +/t males was influenced by female genotype, consistent with postcopulatory female choice for + sperm in +/t females. Analysis of paternity patterns in a wild population of house mice showed that +/t females were more likely than +/+ females to have offspring sired by +/+ males, and unlike +/+ females, paternity of their offspring was not influenced by +/t male frequency, further supporting mate choice for genetic compatibility. As the major histocompatibility complex (MHC) is physically linked to the t, we investigated whether females could potentially use variation at the MHC to identify male genotype at the sperm or individual level. A unique MHC haplotype is linked to the t haplotype. This MHC haplotype could allow the recognition of t and enable pre‐ and postcopulatory mate choice for genetic compatibility. Alternatively, the MHC itself could be the target of mate choice for genetic compatibility. We predict that mate choice for genetic compatibility will be difficult to find in many systems, as only weak fertilization biases were found despite an exceptionally high cost of genetic incompatibility.     

The dynamic of the t-haplotype in wild populations of the house mouse Mus musculus domesticus in Israel

The t-haplotype, a variant of the proximal part of the mouse chromosome 17, is composed of at least four inversions and is inherited as a single genetic unit. The haplotype causes embryonic mortality or male sterility when homozygous. Genes within the complex are responsible for distortion of Mendelian transmission ratio in males. Thus, the t-haplotype in heterozygous males is transferred to over 95% of the progeny. We examined the dynamic and behavior of the t-haplotype in wild populations of the house mouse in Israel. The Israeli populations show high frequency (15%–20%) of both partial and complete t-carrying mice, supporting the suggestion that the t-complex evolved in the M. domesticus line in the Israeli region. In one population that had the highest frequency of t-carrying individuals, we compared the level of gene diversity between t-carrying and normal mice in the marker’s loci: H-2 locus of the major histocompatibility complex (MHC) on the t-haplotype of chromosome 17, three microsatellites on other chromosomes, and the mitochondrial D-loop. Genetic variability was high in all tested loci in both t and (+) mice. All t mice carried the same chromosome and showed the same H-2 haplotype. While t-carrying mice showed significant H-2 heterozygotes access, (+) mice expressed significant H-2 heterozygote deficiency. There were no differences in the level of gene diversity between t and (+) mice in the other loci. Heterozygosity level at the MHC may be an additional factor in the selective forces balancing the t-haplotype polymorphism.     

A Stochastic Model of Gene Evolution with Time Dependent Pseudochaotic Mutations

We develop here a new class of stochastic models of gene evolution in which a random subset of the 64 possible trinucleotides mutates at each evolutionary time t according to some time dependent substitution probabilities. Therefore, at each time t, the numbers and the types of mutable trinucleotides are unknown. Thus, the mutation matrix changes at each time t. This pseudochaotic model developed generalizes the standard model in which all the trinucleotides mutate at each time t. It determines the occurrence probabilities at time t of trinucleotides which pseudochaotically mutate according to 3 time dependent substitution parameters associated with the 3 trinucleotide sites. The main result proves that under suitable assumptions, this pseudochaotic model converges to a uniform probability vector identical to that of the standard model. Furthermore, an application of this pseudochaotic model allows an evolutionary study of the 3 circular codes identified in both eukaryotic and prokaryotic genes. A circular code is a particular set of trinucleotides whose main property is the retrieval of the frames in genes locally, i.e., anywhere in genes and particularly without start codons, and automatically with a window of a few nucleotides. After a certain evolutionary time and with particular time dependent functions for the 3 substitution parameters, precisely an exponential decrease in the 1st and 2nd trinucleotide sites and an exponential increase in the 3rd one, this pseudochaotic model retrieves the main statistical properties of the 3 circular codes observed in genes. Furthermore, it leads to a circular code asymmetry stronger than the standard model (nonpseudochaotic) and, therefore, to a better correlation with the genes.     

Optimizing resolution in multidimensional NMR by three-way decomposition

Resolution depends on the number of points sampled in a FID; in indirectly detected dimensions it is an important determinant of the total experiment time. Based on the high redundancy present in NMR data, we propose the following timesaving scheme for three-dimensional spectra. An extensive grid of discrete t1- and t2-values is used, which increases resolution while preserving the spectral width. Total experiment time is reduced by avoiding the recording of t3-FIDs for selected pairs of t1 and t2; typically the recording is omitted for about 75% of the (t1,t2) combinations. These data sets are referred to as sparse, and post-experimental processing making optimal use of spectral redundancy provides the missing, non-recorded data. We have previously shown that three-way decomposition (TWD) within the MUNIN approach provides a practical way to process dense NMR data sets. Here, a novel TWD algorithm [Ibraghimov, (2002) Numer. Linear Algebra Appl. 9, 551–565] is used to complement a sparselyrecorded time-domain data set by providing the missing FIDs for all (t1,t2) combinations omitted in the experiment. A necessary condition is that for each t1-value at least a few FIDs are recorded, and similar for each t2-value. The method is demonstrated on non-uniformly sampled ¹⁵N-NOESY-HSQC data sets recorded for the 14 kD protein azurin. The spectra obtained by TWD, reconstruction and ordinary transform to frequency-domain are, in spite of the large number of signals and the high dynamic range typical for NOESYs, highly similar to a corresponding reference spectrum, for which all (t1,t2) combinations were recorded.     

Transients of delayed fluorescence induction signal and photosynthetic antennas: A possible relationship. Mathematical modeling approach

A mathematical model was developed for resolved temporal transients of experimentally recorded delayed fluorescence (DF) induction signal. During an intermittent light regime, antennas of the photosynthetic apparatus were treated as targets, repeatedly hit by potentially absorbable photons within a series of consecutive light flashes. Formulas were derived for the number of antennas, cumulatively hit by a specific number of photons, as a function of the flash serial number (time). Model parameters included number of absorbable photons in one flash, antenna sizes, and their number. A series of induction curves were analyzed, obtained from a Zea mays leaf segment and differing in the previous dark period (t _d). Each curve, consisting of the two most prominent DF transients (C and D), was fitted with several model types, differing in the number of absorbed photons. For both transients, the best fitting result was achieved when DF induction was linked to the second absorbed photon. As expected, model parameters related to antenna sizes showed weaker dependence on t _d than those referring to antenna number. With restrictions applied to this model, the two DF induction transients may be related to two classes of photosynthetic antennas. Their different sizes may have a predominant influence on the efficiency of photon absorption and possibly time-dependent appearance of DF transients. Published in Russian in Fiziologiya Rastenii, 2006, Vol. 53, No. 3, pp. 325–335. The text was submitted by the authors in English.     

Kinetics of the assembly of gas vacuoles in the blue-green alga Microcystis aeruginosa Kuetz. emend. Elekin.

Summary In exponentially growing cultures of the blue-green alga Microcystis aeruginosa the number of gas vacuoles per cell decreases and reaches a value of 4.5×10³ at 1.2×10⁷ cells per ml. The assembly of gas vacuoles with respect to number and length was followed after the organelles were caused to collapse by a pulse of ultrasound. The change in the number N of gas vacuoles per cell is N=224.8×t ^0.757, 0<t<24 h. After 24 h 50% of the value of non-sonicated cultures is reached. The changes in the length L of the organelles is expressed by L=87.06 ×t ^0.4084, 0<t<24 h. After 24 h 85% of the control value is reached.Abbreviations used EDTA ethylene diamine tetraacetate - BSA bovine serum albumine - P probability - N number of gas vacuoles - L length of gas vacuoles in nm - t time in hours     

A mathematical model of biological evolution

In order to understand generally how the biological evolution rate depends on relevant parameters such as mutation rate, intensity of selection pressure and its persistence time, the following mathematical model is proposed: dN _n (t)/dt=(m _n (t-)N _n (t)+N _n-1(t) (n=0,1,2,3...), where N _n (t) and m _n (t) are respectively the number and Malthusian parameter of replicons with step number n in a population at time t and is the mutation rate, assumed to be a positive constant. The step number of each replicon is defined as either equal to or larger by one than that of its parent, the latter case occurring when and only when mutation has taken place. The average evolution rate defined by is rigorously obtained for the case (i) m _n (t)=m _n is independent of t (constant fitness model), where m _n is essentially periodic with respect to n, and for the case (ii) (periodic fitness model), together with the long time average m of the average Malthusian parameter . The biological meaning of the results is discussed, comparing them with the features of actual molecular evolution and with some results of computer simulation of the model for finite populations.An early version of this study was read at the International Symposium on Mathematical Topics in Biological held in kyoto, Japan, on September 11–12, 1978, and was published in its Procedings.     

On the empirical choice of the time window for restricted mean survival time

The t-year mean survival or restricted mean survival time (RMST) has been used as an appealing summary of the survival distribution within a time window [0, t]. RMST is the patient's life expectancy until time t and can be estimated nonparametrically by the area under the Kaplan-Meier curve up to t. In a comparative study, the difference or ratio of two RMSTs has been utilized to quantify the between-group-difference as a clinically interpretable alternative summary to the hazard ratio. The choice of the time window [0, t] may be prespecified at the design stage of the study based on clinical considerations. On the other hand, after the survival data have been collected, the choice of time point t could be data-dependent. The standard inferential procedures for the corresponding RMST, which is also data-dependent, ignore this subtle yet important issue. In this paper, we clarify how to make inference about a random “parameter.” Moreover, we demonstrate that under a rather mild condition on the censoring distribution, one can make inference about the RMST up to t, where t is less than or even equal to the largest follow-up time (either observed or censored) in the study. This finding reduces the subjectivity of the choice of t empirically. The proposal is illustrated with the survival data from a primary biliary cirrhosis study, and its finite sample properties are investigated via an extensive simulation study.