Survival Chances of Mutants Starting With One Individual

A simple theoretical model of a Darwinian system (a periodic system with a multiplication phase and a selection phase) of entities (initial form of polymer strand, primary mutant and satellite mutants) is given. First case: one mutant is considered. One individual of the mutant appears in the multiplication phase of the first generation. The probabilities to find N mutants W_n^M(N) after the multiplication phase M of the n-th generation (with probability δ of an error in the replication, where all possible errors are fatal errors) and W_n^S(N) after the following selection phase S (with probability β that one individual survives) are given iteratively. The evolutionary tree is evaluated. Averages from the distributions and the probability of extinction W_∞^S(0) are obtained. Second case: two mutants are considered (primary mutant and new form). One individual of the primary mutant appears in the multiplication phase of the first generation. The probabilities to find N_p primary mutants and N_m of the new form W_n^M(N_p, N_m) after the multiplication phase M of the n-th generation (probability ε of an error in the replication of the primary mutant giving the new form) and W_n^S(N_p, N_m) after the following selection phase S (probabilities β_p and β_m that one individual each of the primary mutant and of the new form survives) are given iteratively. Again the evolutionary tree is evaluated. Averages from the distributions are obtained.     

Entropy and the complexity of graphs: I. An index of the relative complexity of a graph

The structural information content (Rashevsky, 1955; Trucco 1956a, b)I _g (X) of a graphX is defined as the entropy of the finite probability scheme constructed from the orbits of its automorphism groupG(X). The behavior ofI _g on various graph operations—complement, sum, join, cartesian product and composition, is examined. The principal result of the paper is the characterization of a class of graph product operations on whichI _g is semi-additive. That is to say, conditions are found for binary operations o and ∇ defined on graphs and groups, respectively, which are sufficient to insure thatI _g (X o Y)=I _g (X)+I _g (Y)−H _XY , whereH _XY is a certain conditional entropy defined relative to the orbits ofG(X o Y) andG(X) ∇G(Y).     

In vivo cloning of a carboxy-terminalrpoB allele which confers altered transcriptional properties

A 3′-terminal mutation of the gene encoding the β subunit ofEscherichia coli RNA polymerase was isolated using anin vivo polA(Ts) technique. Cloning of the allele was monitored by virtue of the fact that the deletion Δ(rpoB) 1570-1 resulted in an altered-size restriction fragment. DNA sequencing confirmed the predicted nature and location of the mutation: Δ(rpoB) 1570-1 involved an in-frame deletion of 186 bp (62 codons) encoding amino acid residues 967–1028. The phenotype conferred by Δ(rpoB) 1570-1 is discussed with respect to conserved domains within the β polypeptide. Dedicated to Dr. J. Spížek on the occasion of his 60th birthday     

Spatial pattern indices based on distances between individuals on a line-segment with finite length

Let us consider a strip-wise habitat of line-segment, like a corridor, to simplify the subject mathematically, and assume that the length of the habitat is γ and there are n individuals. Here, we assume that the spatial pattern of the individuals is random if the n distances from the left end of the habitat to each individual follow a uniform distribution on the strip. Under such an assumption, the variance of the distances between any two neighbors is represented by the formula nθ²(n+1)⁻²(n+2)⁻¹ and the variance between n+1 distances between n individuals from the left end to the right end to the strip, is represented by the formula nθ²(n+1)⁻²(n+2)⁻¹. These two kinds of variances can be used for determining (1) the spatial pattern of a population on the strip and (2) the spatial structure within the population, by comparison with the variances calculated from the data. Two examples cited from the literature, a cattle population on a pasture and an aphid population on a sycamore leaf, are presented.     

Turing instabilities in general systems

We present necessary and sufficient conditions on the stability matrix of a general n(≥2)-dimensional reaction-diffusion system which guarantee that its uniform steady state can undergo a Turing bifurcation. The necessary (kinetic) condition, requiring that the system be composed of an unstable (or activator) and a stable (or inhibitor) subsystem, and the sufficient condition of sufficiently rapid inhibitor diffusion relative to the activator subsystem are established in three theorems which form the core of our results. Given the possibility that the unstable (activator) subsystem involves several species (dimensions), we present a classification of the analytically deduced Turing bifurcations into p (1 ≤p≤ (n− 1)) different classes. For n = 3 dimensions we illustrate numerically that two types of steady Turing pattern arise in one spatial dimension in a generic reaction-diffusion system. The results confirm the validity of an earlier conjecture [12] and they also characterise the class of so-called strongly stable matrices for which only necessary conditions have been known before [23, 24]. One of the main consequences of the present work is that biological morphogens, which have so far been expected to be single chemical species [1–9], may instead be composed of two or more interacting species forming an unstable subsystem. Received: 21 September 1999 / Revised version: 21 June 2000 / Published online: 24 November 2000     

A note on relations between sets

From the definition of a strong and weakn-ary relation betweenn sets, given in a previous paper (Bulletin of Mathematical Biophysics,27, 477–492), it follows that for a given set ofn sets and givenn-ary relationR between them there can exist only one strong relation, but a large number of weak ones. An expression for the total number of possible weak relations is derived and the notion of the degree of weakness of a relation is introduced and discussed.     

Note on a combinatorial problem

In the first part of this paper we have assembled some properties of the quantitiesR _m ⁿ , whereR _m ⁿ denotes the number of distributions ofn different objects intom indifferent parcels, with no empty parcels allowed. We then discuss the following problem (N. Rashevsky, 1954, 1955 a,b, 1956): to find the total number,G _n , of graphs that can be obtained from the biotopological transformation (T ⁽¹⁾ X) for a given value of the parametern. This is related to the distribution ofn indifferent objects intom different boxes. A formula forG _n is given which, however, is not very convenient for practical computations because it involves a summation over certain “admissible partitions” of the numbermn (m is a second parameter of the transformation). Some theorems are derived; with their help we can simplify the calculation ofG _n to a small extent. The numbersG _n are calculated forn≤9 and estimated forn=10. It is found thatG ₇≈5.4×10⁴,G ₈≈8.3×10⁵,G ₉≈1.4×10⁷, andG ₁₀≈3×10⁸. These values ofn are those which might be used in connection with N. Rashevsky’s work (cf. Rashevsky, 1956).     

A chemometrical study of intermolecular properties of hydrogen-bonded complexes formed by C2H4O⋅⋅⋅HX and C2H5N⋅⋅⋅HX with X = F, CN, NC, and CCH

We presents a chemometrical study of the intermolecular properties of the C₂H₄O⋅⋅⋅HX and C₂H₅N⋅⋅⋅HX hydrogen-bonded complexes with X = F, CN, NC, and CCH. Through the MP2 perturbation theory and B3LYP hybrid functional, as well as modifications on 6-31ijGk basis sets with i = triple-zeta, j = diffuse and k = polarization functions, systematic tendencies in the R_{(n⋅⋅⋅HX)} hydrogen bond distances and υ_{(n⋅⋅⋅HX)} stretch frequencies were determined by the hierarchical cluster analysis, two level factorial designs and principal component analysis. Based on well-fitted math models, not only because polarization functions provide a great variance on statistical analysis, but this basis set reproduces more efficiently the available experimental results. Moreover, independent of whether the quality on basis set is increased, the effects yielded by both DFT and MP2 were not considered important in the statistical analysis.     

Bounds on the total particle number for species governed by reaction-diffusion equations in the infinite spatial domain

Differential inequality methods are developed for establishing upper and lower bounds on the total particle numberN(t)=∫θ(x,t) d³ x associated with solutions to nonlinear reaction-diffusion equations of the form ∂θ/∂t=D∇²θ+fθ-gθ ⁿ⁺¹ , whereD(>0),n(>0),f andg are constant parameters. If finite in a neighborhood oft=0,N(t) is bounded below for allt≥0 by a certain derived function oft for equations withg≥0. An upper bound onN(t) is obtained for equations withn=1,f<0 andg<0. These results provide general preservation and extinction criteria for the total particle number.     

Efficient parallel and out of core algorithms for constructing large bi-directed de Bruijn graphs

Background  

Assembling genomic sequences from a set of overlapping reads is one of the most fundamental problems in computational biology. Algorithms addressing the assembly problem fall into two broad categories - based on the data structures which they employ. The first class uses an overlap/string graph and the second type uses a de Bruijn graph. However with the recent advances in short read sequencing technology, de Bruijn graph based algorithms seem to play a vital role in practice. Efficient algorithms for building these massive de Bruijn graphs are very essential in large sequencing projects based on short reads. In an earlier work, an O(n/p) time parallel algorithm has been given for this problem. Here n is the size of the input and p is the number of processors. This algorithm enumerates all possible bi-directed edges which can overlap with a node and ends up generating Θ(nΣ) messages (Σ being the size of the alphabet).     

Minimum sample sizes for identifying chromosomal fragile sites from individuals: Monte Carlo estimation

A Monte Carlo simulation procedure was used to estimate the exact level of the standardized X ² test statistic (X _s ²) for randomness in the FSM methodology for the identification of fragile sites from chromosomal breakage data for single individuals. A random-number generator was used to simulate 10 000 chromosomal breakage data sets, each corresponding to the null hypothesis of no fragile sites for numbers of chromosomal breaks (n) from 1 to 2000 and at three levels of chromosomal band resolution (k). The reliability of the test was assessed by comparisons of the empirical and nominal α levels for each of the corresponding values of n and k. These analyses indicate that the sparse and discrete nature of chromosomal breakage data results in large and unpredictable discrepancies between the empirical and nominal α levels when fragile site identifications are based on small numbers of breaks (n < 0.5 k). With n≥ 0.5 k, the distribution of X _s ² appears to be stable and non-significant differences in the empirical and nominal α levels are generally obtained. These results are inherent to the nature of the data and are, therefore, relevant to any statistical model for the identification of fragile sites from chromosomal breakage data. For FSM identification of fragile sites at α = 0.05, we suggest that n≥ 0.5 k is the minimum reliable number of mapped chromosomal breaks per individual. Received: 28 April 1997 / Accepted: 1 July 1997     

Computational experience with a parallel algorithm for tetrangle inequality bound smoothing

Determining molecular structure from interatomic distances is an important and challenging problem. Given a molecule with n atoms, lower and upper bounds on interatomic distances can usually be obtained only for a small subset of the atom pairs, using NMR. Given the bounds so obtained on the distances between some of the atom pairs, it is often useful to compute tighter bounds on all the pairwise distances. This process is referred to as bound smoothing. The initial lower and upper bounds for the pairwise distances not measured are usually assumed to be 0 and ∞. One method for bound smoothing is to use the limits imposed by the triangle inequality. The distance bounds so obtained can often be tightened further by applying the tetrangle inequality—the limits imposed on the six pairwise distances among a set of four atoms (instead of three for the triangle inequalities). The tetrangle inequality is expressed by the Cayley—Menger determinants. For every quadruple of atoms, each pass of the tetrangle inequality bound smoothing procedure finds upper and lower limits on each of the six distances in the quadruple. Applying the tetrangle inequalities to each of the ( ₄ ⁿ ) quadruples requires O(n ⁴) time. Here, we propose a parallel algorithm for bound smoothing employing the tetrangle inequality. Each pass of our algorithm requires O(n ³ log n) time on a CREW PRAM (Concurrent Read Exclusive Write Parallel Random Access Machine) with processors. An implementation of this parallel algorithm on the Intel Paragon XP/S and its performance are also discussed.     

Extinction dynamics in mainland-island metapopulations: an N-patch stochastic model

A generalization of the well-known Levins’ model of metapopulations is studied. The generalization consists of (i) the introduction of immigration from a mainland, and (ii) assuming the dynamics is stochastic, rather than deterministic. A master equation, for the probability that n of the patches are occupied, is derived and the stationary probability P _s (n), together with the mean and higher moments in the stationary state, determined. The time-dependence of the probability distribution is also studied: through a Gaussian approximation for general n when the boundary at n = 0 has little effect, and by calculating P(0, t), the probability that no patches are occupied at time t, by using a linearization procedure. These analytic calculations are supplemented by carrying out numerical solutions of the master equation and simulations of the stochastic process. The various approaches are in very good agreement with each other. This allows us to use the forms for P _s 0) and P(0, t) in the linearization approximation as a basis for calculating the mean time for a metapopulation to become extinct. We give an analytical expression for the mean time to extinction derived within a mean field approach. We devise a simple method to apply our mean field approach even to complex patch networks in realistic model metapopulations. After studying two spatially extended versions of this nonspatial metapopulation model—a lattice metapopulation model and a spatially realistic model—we conclude that our analytical formula for the mean extinction time is generally applicable to those metapopulations which are really endangered, where extinction dynamics dominates over local colonization processes. The time evolution and, in particular, the scope of our analytical results, are studied by comparing these different models with the analytical approach for various values of the parameters: the rates of immigration from the mainland, the rates of colonization and extinction, and the number of patches making up the metapopulation.     

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.     

On the maximal cliques in c-max-tolerance graphs and their application in clustering molecular sequences

Given a set S of n locally aligned sequences, it is a needed prerequisite to partition it into groups of very similar sequences to facilitate subsequent computations, such as the generation of a phylogenetic tree. This article introduces a new method of clustering which partitions S into subsets such that the overlap of each pair of sequences within a subset is at least a given percentage c of the lengths of the two sequences. We show that this problem can be reduced to finding all maximal cliques in a special kind of max-tolerance graph which we call a c-max-tolerance graph. Previously we have shown that finding all maximal cliques in general max-tolerance graphs can be done efficiently in O(n ³ + out). Here, using a new kind of sweep-line algorithm, we show that the restriction to c-max-tolerance graphs yields a better runtime of O(n ² log n + out). Furthermore, we present another algorithm which is much easier to implement, and though theoretically slower than the first one, is still running in polynomial time. We then experimentally analyze the number and structure of all maximal cliques in a c-max-tolerance graph, depending on the chosen c-value. We apply our simple algorithm to artificial and biological data and we show that this implementation is much faster than the well-known application Cliquer. By introducing a new heuristic that uses the set of all maximal cliques to partition S, we finally show that the computed partition gives a reasonable clustering for biological data sets.     

Time course solutions of the Sax-Markov binary eurejoining/misrejoining model of DNA double-strand breaks

The Sax-Markov binary eurejoining/misrejoining (SMBE) model is a stochastic representation of Sax’s breakage-and-reunion mechanism of misrejoining DNA double-strand breaks (DSBs). In this model, to approximate DSB misrejoining probabilities that decrease with increasing distance, the nucleus is treated as a collection ofη isolated nuclear subvolumes called sites; DSB free ends within the same site interact with a probability that is independent of distance, and DSB free ends within different sites never interact. In our previous work, SMBE steady-state solutions were used to estimateη from a combination of high-dose PFGE (pulsed-field gel electrophoresis) data and moderate-dose chromosomal aberration data. Here, analytic SMBE transient solutions (i.e., time courses of DSBs and misrejoinings) are derived and used to estimateη from various sets of misrejoining DSB kinetic data. The time courses are multiexponentials with rate constantsκ, 6κ, 15κ, ... j(2j–1)κ corresponding to different nuclear site states and not different types of DSBs. For example, theκ component corresponds to nuclear sites with two DSB free ends and thus only one possible rejoining interaction, and the 6κ component corresponds to sites with four DSB free ends and thus six (four choose two) potential rejoining interactions – four of these six potential interactions lead to a final state of two misrejoinings and the other two of six lead to a final state of correct repair (unrejoinable DSBs are not represented in the SMBE model). The SMBE time course solutions provide site number estimates that fall in the range ofη≈10–100 for premature chromosome condensation (PCC) data andη≈1000 for PFGE data. Received: 23 December 1999 / Accepted: 1 July 2000     

Note on psychophysical discrimination with more than two stimuli

H. D. Landahl's well-known theory of psychophysical discrimination between two stimuli (Psychometrica,3, 107–125, 1938) is generalized to the case ofn mutually inhibiting stimuli, such that all the corresponding reactions are mutually incompatible so that only one response at most can occur at a time. It is shown that while in the two-stimulus case a “no-response” situation does not necessarily need to occur, in the case ofn stimuli andn responses a “no-response” situation always occurs with finite probability. Therefore, there is a probabilityP _i of the occurrence of each responsei as well as a probabilityP _e of no response, with . The probabilitiesP _i andP _e are expressed in terms of the intensities of then stimuliS _i and in terms of then distribution functions of the fluctuations at then corresponding connections. The expressions are in the form of sums ofn-tuple integrals of the products of the distribution functions, the limits of integration being determined by the intensities of then stimuli.     

Telomere Length Distribution and Southern Blot Analysis

Southern blot analysis of terminal restriction fragments (TRFs) is the standard method for quantitative examination of telomere length distributions. Since TRFs contain a subtelomeric component, central parameters of the TRF distributionn(L) such as the arithmetic mean (M) or the median (Me) cannot be derived directly from Southern blot data, i.e. from the optical density distributionOD(L). Several estimates have been applied instead; the seeming arithmetic meanA, the “center of mass”C, and the positions of maximal (P) and half-maximal optical density (P). We show thatC>A>Mfor any non-truncated distributionsn(L), andP>M>P1/2 for any symmetrical unimodaln(L).Symmetric appearance on a Southern blot, however, suggests positive skewness ofn(L). Thus, alognormalform ofn(L) may be considered. Then,C>A>M>P=Me>P. Alternatively, aWeibulldistribution may be assumed. The latter is compatible with negative feedback-regulation of the telomere lengths. Using the maximum likelihood method we compare these distributions with FISH-data on telomere lengths in different cell types. The fit of the lognormal distribution is clearly superior. Lognormal genesis may relate to telomere breakage and recombination.Truncation of the upper end of the TRF distribution is possible due to Southern blot artifacts. Thereby, the order of the estimates may change toP>C>A. Having minimal sensitivity to truncation,Pseems to be the optimal choice. however, the variability ofPis high since peakedness ofOD(L) and DNA length resolution are inversely related.     

Finding a Most Likely Clone Ordering from Oligonucleotide Hybridization Data

Using an extension of a statistical model given by E. Lander and M. Waterman, we define the a posteriori probability of a clone ordering based upon oligonucleotide hybridization data. We give algorithms for computing the likelihood of a clone ordering and for finding a clone ordering of maximum likelihood. The dynamic programming algorithm for computing likelihoods runs in time O(mnc), where m is the number of oligonucleotide probes, n is the number of clones, and c is the coverage of the clone library. We use the Expectation-Maximization technique to maximize likelihoods.     

An illustration of variable precision rough set theory: the gender classification of the European barn swallow (Hirundo rustica)

This paper introduces a new technique in the investigation of object classification and illustrates the potential use of this technique for the analysis of a range of biological data, using avian morphometric data as an example. The nascent variable precision rough sets (VPRS) model is introduced and compared with the decision tree method ID3 (through a ‘leave n out’ approach), using the same dataset of morphometric measures of European barn swallows (Hirundo rustica) and assessing the accuracy of gender classification based on these measures. The results demonstrate that the VPRS model, allied with the use of a modern method of discretization of data, is comparable with the more traditional non-parametric ID3 decision tree method. We show that, particularly in small samples, the VPRS model can improve classification and to a lesser extent prediction aspects over ID3. Furthermore, through the ‘leave n out’ approach, some indication can be produced of the relative importance of the different morphometric measures used in this problem. In this case we suggest that VPRS has advantages over ID3, as it intelligently uses more of the morphometric data available for the data classification, whilst placing less emphasis on variables with low reliability. In biological terms, the results suggest that the gender of swallows can be determined with reasonable accuracy from morphometric data and highlight the most important variables in this process. We suggest that both analysis techniques are potentially useful for the analysis of a range of different types of biological datasets, and that VPRS in particular has potential for application to a range of biological circumstances.