Inconsistencies in standard approximations for selection coefficients at loci affecting a polygenic character

Inconsistencies exist in the standard expansions used to approximate selection coefficients for alleles at a locus underlying a quantitative character. Allelic (marginal) fitnesses obtained from expansions based on average excesses differ from allelic fitnesses obtained from expansions based on genotypic values. Similarly, the mean population fitness based on summing over either allelic or genotypic fitnesses usually differs mean population fitness obtained by averaging over the unrestricted phenotypic distribution. A consistent value of requires no variation in genotypic values. If, as suggested by Nagylaki (1984), expansions are corrected for the decrease in phenotypic variance resulting from conditioning on the presence of a particular allele or genotype, inconsistencies still exist. Unless W(z)[V _z p(z) + zp(z) + p(z)] dz = 0, where p(z) is the phenotypic probability density function, V _z the phenotypic variance, W( _z ) the fitness of phenotypic value z, the primes denote differentiation with respect to z, allelic fitnesses based on average effects differ from allelic fitnesses based on genotypic values. This condition must also be satisfied in order for either expansion to give a consistent , as first shown by Nagylaki. For arbitrary W(z), this is satisfied if and only if phenotypes are normally distributed.     

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.     

Molecular mapping of gene Gm-6(t) which confers resistance against four biotypes of Asian rice gall midge in China

The Chinese rice cultivar Duokang #1 carries a single dominant gene Gm-6(t) that confers resistance to the four biotypes of Asian rice gall midge (Orseolia oryzae Wood-Mason) known in China. Bulked segregant analysis was performed on progeny of a cross between Duokang #1 and the gall midge-susceptible cultivar Feng Yin Zhan using the RAPD method. The RAPD marker OPM06₍₁₄₀₀₎ amplified a locus linked to Gm-6(t). The locus was subsequently mapped to rice chromosome 4 in a region flanked by cloned RFLP markers RG214 and RG163. Fine mapping of Gm-6(t) revealed that markers RG214 and RG476 flanked the gene at distances of 1.0 and 2.3 cM, respectively. Another gall midge resistance gene, Gm-2, mapped previously to chromosome 4, is located about 16 cM from Gm-6(t), to judge by data from a segregating population derived from a cross between Duokang #1 and the Indian cultivar Phalguna that carries Gm-2. We developed a PCR-based marker-assisted selection kit for transfer of the Gm-6(t) gene into Ming Hui 63 and IR50404, two parental lines commonly used in hybrid rice production in China. The kit contains PCR primer pairs based on the terminal sequences of the RG214 and RG476 clones. Polymorphism between Duokang #1 and the hybrid parental lines was found at these markers after digestion of the PCR products with specific restriction endonucleases. The kit will accelerate introduction of gall midge resistance into hybrid rice in China. Received: 18 May 2000 / Accepted: 9 March 2001     

Comparing Membrane Simulations to Scattering Experiments: Introducing the SIMtoEXP Software

SIMtoEXP is a software package designed to facilitate the comparison of biomembrane simulations with experimental X-ray and neutron scattering data. It has the following features: (1) Accepts number density profiles from simulations in a standard but flexible format. (2) Calculates the electron density ε(z) and neutron scattering length density ν(z) profiles along the z direction (i.e., normal to the membrane) and their respective Fourier transforms (i.e., F _ε [q _z ] and F _ν [q _z ]). The resultant four functions are then displayed graphically. (3) Accepts experimental F _ε (q _z ) and F _ν (q _z ) data for graphical comparison with simulations. (4) Allows for lipids and other large molecules to be parsed into component groups by the user and calculates the component volumes following Petrache et al. (Biophys J 72:2237–2242, 1997). The software then calculates and displays the contributions of each component group as volume probability profiles, ρ(z), as well as the contributions of each component to ε(z) and ν(z).     

Cell cycle control at the first restriction point and its effect on tissue growth

Cell cycle is controlled at two restriction points, R ₁ and R ₂. At both points the cell will commit apoptosis if it detects irreparable damage. But at R ₁ an undamaged cell also decides whether to proceed to the S phase or go into a quiescent mode, depending on the environmental conditions (e.g., overpopulation, hypoxia). We consider the effect of this decision at the population level in a spherical tissue {r < R(t)}. We prove that if the cells have full control at R ₁, they can manipulate the size of R(t) to ensure that 0 < c ≤ R(t) ≤ C < ∞; simulations further show that R(t) can be made nearly stationary. In the absence of such control, R(t) will either increase to ∞ or decrease to 0. The mathematical model and analysis involve a system of PDEs in {r < R(t)}.     

Minimum evolution using ordinary least-squares is less robust than neighbor-joining

The method of minimum evolution reconstructs a phylogenetic tree T for n taxa given dissimilarity data d. In principle, for every tree W with these n leaves an estimate for the total length of W is made, and T is selected as the W that yields the minimum total length. Suppose that the ordinary least-squares formula S _W (d) is used to estimate the total length of W. A theorem of Rzhetsky and Nei shows that when d is positively additive on a completely resolved tree T, then for all W ≠ T it will be true that S _W (d) > S _T (d). The same will be true if d is merely sufficiently close to an additive dissimilarity function. This paper proves that as n grows large, even if the shortest branch length in the true tree T remains constant and d is additive on T, then the difference S _W (d)-S _T (d) can go to zero. It is also proved that, as n grows large, there is a tree T with n leaves, an additive distance function d _T on T with shortest edge ε, a distance function d, and a tree W with the same n leaves such that d differs from d _T by only approximately ε/4, yet minimum evolution incorrectly selects the tree W over the tree T. This result contrasts with the method of neighbor-joining, for which Atteson showed that incorrect selection of W required a deviation at least ε/2. It follows that, for large n, minimum evolution with ordinary least-squares can be only half as robust as neighbor-joining.     

Theory of transport in linear biological systems: I. Fundamental integral equation

The conditions under which the output,γ _b (t), of a biological system is related to the input,γ _a (t), by an integral equation of the typeγ _b (t) = ∫ ₀ ^t γ _a (ω)w(t−ω)dω, where ω(t) is a transport functioncharacteristic of the system, are analyzed in detail. Methods of solving this type of integral equation are briefly discussed. The theory is then applied to problems in tracer kinetics in which input and output are sums of exponentials, and explicit formulae, which are applicable whether or not the pool is uniformly mixed, are derived for “turnover time” and “pool” size.     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Current progress of the biological single-ion microbeam at FUDAN

A biological microbeam for precisely positioned single-ion/single cell irradiation is built in the Institute of Modern Physics in Fudan University, Shanghai, China, based on the tandem accelerator (2 × 3MV) in the laboratory. In this paper, the developing progress of the FUDAN microbeam is reported, including the newly constructed beam line, the microbeam collimator, the ion detection system, and the cell-imaging and targeting systems. Statistical models are proposed for evaluating the spatial resolution and dosage precision of the microbeam. By taking the collimated ions as a Gaussian beam, the spatial resolution can be evaluated by the full width at half maximum of the 2-D Gaussian distribution, which is determined by fitting the proportions of peripheral pits outside specific radii in the pit clusters etched on ion track detectors to a 2-D Gaussian distribution. In the preset hitting of defined ion number, by taking the real delivered number of ions as an independent identically distributed random variable (iidrv), according to the Law of Large Numbers and Central Limit Theorem, the expected value μ and standard deviation σ of the real delivered ion number in a preset N-ion hitting can be determined by approaching the normal distribution of N (μ, σ ²/n) with the proportions of the mean counts of pits in multiple pit clusters on ion track detectors. By the values of μ, σ and additional assumptions, statistical dosage precision evaluations can be made on the preset hitting. From the linear fit curve of μ(N) and the power function fit curve of σ(N) on different preset ion numbers, characteristic factors k, b, A, p can be extracted for a precision evaluation independent of the specific preset ion number.     

Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data

In this work we address the problem of the robust identification of unknown parameters of a cell population dynamics model from experimental data on the kinetics of cells labelled with a fluorescence marker defining the division age of the cell. The model is formulated by a first order hyperbolic PDE for the distribution of cells with respect to the structure variable x (or z) being the intensity level (or the log₁₀-transformed intensity level) of the marker. The parameters of the model are the rate functions of cell division, death, label decay and the label dilution factor. We develop a computational approach to the identification of the model parameters with a particular focus on the cell birth rate α(z) as a function of the marker intensity, assuming the other model parameters are scalars to be estimated. To solve the inverse problem numerically, we parameterize α(z) and apply a maximum likelihood approach. The parametrization is based on cubic Hermite splines defined on a coarse mesh with either equally spaced a priori fixed nodes or nodes to be determined in the parameter estimation procedure. Ill-posedness of the inverse problem is indicated by multiple minima. To treat the ill-posed problem, we apply Tikhonov regularization with the regularization parameter determined by the discrepancy principle. We show that the solution of the regularized parameter estimation problem is consistent with the data set with an accuracy within the noise level in the measurements.      

Robustness of Selection Procedures

In the article Bechhofers Indifference-zone formulation for selecting the t populations with the t highest means is considered in a set of non-normal distributions. Selection rules based on the sample mean, the 10% and the 20% trimmed means, two estimators proposed by Tiku (1981) for valuating the smallest and highest accepted sample values higher, the sample median and a linear combination of quantile estimators, two adaptive procedures and a ranksum procedure are investigated in a large scale simulation experiment in respect of their robustness against deviations from an assumed distribution. Robustness is understood as a small percentage of the difference β_A-β between the actual probability of incorrect selection β_A and the nominal β-value. We obtained a relatively good robustness for the classical sample mean selection rule and useful derivations for the employment of other selection rules in an area of practical importance.     

Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

137Cs mobility in soils and its long-term effect on the external radiation exposure

To predict the external gamma-dose rate of Chernobyl-derived ¹³⁷Cs for a period of about 100 years after its deposition, the vertical distribution of radiocesium in several meadow soils in the Chernobyl area and in Germany was determined, and the corresponding residence half-times of this radionuclide in the various soil layers were evaluated using a compartment model. The resulting residence half-times were subsequently used to calculate the vertical distribution of ¹³⁷Cs in the soil as a function of time and finally to predict the external gamma-dose rates in air for these sites at various times. A regression analysis of the data obtained showed that the time dependence of the relative gamma-dose rate in air D(t) at the Chernobyl sites can be described by an exponential equation D(t) = a + b ⋅ exp(–t/c), where t is the time after deposition. For the ten German sites the best fit was obtained using the two-exponential equation D(t) = a ⋅ exp(–t/b) + c ⋅ exp(–t/d). The gamma-dose rate of ¹³⁷Cs at the Chernobyl sites decreases significantly more slowly with time than at the German sites. This means that after e.g. 30 years the mean relative gamma-dose rate at the German sites will have decreased from 100% (corresponding to an infinite plane source on a smooth surface) to 9% (95% confidence interval 8%–10%), while at the sites in the Chernobyl area it will have decreased only to 21% (20%–23%). This difference is the result of the longer residence half-times of ¹³⁷Cs in the soils at the Chernobyl sites. All results are compared with estimates from earlier studies. Received: 16 October 1996 / Accepted in revised form: 28 November 1996     

Exploring optimal current stimuli that provide membrane voltage tracking in a neuron model

Studying neurons from an energy efficiency perspective has produced results in the research literature. This paper presents a method that enables computation of low energy input current stimuli that are able to drive a reduced Hodgkin–Huxley neuron model to approximate a prescribed time-varying reference membrane voltage. An optimal control technique is used to discover an input current that optimally minimizes a user selected balance between the square of the input stimulus current (input current ‘energy’) and the difference between the reference voltage and the membrane voltage (tracking error) over a stimulation period. Selecting reference signals to be membrane voltages produced by the neuron model in response to common types of input currents i(t) enables a comparison between i(t) and the determined optimal current stimulus i*(t). The intent is not to modify neuron dynamics, but through comparison of i(t) and i*(t) provide insight into neuron dynamics. Simulation results for four different bifurcation types demonstrate that this method consistently finds lower energy stimulus currents i*(t) that are able to approximate membrane voltages as produced by higher energy input currents i(t) in this neuron model.     

Effect of oral creatine ingestion on parameters of the work rate-time relationship and time to exhaustion in high-intensity cycling

The relationship between work rate (W˙) and time to exhaustion (t) during intense exercise is commonly described by either a hyperbolic function (NLin), t=W ^′/(W˙−W˙ _cp), or by its linear equivalent (LinW) W _lim =W ^′+W˙ _cp(t). The parameter W˙<INF _</INF>cp (critical power) has been described as an inherent characteristic of the aerobic energy system, while W ′ has been shown to be a ralid estimate of anaerobic work capacity. Recent studies have demonstrated that oral supplementation of creatine monohydrate (CrH₂O) increases total muscle creatine stores, and have linked these increases to improved performances in intense intermittent exercise. This study was conducted to determine the effect of CrH₂O supplementation on estimates of W ′ and W˙<INF _</INF>cp derived from the NLin and LinW equations, and to determine the effect of CrH₂O on t in exhaustive constant power exercise of different intensities. Fifteen active but untrained university students completed three phases of testing on a cycle ergometer: (1) familiarization, three learning trials, (2) baseline determination of W ′ and W˙<INF _</INF>cp, four bouts performed at a W˙ selected to elicit fatigue in 90–600 s, and (3) experimental determination of W ′ and W˙ _cp, four bouts performed at the same W˙ as baseline, but performed after 5 days of ingesting either a placebo (4 × 6 g of glucose/day) or CrH₂O (4 × 5 g of CrH₂O and 1 g glucose/day). Testing was administered in a double-blind manner. Analyses of covariance revealed a significant effect for CrH₂O on both estimates of W ′ (NLin, P=0.04; LinW, P<0.01), but not on estimates of W˙ _cp (NLin, P=0.37; LinW; P=0.30). Within groups, t was significantly different for only CrH₂O at the two highest W˙s (P=0.04). It is concluded that oral ingestion of CrH₂O increases estimates of W ′ due to an improved t at the shorter, more intense exercise bouts. Accepted: 1 September 1997     

The mathematical theory of group selection. I. Full solution of a nonlinear Levins E = E(x) model

The first complete overtime solution is obtained for a group selection model of Levins E = E(x) type with recolonization but no other gene flow between islands. Assuming a subdivided population at carrying capacity, the model describes selection at a biallelic locus (A, a) where a is opposed by Mendelian selection but is favored by a lower rate of extinction of demes having high a frequency. By contrast to the linear diffusion equations encountered in classical mathematical genetics, the PDE governing the dynamics is now nonlinear in the metapopulation gene frequency distribution φ(x, t); furthermore, the initial conditions now heavily influence the equilibrium distribution φ_∞(x). A fully explicit formula (20) expressing this dependence is derived. The results indicate that a fixation is never reached, but (A, a) polymorphism in the metapopulation will result if , where s 1 parametrizes the strength of Mendelian selection, E(x) is the Levins extinction operator, h (typically in the open interval (0, 1)) is the dominance of a, and B is a parameter measuring the flatness of the initial distribution f(x) in the x → 1 limit.     

A model of photosystem II for the analysis of fast fluorescence rise in plant leaves

The polyphasic patterns of fluorescence induction rise in pea leaves in vivo and after the treatment with ionophores have been studied using a Plant Efficiency Analyzer. To analyze in detail photosystem II (PS II) electron transfer processes, an extended PS II model was applied, which included the sums of exponential functions to specify explicitly the light-driven formation of the transmembrane electric potential (ΔΨ(t)) as well as pH in the lumen (pH_L(t)) and stroma (pH_S(t)). PS II model parameters and numerical coefficients in ΔΨ(t), pH_L(t), and pH_S(t) were evaluated to fit fluorescence induction data for different experimental conditions: leaf in vivo or after ionophore treatment at low or high light intensity. The model imitated changes in the pattern of fluorescence induction rise due to the elimination of transmembrane potential in the presence of ionophores, when ΔΨ = 0 and pH_L(t), pH_S(t) changed to small extent relative to control values in vivo, with maximum ΔΨ(t) ∼ 90 mV and ΔΨ(t) ∼ 40 mV for the stationary state at ΔpH ≅ 1.8. As the light intensity was increased from 300 to 1200 μmol m⁻² s⁻¹, the heat dissipation rate constants increased threefold for nonradiative recombination of P680⁺Phe⁻ and by ∼30% for P680⁺Q_A⁻. The parameters ΔΨ, pH_S and pH_L were analyzed as factors of PS II redox state populations and fluorescence yield. The kinetic mechanism of fluorescence quenching is discussed, which is related with light-induced lumen acidification, when ⁺Q_A⁻ and P680⁺ recombination probability increases to regulate the Q_A reduction.     

Chemotactic collapse for the Keller-Segel model

 This work is concerned with the system (S) {u _t =Δu − χ∇ (u∇v) for x∈Ω, t>0Γ v _t =Δv+(u−1) for x∈Ω, t>0 where Γ, χ are positive constants and Ω is a bounded and smooth open set in ℝ². On the boundary ∂Ω, we impose no-flux conditions: (N) ∂u∂n =∂v∂n =0 for x∈∂ Ω, t>0 Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), v(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t) →Aδ(y) as t→T for some T<∞, where A is the total concentration of the species. Received 9 March 1995; received in revised form 25 December 1995     

Cumulative Damage Survival Models for the Randomized Complete Block Design

A continuous time discrete state cumulative damage process {X(t), t ≥ 0} is considered, based on a non‐homogeneous Poisson hit‐count process and discrete distribution of damage per hit, which can be negative binomial, Neyman type A, Polya‐Aeppli or Lagrangian Poisson. Intensity functions considered for the Poisson process comprise a flexible three‐parameter family. The survival function is S(t) = P(X(t) ≤ L) where L is fixed. Individual variation is accounted for within the construction for the initial damage distribution {P(X(0) = x) | x = 0, 1, …,}. This distribution has an essential cut‐off before x = L and the distribution of L – X(0) may be considered a tolerance distribution. A multivariate extension appropriate for the randomized complete block design is developed by constructing dependence in the initial damage distributions. Our multivariate model is applied (via maximum likelihood) to litter‐matched tumorigenesis data for rats. The litter effect accounts for 5.9 percent of the variance of the individual effect. Cumulative damage hazard functions are compared to nonparametric hazard functions and to hazard functions obtained from the PVF‐Weibull frailty model. The cumulative damage model has greater dimensionality for interpretation compared to other models, owing principally to the intensity function part of the model.