The decay of genetic variability in geographically structured populations. II

The ultimate rate of approach to equilibrium in the infinite stepping-stone model is calculated. The analysis is restricted to a single locus in the absence of selection, and every mutant is assumed to be new to the population. Let f(t, x) be the probability that two homologous genes separated by the vector x in generation t are the same allele. It is supposed that $\text{f(}\text{0, x}\text{) = O(x}{}^{\mathrm{?2?\eta }}\text{), η > 0,}\text{as}\text{x ≡ ¦}\text{x}\text{¦ → ∞}$. In the absence of mutation, f(t, x) tends to unity at the rate $\text{t}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ in one dimension and (ln t)^?1 in two dimensions. Thus, the loss of genetic variability in two dimensions is so slow that evolutionary forces not considered in this model would supervene long before a two-dimensional natural population became completely homogeneous. If the mutation rate, u, is not zero f(t, x) asymptotically approaches equilibrium at the rate $\text{(1 ? u)}{}^{\mathrm{2t}}\text{t}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$ in one dimension and (1 ? u)^2tt^?1(lnt)^?2 in two dimensions. Integral formulas are presented for the spatial dependence of the deviation of f(t, x) from its stationary value as t → ∞, and for large separations this dependence is shown to be (const + x) in one dimension and (const + ln x) in two dimensions. All the results are the same for the Malécot model of a continuously distributed population provided the number of individuals per colony is replaced by the population density. The relatively slow algebraic and logarithmic rates of convergence for the infinite habitat contrast sharply with the exponential one for a finite habitat.     

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     

On-line parameter and state estimation of continuous cultivation by extended Kalman filter

Summary The on-line estimation of biomass concentration and of three variable parameters of the non-linear model of continuous cultivation by an extended Kalman filter is demonstrated. Yeast growth in aerobic conditions on an ethanol substrate is represented by an unstructured non-linear stochastic t-variant dynamic model. The filter algorithm uses easily accessible data concerning the input substrate concentration, its concentration in the fermentor and dilution rate, and estimates the biomass concentration, maximum specific growth rate, saturation constant and substrate yield coefficient. The microorganismCandida utilis, strain Vratimov, was cultivated on the ethanol substrate. The filter results obtained with the real data from one cultivation experiment are presented. The practical possibility of using this method for on-line estimation of biomass concentration, which is difficult to measure, is discussed.Nomenclature D dilution rate (h^-1) - DO₂ dissolved oxygen concentration (%) - E identity matrix - F Jacobi matrix of the deterministic part of the system equations g - g continuousn-vector non-linear real function - h m-vector non-linear real function - K Kalman filter gain matrix - K _S saturation constant (kgm^-3) - K_S expectation of the saturation constant estimate - M Jacobi matrix of the deterministic part of the measurement equations h - P(t₀) co-variance matrix of the initial values of the state - P(t_k/t_k) c-variance matrix of the error in (t _k|t _k) - P(t_k+1/t_k) co-variance matrix of the error in (t _k+1|t _k - Q co-variance matrix of the state noise - R co-variance matrix of the output noise - S substrate concentration (kgm^-3) - S _i input substrate concentration - t time - t _k discrete time instant with indexk=0, 1, 2,... - u(t) input vector - v(t_k) measurement (output) noise sequence - w(t) n-vector white Gaussian random process - x(t₀) initial state of the system - (t₀) expectation of the initial state values - x(t) n-dimensional state vector - x(t_k) state vector at the time instantt _k - (t_k|t_k) expectation of the state estimate at timet _k when measurements are known to the timet _k - (t_k+1|t_k) expectation of the state prediction - X biomass concentration (kgm^-3) - expectation of the biomass concentration estimate - y(t_k) m-dimensional output vector at the time instantt _k - Y _XIS substrate yield coefficient - _X|S expectation of the substrate yield coefficient estimate - specific growth rate (h^-1) - _M maximum specific growth rate (h^-1) - expectation of the maximum specific growth rate estimate - state transition matrix     

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.     

Diffusion with attrition

This article treats the problem of the sharp front observed when a diffusing substance interacts irreversibly with binding sites within the medium. The model consists of two simultaneous partial differential equations that are nonlinear and cannot be solved in closed form. The parameters are the diffusion coefficient D in the direction under consideration (x), the interaction constant k, the binding-site concentration μ and the boundary concentration of the diffusing ion c ₀. Our aim is to develop methods to enable the estimation of these parameters from the experimental data. An analytical solution for the case k → ∞, as found by others, is given first and then a finite element analysis package is used to obtain numerical solutions for the general case. Graphs are presented to illustrate the effects of the various parameters. Simple graphical procedures are described to compute μ and c ₀. The position of the advancing front ξ then provides, together with μ, a way to estimate D. A mathematical identity relating D and x and a second one involving D, k and t help to reduce the complexity of the problem. A new, measurable quantity S(t) is defined as where f is the total concentration (free + bound) of the diffusing ion at time t, and detailed plots are furnished that permit the computation of k directly from S(t), μ and D. The accuracy with which such methods can be expected to determine the various parameters of the model is considered at some length. Finally, in a concluding section, we simulate typical experimental data, examine the validity of our methods, and see how their accuracy is affected by controlled amounts of various kinds of noise.     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

The rate of convergence of a generalized stable population

In an age-structured population that grows exponentially, each age groupP _i(t) at periodt is asymptotically equivalent tox ₀ ^t for some positive number x₀. In this paper we show that the speed at which the ith age group reaches its exponential state of equilibrium can be measured by the rate at which the ratio v_i(t)=P_i(t)/p_i(t–1) converges tox ₀. The age specific rate of convergence is determined by considering a quantityr satisfyingv _i(t)-x ₀ ¦ r ^t whent is large;R _i=Infr (over all initial populations,r satisfying the above inequality) is the R-factor used in numerical analysis to measure the rate at which the sequencev _i (t) converges tox ₀;S _i =- In R_i is then defined as the rate of convergence to stability of the ith age group. The case of constant net maternity rates is studied in detail; in this contextS ₀ is compared to the population entropyH, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability.     

Stability of the analytical solution of Penna model of biological aging

There are some analytical solutions of the Penna model of biological aging; here, we discuss the approach by Coe et al. (Phys. Rev. Lett. 89, 288103, 2002), based on the concept of self-consistent solution of a master equation representing the Penna model. The equation describes transition of the population distribution at time t to next time step (t + 1). For the steady state, the population n(a, l, t) at age a and for given genome length l becomes time-independent. In this paper we discuss the stability of the analytical solution at various ranges of the model parameters—the birth rate b or mutation rate m. The map for the transition from n(a, l, t) to the next time step population distribution n(a + 1, l, t + 1) is constructed. Then the fix point (the steady state solution) brings recovery of Coe et al. results. From the analysis of the stability matrix, the Lyapunov coefficients, indicative of the stability of the solutions, are extracted. The results lead to phase diagram of the stable solutions in the space of model parameters (b, m, h), where h is the hunt rate. With increasing birth rate b, we observe critical b ₀ below which population is extinct, followed by non-zero stable single solution. Further increase in b leads to typical series of bifurcations with the cycle doubling until the chaos is reached at some b _c. Limiting cases such as those leading to the logistic model are also discussed.     

Melting curves, denaturation maps, and genetic map of phiX174: their relations and applications.

In this paper we analyze theoretically the observable details of the differential melting curves (DMC) and the denaturation maps (DM) of a DNA. With the help of a mathematical model, we explore their implications, their relation with each other and with the genetic map of the molecule, and discuss possible future applications. ?X174 is used as the example, since its sequence and genetic map are available. We find that each gene section of ?X174 has a characteristic DMC. A reconstruction scheme to get the DMC of a whole piece from those of its constituent genes is shown to be fairly successuful. The relations between the melting curve and the denaturation maps are clarified. We observe that nearly always, the beginning and end of a gene melt at lower temperatures. The sharp features in the DM indicate that despite the long-range cooperative interactions, the DM do reflect the local sequence effect. Denaturation maps (theoretical) of ?X174 and SV40 are presented. From available data of other authors, we estimate that the dependence of the melting temperature t_m on GC, the fraction of (G+C)-content, and on x, the ionic concentration in fractions of the standard saline citrate solution, can be expressed as t_m(x, GC) = -5.2 (log x)GC + 18.4 log x + 41.0GC + 69.4. The first two coefficients are less certain.     

New Na‐Ion Solid Electrolytes Na4−xSn1−xSbxS4 (0.02 ≤ x ≤ 0.33) for All‐Solid‐State Na‐Ion Batteries

Sulfide Na‐ion solid electrolytes (SEs) are key to enable room‐temperature operable all‐solid‐state Na‐ion batteries that are attractive for large‐scale energy storage applications. To date, few sulfide Na‐ion SEs have been developed and most of the SEs developed contain P and suffer from poor chemical stability. Herein, discovery of a new structural class of tetragonal Na_4?xSn_1?xSb_xS₄ (0.02 ≤ x ≤ 0.33) with space group I4₁/acd is described. The evolution of a new phase, distinctly different from Na₄SnS₄ or Na₃SbS₄, allows fast ionic conduction in 3D pathways (0.2–0.5 mS cm^?1 at 30 °C). Moreover, their excellent air stability and reversible dissolution in water and precipitation are highlighted. Specifically, TiS₂/Na–Sn all‐solid‐state Na‐ion batteries using Na_3.75Sn_0.75Sb_0.25S₄ demonstrates high capacity (201 mA h (g of TiS₂)^?1) with excellent reversibility.     

Optimal control of continuous fermentation processes

Three layer control structure is proposed for optimal control of continuous fermentation processes. The start-up optimization problems are solved as a first step for optimization layer building. A steady state optimization problem is solved by a decomposition method using prediction principle. A discrete minimum time optimal control problem with state delay is formulated and a decomposition method, based on an augmented Lagrange's function is proposed to solve it. The problem is decomposed in time domain by a new coordinating vector. The obtained algorithms are used for minimum time optimal control calculation of Baker's Yeast fermentation process.List of Symbols x(t) g/l biomass concentration - s(t) g/l limiting substrate concentration - x ⁰ g/l inlet biomass concentration - s ⁰(t) g/l inlet substrate concentration - D(t) h^–1 dilution rate - (t) h^–1 specific growth rate - Y g/g yield coefficient - (t) h^–1 specific limiting substrate consumption rate - k _D h^–1 disappearing constant - w ₁, w ₂ known constant or piece-wise disturbances - _m h^–1 maximum specific growth rate - k _s g/l Michaelis-Menten's parameter - h time delay - x ₀, s ₀ g/l initial concentrations - ¯x, ¯s, ¯D optimal steady state value - V _min , V _max , v=x,s,d,t bounds of variables - t h sampling period - K number of steps in the optimization horison - Js, J _d performance indexes - L _s Lagrange's function - L _d Lagrange's functional - ₀ weighting coefficient for the amount of the limiting substrate throwing out of the fermentor - ₁, ₂ dual variables of Lagrange's function - steps in steady state coordination procedure - errors values for steady state coordination process - _v , v=x, s conjugate variables of Lagrange's functional - _v , v=x,s penalty coefficients of augmented Lagrange's functional - _v , v=x, s interconnections of the time - e _v , v=x,s, D, _x , _s gradients of Lagrange's functional - j, l indexes of calculation procedures - values of errors in calculations The researches was supported by National Scientific Research Foundation under grants No NITN428/94 and No NITN440/94     

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.     

Optimal Fixing of the Bandwidth-Parameter for the Empirical Regression1,2

The estimator ?₀(x) of the regression r(x) = E (Y | × = x) from measured points (x_i, y_i), i = 1(1) n, of a continuous two-dimensional random variable (X, Y) with unknown continuous density function f(x, y) and with moments up to the second order can be made with the help of a density estimation f?₀(x, y) (see e.g. SCHMERLING and PEIL, 1980). Here f?₀(x, y) still contains free parameters (so-called band-width-parameters), the values of which have to be optimally fixed in the concrete case. This fixing can be done by using a modification of the maximum-likelihood principle including jackknife techniques. The parameter values can be also found from the estimators for r(x). Here the cross-validation principle can be applied. Some numerical aspects of these possibilities for optimally fixing the bandwidth-parameter are discussed by means of examples. If ?₀(x) is used as a smoothing operator for time series the optimal choice of the parameter values is dependent on the purpose of application of the smoothed time series. The fixing will then be done by considering the so-called filter-characteristic of ?C₀(x).     

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.     

A model of the complex response of Staphylococcus aureus to methicillin

It is widely accepted that β-lactam antimicrobials cause cell death through a mechanism that interferes with cell wall synthesis. Later studies have also revealed that β-lactams modify the autolysis function (the natural process of self-exfoliation of the cell wall) of cells. The dynamic equilibrium between growth and autolysis is perturbed by the presence of the antimicrobial. Studies with Staphylococcus aureus to determine the minimum inhibitory concentration (MIC) have revealed complex responses to methicillin exposure. The organism exhibits four qualitatively different responses: homogeneous sensitivity, homogeneous resistance, heterogeneous resistance and the so-called ‘Eagle-effect’. A mathematical model is presented that links antimicrobial action on the molecular level with the overall response of the cell population to antimicrobial exposure. The cell population is modeled as a probability density function F(x,t) that depends on cell wall thickness x and time t. The function F(x,t) is the solution to a Fokker-Planck equation. The fixed point solutions are perturbed by the antimicrobial load and the advection of F(x,t) depends on the rates of cell wall synthesis, autolysis and the antimicrobial concentration. Solutions of the Fokker-Planck model are presented for all four qualitative responses of S. aureus to methicillin exposure.     

Population parameters of stargazer (Uranoscopus scaber Linnaeus, 1758) in the southeastern Black Sea region during the 2011–2012 fishing season

The population parameters of stargazer (Uranoscopus scaber Linnaeus, 1758) were studied regarding age composition, sex ratio, growth, survival and mortality rates, and the exploitation rate in the southeastern Black Sea near the coast of Turkey during the 2011–2012 fishing season. According to ageing analysis of the samples, age group 1 is the most abundant (47.76%), followed by age 2 (41.04%), age 3 (9.33%), age 0 (1.12%) and age 4 (0.75%). Mean totals as well as total lengths and weights for males and females were 14.5 ± 0.22 cm and 12.9 ± 0.19 g, 16.6 ± 0.24 cm and 62.1 ± 2.76 g, and 37.2 ± 1.92 cm and 87.1 ± 3.88 g, respectively. The mean condition factor was K = 0.0167, while the sex ratio was 53.98% female, 38.75% male and 7.27% immature. Length‐weight, age‐length and age‐weight equations were W = 0.014 × L^3.059, L_(t)=44.5*[1?e^{?0.148*(t +1.242)}] and W_(t)=1544.4*[1?e^{?0.148*(t + 1.242)}]^3.059, respectively. The computed survival rate (S), instantaneous total mortality rate (Z), annual mortality rate (A), natural mortality rate (M) and fishing mortality rate (F) were S = 28.94%, Z = 1.24 year^?1, A = 71.06%, M = 0.26 year^?1 and F = 0.98 year^?1, respectively. The exploitation rate was 0.79, which is above the optimum exploitation levels.     

Production of precursors for anti-Alzheimer drugs: Asymmetric bioreduction in a packed-bed bioreactor using immobilized D. carota cells

(S)-1-Phenylethanol derivatives, which are the precursors of many pharmacological products, have also been used as anti-Alzheimer drugs. Bioreduction experiments were performed in a batch and packed-bed bioreactor. Then, the kinetics constants were determined by examining the reaction kinetics in the batch system with free and immobilized carrot cells. Also, the effective diffusion coefficient (D_e) of acetophenone in calcium alginate-immobilized carrot cells was investigated. Kinetics constants for free cells, which are intrinsic values, are reaction rate V_max?=?0.052?mmol?L^?1?min^?1, and constants of the Michaelis–Menten K_M?=?2.31?mmol?L^?1. Kinetics constants for immobilized cells, which are considered apparent values, are V_{max, app}?=?0.0407?mmol?L^?1 min^?1, K_{M, app}?=?3.0472?mmol?L^?1 for 2?mm bead diameter, and V_{max, app}?=?0.0453?mmol?L^?1 min^?1, K_{M, app}?=?4.9383?mmol?L^?1 for 3?mm bead diameter. Average value of effective diffusion coefficient of acetophenone in immobilized beads was determined as 1.97?×?10^?6?cm²?s^?1. Using immobilized carrot cells in an up-flow packed-bed reactor, continuous production of (S)-1-phenylethanol through asymmetric bioreduction of acetophenone was performed. The effects of the residence time and concentrations of substrate were investigated at pH 7.6 and 33°C. Enantiomerically pure (S)-1-phenylethanol (ee?>?99%) was produced with 75% conversion at 4-hr residence time.     

Synthesis of Large Surface‐Area g‐C3N4 Comodified with MnOx and Au‐TiO2 as Efficient Visible‐Light Photocatalysts for Fuel Production

Herein, this study successfully fabricates porous g‐C₃N₄‐based nanocomposites by decorating sheet‐like nanostructured MnO_x and subsequently coupling Au‐modified nanocrystalline TiO₂. It is clearly demonstrated that the as‐prepared amount‐optimized nanocomposite exhibits exceptional visible‐light photocatalytic activities for CO₂ conversion to CH₄ and for H₂ evolution, respectively by ≈28‐time (140 µmol g^?1 h^?1) and ≈31‐time (313 µmol g^?1 h^?1) enhancement compared to the widely accepted outstanding g‐C₃N₄ prepared with urea as the raw material, along with the calculated quantum efficiencies of ≈4.92% and 2.78% at 420 nm wavelength. It is confirmed mainly based on the steady‐state surface photovoltage spectra, transient‐state surface photovoltage responses, fluorescence spectra related to the produced ?OH amount, and electrochemical reduction curves that the exceptional photoactivities are comprehensively attributed to the large surface area (85.5 m² g^?1) due to the porous structure, to the greatly enhanced charge separation and to the introduced catalytic functions to the carrier‐related redox reactions by decorating MnO_x and coupling Au‐TiO₂, respectively, to modulate holes and electrons. Moreover, it is suggested mainly based on the photocatalytic experiments of CO₂ reduction with isotope ¹³CO₂ and D₂O that the produced ?CO₂ and ?H as active radicals would be dominant to initiate the conversion of CO₂ to CH₄.