On the stochastic theory of compartments: Solution forn-compartment systems with irreversible,time-dependent transition probabilities

The bivariate distribution of a two-compartment stochastic system with irreversible, time-dependent transition probabilities is obtained for any point in time. The mean and variance of the number of particles in any compartment and the covariance between the number of particles in each of the two compartments are exhibited and compared to existing results. The two-compartment system is then generalized to ann-compartment catenary and to ann-compartment mammillary system. The multivariate distributions of these two systems are obtained under two sets of initial conditions: (1) the initial distribution is known; and (2) the number of particles in each compartment of the system at timet=0 is determined. The moments of these distributions are also produced and compared with existing results.     

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     

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.     

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.     

A Single Compartment Quantal Response Model with a Renewal Type Input Process and IID Releases

We deal with a single compartment quantal response model, where unlike the previous models, which do not have any input after the administration of a single dose Z(0)=z at time t=0, we allow inputs of doses after time t=0. More precisely, the system uses the (s, S) input policy as in inventory models, and has IID releases. Also when the amount of dose in the subject reaches 0, there is a probability p to stop having input thereafter. Among other results, the probability that the subject never responds and the expressions for some quantities of interest are obtained.     

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.     

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.     

Factors influencing the distribution of Hydrocharis morsus-ranae L. and Rumex hydrolapathum Huds. in a mowed low-lying marshland, Réserve de Cheyres, lac de Neuchâtel, Switzerland

Using environmental parameters we studied the distribution of two endangered species, Hydrocharis morsus-ranae L. and Rumex hydrolapathum Huds., in a low-lying marshland of the Swiss Plateau, a region in which aquatic vegetation is particularly threatened. A large part of the study site is regularly mown by a machine for site management purposes. The caterpillar mower digs ruts, which are especially pronounced along the tracks used to reach the mown compartment. To assess the effects of site management on these species, we have tested six environmental parameters (vegetation unit, water conductivity, water supply indicated by altitude, time elapsed since the last cut, distance from major ruts and disturbance of major ruts) that can potentially influence plant distribution. All the plots of these two species have been found in mowed compartment, which seems to indicate a correlation between site management operations and occurrences. The other factors driving the distribution of these plants are vegetation unit, distance from major ruts and water supply.     

Interaction between cell sodium and the amiloride-sensitive sodium entry step in rabbit colon

Summary Ouabain abolishes the short-circuit current (I _sc ) and decreases the transepithelial conductance (G _t ) of rabbit colon. In contrast, amphotericin B elicits a maximumI _sc and markedly increasesG _t . However, inboth instances the amiloride-sensitive Na entry step is completely blocked, presumably due to an increase in cell Na. Conversely, when Na-depleted tissues are suddenly exposed to 140mm Na, the amiloride-sensitiveI _sc and the amiloride-sensitive component ofG _t ( _a G _Na ) increase abruptly to their maximum values and the decline to steady-state plateaus with a half time of 6 min; throughout the decline (I _sc/a G _Na)=E _Na is constant at a value of 95 mV. In the presence of amphotericin B, theI _sc abruptly rises to the same maximum but does not decline. These findings indicate that in the presence of 140mm Na the conductance of the amiloride-sensitive Na entry step can vary from a maximum value of approximately 1.6 mmhos/cm² when cell Na is depleted, to zero when cell Na is abnormally elevated (e.g., in the presence of ouabain or amphotericin B). Our findings are consistent with a system in which the pathway responsible for transcellular Na transport parallels another cellular compartment with which it communicates. The Na capacity of the active transport pathway appears to be very small so that this compartment fills rapidly after exposure of Na-depleted cells to 140mm Na, and active transepithelial Na transport is initiated and reaches steady-state levels quickly. The Na capacity of the second compartment is much larger; the Na content of this compartment appears to be responsible for the negative feedback effect on the permeability of the amiloride-sensitive entry step.     

Improved LASSO priors for shrinkage quantitative trait loci mapping

Recently, the Bayesian least absolute shrinkage and selection operator (LASSO) has been successfully applied to multiple quantitative trait loci (QTL) mapping, which assigns the double-exponential prior and the Student’s t prior to QTL effect that lead to the shrinkage estimate of QTL effect. However, as reported by many researchers, the Bayesian LASSO usually cannot effectively shrink the effects of zero-effect QTL very close to zero. In this study, the double-exponential prior and Student’s t prior are modified so that the estimate of the effect for zero-effect QTL can be effectively shrunk toward zero. It is also found that the Student’s t prior is virtually the same as the Jeffreys’ prior, since both the shape and scale parameters of the scaled inverse Chi-square prior involved in the Student’s t prior are estimated very close to zero. Besides the two modified Bayesian Markov chain Monte Carlo (MCMC) algorithms, an expectation–maximization (EM) algorithm with use of the modified double-exponential prior is also adapted. The results shows that the three new methods perform similarly on true positive rate and false positive rate for QTL detection, and all of them outperform the Bayesian LASSO.     

On the average cellular volume in synchronized cell populations

As was done by Sinclair and Ross (1969(, we consider a cellular population that consists initially (at time zero) ofN ₀ newborn cells, all with the same volumev _o. It is assumed that the occurrence of cell division is determined only by a cell’s age, and not by its volume. The frequency function of interdivision times, τ, is denoted byf(τ). If cell death is negligible, the expected number of cells,N(t), will increase according to the laws of a simple age-dependent branching process. The expression forN(t) is obtained as a sum over all generations; thevth term of this sum, in turn, is a multiple convolution integral, reflecting the life history ofvth generation cells (i.e., the lengths of thev successive interdivision periods plus the age of the cell at timet). Assuming that cell volume is a given function of cell age, e.g., linear or exponential, and that cellular volume is exactly halved at each division, it is possible to calculate the volume of a cell with a given life history, and thus the average cellular volume of the whole population as a function of time. If at time zero the volumes differ from cell to cell, the final equation must be modified by averaging over initial volumes. In the case of linear volume increase with age, a very simple asymptotic expression is found for the average cellular volume ast→∞. The case of exponential volume increase with age also leads to a simple asymptotic formula, but the resulting volume distribution is unstable. The mean cellular volume at birth and the second moment of the volume distribution can be calculated in a similar manner. Work supported by the U.S. Atomic Energy Commission.     

Two‐compartment processing as a tool to boost recombinant protein production

Pichia pastoris is used extensively as a production platform for many recombinant proteins. The dissolved oxygen (DO) is one of the most important factors influencing protein production. The influence of the DO on productivity has not been studied independent from the feed rate. In this work, various DO levels were investigated independent from the feed rate. The model system was recombinant P. pastoris under the control of methanol‐induced alcohol oxidase promoter, which expressed HRP as the target protein. No significant effect was observed in terms of titer and specific productivity, which is a confirmation of the fact that the DO in a one‐compartment system cannot boost productivity for the model system under study. Hence, a two‐compartment system (a single reactor coupled with a plug flow reactor) was designed and implemented in order to apply oxygen‐related stress in the plug flow reactor and allow the cells to be recovered in the main reactor. Doing so, more than two‐fold increase in the titer and productivity and three‐fold increase in protein‐specific activity were achieved. Hence, partial application of oxygen‐related stress in the two‐compartment system was proposed as a process technology to enhance protein production.     

THE DISCRETE–TIME KINETIC MODEL ANALYSIS OF DNA CONTENT DISTRIBUTIONS IN EXPERIMENTAL TUMOUR CELLS

A method was developed to analyse and characterize FMF measurements of DNA content distribution, utilizing the discrete time kinetic (DTK) model for cell kinetics analysis. The DTK model determines the time sequence of the cell age distribution during the proliferation of a tumor cell population and simulates the distribution pattern of the DNA content of cells in each age compartment of the cell cycle. The cells in one age compartment are distributed and spread into several compartments of the DNA content distribution to allow for different rates of DNA synthesis and instrument dispersion effects. It is assumed that the DNA content of cells in each age compartment has a Gaussian distribution. Thus, for a given cell age distribution the DNA content distribution depends on two parameters of the cells in each age compartment: the average DNA content and its coefficient of variation. As the DTK model generates the best fit DNA content distribution to the FMF measurement data, it enables one to estimate specific values of these two parameters in each stage of the cell cycle and to determine the fraction of cells in each cycle phase. The method was utilized to fit FMF measurements of DNA content distributions and to analyse their relationship to the cell kinetic parameters, namely cell loss rate, cell cycle times and growth fraction of exponentially growing Chinese hamster ovary cells in vitro and, also, with a wide range of coefficients of variation, of the L1210 ascites tumour during the growth period.     

Poly(L–lysine)–DNA interactions in NaCl solutions: B → C and B → ψ Transitions

Thermal denaturation and circular dichroism (CD) properties of poly(L -lysine)–DNA complexes vary greatly when these complexes are prepared differently, that is, whether by NaCl-gradient dialysis starting from 2.0 M NaCl or by direct mixing at low salt. These differing properties were investigated in more detail by examining complexes, made by direct mixing in the presence of various concentrations of NaCl, both before and after the NaCl was dialyzed out of the complex solution. The precipitation curves of DNA due to polylysine binding indicate that such binding is noncooperative at zero salt; from 0.1 up to 1.0 M NaCl they exhibit varying degrees of cooperatively. Starting from zero salt, as the NaCl concentration used for complex formation is increased, both the CD and the melting properties of the complexes are shifted from those of directly mixed at zero salt to those of reconstitution: in the CD spectra there is a gradual shift from a B → C transition to a B → ψ transition; thermal denaturation results show a gradual increase in the melting temperatures of both free DNA (t_m) and polylysine-bound DNA (t′_m). The progressive shift from B → C to B → ψ suggests a close relationship between these two transitions. Large aggregates of the complexes do not warrant the appearance of ψ-type CD spectra: ψ-spectra have been obtained in the supernatants of polylysine–DNA complexes made and measured at 1.0 M NaCl while slightly perturbed CD spectra in B → C transition have been observed in turbid solutions of fully covered complexes made at very low salt. If the complexes are made at intermediate salts and dialyzed to a very low salt, although up to 60% of the DNA is still bound by polylysine, the CD spectra of the complexes are shifted back to the B-type CD characteristic of pure DNA.     

Rhythmic Variations of Pharmacokinetic Processes of Cyproterone Acetate in Rabbits

A study about the influence of administration time on pharmacokinetics of Cyproterone acetate in rabbits was performed. Thirty animals were distributed in six groups, each corresponding to a different time: 2, 6, 10, 14, 18 and 22 hours after light onset (HALO). Rabbits received a single intravenous administration of 4 mg kg^-1 Cyproterone acetate. Blood samples were taken and processed by high performance liquid chromatography. Plasmatic data were fitted to a biexponential equation. Concentration at zero time (Co), concentration at zero time extrapolated from the distribution phase (A), hybrid constant for distribution phase and its half life, volume of the central compartment (V_c), volume of distribution at steady state (V_ss), constants for transferring the drug from the central compartment to the exterior (K₁₀) and from the central to the peripheral compartment presented chronobiological variations (p < 0.05) and were fitted to a cosine equation. The following parameters adjusted to circadian rhythms: Co (Acrophase: 1.72 HALO); A (Acrophase: 4.09 HALO); V_c (Acrophase: 11.92 HALO); V_ss (Acrophase: 10.12 HALO) and K₁₀ (Acrophase: 3.59 HALO). It was concluded that pharmacokinetics of intravenously injected CPA in rabbits would behave in a different, though predictable, manner according to the animal's biological clock.     

Studies of insufficient mixing in bioreactors: Effects of limiting oxygen concentrations and short term oxygen starvation on Penicillium chrysogenum

The effect of oxygen limitation on the respiration rate of Penicillium chrysogenum was studied. The results show that measurements of critical oxygen tensions within a process that on morphological or on physical grounds exhibits an inhomogenous structure are not likely to resemble the Monod model.In order to study the effects of short term oxygen starvation on the respiratory capacity of Penicillium chrysogenum, a two compartment fermenter was constructed. This fermenter consists of one well mixed aerobic part (CSTR) and one minor anaerobic part (CPFR). In the latter the circulation time as well as the volume can be varied. After passage of the whole cell culture volume through the anaerobic part, irreversible inhibition of the respiration was observed. This was caused by a circulation time of 5 and 10 min in the plug flow reactor and with a volume of 6% of the stirred tank reactor volume. However, circulation times of 1 and 2 min with an anaerobic zone of 1% of the stirred tank reactor volume did not give any irreversible effects on the respiratory capacity.This was compared with the results of the previously established model ln(1 — I _OUR//100)^–1 = kt [1]. The I _OUR is the percentage irreversible inhibition of the respiration, t is the anaerobic circulation time and k is a constant. The two compartment fermenter results agree with the earlier model at circulation times of 5 and 10 min, but not with the shorter times, and this suggests that a lag phase exists in the inactivation kinetics.     

Upward ant distribution shift corresponds with minimum,not maximum,temperature tolerance

Rapid climate change may prompt species distribution shifts upward and poleward, but species movement in itself is not sufficient to establish climate causation. Other dynamics, such as disturbance history, may prompt species distribution shifts resembling those expected from rapid climate change. Links between species distributions, regional climate trends and physiological mechanism are needed to convincingly establish climate‐induced species shifts. We examine a 38‐year shift (1974–2012) in an elevation ecotone between two closely related ant species, Aphaenogaster picea and A. rudis. Even though A. picea and A. rudis are closely related with North American distributions that sometimes overlap, they also exhibit local‐ and regional‐scale differences in temperature requirements so that A. rudis is more southerly and inhabits lower elevations whereas A. picea is more northerly and inhabits high elevations. We find considerable movement by the warm‐habitat species upward in elevation between 1974 and 2012 with A. rudis, replacing the cold‐habitat species, A. picea, along the southern edge of the Appalachian Mountain chain in north Georgia, USA. Concomitant with the distribution shifts, regional mean and maximum temperatures remain steady (1974–2012), but minimum temperatures increase. We collect individuals from the study sites and subject them to thermal tolerance testing in a controlled setting and find that maximum and minimum temperature acclimatization occurs along the elevation gradient in both species, but A. rudis consistently becomes physiologically incapacitated at minimum and maximum temperatures 2 °C higher than A. picea. These results indicate that rising minimum temperatures allow A. rudis to move upward in elevation and displace A. picea. Given that Aphaenogaster ants are the dominant woodland seed dispersers in eastern deciduous forests, and that their thermal tolerances drive distinct differences in temperature‐cued synchrony with early blooming plants, these climate responses not only impact ant‐ant interactions, but might have wide implications for ant‐plant interactions.     

A stochastic model for then-compartment irreversible system

This paper deals with a stochasticn-compartment irreversible system with a non-homogeneous Poisson input and arbitrary residence time for each of the compartments. Results relating to the number of particles present in each of the compartments as well as the total number of particles present in the system at any time are derived. Further, explicit expressions for the auto covariance function for each compartment and the cross-covariance function between any two compartments with a given time lag are obtained. As a particular case, then-compartment irreversible system is analyzed with homogeneous Poisson input and exponential residence time distribution for each of the compartments. The possible applications of the model are discussed.     

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.     

Limits of the distal inversion in the t complex of the house mouse: Evidence from linkage disequilibria

The suppression of crossing-over and the consequent linkage disequilibrium of genetic markers within the t complex of the house mouse is caused by two large and two short inversions. The inversions encompass a region that is some 15 centiMorgans (cM) long in the homologous wild-type chromosome. The limits of the proximal inversions are reasonably welldefined, those of the distal inversions much less so. We have recently obtained seven new DNA markers (D17Tu) which in wild-type chromosomes map into the region presumably involved in the distal inversions of the t chromosomes. To find out whether the corresponding loci do indeed reside within the inversions, we have determined their variability among 26 complete and 12 partial t haplotypes. In addition, we also tested the same collection of t haplotypes for their variability at five D17Leh, Hba-ps4, Pim-1, and Crya-1 loci. The results suggest that the distal end of the most distal inversion lies between the loci D17Leh467 and D17Tu26. The proximal end of the large distal inversion was mapped to the region between the D17Tu43 and Hba-ps4 loci, but this assignment is rather ambiguous. The loci Pim-1, Crya-1, and the H-2 complex, which have been mapped between the Hba-sp4 and Grr within the large distal inversion, behave as if they recombine from time to time with their wildtype homologs.