Comparison of mathematical models for the maternal age dependence of Down's syndrome rates

Summary The maternal age dependence of Down's syndrome rates was analyzed by two mathematical models, a discontinuous (DS) slope model which fits different exponential equations to different parts of the 20–49 age interval and a CPE model which fits a function that is the sum of a constant and exponential term over this whole 20–49 range. The CPE model had been considered but rejected by Penrose, who preferred models postulating changes with age assuming either a power function X¹⁰, where X is age or a Poisson model in which accumulation of 17 events was the assumed threshold for the occurrence of Down's syndrome. However, subsequent analyses indicated that the two models preferred by Penrose did not fit recent data sets as well as the DS or CPE model. Here we report analyses of broadened power and Poisson models in which n (the postulated number of independent events) can vary. Five data sets are analyzed. For the power models the range of the optimal n is 11 to 13; for the Poisson it is 17 to 25. The DS, Poisson, and power models each give the best fit to one data set; the CPE, to two sets. No particular model is clearly preferable. It appears unlikely that, with a data set from any single available source, a specific etiologic hypothesis for the maternal age dependence of Down's syndrome can be clearly inferred by the use of these or similar regression models.     

Marcus treatment of endergonic reactions: a commentary

Two forms of the equation for expression of the rate constant for electron transfer through a Marcus-type treatment are discussed. In the first (exergonic) form, the Arrhenius exponential term was replaced by its classical Marcus term; in the second (endergonic) form, the forward rate constant was replaced by the reverse rate constant (the forward rate constant in the exergonic direction), which was expanded to an equivalent Marcus term and multiplied by the equilibrium constant. When the classical Marcus treatment was used, these two forms of the rate equation give identical curves relating the logarithm of the rate constant to the driving force. The Marcus term for the relation between activation free-energy, DeltaG#, reorganization energy, lambda, and driving force, DeltaG(o), derived from parabolas for the reactant and product states, was identical when starting from exergonic or endergonic parabolas. Moser and colleagues introduced a quantum mechanical correction factor to the Marcus term in order to fit experimental data. When the same correction factor was applied in the treatment for the endergonic direction by Page and colleagues, a different curve was obtained from that found with the exergonic form. We show that the difference resulted from an algebraic error in development of the endergonic equation.     

Sister-chromatid exchange in childhood in relation to age and sex

Small children have been found to have a lower SCE/cell than adults and in recent reports females have had higher SCEs/cell than males. We here describe the relationship between SCE/cell and age and sex in 46 girls and 39 boys with an age range of 1.4-19.2 years and 2.6-18.7 years, respectively. For the calculation a transformation y = (sum SCE)1/2 + (sum SCE + 1)1/2 was used. The best fit to our material was represented by the equation y = b0 + b1 X log age. A common slope (b1) could be used for the boys and girls. This slope was significantly different from zero (P less than 0.0005). The levels of the regression lines for the two sexes were different (P = 0.0006). The girls had a 0.55-0.7 higher SCE/cell than the boys, depending on age. The following equations were found: Girls: y = 22.49 + 6.53 X log age. Boys: y = 21.11 + 6.53 X log age. By this model 43% of the variation in y could be explained. As a consequence of the result it is absolutely essential, when planning studies of children, to use age-matched groups to decrease the variability of the test system.     

Frequency of Down syndrome in livebirths by single-year maternal age interval: results of a Massachusetts study.

An analysis of rates of intra-state Down syndrome livebirths to Massachusetts residents by single-year maternal age interval in 1958-1965 inclusive was carried out. A gradual increase of rate of the Down syndrome occurred from age 20 to about age 31, and a steeper increase thereafter. Different regression equations were derived in the 20-31 and the 33-45 age group. The regression equations were ln y = 0.04515 x -1.45759 for those age 20-31 and ln y = 0.24302x-7.57870, for those age 33-45, where y = rate per 1,000 and x = maternal age. The regression-derived rates are slightly lower than those reported in similar analyses of data from Sweden and New York State, but they are not markedly discrepant.     

Developmental causes of allometry: new models and implications for phenotypic plasticity and evolution

Shapes change during development because tissues, organs, and various anatomical features differ in onset, rate, and duration of growth. Allometry is the study of the consequences of differences in the growth of body parts on morphology, although the field of allometry has been surprisingly little concerned with understanding the causes of differential growth. The power-law equation y?=?ax(b), commonly used to describe allometries, is fundamentally an empirical equation whose biological foundation has been little studied. Huxley showed that the power-law equation can be derived if one assumes that body parts grow with exponential kinetics, for exactly the same amount of time. In life, however, the growth of body parts is almost always sigmoidal, and few, if any, grow for exactly the same amount of time during ontogeny. Here, we explore the shapes of allometries that result from real growth patterns and analyze them with new allometric equations derived from sigmoidal growth kinetics. We use an extensive ontogenetic dataset of the growth of internal organs in the rat from birth to adulthood, and show that they grow with Gompertz sigmoid kinetics. Gompertz growth parameters of body and internal organs accurately predict the shapes of their allometries, and that nonlinear regression on allometric data can accurately estimate the underlying kinetics of growth. We also use these data to discuss the developmental relationship between static and ontogenetic allometries. We show that small changes in growth kinetics can produce large and apparently qualitatively different allometries. Large evolutionary changes in allometry can be produced by small and simple changes in growth kinetics, and we show how understanding the development of traits can greatly simplify the interpretation of how they evolved.     

Marcus treatment of endergonic reactions: A commentary

Two forms of the equation for expression of the rate constant for electron transfer through a Marcus-type treatment are discussed. In the first (exergonic) form, the Arrhenius exponential term was replaced by its classical Marcus term; in the second (endergonic) form, the forward rate constant was replaced by the reverse rate constant (the forward rate constant in the exergonic direction), which was expanded to an equivalent Marcus term and multiplied by the equilibrium constant. When the classical Marcus treatment was used, these two forms of the rate equation give identical curves relating the logarithm of the rate constant to the driving force. The Marcus term for the relation between activation free-energy, ΔG^#, reorganization energy, λ, and driving force, ΔG^o, derived from parabolas for the reactant and product states, was identical when starting from exergonic or endergonic parabolas. Moser and colleagues introduced a quantum mechanical correction factor to the Marcus term in order to fit experimental data. When the same correction factor was applied in the treatment for the endergonic direction by Page and colleagues, a different curve was obtained from that found with the exergonic form. We show that the difference resulted from an algebraic error in development of the endergonic equation.     

Generalization of the Monod-Wyman-Changeux model for the case of multisubstrate reactions]

A general equation is derived for the rate of multisubstrate reaction catalyzed by oligomeric enzyme E(R, T) liable to concerted transitions Ro in equilibrium To or Ro in equilibrium 2To. It is shown that with some assumptions about the enzymes the rate equations can be constructed from the rates of corresponding reactions catalyzed by a single active site. These single active site rate equations are known for the majority of catalysis mechanisms, otherwise they can be easily deduced. As an example the rate equation is derived for the reaction S1 + S2 + S3 in equilibrium S4 + S5 catalyzed by an oligomeric enzyme according to the ordered ter-bi mechanism.     

Derivation of an electron-transport rate equation, energy-conservation equations and a luminescence-flux equation of algal and plant photosynthesis.

On the assumption that the photosynthetic electron-transport rate is sometimes limited on the water-splitting side of Q (the oxidized primary electron acceptor), and that Q reduction, as well as primary charge recombination, is not kinetically a monomolecular process, a rate equation, a luminescence-flux equation and several versions of energy-conservation equations are derived. The energy-conservation equations explain most, if not all, observed relationships between rate and fluorescence. In particular, by assuming that the limiting site on the water-splitting side of Q is uncoupler-sensitive, these equations explain the uncoupler-induced simultaneous stimulations of rate and fluorescence as well as inhibition of luminescence without additional assumption ad hoc for each individual phenomenon. A newly introduced parameter central to the derivation of these equations is the specific affinity between two electron carriers.     

Exponentiated exponential model (Gompertz kinetics) of Na+ and K+ conductance changes in squid giant axon

The conductance changes, gK(t) and gNa(t), of squid giant axon under voltage clamp (Hodgkin and Huxley, 1952) may be modeled by exponentiated exponential functions (Gompertz kinetics) from any holding potential VO to any membrane clamp potential V. The equation constants are set by the membrane potential V, and include, for any voltage step in the case of gK, the initial conductance, gO, the asymptote conductance g, and rate constant k: gK = g exp(-be-kt) where b = 1n g/gO. Equations of similar form relate g and k to the voltage V, and govern the corresponding parameters of the gNa system. For the gNa, the fast phase y = y exp (-be-kt) is cut down in proportion to a slow process p = (1 - p)e-k't + p, and thus gNa = py. The expo-exponential functions involve fewer constants than the Hodgkin-Huxley model. In particular, the role of the n, m, h parameters appears to be filled largely by 1n (g/gO) in the case of gK and by 1n (y/yO) in the case of gNa. Membrane action potentials during current clamp may be computed from the conductances generated by use of the appropriate differential forms of the equations; diverse other membrane behaviors may be predicted.     

海南岛南湾半岛猕猴(Macaca mulatta)种群数量动态及分布

自1981年以来,对海南岛南湾半岛猕猴进行了连续5年的调查研究。归纳1965年到1984年底的材料,该种群从100只发展到930只,其间种群的年均增长率为13％,种群增长率受着种群密度的制约,密度与增长率间存在着关系式y=-0.0003x~2+0.0857x+0.3534。该种群生长曲线可表示为指数函数式y=0.1253e~(0.1046x)或为Logistic曲线N=1850/1+e~(11.28-0.13x)。该半岛猕猴最大容纳量为1850只。各猴群的弹性核域受猴群密度和植被质量的影响,绘出了它们之间的关系图。     

On hyperbolic discounting and uncertain hazard rates

The value of a future reward should be discounted where there is a risk that the reward will not be realized. If the risk manifests itself at a known, constant hazard rate, a risk-neutral recipient should discount the reward according to an exponential time-preference function. Experimental subjects, however, exhibit short-term time preferences that differ from the exponential in a manner consistent with a hazard rate that falls with increasing delay. It is shown here that this phenomenon can be explained by uncertainty in the underlying hazard. The time-preference function predicted by this analysis can be calculated by means of either (i) a direct superposition method, or (ii) Bayesian updating of the expected hazard rate. The observed hyperbolic time-preference function is consistent with an exponential prior distribution for the underlying hazard rate. Sensitivity of the predicted time-preference function to variation in the probability distribution of the underlying hazard rate is explored.     

Heterochrony and sexual dimorphism in the pigtailed macaque (Macaca nemestrina)

Somatic growth is not a simple linear process with a constant rate of growth. The most successful attempts to quantify growth as a function of age or size have employed nonlinear techniques. Sexual dimorphism of primate growth, weight vs. age, was examined using nonlinear models with Sirianni and Swindler's ([1985] Growth and Development of the Pigtailed Macaque, Boca Raton, FL: CRC Press) growth data on the pigtailed macaque (Macaca nemestrina). The best fit of several exponential growth models was the Gompertz curve: Different multiple phase models were also fit, where each phase represents a distinct exponential component. The two-phase models proved to be the best (R² = .0.84 for females, 0.91 for males), suggesting that there are two growth spurts, one in infancy and one at puberty. The timing of the beginning and end of the first spurt is the same in males and females, but the rate, and value of the asymptote for this phase, is greater in males. The timing of the second spurt is earlier, and the rate of growth for this spurt is smaller in females than males. The sexual dimorphism in these species is not a simple rate change, but a complex interaction of timing and rate over the entire period of growth. It would be impossible to separate these entities with a linear, polynomial, or single-phase model of the data. While these data and results complement much of the existing work on adult dimorphism, they also emphasize the vital role that ontogenetic data have in elucidating the underlying evolutionary mechanisms that generate sexual dimorphism. © 1994 Wiley-Liss, Inc.     

Modeling transitions in body composition: the approach to steady state for anthropometric measures and physiological functions in the Minnesota human starvation study

Background

This study evaluated whether the changes in several anthropometric and functional measures during caloric restriction combined with walking and treadmill exercise would fit a simple model of approach to steady state (a plateau) that can be solved using spreadsheet software (Microsoft Excel^®). We hypothesized that transitions in waist girth and several body compartments would fit a simple exponential model that approaches a stable steady-state.

Methods

The model (an equation) was applied to outcomes reported in the Minnesota starvation experiment using Microsoft Excel's Solver^® function to derive rate parameters (k) and projected steady state values. However, data for most end-points were available only at t = 0, 12 and 24 weeks of caloric restriction. Therefore, we derived 2 new equations that enable model solutions to be calculated from 3 equally spaced data points.

Results

For the group of male subjects in the Minnesota study, body mass declined with a first order rate constant of about 0.079 wk^-1. The fractional rate of loss of fat free mass, which includes components that remained almost constant during starvation, was 0.064 wk^-1, compared to a rate of loss of fat mass of 0.103 wk^-1. The rate of loss of abdominal fat, as exemplified by the change in the waist girth, was 0.213 wk^-1.On average, 0.77 kg was lost per cm of waist girth. Other girths showed rates of loss between 0.085 and 0.131 wk^-1. Resting energy expenditure (REE) declined at 0.131 wk^-1. Changes in heart volume, hand strength, work capacity and N excretion showed rates of loss in the same range. The group of 32 subjects was close to steady state or had already reached steady state for the variables under consideration at the end of semi-starvation.

Conclusion

When energy intake is changed to new, relatively constant levels, while physical activity is maintained, changes in several anthropometric and physiological measures can be modeled as an exponential approach to steady state using software that is widely available. The 3 point method for parameter estimation provides a criterion for testing whether change in a variable can be usefully modelled with exponential kinetics within the time range for which data are available.

     

The effect of α-factor on the rate of cell-cycle initiation in Saccharomyces cerevisiae : α-Factor modulates transition probability in yeast

The rate of cell-cycle initiation was studied in a-cells of S. cerevisiae in the presence of the synthetic analogue of α-factor [N-Trp, Arg⁷]-α-factor (TA-αF). It was shown that TA-αF lowers the rate constant of cell-cycle initiation (or transition probability) for each separate cell. It was concluded on the basis of these results that the term ‘arrest’ in G1 by α-factor should be interpreted in a quantitative sense as the decrease in the probability of the emergence from ‘start’ per unit time and not as being equivalent to an ‘all-or-none’ response. The dependence of the rate constant of the cell-cycle initiation on the concentration of TA-αF can be described by the Hill equation where n = 1.01 + 0.05 and K = 14.5 ± 2.5 nM (±S.E.). It is demonstrated that after transfer of cells into the medium with a higher or a lower concentration of TA-αF, the rate constant of cell-cycle initiation changes abruptly from one value to another after a lag-period of 30 and 40 min respectively. This suggests a multistep mechanism of action for α-factor. The difference in the lag-periods allows us to suggest that α-factor exerts its action by two independent pathways. Since the Hill coefficient is practically equal to unit, no cooperative interactions are likely to be involved at least in one of these pathways. The inhibition of cell-cycle initiation can be used as a more adequate and sensitive test for biological activity of α-factor as compared to morphological measurements.     

Cell Growth and Division: III. Conditions for Balanced Exponential Growth in a Mathematical Model

In a previous paper, we proposed a model in which the volume growth rate and probability of division of a cell were assumed to be determined by the cell's age and volume. Some further mathematical implications of the model are here explored. In particular we seek properties of the growth and division functions which are required for the balanced exponential growth of a cell population. Integral equations are derived which relate the distribution of birth volumes in successive generations and in which the existence of balanced exponential growth can be treated as an eigenvalue problem. The special case in which all cells divide at the same age is treated in some detail and conditions are derived for the existence of a balanced exponential solution and for its stability or instability. The special case of growth rate proportional to cell volume is seen to have neutral stability. More generally when the division probability depends on age only and growth rate is proportional to cell volume, there is no possibility of balanced exponential growth. Some comparisons are made with experimental results. It is noted that the model permits the appearance of differentiated cells. A generalization of the model is formulated in which cells may be described by many state variables instead of just age and volume.     

The accumulation of radioactive caesium from water by the brown trout (Salmo trutta) and its comparison with plaice and rays

The patterns of accumulation of caesium-137 from water by the tissues and organs of the freshwater teleost, the brown trout ( Salmo trutta ) are described. Estimates of the biological half-times and steady-state concentrations are derived, using a simple exponential equation. In all tissues and organs examined, other than muscle, the rate processes of the trout fall between those of the plaice and the ray. It is concluded that most of the caesium accumulated by the brown trout from water enters other than by the gut, probably through the gills, but as with plaice and ray, the main source of the caesium, possibly 90%, must come from the food. Despite differences in the levels of accumulation, the ratios of the tissue to blood steady state concentrations are very similar in all three species. The steady state caesium concentration of the blood appears to be directly related to the red blood cell count of the fish.     

The interpretation of voltage-concentration relations

The dependence of a cell's membrane potential upon ion concentrations is often described via a Goldman constant field equation (with coefficients called permeabilities) or what may be called the Hodgkin-Horowicz equation. The latter is derived by assuming that different ion species traverse separate pathways of at least formally constant conductance. If one assumes truly constant “permeabilities” or “conductances”, then these two equations have essentially different forms. So the literature has been culled for voltage-concentration data which unambiguously fit an equation of one of these two forms. Eight or nine are found to fit a Hodgkin-Horowicz relation; none, a Goldman one. This adds to the evidence that different ion species traverse relatively separate pathways through cell membranes (and that these pathways are long compared to a Debye length). It also indicates the desirability of generally describing voltage-concentration data with a Hodgkin-Horowicz relation rather than a Goldman one.     

Paternal age and Down's syndrome genotypes diagnosed prenatally: No association in New York State data

Summary An investigation of a paternal age effect independent of maternal age was undertaken for 98 cases of Down's syndrome genotypes diagnosed prenatally compared to 10,329 fetuses with normal genotype diagnosed prenatally in data reported to the New York State Chromosome Registry. The mean of the difference (delta) in paternal age of cases compared to those with normal genotypes after controlling for maternal age, was slightly negative,-0.27 with a 95% confidence interval of-1.59 to +1.06. A regression analysis was also done in which the data were first fit to an equation of the type lny=(bx+c) and then to the equation ln y=(bx+dz+c) where y = rate of Down's syndrome, x = maternal age, z = paternal age, and b, d, and c are parameters. This also revealed no evidence for a paternal age effect. The value of d (the paternal age coefficient) was in fact slightly negative,-0.0058, with an asymptotic 95% confidence interval of-0.0379 to +0.0263. Lastly, multiple applications of the Mantel-Haenszel test considering various boundaries in paternal age also revealed no statistically significant evidence for a paternal age effect independent of maternal age. These results are at variance with claims of others elsewhere of a very strong paternal age effect detected in studies at prenatal diagnoses. Five different hypotheses are suggested which may account for discrepancies among studies to date in findings on paternal age effects for Down's syndrome: (i) there are temporal, geographic, or ethnic variations in paternal age effects, (ii) there is no paternal age effect and statistical fluctuation accounts for all trends to date; (iii) methologic artifacts have obscured a paternal age effect in some studies which did not find a positive outcome; (iv) methodologic artifacts are responsible for the positive results in some studies to date; (v) there is a rather weak paternal age effect independent of maternal age in most if not all populations, but because of statistical fluctuation the results are significant only in some data sets. The results of all data sets to date which we have been able to analyze by one year intervals are consistent with a mean delta of +0.04 to +0.48 and in the value of d (the paternal age coefficient) of +0.006 to +0.017, and it appears the fifth hypothesis cannot be excluded. Projections based on this assumption are presented.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Appendix: A general regression model for analysis of independent maternal and paternal age effects for 47,+21 and other disorders that may arise from mutant gametes from either parent

Summary Biological considerations suggest that regression equations used to model the rate of mutational outcomes as a possible function of maternal age and paternal age (or other parental factors) are most appropriately additive models of the type: r=(h(x)+j(y) or r=(h(x)+(k(x)·j(y)), where r is the rate of the outcome event, x is maternal age, y is paternal age, and h, j and k are functions to be specified. The first, simpler model assumes that there is no independent maternal age effect upon formation of a gamete or zygote with a paternally derived mutation or upon survival of the consequent conceptus. The second more general model relaxes this assumption. These models appear preferable to those used previously, such as log r=(h(x)+j(y)) or equivalently r=exp (h(x)·j(y)), which posit complex relationships closer to a multiplicative interaction for which it is difficult to suggest obvious biological interpretations.