The role of optimal control in assessing the most cost-effective implementation of a vaccination programme: HPV as a case study

Consider a population that develops over units of time labelled by zero and the negative integers. It is assumed that at any time r ? 0 there are respectively N(m, a) males of age a and N(f, b) of females of age b, where a = 1, … , A_m and b = 1, … , A_f. At time 0 a sample of n copies of a gene are assumed to be observed, where n ? min_ab(N(m, a), N(f, b)). It is assumed that at any particular time r any possible mating is equally probable and that numbers of gametes contributed to offspring of age 1 and sex u by parents of sex v are exchangeable within age groups and independently distributed among age-groups. Coalescent theory is then derived, with time measured in multiples 2N_eC of the effective population size N_eC, which depends on a measure T of the generation interval. Theory is developed for both autosomal and sex-linked loci in two special cases.     

An Unbalanced Nested Model with Random Effects Having Unequal Variances

Consider the model Y_ijk=μ + a_i + b_ij + e_ijk (i=1, 2,…, t; j=1,2,…, B_i; k=1,2…,n_ij), where μ is a constant and a¹,b_ij and e_ijk are distributed independently and normally with zero means and variances σ²_ad_ij and σ², respectively, where it is assumed that the d_i's and d_ij's are known. In this paper procedures for estimating the variance components (σ², σ²_a and σ²_b) and for testing the hypothesis σ²_b = 0 and σ²_a = 0 are presented. In the last section the mixed model y_ijk, where x_ijkkm are known constants and β_m's are unknown fixed effects (m = 1, 2,…,p), is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the β_m's.     

Predicting responses of photosynthesis and root fraction to elevated [CO2]a: interactions among carbon,nitrogen, and growth*

At elevated atmospheric CO₂ concentrations ([CO₂]_a), photosynthetic capacity (A_max) and root fraction (η_R, the ratio of root to plant dry mass) increased in some studies and decreased in others. Here, we have explored possible causes of this, focusing on the relative magnitudes of the effects of elevated [CO₂]_a on specific leaf (n_m) and plant (n_p) nitrogen concentrations, leaf mass per unit area (h), and plant nitrogen productivity (α). In our survey of 39 studies with 35 species, we found that elevated [CO₂]_a led to decreased n_m and n_p in all the studies and to increased h and α in most of the studies. The magnitudes of these changes varied with species and with experimental conditions. Based on a model that integrated [CO₂]_a-induced changes in leaf nitrogen into a biochemically based model of leaf photosynthesis, we predicted that, to a first approximation, photosynthesis will be upregulated (A_max will increase) when growth at increased [CO₂]_a leads to increases in h that are larger than decreases in n_m. Photosynthesis will be downregulated (A_max will decrease) when increases in h are smaller than decreases in n_m. The model suggests that photosynthetic capacity increases at elevated [CO₂]_a only when additional leaf mesophyll more than compensates the effects of nitrogen dilution. We considered two kinds of regulatory paradigms that could lead to varying responses of η_R to elevated [CO₂]_a, and compared the predictions of each with the data. A simple static model based on the functional balance concept predicts that η_R should increase when neither n_p nor h is very responsive to elevated [CO₂]_a. The quantitative and qualitative agreement of the predictions with data from the literature, however, is poor. A model that predicts η_R from the relative sensitivities of photosynthesis and relative growth rate to elevated [CO₂]_a corresponds much more closely to the observations. In general, root fraction increases if the response of photosynthesis to [CO₂]_a is greater than that of relative growth rate.     

Approximate Confidence Intervals for Contrasts in the Balanced Three Factor Mixed Model

When conducting a statistical analysis of data from a designed experiment, an investigator is often interested in confidence intervals for contrasts of the fixed effects. If the analysis involves a mixed linear model, exact confidence intervals for contrasts of the fixed effects are not always available. In such cases, confidence intervals with approximate coverage probabilities must be used. As will be shown, this problem may be generalized to that of constructing a confidence interval for the parameter μ, where X is a normal random variable with mean μ and variance ∑ a_qθ_q, where a₁…,a_Q are known constants, U_q = n_qS/θ_q is a chi-squared random variable with n_q degrees of freedom, for each q = 1,…, Q, and X,U₁,…, U_Q are mutually independent. In this paper, we consider the case where Q = 3 and a₃ ≤0.     

Acclimation of photosynthetic capacity to irradiance in tree canopies in relation to leaf nitrogen concentration and leaf mass per unit area

The observation of acclimation in leaf photosynthetic capacity to differences in growth irradiance has been widely used as support for a hypothesis that enables a simplification of some soil‐vegetation‐atmosphere transfer (SVAT) photosynthesis models. The acclimation hypothesis requires that relative leaf nitrogen concentration declines with relative irradiance from the top of a canopy to the bottom, in 1 : 1 proportion. In combination with a light transmission model it enables a simple estimate of the vertical profile in leaf nitrogen concentration (which is assumed to determine maximum carboxylation capacity), and in combination with estimates of the fraction of absorbed radiation it also leads to simple ‘big‐leaf’ analytical solutions for canopy photosynthesis. We tested how forests deviate from this condition in five tree canopies, including four broadleaf stands, and one needle‐leaf stand: a mixed‐species tropical rain forest, oak (Quercus petraea (Matt.) Liebl), birch (Betula pendula Roth), beech (Fagus sylvatica L.) and Sitka spruce (Picea sitchensis (Bong.) Carr). Each canopy was studied when fully developed (mid‐to‐late summer for temperate stands). Irradiance (Q, µmol m^?2 s^?1) was measured for 20 d using quantum sensors placed throughout the vertical canopy profile. Measurements were made to obtain parameters from leaves adjacent to the radiation sensors: maximum carboxylation and electron transfer capacity (V_a, J_a, µmol m^?2 s^?1), day respiration (R_da, µmol m^?2 s^?1), leaf nitrogen concentration (N_m, mg g^?1) and leaf mass per unit area (L_a, g m^?2). Relative to upper‐canopy values, V_a declined linearly in 1 : 1 proportion with N_a. Relative V_a also declined linearly with relative Q, but with a significant intercept at zero irradiance (P < 0·01). This intercept was strongly related to L_a of the lowest leaves in each canopy (P < 0·01, r² = 0·98, n= 5). For each canopy, daily lnQ was also linearly related with lnV_a(P < 0·05), and the intercept was correlated with the value for photosynthetic capacity per unit nitrogen (PUN: V_a/N_a, µmol g^?1 s^?1) of the lowest leaves in each canopy (P < 0·05). V_a was linearly related with L_a and N_a(P < 0·01), but the slope of the V_a : N_a relationship varied widely among sites. Hence, whilst there was a unique V_a : N_a ratio in each stand, acclimation in V_a to Q varied predictably with L_a of the lowest leaves in each canopy. The specific leaf area, L_m(cm² g^?1), of the canopy‐bottom foliage was also found to predict carboxylation capacity (expressed on a mass basis; V_m, µmol g^?1 s^?1) at all sites (P < 0·01). These results invalidate the hypothesis of full acclimation to irradiance, but suggest that L_a and L_m of the most light‐limited leaves in a canopy are widely applicable indicators of the distribution of photosynthetic capacity with height in forests.     

DNA and RNA oligomer thermodynamics: The effect of mismatched bases on double-helix stability

The thermodynamic parameters for the double-strand formation of the molecules rCAmG + rCU_nG, m,n = 5–7, and dCA_mG + dCT_nG, m,n = 5,6, were measured from optical melting curves. Normal helices are formed when m = n. The deoxyoligomers are more stable than the ribo-oligomers, due to a more favorable enthalpy. Double helices with mismatched bases can be formed by mixing oligomers with m ≠ n. Such helices may form several possible structures. A structure with a dangling base is favored over a structure with a bulged base. The destabilization of the double strands by the formation of a bulged base was determined to be greater than 1.6 kcal/mol at 10°C. The extent of aggregation in the oligomer double strand rCA₇G·rCU₇G was determined using ultracentrifugation equilibrium. The possible effects of aggregation on the determination of the thermodynamic parameters for double-strand formation are discussed.     

Generalizations of the Roginsky-Zeldovich (or Elovich) equation for charge transport across biological surfaces

The Roginsky-Zeldovich (or Elovich) equation, which is −dx/dt=m exp (nx) (x=substrate concentration,t=time,m andn=constants), describes the kinetics of various biological electron and ion transport processes, and has been derived from the concept of charge transport across an activation energy barrier at an interface between dissimilar phases, driven by a difference in redox or ion potentials, with the simplifying assumptions that charge carrier concentration is constant, backward current across the interface is zero, and diffusion of substrate is fast. If charge carrier concentration is proportional to substrate concentration, then the kinetic equation is −dx/dt=mx exp (nx). If backward current is not zero, then −dx/dt=m ₁ exp (n ₁x) −m ₂ exp (n ₂ x), wherem ₁,m ₂,n ₁ andn ₂ are constants. Kinetic equations for interfacial charge transport in the presence of a significant substrate diffusion potential are also derived.     

Leaf and environmental parameters influencing transpiration: Theory and field measurements

Summary The influence of variations in the boundary air layer thickness on transpirtion due to changes in leaf dimension or wind speed was evaluated at a given stomatal resistance (r _s) for various combinations of air temperature (T _a) and total absorbed solar energy expressed as a fraction of full sunlight (S _ffs). Predicted transpiration was found to either increase or decrease for increases in leaf size depending on specific combinations of T _a, S _ffs, and r _s. Major reductions in simulated transpiration with increasing leaf size occurred for shaded, highly reflective, or specially oriented leaves (S _ffs=0.1) at relatively high T _a when r _s was below a critical value of near 500 s m^-1. Increases in S _ffs and decreases in T _a lowered this critical resistance to below 50 s m^-1 for S _ffs=0.7 and T _a=20°C. In contrast, when r _s was above this critical value, an increase in leaf dimension (or less wind) resulted in increases in transpiration, especially at high T _a and S _ffs. For several combinations of T _a, S _ffs, and r _s, transpiration was minimal for a specific leaf size. These theoretical results were compared to field measurements on common desert, alpine, and subalpine plants to evaluate the possible interactions of leaf and environmental parameters that may serve to reduce transpiration in xeric habitats.     

Karyological and allozyme survey of the Common shrew, Sorex araneus, from Macedonia

Karyotype and genetic variation of the common shrew (Sorex araneus) from Mt. Pelister in southern Macedonia has been studied. Whereas all autosomes in the chromosomal set (2 n_a, = 28, the only present metacentrics being af, bc, jl, and tu) were of the standard type as well as the sex chromosomes X and Y₂ in males, the Y₁ chromosome was a small metacentric. This chromosomal feature is unique among the common shrew populations studied cytogenetically so far. Three out of 33 loci analysed (Sdh, 6Pgd, Mdh-1) were discriminant between the Mt. Pelister population and Sorex araneus from Slovenia and two loci (Est-3, Ada) were partially discriminant. A relatively high value of Nei's genetic distance (D = 0.137) confirms unique character of the Pelister population.     

Beta-turns in bridged proline-containing cyclic peptide models

The synthesis, CD, ir spectroscopic, and conformational studies of a series of bridged cyclic peptides of the general formula, cyclo[NH-(CH₂)_n-CO-Gly-Pro-Y-Gly] (2a–d, Y = Gly or Ser(OBu^t), n = 4 or 2) is reported. As indicated by difference nuclear Overhauser enhancement and Fourier transform ir experiments, the tetrapeptide sequence of cyclo[NH-(CH₂)₄-CO-Gly-Pro-Gly-Gly] (2a) and cyclo[NH-(CH₂)₂-CO-Gly-Pro-Gly-Gly] (2b) adopts a 1 ← 4 hydrogenbonded type II β-turn conformation in solution, while cyclo[NH-(CH₂)₄-CO-Gly-Pro-Ser(OBu^t) -Gly] (2c) features a type I β-turn, fixed by 1 ← 4 and O^γ … NH intramolecular H bonds. In aqueous solution 2a and 2c show class B and class C CD spectra, respectively. This is the first case reported of a typical class C CD pattern in aqueous solution for a conformationally mobile system having a type I β-turn. Based on the comparison of the band intensities of the bridged models with those of linear and cyclic model systems reported earlier, a set of subspectra with reduced band intensities is suggested for use in the CD analysis of the conformation of polypeptides in solution.     

Analysis of metabolism by mathematical programming techniques

A method is presented for analyzing metabolic interactions by procedures based on mathematical programming techniques. In the procedures described it is assumed that the organism has (through natural selection) maximized (within the constraints imposed on it by its genetic constitution) its fitness to the environment. A practical experimental procedure is described through which the constraints imposed on reaction rates can be observed and from which the metabolic objective function which, it is presumed, metabolism has optimized can be calculated. A method for testing the validity of the objective function is given. Discussion is carried out in terms of a two-dimensional example but the procedures are valid for any number of dimensions. The results of the procedures are expressed by statements of the sort: the metabolic interactions of the cell are such that Q is maximized where Q = a₁x₁ + a₂x₂ + ... + a_nx_n, where a₁, ..., a_n are constants and x₁, ..., x_n are reaction rates. Some possible uses of metabolic objective functions are given.     

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.     

Effects of selection for ethanol preference on gene expression in the nucleus accumbens of HS‐CC mice

Previous studies on changes in murine brain gene expression associated with the selection for ethanol preference have used F₂ intercross or heterogeneous stock (HS) founders, derived from standard laboratory strains. However, these populations represent only a small proportion of the genetic variance available in Mus musculus. To investigate a wider range of genetic diversity, we selected mice for ethanol preference using an HS derived from the eight strains of the collaborative cross. These HS mice were selectively bred (four generations) for high and low ethanol preference. The nucleus accumbens shell of naive S₄ mice was interrogated using RNA sequencing (RNA‐Seq). Gene networks were constructed using the weighted gene coexpression network analysis assessing both coexpression and cosplicing. Selection targeted one of the network coexpression modules (greenyellow) that was significantly enriched in genes associated with receptor signaling activity including Chrna7, Grin2a, Htr2a and Oprd1. Connectivity in the module as measured by changes in the hub nodes was significantly reduced in the low preference line. Of particular interest was the observation that selection had marked effects on a large number of cell adhesion molecules, including cadherins and protocadherins. In addition, the coexpression data showed that selection had marked effects on long non‐coding RNA hub nodes. Analysis of the cosplicing network data showed a significant effect of selection on a large cluster of Ras GTPase‐binding genes including Cdkl5, Cyfip1, Ndrg1, Sod1 and Stxbp5. These data in part support the earlier observation that preference is linked to Ras/Mapk pathways.     

A mathematical model of biological evolution

In order to understand generally how the biological evolution rate depends on relevant parameters such as mutation rate, intensity of selection pressure and its persistence time, the following mathematical model is proposed: dN _n (t)/dt=(m _n (t-)N _n (t)+N _n-1(t) (n=0,1,2,3...), where N _n (t) and m _n (t) are respectively the number and Malthusian parameter of replicons with step number n in a population at time t and is the mutation rate, assumed to be a positive constant. The step number of each replicon is defined as either equal to or larger by one than that of its parent, the latter case occurring when and only when mutation has taken place. The average evolution rate defined by is rigorously obtained for the case (i) m _n (t)=m _n is independent of t (constant fitness model), where m _n is essentially periodic with respect to n, and for the case (ii) (periodic fitness model), together with the long time average m of the average Malthusian parameter . The biological meaning of the results is discussed, comparing them with the features of actual molecular evolution and with some results of computer simulation of the model for finite populations.An early version of this study was read at the International Symposium on Mathematical Topics in Biological held in kyoto, Japan, on September 11–12, 1978, and was published in its Procedings.     

Rules for assortative mating in relation to selection for linear merit functions

Summary This study examined how assortative mating (without selection) based on linear combinations of two traits could be used to change genetic parameters so as to increase efficiency of selection. The efficiency of the Smith-Hazel index for improvement of multiple traits is a function of phenotypic and genetic variances and covariances, and of the relative economic values of the traits involved. Assortative mating is known to change genetic variances and covariances. Recursive formulae were derived to obtain these variances and covariances after t generations of assortative mating on linear combinations (mating rules) of phenotypic values for two traits, with a given correlation between mates. Selection efficiency after t generations of assortative mating without selection was expressed as a function of random mating genetic parameters, economic values, the mating rule, and the correlation between mates. Selection efficiency was maximized with respect to the coefficients in the mating rule. Because the objective function was nonlinear, a computer routine was used for maximizing it. Two cases were considered. When random mating heritabilities for the two traits were h _X ² =0.25 and h _Y ² =0.50, the genetic correlation r_XY=-0.60, and the economic values were a_X=3 and a_Y=1, continued assortative mating based on the optimal mating rule for 31 generations (with a correlation of 0.80 between mates) increased selection efficiency by 29%. Heritabilities changed to 0.38 and 0.66, respectively, and the genetic correlation became – 0.79. When h _X ² =0.60, h _Y ² =0.60, r_XY=– 0.20, a₁=1 and a₂=1, 36 generations of continued assortative mating with the optimal mating rule increased the efficiency of selection by 17%, heritabilities became h _X ² = h _Y ² =0.71, and the genetic correlation changed to 0.25. Only three generations of assortative mating were required to change the sign of the genetic correlation.     

Dynamic light-scattering study of synthetic myosin filaments

Synthetic myosin filaments of rabbit were prepared. Electron microscopy showed that the number-average length (L_n = 470 nm) and sharpness in length distribution (L_w/L_n = 1.03₆) were independent of ionic strengths of 134, 74, and 44 mM, whereas the number ratio of M-filaments (about 15 nm in diameter at the bare zone) to m-filaments (about 10 nm) strongly depended on ionic strength (IS); the major filaments were M-filaments at IS = 134 mM, m-filaments at IS = 74 mM, and almost exclusively m-filaments at IS = 44 mM. Dynamic light scattering showed that the change in diameter with the change in ionic strength by 2-h dialysis was reversible. Combination of dynamic light scattering and sedimentation studies suggested a dynamic equilibrium between M- and m-filaments. Dynamic light-scattering spectra at IS = 134 and 74 mM could be analyzed by a theory for rigid rods, whereas those at IS = 44 mM only by introducing semiflexibility of filaments; m-filaments are more flexible at IS = 44 than at 74 mM.     

Redistribution of hepatocyte chloride during l-alanine uptake

We used ion-sensitive, double-barrel microelectrodes to measure changes in hepatocyte transmembrane potential (V _m), intracellular K⁺, Cl^-, and Na⁺ activities (a ⁱ _k, a _Cl ⁱ and a _Na ⁱ ), and water volume during l-alanine uptake. Mouse liver slices were superfused with control and experimental Krebs physiological salt solutions. The experimental solution contained 20 m l-alanine, and the control solution was adjusted to the same osmolality (305 mOsm) with added sucrose. Hepatocytes also were loaded with 50 mm tetramethylammonium ion (TMA⁺) for 10 min. Changes in cell water volume during l-alanine uptake were determined by changes in intracellular, steady-state TMA⁺ activity measured with the K⁺ electrode. Hepatocyte control V _m was -33±1 mV. l-alanine uptake first depolarized V _m by 2±0.2 mV and then hyperpolarized V _m by 5 mV to-38±1 mV (n = 16) over 6 to 13 min. During this hyperpolarization, a _Na ⁱ increased by 30% from 19±2 to 25±3 mm (P < 0.01), and a _K ⁱ did not change significantly from 83±3 mm. However, with added ouabain (1 mm) l-alanine caused only a 2-mV increase in V _m, but now a _K ⁱ decreased from 61±3 to 54±5 mm (P < 0.05). Hyperpolarization of V _m by l-alanine uptake also resulted in a 38% decrease of a _Cl ⁱ from 20±2 to 12±3 mm (P < 0.001). Changes in V _m and V _Cl — V _m voltage traces were parallel during the time of l-alanine hyperpolarization, which is consistent with passive distribution of intracellular Cl^– with the V _m in hepatocytes. Added Ba²⁺ abolished the l-alanineinduced hyperpolarization, and a _Cl ⁱ remained unchanged. Hepatocyte water volume during l-alanine uptake increased by 12±3%. This swelling did not account for any changes in ion activities following l-alanine uptake. We conclude that hepatocyte a _K ⁱ is regulated by increased Na⁺-K⁺ pump activity during l-alanine uptake in spite of cell swelling and increased V _m due to increased K⁺ conductance. The hyperpolarization of V _m during l-alanine uptake provides electromotive force to decrease a _Cl ⁱ . The latter may contribute to hepatocyte volume regulation during organic solute transport.This work was supported by grant AA-08867 from the Alcohol, Drug Abuse, and Mental Health Association.     

Mapping salt-tolerance genes in tomato (Lycopersicon esculentum) using trait-based marker analysis

The germination responsiveness of an F₂ population derived from the cross Lycopersicon esculentum (UCT5) x L. pennellii (LA716) was evaluated for salt tolerance at two stress levels, 150 mM NaCl + 15 mM CaCl₂ and 200 mM NaCl + 20 mM CaCl₂. Individuals were selected at both tails of the response distribution. The salt-tolerant and salt-sensitive individuals were genotyped at 16 isozyme loci located on 9 of the 12 tomato chromosomes. In addition, an unselected (control) F₂ population was genotyped at the same marker loci, and gene frequencies were estimated in both selected and unselected populations. Trait-based marker analysis was effective in identifying genomic locations (quantitative trait loci, QTLs) affecting salt tolerance in the tomato. Three genomic locations marked by Est-3 on chromosome 1, Prx-7 on chromosome 3, and 6Pgdh-2 and Pgi-1 on chromosome 12 showed significant positive effects, while 2 locations associated with Got-2 on chromosome 7 and Aps-2 on chromosome 8 showed significant negative effects. The identification of genomic locations with both positive and negative effects on this trait suggests the likelihood of recovering transgressive segregants in progeny derived from these parental lines. Similar genomic locations were identified when selection was made either for salt tolerance or salt sensitivity and at both salt-stress treatments. Comparable results were obtained in uni- and bidirectional selection experiments. However, when marker allele gene frequencies in a control population are unknown, bidirectional selection may be more efficient than unidirectional selection in identifying marker-QTL associations. Results from this study are discussed in relationship to the use of molecular markers in developing salt-tolerant tomatoes.     

Sample Sizes for Testing Equality of Location Parameters of Two Exponential Distributions

A table for determining minimal sample sizes n₁ = n₂ = n for testing the hypothesis of equality of location parameters a₄ and a₂ of two two-parametric exponential distributions for a first kind risk α=0,01 and 0,05 are given in such a way that the second kind risk β≦β₀ as long as |a₁–a>₂|>d.     

An Unbalanced Two-way Model With Random Effects Having Unequal Variances

Consider the model y_ijk=u ± a_i ± b_i ± c_ij ± e_ijk i=1, 2,…, t; j=1, 2,…b; k=1, 2,…,n_ij where μ is a constant and a_i, b_i, c_ij are distributed independently and normally with zero means and variances Δ₂ Δ2/bd_ij and δ₂ respectively. It is assumed that d_i's, and d_ij's are known (positive) constants (for all i and j). In this paper procedures for estimating the variance components (Δ₂, Δ²_b and Δ²_a) and for testing the hypothesis H_oc:Δ²_c/Δ² = y₃ and H_oa:Δ²_b/Δ² = y₄ (where y₂, y₃, and y₄, are specified constants) are presented. A generalization for the mixed model case is discussed in the last section.