Tests and model selection for the general growth curve model

The model considered here is a generalized multivariate analysis of variance model useful especially for many types of growth curve problems including biological growth and technology substitution. It is defined as Yp x N = Xp x m tau m x r Ar x N + epsilon p x N, where tau is unknown, and X and A are known design matrices of ranks m less than p and r less than N, respectively. Furthermore, the columns of epsilon are independent p-variate normal with mean vector 0 and common covariance matrix sigma. In general, p is the number of time (or spatial) points observed on each of the N cases, (m - 1) is the degree of polynomial in time, and r is the number of groups. The main focus of this paper is the selection of models for the general growth curve model with regard to the covariance matrix sigma. Likelihood ratio tests and selection procedures based on sample reuse and predictions are proposed. Special emphasis is on the serial covariance structure for sigma, which has been shown to be quite important in the prediction of biological data and technology substitution data. One-population and K-population problems are considered. Some of the results are illustrated with two sets of biological data.     

Mechanical properties of trabecular bone. Dependency on strain rate.

The effect of strain rate (epsilon) and apparent density (rho) on stiffness (E), strength (sigma u), and ultimate strain (epsilon u) was studied in 60 human trabecular bone specimens from the proximal tibia. Testing was performed by uniaxial compression to 5% specimen strain. Six different strain rates were used: 0.0001, 0.001, 0.01, 0.1, 1, and 10 s-1. Apparent density ranged between 0.23 and 0.59 g cm-3. Linear and non-linear regression analyses using strength, stiffness and ultimate strain as dependent variables (Y) and strain rate and apparent density as independent variables were performed using the following models: Y = a rho b epsilon c, Y = rho b(a + c epsilon; Y = (a + b rho)epsilon c, Y = a rho 2 epsilon c, E = a rho 3 epsilon c. The variations of strength and stiffness were explained equally well by the linear and the power function relationship to strain rate. The exponent was 0.07 in the power function relationship between strength and strain rate and 0.05 between stiffness and strain rate. The variation of ultimate strain was explained best using a power function relationship to strain rate (exponent = 0.03). The variation of strength and stiffness was explained equally well by the linear, power function and quadratic relationship to apparent density. The cubic relationship between stiffness and apparent density showed a less good fit. Ultimate strain varied independently of apparent density.     

Plant dispersal, neighbourhood size and isolation by distance

A theoretical relationship between isolation by distance or spatial genetic structure (SGS) and seed and pollen dispersal is tested using extensive spatial-temporal simulations. Although for animals Wright's neighbourhood size N(e) = 4pisigma(2)(t) has been ascertained also, where sigma(2)(t) is the axial variance of distances between parents and offspring, and it was recently confirmed that N(e) = 4pi(sigma(2)(f) + sigma(2)(m))/2 when dispersal of females and males differ, the situation for plants had not been established. This article shows that for a very wide range of conditions, neighbourhood size defined by Crawford's formula N(e) = 4pi(sigma(2)(s) + sigma(2)(p)/2) fully determines SGS, even when dispersal variances of seed (sigma(2)(s)) and pollen sigma(2)(p)) differ strongly. Further, self-fertilization with rate s acts as zero-distance pollen dispersal, and N(e) = 4pi[sigma(2)(s) + sigma(2)(p)(1 - s)/2] fully determines SGS, for most cases where there are both likely parameter values and substantial SGS. Moreover, for most cases, there is a loglinear relationship, I(1) = 0.587 - 0.117 ln(N(e)), between SGS, as measured by I(1), Moran's coefficient for adjacent individuals, and N(e). However, there are several biologically significant exceptions, namely for very low or large N(e), SGS exceeds the loglinear values. There are also important exceptions to Crawford's formula. First, plants with low seed dispersal, high outcross pollen dispersal and high selfing rate show larger SGS than predicted. Second, in plants with very low (near zero) seed dispersal, selfing decreases SGS, opposite expectations. Finally, in some cases seed dispersal is more critical than pollen dispersal, in a manner inconsistent with Crawford's formula.     

Control analysis of transition times in metabolic systems.

The transition time, tau, of a metabolic system is defined as the ratio of the metabolite concentrations in the system, sigma, to the steady-state flux, J. Its value reflects a temporal characteristic of the system as it relaxes towards the steady state. Like other systemic properties, the value of tau will be a function of the enzyme activities in the system. The influence of a particular enzyme activity on tau can be quantified by a Control Coefficient, C tau ei. We show that it is possible to derive a Summation Theorem sigma ni = 1 C tau ei = -1 and a Connectivity Theorem sigma ni = 1 C tau ei.epsilon viSk = -Sk/sigma. We establish a 'sign rule' that predicts the order of positive and negative Control Coefficients in a sequence.     

Optimization of solvation models for predicting the structure of surface loops in proteins

A novel procedure for optimizing the atomic solvation parameters (ASPs) sigma(i) developed recently for cyclic peptides is extended to surface loops in proteins. The loop is free to move, whereas the protein template is held fixed in its X-ray structure. The energy is E(tot) = E(FF)(epsilon = nr) + summation operator sigma(i)A(i), where E(FF)(epsilon = nr) is the force-field energy of the loop-loop and loop-template interactions, epsilon = nr is a distance-dependent dielectric constant, and n is an additional parameter to be optimized. A(i) is the solvent-accessible surface area of atom i. The optimal sigma(i) and n are those for which the loop structure with the global minimum of E(tot)(n, sigma(i)) becomes the experimental X-ray structure. Thus, the ASPs depend on the force field and are optimized in the protein environment, unlike commonly used ASPs such as those of Wesson and Eisenberg (Protein Sci 1992;1:227-235). The latter are based on the free energy of transfer of small molecules from the gas phase to water and have been traditionally combined with various force fields without further calibration. We found that for loops the all-atom AMBER force field performed better than OPLS and CHARMM22. Two sets of ASPs [based on AMBER (n = 2)], optimized independently for loops 64-71 and 89-97 of ribonuclease A, were similar and thus enabled the definition of a best-fit set. All these ASPs were negative (hydrophilic), including those for carbon. Very good (i.e., small) root-mean-square-deviation values from the X-ray loop structure were obtained with the three sets of ASPs, suggesting that the best-fit set would be transferable to loops in other proteins as well. The structure of loop 13-24 is relatively stretched and was insensitive to the effect of the ASPs.     

Calculation of clinical r.b.e. values for neutrons

A method is described for calculating r.b.e. values for normal tissues at risk in clinical neutron beam therapy. This is based on the assumption that with high l.e.t. radiations the slope of the cell survival curve is steeper, mainly in the initial or low-dose region. This effect is quantified by using two coefficients, one (epsilon) to produce a proportionate increase in the initial slope, and a second (eta) determining the change in the terminal slope (D0) of the survival curve. Analysis of published experimental data shows epsilon to be a variable quantity, different for different tissues; epsilon is larger when the survival curve has a large shoulder or slope ratio (rho). By contrast, eta is relatively constant (for a given beam) and less dependent on the tissue or end-point studied. For low doses, the r.b.e. approaches epsilon, which can be calculated given eta (characteristic of the beam) and rho (characteristic of the relevant tissue) [epsilon = eta + rho(eta - 1)]. This provides a useful approximation to the clinical r.b.e. for specific tissues relative to conventionally fractionated low-l.e.t. photons.     

Social selection in human populations: sufficient conditions for protection of deleterious alleles in a subdivided population

Population dynamics of wild type (A1) and the deleterious genes (A2) under social selection have been studied by considering a subdivided population where the i-th subpopulation consists of Ni individuals with relative size ci (= Ni/sigma i Ni, i = 1,2, ..., n). A social selection model is constructed by assuming that the fitness of an individual is determined by its own as well as the parental phenotypes and that the number of migrants (M) from the ith subpopulation is divided equally into other subpopulations including the ith subpopulation itself. It has been shown that the gene frequency change depends on the loss of fitness of an individual due to the trait (gamma), an affected parent in the ith subpopulation (beta i), the probability that the heterozygote develops the trait (h), and the migration rates mi (= M/Ni). For 0 less than h less than or equal to 1, a sufficient condition for protection of the deleterious allele from extinction also depends on all of these parameters. However, when mi much less than 1 for all i, the condition is beta i less than gamma/(1 - gamma) for some i, whereas when mi much greater than h[gamma + beta i(1 - gamma)] for all i it is given by sigma i ci beta i less than -gamma/(1 - gamma). When h = 0, that condition is given by sigma ici beta i less than - gamma/(1 - gamma). Analyses also show that, when the deleterious alleles in a population are rare, the relative fitnesses of A1A1, A1A2, and A2A2 are given approximately by 1, 1-hS, and 1 - S, respectively, where S is the harmonic mean of Si = gamma + beta i(1 - gamma). Thus, under mutation-selection balance, the equilibrium frequency of deleterious alleles in the entire population is given by alpha/hS for 0 less than h less than or equal to 1 and square root alpha/S for h = 0, where alpha is the irreversible mutation rate from A1 to A2 in each generation. Population dynamics of rare deleterious genes under social selection can readily be studied by considering a finite population size.     

Isoforms of the invariant chain regulate transport of MHC class II molecules to antigen processing compartments

Newly synthesized class II molecules of the major histocompatibility complex must be transported to endosomal compartments where antigens are processed for presentation to class II-restricted T cells. The invariant chain (Ii), which assembles with newly synthesized class II alpha- and beta-chains in the endoplasmic reticulum, carries one or more targeting signals for transport to endosomal compartments where Ii dissociates from alpha beta Ii complexes. Here we show that the transport route of alpha beta Ii complexes is regulated selectively by two forms of Ii (p33 and p35) that are generated by the use of alternative translation initiation sites. Using a novel quantitative surface arrival assay based on labeling with [6-3H]-D-galactose combined with biochemical modification at the cell surface with neuraminidase, we demonstrate that newly synthesized alpha beta Ii molecules containing the Ii-p33 isoform can be detected on the cell surface shortly after passage through the Golgi apparatus/trans-Golgi network. A substantial amount of these alpha beta Ii complexes are targeted to early endosomes either directly from the trans-Golgi network or after internalization from the cell surface before their delivery to antigen processing compartments. The fraction of alpha beta Ii complexes containing the p35 isoform of Ii with a longer cytosolic domain was not detected at the cell surface as determined by iodination of intact cells and the lack of susceptibility to neuraminidase trimming on ice. However, treatment with neuraminidase at 37 degrees C did reveal that some of the alpha beta Ii-p35 complexes traversed early endosomes. These results demonstrate that a fraction of newly synthesized class II molecules arrive at the cell surface as alpha beta Ii complexes before delivery to antigen processing compartments and that class II alpha beta Ii complexes associated with the two isoforms of Ii are sorted to these compartments by different transport routes.     

Evidence that the potential antipsychotic agent rimcazole (BW 234U) is a specific, competitive antagonist of sigma sites in brain

Rimcazole (BW 234U) is a potential antipsychotic agent which in open-clinical trials appears to be effective in acute schizophrenic patients. In the present study, rimcazole was found to block the specific binding of [3H]-(+)-SKF 10,047 to sigma sites in rat and guinea pig brain (IC50 = 5.0 X 10(-7) M). The compound was 100 times weaker as a blocker of phencyclidine sites (IC50 = 4.3 X 10(-5) M). At 1 X 10(-5) M, rimcazole had only weak effects on mu, delta, kappa and epsilon opioid receptors. Scatchard analysis of the binding data from guinea pig brain revealed an apparent KD for [3H]-(+)-SKF 10,047 of 85 +/- 5 nM and a Bmax of 824 +/- 27 fmole/mg protein. In the presence of 5 X 10(-7) M BW 234U, the apparent KD was 165 +/- 35 nM, but the Bmax (892 +/- 146 fmoles/mg protein) was not affected. This suggests that rimcazole is a competitive inhibitor of sigma sites. The agent was also capable of blocking sigma sites in vivo (ID50 = 6 mg/kg i.p., mice) as judged by an in vivo sigma receptor binding assay. Thus, if the antipsychotic activity of rimcazole is confirmed in double-blind, placebo-controlled trials, it would be the first compound whose mechanism of antipsychotic activity may best be explained by a direct blockade of sigma sites and not by a direct blockade of dopamine (D2) receptors in brain.     

Comparison of calculation and experiment implicates significant electrostatic contributions to the binding stability of barnase and barstar

The contributions of electrostatic interactions to the binding stability of barnase and barstar were studied by the Poisson-Boltzmann model with three different protocols: a), the dielectric boundary specified as the van der Waals (vdW) surface of the protein along with a protein dielectric constant (epsilon (p)) of 4; b), the dielectric boundary specified as the molecular (i.e., solvent-exclusion (SE)) surface along with epsilon (p) = 4; and c), "SE + epsilon (p) = 20." The "vdW + epsilon (p) = 4" and "SE + epsilon (p) = 20" protocols predicted an overall electrostatic stabilization whereas the "SE + epsilon (p) = 4" protocol predicted an overall electrostatic destabilization. The "vdW + epsilon (p) = 4" protocol was most consistent with experiment. It quantitatively reproduced the observed effects of 17 mutations neutralizing charged residues lining the binding interface and the measured coupling energies of six charge pairs across the interface and reasonably rationalized the experimental ionic strength and pH dependences of the binding constant. In contrast, the "SE + epsilon (p) = 4" protocol predicted significantly larger coupling energies of charge pairs whereas the "SE + epsilon (p) = 20" protocol did not predict any pH dependence. This study calls for further scrutiny of the different Poisson-Boltzmann protocols and demonstrates potential danger in drawing conclusions on electrostatic contributions based on a particular calculation protocol.     

On Wolff's law of trabecular architecture.

Several studies suggest that the yield strain in cancellous bone may be uniformly distributed and isotropic. Yield strain was reported to be independent of textural anisotropy in bovine cancellous bone [Turner, J. biomech. Engng 111, 1-5 (1989)] and it is plausible that yield strain is isotropic in human cancellous bone as well. In this paper, it is hypothesized that uniform, isotropic strain represents a goal of cancellous bone adaptation, i.e. cancellous bone alters its structure to maintain uniform, isotropic peak strains. Therefore, textural anisotropy must exactly cancel the anisotropy of the peak principal stresses imposed upon cancellous bone. When evaluating the relationships between mechanical properties of cancellous bone and trabecular architecture, it was found that over 90% of the variance of yield strength can be explained by one term--rho 2H3 (where rho is apparent density and H is the normalized anisotropy (fabric) constant). Furthermore, this single term explains 70-78% of the variance in Young's modulus of cancellous bone. Based upon these findings, it was postulated that fabric adaptation goes as Hi/Hj = [ sigma i/sigma j[, where Hi and Hj are fabric eigenvalues in the i- and the j-direction and sigma i and sigma j are peak principal stresses.     

The relative gonadal index: an alternative index for quantification of reproductive condition

Gonadal indices (i.e. GSI = gonadal wt/body wt X 100) commonly are used to quantify reproductive condition in fishes. These indices may be inappropriate with specimens of different sizes, however, for gonadal growth often is allometric. A new gonadal index (relative gonadal index, RGI) was developed to quantify the reproductive condition of animals independent of body size. The RGI is based on the underlying model W = alpha i X S beta i, where W is gonadal weight, S is body size (less gonadal weight if body weight is used), and alpha i and beta i are parameters to be estimated for gonadal developmental stage i. Assuming that a multiplicative lognormal error is appropriate, parameter estimates for alpha i and beta i were obtained by linear least squares regression for the log-transformed model ln(W) = beta i X ln(S) + ln(alpha i), where, in this form, beta i is the slope and ln(alpha i) is the intercept. Only if estimates of beta i do not differ significantly among ovarian developmental stages, as in our case, can a pooled estimate of beta be used to obtain the relative gonadal index, RGI = alpha i = W/S beta. Applicability of the RGI was tested using ovaries of three ecologically distinct fish species. The RGI was found to be more appropriate than the gonosomatic index for all three species.     

Solvation parameters for predicting the structure of surface loops in proteins: transferability and entropic effects

A new procedure for optimizing parameters of implicit solvation models introduced by us has been applied successfully first to cyclic peptides and more recently to three surface loops of ribonuclease A (Das and Meirovitch, Proteins 2001;43:303-314) using the simplified model E(tot) = E(FF)(epsilon = nr) + Sigma(i) sigma(i)A(i), where sigma(i) are atomic solvation parameters (ASPs) to be optimized, A(i) is the solvent accessible surface area of atom i, E(FF)(epsilon = nr) is the AMBER force-field energy of the loop-loop and loop-template interactions with a distance-dependent dielectric constant, epsilon = nr, where n is a parameter. The loop is free to move while the protein template is held fixed in its X-ray structure; an extensive conformational search for energy minimized loop structures is carried out with our local torsional deformation method. The optimal ASPs and n are those for which the structure with the lowest minimized energy [E(tot)(n,sigma(i))] becomes the experimental X-ray structure, or less strictly, the energy gap between these structures is within 2-3 kcal/mol. To check if a set of ASPs can be defined, which is transferable to a large number of loops, we optimize individual sets of ASPs (based on n = 2) for 12 surface loops from which an "averaged" best-fit set is defined. This set is then applied to the 12 loops and an independent "test" group of 8 loops leading in most cases to very small RMSD values; thus, this set can be useful for structure prediction of loops in homology modeling. For three loops we also calculate the free energy gaps to find that they are only slightly smaller than their energy counterparts, indicating that only larger n will enable reducing too large gaps. Because of its simplicity, this model allowed carrying out an extensive application of our methodology, providing thereby a large number of benchmark results for comparison with future calculations based on n > 2 as well as on more sophisticated solvation models with as yet unknown performance for loops.     

Peptide-plane flipping in proteins

A peptide-plane flip is a large-scale rotation of the peptide plane that takes the phi,psi angles at residues i and i + 1 to different structural regions in the Ramachandran plot with a comparatively small effect on the relative orientation of their side chains. This phenomenon, which is expected to play an important role during the early stages of protein folding, has been investigated using 76 proteins for which two high-resolution X-ray conformations are available. Peptide-plane flips are identified by looking for those cases where changes in /psi(i)/ + /phi(i + 1)/ are large (>200 degrees), but changes in /psi(i) + phi(i + 1)/ are comparatively small (<50 degrees). Of a total of 23 cases, the most common peptide-plane flip was identified to be the type I to type II beta-turn interconversion. Although individually rarer, there are many other types of flips that are collectively more common. Given the four main accessible regions alpha(R), alpha(L), beta and epsilon, identified from the phi,psi distribution corresponding to non-hydrogen-bonded peptide planes, 32 main types of peptide-plane flip are identified. Only 8 of these are "passive," in that they require only relatively minor adjustments in the orientation of adjacent peptide planes. Of these, only the type I to type II beta-turn interconversion, denoted, beta(i) + alpha(L)(i + 1) <--> alpha(R)(i) + alpha(R)(i + 1), and the rarer alpha(R)(i) + alpha(L)(i + 1) <--> beta(i) + alpha(R)(i + 1), do not involve the epsilon region. "Active" peptide-plane flips affect the orientation of adjacent peptide planes. The flip, alpha(L)(i) + alpha(L)(i + 1) <--> beta(i) + beta(i + 1), of which one example was found, shows how concerted peptide-plane flips can convert the alpha(L) structure to the beta structure without affecting the relative orientations of the side chains.     

The effect of epistasis on the structure of hybrid zones

Within hybrid zones that are maintained by a balance between selection and dispersal, linkage disequilibrium is generated by the mixing of divergent populations. This linkage disequilibrium causes selection on each locus to act on all other loci, thereby steepening clines, and generating a barrier to gene flow. Diffusion models predict simple relations between the strength of linkage disequilibrium and the dispersal rate, sigma, and between the barrier to gene flow, B, and the reduction in mean fitness, W. The aim of this paper is to test the accuracy of these predictions by comparison with an exact deterministic model of unlinked loci (r = 0.5). Disruptive selection acts on the proportion of alleles from the parental populations (p,q): W = exp[-S(4pq)beta], such that the least fit genotype has fitness e-s. Where beta < 1, fitness is reduced for a wide range of intermediate genotypes; where beta > 1, fitness is only reduced for those genotypes close to p = 0.5. Even with strong epistasis, linkage disequilibria are close to sigma 2p'ip'j/rij, where p'i, p'j are the gradients in allele frequency at loci i, j. The barrier to gene flow, which is reflected in the steepening of neutral clines, is given by [formula: see text] where r, the harmonic mean recombination rate between the neural and selected loci, is here 0.5. This is a close approximation for weak selection, but underestimates B for strong selection. The barrier is stronger for small beta, because hybrid fitness is then reduced over a wider range of p. The widths of the selected clines are harder to predict: though simple approximations are accurate for beta = 1, they become inaccurate for extreme beta because, then, fitness changes sharply with p. Estimates of gene number, made from neutral clines on the assumption that selection acts against heterozygotes, are accurate for weak selection when beta = 1; however, for strong selection, gene number is overestimated. For beta > 1, gene number is systematically overestimated and, conversely, when beta < 1, it is underestimated.     

Use of conditional probabilities for determining relationships between amino acid sequence and protein secondary structure.

The conditional probability, P(sigma/x), is a statement of the probability that the value of sigma will be found given the prior information that a value of x has been observed. Here sigma represents any one of the secondary structure types, alpha, beta, tau, and rho for helix, sheet, turn, and random, respectively, and x represents a sequence attribute, including, but not limited to: (1) hydropathy; (2) hydrophobic moments assuming helix and sheet; (3) Richardson and Richardson helical N-cap and C-cap values; (4) Chou-Fasman conformational parameters for helix, P alpha, for sheet, P beta, and for turn, P tau; and (5) Garnier, Osguthorpe, and Robson (GOR) information values for helix, I alpha, for sheet, I beta, for turn, I tau, and for random structure, I rho. Plots of P(sigma/x) vs. x are demonstrated to provide information about the correlation between structure and attribute, sigma and x. The separations between different P(sigma/x) vs. x curves indicate the capacity of a given attribute to discriminate between different secondary structural types and permit comparison of different attributes. P(alpha/x), P(beta/x), P(tau/x) and P(rho/x) vs. x plots show that the most useful attributes for discriminating helix are, in order: hydrophobic moment assuming helix greater than P alpha much greater than N-cap greater than C-cap approximately I alpha approximately I tau. The information value for turns, I tau, was found to discriminate helix better than turns. Discrimination for sheet was found to be in the following order: I beta much greater than P beta approximately hydropathy greater than I rho approximately hydrophobic moment assuming sheet. Three attributes, at their low values, were found to give significant discrimination for the absence of helix: I alpha approximately P alpha approximately hydrophobic moment assuming helix. Also, three other attributes were found to indicate the absence of sheet: P beta much greater than I rho approximately hydropathy. Indications of the absence of sigma could be as useful for some applications as the indication of the presence of sigma.     

Scalability of the surface-based DNA algorithm for 3-SAT

Since Adleman first proposed DNA computing for the Hamiltonian path problem, several authors have reported DNA computing for 3-SAT. Previous research presented DNA computing on surfaces and demonstrated how to solve a four-variable four-clause instance of 3-SAT, and claimed that the surface-based approach was designed to scale up to larger problems. In this paper we establish an error model for the incomplete "mark" and imperfect "destroy" operations. By using the error model we argue that no matter how large the "mark" and "destroy" rates are we can always give satisfiable instances of 3-SAT such that no DNA strands remain on the surface at the end of the computation. By the surface-based approach the satisfiable instances of 3-SAT would be misdetermined to be unsatisfiable. Thus, the error leads to an incorrect result of the SAT computation. Furthermore, given the "mark" rate p and the "not-destroy" rate rho, we find that the approach can only solve at most N-variable instances of 3-SAT problems, where N=[(2+beta(2)+2+2 square root beta (2))/beta(2)] in which beta=1-1/(p+rhoq) and q=1-p and [a] is the greatest integer less than a or equal to a.