Maximum likelihood estimators for incorrect models, with an application to ascertainment bias for continuous characters

Uncertainty about the ascertainment of human family data leads to a need for robust methods for estimating genetic and environmental effects. This in turn leads to a need for efficient techniques for estimating model parameters for data generated under one parametric model but analyzed under a second model. If the two models correspond to different ascertainment schemes for the same exponential family, simple formulas for the asymptotic means and standard errors of both conditional and unconditional MLEs can be derived. In an example for continuous sibship data, these formulas show that estimates derived from conditioning on proband value have greater asymptotic bias than two other estimators. Similarly, either conditioning on proband value or conditioning on the number of affected family members resulted in biases of up to 30% when ascertainment depended on the values of more than one affected family member.     

Replication study supports evidence for linkage to 9p24 in obsessive-compulsive disorder

Obsessive-compulsive disorder (OCD) is a severe psychiatric illness that is characterized by intrusive and senseless thoughts and impulses (obsessions) and by repetitive behaviors (compulsions). Family, twin, and segregation studies support the presence of both genetic and environmental susceptibility factors, and the only published genome scan for OCD identified a candidate region on 9p24 at marker D9S288 that met criteria for suggestive significance (Hanna et al. 2002). In an attempt to replicate this finding, we genotyped 50 pedigrees with OCD, using microsatellite markers spanning the 9p24 candidate region, and analyzed the data, using parametric and nonparametric linkage analyses under both a narrow phenotype model (DSM-IV OCD definite; 41 affected sib pairs) and a broad phenotype model (DSM-IV OCD definite and probable; 50 affected sib pairs). Similar to what was described by Hanna et al. (2002), our strongest findings came with the dominant parameters and the narrow phenotype model: the parametric signal peaked at marker D9S1792 with an HLOD of 2.26 ( alpha =0.59), and the nonparametric linkage signal (NPL) peaked at marker D9S1813 with an NPL of 2.52 (P=.006). These findings are striking in that D9S1813 and D9S1792 lie within 0.5 cM (<350 kb) of the original 9p24 linkage signal at D9S288; furthermore, pedigree-based association analyses also implicated the 9p24 candidate region by identifying two markers (D9S288 and GATA62F03) with modest evidence (P=.046 and .02, respectively) for association.     

A three generations family with blepharo-naso-facial malformations suggestive of Pashayan syndrome

Blepharo-naso-facial syndrome, described by Pashayan et al. (10), is characterized by telecanthus, lateral displacement and stenosis of lacrimal puncta, bulky nose, mask-like facies, trapezo?dal upper lip, torsion dystonia and mental retardation. We report on a family with this rare malformation syndrome, confirming the existence of this syndrome and its dominant inheritance. The proband had a fleshy nose, a prominant nose bridge, an hypoplastic midface, telecanthus with temporal displacement of puncta, lacrimal excretory obstruction. CNS torsion dystonia, increased deep tendon reflexes, Babinski reflexes, poor coordination and joint laxity. The proband's mother, brother and maternal grandfather also showed manifestations of the syndrome. The proband and his brother had delayed developmental milestones. Hearing impairment was present in the proband, his mother and his grandfather but was absent in the proband's brother. The blepharonasofacial syndrome was described by Pashayan et al. (10) in four members of one family, two male and one female sib and their mother. Two other sibs were unaffected. Many of the features of the blepharo-facio-nasal syndrome also occur in other well known syndromes i.e. Waardenburg syndrome. The pedigrees of the family of Pashayan et al. (10) and of our family are compatible with Mendelian dominant inheritance, either autosomal or X-linked. X-linked dominant inheritance cannot be ruled out except by male-to-male transmission, which does not occur in these families. Pashayan et al. (10) suggested that an autosomal gene with variable expressivity appears more likely. More families are needed for defining the transmission of the condition and for mapping the gene involved in the blepharo-naso-facial syndrome.     

A general approach to confidence regions for optimal factor levels of response surfaces

For a response surface experiment, an approximate hypothesis test and an associated confidence region is proposed for the minimizing (or maximizing) factor-level configuration. Carter et al. (1982, Cancer Research 42, 2963-2971) show that confidence regions for optimal conditions provide a way to make decisions about therapeutic synergism. The response surface may be constrained to be within a specified, bounded region. These constraint regions can be quite general. This allows for more realistic constraint modeling and a wide degree of applicability, including constraints occurring in mixture experiments. The usual assumption of a quadratic model is also generalized to include any regression model that is linear in the model parameters. An intimate connection is established between this confidence region and the Box-Hunter (1954, Biometrika 41, 190-199) confidence region for a stationary point. As a byproduct, this methodology also provides a way to construct a confidence interval for the difference between the optimal mean response and the mean response at a specified factor-level configuration. The application of this confidence region is illustrated with two examples. Extensive simulations indicate that this confidence region has good coverage properties.     

Further evidence for the increased power of LOD scores compared with nonparametric methods.

In genetic analysis of diseases in which the underlying model is unknown, "model free" methods-such as affected sib pair (ASP) tests-are often preferred over LOD-score methods, although LOD-score methods under the correct or even approximately correct model are more powerful than ASP tests. However, there might be circumstances in which nonparametric methods will outperform LOD-score methods. Recently, Dizier et al. reported that, in some complex two-locus (2L) models, LOD-score methods with segregation analysis-derived parameters had less power to detect linkage than ASP tests. We investigated whether these particular models, in fact, represent a situation that ASP tests are more powerful than LOD scores. We simulated data according to the parameters specified by Dizier et al. and analyzed the data by using a (a) single locus (SL) LOD-score analysis performed twice, under a simple dominant and a recessive mode of inheritance (MOI), (b) ASP methods, and (c) nonparametric linkage (NPL) analysis. We show that SL analysis performed twice and corrected for the type I-error increase due to multiple testing yields almost as much linkage information as does an analysis under the correct 2L model and is more powerful than either the ASP method or the NPL method. We demonstrate that, even for complex genetic models, the most important condition for linkage analysis is that the assumed MOI at the disease locus being tested is approximately correct, not that the inheritance of the disease per se is correctly specified. In the analysis by Dizier et al., segregation analysis led to estimates of dominance parameters that were grossly misspecified for the locus tested in those models in which ASP tests appeared to be more powerful than LOD-score analyses.     

Contextual constraints on synonymous codon choice

We have studied the statistical constraints on synonymous codon choice to evaluate various proposals regarding the origin of the bias in synonymous codon usage observed by Fiers et al. (1975), Air et al. (1976), Grantham et al. (1980) and others. We have determined the statistical dependence of the degenerate third base on either of its nearest neighbors in mitochondrial, prokaryotic, and eukaryotic coding sequences. We noted an increasing dependence of the third base on its nearest neighbors in moving from mitochrondria to prokaryotes to eukaryotes.A statistical model assuming random equiprobable selection of synonymous codons was found grossly adequate for the mitochondria, but totally indequate for prokaryotes and eukaryotes. A model assuming selection of synonymous codons reflecting a genomic strategy, i.e. the genome hypothesis of Grantham et al. (1980), gave a good approximation of the mitochondrial sequences. A statistical model which exactly maintains codon frequency, but allows the position of corresponding synonymous codons to vary was only grossly adequate for prokaryotes and totally inadequate for eukaryotes. The results of these simulations are consistent with the measures on experimental sequences and suggest that a “frequency constraint” model such as that of Grantham et al. (1980) may be an adequate explanation of the codon usage in mitochondria. However, in addition to this frequency constraint, there may be constraints on synonymous codon choice in prokaryotes due to codon context. Furthermore, any proposal to explain codon usage in eukaryotes must involve a constraint on the context of a codon in the sequence.     

Path analysis of palmar ridge counts.

The methods for path analysis of family resemblance (Rao et al., '74) are employed to test hypotheses concerning the inheritance of a-b, b-c and c-d palmar ridge counts using the correlation data of Pateria ('74). Homogeneity chi-square tests of the various familial correlations provide no evidence for sex-linkage of either kind, and also suggest that maternal effects are absent. The path coefficient model employed here involves heritability (additive) and common sibling environment. Variance components show that both heritability and common environment are significant, and account for most of the variation at each of the three ridge count area; b-c has the highest heritability, significantly higher than that for a-b or c-d.     

Cellular Encapsulation in 3D Hydrogels for Tissue Engineering

The 3D encapsulation of cells within hydrogels represents an increasingly important and popular technique for culturing cells and towards the development of constructs for tissue engineering. This environment better mimics what cells observe in vivo, compared to standard tissue culture, due to the tissue-like properties and 3D environment. Synthetic polymeric hydrogels are water-swollen networks that can be designed to be stable or to degrade through hydrolysis or proteolysis as new tissue is deposited by encapsulated cells. A wide variety of polymers have been explored for these applications, such as poly(ethylene glycol) and hyaluronic acid. Most commonly, the polymer is functionalized with reactive groups such as methacrylates or acrylates capable of undergoing crosslinking through various mechanisms. In the past decade, much progress has been made in engineering these microenvironments - e.g., via the physical or pendant covalent incorporation of biochemical cues - to improve viability and direct cellular phenotype, including the differentiation of encapsulated stem cells (Burdick et al.).The following methods for the 3D encapsulation of cells have been optimized in our and other laboratories to maximize cytocompatibility and minimize the number of hydrogel processing steps. In the following protocols (see Figure 1 for an illustration of the procedure), it is assumed that functionalized polymers capable of undergoing crosslinking are already in hand; excellent reviews of polymer chemistry as applied to the field of tissue engineering may be found elsewhere (Burdick et al.) and these methods are compatible with a range of polymer types. Further, the Michael-type addition (see Lutolf et al.) and light-initiated free radical (see Elisseeff et al.) mechanisms focused on here constitute only a small portion of the reported crosslinking techniques. Mixed mode crosslinking, in which a portion of reactive groups is first consumed by addition crosslinking and followed by a radical mechanism, is another commonly used and powerful paradigm for directing the phenotype of encapsulated cells (Khetan et al., Salinas et al.).     

Regression analysis of doubly censored failure time data using the additive hazards model

Doubly censored failure time data arise when the survival time of interest is the elapsed time between two related events and observations on occurrences of both events could be censored. Regression analysis of doubly censored data has recently attracted considerable attention and for this a few methods have been proposed (Kim et al., 1993, Biometrics 49, 13-22; Sun et al., 1999, Biometrics 55, 909-914; Pan, 2001, Biometrics 57, 1245-1250). However, all of the methods are based on the proportional hazards model and it is well known that the proportional hazards model may not fit failure time data well sometimes. This article investigates regression analysis of such data using the additive hazards model and an estimating equation approach is proposed for inference about regression parameters of interest. The proposed method can be easily implemented and the properties of the proposed estimates of regression parameters are established. The method is applied to a set of doubly censored data from an AIDS cohort study.     

The rescaling method for quantifying the turnover of cell populations

The dynamic nature of immune responses requires the development of appropriate experimental and theoretical tools to quantitatively estimate the division and death rates which determine the turnover of immune cells. A number of papers have used experimental data from BrdU and D-glucose labels together with a simple random birth-death model to quantify the turnover of immune cells focusing on HIV/SIV infections [Mohri et al. 279 (1998) 1223-1227, Hellerstein et al. 5 (1999) 83-89, Bonhoeffer et al. 164 (2000) 5049-5054, Mohri et al. 87 (2001) 1277-1287]. We show how uncertainties in the assumptions of the random birth-death model may lead to substantial errors in the parameters estimated. We then show how more accurate estimates can be obtained from the more recent CFSE data which allow to track the number of divisions each cell has undergone. Specifically, we: (i) describe a general stage-structured model of cell division where the probabilities of division and death are functions of time since the previous division; (ii) develop a rescaling method to identify invariant parameters (i.e. the ones that are independent of the specific functions describing division and death); (iii) show how these invariant parameters can be estimated, and (iv) illustrate this technique by applying it to CFSE data taken from the literature.     

Updating fearful memories with extinction training during reconsolidation: a human study using auditory aversive stimuli

Learning to fear danger in the environment is essential to survival, but dysregulation of the fear system is at the core of many anxiety disorders. As a consequence, a great interest has emerged in developing strategies for suppressing fear memories in maladaptive cases. Recent research has focused in the process of reconsolidation where memories become labile after being retrieved. In a behavioral manipulation, Schiller et al., (2010) reported that extinction training, administrated during memory reconsolidation, could erase fear responses. The implications of this study are crucial for the possible treatment of anxiety disorders without the administration of drugs. However, attempts to replicate this effect by other groups have been so far unsuccessful. We sought out to reproduce Schiller et al., (2010) findings in a different fear conditioning paradigm based on auditory aversive stimuli instead of electric shock. Following a within-subject design, participants were conditioned to two different sounds and skin conductance response (SCR) was recorded as a measure of fear. Our results demonstrated that only the conditioned stimulus that was reminded 10 minutes before extinction training did not reinstate a fear response after a reminder trial consisting of the presentation of the unconditioned stimuli. For the first time, we replicated Schiller et al., (2010) behavioral manipulation and extended it to an auditory fear conditioning paradigm.     

Unbiased and Locally Efficient Estimation of Genetic Effect on Quantitative Trait in the Presence of Population Admixture

Summary Population admixture can be a confounding factor in genetic association studies. Family‐based methods ( Rabinowitz and Larid, 2000 , Human Heredity 50, 211–223) have been proposed in both testing and estimation settings to adjust for this confounding, especially in case‐only association studies. The family‐based methods rely on conditioning on the observed parental genotypes or on the minimal sufficient statistic for the genetic model under the null hypothesis. In some cases, these methods do not capture all the available information due to the conditioning strategy being too stringent. General efficient methods to adjust for population admixture that use all the available information have been proposed ( Rabinowitz, 2002 , Journal of the American Statistical Association 92, 742–758). However these approaches may not be easy to implement in some situations. A previously developed easy‐to‐compute approach adjusts for admixture by adding supplemental covariates to linear models ( Yang et al., 2000 , Human Heredity 50, 227–233). Here is shown that this augmenting linear model with appropriate covariates strategy can be combined with the general efficient methods in Rabinowitz (2002) to provide computationally tractable and locally efficient adjustment. After deriving the optimal covariates, the adjusted analysis can be carried out using standard statistical software packages such as SAS or R . The proposed methods enjoy a local efficiency in a neighborhood of the true model. The simulation studies show that nontrivial efficiency gains can be obtained by using information not accessible to the methods that rely on conditioning on the minimal sufficient statistics. The approaches are illustrated through an analysis of the influence of apolipoprotein E (APOE) genotype on plasma low‐density lipoprotein (LDL) concentration in children.     

Urinary excretion of taurine in epilepsy

Evidence that taurine (2-aminoethanesulfonic acid) is related to the epilepsies is supported by work with both experimental animals and hurmans. It may function as a neurotransmitter or modulator of neurotransmission. Investigators using an automated amino acid analyzer reported lower mean urinary taurine excretion among epileptics. However, Rao et al. reported higher taurine excretion among epileptics using an older method. Analyses of the same epileptic and control urines by both methods coupled with paper and molecular size chromatography indicate that substances in addition to taurine are co-eluted with taurine using the older method, yielding spuriously high values. The resolution of this disparity is important because the urinary excretion of taurine may reflect primarily the influence of taurine transport alleles which may be polygenic components in the idiopathic epilepsies.     

Navigating the unknown: model selection in phylogeography. Models of population structure: tools for thinkers

Despite the widespread use and obvious strengths of model-based methods for phylogeographic study, a persistent concern for such analyses is related to the definition of the model itself. The study by Peter et al. (2010) in this issue of Molecular Ecology demonstrates an approach for overcoming such hurdles. The authors were motivated by a deceptively simple goal; they sought to infer whether a population has remained at a low and stable size or has undergone a decline, and certainly there is no shortage of software packages for such a task (e.g., see list of programs in Excoffier & Heckel 2006). However, each of these software packages makes basic assumptions about the underling population (e.g., is the population subdivided or panmictic); these assumptions are explicit to any model-based approach but can bias parameter estimates and produce misleading inferences if the model does not approximate the actual demographic history in a reasonable manner. Rather than guessing which model might be best for analyzing the data (microsatellite data from samples of chimpanzees), Peter et al. (2010) quantify the relative fit of competing models for estimating the population genetic parameters of interest. Complemented by a revealing simulation study, the authors highlight the peril inherent to model-based inferences that lack a statistical evaluation of the fit of a model to the data, while also demonstrating an approach for model selection with broad applicability to phylogeographic analysis.     

Aorta in vivo parameter identification using an axial force constraint

It was shown in a previous study by Stålhand et al. (2004) that both material and residual strain parameters for an artery can be identified noninvasively from an in vivo clinical pressure–diameter measurement. The only constraints placed on the model parameters in this previous study was a set of simple box constraints. More advanced constraints can also be utilized, however. These constraints restrict the model parameters implicitly by demanding the state of the artery to behave in a specified way. It has been observed in vitro that the axial force is nearly invariant to the pressure at the physiological operation point. In this paper, we study the possibility to include this behaviour as a constraint in the parameter optimization. The method is tested on an in vivo obtained pressure–diameter cycle for a 24-year-old human. Presented results show that the constrained parameter identification procedure proposed here can be used to obtain good results, and we believe that it may be applied to account for other observed behaviours as well.     

Robustness of the latent variable model for correlated binary data

The marginal regression model offers a useful alternative to conditional approaches to analyzing binary data (Liang, Zeger, and Qaqish, 1992, Journal of the Royal Statistical Society, Series B 54, 3-40). Instead of modelling the binary data directly as do Liang and Zeger (1986, Biometrika 73, 13-22), the parametric marginal regression model developed by Qu et al. (1992, Biometrics 48, 1095-1102) assumes that there is an underlying multivariate normal vector that gives rise to the observed correlated binary outcomes. Although this parametric approach provides a flexible way to model different within-cluster correlation structures and does not restrict the parameter space, it is of interest to know how robust the parameter estimates are with respect to choices of the latent distribution. We first extend the latent modelling to include multivariate t-distributed latent vectors and assess the robustness in this class of distributions. Then we show through a simulation that the parameter estimates are robust with respect to the latent distribution even if latent distribution is skewed. In addtion to this empirical evidence for robustness, we show through the iterative algorithm that the robustness of the regression coefficents with respect to misspecifications of covariance structure in Liang and Zeger's model in fact indicates robustness with respect to underlying distributional assumptions of the latent vector in the latent variable model.     

Selection for Chaperone-Like Mediated Genetic Robustness at Low Mutation Rate: Impact of Drift,Epistasis and Complexity

Genetic robustness is defined as the constancy of a phenotype in the face of deleterious mutations. Overexpression of chaperones, to assist the folding of proteins carrying deleterious mutations, is so far one of the most accepted molecular mechanisms enhancing genetic robustness. Most theories on the evolution of robustness have focused on the implications of high mutation rate. Here we show that genetic drift, which is modulated by population size, organism complexity, and epistasis, can be a sufficient force to select for chaperone-mediated genetic robustness. Using an exact analytical solution, we also show that selection for costly genetic robustness leads to a paradox: the decrease of population fitness on long timescales and the long-term dependency on robustness mechanisms. We suggest that selection for genetic robustness could be universal and not restricted to high mutation rate organisms such as RNA viruses. The evolution of the endosymbiont Buchnera illustrates this selection mechanism and its paradox: the increased dependency on chaperones mediating genetic robustness. Our model explains why most chaperones might have become essential even in optimal growth conditions.MUTATIONAL (or genetic) robustness is defined as the constancy of a phenotype in the face of deleterious mutations (Sanjuan et al. 2007). Selection drives populations to adapt to their environment by the fixation of successive advantageous mutations. However, in approaching a fitness optimum—i.e., a genotype that is maximally adapted—they have to cope with an increasing proportion of deleterious mutations and, when at the optimum, they experience only neutral and deleterious mutations (Silander et al. 2007). Therefore any mechanism that would reduce the effect of deleterious mutations, i.e., increase mutational robustness, could be favored by natural selection when at, or near, an optimum of fitness. Indeed, the general observation that for a large range of organisms, mutations have little effect on fitness, suggests that selection for robustness is pervasive (Melton 1994; Winzeler et al. 1999). Three main mechanisms that are not mutually exclusive could explain how genetic robustness has arisen. First, in the “intrinsic hypothesis” (de Visser et al. 2003) robustness could simply be a by-product of some biologically relevant functions. Second, mutational robustness could be a by-product of the selection for nongenetic perturbations such as environment changes or intrinsic noise (Wagner 2005). Third, mutational robustness could be selected for because it is adaptive in itself. In the following we restrict our attention to this “adaptive hypothesis” (de Visser et al. 2003).Chaperone proteins, proteins that help other proteins to fold properly, have been shown to buffer the effect of deleterious mutations in diverse organisms (Rutherford 2003). In lineages that have accumulated deleterious mutations, the overexpression of the chaperone GroESL in Escherichia coli (Fares et al. 2002) or Salmonella typhymurium (Maisnier-Patin et al. 2005) resulted in an improved fitness. However, such robustness appears to come at a cost, as the buffering was visible only in carbon-rich media (Fares et al. 2002), and it is also known that GroESL-mediated refolding of proteins is ATP dependent. Chaperones can also buffer against environmental perturbations (such as heat shock); however, the observation that groESL evolved under positive selection and is overproduced in obligate intracellular endosymbionts (Moran 1996; Fares et al. 2004), for which environmental perturbations are assumed to be very weak, suggests that genetic robustness could be the direct target of selection.Selection for a modifier of genetic robustness, i.e., a gene modulating the effect of mutations, has been mainly studied in the context of high mutation rates, as the effect of the modifier allele affects the fitness of mutants (Wagner 2005). Under some theoretical frameworks, it has been suggested that the intensity of selection acting on a modifier of robustness would be of the order of the mutation rate (Gardner and Kalinka 2006). Therefore it has been presumed that selection for genetic robustness is relevant only in very large populations having a high mutation rate, such as RNA virus populations. In agreement with these ideas, artificial life experiments (Wilke and Adami 2001; Azevedo et al. 2006) and experimental data on viruses (Montville et al. 2005; Sanjuan et al. 2007) have shown that robustness varies between organisms and that it can be selected for under high mutation rates. It has also been shown by Krakauer and Plotkin (2002) that drift, i.e., stochastic effects due to the finite size of populations, can promote selection for robustness even when more robust alleles are costly, as suggested in the case of chaperone overexpression. However, again this effect was examined only under high mutation rates.When mutations are very rare, populations experience at the most the presence of a single mutant. In such conditions, the population fitness at equilibrium does not depend on the mutation rate but only on drift (Sella and Hirsh 2005; Tenaillon et al. 2007). Two factors modulate how drift affects fitness:

1. Epistasis, defined here as a local property of the adaptive landscape, describes how the selective effects of mutations depend on the genetic background in which they arise. Epistasis is negative (positive) if two mutations have a lower (higher) fitness when simultaneously present within a genome than expected if they did not interact. Negative epistasis increases selection against mutation-loaded individuals and therefore reduces the effect of drift on population fitness (Charlesworth 1990; Tenaillon et al. 2007).
2. Phenotypic complexity, defined as the number of independent mutable traits that contribute to fitness (Orr 2000; Tenaillon et al. 2007), also affects population fitness in finite populations: complex organisms are more sensitive to the action of drift (Hartl and Taubes 1998; Poon and Otto 2000; Tenaillon et al. 2007).

In this article, we attempt to further clarify the role of drift on the evolution of chaperone-like genetic robustness and to decouple the effect of drift from the effect of the mutation rate. We use Fisher''s geometric model of adaptation (Fisher 1930), to map phenotype to fitness under an assumption of a vanishing mutation rate and extract exact analytical solutions for the genetic properties of the population at mutation–selection–drift equilibrium (MSDE). We examine how these genetic properties change under various population sizes and epistasis parameters and in organisms ranging in phenotypic complexity.     

On the time dependent diffusion of macromolecules through transient open junctions and their subendothelial spread. 2. Long time model for interaction between leakage sites

In Part 1 of this study (Weinbaum et al., 1988) a short time model has been proposed to describe the initial time dependent leakage of macromolecules at short distances (5 microns or less) from the exit of a transient open junction which the authors have hypothesized as a characteristic feature of endothelial cells in the process of turnover (Weinbaum et al., 1985). This open junction pathway has also been proposed (Weinbaum et al., 1988) to be the primary ultrastructural correlate of the 20 nm diameter large pore suggested by Renkin et al. (1977) using the predictions of cylindrical pore theory. The short time model in (Weinbaum et al., 1988), however, has major limitations in that it neglects the interaction between leakage sites, macromolecular entry through other pathways, the finite thickness of the vessel wall and the curvature of the cell perimeter. The longer time model developed herein will attempt to describe each of these features and also present an improved model and analytic solution for the steady state flux and uptake. In the previous steady state model developed by Weinbaum et al. (1985) the effect of the resistance of the transient open junctions and the non-isotropic diffusion in the underlying tissue due to the internal elastic lamina (IEL) were both neglected. New solutions are first presented which describe the effect of these important model refinements on the steady state macromolecular permeability of the major arteries. Time dependent solutions are then presented to predict the transient longer time labeling following the introduction of tracer macromolecules of varying size. These solutions and the corresponding short time solutions in Weinbaum et al. (1988) are the first solutions to our knowledge to describe the difficult time-dependent boundary value problem to determine how the channel exit concentration and flux at a leaky junction vary with time. This is accomplished by casting the boundary value problem in the form of an integral equation for the unknown flux at the cleft exit and then solving this problem using a specially designed numerical technique. The theoretical predictions are used to interpret the behavior of the localized leaks to HRP and albumin that have been reported in Stemerman et al. (1986) and our own recent experiments (Lin et al., 1988).     

Community robustness in discrete-time periodic environments

The influence of seasonal environmental variation on species coexistence is an ecologically important factor. Its two aspects are how seasonal variation contributes to coexistence mechanisms, and, given a seasonally varying coexistence pattern, how sensitive that coexistence is to nonstationary external influences (such as climate change). Here we develop a formula for calculating the robustness of discrete-time periodic dynamics. Robustness is defined as the sensitivity of the position of the cycle in phase space to varying model parameters. Though the results are different, the main biological conclusions are in line with those from a similar study concerning continuous-time cycles (Barabás et al., 2012a): species segregation in the timing of resource use or predator avoidance increases community robustness in a way that is analogous to the effects of resource partitioning. We also connect this formalism with the widely used and successful framework of Chesson (1994), demonstrating that the merging of these two perspectives yields simplified expressions for robustness more amenable to analytical treatment. As an example, we apply our results to a two-cycle in a model of two competing annual plants with seedbanks, using our formulas to calculate the range of parameters that allow for the coexistence of the competitors. This helps us understand which components of the environmental variation the coexistence is sensitive to; in our case, the model is fairly robust against changing seed survival, moderately so against changing the variance in seed germination, and quite sensitive to changing the mean seed germination rates.     

A simplified approach to derive Cleland model for enzymatic reactions

Metabolic modeling can suggest which is the key enzyme activity that needs to be controlled or its activity enhanced for the required production of a metabolite in a pathway. It also helps to find possible drug targets (enzymes to be inhibited). In metabolic modeling, knowing the kinetics of the enzymes involved in a pathway is mandatory. Most enzymatic reactions involve multi-substrates and follow an ordered sequential or ping–pong mechanism. The kinetic parameters involved in the model are obtained by fitting experimental data using a model based on the mechanism. The Cleland model has been used for some years. The grouping of parameters, such as dissociation constant and Michaelis–Menten constant, makes the strategy meaningful and hence the Cleland model is still in use. Although other alternate methods, e.g., the King-Altman method, are available, derivation by determinants can be used to derive a rate expression for the sequential or ping–pong mechanism, they are tedious. Hence, a meaningful modification is suggested in this communication for deriving the enzyme mechanism which is based on Thilakavathi et al. (Biotech Lett 28:1889–1894, 2006) to obtain the Cleland model in an easier way.