Allowance for effects of thermodynamic nonideality in sedimentation equilibrium distributions reflecting protein dimerization

This reexamination of a high-speed sedimentation equilibrium distribution for α-chymotrypsin under slightly acidic conditions (pH 4.1, I(M) 0.05) has provided experimental support for the adequacy of nearest-neighbor considerations in the allowance for effects of thermodynamic nonideality in the characterization of protein self-association over a moderate concentration range (up to 8 mg/mL). A widely held but previously untested notion about allowance for thermodynamic nonideality effects is thereby verified experimentally. However, it has also been shown that a greater obstacle to better characterization of protein self-association is likely to be the lack of a reliable estimate of monomer net charge, a parameter that has a far more profound effect on the magnitude of the measured equilibrium constant than any deficiency in current procedures for incorporating the effects of thermodynamic nonideality into the analysis of sedimentation equilibrium distributions reflecting reversible protein self-association.     

Determination of reaction volumes and polymer distribution characteristics of tobacco mosaic virus coat protein

A method that allows the quantitative determination of reaction volumes from sedimentation velocity experiments in an analytical ultracentrifuge is presented. Combined with a second method for detecting pressure-induced depolymerization, general characteristics of polymer distributions may be probed. We show that it is possible to determine if a sample is in an equilibrium or metastable state of subunit association. Our approach to probe macromolecular aggregation systems by small pressure perturbations is not restricted to the use of centrifuges. This method has been applied to characterize certain aspects of the polymerization of tobacco mosaic virus coat protein (TMVP). There are at least two helical polymer conformations in RNA-free coat protein rods. The smaller, helix I, polymers are limited to sizes below about 70 subunits (four to five helical turns) and undergo some kind of cooperative conformational change before further subunits may be added indefinitely. In contrast to helix I, the larger helix II polymers occur as broader and skewed size distributions. Under moderately strong polymerization conditions, the equilibrium state can contain both types of helical rods. The reaction volume for the addition of trimers is -220 ml/mol for both types of helical polymers. These results are compared with the results of previous thermodynamic analyses of TMVP polymerization.     

A thermodynamic view of networks

Networks can be described by the frequency distribution of the number of links associated with each node (the degree of the node). Of particular interest are the power law distributions, which give rise to the so-called scale-free networks, and the distributions of the form of the simplified canonical law (SCL) introduced by Mandelbrot, which give what we shall call the Mandelbrot networks. Many dynamical methods have been obtained for the construction of scale-free networks, but no dynamical construction of Mandelbrot networks has been demonstrated. Here we develop a systematic technique to obtain networks with any given distribution of the degrees of the nodes. This is done using a thermodynamic approach in which we maximise the entropy associated with degree distribution of the nodes of the network subject to certain constraints. These constraints can be chosen systematically to produce the desired network architecture. For large networks we therefore replace a dynamical approach to the stationary state by a thermodynamical viewpoint. We use the method to generate scale-free and Mandelbrot networks with arbitrarily chosen parameters. We emphasise that this approach opens the possibility of insights into a thermodynamics of networks by suggesting thermodynamic relations between macroscopic variables for networks.     

Parinaric acids as probes of binding domains in neutrophil elastase

Human neutrophil elastase has an extended hydrophobic substrate binding site which serves as a target for a number of hydrophobic inhibitors. We show here that the parinaric acids, fluorescent-conjugated tetraenoic fatty acids of plant origin, are inhibitors of neutrophil elastase. cis-Parinaric acid (cis-PA) interacts with the enzyme in two inhibitory modes. The high affinity interaction (Ki = 55 +/- 6 nM) results in partial noncompetitive inhibition of amidolytic activity, with 82% residual activity. A lower affinity interaction with cis-PA (Ki = 4 +/- 1 microM) results in competitive inhibition. trans-PA also acts as a high affinity partial noncompetitive inhibitor of elastase with a Ki equal to that for cis-PA but has no low affinity competitive inhibitory action. The endogenous fluorescence from the 3 tryptophan residues in elastase is partially quenched on binding cis- or trans-PA. Dependence of quenching of tryptophan fluorescence on PA concentration is consistent with binding to a single site with an apparent Kd of 26 +/- 3 nM, which may be equivalent to the high affinity partial noncompetitive inhibitory binding mode. Analysis of quenching according to the modified Forster theory of energy transfer developed by Snyder and Freire (Snyder, B., and Freire, E. (1982) Biophys. J. 40, 137-148) leads to an estimate of apparent closest indole-PA distance of 13 +/- 3 A. Fluorescence of either cis- or trans-PA is apparently unperturbed upon binding in the high affinity mode to elastase, but at micromolar cis-PA concentrations, binding to elastase results in a blue shift and 20% increase in intensity of PA emission, suggesting that the lower affinity competitive inhibitory binding mode of binding to elastase provides a hydrophobic environment for cis-PA.     

Biochemical networks with uncertain parameters

The modelling of biochemical networks becomes delicate if kinetic parameters are varying, uncertain or unknown. Facing this situation, we quantify uncertain knowledge or beliefs about parameters by probability distributions. We show how parameter distributions can be used to infer probabilistic statements about dynamic network properties, such as steady-state fluxes and concentrations, signal characteristics or control coefficients. The parameter distributions can also serve as priors in Bayesian statistical analysis. We propose a graphical scheme, the 'dependence graph', to bring out known dependencies between parameters, for instance, due to the equilibrium constants. If a parameter distribution is narrow, the resulting distribution of the variables can be computed by expanding them around a set of mean parameter values. We compute the distributions of concentrations, fluxes and probabilities for qualitative variables such as flux directions. The probabilistic framework allows the study of metabolic correlations, and it provides simple measures of variability and stochastic sensitivity. It also shows clearly how the variability of biological systems is related to the metabolic response coefficients.     

A model for the formation and interconversion of protein A-immunoglobulin G soluble complexes

We present a model for the formation and interconversion of the soluble complexes formed by reacting staphylococcal protein A (SpA) with rabbit immunoglobulin G (IgG) antibodies. The basic elements of the model are developed from reported hydrodynamic and electron microscopic studies of these complexes (see accompanying companion paper), together with established structural and binding properties of IgG and SpA. The model includes specific symmetry and binding requirements for IgG-SpA combination, and a steric constraint between neighboring IgG molecules. We discuss how such a constraint could influence the assembly and distribution of equilibrium complexes. After formulating a convenient symbolism for representing IgG-SpA complexes, the suggested model is used to construct plausible structures for the four predominant complexes observed in moderate SpA excess. Distributions of these stable complexes at different IgG:SpA ratios, together with LeChatelier's principle and a straightforward thermodynamic derivation, are used to predict likely arrangements of equilibrium structures. Also, a scale model of the unique IgG4-SpA2 complex formed in IgG excess is constructed from reported x-ray diffraction and amino acid sequence data. An intuitive thermodynamic argument is used to show that the suggested steric constraint could cause the rather unprecedented reversible transformation of the four 7 to 15S complexes into the unique 17S complex. A computer simulation is used to predict equilibrium concentrations of the various proposed complexes at different IgG:SpA ratios. In support of the suggested structures, the calculated thermodynamic distributions agree surprisingly well with those measured with the ultracentrifuge. We point out how the proposed arrangements of the complexes, and in particular the 17S complex, can account for many of their novel properties, such as antigen-induced conformational changes. Reported differences in complement activation and precipitate formation by SpA complexes formed with antibodies from various species are also discussed with regard to possible differences in structural arrangements of the complexes.     

Maximum-entropy determination of self-association distribution functions; daunorubicin and ATP.

In the present paper we show how one can use the perturbation of some molecular optical property (for example circular dichroism or chemical shift) as a function of concentration to construct cluster distribution functions describing the self-association of molecules in solution. The optical data are first converted into data giving the variation of the average extent of clustering as a function of the total concentration and then, using straightforward thermodynamics, a set of moments of the cluster distribution function can be obtained. Utilizing the maximum-entropy method, the moments are then used to calculate approximate distribution functions, where the more moments that are used the better the approximation obtained. Given the probability distribution for clusters of different sizes one can then calculate the equilibrium constant for each stage of association. Thus one converts average degree of association into equilibrium constants without having to use any specific model. By this method one can clearly tell whether the equilibrium constants remain constant, increase, or decrease with the number of molecules in a cluster. We apply the method to literature data for two systems, namely daunorubicin, which has a strong tendency to cluster in solution, and Mg(ATP)(2-) which forms weaker clusters. We find that the successive equilibrium constants for adding a monomer to a cluster are approximately constant for daunorubicin but clearly decrease as a function of increasing cluster size for Mg(ATP)(2-).     

The Evolution of Virulence in RNA Viruses under a Competition–Colonization Trade-Off

RNA viruses exist in large intra-host populations which display great genotypic and phenotypic diversity. We analyze a model of viral competition between two viruses infecting a constantly replenished cell pool. We assume a trade-off between the ability of the virus to colonize new cells (cell killing rate or virulence) and its local competitiveness (replicative success within coinfected cells). We characterize the conditions that allow for viral spread by means of the basic reproductive number and show that a local coexistence equilibrium exists, which is asymptotically stable. At this equilibrium, the less virulent competitor has a reproductive advantage over the more virulent colonizer reflected by a larger equilibrium population size of the competitor. The equilibria at which one virus outcompetes the other one are unstable, i.e., a second virus is always able to permanently invade. We generalize the two-virus model to multiple viral strains, each displaying a different virulence. To account for the large phenotypic diversity in viral populations, we consider a continuous spectrum of virulences and present a continuum limit of this multiple viral strains model that describes the time evolution of an initial continuous distribution of virulence without mutations. We provide a proof of the existence of solutions of the model equations, analytically assess the properties of stationary solutions, and present numerical approximations of solutions for different initial distributions. Our simulations suggest that initial continuous distributions of virulence evolve toward a distribution that is extremely skewed in favor of competitors. At equilibrium, only the least virulent part of the population survives. The discrepancy of this finding in the continuum limit with the two-virus model is attributed to the skewed equilibrium subpopulation sizes and to the transition to a continuum. Consequently, in viral quasispecies with high virulence diversity, the model predicts collective virulence attenuation. This result may contribute to understanding virulence attenuation, which has been reported in several experimental studies.     

Unimodal Tree Size Distributions Possibly Result from Relatively Strong Conservatism in Intermediate Size Classes

Tree size distributions have long been of interest to ecologists and foresters because they reflect fundamental demographic processes. Previous studies have assumed that size distributions are often associated with population trends or with the degree of shade tolerance. We tested these associations for 31 tree species in a 20 ha plot in a Dinghushan south subtropical forest in China. These species varied widely in growth form and shade-tolerance. We used 2005 and 2010 census data from that plot. We found that 23 species had reversed J shaped size distributions, and eight species had unimodal size distributions in 2005. On average, modal species had lower recruitment rates than reversed J species, while showing no significant difference in mortality rates, per capita population growth rates or shade-tolerance. We compared the observed size distributions with the equilibrium distributions projected from observed size-dependent growth and mortality. We found that observed distributions generally had the same shape as predicted equilibrium distributions in both unimodal and reversed J species, but there were statistically significant, important quantitative differences between observed and projected equilibrium size distributions in most species, suggesting that these populations are not at equilibrium and that this forest is changing over time. Almost all modal species had U-shaped size-dependent mortality and/or growth functions, with turning points of both mortality and growth at intermediate size classes close to the peak in the size distribution. These results show that modal size distributions do not necessarily indicate either population decline or shade-intolerance. Instead, the modal species in our study were characterized by a life history strategy of relatively strong conservatism in an intermediate size class, leading to very low growth and mortality in that size class, and thus to a peak in the size distribution at intermediate sizes.     

Mechanism of Transition from Xanthine Dehydrogenase to Xanthine Oxidase: Effect of Guanidine-HCL or Urea on the Activity

Mammalian xanthine oxidoreductase can be converted from the dehydrogenase to the oxidase form, either reversibly by formation of disulfide bridges or irreversibly by proteolytic cleavage within the xanthine oxidoreductase protein molecule. A tightly packed amino acid cluster stabilizes the dehydrogenase form, and disruption of this cluster is accompanied with rearrangement of the active site loop. Here, we show that the conversion occurs in the presence of guanidine-HCl or urea. We propose that xanthine dehydrogenase and oxidase are in a thermodynamic equilibrium that can be shifted by disruption of the amino acid cluster with a denaturant.     

Mechanism of transition from xanthine dehydrogenase to xanthine oxidase: effect of guanidine-HCL or urea on the activity

Mammalian xanthine oxidoreductase can be converted from the dehydrogenase to the oxidase form, either reversibly by formation of disulfide bridges or irreversibly by proteolytic cleavage within the xanthine oxidoreductase protein molecule. A tightly packed amino acid cluster stabilizes the dehydrogenase form, and disruption of this cluster is accompanied with rearrangement of the active site loop. Here, we show that the conversion occurs in the presence of guanidine-HCl or urea. We propose that xanthine dehydrogenase and oxidase are in a thermodynamic equilibrium that can be shifted by disruption of the amino acid cluster with a denaturant.     

Thermodynamics of anti-sickling agents with hemoglobin S

The effects of oxygen and a second ligand, the anti-sickling agent butylurea, on the hemoglobin S gel-solution phase equilibrium have been studied. The results have been analyzed using thermodynamic properties of the system. In particular, the solubility of deoxy hemoglobin S as a function of butylurea concentration was determined and the thermodynamic analysis shows that there are at least two cooperatively linked butylurea binding sites. Liquid phase oxygen binding studies at various butylurea concentrations show that the linkage between oxygen and butylurea binding is small. The influence of oxygen and butylurea on hemoglobin S solubility was determined by birefringence measurements. The results were interpreted by use of the Gibbs-Duhem equation which combined ligand binding expressions with the non-ideal solution properties and properties of the gel phase. The predicted influence of oxygen and butylurea upon the solubilities of hemoglobin S agrees with experimentally determined values.     

The critical micelle condition revisited.

Several simple alternatives have been examined as a possible basis for micellar size distributions which are internally consistent with the experimentally required concept of a threshold concentration, the critical micelle condition. Among these are the two-state system, monomer in equilibrium with a single high polymer, indefinite self-association, and continuous self-association with an arbitrary upper limit beyond which all further association is absolutely prohibited. Of these, only the last is a possible choice, although lacking experimental support. This report considers in deeper detail a thermodynamic model for micelle distributions, the so-called shell model of the present author, which is in basic agreement with an earlier statistic-mechanical study [Hoeve CA & Benson GC (1957), Colloid Polym Sci, 252, 56] in predicting the possibility of broad distributions of micellar species. A self-consistent distribution model which predicts a critical micelle condition must also predict a location with respect to degree of polymerization having a minimum concentration. Thus, the molar distribution function for the shell model satisfies this requirement, whereas the equivalent concentration distribution function fails to do so, as would the statistical models for multiple ligand binding. Moreover, the position of this minimum is predicted to change with concentration.     

Equilibrium of global amphibian species distributions with climate

A common assumption in bioclimatic envelope modeling is that species distributions are in equilibrium with contemporary climate. A number of studies have measured departures from equilibrium in species distributions in particular regions, but such investigations were never carried out for a complete lineage across its entire distribution. We measure departures of equilibrium with contemporary climate for the distributions of the world amphibian species. Specifically, we fitted bioclimatic envelopes for 5544 species using three presence-only models. We then measured the proportion of the modeled envelope that is currently occupied by the species, as a metric of equilibrium of species distributions with climate. The assumption was that the greater the difference between modeled bioclimatic envelope and the occupied distribution, the greater the likelihood that species distribution would not be at equilibrium with contemporary climate. On average, amphibians occupied 30% to 57% of their potential distributions. Although patterns differed across regions, there were no significant differences among lineages. Species in the Neotropic, Afrotropics, Indo-Malay, and Palaearctic occupied a smaller proportion of their potential distributions than species in the Nearctic, Madagascar, and Australasia. We acknowledge that our models underestimate non equilibrium, and discuss potential reasons for the observed patterns. From a modeling perspective our results support the view that at global scale bioclimatic envelope models might perform similarly across lineages but differently across regions.     

Robustness of downhill folding: guidelines for the analysis of equilibrium folding experiments on small proteins

Previously, we identified the protein BBL as a downhill folder. This conclusion was based on the statistical mechanical analysis of equilibrium experiments performed in two variants of BBL, one with a fluorescent label at the N-terminus, and another one labeled at both ends. A recent report has claimed that our results are an artifact of label-induced aggregation and that BBL with no fluorescent labels and a longer N-terminal tail folds in a two-state fashion. Here, we show that singly and doubly labeled BBL do not aggregate, unfold reversibly, and have the same thermodynamic properties when studied under appropriate experimental conditions (e.g., our original conditions (1)). With an elementary analysis of the available data on the nonlabeled BBL (2), we also show that this slightly more stable BBL variant is not a two-state folder. We discuss the problems that led to its previous misclassification and how they can be avoided. Finally, we investigate the equilibrium unfolding of the singly labeled BBL with both ends protected by acetylation and amidation. This variant has the same thermodynamic stability of the nonlabeled BBL and displays all the equilibrium signatures of downhill folding. From all these observations, we conclude that fluorescent labels do not perturb the thermodynamic properties of BBL, which consistently folds downhill regardless of its stability and specific protein tails. The work on BBL illustrates the shortcomings of applying conventional procedures intended to distinguish between two-state and three-state folding models to small fast-folding proteins.     

Estimation of transmembrane potentials from phase equilibria of hydrophobic paramagnetic ions.

Positively charged hydrophobic spin labels have been synthesized which respond to transmembrane potentials in sonicated liposomes. Electron paramagnetic resonance spectroscopy is used to show that the distribution of these probes between aqueous and membrane phases changes as a function of transmembrane potential. When liposomes are made more inside-negative, the fraction of membrane associated probe increases while the fraction of probe in the aqueous phase decreases. The results are in quantitative agreement with a simple equilibrium thermodynamic theory which allows estimation of absolute transmembrane potentials in phospholipid vesicles.     

The thermodynamics of water-protein interactions

Information about the effects of water on protein structure and function can be obtained from studies on freeze dried protein powders of varying water content. Sorption isotherms of water on proteins can be used to obtain thermodynamic quantities for water-protein interactions. Since such isotherms show hysteresis, there is doubt in regard to their interpretation.General expressions for the thermodynamic quantities of sorption are derived. If isotherms represent data at equilibrium, it is possible to calculate these thermodynamic quantities.There are two types of hysteresis, non-equilibrium hysteresis and equilibrium hysteresis. Absorption and desorption isotherms can show equilibrium hysteresis if different protein conformations, which are only slowly interconvertible, can be present. In this case valid thermodynamic quantities can be obtained. Experimental tests for equilibrium hysteresis are presented. More experiments are needed before definite conclusions can be drawn in regard to isotherms in the literature.If the protein conformation in a protein powder is similar to the protein conformation in aqueous solution, equilibrium data obtained from sorption isotherms can be used to approximate thermodynamic quantities for the interaction of water with proteins in aqueous solution. Examination of what experimental evidence is available indicates that the protein in powders prepared by desorption of water should have a conformation similar to that in solution. Further study of such samples will help to clarify the thermodynamics of water-protein interactions in aqueous solution.     

Analyzing Short-Term Noise Dependencies of Spike-Counts in Macaque Prefrontal Cortex Using Copulas and the Flashlight Transformation

Simultaneous spike-counts of neural populations are typically modeled by a Gaussian distribution. On short time scales, however, this distribution is too restrictive to describe and analyze multivariate distributions of discrete spike-counts. We present an alternative that is based on copulas and can account for arbitrary marginal distributions, including Poisson and negative binomial distributions as well as second and higher-order interactions. We describe maximum likelihood-based procedures for fitting copula-based models to spike-count data, and we derive a so-called flashlight transformation which makes it possible to move the tail dependence of an arbitrary copula into an arbitrary orthant of the multivariate probability distribution. Mixtures of copulas that combine different dependence structures and thereby model different driving processes simultaneously are also introduced. First, we apply copula-based models to populations of integrate-and-fire neurons receiving partially correlated input and show that the best fitting copulas provide information about the functional connectivity of coupled neurons which can be extracted using the flashlight transformation. We then apply the new method to data which were recorded from macaque prefrontal cortex using a multi-tetrode array. We find that copula-based distributions with negative binomial marginals provide an appropriate stochastic model for the multivariate spike-count distributions rather than the multivariate Poisson latent variables distribution and the often used multivariate normal distribution. The dependence structure of these distributions provides evidence for common inhibitory input to all recorded stimulus encoding neurons. Finally, we show that copula-based models can be successfully used to evaluate neural codes, e.g., to characterize stimulus-dependent spike-count distributions with information measures. This demonstrates that copula-based models are not only a versatile class of models for multivariate distributions of spike-counts, but that those models can be exploited to understand functional dependencies.     

The spatial distribution of flocking foragers: disentangling the effects of food availability,interference and conspecific attraction by means of spatial autoregressive modeling

Patch choice of foraging animals is typically assumed to depend positively on food availability and negatively on interference while benefits of the co‐occurrence of conspecifics tend to be ignored. In this paper we integrate a classical functional response model based on resource availability and interference with a conspecific attraction model and use it to simulate spatial distributions of animals in their continuous resource landscapes. We consider both equilibrium and non‐equilibrium distributions. We show that the integrated model produces distributions of foraging animals that closely match the distributions observed in nature. The simulations also show that under information uncertainty the locations of flocks are highly variable when conspecific attraction is strong. We furthermore explain how we can estimate the impacts of conspecific attraction and interference on the distribution of foraging animals by spatial autoregression. On the basis of simulated data we show that the separate impacts of interference and conspecific attraction can be disentangled when prior information on either is available, in addition to information on resource density and predator density, and that the total food effect is given by the spatial multiplier.