Sedimentation equilibrium in macromolecular solutions of arbitrary concentration. I. Self-associating proteins

Relations describing sedimentation equilibrium in solutions of self-associating macromolecules at arbitrary concentration are presented. These relations are obtained by using scaled-particle theory to calculate the thermodynamic activity of each species present at a given radial distance. The results are expected to be valid for solutions of globular proteins under conditions such that interactions between individual solute molecules may be approximated by a hard-particle potential. Sedimentation equilibria in solutions containing either a nonassociating solute or a solute that self-associates according to several different schemes are simulated using the derived relations. The results of these simulations are presented in terms of the dependence of apparent weight-average molecular weight upon solute concentration. Simple empirical relations are presented for estimating the true weight-average molecular weight from the apparent weight-average molecular weight, without reference to any particular self-association scheme. The weight-average molecular weight estimated in this fashion is within a few percent of the true weight-average molecular weight at all experimentally realizable solute concentrations ( < 400 g/L).     

Sedimentation equilibrium in macromolecular solutions of arbitrary concentration. II. Two protein components

Relations describing sedimentation equilibrium in solutions containing two macromolecular solute components are derived for the following cases: (1) two nonassociating proteins at arbitrary concentration, (2) one dilute self-associating protein in the presence of a second inert protein at arbitrary concentration, and (3) two proteins at arbitrary concentration that can associate to form a single heterocomplex of arbitrary composition. As in earlier work (R. C. Chatelier and A. P. Minton (1987) Biopolymers, 26, 507–524), the relations are obtained by using scaled particle theory to calculate the thermodynamic activity of each species present at a given radial distance in the centrifuge. The results of numerical simulations of sedimentation equilibrium are presented as the dependence of apparent molecular weights, or apparent weight-average molecular weights, upon solution composition. Semiempirical methods are presented, by means of which the weight-average molecular weights of self- and heteroassociating proteins in highly nonideal solutions may be estimated from experimental data. It is found that the semiempirical methods yield reasonably accurate estimates of the true weight-average molecular weight over a broad range of experimental conditions, providing that the partial specific volumes of two components in a heteroassociating system do not differ by more than about 0.05 mL/g.     

Static light scattering from concentrated protein solutions, I: General theory for protein mixtures and application to self-associating proteins

Exact expressions for the static light scattering of a solution containing up to three species of point-scattering solutes in highly nonideal solutions at arbitrary concentration are obtained from multicomponent scattering theory. Explicit expressions for thermodynamic interaction between solute molecules, required to evaluate the scattering relations, are obtained using an equivalent hard particle approximation similar to that employed earlier to interpret scattering of a single protein species at high concentration. The dependence of scattering intensity upon total protein concentration is calculated for mixtures of nonassociating proteins and for a single self-associating protein over a range of concentrations up to 200 g/l. An approximate semiempirical analysis of the concentration dependence of scattering intensity is proposed, according to which the contribution of thermodynamic interaction to scattering intensity is modeled as that of a single average hard spherical species. Simulated data containing pseudo-noise comparable in magnitude to actual experimental uncertainty are modeled using relations obtained from the proposed semiempirical analysis. It is shown that by using these relations one can extract from the data reasonably reliable information about underlying weak associations that are manifested only at very high total protein concentration.     

SANS/SAXS study of the BSA solvation properties in aqueous urea solutions via a global fit approach

We report on the solvation properties and intermolecular interactions of a model protein (bovine serum albumine, BSA) in urea aqueous solutions, as obtained by combining small-angle neutron and X-ray scattering experiments. According to a global fit strategy, all the whole set of scattering curves are analysed by considering a unique model which includes the BSA structure, the protein-protein interactions and the thermodynamic exchange process of water/urea molecules at the protein solvent interface. As a main result, the equilibrium constant that accounts for the difference in composition between the bulk solvent and the protein solvation layer is derived. Results confirm that urea preferentially sticks to the protein surface, inducing a noticeable change in both the repulsive and the attractive interaction potentials.     

Interspecific Competition, Environmental Gradients, Gene Flow, and the Coevolution of Species' Borders

Darwin viewed species range limits as chiefly determined by an interplay between the abiotic environment and interspecific interactions. Haldane argued that species' ranges could be set intraspecifically when gene flow from a species' populous center overwhelms local adaptation at the periphery. Recently, Kirkpatrick and Barton have modeled Haldane's process with a quantitative genetic model that combines density-dependent local population growth with dispersal and gene flow across a linear environmental gradient in optimum phenotype. To address Darwin's ideas, we have extended the Kirkpatrick and Barton model to include interspecific competition and the frequency-dependent selection that it generates, as well as stabilizing selection on a quantitative character. Our model includes local population growth, movements over space, natural selection, and gene flow. It simultaneously addresses the evolution of character displacement and species borders. It reproduces the Kirkpatrick and Barton single-species result that limited ranges can be produced with sufficiently steep environmental gradients and strong dispersal. Further, in the absence of environmental gradients or barriers to dispersal, interspecific competition will not limit species ranges at evolutionary equilibrium. However, interspecific competition can interact with environmental gradients and gene flow to generate limited ranges with much less extreme gradient and dispersal parameters than in the single-species case. Species display character displacement in sympatry, yet the reduction in competition that results from this displacement does not necessarily allow the two species to become sympatric everywhere. When species meet, competition reduces population densities in the region of overlap, which, in turn, intensifies the asymmetry in gene flow from center to margin. This reduces the ability of each species to adapt to local physical conditions at their range limits. If environmental gradients are monotonic but not linear, the transition zone between species at coevolutionary equilibrium occurs where the environmental gradient is steepest. If productivity gradients are also introduced into the model, then patterns similar to Rapoport's rule emerge. Interacting species respond to climate change, as it affects the optimal phenotype over space, by a combination of range shifts and local evolution in mean phenotype, while solitary species respond solely by range shifts. Finally, we compare empirical estimates for intrinsic growth rates and diffusion coefficients for several species to those needed by the single-species model to produce a stable limited range. These empirical values are generally insufficient to produce limited ranges in the model suggesting a role for interspecific interactions.     

Quantitative characterization of weak self-association in concentrated solutions of immunoglobulin G via the measurement of sedimentation equilibrium and osmotic pressure

The sedimentation equilibrium of solutions of immunoglobulin G in saline buffer, over a concentration range up to 125 g/L, was measured and analyzed in the context of a model that takes into account the possibility of attractive intermolecular interaction leading to the reversible formation of oligomeric species and repulsive intermolecular interaction leading to nonideal solution behavior. Additionally, previously published data on the concentration dependence of the osmotic pressure of immunoglobulin G under similar conditions, over a concentration range up to 400 g/L, were analyzed in the context of a newly developed thermodynamic formalism describing the osmotic pressure of a solution containing multiple nondiffusible solute species at an arbitrary concentration. Both sets of data are quantitatively accounted for by a model in which IgG self-associates at very high concentration to form (predominantly) trimers under the conditions of these experiments.     

Interactions between globular proteins and F-actin in isotonic saline solution.

Solutions of each of three different globular proteins (cytochrome c, chromophorically labeled serum albumin, and chromophorically labeled aldolase), mixed with another unlabeled globular protein or with fibrous actin, were prepared in pH 8.0 Tris-HCl buffer containing 0.15 M NaCl. Each solution was centrifuged at low speed, at 5 degrees C, until unassociated globular protein in solution achieved sedimentation equilibrium. Individual absorbance gradients of both macrosolutes in the mixtures subsequent to centrifugation were obtained via optical scans of the centrifuge tubes at two wavelengths. The gradients of each macrosolute in mixtures of two globular proteins revealed no association of globular proteins under the conditions of these experiments, but perturbation of the gradients of serum albumin, aldolase, and cytochrome c in the presence of F-actin indicated association of all three globular proteins with F-actin. Perturbation of actin gradients in the presence of serum albumin and aldolase suggested partial depolymerization of the F-actin by the globular protein. Analysis of the data with a simple phenomenological model relating free globular protein, bound globular protein, and total actin concentration provided estimates of the respective equilibrium constants for association of serum albumin and aldolase with F-actin, under the conditions of these experiments, of the order of 0.1 microM-1.     

Static Light Scattering From Concentrated Protein Solutions II: Experimental Test of Theory for Protein Mixtures and Weakly Self-Associating Proteins

Using an experimental technique recently developed in this laboratory (Fernández C. and A. P. Minton. 2008. Anal. Biochem. 381:254-257), the Rayleigh light scattering of solutions of bovine serum albumin, hen egg white ovalbumin, hen egg white ovomucoid, and binary mixtures of these three proteins was measured as a function of concentration at concentrations up to 125 g/L. The measured concentration dependence of scattering of both pure proteins and binary mixtures is accounted for nearly quantitatively by an effective hard particle model (Minton A. P. 2007. Biophys. J. 93:1321-1328) in which each protein species is represented by an equivalent hard sphere, the size of which is determined by the nature of repulsive interactions between like molecules under a given set of experimental conditions. The light scattering of solutions of chymotrypsin A was measured as a function of concentration at concentrations up to 70 g/L at pH 4.1, 5.4, and 7.2. At each pH, the measured concentration dependence is accounted for quantitatively by an effective hard particle model, according to which monomeric protein may self-associate to form an equilibrium dimer and, depending upon pH, an equilibrium pentamer or hexamer.     

EQUIL: Simulation and data analysis of binding reactions with arbitrary chemical models

We have developed an algorithm for simulation and analysis of arbitrary chemical systems in equilibrium, with emphasis on ligand binding reactions. The program EQUIL can treat reactions involving multiple ligands, multiple binding sites, ternary complex models, allosteric effectors, competitive and noncompetitive binding, conformational changes, cooperativity, and generally any scheme that can be represented as a set of chemical equations. EQUIL is based on a general thermodynamic model of chemical equilibria; it does not involve nonlinear transformation of experimental data, but it does require the user to define the model of interaction between ligands and receptors by writing down the appropriate chemical reactions. EQUIL contains features of particular importance to ligand binding experiments: variable binding capacities, nonspecific binding, and the ability to simultaneously analyze data from different types of experiments. Furthermore, the simulation feature of EQUIL allows the user to investigate the feasibility of experiments that could possibly distinguish between different reaction models. We illustrate the use of this program on personal computers to analyze and simulate simple and complicated interactions between ligands and receptors.     

Niche and Neutral Processes Together Determine Diversity Loss in Response to Fertilization in an Alpine Meadow Community

Fertilization via nutrient deposition and agricultural inputs is one of the most important factors driving decreases in plant diversity. However, we still do not fully understand which processes (niche process or neutral process) are more important in leading to decreases in plant diversity caused by fertilization. A hypothesis-based approach was used to test the relative importance of niche versus neutral processes along a fertilization gradient in an alpine meadow community on the eastern Tibetan plateau, China. Niche overlap values were calculated for species biomass, and the null model was used to generate the values of niche overlap expected at random. A linear regression modeling was used to evaluate the relationship between functional traits (specific leaf area, leaf dry matter content, and leaf total nitrogen concentration) and species relative abundance. Our results demonstrated that observed niche overlap for species biomass was significantly higher than expected at lower fertilization gradients. Moreover, we also found a significantly negative correlation between species relative abundance and specific leaf area and leaf dry matter content, but a significantly positive correlation between relative abundance and leaf nitrogen concentration at lower fertilization gradients. However, these relationships were not significant at higher fertilization gradients. We concluded that community assembly is dynamic progression along the environmental gradients, and niche and neutral processes may together determine species diversity loss in response to fertilization.     

Migration of connective tissue-derived cells is mediated by ultra-low concentration gradient fields of EGF

The directed migration of cells towards chemical stimuli incorporates simultaneous changes in both the concentration of a chemotactic agent and its concentration gradient, each of which may influence cell migratory response. In this study, we utilized a microfluidic system to examine the interactions between epidermal growth factor (EGF) concentration and EGF gradient in stimulating the chemotaxis of connective tissue-derived fibroblast cells. Cells seeded within microfluidic devices were exposed to concentration gradients established by EGF concentrations that matched or exceeded those required for maximum chemotactic responses seen in transfilter migration assays. The migration of individual cells within the device was measured optically after steady-state gradients had been experimentally established. Results illustrate that motility was maximal at EGF concentration gradients between .01- and 0.1-ng/(mL.mm) for all concentrations used. In contrast, the number of motile cells continually increased with increasing gradient steepness for all concentrations examined. Microfluidics-based experiments exposed cells to minute changes in EGF concentration and gradient that were in line with the acute EGFR phosphorylation measured. Correlation of experimental data with established mathematical models illustrated that the fibroblasts studied exhibit an unreported chemosensitivity to minute changes in EGF concentration, similar to that reported for highly motile cells, such as macrophages. Our results demonstrate that shallow chemotactic gradients, while previously unexplored, are necessary to induce the rate of directed cellular migration and the number of motile cells in the connective tissue-derived cells examined.     

Effects of excluded surface area and adsorbate clustering on surface adsorption of proteins I. Equilibrium models

Statistical-thermodynamic models for the equilibrium adsorption of proteins onto homogeneous, locally planar surfaces are presented. An extension of earlier work [R.C. Chatelier, A.P. Minton, Biophys. J. 71 (1996) 2367], the models presented here allow for the formation of a broadly heterogeneous population of adsorbate clusters in addition to excluded volume interactions between all adsorbate species. Calculations are carried out for three simple models for the structure of adsorbate, illustrating similarities and differences in the equilibrium properties of maximally compact clusters, minimally compact clusters and isomerizing clusters. Depending upon the strength of attractive interactions between adsorbate molecules, the resulting equilibrium isotherms may exhibit negative cooperativity, positive cooperativity, essentially no apparent cooperativity, or a mixture of positive cooperativity at low surface density and negative cooperativity at high surface density of adsorbate. The condition of apparent lack of cooperativity, which might naively be interpreted as evidence of a lack of interaction between adsorbate molecules, actually conceals a balance between attractive and repulsive interactions and extensive clustering of adsorbate.     

Stability and nuclear dynamics of the bicoid morphogen gradient

Patterning in multicellular organisms results from spatial gradients in morphogen concentration, but the dynamics of these gradients remain largely unexplored. We characterize, through in vivo optical imaging, the development and stability of the Bicoid morphogen gradient in Drosophila embryos that express a Bicoid-eGFP fusion protein. The gradient is established rapidly (approximately 1 hr after fertilization), with nuclear Bicoid concentration rising and falling during mitosis. Interphase levels result from a rapid equilibrium between Bicoid uptake and removal. Initial interphase concentration in nuclei in successive cycles is constant (+/-10%), demonstrating a form of gradient stability, but it subsequently decays by approximately 30%. Both direct photobleaching measurements and indirect estimates of Bicoid-eGFP diffusion constants (D < or = 1 microm(2)/s) provide a consistent picture of Bicoid transport on short ( approximately min) time scales but challenge traditional models of long-range gradient formation. A new model is presented emphasizing the possible role of nuclear dynamics in shaping and scaling the gradient.     

Influence of the NaCl or CaCl2 concentration on the structure of heat-set bovine serum albumin gels at pH 7

The structure of heat-set systems of the globular protein bovine serum albumin (BSA) was investigated at pH 7 in different salt conditions (NaCl or CaCl(2)) using light scattering. Cross-correlation dynamic light scattering was used to correct for multiple scattering from turbid samples. After heat treatment, aggregates are formed whose size increases as the protein concentration increases. Beyond a critical concentration that decreases with increasing salt concentration, gels are formed. The heterogeneity and the reduced turbidity of the gels were found to increase with increasing salt concentration and to decrease with increasing protein concentration. The structure of the gels is determined by the strength of the repulsive electrostatic interactions between the aggregated proteins. The results obtained in NaCl are similar to those reported in previous studies for other globular proteins. CaCl(2) was found to be much more efficient in reducing electrostatic interactions than NaCl at the same ionic strength.     

Interspecific interactions and range limits: contrasts among interaction types

There is a great deal of interest in the effects of biotic interactions on geographic distributions. Nature contains many different types of biotic interactions (notably mutualism, commensalism, predation, amensalism, and competition), and it is difficult to compare the effects of multiple interaction types on species’ distributions. To resolve this problem, we analyze a general, flexible model of pairwise biotic interactions that can describe all interaction types. In the absence of strong positive feedback, a species’ ability to be present depends on its ability to increase in numbers when it is rare and the species it is interacting with is at equilibrium. This insight leads to counterintuitive conclusions. Notably, we often predict the same range limit when the focal species experiences competition, predation, or amensalism. Similarly, we often predict the same range margin or when the species experiences mutualism, commensalism, or benefits from prey. In the presence of strong positive density-dependent feedback, different species interactions produce different range limits in our model. In all cases, the abiotic environment can indirectly influence the impact of biotic interactions on range limits. We illustrate the implications of this observation by analyzing a stress gradient where biotic interactions are harmful in benign environments but beneficial in stressful environments. Our results emphasize the need to consider the effects of all biotic interactions on species’ range limits and provide a systematic comparison of when biotic interactions affect distributions.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

Nonlinear gradient isotherm parameter estimation for proteins with consideration of salt competition and multiple forms.

Salt gradients in ion-exchange chromatography are routinely used to speed separation of proteins and to concentrate products, but systematic optimization of these gradients requires protein equilibrium data as a function of salt concentration. An understanding of conformational changes, aggregation, and salt effects, which include both competition and affinity modulation, is important for equilibrium isotherm parameter estimation. In this study, gradient elution of bovine serum albumin (BSA) in anion exchange was well predicted by a salt-modulated nonlinear isotherm which considers salt competition. The isotherm was able to predict BSA gradient elution from batch equilibrium data. The same isotherm was also able to predict elution for various gradient slopes when fitted to an intermediate slope gradient experiment. If multiple forms due to aggregation or denaturation exist, isotherm parameters are readily averaged in batch experiments because of the long equilibration times. Similarly, gradient experiments yield averaged parameters because the salt gradient tends to merge the closely eluting forms. However, in isocractic elution, if the reaction rate is not rapid enough to give a merged peak, the estimated isotherm parameters are only fair predictors of gradient behavior and vice versa. Slower flow rates in isocratic elution can help reduce the discrepancy by allowing forms to merge through interconversion. As an alternative to determining averaged parameters, consideration of two binding forms, using VERSE-LC, an advanced rate model, gave good agreement with experimental data over the entire range of salt gradient durations.     

伊犁河谷山地植物群落物种多样性分布格局及环境解释

结合对新疆伊犁河两岸科古琴山南坡(河谷北坡)和乌孙山北坡(河谷南坡) 94个样地的调查资料, 采用DCCA (detrended canonical correspondence analysis)排序法, 分析了物种多样性指数与环境因子之间的关系, 运用广义可加模型拟合植物群落总体多样性指数对海拔梯度的响应曲线, 探讨了伊犁河谷山地植物群落物种多样性的垂直分布格局。结果表明: 在调查的94个样地中, 共出现259种植物, 其中, 草本植物的种类极其丰富, 多达235种, 木本植物的种类极其有限; 垂直结构完整的植物群落具有较高的多样性指数; 河谷北坡植物群落物种多样性的分布格局受海拔、坡度、坡向以及土壤全氮、全钾、有机质、含水量等环境因子的影响较大, 而在河谷南坡, 物种多样性分布格局主要受坡度、海拔、有效磷含量和土壤含水量等环境因子的影响; 在河谷北坡, 植物群落的Patrick丰富度指数与Shannon-Wiener指数与海拔呈明显的双峰曲线关系, Simpson指数与Pielou均匀度指数呈不对称的单峰格局, 而河谷南坡的物种多样性指数随海拔均呈双峰格局, 尽管Patrick丰富度指数不甚明显。山地植物群落物种多样性的垂直分布格局是由海拔为主的多种环境因子综合作用的结果。     

Changes in plant interactions along a gradient of environmental stress

A combination of competition and facilitation effects operating simultaneously among plant species appears to be the rule in nature, where these effects change along productivity gradients often in a non-proportional manner. We investigated changes in competition and facilitation between a leguminous shrub, Retama sphaerocarpa , and its associate understorey species along an environmental gradient in semi-arid southeast Spain. Our results show a change in the net balance of the interaction between the shrub and several of its associated species, from clearly positive in the water-stressed, infertile environment to neutral or even negative in the more fertile habitat. There was a weakening of facilitation along the fertility gradient as a consequence of improved abiotic conditions. Competition was the most intense for below-ground resources in the less fertile environment while total competition tended to increase towards the more productive end of the gradient. Changes in the balance of the interaction between and among different plant species along the gradient of stress were caused by a decline in facilitation rather than by a change in competition. As both competition intensity and facilitation change along gradients of resource availability, plant interactions are best viewed as dynamic relationships, the outcome of which depends on abiotic conditions.     

Ion transport and the vibrating probe.

The theory of ion transport in the vicinity of a vibrating probe is developed. It is shown that the convection loops produced by the probe will not affect the electrical current density, assuming that the action of the probe does not affect the sources of the current in the biological system. However, the convection loops will significantly alter the ion concentration gradients in the unstirred layer near a tissue or cell surface. The concentration gradients within each convection loop will be reduced, while the concentration gradients between the loops and outside of the loops will be increased relative to the gradients existing without the probe. As a consequence, the electrical potential gradients can be changed relative to the potential gradients existing in the absence of the convection caused by the probe. If the mobility of the ion species carrying the electrical current is greater than the average ion mobility in the medium, then a decrease in ion concentration gradient will be accompanied by an increase in electrical potential gradient, while an increase in concentration gradient will be accompanied by a decrease or even a reversal of electrical potential gradient. Thus, the electrical potential gradient measured by the probe will depend on the concentration gradient in the vicinity of the probe, which will depend in turn on the spatial relation of the convection loops to the probe. An example of the effect of the convection loops on ion concentration and electrical potential is obtained from the theory via a numerical computer calculation. Experimental tests of this theory are discussed.