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During the last two decades the measurement and analysis of sedimentation equilibrium has been used increasingly as a method for characterizing intermolecular interactions in solution. More recently, a variant of this technique has been developed, in which a trace amount of a single solute component is labelled so that the concentration gradient of that component at sedimentation equilibrium may be measured independently of the gradients of all other components. The dependence of equilibrium tracer gradients upon solution composition may be readily interpreted in the context of models for self- and hetero-association of the labelled component, and, in the case of concentrated solutions, repulsive as well as attractive solute-solute interactions. We present a summary of experimental and analytic methods, a brief review of some previously published applications, and a preview of new applications demonstrating capabilities beyond those afforded by other current techniques for characterizing macromolecular associations in solution.

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Models for equilibrium surface adsorption of proteins have been recently proposed (Minton, A. P., 2000. Biophys. Chem. 86:239-247) in which negative cooperativity due to area exclusion by adsorbate molecules is compensated to a variable extent by the formation of a heterogeneous population of monolayer surface clusters of adsorbed protein molecules. In the present work this concept is extended to treat the kinetics of protein adsorption. It is postulated that clusters may grow via two distinct kinetic pathways. The first pathway is the diffusion of adsorbed monomer to the edge of a preexisting cluster and subsequent accretion. The second pathway consists of direct deposition of a monomer in solution onto the upper (solution-facing) surface of a preexisting cluster ("piggyback" deposition) and subsequent incorporation into the cluster. Results of calculations of the time course of adsorption, carried out for two different limiting models of cluster structure and energetics, show that in the absence of piggyback deposition, enhancement of the tendency of adsorbate to cluster can reduce, but not eliminate, the negative kinetic cooperativity due to surface area exclusion by adsorbate. Apparently noncooperative (Langmuir-like) and positively cooperative adsorption progress curves, qualitatively similar to those reported in several published experimental studies, require a significant fraction of total adsorption flux through the piggyback deposition pathway. According to the model developed here and in the above-mentioned reference, the formation of surface clusters should be a common concomitant of non-site-specific surface adsorption of proteins, and may provide an important mechanism for assembly of organized "protein machines" in vivo.

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Recent measurements of sedimentation equilibrium and sedimentation velocity have shown that the bacterial cell division protein FtsZ self-associates to form indefinitely long rod-like linear aggregates in the presence of GDP and Mg(2+). In the present study, the newly developed technique of non-ideal tracer sedimentation equilibrium was used to measure the effect of high concentrations-up to 150 g/liter-of each of two inert "crowder" proteins, cyanmethemoglobin or BSA, on the thermodynamic activity and state of association of dilute FtsZ under conditions inhibiting (-Mg(2+)) and promoting (+Mg(2+)) FtsZ self-association. Analysis of equilibrium gradients of both FtsZ and crowder proteins indicates that, under the conditions of the present experiment, FtsZ interacts with each of the two crowder proteins essentially entirely via steric repulsion, which may be accounted for quantitatively by a simple model in which hemoglobin, albumin, and monomeric FtsZ are modeled as effective spherical hard particles, and each oligomeric species of FtsZ is modeled as an effective hard spherocylinder. The functional dependence of the sedimentation of FtsZ on the concentrations of FtsZ and either crowder indicates that, in the presence of high concentrations of crowder, both the weight-average degree of FtsZ self-association and the range of FtsZ oligomer sizes present in significant abundance are increased substantially.

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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.

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The measurement and analysis of sedimentation equilibrium provides one of the most powerful and widely applicable methods for the characterization of reversible associations of macromolecules in solution. Recent developments in instrumentation, experimental design, and data analysis have substantially broadened the range of systems to which this technique may be applied, simplified its application, and reduced the cost of acquiring analytical capability.

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The bacterial cell division protein FtsZ from Escherichia coli has been purified with a new calcium precipitation method. The protein contains one GDP and one Mg(2+) bound, it shows GTPase activity, and requires GTP and Mg(2+) to polymerize into long thin filaments at pH 6.5. FtsZ, with moderate ionic strength and low Mg(2+) concentrations, at pH 7.5, is a compact and globular monomer. Mg(2+) induces FtsZ self-association into oligomers, which has been studied by sedimentation equilibrium over a wide range of Mg(2+) and FtsZ concentrations. The oligomer formation mechanism is best described as an indefinite self-association, with binding of an additional Mg(2+) for each FtsZ monomer added to the growing oligomer, and a slight gradual decrease of the affinity of addition of a protomer with increasing oligomer size. The sedimentation velocity of FtsZ oligomer populations is compatible with a linear single-stranded arrangement of FtsZ monomers and a spacing of 4 nm. It is proposed that these FtsZ oligomers and the polymers formed under assembly conditions share a similar axial interaction between monomers (like in the case of tubulin, the eukaryotic homolog of FtsZ). Similar mechanisms may apply to FtsZ assembly in vivo, but additional factors, such as macromolecular crowding, nucleoid occlusion, or specific interactions with other cellular components active in septation have to be invoked to explain FtsZ assembly into a division ring.

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The dispersity, size, and self-interaction of generation 5 polyamidoamine dendrimeric polymers with different terminal groups (surfaces) were characterized using several physicochemical techniques. Amino-surface dendrimers form oligomeric aggregates in aqueous solution, even in the presence of high salt concentrations (0.6M sodium phosphate). In contrast, the hydroxyl-surface polymer G5-OH behaves as a single homogeneous (or paucidisperse) species at low concentration. Measurements of density increment and the sedimentation and diffusion coefficients of G5-OH suggest a more swollen, porous structure than a globular protein of comparable mass. Measurements of the concentration dependence of sedimentation equilibrium of G5-OH in pH 7.2 phosphate buffer indicate the presence of significant electrostatic repulsion overlaid on weakly attractive interactions, leading to the formation of nonspecific aggregates at sufficiently high dendrimer concentration.

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Recent results support the notion that macromolecular 'crowding' enhances protein aggregation, at the expense of correct folding. The results can be rationalised in terms of kinetic competition between distinct processes, taking into account the relative influence of crowding on each process.

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Recent studies have led to increased appreciation of the influence of excluded volume in solutions of high total macromolecular content ('macromolecular crowding') upon the various classes of reaction that lead to the assembly of proteins and protein complexes. In general, crowding is expected to stabilize native protein structure relative to less compact non-native structures and to favor the formation of functional complexes of native proteins. Under certain pathological conditions, 'overcrowding' may enhance the formation of nonfunctional aggregates of non-native protein (e.g. amyloid and inclusion bodies).

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An equilibrium statistical-thermodynamic model for the effect of volume exclusion arising from high concentrations of stable macromolecules upon the stability of a trace globular protein with respect to denaturation by heat and by chaotropes is presented. The stable cosolute and the native form of the trace protein are modeled by effective hard spherical particles. The denatured state of the trace protein is represented as an ensemble of substates modeled by random coils having the same contour length but different rms end-to-end distances (i.e., different degrees of compaction). The excess or nonideal chemical potential of the native state and of each denatured substate is calculated as a function of the concentration of stable cosolute, leading to an estimate of the relative abundance of each state and substate, and the ensemble average free energy of the transition between native and denatured protein. The effect of the addition of stable cosolute upon the temperature of half-denaturation and upon the concentration of chaotrope required to half-denature the tracer at constant temperature is then estimated. At high cosolute concentration (>100 g/l) these effects are predicted to be large and readily measurable experimentally, provided that an experimental system exhibiting a fully reversible unfolding equilibrium at high total macromolecular concentration can be developed.

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Methods for quantitative characterization of heterologous protein-protein interactions by means of analytical ultracentrifugation (AUC) include sedimentation equilibrium, tracer sedimentation equilibrium, sedimentation velocity, and analytical band sedimentation. Fundamental principles governing the behavior of macromolecules in a centrifugal field are summarized, and the application of these principles to the interpretation of data obtained from each type of experiment is reviewed. Instrumentation and software for the acquisition and analysis of data obtained from different types of AUC experiments are described.

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The technique of tracer sedimentation equilibrium [Rivas, G., et al. (1994) Biochemistry, 2341-2348 (1); Rivas, G., et al. (1996) J. Mol. Recognit. 9, 31-38 (2)] is utilized, together with an extension of the theory of sedimentation equilibrium of highly nonideal solutions [Chatelier and Minton, (1987) Biopolymers 26, 1097-1113 (3)], to characterize the thermodynamic activity and/or the state of association of a dilute, labeled macromolecular solute in the presence of an arbitary concentration of a second, unlabeled macromolecular solute. Experiments are performed on solutions of labeled fibrinogen (0.25-1 g/L) in bovine serum albumin (0-100 g/L) in the presence and absence of divalent cations (Ca(2+), Mg(2+)), and on solutions of labeled tubulin (0.2-0.6 g/L) in dextran (0-100 g/L). It is found that in the absence of the divalent cations, the large dependence of the thermodynamic activity of fibrinogen on BSA concentration is well accounted for by a simple model for steric repulsion. In the presence of the cations and sufficiently large concentrations of BSA (>30 g/L), fibrinogen appears to self-associate to a weight-average molar mass approximately twice that of monomeric fibrinogen. Tubulin appears to self-associate to an extent that increases monotonically with increasing dextran concentration, reaching a weight-average molar mass almost 3 times that of the alphabeta dimer in the presence of 100 g/L dextran. Possible biological ramifications are discussed.

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Equilibrium and kinetic models for nonspecific adsorption of proteins to planar surfaces are presented. These models allow for the possibility of multiple interconvertible surface conformations of adsorbed protein. Steric repulsion resulting in area exclusion by adsorbed molecules is taken into account by treating the adsorbate as a thermodynamically nonideal two-dimensional fluid. In the equilibrium model, the possibility of attractive interactions between adsorbed molecules is taken into account in a limited fashion by permitting one of the adsorbed species to self-associate. Calculated equilibrium adsorption isotherms exhibit apparent high-affinity and low-affinity binding regions, corresponding respectively to adsorption of ligand at low fractional area occupancy in an energetically favorable side-on conformation and conversion at higher fractional area occupancy of the side-on conformation to an entropically favored end-on conformation. Adsorbate self-association may lead to considerable steepening of the adsorption isotherm, compensating to a variable extent for the broadening effect of steric repulsion. Kinetic calculations suggest that in the absence of attractive interactions between adsorbate molecules, the process of adsorption may be highly "stretched" along the time axis, rendering the attainment of adsorption equilibrium in the context of conventional experiments problematic.

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Results of experimental studies published since the last major review of excluded volume effects in biopolymer solutions in 1993 add to our appreciation of the scope and magnitude of such effects. Recent theoretical studies have improved incrementally our ability to understand and model excluded volume effects in simple model systems.

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The dissociation equilibria of Salmonella typhimurium tryptophan synthase alpha 2 beta 2 complex were studied via centrifugation of the complex to sedimentation equilibrium in neutral saline buffers containing 0 to 137 mM guanidine isothiocyanate (GuSCN). The resulting concentration gradients were analyzed in the context of an equilibrium model for sequential dissociation of two alpha subunits from a stable beta 2 subunit. Under the conditions of these experiments, the first dissociation constant alone could be evaluated at GuSCN concentrations < or = 100 mM, and the second dissociation constant alone could be evaluated at GuSCN = 137 mM. At intermediate GuSCN, both dissociation constants were sufficiently well defined to rule out the presence of a large equilibrium cooperative effect in the stepwise dissociation of the alpha subunits.

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This paper concerns the development of a short measure of perceived empathy. Developed in the context of research on sales performance, the scale provided indicators for perceived empathy as both a cognitive as well as an affective construct. While not being able to differentiate cleanly between affective and cognitive empathy, the resulting single-factor scale demonstrates good internal consistency and validity, being predictive of successful versus unsuccessful sales encounters.

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The validity of the most common kinetic interpretation of biosensor data can be quickly assessed with the aid of two simple tests for self-consistency, requiring only back-of-the-envelope calculations. A search of the recent literature reveals that many published results fail these tests qualitatively.

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