Kinetic isotope effects significantly influence intracellular metabolite 13C labeling patterns and flux determination

Rigorous mathematical modeling of carbon-labeling experiments allows estimation of fluxes through the pathways of central carbon metabolism, yielding powerful information for basic scientific studies as well as for a wide range of applications. However, the mathematical models that have been developed for flux determination from ¹³C labeling data have commonly neglected the influence of kinetic isotope effects on the distribution of ¹³C label in intracellular metabolites, as these effects have often been assumed to be inconsequential. We have used measurements of the ¹³C isotope effects on the pyruvate dehydrogenase enzyme from the literature to model isotopic fractionation at the pyruvate node and quantify the modeling errors expected to result from the assumption that isotope effects are negligible. We show that under some conditions kinetic isotope effects have a significant impact on the ¹³C labeling patterns of intracellular metabolites, and the errors associated with neglecting isotope effects in ¹³C-metabolic flux analysis models can be comparable in size to measurement errors associated with GC–MS. Thus, kinetic isotope effects must be considered in any rigorous assessment of errors in ¹³C labeling data, goodness-of-fit between model and data, confidence intervals of estimated metabolic fluxes, and statistical significance of differences between estimated metabolic flux distributions.     

A Thermodynamic Model of the Cardiac Sarcoplasmic/Endoplasmic Ca (SERCA) Pump

We present a biophysically based kinetic model of the cardiac SERCA pump that consolidates a range of experimental data into a consistent and thermodynamically constrained framework. The SERCA model consists of a number of sub-states with partial reactions that are sensitive to Ca²⁺ and pH, and to the metabolites MgATP, MgADP, and Pi. Optimization of model parameters to fit experimental data favors a fully cooperative Ca²⁺-binding mechanism and predicts a Ca²⁺/H⁺ counter-transport stoichiometry of 2. Moreover, the order of binding of the partial reactions, particularly the binding of MgATP, proves to be a strong determinant of the ability of the model to fit the data. A thermodynamic investigation of the model indicates that the binding of MgATP has a large inhibitory effect on the maximal reverse rate of the pump. The model is suitable for integrating into whole-cell models of cardiac electrophysiology and Ca²⁺ dynamics to simulate the effects on the cell of compromised metabolism arising in ischemia and hypoxia.     

A Biophysically Based Mathematical Model for the Kinetics of Mitochondrial Na-Ca Antiporter

Sodium-calcium antiporter is the primary efflux pathway for Ca²⁺ in respiring mitochondria, and hence plays an important role in mitochondrial Ca²⁺ homeostasis. Although experimental data on the kinetics of Na⁺-Ca²⁺ antiporter are available, the structure and composition of its functional unit and kinetic mechanisms associated with the Na⁺-Ca²⁺ exchange (including the stoichiometry) remains unclear. To gain a quantitative understanding of mitochondrial Ca²⁺ homeostasis, a biophysical model of Na⁺-Ca²⁺ antiporter is introduced that is thermodynamically balanced and satisfactorily describes a number of independent data sets under a variety of experimental conditions. The model is based on a multistate catalytic binding mechanism for carrier-mediated facilitated transport and Eyring's free energy barrier theory for interconversion and electrodiffusion. The model predicts the activating effect of membrane potential on the antiporter function for a 3Na⁺:1Ca²⁺ electrogenic exchange as well as the inhibitory effects of both high and low pH seen experimentally. The model is useful for further development of mechanistic integrated models of mitochondrial Ca²⁺ handling and bioenergetics to understand the mechanisms by which Ca²⁺ plays a role in mitochondrial signaling pathways and energy metabolism.     

Strontium isotopes (87Sr/86Sr) in terrestrial ecological and palaeoecological research: empirical efforts and recent advances in continental‐scale models

Strontium (Sr) isotope analysis can provide detailed biogeographical and ecological information about modern and ancient organisms. Because Sr isotope ratios (⁸⁷Sr/⁸⁶Sr) in biologically relevant materials such as water, soil, vegetation, and animal tissues predominantly reflect local geology, they can be used to distinguish geologically distinct regions as well as identify highly mobile individuals or populations. While the application of Sr isotope analysis to biological research has been steadily increasing, high analytical costs have prohibited more widespread use. Additionally, accessibility of this geochemical tool has been hampered due to limited understanding of (i) the degree to which biologically relevant materials differ in their spatial averaging of ⁸⁷Sr/⁸⁶Sr ratios, and (ii) how these differences may be affected by lithologic complexity. A recently developed continental‐scale model that accounts for variability in bedrock weathering rates and predicts Sr isotope ratios of surface water could help resolve these questions. In addition, if this ‘local water’ model can accurately predict ⁸⁷Sr/⁸⁶Sr ratios for other biologically relevant materials, there would be reduced need for researchers to assess regional Sr isotope patterns empirically. Here, we compile ⁸⁷Sr/⁸⁶Sr data for surface water, soil, vegetation, and mammalian and fish skeletal tissues from the literature and compare the accuracy with which the local water model predicts Sr isotope data among these five materials across the contiguous USA. We find that measured Sr isotope ratios for all five materials are generally close to those predicted by the local water model, although not with uniform accuracy. Mammal skeletal tissues are most accurately predicted, particularly in regions with low variability in ⁸⁷Sr/⁸⁶Sr predicted by the local water model. Increasing regional geologic heterogeneity increases both the offset and variance between modelled and empirical Sr isotope ratios, but its effects are broadly similar across materials. The local water model thus provides a readily available source of background data for predicting ⁸⁷Sr/⁸⁶Sr for biologically relevant materials in places where empirical data are lacking. The availability of increasingly high‐quality modelled Sr data will dramatically expand the accessibility of this geochemical tool to ecological applications.     

A statistical mechanical treatment of fatty acyl chain order in phospholipid bilayers and correlation with experimental data. B. Dipalmitoyl-3-sn-phosphatidylcholine

In order to help bridge the conceptual gap between experimental data on chains of phospholipid molecules and their microscopic organization, a theoretical model has been proposed in a preceding paper. The intentions associated with the new theory were to describe a model able to reproduce accurately the experimental data. This capability is essential to monitor some of the mechanisms behind the physical data. The results presented here show first that, provided a suitable fitting of the phenomenological parameters entailed in the model, the theory indeed gives good agreement with experimental data (²H-NMR, neutron scattering, calorimetry) obtained for a $\text{dipalmitoyl}\text{-3-sn-}\text{phosphatidylcholine}$ bilayer. This property of the model is then specifically used to describe the nature of the perturbing effects of local anaesthetics and cholesterol on the organization of the acyl chains and to correlate these effects with the experimental data. Finally the theoretical model is used to supplement experimental data by describing the acyl chain organization in terms of the most probable spectrum of chain conformations. Predictions are made about the one-, two- and three-dimensional mean spatial characteristics of the acyl chains.     

基于辐热积法模拟烤烟叶面积与烟叶干物质产量

烟叶叶面积增长与干物质累积是烤烟产量形成的主要部分,对品质的形成也有影响。本研究根据气温和光照对烤烟叶片生长和干物质累积的影响,基于辐热积理论建立了适用于不同烟区的烤烟叶面积模型和干物质累积模型,分别使用独立的试验数据建模及对模型进行检验,再通过多年次烟叶干重试验数据对模型进行检验。结果表明,与传统的预测方法相比,用辐热积模型获得的叶面积模拟值与实测值间1:1线的决定系数（R2）和RMSE值为0.9634和0.1653 m2/株,预测精度比SLA法和GDD法分别提高了93%和82%。模型对叶干重模拟的RMSE值为27.1 g/m2,用历年玉溪试验数据检验的RE值为24.5%,说明模型的拟合度和可靠性较好。本研究所建立的模型能够利用气温、日照等常规气象观测数据,动态预测烤烟叶面积增长和干物质累积,且模型参数少,符合度好,实用性强,可以为烤烟生产中的产量预测提供理论依据和决策支持。     

Creating Dynamic Images of Short-lived Dopamine Fluctuations with lp-ntPET: Dopamine Movies of Cigarette Smoking

We describe experimental and statistical steps for creating dopamine movies of the brain from dynamic PET data. The movies represent minute-to-minute fluctuations of dopamine induced by smoking a cigarette. The smoker is imaged during a natural smoking experience while other possible confounding effects (such as head motion, expectation, novelty, or aversion to smoking repeatedly) are minimized.We present the details of our unique analysis. Conventional methods for PET analysis estimate time-invariant kinetic model parameters which cannot capture short-term fluctuations in neurotransmitter release. Our analysis - yielding a dopamine movie - is based on our work with kinetic models and other decomposition techniques that allow for time-varying parameters ^1-7. This aspect of the analysis - temporal-variation - is key to our work. Because our model is also linear in parameters, it is practical, computationally, to apply at the voxel level. The analysis technique is comprised of five main steps: pre-processing, modeling, statistical comparison, masking and visualization. Preprocessing is applied to the PET data with a unique ''HYPR'' spatial filter ⁸ that reduces spatial noise but preserves critical temporal information. Modeling identifies the time-varying function that best describes the dopamine effect on ¹¹C-raclopride uptake. The statistical step compares the fit of our (lp-ntPET) model ⁷ to a conventional model ⁹. Masking restricts treatment to those voxels best described by the new model. Visualization maps the dopamine function at each voxel to a color scale and produces a dopamine movie. Interim results and sample dopamine movies of cigarette smoking are presented.     

单木生物量模型估计区域尺度生物量的不确定性

采用系统抽样体系江西省固定样地杉木连续观测数据和生物量数据,通过Monte Carlo法反复模拟由单木生物量模型推算区域尺度地上生物量的过程,估计了江西省杉木地上总生物量。基于不同水平建模样本量n及不同决定系数R~2的设计,分别研究了单木生物量模型参数变异性及模型残差变异性对区域尺度生物量估计不确定性的影响。研究结果表明:2009年江西省杉木地上生物量估计值为(19.84±1.27)t/hm~2,不确定性占生物量估计值约6.41%。生物量估计值和不确定性值达到平稳状态所需的运算时间随建模样本量及决定系数R~2的增大而减小;相对于模型参数变异性,残差变异性对不确定性的影响更小。     

Computational Model of Ca2+ Wave Propagation in Human Retinal Pigment Epithelial ARPE-19 Cells

Objective

Computational models of calcium (Ca²⁺) signaling have been constructed for several cell types. There are, however, no such models for retinal pigment epithelium (RPE). Our aim was to construct a Ca²⁺ signaling model for RPE based on our experimental data of mechanically induced Ca²⁺ wave in the in vitro model of RPE, the ARPE-19 monolayer.

Methods

We combined six essential Ca²⁺ signaling components into a model: stretch-sensitive Ca²⁺ channels (SSCCs), P₂Y₂ receptors, IP₃ receptors, ryanodine receptors, Ca²⁺ pumps, and gap junctions. The cells in our epithelial model are connected to each other to enable transport of signaling molecules. Parameterization was done by tuning the above model components so that the simulated Ca²⁺ waves reproduced our control experimental data and data where gap junctions were blocked.

Results

Our model was able to explain Ca²⁺ signaling in ARPE-19 cells, and the basic mechanism was found to be as follows: 1) Cells near the stimulus site are likely to conduct Ca²⁺ through plasma membrane SSCCs and gap junctions conduct the Ca²⁺ and IP³ between cells further away. 2) Most likely the stimulated cell secretes ligand to the extracellular space where the ligand diffusion mediates the Ca²⁺ signal so that the ligand concentration decreases with distance. 3) The phosphorylation of the IP₃ receptor defines the cell’s sensitivity to the extracellular ligand attenuating the Ca²⁺ signal in the distance.

Conclusions

The developed model was able to simulate an array of experimental data including drug effects. Furthermore, our simulations predict that suramin may interfere ligand binding on P₂Y₂ receptors or accelerate P₂Y₂ receptor phosphorylation, which may partially be the reason for Ca²⁺ wave attenuation by suramin. Being the first RPE Ca²⁺ signaling model created based on experimental data on ARPE-19 cell line, the model offers a platform for further modeling of native RPE functions.     

Understanding drug resistance for monotherapy treatment of HIV infection

The purpose of this study was to investigate strategies in the monotherapy treatment of HIV infection in the presence of drug-resistant (mutant) strains. A mathematical system is developed to model resistance in HIV chemotherapy. It includes the key players in the immune response to HIV infection: virus and both uninfected CD4⁺ and infected CD4⁺ T-cell populations. We model the latent and progressive stages of the disease, and then introduce monotherapy treatment. The model is a system of differential equations describing the interaction of two distinct classes of HIV—drug-sensitive (wild type) and drug-resistant (mutant)—with lymphocytes in the peripheral blood. We then introduce chemotherapy effects. In the absence of treatment, the model produces the three types of qualitative clinical behavior—anuninfected steady state, andinfected steady state (latency), andprogression to AIDS. Simulation of treatment is provided for monotherapy, during theprogression to AIDS state, in the consideration of resistance effects. Treatment benefit is based on an increase or retention in CD4⁺ T-cell counts together with a low viral titer. We explore the following treatment approaches: an antiviral drug which reduces viral infectivity that is administered early—when the CD4⁺ T-cell count is ≥300/mm³, and late—when the CD4⁺ T-cell count is less than 300/mm³. We compare all results with data. When treatment is initiated during the progression to AIDS state, treatment prevents T-cell collapse, but gradually loses effectiveness due to drug resistance. We hypothesize that it is the careful balance of mutant and wild-type HIV strains which provides the greatest prolonged benefit from treatment. This is best achieved when treatment is initiated when the CD4⁺ T-cell counts are greater than 250/mm³, but less than 400/mm³ in this model (i.e. not too early, not too late). These results are supported by clinical data. The work is novel in that it is the first model to accurately simultate data before, during and after monotherapy treatment. Our model also provides insight into recent clinical results, as well as suggests plausible guidelines for clinical testing in the monotherapy of HIV infection.     

Model predictions of myoelectrical activity of the small bowel

A mathematical model for the periodic electrical activity of a functional unit of the small intestine is developed. Based on real morphological and electrophysiological data, the model assumes that: the functional unit is an electromyogenic syncytium; the kinetics of L, T-type Ca²⁺, mixed Ca²⁺-dependent K⁺, potential sensitive K⁺ and Cl⁻ channels determines electrical activity of the functional unit; the basic neural circuit, represented by a single cholinergic neurone, provides an excitatory input to the functional unit via receptor-linked L-type Ca²⁺ channels. Numerical simulation of the model has shown that it is capable of displaying the slow waves and that slight modifications of some of the parameters result in different electrical responses. The effects of the variations of the main parameters have been analyzed for their ability to reproduce various electrical patterns. The results are in good qualitative and quantitative agreement with results of experiments conducted on the small intestine.     

Modeling Plasmalemma Ion Transport of the Aquatic Plant Egeria densa

Fresh-water plants generate extraordinarily high electric potential differences at the plasma membrane. For a deeper understanding of the underlying transport processes a mathematical model of the electrogenic plasmalemma ion transport was developed based on experimental data mainly obtained from Egeria densa. The model uses a general nonlinear network approach and assumes coupling of the transporters via membrane potential. A proton pump, an outward-rectifying K⁺ channel, an inward-rectifying K⁺ channel, a Cl⁻ channel and a (2H-Cl)⁺ symporter are considered to be elements of the system. The model takes into consideration the effects of light, external pH and ionic content of the bath medium on ion transport. As a result it does not only satisfactorily describe the membrane potential as a function of these external physiological factors but also succeeds in simulating the effects of specific inhibitors as well as I-V-curves obtained with the patch-clamp technique in the whole cell mode. The quality of the model was checked by stability and sensitivity analyses. Received: 18 March 1996/Revised: 17 July 1996     

A Four-Compartment Model for Ca2+ Dynamics: An Interpretation of Ca2+ Decay after Repetitive Firing of Intact Nerve Terminals

In the presynaptic nerve terminals of the bullfrog sympathetic ganglia, repetitive nerve firing evokes [Ca²⁺] transients that decay monotonically. An algorithm based on an eigenfunction expansion method was used for fitting these [Ca²⁺] decay records. The data were fitted by a linear combination of two to four exponential functions. A mathematical model with three intraterminal membrane-bound compartments was developed to describe the observed Ca²⁺ decay. The model predicts that the number of exponential functions, n, contained in the decay data corresponds to n – 1 intraterminal Ca²⁺ stores that release Ca²⁺ during the decay. Moreover, when a store stops releasing or starts to release Ca²⁺, the decay data should be fitted by functions that contain one less exponential component for the former and one more for the latter than do the fitting functions for control data. Because of the current lack of a parameter by which quantitative comparisons can be made between two decay processes when at least one of them contained more than one exponential components, we defined a parameter, the overall rate (OR) of decay, as the trace of the coefficient matrix of the differential equation systems of our model. We used the mathematical properties of the model and of the OR to interpret effects of ryanodine and of a mitochondria uncoupler on Ca²⁺ decay. The results of the analysis were consistent with the ryanodine-sensitive store, mitochondria, and another, yet unidentified store release Ca²⁺ into the cytosol of the presynaptic nerve terminals during Ca²⁺ decay. Our model also predicts that mitochondrial Ca²⁺ buffering accounted for more than 86% of all the flux rates across various membranes combined and that there are type 3 and type 1 and/or type 2 ryanodine receptors in these terminals.     

Salinity,water use and yield of maize: Testing of the mathematical model ecosys

There is a need to establish how root water uptake should be calculated under saline conditions, and to test calculated uptake against experimental data recorded under documented site conditions. In this study, the ecosystem simulation model ecosys was expanded to include an ion transfer-equilibrium-exchange model used to calculated electrical conductivity and osmotic potential. This expanded model was tested against experimental data for maize growth and water use reported under different irrigation and salinity levels at four different sites in the western U.S. to determine if salinity effects on crop growth and water use could be modelled from the effects of salinity on soil osmotic potential. The model was able to reproduce reductions in water use and phytomass yields on salinized (10 g total salts kg^–1 water) soils that ranged from 10 to 50% of those on non-salinized controls. In general, these reductions increased with increasing irrigation deficits. These reductions arose in the model from reduced canopy water potentials and conductances caused by reduced osmotic potentials in the saline soils. The hypothesis that salinity effects on crop growth and water use are caused by salinity effects on soil osmotic potential appear to be supported under the range of conditions included in this study. Models such as ecosys that are based on general hypotheses for the effects of salinity upon biological activity may be well adapted for general use in assessing the effects of salinity on crop growth and water use with different soils, managements and climates.     

Theory of phase-modulation fluorescence spectroscopy for excited-state processes

Theory is presented for the analysis of excited-state reactions by fluorescence phase shift and demodulation methods. Initially, a two-state model with spectral overlap is considered to illustrate most simply the effects of excited-state reactions on the expected phase and modulation values. Secondly, a multistate model is described to illustrate the probable effects of a fluorophore interacting with several solvent molecules. We note the following unique features of phase-modulation data expected from a fluorophore whose emission spectrum shifts during the lifetime of the excited state: (1) The modulation frequency dependence of the apparent phase (τ^p) and modulation (τ^m) lifetimes of the reacted species is opposite to that of a heterogeneous population of fluorophores. (2) For the reacted species τ^p > τ^m. For a heterogeneous sample τ^p < τ^m. (3) The phase angle of the reacted species can exceed 90°. For a heterogeneous sample phase angles are always less than 90°. Thus, phase and modulation measurements can distinguish between time-dependent processes and spectral heterogeneity by observation of any feature described above. Additionally: (4) The lifetime of the product species can be measured directly. (5) Reverse relaxation can be identified, and the reverse relaxation rates calculated. (6) The wavelength-dependent phase and modulation data can be used to resolve the individual spectra from a two-state reaction. (7) And finally, under favorable conditions, a two-state excited-stale process can be distinguished from a continuous multiple-state process. In each instance, model calculations are presented to illustrate the unique potentials of phase-modulation fluorometry for investigations of excited-state processes.     

The nature of 2-locus epistatic interactions in animals: evidence from Sewall Wright's guinea pig data

Summary The nature of epistatic interactions affects covariance between relatives and the expression of heterosis in various crossbred genotypes. The investigation of these interactions for metric traits requires large data sets of a suitable type. Data from Sewall Wright's early work with guinea pigs are used to compare the goodness-of-fit of seven biological models of 2-locus interaction for the six out of eleven traits in which epistatic effects are apparent. The model equivalent to additive x additive epistasis gives the best general fit over traits, with an average transformed R² value significantly greater than that of the next best fitting model (P<0.05). This result is compatible with results from the one other study in this area, using data from mice. It is concluded that, based on results available to date, the additive x additive 2-locus model of epistatic interaction appears most suitable for reduced genetic models.     

Coupling biochemistry and hydrodynamics captures hyperactivated sperm motility in a simple flagellar model

Hyperactivation in mammalian sperm is characterized by highly asymmetrical waveforms and an increase in the amplitude of flagellar bends. It is important for the sperm to be able to achieve hyperactivated motility in order to reach and fertilize the egg. Calcium (Ca²⁺) dynamics are known to play a large role in the initiation and maintenance of hyperactivated motility. Here we present an integrative model that couples the CatSper channel mediated Ca²⁺ dynamics of hyperactivation to a mechanical model of an idealized sperm flagellum in a 3-d viscous, incompressible fluid. The mechanical forces are due to passive stiffness properties and active bending moments that are a function of the local Ca²⁺ concentration along the length of the flagellum. By including an asymmetry in bending moments to reflect an asymmetry in the axoneme's response to Ca²⁺, we capture the transition from activated motility to hyperactivated motility. We examine the effects of elastic properties of the flagellum and the Ca²⁺ dynamics on the overall swimming patterns. The swimming velocities of the model flagellum compare well with data for hyperactivated mouse sperm.     

Macroscopic orientation effects in broadline NMR-spectra of model membranes at high magnetic field strength: A method preventing such effects

The partial orientation of multilamellar vesicles (MLV) in high magnetic fields has been studied and a method to prevent such effects is herewith proposed. The orientation effect was measured with ²H-, ³¹P-NMR and electron microscopy on MLVs of dipalmitoyl phosphatidylcholine with 30 mol% cholesterol. We present the first freeze—etch electron microscopy data obtained from MLV samples that were frozen directly in the NMR magnet at a field strength of 9.4 Tesla. These experiments clearly show that the MLVs adopt an ellipsoidal (but not a cylindrical) shape in the magnetic field. Best fit ³¹P-NMR lineshape calculations assuming an ellipsoidal distribution of molecular director axes to the experimentally obtained spectra provide a quantitative measure of the average semiaxis ratio of the ellipsoidal MLVs and its change with temperature. The application of so-called spherical supported vesicles (SSV) is found to prevent any partial orientation effects so that undistorted NMR powder pattern of the bilayer can be measured independently of magnetic field strength and temperature.

The usefulness of SSVs is further demonstrated by a direct comparison of spectral data such as ³¹P-and ²H-NMR lineshapes and relaxation times as well as ²H-NMR dePaked spectra obtained for both model systems. These experiments show that spectral data obtained from partially oriented MLVs are not unambiguous to interpret, in particular, if an external parameter such as temperature is varied.

     

Determinants and patterns of population growth in water hyacinth

Mathematical models of the growth of water hyacinth are developed to provide a sound basis for assessing existing and potential control options. We show that under constant experimental conditions, water hyacinth shows logistic growth (r² of 0.69–1.00). The effects of nutrients and temperature on model parameters are explored using data from the literature. The model is verified against growth in a natural infestation. The resulting model incorporates two of the most important factors (temperature and water nutrient level) that determine whether water hyacinth is an important environmental problem at a given site. These relationships form a robust basis for further model development, and can be readily used to evaluate how the plant will respond to changes in nutrient inputs.     

Self Crowding of Globular Proteins Studied by Small-Angle X-Ray Scattering

Small-angle x-ray scattering (SAXS) was used to study the behavior of equine metmyoglobin (Mb) and bovine pancreatic trypsin inhibitor (BPTI) at concentrations up to 0.4 and 0.15 g/mL, respectively, in solutions also containing 50% D₂O and 1 M urea. For both proteins, significant effects because of interference between x-rays scattered by different molecules (interparticle interference) were observed, indicating nonideal behavior at high concentrations. The experimental data were analyzed by comparison of the observed scattering profiles with those predicted by crystal structures of the proteins and a hard-sphere fluid model used to represent steric exclusion effects. The Mb scattering data were well fit by the hard-sphere model using a sphere radius of 18 Å, only slightly smaller than that estimated from the three-dimensional structure (20 Å). In contrast, the scattering profiles for BPTI in phosphate buffer displayed substantially less pronounced interparticle interference than predicted by the hard-sphere model and the radius estimated from the known structure of the protein (15 Å). Replacing the phosphate buffer with 3-(N-morpolino)propane sulfonic acid (MOPS) led to increased interparticle interference, consistent with a larger effective radius and suggesting that phosphate ions may mediate attractive intermolecular interactions, as observed in some BPTI crystal structures, without the formation of stable oligomers. The scattering data were also used to estimate second virial coefficients for the two proteins: 2.0 ×10^-4 cm³mol/g² for Mb in phosphate buffer, 1.6 ×10^-4 cm³mol/g² for BPTI in phosphate buffer and 9.2 ×10^-4 cm³mol/g² for BPTI in MOPS. The results indicate that the behavior of Mb, which is nearly isoelectric under the conditions used, is well described by the hard-sphere model, but that of BPTI is considerably more complex and is likely influenced by both repulsive and attractive electrostatic interactions. The hard-sphere model may be a generally useful tool for the analysis of small-angle scattering data from concentrated macromolecular solutions.