Effects of diffusion on the electrophoretic behavior of associating systems: The Gilbert-Jenkins theory revisited

The Gilbert-Jenkins theory predicts the asymptotic shape of moving-boundary sedimentation and electrophoretic patterns and broad zone molecular sieve chromatographic elution profiles for the class of interacting systems, A + B ⇄ C, in which two dissimilar macromolecules react reversibly to form a complex. A particularly provocative case is the one in which the complex has a greater migration velocity than that of either reactant, each of which has a different velocity. Depending upon conditions, this case predicts, for example, that in the asymptotic limit an ascending electrophoretic pattern or a frontal gel chromatographic elution profile can show two hypersharp reaction boundaries separated by a plateau. This prediction is now confirmed by numerical solution of transport equations which retain the second-order diffusional term and extrapolation of the computed patterns to zero diffusion coefficient. For finite diffusion coefficient, however, the two hypersharp reaction boundaries are separated by a weak negative gradient. These calculations are extended to an examination of the transitions between the three types of patterns admitted by the case under consideration in order to gain physical understanding and to define criteria for recognizing the transitions. Studies of this kind not only establish confidence in the Gilbert-Jenkins theory, but, in addition, they provide new insights which make for more effective application of the theory to real systems.     

Frontal gel chromatography of interacting systems: theoretical and experimental evaluation of the shapes of elution profiles for systems of the type A + B in equilibrium C

A theoretical and experimental study has been made of the advancing elution profile in frontal gel chromatography of interacting systems for which the elution volume of the complex is smaller than that of the larger reactant. First, Gilbert-Jenkins theory is used to delineate the form of the elution profile from the magnitudes of the elution volumes and concentrations of reacting species. This procedure resulted in the detection of a misinterpretation of certain patterns obtained in a gel chromatographic study of the interaction between myoglobin and ovalbumin. Second, a numerical computational procedure, which incorporates both axial dispersion and concentration-dependence of species elution volumes, is used to establish the influence of these two factors on boundary shapes for such systems. Third, frontal gel chromatography on Sephadex G-75 is used to compare experimental behavior with theoretical profiles predicted for the electrostatic interaction between cytochrome c and soybean trypsin inhibitor (pH 6.8, I 0.01). Results of these experiments serve as a guide for future conduct of experiments aimed at characterization of biologically important, reversible complex formation between proteins and/or other macromolecules.     

Sedimentation velocity analysis of heterogeneous protein-protein interactions: sedimentation coefficient distributions c(s) and asymptotic boundary profiles from Gilbert-Jenkins theory

Interacting proteins in rapid association equilibrium exhibit coupled migration under the influence of an external force. In sedimentation, two-component systems can exhibit bimodal boundaries, consisting of the undisturbed sedimentation of a fraction of the population of one component, and the coupled sedimentation of a mixture of both free and complex species in the reaction boundary. For the theoretical limit of diffusion-free sedimentation after infinite time, the shapes of the reaction boundaries and the sedimentation velocity gradients have been predicted by Gilbert and Jenkins. We compare these asymptotic gradients with sedimentation coefficient distributions, c(s), extracted from experimental sedimentation profiles by direct modeling with superpositions of Lamm equation solutions. The overall shapes are qualitatively consistent and the amplitudes and weight-average s-values of the different boundary components are quantitatively in good agreement. We propose that the concentration dependence of the area and weight-average s-value of the c(s) peaks can be modeled by isotherms based on Gilbert-Jenkins theory, providing a robust approach to exploit the bimodal structure of the reaction boundary for the analysis of experimental data. This can significantly improve the estimates for the determination of binding constants and hydrodynamic parameters of the complexes.     

Ion exchange chromatography of proteins-predictions of elution curves and operating conditions. II. Experimental verification

The applicability and validity of the model developed in Part I were confirmed experimentally. In this article, various proteins were eluted both by stepwise and linear gradient elution on DEAE ion exchangers under a variety of experimental conditions. Adsorption isotherms were measured as a function of ionic strength in batch experiments. The moment method was empolyed for the determination of various parameteres such as the gel-phase diffusion coefficient and the longitudinal dispersion coefficient. By use of these parameters and the experimentally measured ionic strength of the peak position, the number opf plates was determined according to the method described in Part I. Theoretical elution curves were calculated with the experimentally measured adsorption eqluilibria and the number of plates. Good agreement was observed between theory an experiments. Various factors affecting the separation were investigated. It was found that the effect of the number of plates for salts, N'(p), was negligible except the case of stepwise elution of high ionic strength buffer. When elution curves were symmetrical, the widths of the elution curves were inversely proportional to the square root of the number of plates of proteins, N(p), as in other chromatographic techniques. A simple graphical method for prediction of the peak position in linear gradient elution described in Part I was found applicable when the elution curves were symmetrical. A useful correlation of prediction of the peak width in a linear gradient elution was proposed on the basis of the approximate solution derived in Part I of this study. This graphical method and correlation permit easy prediction of the peak position and peak width in linear gradient elution in the case of symmetrical elution curves.     

Isohormones: molecular forms of the human chorionic somatomammotropic hormone.

The chorionic somatommaotropic hormone extracted from the human placenta exists in several molecular forms. Analytical electrophoresis in polyacrylamide gel permits separation of a highly anodic migration form, form 1, and another form migrating slightly faster than albumin, form 2. These two forms are active as measured by radioactive immunological analysis, form 2 being about 25 times more active than form 1. The two forms are mutually interconvertible. The two forms may also be separated by filtration on Sephadex G-50. However, they do not differ in molecular weight, they have the same coefficient of sedimentation and the same coefficient of apparent diffusion, measured by analytic ultracentrifugation. Glutaraldehyde and 8 M urea do not modify their electrophoretic or chromatographic behavior. On the hand, the two forms differ in their tertiary structure, with the modification depending on the greater or lesser degree of oxidation of the intra-chain disulfide groups. The two forms also exist in placental culture media and the incorporation of tritiated leucine occurs preferably in form 1. The physiological significance of the two hormone pools is not clarified.     

Diffusion of the Reaction Boundary of Rapidly Interacting Macromolecules in Sedimentation Velocity

Sedimentation velocity analytical ultracentrifugation combines relatively high hydrodynamic resolution of macromolecular species with the ability to study macromolecular interactions, which has great potential for studying dynamically assembled multiprotein complexes. Complicated sedimentation boundary shapes appear in multicomponent mixtures when the timescale of the chemical reaction is short relative to the timescale of sedimentation. Although the Lamm partial differential equation rigorously predicts the evolution of concentration profiles for given reaction schemes and parameter sets, this approach is often not directly applicable to data analysis due to experimental and sample imperfections, and/or due to unknown reaction pathways. Recently, we have introduced the effective particle theory, which explains quantitatively and in a simple physical picture the sedimentation boundary patterns arising in the sedimentation of rapidly interacting systems. However, it does not address the diffusional spread of the reaction boundary from the cosedimentation of interacting macromolecules, which also has been of long-standing interest in the theory of sedimentation velocity analytical ultracentrifugation. Here, effective particle theory is exploited to approximate the concentration gradients during the sedimentation process, and to predict the overall, gradient-average diffusion coefficient of the reaction boundary. The analysis of the heterogeneity of the sedimentation and diffusion coefficients across the reaction boundary shows that both are relatively uniform. These results support the application of diffusion-deconvoluting sedimentation coefficient distributions c(s) to the analysis of rapidly interacting systems, and provide a framework for the quantitative interpretation of the diffusional broadening and the apparent molar mass values of the effective sedimenting particle in dynamically associating systems.     

Pore network modelling of affinity chromatography: determination of the dynamic profiles of the pore diffusivity of beta-galactosidase and its effect on column performance as the loading of beta-galactosidase onto anti-beta-galactosidase varies with time.

A three-dimensional pore network model for diffusion in porous adsorbent particles was employed in a dynamic adsorption model that simulates the adsorption of a solute in porous particles packed in a chromatographic column. The solution of the combined model yielded the dynamic profiles of the pore diffusion coefficient of beta-galactosidase along the radius of porous adsorbent particles and along the length of the column as the loading of beta-galactosidase onto anti-beta-galactosidase immobilized on the surface of the pores of the particles occurred, and, the dynamic adsorptive capacity of the chromatographic column as a function of the design and operational parameters of the chromatographic system. It was found that for a given column length the dynamic profiles of the pore diffusion coefficient were influenced by (a) the superficial fluid velocity in the column, (b) the diameter of the adsorbent particles, and (c) the pore connectivity of the porous structure of the adsorbent particles. The effect of the magnitude of the pore connectivity on the dynamic profiles of the pore diffusion coefficient of beta-galactosidase increased as the diameter of the adsorbent particles and the superficial fluid velocity in the column increased. The dynamic adsorptive capacity of the column increased as (i) the particle diameter and the superficial fluid velocity in the column decreased, and (ii) the column length and the pore connectivity increased. In preparative affinity chromatography, it is desirable to obtain high throughputs within acceptable pressure gradients, and this may require the employment of larger diameter adsorbent particles. In such a case, longer column lengths satisfying acceptable pressure gradients with adsorbent particles having higher pore connectivity values could provide high dynamic adsorptive capacities. An alternative chromatographic system could be comprised of a long column packed with large particles which have fractal pores (fractal particles) that have high pore connectivities and which allow high intraparticle diffusional and convective flow mass transfer rates providing high throughputs and high dynamic adsorptive capacities. If large scale monoliths could be made to be reproducible and operationally stable, they could also offer an alternative mode of operation that could provide high throughputs and high dynamic adsorptive capacities.     

Effects of starvation and different culture conditions on the phospholipid content of isolated pancreatic islets

Three forms of the normal human plasma fibrinogen gamma-chain which differ in molecular weight have been purified. Plasma fibrinogen was separated by ion exchange chromatography on DEAE-Sephacel into three populations of molecules, each with a unique gamma-chain composition. Following reduction and S-carboxymethylation, the fibrinogen polypeptide chains in each chromatographic peak were separated by ion exchange chromatography on DEAE-Sephacel and identified following sodium dodecyl sulfate-polyacrylamide gel electrophoresis. The A alpha, B beta and smallest gamma-chain (gamma 50) eluted at progressively higher ionic strengths, but the elution positions of A alpha, B beta and gamma 50 chains were identical for fibrinogen from each of the three different chromatographic fractions. The unique gamma chain of fibrinogen in the second chromatographic peak (gamma 55) eluted at an ionic strength higher than that of the gamma 50 chain, while the largest gamma-chain (gamma 57.5), which was contained only in the third chromatographic peak of fibrinogen, eluted at the highest ionic strength. The higher ionic strengths needed to elute fibrinogen in the second and third peaks was paralleled by the higher ionic strengths needed to elute the gamma-chains unique to them, suggesting that the gamma-chain composition of the three fibrinogen fractions accounted for their differential binding to the ion exchange resin. Following desialation with neuraminidase, the differences in electrophoretic mobilities between the three gamma-chain forms was maintained, indicating that differential migration on SDS-polyacrylamide gel electrophoresis was not due to variation in sialic acid content.     

Fractionation of macromolecules in an alternating transverse electric field: simulation of the method

An electric field of alternating polarity applied in a direction transverse to the direction of solute transport is used as the basis of a method for the separation of biological macromolecules. The method derives directly from the ability of an electric field to induce movement of a charged macromolecule and from the physics of laminar fluid flow; no adsorptive immobile phase component is involved.

The method is simulated by computer for the case of solute molecules in a solvent flowing through a narrow chamber of recta generates an electric field orthogonal to the direction of solvent flow. Solute molecules repetitively traverse the solvent channel at rates determined by their electrophoretic mobility. During the transit across the channel, solute molecules are transported in the direction of solvent flow; at the channel wall, solvent velocity is negligible and solute transport is limited to that provided by transient diffusion into a mobile solvent zone. Molecules of different intrinsic electrophoretic mobility are separated.

The computer model was used to illustrate the process and to demonstrate the ‘tunability’ of the method as a function of the oscillation frequency and voltage wave form. Because of this tunability, a single instrument can function as the equivalent of several different chromatographic systems. Because fractionation is effected by direct physicochemical phenomena rather than via interaction with chromatographic sites, variations in fractionation results arising from formation of polymers for gel electrophoresis, packing of chromatography columns, or deterioration of columns with use are avoided. This method may be of particular use for the purification of nucleic acid fragments and for the analysis of protei: nucleic acid interactions.     

Ion exchange chromatography of proteins-prediction of elution curves and operating conditions. I. Theoretical considerations

A mathematical model is proposed for the elution of proteins on ion exchange columns by a linear gradient increase and stepwise increase of ionic strength in order to predict relationships between the elution characteristics (the peak position, the peak width, etc.) and the operating conditions (the flow rate, the slope of gradient, etc). This model is in principle based on the continuous-flow plate theory, in which the protein concentration and ionic strength dependent distibution coefficient between proteins and ion exchangers and zone sperading effects are taken into consideration. The advantage of this model is its simplicity since it requires only two parameters: The distribution coefficient and the number of plates. Since the distribution coefficient of proteins depends on both the protein concentration and ionic strength of the elution buffer, the number of plates should vary with time. However, it is extremely difficult to take into consideration the time-dependent number of plates. Therefore, we assume that the number of plates is constant and related to that number derived from a mass balance model which includes longitudinal dispersion and gel phase diffusion. On the basis of these assumptions, a method for determining the number of plates by the moment method is presented. Although the dependencies of the peak position and peak width on the slope of linear gradient are predictable by numerical calculations of the present model, simpler methods for prediction of these dependencies are desirable. A graphical method is proposed for prediction of the peak position. For prediction of the peak width, an asymptotic solution is derived from a quasi-steady-state model.     

Genetic variability due to geographical inhomogeneity

The frequency of one of two alleles is studied as a function of position and time in a one, two, or three dimensional region. A nonlinear diffusion equation is employed. Each allele is assumed to have a selective advantage in some part of the region. An asymptotic solution is constructed for the case when the selection coefficient is large compared to the diffusion coefficient, i.e. when selection acts more rapidly than diffusion. Then as time increases, the solution tends to a cline, i.e. an equilibrium distribution in which both alleles are present everywhere, each predominating where it has the advantage. In a narrow region around the boundary where the selective advantage switches from one allele to the other, both alleles are present with comparable frequencies. Along a line normal to this boundary, the frequency varies as in a one dimensional habitat with a simple variation in selective advantage. The asymptotic solution is compared with the numerical solution for a special two dimensional case, and the agreement is found to be good.Research supported by the National Science Foundation.     

An electrochemical model of the transport of charged molecules through the capillary glycocalyx

An electrochemical theory of the glycocalyx surface layer on capillary endothelial cells is developed as a model to study the electrochemical dynamics of anionic molecular transport within capillaries. Combining a constitutive relationship for electrochemical transport, derived from Fick's and Ohm's laws, with the conservation of mass and Gauss's law from electrostatics, a system of three nonlinear, coupled, second-order, partial, integro-differential equations is obtained for the concentrations of the diffusing anionic molecules and the cations and anions in the blood. With the exception of small departures from electroneutrality that arise locally near the apical region of the glycocalyx, the model assumes that cations in the blood counterbalance the fixed negative charges bound to the macromolecular matrix of the glycocalyx in equilibrium. In the presence of anionic molecular tracers injected into the capillary lumen, the model predicts the size- and charge-dependent electrophoretic mobility of ions and tracers within the layer. In particular, the model predicts that anionic molecules are excluded from the glycocalyx at equilibrium and that the extent of this exclusion, which increases with increasing tracer and/or glycocalyx electronegativity, is a fundamental determinant of anionic molecular transport through the layer. The model equations were integrated numerically using a Crank-Nicolson finite-difference scheme and Newton-Raphson iteration. When the concentration of the anionic molecular tracer is small compared with the concentration of ions in the blood, a linearized version of the model can be obtained and solved as an eigenvalue problem. The results of the linear and nonlinear models were found to be in good agreement for this physiologically important case. Furthermore, if the fixed-charge density of the glycocalyx is of the order of the concentration of ions in the blood, or larger, or if the magnitude of the anionic molecular valence is large, a closed-form asymptotic solution for the diffusion time can be obtained from the eigenvalue problem that compares favorably with the numerical solution. In either case, if leakage of anionic molecules out of the capillary occurs, diffusion time is seen to vary exponentially with anionic valence and in inverse proportion to the steady-state anionic tracer concentration in the layer relative to the lumen. These findings suggest several methods for obtaining an estimate of the glycocalyx fixed-charge density in vivo.     

Determination of the asymptotic shapes of sedimentation velocity patterns for reversibly polymerizing solutes

A simple method is described for the determination of the asymptotic boundary shape from a series of schlieren patterns taken during a sedimentation velocity experiment on a rapidly and reversibly associating solute. The form of the boundary so obtained reflects effects of sedimentation and chemical reaction but is free from effects of diffusion. The procedure is illustrated with analyses of experiments on diisopropyl fluorophosphate-inhibited α-chymotrypsin in 0.29 I phosphate, pH 7.9 (a monomer-dimer system), and on β-lactoglobulin A in 0.1 I acetate, pH 4.65 (a monomer-dimer-trimer-tetramer system). Asymptotic patterns so determined exhibit close agreement with those predicted by Gilbert theory.     

Characterization of rat bone marrow lymphoid cells. I. A study of the distribution parameters of sedimentation velocity, volume and electrophoretic mobility

Various cell populations in rat bone marrow were characterized by means of a two dimensional separation using velocity sedimentation and free flow electrophoresis and by electrical sizing of the separated cells. Up to 4.5 mm/hr five different populations with discrete distributions in volume (coefficient of variation 10% to 13%) and sedimentation velocity (coefficient of variation 6% to 10%) were observed. Three of the small sized populations represented lymphocytes and small normoblasts and two of the larger sized populations represented myeloid cells. Almost all of these cells were in the G0/G1 cycle phase. In the faster sedimenting fractions which contained immature myeloid, erythroid and undefined blast cells and two S phase populations, discrete volume distributions were not evaluated. The cell populations with homogeneous volume (particularly the small lymphocytes) showed high density variations which condiserably impair the separation resolution. The cells sedimenting slower than 3.5 mm/hr were further separated by means of free flow electrophoresis into three peaks differing in electrophoretic mobility (EPM). The peaks of low and high EPM contained two populations and the peak of medium EPM contained three populations all characterized by normal volume distributions of uniform coefficient of variation between 11% and 14%. The small cells in the peaks of high and medium EPM were normolblasts and the other cells were lymphocytes. The biological significance of these results is discussed.     

Size-dependent diffusion of membrane inclusions

Experimentally determined diffusion constants are often used to elucidate the size and oligomeric state of membrane proteins and domains. This approach critically relies on the knowledge of the size-dependence of diffusion. We have used mesoscopic simulations to thoroughly quantify the size-dependent diffusion properties of membrane inclusions. For small radii R, we find that the lateral diffusion coefficient D is well described by the Saffman-Delbrück relation, which predicts a logarithmic decrease of D with R. However, beyond a critical radius Rc approximately hetam/(2etac) (h, bilayer thickness; etam/c, viscosity of the membrane/surrounding solvent) we observe significant deviations and the emergence of an asymptotic scaling D approximately 1/R2. The latter originates from the asymptotic hydrodynamics and the inclusion's internal degrees of freedom that become particularly relevant on short timescales. In contrast to the lateral diffusion, the size dependence of the rotational diffusion constant Dr follows the predicted hydrodynamic scaling Dr approximately 1/R2 over the entire range of sizes studied here.     

A theoretical model of the pinocytotic vesicular transport process in endothelial cells

A new theoretical model for vesicular transport in single endothelial cells is described using a kinetic molecular approach in which the vesicle diffusion process is coupled with the vesicle attachment/detachment process occurring at the cell plasmalemmal boundaries. Rate constants k^d_i, k_i characterizing a two stage reaction sequence in the attachment/detachment region and the vesicle diffusion coefficient D are obtained by comparison of the theory with the results of tracer studies. For the condition of rapid vesicle loading/discharge of macromolecules it is found that the permeability of endothelial cells to macromolecules tends to be controlled by the vesicular attachment/detachment process rather than the vesicle diffusion process. The rate limiting step in the vesicle attachment/detachment process tends to be the reaction process involving the rate at which a vesicle and the plasmalemmal membrane are brought into/separated from intimate contact rather than that involving the rate of formation/dissolution of the membrane diaphragm of an attached vesicle. Estimated relaxation times for processes occurring in the attachment/detachment region and in the diffusion region, the vesicle transit time in the diffusion region, and the viscosity of the cytoplasm in the diffusion region are deduced. Fair agreement is obtained between the predicted and the observed temperature dependence of the permeability.     

Inert gas diffusion in heterogeneous tissue I: Without perfusion

A mathematical model is proposed to describe transient gas diffusion into a block of heterogenous tissue placed on an impermeable base. The corresponding asymptotic sultion of mass uptake of the gas is derived on the assumption that the diffusion constant is very much smaller in the cellular phase. It is expected that this will be useful in evaluating the diffusion constant in cellular material, and the volume fraction of extracellular fluid, providing the partition coefficient is known. The phenomenon of mutual interaction and multiple feedback between cellular and extracellular fluid is clearly seen in the overall response of the tissue. In this regard it is shown that the extraction of the two least dominant time constants, by backward projection of the experimental data curve of gas uptake, is likely to confuse the numerical evaluation of the physical parameters of the system. In an appendix, the problem of diffusion straight through a tissue slice is solved at the asymptotic stage, before steady state is reached. The resulting expression predicts the by-passing of cells by the diffusing gas and shows how the parameters cannot reliably be, determined.     

Improved and simplified liquid chromatography/electrospray ionization mass spectrometry method for the analysis of underivatized glucosamine in human plasma

Glucosamine is an amino monosaccharide reagent. It is difficult to assay using typical reversed-phase column due to the early elution, by optimizing the chromatographic conditions, especially the analytical column and the mobile phase composition, an improved analytical method was developed and validated, which offers rapid, sensitive and specific determination of glucosamine in human plasma. Following protein precipitation, the analyte and internal standard (valibose) were separated using an isocratic mobile phase on an Inertsil CN-3 column and detected by mass spectrometry in the multiple reaction monitoring mode using the respective precursor to product ion combinations of m/z 180/72 for glucosamine and m/z 252/198 for valibose. The chromatographic time was just 4.2 min for each sample, which made it possible to analyze more than 120 human plasma samples per day. The method exhibited a linear dynamic range of 4.00-4000 ng/mL for glucosamine in human plasma. The lower limit of quantification (LLOQ) was 4.00 ng/mL with a relative standard deviation of less than 10.9%. Acceptable precision and accuracy were obtained for the plasma concentrations over the standard curve range. By monitoring the two different MRM transitions, it was proved that no endogenous glucosamine was found in human plasma. The validated method has been successfully used to analyze human plasma samples for application in a bioequivalence study.     

Kinetic cooperativity in the concerted model for allosteric enzymes.

The cooperativity of enzyme-substrate interactions is investigated in the concerted allosteric model of Monod, Wyman and Changeux. The general case of K-V systems is considered, in which the two protomer conformational states R and T postulated in the theory differ in catalytic and binding properties. An expression for the Hill coefficient nH defined with respect to the asymptotic velocity V infinity to is analyzed in conditions which exclude substrate inhibition. Kinetic cooperativity is always positive (nH greater than 1) in the case of a dimer enzyme, and in the case of an inactive T state. Slight kinetic negative cooperativity (nH less than 1) occurs under restrictive conditions for larger numbers of protomers when the substrate binds significantly to the less active state of the enzyme, but the phenomenon remains negligible for trimers and tetramers. These conclusions differ from those obtained [A. Goldbeter, J. Mol.Biol.90 (1974) 185] with the Hill coefficient based on the absolute maximum velocity, which may exceed the experimental value V infinity to in K-V systems. The results extend those of Paulus and DeRiel [J. Mol. Biol. 97 (1975) 667] and support the view that in most cases, negative cooperativity is not compatible with a mechanism based on a concerted and conservative allosteric transition. The Hill coefficients for binding and catalysis are compared in K-V systems.