The Gilbert-Jenkins theory (GJT) has had a profound impact upon the understanding of migration experiments of interacting systems. For the sedimentation of two-component solutions, it predicts the existence of two boundaries: the undisturbed sedimentation of the free species of one of the components, and the reaction boundary exhibiting the coupled sedimentation of a mixture of complex and free forms of both components. The undisturbed boundary sediments with a single sedimentation coefficient, whereas the reaction boundary extends over a range of sedimentation coefficients that correspond to different ratios of free and complex species. The amplitude of each boundary, as well as the asymptotic shape of the reaction boundary can be calculated (3). The predicted features have been experimentally verified (5). They have been used as qualitative guides in the interpretation of experimental boundaries and for the quantitative analysis of transport experiments (6,7). In analytical ultracentrifugation, however, in comparison to the frequent use of weight-average sedimentation coefficients for the determination of association constants (8–10), apparently only relatively few applications have made quantitative use of the boundary structure, for example, the amplitudes and s-values predicted by GJT (see, e.g., Singer et al. and others (11–13)). This may be partially due to the numerical complexity of the calculations at the time, and/or the difficulty in precise determination of the boundary amplitudes and shapes in the presence of diffusion, as described, for example, in Palmer and Neet (14).

Subsequently, much interest has been devoted to the numerical solution of Lamm equations, the partial-differential equations of sedimentation, including chemical reaction kinetics as well as diffusion terms (6,15–22), and similar approaches have recently been implemented for the fitting of experimental sedimentation experiments (23–26). However, although this approach is very powerful, these models may not always be suitable in practice because they imply a detailed interpretation of the shape of the sedimentation boundary, which is exquisitely sensitive to any heterogeneity of the sample, such as impurities or microheterogeneity (26–29). Therefore, the more robust analysis of the concentration dependence of weight-average sedimentation coefficients continues to be of practical importance (for examples of applications, see Frigon and Timasheff and others (31–38); for methodological analyses, see Correia (39) and Schuck (24)). The weight-average s-value of the sedimenting mixture is based solely on mass balance considerations and is completely independent of the boundary shape. This study is focused on the question of how one can extract more robust information from sedimentation experiments beyond the overall mass balance, by exploiting the characteristic bimodal shapes of the boundaries for the quantitative analysis of heterogeneous protein interactions.

Recently, we have introduced a method for the computation of diffusion-deconvoluted sedimentation coefficient distributions c(s) from noisy experimental sedimentation data (40,41). It is based on the direct modeling of the sedimentation data with superpositions of Lamm equation solutions for noninteracting species, and is combined with maximum entropy regularization to result in the simplest distribution consistent with the experimental sedimentation data. The diffusion is approximated by means of a hydrodynamic scale relationship of sedimentation and diffusion and is based on a weight-average frictional ratio of the sedimenting macromolecules, extracted from the experimental data. This approximation takes advantage both of the weak shape dependence of the frictional ratio, and the lower size dependence of diffusion relative to sedimentation. Many applications have verified the high resolution and sensitivity of the resulting c(s) distributions (42). Although the interpretation of c(s) is straightforward for mixtures of noninteracting proteins, interacting systems show concentration-dependent peak positions and areas (29), as can be expected from GJT. For interacting systems, we have previously shown that the integration of c(s) over all peaks of the interacting system allows a rigorous determination of the weight-average s-value of the system, that is essentially independent of the kinetics of the interaction (24). In the accompanying article, we have described the shapes of c(s) distributions obtained at different reaction rate constants, and characterized the transition from c(s) resolving slowly interacting sedimenting species to c(s) representing the sedimentation/reaction boundaries of rapidly interacting systems. One goal of the present work is to provide a more general theoretical framework for the latter case of rapid interactions, that will allow a quantitative analysis of the c(s) peaks, utilize the deconvolution of diffusion, and exploit the structure of the underlying sedimentation boundaries of rapidly reacting systems.

A more quantitative comparison is possible by calculating the signal-average s-value of the reaction boundary, s_fast (Fig. 1 B, solid circles), as well as the amplitudes of the undisturbed boundary, c_slow (Fig. 1 C, solid circles), and the reaction boundary, c_fast (open circles), determined from integration of the slow and fast peaks of the c(s) profiles, respectively (Eq. 9). The concentration dependence of these values forms an isotherm that can be compared with the theoretical isotherm determined from the analogous integration of the dc/dv boundaries predicted by GJT (Eq. 4). In Fig. 1, B and C, the solid lines represent the GJT isotherms based on the parameter values underlying the simulations. The comparison with the c(s)-derived data (solid and open circles) shows excellent agreement. As predicted by Gilbert and Jenkins, the signal amplitudes of the fast and slow boundary components do not correspond to the populations of free and bound species calculated by the mass action law for the initial composition of the system before migration (short-dashed lines in Fig. 1 C), due to the coexistence of free A and B in the reaction boundary.

Although the curves shown in Fig. 1, B and C, do not involve fitting of any parameter, these isotherm models based on GJT were implemented in SEDPHAT for nonlinear regression of experimental data extracted from c(s) analysis for estimating the s-value of the complex and the binding constant. The concentration dependence of s_fast, c_slow, and c_fast can be analyzed globally and jointly with the weight-average s-values, s_w (shaded squares in Fig. 1 B). In this configuration of equimolar loading concentrations, the global analysis of all four data sets essentially halves the error estimates of the binding constant, as compared to the analysis of s_w alone

Clearly, the shapes of the sedimentation boundaries will depend on total loading concentrations of both A and B (c_A,tot and c_B,tot, respectively). Therefore, the value of s_fast(c_A,tot, c_B,tot) forms a two-dimensional isotherm and it is of interest to explore its shape. Fig. 2 shows the shape of the isotherm of s_fast(c_A,tot, c_B,tot) as predicted by GJT for the parameters used in Fig. 1, in comparison with the well-known isotherm s_w(c_A,tot, c_B,tot). For a data analysis, in practice, any combination of loading concentrations can be used that permits sampling the isotherm surface in several points sufficient to characterize its shape. Solely to systematically explore the properties of s_fast(c_A,tot, c_B,tot) in this work, we examine the relationship between GJT and c(s) for combinations of loading concentrations that follow three lines: the diagonal (equimolar dilution series corresponding to Fig. 1; see black lines in Fig. 2), a line of constant c_B,tot (titration with the smaller species; red lines in Fig. 2), and a line of constant c_A,tot (titration with the larger species; magenta lines in Fig. 2). These series of loading concentrations also highlight the properties of c(s) distributions for different variations of loading concentrations in different regimes. However, it should be noted that this does not imply constraints in the different experimental designs in practice, which may be chosen differently, for example, to accommodate practical limitations in the amounts of material. (Also, the implementation in SEDPHAT does not require following any particular experimental design.)

Next, we examined the c(s) and dc/dv traces for a titration of a constant loading concentration of the larger 10-S component (c_B,tot = K_D) with varying concentrations of the smaller 7-S component (with c_A,tot ranging from 0.1-fold K_D to 10-fold K_D), under otherwise identical conditions. Fig. 3 A shows the c(s) curves shifting in the peak position of the reaction boundary, starting at slightly above 10 S at the lowest concentration of A, and increasing monotonically with increasing loading concentrations of A. This again mimics the behavior of the asymptotic boundary shapes, and we find excellent quantitative agreement with regard to s_fast (Fig. 3 B) as well as the signal amplitudes c_slow and c_fast of the boundary components (Fig. 3 C). Here, the amplitude of the fast boundary does not change very much because the constant concentration of B is limiting the complex formation.

In contrast to the equimolar configuration shown in Fig. 1, the titration series appears to have a significant advantage for determining the s-value of the complex. At the highest concentrations of 10-fold K_D, the peak s-value is 12.7 S, significantly closer to that of the complex species at 13 S. The isotherm of s_fast shows a qualitatively different behavior than the isotherm of the weight-average s-value, s_w, of the whole system (Fig. 3 B): s_w has a shallow maximum and then decreases due to excess A, whereas s_fast increases throughout, approaching the complex s-value, because the excess of A sediments largely in the undisturbed boundary. This highlights the advantage of exploiting the hydrodynamic separation of boundary components, as opposed to restricting the analysis to the overall mass balance reflected in s_w. If the complete set of four isotherms shown in Fig. 3, B and C, is used in nonlinear regression to estimate the complex s-value and binding constants, an error analysis based on the covariance matrix shows a 4.8-fold reduction of the error for s_AB and 2.7-fold reduction of the error of log₁₀(K_D) as compared to the analysis of s_w alone.

The reverse titration of a constant loading concentration of the smaller 7-S component (c_A,tot = K_D) with an increasing concentration of the larger 10-S component is shown in Fig. 4. Qualitatively different effects are observed, in that the reaction boundary, even at the lowest concentration of B, is well in between the s-value of free B and that of the complex. This is due to the fact that a substantial fraction of total B is already present in the complex form. Although the asymptotic boundaries are sharp at low concentrations of B, the c(s) distribution is broad due to the limited signal/noise ratio at the low concentrations. Adding more B shifts the s-value of the reaction boundary toward free B, which is a result of the limited (constant) concentration of A available for complex formation, and correspondingly the fraction of free A decreases. Importantly, at a certain concentration where B is sufficiently in excess over A, all of A will participate in the reaction boundary, and the excess of species B will constitute the undisturbed boundary (gray and green circles in Fig. 4 A). The transition point under the conditions of Fig. 4 is at approximately threefold K_D. The transition point was found to depend on the s-values of A and B, and also on the stoichiometry of the interaction. For concentrations exactly at the transition point, the undisturbed boundary vanishes and the two-component mixture sediments in a single boundary. Slightly above the transition point, the asymptotic boundary is very broad, which is well reflected by c(s) (green curve). At a much higher excess of B, a sharp peak in both dc/dv and c(s) is observed closer to the complex s-value. It is important to note that, initially at low concentration of B, the position of the c(s) peaks as well as those of dc/dv do not change very much with increasing concentrations of B until the transition point is passed.

The isotherms of s_fast, c_slow, and c_fast determined by integration of c(s) are in reasonable quantitative agreement with the predictions of dc/dv, reflecting the transition of the undisturbed boundary. A small deviation of the theoretical and c(s)-derived value of s_fast can be discerned in Fig. 4 B, which is due to the difficulty in distinguishing the undisturbed boundary from the reaction boundary close to and slightly above the transition point (green circle and bar in Fig. 4 A). In the analysis of experimental data, this data point close to the transition point should be excluded from the isotherms (it can be included in the isotherm of s_w, for which the ambiguity of the boundary interpretation is irrelevant (24)). The global analysis of the remaining isotherms provides three- to fourfold reduced error estimates when compared to the analysis of s_w alone.

The asymptotic boundaries obtained with GJT are derived under the assumption of a rectangular cell geometry and at infinite time. To test how well this limiting case does describe sedimentation boundaries in the absence of diffusion in sector-shaped cells, we have simulated the sedimentation of particles with very small diffusion coefficients. Because the finite element Lamm equation solution is numerically unstable at a value of D = 0, simulations were performed in a series of sequentially 10-fold lower diffusion coefficients (Fig. 5). Except for some minor oscillations at values of D < 10⁻¹⁵ m²/s, the boundary profiles (transformed to apparent sedimentation coefficient distributions g*(s) with the ls-g*(s) method) approached a limiting value. This was examined for the case of equimolar loading concentration corresponding to the red trace in Fig. 1 A, and the broader distribution shown in the green trace of Fig. 4 A at conditions close to the transition point. In both cases, when the ls-g*(s) distributions are compared to the dc/dv traces of GJT, only minor differences are visible, primarily the lack of the sharp peak appearing at the maximum of dc/dv. Closer inspection revealed a slight overestimation of the s-values in GJT, which is reflected in an s_fast value exceeding that of the limiting ls-g*(s) trace by ~0.26%. If this is attributed to the lack of radial dilution in the rectangular geometry of GJT, corrections can be applied using the approximation for the effective time-average radial dilution during the sedimentation (Eq. 8 in (24)). The basis for calculating the radial dilution was taken as the s-value of the undisturbed boundary for component A and the s-value of the reaction boundary for component B. This reduced the deviation between s_fast values between GJT and the limiting s-values from solving the Lamm partial-differential equations (PDE) to 0.05%.

So far, the comparison of c(s) and dc/dv has been made for relatively large proteins, where, under most conditions, the undisturbed and the reaction boundaries are clearly visible in the raw data. A very stringent test for the performance of c(s) for the deconvolution of diffusion from the reaction boundaries is its application to small molecules, where the appearance of the experimental concentration profiles allows the visual discernment of only a single, diffusionally broadened boundary. This case was examined in a simulation equivalent to that shown above (in the equimolar case), but with a 2.5 S species A and a 3.5 S species B forming a 5 S complex with 1:1 stoichiometry. The dashed lines in Fig. 6 A show the g*(s) profiles, calculated as ls-g*(s) (50), which verifies that the raw sedimentation profiles appear as only a single broad boundary. In contrast, the c(s) curves resolve the undisturbed and the reaction boundaries, except for the lowest concentration where the signal/noise ratio is the limiting factor. The agreement between the peak positions of c(s) and dc/dv is good. However, an indication of too strong deconvolution is observed at the highest concentration in the form of a small secondary peak of the reaction boundary (black line at ~4 S). Nevertheless, the isotherms of s_fast, c_slow, and c_fast are in excellent agreement (Fig. 6, B and C). (The data points for the lowest concentration were omitted due to the lack of resolution.) In this case, only a ~1.5-fold improvement of the global isotherm analysis was found when compared to the analysis of s_w alone.

An alternative sedimentation velocity analysis approach is the extrapolation method by van Holde and Weischet to determine integral sedimentation coefficient distributions G(s) (51), which, to some extent, can unravel the effects of diffusion from sedimentation. The inset in Fig. 6 A shows G(s) distributions calculated on the basis of the least-squares algorithm described in Schuck et al. (41). In the absence of smoothing of the data, which may introduce bias in the subsequent analysis, the extrapolation of the high boundary fractions has the property of being very sensitive to noise and results in too small s-values. This is a limitation inherent in the extrapolation method requiring to locate the boundary fraction. This is particularly difficult at low signal/noise ratio and for regions of the sedimentation profiles with small gradient. Clearly, the integral sedimentation coefficient distributions would allow the correct diagnosis of the presence of the interaction, but would not permit a quantitative analysis. As described previously for the study of noninteracting mixtures, the van Holde-Weischet method cannot deconvolute diffusion from species that do not exhibit clearly separating boundaries (i.e., when the rms displacement from diffusion is smaller than the distance between boundary midpoints), and instead produces sloping G(s) profiles with intermediate s-values (41). This is due to the property of the inverse error function, on which the extrapolation is theoretically based, not being linear in its parameters. This problem is not addressed by the additional layer of extrapolation recently proposed (52). In theory, the average position of G(s) also does not lend itself to the analysis of weight-average sedimentation coefficients, as no rigorous relationship to mass conservation considerations are known (24). Similarly, in this case, no undisturbed boundary can be discerned, except for the presence of slower-sedimenting boundary contributions indicated by G(s) sloping to lower s-values. Further, even the maximum s-values of the G(s) distributions do not approach those expected for the fast boundary component (open triangles in Fig. 6 B).

The last aspect studied on the analysis of a 1:1 reaction of the type A + B ↔ AB was the performance of the multicomponent c_k(s) analysis from global multisignal analysis of the sedimentation profiles. For the conditions of Fig. 1, assuming equimolar concentrations, we simulated sedimentation profiles at two signals with twofold different extinction coefficients for A and B at both signals. As shown previously, the multicomponent c_k(s) analysis permits the determination of the separate sedimentation coefficient distributions of components A and B, and the determination of the molar ratio of the complexes formed (48). The component c_k(s) distributions are in excellent agreement with the composition of the asymptotic boundary, calculated as component boundaries dm_A,tot/dv and dm_B,tot/dv predicted via GJT (Fig. 7). If the ratio of the concentrations of A and B in the reaction boundary is calculated for this instantaneous reaction, equimolar concentrations 10-fold higher than K_D are required to achieve an 85% average saturation of the complex in the reaction boundary (inset in Fig. 7 B). However, if the concentration of B is lower (for example, c_B,tot = K_D), a concentration of A at c_A,tot = 10 K_D leads to a significantly higher saturation of the reaction boundary (95% for c_B,tot = K_D).

The motivation for examining the application of the c(s) analysis to the analysis of reaction boundaries was the observation that the reaction boundary sediments approximately with a single s- and D-value, which was predicted from the “constant bath” approximation. Interestingly, it has been shown that this also holds for reactions with stoichiometry >1:1 (43,44,53). Therefore, we also examined the application of c(s) to the analysis of sedimentation profiles from a two-site reaction A + 2B ↔ AB + B ↔ ABB. As before, we first generated sedimentation profiles by solving the Lamm equation with explicit reaction terms for a two-site reaction. The sedimentation profiles were calculated for an instantaneous reaction between molecule A (100 kDa, 6 S) with two identical noncooperative sites for a smaller ligand molecule B (50 kDa, 4 S), resulting in 8-S and 10-S complexes. The comparison of the c(s) analyses at different concentrations with the asymptotic boundaries for this reaction is shown in Fig. 8. Similar to the case of the reaction with 1:1 stoichiometry, the peaks of c(s) provide a good approximation of the undisturbed boundary and the reaction boundary predicted from GJT. Interestingly, here the broader sedimentation boundaries at concentrations >K_D,1 (the macroscopic binding constant of the first site) result in a double peak structure of c(s) (Fig. 8 A, black and green solid line). This appears to be caused by an overcompensation of diffusion. However, if the frictional ratio in the c(s) modeling is fixed to 1.3, broader structures consistent with dc/dv appear (short-dashed lines in Fig. 8 A). Independent of the frictional ratio value, the integral over the c(s) peaks of the reaction boundary (taken over the double peak structure) does reflect the correct weight-average s-value and amplitude of the reaction boundary, as shown in Fig. 8, B and C. (It should be noted that the double peak structure in Fig. 8 A can be easily distinguished from the peaks of a slow reaction A + B ↔ AB, which would also be expected to produce a total of three peaks, but with all peaks at constant position.) In comparison with the analysis of the overall weight-average s-value alone, the error analysis for the global isotherm model indicates an improvement in the statistical precision of the s-value of the 1:1 and 2:1 complexes by a factor of 100 and 5, respectively, and an improvement in the value of the binding constant by a factor of 50.

A more stringent test for the GJT-based analysis of c(s) curves is a 2:1 reaction of smaller molecules, where the deconvolution of diffusion will be significantly more important. To examine a configuration with even broader reaction boundaries, we have simulated a 2:1 system where the bivalent species A is smaller than the ligand B. Further, we assume a titration series of constant concentration of A with an increasing concentration of B, which exhibits a switch in the species of the undisturbed boundary and the corresponding broadening of the reaction boundary (compare Fig. 4). Such sedimentation profiles were simulated with Lamm equation solutions for a receptor A (31 kDa, 2.66 S) with two indistinguishable and noncooperative sites for binding of a larger ligand B (45 kDa, 3.56 S), forming 4.96 S and 6.11 S complexes in instantaneous local equilibrium. The macroscopic K_D for site I (K_D,1) underlying the simulations is 1.7 μM. As shown in Fig. 9, dc/dv exhibits extremely broad reaction boundaries close to the transition point (black and blue bars in Fig. 9 A), similar to the situation encountered in Fig. 4 (green bar). Also, as in Fig. 4, at concentrations lower than the transition point the peak positions of neither c(s) nor dc/dv change very much with increasing concentration of B.

Overall, good agreement of c(s) with dc/dv is observed. Without additional knowledge, the width of the boundaries close to the transition point (black bar extending from 3.5 to 5 S) may result in a misinterpretation of the c(s) peak at 3.8 S (black line) as representing the new undisturbed boundary, shifted slightly to higher s-values. However, it may be possible to recognize, either from the superposition of c(s) traces at different concentrations, or by comparison with the expected s-value of the free ligand, that the 3.8- and 5-S peaks jointly reflect the reaction boundary. In any case, the trace in question was excluded from the GJT isotherm analysis shown in panels B and C. (It should be noted that in the alternative titration series in the design of Figs. 1 and 3, there is no transition, and therefore no ambiguity about this point.) The global fit resulted in parameter estimates for the s-values of the 1:1 and 2:1 complexes of 5.23 and 6.20 S, and a binding constant of 2.4 μM for K_D,1 (short-dashed red lines). As can be expected, the s-value of the 1:1 intermediate is the most difficult to determine from the sedimentation data. Nevertheless, using the weight-average s-values and the GJT isotherms jointly reduced the error estimates by factors of 10 (s_AB), 50 (s_ABB), and 20 (K_D), respectively, as compared to the analysis of s_w alone.

Because, in practice, it may not be known if the reaction rate of a given system of interacting proteins can be considered infinitely fast on the timescale of sedimentation, we have studied the effect of finite reaction rate constants. Fig. 10 shows the isotherms of a simulated sedimentation system equivalent to Fig. 9, except for having a finite rate of chemical reaction with an off-rate constant of 2.5 × 10⁻³/s. To make the analysis of the GJT isotherms realistic, the s-value of the fast boundary component at the transition point is not included. The effect of the finite reaction rate can be observed, for example, in the higher s-value of the reaction boundary (Fig. 10 A, circles compared to black solid line), and an underestimation of the signal amplitude of the reaction boundary (Fig. 10 B, open circles) combined with an overestimation of the signal of the undisturbed boundary (Fig. 10 B, solid circles). This can be understood considering that the finite reaction will lead to a longer persistence of the complex during sedimentation and lower concentrations of free species cosedimenting in the reaction boundary. If these deviations are ignored (because the reaction kinetics may not be known) the parameters derived from a fit of the GJT isotherms are K_D,1 = 2.3 μM, s_AB = 4.84 S, and s_ABB = 6.14 S, as compared to the values underlying the simulations of K_D,1 = 1.7 μM, s_AB = 4.53 S, and s_ABB = 6.08 S, respectively. The largest error appears in the s-value of the reaction intermediate, the 1:1 complex. Despite the deviations of the parameter estimates, the GJT isotherms still permit unequivocal assignment of the reaction scheme. The best fit with an impostor 1:1 interaction model is depicted in Fig. 10 as blue lines. It leads to sevenfold higher rms errors for the fit. Qualitatively similar results were observed with slower reactions with k_off < 10⁻³/s (data not shown).

Finally, we applied the GJT-based analysis of the c(s) traces to the experimental data of a natural killer cell receptor Ly49C (31 kDa) interacting with a MHC molecule H-2Kb (45 kDa) sedimenting at 50,000 rpm (35). This interaction was previously shown to have a 2:1 stoichiometry with equivalent sites (35), and the experimental parameters and best-fit sedimentation parameters from Lamm equation modeling (26) are equivalent to those simulated in Fig. 10. The additional aspects of analyzing real experimental data were the appearance of c(s) peaks of small and very large species, most likely traces of impurities or degradation products. Further, occasionally multiple small peak structures appeared in the s-range of the free species, in addition to the main peak of the undisturbed boundary. As a practical approach to arrive at c_slow, we integrated the c(s) regions of the free species, as determined in the experiments with each component alone. Data sets where the reaction boundary could not be distinguished well from the undisturbed boundary were excluded from the GJT analysis, but included in the isotherm of s_w. Similarly, for data sets where the undisturbed boundary could not be satisfactorily assigned, the corresponding data point was excluded from the GJT isotherm analysis, but included in the s_w isotherm.

A popular and generally much more robust approach is the interpretation of the weight-average sedimentation coefficient as a function of loading concentration (8,31–38). The analysis of this isotherm does not depend on the reaction kinetics (except for a small influence caused by radial dilution), and permits a rigorous thermodynamic analysis of the interaction without requiring detailed interpretation of the shape of the sedimentation profiles, as long as a good empirical description of the sedimentation boundary can be found for the implicit mass conservation considerations (24). Although the isotherm analysis is well conditioned only if a large concentration range can be covered or if the sedimentation coefficients of the complexes are known, it is computationally considerably simpler and may serve to discriminate between alternative models for different reaction schemes. A practical advantage is that the weight-average s-values can be obtained by integration of the differential sedimentation coefficient distributions, which, in particular in conjunction with the c(s) analysis, may allow excluding from the analysis any sedimenting species not participating in the interaction of interest (24). So far, an open question has been how more information can be extracted from the structure of the sedimentation coefficient distributions, and in particular, how the details of the peak structure of the c(s) distribution can be interpreted for the interacting systems, other than representing the overall weight-average s-value. We have addressed this problem in this article and have proposed a new analytical approach for interacting systems that does not require the full Lamm equation modeling but nevertheless takes advantage of the structure of the sedimentation boundary.

Beyond the rigorous detailed description of the sedimentation with the Lamm PDE and the rigorous mass balance analysis in the form of analyzing the weight-average s-values, only approximate descriptions of the reaction-diffusion-sedimentation process are known. For reactions that can be considered essentially instantaneous on the timescale of sedimentation, Gilbert and Jenkins have described the characteristic shapes of asymptotic sedimentation boundaries obtained at infinite time (i.e., in the absence of diffusion), in a rectangular cell geometry, and with constant force (1,3). For two-component mixtures, the GJT predicts the presence of an undisturbed boundary component sedimenting with the s-values of one free species, and the velocity profile of the reaction boundary. This work has been very influential in the development and understanding of transport experiments of reacting systems. Although the asymptotic boundaries can, in principle, be experimentally determined by differential sedimentation approaches (45), this experimental technique has been dormant, as well as the quantitative use of GJT for data analysis.

In preparation for the quantitative use of GJT, in this work we have examined the influence of the assumption of rectangular geometry on the asymptotic boundary profiles, with the use of Lamm equation solutions of reacting particles in the limiting case of negligible diffusion. Our results show that the cell geometry has only small effects on the shape of dc/dv and on the weight-average s-value of the reaction boundary. A small overestimation of the s-value in GJT can be attributed to the effect of radial dilution, as discussed for self-associating systems in (24). Although this deviation is small and perhaps negligible in most cases, corrections can be applied by considering the time-averaged radial dilution introduced previously (24). As expected, the limiting case of infinite time in GJT corresponded to the limiting case of negligible diffusion at finite time.

A theory for a different special case of the reaction/diffusion/sedimentation process was described initially by Krauss and co-workers (43) and reintroduced by Urbanke and colleagues (20,44). It treats the sedimentation process of a reactive system in the presence of diffusion, at finite times, and in sector-shaped solution columns under the influence of a radial-dependent force, but assuming the concentration gradient of one free species A to vanish in the reaction boundary, such that a larger species B and the complexes AB sediment in a “constant bath” of the smaller species. Although this situation cannot be physically realized strictly, it is conceived as a special case for the interaction of small ligands with larger proteins. In the accompanying article, we have shown that it provides a useful and robust description also for proteins differing in size by only twofold (26). An important result of the “constant bath” approximation is that the reaction boundary (B+C) can be described by a single sedimentation s* and diffusion coefficient D*, a prediction that was shown to yield an excellent description of the sedimentation process (26). In this framework, the deconvolution of diffusion achieved in the c(s) analysis of reacting systems can be understood. Further, it can be argued that the finite gradients of the first free species A (which are originally neglected in the “constant bath” approach) will lead to a variation of the fractional occupation of the second species (C/(B+C)), which results in a dispersion of the sedimentation velocity of the reacting mixture that is represented in the shape and finite width of the c(s) profiles. To the extent that the effective diffusion D* of the reactive system does not vary greatly within the range of composition of the reaction boundary (compare Fig. 3 in (26)), and as long as the dispersion of the sedimentation velocity is not too large, the error in the deconvolution of diffusion based on a hydrodynamic scaling law of c(s) (D* ~ s*^−1/2), as opposed to a composition-dependent value of D*, should remain small. To study if this approximation captures the essential qualitative and quantitative features of the sedimentation process, we solved the Lamm PDE for different configurations, and systematically compared the results of the c(s) analysis with the asymptotic boundaries of the GJT, thus testing the hypothesis that c(s) can be regarded as an approximation of the diffusion-free sedimentation coefficient distributions of the reacting system.

Despite the different theoretical framework and the different approximations made in the GJT (constant force, linear geometry, infinite time) and this c(s) interpretation (diffusion of the reaction boundary proceeds with D* ~ s*^−1/2), very good agreement was observed overall, after making allowance for the fact that a finite signal/noise ratio does not permit sharp features in the sedimentation coefficient distributions to be resolved. This was true for both the conventional c(s) profiles as well as for the multicomponent c_k(s) distributions from global multisignal analysis (48). Deviations were found mainly for data with low signal/noise ratios, where the maximum entropy regularization employed in the c(s) approach is known to cause broadening and merging of adjacent peaks, as well as for configurations generating very broad reaction boundaries, such as those where the undisturbed boundary vanishes. These situations can be easily identified when considering a series of experiments at different loading concentrations. Some overcompensation of diffusion was observed for the two-site binding model, which could be identified by a drop in the apparent weight-average frictional coefficient obtained in the c(s) model, and could be eliminated by constraining this parameter. Further, some differences were observed in the two-site interaction of small molecules. It is unclear if the latter are a result of incorrect deconvolution of the diffusion, or simply a result of the boundaries not being sufficiently developed under these conditions to resemble the limit of infinite time assumed in the GJT. In any case, these deviations did not affect the excellent consistency of the amplitude of the undisturbed and both the amplitude and s-value of the reaction boundary from integration of c(s) with the predictions of GJT.

Therefore, we propose to use the isotherms of the amplitudes of both the undisturbed and the reaction boundary, as well as the average s-value of the reaction boundary as an analytical tool. This makes quantitative use of the bimodal shape of the reaction boundary of interacting two-component mixtures, thereby significantly improving the determination of the equilibrium constant of the interaction as well as the sedimentation coefficient (and therefore the hydrodynamic shape) of the complex. These isotherms are more generally applicable than the isotherms of the effective s-value of the “constant bath” theory, which we have examined previously (26), and additionally permit the interpretation of the boundary amplitudes (peak areas in c(s)). The new isotherms based on GJT modeled to the integrals of c(s) regions were found to be not very sensitive to deviations in the underlying assumption of an infinitely fast reaction. In fact, even a reaction that is very slow on the timescale of sedimentation appeared to provide satisfactory results (data not shown). Similarly, the isotherm analysis is not very sensitive to imperfect knowledge of the loading concentrations as would be encountered in the analysis of experimental data. Like the analysis of the overall weight-average s-values, it has the advantage that trace contaminations of material not participating in the reaction can be easily excluded from consideration. In the implementation in SEDPHAT, the global analysis of the isotherms of the overall weight-average sedimentation coefficients and that of the amplitudes and s-values of the reaction boundaries can be performed jointly, thus maximizing the information extracted from the experiment. Data points from experiments where the undisturbed and the reaction boundary components cannot be easily discerned can be excluded from the GJT-based isotherms.

We have systematically explored different regimes of loading concentrations that lead to qualitatively different behavior. Interestingly, at concentrations below K_D, the titration of a smaller species at constant concentration with increasing concentration of the larger species can lead to c(s) and dc/dv traces with very little change in peak position, potentially mimicking the concentration-independent peak positions of interacting mixtures with slow kinetics. Although the difference should be apparent from the lack of a third peak for the free larger species, and from the different s-value of the reaction boundary as compared to the s-value of the individual species, this highlights the importance of conducting experiments over a sufficient range of concentrations, including concentrations both significantly higher and lower than K_D. Another interesting feature of this titration design is the appearance of a transition point of the undisturbed boundary and a characteristic shape of the s_fast isotherm dependent on the reaction scheme.

Loading configurations with the concentration of the smaller component greater than K_D and in significant excess of the larger component, enable the closest approximation of the complex s-value by the reaction boundary, as well as the best approximation of the molar ratio for multicomponent c_k(s) analysis (55). Under these conditions, the s-value of the reaction boundary allows a significantly better representation of the s-value of the complex than is obtained from the weight-average s-value.

Regarding the analysis of the concentration dependence of the boundary amplitudes, it is well known that the undisturbed boundary does not accurately reflect the free concentration of the smaller species in the loading mixture. Although the contribution may be small, some free species of both components are always comigrating in the reaction boundary, and it is significant in this regard that the transition point of the GJT isotherms of Figs. 4 and 7 does not occur at equimolar loading concentration of the components. These effects are considered in the “population isotherms” model of the SEDPHAT software based on GJT, assuming a fast reaction on the timescale of sedimentation.

Beyond the original development and direct application of GJT (11,12,14), other approaches have been proposed to take advantage of boundary structure beyond the determination of the weight-average s_w. Correia suggested the use of higher averages, for example, z-averages and z + 1 averages as determined from integration of the apparent sedimentation coefficient distributions g(s*) derived from the dc/dt transformation (39). These averages were used previously to deduce qualitatively the presence of self-association (56), but the theoretical concept for their quantitative analysis was developed on the basis of s_z as a derivative s_z = d(cs_w)/dc of the isotherm s_w(c), with the values of s_w(c) being the experimentally determined quantities (57,58), and thus not reflecting on the boundary shapes. However, with regard to the moments of g(s*), it is unclear how the effects of diffusion could be taken into account, short of calculating full Lamm PDE solutions to derive s_z. The GJT-based isotherm analysis introduced in this work may also provide an alternative approach for the robust analysis of g*(s) distributions. Although the deconvolution of diffusion in c(s) is a key for the theoretical link between GJT and the analysis of the boundary structure via the experimentally determined sedimentation coefficient distributions, in many cases the boundary components may be discerned even from g*(s) traces (from either the least-squares method ls-g*(s) (50) or using the dc/dt transformation (59)), and the corresponding s_fast, c_fast, and c_slow values may be determined by integration. It could be envisioned that they be modeled with GJT-based isotherm presented here. On the other hand, it is not apparent to us that this would provide any advantage over the c(s)-based approach, as this would sacrifice resolution, could utilize only a fraction of the raw sedimentation data, and potentially introduce bias (60).

For self-associating systems, a different approach was suggested by Winzor and colleagues (13,61). It consists of the approximate elimination of diffusion contributions to the sedimentation boundary by extrapolation of boundary divisions to infinite time on an inverse timescale, similar to the van Holde-Weischet extrapolation (51). The resulting sedimentation coefficient distribution was interpreted in the framework of asymptotic boundaries of the GJT (13). This approach clearly displayed the presence of the self-association and the dispersion of the sedimentation coefficients in the reaction boundary. A limitation of this approximation is that unraveling the diffusion through extrapolation to infinite time is successful only for single species or where multiple species show clearly separating sedimentation boundaries (41). It is not clear that this approach is sufficiently well defined for reaction boundaries to permit quantitative modeling. Krauss and co-workers have taken diffusion into account more rigorously for heterogeneous interactions of a macromolecule with a small ligand (43). They have introduced the “constant bath” approximation of the Lamm equation, which allows making full use of the average s-value of the reaction boundaries (26,43,44). As described in the accompanying article (26), the isotherm of s_fast(c) from the “constant bath” approximation is virtually identical to that of GJT, and can be extended to take advantage of diffusional deconvolution of c(s), but it is restricted to the experimental configurations for which the “constant bath” approximation holds. Thus, to fully exploit the structure of the reaction boundary as an analytical tool in addition to the weight-average s-value, we suggest that the combination of diffusional deconvolution of c(s) in combination with the general and rigorous theoretical framework of GJT could be advantageous.