Typically, the resolution in the frictional ratio dimension cannot be expected to be very high, even in combination with data from SV experiments at a range of rotor speeds (Schuck 2002) . Therefore, a discretization providing F = 10 values between 1.0 and 2.5 (ranging from extremely compact to very extended protein structures) should be a sufficiently flexible basis to describe the actual frictional ratio for each species (knowing that we have inserted folded proteins into the sample solution, and keeping in mind the average frictional ratio of 1.68 estimated from the c(s) analysis). The resulting 10 9 100 grid with a total of K = 1,000 species was fitted with the standard algorithm to the data in Fig. 1a , leading to virtually random distribution of residuals (1b), with a root-mean-square deviation (rmsd) of 0.00672, consistent with the noise in the data acquisition. The resulting distribution is shown with and without regularization in Fig. 1d and c, respectively. As expected, while the s-values of the species are well defined, the shape dimension is highly underdetermined, resulting in the Fig. 1 Illustration of the standard algorithm for size-and-shape distributions applied to the experimental data of an immunoglobulin G sample, sedimenting at 45,000 rpm. a Experimental data acquired with the absorbance optical system (solid lines). The color temperature indicates the temporal order of the scans, with blue for the early and red for the late scans. The dotted lines, virtually overlapping the experimental data, are the best-fit distribution from modeling with Eq. (3) for a grid of K = 10 9 100 (f r , s)-values ranging from f r -values of 1.0 to 2.5 in 10 equidistant steps, and from s-values of 0.1 to 15.0 S in 100 equidistant steps. b Residuals of the fit, presented as a bitmap (Dam and Schuck 2004) and as an overlay plot for all traces. The root-mean-square deviation is 0.00672 OD. c Raw size-and-shape distribution without regularization. As in (Brown and Schuck 2006) , the 2D grid of (f r , s)-values is indicated by solid lines, combined with a color temperature contour map in the plane below. The solution is a series of spikes in f r -dimension, with a comparatively well-defined s-value of *5.8 S for the main species. An observation familiar in the study of IgG (and many other protein) samples is the low-level population of dimeric species at *9 S, as well as trimeric traces at *11-12 S. d Tikhonov-Phillips regularization applied to produce the most parsimonious size-and-shape distribution of all that fit the data statistically indistinguishably well at a P = 0.95 confidence level (i.e., that produce a rmsd value of 0.00677 OD or better) c typical series of peaks in 1c, and in a smooth relatively uniform distribution after regularization in 1d. (This justifies, in retrospect, the choice of F = 10 values as a sufficiently detailed discretization of the frictional ratio dimension.)