Supporting information for: Norovirus capsid proteins self-assemble through. biphasic kinetics via long-lived stave-like.

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1 Supporting information for: Norovirus capsid proteins self-assemble through biphasic kinetics via long-lived stave-like intermediates Guillaume Tresset,, Clémence Le Cœur, Jean-François Bryche, Mouna Tatou, Mehdi Zeghal, Annie Charpilienne, Didier Poncet, Doru Constantin, and Stéphane Bressanelli, Laboratoire de Physique des Solides, Université Paris-Sud, CNRS, Orsay, France, Institut de Chimie et des Matériaux Paris-Est, Université Paris-Est, CNRS, Thiais, France, and Laboratoire de Virologie Moléculaire et Structurale, CNRS, INRA, Gif-sur-Yvette, France Phone: +33 (0) ; +33 (0) Fax: +33 (0) ; +33 (0) NB2-VP1 State in Dissociation Buffer The scattered intensity at q = 0 is given by I 0 I(q = 0) = K(v, b)cm where K(v, b) is a constant depending upon the specific volume v and the scattering length density b of the protein, To whom correspondence should be addressed Laboratoire de Physique des Solides Institut de Chimie et des Matériaux Paris-Est Laboratoire de Virologie Moléculaire et Structurale S1

2 c is the mass concentration and M the molecular weight. K is computed from K = (v b) 2 /N A with N A being Avogadro s number. The specific volume was determined by CRYSOL S1 for NV- VP1 and reads v = 0.76 cm 3 /g. b = r el (ρ protein ρ solvent ) where r el = nm is the Thomson scattering length, and ρ solvent = 334 nm 3 and ρ protein = 426 nm 3 (after CRYSOL for NV-VP1) are the electron densities of the water solvent and of the protein respectively. Note that the weak contrast b with respect to water is typical for proteins. With a concentration c = g/cm 3 measured by spectrophotometry at 280 nm (extinction coefficient ε percent = 8.1) and I 0 = cm 1 (Figure SS1) estimated from the GNOM reconstruction, S2 we arrive at M = kda. Figure S1: SAXS pattern of NB2-VP1 dimers in dissociation buffer. A reconstruction performed with the program GNOM (red line) is compared to experimental data (black symbols). The dimer concentration was 28 µm. The inset shows the pair distribution function p(r) computed by GNOM along with the calculated radius of gyration R g and the maximum dimension D max. S2

3 Singular Value Decomposition (SVD) of the Matrix of Intensities The M N (M > N) matrix of intensities I can be expressed as the product I = UΣV T (1) where U is a M N matrix and V is a N N matrix, the columns of each comprising orthonormal sets of vectors (i.e. U T U = V T V = I N the N N identity matrix), and Σ is a N N diagonal matrix with non-negative diagonal elements s i called the singular values of I. If I is a matrix of spatio-temporal intensities where the spectra are arranged in columns, then the columns of U are themselves spectra of the same nature, and called the normalized basis spectra of I. It should be noted that these columns have no direct significance, and a N-dimensional rotation is required to draw a meaningful interpretation. The singular values s i are sorted in a rapidly decreasing order so that the matrix of intensities can be approximated at varying levels by a product of truncated matrices Ũ, Σ and Ṽ. Indeed, if we retain the first r columns of U and V to build Ũ and Ṽ respectively, and we set Σ as a diagonal matrix whose diagonal elements are the r largest singular values s 1 s r, the matrix Ĩ = Ũ ΣṼ provides the best least-squares approximation of rank r of the matrix I. Given an estimate of the measurement uncertainties, it is then possible to use the singular values to determine the rank sufficient to describe the intensities to within the experimental uncertainties via the relation I Ĩ 2 = N s 2 i µνσ 2 (2) i=r+1 with σ 2 being the variance of data. A good choice for parameters µ and ν is the number of degrees of freedom for the representation which remain after the selection of r eigenvectors, i.e. µ = M r and ν = N r. S3 As mentionned earlier, the eigenvectors U and V do not hold any physical meaning and remain purely mathematical quantities. Even with a large associated eigenvalue, the signal-to-noise ratio of a given vector can be rather low. The content of relevant information is better quantified by the S3

4 autocorrelations defined by S3 C(U i ) = C(V i ) = M 1 j=1 U j,i U j+1,i N 1 (3) V j,i V j+1,i j=1 An autocorrelation value close to one indicates a high signal-to-noise ratio for the given vector (good information content), while a noisy vector (poor information content) will have rapid rowto-row variations and consequently an autocorrelation much less than one, possibly negative. Figure SS2 illustrates a SVD analysis performed on the matrix of intensities I. The noise variance was taken as the mean value of the variance over q. Clearly, a rank of 3 was sufficient to reliably describe the data both in terms of noise variance and autocorrelation of eigenvectors. Figure S2: SVD analysis of the matrix of intensities. The graph shows the residual of the truncated matrix of rank r with respect to the initial matrix as explicited by equation (2) (blue dots). The residual is compared to the experimental noise variance weighted by the number of remaining degrees of freedom (red line). The figures on the right side are the five largest singular values and the autocorrelations of the associated eigenvectors. S4

5 Nucleation-Elongation Kinetic Model A classical nucleation-elongation model was tested on our experimental data. The kinetic scheme is depicted on Figure SS3. It consisted of a nucleation step of order α assembling free dimers into a nucleus. Afterwards, each of these nuclei behaved as a site for a polymerization-like reaction in which remaining free dimers were captured to complete the capsid. Stoichiometric considerations impose the elongation step be made of 90 α successive additions of free dimers. Figure S3: Nucleation-elongation kinetic scheme. D and C denote dimers and capsids respectively, while k + and k are the forward and backward reaction rates. I α is an intermediate made up of α dimers. Global fitting was carried out with vaying values of α, and the form factors of dimers and capsids were fixed. Table S1 summarizes the results: for a wide range of α values, the figures of merit χ 2 and R were both above 5 and 20% respectively. The fits were therefore deemed not acceptable and this kinetic model was ruled out. Table S1: Results of global fitting for a nucleation-elongation kinetic model. α χ 2 R (%) ρ M ρ V S5

6 Figure S4: Schematic representation of the intermediate species in the hypothesis of an assembly around the fivefold axes. The number α of dimers constituting each intermediate is indicated above its structure. The dimers assembled around the fivefold axes are labelled in blue and the dimers connecting the pentamers of dimers are colored in red. S6

7 Figure S5: Comparison between the extracted form factor of intermediates and models constructed from the crystal structure of NV-VLPs (PDB reference 1IHM). Extracted data (symbols) were smoothed by a GNOM reconstruction (blue). The form factors of the models of 22-mer (red), 20-mer (green) and 10-mer (black) were calculated by the program CRYSOL. The corresponding structures are depicted on the right with the conventions used on Figure SS4. As we can see, the form factor of intermediates is close to that of the 22-mer model both in terms of radius of gyration (108 Å for NB2 and 96 Å for NV) and shape. The radii of gyration for the 20-mer and 10-mer models are 77 Å and 60 Å respectively. S7

8 References (S1) Svergun, D. I.; Barberato, C.; Koch, M. H. J. J. Appl. Crystallogr. 1995, 28, (S2) Svergun, D. I. J. Appl. Crystallogr. 1992, 25, (S3) Henry, E. R.; Hofrichter, J. Methods Enzymol. 1992, 210, S8