Supporting Information for: Complexation of β-lactoglobulin Fibrils and Sulfated Polysaccharides

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1 Supporting Information for: Complexation of β-lactoglobulin Fibrils and Sulfated Polysaccharides Owen G Jones 1, Stephaandschin 1, Jozef Adamcik 1, Ludger Harnau 2, Sreenath Bolisetty 1, and Raffaele Mezzenga 1 1 ETH Zurich, Food and Soft Materials Laboratory, Institute of Food, Nutrition & Health, Schmelzbergstr. 9, LFO E22, 8092 Zurich, Switzerland 2 Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, Stuttgart, Germany, and Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, Germany 1

2 Scattering intensity We consider a multicomponent system involving ν species of particles with particle numbers N H in the volume V. Each particle of a species H (1 H ν) carries scattering units. The coherent contribution to the scattering intensity in the ν-component system is given by I(q) = with the partial scattering intensities I HD (q) = b Hb D nh V Here r (α) ih ν H=1 D=1 ν I HD (q), (1) n D N H N D j=1 α=1 γ=1 iq ( ) r e (α) ih r(γ) jd. (2) is the position vector of the i-th scattering unit (1 i ) of the α-th particle (1 α N H ) of species H. The difference of the scattering length of this scattering unit and the average scattering length of the solvent is denoted as b H, and is an ensemble average. It proves convenient to decompose the partial scattering intensities according to where I HD (q) = ρ H b 2 Hω H (q)δ HD + ρ H ρ D b H b D h HD (q), (3) h HD (q) = V N H N D n D nh n D N H N D j=1 α=1 γ=1 γ α iq ( ) r e (α) ih r(γ) jd (4) is a particle-averaged total correlation function for pairs of particles of species H and D. The scattering unit number density of particles of species H is designated by ρ H = N H /V. The particle-averaged intramolecular correlation function ω H (q) = 1 nh N H iq ( ) r e (α) ih r(α) jh (5) N H j=1 α=1 characterizes the scattering unit distribution, and hence also the geometric shape of particles of species H. While the particle-averaged intramolecular correlations functions account for the interference of radiation scattered from different parts of the same particle in a scattering experiment, the local order in the fluid is characterized by the total correlation functions. 2

3 These functions are related to a set of direct correlation functions c HD (q) by generalized Ornstein-Zernike equations, which in Fourier space read [1, 2] h HD (q)= ν ω H (q)c HA (q) (ω A (q)δ AD + ρ A h AD (q)). (6) A=1 This set of generalized Ornstein-Zernike equations must be supplemented by a set of closure equations. The resulting integral equation theory has been successfully applied to various experimental systems such as binary mixtures (ν = 2) of charged colloids [3, 4], three-component mixtures (ν = 3) of charged colloids and salt ions [5], as well as charged polydisperse (ν 1) nanoparticles [6] and polyelectrolyte brushes [7]. Representing β-lactoglobulin fibrils as a sum of N H rigidly arranged spherical scattering units of radius R i leads to the particle-averaged intramolecular correlations [8] where ω H (q) = 1 F (q, R i ) 2 + j=1 j i F (q, R i )F (q, R j ) sin(qd ij) qd ij, (7) F (q, R i ) = 3 (qr i ) 3 (sin(qr i) qr i cos(qr i )) (8) is the scattering amplitude of the i-th scattering unit and d ij is the distance between the centers of the i-th and j-th scattering unit. Moreover, q is the magnitude of the scattering vector q. As suggested by the imaging data, the particle-averaged intramolecular correlations may be best described by a fibril which has a multi-stranded helical shape with a twisted ribbon-like structure. In the case of a rigid three-stranded twisting-ribbon helix the position vectors of the scattering units are given by with r ih = b(0, 0, i), i = 1,..., /3, (9) r ih = b(cos φ i, sin φ i, i), i = /3 + 1,..., 2 /3, (10) r ih = b( cos φ i, sin φ i, i), i = 2 /3 + 1,...,, (11) φ i = 2π(i 1) p (12). (13) 3

4 Here (p + 1)b is the pitch length. The blue line in Figure 1 shows the scattering intensity I(q) multiplied with the magnitude of the scattering vector q calculated for a one-component system (ν = 1) consisting of non-interacting (i.e., h HH (q) = 0) rigid three-stranded twistingribbon helices with R i = 0.94 nm, (p + 1)b = nm, and b = 1.94 nm. Hence the maximum diameter of the cross-section of an individual helix is 2(a + b) = 5.76 nm. A comparison with the measured scattering intensity of a fibril solution (pd=3) in the presence of 75 mm salt (blue squares in Figure 1) reveals that a rigid three-stranded twisting-ribbon model may be used as an appropriate approximation. However, it is not possible to determine the length L = ( /3 1)b+2a and the pitch length (p+1)b of the fibrils uniquely in the q range accessible to small angle neutron scattering. The scattering intensity is rather independent of geometrical features on length scales 100 nm. The blue circles in Figure 1 display the measured scattering intensity of a β-lactoglobulin monomer solution (pd=2) without added salt. For this system experimental data are available for higher scattering vectors. The pronounced suppression of the measured scattering intensity for small scattering vectors is due to both electrostatic repulsive interaction and the absence of fibrils in the monomer solution. For comparison, the red and green line display the calculated scattering intensity for rigid two-stranded and four-stranded twisting-ribbon helices with maximum cross-section diameters [pitch lengths] of 4 nm [74 nm] and 8 nm [200 nm], respectively. In both cases the radius of a scattering unit has been fixed to R i = 1 nm. The particle-averaged intramolecular correlations ω D (q) of coagulated κ-carrageenan can be represented as a sum of N D spherical scattering units similar to equation (7). As suggested by the imaging data, coagulated κ-carrageenan forms globular clusters along the fibrils. Therefore, it proves convenient to discuss the arrangement of scattering units inside the clusters in terms of a scattering unit distribution. The theoretical analysis of the small angle neutron scattering data revealed an inhomogeneous scattering unit distribution which decreases upon increasing the radial distance r from the center of a globular cluster located at r = 0 according to exp( r 2 /(2σ 2 )), where σ = 11 nm. Figure 2 shows the corresponding normalized scattering unit distribution (blue line). For comparison the black curve shows a homogeneous scattering unit distribution which is valid for a homogeneous sphere of radius R = 30 nm. One reason for the inhomogeneous scattering unit distribution is that one-dimensionally connected objects such as polyelectrolyte chains do not fill a volume corresponding to their overall size homogeneously due to molecular stiffness. 4

5 FIG. 1. Experimentally determined scattering intensity I(q) multiplied with the magnitude of the scattering vector q of a fibril solution (pd=3, 0.3 wt %) in the presence of 75 mm salt (blue squares) and a β-lactoglobulin monomer solution (pd=2, 2.0 wt %) without added salt (blue circles). The data for the monomer solution have been shifted down to account for the different weight fraction. The lines depict the calculated scattering intensities for rigid multi-stranded twisting-ribbon helices (red line: two-stranded helices with 4 nm maximum cross-section diameter; blue line: three-stranded helices with 5.76 nm maximum cross-section diameter; green line: fourstranded helices with 8 nm maximum cross-section diameter). 5

6 FIG. 2. Normalized scattering unit distribution inside a κ-carrageenan cluster (blue line) together with the scattering unit distribution of a homogeneous sphere of radius R = 30 nm (black line). Here the origin of the spherical coordinates is taken to be the center of the cluster and the sphere. 6

7 [1] Schweizer, K. S.; Curro J. G., Integral equation theories of the structure, thermodynamics, and phase transitions of polymer fluids. Advances in Chemical Physics 1997, 98, [2] Yethiraj, A., Liquid state theory of polyelectrolyte solutions. Journal of Physical Chemistry 2009, 113, [3] Yethiraj, A.; Shew C.-Y., Ion binding in tobacco mosaic virus solutions, Journal of Chemical Physics 1998, 109, [4] Harnau, L.; Hansen, J.-P., Colloid aggregation induced by oppositely charged polyions, Journal of Chemical Physics 2002, 116, [5] Harnau, L.; Reineker, P., Integral equation theory for polyelectrolyte solutions containing counterions and coions, Journal of Chemical Physics 2000, 112, [6] Weber, C. H. M.; Chiche, A.; Krausch, G.; Rosenfeldt, S.; Ballauff, M.; Harnau, L.; Göttker- Schnetmann, I.; Tong, Q.; Mecking, S., Single lamella nanoparticles of polyethylene Nano Letters 2007, 7, [7] Henzler, K.; Rosenfeldt, S.; Wittemann, A.; Harnau, L.; Ballauff, M.; Narayanan, T., Directed motion of proteins along tethered polyelectrolytes, Physical Review Letters 2008, 100, (1) (4). [8] Debye, P., Zerstreuung von Roentgenstrahlen, Annalen der Physik 1915, 46,

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