Measuring the Length Distribution of a Fibril System: a flow birefringence technique applied to amyloid fibrils 2. Motivation The fibrils, which self-assemble from  -lactoglobulin monomers, are examples of amyloid fibrils. Other amyloid fibrils are involved in various degenerative diseases, and an understanding of the assembly kinetics of the fibrils would be important in the study of these diseases. A versatile technique of measuring the length distribution would provide a powerful experimental approach to the assembly kinetics. There is interest in using  -lactoglobulin fibrils in food or biomaterials. The mechanical properties of such materials would depend crucially on the length of the fibrils. There are no easy methods of measuring a length distribution of fibrils like these. Abstract Relaxation of flow birefringence can give a direct measure of the rotational diffusion of rod-like objects in solution. With a suitable model of the rotational diffusivity, a length distribution can be sought by fitting the decay curve. We have measured the flow birefringence decay from solutions of amyloid fibrils composed of  -lactoglobulin, and extracted a length distribution using the Doi- Edwards-Marrucci-Grizzuti theory of semidilute rotational diffusion. The concentration scaling of the results show that the fibrils diffuse as free rods: they cannot be significantly branched, sticky or break up under dilution. The length distribution obtained shows a single broad peak, consistent with measurements of the fibrils by electron microscopy. This comparison, and combination of the experiment with an assay to find the total concentration of fibrils, allows calibration of the length scale and concentration scale of the length distribution. It is our hope that this method can be used for following the growth kinetics of amyloid fibrils in vitro, and for studying the length distribution of rod-like systems in general. Salman S Rogers, 1* Paul Venema, 2 Leonard Sagis, 2 Erik van der Linden, 2 and Athene M Donald 1 1 BSS Sector, Cavendish Laboratory, Cambridge University, Cambridge CB3 0HE, UK. 2 Laboratory of Food Physics, Wageningen University, PO Box 8129, 6700EV Wageningen, The Netherlands. *Correspondence Now published: Macromolecules; 2005, 38(7) 2948–2958; DOI: /ma – on web since 3 March 2005 (Background image: -lactoglobulin fibrils imaged by TEM, coloured digitally) 4. Inverting the data The scaled decay curve can be fitted using the DEMG model. First we estimate the length distribution using a linearly regularised inverse Laplace transform. Then we iteratively adjust the distribution until it fits the measured birefringence decay. Fig 2. Fitting the decay with various initial estimates shows fitting error due to noise is modest. (a) measured/fitted decays (b) length dist. 1. Aim: to measure the length distribution of a solution of  - lactoglobulin fibrils, which are: very polydisperse in length ~1  m semiflexible in semidilute solution (i.e. entangled) 7. Conclusion We have developed a promising technique for measuring a fibril length distribution, which can potentially be applied to any fibril system whose orientational relaxation can be resolved in time. The errors in the distribution can be evaluated quantitatively. Coming soon: electric birefringence measure of the short end of the distribution,  -lactoglobulin length distributions in different solution conditions. 3. Approaching the problem – rheo-optics In shear flow, the fibrils align, leading to measurable flow birefringence. When the flow is stopped, the birefringence decays on a spectrum of time scales, as the fibrils diffuse rotationally. Short fibrils diffuse faster than longer ones, so the decay curve contains information about the range of lengths in the system. The decay curve can be fitted quantitatively using a theory of diffusivity of rods: we use the Doi-Edwards-Marrucci-Gruzzuti (DEMG) model, which describes a polydisperse, semidilute solution. Fig 1. Scaling of decay time with (conc.) 2 shows the applicability of DEMG 6. Comparison with TEM The fitted length distribution has two missing parameters: M – optical anisotropy per unit length concentration of the fibrils,  – rotational diffusivity prefactor, which can be determined with other experiments: we determined the fractional conversion of monomers to fibrils with sedimentation, and compared the fibril lengths with TEM measurements, to find M=1.74  and  =6.5  Fig 3. Rheo-optics and TEM distributions match with M, as above Fig 4. Rheo-optics length distribution with errors from fitting, fibril stretching, sample inertia and incomplete alignment considered