Morphology selection via geometric frustration in chiral filament bundles Douglas M. Hall, Isaac R. Bruss, Justin R. Barone and Gregory M.

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1 SUPPLEMENTARY INFORMATION DOI: /NMAT4598 Morphology selection via geometric frustration in chiral filament bundles Douglas M. Hall, Isaac R. Bruss, Justin R. Barone and Gregory M. Grason Supplementary Method 1: Continuum model of morphology selection 1.1 Continuum theory Here we describe the continuum model of frustration-induced morphology selection in twisted filament bundles. The model makes the following assumptions. First, that for a given filament chemistry and solution conditions, interactions between chiral filaments generate assemblies of final helical pitch,. For simplicity of notation, here we describe twist in terms of rate of torsion along bundle length,, which is proportional to the handed skew angle,, between successive layers of chiral filament layers (shown in Fig. S1). Second, we consider the condensed state of filaments where bundles have formed via an equilibrium assembly process to a state where all but a negligible number of filaments remain dispersed in the solution. Finally, we consider no length-dependent thermodynamic costs of bundle formation and bundles are assumed to have sufficient length that the contribution of fiber ends to the free energy of fiber formation is negligible. The model calculates the free energy of bundle formation considering three morphology-dependent contributions: 1) filament cohesion; 2) intra-filament bending energy; and 3) inter-filament elastic energy. Specifically, our model of free energy has the form (S1) The first term represents the bulk free energy gain for assembly ( per unit volume) of a bundle of volume, while the second term represents the excess surface energy due to the sides of the bundle, of area, and parameterizes the deficit of inter-filament cohesive binding at the bundle sides. The third term describes the mechanical cost of intra-filament bending in a twisted bundle where is the bending modulus of filaments, is the areal density of filaments in the bundle cross-section and is the local curvature of a filament segment at position in the bundle. In twisted bundles, where filaments are bent into helical curves, the curvature is a function of the radial distance r from the bundle center,. The final term above models the elastic cost of distortions of inter-filament spacing in the hexagonally packed array in the bundle cross-section. To describe the generic cost of geometrically imposed SI Figure 1, Chiral geometry of bundles Schematic of chiral packing of filaments in twisted, multi-filament bundles, showing the (constant) skew angle between neighbor filaments packed along the radial direction in the bundle. NATURE MATERIALS 1

2 variations in inter-filament spacing, we model the bundle volume as a 2D columnar material, which assumes no cost for inter-filament sliding shear deformations 1. The elastic energy density of a twisted bundle derives from the 2D strain tensor (components in the x-y directions with the mean filament orientation along the z-axis), which, assuming an isotropic (i.e. hexagonal) packing symmetry of the bundle, is related to the inter-filament stress via, where and are Lame elastic constants for the filament array. Inter-filament strain derives from 1) the 2D displacement of filament position (relative to the initial position) in the xy plane perpendicular to the long-axis of fiber and 2) the inplane components of filament tangent vector (S2) where the non-linear contribution to strain from in-plane tangent components reflects the ability to reduce the true distance between filaments of fixed separation in the x-y plane through simple tilt deformations. Choosing coordinates where the 2D positions of filaments rotate around the z-axis, the inplane components of filament tilt increase with distance from the bundle center as. As described previously 1, the equilibrium state of stress (in the limit of small tilts) satisfies the in-plane force balance,, as well as a generalized compatibility relation that links in-plane stress to the outof-plane texture of twist, (S3) where is a 2D Young s modulus for the array and denotes the 2D Laplacian in the plane perpendicular to the pitch axis. These equations are to be solved for stress-free boundary conditions, at the surface of the bundle, where describes the local surface normal. For these boundary conditions, the inter-filament elastic energy reduces to the simplified form 1. Defining as the lateral pressure of the filament array yields the compatibility relation and inter-filament elastic energy described in the main text. Assuming a nearly monodisperse size distribution of bundles, equilibrium properties of filament assembly derive from the free energy density and its dependence on cross-sectional dimensions. Expressions for derive from a dependence on surface geometry, mean filament bending and interfilament stress on cross-sectional shapes of bundles, as summarized below. 1.2 Isotropic cylinders For cylindrical bundles of radius, inter-filament stresses are axisymmetric and conveniently expressed in polar coordinates. Combining the radial force balance with the compatibility yields (S4) which satisfies. Integrating inter-filament stresses and bending energy (retaining lowest order in ) over the cylindrical volume and combining with the surface energy of bundle sides yields the free energy density (S5) Optimizing with respect to radius according to yields the equation of state (S6)

3 where equilibrium radius can be written as. The solution to eq. (S5) for defines the one-to-one mapping between the ratio of inter- to intra-filament elastic energy,, and the reduced surface energy,. Solutions for the equilibrium radius of cylindrical bundles (or equivalently ) are plotted in Fig. 2B of the main text. 1.3 Anisotropic tapes Modeling the bundle cross-section as a rectangular strip with width and thickness axes, respectively, along the x and y axes, we compute the bending energy using, where is the central axis of the bundle. Integrating over the 2D cross section we have the bending energy density (S7) In the limit of high aspect ratio, we neglect the thickness dependence of bending energy, an assumption that is consistent with the large limit where the inter-filament elastic cost dominates over the intra-filament bending energy. For, inter-filament elastic stresses are determined assuming the ribbon to be sufficiently wide such that far away from the ends at, stresses are independent of position along the wide direction and x-derivatives of stress vanish 2. Hence, the equations of force balance then require that and are constants, and therefore, must vanish due to stress-free boundary conditions at. The remaining condition for the stress along the width axis comes from the compatibility equation, which takes the form and has the solution. The stress along the mid-line is determined by minimizing the elastic energy density, yielding. Integrating the elastic energy density over the cross-section and combining this with bending energy (in the limit) and surface energy cost for a rectangular cross-section yields the free energy density (S8) Optimizing with respect to width and thickness of the tapes gives equations of state written in terms of the reduced surface energy, (S9) Hence, using eq. (S5) and (S7), the equilibrium width and thickness of tapes can be written directly as functions of, or equivalently the equilibrium radius of cylindrical bundles, which are stable or metastable at the same thermodynamic conditions. 1.4 Cylinder-to-tape transition Using the equations of state derived above, (S6) and (S9), free energy densities for equilibrium cylinders and tapes can be derived in terms of the common variable of (S8) and (S9) The root of the equation at defines the predicted transition between cylinders at small and tapes at large. Note that the second root occurring at smaller corresponds to

4 a predicted aspect ratio outside of the range, and therefore, outside of the range of validity of the asymptotic limit of large aspect ratio assumed in its derivation. 1.5 Equilibration of bundle twist Our primary aim is to derive the equations of state relating lateral dimensions of the bundle to the equilibrium (post-relaxation) value of bundle pitch. It is straightforward to generalize the constant- ensemble above, to describe the physics of twist relaxation for bundles whose chiral interactions set a preferred value of inter-filament skew angle. The free energy deriving from chiral forces between rod-like elements is described at the coarse-grained level by the twist terms of the elastic Frank energy 3, (S9) where is the filament orientation in the bundle. The Frank constant parameterizes the elastic penalty for deviations from preferred chiral packing between neighbors and is the preferred pitch of (uniaxial) cholesteric ordering, a coarse-grained measure of the preferred inter-filament skewangle between neighbors. Evaluating the twist free energy density, we find (S10) where defines the helical bundle twist preferred by chiral filament forces. Minimizing the sum of and the non-chiral free energy density,, with respect to, we derive the equation of state relating the final twist to the value preferred by chiral interactions, (S11) Because this equation has a single real solution for all, corresponding to a unique energy minimum, equilibrium twist is finite for all finite-size bundles. Supplementary Method 2: Discrete filament simulations Here we describe the method for determining the equilibrium morphology of twisted bundles in our discrete-filament model. To maintain a uniform inter-filament twist in the bundle, filaments in the bundle have a pitch. This ensures that their in-plane positions rotate around a central axis at a rate while moving along the central bundle axis (e.g. the helical axis is fixed to the center of the bundle). Assuming no defects along the length of the bundle, cross sections are therefore identical up to a rigid rotation around the bundle center and filament positions are characterized by the set of 2D position vectors, where labels distinct filaments in a N-filament bundle. The free energy density for the bundle is computed with a transverse section of the bundle of height,. The mechanical energy contribution of filaments in this section derives from the filament curvatures and their contour length per section (S12) The cohesive energy contribution from the section takes the form (S13)

5 where is the cohesive interaction of a segment of filament i with filament j and is the distance of closest approach between the filaments. In ref. 4 it was shown that the case of the interfilament potential corresponds to the integrated effect of pairwise Lennard-Jones interactions between the filament center lines 4. The generalization of this form to is used to independently vary the equilibrium spacing d and cohesive range of the attractive minimum; the ranges of and correspond to effectively ductile and brittle interactions respectively, and is shown in Fig. 3A. We use a numerical approximation for the distance of closest approach between filaments derived in ref. 3 (S14) where. Eq. (S12) takes into account the out-of-plane geometry of the contact between coaxial helical curves, which are generically nearer in separation than their spacing in the x-y plane, and therefore, encodes the intrinsic frustration of metric geometry in twisted bundles, with the out-of-plane component of inter-filament contact. The total free energy density per unit of filament length is computed as (S15) where is the total contour length of filaments in the bundle section. To model the optimal cross sections of bundles we generate initial configurations as subsets of the hexagonal lattice with a predetermined anisotropy. We parameterize initial configurations by and, which correspond to the transverse dimensions of a 2D mask used to define the set of initial in-plane coordinates (see Fig. S2). The mask is defined by a stadium curve two semi-circular ends, of radius connected by parallel straight lines of length where the dimension is aligned along the nearest-neighbor direction of the lattice. The stadium mask is centered on one of three points of symmetry in the hexagonal lattice (Fig. S2): A) a lattice vertex; B) the mid-point between adjacent vertex points and C) the center of a triangular plaquette. For given dimensions and, interaction range, filament stiffness and bundle twist, an initialization step selects the mask centering that corresponds to the maximal cohesive energy at zero-twist ( =0), typically consistent with the maximum number of nearest-neighbor bonds for the selected points. Note that the total number of filaments N is not fixed in the simulations, as this varies with, and mask center. From this point the twist is incremented to its chosen value through a series (between 10-15) of small steps, where at each intermediate twist, 2D filament positions are relaxed via steepest descent. By annealing bundle twist sufficiently slowly, simulations avoid the introduction of additional defects associated with large changes of inter-filament spacing, which may result in plastic deformation of the 2D filament array. 5-fold disclinations have been shown to be stable for highly twisted cylindrical bundles 5,6 above a threshold twist of roughly. Notably, present simulations are carried for a range of twist angles below this value, and therefore, topological defects are not anticipated features of the ground state packings. SI Figure 2, Initialization of anisotropic bundle cross sections Schematics of algorithm that generates initial filament positions for anisotropic bundles. For chosen values of and, the transverse dimensions of the bundle cross section, initial 2D filament positions are selected from a hexagonal lattice of spacing d (shown as filled circles) for lattice points that fall within a stadium curve, or mask, of transverse dimensions X and Y (lattice points not chosen are shown as open circles). A, B and C show, respectively, the trial center positions considered in the optimization scheme, each of which slightly varies the anisotropic geometry of the boundary filaments.

6 For selected values of, and, free energy density landscapes are produced by optimizing the fixed dimension bundles over a range of and (typically from to roughly twice the equilibrium dimension) in increments of. The equilibrium bundle and its corresponding dimensions are identified from the minimal-energy structure. To explore the broad influence of filament-scale stiffness, interactions and (chiral) twist, simulations were carried out over a range of parameters: ; ; and. Results are reported in main text Figs. 3 and 4, where the simulation resolved well-defined minima in the landscape for bundles possessing up to filaments. To map the discrete filament model parameters onto the continuum theory, we use the areal density of a hexagonal lattice of spacing d. The inter-filament elastic parameters are determined by considering the elasticity of an infinite, regular hexagonal lattice possessing interactions, and approximating the net effect of cohesive interactions as nearest-neighbor springs that prefer inter-filament spacing d. The spring constant of a nearest neighbor bond is simply, from which the Lame constants and 2D Young s modulus of the hexagonal array may be directly computed (S16) Likewise, the surface energy of the bundle may be estimated from deficit of (nearest neighbor) cohesive bonds along an interface normal to the next nearest neighbor direction (a low energy sector) in a hexagonal array, e.g. 2 bonds per inter-filament distance d, from which we have the bundle surface energy (S17) From these relations and eq. (S5), each discrete-filament simulation, characterized by values, and, is assigned a value of, the predicted ratio of inter-filament to intra-filament elastic energy and tested against continuum theory predictions in the main text Fig. 3. The continuum model predicts a relation between the equilibrium dimensions of tapes their width, thickness and pitch and the characteristic scales that compare bending stiffness and surface energy to inter-filament modulus. We test this prediction in Fig. S3 using our simulation results. Here, we compare the exact value of and for a given discrete filament simulation, to the values that SI Figure 3, Exact vs. extracted characteristic lengths The predicted relationship between equilibrium tape dimensions and characteristic material lengths, and, are tested for simulated tapes in (A) and (B), respectively. Color scale indicates the number of filaments in the simulated bundle (with larger filament numbers shown as blue-purple), and the dashed line shows the predicted equality.

7 can be extracted from their simulated equilibrium dimension according to the relations and. Though we observe some scatter for bundles of fairly small numbers of filaments, as bundle sizes become larger (from 100s-1000s of filaments), results converge to the predicted equalities, indicating good agreement for bundles in continuum limit. Supplementary Method 3: Preparation of ultra-large amyloid fibers and characterization of morphology Large amyloid fibers were self-assembled from individual proteins and protein mixtures in aqueous solution. The preparation conditions and individual protein amino acid sequences and properties have been described in detail elsewhere Briefly, proteins at the appropriate molar ratio were incubated under moderate mixing with a stirring bar at the desired concentration, ph, and temperature for at least 20 days (Table S1). Solutions were monitored with Fourier transform infrared (FTIR) spectroscopy to characterize the molecular interactions during self-assembly and the FTIR spectrum for all systems was done changing within 20 days. P4, P7, and THGd:P7 displayed very rapid changes in the FTIR spectrum with time compared to other self-assembling systems and were monitored for 35 days to make sure they were done changing. While these large amyloid fibers were obtained at 35 days there was no indication of further self-assembly after several days. Properties of individual proteins, type and molar ratio of proteins in the mixture, and solution self-assembly conditions all influence the final large fiber morphology obtained. Therefore, the final large amyloid fiber morphology and properties can be anticipated by the characteristics of the self-assembling system. Self-assembly proceeds hierarchically where proteins selfassemble into amyloid protofibrils 2-4 nm high and nm wide that then bundle into fibrils 7-10 nm high and nm wide. Fibrils further assemble into either cylindrical large fibrils nm in diameter or rectangular large fibrils nm high and nm wide. It is the large fibrils that are the filaments that bundle into the large amyloid fibers as described in this paper. Large amyloid fibers separated from solution were mounted onto aluminum SEM stubs with double-sided tape. Scanning electron micrographs were obtained using a LEO 1550 field-emission SEM (Zeiss, Peabody, MA) with a 4-6 mm working distance, 5 kv accelerating voltage, and an In-lens SE-detector. Fiber morphology measurements were obtained as described in Fig. 4A,B on at least 3 fibers for samples with template and adder proteins in the mixture, which includes THWG, a mixture of trypsin-hydrolyzed wheat gluten (THWG) proteins. Fibers formed from single proteins were not prolific fiber formers in the absence of a complementary template or adder protein. In the case of fibers formed from single proteins, measurements were based on 1 or 2 fibers. Large fiber diameter (for cylinders, 2R), and width (w) and thickness (t, for tapes) are measured from apparent caliper of fiber cross sections. Filament diameter (d) is measured from thickness of apparent surface grooves on larger fiber surfaces (highlighted in main text Fig. 4A). Twist angle () is measured as the inclination of surface grooves with respect to the larger fiber axis (for anisotropic tapes, the angle is measured on the center of the wide face of the fiber). The helical pitch of filaments P is inferred from twist angle via the relation for cylinders and for tapes. Morphological measurements are reported as averages ± standard error in Table S2.

8 Table S1. Self-assembly conditions. Template:Adder mol:mol Conc. (mg/ml) ph T ( o C) Incubation time (days) THWG ph 4 N/A THWG ph 6 N/A THWG ph 8 N/A THWG ph 10 N/A THWG-95 o C N/A /95 20 THWG 100 mm NaCl N/A THGd:My 0.63: THGd:My 0.36: THGd:My 0.16: THGd:P4An 0.07: Gd20KK:P4An 0.50: Gd20KK:My 0.36: CB4:My 0.36: THGd:My ph : THGd:My ph : Am N/A THGd:My 100 mm NaCl 0.36: In N/A P4:My 0.50: THWG 22 o C N/A THWG 60 o C N/A THGd:My 22 o C 0.36: P7 N/A THGd:P7 0.07: P4An N/A P7:My 0.50: P4An:My ph : THGd:My 60 o C 0.36: THGd:My ph : THGd:In 0.36: P4An:My 0.50:

9 Table S2. Ultra-large amyloid fiber morphology measurements. Template:Adder Morphology d (μm) θ ( o ) 2R or w t (μm) (μm) THWG ph 4 Cylinder 0.68 ± ± ± 0.5 N/A THWG ph 6 Cylinder 0.84 ± ± ± 1.0 N/A THWG ph 8 Cylinder 0.61 ± ± ± 0.5 N/A THWG ph 10 Cylinder 0.58 ± ± ± 0.7 N/A THWG-95 o C Cylinder 0.55 ± ± ± 0.1 N/A THWG 100 mm Tape 0.09 ± ± ± ± 0.2 NaCl THGd:My 0.63:0.37 Tape 0.52 ± ± ± ± 0.2 mol:mol THGd:My 0.36:0.64 Tape 0.26 ± ± ± ± 0.3 mol:mol THGd:My 0.16:0.84 Tape 0.50 ± ± ± ± 0.1 mol:mol THGd:P4An Tape 0.32 ± ± ± ± 0.3 Gd20KK:P4An Tape 0.27 ± ± ± ± 0.1 Gd20KK:My Tape 0.30 ± ± ± ± 0.1 CB4:My A1 Tape 0.31 ± ± ± ± 0.1 THGd:My ph 4 Tape 0.18 ± ± ± ± 0.2 THGd:My ph 10 Tape 0.26 ± ± ± ± 0.1 Am Tape 0.27 ± ± ± ± 0.5 THGd:My 100 mm Tape 0.05 ± ± ± ± 0.2 NaCl In Tape 0.21 ± ± ± ± 1.2 P4:My Tape 0.18 ± ± ± ± 0.3 THWG 22 o C Tape 0.12 ± ± ± ± 0.5 THWG 60 o C Tape 0.24 ± ± ± ± 0.3 THGd:My 22 o C Tape 0.13 ± ± ± ± 0.6 P7 Tape 0.49 ± ± ± ± 0.1 THGd:P7 Tape 0.24 ± ± ± ± 1.3 P4An Tape 0.21 ± ± ± ± 0.5 P7:My Tape 0.48 ± ± ± ± 0.6 P4An:My ph 7 Tape 0.32 ± ± ± ± 0.2 THGd:My 60 o C Tape 0.49 ± ± ± ± 0.9 THGd:My ph 6 Tape 0.38 ± ± ± ± 0.4 THGd:In B1 Tape 0.36 ± ± ± ± 0.3 P4An:My Tape 0.32 ± ± ± ± 0.3

10 Supplementary Method 4: Cohesive cylinder model & amyloid tape assembly properties For large filaments where the range of inter-surface cohesive forces (e.g. van der Waals, screened electrostatic) is very short ranged in comparison to diameter, the effective inter-filament potential, and consequently, the in-plane elastic response of an ordered array can be derived from the so-called JKR model of cohesive contact between elastic cylinders 12. Here, we consider filaments to be transversely isotropic elastic bodies with an isotropic response in the plane perpendicular to the long axis characterized by the in-plane modulus and Poisson ratio with the extensional modulus along the filament axis 13. In the JKR limit, the relation between force F (per unit length) and width of contact zone a (see Fig. S4A), has been derived by Barquins 14, assuming cohesive gain per area of inter-filament surface contact between cylinders (S18) where d is the (undeformed) diameter of the cylinders. The equilibrium contact is determined by from which. To estimate the (per unit length) stiffness k of intercylinder bonds, we assume that the geometric relation between inter-cylinder indentation depth to a is the same as in Hertzian contact 15, i.e., from which we have (S19) By arranging cohesive inter-cylinder bonds in a hexagonal lattice (see Fig. S4B), we may relate the bond stiffness to the respective Lame elastic constants and 2D Young s modulus of the columnar array (S20) which shows that for cohesive contact between large filaments, elasticity of the filament array directly SI Figure 4, Cohesive cylinder model of filament interactions (A) A schematic of the contact geometry for the pair-wise cohesion between elastic cylinders. (B) A schematic of approximately pair-wise cohesive binding between filaments in a hexagonal array, which is used as a model of inter-filament binding and elasticity in bundles of larger (micron-scale) filaments.

11 reflects elastic cost to deform constituent filaments. To determine the characteristic length we use the bending stiffness for a circular elastic beam and the cross-sectional density of close-packed cylinders, from which we have (S21) Consistent with values of extracted for amyloid tapes, this model shows this length scale is comparable to filament diameter. However, the proportionality constant relating to d is shown to vary with the ratio of longitudinal to transverse intra-filament stiffness,. To examine the variation of this proportionality constant for the tape-forming amyloids, the ratio is plotted versus observed tape aspect ratio in Fig. S5, showing the proportionality factor to vary between ~0.25 and ~10. There is a significant correlation between values of and, with higher (lower) aspect ratios exhibiting relatively smaller (larger) values of the ratio, consistent with a power-law relationship (Fig. S5 inset). Taken in combination with the models of cohesive cylinders and geometrically frustrated bundles, these data suggest that the ratio of longitudinal and transverse moduli of constituent filaments (e.g. microfibrils) varies by a factor of ~100 across the observed fibers, and further, that the ratio is a critical factor to guide the ultimate anisotropic morphologies of these self-assembling amyloids. Given the significant variation of for these materials, an accurate estimate for the value of for the isotropic cylinders based on the observed microfibril diameter is not possible. Nevertheless, based on the nearly-isotropic ( ) range of the tape data we estimate the proportionality for cylinders to be to provide an approximate, albeit rough, placement of observed isotropic cylinders on the phase diagram in main text Fig. 4. To assess the expected magnitude of the length scale that compares cohesive interactions to interfilament elastic stiffness, we calculate the strength of inter-filament bonds based on the cohesive cylinder mode. Integrating the contact force F to determine the work done by inter-surface cohesion, we have the cohesive energy per unit length contacting filament pair (S22) To estimate the surface energy, we consider the deficit of cohesive bonds along an interface normal to SI Figure 5, Proportionality of to filament aspect ratio The ratio of extracted material length scale and measured diameter of constituent filaments (micro-fibrils) from observations of tape-like bundle forming amyloids, plotted vs. observed aspect ratio. The inset shows the same data on a log-log scale, with the solid line showing a power law relationship of -3.

12 the next nearest neighbor direction (a low energy sector) in a hexagonal array, e.g. 2 bonds per interfilament distance d, from which we have the bundle surface energy (S23) Taking the ratio of surface energy to 2D modulus of the array we estimate the elasto-cohesion length of the bundle (S24) Hence, is smaller than d by a factor of. Assuming values for the filament modulus 8 and surface energy 16 the ratio is expected to be in the range of a few Angstroms, two or three orders of magnitude below the microfibril diameter, consistent with the observation in main text Fig. 4C that. References 1. Grason, G. M. Defects in crystalline packings of twisted filament bundles. I. Continuum theory of disclinations. Phys. Rev. E 85, (2012). 2. Ghafouri, R. & Bruinsma, R. Helicoid to Spiral Ribbon Transition. Phys. Rev. Lett. 94, (2005). 3. Grason, G. M. Braided bundles and compact coils: The structure and thermodynamics of hexagonally packed chiral filament assemblies. Phys. Rev. E 79, (2009). 4. Bruss, I. R. & Grason, G. M. Topological defects, surface geometry and cohesive energy of twisted filament bundles. Soft Matter 9, 8327 (2013). 5. Bruss, I. R. & Grason, G. M. Non-Euclidean geometry of twisted filament bundle packing. Proc. Natl. Acad. Sci. 109, (2012). 6. Grason, G. M. Topological Defects in Twisted Bundles of Two-Dimensionally Ordered Filaments. Phys. Rev. Lett. 105, (2010). 7. Ridgley, D. M., Ebanks, K. C. & Barone, J. R. Peptide mixtures can self-assemble into large amyloid fibers of varying size and morphology. Biomacromolecules 12, (2011). 8. Ridgley, D. M., Claunch, E. C. & Barone, J. R. The effect of processing on large, self-assembled amyloid fibers. Soft Matter 8, (2012). 9. Ridgley, D. M. & Barone, J. R. Evolution of the Amyloid Fiber over Multiple Length Scales. ACS Nano 7, (2013). 10. Ridgley, D. M., Claunch, E. C., Lee, P. W. & Barone, J. R. The Role of Protein Hydrophobicity in Conformation Change and Self-Assembly into Large Amyloid Fibers. Biomacromolecules 15, (2014). 11. Ridgley, D., Rippner, C. & Barone, J. Design and Construction of Large Amyloid Fibers. Fibers 3, (2015). 12. Baney, J. M. & Hui, C.-Y. A cohesive zone model for the adhesion of cylinders. J. Adhes. Sci. Technol. 11, (1997). 13. Bower, A. F. Applied Mechanics of Solids. (CRC Press, 2010). 14. Barquins, M. Adherence and Rolling Kinetics of a Rigid Cylinder in Contact with a Natural Rubber Surface. J. Adhes. 26, 1 12 (1988).

13 15. Johnson, K. L. Contact Mechanics. (Cambridge University Press, 1985). 16. Nicholls, A., Sharp, K. A. & Honig, B. Protein folding and association: Insights from the interfacial and thermodynamic properties of hydrocarbons. Proteins Struct. Funct. Genet. 11, (1991).