A minimal conformational switching-dependent model for amyloid self-assembly

Amyloid formation is associated with various pathophysiological conditions like Alzheimer’s and Parkinson’s diseases as well as many useful functions. The hallmark of amyloid assemblies is a conformational transition of the constituent proteins into a β - sheet rich filament. Accounting for this conformational transition in amyloidogenic proteins, we develop an analytically solvable model that can probe the dynamics of an ensemble of single filaments. Using the theory and Monte Carlo simulations, we show the presence of two kinetic regimes for the growth of a self-assembling filament – switching-dependent and –independent growth regimes. We observe a saturation in fibril elongation velocities at higher concentrations in the first regime, providing a novel explanation to the concentration-independence of growth velocities observed experimentally. We also compute the length fluctuation of the filaments to characterize aggregate heterogeneity. From the early velocities and length fluctuation, we propose a novel way of estimating the conformational switching rate. Our theory predicts a kinetic phase diagram that has three distinct phases – short oligomers/monomers, disordered aggregates and β -rich filaments. The model also predicts the force generation potential and the intermittent growth of amyloid fibrils evident from single molecular experiments. Our model could contribute significantly to the physical understanding of amyloid aggregation.


Supplementary Text S1
Analytical solution for filament and β-sheet growth velocities for k > 1, P 0 + P 1 + P 2 + P 3 + ..P ∞ = 1 (4) P 0 + P 1 + A n k=2 q k = 1 (5) where, n k=2 q k = q 2 1 − q Note that, the above equation is valid only if k − cc + k s cb >= k + cc C s . We set k s bc = 0 in order to neglect the contribution of backward switching from state B to state C.
Now, at steady state, dP 0 dt = dP 1 dt = 0 (9) Solving for P 0 and P 1 we get, Expressions for velocities in the two regimes: Regime 1 (for q < 1): The filament growth velocities in the first regime (q < 1), under the steady state assumption (dP k /dt = 0) is given by: and Regime 2 (q = 1 and k + cc C s ≥ k − cc + k s cb ) In the second regime (q=1), dP k /dt = 0. Therefore, we cannot use the earlier steady state assumption that was valid for regime 1. The filament growth velocities in this regime can be derived without any steady-state assumption. In this regime, the filament growth depends only on the growth of the 'C' monomers. Therefore, one can write a master equation for G k which is the probablity of finding a filament with total length k: The above expression can be solved by performing a transformation (also see reference [1] below) given by: Doing this kind of a z-transformation, we obtain, The above expression can be solved under to obtain a relationship for the velocity such that,

Computing the force-velocity relationship
In order to compute the effect of external force on the growth velocity, we use modified forms of the velocity equations to account for the effect of the external force on the polymerization and depolymerization rates.
Let v f (F ) be the growth velocity in the presence of an external force, 'F'. The modified form of the velocity equation are shown in expressions 20 and 21. and Here, k + cc (F ),k − cc (F ),k + cb (F ) and k − cb (F ) are rescaled rate constants in the response to external force, and the expressions for the same are given in equations 10-13 in the main text.

Model setup for the fibril breakage simulations
Two-state model with fibril breakage: In addition to the aforementioned model with events like polymerization, depolymerization and conformational switching, we also performed simulations wherein any growing filament could also undergo breakage with certain rate, k break . Upon breaking, the two resultant filaments would both continue to grow as per the our model setup described before. The location of the break in our simulations was close to the center of the fibril. The setup of the fibril breakage simulations is schematically depicted in Supplementary Figure S9.A. simulations involving fibril breakage were performed under the constant free monomer concentration conditions. These simulations were performed to emphasize the fact that under these conditions of constant free monomer concentration, any resulting seed formed as a result of fibril breakage would continue to grow at velocities that can be predicted by our analytical theory.

Fitting the velocity-concentration curve to experiment
In order to fit the concentration-velocity profile to experiment ( Figure. 3 of the main text), the following rationale was used. Firstly, not all 5 parameters are mandatory to obtain a good fit. However, a certain set of relations between the magnitude of rate constants must be satisfied in order to obtain the fit. More importantly, irrespective of the value of other parameters, the saturation value of the velocity is dictated by just one parameter-the rate of conformational switching. Therefore, we can equate the saturation velocity to the switching rate and obtain k s cb from the saturation values. Since, the saturation of growth velocities is a phenomenon that emerges in the first regime of growth (C s < [C] * ), the ratio of magnitudes of rates is such that, k + cc /k − cc << k + cb /k − cb . Upon satisfying this condition, a large range of values could be used to fit the curve to the experiment. However, the ratio of magnitudes of the rates, k + cb /k − cb must be a constant value.
The emphasis of the plot ( Figure. 3 of the main text) is therefore to ascertain the switching rate, which dictates the maximum attainable velocity in the first regime. In order to obtain the absolute values of model parameters like k c c + and k c c − , one would need to perform experiments spanning both the growth regimes (C s < [C] * , and C s > [C] * ).

Amyloid fibril formation
The aggregation kinetics was initiated with LMW synuclein at a concentration of 300 M in 1.5 ml eppendorf tube in 20 mM Gly-NaOH, pH 7.4, 0.01% sodium azide. The eppendorf tubes containing protein solutions were placed into an Echo Therm model RT11 rotating mixture (Torrey Pines Scientific, USA) with a speed corresponding to 50 r.p.m. inside a 37 degrees C incubator.
ThT fluorescence assay 1 mM ThT was prepared in Tris-HCl buffer, pH 8.0, 0.01% sodium azide. 3.3 µl of the protein solution was taken and diluted to 200 µl Gly-NaOH buffer, pH 7.4, 0.01% sodium azide such that final protein concentration becomes 5 µM. 2 µl of 1 mM ThT solution was added to the 5 µM protein solution in 200 µl Gly-NaOH buffer, pH 7.4, 0.01% sodium azide. Immediately after addition, ThT fluorescence assay was measured using Horiba-Jobin Yvon (Fluomax4) with excitation at 450 nm and emission in the range of 460-500 nm. The slit width for both excitation and emission were kept at 5 nm. ThT fluorescence obtained at 480 nm was plotted against incubation time and the data were fitted to a sigmoidal curve.

Atomic force microscopy (AFM)
Time dependent morphological changes during synuclein aggregation kinetics was performed using AFM (Asylum research, USA). At three different time points (i.e 0h, beginning of lag phase; 96h, middle of elongation phase and 180h, saturation phase) during aggregation, a small aliquot of incubated samples were taken out, diluted to a final concentration of 30 µM and spotted on a freshly cleaved mica sheet and incubated at room temperature for 5 minutes. Unbound proteins/aggregates were washed by distilled water and dried under vacuum. The imaging was performed in tapping mode with silicon cantilever with a scan rate 1 Hz and around 6 different areas were scanned randomly. Beta sheet growth velocity Figure S2. Comparison of beta-sheet and overall aggregate growth velocities. In the first regime, the filament extension is mainly the growth of the beta sheet which is evident by the overlapping curves (solid blue and dotted green). However, in the second regime the aggregate grows mainly by polymerizing C monomers while the rate of β-sheet development is constant and equal to the switching rate, k s cb .  Figure S4. Behavior of probabilities and individual terms in the velocity expression in response to concentration. A) The probability of finding 0, 1 and more than one 'C' monomers at the tip of a growing filament and the effect of concentration on the probabilities. At lower concentrations, there is a high probability of encountering filaments which are entirely in state B. At higher concentrations, P 1 and 1 − P 0 − P 1 start dominating over P 0 , and B) Behavior of the P 0 ,P 1 and 1 − P 0 − P 1 terms in the velocity expression with increasing free monomer concentration, C s . Figure S5. Filament growth against external force. Schematic representation of the Brownian rachet principle for force-generating polymerizing amyloid fibrils. The filament growth occurs against an external force 'F' applied by a wall which is constantly undergoing thermal fluctuations. The fibril can grow as long as the space between the tip of the fibril and the wall is greater than the size of a subunit 'd'. The rates of polymerization and depolymerization are k + cb and k − cb respectively. Thermal fluctuation of the wall could occassionally allow for the extension of the fibril by addition of monomers.