Inherent Variability in the Kinetics of Autocatalytic Protein Self-Assembly

In small volumes, the kinetics of filamentous protein self-assembly is expected to show significant variability, arising from intrinsic molecular noise. This is not accounted for in existing deterministic models. We introduce a simple stochastic model including nucleation and autocatalytic growth via elongation and fragmentation, which allows us to predict the effects of molecular noise on the kinetics of autocatalytic self-assembly. We derive an analytic expression for the lag-time distribution, which agrees well with experimental results for the fibrillation of bovine insulin. Our expression decomposes the lag-time variability into contributions from primary nucleation and autocatalytic growth and reveals how each of these scales with the key kinetic parameters. Our analysis shows that significant lag-time variability can arise from both primary nucleation and from autocatalytic growth and should provide a way to extract mechanistic information on early-stage aggregation from small-volume experiments.

Figure 1

(a) Experimental kinetic curve (black line) for the aggregation of bovine insulin in a volume of $100\mu \mathrm{l}$ from our own experiments, fitted to the theoretical prediction of a model [7] involving primary nucleation, elongation, and fragmentation (dashed red line); for full experimental details, see the Supplemental Material [16]. (b) Kinetic curves obtained from kinetic Monte Carlo simulations of a stochastic version of the same model [7, 17] and the fit parameters extracted from (a), but for a much smaller volume of 830 fl. (c) Schematic illustration of (I) primary nucleation, (II) elongation via polymerization, and (III) fragmentation. The critical nucleus size for primary nucleation is denoted by $n_c$.

Figure 2

The lag-time distribution $L(t)$ for several values of $\alpha$, compared to that obtained by running 1000 independent kinetic Monte Carlo simulations of the full stochastic model (in which individual fibril lengths are resolved) [17]. From left to right: $\alpha =50$ (full line), 5 (dashed line), and 1.5 (dot-dashed line), all in units of ${10}^{-15}\mathrm{mol}/(\mathrm{ls})$. Inset: $L_1(t)$ (dashed line) compared to simulations for $\alpha =5\times {10}^{-17}\mathrm{mol}/(\mathrm{ls})$. The other parameters are $V=830\mathrm{fl}$, $M_T=10\%$ of $c_{\mathrm{tot}}$, $c_{\mathrm{tot}}=100\mu \mathrm{mol}/\mathrm{l}$, $n_c=2$, $k_{+}=5\times {10}^4\mathrm{l}/(\mathrm{mol}\mathrm{s})$, and $k_f=3\times {10}^{-8}{\mathrm{s}}^{-1}$.

Figure 3

(a) Volume dependence of the lag time for the aggregation of bovine insulin in microdroplets of varying volume (red dots) [18], compared to the mean (solid line) and the standard deviation (error bars) from Eq. (17), using the following values, which were obtained from Ref. [18]: $T=104\min$, $\alpha =1/(1.7\times {10}^{-7}{N}_A)\mathrm{mol}/(\mathrm{l}\mathrm{s})$, $k_{+}=8.9\times {10}^4\mathrm{l}/(\mathrm{mol}\mathrm{s})$, $k_f=2\times {10}^{-8}{\mathrm{s}}^{-1}$, and assuming $n_c=2$. (b) Corresponding volume dependence of the standard deviation; the red dots are six-point moving standard deviations from the experimental data, and the solid green line is from Eq. (17). (c) Theoretical predictions for the standard deviation ${\sigma }_1$ as a function of volume, relative to $\epsilon /\alpha$. The green (lower) lines correspond to the protein concentration $30\mathrm{mg}/\mathrm{ml}$ used in Ref. [18], while the blue (upper) lines are for a higher protein concentration, $100\mathrm{mg}/\mathrm{ml}$, assuming that $\alpha (c_{\mathrm{tot}})\propto {c_{\text{tot}}}^{n_c}$ [7]. In both cases, the dashed lines correspond to (17), while the solid lines are calculated numerically from (16).