Scaling behaviour and rate-determining steps in filamentous self-assembly

Abstract

The formation of filaments from naturally occurring protein molecules is a process at the core of a range of functional and aberrant biological phenomena, such as the assembly of the cytoskeleton or the appearance of aggregates in Alzheimer's disease. The macroscopic behaviour associated with such processes is remarkably diverse, ranging from simple nucleated growth to highly cooperative processes with a well-defined lagtime. Thus, conventionally, different molecular mechanisms have been used to explain the self-assembly of different proteins. Here we show that this range of behaviour can be quantitatively captured by a single unifying Petri net that describes filamentous growth in terms of aggregate number and aggregate mass concentrations. By considering general features associated with a particular network connectivity, we are able to establish directly the rate-determining steps of the overall aggregation reaction from the system's scaling behaviour. We illustrate the power of this framework on a range of different experimental and simulated aggregating systems. The approach is general and will be applicable to any future extensions of the reaction network of filamentous self-assembly.

Fig. 1. Full reaction network of fibrillar self-assembly. The reactive fluxes for an aggregate of size n are shown schematically. Fluxes to and from the species directly connected to the aggregate of size n are hinted at where space permits. Elongation of aggregates of size n – 1 and fragmentation of all aggregates larger than n results in an increase in the concentration of aggregates of size n. By contrast, elongation and fragmentation of aggregates of size n, results in a decrease in their concentration. Moreover, aggregates of size n are also involved in the production of secondary nuclei which, however, leaves their concentration unchanged. Additionally, primary nuclei are produced directly from monomers (not shown). The full reaction network consists of an equivalent scheme for every single size of aggregate, all connected by elongation and fragmentation to yield a complex interconnected system.

Fig. 2. Reaction network of fibrillar self-assembly in terms of moments. The full reaction network, shown in Fig. 1, is significantly simplified by considering the moments of the fibril distribution rather than the concentration of species of all sizes individually. A closed reaction network can be formulated as a Petri net in terms of the first two moments of the fibril length distribution, the mass concentration (M) and the number concentration (P). Circles represent populations of different species, and squares represent the processes that inter-convert them. Note that fragmentation and monomer-dependent secondary nucleation have arcs going in both directions, to and from the fibril mass, because fibril mass is not removed when new ends are created in these processes.

Fig. 3. Processes of fibrillar self-assembly. The processes that make up the reaction network of filamentous self-assembly fall into two categories, nucleation/multiplication processes, which produce new fibril ends, P, and only a negligible quantity of new fibril mass, and growth processes, which produce fibril mass, M, but leave the fibril number unchanged. The contributions to the rates of growth and nucleation/multiplication are given below the schematic representation for each process. For elongation and secondary nucleation we use two alternative schemes, a single-step and a two-step one. Under certain conditions the single-step description may be sufficient, as discussed in the main text. The rate constants and reaction rates are defined in Table 1.

Fig. 4. Rate-determining steps and scaling in serial and parallel reactions. The saturating serial process (left) consists of two steps, S1 and S2, in series where the intermediate (yellow pentagon) is bound to a catalyst which is released again upon formation of the product (blue square). The parallel process consists of two steps, P1 and P2, that both yield the same product, directly from monomer (blue circles) in the case of P1 and from a different species (red triangle) in the case of P2. The mechanisms are considered in the limits of high and low concentrations of starting material (blue circles) and the rate-determining step is highlighted in each case. The behaviour of each system over the range of monomer concentrations is summarized in the scaling plots, i.e. the double logarithmic plots of the half time versus monomer concentration (the equations used to generate these plots are given in the ESI Sections 4.1 and 4.2†). Note that for the parallel network the scaling exponent increases in magnitude with increasing concentration (i.e. the faster process dominates the reaction), whereas for the serial network the magnitude of the scaling exponent decreases with increasing concentration (i.e. the slower process limits the overall rate).

Fig. 5. Serial processes. The saturation of elongation (left column) and the saturation of secondary nucleation (right column) are two examples of systems where a variation in monomer concentration leads to a change in the rate-determining step through a switch between two steps in series. (A) Petri net with two-step elongation. The relevant region is highlighted. (B) Petri net with two-step secondary nucleation. Again, the relevant region is highlighted. (C) Saturation of elongation occurs in the aggregation of bovine insulin, here starting from monomeric protein in solution, in triplicate repeats, at concentrations between 5 and 86 μM. The solid lines represent the best global fit to the model, using 3 free parameters. (The data at the lowest two concentrations are noisy and were therefore excluded from the fit, but they are shown in the half time plots to lie on the curve predicted from the fit.) (D) Saturation of secondary nucleation is observed for the aggregation of Aβ40 (pH 7.4). Solid lines are the global best fit using 3 free fitting parameters. Only curves at every other concentration are shown for clarity, however all the data were used to obtain the fit. (E) Half time versus initial monomer concentration for insulin. The solid line is plotted using the parameters obtained through the fit in (C). Note the positive curvature due to a decrease in the monomer dependence of the elongation step at high monomer concentrations. (F) Half time versus initial monomer concentration for Aβ40. The solid line is plotted using the parameters obtained from the fit in (D). Again positive curvature is evident, in this case due to a saturation of secondary nucleation at high monomer concentrations.

Fig. 6. Parallel processes. Competition of two processes in parallel can be observed between fragmentation and secondary nucleation (left) and in simulations also between primary and secondary nucleation (right). (A) Petri net with both secondary processes active. The relevant region is highlighted. (B) Petri net with primary and secondary nucleation. Although primary nucleation was also present in all other nets presented, it was there assumed to be significant only at very early times. (C) Competition of secondary nucleation and fragmentation appears in the aggregation of Aβ42 at low ionic strengths (4 mM sodium phosphate buffer at pH 8.0, 5 mM NaCl). Points are the relative aggregate mass, measured by ThT fluorescence, solid lines are the global best fit using 4 free fitting parameters. Some curves at low monomer concentrations are omitted for clarity although all the data were used to obtain the fit. (D) A minimal model of secondary nucleation was used in a Monte-Carlo algorithm where proteins were modeled as rods with attractive patches; a visualization of the primary and secondary nucleation processes is shown.^, (E) Half time versus initial monomer concentration. The solid line is plotted using the parameters obtained through the fit in (C). Note the negative curvature due to an increase in the monomer dependence when secondary nucleation dominates over fragmentation at high monomer concentrations. (F) The overall rate of nucleus formation (blue circles), the rate of nucleus formation from primary nucleation (grey crosses) and from secondary nucleation (red pluses) as obtained from the Monte-Carlo simulations (D). The negative curvature agrees with the predicted behaviour and moreover the simulations allow one to directly establish the origin of the nuclei, again confirming the prediction that the more monomer-dependent process dominates at high monomer concentrations.

Fig. 7. Summary of scaling exponents and mechanisms of aggregation. Top, the scaling exponent for non-saturated systems with only one of the three different processes dominating the production of new nuclei. The approximate increase in aggregate mass, as a function of the time t and initial monomer concentration m ₀, is also given, showing the quadratic increase in the absence of secondary processes and the more sudden, exponential increase in their presence. Bottom, the change in scaling exponent (+1/2 meaning an increase by 1/2) upon an increase in the monomer concentration, going from one extreme (e.g. non-saturated elongation) to the other extreme (e.g. fully saturated elongation). For each scenario, the effect is either due to a change in the rate-determining step of growth or in the rate-determining step of nucleation/multiplication. The associated topology of the reaction network and schematic expected half time plots are also shown.