Initial condition of stochastic self-assembly

The formation of a stable protein aggregate is regarded as the rate limiting step in the establishment of prion diseases. In these systems, once aggregates reach a critical size the growth process accelerates and thus the waiting time until the appearance of the first critically sized aggregate is a key determinant of disease onset. In addition to prion diseases, aggregation and nucleation is a central step of many physical, chemical, and biological process. Previous studies have examined the first-arrival time at a critical nucleus size during homogeneous self-assembly under the assumption that at time $t=0$ the system was in the all-monomer state. However, in order to compare to in vivo biological experiments where protein constituents inherited by a newly born cell likely contain intermediate aggregates, other possibilities must be considered. We consider one such possibility by conditioning the unique ergodic size distribution on subcritical aggregate sizes; this least-informed distribution is then used as an initial condition. We make the claim that this initial condition carries fewer assumptions than an all-monomer one and verify that it can yield significantly different averaged waiting times relative to the all-monomer condition under various models of assembly.

Figure 1

Graphic depiction of the coagulation-fragmentation-type assembly. Under this assembly model polymers of any size may join and fragmentation may occur between any two adjacent constituent monomers.

Figure 2

Graphic depiction of the Becker-Döring-type assembly. Under this assembly model only monomer polymerization and depolymerization may occur.

Figure 3

Nondimensionalized mean assembly times $\beta \mu$ for the coagulation-fragmentation model with $c=5$ and $m=50$.

Figure 4

Nondimensionalized mean assembly times $\beta \mu$ for the coagulation-fragmentation model with $c=7$ and $m=30$.

Figure 5

Nondimensionalized mean assembly times $\beta \mu$ for the Becker-Döring model with $c=5$ and $m=50$.

Figure 6

Nondimensionalized mean assembly times $\beta \mu$ for the Becker-Döring model with $c=7$ and $m=30$.

Figure 7

Relative difference of mean assembly times under the different IC (least-informed and all-monomer) $(1-{\mu }_{\mathrm{LI}}/{\mu }_{\mathrm{AM}})\times 100\%$, computed with $c=5$ and $m=100$ with the coagulation-fragmentation assembly.

Figure 8

Same as Fig. 7, but with the Becker-Döring assembly.