Stochastic yield catastrophes and robustness in self-assembly

Abstract

A guiding principle in self-assembly is that, for high production yield, nucleation of structures must be significantly slower than their growth. However, details of the mechanism that impedes nucleation are broadly considered irrelevant. Here, we analyze self-assembly into finite-sized target structures employing mathematical modeling. We investigate two key scenarios to delay nucleation: (i) by introducing a slow activation step for the assembling constituents and, (ii) by decreasing the dimerization rate. These scenarios have widely different characteristics. While the dimerization scenario exhibits robust behavior, the activation scenario is highly sensitive to demographic fluctuations. These demographic fluctuations ultimately disfavor growth compared to nucleation and can suppress yield completely. The occurrence of this stochastic yield catastrophe does not depend on model details but is generic as soon as number fluctuations between constituents are taken into account. On a broader perspective, our results reveal that stochasticity is an important limiting factor for self-assembly and that the specific implementation of the nucleation process plays a significant role in determining the yield. arXiv:1905.09912v2 [physics.bio-ph] 18 Mar 2020 2 Model definitionWe model the assembly of a fixed number of well-defined target structures from limited resources. Specifically, we consider a set of S different species of constituents denoted by 1, . . . , S which assemble into rings of size L. The cases S = 1 and 1 < S ≤ L (S = L) are denoted as homogeneous and partially (fully) heterogeneous, respectively. The homogeneous model builds on previous work on virus capsid [3,17], linear protein filament assembly [7,27,28] and aggregation and polymerization models [24]. The heterogeneous model in turn links to previous model systems used to study, for example, DNA-brick-based assembly of heterogeneous structures [6,19,30]. We emphasize that, even though strikingly similar experimental realizations of our model exist [11,32,36], it is not intended to describe any particular system. The ring structure represents a general linear assembly process involving building

The Time Complexity of Self-Assembly

“…The dimerization scenario can be simulated most efficiently, because far fewer steps are needed due to the irreversibility of binding reactions. Furthermore, stochastic effects do not play an important role [12], so N can be chosen to be relatively small. Conversely, fluctuations in the assembly time between independent runs decrease with increasing N , allowing for greater accuracy in the determination of the exponents.…”

“…We are interested in the asymptotic dependence of on the structure size S for S, N ≫ 1. In particular, while we have shown previously [12] that for small copy numbers N , the activation scenario is strongly influenced by stochastic effects, we here assume N to be large enough so that stochastic effects can be considered irrelevant. In this deterministic regime, we find that both the optimal control parameter and the minimal assembly time exhibit power-law dependencies on the size S of the target structure (Fig.…”

Thresholds in Origin of Life Scenarios

“…It has been studied for the formation of vesicles ( Bachmann et al., 1992 ), coacervates ( Jiang et al., 2015 ), or microdomains in liquid crystals ( Todisco et al., 2018 ) ( Figure 1 C). For finite-size structures, such as peptide nanospheres ( Carny and Gazit, 2005 ), mathematical models have established a threshold separating the self-assembled state from a “yield catastrophe phase” where constituents nucleate without stabilizing into final structures, controlled by activation and dimerization rates ( Gartner et al., 2020 ).…”

“…The transition from dominant HGT to lineages where genetic polymers and compartments are strongly correlated has been coined the ''Darwinian threshold'' (Woese, 2002) (Figure 1F). Mathematical models (Arnoldt et al, 2015;Goldenfeld et al, 2017) show that such a threshold can be crossed when the progenotes HGT rate decreases below a specific value. This decrease might result from mutations and environmental changes such as protocell density, alkaline shifts, or nutrient limitation (Claverys et al, 2006;Ková cs et al, 2009).…”

“…One of the most studied thresholds is Eigen's ''error threshold,'' which constrains replication-based scenarios of early genetic polymers (Eigen, 1971;Kun et al, 2015;Takeuchi et al, 2017). A number of other prebiotic transitions invoke thresholds explicitly or implicitly: for instance, the appearance of homochirality (Budin and Szostak, 2010;Hawbaker and Blackmond, 2019), replication (Kauffman, 1986;Szathmá ry, 2006), compartmentalization (Hargreaves et al, 1977), or Darwinian evolution (Goldenfeld et al, 2017).…”