Stochastic yield catastrophes and robustness in self-assembly

Florian M. Gartner*, Isabella R. Graf*, Patrick Wilke*, Philipp M. Geiger, Erwin Frey

Self-assembly and self-organisation are two fundamental concepts used to explain the astonishing ability of natural systems to autonomously generate complex structures and patterns. Moreover, the artificial fabrication of complex nanostructures via self-assembly is assumed to be of great relevance for future technologies in medicine as well as in engineering, biology and nanotechnology. However, it remains unclear what conditions must be met for such processes to function in a robust and resilient way, and which generic mechanisms apply. A better understanding of the principles and pitfalls of the processes involved might therefore be essential and indicatory for the future development of these fields.

In this work, we show employing theory and mathematical modelling that self-assembling heterogeneous systems are generally subject to stochastic effects, which can dominate the entire dynamics, completely undermining the creation of target structures. Interestingly, these stochastic effects are not captured by the chemical rate equations but can only be revealed in a stochastic description of the assembly process. In other words, with a naïve implementation of the assembly process, the number of correctly and completely assembled structures may be zero even though the chemical rate equations predict perfect yield. Our finding of this stochastic yield catastrophe implies the necessity of additional regulation mechanisms in natural and artificial self-assembling systems, or alternatively, a stochastically robust implementation of the assembly process such as the dimerization scenario we discuss in this work. For homogeneous target structures (i.e. consisting only of a single type of building block) stochasticity does not deteriorate the yield and the system is robust to stochastic effects.

The central results of this work are:

• For biologically relevant subunit numbers, stochastic effects play an important role in self-assembly, and can lead to stochastic yield catastrophes if the target structure is heterogeneous. We specify conditions for the loss of robustness against stochastic effects and discuss their implications.
• Due to stochastic effects, the assembly yield can be a non-monotonic or even an increasing function of the deterministic nucleation speed. As a result, a deterministically slow nucleation speed is not sufficient to obtain good yield. The slow-nucleation principle which states that yield increases if the nucleation speed (i.e. initiation of new structures) is decreased relative to the growth rate, thus, has to be interpreted in terms of the corresponding stochastic framework. These novel and counterintuitive phenomena are due to an increase in nucleation events due to fluctuations between the species.
• We also show that heterogeneity of the target structure is irrelevant if resources are abundant, but plays an essential role if resources are scarce and stochastic effects become important. These findings establish an important connection between hitherto separate fields of research – those concerned with the assembly of homogeneous structures (e.g. virus capsid assembly) and those that focus on heterogeneous structures (e.g. DNA-brick-based assembly).

These findings suggest that, for self-assembling systems, stochastic effects play a vital role and a description based purely on chemical rate equations is, in general, inadequate. We believe that a solid understanding of these stochastic effects is essential for the establishment of a concise theoretical framework for self-assembly.