Large molecules such as proteins and nucleic acids are crucial for life, yet their primordial origin remains a major puzzle. The production of large molecules, as we know it today, requires good catalysts, and the only good catalysts we know that can accomplish this task consist of large molecules. Thus the origin of large molecules is a chicken and egg problem in chemistry. Here we present a mechanism, based on autocatalytic sets (ACSs), that is a possible solution to this problem. We discuss a mathematical model describing the population dynamics of molecules in a stylized but prebiotically plausible chemistry. Large molecules can be produced in this chemistry by the coalescing of smaller ones, with the smallest molecules, the `food set, being buffered. Some of the reactions can be catalyzed by molecules within the chemistry with varying catalytic strengths. Normally the concentrations of large molecules in such a scenario are very small, diminishing exponentially with their size. ACSs, if present in the catalytic network, can focus the resources of the system into a sparse set of molecules. ACSs can produce a bistability in the population dynamics and, in particular, steady states wherein the ACS molecules dominate the population. However to reach these steady states from initial conditions that contain only the food set typically requires very large catalytic strengths, growing exponentially with the size of the catalyst molecule. We present a solution to this problem by studying `nested ACSs, a structure in which a small ACS is connected to a larger one and reinforces it. We show that when the network contains a cascade of nested ACSs with the catalytic strengths of molecules increasing gradually with their size (e.g., as a power law), a sparse subset of molecules including some very large molecules can come to dominate the system.
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