Discovering optimal kinetic pathways for self-assembly using automatic differentiation.

Macromolecular complexes are often composed of diverse subunits. The self-assembly of these subunits is inherently nonequilibrium and must avoid kinetic traps to achieve high yield over feasible timescales. We show how the kinetics of self-assembly benefits from diversity in subunits because it generates an expansive parameter space that naturally improves the "expressivity" of self-assembly, much like a deeper neural network. By using automatic differentiation algorithms commonly used in deep learning, we searched the parameter spaces of mass-action kinetic models to identify classes of kinetic protocols that mimic biological solutions for productive self-assembly. Our results reveal how high-yield complexes that easily become kinetically trapped in incomplete intermediates can instead be steered by internal design of rate-constants or external and active control of subunits to efficiently assemble. Internal design of a hierarchy of subunit binding rates generates self-assembly that can robustly avoid kinetic traps for all concentrations and energetics, but it places strict constraints on selection of relative rates. External control via subunit titration is more versatile, avoiding kinetic traps for any system without requiring molecular engineering of binding rates, albeit less efficiently and robustly. We derive theoretical expressions for the timescales of kinetic traps, and we demonstrate our optimization method applies not just for design but inference, extracting intersubunit binding rates from observations of yield-vs.-time for a heterotetramer. Overall, we identify optimal kinetic protocols for self-assembly as a powerful mechanism to achieve efficient and high-yield assembly in synthetic systems whether robustness or ease of "designability" is preferred.