  The promise of self-assembly to enable bottom-up formation of materials with
prescribed architectures and functions has driven intensive efforts to uncover
rational design principles for maximizing the yield of a target structure. Yet,
despite many successful examples of self-assembly, ensuring kinetic
accessibility of the target structure remains an unsolved problem in many
systems. In particular, long-lived kinetic traps can result in assembly times
that vastly exceed experimentally accessible timescales. One proposed solution
is to design non-equilibrium assembly protocols in which system parameters
change over time to avoid such kinetic traps. Here, we develop a framework to
combine Markov state model (MSM) analysis with optimal control theory to
compute a time-dependent protocol that maximizes the yield of the target
structure at a finite-time. We present an adjoint-based gradient descent method
that, in conjunction with MSMs for a system as a function of its control
parameters, enables efficiently optimizing the assembly protocol. We also
describe an interpolation approach to significantly reduce the number of
simulations required to construct the MSMs. We demonstrate our approach on two
examples; a simple semi-analytic model for the folding of a polymer of
colloidal particles, and a more complex model for capsid assembly. Our results
show that optimizing time-dependent protocols can achieve significant
improvements in yields of selected structures, including equilibrium free
energy minima, long-lived metastable structures, and transient states.
