  We perform a spatially resolved simulation study of an AND gate based on DNA
strand displacement using several lengths of the toehold and the adjacent
domains. DNA strands are modelled using a coarse-grained dynamic bonding model
{[}C. Svaneborg, Comp. Phys. Comm. 183, 1793 (2012){]}. We observe a complex
transition path from the initial state to the final state of the AND gate. This
path is strongly influenced by non-ideal effects due to transient bubbles
revealing undesired toeholds and thermal melting of whole strands. We have also
characterized the bound and unbound kinetics of single strands, and in
particular the kinetics of the total AND operation and the three distinct
distinct DNA transitions that it is based on. We observe a exponential kinetic
dependence on the toehold length of the competitive displacement operation, but
that the gate operation time is only weakly dependent on both the toehold and
adjacent domain length. Our gate displays excellent logical fidelity in three
input states, and quite poor fidelity in the fourth input state. This
illustrates how non-ideality can have very selective effects on fidelity.
Simulations and detailed analysis such as those presented here provide
molecular insights into strand displacement computation, that can be also be
expected in chemical implementations.
