We derive a coarse-grained model that captures the temporal evolution of DNA nanotube length distribution during growth experiments. The model takes into account nucleation, polymerization, joining, and fragmentation processes in the nanotube population. The continuous length distribution is segmented, and the time evolution of the nanotube concentration in each length bin is modeled using ordinary differential equations. The binning choice determines the level of coarse graining. This model can handle time varying concentration of tiles, and we foresee that it will be useful to model dynamic behaviors in other types of biomolecular polymers.

We design a new biomolecular circuit with the potential for bistable behavior. We study this candidate toggle switch and find sufficient conditions on key parameters that guarantee bistability. The circuit structure is based on a positive feedback loop created by the mutual repression of two synthetic genes. Repression is generated not by direct inactivation of the promoter region, but by inactivation of the enzymes performing transcription. Inactivation is experimentally possible by designing the RNA outputs of the two genes to be inhibitory aptamers for the enzymes.

In a synthetic biological network it may often be desirable to maximize or minimize parameters such as reaction rates, fluxes and total concentrations of reagents, while preserving a given dynamic behavior. We consider the problem of parameter optimization in biomolecular bistable circuits. We show that, under some assumptions often satisfied by bistable biological networks, it is possible to derive algebraic conditions on the parameters that determine when bistability occurs. These (global) algebraic conditions can be included as nonlinear constraints in a parameter optimization problem. We consider two simple case studies for which we derive bistability conditions using Sturm's theorem, and optimize their nominal parameters to improve switching speed and robustness to a subset of uncertain parameters.