Less haste, less waste: on recycling and its limits in strand displacement systems

We study the potential for molecule recycling in chemical reaction systems and their DNA strand displacement realizations. Recycling happens when a product of one reaction is a reactant in a later reaction. Recycling has the benefits of reducing consumption, or waste, of molecules and of avoiding fuel depletion. We present a binary counter that recycles molecules efficiently while incurring just a moderate slowdown compared with alternative counters that do not recycle strands. This counter is an n-bit binary reflecting Gray code counter that advances through 2(n) states. In the strand displacement realization of this counter, the waste-total number of nucleotides of the DNA strands consumed-is polynomial in n, the number of bits of the counter, while the waste of alternative counters grows exponentially in n. We also show that our n-bit counter fails to work correctly when many (Θ(n)) copies of the species that represent the bits of the counter are present initially. The proof applies more generally to show that in chemical reaction systems where all but one reactant of each reaction are catalysts, computations longer than a polynomial function of the size of the system are not possible when there are polynomially many copies of the system present.