We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.

Toxin-antitoxin modules are gene pairs encoding a toxin and its antitoxin, and are found on the chromosomes of many bacteria, including pathogens. Here, we characterize the specific contribution of the TxpA and YqcG toxins in elimination of defective cells from developing Bacillus subtilis biofilms. On nutrient limitation, defective cells accumulated in the biofilm breaking its symmetry. Deletion of the toxins resulted in accumulation of morphologically abnormal cells, and interfered with the proper development of the multicellular community. Dual physiological responses are of significance for TxpA and YqcG activation: nitrogen deprivation enhances the transcription of both TxpA and YqcG toxins, and simultaneously sensitizes the biofilm cells to their activity. Furthermore, we demonstrate that while both toxins when overexpressed affect the morphology of the developing biofilm, the toxin TxpA can act to lyse and dissolve pre-established B. subtilis biofilms.