Optimal choice between feedforward and feedback control in gene expression to cope with unpredictable danger

All organisms face risks of unpredictable danger caused by harmful physical environments, pathogens, parasites or predators. Organisms may have several alternative ways of coping with such dangers. These differ in cost, effectiveness and activation time. We study the conditions under which it is optimal to use different alternatives for damage control. As an example we consider a microbe (such as E. coli), which may experience heat shocks that cause denaturation of proteins in the cell. To restore the denatured proteins the organism produces heat-shock proteins (HSP). There are two different pathways for production of HSP. Some HSP are produced immediately after a heat shock (feedforward control), but additional HSP may be produced thereafter, stimulated by the presence of denatured proteins (feedback control). Feedforward is based solely on heat-shock intensity without accurate information on the resulting amount of denatured proteins. We examine the optimal combination of the two pathways that minimizes the sum of the damage caused by the presence of untreated denatured proteins and the production cost of HSP. The optimal response depends on the time delay for feedback control, the effectiveness of HSP in processing denatured proteins, the production cost of HSP, the severity of damage by denatured proteins and the probability distribution of the abundance of denatured protein conditional on heat-shock intensity. We find that feedforward control should always be used. Additional HSP may be produced by feedback control when the abundance of denatured protein is large whilst no feedback control should be used when it is small. All the HSP are produced by feedforward control when the maximum is close to the mean of denatured protein abundance conditional on the heat-shock intensity.