Short time extremal response to step stimulus for a single cell {\sl E. coli}

After application of a step stimulus, in the form of a sudden change in attractant environment, the receptor activity and tumbling bias of an {\sl E. coli} cell change sharply to reach their extremal values before they gradually relax to their post-stimulus adapted levels in the long time limit. We perform numerical simulations and exact calculations to investigate the short time response of the cell. For both activity and tumbling bias, we exactly derive the condition for extremal response and find good agreement with simulations. We also make experimentally verifiable prediction that there is an optimum size of the step stimulus at which the extremal response is reached in the shortest possible time.