Short time extremal response to step stimulus for a single cell E. coli

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