A Universal Control Budget for First-Passage Kinetics

Abstract

The first-passage time is the natural observable of reaction completion, yet how its mean responds to a rate change has lacked a general constraint. We show that the logarithmic sensitivity of the mean first-passage time of any finite Markov chain to any rate is bounded by one in magnitude, and that these sensitivities sum to -1. Together the two laws form a conserved control budget: speeding completion through some transitions must be paid for by others, and a coordinated change shifts the completion time only as far as the budget allows. Raising an activation barrier or shifting the depth of a well moves many rates at once, yet neither can shift the completion time further than a single rate could. The budget caps kinetic-proofreading discrimination at the checkpoint count, and prices it in sensitivity to substrate concentration.