  This paper deals with temporal enzyme distribution in the activation of
biochemical pathways. Pathway activation arises when production of a certain
biomolecule is required due to changing environmental conditions. Under the
premise that biological systems have been optimized through evolutionary
processes, a biologically meaningful optimal control problem is posed. In this
setup, the enzyme concentrations are assumed to be time dependent and
constrained by a limited overall enzyme production capacity, while the
optimization criterion accounts for both time and resource usage.
  Using geometric arguments we establish the bang-bang nature of the solution
and reveal that each reaction must be sequentially activated in the same order
as they appear in the pathway. The results hold for a broad range of enzyme
dynamics which includes, but is not limited to, Mass Action, Michaelis-Menten
and Hill Equation kinetics.
