Control analysis of stationary forced oscillations

Metabolic control analysis is here extended to stationary dynamic phenomena arising from a steadily oscillating external force. The extension focuses on the form of the oscillations, in terms of the discrete spectrum of the frequencies, as obtained by expansion into a Fourier series. The control of each oscillating metabolite concentration (or reaction rate) by any enzyme in the system is described by (i) periodic control coefficients referring to the control on the time dependence of that concentration and (ii) Fourier control coefficients. One for each Fourier frequency, the latter refer to the control of the waveform (and total amplitude) of the oscillations. It is shown how both types of control coefficient can be expressed in terms of elasticity coefficients (which comprise the relevant enzyme kinetics) and network structure. Importantly, integrals of the elasticity coefficients and reaction rates enter the expressions for the control coefficients; enzyme kinetic information along the entire oscillation route is important for the distribution of the control over the pathway enzymes. For both types of control coefficient, summation and connectivity theorems are derived. Including the control by the external frequency in the summation, the sums equal 0 and 1 for all the Fourier components of concentrations and reaction rates. An example illustrates the application of this control analysis.