Friction analysis of kinetic schemes: the friction coefficient.

Friction analysis is proposed as the application of general control analysis to single enzymes to describe the control of elementary kinetic steps on the overall catalytic rate. For each transition, a friction coefficient is defined that measures the sensitivity of the turnover rate to the free energy of the transition state complex of the transition. The latter is captured in a single property of the transition, termed friction, as the geometrical mean of the inverse of the forward and backward rate constants. By definition, the friction coefficient measures the relative change in the turnover rate in response to a small change in the friction. The friction coefficient is the sum of the flux control coefficients of the forward and backward rate constants from general control theory and measures the extent to which an elementary step is rate determining. Two basic rules apply to the friction coefficients: (i) the summation theorem states that summation of the friction coefficients over all the steps in a scheme results in a value of 1, and (ii) the group rule states that grouping of rate constants of similar transitions results in a friction coefficient for the group that is the sum of the friction coefficients of the individual steps in the group. The friction coefficients are derived for a number a kinetic schemes taking the rate equations as the starting point and both rules are demonstrated. In fully coupled systems the friction coefficients of individual steps lie between 0 and 1. In partially uncoupled systems the summation theorem applies to all the rates in the system, however, the summation of subsets of friction coefficients may exceed the value of one, implying negative values for other steps in the scheme. The values of individual friction coefficients lie between -1 and 1. The friction coefficient is redefined in a numerical treatment of the steady state of more complex enzymatic schemes.