In order to determine the bimolecular rate constants for the protonation processes we have conducted concentration dependent experiments in which the concentration of the protein was varied at fixed pH.  The concentration of the protein was much larger than the concentration of the photodetached protons and the reactions could be considered pseudo first order.

From the slope of the apparent rate constant of the expansion accompanying the protonation of carboxylic acids as a function of the concentration of the protein we determine the apparent bimolecular rate constant at each pH.   Data reported are at prepulse pH =  7 (rate constant for the expansion and for the contraction) and pH = 4.5 (only the expansion is present).

The contraction shows a concentration dependence at low protein (bimolecular process) concentrations and a saturation at higher concentrations (sequential unimolecular process), indicating that the diffusion  mediated binding of the protons is not the rate limiting step in this range.  This transient is not observed below pH 5.5. The values of the lifetimes are close to each other and the two sequential transients cannot be separated by our experimental setup, also due to the small volume change (-1 ml/mol)  accompanying the protonation of his residues.  The volume change we detect is essentially determined by the following unimolecular, rate limiting step.

Results are summarized in table1. In this table we have reported the values of the bimolecular rate constants (k_2b and k_3b) for proton binding, whereas the plateau value of k_3b, k₃^p, has been used as an estimate of the rate of the sequential unimolecular process.

Reaction volumes DV_R,i have been calculated using the effciency of each process, considering the protonation reactions of carboxyls and histidines as parallel reactions.

The  value of DV_R,2  is lower than the values observed for the protonation of carboxylates in simple molecules, which ranges around 15 ml/mol.  However, as evidenced by Kauzmann (J. Rasper and W. Kauzmann, J. Am. Chem. Soc. (1962), 84, 1771-1777), the environment of the carboxyl groups strongly influences the value of the observed reaction volume.  The small value of DV_R,2 indicates that the groups that are being protonated are close to positively charged residues.

The extent of the slower contraction (DV_R,3) indicates that a substantial solvation is occurring when the involved histidine residues bind a proton.