Fractal analysis of proton exchange kinetics in lysozyme

Experimental data for the exchange of protons in tritiated lysozyme is reexamined by using a fractal model. The fraction of protons unexchanged, f, is seen to follow a stretched exponential, f infinity exp[(-t/tau)alpha], in the long time limit. Data over a range of temperatures are considered, and accurate fits are obtained with a single, unadjusted scaling exponent, alpha. The time constant, tau, follows an Arrhenius law and gives an activation energy comparable to that obtained for free peptide exchange. A model is proposed where proton exchange occurs as a result of solvent reacting with protein side groups in a restricted volume surrounding the protein. Dynamic fluctuations of the protein allow the protonated groups to enter this volume. Solvent also penetrates this volume, allowing proton exchange to occur. The fluctuations of reactants in this restricted volume dominate the kinetics and result in anomalous behavior. The topology of this reaction volume can be characterized by its fractal dimension. The fractal dimension of the space excluded by the protein is equal to 3-ds, where ds is the fractal dimension of the protein surface. The dimensionality of this "reaction space" can be used to predict the value of the exponent alpha. When the problem is treated as a reaction of the type A + B-->C in a confined region, the exponent is given by alpha = (3-ds)/4. By using the value of 2.17 previously established for the surface dimension of lysozyme [Pfeifer, P., Welz, U. & Wippermann, H. (1985) Chem. Phys. Lett. 113, 535-540], a corresponding alpha of 0.21 is calculated. Data for lysozyme at six different temperatures could be accurately fit by using this unadjusted value for alpha. These results show how the surface morphology of a protein influences diffusional processes of small molecules associating with the protein.