Ultrametric theory of conformational dynamics of protein molecules in a functional state and the description of experiments on the kinetics of CO binding to myoglobin

Abstract

The paper is devoted to a systematic account of the theory of conformational dynamics of protein molecules. As an example of application of this theory, we provide a complete analytical description of experiments on the kinetics of CO binding to myoglobin, which were carried out by the group of Frauenfelder more than 30 years ago and acquired the status of base experiments for studying the properties of the fluctuation dynamic mobility of protein molecules. As early as 2001, the authors could demonstrate that, within the model of ultrametric random walk with a reaction sink, the experimental curves of CO binding to myoglobin can be reproduced in the high-temperature region. Later, in 2010, the authors proposed a modified model and, based on its numerical analysis, demonstrated that this model can reproduces the experimental results over the whole temperature range covered in the experiment. In the present study, based on the previously proposed model, we formulate a rigorous mathematical theory of conformational dynamics of protein molecules. We demonstrate that the proposed theory provides not only a complete description of the experiment over the whole temperature range of $\left(60\div300\right)$ K and in the observation time window of $\left(10^{-7}\div10^{2}\right)$ s but also a unified picture of the conformational mobility of a protein molecule, as well as allows one to realize the fact that the mobility changes in a self-similar way. This specific feature of protein molecules, which has remained hidden to date, significantly expands the ideas of dynamic symmetry that proteins apparently possess. In addition, we show that the model provides a prediction for the behavior of the kinetic curves of the experiment in the low-temperature range of $\left(60\div180\right)$ K at times not covered by the experiment (more than $10^{2}$ s).