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

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 were recognized as a basis for studying the properties of the fluctuation dynamic mobility of protein molecules. As early as 2001, the authors demonstrated that the model of ultrametric random walk with a reaction sink well reproduces the experimental curves of CO binding to myoglobin in the high-temperature region. Later, in 2014, the authors proposed a modified model and demonstrated that it can reproduce the experimental results over the whole temperature range covered in the experiment. In the present study, we formulate a rigorous mathematical theory of conformational dynamics of protein molecules on the basis of this model. We demonstrate that this theory provides not only a complete description of the experiment over the whole temperature range of 60÷300 K and in the observation time window of 10−7÷102 s, but also a unified picture of the conformational mobility of a protein molecule; moreover, it demonstrates that the mobility changes in a self-similar way. This specific feature of protein molecules, which has remained hidden to date, significantly expands our view of dynamic symmetry that proteins apparently possess. In addition, we show that the model can predict the behavior of the kinetic curves of the experiment in the low-temperature range of 60÷180 K at times not covered by the experiment (greater than 102 s).