## Description

This is a book about the physical processes in reacting complex molecules, in particular in biomolecules. In the last decade scientists from different fields as medicine, biology, chemistry, and physics collected a huge amount of data about the structure, the dynamics and the functioning of biomolecules. Great progress has been achieved in exploring the structure of complex molecules. The knowledge of the structure of complex molecucules is of course a ‘conditio sine qua non’ for the understanding of their functioning, however the understanding of the dynamics is as important [Prauenfelder & Wolynes, 1985; Preissner, Goede & Froemmel, 1986; Mc- Cammon & Harvey, 1987; Havsteen, 1989; Froemmel & Sander, 1989]. Without a deep analysis of the physical mechanisms of the dynamics it seems to be impossible to understand the all details of the functioning of biological macromolecules. In particular this refers to the functioning of enzymes, which are the basic molecular machines working in living systems. Since this molecules operate on many thousands of degrees of freedom we have to start to analyse the physical mechanisms e.g. the dynamics of clusters consisting of a many atomic units. Further we have to study the dynamics of conformations, the dynamics of transitions between conformations etc.. In order to give an example, we want to understand the dynamics and the physical mechanism of enzyme-catalyzed bond breaking in substrate molecules. In particular we want to find out what determines the high rate of bond breaking in complex molecules. However to explore the dynamics of this or other complex processes we have to pay a price, only very simple structures allow a investigation of the dynamical phenomena. This is why we have to restrict our studies to rather simple models. In this context we will analyse simple mechanisms as the transitions beween two potential wells, the nonlinear coupling between oscillatory modes, the Fermi resonance, the excitation of solitons in chains of nonlinear springs etc..

The analysis of the complex processes developed in this book is based on methods of nonlinear dynamics, stochastics and molecular dynamics. In the first part of the book we start from the classical stochastic reaction theory. We intended to show how the famous Kramers expression for the chemical reaction rate is to be modified in the case of the more complicated processes occuring during the enzymatic catalysis. Kramers’ classical reaction theory describes reactions as transitions over a potential barrier (activation processes) by studying Langevin equations and solving the corresponding Fokker-Planck equations. The basic assumption of Kramers model is that transitions over a potential barrier are due to stochastic forces. Kramers’ model is based on the assumption of uncorrelated stochastic forces. For the case of reactions with simple molecules this model has been very successful [Hanggi, Talkner and Borkovec, 1991; Popielavski & Gorecki, 1991]. On the other hand there are specific reaction effects, which cannot be understood on the basis of Kramers model [Troe, 1991]. In particular this is true for enzymatic reactions which show reaction rates which are by orders of magnitude higher than the simple estimates provided by the Kramers theory [Chernavsky, Khurgin and Schnol, 1967; Volkenstein, 1981,; Somogyi, Welch and Damjanovich, 1984; Ebeling and Romanovsky, 1985; McCammon and Harvey, 1987; Havsteen, 1989, 1991].