Introduction: Historical Background and Recent Developments that Motivate this Book

This book grew from the lectures given at the summer schools that the three of us organized in Telluride and in Lausanne. The purpose of the schools was to introduce young researchers to modern kinetics, with an emphasis to applications in life sciences, in which rigorous methods rooted in statistical physics are playing an increasing role. Indeed, molecular-level understanding of virtually any process that occurs in cellular environment requires a kinetic description as well as the ability to measure and/or compute the associated timescales. Similarly, dynamic phenomena occurring in materials, such as nucleation or fracture growth, require a kinetic framework. Experimental and theoretical developments of the past two decades, which are briefly outlined below, rejuvenated the well-established field of kinetics and placed it at the juncture of biophysics, chemistry, molecular biology, and materials science; the composition of the class attending the schools, which included graduate students and postdocs working in diverse areas, reflected this renewed interest in kinetic phenomena. In this book we strive to introduce a diverse audience to the modern toolkit of chemical kinetics. This toolkit enables prediction of kinetic phenomena through computer simulations, as well as interpretation of experimental kinetic data; it involves methods that range from atomistic simulations to physical theories of stochastic phenomena to data analysis.

The material presented in this book can be loosely divided into three interconnected parts. The first part consists of Chapter 1-7 and provides a comprehensive account of stochastic dynamics based on the model where the dynamics of the relevant degree(s) of freedom is described by the Langevin equation. The Langevin model is especially important for several reasons: First, it is the simplest model that captures the essential features of any proper kinetic theory, thereby offering important insights. Second, it often offers a minimal, low-dimensional description of experimental data. Third, many problems formulated within this model can be solved analytically. Theoretical approaches to Langevin dynamics described in this book include the Fokker-Planck equation, mapping between stochastic dynamics and quantum mechanics, and Path Integrals.

The second part of the book, comprised of Chapter 8-12, describes rate theories and explains our current understanding of what a "reaction rate" is and how it is connected to the underlying microscopic dynamics. The rate theories discussed in this part range from the simple transition state theory to multidimensional theories of diffusive barrier crossing. While they were originally formulated in different languages and in application to different phenomena, we explain their interrelationship and formulate them using a unified language.

Finally, the third part, formed by Chapter 13-19, focuses on atomistic simulations and details approaches that aim to: (i) perform such simulations in the first place, (ii) predict kinetic phenomena occurring at long time timescales that cannot be reached via brute force simulations, and (iii) obtain insights from the simulation data (e.g. by inferring the complex dynamical networks or transition pathways in large-scale biomolecular rearrangements).

We start with a brief historical overview of the field. To most chemists, the centerpiece of chemical kinetics is the Arrhenius law. This phenomenological rule states that the rate coefficient of a chemical reaction, which quantifies how fast some chemical species called reactant(s) interconvert into different species, the product(s), can be written in the form

To someone with a traditional physics background, however, this rule may require much explaining. In classical mechanics, the state of a molecular system consisting of atoms is described by its position in phase space , where x is the 3-dimensional vector comprised of the coordinates of its atoms and p the corresponding vector of their momenta. The time evolution in phase space is governed by Newton's second law, which results in a system of 6 differential equations. What exactly does a chemist mean by "reactants" or "products"? When, why, and how can the complex Newtonian dynamics (or the dynamics governed by the laws of quantum mechanics) be reduced to a simple rate law, what is the physical meaning of and , and how can we predict their values?

Early systematic attempts to answer these questions started in the early twentieth century with work by Rene Marcelin [1] and culminated with the ideas of transition state theory formulated by Wigner [2], Eyring [3] and others (see, e.g., the review [4]. In the modern language (closest to Wigner's formulation) "transition state" is a hypersurface that divides the phase space into the reactants and products; it further possesses the hypothetic property that any trajectory crossing this hypersurface heading from the reactants to the products will not turn back. Interestingly, the existence (in principle) of such a hypersurface has remained an unsettled issue even in the recent past [5]. In practice, transition state theory is often a good approximation for gas-phase reactions involving few atoms, or for transitions in solids, which have high symmetry. But practical transition-state theory estimates of the prefactor for biochemical phenomena that take place in solution are usually off by many orders of magnitude.

Several important developments advanced our understanding of the Arrhenius law later in the twentieth century. First, Kramers in his seminal paper [6] approached the problem from a different starting point: instead of considering the dynamics in the high-dimensional space involving all the degrees of freedom, he proposed a model where one important degree of freedom characterizing the reaction is treated explicitly, while the effect of the remaining degrees of freedom is captured phenomenologically using theory of Brownian motion. Kramers's solution of this problem described in Chapter 6 of this book has had a tremendous impact, particularly, on biophysics.

Second, Keck and collaborators [7] showed that, even if the hypersurface separating the products from the reactants does not provide a point of no return and can be recrossed by molecular trajectories, it is still possible to correct for such recrossings, and - if there are not too many - practical calculation methods exist that will yield the exact rate [8-10].

Third, Kramers's ideas were extended beyond simple, one-dimensional Brownian dynamics to include multidimensional effects [11] and conformational memory [12,13]. Moreover, a unification of various rate theories was achieved [4,14,15] using the idea that Brownian dynamics can be obtained from the conservative dynamics of an extended system where the degree of freedom of interest is coupled bilinearly to a continuum of harmonic oscillators [16,17]. This unified perspective on rate theories will be explored in detail in Chapter 9-11 of this book.

There is, however, more to kinetics than calculating the rate of an elementary chemical step. Many biophysical transport phenomena cannot be characterized by a single rate coefficient as in Eq. (1). Yet a description of such phenomena in full atomistic detail would both lack insight (i.e. fail to provide their salient features or mechanisms) and be prohibitive computationally. The challenge is to find a middle ground between simple phenomenological theories and expensive molecular simulations, a task that has been tackled by a host of new methods that have emerged in the last decade and that employ "celling" strategies described in Chapters 18-19 of this book.

Another recent development in molecular kinetics has been driven by single-molecule experiments, whose time resolution has been steadily improving: it has become possible to observe properties of molecular transition paths by catching molecules en route from reactants to products [18,19] as they cross activation barriers. Such measurements provide critical tests of various rate theories described in Chapters 6, 10, 11, and inform us about elusive reaction mechanisms. Properties of transition paths, such as their temporal duration and dominant shape are discussed in Chapter 6-7.

Given that the focus of this book is on the dynamic phenomena in condensed phases, and especially on biophysical applications, a number of topics were left out. Those, for example, include theories of reaction rate in the gas phase. Likewise, the low-friction regime (also known as the energy diffusion regime) and the related Kramers turnover problem [15] for barrier crossing are not covered here, as those are rarely pertinent for chemical dynamics in solution. Throughout most of this book, it was assumed that the dynamics of molecules can be described by the laws of Newtonian mechanics, with quantum mechanics only entering implicitly through the governing inter- and intramolecular interactions. This assumption holds in many cases, but the laws of quantum mechanics become important when a chemical reaction involves the transfer of a light particle, such as proton or electron, and/or when it occurs at a very low temperature. The...