Statistics and Related Topics in Single-Molecule Biophysics

Abstract

Since the universal acceptance of atoms and molecules as the fundamental constituents of matter in the early twentieth century, molecular physics, chemistry and molecular biology have all experienced major theoretical breakthroughs. To be able to actually "see" biological macromolecules, one at a time in action, one has to wait until the 1970s. Since then the field of single-molecule biophysics has witnessed extensive growth both in experiments and theory. A distinct feature of single-molecule biophysics is that the motions and interactions of molecules and the transformation of molecular species are necessarily described in the language of stochastic processes, whether one investigates equilibrium or nonequilibrium living behavior. For laboratory measurements following a biological process, if it is sampled over time on individual participating molecules, then the analysis of experimental data naturally calls for the inference of stochastic processes. The theoretical and experimental developments of single-molecule biophysics thus present interesting questions and unique opportunity for applied statisticians and probabilists. In this article, we review some important statistical developments in connection to single-molecule biophysics, emphasizing the application of stochastic-process theory and the statistical questions arising from modeling and analyzing experimental data.

Figure 1

A typical enzyme kinetics can be written as a sequence of biochemical steps as in Eq. 19, or from a single enzyme perspective, a cycle as illustrated here. Note that the second order rate constants $k_1^o$ and $k_3^o$ in (19) are replaced by pseudo-first-order rate constants $k_1^{+}$ and $k_3^{-}$, respective. The simplest statistical kinetic model is to consider this system as a continuous-time, discrete-state Markov process. More sophisticated model, when there are sufficient data, could be a semi-Markov model with arbitrary, non-exponential sojourn time for each of the three states [63].

Figure 2

A discrete schematic illustrating the Markovian kinetics of a single enzyme molecule with conformational fluctuations.