Fluctuation Theories in Model Photosynthetic Systems
Electronic energy transfer (EET) is the physical process responsible for directing and trapping photonic energy in the majority of autotrophic life. These autotrophs allow make life on earth possible by providing dense chemical energy sources for higher organisms. Over evolutionary time scales, autotrophs have evolved complex systems that robustly and re- siliently convert molecular excitations into stored chemical energy. We will analyze one such system in depth: the Fenna-Matthews-Olson pigment protein complex of the green sulfur bacteria.
Although progress has been made towards accurate scientific models to describe EET, proteins are still treated only at the level of harmonic approximations. There are two main precedents for these approximations: First, it is well established that polymer energetic land- scapes are not harmonic, but thanks to the central limit theorem, the collective motions of polymers can often be statistically approximated as a Gaussian-distributed liquid-like sys- tem. Second, folded proteins are often treated as energetically solid in analogy to crystalline solids by treating fluctuations as elastic motions about the native state. The fact that pro- teins fall in the intermediate regime between highly mobile polymers and elastic solids calls into question the harmonic approximations employed in EET modeling. We must validate any approximations made about the system as there is no a priori justification for these approximations in energetics or in a central limit theorem argument. In fact, a growing body of protein research suggests that anharmonic fluctuations and long range correlations are generic features of protein dynamics.
Anharmonic fluctuations in proteins are di cult to describe in a uniform way because they occur heterogeneously across time scales. To begin with, transitions occur over sev- eral picoseconds, but waiting times can be much greater than nanoseconds. Furthermore, proteins may experience periods of high activity followed by periods of no activity at all. This heterogeneity suggests a separation into what are called static fluctuations (comprising activity with duration longer than a few hundred picoseconds) and dynamic fluctuations (comprising activity with duration shorter than 100 ps at which scale the protein is well approximated by elastic motion). Therefore, if we try to use equilibrium methods directly by averaging over long times, we fail to respect the unique ability of the fast fluctuations to respond to impulses. However, if we abandon equilibrium averaging, our results fail to ac- knowledge the conformational diversity inherent to proteins and the possibility for dynamic fluctuations to vary significantly as the conformational space is explored.
We propose a framework that reconciles these disparate limits. By analyzing chromophore electrochromic shifts in the Fenna-Matthews-Olson pigment protein complex on much longer time scales than reported in the literature, we can build a multiscale description of the pigment dynamics. The observed electrochromic shift dynamics are heterogeneous in time. and the protein itself is resistant to symmetric equilibration. Through a decomposition of time scales, we show that while fluctuations slower than 1ns are sluggish and heterogeneous in time, fluctuations faster than 1ns are unchanged by conformational exploration, validating the methods that have been published to estimate spectral densities from simulation. Fast fluctuations are protected from slow conformational heterogeneity by spatial proximity to the chromophore and non-collectivity of contributions from residues (except for a scant few exceptions). We assess the validity and extent to which fluctuations obey Gaussian statistics and linear response with a statistical methodology directly based on the use of a spectral density as a generative model. Slow fluctuations are notably heterogeneous, on the other hand. The conformational changes that correlate to slow electrochromic activity are widely spatially distributed around the protein, making sense of the breadth of time scales on which fluctuations occur.
In many areas of statistical mechanics, the central limit theorem is used as an explanation for the validity of harmonic assumptions. It is thus notable that proteins frequently fail to obey harmonic approximations as shown in experiments and simulations of increasing spatiotemporal resolution. There are likely many relationships between anharmonic protein fluctuations and biologically important observables yet to be discovered. In this vein, the analysis of electrochromic shifts fluctuations are a novel way to understand the coupling between anharmonic protein fluctuations and biological electronic energy transfer. With a better foundational understanding of protein’s impact on EET, we can carry on with greater certainty about the limits of approximate harmonic approaches. Furthermore, with improved protein modeling in EET, it may be possible for EET research to give back to protein research though the deployment and study of natural and artificial optical probes of protein environments.