Protein-Water Energy Transfer via Anharmonic Low-Frequency Vibrations

TL;DR

This work investigates how a solvated protein dissipates excess thermal energy into its aqueous environment by dissecting vibrational energy transfer across frequencies using an anharmonic FRESEAN framework. By comparing equilibrium and steady‑state non‑equilibrium MD simulations, it identifies two main channels: a fast, friction‑driven transfer from zero‑frequency, diffusive/damped modes, and a slower, yet overall dominant transfer from the large number of far‑infrared vibrations around ~70 cm$^{-1}$. The study shows that efficient transfer strongly correlates with zero‑frequency VDoS contributions and solvent coupling, yet large mode counts at finite frequencies ultimately drive most energy dissipation; spectral overlap between protein and water is necessary but not sufficient. Additionally, the FRESEAN analysis provides a sensitive, non‑ergodicity metric via $ extbf{C}( au=0)$ and highlights that finite simulation times can mask true energy equi‑partition in high‑dimensional systems, with implications for understanding heat management in biomolecules.

Abstract

Heat dissipation is ubiquitous in living systems, which constantly convert distinct forms of energy into each other. The transport of thermal energy in liquids and even within proteins is well understood but kinetic energy transfer across a heterogeneous molecular boundary provides additional challenges. Here, we use atomistic molecular dynamics simulations under steady-state conditions to analyze how a protein dissipates surplus thermal energy into the surrounding solvent. We specifically focus on collective degrees of freedom that govern the dynamics of the system from the diffusive regime to mid-infrared frequencies. Using a fully anharmonic analysis of molecular vibrations, we analyzed their vibrational spectra, temperatures, and heat transport efficiencies. We find that the most efficient energy transfer mechanisms are associated with solvent-mediated friction. However, this mechanism only applies to a small number of degrees of freedom of a protein. Instead, less efficient vibrational energy transfer in the far-infrared dominates heat transfer overall due to a large number of vibrations in this frequency range. A notable by-product of this work is a highly sensitive measure of deviations from energy equi-partition in equilibrium systems, which can be used to analyze non-ergodic properties.

Figures (9)

• Figure 1: Vibrational density of states (VDoS) of ubiquitin. (A) Comparison of equilibrium (black) and steady-state non-equilibrium (red) simulations with constant heat flux from the protein to the solvent. (B) Difference between the steady-state and equilibrium VDoS from (A). (C) VDoS of (pure) water simulated with the flexible flexible TIP4P/2005 model used as a solvent in the protein simulations. Error bars are shown as shaded regions but are only faintly visible in the difference spectrum in panel B.
• Figure 2: Average temperature of collective DOFs obtained from FRESEAN mode analysis of the equilibrium system. (A) Temperatures obtained after projecting the equilibrium (black) and steady-state (red) simulations on all eigenvectors of $\mathbf{C}(\nu)$ at zero frequency with statistical error bars. Dotted horizontal lines in black and red indicate average equilibrium and steady-state protein temperatures. Temperatures of thermostats coupled to the protein (310 K) and water (290 K) in the steady-state system are indicated by dashed lines in gray. Eigenvectors are sorted by descending eigenvalues (blue) indicated on the alternative y-axis. (B) Temperature obtained after projecting the steady-state simulation on all eigenvectors of $\mathbf{C}(\nu)$ at all sampled frequencies. The color code describes temperatures within $\pm$10 K of the average protein temperature (306 K); superimposed white lines indicate the number of eigenvalues needed to describe 50%, 75%, 90%, and 99% of the VDoS in Eq. \ref{['e:lambda']} for each frequency. The total VDoS as a function of frequency is shown as an inset on the left; shaded in gray are frequencies with insignificant total VDoS intensities.
• Figure 3: Average temperature of collective DOFs constructed specifically based on differences in kinetic energy as eigenvectors of the static velocity correlation matrix $\mathbf{C}(\tau = 0)$. Results for the equilibrium and steady-state simulations are shown as black and red dots, respectively. Statistical errors are shown as shaded areas but negligible on the scale shown. Average protein temperatures in the equilibrium and steady-state simulations are indicated as dotted horizontal lines in the respective colors. The temperatures of the thermostats coupled to the protein (310 K) and water (290 K) in the steady-state simulation are indicated by horizontal dashed lines in gray.
• Figure 4: VDoS of single collective DOFs. (A) VDoS of fluctuations along eigenvectors of the static velocity correlation matrix $\mathbf{C}(\tau = 0)$ obtained from the equilibrium (top panel) and steady-state (lower panel) simulations. (B) Difference of the single DOF VDoS with respect to the average VDoS over all DOF (the average is equivalent to the protein VDoS for the equilibrium system shown in Figure \ref{['f:vdos']}).
• Figure 5: Protein-solvent heat flux under steady-state conditions. (A) Frequency-resolved heat flux as described by $\Theta(\nu)$ defined in Eq. \ref{['e:flux']}. (B) Frequency-resolved heat flux per degree of freedom obtained by dividing $\Theta(\nu)$ by the protein VDoS. Insets: Frequency-resolved heat flux (defined as in corresponding main panel) at low frequencies (dashed boxes) with comparison to scaled protein and water VDoS (scaled to match peak intensity).