Hamiltonian model for energy condensation in classical systems: Relevance to proteins

Abstract

Recent experimental evidence for collective protein vibrations in the terahertz (THz) domain indicates that energy in biomolecular systems can self-organize in an orderly manner, as anticipated by Fröhlich's theory of condensates within a quantum framework. As a first step to bridge THz experiments with theory, we study the Hamiltonian dynamics of a classical network of coupled normal modes representing Fröhlich-type systems. Our results demonstrate that biologically relevant condensates can emerge at room temperature under appropriate nonlinear coupling schemes. The condensation mechanism remains robust also when the original Fröhlich resonance conditions are relaxed.

Figures (5)

• Figure 1: Illustration of a Fröhlich system consisting of a set of oscillators, or modes, in contact with a heat bath at temperature $T$ and a source at temperature $T_S$. More details are given in the main text. Figure was inspired by Fig. 1 in reimers2009weak.
• Figure 2: Time evolution of harmonic energies in a Fröhlich system made of $9$ protein modes; only the first 5 modes are shown. Results were obtained by solving Eqs \ref{['eq:1']} numerically with frequencies uniformly distributed ranging from $\omega_1 = 0.2\ \text{THz}$ to $\omega_9 = 1 \ \text{THz}$ with a $0.1$-THz increment [While frequencies are typical of low-frequency modes in proteins (see for example ][)rate constant values were essentially chosen to show condensation within a time frame accessible to MD simulations.]go1983dynamics. Other parameters are $\Phi_i = 5\cdot10^{-5} \, \text{ps}^{-1}$, $\Lambda_{ij}= 5\cdot10^{-5} \, \text{ps}^{-1}$, $T=300 \ \text{K}$, $T_S=3000 \ \text{K}$. $\Xi_i$ was set to $5\cdot10^{-6} \, \text{ps}^{-1}$ at $50$ ns ($\Xi_i = 0$ before $50$ ns).
• Figure 3: Time evolution of the total energies, i.e., the sum of kinetic and potential energies, in a Fröhlich system made of $9$ protein modes; only the first 5 modes are shown. Energies were computed as moving averages over $300$ ns. Results were obtained from Hamiltonian dynamics by keeping the bath at $T=300 \ \text{K}$ and the source at $T_S=3000 \ \text{K}$. Protein frequencies were set uniformly from $\omega_1 = 0.2$ to $\omega_9 = 1$ THz with $0.1$-THz increment. $\phi_{ik}$, $\xi_{il}$ and $\lambda_{ijk}$ were taken from Eqs. \ref{['eq:5']} with $\phi=1.0$ and $\xi=0.4$ (from $100$ ns). Top: $\lambda=0.0$, bottom: $\lambda=0.95 \ \text{K}^{-1/2}$. Masses were all set to unity. Friction coefficients for the bath and source thermostats were both set to 0.1 $\mathrm{ps}^{-1}$. Curves in black correspond to the predictions of the FREs using Eqs. \ref{['eq:4']} with $\alpha=0.02$ ps; the solid black line shows mode 1 while dashed curves correspond to secondary modes. Predictions of FREs are similar to those expected based on ref. reimers2009weak
• Figure 4: Condensation index $\rho$ of a Fröhlich system as deduced from Hamiltonian dynamics for different $\lambda$ values. $\lambda_{ijk}$ coefficients were calculated from Eq. \ref{['eq:5c']} under different types of resonance: $\omega_i - \omega_j \pm \omega_k^{ (B)} = 0$ (Fröhlich), $\omega_i + \omega_j - \omega_k^{ (B)} = 0$ (Lifshits) and $\omega_i \pm \omega_j \pm \omega_k^{ (B)} = 0$ (combination of both). Other parameters remain the same as Fig. \ref{['fig:epsart3']}.
• Figure 5: Blue: condensation ratio $\rho$ as a function of the scaling factor $\kappa$ multiplying the coupling coefficients $\phi$, $\xi$, and $\lambda$ and the friction coefficients $\gamma$ and $\gamma_S$ simultaneously. $\kappa=1.0$ corresponds to the parameters used in Fig. \ref{['fig:epsart3']}. Except for $\phi$, $\xi$, $\lambda$, $\gamma$, and $\gamma_S$, all other parameters are the same as in Fig. \ref{['fig:epsart3']} for all $\kappa$ values. Green: condensation time (see main text for details).