A molecular theory of the structural dynamics of protein induced by a perturbation

An equation to describe the structural dynamics of protein molecule induced by a perturbation such as a photo-excitation is derived based on the linear response theory, which reads 𝐑_α(t)=𝐑_α(t=∞)-1k_BT∑γ⟨Δ𝐑_α(t)Δ𝐑_γ⟩_eq⁽⁰⁾⋅𝐟_γ(0). In the equation, α and γ distinguish atoms in protein, 𝐟_γ(0) denotes a perturbation at time t = 0, 𝐑_α(t) the average position (or structure) of protein atom α at time t after the perturbation being applied, and 𝐑_a(t=∞) the position at t=∞. ⟨Δ𝐑_α(t)Δ𝐑_γ⟩_eq⁽⁰⁾ is a response function in which Δ𝐑_α(t) is the fluctuation of atom α at time t in the equilibrium system. The perturbation is defined in terms of the free energy difference between perturbed and unperturbed equilibrium-states, which includes interactions between solute and solvent as well as those among solvent molecules in a renormalized manner. The response function signifies the time evolution of the variance-covariance matrix of the structural fluctuation for the unperturbed system. A theory to evaluate the response function ⟨Δ𝐑_α(t)Δ𝐑_γ⟩_eq⁽⁰⁾ is also proposed based on the Kim-Hirata theory for the structural fluctuation of protein [B. Kim and F. Hirata, J. Chem. Phys. 138, 054108 (2013)]. The problem reduces to a simple eigenvalue problem for a matrix which includes the friction and the second derivative of the free energy surface of protein with respect to its atomic coordinates.