Solvent viscosity dependence for enzymatic reactions

A mechanism for relationship of solvent viscosity with reaction rate constant at enzyme action is suggested. It is based on fluctuations of electric field in enzyme active site produced by thermally equilibrium rocking (cranckshaft motion) of the rigid plane (in which the dipole moment $\approx 3.6 D$ lies) of a favourably located and oriented peptide group (or may be a few of them). Thus the rocking of the plane leads to fluctuations of the electric field of the dipole moment. These fluctuations can interact with the reaction coordinate because the latter in its turn has transition dipole moment due to separation of charges at movement of the reacting system along it. The rocking of the plane of the peptide group is sensitive to the microviscosity of its environment in protein interior and the latter is a function of the solvent viscosity. Thus we obtain an additional factor of interrelationship for these characteristics with the reaction rate constant. We argue that due to the properties of the cranckshaft motion the frequency spectrum of the electric field fluctuations has a sharp resonance peak at some frequency and the corresponding Fourier mode can be approximated as oscillations. We employ a known result from the theory of thermally activated escape with periodic driving to obtain the reaction rate constant and argue that it yields reliable description of the preexponent where the dependence on solvent viscosity manifests itself. The suggested mechanism is shown to grasp the main feature of this dependence known from the experiment and satisfactorily yields the upper limit of the fractional index of a power in it.

💡 Research Summary

The paper proposes a novel microscopic mechanism to explain the well‑known empirical relationship between solvent viscosity (η) and the rate constant (k) of enzymatic reactions, typically expressed as k ∝ η⁻ᵅ with 0 < α < 1. The authors focus on a specific peptide bond located in the active site that carries a permanent dipole moment of roughly 3.6 D and lies in a relatively rigid planar configuration. At thermal equilibrium this plane undergoes a “crankshaft” motion – a small‑amplitude rocking or torsional oscillation – whose amplitude is damped by the microviscosity (μ) of the surrounding protein matrix. Because μ is itself a function of the bulk solvent viscosity η, the crankshaft dynamics become indirectly η‑dependent.

The rocking of the dipolar plane generates time‑dependent fluctuations of the local electric field, δE(t), at the active site. By treating the angular displacement θ(t) as a harmonic oscillation, θ(t) ≈ θ₀ cos(ω₀t), the field fluctuation can be written as δE(t) ∝ θ₀ cos(ω₀t). Spectroscopic data (e.g., micro‑Raman) suggest that ω₀ lies in the terahertz range (10¹²–10¹³ Hz), and the power spectrum of δE(t) exhibits a sharp resonance peak because the rigid peptide plane limits dissipative broadening.

During the chemical step the reaction coordinate acquires a transition dipole moment μ‡ as charges separate along the pathway. The interaction energy –μ‡·δE(t) therefore adds a periodic driving term to the reaction free‑energy surface. The authors invoke the theory of thermally activated escape over a barrier under periodic forcing (a Kramers‑type framework with a time‑periodic perturbation). In this formalism the rate constant can be expressed as

k = A · exp