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.
Deep Dive into 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 proper
The functional dependence of the rate limiting stage k cat for enzymatic and protein (ligand binding/rebinding) reactions on solvent viscosity η of the type
where 0 < β < 1 (usually β ≈ 0.4 ÷ 0.8) has been known for a long time [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. More detailed studies revealed that in fact the fractional index of a power β is a function of cosolvent molecular weight M (i.e., the mass of a cosolvent molecule expessed in atomic units and measured in Daltons) [8]
If one varies the solvent viscosity by large cosolvent molecules with high molecular weight that do not penetrate into enzyme then one obtains that the fractional exponent β → 0, i.e., the reaction rate constant does not depend on solvent viscosity. With the decrease of cosolvent molecular weight the fractional exponent β increases. In the limit of hypothetical “ideal” cosolvent with infinitely small molecular weight (cosolvent molecules freely penetrate into enzyme and are distributed there homogeneously) it tends to a limit value β max = lim M →0 β(M) ≈ 0.79. The latter is neither experimental value nor a calculated one. It is an extrapolated number (see [8] for details).
The functional dependence (1) also takes place for folding of proteins (see [12], [13] and refs therein). In our opinion enzyme catalysis and folding are quite different phenomena proceeding on different timescales (an enzyme turnover is typically 10 -4 ÷ 10 -3 s while “the time of folding varies from microseconds to hours” [13]). An enzymatic or protein reaction typically has a distinct rate limiting stage and can be perceived as an elementary step. Folding is a complicated process that involves huge number of elementary steps of commensurable importance. That is why we suppose that enzyme catalysis and folding involve different origin of the dependence (1). In the present paper we deal only with enzymatic and protein reactions (i.e., those of bond breaking/bond making in protein interior) and do not touch upon folding. For an unprejudiced observer folding seems to be an overworked issue in the literature while physical aspects of enzyme action still remain in a deep shadow of their chemical counterparts for this phenomenon. Overwhwelming majority of researches from physical community perceive enzyme catalysis as some “chemistry” or “biology”. That is why the aim to attract their attention to it as to a physical problem initiated by the the collection [14] and continued by the review article [11] seems to remain as urgent as it was in the previous century.
The attempts to explain the functional dependence (1) can be roughly divided into “phenomenological” and “theoretical”. The former suggest that the fractional exponent β is the degree with which solvent viscosity is coupled with (frequency dependent friction) [15] or penetrates into (position dependent friction) [16], [17] the protein interior. The latter try to derive it from the first principles [18], [19]. However Zwanzig model yields too small value for the fractional exponent β = 0.5 [19]. Grote-Hynes theory [18] gives that the rate dependence on solvent viscosity should be weaker than that predicted by Kramers’ one (the latter yields k ∝ 1/η in the high friction limit [20]). However no explicit derivation of expression (1) from the Grote-Hynes theory has been achieved. As the authors of [8] conclude “there seems to be no general agreement yet about the origin of the fractional β value in Eq.1”. The authors of [10] draw to a similar conclusion. In our opinion little has changed in this issue (as applied to enzyme catalysis only because there is certain progress in understanding of viscosity dependence for folding [13]) since the date of the cited papers. The aim of the present paper is to provide theoretical interpretation of the functional dependence (1) and to “explain” the limit value β max ≈ 0.79.
There seems to be a consensus among researhes in understanding that the dependence of an enzymatic reaction rate constant on solvent viscosity is mediated by internal protein dynamics. This undersanding goes back to the so called transient strain model. The latter is based on the idea of overcoming the energy barrier of an enzymic reaction by structural fluctuations whose frequency is inversely proportional to the viscosity of the medium [21], [22], [23]. That is why any theory of the phenomenon should be a part of the mainstream of modern enzymology to study the role of dynamical contribution into enzyme catalysis (see the materials of a recent conference in the subject issue of Phil. Trans. R. Soc. B (2006) 361). There are different sonorous names for such dynamical mechanism: “rate promoting vibration” (RPV) [24], the “protein promoting modes” [25], [26], etc. In the present paper the name RPV is used as the most appropriate one for the concept under consideration that some conformational motion of vibrational character in protein is coupled somehow to the reaction coordinate. Howe
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