Model for solvent viscosity effect on enzymatic reactions
aa r X i v : . [ q - b i o . B M ] O c t Model for solvent viscosity effect onenzymatic reactions
A.E. Sitnitsky,
Institute of Biochemistry and Biophysics, P.O.B. 30, Kazan 420111, Russia.e-mail: [email protected]
Abstract
Why reaction rate constants for enzymatic reactions are typically inversely propor-tional to fractional power exponents of solvent viscosity remains to be already athirty years old puzzle. Available interpretations of the phenomenon have not ledto consensus among researches about its origin. They invoke to either a modifica-tion of 1. the conventional Kramers’ theory or that of 2. the Stokes law. We showthat there is an alternative interpretation of the phenomenon at which neither ofthese modifications is in fact indispensable. Basing on an analogy from the theoryof adsorption on heterogeneous surfaces we reconcile 1. and 2. with the experimen-tally observable dependence. We assume that an enzyme solution in solvent withor without cosolvent molecules is an ensemble of samples with different values ofthe viscosity for the movement of the system along the reaction coordinate. We as-sume that this viscosity consists of the contribution with the weight q from cosolventmolecules and that with the weight 1 − q from protein matrix and solvent molecules.We introduce heterogeneity in our system with the help of a distribution over theweight q . The function of the distribution is a unique characteristic of the solutionof enzymes (type of the enzyme, cosolvent molecular weight, pH, temperature, etc.).We verify the obtained solution of the integral equation for the unknown function ofthe distribution by direct substitution. We conclude that even at linear relationshipbetween the solvent viscosity and that for the movement of the system along thereaction coordinate our approach enables us to obtain the required dependence. Allparameters of the model are related to experimentally observable values. The mean-ing of fractional exponents appears to be the characteristic for the behavior of thedistribution with the variation of the weight q . Our approach yields the existenceof the limit value for the fractional power exponent with the decrease of cosolventmolecular weight that is in agreement experimental data known from the literature.This limit value is determined by the properties of the protein structure and thusis a unique characteristic of the type of the enzyme only rather than that of thesolution. General formalism is exemplified by the analysis of literature experimentaldata for oxygen escape from hemerythin. Key words: enzyme catalysis, solvent viscosity, Kramers’ theory, proteindynamics.
Preprint submitted to 16 October 2018
Introduction
Viscosity dependence of enzymatic and protein (ligand binding/rebinding)reactions is a long standing problem for biophysics [1], [2], [3], [4], [5], [6], [7],[8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. For such reactionsthe functional dependence of the reaction rate constant for the rate limitingstage k on solvent viscosity η has the form k ∝ η/η ) β (1)where η is the viscosity of pure solvent (for water η = 1 cP at room tempera-ture) and 0 < β < β ≈ . ÷ . η < cP . Similar dependence also takes place for folding of proteins(see [20], [21], [22] and refs. therein) and at formation of protein structure [23].However we will not touch upon these processes in the present paper.Much efforts were devoted to explaining the functional dependence (1) in theprevious century with no obvious consensus and commonly accepted mecha-nism of the phenomenon. For instance the authors of [13] conclude that ”thereseems to be no general agreement yet about the origin of the fractional β valuein Eq.1”. The authors of [16] draw to a similar conclusion: ”At present thereis no general agreement on the meaning of fractional exponents implying abreakdown of Stokes law”. Despite steady growth of experimental data littlehas changed in this issue since the date of the cited papers (cf., e.g.,: ”Themeaning of fractional exponents implying that the friction coefficient does notvary linearly with the solvent viscosity, essentially a violation of Stokes law( f ∝ η s ), is not clear at present” [22]). Detailed studies revealed that in factthe 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 measuredin Daltons) β = β ( M ) [13]. If one varies the solvent viscosity by large cosolventmolecules with high molecular weight that do not penetrate into enzyme thenone obtains that the fractional exponent β →
0, i.e., the reaction rate constantdoes not depend on solvent viscosity. With the decrease of cosolvent molec-ular weight the fractional exponent β increases. In the limit of hypothetical”ideal” cosolvent with very small molecular weight (cosolvent molecules freely Email address: [email protected] (A.E. Sitnitsky). β max = lim M → M min β ( M ) (2)For oxygen escape from hemerythin this value is β max ≈ .
79. The latter isneither experimental value nor a calculated one. It is an extrapolated number(see [13] for details). The existence of enzymatic reactions with β = 1 [17],[21] suggests that the limit value is a unique characteristic of the type of theenzyme and can take the values up to 1 as well.The main tool to describe the viscosity dependence of a reaction rate constantis the high friction limit (also called strong damping or overdamped regime) ofthe Kramers’ theory [24]. In it the reaction is conceived as a diffusion processof a particle with some effective mass along a reaction coordinate over somepotential surface. The friction coefficient for the particle ν is supposed to obeythe Stokes law ν ∝ η that yields the well known dependence k K ∝ η/η (3)As was stated above for enzymatic reaction this dependence is inconsistentwith observations. There are two strategies to interpret the experimentallyobserved dependence (1) at present: 1. to modify the Kramers’ theory and2. to modify the Stokes law. The first approach usually leads to rather com-plicated theoretical constructions that have not proved to yield a universaland commonly accepted resolution of the problem. It is realized in, e.g., theGrote-Hynes theory (GH) [25], the Zwanzig model [26] or the model suggestedin [27], [19]. GH gives that the rate dependence on solvent viscosity shouldbe weaker than that predicted by Kramers’ one. The authors of this modelargue that the friction coefficient should be proportional to a high-frequencyviscosity at an appropriately renormalized frequency [25]. Assuming a pos-itive power dependence of viscosity on frequency, they find that the renor-malized high frequency viscosity has a fractional power dependence on thelow-frequency viscosity which can be measured as the usual viscosity η . Thustheir assumption is in fact introducing the fractional exponent ”by hand” ina phenomenological manner. The Zwanzig model infers the reaction rate con-stant from the first principles but yields too small value for the fractionalexponent β = 0 .
5. The model [27], [19] deals only with the limiting case of thehypothetical ”ideal” cosolvent with very small molecular weight and yields forthe fractional exponent the limit value β max ≈ .
75 that is in good agreementwith the extrapolated one β max ≈ .
79 from the paper [13]. No extension ofthis model to the case of realistic cosolvent with finite molecular weight hasbeen suggested yet. The second approach takes into account that the value of3iscosity in enzyme active site differs from that of the solvent [2] and may bea nonlinear function of the latter. It is suggested that the fractional exponent β is the degree with which solvent viscosity is coupled with (frequency depen-dent friction) [5], [28], [16], or penetrates into (position dependent friction)[4], [14] the protein interior. Any of these variants yields the modification ofthe Stokes law ν ∝ η β . However the fractional exponent β in these approachesstill appears as an empirical parameter. It accounts for the nonlinearity of therelationship between the values of the solvent viscosity and that in the enzymeactive site but its value is not calculated and its origin remains mysterious.In our opinion there is another way to interpret the experimental data that re-quires neither modification of the Kramers’ model nor that of the Stokes law.Thus the aim of the present paper is to reconcile the conventional Kramers’model and the Stokes law with the experimentally observed dependence (1).The main premise of our approach is in the fact that a realistic enzyme solu-tion in solvent with or without cosolvent molecules is actually an ensemble ofenzymes with different conditions (values of viscosity for the movement of thesystem along the reaction coordinate). We show that even at linear relation-ship between solvent viscosity and that for the movement of the system alongthe reaction coordinate we can obtain (1) if we take into account heterogeneityof conditions in the ensemble. It should be stressed that the present approachcan be generalized to be based not on the Kramers’ model but on the mod-els [25], [26] or [27], [19]. However for the sake of definiteness we choose theKramers’ model and reconcile the simple dependence (3) with the experimen-tally observable expression for the reaction rate constant (1). The latter isobtained by averaging of individual Kramers’ rate constants over the distri-bution. The aim of the present paper is to show that the idea of heterogeneityenables one to resolve the thirty years old puzzle of solvent viscosity effecton enzymatic reaction rate constant in a conceptually much more simple waythan modification of either the Kramers’ theory or that of the Stokes law.Also we stress that our distribution has nothing to do with the nonthermallyequilibrium form of the reactant distribution during the reaction employed inthe approach of [29].There is a noteworthy analogy between the problem under consideration andthat of adsorption on heterogeneous surfaces. In the latter one reconciles theLangmuir isotherm grounded by statistical mechanics and giving the num-ber of adsorbed molecules θ proportional to pressure P in the low pressurelimit θ ∝ P with the experimentally observable phenomenological Freundlichisotherm giving the dependence θ ∝ P δ where 0 < δ <
1. One attains this aimvia introducing a distribution over energy due to heterogeneity of the surface.As is well known the surface heterogeneity in adsorption theory is taken intoaccount by integral equation approach (see, e.g., [30], [31] and refs. therein).This line in adsorption theory has a long history. It was initiated by the paperof Zel’dovich [32] (see, e.g., [33] for authoritative discussion of priority ques-4ions). The next important steps were made by Sips [34], [35] and Misra [36]who used Stieltjes transform. Detailed discussion of related problems is givenin [37], [38], [39]. For the solution of the corresponding integral equations theauthors of these papers used the so called condensation approximation [37],[31]. Finally this line was culminated by the paper of Landman and Montroll[40] who applied powerful Wiener-Hopf technique for the solution of the corre-sponding integral equations. The enormous preceding experience accumulatedin this field of chemical physics provides an invaluable source of informationfor our problem both from the side of conceptual aspects and that of mathe-matical technique.The paper is organized as follows. In Sec. 2 the character of the relation-ship between the viscosity for the movement of the system along the reactioncoordinate and that of solvent viscosity is discussed. In Sec. 3 we formulatethe integral equation for taking into account the heterogeneity of conditionsin the enzymes from the ensemble via introducing the distribution over theweight q . In Sec. 4 we cast this equation to the form solvable with the help ofFourier transform and verify our solution by direct substitution. In Sec.5 weobtain the relationship of the all parameters of the model with experimentallyobservable values. In Sec. 6 the results are discussed and the conclusions aresummarized. In Appendix some known mathematical formulas are collectedfor convenience. The reaction coordinate in the Kramers’ theory is a notion that enables one toreduce very complicated dynamics of the system in a multidimensional config-urational space to the movement of a particle with some effective mass alongone effective dimension. For the case of an enzymatic reaction the movementof the system along the reaction coordinate is determined by: processes in thesubstrate molecule (e.g., stretching of the bond to be cleaved); friction forthese processes by the environment in the active site of the enzyme (that isstipulated by solvent and cosolvent molecules and also protein side chains lo-cated there); relevant motion for catalysis of side chains and larger fragmentsof structure from protein interior (such as, e.g., α -helicies and β -sheets) andfriction to it by solvent and cosolvent molecules and also by protein matrixin their environment; etc. We denote the viscosity for the movement of thesystem along the reaction coordinate as ζ and suppose the conventional Stokeslaw for the friction coefficient of the particle ν ∝ ζ . The solvent viscosity wedenote as earlier η . The linear relationship between η and ζ can be motivatedas follows. 5here seems to be consensus among researches that the effect of solvent vis-cosity on the enzymatic reaction is mediated by protein dynamics [2], [3], [5],[6], [8], [9], [10], [11], [12], [15]. That is why we further identify the viscosity forthe movement of the system along the reaction coordinate with the viscosityat relevant motion for catalysis of individual fragments in protein interior andin enzyme active site. Experiment yields that protein dynamics over a widerange of temperatures obey the relationship between solvent viscosity η andthe mean square displacement (MSD) < u > [41], [42] < u > = g ln ( η/η ) (4)where g is a constant. We notice that firstly (4) is divergent at η/η = 1and secondly nobody verifies (1) experimentally in the limit η/η → η/η ) min ≈ .
6. That is why further on we consider the range of solventviscosity ηη ≥ < u > = k B Tmω (6)where k B is the Boltzman constant, T is the temperature, m is the mass ofthe moving fragment and ω is the frequency defined by the curvature of thepotential well near its bottom for the motion of the fragment. This motionis determined by the value of the local viscosity ζ of the environment for themotion of the fragment in protein interior.First we notice that a nonlinear relationship between ζ and η is compatiblewith (4) and (6) with the linear one being a particular case. We assume that ζ /η consists of the contribution with the weight 1 − q determined by proteinmatrix and solvent molecules and the relative contribution with the weight q from cosolvent molecules in the fragment environment. The weight q may takevalues from the range0 < q < E/ ( k B T )]subtracted by 1 (because for the case of pure solvent η/η = 1 or equivalentlyat E = 0 the exponent is 1). Here E is the activation energy ( E ≥ ζ /η = a (cid:26) (1 − q ) + q (cid:20) exp (cid:18) Ek B T (cid:19) − (cid:21)(cid:27) (8)The parameter a is related to the fractional power exponent β that will beobtained later on theoretical grounds. Also below the explicit dependence a = a ( M ) of the parameter on the cosolvent molecular weight M will beobtained from the analysis of experimental data.One can easily see that the nonlinear relationship between the solvent viscosity η and the local viscosity for the motion of the fragment in protein interior ζ of the form ζη = a ( (1 − q ) + q " ηη ! κ − (9)is compatible with the relationships (4), (6) and (7) at any κ > g = Emω κ (10)At κ = 1 we have the particular linear case ζη = a " (1 − q ) + q ηη − ! (11)There in no wonder that we can obtain the experimentally observable depen-dence (1) for the nonlinear case 0 < κ < β = κ . Our aim isto stress that even the linear case κ = 1 can lead to the required dependence(1) as well. At the same time it should be emphasized that the assumptionabout linearity is by no means crucial for the validity of our approach. Thelatter can still remain workable for the general nonlinear case even at κ = β .However we will not explore this option in the present paper and restrict our-selves by the linear case κ = 1. In our opinion namely this case can elucidatethe essence of our approach with full clarity. We pursue this line in the nextSec. 7 Formulation of the integral equation
The value of viscosity for the movement of the system along the reactioncoordinate may be different for different enzyme molecules (samples) from theensemble of enzymes. In the individual samples from the ensemble we haveheterogeneity of conditions both in the enzyme active sites and in proteininterior. That is why the experimentally observable dependence of the reactionrate constant on solvent viscosity should be determined by some averaged valueof the viscosity for the movement of the system along the reaction coordinate.We consider the following way to take into account this heterogeneity byaveraging with the help of some integral equation.We assume that due to heterogeneity of conditions in the individual samplesfrom the ensemble of enzymes the weight q in (8) and consequently in (11)can take different values from the range 0 < q <
1. Thus the ensemble ischaracterized by the distribution ρ ( q ) over the values of the weight q . We alsoassume that the reaction rate constant for the individual sample from the en-semble of enzymes firstly obeys the Stokes law ν ∝ ζ /η and secondly is givenby the Kramers’ formula (3) that is k K ∝ ( ζ /η ) − . Then the experimentallyobservable reaction rate constant is the average over the distribution k ηη ! = Z dq ρ ( q ) k K ζη q, ηη !! (12)The distribution ρ ( q ) must be normalized Z dq ρ ( q ) = 1 (13)As we want the reaction rate constant to be the experimentally observable (1)and the local viscosity ζ to be (11) then the relationship (12) takes the form1( η/η ) β = 1 a Z dq ρ ( q )(1 − q ) + q ( η/η −
1) (14)The latter is a Fredholm integral equation of the first kind for the unknownfunction of the distribution ρ ( q ). It is analyzed in the next Sec.8 Analysis of the integral equation (14)
We introduce the critical value of the weight qq c = 12 (15)Equation (14) can be cast into the form solvable with the help of Fouriertransform. Making use of N2.2.6.15. from [45] and the properties of the hyper-geometric function (see Appendix) one can easily verify by direct substitutioninto (14) that in the range (5) the normalized solution of (14) is ρ ( q ) = sin ( πβ ) πq − β ( q c − q ) β (16)for 0 ≤ q ≤ q c and ρ ( q ) = 0 (17)for q c < q <
1. At that the parameter a and the the fractional power exponent β must obey the relationship a = 2 β (18)From here we have β = ln aln ρ ( q ) is depicted at several values of β . The main experimental facts on the effect of solvent viscosity on the enzymaticreaction rate constant are: 1. the inverse proportionality to fractional powerexponents of solvent viscosity given by (1) and 2. the existence of the limitvalue β max of the fractional power exponent with the decrease of cosolventmolecular weight given by (2). The typical dependence of the fractional powerexponent β on the cosolvent molecular weight M has the form [13] β ( M ) = bM δ (20)9t M min ≥ M ≥ M max so that β max = β ( M min ). The dependence (1) isexperimentally verified at values of solvent viscosity higher that some minimalvalue ( η/η ) min . For the oxygen escape from hemerythin the experimentalvalues are: β max = 0 . b = 1 . δ = 0 . M min = 18, M max = 500000 and( η/η ) min ≈ . a on the cosolvent molecular weight a ( M, b, δ ) = 2 b/M δ (21)In Fig.2 the dependence of the parameter a ( M, b, .
23) as the function of thelogarithm of the cosolvent molecular weight M is plotted for several valuesof the parameter b at δ = 0 .
23. In Fig.3 the dependence of the parameter a ( M, . , δ ) as the function of the logarithm of the cosolvent molecular weight M is plotted for several values of the parameter δ at b = 1 .
52. At M → M min we have a ( M ) → a max = a ( M min , b, δ ). The behavior of a max as the functionof the parameters b and δ is plotted in Fig.4. In the limit a ( M, b, δ ) → a max the value of the fractional power exponent tends to the limit value β max . From(19) we have β max = ln a max ln β max given by (22) is plotted as thefunction of a max . The experimental value β max = 0 .
79 for the oxygen escapefrom hemerythin from [13] is obtained at a max ≈ . A theoretical interpretation of the phenomenon of solvent viscosity dependencefor the enzymatic reaction rate constant must account for two undisputableexperimental facts: 1. the inverse proportionality to fractional power exponentsof solvent viscosity given by (1) and 2. the existence of the limit value β max ofthe fractional power exponent with the decrease of cosolvent molecular weightgiven by (2). Up to now suggested models were mainly concerned with theexplanation of the first fact while the second one was overlooked or ignored.On the other hand the model [19] suggests the explanation of the secondfact but encounters difficulties at general description of the first one. Thepresent approach is an alternative to all previous attempts that gives combinedinterpretation of both experimental facts in a self-consistent manner.The basis of our approach is the relationship (8). The latter is compatible withboth nonlinear relationship of the viscosity for the movement of the system10long the reaction coordinate with solvent viscosity (9) and the linear one(11). The choice among these options is by no means crucial for our approach.However we choose the linear relationship (11) because in this case the ap-pearance of the fractional dependence (1) seems to be more nontrivial thanin the case of nonlinear relationship. In the latter case we introduce in themodel the fractional exponent κ = 1 from the very beginning by hands. Thisfact obscures the essence of our approach that in fact all additional fractionalexponents and nonlinearities are superfluous. In (8) we factorize the viscosityfor the movement of the system along the reaction coordinate ζ /η onto thecontribution from protein matrix and solvent molecules and that from cosol-vent molecules. The latter enters in ζ /η with the weight q . This weight maytake different values from the range 0 < q < ρ ( q ). We show that the very existence of this distribution isquite sufficient for the appearance of the required dependence of the reactionrate constant on solvent viscosity (1) consistent with observations. Neitherthe modification of the Kramers’ theory nor that of the Stokes law is in factindispensable at such approach. In this regard our approach is conceptuallymuch more simple than other interpretations of the phenomenon under con-sideration. The approach is motivated by vivid analogy of our problem withthat of adsorption on heterogeneous surfaces and thus has apparent roots inthe preceding experience of chemical physics. In this connection we note thatwe can cast (11) into the form ζη = a " q ηη − ! (23)and call q the ”shielding parameter” that shows to what extent protein struc-ture shields and screens the reaction coordinate from the solvent viscosity at η/η ≥
2. This parameter is less than 1 (0 < q <
1) because in experiment thesolvent viscosity is varied by the cosolvent of the type of sugars, such as tre-halose. It is known that ”in trehalose solutions, there is generally a deficiencyof trehalose and an excess of water in the vicinity of the protein” [46]. Thedeficiency of the cosolvent compared with the bulk in this case is by far takesplace for protein interior because the larger the cosolvent molecular weightso much the worse it can be transmitted by protein structure to the reactioncoordinate. Then the shielding parameter quantifies the measure of the de-ficiency of the cosolvent molecules in the vicinity of the reaction coordinatecompared with the bulk. We can represent the shielding parameter q in theenergetic form exp [ − ǫ/ ( k B T )] where ǫ ≥
0. In this case instead of the distri-bution ρ ( q ) over the weight q we obtain the one ψ ( ǫ ) over the energy ǫ thathas the form identical to the well known Sips’s distribution from the theoryof adsorption on heterogeneous surfaces. However in our problem the physicalmeaning of the weight q seems to be more lucid than that of the shielding11arameter. That is why we prefer to deal with the former notion rather thanwith the latter one.Our approach takes into account the experimental fact that the fractionalpower exponent β tends to some limit value β max with the decrease of thecosolvent molecular weight M . This fact is incorporated in the model as theinherent one. The limit value β max is the function of the parameters b and δ .These parameters characterize the protein structure itself and its ability totransmit cosolvent molecules to the reaction coordinate. We conclude that thelimit value of the fractional power exponent β max is a unique characteristic ofthe type of the enzyme (that determines the protein structure) rather thanthat of the solution of the enzymes. This conclusion is in agreement withexperimental data (see Introduction).The equation (17) and Fig.1 show that the distribution ρ ( q ) is zero abovethe critical value of the weight q c = 0 . ζ /η can not enter with the weightgreater that one half of that from protein matrix and solvent molecules tobe compatible with observations expressed by (1). For small values of thefractional power exponent β the distribution is skewed to the upper bound q = q c of the range while for large values of β it is skewed to the lower bound q = 0. In between 0 < q < q c the distribution ρ ( q ) tends to infinity at both q →
0+ and q → q c − . Thus there are two main fractions of the samples inthe ensemble of enzymes: those with pure contribution from protein matrixand solvent molecules (when q ≈
0) and those with the contribution fromcosolvent molecules having the weight q ≈ q c and the contribution from proteinmatrix and solvent molecules having the weight ≈ − q c . These fractionsare the most representative in the ensemble of enzymes. However there arealways fractions of the samples with intermediate values of the weight q inbetween 0 < q < q c . The behavior of the distribution ρ ( q ) in this region isquantified by the fractional power exponent β (see Fig.1). Thus the physicalmeaning of the fractional power exponent β in the experimentally observabledependence of the enzymatic reaction rate constant on solvent viscosity (1)is the characteristic of the behavior of the distribution ρ ( q ) over the weight q in the ensemble of enzymes. This distribution characterizes the solutionof enzymes, i.e., is determined among with the type of the enzyme by suchcharacteristics as the cosolvent molecular weight, pH, temperature, etc. Thedistribution ρ ( q ) over the weight q acquires at our approach the status of theunique characteristic for the solution of enzymes.We conclude that our approach yields conceptually simple interpretation andquantitative description of the main experimental data on solvent viscositydependence for enzymatic reactions. 12 Appendix
The formula N2.2.6.15. from [45] is a Z dx x α − ( a − x ) β − ( x + z ) − ρ = a α + β − z − ρ B ( α, β ) F ( α, ρ ; α + β ; − a/z ) (24)where | arg z | < π ; a > , Re α > , Re β > F ( a, b ; c ; z ) = (1 − z ) − a F (cid:18) a, c − b ; c ; zz − (cid:19) (25)see N7.2.1.8. in [47] F (0 , b ; c ; z ) = F ( a, c ; z ) = 1 (26)These properties are used at obtaining (16).Acknowledgements. The author is grateful to Dr. Yu.F. Zuev for helpful dis-cussions. The work was supported by the grant from RFBR and by the pro-gramme ”Molecular and Cellular Biology” of RAS.13 eferences [1] B. Gavish, The role of geometry and elastic strains in dynamic states of proteins,Biophys. Struc. Mech. 4 (1978) 37-52.[2] B. Gavish, M.M. Werber, Viscosity-Dependent Structural Fluctuations inEnzyme Catalysis, Biochemistry 18 (1979) 1269-1275.[3] D. Beece, L. Eisenstein, H. Frauenfelder, D. Good, M. C. Marden, L. Reinisch,A. H. Reynolds, L. B. Sorensen and K. T. Yue, Solvent Viscosity and ProteinDynamics, Biochemistry, 19 (1980) 5147- 5157.[4] B. Gavish, Position-dependent viscosity effects on rate coefficient,Phys.Rev.Lett. 44 (1980) 1160-1163.[5] W. Doster, Viscosity scaling and protein dynamics, Biophys. Chem. 17 (1983)97-103.[6] B. Gavish, Molecular dynamics and the transient strain model of enzymecatalysis, in: C.R. Welsh, ed., The fluctuating enzyme, Wiley, N.Y., 1986.[7] H. Frauenfelder, F. Parak, R.D. Young, Conformational substates in proteins,Ann.Rev.Biophys.Chem. 17 (1988) 451-479.[8] A.P. Demchenco, C.I. Rusyn, E.A. Saburova, Kinetics of the lactatedehydrogenase reaction in high-viscosity media, Biochem. et Biophys. Acta 998(1989) 196-203.[9] K. Ng, A. Rosenberg, Possible coupling of chemical to structural dynamics insubtilisin BPN’ catalyzed hydrolysis, Biophys Chem. 39 (1991) 57-68.[10] K. Ng, A. Rosenberg, The coupling of catalytically relevant conformationalfluctuations in subtilisin BPN’ to solution viscosity revealed by hydrogen isotopeexchange and inhibitor binding, Biophys Chem. 41 (1991) 289-99.[11] B. Gavish, S. Yedgar, Solvent viscosity effects on protein dynamics: updatingthe concepts, in: Protein-solvent interactions, Ed. R. B. Gregory, Dekker, N.Y., 1994.[12] W. Doster, Th. Kleinert, F. Post, M. Settles, Effect of solvent on protein internaldynamics, in: Protein-solvent interactions, Ed. R. B. Gregory, Dekker, N. Y.,1994.[13] S. Yedgar, C. Tetreau, B. Gavish and D. Lavalette, Viscosity dependence of O escape from respiratory proteins as a function of cosolvent molecular weight,Biophys. J. 68 (1995) 665-670.[14] G. Barshtein, A. Almagor, S. Yedgar, B. Gavish, Inhomogeneity of viscousaqueous solutions, Phys. Rev. E52 (1995) 555-557.[15] H. Oh-oka, M. Iwaki, S. Itoh, Viscosity dependence of the electron transfer tatefrom bound cytochrome c to P840 in the photosynthetic reaction center of thegreen sulfur bacterium Chlorobium tepidum, Biochemistry 36 (1997) 9267-9272.
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23 for the values of theparameter b from the down line to the upper one: b = 1, b = 1 . b = 1 . b = 1 . b = 1 . b = 2. (cid:22) (cid:23) (cid:24) ORJ 0(cid:20)(cid:17)(cid:21)(cid:20)(cid:17)(cid:23)(cid:20)(cid:17)(cid:25)(cid:20)(cid:17)(cid:27)(cid:21)(cid:21)(cid:17)(cid:21)D+0(cid:15)(cid:20)(cid:17)(cid:24)(cid:21)(cid:15)G/ Fig. 3. The dependence of the parameter a ( M, b, δ ) (eq.(21)) as the function of thelogarithm of the cosolvent molecular weight M at b = 1 .
52 for the values of theparameter δ from the down line to the upper one: δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ = 0 . (cid:20)(cid:17)(cid:21)(cid:20)(cid:17)(cid:23)(cid:20)(cid:17)(cid:25)(cid:20)(cid:17)(cid:27)(cid:21) E(cid:19) (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24)G(cid:21)(cid:22)(cid:23)D PD[ (cid:21)(cid:22)(cid:23)D
PD[
Fig. 4. The dependence of the parameter a max (eq. (21) at M min = 18) as thefunction of the parameters b and δ . (cid:17)(cid:21) (cid:20)(cid:17)(cid:23) (cid:20)(cid:17)(cid:25) (cid:20)(cid:17)(cid:27) (cid:21) D PD[ (cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:27)(cid:20)E
PD[ +D PD[ / Fig. 5. The dependence of the limit value for the fractional power exponent β max (eq.(22)) as the function of the parameter a max ..