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The effect of viscosity on the apparent decomposition rate on enzyme-ligand complexes
J. theor. Bioi. (1978) 74, 209-216
The Effect of Viscosity on the Apparent Decomposition Rate on Enzyme-ligand Complexes BELA SOMOGYI, FRANK E. KARASZ~, LAJOS TRON AND PETER R. COUCHMAN? Department
of Biophysics, Medical University, 40 12 Hungary
(Received 31 January 1978 and in revisedform
Debrecen,
22 March 1978)
A theoretical model is presented for describing a previously untreated effect of viscosity on the apparent decomposition rate of enzyme-ligand complexes. Since the translational diffusion is hindered by the viscosity, its increased value results in an enlarged portion of ligands which can be rebound by the enzyme immediately after the dissociation of the complex. The model accounts for the experimentally observed decrease in maximal velocity of enzymic reactions at high viscosity. At the same time, it serves as a tool to obtain new information about the energetic processes of enzyme action.
1. Introduction It is widely accepted that the kinetic parameters of an enzyme working in vivo should differ from those determined by the usual in vitro conditions. Therefore the examinations of the specific kinetic and regulatory features of enzymes tend to approach the rather complex circumstances characteristic of the in vivo milieu of enzymes. One of the possible ways of this approach is arising theviscosity of reaction mixtures, by the use of different polymers, glycerin, etc., to a value comparable with that of the cellular interior. According to the related experimental data showing relatively strong effects of inert viscous materials on kinetic parameters of enzymes (Ceska, 1971, 1972; Damjanovich, Bot, Somogyi & Siimegi, 1972; Jancsik et al., 1975,1976; Tr6n, Somogyi, Papp & Novak, 1976), this way seems to be a successful one. However, even in the simple case of approaching the in vivo conditions by increasing viscosity alone, one has to face some many-sided phenomena. Alterations in the concentration of the applied inert materials change the mass distribution of f Polymer Scienceand Engineering, University of Massachusetts, Amherst, Massachusetts 01003, U.S.A. 209
0022~5193/78/180209+08 $02.00/O
0 1978 Academic Press Inc. (London) Ltd.
210
B.
SOMOGYI
ET
AL.
the aqueous environment, and, consequently, the activation energy as well as the pre-exponential factor in the expression describing the enzyme-ligand (EL) complex decomposition rate (Damjanovich & Somogyi, 1973; Somogyi & Damjanovich, 1975). Another possible effect is the change of equilibrium constants owing to the different extent of exclusion of reactants, caused by the applied polymer (Laurent, 1971). The “sieve effect” (i.e. the decrease in the translational diffusion of the reaction components in viscous media) should also result in different alterations of reaction parameters (Laurent, 1971). Regarding further influences of viscosity on reaction parameters, one should refer to some review articles (Noyes, 1961; Berth, 1964; Welch, 1977). The diversity of interactions between the enzyme and its environment raises difficulties in the analysis of experimental data. At the same time, however, the wide range of information to be obtained with the aid of kinetic studies of enzymes in viscous media underlines the importance of investigations of this type. In order to help a better design and a more successful analysis of these experiments, we present a theoretical model describing a new effect of viscosity on the decomposition rate of EL complexes. This effect is an apparent one and arises from the decrease of the translational diffusion rates of reactants at high viscosity. After the decomposition of EL complex the portion of dissociated molecule pairs which do not have enough kinetic energy to cover more than a critical distance might reassociate without allowing the enzyme molecule to bind any other ligand having affinity for the same binding site. EL complexes reassociated as outlined above kinetically appear as complexes having extended lifetime compared with the intrinsic one. 2. The Model To describe the effect of viscosity on the decomposition complex, the E+L&L
rate of EL
(1) k-1 reaction scheme will be considered. The rate constants k, and k-, can be expressed in terms of molecular parameters as follows (Somogyi & Damjanovich, 1971) :
k -1=;,
1
VISCOSITY
AND
EL
COMPLEX
211
LIFETIME
where: c is the ligand concentration; V is the recognition volume which is assigned to the binding site of the enzyme molecule so that within this volume the ligand can be bound by the enzyme molecule as a result of specific forces acting between the two molecules. If the ligand falls out of this volume, the specific forces are ineffective; I is the recognition time, i.e. the mean time spent by a free ligand molecule in the recognition volume before leaving it; 4 is the probability that the ligand becomes bound by the enzyme during the recognition time; t’ is the intrinsic lifetime of the EL complex. Now, turning to the energetics of the processes, which occur after the dissociation of EL complexes, let E, be the total initial kinetic energy of a pair of dissociated enzyme and ligand molecules in a coordinate system fixed to the mass center (MC) of these molecules: E, = +(mz$, + Mw;),
(4) where m and M are the masses, co and wO are the initial velocities of the ligand and enzyme molecules, respectively. This energy originates from that part of the intramolecular vibrational and rotational motion which is closely related to the chemical bond (or bonds) breaking down during the dissociation events. Further, one can write: jrnv,i = jMw,l.
(5) Assuming that just before the dissociation the velocity of the EL complex can be neglected as compared to zi,, and we, the velocities of the recoils are the same in the coordinate system fixed to the MC or the laboratory. Consequently, for the motion of the enzyme and ligand molecules the following balances will be satisfied: dv - -m = 6ntpv dt dw - dt M = 6nqRw, where q is the viscosity, r and R are the radii of the ligand and enzyme molecules (the equations contain the assumption that both the enzyme and ligand molecules are sphere-like). The separation distance can be obtained by integration of equations (6) and (7):
s=&&+~wdt=~67cry +e” 6nqR 0
Introducing
0
(8) l
p, the reduced radius, as
1 1 - =;fjp P
1
(9)
212
B.
SOMOGYI
ET
AL.
the separation distance is : mu0 s=G&p*
(10)
In the case of s > d, where d is a critical distance characteristic of the recognition volume (Somogyi & Damjanovich, 1973; Somogyi, 1974), the reassociation has only very low probability (4’) because of the following reasons : The dissociation, being an asymmetric process, results in a complex motion of the recoiling molecules. This kind of motion is a stochastic one because it possesses a rotational part determined by the energies belonging to different degrees of freedom of the EL complex. This rotational motion of the recoiling molecules has, even initially, a random character and will also be randomly affected by interactions with the surrounding molecules. During the short time while the ligand moves through the recognition volume, its orientation is randomized to a high degree. So the ligand has practically no chance to be in the right steric position for rebinding. When s Q d, the ligand loses its 4 rnvi kinetic energy within the recognition volume and performs, more or less, free diffusion motion. Thus the enzyme can rebind it with a q probability which is the same as the binding probability for another ligand molecule entering the recognition volume from outside. This should become clear considering the random orientation of the binding site of the ligand after the dissociation. Accordingly, the p probability of reassociation of enzyme and ligand molecules is : P = PEc7,
(11)
where pa=P(s
(12)
is the probability that the separation distance is smaller than, or equal to d. It is obvious that pE depends on Eo. If n successive rebindings occur and the ligand leaves the recognition volume after the (n + 1)th dissociation the apparent lifetime of the EL complex is: t;+1 = (n+l)t’,
(13)
where t’ is the intrinsic lifetime of the complex?. For the probability that the apparent lifetime of a given EL complex equals to (n + l)t’, we have:
PiIf1 In fact, experimental t In this approximation
smaller than t’.
=
(I-PIP”.
(14)
conditions involve a mixture of EL complexes having it is necessarily assumed that the f recognition
time is much
VISCOSITY
AND
different apparent lifetimes. average of these lifetimes:
EL
COMPLEX
Therefore,
213
LIFETIME
the observable lifetime,
r, is an
(15)
Substituting
equation (14) into equation (15) :
z=t,i~o(i+l~pi
(16)
f Pi
i=O
The nominator
in equation (16) can be rearranged as :
~~o~~+l)Pi=i~oP’+Pi~oPi+P2 f I++..*=(f Pl)l. i=O
(17)
i=O
By the use of equation (17) and the
ijIoPi=l&p, if OGp
for the value of z we have:
According to equations (3) and (18), the value of k’- 1 observable decomposition rate constant of the EL complex is: kLl =;
= (l-p)k-,
(19)
where km1 is the intrinsic decomposition rate constant characteristic of the reaction when there is no viscosity effect. In order to elaborate the relationship between k’-, and the viscosity, a detailed form of the reassociation probability (p) is required. Iff(E)o is the density function for the distribution of the kinetic energy, the pE probability can be expressed as: PE = $f(Eol
dEo,
(20)
where E, is the critical energy resulting in separation distance d. The value of E. can be obtained using equations (9), (5) and (10) as: n2p2d2 2 ? = w2,
E, = 18- ~
(21)
214
B.
SOMOGYI
ET
AL.
where h is the reduced mass: 1 1 1 -P =m+-’ M Substituting equations (1 l), (20) and (21) into equation observable decomposition rate constant we obtain: kI_, = k-,
I-q&$,)dE].
(22)
(19), for the k’- 1
(23)
0
3. Discussion The presented model predicts an apparent decrease of decomposition rate constants of EL complexes [equation (23)] along with viscosity increase. There are several experimental data showing the decline of maximum velocity, or activity, of enzymes at high viscosity (Ceska, 1971, 1972; Ruwart & Suelter, 1971; Laurent, 1971; Damjanovich et al., 1972; Cercek, 1972; Cercek & Cercek, 1973 a$; Jancsik et al., 1975, 1976; Tron et al., 1976). Considering that both the chemical nature of viscosity elevating agents used in these experiments and the reactions are different, the above results are very likely due to the change in the only common parameter, the viscosity. This is in accordance with our prediction. For the examination of the above viscosity effect, it will be useful to introduce an a parameter, defined as a=&k’_, k-l’
From the comparison
(24)
of equations (23) and (24) we have: a= 4 ‘ff(E3
dEo = 4 %#>
(25)
where F($) is the energy distribution function. As it has been shown (Somogyi & Damjanovich, 1973; Somogyi, 1974), the 4 recognition probability is independent of the viscosity, i.e. the a parameter contains only F(m2) as a viscosity dependent factor. Therefore, the experimental examination of the relationship between a and the viscosity should give new information about the energetics of enzyme action. For this purpose, equation (25) can be used in different ways: (a) The values of p, d, p and q can be estimated or calculated (Somogyi & Damjanovich, 1973; Somogyi, 1974). Furthermore, it can be accepted that under usual conditions the value of k’- 1 measured in the absence of viscosity elevating agents gives a good approximation for the value of k- 1. Having the values for the above parameters, the plot of et/q against yq2 can be obtained giving an approximation for the I;(Eo) distribution function. By numerical
VISCOSITY
AND
EL
COMPLEX
LIFETIME
215
derivation one can get a similar approximation for thef(Ee) density function as well. (b) If the experiment is performed over a wide range of viscosity, because of lim F(yr2) = 1, (26) l-‘W
an approximate value of q becomes experimentally determinable by extrapolation. (2) According to the absolute reaction rate theory activated complexes are in thermodynamic equilibrium (Johnson, Eyring & Polissar, 1954), thus the f(E,) distribution function is of Maxwell-Boltzmann (M-B) type. Let us assume that a single bond of EL complex is broken for dissociation and the interatomic vibration energy belonging to this bond follows M-B distribution. If U denotes the binding energy of this bond, the pE probability can be written in the following form: U+YrZ 1 j G emEikT dE p,=P(E< U+yy2/E> V)= +I (27) L kT eaEtkT dE where pE is the probability that the E < U-l- yq2 inequality is satisfied when E > U. T is the absolute temperature and k is the Boltzmann constant. After integration we get: pE = 1 -e-Y~‘lkT.
(28)
The value of a can be obtained using equations (1 I), (19), (24) and (28) as : g = q (1 -e-y’PJkT > ( 29) Choosing an ql, q2, . . . , r, set of increasing viscosity values such a way that rf+ 1 -$ = A is constant for i = 1,2, . . . , y1- 1, the appropriate a, values can be evaluated from experimental data. For ,329 ***, %, log (~%~-a~+,) one can write: log(a, - ai+ 1) = log 4 f log (1 - emyAlkT) - ?-J . Plotting log(a,-ai+l) vs. Q$ one should get a straight line, allowing one to determine the values of y and q. In conclusion, the procedures described under 1. are useful for obtaining the shapes off(E,) and F(E,) functions and all [case l(b)], or at least some [case l(a)], of their characteristic parameters. When our assumptions are fulfilled, the derivations of functions f(E,) and F(E,) from the M-B type distribution should mean that the basic condition for use of M-B statistic, i.e. the thermal equilibrium is absent in that
216
B.
SOMOGYI
ET
AL.
particular case. In other words, some kind of non-equilibrium processes are taking place during the events of EL complex decomposition. Therefore, via the study of these deviations one might decide whether a given activated complex can or cannot be considered as being in thermodynamic equilibrium, which is one of the central questions of enzymology. If the energy distribution follows M-B statistics, the procedure described under section 2. can be followed for getting more detailed information about the molecular parameters composing the individual rate constants of the enzyme-ligand systems. The presented model deals only with the simple reaction described by section 1. The results are, however, applicable for any enzymic reaction in which the rate limiting step is the decomposition of an EL complex. In this case, the model is valid as long as the rate limiting step remains the same. Finally, it should also be mentioned that under in viuo conditions the described effect of viscosity can play an important role in the action of multienzyme complexes by affecting the transfer of intermedier molecules from one enzyme to the next (Welch, 1977b). The authors wish to thank Professor S. Damjanovich during the preparation of this material.
for his helpful discussions
REFERENCES L. (1972). Curr. Top. Radiat. Res. 7, 355. L. & CERCEK, B. (1973a). Biophysik 9,105. L. & CERCEK, B. (19736). Biophysik 9,109. C&SKA, M. (1971). Experientia 27, 767. CESKA, M. (1972). Experientia 28, 146. DAMJANOVICH, S., BOT, J., SOMOGYI, B. & SMEGI, J. (1972). Biochim. biophys. Acta 284 345. DAMJANOVICH, S. & SOMOGYI, B. (1973). J. theor. Biol. 41,567. JANCSIK, V., KIILETI, T., BICZ~K, GY., NAGY, M., SZAB~~, Z. & WOLFRAM, E. (1975/76). J. mol. Catalysis 1, 137. JOHNSON, F. K., EYRING, H. & POLISSAR, M. J. (1954). The Kinetic Basis of Molecular Biology. New York, London: John Wiley & Sons. LAURENT, T. C. (1971). Eur. J. Biochem. 21, 498. MYERS, J. S. & JAKOBY, W. B. (1975). Biochem. Biophys. Res. Commun. 51,631. NOR=, A. M. (1964). The Collisioe Theory of Chemical Reactions in Liquids. London: Methuen & Co. Ltd. New York: John Wiley & Sons. NOYES, R. M. (1961). Progr. React. Kinet. 1, 151. RIJWART, M. J. & SUELTER, C. H. (1971). J. biol. Chem. 246,599O. SOMOGYI, B. (1974). Acta Biochim. Biophys. Acad. Sci. Hung. 9, 185. SOMOGYI, B. & DAMJANOVICH, S. (1971). Acta Biochim. Biophys. Acad. Sci. Hung. 6, 353. SOMOGYI, B. & DAMJANOVICH, S. (1973). Acta Biochim. Biophys. Acad. Sci. Hung. 8,153. SOMOGYI, B. & DAMJANO~~H, S. (1975). J. theor. Biol. 48, 393. TR~N, L., SOMOGYI, B., PAPP, S. & Novp;K, M. (1976). Studia Biophysics 60, 157. WELCH, G. R. (1977a) Progr. Biophys. mol. Biol. 32,103. WELCH, G. R. (1977b). J. theor. Biol. 68, 267. CERCEK, CERCEK, CERCEK,
The Effect of Viscosity on the Apparent Decomposition Rate on Enzyme-ligand Complexes BELA SOMOGYI, FRANK E. KARASZ~, LAJOS TRON AND PETER R. COUCHMAN? Department
of Biophysics, Medical University, 40 12 Hungary
(Received 31 January 1978 and in revisedform
Debrecen,
22 March 1978)
A theoretical model is presented for describing a previously untreated effect of viscosity on the apparent decomposition rate of enzyme-ligand complexes. Since the translational diffusion is hindered by the viscosity, its increased value results in an enlarged portion of ligands which can be rebound by the enzyme immediately after the dissociation of the complex. The model accounts for the experimentally observed decrease in maximal velocity of enzymic reactions at high viscosity. At the same time, it serves as a tool to obtain new information about the energetic processes of enzyme action.
1. Introduction It is widely accepted that the kinetic parameters of an enzyme working in vivo should differ from those determined by the usual in vitro conditions. Therefore the examinations of the specific kinetic and regulatory features of enzymes tend to approach the rather complex circumstances characteristic of the in vivo milieu of enzymes. One of the possible ways of this approach is arising theviscosity of reaction mixtures, by the use of different polymers, glycerin, etc., to a value comparable with that of the cellular interior. According to the related experimental data showing relatively strong effects of inert viscous materials on kinetic parameters of enzymes (Ceska, 1971, 1972; Damjanovich, Bot, Somogyi & Siimegi, 1972; Jancsik et al., 1975,1976; Tr6n, Somogyi, Papp & Novak, 1976), this way seems to be a successful one. However, even in the simple case of approaching the in vivo conditions by increasing viscosity alone, one has to face some many-sided phenomena. Alterations in the concentration of the applied inert materials change the mass distribution of f Polymer Scienceand Engineering, University of Massachusetts, Amherst, Massachusetts 01003, U.S.A. 209
0022~5193/78/180209+08 $02.00/O
0 1978 Academic Press Inc. (London) Ltd.
210
B.
SOMOGYI
ET
AL.
the aqueous environment, and, consequently, the activation energy as well as the pre-exponential factor in the expression describing the enzyme-ligand (EL) complex decomposition rate (Damjanovich & Somogyi, 1973; Somogyi & Damjanovich, 1975). Another possible effect is the change of equilibrium constants owing to the different extent of exclusion of reactants, caused by the applied polymer (Laurent, 1971). The “sieve effect” (i.e. the decrease in the translational diffusion of the reaction components in viscous media) should also result in different alterations of reaction parameters (Laurent, 1971). Regarding further influences of viscosity on reaction parameters, one should refer to some review articles (Noyes, 1961; Berth, 1964; Welch, 1977). The diversity of interactions between the enzyme and its environment raises difficulties in the analysis of experimental data. At the same time, however, the wide range of information to be obtained with the aid of kinetic studies of enzymes in viscous media underlines the importance of investigations of this type. In order to help a better design and a more successful analysis of these experiments, we present a theoretical model describing a new effect of viscosity on the decomposition rate of EL complexes. This effect is an apparent one and arises from the decrease of the translational diffusion rates of reactants at high viscosity. After the decomposition of EL complex the portion of dissociated molecule pairs which do not have enough kinetic energy to cover more than a critical distance might reassociate without allowing the enzyme molecule to bind any other ligand having affinity for the same binding site. EL complexes reassociated as outlined above kinetically appear as complexes having extended lifetime compared with the intrinsic one. 2. The Model To describe the effect of viscosity on the decomposition complex, the E+L&L
rate of EL
(1) k-1 reaction scheme will be considered. The rate constants k, and k-, can be expressed in terms of molecular parameters as follows (Somogyi & Damjanovich, 1971) :
k -1=;,
1
VISCOSITY
AND
EL
COMPLEX
211
LIFETIME
where: c is the ligand concentration; V is the recognition volume which is assigned to the binding site of the enzyme molecule so that within this volume the ligand can be bound by the enzyme molecule as a result of specific forces acting between the two molecules. If the ligand falls out of this volume, the specific forces are ineffective; I is the recognition time, i.e. the mean time spent by a free ligand molecule in the recognition volume before leaving it; 4 is the probability that the ligand becomes bound by the enzyme during the recognition time; t’ is the intrinsic lifetime of the EL complex. Now, turning to the energetics of the processes, which occur after the dissociation of EL complexes, let E, be the total initial kinetic energy of a pair of dissociated enzyme and ligand molecules in a coordinate system fixed to the mass center (MC) of these molecules: E, = +(mz$, + Mw;),
(4) where m and M are the masses, co and wO are the initial velocities of the ligand and enzyme molecules, respectively. This energy originates from that part of the intramolecular vibrational and rotational motion which is closely related to the chemical bond (or bonds) breaking down during the dissociation events. Further, one can write: jrnv,i = jMw,l.
(5) Assuming that just before the dissociation the velocity of the EL complex can be neglected as compared to zi,, and we, the velocities of the recoils are the same in the coordinate system fixed to the MC or the laboratory. Consequently, for the motion of the enzyme and ligand molecules the following balances will be satisfied: dv - -m = 6ntpv dt dw - dt M = 6nqRw, where q is the viscosity, r and R are the radii of the ligand and enzyme molecules (the equations contain the assumption that both the enzyme and ligand molecules are sphere-like). The separation distance can be obtained by integration of equations (6) and (7):
s=&&+~wdt=~67cry +e” 6nqR 0
Introducing
0
(8) l
p, the reduced radius, as
1 1 - =;fjp P
1
(9)
212
B.
SOMOGYI
ET
AL.
the separation distance is : mu0 s=G&p*
(10)
In the case of s > d, where d is a critical distance characteristic of the recognition volume (Somogyi & Damjanovich, 1973; Somogyi, 1974), the reassociation has only very low probability (4’) because of the following reasons : The dissociation, being an asymmetric process, results in a complex motion of the recoiling molecules. This kind of motion is a stochastic one because it possesses a rotational part determined by the energies belonging to different degrees of freedom of the EL complex. This rotational motion of the recoiling molecules has, even initially, a random character and will also be randomly affected by interactions with the surrounding molecules. During the short time while the ligand moves through the recognition volume, its orientation is randomized to a high degree. So the ligand has practically no chance to be in the right steric position for rebinding. When s Q d, the ligand loses its 4 rnvi kinetic energy within the recognition volume and performs, more or less, free diffusion motion. Thus the enzyme can rebind it with a q probability which is the same as the binding probability for another ligand molecule entering the recognition volume from outside. This should become clear considering the random orientation of the binding site of the ligand after the dissociation. Accordingly, the p probability of reassociation of enzyme and ligand molecules is : P = PEc7,
(11)
where pa=P(s
(12)
is the probability that the separation distance is smaller than, or equal to d. It is obvious that pE depends on Eo. If n successive rebindings occur and the ligand leaves the recognition volume after the (n + 1)th dissociation the apparent lifetime of the EL complex is: t;+1 = (n+l)t’,
(13)
where t’ is the intrinsic lifetime of the complex?. For the probability that the apparent lifetime of a given EL complex equals to (n + l)t’, we have:
PiIf1 In fact, experimental t In this approximation
smaller than t’.
=
(I-PIP”.
(14)
conditions involve a mixture of EL complexes having it is necessarily assumed that the f recognition
time is much
VISCOSITY
AND
different apparent lifetimes. average of these lifetimes:
EL
COMPLEX
Therefore,
213
LIFETIME
the observable lifetime,
r, is an
(15)
Substituting
equation (14) into equation (15) :
z=t,i~o(i+l~pi
(16)
f Pi
i=O
The nominator
in equation (16) can be rearranged as :
~~o~~+l)Pi=i~oP’+Pi~oPi+P2 f I++..*=(f Pl)l. i=O
(17)
i=O
By the use of equation (17) and the
ijIoPi=l&p, if OGp
for the value of z we have:
According to equations (3) and (18), the value of k’- 1 observable decomposition rate constant of the EL complex is: kLl =;
= (l-p)k-,
(19)
where km1 is the intrinsic decomposition rate constant characteristic of the reaction when there is no viscosity effect. In order to elaborate the relationship between k’-, and the viscosity, a detailed form of the reassociation probability (p) is required. Iff(E)o is the density function for the distribution of the kinetic energy, the pE probability can be expressed as: PE = $f(Eol
dEo,
(20)
where E, is the critical energy resulting in separation distance d. The value of E. can be obtained using equations (9), (5) and (10) as: n2p2d2 2 ? = w2,
E, = 18- ~
(21)
214
B.
SOMOGYI
ET
AL.
where h is the reduced mass: 1 1 1 -P =m+-’ M Substituting equations (1 l), (20) and (21) into equation observable decomposition rate constant we obtain: kI_, = k-,
I-q&$,)dE].
(22)
(19), for the k’- 1
(23)
0
3. Discussion The presented model predicts an apparent decrease of decomposition rate constants of EL complexes [equation (23)] along with viscosity increase. There are several experimental data showing the decline of maximum velocity, or activity, of enzymes at high viscosity (Ceska, 1971, 1972; Ruwart & Suelter, 1971; Laurent, 1971; Damjanovich et al., 1972; Cercek, 1972; Cercek & Cercek, 1973 a$; Jancsik et al., 1975, 1976; Tron et al., 1976). Considering that both the chemical nature of viscosity elevating agents used in these experiments and the reactions are different, the above results are very likely due to the change in the only common parameter, the viscosity. This is in accordance with our prediction. For the examination of the above viscosity effect, it will be useful to introduce an a parameter, defined as a=&k’_, k-l’
From the comparison
(24)
of equations (23) and (24) we have: a= 4 ‘ff(E3
dEo = 4 %#>
(25)
where F($) is the energy distribution function. As it has been shown (Somogyi & Damjanovich, 1973; Somogyi, 1974), the 4 recognition probability is independent of the viscosity, i.e. the a parameter contains only F(m2) as a viscosity dependent factor. Therefore, the experimental examination of the relationship between a and the viscosity should give new information about the energetics of enzyme action. For this purpose, equation (25) can be used in different ways: (a) The values of p, d, p and q can be estimated or calculated (Somogyi & Damjanovich, 1973; Somogyi, 1974). Furthermore, it can be accepted that under usual conditions the value of k’- 1 measured in the absence of viscosity elevating agents gives a good approximation for the value of k- 1. Having the values for the above parameters, the plot of et/q against yq2 can be obtained giving an approximation for the I;(Eo) distribution function. By numerical
VISCOSITY
AND
EL
COMPLEX
LIFETIME
215
derivation one can get a similar approximation for thef(Ee) density function as well. (b) If the experiment is performed over a wide range of viscosity, because of lim F(yr2) = 1, (26) l-‘W
an approximate value of q becomes experimentally determinable by extrapolation. (2) According to the absolute reaction rate theory activated complexes are in thermodynamic equilibrium (Johnson, Eyring & Polissar, 1954), thus the f(E,) distribution function is of Maxwell-Boltzmann (M-B) type. Let us assume that a single bond of EL complex is broken for dissociation and the interatomic vibration energy belonging to this bond follows M-B distribution. If U denotes the binding energy of this bond, the pE probability can be written in the following form: U+YrZ 1 j G emEikT dE p,=P(E< U+yy2/E> V)= +I (27) L kT eaEtkT dE where pE is the probability that the E < U-l- yq2 inequality is satisfied when E > U. T is the absolute temperature and k is the Boltzmann constant. After integration we get: pE = 1 -e-Y~‘lkT.
(28)
The value of a can be obtained using equations (1 I), (19), (24) and (28) as : g = q (1 -e-y’PJkT > ( 29) Choosing an ql, q2, . . . , r, set of increasing viscosity values such a way that rf+ 1 -$ = A is constant for i = 1,2, . . . , y1- 1, the appropriate a, values can be evaluated from experimental data. For ,329 ***, %, log (~%~-a~+,) one can write: log(a, - ai+ 1) = log 4 f log (1 - emyAlkT) - ?-J . Plotting log(a,-ai+l) vs. Q$ one should get a straight line, allowing one to determine the values of y and q. In conclusion, the procedures described under 1. are useful for obtaining the shapes off(E,) and F(E,) functions and all [case l(b)], or at least some [case l(a)], of their characteristic parameters. When our assumptions are fulfilled, the derivations of functions f(E,) and F(E,) from the M-B type distribution should mean that the basic condition for use of M-B statistic, i.e. the thermal equilibrium is absent in that
216
B.
SOMOGYI
ET
AL.
particular case. In other words, some kind of non-equilibrium processes are taking place during the events of EL complex decomposition. Therefore, via the study of these deviations one might decide whether a given activated complex can or cannot be considered as being in thermodynamic equilibrium, which is one of the central questions of enzymology. If the energy distribution follows M-B statistics, the procedure described under section 2. can be followed for getting more detailed information about the molecular parameters composing the individual rate constants of the enzyme-ligand systems. The presented model deals only with the simple reaction described by section 1. The results are, however, applicable for any enzymic reaction in which the rate limiting step is the decomposition of an EL complex. In this case, the model is valid as long as the rate limiting step remains the same. Finally, it should also be mentioned that under in viuo conditions the described effect of viscosity can play an important role in the action of multienzyme complexes by affecting the transfer of intermedier molecules from one enzyme to the next (Welch, 1977b). The authors wish to thank Professor S. Damjanovich during the preparation of this material.
for his helpful discussions
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