Dynamics and Kinetics of Enzymes Kinetic Equilibrium of Forces in Biochemistry Erwin W. Becker* Institut für MikroStrukturtechnik der Universität Karlsruhe, Bundesrepublik Deutschland Z. Naturforsch. 47c, 628-633 (1992); received May 18, 1992 Enzyme, Mechanism, Free Energy Transfer, ATPase, Calmodulin, Muscle Contraction To explain the high specificity, high reaction rate, and high thermodynamic efficiency in enzymatic processes, cooperation of the enzyme with a molecular transfer unit is assumed. A “kinetic equilibrium of forces” is suggested, which enables high reaction rates to occur under equilibrium conditions and a thorough examination of the substrate to be made without con sumption of free energy. In case o f ATPases, ion-binding proteins are the most probable trans fer units. By analyzing the elementary effect in muscle contraction it is shown that the new theorem may be of substantial value in elucidating biochemical processes.
Introduction It is well know n th a t enzym atic processes m ay proceed w ith high specificity, high reaction rate, and high therm odynam ic efficiency. T o explain the specificity, Emil Fischer an d Paul E hrlich in troduced their fam ous lock-and-key hypothesis. A ccording to a suggestion by L. Pauling, the en zyme has to fit the tran sitio n state ra th e r th a n the g round state o f the substrate [1], D. E. K o shland [2 ] com plem ented this m odel by the so-called in duced-fit theory, w hich assum es an a d a p ta tio n o f the enzyme to the substrate. M . Eigen et al. [3] have dem on strated th a t the v arious steps o f en zym atic reactions m ay proceed a t considerable ve locity. W ith the prim ary objective to explain the high therm odynam ic efficiency o f enzym atic reac tions involving m em branes, P. M itchell in tro duced the so-called “chem iosm otic equilib riu m ” [4]. In spite o f the vast know ledge existing a b o u t the chem istry o f enzym atic reactions, the b ack g round for the surprising capabilities o f the enzym es is still a m atter o f discussion [5]. In the present p ap er an attem p t is m ade to su b stan tiate a biom echanical theorem called “kinetic equilibrium o f forces” w hich could be o f som e help in answ ering open questions. It is based on an assum ed co o p eratio n o f the enzyme w ith a m olecular tran sfer unit. U n der the condition th a t the co o p eratin g p artn ers ex hibit opposite relations betw een their free energy * Reprint requests and correspondence to Prof. Dr. E. W. Becker, Strählerweg 18, D-W-7500 Karlsruhe 41. Verlag der Zeitschrift für Naturforschung, D-W-7400 Tübingen 0939-5075/92/0700-0628 $01.30/0
contents and their forces exerted on each other, a reversible exchange o f free energy by therm al m o lecular m otion can be expected to take place. The suggested “kinetic equilibrium o f forces” enables high reaction rates to occur under equilibrium conditions and a th o ro u g h exam ination o f the sub strate to be m ade w ithout consum ption o f free energy. In case o f A TPases, ion-binding proteins are the m ost probable transfer units. By analyzing the elem entary effect in muscle co n traction it is show n th a t the new theorem m ay be o f substantial value in elucidating biochem ical processes. Cooperation of an enzyme with a transfer unit C onsidering their im portance to the econom y o f the cell, A T Pases are taken as exam ples in estab lishing the new theorem . F o r the reaction A T P + H 20 = A D P + Pi, leading to the products A D P and p h o sphate ion (Pi), Fig. 1 indicates in a schem atic m anner the de pendence o f free energy G on the distance x 1 be tween A T P and H 20 and on the distance x2 be tween A D P and Pi. The com m on origin o f the dis tances lies roughly at the center o f the transition state. T he difference in the free energies o f activa tion, G a 2 - G a l, is the free energy o f reaction G r. Except in close proxim ity to the transition state, where intram olecular changes take place, free en ergy is based on repulsive interm olecular forces. D uring the m ain p a rt o f the interaction, therefore, an increase in free energy is connected w ith an in crease in force. U n d er physiological conditions, the th erm o dynam ic equilibrium o f the reaction is far to the
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the A T Pase cleft w ithout im pairing the m aterial flow. Prerequisites of a reversible transfer of free energy
Fig. 1. Schematic representation of the free energy G of ATP hydrolysis as a function of the distance x 1 between ATP and H20 on the one hand and of the distance x2 between ADP and phosphate ion (Pi) on the other hand.
right. In case o f m uscle contraction, how ever, a re versible A TP-hydrolysis far from therm odynam ic equilibrium is observed [6 ]. So, co o p eratio n be tween A T Pase and a m olecular “transfer u n it” m ust be assum ed. D uring the exergonic phase o f the reaction, the transfer unit is expected to store the free energy released so that, during the endergonic phase, it can be fed back to the A T Pase. In spite o f the fact th a t the transfer unit m ay be an integral p a rt o f the enzyme it will be treated as a separate entity. T o illustrate a possible way o f co o p eratio n , the hinge-cleft m odel o f an enzyme [7] is assum ed. A c cording to Fig. 2, the enzyme is to consist o f the globular units S 1 a an d S 1 b connected by a hinge like flexible section so th a t a pair o f tongs is form ed. The tran sfer unit T, the specifications o f w hich are to be established, is expected to bridge
Fig. 2. Hinge-cleft model of the cooperation between a transfer unit T and an ATPase consisting of the globular units S 1a and S 1b carrying the reactants ATP and H20 .
T he prerequisites o f a reversible transfer o f free energy between an enzym e and a transfer unit, co operating according to Fig. 2, are illustrated by the m odel o f a m o u n tain railw ay show n in Fig. 3. The co operating units are represented by two cars connected by a weightless rope and running on two curved slopes o f the m ountain. T o sim ulate the conditions in the cell it is assum ed th at the cars m ove w ithout m echanical friction, b u t w ithin a m edium , the viscosity o f which prevents consider able kinetic energy from occurring. U nder these conditions, the free energies o f the cars are identi cal w ith their poten tial energies. In order to m ake the dynam ic situ atio n m ore surveyable, it is fur ther assum ed th a t a frictionless guide forces the rope to run parallel to the slopes. So, the gradients o f the slopes at the positions o f the cars are m eas ures o f the forces exerted upon the rope. The car on the left side, together with the corresponding
Fig. 3. Model of a mountain railway for illustrating the cooperation between an enzyme (left) and a transfer unit (right). Upper case: Equilibrium of forces in one posi tion o f the cars only. Lower case: Equilibrium of forces in any position.
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slope, represents the A T Pase, w hereas the car on the right side represents the transfer unit. In the u p p er p a rt o f the figure, b o th p artn ers exhibit the relation between free energy and force typical o f A T Pases (see above). In the lower p a rt, the o p p o site relation is assum ed for the transfer unit. In the upp er case, equilibrium o f forces is estab lished in one position o f the cars only. W ith in creasing deviation from th a t p oint, a c o rresp o n d ing force builds up w hich pulls the cars back to the equilibrium state. In p ro p o rtio n to the resulting velocity entropy is generated. In the low er case, the equilibrium o f forces is m aintained in any position o f the cars. In such “extended” equilibrium o f forces, a small overw eight o f one c a r pulls the o th er car o n to the to p o f the m ountain. The rate o f the corresponding transfer o f free energy is p ro p o rtio n al to the unbalance o f forces, i.e. to the en tro p y generation. So, in a m acroscopic system o perating w ithin a viscous m edium , there is a rigor ous co rrelation betw een the “reaction ra te ” and the entropy generation. A m icroscopic system cooperating w ith a tra n s fer u n it in an extended equilibrium o f forces, how ever, m ay avoid the detrim ental co rrelatio n ju st m entioned by profiting from th erm al m olecular m otion. O f course, the resulting statistical transfer o f free energy is n o t directed. O n the o th er hand, u n d er an extended equilibrium o f forces, every p o in t on the reaction coord in ate p ertin en t to the next step o f the reaction is reached in due time. It is concluded, therefore, th a t a “ kinetic equilibrium o f forces” is the basis o f the surprising co m bina tion o f high specificity, high reactio n rate, and high therm odynam ic efficiency in enzym atic p ro c esses. H ere “kinetic” m eans subject to therm al m o tion. A prerequisite o f the supposed “kin etic” equili brium o f forces is the “extended” equilibrium o f forces which is d em onstrated for m acroscopic sys tem s in the lower p a rt o f Fig. 3.
Conceivable realization o f a kinetic equilibrium of forces Probable p artners o f enzymes in establishing a kinetic equilibrium o f forces are ion-binding p ro teins. In the presence o f a target p ro tein , binding o f calcium ions to calm odulin, e.g., tran sfo rm s the calm odulin chain into a m ore co m p act co n figura
E. W. Becker • Dynamics and Kinetics of Enzymes
tion [8 ]. U n d er the influence o f the target protein, binding o f ions proceeds in a pronounced co o pera tive m anner [9]. So, an increasing tractio n force should result. As the force is based m ainly on ionic attractio n , the free energy content o f the ion-bind ing protein is expected to decrease with increasing traction force. In the cooperation, according to Fig. 2, o f an A T Pase w ith a calm odulin-like p ro tein, therefore, the prerequisites o f a kinetic equili b rium o f forces should be fulfilled. U n d er such cooperation, A T P hydrolysis will result in a revers ible stretching o f the protein chain which is accom panied by a release o f ions. By reab sorption o f ions, A T P can be resynthesized. Like calm odulin, m any other proteins com bine preferably w ith calcium ions [10]. O ther ions, e.g. m agnesium ions, on the other hand, are far m ore ab u n d an t in the cell. So, a considerable num ber o f possibilities exist for achieving a kinetic equili brium o f forces in enzym atic reactions. As already m entioned, the transfer unit m ay be a separate entity as well as an integral part o f the enzyme.
Relevance of the kinetic equilibrium of forces to the specificity of enzymes In Fig. 4 it is show n how the high specificity o f enzymes can be explained on the basis o f the kinet ic equilibrium o f forces. T o m ake the essential a r gum ents m ore tran sp aren t, a m onom olecular reac tion w ithout a net free energy o f reaction is as sumed. T he tasks o f the enzyme, therefore, are restricted to selecting the proper substrate S and to starting the reaction m aking use o f the highly effi cient storage o f free energy by the transfer u nit T. In Fig. 4 a, the cleft o f the enzyme is em pty. It is kept open by the m utual repulsion o f the internal negative charges o f the ion-binding protein chain which acts as a transfer unit. U nder th a t condition, the affinity o f the positive ions to the protein chain is n ot sufficient to start the cooperative ion-binding process. The situation is different w hen a sub strate enters the cleft. By m eans o f its non-specific attractive forces, the substrate tries to narrow the cleft. The corresponding shortening o f the protein chain starts the cooperative binding o f positive ions which results in squeezing o f the substrate. In a first approxim ation, the squeezed substrate will react as an elastic body. As such, it exhibits a rela tion betw een the free energy and the force opposite
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be done, e.g. by phosphorylation or deph o sp h o ry lation o f a p articipant in the reaction.
Elementary effect in muscle contraction
Fig. 4. Different phases o f the examination of the sub strate S by the combination o f an enzyme and a transfer unit T.
to th a t o f the transfer unit T. So, in a kinetic equili brium o f forces a th o ro u g h exam ination o f the s u b strate by the enzyme is feasible. If, according to Fig. 4 c, the transition state o f the su b strate fits the enzyme, the protein chain T becom es shorter. The corresponding im provem ent o f ion-binding causes the free energy o f activation to be supplied in a kinetic equilibrium o f forces, and the reaction is started. The repulsive forces o f the reaction p ro d ucts force the two g lobular units o f the enzym e ap art. In this way, the ions b o und to the p ro tein chain are liberated, and the free energy o f ac tivation is restored. A fter the reaction p ro d u cts have been released the system again becom es func tional. The reaction rate, i.e. the p ro b ab ility o f the reac tion cycle occurring, m ay be regulated by the c o n centratio n o f the ions p ertin en t to the tran sfer unit. A nother possibility for regulation seems to be the supply o r the w ithdraw al o f the free energy o f acti vation stored within the tran sfer unit. T his could
It is well know n th a t muscle contraction is based on the relative sliding o f m yosin and actin fila m ents, the necessary free energy being supplied by the hydrolysis o f A T P [11-13]. A lthough muscle co n tractio n is one o f the principal them es o f bio logical research, the exact m echanism w hereby the chem ical free energy o f A T P hydrolysis is co n v ert ed into m echanical w ork is still a m atter o f discus sion [14]. A bove all, the high therm odynam ic effi ciency o f the process [15], and the precise a d a p ta tion o f A T P hydrolysis to the m echanical require m ents are difficult to explain on the basis o f present theories. R ecently, the a u th o r proposed a m odel o f the elem entary effect in m uscle contraction based on the assum ption o f free energy transfer by the ionbinding m yosin light chains [16]. T aking into con sideration the experim ental results described in the literature, it was concluded th at the d eto u r o f free energy via the light chains, under a so-called chem im echanic equilibrium , accounts for the high therm odynam ic efficiency. In the present paper, the term “chem im echanic equilibrium ” has been su b stituted by the m ore com prehensive “kinetic equilibrium o f forces” . The em phasis o f the fu n d a m ental role o f therm al m olecular m otion allows a m ore precise in terpretation o f kinetic details. This will be d em onstrated along the line o f Fig. 5. T he left h and colum n o f Fig. 5, in a schem atic m anner, indicates the different phases o f m uscu lar co n tractio n [16]. In the nom enclature o f Fig. 1 , the right-hand colum n denotes the distribution o f free energy am ong the com ponents involved. In Fig. 5 A, the free energy G a 2 is concentrated on the tran sitio n state o f the A T P /H 20 system. T he light chains LC and the elastic elem ent E do n ot co n tain free energy. In Fig. 5 B, the reaction A T P + H 2 0 - ^ A D P + Pi has taken place and G a 2 is tak en over by the light chains. In Fig. 5 C, loss o f the Pi has caused the so-called weak binding state to pass in to the strong binding state [17]. U n d er the new condition, the light chains LC pass the free energy o f reaction, G r, to the elastic elem ent E, while retaining the free energy o f activation, G a 1. Sim ultaneously, A D P is exchanged for A T P + H 2 0 .
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E. W. Becker • Dynamics and Kinetics of Enzymes
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-A -
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an actin unit (Fig. 5E ). The system thus becomes functional again. T he oscillation betw een states A and B under a kinetic equilibrium o f forces allows relative sliding o f the filam ents to take place w ith reduced A TP hydrolysis observed in a fast, unloaded contrac tion [18]. As free energy o f a ctivation is not deliv ered to the A T P /H 20 system before the muscle stran d s have taken over the free energy stored in the elastic elem ent E (transition C —D in Fig. 5), the m uscle can m aintain isom etric tension with rel atively low energy consum ption (“ F enn effect”). W ithin the proposed m echanism , the theory o f cal cium regulation rem ains essentially unaffected [16]. The analysis indicates th a t the kinetic equili b rium o f forces is essential n o t only for the effi ciency o f free energy transfer betw een the muscle com ponents b u t also for the precise ad ap tation o f A T P hydrolysis to the m echanical requirem ents.
Summary
k> Fig. 5. Mechanical diagrams representing the different phases of muscle contraction according to the proposed model.
W hen, according to Fig. 5D , the free energy co n tent G r o f the elastic elem ent E is taken over by the m uscle filam ents, the A T Pase cleft is fu rth er re duced, and the free energy o f activation, G a l , is transferred from LC to the A T Pase. In this way, the tran sitio n state o f the fresh A T P /H 20 load is reached. The pho sp h ate ion Pi form ed by A T P-hydrolysis transfers the strong binding state back to the w eak binding state. Thus, the m yosin head can follow the shift o f the myosin stran d and leap over
In the paper, the astonishing capabilities o f en zymes are explained by the assum ption o f a specif ic therm odynam ic state o f living m atter, called “ki netic equilibrium o f forces” . It is characterized by the reversible tra n sp o rt o f free energy under the action o f therm al m olecular m otion. The ability to avoid en tro p y generation in spite o f high reaction rates seems to be a peculiarity o f biological m a crom olecules. If the suggested theorem o f kinetic equilibrium o f forces turns o u t to be o f general sig nificance to living m atter, it should be an im por ta n t guide in elucidating biochem ical processes.
A cknowledgemen ts The a u th o r is grateful to Professors W. H as selbach, B. Hess, K. C. H olm es, L. Jaenicke, U. Schindew olf, D. M . Taylor, F. W ieland, and Th. W ieland for illum inating discussions and critically reading the m anuscript.
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E. W. Becker • Dynamics and Kinetics o f Enzymes [1] L. Pauling, Chem. Eng. News 24, 1375 (1946). [2] D. E. Koshland, Jr., Sei. Am. 229/4, 5 2 -6 4 (1973). [3] M. Eigen and G. G. Hammes, Adv. Enzym. 25, 1-38(1963). [4] P. Mitchell, Nature 191, 144-148 (1961); J. Biochem. 97, 1-18(1985). [5] L. Kovac, Curr. Topics in Bioenergetics 15, 331 — 372(1987). [6] C. R. Bagshaw and D. R. Trentham, Biochem. J. 133, 323-328(1973). [7] B. Mao, M. R. Pear, J. A. McCammon, and F. A. Quiocho, J. Biol. Chem. 257, 1131-1133(1982). [8] M. Kataoka, J. F. Head, B. A. Seaton, and D. M. Engelman, Proc. Natl. Acad. Sei. U.S.A. 86, 69446948 (1989). [9] J. A. Cox, M. Comte, A. Mamar-Bachi, M. Milos, and J.-J. Schaer, Cation Binding to Calmodulin and Relation to Function, in: Calcium and Calcium Binding Proteins (Ch. Gerday, R. Gilles, and L. Bolis, eds.), pp. 141 -1 6 2 , Springer, Heidelberg 1988.
633 [10] C. W. Heizmann and W. Hunziker, TIBS 16/3, 98-103(1991). [11] H. E. Huxley, Science 164, 1356-1366(1969); H. E. Huxley and M. Kress, J. Muse. Res. Cell Motil. 6, 153-161 (1985). [12] A. F. Huxley, J. Physiol. 243, 1 -4 3 (1974); A. F. Huxley, Ann. Rev. Physiol. 50, 1-16(1988). [13] R. S. Goody and K. C. Holmes, Biochim. Biophys. Acta 726, 13-39(1983). [14] M. A. Geeves, Biochem. J. 274, 1- 1 4 (1991). [15] R. C. Woledge, Nature 334, 655 (1988). [16] E. W. Becker, Naturwissenschaften 78, 445-449 (1991). [17] B. Brenner, Muscle Mechanics and Biochemical Ki netics, in: Molecular Mechanisms in Muscular Con traction (J. M. Squire, ed.), pp. 77-149, Macmillan Press, London 1990. [18] T. Yanagida, T. Arata, and F. Oosawa, Nature 316, 366-369(1985).
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Introduction It is well know n th a t enzym atic processes m ay proceed w ith high specificity, high reaction rate, and high therm odynam ic efficiency. T o explain the specificity, Emil Fischer an d Paul E hrlich in troduced their fam ous lock-and-key hypothesis. A ccording to a suggestion by L. Pauling, the en zyme has to fit the tran sitio n state ra th e r th a n the g round state o f the substrate [1], D. E. K o shland [2 ] com plem ented this m odel by the so-called in duced-fit theory, w hich assum es an a d a p ta tio n o f the enzyme to the substrate. M . Eigen et al. [3] have dem on strated th a t the v arious steps o f en zym atic reactions m ay proceed a t considerable ve locity. W ith the prim ary objective to explain the high therm odynam ic efficiency o f enzym atic reac tions involving m em branes, P. M itchell in tro duced the so-called “chem iosm otic equilib riu m ” [4]. In spite o f the vast know ledge existing a b o u t the chem istry o f enzym atic reactions, the b ack g round for the surprising capabilities o f the enzym es is still a m atter o f discussion [5]. In the present p ap er an attem p t is m ade to su b stan tiate a biom echanical theorem called “kinetic equilibrium o f forces” w hich could be o f som e help in answ ering open questions. It is based on an assum ed co o p eratio n o f the enzyme w ith a m olecular tran sfer unit. U n der the condition th a t the co o p eratin g p artn ers ex hibit opposite relations betw een their free energy * Reprint requests and correspondence to Prof. Dr. E. W. Becker, Strählerweg 18, D-W-7500 Karlsruhe 41. Verlag der Zeitschrift für Naturforschung, D-W-7400 Tübingen 0939-5075/92/0700-0628 $01.30/0
contents and their forces exerted on each other, a reversible exchange o f free energy by therm al m o lecular m otion can be expected to take place. The suggested “kinetic equilibrium o f forces” enables high reaction rates to occur under equilibrium conditions and a th o ro u g h exam ination o f the sub strate to be m ade w ithout consum ption o f free energy. In case o f A TPases, ion-binding proteins are the m ost probable transfer units. By analyzing the elem entary effect in muscle co n traction it is show n th a t the new theorem m ay be o f substantial value in elucidating biochem ical processes. Cooperation of an enzyme with a transfer unit C onsidering their im portance to the econom y o f the cell, A T Pases are taken as exam ples in estab lishing the new theorem . F o r the reaction A T P + H 20 = A D P + Pi, leading to the products A D P and p h o sphate ion (Pi), Fig. 1 indicates in a schem atic m anner the de pendence o f free energy G on the distance x 1 be tween A T P and H 20 and on the distance x2 be tween A D P and Pi. The com m on origin o f the dis tances lies roughly at the center o f the transition state. T he difference in the free energies o f activa tion, G a 2 - G a l, is the free energy o f reaction G r. Except in close proxim ity to the transition state, where intram olecular changes take place, free en ergy is based on repulsive interm olecular forces. D uring the m ain p a rt o f the interaction, therefore, an increase in free energy is connected w ith an in crease in force. U n d er physiological conditions, the th erm o dynam ic equilibrium o f the reaction is far to the
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E. W. Becker ■Dynamics and Kinetics of Enzymes
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the A T Pase cleft w ithout im pairing the m aterial flow. Prerequisites of a reversible transfer of free energy
Fig. 1. Schematic representation of the free energy G of ATP hydrolysis as a function of the distance x 1 between ATP and H20 on the one hand and of the distance x2 between ADP and phosphate ion (Pi) on the other hand.
right. In case o f m uscle contraction, how ever, a re versible A TP-hydrolysis far from therm odynam ic equilibrium is observed [6 ]. So, co o p eratio n be tween A T Pase and a m olecular “transfer u n it” m ust be assum ed. D uring the exergonic phase o f the reaction, the transfer unit is expected to store the free energy released so that, during the endergonic phase, it can be fed back to the A T Pase. In spite o f the fact th a t the transfer unit m ay be an integral p a rt o f the enzyme it will be treated as a separate entity. T o illustrate a possible way o f co o p eratio n , the hinge-cleft m odel o f an enzyme [7] is assum ed. A c cording to Fig. 2, the enzyme is to consist o f the globular units S 1 a an d S 1 b connected by a hinge like flexible section so th a t a pair o f tongs is form ed. The tran sfer unit T, the specifications o f w hich are to be established, is expected to bridge
Fig. 2. Hinge-cleft model of the cooperation between a transfer unit T and an ATPase consisting of the globular units S 1a and S 1b carrying the reactants ATP and H20 .
T he prerequisites o f a reversible transfer o f free energy between an enzym e and a transfer unit, co operating according to Fig. 2, are illustrated by the m odel o f a m o u n tain railw ay show n in Fig. 3. The co operating units are represented by two cars connected by a weightless rope and running on two curved slopes o f the m ountain. T o sim ulate the conditions in the cell it is assum ed th at the cars m ove w ithout m echanical friction, b u t w ithin a m edium , the viscosity o f which prevents consider able kinetic energy from occurring. U nder these conditions, the free energies o f the cars are identi cal w ith their poten tial energies. In order to m ake the dynam ic situ atio n m ore surveyable, it is fur ther assum ed th a t a frictionless guide forces the rope to run parallel to the slopes. So, the gradients o f the slopes at the positions o f the cars are m eas ures o f the forces exerted upon the rope. The car on the left side, together with the corresponding
Fig. 3. Model of a mountain railway for illustrating the cooperation between an enzyme (left) and a transfer unit (right). Upper case: Equilibrium of forces in one posi tion o f the cars only. Lower case: Equilibrium of forces in any position.
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slope, represents the A T Pase, w hereas the car on the right side represents the transfer unit. In the u p p er p a rt o f the figure, b o th p artn ers exhibit the relation between free energy and force typical o f A T Pases (see above). In the lower p a rt, the o p p o site relation is assum ed for the transfer unit. In the upp er case, equilibrium o f forces is estab lished in one position o f the cars only. W ith in creasing deviation from th a t p oint, a c o rresp o n d ing force builds up w hich pulls the cars back to the equilibrium state. In p ro p o rtio n to the resulting velocity entropy is generated. In the low er case, the equilibrium o f forces is m aintained in any position o f the cars. In such “extended” equilibrium o f forces, a small overw eight o f one c a r pulls the o th er car o n to the to p o f the m ountain. The rate o f the corresponding transfer o f free energy is p ro p o rtio n al to the unbalance o f forces, i.e. to the en tro p y generation. So, in a m acroscopic system o perating w ithin a viscous m edium , there is a rigor ous co rrelation betw een the “reaction ra te ” and the entropy generation. A m icroscopic system cooperating w ith a tra n s fer u n it in an extended equilibrium o f forces, how ever, m ay avoid the detrim ental co rrelatio n ju st m entioned by profiting from th erm al m olecular m otion. O f course, the resulting statistical transfer o f free energy is n o t directed. O n the o th er hand, u n d er an extended equilibrium o f forces, every p o in t on the reaction coord in ate p ertin en t to the next step o f the reaction is reached in due time. It is concluded, therefore, th a t a “ kinetic equilibrium o f forces” is the basis o f the surprising co m bina tion o f high specificity, high reactio n rate, and high therm odynam ic efficiency in enzym atic p ro c esses. H ere “kinetic” m eans subject to therm al m o tion. A prerequisite o f the supposed “kin etic” equili brium o f forces is the “extended” equilibrium o f forces which is d em onstrated for m acroscopic sys tem s in the lower p a rt o f Fig. 3.
Conceivable realization o f a kinetic equilibrium of forces Probable p artners o f enzymes in establishing a kinetic equilibrium o f forces are ion-binding p ro teins. In the presence o f a target p ro tein , binding o f calcium ions to calm odulin, e.g., tran sfo rm s the calm odulin chain into a m ore co m p act co n figura
E. W. Becker • Dynamics and Kinetics of Enzymes
tion [8 ]. U n d er the influence o f the target protein, binding o f ions proceeds in a pronounced co o pera tive m anner [9]. So, an increasing tractio n force should result. As the force is based m ainly on ionic attractio n , the free energy content o f the ion-bind ing protein is expected to decrease with increasing traction force. In the cooperation, according to Fig. 2, o f an A T Pase w ith a calm odulin-like p ro tein, therefore, the prerequisites o f a kinetic equili b rium o f forces should be fulfilled. U n d er such cooperation, A T P hydrolysis will result in a revers ible stretching o f the protein chain which is accom panied by a release o f ions. By reab sorption o f ions, A T P can be resynthesized. Like calm odulin, m any other proteins com bine preferably w ith calcium ions [10]. O ther ions, e.g. m agnesium ions, on the other hand, are far m ore ab u n d an t in the cell. So, a considerable num ber o f possibilities exist for achieving a kinetic equili brium o f forces in enzym atic reactions. As already m entioned, the transfer unit m ay be a separate entity as well as an integral part o f the enzyme.
Relevance of the kinetic equilibrium of forces to the specificity of enzymes In Fig. 4 it is show n how the high specificity o f enzymes can be explained on the basis o f the kinet ic equilibrium o f forces. T o m ake the essential a r gum ents m ore tran sp aren t, a m onom olecular reac tion w ithout a net free energy o f reaction is as sumed. T he tasks o f the enzyme, therefore, are restricted to selecting the proper substrate S and to starting the reaction m aking use o f the highly effi cient storage o f free energy by the transfer u nit T. In Fig. 4 a, the cleft o f the enzyme is em pty. It is kept open by the m utual repulsion o f the internal negative charges o f the ion-binding protein chain which acts as a transfer unit. U nder th a t condition, the affinity o f the positive ions to the protein chain is n ot sufficient to start the cooperative ion-binding process. The situation is different w hen a sub strate enters the cleft. By m eans o f its non-specific attractive forces, the substrate tries to narrow the cleft. The corresponding shortening o f the protein chain starts the cooperative binding o f positive ions which results in squeezing o f the substrate. In a first approxim ation, the squeezed substrate will react as an elastic body. As such, it exhibits a rela tion betw een the free energy and the force opposite
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E. W. Becker • Dynamics and Kinetics o f Enzymes
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be done, e.g. by phosphorylation or deph o sp h o ry lation o f a p articipant in the reaction.
Elementary effect in muscle contraction
Fig. 4. Different phases o f the examination of the sub strate S by the combination o f an enzyme and a transfer unit T.
to th a t o f the transfer unit T. So, in a kinetic equili brium o f forces a th o ro u g h exam ination o f the s u b strate by the enzyme is feasible. If, according to Fig. 4 c, the transition state o f the su b strate fits the enzyme, the protein chain T becom es shorter. The corresponding im provem ent o f ion-binding causes the free energy o f activation to be supplied in a kinetic equilibrium o f forces, and the reaction is started. The repulsive forces o f the reaction p ro d ucts force the two g lobular units o f the enzym e ap art. In this way, the ions b o und to the p ro tein chain are liberated, and the free energy o f ac tivation is restored. A fter the reaction p ro d u cts have been released the system again becom es func tional. The reaction rate, i.e. the p ro b ab ility o f the reac tion cycle occurring, m ay be regulated by the c o n centratio n o f the ions p ertin en t to the tran sfer unit. A nother possibility for regulation seems to be the supply o r the w ithdraw al o f the free energy o f acti vation stored within the tran sfer unit. T his could
It is well know n th a t muscle contraction is based on the relative sliding o f m yosin and actin fila m ents, the necessary free energy being supplied by the hydrolysis o f A T P [11-13]. A lthough muscle co n tractio n is one o f the principal them es o f bio logical research, the exact m echanism w hereby the chem ical free energy o f A T P hydrolysis is co n v ert ed into m echanical w ork is still a m atter o f discus sion [14]. A bove all, the high therm odynam ic effi ciency o f the process [15], and the precise a d a p ta tion o f A T P hydrolysis to the m echanical require m ents are difficult to explain on the basis o f present theories. R ecently, the a u th o r proposed a m odel o f the elem entary effect in m uscle contraction based on the assum ption o f free energy transfer by the ionbinding m yosin light chains [16]. T aking into con sideration the experim ental results described in the literature, it was concluded th at the d eto u r o f free energy via the light chains, under a so-called chem im echanic equilibrium , accounts for the high therm odynam ic efficiency. In the present paper, the term “chem im echanic equilibrium ” has been su b stituted by the m ore com prehensive “kinetic equilibrium o f forces” . The em phasis o f the fu n d a m ental role o f therm al m olecular m otion allows a m ore precise in terpretation o f kinetic details. This will be d em onstrated along the line o f Fig. 5. T he left h and colum n o f Fig. 5, in a schem atic m anner, indicates the different phases o f m uscu lar co n tractio n [16]. In the nom enclature o f Fig. 1 , the right-hand colum n denotes the distribution o f free energy am ong the com ponents involved. In Fig. 5 A, the free energy G a 2 is concentrated on the tran sitio n state o f the A T P /H 20 system. T he light chains LC and the elastic elem ent E do n ot co n tain free energy. In Fig. 5 B, the reaction A T P + H 2 0 - ^ A D P + Pi has taken place and G a 2 is tak en over by the light chains. In Fig. 5 C, loss o f the Pi has caused the so-called weak binding state to pass in to the strong binding state [17]. U n d er the new condition, the light chains LC pass the free energy o f reaction, G r, to the elastic elem ent E, while retaining the free energy o f activation, G a 1. Sim ultaneously, A D P is exchanged for A T P + H 2 0 .
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E. W. Becker • Dynamics and Kinetics of Enzymes
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r9r
k }—
A -
-A -
k>
K>
an actin unit (Fig. 5E ). The system thus becomes functional again. T he oscillation betw een states A and B under a kinetic equilibrium o f forces allows relative sliding o f the filam ents to take place w ith reduced A TP hydrolysis observed in a fast, unloaded contrac tion [18]. As free energy o f a ctivation is not deliv ered to the A T P /H 20 system before the muscle stran d s have taken over the free energy stored in the elastic elem ent E (transition C —D in Fig. 5), the m uscle can m aintain isom etric tension with rel atively low energy consum ption (“ F enn effect”). W ithin the proposed m echanism , the theory o f cal cium regulation rem ains essentially unaffected [16]. The analysis indicates th a t the kinetic equili b rium o f forces is essential n o t only for the effi ciency o f free energy transfer betw een the muscle com ponents b u t also for the precise ad ap tation o f A T P hydrolysis to the m echanical requirem ents.
Summary
k> Fig. 5. Mechanical diagrams representing the different phases of muscle contraction according to the proposed model.
W hen, according to Fig. 5D , the free energy co n tent G r o f the elastic elem ent E is taken over by the m uscle filam ents, the A T Pase cleft is fu rth er re duced, and the free energy o f activation, G a l , is transferred from LC to the A T Pase. In this way, the tran sitio n state o f the fresh A T P /H 20 load is reached. The pho sp h ate ion Pi form ed by A T P-hydrolysis transfers the strong binding state back to the w eak binding state. Thus, the m yosin head can follow the shift o f the myosin stran d and leap over
In the paper, the astonishing capabilities o f en zymes are explained by the assum ption o f a specif ic therm odynam ic state o f living m atter, called “ki netic equilibrium o f forces” . It is characterized by the reversible tra n sp o rt o f free energy under the action o f therm al m olecular m otion. The ability to avoid en tro p y generation in spite o f high reaction rates seems to be a peculiarity o f biological m a crom olecules. If the suggested theorem o f kinetic equilibrium o f forces turns o u t to be o f general sig nificance to living m atter, it should be an im por ta n t guide in elucidating biochem ical processes.
A cknowledgemen ts The a u th o r is grateful to Professors W. H as selbach, B. Hess, K. C. H olm es, L. Jaenicke, U. Schindew olf, D. M . Taylor, F. W ieland, and Th. W ieland for illum inating discussions and critically reading the m anuscript.
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E. W. Becker • Dynamics and Kinetics o f Enzymes [1] L. Pauling, Chem. Eng. News 24, 1375 (1946). [2] D. E. Koshland, Jr., Sei. Am. 229/4, 5 2 -6 4 (1973). [3] M. Eigen and G. G. Hammes, Adv. Enzym. 25, 1-38(1963). [4] P. Mitchell, Nature 191, 144-148 (1961); J. Biochem. 97, 1-18(1985). [5] L. Kovac, Curr. Topics in Bioenergetics 15, 331 — 372(1987). [6] C. R. Bagshaw and D. R. Trentham, Biochem. J. 133, 323-328(1973). [7] B. Mao, M. R. Pear, J. A. McCammon, and F. A. Quiocho, J. Biol. Chem. 257, 1131-1133(1982). [8] M. Kataoka, J. F. Head, B. A. Seaton, and D. M. Engelman, Proc. Natl. Acad. Sei. U.S.A. 86, 69446948 (1989). [9] J. A. Cox, M. Comte, A. Mamar-Bachi, M. Milos, and J.-J. Schaer, Cation Binding to Calmodulin and Relation to Function, in: Calcium and Calcium Binding Proteins (Ch. Gerday, R. Gilles, and L. Bolis, eds.), pp. 141 -1 6 2 , Springer, Heidelberg 1988.
633 [10] C. W. Heizmann and W. Hunziker, TIBS 16/3, 98-103(1991). [11] H. E. Huxley, Science 164, 1356-1366(1969); H. E. Huxley and M. Kress, J. Muse. Res. Cell Motil. 6, 153-161 (1985). [12] A. F. Huxley, J. Physiol. 243, 1 -4 3 (1974); A. F. Huxley, Ann. Rev. Physiol. 50, 1-16(1988). [13] R. S. Goody and K. C. Holmes, Biochim. Biophys. Acta 726, 13-39(1983). [14] M. A. Geeves, Biochem. J. 274, 1- 1 4 (1991). [15] R. C. Woledge, Nature 334, 655 (1988). [16] E. W. Becker, Naturwissenschaften 78, 445-449 (1991). [17] B. Brenner, Muscle Mechanics and Biochemical Ki netics, in: Molecular Mechanisms in Muscular Con traction (J. M. Squire, ed.), pp. 77-149, Macmillan Press, London 1990. [18] T. Yanagida, T. Arata, and F. Oosawa, Nature 316, 366-369(1985).
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