Plausible View on the Biological Molecular Energy Machines ALEXANDR KREMEN‘

Institute of Microbiology, Czechoslovak Academy of Sciences, VidekkA 1083, 142 20 Praha 4, Czechoslovakia, and Division of Biochemistry, World Open University, U tie5Aovky 4, 182 00 Prague 8, Czechoslovakia

SYNOPSIS

A qualitative picture of operation modes of biological molecular energy machines is presented. It is suggested that there is mutual control between the flow of molecular energy stored in a biological molecular energy machine and the sequence of nonequilibrium conformational states through which the machine passes in doing work. If the structure of the conformational space is favorable, the set of trajectories in this space decomposes into two families, each of which accomplishes another task. This divergence of trajectories enables to distinguish molecular objects according to differences in interaction between the machine and the object, i.e., to perform a measurement on a molecular object and process the object according to the result of that measurement.

INTRODUCTION In a preceding article the “refutations” (published by several authors in the 1970s) of McClare’s of biological molecular energy machines (BM in the following) have been disproved. Nonetheless, a satisfactory up-to-date reevaluation of the nearly 20year-old physical picture presented by McClare requires more than a disproof of objections. This article suggests an area (different from that envisaged by McClare) where the idea of the BMs might be fruitful. The next section therefore presents a physical description on the level of processes in the conformational space of a BM (the recent article by Welch and Kel15 was’not concerned with this point of view). The third section describes the operation of a BM as a measuring device. Biological implications are mentioned where appropriate.

OPERATING MODES OF A B M As with other macromolecular structures, the dynamics of the BMs is based on the presence of multiple conformational states. In proteins, for example, local minima of potential energy with respect to the main- or side-chain dihedral angles of individual Biopolymers, Vol. 32, 471-475 (1992) 0 1992 John Wiley & Sons, Inc.

CCC 0006-3525/92/050471-05$04.00

* Present address: Division of Biochemistry and Biophysics, World Open University, U tiegiiovky 4,182 00 Prague 8, Czechoslovakia

amino acid residues compose a large set of states of one order. Each of these minima contains many finer minima separated by lower potential energy barriers. These minima result from more subtle structural rearrangements than the dihedral transitions and are said to be of lower orders. The minima due to the dihedral angles themselves are embedded in minima of higher orders separated by correspondingly higher and higher potential energy barriers. They are due to structural changes in larger and larger segments of the protein structure. The number of experimental, as well as theoretical, works dealing with conformational states of different orders or with protein (mostly equilibrium) dynamics based on their presence is growing (see, e.g., ref. 6-11). With nucleic acids, in general, qualitatively the same situation will be encountered, although details will be different, of course, This conclusion follows from the fact that conformational changes in nucleic acids likewise occur on a wide scale, from subtle rearrangements within individual bases up to changes in large segments of the macromolecule. Standard enzymes and the BMs differ essentially in which conformational states they visit and, accordingly, in the significance of thermal fluctuations in their dynamics. Both types of units operate in cycles and each cycle consists of a sequence of transitions between conformational states. In the following, only the “forward” parts of the cycles will be compared, whereas resetting of the enzymes or 471

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BMs to the conditions from which they commence next cycles will be omitted. The transitions of a standard enzyme are governed by thermal fluctuations. In other words, standard enzymes are random walkers on a lattice of accessible conformational states (note that the dimension of this lattice and therefore of the random walk may be higher than unity even if the kinetic scheme of the enzyme action is linear). The BMs, on the contrary, do not operate at equilibrium (both internal and with the surroundings) because they gain molecular energy in a chemical reaction or by absorption of light and store it temporarily within the BM. An operating cycle of a BM, in contrast to a standard enzyme, thus consists of excitation to a metastable state, work, and canceling the metastable state by equilibration ( internal and with the surroundings). This sequence is essential since the work must be done before the stored energy degrades to heat during equilibration.’ Nonequilibrium conformational states are well suited for energy storage since they can provide a wide range of lifetimes and transitions between them offer a rich choice of pathways (see later ) .The present author suggests that the BMs direct the flow of stored molecular energy so that transitions between those nonequilibrium conformational states be facilitated that in turn drain the molecular energy flow into the most effective sequence of these states. This mutual control of molecular energy flow and conformational transitions may either merely weaken or completely override the effect of thermal fluctuations in dependence on the amount of the stored energy (it determines which nonequilibrium states become occupied) and the barrier heights between the states in the conformational space (note that the average energy of thermal fluctuations at room temperature is about 15-20 times lower than the energy gained when one molecule of ATP or GTP is cleaved; the corresponding Boltzmann factor The reader will notice that exp (-15) = 3 X this idea is an extension of Blumenfeld’s conformational relaxation? This concept of relaxation does not necessarily imply that the process proceeds always downhill on a potential energy scale.’’ Certain transitions may comply with McClare’s req~irernent”~ that the BMs utilize resonance transitions (since these are fast and nondissipative, even when they occur between nonequilibrium states). On invalidating (in the preceding paper’ ) Gray’s objection l2against reversible operation of the BMs, resonance transitions regain their attractiveness and physical justification in this context. In the conformational space, then, the trajectory

representing the actual motion of the BM only “exp l o r e ~selected ~’ states (of appropriate orders in their hierarchy) and thus simulates more or less closely a dynamic trajectory of a deterministic system rather than a trajectory of a random walker. In favorable cases, the structure of the conformational space may even cause one family of trajectories to eventually reach one particular subset of states (during this motion, the BM performs certain function), whereas the rest of the trajectories perform a different function, reaching another distinctive subset of states. If so, the BM can effectively distinguish between at least two possibilities, which is, in fact, the simplest implementation of a measurement. The measurement process, even if the discussion is confined to biological macromolecules, can be discussed from various aspects. This study deals with three of them: intramolecular dynamics, dissipation, and analogy with computation. It intentionally omits the quantum mechanical aspects likely to be difficult to readers not interested in this field. Readers missing this and other points may consult ref. 13 to compare relevant features of the BMs with Pattee’s constraints and verify that they are satisfied (e.g., choice between sets of alternative metastable states, reduction of the number of possibilities, impossibility of simultaneous classification and time-symmetric causal description, etc.) .

BMs AS MEASURING AND DATA PROCESSING DEVICES Bennett l 4 discussed results showing that any measurement cycle unavoidably contains at least one irreversible step: information destruction resetting the measuring apparatus in preparation for the next measurement. As an operating cycle of a BM likewise contains a resetting dissipative step, this analogy supports Gray’s1’ view that at least certain BMs may be suspected to perform a measurement (it does not matter that Gray used this approach to refute the BMs). Also, this analogy is tempting if one thinks of enzymatic units performing accurate recognition (e.g., the ribosomes). In the following, the idea that the BMs can operate as measuring devices will be discussed in some detail. First, a plausible mechanism in terms of molecular dynamics will be explained. Then, the connection to the ideas of reversible computation will be discussed briefly. If a BM operates as a measuring device, it can hardly store the data for later use as a computer coupled to a measuring device can do. Clearly, a BM must combine data acquisition and their processing.

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The corresponding action of a BM can be translated into the language of physics as follows. The molecular energy excites the active site of the macromolecular unit (where it was transferred to the B M ) into a nonequilibrium conformational state. T h e energy is then transferred t o other parts of the BM (this is called a relaxation process) and eventually excites the measuring site (e.g., a recognition region) into a particular (out of many possible) nonequilibrium conformational state. Which state it is depends on the actual interactions with the measured object (i.e., on their types and strengths) and these in turn may depend, among others, even on the history of the combined unit. While the two parts (measuring site measured object) might have been interacting before excitation, the transition into a metastable state establishes stronger correlation between them because they are now excited as a single subsystem with (temporarily) no or insignificant interaction both with the nonexcited remainder of the BM and with the surrounding. This effective (although temporary) isolation of the excited subsystem changes ( a t least partly) the kinds of degrees of freedom available to the subsystem and also lowers their number ( t h e correlation restricts the number of allowed motions). As a result, the subsystem occupies another part of the conformational space, the volume of which is smaller than the volume occupied before excitation. I t is this compression in the conformational space and the accompanying entropy reduction that represents the “data acquisition” and is a prerequisite for converting the original (near to equilibrium) interaction into a measurement event. ( A simple paradigm of this reasoning may be found in Bennett’s discussion l4 on Maxwell’s demon.) Since the relevant part of the B M is effectively decoupled from the rest of the BM, a s well as from the surroundings ( a t least on the significant levels of the state hierarchy), we may regard it a s a n isolated subsystem so each of the feasible states represents particular initial conditions defining a solution of the equations of motion (which need not be specified here) and therefore defines a particular sequence of conformational states. (If states of lower orders are taken into account, each single sequence expands into a family of lower-order sequences; see, e.g., ref. 7 ) . One of these sequences includes that initial state that minimizes the energy of the initial nonequilibrium interactions. If thermal fluctuations could be avoided completely, motion would only start and proceed along this optimal sequence. The fluctuations are, however, always present, and may in principle cause random transitions both between states within one sequence and between different

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sequences. As already mentioned, relative importance of thermal fluctuations depends on the heights of the barriers between states and on the amount of molecular energy stored in the B M and controlling the motion through the sequences of nonequilibrium conformational states. Considered as a result of “measurement data processing,” the actual conformational state occupied a t any later time thus may-but need not-project correctly the true interactions a t the initial instant t o the actual time. While then a BM necessarily operates, for physical reasons, with a positive chance of error, it can diminish this chance in the course of the motion away from the initial state by draining the motion (“data processing”) as near as possible t o the optimal sequence of states. This will be so if the structure of the conformational space facilitates transitions to the optimal sequence and hampers those out of it. This process (in general, any kind of work done by the B M ) may comprise dissipative steps, but not necessarily. T h e term “data processing” then denotes both dissipative and nondissipative processes. On terminating the task, the BM may dissipate the unused part of molecular energy. This relaxation process results in resetting the BM into that equilibrium conformational state from which it can commence another operating cycle. ( I t is feasible, however, that the unused part of molecular energy may remain stored in the BM, being added to the amount of molecular energy gained in the next cycle. If so, the initial states of subsequent cycles may not coincide and the actual operation of the BM depends, t o some extent, on the preceding cycles. This may be the case of elongation cycles in protein synthesis. Complete relaxation to equilibrium then would only occur after releasing the terminated polypeptide from the ribosome.) On this general level, the dynamics of a BM’s measurement cycle necessarily comprises the dissipative step during the final resetting t o e q ~ i l i b r i u m ’and ~ either accidental or principal dissipative steps during the preceding process. As to the dissipation during the resetting step, it is essentially determined by the conditions imposed by the kind of work done during the operating cycle. The work can only be performed by a n appropriate sequence of states and the minimum distance from equilibrium among the substates comprising the last state in the sequence determines the amount of energy dissipated during resetting. If we believe t h a t the operating cycle of a BM is optimized with respect t o dissipation, we can only consider nonaccidental dissipative steps. A comparison with the principles of reversible computation

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(see, e.g., ref. 14) will indicate their necessity. Note that the mechanism of reversible computation was invented to show that in principle there is no lower limit t o dissipation. Since reversible computation requires an enormous amount of computational space and time, the operational lower limit of dissipation is determined by the requirement of “forgetting” the unnecessary part of the history of the computational process. By analogy, if a viable BM is optimized with respect to dissipation, it utilizes only those dissipative steps that could not be made reversible without additional resources of processing time, as well as without the necessity to remember exactly the dynamic pathway, which would also have to include complete insensitivity to thermal noise. This argument points again to the advantages of utilizing truly reversible steps like resonance processes whenever possible.

A COMMENT ON FREE ENERGY There are good reasons why in the preceding discussions the notion of free energy has been avoided. Statistical mechanics defines free energy F in terms of the partition function Z of the system F=-kT-lnZ Unfortunately, due to the hierarchical structure of the conformational space of a protein described in the foregoing, the partition function Z need not be well defined because a t any moment not all allowed conformations are accessible (the system is not ergodic, i.e., meaningful time averages cannot be calculated as averages over the conformational space). The excellent report by Palmer l5 allows even to suspect that the conformational spaces of the proteins often cannot be decomposed to components, each of which, taken separately, could be regarded as an ergodic subregion of the respective conformational space.

In this way, nearby (in the conformational space ) sequences of nonequilibrium states may be optimal with respect to differenttasks and external conditions. On the contrary, standard enzymes can hardly prevent visiting nearby states during their random walk on them under the influence of fluctuations. Their nearby states therefore cannot participate in fulfilling different tasks. (Standard enzymes can select between two possibilities in another way: They can either complete the operating cycle or cancel the action by returning to the starting point.) This superiority of the BMs is paid for by more complex structure and dependence on molecular energy supply. 2. Minimizing or avoiding random walking in the conformational space, the BMs operate relatively faster than if their operation were controlled by thermal fluctuations. (The difference between in vivo and in vitro rates of certain processes, e.g., protein synthesis, may well be due to inadvertent modifications of the BMs and their operation modes during the preparation procedures. A simple model shows that profound changes in the operation modes may require changes in as little as a single transition probability.) These differences can be interpreted in terms of “molecular computers.” The standard enzymes operate as Brownian reversible computers l4 dragged toward completion of their random walk by the actual affinity of the catalyzed reaction, whereas the BMs simulate more or less closely ballistic computers, l4 creating in the conformational space a (nearly) dynamical trajectory isomorphic to the desired “computation” (which is the physical way to accomplish the desired task). Encouraging discussions with Professor Witold Brostow (North Texas University, Denton) on related topics are gratefully acknowledged.

CONCLUSIONS The BMs are superior to standard enzymes at least in two respects: 1. They can accommodate their operation modes to more diverse conditions and more exacting tasks because they can diminish or override the influence of thermal fluctuations.

REFERENCES 1. KFemen, A. Biopolymers 32,467-470. 2 . McClare, C. W. F. (1971) J. Theor. Biol. 3 0 , 1-34. 3. McClare, C. W. F. (1972) J. Theor. Biol. 35, 569595. 4. McClare, C. W. F. (1974) Ann N Y Acad. Sci. 2 2 7 , 74-97.

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5. Welch, G. R. & Kell, B. (1986) in The Fluctuating Enzyme, Welch, G. R., Ed., Wiley Interscience, New York, pp. 451-492. 6. Blumenfeld, L. A. (1983) Physics of Bioenergetic Processes, Springer, Berlin. 7. Ansari, A., Berendzen, J., Bowne, S. F., Frauenfelder, H., Iben, I. E. T., Sauke, T. B., Shyamsunder, E. & Young, R. D. (1985) Proc. Natl. Acad. Sci. U S A 8 2 , 5000-5004. 8. Parak, F. (1985) in Structure and Motion: Membranes, Nucleic Acids and Proteins, Clementi, E., Corongiu, G., Sarma, M. H., & Sarma, R. H., Eds., Adenine Press, pp. 243-250. 9. Blumenfeld, L. A., Burbajev, D. S. & Davydov, R. M. (1986) in The Fluctuating Enzyme, Welch, G. R., Ed., Wiley Interscience, New York, pp. 369-402.

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10. Cartling, B. (1989) J. Chem. Phys. 9 1 , 427-438. 11. Frauenfelder, H., Alberding, N. A., Ansari, A., Braun-

12. 13. 14. 15.

stein, D., Cowen, B. R., Hong, M. K., Iben, I. E. T., Johnson, J. B., Luck, S., Marden, M. C., Mourant, J. R., Ormos, P., Reinisch, L., Scholl, R., Schulte, A., Shyamsunder, E., Sorensen, L. B., Steinbach, P. J., Xie, A., Young, R. D. & Yue, K. T. (1990) J. Phys. Chem. 9 4 , 1024-1037. Gray, B. F. (1975) Nature 253,436-437. Pattee, H. H. (1967) J. Theor. Biol. 17, 410-420. Bennett, C. H. (1988) IBM J. Res. Den 3 2 , 16-23. Palmer, R. G. (1982) Adv. Physics 3 1 , 669-735.

Received February 25, 1991 Accepted August 5, 1991

Plausible view on the biological molecular energy machines.

A qualitative picture of operation modes of biological molecular energy machines is presented. It is suggested that there is mutual control between th...
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