Free Energy Transduction and Biochemical Cycle Kinetics

Free Energy Transduction and Biochemical Cycle Kinetics

by Terrell L. Hill

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This three-part treatment translates the technical language of research monographs on the theory of free energy transfer in biology, making the subject more accessible to those entering the field. Designed for upper-level classes in biochemistry or biophysics, it can also be used for independent study. 36 figures. 1989 edition.See more details below


This three-part treatment translates the technical language of research monographs on the theory of free energy transfer in biology, making the subject more accessible to those entering the field. Designed for upper-level classes in biochemistry or biophysics, it can also be used for independent study. 36 figures. 1989 edition.

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Simplified, abbreviated, and updated version of: Free energy transduction in biology, Academic Press, 1977. Textbook for use by advanced undergraduates in a classroom setting or for self-study. Annotation c. Book News, Inc., Portland, OR (

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Free Energy Transduction and Biochemical Cycle Kinetics

By Terrell L. Hill

Dover Publications, Inc.

Copyright © 1989 Springer-Verlag New York, Inc.
All rights reserved.
ISBN: 978-0-486-15065-9


Survey of the Elements of Free Energy Transduction

The primary purpose of this book is to explain the basic principles of free energy transduction in biology in as simple a way as possible. A secondary purpose is to study biochemical kinetic diagrams and cycles. These topics are approached by a consideration of hypothetical model systems, at steady state, that are no more complicated than necessary to bring out the essential points. Also, certain refinements and special topics are intentionally omitted in order to keep the discussion at an elementary level. Thus the limited objective here is to provide the reader with an introductory foundation in these subjects. He or she will then be in a good position to study more sophisticated theory and applications to real systems, as summarized in the more advanced books of Hill (1977), Caplan and Essig (1983), and Westerhoff and van Dam (1987).

A related feature of the present book is that any combination of successive chapters, 1, 12, or 123, provides a coherent story. Chapter 1 contains a survey that uses only very simple mathematics and then the two succeeding chapters add somewhat more detailed quantitative aspects. In fact, Section 1 itself introduces many of the main ideas in a descriptive way.

1. States, Diagrams, Cycles, and Free Energy Transduction

We introduce general concepts here by means of an extended example. Consider a cell that is surrounded by a membrane that separates the cell's interior (inside) from its environment (outside). Suppose that a small molecule M has a much larger concentration inside than outside, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Another small molecule L has a somewhat larger concentration outside than inside, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The inequalities as used here mean, explicitly, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. These concentrations are maintained constant (i.e., independent of time). Given a mechanism or pathway across the membrane, M molecules would tend to move spontaneously from inside to outside whereas L molecules would tend to move in the opposite direction. Is it possible for the larger M concentration gradient to be utilized somehow to drive molecules of L from inside to outside against the smaller L concentration gradient? If this is accomplished, it is an example of "free energy transduction" (some of the free energy of M is used to increase the free energy of L, as explained in Section 2).

Because of the nature of cell membranes, it is hard to imagine how M and L could perform this trick on their own, using a realistic mechanism. However, this type of activity is commonplace in cells through the mediation of a large protein molecule, or complex of protein molecules, that spans the membrane and interacts suitably with both M and L. Let us denote this large protein molecule or complex by E. The net process accomplished involves M and L only, but the process is made possible only through the intervention or participation of E: E plays the role of an "agent" or "broker" or "middleman."

To illustrate, let us consider a hypothetical but possible model that will accomplish the job specified above. Suppose E exists in two different structures or "conformations," denoted E and E*, which are interconvertible. There is one binding site for L and one for M on both E and E*. However, these sites are accessible only to inside L and M molecules in conformation E, and only to outside L and M molecules in conformation E*. This is shown very schematically in Fig. 1.1(a). Suppose further that L can be bound on its site only if M is already bound on its neighboring site (i.e., the presence of M stabilizes the binding of L). The notation we use for this step is

L + EM -> LEM

L + E*M -> LE*M

Also, we assume that the binding of L to EM induces the conformation change E -> E*. That is,

L + EM -> LEM -> LE*M.

L and M are bound less strongly to E* than to E so they can now be released to the outside.

This model or mechanism is completed and summarized in Fig. 1.1(b). E exists in six states, numbered for later convenience in Fig. 1.1(c). Small molecules are attached to E in some of these states. The lines between pairs of states indicate possible transitions in either direction. The counterclockwise central arrow shows the dominant (but not the only) direction of the transitions.

If one E complex completes one cycle in the counterclockwise direction, the net effect is to transport one M molecule and one L molecule across the membrane from inside to outside. M moves in the direction (downhill) of its concentration gradient, but L is moved against its gradient. Thus, through the participation of E, which is not itself altered by the complete cycle, free energy associated with the concentration gradient of M is used to move L "uphill" in concentration or free energy. Hence the model illustrates transduction of free energy from M to L, or the "active transport" of L, "active" referring to the uphill aspect just mentioned.

With the use of only one cycle in Fig. 1.1(b), there is so-called tight or complete coupling between the movements of M and L. That is, the stoichiometry is exactly one-to-one: each complete cycle moves one M and one L across the membrane.

Figure 1.2(a) is a generalization of the model in Fig. 1.1(b): possible transitions between EM and E*M are now included. This small modification introduces new complexities. The diagram of states, or "kinetic diagram," Fig. 1.2(a) or 1.2(b), now has three possible cycles, shown in Fig. 1.2(c). The positive direction is arbitrarily chosen as counterclockwise in all three cycles. Because cMi >cMo, cycle a would operate spontaneously in the positive direction, transporting M from inside to outside. By means of this cycle, E provides a mechanism for the transport of M only. To the extent that cycle a is used, M moves from inside to outside without assisting in the transport of L. From the point of view of free energy transduction (from M to L), this cycle does not contribute and simply dissipates some of the free energy of M.

Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], cycle b would operate spontaneously in the negative (clockwise) direction, moving L from outside to inside. M is not involved (except that it is bound to E in every state of cycle b). Because the object, in this model, is to transport L from inside to outside, clearly cycle b allows movement of L in the wrong direction, and is also a wasteful cycle.

Participation of cycles a and b, as just described, is often referred to as "slippage."

Cycle c in Fig. 1.2(c) is the same cycle as in Fig. 1.1(b). This cycle operates spontaneously in the positive direction and transports L against its concentration gradient at the expense of some of the free energy of M. This is the only cycle in which free energy transduction occurs. Cycles a and b both reduce the efficiency of the free energy transduction. Note also that participation of cycles a and b spoils and exact one-to-one stoichiometry (tight coupling between M and L) that would obtain if cycle c were the only cycle. Thus the transitions EM [??] E*M, if they occur, have significant effects.

Let us digress to note Fig. 1.3. This is a considerable generalization of Fig. 1.2(a), with eight states for E and many possible cycles. The heavy cycle is the same as cycle c in Fig. 1.2(c); the arrows show the predominant direction. States in the top square have conformation E and access to the inside; states in the bottom square have conformation E* and access to the outside. The dashed cycle was used as the main cycle in another (more detailed) discussion. In the dashed cycle, L is bound before M, and the subsequent binding of M induces the conformation change E -> E*. Of course, the full diagram in Fig. 1.3 allows for the possible participation of every cycle in the overall activity of E, interacting with M and L.

Rate Constants and the Stochastic Nature of the Kinetics

We continue to use Fig. 1.2 as an aid in introducing some further important general concepts.

In Figs. 1.2(a) and 1.2(b), there are seven lines in the kinetic diagram and therefore 14 possible transitions. Each transition has a first-order rate constant associated with it, denoted αij [here we need the numbering of states in Fig. 1.2(b)]. For example, the meaning of α12dt is the following: if a complex is in state 1 (E) at a certain time, the probability that the complex will make the particular transition 1 -> 2 (E -> E*) in an infinitesimal time interval dt is α12dt. Actually, four of the 14 rate constants in Fig. 1.2(a) are "pseudo-first-order." These four are for the transitions 1 -> 3, 3 -> 5, 2 -> 4, and 4 -> 6. In each of these cases a small molecule is added to E. For example, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where α13 is the pseudo-first-order rate constant included among the 14 αij mentioned above, α*13 is the second-order rate constant for the process E + M (at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) -> EM, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the concentration of M on the inside. All concentrations and all the αij are considered to be constants, independent of time. Typical units employed are ms-1 for the αij and μM-1 ms-1 or mM-1 ms-1 for the α*ij. The concentrations usually have units μM or mM.

Ordinarily we would be interested in a very large ensemble of independent and equivalent E complexes in a cell membrane. If we could select any one of these complexes and follow the detailed succession of states it goes through on the diagram of Fig. 1.2(b), we would observe a random walk from state to state along the lines of the diagram. The complex would spend a random amount of time in a given state (see below) and then jump instantaneously to a neighboring state. Of course, any other E of the ensemble would be doing its own independent random walk, quite oblivious of the activity of its companions. A typical sequence of states in a random walk might be


In this particular sequence, three cycles are completed as indicated: b (+ direction), a +, and b -. Cycle b + moves one L from inside to outside, cycle a + moves one M from inside to outside, and cycle b - moves one L from outside to inside.

Whenever the random walk reaches, say, state 1, there are two outgoing rate constants, α12 and α13. The rate constant for any out going transition is then α12 + α13. (States 3 and 4 would have a sum of three outgoing rate constants.) The probability that no transition has occurred after a time t in state 1 is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The probability that a transition does occur in the next interval dt is (α12 + α13)dt. Thus, the probability that a transition first occurs between t and t + dt is


From this, the mean time [bar.t]1 (the subscript refers to state 1) at which a transition first occurs is


This is the mean or "expected" time that the random walk spends in state 1, before an outgoing transition, every time state 1 is visited on the walk. When an outgoing transition from state 1 does occur, the probability that it is 1 -> 2 is α12/(α12 + α13) and the probability that it is 1 -> 3 is α13/(α12 + α13). Similar comments can be made about the other five states in the diagram.

If a single complex could be followed in its steady-state random walk over a huge number of transitions (see Section 7), we could observe the fraction of time pi spent in each state i = 1, 2,..., 6. The fraction pi for state i depends not only on but also on the relative frequency that state i is visited during the walk. Also, we could observe the mean frequency or mean rate Ja+, ..., Jc- at which each of the six cycle types a + , ..., c - are completed in the course of the walk. These are positive pure numbers per unit time. Typical units for the Jκ± (κ = a, b, c) are ms-1 or s-1 (cycle completions are less frequent than transitions). Methods of calculating the pi and Jκ± from a diagram and its associated αij will be discussed in Sections 6 and 7.

The net cycle fluxes (+ direction) are defined as Jκ = Jκ+ - Jκ- for any cycle κ. Jκ may be positive or negative.

If we had a large ensemble of E complexes and could observe the particular state i = 1, 2, ..., 6 of each complex at the same time t, the fraction of complexes in state i would be the same pi as above.

To be more precise, if time averages (above) for a single E are taken over infinite time and instantaneous ensemble averages are taken over an infinite ensemble, then the same pi would be found by the two methods. The use of finite samples will lead to small differences that can be attributed to fluctuations.

It is interesting that the mean rate of cycle completions (Jκ±) cannot be obtained from an instantaneous ensemble average (as above, for the pi). However, this is not true if an expanded diagram, with more details and states, is used (Section 8).

Other Examples of Kinetic Diagrams

We have used Fig. 1.2 to introduce a number of concepts that are actually applicable to a vast array of kinetic systems and diagrams. A few further examples are mentioned here to illustrate this point.

Kinetic diagrams, as in Figs. 1.1, 1.2, and 1.3, are by no means limited to systems involved in free energy transduction or to systems in a membrane. Three extremely simple illustrations of this are shown in Fig. 1.4. Figure 1.4(a) is the diagram for binding a ligand L from solution (concentration cL) onto a macromolecule E that may be free in solution, or in a membrane, or on a surface. This diagram does not have any cycles. Figure 1.4(b) is the diagram for an enzyme E, possibly free in solution, that binds a substrate S (concentration cS) and then catalyzes its reaction, S -> P, to product (concentration cP). The dominant direction in the cycle is counterclockwise. The net effect of one cycle by one E, in the counterclockwise direction, is the conversion of one S in solution to one P in solution; E is unaffected by a complete cycle. Figure 1.4(c) is an explicit and more elaborate diagram of the same type. Here E is an ATPase, possibly in solution. First ATP is bound to E; then ATP is hydrolyzed on E, to products; then Pi (inorganic phosphate) is released; and finally ADP is released.

In all of these diagrams, transitions can occur in either direction along each line of the diagram, and the transitions are stochastic (random), as governed by the first-order rate constants αij. Also, characteristically, a macromolecule is the central figure: the diagram enumerates the possible discrete states and transitions of the macromolecule, including small molecules that interact directly with the macromolecule.

Figure 1.5(a) is a modification of Fig. 1.2(a) in which the spontaneous chemical reaction S -> P (both species, say, on the inside of the membrane, with concentrations cS and cP) is the free energy source and drives L from inside to outside against its concentration gradient. Slippage may occur because of the possible transitions ES [??] E*S: the reaction S -> P in the top small cycle accomplishes nothing; the bottom small cycle, running clockwise, allows L to move the wrong way, from outside to inside. Free energy transduction occurs only in the large cycle.

Figure 1.5(b) generalizes Fig. 1.5(a), and is the analogue of Fig. 1.3. The main (large) cycle in Fig. 1.5(a) is marked in Fig. 1.5(b) with heavy lines and arrows. The dashed transitions provide an alternate plausible path (used in ref. 4) between states E and LES.


Excerpted from Free Energy Transduction and Biochemical Cycle Kinetics by Terrell L. Hill. Copyright © 1989 Springer-Verlag New York, Inc.. Excerpted by permission of Dover Publications, Inc..
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