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be the result of dimerization of PM-BME molecules through the pyrene group. However, although putative transient dimers of pyrene have been p r o p o ~ e d , ’ ~none J~ have been isolated. Far-red shifts have been reported for pyrene derivatives placed in strongly basic environments,20-22these are believed to be the result of simple proton abstraction. To determine if PM-BME could be used as a probe of electron mobility and protein structure, PM-BME fluorescence was studied in a redox system in the presence or absence of cytochrome c (Figure 8). PM-BME sensitivity to redox potential was markedly enhanced in the presence of cytochrome e. The addition of cytochrome e, which should facilitate electron transfer between the redox species, may act to enhance electron mobility near the microenvironment of PM-BME. Increased electron mobility is observed to enhance PM-BME fluorescence (Figure 6). In this system, PM-BME may be bound to, or associated with, a segment or portion of cytochrome c. Any association must be relatively weak, however, since ultracentrifugation did not lead to a fluorescent pellet. In future studies it may be possible to conjugate directly, or indirectly after thiolation, cytochrome c and other electron transport proteins to study electron mobility influences a t different locations on the protein. Acknowledgment. This work was conducted under the guidance of John C. Mitchell, Chief, Radiation Physics Branch, Radiation Sciences Division, USAF School of Aerospace Medicine, Aerospace Medical Division, Brooks
A. Ben-Naim
Air Force Base, Texas.
References and Notes (1) L. Brand and J. K. Gohelke, Anal. Rev. Biochem., 41, 843 (1972). (2) L. Stryer, Science, 162, 526 (1968). (3) E. L. Wehry, “Modern Fluorescence Spectroscopy”, Vol. 2, Plenum Press, New York, N.Y., 1976, p 98. (4) J. Weltman, R. Szaro, A. Frackekon, R. Dowben, J. Bunting, and K. Cathou, J. Biol. Chem., 248, 3173 (1973). (5) R. P. Liburdy, and J. K. Weltman, presented at the 166th National Meeting of the American Chemical Society, Chicago, Ill., 1976, Abstract BIOI 97. (6) R. P. Liburdy, and J. K. Wekman, J. Mechanochem. Cell Motil., 3, 299 (1976). (7) R. P. Liburdy, Ph.D., Thesis, Brown University, 1975. (8) N. Schechter, D. Elson, and P. Spitnik-Elson, FEBS Left., 57, 149 (1975). (9) R. P. Liburdy, Patent pending, USAF, Brooks AFB, Tex., 1976. (10) R. P. Liburdy, Fed. Proc., Fed. Am. SOC.Exp. Biol., 35, ABS 2195, 604 (1975). (11) C. A. Parker and W. T. Reese, Analysf, 85, 587 (1960). (12) C. E. White and R. J. Arganuer, “Fluorescence Analysis“, Marcel Dekker, New Ysrk, N.Y., 1970, p 50. (13) J. R. Heitz, C. D. Anderson, and 6. M. Anderson, Arch. Biochem. Blophys., 127, 627 (1968). (14) D. G. Smyth and H. Tuppy, Biochem. Biophys. Acta, 168, 173 (1968). (15) D. G. Smyth, A. Nagamatsu, and J. S.Fruton, J. Am. Chem. Soc., 83, 4600 (1960). (16) Y. Kanoaka, M. Machda, Y. Ban, and T. Sekine, Chem. pharm. Bull., 15, 1738 (1967). (17) C. W. Wu, L. R. Yarbrough, and F. Y. Wu, Biochemisfry, 15, 2863 (1976). (18) E. Doeller, and T. Forster, Z. Phys. Chem., 34, 132 (1962). (19) E. Doeller, Z. Phys. Chem., 34, 151 (1962). (20) T. Forster, Z. Necfrochem., 54, 42 (1950). (21) T. Forster, Z. Electrochem., 54, 531 (1950). (22) S.Peterson, Angew. Chem., 61, 71 (1949).
A Simple Model for Demonstrating the Relation between Solubility, Hydrophobic Interaction, and Structural Changes in the Solvent A. Ben-Nalm Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel (Received May 3 1, 1977)
A simple model is solved to demonstrate the relationship between various thermodynamic quantities associated with certain processes in solutions and the corresponding structural changes in the solvent induced by these processes. In this specific model it is possible to derive exact expressions for the analogues of the dissolution, and of the hydrophobic interaction processes. Also the exact effect of these processes on the equilibrium composition of the solvent is computed. Following this, we proceed to treat the general case of processes in aqueous solution. The conclusion reached from both the specific model and from the general formalism is the following: If one defines the concept of the “structure of the solvent” (e.g., water) as an ensemble average, then the structural changes in the solvent induced by processes taking place in dilute aqueous solutions (such as solubility, dimerization, or any other association process) will not contribute to the standard free energy of the process, but may well contribute and even significantlyto other standard quantities associated with that process. 1. Introduction
A few years ago’s2 we examined the relation between “structural changes in the solvent” (SCIS), induced by various processes in dilute aqueous solutions, and the various thermodynamic quantities associated with these processes. This relationship is of fundamental importance in understanding the anomalous nature of aqueous solutions, and in particular its role in biochemical proces~es.~-’The main conclusion that has been reachedlJ is the following: if we adopt a definition of the “structure of water”, as an ensemble average (a specific example is given in ref 2, chapter 6, and in ref 8 and will be briefly repeated in section 3 of this paper), and if we examine the 0022-365417812082-0874$0 1.OO/O
thermodynamics of a simple process in very dilute aqueous solutions (such as solubility, hydrophobic interaction (HI), or any other aggregation process), then the SCIS that may be induced by the process itself will have no effect on the standard free energy of the process, but at the same time may well affect other standard thermodynamic quantities associated with that process. Although some details of the argumentation may depend on the precise definition of the concept of the “structure of water”, the general statement (and its proof) were given in such a way as to encompass a wide family of possible definitions for this concept. However, since publication of ref 1,numerous questions have been addressed to the 0 1978 American Chemical Society
Structural Changes in Solvent
author expressing difficulties in grasping the implications of that paper. In a recent paper, Marcelja et aLgahave explicitly controverted (but without offering a proof) our conclusion by stating:
“In fact, it is clear that AGr cannot be zero. I f it were true, there would be no reason for the relaxation from NH, NL t o ”’, NL’” (1.1) The notation in (1.1) is from ref 1 and the meaning of its content will be clear from the content of the following sections. Though a full proof of our contention has been given in ref 1 and 2, we admit that the main conclusion is difficult to accept, not so much because of its logical depth, but more importantly because there exists a sort of dogma which is prevalent in the literature on aqueous solutions. One can easily encounter in many articles and in textbooks of biochemistry, arguments of the following nature (for more details see ref 1): “One finds experimentally that an anomalously large AGO is mainly due to the standard entropy change AS”; the latter is presumed to be dependent on the SCIS induced by the process-hence the anomalously large AGO is explainable in terms of SCIS” (1.2) This kind of reasoning has been propagated in the literature and was accepted without any supplementary proof. Therefore, it was quite an unpopular event when we published a paper proving that statement (1.2) is actually wrong, while admitting that its content is very appealing on qualitative grounds. Perhaps it is worthwhile mentioning that the kind of reasoning presented above is very characteristic of the field of aqueous solution. The reason why many unsound and often incorrect speculations have survived for a long time is simply because it is very difficult to either prove or disprove a contention pertaining to aqueous solution. Therefore it is sometimes advantageous to work out a model which contains all the elements of a selected problem and which is also solvable so that the specific question may be examined, and given an exact answer. The main purpose of this paper is in fact a clarification by details and by specific example of the content of ref 1. The tactics chosen here sharply contrast the ones used before, i.e., instead of using.genera1, and difficult to grasp, arguments, we shall work out a specific model in great detail. Although one outcome of the present paper will be a clear-cut disproof of the content of (1.1) and (1.2), we stress a t the outset that this is not the sole or main purpose of the paper. Our conviction is still firm that the necessary proof has already been given before.lS2 However, since many doubts have been raised and since we admit that the conclusion is indeed difficult to accept, we feel that there is room for further elaboration on the problem. The general nature of the next section will be more pedagogical, rather than providing any new ideas. The level of the solution of the model will be well within any elementary textbook on statistical mechanics.1° We shall show explicitly and exactly that a process may cause a nonzero SCIS yet the SCIS does not effect its free energy change. Furthermore we shall show how the contribution to the entropy and enthalpy change, due to SCIS, exactly compensate each other when combined to form the standard free energy change-hence proving in an explicit fashion that both statements (1.1)and (1.2) are incorrect.
The Journal of Physical Chemistry, Vol. 82, No. 8, 1978 875
The model worked out in section 2 is in fact of some interest outside the field of aqueous solutions. It is relevant to any system of chemical equilibrium of the form A F! B in which we add a third component C and we are interested in the effect of C on the equilibrium composition of the system. This problem is not solvable in the general case of association equilibrium,’l therefore it is of interest to provide some specific examples which are exactly solvable. We also feel that working out the details of the simple model provides some insight into the more intricate problem that arises in processes in aqueous solution. The model treated in section 2 is the simplest one that contains all the relevant elements for the formulation of our question, but it is admittedly quite far from real liquid water. A similar solution may be obtained for a lattice model for water. This model, though still a primitive one, has been used in the past to study various aspects of the properties of water and aqueous s o l ~ t i o n s . ~ J ~ J ~ In section 3 we turn to the general treatment of solubility and hydrophobic interaction in liquid water. We shall stress the analogy between the general result, on one hand, and the specific result obtained from the model, on the other. Finally in section 4 we formulate our general conclusions of this paper. 2. An Adsorption Model (a) The Model and Its Solution. Consider a system of M independent, identical, and localized adsorbing sites, each of which may attain one of two states, L or H, with corresponding energy levels EL and EH, respectively, and we assume that EL C EH. Each site can adsorb a single gas molecule G; the adsorption energies will be UL and UH, according to whether the site is in the state L or H, respectively. Thus in essence we have a system of two states in chemical equilibrium L ZH
(2.1)
and a third agent G interacts with both L and H in the most elementary fashion. As a result of the interaction a shift in the equilibrium composition of L and H may be expected upon adsorption of G. This model may be viewed as a prototype of an equilibrium between two states of a molecule (e.g., helix-coil transformation in biopolymers) and we are interested in the effect of an interacting agent G on this equilibrium. We have chosen localized sites to simplify slightly the mathematical treatment (i.e., we do not consider any translational or rotational degrees of freedom of the sites), but this assumption is not essential for the results of this section. Let us denote by where fl = (hT)-l, with h the Boltzmann constant and T the absolute temperature. The canonical partition function of the empty system, Le., of M site at temperature T , islo
Q(M, T ) =
=
Z: Q * ( M L , M HTI ,
ML
(2.3)
where Q*(ML,MH, T )may be interpreted as the partition function of the same system with fixed values of ML and MH. The equilibrium concentrations of L and M is obtained from the condition
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A. Ben-Nalm
The Journal of Physical Chemistry, Vol. 82, No. 8, 1978
a In Q*/aML = 0
(2.4)
which leads to the solution
=K
(2.5) which is the equilibrium condition for the “reaction” 2.1. We use a bar over ML and MH to denote the values of ML and MH that maximize Q* in (2.3). We also introduce the mole fractions of sites in the two states by = QH/QL
where h = exp(Pp) is the absolute activity of the gas and p its chemical potential. The last form of the partition function is typical to a system of independent sites, where 4 may be viewed as the grand partition function of a single site.1° The four terms of 4 correspond to the four possible states of a site, i.e., empty L, empty H,occupied L, and occupied H. The corresponding probabilities of finding the system in each of these states are given by
-
For later applications it will be also useful to introduce the conditional probability P(L/G) of finding a site in state L when it is known to be occupied by a gas molecule, i.e. These are also the probabilities of finding a specific site in the L or H state for the empty system. We next turn to the case where in addition to the M sites we have also N adsorbed molecules distributed over the sites. For simplicity we assign no internal degrees of freedom to these molecules; they are characterized solely by their adsorption energies UL and UH. The canonical partition function for such a system is
(2.12) where P(G) is the probability of finding an occupied site. Similarly the conditional probability of finding a site in state H, given that it is occupied, is
(2.13) & L ~ ~ Q MH H qLNL4H“
(2.7)
where we introduced the notation qL = exp[-PUL] and qH = exp[-PUH] and NL and NH are the number of molecules adsorbed on L and H sites, respectively, and the summation in (2.7) extends over all possible values of ML,MH, NL, and NH with the obvious conditions ML+MH=M NL+NH=N NL d ML N H < MH (2.8) Clearly the auxiliary parameters NL, NH, ML, and MH serve as intermediate quantities which determine the “energy levels” of the system, namely
E(NL,N H ,M L , M H ) = MLEL + M H E H + NLUL + N H U H
(2.9)
The degeneracy of this energy level is given by the product of the three combinatorial factors in (2.7). The summation in (2.7) cannot be carried out to obtain a closed form of the partition function. This is easily achieved, however, by transforming to an open system with respect to the gas molecules, Le., we define the Grand partition function by
The average number of gas molecules in the system is given by the standard relationlo
(2.14) Denote by x = N/M, the average fraction of occupied sites, we obtain
x = P(L, G) t P(H, G)
(2.15)
Elimination of h from (2.14) gives (2.16) from which we can obtain all the partial thermodynamic quantities of the adsorbed gas. First the chemical potential is p = hT i n
(“)l - x
-
hT l n P L QL
’
q H QH] QL + QH
(2.17)
Using the distribution of the two states in the empty system given in (2.6) we may rewrite (2.17) as
M
Z ( h , M, 2’) = N =.ZO ANQ(N, M, T ) = hTln
(x) l-x
- hT In (exp(-pBG))o
(2.18)
where BG represents the “binding energy” of the gas to the site. This may attain one of the two possible values UL and UH.The notation ( )o represents an average using the probability distribution (2.6), of a system containing no gas molecules. It is important to understand that for any x the second term of (2.18) is always an average over a probability distribution taken in the empty system. In most of the discussions in this paper we shall be interested only in the limit of dilute system, where x
