324
TERRELL L. HILL
Vol. 57
ADSORPTION ON PROTEINS, THE GRAND PARTITION FUNCTION AKD FIRST-ORDER PHASE CHANGES, ACCORDING T O APPROXIMATE STATISTICAL MECHANICAL THEORIES BY TERRELL L. HILL Naval Medical Research Institute, Bethesda, &Id. Received July 99,1969
Adsorption on protein molecules in solution is an actual illustration of Gibbs’ grand canonical ensemble. The transition in properties that occurs as the number of adsorption sites per protein molecule, B, gets very large is examined using the Bragg-Williams approximation. With sufficiently large attractive interactions between adsorbed molecules, the usual partition function, &, based on an ap roximate theory (e.g., Bragg-Williams), predicts a loop and first-order phase change for B + m . A rigorous theory woufd give the thermodynamically stable equilibrium path instead of a loop in the two-phase region. Employment of the complete grand partition function, E, instead of Q eliminates the loop and gives the stable equilibrium path even with an approximate theory. However, in this case a related deficiency in the theory replaces the loop difficulty, as must be expected. Fluctuations and the critical point are discussed. The above considerations apply to the Bragg-Williams approximation, the van der Waals equation, the Lennard-Jones and Devonshire theory of liquids, the quasi-chemical approximation for regular solutions, etc.
I. Introduction Adsorption (or “binding”) on protein molecules in solution is not only an important problem in its own right but is also an interesting actual illustration of Gibbs’ grand canonical ensemble.’ We examine in this paper the relations between protein adsorption with attractive interactions,2 the grand partition function and first-order phase changes, using approximate statistical mechanical theories. Approximate theories of condensation of the van der Waals and Lennard-Jones and Devonshire type, phase splitting in “regular” solutions, etc., are related systems to which essentially the same considerations apply. However, it should be understood throughout the paper that while individual systems of the grand ensemble happen to have a physical significance in the protein adsorption problem, in referring to liquids, solutions, etc., the systems of the grand ensemble have only the usual abstract meaning of mental replicas of the single physical system. While the language of this paper is that appropriate to adsorption on proteins, most of the discussion is in fact concerned with certain properties of approximate partition functions in relation to phase changes. I n this connection, the partition functions written below do not include explicit contributions associated with the protein molecules themselves (e.g., translation and rotation)-that is, the protein molecules are viewed as independent of each other and simply as furnishing sites for adsorption of solute molecules. We shall reserve for the future a more detailed discussion of protein binding, including interactions between protein molecules which depend on the amount of solute bound. For recent papers using the grand canonical ensemble, see for example the work of Mayer, Kirkwood and others.3-6 (1) T. L. Hill, J . Chsm. Phys., 18, 988 (1950). (2) Adsorption of a gas onto very small particles (a.g., dust in t h e
atmosphere) is an equivalent system. (3) W. G. McMillan and J. E. Mayer, J . Chem. Phys.. 13, 2 7 6 (1945); J. E. Mayer, ibid., 10, 629 (1942); 19, 1024 (1951); B. H. Zirnm. ibid., 19, 1019 (1951). (4) J. G. Kirkwood and R. J. Goldberg, ibid., 18, 54 (1950); J. G. Kirkwood and F. P. Buff, ibid., 19, 774 (1951). (5) W. H. Stockmayer, ibid., 18, 58 (1950). (6) H. C. Brinkman and J. J. Hermans. ibid., 11, 574 (1949).
11. Adsorption on Proteins and the Grand Partition Function Consider a solution containing a very large number of protein molecules. Each protein molecule contains B 3 1 equivalent sitesfor thelocalized adsorption of a solute from the solution. The chemical potential of the solute in solution is p , its activity is a and its absolute activity is h p =
=
+
p’J(T) kT In a kT I n A
(1)
p o ( T ) , a standard free energy, is a function of temperature only (the hydrostatic pressure on the solution is kept constant). We may consider each protein molecule as presenting an area a (proportional to B ) . Then in this case the grand canonical ensemble consists of a large number of open “systems” (“system” = a protein molecule of area a, a t temperature T , and with N adsorbed molecules) in equilibrium with a reservoir of adsorbate molecules a t T and p (solute molecules in solution). Let Q(B,N,T) be the partition function for N molecules adsorbed on B sit.es, a t T. Then the grand partition function’ is B
Q(B,N,T)AN
E =
(2)
N-0
The average number of adsorbed molecules per protein molecule is then’ (3)
Equation (3) is the adsorption isotherm, as it gives N(B,h,T)* Special cases of eq. (3), derived from step-bystep equilibrium constants, have in fact been used for a long time8-10 in treating the acid and base dissociation of, for example, polybasic acids and proteins, binding of ions and molecules on proteins, etc. T o verify this connection we can write eq. (7) R. H . Fowler and E. A. Guggenheim, “Statistical Thermodynamics,” Cambridge University Press, 1939; C. 8. Rushbrooke, “Introduction t o Statistical Mechanics,” Oxford University Press, 1949. (8) E. J. Cohn and J. T. Edsall, “Proteins. Amino Acids and Pep tides,” Reinhold Publ. Corp., New York, N. Y., 1943, Chapter 20. (9) G. Scatchard, Ann. N. Y. Acad. Sci., 61, 660 (1949). (10) I. M. Klote in “Modern Trends in Physiology and Biochemistry,” Academic Press, New York. N Y.,1952, edited by E. 8. G. Barron.
ADSORPTION ON PROTEINS
Mar., 1953
(3) in conventional equilibrium constant form. Let us define P ( N ) as &AN. Then P is proportional to the probability that a given protein molecule will have N adsorbed molecules, or to the fraction or number of protein molecules with N adsorbed molecules. Then one finds, for example P ( N ) u N ‘ - ~- Q(N)[exp ( - p ” / k T ) ] N ’ - N P(”)
(4)
Equation (4) has the usual equilibrium constant form with the proper relation to partition functions. In the absence of interactions B!jN - N)I
325
or jX
= e exp
1 - e
(zae)
(9)
where 0 = N / B . Equation (9) may also be found by ‘(picking out the maximum term” in E ; that is, from
(a In Q h N / W ) ~ . =~ ,0~
(10)
Equations (10) and (8) are identical. For sufficiently negative a, eq. (9) predicts a first-order phase ~ h a n g e . ~For example, eq. (9) gives the characteristic loop ACDEFG in Fig. 1 for a = - 3 and the critical curve in Fig. 2 for a = - 2.
(5)
= NI(B
where j ( T ) is the “intrinsic” partition function for adsorption on a particular site. Equations (3) and (5) give, of course, the Langmuir equation’ 1, for any B
>
= Z / B = jX/( 1
+jh)
(6)
111. Adsorption on Proteins Using the BraggWilliams Approximation Even when there are strong attractive interactions between adsorbed molecules, it is easy to see that critical phenomena cannot occur if each protein molecule has only a few sites B. However, they can occur for B of the order of, say, loz3, as is well known.’ In this section we examine this rather instructive transition, using the grand partition function and the Bragg-Williams (B-W) approximation. As will be clear below, other I a = - 3 ,,L ,x,=-3 approximate but more refined and complicated 1 theories (e.g., the quasi-chemical or Bethe approxi- Fig. 1.-Adsorption isotherms in the form In jX versus 8. mation7)will lead to the same general results. Experimental examples of protein binding with attractive interactions have been relatively rare; however, Colvinl’ has recently encountered a - 1.7 number of excellent examples of this type of behavior, with adsorption isotherms of the same form as those shown below. We must mention first the essential features in the B-W approximation. Suppose each site has z nearest neighbor sites and the interaction free energylZbetween two adsorbed molecules on nearest iieighbor sites is w. Then in this approximation’
,I
Q
=
B!jN N I ( B - NyexP (-ffNz/B) cy
(7)
zw/2kT
There will be appreciable geometrical difficulties13 in arranging each site with z nearest neighbors for B too small (say B < 10). A partition function of this same form may also represent, approximately, electrostatic interactions8-10 (a positive), interacting hydration effects,“ etc. The conventional treatment’ of eq. (7) for B very large is to write p / k T = In X = -
(a In Q / ~ N ) B , T
(8)
(11) J. R. Colvin, Canadian J . Chem., 80,320 (1952). (12) E. A. Guggenheim, Trans. Faraday Soc., 44, 1007 (1948). (13) The exact partition function for certain cases with B 12 has
