2907
Equilibrium Studies on Indefinitely Self-Associating Systems
Sedimentation Equilibrium Studies on Indefinitely Self-Associating Systems. N-Methylacetamide in Carbon Tetrachloride G. J. Howlett, L. W. Nichol," and P. R. Andrews Department of Physical Biochemistry, John Curtin School of Medical Research, Austraiian National University. Canberra. A.C. T . , 2601. Australia (Received December 4, 1972; Revised Manuscript Received June 4, 7973)
Theory is developed to permit simulation by numerical integration of sedimentation equilibrium distributions pertaining to systems undergoing indefinite self-association. The treatment includes consideration of volume changes on reaction, nonideality effects, and density variations with radial distance arising as a result of solute redistribution and compressibility effects. The development, initially in terms of an isodesmic association, is extended to cases where two equilibrium constants apply. For both models, expressions for the second derivative of the concentration distribution are obtained and discussed in relation to the evaluation of nonideality coefficients corresponding to t h e conventional second and third viria1 coefficients, and to the appearance of maxima in schlieren records. The theory is applied to the analysis of sedimentation equilibrium results obtained with N-methylacetamide in carbon tetrachloride. It is shown that the distributions may be described in terms of an isodesmic indefinite association characterized by an equilibrium constant of 2 X 103 M - 1, in agreement with previous findings. However, it is also shown that this solution is not unique and, indeed, the same experimental results are fitted with a distribution computed on the basis of a model in which dimer formation is governed by an equilibrium constant of 1M - l and addition of monomer to higher polymers by a n association constant of 50 M-1. These values are in substantial agreement with those used to interpret previous infrared and nuclear magnetic resonance results obtained with the same system. Accordingly, the work resolves the apparent controversy between previous interpretations given to spectrophotometric and sedimentation equilibrium results.
Introduction Sedimentation equilibrium results obtained with several systemsl-5 have been analyzed successfully in terms of an indefinite self-association, for which a single equilibrium constant suffices to describe the formation of a series of polymers. The treatments have included the dependence of the activity coefficients of the solute species on total concentration, but have not accounted for possible volume changes on reaction or effects due to density variation with radial distance. One purpose of the present work is to present equations which account for these effects and which permit computer simulation of equilibrium distributions pertaining to indefinitely self-associating systems. Such simulations have proved useful in analyzing results obtained with systems undergoing discrete selfassociations6 and heterogeneous associations.7.s In addition, an expression is obtained defining the appearance of a maximum in the schlieren record of a sedimentation equilibrium experiment conducted with an indefinitely self-associating system. In previous work,g with the single solute hyaluronic acid, the position of a maximum of this type proved useful in estimating the nonideality coefficient corresponding to the conventional third virial coefficient. Although the relations to be presented are applicable to any indefinite self-association, they are used herein to reexamine the association of N-methylacetamide in carbon tetrachloride under conditions where the schlieren plot of the equilibrium distribution exhibits a maximum.1° The self-association is thought to occur by linear hydrogen bonding, the system serving as a useful model in the investigation of the contribution of peptide hydrogen bonds to protein and polypeptide stability.ll-l3 Previous sedimentation equilibrium resultslO have been interpreted
in terms of an indefinite self-association described by a single equilibrium constant (an isodesmic association*) of approximately 2 X 103 M - I , in marked contrast to lower values found from infrared14 and nuclear magnetic resonancel5 studies. In the latter work, a model was preferred in which dimer formation was characterized by an equilibrium constant of -1-1.5 M-1 and higher polymer formation by a constant of -24-38 M-1. In an attempt to correlate these findings, the present experimental results are analyzed in terms of both models, with allowance for nonideality, density variation, and possible volume changes in the isodesmic case. Theory
Isodesmic Indefinite Self-Association. Consider first the A SA, (i = 1, 2, . . .), each equilibrium system, A,-1 being governed by the equilibrium constant
+
where m denotes the molar concentration and y the activity coefficient on the same scale. The activity coefficient of each species A, may be expressed as a virial expansion of the formgJ6
in -y4
=
il.I,z(Bc+ C?
+ ...)
(2) where c is the total weight concentration (g l.-I), MA,is the molecular weight ( = MA) of species A, and the nonideality coefficients, B, C, etc., are assumed identical for each species. It follows that ya,/-yA,-lyA = 1 and from eq 1 that mA = K,m4&-?mA.If all K , are set equal, K , = K (an isodesmic indefinite self-association) the latter relation becomes maL= KL-lm41. This equation has been derived on the basis of a model involving successive addiThe Journal of Physical Chemistry, Vol. 77, No. 24, 1973
2908
G.J. Howlett, L. W. Nichol, and P. R. Andrews
tions of monomer to polymeric species, but is valid regardless of the pathway of formation of A, provided any polymer-polymer interaction is also described by K The following useful relations may now be derived2 on the basis of sums of geometrical progressions since the common ratio KmA < 1.
AV
= zMAU4(r)- ( I
or
-l)MAEA (r) - M4UA(r) (9a)
+
Ca,(r) = E,(r) / ( I - l)AV/zM\i (9bi To complete the basic equations, it need only be noted that if A V # 0, K(r) varies with r according
n
ca/MA -
=
m4/(l-Km4)
(3a)
d K(r) / d(r2)
=
-AVdp(r)K(r)/2RT
(10)
j-.
Combination of eq 3,4b, 6, and 9b yields =
MAm,/(l-KmJ2
(3b)
-1
di?(r)/ d(r2)
=
$
=
a/B
(Ila)
m
ciV4ck = i=1
MA*mk(l-t-KmJ/(l -KmA)'
(3c)
At equilibrium in a sedimentation equilibrium experiment conducted at constant angular velocity and temperature T each species, A,, distributes with radial distance r according to17
d In a,&>/d(r*>
4,(r)
= $4(r)M4
= (1- EA(r)p(r))w2/2RT
(4a)
(4b)
where a4L(r)and 04L(r)are the activity and partia: specific volume of A,. respectively, and p(r) is the solution density a t r Equation 4 may be rewritten as
d In cq (r) d In m,(r) d In YA ( r ) _____- = $4,jr)M
