4012
MATTHEW HUSSEY AND PETER D. EDNONDS
Proton-Transfer Reactions.
A Mechanism for the Absorption of
Ultrasound in Aqueous Solutions of Proteins
by Matthew Hussey*l and Peter D. Edmonds Department of Biomedical Engineering, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (Received June 16, 1971) Publication costs assisted by the Department of Biomedical Engineering, Moore School of Electrical Engineering, University of Pennsylvania
This paper presents a first-order method of calculating the excess ultrasonic absorption due to proton-transfer reactions for solutioiis of proteins and polypeptides. The calculated results for three proteins-bovine serum albumin, human hemoglobin, aiid gelatin-and two polypeptides-polylysine aiid polyglutamic acid-are plotted and are s h o w to resemble qualitatively the experimental data from other authors. It is concluded that this mechaiiism is appreciable in the ranges pH 8. However, the calculation does need refiiiemeiit and some ways to do so are recommended.
Introduction Iiessler and Dunn2 have reported results of ultrasonic investigations on aqueous solutions of bovine serum albumin (BSA) at 20" over the frequency range 0.3-163 RIHz and over the pH range 2.3-11.8. They found a sharp increase in the excess absorption outside the range 4.3 < p H < 10.5, This effect was reversible and was more pronounced a t the lower frequencies. They attributed the increase in absorption belo\y p H 4.3 to the intermrdiate N-B' transition discusscd by Foster3 and that above pH 10.5 to cxpansion of the molecule, Both sets of mechanisms represent conformational changes in the molecule. Subsequently, Zana and Lang4 reported results of their ultrasonic experiments on aqueous solutions of BSA, p-lactoglobulin, and lysozyme at 25" a t 2.82 RIHz and over the p H range 1.0-13.3. They concluded that proton-transfer reactions at the basic and acidic side chains constituted the mechanism most likely giving rise to the observed excess absorption in the ranges pH 10.0. Some recent work by the present authors should help to shed some more light on this p r ~ b l e m . ~ -A~ theory of the pH dependence of the excess absorption due to proton-transfer reactions in aqueous solution was worked out and applied to the results of extensive ultrasonic measurements on solutions of glycine5 and of Issine and arginineU6 The measurements on glycine were at 22, 30, and 37") covered the frequency range 10-130 MHz, and involved initial concentrations 0.25, 0.5, and 1.0 M at pH 8. The studies on lysine and arginine were at 2 2 O , 0.25 M , pH >9, and frequencies between 10 and 130 MHz. In this work it is proposed to apply the theory in a predictive manner to solutions of the following polypeptides and proteins for which ultrasonic absorption T h e Journal of Physical Chemistry, Vol. 76, N o . 26, 1971
data already exist: BSA,2r4h e m o g l ~ b i n ,gelatin,lO)ll ~,~ polylysine, l 2 and polyglutamic acid.I3
Summary of the Theory The general form for the proton-transfer reaction between species A and B is 1212
A + X Z B
(1)
k2l
Here 1q2 and k21are rate constants, forward and reverse, respectively. I n aqueous solution X may be the hydrated hydrogen ion, HaO+, or the hydroxyl ion, OH-. For small displacements from equilibrium this reaction has a single relaxation time 7 described as14
* To whom correspondence should be addressed at the Western Re-
gional Hospital Board, Depart,ment of Clinical Physics and Bioengineering, Glasgow, C.4, Scotland. (1) This work was done in partial fulfillment, of the requirements for the Ph.D. degree in Biomedical Engineering at the University of Pennsylvania. (2) L. W. Xessler and F. Dunn, J. Phys. Chem., 73,4286 (1969). (3) J. F. Foster, "Plasma Proteins," F. IT. Putnam, Ed., Academic Press, New York, N . Y . ,1960, Chapter 6. (4) R. Zana and J. Lang, J . Phys. Chem., 74,2734 (1970). ( 5 ) M. Hussey and P. D. Edmonds, J . Acoust. Soc. Amer., 49, 1309 (1971).
(6) M. Hussey and P. D. Edmonds, ibid., 40, 1908 (1971). (7) M. Hussey, Ph.D. Dissertation, University of Pennsylvania, Philadelphia, Pa., 1970. (8) P. D. Edmonds, T . J. Bauld, J. F. Dyro, and M. Hussey, Biochim. Biophys. Acta, 200, 174 (1970). (9) W.D. O'Brien, private communication, 1969. (10) H. Pauly and H. P. Schwan, accepted for publication in J .
Acoust. Soc. A m e r . (11) Y . Wada, H. Sasabe and 31.Tomono, Biopolymers, 5 , 887 (1967). (12) R. C. Parker, L. J. Slutsky, and K . IZ. Applegate, J . P h y s . Chem., 72,3177 (1968) (13) J. J. Burke, G . G. Hammes, and T. B. Lewis, J . Chem. Phys., 42,3520 (1965). I
ABSORPTION OF ULTRASOUNU IN AQUEOUS SOLUTIONS OF PROTEINS 7 =
[k12(6A
+ + 1/K)]-‘ ZX
4013
(2)
whcrc ai is activity of spc.eirs ‘5” at cquilibrium and K is thc cquilibrium constant (dcz1/7cln). Also 8;=
(3)
yizi
whcrc F; is equilibrium valuc of concentration of species “i” and y; is thc corrcsponding activity cocfficicnt. Whcn a solution containing cquilibrium 1 is irradiatcd by ultrasound, thc cquilibrium is disturbcd. If thc rathe displaccdiant power lcvcl is low (((1 ’CV mcnt from equilibrium is small and the restoration of equilibrium is spccificd by the characteristic time given by cq 2. I n thc frcqucncy domain this rclaxational bchavior is manifested as an cxccss absorption characterized by
where ach is the exccss amplitude absorption coefficient due to the reaction, f is frequency, fo is relaxation frequency, and A is relaxation amplitude.
Here D represents the term in parentheses and /
where p is solution density, R is the gas constant, T is absolute temperature
(7) and
AU
=
AV - mAH
(8)
Here AV and A H are the partial molar volume and enthalpy, respectively, for the reaction and m is a constant for the medium a t any temperature.14 If activity coefficients be assumed to equal unity, the D and G parameters may be calculated for any value of of pX (=-log E X ) from eq 5 and 7 when one knows the equilibrium constant and the initial concentration Co, where
CO
EA
+
CB
(9)
Figure 1 illustrates the dependencies of the D and G parameters (and hence fo and A , respectively) on pX for three different values of Cos Typical values for 1~12,AU, and K were assumed in making the calculations for this figure. Certain important features of the curves in Figure 1 should be noted. The range of pX in which appreciable excess ultrasonic absorption occurs covers about 1.2 pX units and falls on the low side of p K . The absorption
I
I
I
0
2
I. 3
PX
Figure 1. Dependence of the D and G parameters on pX for three different values of initial concentration, CO.
magnitude parameter, G, peaks a t a particular pX a t which, for the same initial concentration value the relaxation frequency parameter, D ,troughs. This special pX value, pX, is given approximately by
1 p x , = --(log K 2
+ log CO)
=
1 -(pK - log CO)
2
(10)
Decreasing Co shifts pX, toward pK and diminishes, but not linearly, the peak value of G and the trough value of D. The introduction of activity coefficients5 adds the term - l / 2 log Y B t o the right-hand side of eq 10. Conditions under which the activity coefficients would be appreciably less than unity (e. g., high ionic strength) could thus introduce further shifts in the location of pX,. For the particular case of the proton-transfer reaction at a carboxyl group in aqueous solution, eq 1 becomes
-COO-
+ H3O+
-COOH*H20
(11)
Here pX becomes pH and the behavior of the appropriate D and G parameters is cxactly as in Figure 1. (14) M. Eigen and L. De Maeyer, in “Technique of Organic Chemistry,” Vol. VIII, 2nd ed, Wiley-Interscience, New York, N . Y., 1963, Chapter XVIII. T h e Journal of Physical Chemistry, Vol. 76, N o . 26, 1971
MATTHEW HUSSEYAKD PETERI>.EDDJIONDS
40 14
For the reaction a t an amino group equation 1 becomes
-XHg+
+ OH- --i\THz.HaO A
(12)
Here p X becomes pOH, so that on the pH scale (whcrc pH pK, - pOH and nhcrc K,v is thc ionization constant of water) the behavior of the D and G parameters is a mirror image of the curvm in Figure 1rcflcctcd through the line pH = pK (where K is the appropriate acid equilibrium constant),
Application of This Theory to Polypeptide and Protein Solutions
A polypepbide consists of a long chain of amino acids artificially constructed, the sequence of which may be known. Thus polylysine consists of a backbone from which protrude side chains all terminated by +amino groups. Polyglutamic acid has a series of y-carboxyl side-chain groups. The naturally occurring protein has a number of structural features but its primary structure is a chain of amino acids with side chains, some of which are carboxyl or amino or other ionizable types and some of which are not ionizable. The secondary, tertiary, and quaternary structures of the protcin determine the location and orientation in space of the basic aniino acid chain. To apply eq 4-12 in a predictive manner to any general case, one must have fairly good approximate values a t least for the following unknowns: k12, AC, K , and Co. I n applying them, even in a strictly linear fashion to polypeptides and proteins, as it is proposed to do here, there are many difficulties to overcome. One must know first what side-chain types are present and how many of each type. For most common proteins this information is now available from amino acid sequence studies.l6 However, since proteins are such complex three-dimensional structures, some of these side chains may be masked and so not available for proton transfer. Titration studies usually can reveal the number, n, of “available” side hai ins.^^-^^ Titrimetric data can also provide mean pK values for particular types of ionizable side ~ h a i n s . I ~ -Rate ~ ~ constant values for individual side-chain types may be estimated AU values from those of similar model compound.6~e~20 may be estimated, chiefly from dilatometric data on similar model compounds and on protein^^^-^^ or from ultrasonic data on similar model compound^.^^^^'^ Knowing these parameter values for each side-chain type, one can apply eq 4-12 to calculate the contribution of this mechanism at each side chain to the excess ultrasonic absorption. Simple summation provides a total predicted excess absorption. This calculation assumes independence of the different reactions and ignores possible variation in the environment around and parameters applying to side chains of the same type. These are major limitaThe Journal of Physical Chemistry, Vol. 76, N o . 16,1971
tions to the validity of the cnlculn tiori but it should allow a reasonably good qunlitativc picture to cmcrgc. Tablc I shons thc paramctcr valucs uscd in the sample calculation for thc aqucous solution of BSA, togcthcr with the sourccs of thcsc paramctcr vnlucs. Tables I1 and I11 similarly trcat solutions of human hcmoglobin and gelatin, respectively. Tablc IV lists thc pertinrnt parameters for polylysinc and polygl(itamic acid solutions.
Table I: Parameter Valiies IJsed to Calcidate Conliibiitions t o Excess Absorption from Proton-Transfer Reactions at Side Chains in a 0.042 g/crn3 BSA Solution a t 2 5 ” ~ 1010
X
f~12,a!a~lrlzo
Group
a-Carboxyl s-Carboxylb Imidazole a-Amino €-Amino Phenolic Guanidine a
pKiG
I
17
3.75 3.93 6.90 7.75 9.80 10.35 12.50
M-l
A~~,a,e,zl-za
n16117
sec-1
cm3 mol-1
1 99 16 1
3 3 3 6
57
5 2 5
19 22
10 10 18 15 15 17 15
Molecular weight of BSA 65,000; CO = 0.00063 M.
* s re-
fers to side chain.
Figures 2-6 show calculated excem absorption due to proton transfer at side chains and fairly directly comparable experimentally observed absorption (in excess of that in the region of p H 7 ) in the pertinent pH ranges for solutions of BSA, human hemoglobin, gelatin, polylysine, and polyglutamic acid, respectively.
Discussion Qualitatively the calculated ultrasonic absorption behavior due to proton-transfer reaction at side chains in proteins is similar to that calculated and observed for glycine:5 excess absorption occurs at high pH (>8) and at low pH (
