11. LOW MOLECULAR WEIGHT PROTEINS THERMODYNAMICS OF THE ASSOCIATION OF INSULIN MOLECULES BY PAUL DOTY AND GEORGE E. MYERS* Gi bbs Laboratory, Harvard University, Cambridge, Mass. Received 19th May, 1952 Light scattering studies of insulin solutions at pH 1.9 and 2.6 in 0.1 M NaCl and NaHzP04 permit the evaluation of equilibrium constants governing the equilibria among monomers, dimers, and trimers. The variation of the constants appears to be explicable in terms of changes in the electrostatic repulsion. Measurements in the range of 20" to 40" C lead to the determination of the corresponding heats and entropies. These are found to be surprisingly low and suggest that increased solvation of the monomer with respect to the dimer is taking place. In a previous publication 1 it has been shown that the variable particle weight exhibited by insulin in acidic solutions is due to the existence of simultaneous equilibria involving monomer, dimer, trimer, tetramer and possibly higher polymers.The monomer is taken as the 12,000 molecular weight unit. Below a pH of 2.2 it was found that dissociation of the polymers was virtually complete except for dimer. Thus in this region the monomer-dimer equilibrium was isolated and its behaviour could be characterized by a single equilibrium constant. In this paper we report more precise determinations of this equilibrium constant at an ionic strength of 0.1 in NaCl and NaH2P04 and at pH values of 1.9 and 2-6 in order to define quantitatively the influence of anion type and pH on the equilibrium. Under these same conditions we have also determined the temper- ature dependence of the equilibrium constant providing values of the heat and entropy of dissociation of the dimer.At the higher pH in NaCl solution, some trimer appeared and consequently information on the equilibrium involving the trimer was obtained. This investigation, like the one just summarized, has been carried out using the measurement of the reduced intensity of scattered light R90 for the deter- mination of the weight average molecular weight. The relation between these two quantities is given by the basic light scattering equation : Kc/R90 = 1/M + ~ B c , (1) where K = 2712n02 (dn/dc)2/Nh4, no being the refractive index of the solvent, dn/dc the refractive index increment of the solution, N Avogardo's number, and h the wave length of light.The concentration in g/ml is given by c ; B represents the second virial coefficient which characterizes the extent of deviation from ideal solution behaviour. Although reference must be made to reviews 2 for detailed discussion of this method two points deserve emphasis here. First is recognition that the insulin particles met with in this investigation are sufficiently small to scatter symmetrically about 90" thereby requiring only measurements at this one angle to characterize the scattering. The second point is that with the average particle weight being a function of concentration it can only be determined from *Public Health Service Research Fellow of the National Tnstitutes of Health, 195 1-2. 5152 ASSOCIATION OF INSULIN eqn (1) within the limits within which B is known to approach zero.If there were no attractive forces operating among insulin molecules in acid solution, the upper limit of B could be estimated from the well-known Donnan term. How- ever, the limited solubility of insulin as well as the observation that the mono- meric units attract sufficiently to form aggregates indicates that B must be quite small and perhaps slightly negative. If this assumption is not correct, it will show up by the slope at high concentrations taking on a value that is incom- patible with the equilibrium constants deduced from the data at lower con- centrations. EXPERIMENTAL INSuLIN.-The insulin used in these experiments was pancreatic beef insulin 5 times recrystallized (Lot T-2344) obtained from the Eli Lilly Co., Indianapolis.This is the same material as that reported on previously.1 Its zinc content, as reported by the Lilly Co., is 0.59 % ; it was not removed. EXTINCTION COEFFICIENT.-The value of the extinction coefficient determined on the basis of dry weight was found to be 10.9 at the maximum of the 280 mp absorption band. Our concentration measurements are based on this value rather than the previous value 1 of 11.3 since the latter was measured on a different insulin sample. REFRACTIVE INDEX 1NCREMENT.The refractive index increment at A = 436 m p was determined with a Phoenix differential refractometer. The concentration is based on dry weight determinations at 80" C in vacuo. The values reported here are for an approxim- ately 0.8 % insulin solution which was dialyzed against NaCl, ionic strength 0.1, pH 2.6 to remove zinc and possible impurities.Such a procedure necessitated a correction for the refraction of the chloride ions, which are present in excess of those in the salt solution in order to compensate for the positive charge of the protein. Using a value of 9-04 for the molar refraction of chloride ion, the correction applied to the measured refractive index increment is about 4 %. The refractive index increment of undialzyed insulin in NaCl and in NaHzP04 at the same pH and ionic strength were found to be the same. The evaluation of the constant K in eqn. (1) at different temperatures requires know- ledge of the values of the refractive index increment at those temperatures.Consequently, the temperature dependence of the refractive index increment was measured for the dialyzed solutions in sodium chloride and the results are given in table 1. The tem- perature variation is only slightly greater than the probable experimental error and is about the same as has been found for other proteins.3 The values of dn/dc for tem- peratures other than 25" C were taken from the best straight line through the points in table 1 . The value of 0.188 at 25" differs considerably from 0.202 used previously,l but the present value is believed more reliable from the point of view of experimental technique and instrumental calibration and from the fact that the solutions were not dialyzed for the earlier determination. TABLE 1 temperature ("C) 16-0 & 0.3 25.0 0.1 30.0 & 0-3 dn/dc (A = 436 mp) 0-1891 0.1879 0.1874 PREPARATION OF soLuTIoN.-Salt solutions were prepared by dissolving the appro- priate quantity of recrystallized salt in doubly glass-distilled water and adjusting the pH to the desired value.To avoid contact of the protein with concentrated acid or base, sufficient additional acid was added to part of the solvent to compensate for the acid bind- ing of the protein and any slight pH adjustments were then made with dilute acid or base. Both solution and solvent were then centrifuged at 80,000 g for 1 h. LIGHT SCATTERING PHo-roMErER.-Scattering measurements were made on a slightly modified Brice-Speiser light scattering photometer 4 using the blue mercury line ( A = 436 mp). The modi- fications were those involved in thermostating the instrument and in the use of a narrow slit system for the special cells employed. Centrifuged solvent was added to the optically clean photometer cell and the reduced intensity at 90°, R90, was determined.For the dilute (0-0.3 %) range, the centrifuged solution of about 1 % concentration was added stepwise to the solvent, the concentration being determined The calibration of the instrument has been reported previ0usly.4~ 5 Measurements were made in the following manner.PAUL DOTY A N D GEORGE E . MYEKS 53 gravimetrically, the concentration of the centrifuged solution having been determined by extinction measurements. In the more concentrated range the reverse procedure was followed, the solvent being added stepwise to the solution.In every case, stirring was accomplished magnetically using a small glass-encased piece of iron within the cell and the solution in the cell was examined critically at low angles to a strong light beam and rejected if significant optical impurities were present. DESIGN AND OPERATION OF CELLS.-TWO light scattering cells were employed. The first was an optically polished square cell of height 10 cm and internal width 13 mm, obtained from Fisher and Porter Co., Hatboro, Pa. This was modified by sealing an inner ground glass joint to the top, the outer joint being then used as a cap to prevent evaporation and the entrance of dust. The cell was cemented to a square metal base which fitted tightly into the central table of the light scattering photometer.The centring of the cell was checked by measuring the 90" light intensity of a fluorescein solution from all four 90" orientations of the cell. This cell was used for measurements at a fixed temperature ; in this case the photometer and the room were thermostated at 25.0f0-3" C . The second cell permitted relatively rapid measurements at several temperatures. It consisted basically of two cells, one within the other. The inner cell was one of the 13 mm cells described above without the metal base. The outer cell was one of the standard square cells for the Brice-Speiser photometers, 70 mm in height and 30 mm internal width. The smaller cell was placed in the centre of the larger, with the walls of the two parallel and held firmly in position by a Lucite plate at the base and by a Lucite top which fitted tightly into the top of the large cell and around the small cell.All plastic to glass surfaces were cemented by plasticizing the Lucite with a little organic solvent. Two 3 mm copper tubes were screwed through the back two corners of the Lucite top, allowing for the circulation of thermostated water through the space between the inner and outer cells. With the optical slit system employed it was found that the 90" light intensity was inde- pendent of the rate of flow of thermostat water through the cell and of reasonable amounts of dust in that water ; excessive dust and bubbles were removed by passage through a sintered glass filter. Scattering measurements were made on insulin solutions with this cell between 20" and 40" C and they appeared to be completely reversible and reproducible. Prolonged exposure to high temperatures was not necessary since the equilibrium shifts almost instantaneously.Employment of these small cells necessitated the use of a 3 mm slit system rather than the 12 mm system standard with this photometer. Consequently, in order to deter- mine reduced intensities on an absolute scale, it was necessary to calibrate the cells against the standard 30 mm square cells and 12 mm slit system. This is accomplished simply by comparing the 90" scattering of the same solution in both the small and the standard cell ; the ratio of the excess turbidities gives the cell calibration constant. RESULTS EQUILIBRIUM coNsTANTs.-Proceeding in the manner just outlined, we have determined the value of Kc/R90 as a function of concentration at 25" C under the four sets of con- ditions listed in table 2.The data for pH 1.9 and 2.6 in NaH2P04 solution are shown in fig. 1. The data for pH 2.6 in NaCl are shown as the lower points in fig. 2. The lengths of the vertical lines through these points are a measure of the probably experimental error. Although some allowance has been made for the possible contribution of dust and other randomly occurring impurities, in assigning the probable error we must essenti- ally rely upon the reproducibility of our data and the visual inspection of the solutions at very low angles to the incident beam to insure that dust is not contributing significantly. The data can be fitted fairly well by trial arid error assignment of the equilibrium constant as was done previously.However, a more objective treatment is possible.6 If we let x equal the mole fraction of monomer existing as monomer and M be the molecular weight of the monomer (12,000) one can show that the weight average particle weight Mw is related to the concentration by the relation M d In x ---=1+- MW d In c' Consequent I y , In x = Ji (g - 1) d In c. (3)54 ASSUCIATiON OF INSULIN Thus, by nieans of graphical integration x may be determined from the experimental data as a function of c. With the data in this form, it may be used to implement the irdsuLiN T - 2344 SODIUM PHOSPHPiTE +=OI TEMP = 25'C - PH = 1.9 ; K~~ = 7 2 xto-4 MOLESIL 8 0 t - PH = 2.6; K Z I =40 XtO-4 MOLES/L 1 0 5 ~ g ~ - 90 70 6.0 50 0 2 4 10 12 FIG.1. following relation in which K21 is the dissociation constant of the dimer and K31 is the dissociation constant of the trimer into the monomer. 105 x 1 INSULIN T-2344 SODIUM CHLORIDE p = O . l PI{ =2.6 1 - T = 25°C 0 - T = 33°C : _ I Y \ \ 0 - T = 4 0 ° C I I I I I I I I I I - 2 4 1 0 ~ c i 9 / r n l ? 3.5 1 10 FIG. 2. Thus plotting the quotient on the left against x(c/M) should prcduce a line whose inter- cept is 4/&1 and whose initial slope is 9/K31. The appearance of curvature would indicate the existence of polymers higher than the trimer and, providing the precision of the dataP A U L DOTY A N D GEORGE E . MYERS 55 warrant, the equilibrium constants for higher species could be deduced by evaluating the coefficients of higher terms in the series shown in eqn.(4). Treating the data in this manner we find that for the first two cases listed in table 2 a horizontal straight line is obtained confirming our view that only monomer and dimer are present and are in dynamic equilibrium. The values of K21 so obtained are listed. TABLE 2.THE EQUILIBRIUM CONSTANTS FOR THE DISSOCIATION OF INSULIN DIMERS AND TRIMERS AT 25" C (Ic = 0.1) moles/l. cal/mole K321 G l 1-90 NaH2P04 7.2 x 10-4 4320 i 100 - - 2.60 NaH2P04 4.0 x 10-4 4690 & 100 - - 2.00 NaCl 2.5 x 10-4 4930 f 100 10.7 X 10-4 4050 f 100 2.60 NaCl 1.7 X 10-4 5190 k 100 3.4 X 10-4 4770 f 100 moles/l. cal/mole salt KZl Wl PH In the last two cases pH 2.0 and 2.6 in NaCl, the plot of eqn. (4) yields a straight line with a pronounced positive slope.The value of K31 is determined therefrom but in table 2 we list the value of K321 = for use in the discussion. The values for NaCl, pH 2.0, supersede those previously reported,l the difference being due to more stringent pre- cautions in removing dust. The full lines in fig. 1 and the lower curve in fig. 2 are drawn using the values of K21 and K321 from table 2. THE HEATS AND ENTROPIES OF DISSOCIATION.-BY means of the thermostated cell de- scribed above, we have at several temperatures measured R90 at different concentrations for each of the four conditions listed in table 2. When the same solution is measured at various temperatures each term in the quantity Kc/&, varies. Consequently, we have listed these terms separately in table 3 which summarizes the data in NaH2P04 solution.Since it was known from the treatment of the 25" C data in the previous section that only the monomer-dimer equilibrium data were involved here the equilibrium constant for each temperature was obtained in the more direct fashion of calculating a value of the constant for each concentration and averaging the values so obtained at each temperature to provide a mean equilibrium constant for that temperature. This is the way in which the constants in the last column of table 3 have been obtained. A plot of the logarithm TABLE 3.-DATA ON THE TEMPERATURE DEPENDENCE OF THE INSULIN DISSOCIATION pH 1.90 ; NaHzP04 ; p = 0.1 T O C c (moles/l.) x 103 20" 2-07 (K= 5-80 x 10-7) 3.62 6.50 10.09 30" 2-07 ( K = 5.73 x 10-7) 3.60 6.48 10.07 40" 2-07 6-45 10.01 ( K = 5.67 x 10-7) 3.59 Rgo X 105 1-947 3-40 6.57 10.72 1.82 3-40 6.28 10-51 1.731 3.23 5.98 9-86 20" 25" 30" 40" pH 2-60 ; NaH2P04 ; p = 0.1 1-41 1-310 2.70 2.74 1-41 1 -276 2-70 2.63 1-41 1.230 2.70 2.60 1 -40 1-180 2-69 2-50 Kc/Rgo x 105 K21 x 104 6-16 - 5.84 1 5.74 - 5-40 5.82 6.5 1 - 6-05 - 5.90 - 5-50 7.54 6-75 - 6.30 - 6.12 - 5.76 10.6 4-25 - 5-92 3.01 6-40 - 5-92 3-82 6.56 - 5.95 4.28 6.74 - 6-10 5.4356 ASSOCIATlON OF INSULIN of these constants against reciprocal absolute temperature leads to the values of the standard heats of dissociation listed in table 4.TABLE 4.-HEATS AND ENTROPIES OF DISSOCIATION AS,.,, 1.90 NaH2P04 5200 400 3.0 3.0 - - 2.60 NaH2P04 4900 k 500 0.7 + 2.0 - - cal/mole deg. AHL cal/mole salt -4% AS,O, pH (p =0.1) cal/mole cal/mole deg.2.00 NaCl 8100 & 600 12.1 1 2 . 4 8700 & 1000 15.4 + 5 2.60 NaCl 7700 _+- 1000 9.0 i 5.0 8100 -I 1100 19.2 f 14 The corresponding data for NaCl solutions at pH 2.6 are shown directly in fig. 2. Due to the presence of trimer the analytical procedure of the previous section was required. The values of K21 and K321 are plotted as logarithms in fig. 3 and the values of these constants have been used to draw the lines through the constant temperature points in fig. 2. The points in fig. 3 scatter somewhat more than other corresponding plots, partly because a smaller temperature range was employed and because two constants rather than one had to be determined. FIG. 3. The values of the standard heats of dissociation obtained from plots such as that shown in fig.3 and the corresponding values of the entropies obtained from the heats and the free energies listed in table 2 are listed in table 4 together with the precision measures we have estimated for each value. DISCUSSION If we consider first the values of the equilibrium constants listed in table 2, we note that in the presence of both salts the constant is decreased by half when the pH is increased to 2.6 and that at the same pH the constant is twice as great in NaH2P04 as in NaCl solution. These two shifts in the equilibria seem lo be most simply explained in terms of the alteration of the electrostatic repulsion between the like-charged monomers, a point first suggested by Oncley and Ellen- bogen.7 Thus, if we employ the Debye-Hiickel expression for the electricalPAUL DOTY AND GEORGE E.MYERS 57 free energy or the nearly equivalent tabulation provided by Verwey and Overbeek,* we find that for two spheres 20 A in radius carrying 12 charges each (the maximum charge for a monomer unit) the contribution to AGO is about - 4800 cal. Reducing the charge to 11 changes this contribution to - 4100 cal. Thus we estimate that AGO would be increased by about 700 cal if one charge is removed from each monomer unit. The titration curve of insulin at this ionic strength 7 shows that there is a difference of about 0.8 charges between pH values of 1.9 and 2.6. Thus one predicts an increase of 560 cal between the first and second and the third and fourth AGO values in table 2. The agree- ment between these two figures is as good as can be expected taking into account the probable experimental error and the errors inherent in the calculation.The errors in the calculation arise from the failure of the Debye-Huckel limiting law at this ionic strength, the assumption that the charge distribution is spherical and the use of the dielectric constant of water. The differences in dissociation in the two salt solutions at the same pH values can be tentatively assigned to greater chloride ion binding9 In view of the bind- ing of chloride ion observed with other proteins, it would not be unexpected lo find that at this ionic strength about one chloride ion per monomer unit is bound, there by explaining the observed difference. In view of the electrostatic repulsion the trimer would be expected to be less stable relative to the dimer and monomer than the dimer with respect to the monomer.This is borne out by finding that the former dissociation constant is about twice as great as the latter. Although the various geometrical forms which the trimer could take would give rise to different electrostatic contributions, the uncertainties mentioned above in calculating such contributions are so great that an attempt to discriminate among various possible trimer structures does not appear to be justified. Turning next to the heats and entropies listed in table 4, we note at once that the heat of dissociation is remarkably low for an association which persists to such low concentrations. However, further consideration shows that it is the values of the entropies of dissociation that are unexpected.As a basis of com- parison we can calculate the entropy of dissociation of a structureless dimer based upon the gain of translational and rotational freedom. Using the Sackur-Tetrode equation the translational contribution to AS' is found to be 77 cal/mole deg. and taking as a model for the monomer a cylinder 30A in height and 30A in diameter, a contribution of 47 is calculated for the rotational contribution. Upon comparing the sum of these, 122, with the values shown in table 4, it is at once apparent that the dissociation of the insulin dimer is very much different than the simple breaking apart of two monomer units. Although the explanation of thermodynamic data in terms of molecular models is usually hazardous this unusually large discrepancy may possibly justify the speculation that such a large loss of entropy upon dissociation could only come from the simultaneous immobilization of water molecules.The entropy loss due to water being bound can be estimated either directly from the entropy change in the freezing of water or indirectly from the entropy changes found upon charge neutralization by ion binding which is considered to set free bound water.10 By either route one finds that the entropy loss per mole of water bound is about 4.8 cal/mole deg. Thus the observed entropy change could be accounted for by assuming the dissociation reaction to be 12 -1 24 H20 = 2 I x 12 H20. In other words we suggest that upon dissoci- ation the areas of the insulin monomer which had previously been in contact become solvated with about 12 water molecules each. Recognizing that a mono- valent anion can be solvated by about four water molecules,ll about three anionic groups would be required for each monomer. Since 12 such charges exist on the insulin monomer in acidic solution, this requirement could be easily met. The observed difference is about 300.58 ‘TRY PSIN AND RELATED PROTEINS 1 Doty, Gellert and Rabinovitch, J. Amer. Cherrr. SOC., 1952, 74, 2065. 2 Doty and Edsall, Adv. in Protein Chem., 1951, 6, 35. 3 Perlmann and Longsworth, J. Amer. Chem. Soc., 1948, 70, 2719. 4 Brice, Halwer and Speiser, J. Opt. SOC., 1950, 40, 768. 5 Doty and Steiner, J. Chem. Physics, 1950, 18, 121 1. 6 Steiner, private communication. 7 Oncley and Ellenbogen, J, Physic. Chem., 1952, 56, 85. 8 Verwey and Overbeek, Theory of the Stability of Lyophobic Colloids (Amsterdam, 9 Fredericq and Neurath, J. Amer. Chem. SOC., 1950, 72,2684. 10 Klotz, Cold Spring Harbor Symp. on Quantitative Biology, 1949, 14, 97. 11 Bockris, Quart. Rev., 1949, 3, 173. Elsevier, 1948).