1054
Titration and Temperature-Dependent Properties of Homodinucleoside Monophosphates. Evaluation of Stacking Equilibrium Quotients for Neutral and Half-Ionized ApA, CpC, GpG, and UpU Naotake Ogasawara and Yasuo Inoue* Contributionf r o m the Department of Biophysics and Biochemistry, Faculty of Science, University of Tokyo, Hongo, Tokyo, Japan. Received February 4, 1976
Abstract: The temperature dependence of the ultraviolet difference spectra of ApA, CpC, and GpG at neutral pH was analyzed by two-state model. From these thermal denaturation data, apparent thermodynamic parameters, A H o , ASo, or AGO, were determined by a method of iterative least-squares treatment. The thermal denaturation data were used together with the overlapping p K values of ApA, CpC, GpG, and UpU to throw light on the degree of stacking for the half-ionized homodinucleotides. The comparison of the intramolecular stacking association quotients between XpX and the corresponding half-protonated species indicated that XpX can be classified into those with a relation of SO = s I and SO < s I at 25 OC, the former including ApA and CpC, while the latter is GpG. The microscopic stacking equilibrium quotients for the half-protonated GpG were evaluated from the data based on the determination of the microscopic ionization constants of GpG at 25 "C. The results indicated that the enhanced stacking observed for (GpG)+ as compared to GpG at neutral pH was due to the protonation on the guanine base in the 3'-linked nucleoside.
It is well known that poly(A)' and poly(C) a t neutral p H exist in a single-stranded form with stacked bases arranged in a helical f a ~ h i o n . ~Dinucleoside -~ monophosphates are the shortest chain-length oligomers having the ability of such base-base intramolecular stacking interactions. It is also well recognized that single-stranded oligo- and polynucleotides approach the unstacked conformation a t high temperatures and stacked conformation is favored a t lower temperat u r e ~ . ~ The , ~ , lack ~ - ~of~ cooperativity gives rise to a broad thermal denaturation curve, associated with the gradual diminish of stacking interactions as the temperature is raised from 0 to 85 O C . Among dinucleotides, ApA has been most extensively studied and there have been a t least eight experimental determination^^^^^-'^^^^-'^^^^^^^ of the enthalpy and entropy for the conformational equilibrium (ApA)unstacke,j + (APA)stacked. It has been suggested and confirmed by experiments that "intramolecular stacking" tends to suppress the ionization of the base residue^.^^-^^ Since there is a lack of experimental information about stacking properties of half-ionized molecular species of simple homooligonucleotides, we have undertaken the present investigation concerning the effect of ionization on the intramolecular stacking interactions of four homodinucleotides made up of the four principal nucleosides. Measurements of the stepwise ionization constants of ApA, CpC, GpG, and UpU together with thermal denaturation data have enabled an estimation to be made of the values of pKo and stacking equilibrium quotients for their half-ionized species. In a previous2* and the preceding paper,27 we have already shown that ApA, CpC, and GpG undergo stepwise protonation and determined the apparent pK values of ApA, CpC, GpG, and UpU for the gain and/or the loss of two protons. W e have also estimated, merely based on the titration data of homodinucleotides and the component monomers, the extent to which these dimers and their half-ionized form exist in the stacked conf~rmation.~~ As early as 1965, Warshaw and T i n ~ c pointed o ~ ~ out that the ORD of ApA is the same as that of an equimolar mixture of adenylic acid and adenosine at pH 1 within the experimental certainty. They have concluded from these observations that ApA is unstacked a t p H 1. Now, there is a general agreement that doubly protonated dinucleoside monophosphates are unstacked, although the origin of the low pH induced unJournal of the American Chemical Society
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98:22
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stacking is not fully u n d e r ~ t o o d . 'However, ~ little has been known concerning the effect of half-ionization of dinucleotide upon the stacking interactions. Furthermore, for this problem, the experimental results based on the p H titration26and the theoretical predicti01-1~~ are conflicting; Simpkins and Richardsz6 have concluded that the singly ionized form of ApA is unstacked, while Jordan and Sostman'O have reported the half-protonated dimers are expected to be even more stable than unprotonated ones in stacking mode. In favor of the latter view, Guschlbauer et aL3I have recently reported that G p U stacks only a t low pH. In order to settle the problem of literature discrepancies with regard to the intramolecular stacking interactions of half-protonated homodinucleotides, this and the preceding paper record the determination of the stacking equilibrium quotient for (XpX),+ + (XpX),+ by means of the two alternative methods. No information is available on the stacking equilibrium of GpG at neutral pH, and there is a range of values of AHo and ASofor ApA and CpC in the literature. Hence, we have examined the temperature dependence of the optical properties of ApA, CpC, and GpG in aqueous solution. This information, together with values of the overlapping ionization constants determined in the preceding paper,27 enables us to estimate the stacking equilibrium quotient, s 1, of the homodinucleotides named in the title and to compare the present results with those in the preceding paper. In this paper, we also present the results of the determination of microscopic stacking equilibrium quotients for GpG.
Experimental Section Materials. Materials were as given in a preceding paper." Methods. (a) Thermal Denaturation. The thermal denaturation of XpX at neutral pH was followed spectrophotometrically. Previous s t u d i e ~ ' ~ have J * shown that increasing temperature does not appreciably alter the conformational features nor the absorption spectra of nucleosides and mononucleotides, so that the spectral changes, which occurred upon thermal denaturation of XpX, were measured by the difference spectra obtained for solutions Y m M per residue in XpX vs. Y mM component monomers (RNase Tl hydrolysate of XpX), both at pH 7 (0.1 3 < Y < 0.23). The advantage of the use of difference spectrophotometry was the increased differences during the thermal denaturation by the use of a higher concentration of XpX. This was particularly so for CpC which showed only small differences in the extinction coefficient (see Figure 1). Since the analysis of the
October 27, 1976
7055 amplitude (A273 5 -.A290) gave the best fit (smallest rms), in our investigationswe have preferred to use amplitude of difference spectra. This eliminates errors due to base-line shifts between readings. The variations of the spectra with temperature were measured on a Hitachi Model 124 spectrophotometer. In order to prevent evaporation the cells used were stoppered securely and placed in the thermostatically controlled cell holder (Komatsu Electronics Inc., Tokyo). Temperature was measured to f 0 . 2 "C by a standard thermometer connected to a calibrated thermistor inserted in the sample cell. Solutions were stirred during measurements with a magnetic stirring bar to prevent bubbles (solutionswere degassed under vacuum before use for spectral measurements).At low temperatures, dry nitrogen gas was introduced through the cell compartment to prevent condensation. Absorbance readings were corrected for volume expansion. (b) Data Analysis for Determinationof AH" and AS". Data analysis was performed by the iterative least-squares method (see Appendix), and actual computations were carried out at the Computer Centre of the University on a HITAC 8800/8700 computer.
Results and Discussion Analysis of Thermal Denaturation Data for ApA, CpC, and G p G It is well known that dinucleotides have a strong tendency to form intramolecularly stacked base helices that exhibit either a hypo- or a hyperchromic effect, depending on wavelength, relative to the unstacked form.25There are several prior determinations of the values of AH" and AS" for ApA and CpC, but disagreement is seen among these values of various authors. Besides this, no experimental determinations of AHo and AS" for ( G P G ) ~s ( G P G ) ~have been made until W e have thus reexamined the temperature dependence of the ultraviolet absorption spectra of ApA and CpC together with GpG at neutral pH. Since the rate a t which (XpX), and (XpX), interconvert is slow compared to the time scale of uv spectroscopy, it is possible to observe superposition spectra for the equilibrium mixture at temperatures between 0 and 85 "C. In line with the previous suggestion by Powell et al., the stacking equilibrium can be operationally regarded as a twostate system, and thermodynamic parameters can be extracted by the iterative least-squares computer method of the thermal denaturation profiles. (a) Effect of Temperature on the Absorption Spectra of XpX. Absorption spectra a t a series of temperatures for ApA, CpC, and GpG give excellent isosbestic points. The thermal denaturation spectra of ApA, CpC, and GpG are illustrated as the difference in molar extinction coefficient per base residue for the dimer at a lower and a higher temperature as denoted over the wavelength range 220-300 nm (Figure 1). Uv difference spectra of ApA a t different temperatures displayed isosbestic points a t 217.5, 229.7, 274.3 nm. The CpC vs. component monomer thermal difference spectra taken over the range 0-85 "C showed that CpC changed progressively to the unstacked form with a clear isosbestic point a t 286.6 nm. GpG has exhibited changes in the GpG vs. component monomer difference spectrum on heating which are similar to those observed with CpC; however, the magnitude of change is somewhat less than CpC. The relatively large experimental error in this case renders a precise analysis of data difficult. There is a hyperchromic effect on band(s) above 292 nm as the temperature is raised. This gives a positive feature above 292 nm in the GpG vs. component monomer thermal difference spectrum (Figure 1). W e have found a definite isosbestic point a t 291.5 nm in a set of the GpG vs. component monomer thermal difference spectra. All of these results are again considered as a strong indication of the validity of the so-called "two-state model" for the stacking-destacking equilibrium system.18,'9,21 (b) Determination of the Thermodynamic Parameters, AH" and AS", for Stacking Equilibrium, (XpX),,s (XpXk. Consider intensive parameters tu and t, of extreme conformers (XpX), and (XpX), under mobile conformational equilibrium in aqueous solution at neutral pH. At the ith of n temperatures,
AH" ASo (4) RT; R Besides the assumption of two optical states, the following assumptions have to be made before the analysis of the thermal denaturation data by the van't Hoff procedure. (1) tu and t, are temperature independent, and (2) AH" is also temperature independent, which is equivalent to the standard heat capacity change AC," being close to zero for the reaction (XpX), s (XpX),. Then, the temperature dependence of ti simply reflects the temperature dependence off,i andf,i at constant AH" and ASo.Equation 4 can now be solved, in principle, for unknowns tu, ts, AH", and AS" by the use of the thermal denaturation data (see Appendix). Since tu and ts occur above 85 and below 0 "C, these limiting values are not determined directly. Consequently, a method of computing AH" and AS", which does not rely on the tu and ts, has to be adopted. In this work, the iterative least-squares method was applied to the analysis of the uv absorption data over the temperature range 0-85 "C. Of several factors which will determine the reliability of the thermodynamic parameters,21 the accuracy of the experimental readings is most im-
Ogasawara, Inque
/ Stacking Equilibrium Quotients of Four Homodinucleotides
0 250 300 n m Figure 1. Thermal denaturation spectra of ApA(-), CpC(- - -), and GpG(. at neutral pH: ApA, C86.6- - C 2 4 . 9 O ; CpC, E78.00 C 2 3 . 3 O ; GpG, ( 7 8 . 5 9 - ~ 2 3 . 3 (Correction ~. for a change in concentration of XpX due to the thermal expansion of water was made.)
-
-
a)
the observed parameter, tem
e;,
(XPX),
of the stacking equilibrium sys+
GPXh
(1)
reflects an average of tu and e, weighted according to the relative populations of the two conformers, f u i andfsi. The essence of the two-state model is that t; can be expressed by means of a linear combination of tu and ts with temperature coefficient. Ei = f u i € u
+fsits
= (1 -fsi)tu
+fsits
(2)
The stacking equilibrium quotient a t a given temperature, i , may be written as (3) or In(-)=-ti
- t,
+-
7056 Table I. Thermodynamic Parameters and Stacking Equilibrium Quotients (SO at 25 "C) of ApA, CpC, GpG, and UpU Dimer
AH", kcal/mol ( u ) ~AS",eu (a)" -5.5 -4.4 -4.5 -6.1
APA CPC GPG UpUb
(0.30) (1.05) (0.78) (0.7)
-18 (0.95) -16 (3.16) -17 (2.19) -22.2 (1.9)
SO
Table 11. Stacking Equilibrium Quotient
(SI = [ ( X P X ) ~ + ] / [(XpX)"+]) and Intrinsic Basic pKo for ApA, CpC, and GpG at 25 "Ca
at 25 " C (u)" 1.OO (0.04) 0.63 (0.12) 0.45 (0.10) 0.44
a The standard deviations ( u ) were computed from the root mean square (rms) values between experimental and calculated melting 4.6 X and 5.1 X for ApA, profiles [rms are 2.7 X CpC, and GpG, respectively, when differences between the values of the optical (absorbance) parameters for the fully stacked and fully unstacked species are normalized to unity (see also Appendix c)]. Taken from ref 21.
Dimer
SI
APA CPC GPG
0.93 f 0.22 0.71 f 0.17 0.95 f 0.13
D K ~ 3.611 f 0.086 4.198 f 0,066 2.076 f 0.042
" The probable errors given for S Iand PKn are based on estimated
-1 00
-50
1
0
100
50
*c
150
1
1. I
'
I
fsi
0.
I
Figure 2. Thermal denaturation of ApA (0;, 4 2 6 2 5 vs. temperature at 1.27 x 10-4 M), c p c ( 0 ;A 273 5 - A290 vs. temperature at 1.97 X M), and GpG ( 0 ;A 2 7 7 5 - A 2 9 5 vs. temperature at 2.28 X IOm4 M) at pH 7 and I = 0.1. The curves are calculated by the iterative least-squaresmethod [see text and Appendix (a)], T , ( T , is the temperatureat which 50% of the total increase in extinction due to thermal denaturation is observed) is indicated by an arrow. The spectral extremes for the fully stacked and fully unstacked states are ( A 2 6 2 & , s t a c k e d = 1.881 and ( A 2 6 2 5)stacked = 1.383 for ApA; ( A 2 7 3 5 - A29O)unstacked = 1.375 and (A273 5 - A29O)stacked = 0.817 for CpC; and ( A 2 7 7 5 - A295)unstacke,3 = 1.616 and ( A 2 7 7 5 A 2 9 5 ) s t a c k e d = 1.046 for GpG.
portant; when the change in absorbance is small over the temperature range employed, sizable errors would occur. However, in the present investigation, the error was minimized by taking spectra at more than 20 temperatures in the range 0-85 "C. In order to determine the effect of precision in the melting data on the accuracy of the thermodynamic parameters obtained, synthetic t i values in a temperature range used were processed by the same computer program [see Appendix (b)]. A summary of the average best values of AH", AS", and stacking equilibrium quotient so at 25 "C, for more than two runs, is shown in Table I. The uncertainties are the standard deviations. Our results for the apparent thermodynamic parameters for ApA seem to be in general agreement with prior determinations ranging from - 10 to -4.9 kcal/mol in AH" and from -30 to -17 eu in AS", though the IAHOI and I A S O I values are somewhat lower than the literature values clustering at I AH" I N 8.5 kcal/mol and at IAS"I N 30 eu. Temperatureabsorbance profiles of ApA, CpC, and GpG a t the analytical wavelengths are obtained from a series of the XpX vs. component monomer thermal difference spectra and shown in
Journal of the American Chemical Society
150
200
250
350
300
400
eK
4
us,)
Figure 3. The fraction of stacked bases as a function of temperature: ApA (e),CpC ( e ) ,and GpG ( 0 ) .Curves represent the best fits to the data, calculated by the iterative least-squares method.
Figure 2, together with curves representing the best fits to the data. Evaluation of Values of pK0 and SIat 25 "Cfrom the Data Based on SO and Overlapping pK Measurements. At each temperature, the fraction of molecules in the fully stacked state can be estimated using the van't Hoff equations and the AH" and values derived as described above. Figure 3 illustrates the variation of fsi of ApA, CpC, and GpG with temperature. Our values of so at 25 "C are listed in Table I, together with relevant data for UpU from the literature.2' According to Lowe and Schellman17 and Catlin and Guschlbauer,21UpU does stack at 25 "C to some extent. At 25 "C, GpG is also found to exist in a stacked conformation to an appreciable extent. The so values for ApA, CpC, GpG, and UpU are now compared with those estimated from pK measurement^.^^ The values of so listed in Table I are somewhat smaller than those obtained by titration experiments though the former has been found to agree, to within an order of magnitude, with the latter. Based on the ORD and CD studies of ApA, CpC, and GpG, the diprotonated species are considered to be unstacked, Le., s2 N 0. Thus, we have combined the overlapping basic ionization constant data and the measured values of s~ and obtained values of s 1 and basic pKo. The estimated values of s 1 and basic pKo at 2.5 "C are listed in Table 11. The basic pKo values listed in Table I1 compare favorably with the corresponding values taken for ApA, CpC, GpG as pKo =
/ 98:22 / October 27, 1976
7057 Table 111. Ionization Constants (as pK) of ApA, CpC, and GpG at 25 ‘ C n
Dimer APA
CPC GDG
PKU,
PKU,
3.91I f 0.086 4.498 f 0.066 2.37~ f 0.042
PKS,
3.31 I f 0.086 3.898 f 0.066 1.776 f 0.042
3.880 f 0.13 4.551f 0.13 2.707 f 0.12
The probable errors of pK,,, pK,,, pK,,, and pK,,, are estimated from the following relationships: ( u s , ) 2 /5.29~,*]]/2. 2.3sos1)[5.29so~s1~(u~~~)* + s ~ ~ ( u , JS ~O ~ ( U , , ) ~ ]and ~ / ~ U, ~ K , ,=~ [(a .pKo)2
+
+
PKSU,
3.342.f0.14 4.046f 0.13 1.797f 0.08 up^,, = u P ~ ,= > up,yO,up^,, =
(1/
1/4(PK3’-mononucleotide PK~’-mononuc~eotide + 2PKnuc1eoside)r 3.67, basic pK comparisons with methylated derivatives.2s The present work, together with the data reported previously for 4.21, and 2.19. We consider these agreements sufficiently good. GpG, has enabled an estimation of various microscopic conThough the slightly higher values of pKo, as compared to the stants involved in stacking and protonation equilibrium values in Table 11, cause SO and S I to be somewhat greater than schemes. those derived from thermal denaturation data, s 1 values which (a) Ionization Constants, K,,,, K,,, K,,, and KSU2.Prototropic have been estimated on the basis of titration experiments alone, and stacking equilibria of XpX can be depicted by Scheme I seem to be in reasonable agreement with those given in Table 11. It is therefore gratifying to note that conformational free and AGo(xpx),+,(xpx),+, evaluenergies, AGo(xp~),e(xpx), ated from two approaches are reasonably good. The s1 values of ApA and CpC now obtained show that the half-protonated species have stabilities comparable to their un-ionized counterparts. These findings are again in disagreement with the conclusion of S I = 0 for ApA deduced by Simpkins and Richards.26The conclusion reached for ApA and CpC, SO = S I , seems to represent a general relationship as judged from other work carried out by our own group on homodimers having methylated bases (Tazawa and Inoue, unpublished results). By contrast, the abnormality (SI> S O ) provided that doubly protonated dinucleotides are unstacked, found for GpG may be attributed to markedly different overlapping of the two bases in GpG and (GpG)+ (see ref 28 and Le., SO. S I >> s2 N 0. By definition, pK,, = pKo log 2; pK,, the next section). The relative stability order, as measured by = PKO- log 2; pK,, = pK,, - log (SO/SI);and pK,,, = pK,, stacking equilibrium quotients, is found to be (XpX) - log s 1. Since the values of so and s I are known for the three (XpX)+ compared with the order (XpX) < (XpX)+ predicted homodinucleotides, by applying the above relationships the pK by Jordan and S ~ s t m a nTheir . ~ ~ calculations are purely envalues of completely unstacked and stacked species have been ergetic so that the difference may, in some part, be attributed determined and are set out in Table 111. The larger percentage to a contribution of the enhanced solvation in (XpX)+ to the of stacked half-protonated GpG is reflected in the higher basic overall stacking interactions. Similar associations between strength (pK,,) of the stacked GpG relative to pK,,: if pK,, > nitrogenous cations and neutral hydrophobic molecules have pK,,, then S I > S O ; if pK,, < pK,,, S I < S O . been established for both m i ~ e l l a r and ~ ~ n, ~~ ~n m i c e l l a r ~ ~ ~(b) ~ Microscopic ~ Stacking Equilibrium Quotients, s13’and s15‘ systems of quarternary ammonium ions such as cetyltriof GpG. The empirical stacking equilibrium quotient (s I ) of methylammonium ion and aromatic compounds and are XpX is composite, and it is related to the true constants of the probably very common. XpX+ (s 13’) and the +XpX tautomer (s I s f ) by the expressions, Using the known values of so and the overlapping acidic pK, s13’ = [(XpX+),]/[(XpX+),] = (1 - Kt 2sl)/(l K,) and we have computed s-1 = 0.20 f 0.09 and acidic pKo = 9.37 s15’=[ ( + X P X ) ~ I / [ ( + X P X ) ~=I (2s1Ki+ KI - 1)/(1 + Kt). f 0.05 1 for GpG. W e have used the literature value of so for From the Kt value and S I value of GpG, the microscopic UpU, and values of s-1 and acidic pKo have been estimated stacking equilibrium quotients can be calculated to be s 13’ = as s-1 = 0.20 f 0.07 and pKo = 9.23 f 0.035 for UpU. 0.44 f 0.15 and sI5’= 1.45 f 0.21, the latter being more than Compared with these pKo values we obtained in the preceding three times greater than s13’,which is almost identical with the paper27 by the empirical choice of pKo = 1/4(PK3~-nuc~eotide + S O value at 25 O C . This fact favors the hypothesis of anti syn conversion of the guanosine residue on protonation.2s PK~’-nucleotide-I-2PKnucleoside) values of 9.22 for GpG and 9.19 for UpU. (c) Microscopic Ionization Constants and Tautomeric The principal conclusion from the results is that half-proQuotients of GpG. The pK,, and pK,,, are now seen to be tonation to a homodinucleotide does not reduce the extent of composite constants, namely, K,, = aH [(XpX),] / [ [(+XpX),] intramolecular stacking i n t e r a ~ t i o n although ,~~ a t present it + [(xPx+)sll and Ksu, = aH[[(+XpX)s] + [(xpx+)s]1/ is not certain what determines the equilibrium conformation [(+XpX+),], determined under experimental conditions where of half-protonated stacked dinucleotides. It is hoped to extract it is likely that (+XpX), and (XpX+), are in equilibrium. The thermodynamic parameters for the (XpX),+ @ (XpX),+ true pK of species (+XpX), and (XpX+), can be obtained from process by studying the temperature dependence of ionization P K , , ~ ’= pKo - log (so/sI”); pKSl5’= pKo - log (so/sl5’); of homodinucleotides. 3 9 pK,,:’ = pKo - log s 15’; and pK,,25’ = pKo - log s 3’. Since Determination of Microscopic Stacking Equilibrium Quowe have stipulated that the two sites are equivalent and nontients and Intrinsic Tautomeric Quotients. Site of protonation interacting in (XpX),, it follows that KUl3’= K U I s = K u 2 3‘ = in XpX is of course expected to influence the stacking of KU25‘= KO.W e have determined the microscopic ionization half-protonated dimers. The degree to which half-protonated constants for the stacked conformational isomers of GpG and XpX would exist in +XpX or XpX+ is, a t present, unknown the results are included in Table IV. These constants involve except for (GpG)+. For the specific case of GpG, we have reonly one species and its conjugate acid and can therefore be ported28 the tautomeric quotient K t (= [+GpG]/[GpG+]) to used for purpose of molecular interpretation. In the stacked be approximately 1.7 a t 25 O C . This was obtained by C D and GpG conformation the difference betw’een pK,, 5’ and pKSl3’
+
+
+
-
Ogasawara, Inoue
/ Stacking Equilibrium Quotients of Four Homodinucleotides
7058 Table IV. Summary of Ionization Constants and Stacking and Tautomeric Equilibrium Quotients of GpG at 25 " C Overall constants
+
Microscopic constants
+
+
Ionization constants (composite: s u) pKI3' = 2.07; pKI5' = 2.30; = 1.68; P K ~=~ 1.91 ' Ionization constants ' 2.58 f 0.12; P K ~ " , = ~ ' 1.91 f 0.13; P K , , ~ '= 2.06 f 0.18; P K , , ~= PK,,~' = 2.43 f 0.19; P K , , ~ ' = pKUl5'= PK,,,~' = pK,," = pKo = 2.076 f 0.042 Stacking equilibrium quotients ~ o = 0 . 4 5 f 0 . 1 0 ; ~ 1 3 ' = 0 . 4 4 f 0 . 1 5 ; ~1 .~455 'f=0 . 2 1 Tautomeric equilibrium quotients Kt, = 3.3 f 1.8; Ktu = 1.0
Ionization constants (composite: s u; 3' 5') pK1 = 2.50; pK2 = 1.48 Ionization constants (composite: 3' 5') pKsl = 2.702 f 0.12; pKsU2 = 1.797 f 0.08; PKu, = 2.376 f 0.042; pKu2 = 1.776 f 0.042
+
+
Stacking equilibrium quotient (composite: 3' 5') S I = 0.95 f 0.13 Tautomeric equilibrium quotient (composite: 3' 5') Kt = 1.7 f 0.3
+
is 0.52 pK units, considered to be due solely to the conformational stabilization of the stacked +GpG species relative to the stacked GpG+. This could be rationalized on the basis of a preferred geometry of right-handed (syn)+Gp(anti)G which would be especially suited for intramolecular overlapping of the two bases. Microscopic stacking equilibrium constants also enabled the microscopic tautomeric quotients, K,, and Kts, of completely stacked and unstacked species of half-protonated GpG to be obtained from the following relations: Ktu =, [(+XpX),]/[(XpX+),] = (sI3'- S I ) / ( S I - s i 5 ' ) = K 3'/K and K t , = [(+XpX),]/[(XpX+),] = (sI3' - sl)s19/(sluL s15')s13' = K,,3'/K,,5'. W e have arrived at the conclusion that GpG undergoes preferred 3'-linked nucleoside base protonation. This conclusion was apparently backed by both CD and pK evidence,** but exact analysis of the geometry of halfprotonated GpG is beyond the present scope of conformational analysis.
Acknowledgments. W e thank our collaborators I. Tazawa and M. Sakurai for their help with the statistical calculations. We are particularly grateful to a referee for his suggestions which are embodied in our estimation of the possible errors in the derived equilibrium quotients. Appendix (a) Extraction of Thermodynamic Parameters from the Thermal Denaturation Data.18,40-43 As a result of the assumptions made in text, the conformational equilibrium quotient SO;, which is given by (tu - €,)/(e; - ts), is related to AHo and A S o by the familiar expression (eq 4). In solving eq 4 we consider solution values of the unknowns to be those which make the sum of the squared residues (Eres)of the molar extinctions a minimum res
=
5
- ticalcd )2
i= I
(5)
where ticalcd may be written as ficalcd
=
+
tu
-
ES
+ e, + (ASo/R)]
(6)
1 exp[(-AHo/RT;) W e adopted the well-known Newton-Raphson iterative least-squares approach43to determine values of the unknowns that result in a minimization of E,,, and, hence, in the solution of eq 4. If the difference between the observed and calculated extinctions for the ith one ( t i - cicaicd) is approximately linearly related to corrections to the four unknowns, AU, (s = 1, 2 , 3,4), according to (7)
for i = 1,2, . . . , n, AU, may be obtained by a method of least squares, which, after application to the initial values of the unknowns, OU,, provides new values for unknowns, ' U s . and Journal of the American Chemical Society
for ticalcd that will reduce Eres.These new values of are returned to eq 7 and the process is repeated and continued until all values of AU, are less than 0.001 units. Thus, a practical minimization of Z, is realized. If the calculated experimental parameters, t;calcd,are assumed to be linear functions of the unknown parameters, Us, small changes in the unknown parameters, €Us,result in linear changes in the calculated parameters, Aqcalcd. Thus, one may write
where Us refers to any particular unknown parameter. When Aticaicd = ci - ticalcd, AU, becomes the correction to the unknown, Us. In addition AU,, = IUS - 'Us
(9)
where OUSis the value of the sth unknown in a given iteration and IUS is the corresponding value in the next iteration. One would like to find changes in all the unknown parameters such that tiCaiCdare brought into coincidence with t i , i.e. i= I
AU, = AtiCalCd (i = 1, 2 , . . . , n)
(10)
where the number of unknowns is 4 (tu, tS, AHo, and &.To) and n is the total number of observed parameters, e;. The above equations of condition can be written in matrix notation
DU=t (11) where D is the n X 4 matrix of partial differentials with elements tis = aticaicd/dUsobtained by differentiating eq 6 at;calcd 1 -1 exp[(-AH"/RTi) (AS"/R)] atu at ;calcd 1 -- - 1 1 exp[(-AH"/RT;) (AS"/R)] ats exp[(-AH"/RT;) + (Aso/R)l - t u - e,. aAHo RT; [ l exp[(-AH"/RTi) (ASo/R)]]2 aticalcd exp[(-AH"/RTi) (AS"/R)] tu - e, a w R [ l exp[(-AH"/RT;) (ASo/R)]I2 U is the four-dimensional vector of corrections to the unknowns and E is the n-dimensional vector of the residuals in the molar extinction coefficients. Standard least-squares procedure for minimizing E,,, is to form the normal equations
+
+
---.
+
+ +
+
+ + + +
DTDU = DTe (12) which are solved to yield the corrections to the unknown parameters, U.
/ 98:22 / October 27, 1976
U = (DTD)-'DTt
(13)
7059 0
0.5
0.4 fUi
0.1
0.5
4
1.2.
*
"
2
1.0-
G
2 b
D
150
200
250
350
300 Temperature
400
0.8-
0
1.0 450
-u .
6
1
( O K 1
Figure 4. Normalized melting profiles for various values of A H o with a fixed value of ASo (-20 eu): a, A H o = -3.0 kcal/mol; b, AHo = -4.0 kcal/mol; c, A H o = -4.5 kcal/mol; d, AH" = -5.0 kcal/mol; e, AHo = -5.5 kcal/mol;*f, A H o = -6.0 kcal/mol; g, A H o = -6.5 kcal/mol; h, A H o = -7.0 kcal/mol; i, AHo = -7.5 kcal/mol; j , AHo = -8.0 kcal/mol; k, AHo = -8.5 kcal/mol; 1, A H o = -9.0 kcal/mol; m, A H o = -10.0 kcal/mol. The dashed lines are upper and lower temperature limits between which e , may be determined i n the present experiments. 8-02.0
-5
AHcorrect (Kcal/molei
The corrections are applied to the unknowns and the iterative loop is repeated until the rms (root mean square) changed by less than lo-' in the last cycle. (b) Reliability of Results. To test the reliability of the results obtained by the present approach, computation of the errors associated with the parameters was made by introducing synthetic data containing Gaussian distributed random errors for a series of assumed values of the parameters into the present computer program. W e have applied standard deviations ranging from 2 X to 5 X to construct synthetic data in solving eq 4 with assumed values of the parameters, because the smallest root mean square (rms) values in fitting of experimental and calculated melting profiles have been found to be in this range after several (less than 10) iterative cycles (see footnote a in Table I). In Figure 4 typical normalized melting curves are shown for different values of AHowith ASo = 20 eu. Because the temperature range over which t i is determined is limited (0-80 "C), in intervals of about 3 OC, synthetic Q values obtained for various values of AHoand ASo in this temperature range were processed by our program. I n order to demonstrate the effect of precision in t i on the accuracy of the solutions of eq 4, the probable errors in the calculated AH', ASo,SO a t 25 OC, and T , are plotted against the correct AHo for systems in which a fixed value of ASo (-20 eu) has been assigned (Figure 5). It was again found that for the synthetic values of ti having 2 X
