iM are report,ed and discussed. The electrost.atic free energy change in the helix-coil transition, AF,, is related to the potential, $, which represents the ...
VOL. 3, PI’. 195-208 (1965)
BIOPOLY MERS
Dependence of the Melting Temperature of DNA on Salt Concentration CARL SCHILDKRAUT, Department of Cell Biology and the Unit for Research in Aging, Albert Einstein College of dfedicine, N e w Yo&, N e w York, and SHNEIOR LIFSON, W e i z m a n n Institute of Science, Rehovot, Israel
Synopsis Data on the decrease of the DNA melting temperature T , with the salt concentration
iM are report,ed and discussed. The electrost.atic free energy change in the helix-coil ~~
transition, AF,, is related to the potential, $, which represents the electrostatic repulsion between the phosplmte charges; $ is calculated as a function of M and of the dist,ances bet,ween the charges of the two strands. The Debye-Huckel approximation is shown to overest,imate $. I t is suggested that the high local concentration of the counterione in the immediate vicinity of the fixed charges screen these charges from interacting with other fixed charges, to the extent, that the system behaves as if the fixed ions carry :L redriced charge. The notioii of a redriced charge represents in a single parameter the deviation-of the 1)ehye-Hiickel approximation from the true potential. A plot, of gives a straight line as predict,ed. A H u is Calculated from the slope and T , versris erit with experimentally determined vahies. Our calculations support t,he hypothesis that the change of T , with salt concentration is due to changes in the screened iriteractioris between the fixed phosphate charges. In analyzing the results of these caldations, we are able on the one hand t,o indicate some of the limitations of the theoretical model and, on the other hand, draw some conchisions about the order of magnitude of the nonelectrostat,ic interaction energy of format,ion of the double helix. ~~
INTRODUCTION The absorbancy of DXA at 260 inp is about 60Yc of the sun1 of the absorbancies of the constituent iiucleotides. When DNA undergoes mild treatiiieiits (decrease in Sa+ i o i ~concentration, application of heat, increase or decrease in pH) the absorbancy increases about 40%,.’ The midpoint of the absorbancy rise with increasing temperature has been called the melting teinperature T,,L. This hypochromicity of DNA has been e x p l a i i ~ e d in ~ , ~terins of interactions of the bases when they are stacked in the double helical array as delineated by the Watson-Crick model. Denaturing agents disrupt the structure, unstack the bases, and thus produce the increase in absorbancy. If denaturation is carried on to coinpletion the base pairing is sufficiently disrupted to cause strand separat ion.5 The T , decreases with decreasing salt c ~ ~ n c e n t r a t i o n .Similarly, ~.~~~ l o ~ v e r ~of i i ~salt conrcntmtion at constant tcinperaturc below a critical 195
196
C. SCIIILDKRAUT AND S. LIFSON
point causes strand separation.* Part of the energy necessary to break the hydrogen bonds and overcoiiie any other interaction forces among the bases may therefore be provided by mutual repulsion of the negatively charged phosphate groups which are regularly placed along the DNA chains. Since the phosphate charges are less shielded by counterions a t lower salt concentration, there is more repulsive potential energy. Therefore less thermal energy is necessary to separate the strands. I n this paper we present a semiquantitative discussion of the relation between T , and the free energy of electrostatic interactions. We first show that T , varies linearly with where is the change of electrostatic free energy per mole of nucleotide pair in the transition. We then proceed to obtain AT, froiii a simple iiiodel of electrostatic interactions. Our calculations support thc hypothesis that the change of ?,' with salt concentration is due to changes in the screened interactions between the fixed phosphate charges. I n analyzing the results of these calculations we are able on the one hand to indicate some of tlhe limitations of the theoretical niodel and, on the other hand, draw some conclusions about the order of magnitude of the nonelectrostatic interaction energy of formation of the double helix. The experimental test of the theoretical considerations is based niainly on studies performed (in collaboration with Drs. J. Marmur and P. Doty) on the salt concentration dependence of the T , of samples of DNA having different guanine plus cytosine (G-C) contents. Some of these results have been published elsewhere.6
me,
me
MATERIALS AND METHODS The three bacterial DNA samples of different G-C content used in these studies were isolated by the method of ?tlarmurgfrom Diplococcus pneumoniae, Escherichia coli, and Pseudoinonas aeruginosa. I n the case of E. coli DNA, the product of Marniur's final isopropanol precipitation step was dissolved in 0.015M NaC1 plus O.OOl5~W sodium citrate to make a stork solution of 2 nig. of DNA/ml. The other samples were similarly dissolved in this same solvent. A very small volume of the concentrated DNA stock solution coiitainirig about 54 pg. of DNA was placed in a cuvet and then diluted with the appropriate solvent (KCl, sodium citrate, or sodiuni phosphate) to give a filial concentration of 18 pg./nil. Dialysis was avoided by this dilution procedure. Thus, while some chelating agent (citrate) was always present , it made a negligible contribution to the total ionic strength. By use of 3-inl. glass-stoppered quartz absorption cells (Pyrocell Mfg. Co., IT. Y.) absorbancy was measured a t 260 nip with a therniostated Beckman DU spectrophotonieter. The complete details of the thermostating of the cell chamber and temperature measurement are given by Haselkorn.*O
MELTING TEMPERATURE OF D SA
197
EXPERIMENTAL RESULTS Variation of T , with Salt Concentration Intermediate Values of Salt Concentration. For each DNA solution the absorbancy (corrected for the volume increase of water) was determined as a function of teinperature. The results for Ps. aeruginosa DFA dis-
40 -
I
E 0
W
N
+ 4
> 30-
U
z 4
m LT v) 0
m 4
z -
20 -
W (0
W 4 (r
U
z
- 10-
c z W U
X
Fig. 1. Temperature depeiideiice of the absorbancy of Ps. aerugznosa DNA a t salt concentration between 0.01 atid 0.6M. The per cent increase is relative to the absorbancy a t 25°C. TABLE I Tmof DNA Samples as a Fiinction of Salt Coiiceiitratioii" T,lL,OC.
Salt co~icent,ratioii,
11
KCl
0.010 0.015 0.020 0.035 0.050 0.075 0.10
69 8
0.12
0.195 0.60 a
b
E. cola
75 79 81 84 86 87 90 95
2 0
5 5
8 3 8 7
Na citrat,eh 65 67 72 76 79 81 83 84 88 93
4 6 2
4 1 8 7 7 6 6
D. Na phosphate 68.7 70.7 73.4 i(.l 80.0 83.3 86.5 86.0 88.7 93.9
--
For details of buffers used, see legends t o Fig. 2. For this buffer the first column gives the ionic strength.
PS .
pnez~moniae, aeruginosa, KC1 KC1 68.3 70.3 73.8 76.5 79.0 79.7 80.8 85.1 90.3
78.6 79.5 80.2 84.7 87.7 88.9 92.2 93.2 95.0 98.7
198
C. SCIIILDKRAU'I' AKD S. LIFSOX
Fig. 2. Variation of the Z ', with the logarithm of the salt concentration (molar). The sodium citrate buffer was prepared by diluting a stock solution of 1.50M NaCl 0.15M sodium citrate to give the desired sodium ion concentration. The sodium phos0.30211 phate buffer was prepared by diluting a stock solution 0.15M Na2HP04 NaH2P04 to give the desired sodium ion concentration. The final p H of both types of buffer solntions was approximately 7. I n the case of E. coli DNA in the sodium citrate solvent, the values of the ionic strength listed in Table I were first converted to salt concentration, and the T, was plotted as a function of this latter quantity.
+
+
solved in different concentrations of KC1 are shown in Figure 1. The melting temperatures of all three DNA samples in KC1 solution obtained from such curves are summarized in Table I, which also includes the values for E. coli DNA in two buffered solutions. The T, values listed in Table I for E. coli DNA and D. pneumoniae DNA in KC1 were obtained from absorbancy-temperature profiles that have been published elsewhere.6 I n Figure 2 , T , is plotted as a function of the logarithm of the salt conceiitration. As was first, suggested for synthetic polynucleotides by HaselkornL0and for DNA by Doty and collaborators,2 a linear relation is obtained except a t high salt concentrations. I n the case of E. coli DNA, for example, in the range of 0.014.20111 salt concentration in a buffered solution, the data of Figure 2 can be expressed as
T,
=
16.6 log
+ 102
(1)
where T , is the melting temperature in degrees Centigrade and M is the salt coiicentration (molar). The equation was obtained by drawing the best fitting straight line through the points for sodium citrate and sodium phosphate buffer solutions as shown in Figure 2. By combining eq. (1) with the equation given by Marmur and Doty6 for the dependence of T , on G-C content, one obtains the equation
T,
=
16.6 log ilf
+ 0.41(G-C) + 81.5
(2)
MELL'IhG TEMPERATURE OP' D N A
199
91
9(
Trn
81
8C
I
I
I
I
log M
Fig. 3. T , a t very high salt concentrations. The T , of D . pneumoniae DNA was determined in CsCl solutions having concentrations up to 6 M . These data were kindly supplied by Professor J. Marmnr.
where (G-C) is the mole percentage of guanine plus cytosine. Since the presence of very small amounts of containinating divalent cations may influence the ?Im, this relation is only an approximation. Dove and Davidson7 were able to obtain very low ineltirig teinperatures by removing divaleiit cations. The T , values predicted by eq. ( 2 ) agree within 3°C. for all their determinations at low salt concentration. That eq. ( 2 ) is in fact consistent with all the T,, values given in this report as well as three other^,^*^*" should be useful in predicting approxiinate T , values in the and 0.2M. range of salt concentration between 3 X The BCI solvent was unbuffered except for the trace amount of citrate present as a result of the dilution procedure described in the section on materials and niethods. Since similar results were obtained with E. coZi DNA in two different buffered solutions, the absence of buffer does not appear to coniplicate the results. High Values of Salt Concentration. E. coli DKA solutions in 1.0, 2.0, and 4.0flT KC1 gave T , values of 98.5, 98.0, and 98.fL0C., respectively. Thus, a logarithmic plot lcvcls off at these high salt concentrations. In the case of D. pneumoniue DNA, the T , decreases in concentrated CsCl solutions. This IS probably due to denaturation by C1- a t high concentrations (Fig. 3), as described by Hamaguchi and Geiduscheli,12 who suggested that C1- ion acts as a hydrophobic bond-breaking agent.
THEORY The dec.rease in melting temperature at low salt concentrations is due to the increased electrostatic repulsion between the negative phosphate
C. SCIIILDKRAU'I' AND S. LIFSON
200
ions of the two strands, which favors their separation. A quantitative analysis of the dependence of the DNA melting temperature on salt concentration requires the generalization of existing theories on the helix-coil transition of double he lice^'^-^^ to include the electrostatic contribution and is beyond the scope of the present discussion. A much siinpler model will instead be discussed which, though neglecting many aspects of the order-disorder transition, nevertheless demonstrates some of the essential features.
Thermodynamic Relation between T , and Electrostatic Free Energy The helix-coil transition is considered to be the result of two opposing tendencies. One is the tendency of a system to go from states of high energies to states of low energies. Since hydrogen bonds between purines and pyrimidines, as well as stacking of neighboring puriiie-pyriniidine pairs, have negative energies, the more base pairs bonded, the more stable the system. The other is the tendency to go from states of low entropy to states of high entropy. Since hydrogen bonds restrict the freedom of internal rotations around the various skeletal bonds of which the polynucleotide chain is made, breaking them mould increase the entropy of the system and thus favor the randoiii chain foriil. The free energy F of the system reflects both tendencies, and the stable state is that with the lowest free energy. Let AF = Fco,l- F h e l l x denote the differencc between the free energies of the coiled and helical states; then when AF > 0 the helix is stable, when AF < 0 the coil is preferred, while AF = 0 indicates that the two are in equilibrium. The temperature a t which AF = 0 is the melting temperature. The free energy change AF will be considered as the sum of two terms, AF = AFo AF,, where AF, is the part due to electrostatic interactions between the phosphate charges fixed on the DKA molecule; AFo represents all the other nonelectrostatic contributions to AF, which are approximately independent of the salt concentration. AF, decreases with increasing salt roncentration, because of the screening of electrostatic interactions a t high salt concentrations. I n the complete absence of such electrostatic interactions ( A F , = O j , the melting temperature is determined by the balance between the nonelectrostatic energy change AH0 and the corresponding entropy change ASo according to
+
AFo = AH0 - T,r60ASo= 0
where T,,o denotes T , at the limit of AF,
T,o
=
(3)
0, or
(4)
= AHo/ASo
At lower salt concentrations, however, AF, # 0 and then AF AFo = AH0 - T,,ASo
=
-AF,
=
0 implies
(51
In the following, we shall find it convenient to express T , in terms of
MELTING TEMPERATURE OF DNA __
201
--
A H O , A&, and T F , which are the corresponding quantities per mole of nucleotide pair, namely,
T,
=
(nHo + A F , ) / L &
(6)
LF, is negative, i.e., F , (helix) > F , (coil), because of the repulsions between the fixed charges. As these repulsions increase, T , must decrease with decreasing salt conceiitration. Relation between Pp and the Electrostatic Potential The exact calculation of AF,, including the contributions of both the helix and the coil is rather difirult , but one can obtain some useful information by making the followiug assumptions. Assumption 1 : The superposition of electrostatic potentials. At a point occupied by a specified phosphate charge fixed on one of the strands of the DNA molecule, there will be an electrostatic potential, induced by all other phosphate charges of the two strands, as well as by all other free ions which surround the phosphate charges as ionic atmospheres. The contribution of the ionic atmosphere is represented according to Debye and Hiickel as a screening potential induced by the phosphate charges. Assumption 1 states that the screened potential at the point of a specified phosphate charge is ihe sum of the potentials induced by the charges of each strand separately. This assuniption makes a good approximation only i f the energy of the specified ion due to the total electrostatic potential is less than IzT. We shall discuss the validity of this assumption in more detail later, after we shall calculate the potential and see whether the results are consistent with the assumption. Assumption 2. The electrostatic potential a t a point occupied by a phosphate charge due to the other phosphate charges of the same strand is approximately the same in the helical and in the coiled conformations. According to this assumption, the contributions of AT, from the electrostatic interactions between phosphate charges of the same strand in the coiled and the helical conformations cancel each other, so that we have to consider only electrostatic interactions between the phosphate charges of one strand and those of the other strand. I n other words, we shall be concerned oiily with the electrostatic energy involved in the strand separation and neglect the accoiiipaiiyiiig changes in the electrostatic energy of the two separated strands. While the neglected part is not necessarily negligible, it is believed to be considerably smaller than the electrostatic energy of strand separation. A iiiore detailed consideration of the coiled and helical contributions to separately is not warranted, in our opinion, due to the generally approximate and semiquantitative nature of the present discussion and in view of the absence of sufficiently good theories of both helical and randomly coiled polyelectrolytes. I n support of our approximation, it may be pointed out that any two adjacent phosphates on the same strand in the Watson-Crick model are almost as far apart as is sterically possible. The transition to the coil form cannot significantly
z,
C . SCIIIT~DKRAU’I’A Y D S. LIFSOY
202
decrease the electrostatic potential energy of the molecule by allowing a greater separation of neighboring phosphates. On the other hand, the coil form cannot have a much higher electrostatic energy, since the conformations of higher energy have a low probability and, as a result, electrostatically charged coils are considerably extended. According to assuniption 2, AF, is the work done by renioving, isothermally and reversibly, a phosphate of charge e on one strand against the electrostatic potential, +, induced by the other strand. Since is zero when the strands are separated, is related to by
+
me
A F , = -&
+
(7)
Calculation of + for DNA
+
It now remains to calculate the potential due to one of the DXA strands at the position of any of the charges of the other strand. This could be done by solving the Poisson-Boltzmaiin equation for the particular charge distribution of the DNA molecule. One ronsiders here the electrostatic potential, +(r), a t a point r in an electrolyte solution as determined by a set of fixed charges a t points rl, . . . , rn (these are the charges of the phosphate ions of one of the strands of the D S A niolecule) and by all the other mobile ions of the system. The iuobile ions are assumed to form a n average net charge density p(r) which is related to +(r) according to both the Poisson differential equation and the Roltzinann distribution law. These relations are vombmed in the well known I’oisson-Boltzmaiin equation, which may he mritteii (for inonovalent ions) as V 2 [ e + ( r ) / k T ]= - (47re/DliT)p(r)
=
K~
sinh [e+(r)/kT]
(8)
where K~ = 8ae2NM/1000 Dk?‘, e is the electronic charge, N the Avogadro number, D the dielectric constant, k the Boltziiiann constant, and T the absolute temperature. The boundary conditions require that y5 should vanish a t large distances from the fixed charges and that it should tend to infinity like e/Dlr - r31 in the close neighborhood of any one of the fixed ions. Equation (8) is difficult to solve for such boundary conditions, however, in view of assurnption 1, it may be replaced by its linear approximation, the Debye-Hurkel differential equation,
v2+ =
(9)
K2+
A solution of eq. (9) for the above-mentioned boundary conditions is n
*(r)
=
(e/U)
c CXP
3=1
--KIT
-
r3l l/lr - rll
(10)
If r is chosen to be the location of a specified ion on the other strand, the distance / r - r,l is given by / r - r,/2 = ( z - zj)2
+ 4r2 sin2
(‘p
-
(.,>/a
(11)
MELTING TEMPERATURE OF DNA
203
where r , z,, (p3 and r , z, cp are, respectively, the cylindrical coordinates of tlhe vectors r3and r. Equations (10) and (11) give the required potential induced by one strand a t a point occupied by a phosphate group on the other strand as a function of the ionic strength of the solution and the structure of the double-stranded DNA molecule.
RESULTS AND DISCUSSION The coordinates of the vectors r, were obtained froin Table I11 of Langridge et aLZ1 (After our coniputations had been made, niore accurate values were published by the same group.22 Use of the niore recent values in our calculations would not, however, ilialie a significant change in $.) AF,was calculated from eqs. (7), (lo), and (11) for the set of values of K given in Table 11. It was necessary to go as far as 93 A. away from the TABLE I1 Electrostatic Potential Energy Due to Interchain Phosphate Repulsions ~
-
~~
- AFe, Salt roncentration, ,If
K X 1 0 '
kcal./mole nucleotide pair
0.010 0.015 0.020 0.035 0.050 0.075 0.10 0.12 0.195 0.60
0.0339 0.0412 0.0481 0.0639 0.0766 0.0939 0.109 0.118 0.152 0.269
0.16 0.14 0.12 0.087 0.068 0.049 0.038 0.033 0.019
0.003
The electrostatic potential energy due to interchain phosphate repulsion was calculated by using a modified Debye-Huckel potential. Equations (10) and (12) were used to calculate this energy, and it is assumed t,hat t,he charge on each phosphate is 0.22e.
location of the specified phosphate group in order to include in our summation all terms which iiiade any significant contribution. This involved a total of 56 phosphate groups on the opposite chain. The region coiisidered was over five coinplete turns of the double helix. The T , values for the three DKA samples were plotted as a function of the corresponding values of KFe obtained by using equation (10). The curves were seen to be fairly linear in the range of 0.01-0.211.1 salt concentration. This seemed, a t first glance, to substantiate the validity of the relation between T , and eq. (6), as well as the approximation of linearization, eq. (9). A closer exaniiriatioii of the results, however, reveals a difficulty. The absolute values of ell. obtained froin eqs. (10) and (11) are considerably larger than kT for the lower values of ionic st,rength, while the Debye-Huckel approximation, eq. (9), requires I e $ , < kT.
v,,
204
C. SCHILDKRAUT A N D S. LIFSON
It is therefore evident that eq. (10) is inadequate and has to be replaced by a better approximation. Unfortunately, a solution of eq. (8) with the DNA boundary conditions is not available. However, both numerical and a n a l y t i ~ a l studies ~ ~ - ~ ~of the Poisson-Boltzmann equation with spherically or cylindrically symmetric boundary conditions show that the linear approximation always gives higher potentials and smaller potential gradients at each point than the exact solution, and that the deviation increases rapidly with increasing density of the source charges. The qualitative rationale for this behavior is as follows. The absolute value of e$ is highest in the region close to the fixed ions. If in this region le$/kTI >> 1, the counterion coiicentratioii is very much higher than assumed by eq. (9), varying exponentially with $, i.e., p siiih ( e $ / k T ) E exp ( e $ / k T ] when le$//cTi>>l, rather than linearly. Since p and $ have necessarily opposite signs, the effect of large p is to reduce $ very rapidly within a short distance (the screening effect of the ionic atmosphere). Under these circumstances, the linear approximation seriously underestimates p and its screening effect. Thus, the actual potential very near to the fixed charges is smaller and, moreover, falls off much faster than predicted by eq. (10). Therefore, the potential $(ro) induced by one strand a t a point occupied by a phosphate charge of the other strand niust also be smaller, in absolute value, than that predicted by eqs. (10) and (11). These considerations indicate a way out of the difficulties encountered in using eq. (10). For this purpose it is necessary to make a third assuniption, namely, that the region of high electrostatic potential around one strand is actually so narrow that the ions on the other strand are already well inside the region where le$l is snialler than k T . I n the latter region the linear eq. (9) is a valid approximation. Since we are not interested in the exact behavior of $ in the narrow nonlinear region, we may consider the fixed charges together with the counterions in their iinniediate vicinity as constituting together the source of the linear potential at the regions farther from the fixed charges. From the niatheniatical point of view, this assumption amounts to a change in the boundary condition for the solution of eq. (9). Similar reasoning was suggested by Kotin and Kagasawa in their discussion of ion binding in polyelectrolytes. 26 Thc boundary has to be chosen just broad enough to make the solution of eq. (9) outside this boundary such that le$l < /cT. Then the application of eq. (9) as a n approximation of eq. (8) becomes self-consistent. To simplify matters even further, we may regard the screening effect of the counterions in this region as if it were a partial association, so that each fixed phosphate ion is assumed to carry a reduced charge Xe. The solution of equation (9) is now given by eq. (10) in whirh e is replaced by Xe. Thus, the potential reduces to A$. The electrostatic free energy change, eq. (7), reduces to
-
~
AFe = -(Xe)(X$) = -X2e$
(12)
if we also replace the specified charge on the other strand by he. A value of X in the range 0.1-0.25 would be sufficient to make le+l 5 k T , justifying
M m ’ r n C r TEMPERATURE OF DUA
205
0.4
0.3
-
-AFe
0.2 I
0.1
C
I
0.1
0.
M
Fig. 4. Interchaiii electrostatic potential energy of DNA as a fiinctiori of the molar salt coiicentratiori ( M ) . Equations (10) and (12) were wed to calciilate this energy, and it is assumed that the charge on each phosphate is 0 . 2 2 ~ .
the use of assumption 1 and, at the same time, preserve the linear relation between the measured T , and the calculated Such values of X are within the range of “charge fractions” of DKA reported by several author^.^^'^^ We chose 0.22 as a value for In accordance with experiiiierital results,28X is constant over a wide range of salt concentrations. The values calculated for AF, are given in Table 11. In Figure 4, these values are plotted as a function of salt concentration. The dependence of of the T , values for each of the three samples of DXA used in this study is given in Figure 5. In addition, results are shown for dAT and dGdC polyiners, based 011 the T , values of Inniari and B a l d w i ~ i . ~In~ each case a linear relation is obtained as predicted by eq. (6). It is possible to obtain the value of froin these graphs. By coiiibining eys. (4) and (6) we obtain
n,.
z,
__
AH,
a
= - Tm,oAF,/(T, o -
T,)
(13)
Assuming aHoand are independent of temperature, we can obtain AH0 from the slopes of the lines ill Figure 5. The values of obtained are listed in Table 111. Due to the semiquantitative nature of our treatment, we would not necessarily expect to increase with the G-C content of the samples. We thus consider 2.5 kcal./niole of nucleotide pairs to be an approximate average value obtained from our studies. The reported experiinental
m,
C . SCHILDKRAUT AND S. TATFSOU
206
Fig. 5 . 7', as a function of interchairi electrostatic potential energy. A modified Debye-Huckel potent,ial was nsed to calculate the energies a t the KC1 concentrations specified in Table TI. The values of ZF, listed in Table I1 were calculated from eqs. (10) and (12) for D = 60. For Fig. 5, was calculated by using t,he valiie of D for water a t the corresponding l',n. The T , valiies for dAT and dGdC were obtained from the datsa of Inman and B a l d w i i ~ ~ 9
a,
rai
values of are between 5 and 10 kcal./inole of iiucleotide pair~.3~-3:< Considering the scatter of the experimental values, the differences in the detailed experiiiieiital c.onditions, and the seiiiiquant itative nature of the parameter A, we feel one could not expect inore than this seiiiiquantitative agreement. The charge fraction A is closely related to the hypothesis of counterioii TABLE I11 Energy Necessary to Separate the Strands of the DNA Double Helix in the Absence of Phosphate Electrostatic Repulsiona AHo,
Sample
kcal./mole nucleotide pair
dAT
1.5
D. pneunzoniae E. coli Ps. aeruginosa
2.8 2.3 2.4 2 7
dGdC
a Equations (10) and (12) were used to calculate AF,, the charge on each phosphate being assumed to be 0.22e. Fig. 5 shows the linear relation between Ym and AF,, the intercepts giving AHo. The values for dAT (enzymatically synthesized copolymer of deoxyadenylate and deoxythgmidylate) and for dGdC (enzymatically synthesized homopolymeric strands of polydeoxyguanylate hydrogen bonded to strands of polydeoxycytidylate) were obtained from the data of Inman and Raldwin.*g
MELTING TEMPERATURE O F DNA
207
binding which has been suggested by a number of a ~ t h o r s to ~ ~explain -~~ electrochemical, thermodynamic, and other properties of various polyelectrolytes. It was pointed o ~ t that ~ ~very v often ~ ~ the terms "counterion association" or "ion binding" are used in reference to experiments where no distinction can be made between true pairwise associations and a large iiuniber of other factors which may contribute t'o t,he decrease of couiiterion activity. The same is certainly t'rue with respect to t'he physical nieaning of the charge fract'ion as used in our present' discussion. While t'he nonlinear nature of eq. (8) necessarily contributes to X as a correct'ion for tJhe Debye-Huckel approximation, there are a iiuniber of other factors which should not be overlooked, e g . , the fact t>hatthe use of an average pot,entzialin place of a potent,ial of average force in eq. (8) is in itself a n approximation which becomes worse as the average potential increases; that' the separation of T F into a suni of two terms and AF; is also an approximation; the low dielectric constant of the bulk of the DNA molecule ; the different solution properties of different ions; and finally the possibility of true ion binding between the cat'ioiiic countI. Cutler, and H. Samelson, J . Phys. ('hem., 56, 57 (1952). 25. Lifson, S., J. ('hem. Phys., 27, 700 (1957). 26. Kotin, L., and M. Nagasawa, J . Chcm. Phys., 36, 873 (1962). 27. Ross, P. D., Biopolymers, 2, 9 (1964). 28. Jordan, 11. O., The Chernzstry of the Nucleic Aczds, Rutterworths, London, 1960, pp. 216-221. 29. Inman, R. B., and R. L. Baldwin, J . Mol. Biol., 8 , 452 (1964). 30. Sturtevant, J. RI., S. A. Rice, and E. P. Geiduschek, Discussions Faruday SOC.,25, 138 (1958). 31. Rawitscher, >I., and J. M. Sturtevant, J . 4 m . ('hem. Soc., 82, 3739 (1960). 32. Warner, R. C., and E. Breslow, Proc. Fourth Intern. C m g r . Biochem., 9, 157 (1958). 33. Steiner, R., and C. Kitzinger, hratccre, 194, 1172 (1962). 34. Harris, F. E., and S. A. Rice, J . Phys. Chem., 58, 725, 733 (1954). 35. Ross, P. I)., and IT. P. Straws, J . -4m. Chem. Soc., 82, 1311 (1960). 36. Rice, S. A,, and AT. Nagasawa, Polyelectrol?jte Solutions, Academic Press, New York, 1961. 37. Lifson, S., J. Chem. Phys., 26, 7'27 (1957). 38. Kotin, L., and XI. Nagasawa, J . Am. C'hem. Soc., 83, 1026 (1961). 39. Kotin, L., J. Mol. Biol.,7 , 309 (1963).
Received November 12, 1964
