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Approximate calculation of the electrostatic free energy of nucleic acids and other cylindrical macromolecules
Approximate Calculation of the Electrostatic Free Energy of Nucleic Acids and Other Cylindrical Macromolecules Terre11 L. Hill From lhe N~cul Medical
Research Institute,
Bethcstla,
~llc~rylu71tl
Received February 2, 1955
I. INTRODUCTION The \vell-known
expression (1)
I,$7= I ZE 1 2 Db
Kb
1 + KU>
(1)
for the electrostat’ic free energy of a sphere of radius b with total uniform surface caharge zc immersed in an electSrolyte whose ions can approach the center of the sphere to within a distance CIhas found extcnsiw use ill discussions of titration curves, ion binding, &., on spherical or approximately spherical protein molecules. In view of the growing interest in macromolecules wit,h approximately caylindrical symmet*ry (e.g., helical nucleic acid and protein molecules, viruses, extended polyelectrolytes, etc.), it would seem wort.h while to provide corresponding expressions for certain cylindrical models. Ahhough this is t,he primary purpose of the present paper, we take this opport,unit,y to make a few additional remarks concerning electrostatic effects in adenosine triphosphat’e (ATP) (2). The two subjects are not unrelat,ed in view of the fact that nucleic acids are composed of nucleotides: it is of some interest to compare t’he electrost,atic free energy associated with the charged phosphate groups in the two cases. ,\ttention should be drawn to two important recent papers in this general area, though the particular models considered iu these papers [cylindrical (3), spherical (4), and spheroidal (4)] are not, direct#ly related to the main argument of the present work. II. SPHERICAL MODELS A. Consider a sphere of radius b and dielectric constant Di immersed in a dilute electrolyte solution (solvent dielectric constant D) with 229
230
TERRELL
FIQ. 1. Geometry
L. HILL
for spherical
and cylindrical
mod&.
Debye-Hiickel constant K and with ions which can approag.:hto within a distance a of the center of the sphere (a > b). If z charges E are distributed (smeared) uniformly over the surface of the sphere, the work required to charge up these charges from E = 0 to the Ml value E, in the presence of electrolyte, is given by Eq. (1). This msult is independent of Di . B. On the other hand, suppose the charge ze is distribt ted uniformly throughout the sphere, giving a uniform volume charge density p = 3zq’4?rb3. Then we have (Fig. 1) V”& = 0,
V”#I = - 4?rp/Di,
v2#3
=
K2#3
with boundary conditions
D&‘(b) = Qh’(b)
h(b) = h(b), hb>
=
$2’(a)
h(a),
=
(3)
#3’(a)
The solutions of Eqs. (2), finite at r = 0 and vanishing a.t r = CQ,are #I = CI - 2g
)
fi3
-
C4e-“’
(4) T i The constants are evaluated by use of Eqs. (3), and finally the work of *2
=
;
+
co,
FREE
ENERGY
OF NUCLEIC
231
ACIDS
charging the sphere from the equation w = I’
Lb X&(r)47rr2 czrpdX
where X is the charging parameter. The result is (-5) If we take (5), say, D = 74 and Di = 2, the term D/5Di is 7.4 compared to the other terms which are of order unity. Thus the electrostatic free energy is much larger as expected, when a given amount’ of charge is distributed throughout the sphere rather t’han on the surface. III.
CYLINDRICAL
MODELS
A. Consider an infinite cylinder of radius b and dielectric constant Di immersed in a dilute electrolyte solution as above. Figure 1 is applicable here also if it is regarded as a cross section of t.he cylinder. In the model under consideration here, let there be a uniform surface charge density u at r = b (i.e., on the surface of the cylinder). Then we have v2+1 = 0,
v2+2
= 0,
Id v2 = ; Y&
v2+3 = K2#3
(
with boundary h(b)
(6)
cl r -& >
conditions =
D#z’(b) - D&‘(b)
#z(b),
= -4m (7?
h(a)
=
$2’(a)
#s(a),
=
#a’(n)
The required solutions of Eqs. (6) are $1 = Cl ,
*2
= C2
In r +
C3 ,
$3
=
C.&O(Kr)
(8)
where K,(z) is a modified Bessel function of the second kind. The constants can be found from Eqs. (7). Then the work 117of charging a lengt,h of cylinder 1 is
1
232
TERRELL
L. HILL
where z’ is the number of (smeared) charges c per unit leng >h of cylinder, That is, n = z’ej2ab. The work W in Eq. (9) is also independent of Di. B. Now suppose the cylinder has a uniform volume charge density with a total charge Z’E per unit length; that is p = z’~/at”. The first of Eqs. (6) becomes V2#1 = -4sp/D
With the appropriate
boundary
conditions
j
[Eqs. (3)] we jind here
+lnz+-& z
1
(10)
Again the other terms are of order unity so that the term D/4D; results in general in the free energy of a volume distribution [Eq. (lo)] being considerably larger than the corresponding free energy for a surface distribution [Eq. (9)]. C. Some helical structures are of interest in which thtrre is a cavity down the middle, which we assume to be filled with el xtrolyte. The cross-sectional geometry is shown in Fig. 2 where c - d = a - b. The
FIG. 2. Geometry
for cylindrical
models.
FREE
ENERGY
OF NUCLEIC
233
-4CIDP
region 3 is considered filled with protein, nucleic acid, em., and has a dielectric constam Di . Electrolyte occupies regions 1 and 5 but the ionic centers are excluded from regions 2 and 4 because of t,he finite size of t,he ions. T,et there be a uniform surface charge density g = z'42d at r = b (i.e., on the outside surface of t’he cylinder). We proceed here from -?‘I)* = KQ )
v*q3 = 0,
v*+* = 0,
V2& = ii*+6
v*+4 = 0,
essentially as in sec. ITIA above, except that
where In(z) is a modified Bessel function of t,he first kind, and that there are no\v eight constants to evaluate. We find
n DiIo(dI[ I Ko(KU)
l1.
=
fT*!
[
KU&(KU)
’
“I g
DiIo(Kd)
D~K~(Kcx)
DKdIl(Kd)
+
DKoK~(KO)
D.
c
D
d
1(1l‘i ___~ 1 b
+ 2 ln - + It1 -
D~dI,(td)
Iii
UC
ll
bd
c
b
+ -- ln -
+ 111-
c
This is always less than II’ in Eq. (9), as expected, because of the additional elect,rolyte in the “core” of the cylinder. D. If t’he uniform surface charge is on the “inside” rather than the “outside” of the cylinder, that is at T = c with q = 2’~/2nc, the work is Iobd IT7 = x’“$ D
[ KdI&d) r DJohd) L ~,&l(Kd)
+
111 ”
d
DiKo(d I[ DKuK~(KU)
D.
DiKoha)
’
+~ln~+lnb
DKCZK~(KU)
+
ac
n’” In w
’
1 (12) 1 c
b “’
2
E. We suppose here that there is a uniform volume charge densit,y p in region 3 of Fig. 2, with p = x’q’vr(b2 - 3). We find in this case z
)
+ B
[
hi b - $ In c (13)
A(1 - $) - &(1 - $1)
234
TERRELL L. HILL
where A=&+ln;+
&(td -B Ka&(KU)
I
lnb
D&&a) hU&(KU)
+$n;+
1
B=
Eqs. (II),
(12), and (13) all reduce to
w= i2e21 D
(14)
in the special case c + b (that is, region 3 has zero thickness). IV. ADENOSINE TRIPHOSPHATE
(ATP)
By choosing (2) an effective dielectric constant De of 49.6 for all interactions in ADPT3 (adenosine diphosphate), the total electrostatic free energy associated with interactions between negal,ively charged oxygen atoms attached to different phosphorus atoms in .4DP-3 can be made to agree with the same quantity found earlier in :I. more refined treatment of pyrophosphoric acid (6). We can have considerable confidence in the pyrophosphoric acid calculations because ofj the excellent agreement with experiment (2, 6). Using DE = 49.6, as ex:plained earlier (2), we find in the reaction ATP-4 + Hz0 + ADP-2 + HP04”
(15)
that W(ATP-4)
= 7.12 kcal./mole
W(ADPm2) = 1.67 kcal./mole AW,,
= 1.67 - 7.12 = -5.45 kcal./mole
(16)
where W includes all charge interactions except between P+ and Obonded to each other [these interactions cancel in Eq. (15)]. Also, for the
FREE
ENERGY
OF
NUCLEIC
ACIDS
235
+ HPOd-2
(17)
ADP reaction ADP-3 + Hz0 -+ AMP-I we have W(ADP-3)
= 3.16 kcal./mole,
AWmp = -3.16
W(AMP’)
kcal./mole
= 0 (18)
The values in Eqs. (16) and (18) are the ones referred to in the last paragraph of sec. II, Ref. (2). The results in Eqs. (16) and (18) are based ultimately on the experimental dissociation constants of pyrophosphoric acid at zero ionic strength. Although salt effects were mentioned in Ref. (2), no quantitative correction was attempted. We try to estimate this correction here. From a model of tripolyphosphate (P,O,O , the “business end” of ATP), the semimajor and semiminor axes of the ellipsoid of revolution roughly filled by the molecule are estimated as 4.5 and 2.25 A., respectively. If we assume that the net charge on ATP of -4 is smeared uniformly over the surface of this ellip,soid, a treatment for an ellipsoid such as that in sec. IL4 for a sphere could be used to estimate the salt correction. Smearing a small number of charges is a serious approximation, but it should be much less serious for the salt correction,, which is all we need, than for the absolute value of W. Actually, the treatment of an ellipsoid is relatively complicated mathematically (4), so we use a spherical model with the same surface area (and surface charge density). The radius of t,he equivalent sphere is b = 2.95 A. The salt correction factor in Eq. (1) is l--
Kb 1 +
KU
(19)
We choose, somewhat arbitrarily, a = 3.95 A. (the correction given by Eq. (19) is quite insensitive to the exact choice of a - b). with the choice I/K = 7.79 A. (0.15 &f NaCl, 37”C.), the electrolyte correction factor for ATP is 0.75 so t’hat W(ATPw4) = 7.12 X .75 = 5.3 kcal./mole An analogous calculation gives
for ADP, using b = 2.60 A. and a = 3.60 A.,
236
TERRELL L. HILL
W(ADPm2) = 1.67 X .77 = 1.3 kcal./mole W(ADPe3)
= 3.16 X .77 = 2.4 kcal./molc:
and hence AWARD = 1.3 - 5.3 = -4.0
AW
ADP
=
kcal./mole
- 2.4 kcal./mole
(20) (21)
in 0.15 M NaCl. The value of 4.0 kcal./mole in Eq. (20) thus accounts for two thirds of the current best est.imate of the free ent rgy of hydrolysis of ATP; -AF” = 6 kcal./mole (7). Although we have already pointed out t’hat the abs Jute value of WATp for Eq. (15) calculated from Eq. (1) is unreliable, thl: extent of the unreliability may be of incidental interest. We find from Eq. (1) -W(z=4, b=2.95,a =3.95) AWATP= W(z=2,b=2.60,~=3.60) (22) = 2.7 - 9.1 = -6.4 kc: I./mole compared to the calculation on the basis of discrete charg YS,Eq. (20). V. DEOXYRIBONUCLEIC
ACID (DNA)
The structure of ribonucleic acid (RNA) is unknown as If this writing, so we confine our remarks here primarily to DNA. Eq. ( j) should be a fairly good approximation when applied to DNA, which las two singly charged phosphate groups (at pH 7) on the outside of t double helix for every 3.4 A. along the axis of the double helix (8). A~ually, Eq. (9) will probably give an upper limit to W because (a) we art smearing the surface charge in this equation and (b) some electrolyte can penetrate inside the helix (see below). aside from applications to ion binding and titration CLrves of DNA, Eq. (9) is of some interest in connection with the free enerj’y of synthesis (nucleotide polymerization) of DNA. The sign of the free energy of DNA synthesis has an important bearing on the mechanism of the synthesis. For example, if the free energy turns out to be positive, a self-duplicating mechanism without coupling to an outside free energy source is not possible. Assuming RNA is involved in protein synthesis, t:le free energy of synthesis of RNA has a bearing on the mechanism of both RNA and protein synthesis. For example, if this free energy is positive, RNA (as above for DNA) cannot be synthesized by a self-duplicating mechanism without outside coupling; and, if the free energy is sufficiently positive,
FREE
EXERGY
OF
NUCLEIC
APITS
23i
it, is possil)lc that depolymerization of RXA would be coupled with prot,c:ili synt’hrsis (i.e., RNA would be the free-w~crgy donor for protein synt,hesis). The replacement, of the depolymerizecl RN.\ \\ould the11 :monntj for I 11~reported link lwt~wecn the trlrtw\w of RX.\ and prot,ctill (!j). The olwtrostatic free energies of 11X,1 and RX21 are positive due to the repulsion of tlegabively charged phosphate groups, just as in h’l’P escept thatJ there are more phosphate groups in 1he nucleic acids but. these are fart,her apart. Thus electrostatic effects (*ontribute to a positive free energy of synthesis of Dr\‘A and RKA. The analog here (for I>K&Z) of I?q. (15) is the depolymerization reaction (schematic)
where R = ribose and B = base. XVic can estimate (see belon-) the electrostatic contribution to the free energy of t,his process for DKA, lvhosc structure is known, and compare it with Eq. (20) for ATP. Othet cont,ribut,ions to the free-energy change in Eq. (23) for I)SL2 could also be estimated: e.g., net hydrogen bond energies, van der \Vaals’ interartions (especially between bases), gain of t~ranslational and rotational (internal and ext,ernal) free energy of t.hc nucleotide split off, split,ting of the phosphate ester bond, etc. But there remains a presumably large cont,ribut>ion which is very difficult to assess, that of the solvent molecules and ions. Because of t’his difficulty standing in the way of an estimat,e of the complete AF”, wc confine ourselves here t#othe electrostatic contribution only. Let AlV be the electrostatic contribut#ion to AF” in Eq. (23). ‘I’hc elect,rostatic work necessary to bring a singly charged nucleotide, P-It-B, from infinity up to the end of a long DXA cylinder is then -AlV/2. Now the net effect of adding one P-l--R--B to (PI-R-B), , where n is very large, is to increase t’he central portion of the molecule (where Eq. (9) is applicable) by one unit’, leaving the ends unchanged. Therefore if w-e denote W in Eq. (9) by W(Z), we have -- AW = 2
- W(Z) = 11’ f 0
238
TERRELL L. HILL
or
Aw= e2$ We take (8) b = 11 A., a = 12 A., find from Eq. (24)
+ In;] l/~
(24)
= 7.79 A., and z’ = 2/3.4A., and
AW = -3.3 kcal./mole
(25)
which is to be compared with Eq. (20) for ATP. As aln ady mentioned, this computation probably gives an upper limit’ to -AW. But the electrostatic contribution to the free energy of synthl sis of DNA is definitely significant. ACKNOWLEDGMENTS The author is indebted to Drs. Sidney Bernhard, Dunitz for helpful discussions.
Alexandr : Rich, and Jack
SUMMARY
Equations are derived for the electrostatic free en#:rgy of certain spherical and especially cylindrical models of charged Inacromolecules immersed in an electrolyte solution. The equations shou d be useful, for example, in the interpretation of ion binding and titr cbion curves of proteins, nucleic acids, etc. Another application, which k discussed here, is the calculation of the electrostatic contribution to th’ free energy of hydrolysis of ATP and synthesis of DNA. The possible relation of the present work to the mechanisms of nucleic acid and prot :in syntheses is pointed out. REFERENCES 1. See, for example, COHN, E. J., AND EDSALL, J. T., “Proteins, Amino Acids and Peptides.” Reinhold Publishing Corp., New York, N. Y., L943. 1 Electrolyte can penetrate inside DNA, but in an unsymme trical way. However, the order of magnitude of the appropriate correction can be obtained from the symmetrical model which leads to Eq. (11). From atomic mc iels, we estimate that a single average nucleotide residue of DNA occupies about 250 cu. A. The volume of 3.4 A. of a cylinder of radius 11 A. is 1292 cu. A. Hence we take the outer nucleotide occupied shell (b > T > c) as having a volume 530 cu. A. and the inside of the cylinder (r < c) as having a volume 792 cu. A. ThL gives a = 12 A., b = 11 A., c = 8.6 A., d = 7.6 A., and a correction factor [corn Bare Eqs. (9) and (ll)] of 0.95, i.e., only a 5% correction owing to the penetration of electrolyte inside the cylinder.
FREE
ENERGY
OF
NUCLEIC
ACIDS
239
2. HILL, T. L., AND MORALES, M. F., J. Am. Chem. Sot. 73, 1656 (1951). 3. Fuoss, R. M. (with KATCHALSKY, A., AND LIFSON, S.), PTOC. Natl. Acad. Sri. u. s. 37,579 (1951). 4. LINDERSTROM-LANG, K., Compt. rend. true. lab. Car&berg 28, 281 (1953). 5. KIRKWOOD, J. G., AND WESTHEIJIER, F., J. Chem. Phys. 8, 506, 513 (1938). 6. HILL, T. L., J. Chem. Phys. 12, 147 (1944). 7. LEVINTOW, L., AND MEISTER, A. (with MORALES, RI. F.), J. Biol. Chum. 209, 265 (1954); PODOLSKY, R. J., AND MORALES, M., J. Biol. Chem., in press. S. WATSON, J. D., AND CRICK, F. H. C., Nature 171, 737 (1953). 9. See, for example, DAVIDSON, .J. N., “Biochemistry of the Nucleic Acids,” p. 163. Methuen and Co., London, 1953.
Research Institute,
Bethcstla,
~llc~rylu71tl
Received February 2, 1955
I. INTRODUCTION The \vell-known
expression (1)
I,$7= I ZE 1 2 Db
Kb
1 + KU>
(1)
for the electrostat’ic free energy of a sphere of radius b with total uniform surface caharge zc immersed in an electSrolyte whose ions can approach the center of the sphere to within a distance CIhas found extcnsiw use ill discussions of titration curves, ion binding, &., on spherical or approximately spherical protein molecules. In view of the growing interest in macromolecules wit,h approximately caylindrical symmet*ry (e.g., helical nucleic acid and protein molecules, viruses, extended polyelectrolytes, etc.), it would seem wort.h while to provide corresponding expressions for certain cylindrical models. Ahhough this is t,he primary purpose of the present paper, we take this opport,unit,y to make a few additional remarks concerning electrostatic effects in adenosine triphosphat’e (ATP) (2). The two subjects are not unrelat,ed in view of the fact that nucleic acids are composed of nucleotides: it is of some interest to compare t’he electrost,atic free energy associated with the charged phosphate groups in the two cases. ,\ttention should be drawn to two important recent papers in this general area, though the particular models considered iu these papers [cylindrical (3), spherical (4), and spheroidal (4)] are not, direct#ly related to the main argument of the present work. II. SPHERICAL MODELS A. Consider a sphere of radius b and dielectric constant Di immersed in a dilute electrolyte solution (solvent dielectric constant D) with 229
230
TERRELL
FIQ. 1. Geometry
L. HILL
for spherical
and cylindrical
mod&.
Debye-Hiickel constant K and with ions which can approag.:hto within a distance a of the center of the sphere (a > b). If z charges E are distributed (smeared) uniformly over the surface of the sphere, the work required to charge up these charges from E = 0 to the Ml value E, in the presence of electrolyte, is given by Eq. (1). This msult is independent of Di . B. On the other hand, suppose the charge ze is distribt ted uniformly throughout the sphere, giving a uniform volume charge density p = 3zq’4?rb3. Then we have (Fig. 1) V”& = 0,
V”#I = - 4?rp/Di,
v2#3
=
K2#3
with boundary conditions
D&‘(b) = Qh’(b)
h(b) = h(b), hb>
=
$2’(a)
h(a),
=
(3)
#3’(a)
The solutions of Eqs. (2), finite at r = 0 and vanishing a.t r = CQ,are #I = CI - 2g
)
fi3
-
C4e-“’
(4) T i The constants are evaluated by use of Eqs. (3), and finally the work of *2
=
;
+
co,
FREE
ENERGY
OF NUCLEIC
231
ACIDS
charging the sphere from the equation w = I’
Lb X&(r)47rr2 czrpdX
where X is the charging parameter. The result is (-5) If we take (5), say, D = 74 and Di = 2, the term D/5Di is 7.4 compared to the other terms which are of order unity. Thus the electrostatic free energy is much larger as expected, when a given amount’ of charge is distributed throughout the sphere rather t’han on the surface. III.
CYLINDRICAL
MODELS
A. Consider an infinite cylinder of radius b and dielectric constant Di immersed in a dilute electrolyte solution as above. Figure 1 is applicable here also if it is regarded as a cross section of t.he cylinder. In the model under consideration here, let there be a uniform surface charge density u at r = b (i.e., on the surface of the cylinder). Then we have v2+1 = 0,
v2+2
= 0,
Id v2 = ; Y&
v2+3 = K2#3
(
with boundary h(b)
(6)
cl r -& >
conditions =
D#z’(b) - D&‘(b)
#z(b),
= -4m (7?
h(a)
=
$2’(a)
#s(a),
=
#a’(n)
The required solutions of Eqs. (6) are $1 = Cl ,
*2
= C2
In r +
C3 ,
$3
=
C.&O(Kr)
(8)
where K,(z) is a modified Bessel function of the second kind. The constants can be found from Eqs. (7). Then the work 117of charging a lengt,h of cylinder 1 is
1
232
TERRELL
L. HILL
where z’ is the number of (smeared) charges c per unit leng >h of cylinder, That is, n = z’ej2ab. The work W in Eq. (9) is also independent of Di. B. Now suppose the cylinder has a uniform volume charge density with a total charge Z’E per unit length; that is p = z’~/at”. The first of Eqs. (6) becomes V2#1 = -4sp/D
With the appropriate
boundary
conditions
j
[Eqs. (3)] we jind here
+lnz+-& z
1
(10)
Again the other terms are of order unity so that the term D/4D; results in general in the free energy of a volume distribution [Eq. (lo)] being considerably larger than the corresponding free energy for a surface distribution [Eq. (9)]. C. Some helical structures are of interest in which thtrre is a cavity down the middle, which we assume to be filled with el xtrolyte. The cross-sectional geometry is shown in Fig. 2 where c - d = a - b. The
FIG. 2. Geometry
for cylindrical
models.
FREE
ENERGY
OF NUCLEIC
233
-4CIDP
region 3 is considered filled with protein, nucleic acid, em., and has a dielectric constam Di . Electrolyte occupies regions 1 and 5 but the ionic centers are excluded from regions 2 and 4 because of t,he finite size of t,he ions. T,et there be a uniform surface charge density g = z'42d at r = b (i.e., on the outside surface of t’he cylinder). We proceed here from -?‘I)* = KQ )
v*q3 = 0,
v*+* = 0,
V2& = ii*+6
v*+4 = 0,
essentially as in sec. ITIA above, except that
where In(z) is a modified Bessel function of t,he first kind, and that there are no\v eight constants to evaluate. We find
n DiIo(dI[ I Ko(KU)
l1.
=
fT*!
[
KU&(KU)
’
“I g
DiIo(Kd)
D~K~(Kcx)
DKdIl(Kd)
+
DKoK~(KO)
D.
c
D
d
1(1l‘i ___~ 1 b
+ 2 ln - + It1 -
D~dI,(td)
Iii
UC
ll
bd
c
b
+ -- ln -
+ 111-
c
This is always less than II’ in Eq. (9), as expected, because of the additional elect,rolyte in the “core” of the cylinder. D. If t’he uniform surface charge is on the “inside” rather than the “outside” of the cylinder, that is at T = c with q = 2’~/2nc, the work is Iobd IT7 = x’“$ D
[ KdI&d) r DJohd) L ~,&l(Kd)
+
111 ”
d
DiKo(d I[ DKuK~(KU)
D.
DiKoha)
’
+~ln~+lnb
DKCZK~(KU)
+
ac
n’” In w
’
1 (12) 1 c
b “’
2
E. We suppose here that there is a uniform volume charge densit,y p in region 3 of Fig. 2, with p = x’q’vr(b2 - 3). We find in this case z
)
+ B
[
hi b - $ In c (13)
A(1 - $) - &(1 - $1)
234
TERRELL L. HILL
where A=&+ln;+
&(td -B Ka&(KU)
I
lnb
D&&a) hU&(KU)
+$n;+
1
B=
Eqs. (II),
(12), and (13) all reduce to
w= i2e21 D
(14)
in the special case c + b (that is, region 3 has zero thickness). IV. ADENOSINE TRIPHOSPHATE
(ATP)
By choosing (2) an effective dielectric constant De of 49.6 for all interactions in ADPT3 (adenosine diphosphate), the total electrostatic free energy associated with interactions between negal,ively charged oxygen atoms attached to different phosphorus atoms in .4DP-3 can be made to agree with the same quantity found earlier in :I. more refined treatment of pyrophosphoric acid (6). We can have considerable confidence in the pyrophosphoric acid calculations because ofj the excellent agreement with experiment (2, 6). Using DE = 49.6, as ex:plained earlier (2), we find in the reaction ATP-4 + Hz0 + ADP-2 + HP04”
(15)
that W(ATP-4)
= 7.12 kcal./mole
W(ADPm2) = 1.67 kcal./mole AW,,
= 1.67 - 7.12 = -5.45 kcal./mole
(16)
where W includes all charge interactions except between P+ and Obonded to each other [these interactions cancel in Eq. (15)]. Also, for the
FREE
ENERGY
OF
NUCLEIC
ACIDS
235
+ HPOd-2
(17)
ADP reaction ADP-3 + Hz0 -+ AMP-I we have W(ADP-3)
= 3.16 kcal./mole,
AWmp = -3.16
W(AMP’)
kcal./mole
= 0 (18)
The values in Eqs. (16) and (18) are the ones referred to in the last paragraph of sec. II, Ref. (2). The results in Eqs. (16) and (18) are based ultimately on the experimental dissociation constants of pyrophosphoric acid at zero ionic strength. Although salt effects were mentioned in Ref. (2), no quantitative correction was attempted. We try to estimate this correction here. From a model of tripolyphosphate (P,O,O , the “business end” of ATP), the semimajor and semiminor axes of the ellipsoid of revolution roughly filled by the molecule are estimated as 4.5 and 2.25 A., respectively. If we assume that the net charge on ATP of -4 is smeared uniformly over the surface of this ellip,soid, a treatment for an ellipsoid such as that in sec. IL4 for a sphere could be used to estimate the salt correction. Smearing a small number of charges is a serious approximation, but it should be much less serious for the salt correction,, which is all we need, than for the absolute value of W. Actually, the treatment of an ellipsoid is relatively complicated mathematically (4), so we use a spherical model with the same surface area (and surface charge density). The radius of t,he equivalent sphere is b = 2.95 A. The salt correction factor in Eq. (1) is l--
Kb 1 +
KU
(19)
We choose, somewhat arbitrarily, a = 3.95 A. (the correction given by Eq. (19) is quite insensitive to the exact choice of a - b). with the choice I/K = 7.79 A. (0.15 &f NaCl, 37”C.), the electrolyte correction factor for ATP is 0.75 so t’hat W(ATPw4) = 7.12 X .75 = 5.3 kcal./mole An analogous calculation gives
for ADP, using b = 2.60 A. and a = 3.60 A.,
236
TERRELL L. HILL
W(ADPm2) = 1.67 X .77 = 1.3 kcal./mole W(ADPe3)
= 3.16 X .77 = 2.4 kcal./molc:
and hence AWARD = 1.3 - 5.3 = -4.0
AW
ADP
=
kcal./mole
- 2.4 kcal./mole
(20) (21)
in 0.15 M NaCl. The value of 4.0 kcal./mole in Eq. (20) thus accounts for two thirds of the current best est.imate of the free ent rgy of hydrolysis of ATP; -AF” = 6 kcal./mole (7). Although we have already pointed out t’hat the abs Jute value of WATp for Eq. (15) calculated from Eq. (1) is unreliable, thl: extent of the unreliability may be of incidental interest. We find from Eq. (1) -W(z=4, b=2.95,a =3.95) AWATP= W(z=2,b=2.60,~=3.60) (22) = 2.7 - 9.1 = -6.4 kc: I./mole compared to the calculation on the basis of discrete charg YS,Eq. (20). V. DEOXYRIBONUCLEIC
ACID (DNA)
The structure of ribonucleic acid (RNA) is unknown as If this writing, so we confine our remarks here primarily to DNA. Eq. ( j) should be a fairly good approximation when applied to DNA, which las two singly charged phosphate groups (at pH 7) on the outside of t double helix for every 3.4 A. along the axis of the double helix (8). A~ually, Eq. (9) will probably give an upper limit to W because (a) we art smearing the surface charge in this equation and (b) some electrolyte can penetrate inside the helix (see below). aside from applications to ion binding and titration CLrves of DNA, Eq. (9) is of some interest in connection with the free enerj’y of synthesis (nucleotide polymerization) of DNA. The sign of the free energy of DNA synthesis has an important bearing on the mechanism of the synthesis. For example, if the free energy turns out to be positive, a self-duplicating mechanism without coupling to an outside free energy source is not possible. Assuming RNA is involved in protein synthesis, t:le free energy of synthesis of RNA has a bearing on the mechanism of both RNA and protein synthesis. For example, if this free energy is positive, RNA (as above for DNA) cannot be synthesized by a self-duplicating mechanism without outside coupling; and, if the free energy is sufficiently positive,
FREE
EXERGY
OF
NUCLEIC
APITS
23i
it, is possil)lc that depolymerization of RXA would be coupled with prot,c:ili synt’hrsis (i.e., RNA would be the free-w~crgy donor for protein synt,hesis). The replacement, of the depolymerizecl RN.\ \\ould the11 :monntj for I 11~reported link lwt~wecn the trlrtw\w of RX.\ and prot,ctill (!j). The olwtrostatic free energies of 11X,1 and RX21 are positive due to the repulsion of tlegabively charged phosphate groups, just as in h’l’P escept thatJ there are more phosphate groups in 1he nucleic acids but. these are fart,her apart. Thus electrostatic effects (*ontribute to a positive free energy of synthesis of Dr\‘A and RKA. The analog here (for I>K&Z) of I?q. (15) is the depolymerization reaction (schematic)
where R = ribose and B = base. XVic can estimate (see belon-) the electrostatic contribution to the free energy of t,his process for DKA, lvhosc structure is known, and compare it with Eq. (20) for ATP. Othet cont,ribut,ions to the free-energy change in Eq. (23) for I)SL2 could also be estimated: e.g., net hydrogen bond energies, van der \Vaals’ interartions (especially between bases), gain of t~ranslational and rotational (internal and ext,ernal) free energy of t.hc nucleotide split off, split,ting of the phosphate ester bond, etc. But there remains a presumably large cont,ribut>ion which is very difficult to assess, that of the solvent molecules and ions. Because of t’his difficulty standing in the way of an estimat,e of the complete AF”, wc confine ourselves here t#othe electrostatic contribution only. Let AlV be the electrostatic contribut#ion to AF” in Eq. (23). ‘I’hc elect,rostatic work necessary to bring a singly charged nucleotide, P-It-B, from infinity up to the end of a long DXA cylinder is then -AlV/2. Now the net effect of adding one P-l--R--B to (PI-R-B), , where n is very large, is to increase t’he central portion of the molecule (where Eq. (9) is applicable) by one unit’, leaving the ends unchanged. Therefore if w-e denote W in Eq. (9) by W(Z), we have -- AW = 2
- W(Z) = 11’ f 0
238
TERRELL L. HILL
or
Aw= e2$ We take (8) b = 11 A., a = 12 A., find from Eq. (24)
+ In;] l/~
(24)
= 7.79 A., and z’ = 2/3.4A., and
AW = -3.3 kcal./mole
(25)
which is to be compared with Eq. (20) for ATP. As aln ady mentioned, this computation probably gives an upper limit’ to -AW. But the electrostatic contribution to the free energy of synthl sis of DNA is definitely significant. ACKNOWLEDGMENTS The author is indebted to Drs. Sidney Bernhard, Dunitz for helpful discussions.
Alexandr : Rich, and Jack
SUMMARY
Equations are derived for the electrostatic free en#:rgy of certain spherical and especially cylindrical models of charged Inacromolecules immersed in an electrolyte solution. The equations shou d be useful, for example, in the interpretation of ion binding and titr cbion curves of proteins, nucleic acids, etc. Another application, which k discussed here, is the calculation of the electrostatic contribution to th’ free energy of hydrolysis of ATP and synthesis of DNA. The possible relation of the present work to the mechanisms of nucleic acid and prot :in syntheses is pointed out. REFERENCES 1. See, for example, COHN, E. J., AND EDSALL, J. T., “Proteins, Amino Acids and Peptides.” Reinhold Publishing Corp., New York, N. Y., L943. 1 Electrolyte can penetrate inside DNA, but in an unsymme trical way. However, the order of magnitude of the appropriate correction can be obtained from the symmetrical model which leads to Eq. (11). From atomic mc iels, we estimate that a single average nucleotide residue of DNA occupies about 250 cu. A. The volume of 3.4 A. of a cylinder of radius 11 A. is 1292 cu. A. Hence we take the outer nucleotide occupied shell (b > T > c) as having a volume 530 cu. A. and the inside of the cylinder (r < c) as having a volume 792 cu. A. ThL gives a = 12 A., b = 11 A., c = 8.6 A., d = 7.6 A., and a correction factor [corn Bare Eqs. (9) and (ll)] of 0.95, i.e., only a 5% correction owing to the penetration of electrolyte inside the cylinder.
FREE
ENERGY
OF
NUCLEIC
ACIDS
239
2. HILL, T. L., AND MORALES, M. F., J. Am. Chem. Sot. 73, 1656 (1951). 3. Fuoss, R. M. (with KATCHALSKY, A., AND LIFSON, S.), PTOC. Natl. Acad. Sri. u. s. 37,579 (1951). 4. LINDERSTROM-LANG, K., Compt. rend. true. lab. Car&berg 28, 281 (1953). 5. KIRKWOOD, J. G., AND WESTHEIJIER, F., J. Chem. Phys. 8, 506, 513 (1938). 6. HILL, T. L., J. Chem. Phys. 12, 147 (1944). 7. LEVINTOW, L., AND MEISTER, A. (with MORALES, RI. F.), J. Biol. Chum. 209, 265 (1954); PODOLSKY, R. J., AND MORALES, M., J. Biol. Chem., in press. S. WATSON, J. D., AND CRICK, F. H. C., Nature 171, 737 (1953). 9. See, for example, DAVIDSON, .J. N., “Biochemistry of the Nucleic Acids,” p. 163. Methuen and Co., London, 1953.