Polyelectrolyte Properties of Biopolymers: Conductivity and Secondary Structure of Polyriboadenylic Acid and Its Salts in Solutions 1. A. KUZNETSOV, 0. V. VORONTSOVA, and A. G. KOZLOV Chemical Department, M. V. Lomonosov Moscow State University, Moscow 11 7234, USSR
SYNOPSIS
Polyriboadenylates of alkali metals were obtained from (1)K+-poly(A) (salts I ) and ( 2 ) H+-poly(A) (salts 11) by the ion-exchange method. The conductivity of these salts as well as of H+-poly(A)were studied. Salts I and I1 of the same counterion were shown to have significantly different conductivity coefficients ( f ) and polyion conductances (A:). The charge density parameter ( E ) was 1.3 and 2.5, respectively, with A: equal to 44 and 83 ohm-’ cm2 mole-’ for poly( A ) - I and poly( A) -11 salts, respectively. This is credited to the difference in the conformations of corresponding polyions. The linear dependence of equivalent conductivity on the square root of polymer concentration ( Kohlrausch coordinates), earlier obtained for DNA, is also satisfied for the studied polynucleotides. A comparison of the slopes of straight lines in Kohlrausch coordinates for poly ( A ) , simple electrolytes, and for earlier studied polyribouridylic acid salts lends credence to the concepts, developed by a number of authors, that DNA can act as a “buffer” against the ion-ion interaction in concentrated electrolyte solutions. Using the approximation that the polyion conductance is independent of the counterion nature, parameter f (agreeing in this case with Eisenberg parameter 4 ) has been shown to decrease as the polynucleotide concentration is increased; the decrease is caused by the relaxation effect. The transference numbers of counterions, which have negative values in poly ( A ) -11 solutions, grow with the increase in polymer concentration; the higher the l , the more apparent is this increase. This is explained by the increase in the fraction of conductivity along the polyion chains (“surface” conductivity) with the growth of polyelectrolyte concentration.
INTRODUCTION The study of polyelectrolyte properties of synthetic polynucleotides, which are simple models of nucleic acids, is of great interest for molecular biology. The behavior of DNA in living organisms is largely conditioned by the fact that DNA is a polyelectrolyte. This is why increased interest is being shown in the past few years in the polyelectrolytic properties of DNA.’-20 In a number of modern theories of polyelectrolytes it has been ascertained that the conductance of a polyelectrolyte is definitely related to its charge
Biopolyrners. Vol. 31,65-76 (1991) 0 1991 John Wiley & Sons, Inc.
CCC 0006-3525/91/010065-12$04.00
density. This enables conductivity measurements to be used for studying the structure of polyion in solution. According to the present-day concepts
where A is the equivalent conductance of the polyelectrolyte in salt-free solution, A, is the polyion conductance, and A: is the limiting conductance of the counterion. The dependence of X, on concentration is expressed as
where K is the Debye-Huckel screening parameter and r is the radius of the cylindrical p o l y i ~ n . ~ ~ - ~ ~ 65
66
KUZNETSOV, VORONTSOVA, AND KOZLOV
The coefficient f appearing in eq. ( 1 ) is determined by the relationships
DC
f = -
D:
(3)
where D, and D: are the diffusion coefficients of counterions in the solution of polyelectrolyte and pure solvent, r e ~ p e c t i v e l y .I~t ~can be seen that f reflects the variation in the mobility of counterions in a polyelectrolyte solution compared to the mobility in a low molecular salt solution. Neglecting the mobility of condensed counterions, Manning derived a n equation connecting the parameters f and [ for the case 5 > 1 in salt-free polyelectrolyte solutions:
Here, e is the proton charge, k is the Boltzmann constant, T is the temperature, b is the distance between the ionogenic groups along the polyion chain, and 2, is the valence of the counterion. Good agreement of polyelectrolyte conductance with the Manning theory has been obtained in a number of ~ o r k s . ' ~ Nevertheless, ,~~ often the theoretical dependence of A on concentration ( C , ) does not correlate with the experimental The possible reason is that the model of a free-draining cylinder does not serve as good approximation for all polyelectrolytes. It has been shown that the linear dependence of A on 12;'' holds for salt-free DNA and poly ( U ) solutions. This enables the limiting conductance of polynucleotides A h p to be estimated. T h e values of ALP obtained by extrapolation are proportional to the mobilities of counterions. This made it possible t o estimate quite reasonable values of $, for the enumerated polymer^.^^^^^ However, the shape of the dependence of conductance on concentration does not agree with the Manning theory. T h e present work studies the dependence of conductance of poly ( A ) on the concentration and nature of the counterion. I t also compares the slopes of concentration dependencies of conductance for poly(A) ( t h e present work) and poly(U) (data taken from Ref. 31 ) , and for low molecular electrolytes. The polyelectrolyte properties of poly( A ) and poly ( U ) have been found to resemble the earlier studied properties of DNA in salt-free solutions.
EXPERIMENTAL A potassium salt of polyriboadenylic acid supplied by Reanal of Hungary was used in the experiment. T h e isoionic solution of poly ( A ) was obtained by passing a n aqueous solution of K+-poly( A ) through a column packed with a mixed bed of the ion exchangers. T h e procedure of obtaining the isoionic solution and the solution's characteristics, as well as the spectrophotometric technique of determining the concentration of poly ( A ) , are discussed in detail in Ref. 32. T h e concentration of isoionic solution was taken as the arithmetic mean of concentrations determined spectrophotometrically and by potentiometric titration of H+-poly(A ) solution with NaOH.32In isoionic solutions of poly ( A ) the double helix is more stable than in the presence of salts because the favorable interactions of the positively charged adenylic base of one chain and of the negatively charged phosphate group of the other chain are not screened. About 20% poly ( A ) is present in the single-stranded form.32 Obtaining Solutions of Ribonucleates of Alkali Metals
T h e salts of alkali metals of poly ( A ) were obtained by filtering through a column packed with cation exchangers in the corresponding M+-form (here, M + stands for Li+, Na', Rbt, K + , or Cs') (1) of the starting solution of K+-poly(A ) (salts I ) and ( 2 ) of the isoionic solution of H +-poly( A ) (salts 11). The completeness of transfer of K +-poly( A ) into a corresponding M + form was controlled by the flame photometry method using the FLAPHO ( G D R ) emission flame photometer. At a sufficiently slow rate of filtration, the filtrate was found to be free of K + ions. The completeness of transfer of H +-poly ( A ) into the M + form (salts 11) was controlled by potentiometric titration of solutions of the obtained salts with a 0.09M NaOH solution. Measurement of Conductance
The resistance of solutions was measured in a flow cell (platinized platinum electrodes) with a constant 0.049 cm-' a t 25°C using the LKB (Sweden) conductometer, the alternating current frequency being 2 kHz. CD Measurements
Ultraviolet CD measurements were made in 1.0-mm cells, using a Jasco J-500C ( J a p a n ) recording spec-
POLYELECTROLYTE PROPERTIES OF BIOPOLYMERS
tropolarimeter. The instrument was calibrated with a n aqueous solution of d-10-camphor sulfonic acid. Spectrally pure salts and deionized water were used in the experiments.
67
any electrolyte for which the shape of the function F ( C ) is not known. It has been shown that for isoionic solutions of DNA and poly ( U ) and their salts the linear dependence between A and C:l2 is satisfied, i.e., F ( C ) = - A * C;", where A is a In the presRESULTS AND DISCUSSION ent work we have found that the A - C;I2 dependence is also linear for H+-poly(A) and its salts. Usually for calculating the equivalent conductance Figure 1( a and b ) shows the equivalent conductance of a solution ( A ) , use is made of K~ of pure solvent. dependence in Kohlrausch's coordinates for salts I For DNA and polynucleotides, K' is determined by and 11, respectively. This dependence is defined by extrapolating the linear portion of the K - C, dethe expression pendence to zero concentration of the ~ o l y m e r . ~ ~ - ~ l Recently, using a n analogous approach, Vink33g34has shown that A is the sum of limiting conductance A' and some function of concentration F ( C) : From Figure 1 it is seen that the conductance of polyadenylates obtained by the first method is much K - Kg higher than for salts 11, and its dependence on the A== ha F ( C ) nature of counterion in salts I is stronger than in c salts 11. Table I lists the values of limiting equivalent Proceeding from eq. ( 6 ) , the specific conductance conductance A; obtained by extrapolating eq. ( 9 ) of a solution is expressed as to zero concentration of the polymer for polyriboadenylates I and I1 (Figure l a and b ) , and also for polyribouridylates (reproduced from Ref. 31 ) . For all salts, the values of A; are in good agreement It is clear that when C, tends to zero, the term with A t , which have been determined as slopes of C - F ( C ) tends to zero faster than 11'. C. Thus, nestraight portions of the K - C, dependence for every glecting the last term of eq. ( 7 ) we obtain salt a s per eq. ( 8 ) . T h e K - C, dependence for poly(A)-I and -11 salts as well as for poly(U) are shown respectively in Figure 2 ( a , b, and c ) . The Vink has proposed a method of estimating A' for coincidence of A; and A t shows that our way28-31
+
Figure 1. Dependence of equivalent conductance ( A ) of poly ( A ) -I ( a ) and poly ( A ) -11 ( b ) on the nature of counterion and polymer concentration at 298 K.
Average
a
27 25 30 28
83
78 75
77 80 80
66 46 65 76
61 47 62 56 50 61
26 28 32 27 28f 2
83
78 76
78 80 77
66 46 62 68
58 45 57 57 47 56
Poly(A)-I
A8
178 176 170 179 176 f 3
H+-Poly(A)"
77f 2 82 83f 1
78t 1 79 76
61f 8 76 80 78
64 44 61 73
60 47 63 58 49 59 55+ 5
A%
2.78 2.83 2.70 2.95 2.82 f 0.07
1.97 f 0.13 1.60
1.67 f 0.13 2.06 1.88
1.85 0.16 1.62 1.86 1.52
*
2.15 1.84 1.64 1.75
1.50 1.64 1.70 2.20 1.60 2.13 1.79 f 0.24
PK
57
58 62 60 45 58
65 58 43 56 55
39 38 46 54
46 36 47 47
AE
54 f 6 56 60 59 45 57 55 f 5 54 56 f 2
44 2 6 65 57 41 53 52
38 39 47 53
43 2 4
50 34 47 45
A%
57
54 57 58 45 58
64 56 40 55 52
39 38 45 56
46 33 42 44
Poly(A)-I1
A8
Values of A: have been computed with consideration for concentration of single-stranded structures of H+-poly(A).
Average
Rb'
Average
cs+
K'
Average
Na+
Average
Li+
Ion
+
1.88 f 0.22 1.90 2.06 1.90 1.86 1.50 1.84 0.13 2.06
*
1.73 0.08 1.61 2.15 2.13 1.91 1.60
1.71 1.60 1.85 1.78
1.85 f 0.15
1.60 1.88 2.13 1.80
PK
89 73 83+ 4 89 106 82 95 118 86 113 112 100 f 11 106 100 89 116 87 101 f 10
90 75
108 104 92 117 88
91 108 84 96 119 91 115 110
84 78 87 84
72+ 7
72 79 61 79
A%
84 81 87 86
74 79 62 80
A$
104 88 119 89
111
91 108 80 91 121 89 99 110
89 76
82 77 90 84
2.30 1.81 1.60
64 62 82
+
*
1.73 1.81 1.85 0.21 1.63 1.47 1.88 1.94 1.90 1.86 1.51 1.60 1.72 k 0.17 1.64 1.77 1.80 1.67 1.72 1.72 0.05
1.55 1.86 2.33 2.17
1.85 f 0.22
1.71
PK
POlY(U) 74
A;
Table I Limiting Conductances (ohm-' cm2 mole-') Determined from Kohlrausch's (A:), Vink's ( A t ) , and Ostwald's (A:) Dependencies [eqs. (9), (€9,and (14), respectively] and pK Values Computed by eq. (14) for Poly(A) Salts Obtained by the First and Second Methods, and for Poly(U) at 298 K
c
0
r
x
U
z
P
0 c "P
rn
5
0
z
c 0
"C
;
3
s
x
5 2 OD
POLYELECTROLYTE PROPERTIES OF BIOPOLY MERS
Figure 2.
The
K
vs
69
C, (298 K ) . (a) poly (U), (b) poly (AbI, (c) poly (A)-II
of' taking account of the contribution of solvent is right. Significant discrepancy between the conductances of' poly ( A ) -I and -11 salts is obviously due to the diff'erence in the conformations of these two samples. In a neutral medium, poly ( A ) is known to exist in the single-stranded form with some stacking.:jS O n making the solution slightly acidic, the transit ion to double-stranded conformation takes place --first a semiprotonated and then a less compact, c,ompletely protonated form appears. In the literature, these two structures are referred to as B and A forms of poly ( A ) , respectively.36337 Figure 3 shows the potentiometric titration curve for the Ka . -poly ( A ) -11 solution. It can be seen that the equivalence point is observed only upon adding 0.40 mole NaOH to one mole of phosphorus. It has been shown that in an isoionic solution of poly(A) double-stranded structures coexist with singlestranded ones. The latter contain about 20% of all nucleotides of poly(A) .32 Taking this into consideration, i t can be assumed that in the poly( A ) -11 salts atiout 80% nucleotides are present in the form of double-stranded semiprotonated structure, that is, as structure of t,ype B (after Finch et al.36 and Adler et al.'") . This is also evidenced by the CD spectra of Figure 4,which reveal that the secondary structure of N a + poly ( A ) -11 in a solution is more pronounced than the structure of Na+-poly(A ) -I and H + - p o l y ( A ) . T h e double-stranded structure of poly ( A ) in the A and R (after Ref. 3 7 ) forms are known to differ in
neighboring bases. In the B form they are disposed on alternate bases, which ensures a minimum repulsion between the positively charged bases. The appearance of protons on the neighboring bases causes the repulsive forces to increase abruptly. That is why the B form is more compact than the A form. Upon filtering an isoionic solution of H +-poly( A ) through a M+-form cation exchanger the amount of unexchanged protons equals O.4Cp (Figure 3 ) . This
.60
Figure 3. Potentiometric titration curve [ N a + poly(A)-11, C, = 2.2 X 10-'M, titrated with 0.09M NaOH] . The r is the number of moles of NaOH added to 1 mole of nucleic phosphorus a t 298 K.
70
KUZNETSOV, VORONTSOVA, AND KOZLOV
I
no
I
240
2a
280
320
320
Figure 4. Ultraviolet CD spectra of poly(A) ( 1 ) Na'p o b ( A ) - I , ( 2 ) H + - p o l y ( A ) , and ( 3 ) Na+-poly(A)-11.
amount of protons is present on the semiprotonated double-stranded structure of poly ( A ) . These are the protons that are more strongly bonded: pK values of adenine equal, respectively, 3.7, 5.9, and 7.1 for single-stranded, double-stranded fully protonated, and double-stranded semiprotonated structure^.^^ Structures of this type have been studied for poly (dC ) . It has been noted that in poly (dC ) a double-stranded poly (dC : d C + ) is formed upon protonating half of the bases. This "acidic" form is stable at pH 7.0 and at 20°C.38 Table I1 lists the average values (of all the measurements for every salt) of the slopes of A - C;/' dependencies for polyadenylates I and I1 and for polyuridylates. Therein, for the purpose of comparison, are given the slopes of chlorides and sulfates of these very cations. These slopes have been calculated by formula ( 10 1, 39
where t and T have the same meaning as in eq. ( 5 ) , 77 is the viscosity of water, and 2, and Zz are the cation and anion charges. The values of limiting ion mobilities have been reproduced from Ref. 39. The first term of formula ( 1 0 ) derived in the theory of conductance of low-molecular electrolytes takes account of the electrophoretic retardation of ions. The second term allows for the relaxation effect related to retardation of a moving ion due to the asymmetry of the ionic atmosphere, which (asymmetry) appears with the application of an electric field. From Table I1 it is evident that the absolute values of slopes for polynucleotides are more than for low molecular electrolytes. However, with an increase in the limiting conductivity of salts, a linear increase in the values of A is observed both for the former and the latter:
A
=
140
+ 0.81.A"
+ 0.74 - A" A = 146 + 0.46-A' A
=
156
( r = 0.99) for poly(U) ( r = 0.99)
for poly( A ) -I
( r = 0.99) for poly(A)-I1
where r is the correlation coefficient. It can be seen that for the studied polynucleotides that the constants A , depending on the nature of counterion, are expressed as
Here, a and ,&-constants for the given polyelectrolyte-are proportional to the electrophoretic and
Table I1 Constants A Appearing in eq. (9) (ohm-' cm2 L'/') for Poly(U) (after Ref. 31), Poly(A)-I, and Poly(A)-11, and Also for Chlorides and Sulfates of Alkali Metals at 298 K Cation
POlY(U)
Poly(A)- I
Li+ Na+ K+
187 +- 13 199 f 17 219 -t 19 223 +- 17 -
198 f 201 f 212 f 213 f 220 f
cs+ Rb+
12
14 12 19 19
Poly(A)-11
c1-
soq-
165 f 12 167 f 11 170 f 15 173 f 13 171 f 13
87 89 95 95 95
177 181 190 192 192
POLYELECTROLYTE PROPERTIES OF BIOPOLYMERS
relaxation effects, respectively, and A’ is the limiting conductance of the given salt of polynucleotide. It is known that the stronger the association in the electrolyte solution the greater the values of slopes in the Kohlrausch coordinate^.^' Since strong bonding of counterions is noticed in polyelectrolyte solutions, one can expect that the values of A would turn out to be much more than those for simple electrolytes. Inspite of this, the values of slopes of the A - C;/’ dependencies for polyuridylates and polyadenylates I and I1 were found t o be comparable to those for simple electrolytes. Possibly, this is partly due to the contribution of surface conductance (see below) whose effect grows with increase in the concentration of polynucleotide. On the other hand, such a behavior of the polynucleotides studied by us correlates with the strongly charged polyions exerting a “buffer” effect against the interactions between small ion^.',^^^^ Estimating Charge Density Parameter for Poly(A)-l and Poly(A)-ll
As for DNA salts” and poly ( U ),31 the limiting conductances of polyadenylates I and I1 were found to be proportional to the conductances of counterions (Figure 5 ) . This makes it possible to estimate the values o f f , A,, [, and b from eqs. ( l ) , (41, and ( 5 ) . For poly ( A ) -I and -11 the values o f f , A,, and b are given in Table 111; therein, the values of these parameters for poly(U) are reproduced from Ref. 31.
0
20
7I
For the purpose of comparison, this table contains the corresponding values of b and [ obtained by other methods in Ref. 21. The value [ = 1.6 ( a s per the data of Ref. 21) for poly(U) seems to be overestimated if we take into consideration numerous proofs of the absence of a n ordered structure in poly ( U ) a t room t e m p e r a t ~ r eArcher .~~~~ and ~ his colleagues43 have found [ = 1.05 from the measurements of activity of sodium ions in a Na+-poly ( U ) solution. This value correlates with that ([ = 1.13) obtained by the conductometric method.31 Table I11 lists the values of b-average distance between the projections of phosphates on the spiral axis-computed from eq. ( 5 ) for the obtained values of [. A comparison of values of b for poly ( U ) and poly ( A ) -I and I1 permits us to conclude that in saltfree solutions the conformation of poly(A) is far from the maximum stretched one for which b = 6.8 A, unlike poly ( U ). This is in agreement with the data on small-angle scattering of x-rays, according to which the distance between the phosphates in single-stranded poly( A ) equals about 3.5 A.44 According to our results obtained from the measurements of conductance, the poly ( A )-I and poly ( U ) salts are in the single-stranded conformation, but have different charge density parameters. Manning, in one of his works, has uttered a n opinion according to which the value of b for singlestranded polynucleotides is independent of temperature and the nucleotide composition, and lies between 3 and 4 A. T h a t is, it is not related to the
40
60
A,,
c c.,*-oIL-’
80
100
Figure 5. Limiting equivalent conductance of poly ( A ) -I ( a ) and poly ( A ) -11 ( b ) salts vs limiting conductance of counterion at 298 K.
72
KUZNETSOV, VORONTSOVA, AND KOZLOV
Table I11 Parameters f , [, A:, and b for Poly(A)-I and Poly(A)-I1 and Also for Poly(U) (298 K), Obtained in the Present Work and in Ref. 31 for Extremely Dilute Solutions, and the Values o f t and b after Ref. 21
Obtained from the data on conductance From the data of Ref. 21
A:,
Polynucleotide
f
b, A
i
ohm-’ cm2 mol-’
POlYiU) Poly(A)-I PolyiA) -11 POlY(U) PMA) Native DNA Denatured DNA
0.77 0.66 0.35
6.3 5.4 2.9 4.5 3.1 1.7 4.0
1.1 1.3 2.5 1.6 2.3 4.2 1.8
56 44 83
differences in the stacking of bases in different polynucleotide~.~’ He has based his views on the results of Archer and his colleague^,^^ who from the potentiometric measurements have obtained similar values of the bonding degree of N a + ion for poly(A) and poly ( U ) . In a much later work, Manning has reported different values of [ for poly(A) and poly ( U ) (see Table I11 ) .‘l Our results suggest that the nature of the polynucleotide strand largely determines the polyion charge density. Probably, the disagreement with the results of Ref. 43 is due to transport methods (conductance being one of them) that are more sensitive t o the type of polyelectrolyte skeleton than the equilibrium method^.^^^^' According to the Manning theory, A, depends on the nature of counterion.’* However, the constancy of A, is a reasonable approximation in obtaining f from eq. ( 1).47 For the salts of polymetacrylic and polyacrylic acids, Eisenberg has found that 4 equals
and remains constant for different pairs of M and N ( A M P and ANp are equivalent conductances of polymer salts, and A h and A; are limiting conductances of the counterions M and N.4s It has been shown that for concentrated solutions of polyacrylic acid, (#) depends on the nature of c~unterion.~’ Nelson and Ander5’ have found that (#) increases linearly with the growth of C, in gum arabic salt solutions in the concentration range of 0-0.01M. For carrageenanes, (#) decreases linearly with the growth of C,. T h e authors have attributed these results to different charge densities of polyelectrolytes. From our data it follows that (#)’ = (ALP - AtP)/(1 ; - A t ) is independent of the nature of the counterion. Also, it can be shown that a t a constant concentration of the polyelectrolyte (#) is constant for all pairs of ions. Having denoted AMP by
A iand ANP by A,, and expressing their dependences on C, by eqs. (9), ( 11) , and ( lOa), we write (#) Al-Al _(A~-AlC”2)-(A,”-A,C”2)
(#)=--
xp - A,”
4=
(A:
-
xp - A,” A:)
( CY (#)=(#)”-
(A, - A,)C1” - A,”
-
xp
+ pAp)C”’
-
( (Y
+
xp - A,”
( # ) = ( # ) ”@(A: -
-
A,”)C”‘
xp - A,” i.e.,
Thus, it can be seen that 4,and hence f for polynucleotides, are independent of the nature of counterions, but vary with polymer concentration; this dependence on concentration is entirely caused by the relaxation effect. Equation ( 1 2 ) is consistent with the dependence of f on concentration for carrageenanes.” Carrageenanes have higher charge density than gum arabic and are close, in this respect, t o polynucleotides. From the analysis of eq. ( 1 2 ) it follows that the charge density decreases with dilution (provided A, is independent of the nature of counterions). This dependence of [ on C, is not considered in the Manning theory. Estimating the Transference Numbers in Solutions
of Polynucleotides Assuming that only two types of ions exist in a saltfree polyelectrolyte solution and that the t, part of electricity is transferred by counterions and the tp
POLYELECTROLYTE PROPERTIES OF BIOPOLYMERS
part by polyions ( t, and t, are the transference numbers and t, t, = 1) , we write eq. ( 1) as
73
ative transference numbers are met with in low molecular electrolyte solutions, and this points to strong bonding of, say, complex formation. In highly charged polyelectrolyte solutions, polyion transference numbers (more than 1.0) are quite often observed; this is indicative of strong bonding of counterions by the polyion. This manifests itself in that part of the counterions’ (cations) travel toward the anode together with the polyion. The values of A, for some polymer concentrations are listed in Table IV. It can be seen that X, grows with the increase in C , . An analogous picture was noticed in the isoionic solution of DNA.30 The authors have attributed these results to the increase in “surface” conductance along the chains of polyions. As is evident from Table IV, the increase in X, with polymer concentration is particularly pronounced for polynucleotides having high charge density. Apparently, this is due to the increased rigidity of strongly
+
-
Whence t, = ( A - f A:) / ( f A ) . Substituting into this equation the values of f and A calculated from eqs. ( 12 ) and ( 9 1, respectively [ t h e values of A used in the calculation A were taken from Table I1 and the values of P were extracted by a n equation of type ( 10a ) 1, we find the values of t, for several polymer concentrations. Table IV lists the values of t, for poly ( A ) -I and -11 and poly ( U ) a t different concentrations. From this table it can be seen that, even in the poly( U ) solutions, a large part of electricity is transferred by the polyion, and the higher its charge density the more the value of t,. For the poly ( A ) -11 salts, negative transference numbers of counterions are observed. It must be noted that neg-
Table IV Dependence of Charge Density Parameter ( t ) ,Conductance (A), Polyion Transference Number (&), and Counterion Conductance (A,) on Polymer Concentration at 298 K (A and X, are in ohm-’ cm2 mole-’)
Li+-poly(U ) 50 30 10 5 2 1 0
1.20 1.18 1.16 1.15 1.14 1.14 1.13
59.1 61.8 65.1 67.8 69.3 70.1 71.8
0.71 0.72 0.74 0.75 0.76 0.76 0.76
17.1 17.3 16.9 17.0 16.6 16.8 17.2
69.0 72.1 76.7 78.5 80.2 81.0 83.1
Li+-poly(A)-I 50 30 13 5 2 1 0
1.37 1.36 1.34 1.33 1.33 1.33 1.32
41.5 44.2 48.8 50.6 52.2 53.0 55.0
2.57 2.57 2.57 2.50 2.50 2.50 2.50
31.3 34.0 37.8 39.3 40.7 41.4 43.0
0.63 0.69 0.74 0.77 0.77 0.79 0.81
1.7 1.8 1.8 1.9 1.9 1.9 2.0
24.8 24.5 24.5 25.1 24.8 24.3 25.8
0.64 0.66 0.68 0.68 0.69 0.70 0.69
85.5 89.0 94.1 96.1 97.9 98.8 100.5
15.4 13.7 12.7 11.6 12.0 11.1 10.5
46.8 50.0 54.6 56.5 58.2 59.0 61.0
-21.9 -27.2 -30.2 -35.4 -36.6 -37.3 -43.0
23.9 22.0 20.8 19.8 20.4 19.5 18.9
0.49 0.56 0.62 0.65 0.65 0.67 0.69
62.9 66.3 71.3 73.2 75.0 75.9 78.0
Na+-poly(A)-I1 32.2 34.8 38.7 40.3 41.6 42.3 44.2
1.4 1.5 1.6 1.6 1.7 1.7 1.7
0.51 0.53 0.55 0.55 0.56 0.57 0.57
41.9 41.8 42.3 43.2 43.1 42.5 43.2
K+-poly(A) -I
Na+-poly(A)-I
Li+-poly(A)-I1 50 30 10 5 2 1 0
K+-poly(U)
Na+-poly(U)
-12.9 -17.4 -23.2 -24.2 -29.1 -29.6 -30.9
0.39 0.45 0.51 0.53 0.53 0.55 0.57
38.4 36.5 34.9 34.4 35.3 34.2 33.5
K+-poly(A)-I1 42.0 44.7 48.6 50.2 51.6 52.3 54.0
1.2 1.3 1.4 1.4 1.4 1.5 1.5
-8.4 -13.4 -19.4 -20.1 -20.6 -26.2 -27.0
74
KUZNETSOV, VORONTSOVA, AND KOZLOV
charged polyions, which results in the increase of the probability of random contacts between the polyions and therefore causes the fraction of surface conductivity t o grow.
Determining the Dissociation Constants of Phosphate Groups from Conductometric Data For poly ( A ) -I and poly ( A ) -11 salts is observed a linear dependence of conductance in the coordinates conforming to the linear form of the Ostwald's dilution law
C*A= -K*Ao
+ K*(A0)2.A-1
(14)
where K is the dissociation constant. Earlier the linear dependence has been found to hold for isoionic solutions of DNA3' and poly ( U ).31 At the same time, the Ostwald's dilution law is not satisfied for the copolymer of vinyl alcohol and methylimidazol, which was proposed to be regarded as 1 : 1-valent e l e ~ t r o l y t eFigure . ~ ~ 6 shows the dependence of conductance of poly ( A ) -I and -11 salts in Ostwald coordinates. Table I lists the values of A: and pK computed from the parameters of linear dependencies (shown in Figure 6 ) by eq. ( 1 4 ) . Possibly the simultaneous fulfillment of linear dependencies in Kohlrausch's and Ostwald's coordinates is a peculiarity of polynucleotides. Note that the dependence [ eq. ( 1 4 ) ] is satisfied only approximately for the systems studied by us. Thus, for one and the same
10
12
14 1 /A. 103,
16
s-l CI-~YOLE
18
20
counterion the scatter in the pK values of primary phosphate groups is quite large. However, very good convergence of the values of A' computed by three different methods for one and the same sample of polynucleotide (see Table I ) must be stressed. This is indicative of the legitimacy of extrapolation of eq. ( 9 ) to zero concentration for all the studied polynucleotides.
Peculiarities of Conductance of lsoionic Solution of Poly( A ) T h e pK values of adenine bases equal respectively 3.7 and 5.9 in single- and double-stranded structures of H+-poly( A ) .32 Therefore, a t a zero degree of neutralization one can neglect the dissociation of groups with pK 6, and poly ( A ) can be regarded as monobasic polyacid with pK 3.7. Figure 7 shows the dependence of conductance of isoionic solution of poly ( A ) on concentration. The limiting value of conductance of H+-poly(A) (Table I ) is appreciably less than A' for poly ( A ) salts and is about 10 times less than A' for H+-poly(U ) .31 This is understood to be due to a high degree of protonation of the bases in the isoionic poly ( A ) solution. Only polyions and hydronium ions that are formed upon dissociation of the adenine bases of single-stranded structures, whose pK is about three units less than that of adenine bases of double-stranded structures, participate in the transfer of electricity. T h a t is why the conductance of H+-poly( A ) , shown in Figure 7 in
15
20
25 1 /A.103,
30
35
40
s-l cpl-*t,TooLF:
Figure 6. Conductance of poly(A)-I ( a ) and poly(A)-II ( b ) at 298 K in Ostwald's coordinates.
45
POLYELECTROLYTE PROPERTIES OF BIOPOLYMERS
80
60
c
‘F 40 w
N
2
5.0
5.5
6.0 1LA .103,
6.5
7.0
s-’ CM-~POLE
Figure 7. Conductance of H+-poly(A)at 298 K in Ostwald’s coordinates.
Ostwald’s coordinates, has been calculated with consideration for the fraction of single-stranded poly ( A ) (20% ) . T h e value of A’ computed by eq. ( 1 4 ) equals 177 ohm-’ c m 2 mole-’; pK equals 2.8. T h e value of pK determined from the conductometric data is substantially more than the pK of poly( A ) salts, but is less than that determined from the potentiometric titration curve of H+-poly( A ) . Such a discrepancy in the values of pK determined by these two methods is also noticed for simple acids. For H +-poly ( U ) a s well a s for the salts of polynucleotides, the value of pK computed from the conductometric data equals about 2, which reflects the fact that the bases in poly( U ) are not protonated. In conclusion, it may be noted that the negative mobilities of counterions, listed in Table IV, can be foreseen by analyzing eq. ( 1).T o do so, one has t o rewrite this equation as5’ A
=
A,
+ A,
=
f ( A E i- A,)
where A, is the conductance of a polyion and A, is the conductance of a counterion in the polynucleotide solution. T h e remaining notations have the same meaning a s in eq. ( 1 ) . Then A, = f *A: - (1 - f ) . A ,
From this equation it is evident that part of the counterions, equal t o ( 1 - f ) , has a mobility equal in sign and absolute value to that of the polyion,
75
because the counterions travel together with the polyion. At definite ratios of f , A;, and X, the total conductance of counterions (A,) may become negative. This is what is observed in poly( A ) -11 solutions. A transition from positive to negative equivalent conductances of N a + ions was noticed by Nagasawa and c o - ~ o r k e rin s ~polyacrylic ~ acid solutions (neutralized to different extents) by decreasing the concentration of NaC1. From Table IV it also follows that the conductance coefficient f decreases very slowly with the increase in concentration, because [ grows very weakly. According t o the present-day concepts, the fraction of free counterions is proportional to f . This permits us t o assume that this fraction remains practically unchanged under changes of concentration of polynucleotides. A significant increase in the fraction of current carried by a counterion ( t h a t is, the decrease in the transference number of a polyion with the increase in the concentration of polynucleotides) is due not to the increase in the number of free counterions, as is observed in d e ~ t r a s u l f a t e , ~ ~ polystyrene sulfonic acid, 55 and its sodium salt sol u t i o n ~In . ~the ~ enumerated solutions ( i n contrast t o those studied by u s ) , f appreciably grows with the increase in the polymer concentration. The increase in the transference numbers of counterions with the increase of the concentration of polynucleotides can be explained by the rise in their mobility a t the cost of “surface” conductance and the decrease in the mobility of the polyion. In its turn, the decrease in the mobility of the polyion with the increase in the concentration of polynucleotides may apparently be due to the decrease in the electrostatic potential of the polyion owing t o the shielding of its charge by free counterions. In concentrated salt-free solutions of DNA such a shielding, known as selfprotective effect,57causes the melting point of DNA t o rise with the increase in the concentration of DNA.57,58
REFERENCES Anderson, C. F. & Record, M. Th., Jr. (1982) Ann. Rev. Phys. Chem. 33, 191-222. Klein, B. K., Anderson, C. F. & Record, M. Th., Jr. (1981) Biopolymers 20,2263-2280. Record, M. Th., Jr., Mazur, S. J., Melancon, P., Roe, J.-H., Shaner, S. L. & Unger, L. (1981) Ann. Rev. Biochem. 50,997-1024. Fried, M. G. & Bloomfield, V. A. (1984) Biopolymers 23, 2141-2155.
76
KUZNETSOV, VORONTSOVA, AND KOZLOV
5. Bleam, M. L., Anderson, C. F. & Record, M. Th., Jr. (1983) Biochemistry 22, 5418-5425. 6. Frank-Kamenetskii, M. D., Lukashin, A. V. & Anshelevich, V. V. (1985) J. Biomol. Struct. Dynam. 3, 35-42. 7. Richard, A. J. (1984) Biopolymers 23,1307-1313. 8. Post, C. B. & Zimm, B. H. (1982) Biopolymers 21, 2133-2137. 9. Manning, G. S. (1983) in Structure and Dynamics: Nucleic Acids and Proteins, Clementi, E. & Sarma, R. H., Eds., Adenine Press, New York, pp. 289-300. 10. Anshelevich, V. V., Lukashin, A. V. & Frank-Kamenetskii, M. D. ( 1984) Chem. Phys. 31, 225-236. 11. Grosberg, A. Yu. & Zhestkov, A. V. (1986) J. Biomol. Struct. Dynam. 3,859-872. 12. Frank-Kamenetskii, M. D. (1983) Mol. Biol. (Moscow) 17,639-652. 13. Frank-Kamenetskii, M. D. & Chogovadze, G. I. (1984) J . Biomol. Struct. Dynam. 6,1516-1523. 14. Frank-Kamenetskii, M. D., Anshelevich, V. V., Lukashin, A. V. (1987) Usp. Fiz. Nauk 151, 595-618. 15. Manning, G. S. (1985) Cell Biophys. 7,57-89. 16. Widom, J. & Baldwin, R. L. (1983) Biopolymers 22, 1595-1620. 17. Grosberg, A. Yu., Erukhimovitch, I. Ya. & Shakhnovitch, E. I. (1982) Biopolymers 21, 2413-2432. 18. Clementi, E. (1983) in Structure and Dynamics: N u cleic Acids and Proteins, Clementi, E. & Sarma, R. H., Eds., Adenine Press, New York, pp. 321-364. 19. Klein, B. J. & Pack, G. R. (1983) Biopolymers 22, 2331-2352. 20. Lee, W. K. & Prohofsky, E. W. (1984) Biopolymers 23, 257-270. 21. Manning, G. S. (1978) Quart. Rev. Biophys. 11,179242. 22. Manning, G. S. (1975) J . Phys. Chem. 79, 262-265. 23. Manning, G. S. (1972) Ann. Rev. Phys. Chem. 23, 117-140. 24. Varoqui, R. & Schmitt, A. ( 1973) J. Chem. SOC.Faraday Trans. 2 69, 1087-1103. 25. Szymczak, J., Holyk, P. & Ander, P. (1975) J . Phys. Chem. 79, 269-272. 26. Kwak, J. C. T. & Johnston, A. J. (1975) Can. J . Chem. 53, 792-796. 27. Kwak, J. C. T., Murphy, G. F. & Spiro, E. J. (1978) Biophys. Chem. 7, 379-386. 28. Kuznetsov, I. A,, Apolonnik, N. V. & Maznikova, L. F. (1978) Mol. Biol. (Moscow) 12, 546-551. 29. Kuznetsov, I. A. & Apolonnik, N. V. (1981) Biopolymers 20,2083-2093. 30. Kuznetsov, I. A., Filippov, S. M. & Vorontsova, 0. V. (1979) Mol. Biol. (Moscow) 13,543-549. 31. Kuznetsov, I. A., Vorontsova, 0. V. & Fabrichnaya, 0. B. (1983) Mol. Biol. (Moscow) 17,392-402.
32. Kuznetsov, I. A. & Vorontsova, 0. V. (1984) Mol. Bwl. (Moscow) 18,1032-1041. 33. Vink, H. ( 1981) J . Chem. SOC.Faraday Trans. I 77, 2439-2449. 34. Vink, H. (1982) Makromol. Chem. 183, 2273-2283. 35. Karpetsky, T., Bogyski, M. & Levy, C. ( 1979) in Subcellular Biochemistry Vol. 6, Roodyn, D., Ed., Plenum Press, New York and London, pp. 1-116. 36. Finch, J. T. & Klug, A. (1969) J . Mol. Biol. 46, 597598. 37. Adler, A. I., Grossman, L. & Fasman, C. D. (1969) Biochemistry 8,3846-3859. 38. Gray, D. M. & Bollum, F. J. (1974) Biopolymers 13, 2087-2102. 39. Glasstone, S. ( 1947) A n Introduction to Electrochemistry, Van Nostzand, New York. 40. Fixman, M. ( 1979) J . Chem. Phys. 70,4995-5005. 41. Richards, E. G., Flessel, C. P. & Fresco, J. R. (1963) Biopolymers 1, 431-446. 42. Wits, J. & Luzzati, V. ( 1965) J. Mol. Biol. 11, 620630. 43. Archer, B. G., Craney, C. Z. & Krakauer, H. (1972) Biopolymers 11, 781-809. 44. Stannard, B. S. & Felsenfeld, G. ( 1975) Biopolymers 14, 299-307. 45. Manning, G. S. (1976) Biopolymers 15, 2385-2390. 46. Kwak, J. C. T. & Hayes, R. C. (1975) J . Phys. Chem. 79, 265-269. 47. Katchalsky, A. (1971) Pure Appl. Chem. 26, 327374. 48. Eisenberg, H. (1958) J . Polym. Sci. 30,47-66. 49. Shavit, N. (1973) Zsr. J. Chem. 11, 235-242. 50. Nelson, R. E. & Ander, P. (1971) J. Phys. Chem. 75, 1691-1697. 51. Gregor, H. P. & Gold, D. H. (1957) J . Phys. Chem. 61, 1347-1352. 52. Devore, D. J. & Manning, G. S. (1974) J. Phys. Chem. 78, 1242-1244. 53. Nagasawa, M., Noda, J., Takahashi, T. & Shimamoto, N. (1972) J . Phys. Chem. 76, 2286-2294. 54. Joshi, Y. M. & Kwak, J. C. T. (1980) Biophys. Chem. 12, 323-328. 55. Dolar, D., Span, J. & Pretnar, A. (1968) J . Polym. Sci. Part C 16, 3557-3564. 56. Dolar, D., Span, J. & Isakovic, S. (1974) Biophys. Chem. 1, 312-317. 57. Auer, H. E. & Alexandrowicz, Z. (1969) Biopolymers 8,1-20. 58. Akhrem, A. A., Andrianov, V. T., Vlasov, A. P., Korolyev, N. I. & Kuznetsov, I. A. (1985) Mol. Biol. (MOSCOW) 19,623-628. Received November 11, 1988 Accepted July 3, 1990
SYNOPSIS
Polyriboadenylates of alkali metals were obtained from (1)K+-poly(A) (salts I ) and ( 2 ) H+-poly(A) (salts 11) by the ion-exchange method. The conductivity of these salts as well as of H+-poly(A)were studied. Salts I and I1 of the same counterion were shown to have significantly different conductivity coefficients ( f ) and polyion conductances (A:). The charge density parameter ( E ) was 1.3 and 2.5, respectively, with A: equal to 44 and 83 ohm-’ cm2 mole-’ for poly( A ) - I and poly( A) -11 salts, respectively. This is credited to the difference in the conformations of corresponding polyions. The linear dependence of equivalent conductivity on the square root of polymer concentration ( Kohlrausch coordinates), earlier obtained for DNA, is also satisfied for the studied polynucleotides. A comparison of the slopes of straight lines in Kohlrausch coordinates for poly ( A ) , simple electrolytes, and for earlier studied polyribouridylic acid salts lends credence to the concepts, developed by a number of authors, that DNA can act as a “buffer” against the ion-ion interaction in concentrated electrolyte solutions. Using the approximation that the polyion conductance is independent of the counterion nature, parameter f (agreeing in this case with Eisenberg parameter 4 ) has been shown to decrease as the polynucleotide concentration is increased; the decrease is caused by the relaxation effect. The transference numbers of counterions, which have negative values in poly ( A ) -11 solutions, grow with the increase in polymer concentration; the higher the l , the more apparent is this increase. This is explained by the increase in the fraction of conductivity along the polyion chains (“surface” conductivity) with the growth of polyelectrolyte concentration.
INTRODUCTION The study of polyelectrolyte properties of synthetic polynucleotides, which are simple models of nucleic acids, is of great interest for molecular biology. The behavior of DNA in living organisms is largely conditioned by the fact that DNA is a polyelectrolyte. This is why increased interest is being shown in the past few years in the polyelectrolytic properties of DNA.’-20 In a number of modern theories of polyelectrolytes it has been ascertained that the conductance of a polyelectrolyte is definitely related to its charge
Biopolyrners. Vol. 31,65-76 (1991) 0 1991 John Wiley & Sons, Inc.
CCC 0006-3525/91/010065-12$04.00
density. This enables conductivity measurements to be used for studying the structure of polyion in solution. According to the present-day concepts
where A is the equivalent conductance of the polyelectrolyte in salt-free solution, A, is the polyion conductance, and A: is the limiting conductance of the counterion. The dependence of X, on concentration is expressed as
where K is the Debye-Huckel screening parameter and r is the radius of the cylindrical p o l y i ~ n . ~ ~ - ~ ~ 65
66
KUZNETSOV, VORONTSOVA, AND KOZLOV
The coefficient f appearing in eq. ( 1 ) is determined by the relationships
DC
f = -
D:
(3)
where D, and D: are the diffusion coefficients of counterions in the solution of polyelectrolyte and pure solvent, r e ~ p e c t i v e l y .I~t ~can be seen that f reflects the variation in the mobility of counterions in a polyelectrolyte solution compared to the mobility in a low molecular salt solution. Neglecting the mobility of condensed counterions, Manning derived a n equation connecting the parameters f and [ for the case 5 > 1 in salt-free polyelectrolyte solutions:
Here, e is the proton charge, k is the Boltzmann constant, T is the temperature, b is the distance between the ionogenic groups along the polyion chain, and 2, is the valence of the counterion. Good agreement of polyelectrolyte conductance with the Manning theory has been obtained in a number of ~ o r k s . ' ~ Nevertheless, ,~~ often the theoretical dependence of A on concentration ( C , ) does not correlate with the experimental The possible reason is that the model of a free-draining cylinder does not serve as good approximation for all polyelectrolytes. It has been shown that the linear dependence of A on 12;'' holds for salt-free DNA and poly ( U ) solutions. This enables the limiting conductance of polynucleotides A h p to be estimated. T h e values of ALP obtained by extrapolation are proportional to the mobilities of counterions. This made it possible t o estimate quite reasonable values of $, for the enumerated polymer^.^^^^^ However, the shape of the dependence of conductance on concentration does not agree with the Manning theory. T h e present work studies the dependence of conductance of poly ( A ) on the concentration and nature of the counterion. I t also compares the slopes of concentration dependencies of conductance for poly(A) ( t h e present work) and poly(U) (data taken from Ref. 31 ) , and for low molecular electrolytes. The polyelectrolyte properties of poly( A ) and poly ( U ) have been found to resemble the earlier studied properties of DNA in salt-free solutions.
EXPERIMENTAL A potassium salt of polyriboadenylic acid supplied by Reanal of Hungary was used in the experiment. T h e isoionic solution of poly ( A ) was obtained by passing a n aqueous solution of K+-poly( A ) through a column packed with a mixed bed of the ion exchangers. T h e procedure of obtaining the isoionic solution and the solution's characteristics, as well as the spectrophotometric technique of determining the concentration of poly ( A ) , are discussed in detail in Ref. 32. T h e concentration of isoionic solution was taken as the arithmetic mean of concentrations determined spectrophotometrically and by potentiometric titration of H+-poly(A ) solution with NaOH.32In isoionic solutions of poly ( A ) the double helix is more stable than in the presence of salts because the favorable interactions of the positively charged adenylic base of one chain and of the negatively charged phosphate group of the other chain are not screened. About 20% poly ( A ) is present in the single-stranded form.32 Obtaining Solutions of Ribonucleates of Alkali Metals
T h e salts of alkali metals of poly ( A ) were obtained by filtering through a column packed with cation exchangers in the corresponding M+-form (here, M + stands for Li+, Na', Rbt, K + , or Cs') (1) of the starting solution of K+-poly(A ) (salts I ) and ( 2 ) of the isoionic solution of H +-poly( A ) (salts 11). The completeness of transfer of K +-poly( A ) into a corresponding M + form was controlled by the flame photometry method using the FLAPHO ( G D R ) emission flame photometer. At a sufficiently slow rate of filtration, the filtrate was found to be free of K + ions. The completeness of transfer of H +-poly ( A ) into the M + form (salts 11) was controlled by potentiometric titration of solutions of the obtained salts with a 0.09M NaOH solution. Measurement of Conductance
The resistance of solutions was measured in a flow cell (platinized platinum electrodes) with a constant 0.049 cm-' a t 25°C using the LKB (Sweden) conductometer, the alternating current frequency being 2 kHz. CD Measurements
Ultraviolet CD measurements were made in 1.0-mm cells, using a Jasco J-500C ( J a p a n ) recording spec-
POLYELECTROLYTE PROPERTIES OF BIOPOLYMERS
tropolarimeter. The instrument was calibrated with a n aqueous solution of d-10-camphor sulfonic acid. Spectrally pure salts and deionized water were used in the experiments.
67
any electrolyte for which the shape of the function F ( C ) is not known. It has been shown that for isoionic solutions of DNA and poly ( U ) and their salts the linear dependence between A and C:l2 is satisfied, i.e., F ( C ) = - A * C;", where A is a In the presRESULTS AND DISCUSSION ent work we have found that the A - C;I2 dependence is also linear for H+-poly(A) and its salts. Usually for calculating the equivalent conductance Figure 1( a and b ) shows the equivalent conductance of a solution ( A ) , use is made of K~ of pure solvent. dependence in Kohlrausch's coordinates for salts I For DNA and polynucleotides, K' is determined by and 11, respectively. This dependence is defined by extrapolating the linear portion of the K - C, dethe expression pendence to zero concentration of the ~ o l y m e r . ~ ~ - ~ l Recently, using a n analogous approach, Vink33g34has shown that A is the sum of limiting conductance A' and some function of concentration F ( C) : From Figure 1 it is seen that the conductance of polyadenylates obtained by the first method is much K - Kg higher than for salts 11, and its dependence on the A== ha F ( C ) nature of counterion in salts I is stronger than in c salts 11. Table I lists the values of limiting equivalent Proceeding from eq. ( 6 ) , the specific conductance conductance A; obtained by extrapolating eq. ( 9 ) of a solution is expressed as to zero concentration of the polymer for polyriboadenylates I and I1 (Figure l a and b ) , and also for polyribouridylates (reproduced from Ref. 31 ) . For all salts, the values of A; are in good agreement It is clear that when C, tends to zero, the term with A t , which have been determined as slopes of C - F ( C ) tends to zero faster than 11'. C. Thus, nestraight portions of the K - C, dependence for every glecting the last term of eq. ( 7 ) we obtain salt a s per eq. ( 8 ) . T h e K - C, dependence for poly(A)-I and -11 salts as well as for poly(U) are shown respectively in Figure 2 ( a , b, and c ) . The Vink has proposed a method of estimating A' for coincidence of A; and A t shows that our way28-31
+
Figure 1. Dependence of equivalent conductance ( A ) of poly ( A ) -I ( a ) and poly ( A ) -11 ( b ) on the nature of counterion and polymer concentration at 298 K.
Average
a
27 25 30 28
83
78 75
77 80 80
66 46 65 76
61 47 62 56 50 61
26 28 32 27 28f 2
83
78 76
78 80 77
66 46 62 68
58 45 57 57 47 56
Poly(A)-I
A8
178 176 170 179 176 f 3
H+-Poly(A)"
77f 2 82 83f 1
78t 1 79 76
61f 8 76 80 78
64 44 61 73
60 47 63 58 49 59 55+ 5
A%
2.78 2.83 2.70 2.95 2.82 f 0.07
1.97 f 0.13 1.60
1.67 f 0.13 2.06 1.88
1.85 0.16 1.62 1.86 1.52
*
2.15 1.84 1.64 1.75
1.50 1.64 1.70 2.20 1.60 2.13 1.79 f 0.24
PK
57
58 62 60 45 58
65 58 43 56 55
39 38 46 54
46 36 47 47
AE
54 f 6 56 60 59 45 57 55 f 5 54 56 f 2
44 2 6 65 57 41 53 52
38 39 47 53
43 2 4
50 34 47 45
A%
57
54 57 58 45 58
64 56 40 55 52
39 38 45 56
46 33 42 44
Poly(A)-I1
A8
Values of A: have been computed with consideration for concentration of single-stranded structures of H+-poly(A).
Average
Rb'
Average
cs+
K'
Average
Na+
Average
Li+
Ion
+
1.88 f 0.22 1.90 2.06 1.90 1.86 1.50 1.84 0.13 2.06
*
1.73 0.08 1.61 2.15 2.13 1.91 1.60
1.71 1.60 1.85 1.78
1.85 f 0.15
1.60 1.88 2.13 1.80
PK
89 73 83+ 4 89 106 82 95 118 86 113 112 100 f 11 106 100 89 116 87 101 f 10
90 75
108 104 92 117 88
91 108 84 96 119 91 115 110
84 78 87 84
72+ 7
72 79 61 79
A%
84 81 87 86
74 79 62 80
A$
104 88 119 89
111
91 108 80 91 121 89 99 110
89 76
82 77 90 84
2.30 1.81 1.60
64 62 82
+
*
1.73 1.81 1.85 0.21 1.63 1.47 1.88 1.94 1.90 1.86 1.51 1.60 1.72 k 0.17 1.64 1.77 1.80 1.67 1.72 1.72 0.05
1.55 1.86 2.33 2.17
1.85 f 0.22
1.71
PK
POlY(U) 74
A;
Table I Limiting Conductances (ohm-' cm2 mole-') Determined from Kohlrausch's (A:), Vink's ( A t ) , and Ostwald's (A:) Dependencies [eqs. (9), (€9,and (14), respectively] and pK Values Computed by eq. (14) for Poly(A) Salts Obtained by the First and Second Methods, and for Poly(U) at 298 K
c
0
r
x
U
z
P
0 c "P
rn
5
0
z
c 0
"C
;
3
s
x
5 2 OD
POLYELECTROLYTE PROPERTIES OF BIOPOLY MERS
Figure 2.
The
K
vs
69
C, (298 K ) . (a) poly (U), (b) poly (AbI, (c) poly (A)-II
of' taking account of the contribution of solvent is right. Significant discrepancy between the conductances of' poly ( A ) -I and -11 salts is obviously due to the diff'erence in the conformations of these two samples. In a neutral medium, poly ( A ) is known to exist in the single-stranded form with some stacking.:jS O n making the solution slightly acidic, the transit ion to double-stranded conformation takes place --first a semiprotonated and then a less compact, c,ompletely protonated form appears. In the literature, these two structures are referred to as B and A forms of poly ( A ) , respectively.36337 Figure 3 shows the potentiometric titration curve for the Ka . -poly ( A ) -11 solution. It can be seen that the equivalence point is observed only upon adding 0.40 mole NaOH to one mole of phosphorus. It has been shown that in an isoionic solution of poly(A) double-stranded structures coexist with singlestranded ones. The latter contain about 20% of all nucleotides of poly(A) .32 Taking this into consideration, i t can be assumed that in the poly( A ) -11 salts atiout 80% nucleotides are present in the form of double-stranded semiprotonated structure, that is, as structure of t,ype B (after Finch et al.36 and Adler et al.'") . This is also evidenced by the CD spectra of Figure 4,which reveal that the secondary structure of N a + poly ( A ) -11 in a solution is more pronounced than the structure of Na+-poly(A ) -I and H + - p o l y ( A ) . T h e double-stranded structure of poly ( A ) in the A and R (after Ref. 3 7 ) forms are known to differ in
neighboring bases. In the B form they are disposed on alternate bases, which ensures a minimum repulsion between the positively charged bases. The appearance of protons on the neighboring bases causes the repulsive forces to increase abruptly. That is why the B form is more compact than the A form. Upon filtering an isoionic solution of H +-poly( A ) through a M+-form cation exchanger the amount of unexchanged protons equals O.4Cp (Figure 3 ) . This
.60
Figure 3. Potentiometric titration curve [ N a + poly(A)-11, C, = 2.2 X 10-'M, titrated with 0.09M NaOH] . The r is the number of moles of NaOH added to 1 mole of nucleic phosphorus a t 298 K.
70
KUZNETSOV, VORONTSOVA, AND KOZLOV
I
no
I
240
2a
280
320
320
Figure 4. Ultraviolet CD spectra of poly(A) ( 1 ) Na'p o b ( A ) - I , ( 2 ) H + - p o l y ( A ) , and ( 3 ) Na+-poly(A)-11.
amount of protons is present on the semiprotonated double-stranded structure of poly ( A ) . These are the protons that are more strongly bonded: pK values of adenine equal, respectively, 3.7, 5.9, and 7.1 for single-stranded, double-stranded fully protonated, and double-stranded semiprotonated structure^.^^ Structures of this type have been studied for poly (dC ) . It has been noted that in poly (dC ) a double-stranded poly (dC : d C + ) is formed upon protonating half of the bases. This "acidic" form is stable at pH 7.0 and at 20°C.38 Table I1 lists the average values (of all the measurements for every salt) of the slopes of A - C;/' dependencies for polyadenylates I and I1 and for polyuridylates. Therein, for the purpose of comparison, are given the slopes of chlorides and sulfates of these very cations. These slopes have been calculated by formula ( 10 1, 39
where t and T have the same meaning as in eq. ( 5 ) , 77 is the viscosity of water, and 2, and Zz are the cation and anion charges. The values of limiting ion mobilities have been reproduced from Ref. 39. The first term of formula ( 1 0 ) derived in the theory of conductance of low-molecular electrolytes takes account of the electrophoretic retardation of ions. The second term allows for the relaxation effect related to retardation of a moving ion due to the asymmetry of the ionic atmosphere, which (asymmetry) appears with the application of an electric field. From Table I1 it is evident that the absolute values of slopes for polynucleotides are more than for low molecular electrolytes. However, with an increase in the limiting conductivity of salts, a linear increase in the values of A is observed both for the former and the latter:
A
=
140
+ 0.81.A"
+ 0.74 - A" A = 146 + 0.46-A' A
=
156
( r = 0.99) for poly(U) ( r = 0.99)
for poly( A ) -I
( r = 0.99) for poly(A)-I1
where r is the correlation coefficient. It can be seen that for the studied polynucleotides that the constants A , depending on the nature of counterion, are expressed as
Here, a and ,&-constants for the given polyelectrolyte-are proportional to the electrophoretic and
Table I1 Constants A Appearing in eq. (9) (ohm-' cm2 L'/') for Poly(U) (after Ref. 31), Poly(A)-I, and Poly(A)-11, and Also for Chlorides and Sulfates of Alkali Metals at 298 K Cation
POlY(U)
Poly(A)- I
Li+ Na+ K+
187 +- 13 199 f 17 219 -t 19 223 +- 17 -
198 f 201 f 212 f 213 f 220 f
cs+ Rb+
12
14 12 19 19
Poly(A)-11
c1-
soq-
165 f 12 167 f 11 170 f 15 173 f 13 171 f 13
87 89 95 95 95
177 181 190 192 192
POLYELECTROLYTE PROPERTIES OF BIOPOLYMERS
relaxation effects, respectively, and A’ is the limiting conductance of the given salt of polynucleotide. It is known that the stronger the association in the electrolyte solution the greater the values of slopes in the Kohlrausch coordinate^.^' Since strong bonding of counterions is noticed in polyelectrolyte solutions, one can expect that the values of A would turn out to be much more than those for simple electrolytes. Inspite of this, the values of slopes of the A - C;/’ dependencies for polyuridylates and polyadenylates I and I1 were found t o be comparable to those for simple electrolytes. Possibly, this is partly due to the contribution of surface conductance (see below) whose effect grows with increase in the concentration of polynucleotide. On the other hand, such a behavior of the polynucleotides studied by us correlates with the strongly charged polyions exerting a “buffer” effect against the interactions between small ion^.',^^^^ Estimating Charge Density Parameter for Poly(A)-l and Poly(A)-ll
As for DNA salts” and poly ( U ),31 the limiting conductances of polyadenylates I and I1 were found to be proportional to the conductances of counterions (Figure 5 ) . This makes it possible to estimate the values o f f , A,, [, and b from eqs. ( l ) , (41, and ( 5 ) . For poly ( A ) -I and -11 the values o f f , A,, and b are given in Table 111; therein, the values of these parameters for poly(U) are reproduced from Ref. 31.
0
20
7I
For the purpose of comparison, this table contains the corresponding values of b and [ obtained by other methods in Ref. 21. The value [ = 1.6 ( a s per the data of Ref. 21) for poly(U) seems to be overestimated if we take into consideration numerous proofs of the absence of a n ordered structure in poly ( U ) a t room t e m p e r a t ~ r eArcher .~~~~ and ~ his colleagues43 have found [ = 1.05 from the measurements of activity of sodium ions in a Na+-poly ( U ) solution. This value correlates with that ([ = 1.13) obtained by the conductometric method.31 Table I11 lists the values of b-average distance between the projections of phosphates on the spiral axis-computed from eq. ( 5 ) for the obtained values of [. A comparison of values of b for poly ( U ) and poly ( A ) -I and I1 permits us to conclude that in saltfree solutions the conformation of poly(A) is far from the maximum stretched one for which b = 6.8 A, unlike poly ( U ). This is in agreement with the data on small-angle scattering of x-rays, according to which the distance between the phosphates in single-stranded poly( A ) equals about 3.5 A.44 According to our results obtained from the measurements of conductance, the poly ( A )-I and poly ( U ) salts are in the single-stranded conformation, but have different charge density parameters. Manning, in one of his works, has uttered a n opinion according to which the value of b for singlestranded polynucleotides is independent of temperature and the nucleotide composition, and lies between 3 and 4 A. T h a t is, it is not related to the
40
60
A,,
c c.,*-oIL-’
80
100
Figure 5. Limiting equivalent conductance of poly ( A ) -I ( a ) and poly ( A ) -11 ( b ) salts vs limiting conductance of counterion at 298 K.
72
KUZNETSOV, VORONTSOVA, AND KOZLOV
Table I11 Parameters f , [, A:, and b for Poly(A)-I and Poly(A)-I1 and Also for Poly(U) (298 K), Obtained in the Present Work and in Ref. 31 for Extremely Dilute Solutions, and the Values o f t and b after Ref. 21
Obtained from the data on conductance From the data of Ref. 21
A:,
Polynucleotide
f
b, A
i
ohm-’ cm2 mol-’
POlYiU) Poly(A)-I PolyiA) -11 POlY(U) PMA) Native DNA Denatured DNA
0.77 0.66 0.35
6.3 5.4 2.9 4.5 3.1 1.7 4.0
1.1 1.3 2.5 1.6 2.3 4.2 1.8
56 44 83
differences in the stacking of bases in different polynucleotide~.~’ He has based his views on the results of Archer and his colleague^,^^ who from the potentiometric measurements have obtained similar values of the bonding degree of N a + ion for poly(A) and poly ( U ) . In a much later work, Manning has reported different values of [ for poly(A) and poly ( U ) (see Table I11 ) .‘l Our results suggest that the nature of the polynucleotide strand largely determines the polyion charge density. Probably, the disagreement with the results of Ref. 43 is due to transport methods (conductance being one of them) that are more sensitive t o the type of polyelectrolyte skeleton than the equilibrium method^.^^^^' According to the Manning theory, A, depends on the nature of counterion.’* However, the constancy of A, is a reasonable approximation in obtaining f from eq. ( 1).47 For the salts of polymetacrylic and polyacrylic acids, Eisenberg has found that 4 equals
and remains constant for different pairs of M and N ( A M P and ANp are equivalent conductances of polymer salts, and A h and A; are limiting conductances of the counterions M and N.4s It has been shown that for concentrated solutions of polyacrylic acid, (#) depends on the nature of c~unterion.~’ Nelson and Ander5’ have found that (#) increases linearly with the growth of C, in gum arabic salt solutions in the concentration range of 0-0.01M. For carrageenanes, (#) decreases linearly with the growth of C,. T h e authors have attributed these results to different charge densities of polyelectrolytes. From our data it follows that (#)’ = (ALP - AtP)/(1 ; - A t ) is independent of the nature of the counterion. Also, it can be shown that a t a constant concentration of the polyelectrolyte (#) is constant for all pairs of ions. Having denoted AMP by
A iand ANP by A,, and expressing their dependences on C, by eqs. (9), ( 11) , and ( lOa), we write (#) Al-Al _(A~-AlC”2)-(A,”-A,C”2)
(#)=--
xp - A,”
4=
(A:
-
xp - A,” A:)
( CY (#)=(#)”-
(A, - A,)C1” - A,”
-
xp
+ pAp)C”’
-
( (Y
+
xp - A,”
( # ) = ( # ) ”@(A: -
-
A,”)C”‘
xp - A,” i.e.,
Thus, it can be seen that 4,and hence f for polynucleotides, are independent of the nature of counterions, but vary with polymer concentration; this dependence on concentration is entirely caused by the relaxation effect. Equation ( 1 2 ) is consistent with the dependence of f on concentration for carrageenanes.” Carrageenanes have higher charge density than gum arabic and are close, in this respect, t o polynucleotides. From the analysis of eq. ( 1 2 ) it follows that the charge density decreases with dilution (provided A, is independent of the nature of counterions). This dependence of [ on C, is not considered in the Manning theory. Estimating the Transference Numbers in Solutions
of Polynucleotides Assuming that only two types of ions exist in a saltfree polyelectrolyte solution and that the t, part of electricity is transferred by counterions and the tp
POLYELECTROLYTE PROPERTIES OF BIOPOLYMERS
part by polyions ( t, and t, are the transference numbers and t, t, = 1) , we write eq. ( 1) as
73
ative transference numbers are met with in low molecular electrolyte solutions, and this points to strong bonding of, say, complex formation. In highly charged polyelectrolyte solutions, polyion transference numbers (more than 1.0) are quite often observed; this is indicative of strong bonding of counterions by the polyion. This manifests itself in that part of the counterions’ (cations) travel toward the anode together with the polyion. The values of A, for some polymer concentrations are listed in Table IV. It can be seen that X, grows with the increase in C , . An analogous picture was noticed in the isoionic solution of DNA.30 The authors have attributed these results to the increase in “surface” conductance along the chains of polyions. As is evident from Table IV, the increase in X, with polymer concentration is particularly pronounced for polynucleotides having high charge density. Apparently, this is due to the increased rigidity of strongly
+
-
Whence t, = ( A - f A:) / ( f A ) . Substituting into this equation the values of f and A calculated from eqs. ( 12 ) and ( 9 1, respectively [ t h e values of A used in the calculation A were taken from Table I1 and the values of P were extracted by a n equation of type ( 10a ) 1, we find the values of t, for several polymer concentrations. Table IV lists the values of t, for poly ( A ) -I and -11 and poly ( U ) a t different concentrations. From this table it can be seen that, even in the poly( U ) solutions, a large part of electricity is transferred by the polyion, and the higher its charge density the more the value of t,. For the poly ( A ) -11 salts, negative transference numbers of counterions are observed. It must be noted that neg-
Table IV Dependence of Charge Density Parameter ( t ) ,Conductance (A), Polyion Transference Number (&), and Counterion Conductance (A,) on Polymer Concentration at 298 K (A and X, are in ohm-’ cm2 mole-’)
Li+-poly(U ) 50 30 10 5 2 1 0
1.20 1.18 1.16 1.15 1.14 1.14 1.13
59.1 61.8 65.1 67.8 69.3 70.1 71.8
0.71 0.72 0.74 0.75 0.76 0.76 0.76
17.1 17.3 16.9 17.0 16.6 16.8 17.2
69.0 72.1 76.7 78.5 80.2 81.0 83.1
Li+-poly(A)-I 50 30 13 5 2 1 0
1.37 1.36 1.34 1.33 1.33 1.33 1.32
41.5 44.2 48.8 50.6 52.2 53.0 55.0
2.57 2.57 2.57 2.50 2.50 2.50 2.50
31.3 34.0 37.8 39.3 40.7 41.4 43.0
0.63 0.69 0.74 0.77 0.77 0.79 0.81
1.7 1.8 1.8 1.9 1.9 1.9 2.0
24.8 24.5 24.5 25.1 24.8 24.3 25.8
0.64 0.66 0.68 0.68 0.69 0.70 0.69
85.5 89.0 94.1 96.1 97.9 98.8 100.5
15.4 13.7 12.7 11.6 12.0 11.1 10.5
46.8 50.0 54.6 56.5 58.2 59.0 61.0
-21.9 -27.2 -30.2 -35.4 -36.6 -37.3 -43.0
23.9 22.0 20.8 19.8 20.4 19.5 18.9
0.49 0.56 0.62 0.65 0.65 0.67 0.69
62.9 66.3 71.3 73.2 75.0 75.9 78.0
Na+-poly(A)-I1 32.2 34.8 38.7 40.3 41.6 42.3 44.2
1.4 1.5 1.6 1.6 1.7 1.7 1.7
0.51 0.53 0.55 0.55 0.56 0.57 0.57
41.9 41.8 42.3 43.2 43.1 42.5 43.2
K+-poly(A) -I
Na+-poly(A)-I
Li+-poly(A)-I1 50 30 10 5 2 1 0
K+-poly(U)
Na+-poly(U)
-12.9 -17.4 -23.2 -24.2 -29.1 -29.6 -30.9
0.39 0.45 0.51 0.53 0.53 0.55 0.57
38.4 36.5 34.9 34.4 35.3 34.2 33.5
K+-poly(A)-I1 42.0 44.7 48.6 50.2 51.6 52.3 54.0
1.2 1.3 1.4 1.4 1.4 1.5 1.5
-8.4 -13.4 -19.4 -20.1 -20.6 -26.2 -27.0
74
KUZNETSOV, VORONTSOVA, AND KOZLOV
charged polyions, which results in the increase of the probability of random contacts between the polyions and therefore causes the fraction of surface conductivity t o grow.
Determining the Dissociation Constants of Phosphate Groups from Conductometric Data For poly ( A ) -I and poly ( A ) -11 salts is observed a linear dependence of conductance in the coordinates conforming to the linear form of the Ostwald's dilution law
C*A= -K*Ao
+ K*(A0)2.A-1
(14)
where K is the dissociation constant. Earlier the linear dependence has been found to hold for isoionic solutions of DNA3' and poly ( U ).31 At the same time, the Ostwald's dilution law is not satisfied for the copolymer of vinyl alcohol and methylimidazol, which was proposed to be regarded as 1 : 1-valent e l e ~ t r o l y t eFigure . ~ ~ 6 shows the dependence of conductance of poly ( A ) -I and -11 salts in Ostwald coordinates. Table I lists the values of A: and pK computed from the parameters of linear dependencies (shown in Figure 6 ) by eq. ( 1 4 ) . Possibly the simultaneous fulfillment of linear dependencies in Kohlrausch's and Ostwald's coordinates is a peculiarity of polynucleotides. Note that the dependence [ eq. ( 1 4 ) ] is satisfied only approximately for the systems studied by us. Thus, for one and the same
10
12
14 1 /A. 103,
16
s-l CI-~YOLE
18
20
counterion the scatter in the pK values of primary phosphate groups is quite large. However, very good convergence of the values of A' computed by three different methods for one and the same sample of polynucleotide (see Table I ) must be stressed. This is indicative of the legitimacy of extrapolation of eq. ( 9 ) to zero concentration for all the studied polynucleotides.
Peculiarities of Conductance of lsoionic Solution of Poly( A ) T h e pK values of adenine bases equal respectively 3.7 and 5.9 in single- and double-stranded structures of H+-poly( A ) .32 Therefore, a t a zero degree of neutralization one can neglect the dissociation of groups with pK 6, and poly ( A ) can be regarded as monobasic polyacid with pK 3.7. Figure 7 shows the dependence of conductance of isoionic solution of poly ( A ) on concentration. The limiting value of conductance of H+-poly(A) (Table I ) is appreciably less than A' for poly ( A ) salts and is about 10 times less than A' for H+-poly(U ) .31 This is understood to be due to a high degree of protonation of the bases in the isoionic poly ( A ) solution. Only polyions and hydronium ions that are formed upon dissociation of the adenine bases of single-stranded structures, whose pK is about three units less than that of adenine bases of double-stranded structures, participate in the transfer of electricity. T h a t is why the conductance of H+-poly( A ) , shown in Figure 7 in
15
20
25 1 /A.103,
30
35
40
s-l cpl-*t,TooLF:
Figure 6. Conductance of poly(A)-I ( a ) and poly(A)-II ( b ) at 298 K in Ostwald's coordinates.
45
POLYELECTROLYTE PROPERTIES OF BIOPOLYMERS
80
60
c
‘F 40 w
N
2
5.0
5.5
6.0 1LA .103,
6.5
7.0
s-’ CM-~POLE
Figure 7. Conductance of H+-poly(A)at 298 K in Ostwald’s coordinates.
Ostwald’s coordinates, has been calculated with consideration for the fraction of single-stranded poly ( A ) (20% ) . T h e value of A’ computed by eq. ( 1 4 ) equals 177 ohm-’ c m 2 mole-’; pK equals 2.8. T h e value of pK determined from the conductometric data is substantially more than the pK of poly( A ) salts, but is less than that determined from the potentiometric titration curve of H+-poly( A ) . Such a discrepancy in the values of pK determined by these two methods is also noticed for simple acids. For H +-poly ( U ) a s well a s for the salts of polynucleotides, the value of pK computed from the conductometric data equals about 2, which reflects the fact that the bases in poly( U ) are not protonated. In conclusion, it may be noted that the negative mobilities of counterions, listed in Table IV, can be foreseen by analyzing eq. ( 1).T o do so, one has t o rewrite this equation as5’ A
=
A,
+ A,
=
f ( A E i- A,)
where A, is the conductance of a polyion and A, is the conductance of a counterion in the polynucleotide solution. T h e remaining notations have the same meaning a s in eq. ( 1 ) . Then A, = f *A: - (1 - f ) . A ,
From this equation it is evident that part of the counterions, equal t o ( 1 - f ) , has a mobility equal in sign and absolute value to that of the polyion,
75
because the counterions travel together with the polyion. At definite ratios of f , A;, and X, the total conductance of counterions (A,) may become negative. This is what is observed in poly( A ) -11 solutions. A transition from positive to negative equivalent conductances of N a + ions was noticed by Nagasawa and c o - ~ o r k e rin s ~polyacrylic ~ acid solutions (neutralized to different extents) by decreasing the concentration of NaC1. From Table IV it also follows that the conductance coefficient f decreases very slowly with the increase in concentration, because [ grows very weakly. According t o the present-day concepts, the fraction of free counterions is proportional to f . This permits us t o assume that this fraction remains practically unchanged under changes of concentration of polynucleotides. A significant increase in the fraction of current carried by a counterion ( t h a t is, the decrease in the transference number of a polyion with the increase in the concentration of polynucleotides) is due not to the increase in the number of free counterions, as is observed in d e ~ t r a s u l f a t e , ~ ~ polystyrene sulfonic acid, 55 and its sodium salt sol u t i o n ~In . ~the ~ enumerated solutions ( i n contrast t o those studied by u s ) , f appreciably grows with the increase in the polymer concentration. The increase in the transference numbers of counterions with the increase of the concentration of polynucleotides can be explained by the rise in their mobility a t the cost of “surface” conductance and the decrease in the mobility of the polyion. In its turn, the decrease in the mobility of the polyion with the increase in the concentration of polynucleotides may apparently be due to the decrease in the electrostatic potential of the polyion owing t o the shielding of its charge by free counterions. In concentrated salt-free solutions of DNA such a shielding, known as selfprotective effect,57causes the melting point of DNA t o rise with the increase in the concentration of DNA.57,58
REFERENCES Anderson, C. F. & Record, M. Th., Jr. (1982) Ann. Rev. Phys. Chem. 33, 191-222. Klein, B. K., Anderson, C. F. & Record, M. Th., Jr. (1981) Biopolymers 20,2263-2280. Record, M. Th., Jr., Mazur, S. J., Melancon, P., Roe, J.-H., Shaner, S. L. & Unger, L. (1981) Ann. Rev. Biochem. 50,997-1024. Fried, M. G. & Bloomfield, V. A. (1984) Biopolymers 23, 2141-2155.
76
KUZNETSOV, VORONTSOVA, AND KOZLOV
5. Bleam, M. L., Anderson, C. F. & Record, M. Th., Jr. (1983) Biochemistry 22, 5418-5425. 6. Frank-Kamenetskii, M. D., Lukashin, A. V. & Anshelevich, V. V. (1985) J. Biomol. Struct. Dynam. 3, 35-42. 7. Richard, A. J. (1984) Biopolymers 23,1307-1313. 8. Post, C. B. & Zimm, B. H. (1982) Biopolymers 21, 2133-2137. 9. Manning, G. S. (1983) in Structure and Dynamics: Nucleic Acids and Proteins, Clementi, E. & Sarma, R. H., Eds., Adenine Press, New York, pp. 289-300. 10. Anshelevich, V. V., Lukashin, A. V. & Frank-Kamenetskii, M. D. ( 1984) Chem. Phys. 31, 225-236. 11. Grosberg, A. Yu. & Zhestkov, A. V. (1986) J. Biomol. Struct. Dynam. 3,859-872. 12. Frank-Kamenetskii, M. D. (1983) Mol. Biol. (Moscow) 17,639-652. 13. Frank-Kamenetskii, M. D. & Chogovadze, G. I. (1984) J . Biomol. Struct. Dynam. 6,1516-1523. 14. Frank-Kamenetskii, M. D., Anshelevich, V. V., Lukashin, A. V. (1987) Usp. Fiz. Nauk 151, 595-618. 15. Manning, G. S. (1985) Cell Biophys. 7,57-89. 16. Widom, J. & Baldwin, R. L. (1983) Biopolymers 22, 1595-1620. 17. Grosberg, A. Yu., Erukhimovitch, I. Ya. & Shakhnovitch, E. I. (1982) Biopolymers 21, 2413-2432. 18. Clementi, E. (1983) in Structure and Dynamics: N u cleic Acids and Proteins, Clementi, E. & Sarma, R. H., Eds., Adenine Press, New York, pp. 321-364. 19. Klein, B. J. & Pack, G. R. (1983) Biopolymers 22, 2331-2352. 20. Lee, W. K. & Prohofsky, E. W. (1984) Biopolymers 23, 257-270. 21. Manning, G. S. (1978) Quart. Rev. Biophys. 11,179242. 22. Manning, G. S. (1975) J . Phys. Chem. 79, 262-265. 23. Manning, G. S. (1972) Ann. Rev. Phys. Chem. 23, 117-140. 24. Varoqui, R. & Schmitt, A. ( 1973) J. Chem. SOC.Faraday Trans. 2 69, 1087-1103. 25. Szymczak, J., Holyk, P. & Ander, P. (1975) J . Phys. Chem. 79, 269-272. 26. Kwak, J. C. T. & Johnston, A. J. (1975) Can. J . Chem. 53, 792-796. 27. Kwak, J. C. T., Murphy, G. F. & Spiro, E. J. (1978) Biophys. Chem. 7, 379-386. 28. Kuznetsov, I. A,, Apolonnik, N. V. & Maznikova, L. F. (1978) Mol. Biol. (Moscow) 12, 546-551. 29. Kuznetsov, I. A. & Apolonnik, N. V. (1981) Biopolymers 20,2083-2093. 30. Kuznetsov, I. A., Filippov, S. M. & Vorontsova, 0. V. (1979) Mol. Biol. (Moscow) 13,543-549. 31. Kuznetsov, I. A., Vorontsova, 0. V. & Fabrichnaya, 0. B. (1983) Mol. Biol. (Moscow) 17,392-402.
32. Kuznetsov, I. A. & Vorontsova, 0. V. (1984) Mol. Bwl. (Moscow) 18,1032-1041. 33. Vink, H. ( 1981) J . Chem. SOC.Faraday Trans. I 77, 2439-2449. 34. Vink, H. (1982) Makromol. Chem. 183, 2273-2283. 35. Karpetsky, T., Bogyski, M. & Levy, C. ( 1979) in Subcellular Biochemistry Vol. 6, Roodyn, D., Ed., Plenum Press, New York and London, pp. 1-116. 36. Finch, J. T. & Klug, A. (1969) J . Mol. Biol. 46, 597598. 37. Adler, A. I., Grossman, L. & Fasman, C. D. (1969) Biochemistry 8,3846-3859. 38. Gray, D. M. & Bollum, F. J. (1974) Biopolymers 13, 2087-2102. 39. Glasstone, S. ( 1947) A n Introduction to Electrochemistry, Van Nostzand, New York. 40. Fixman, M. ( 1979) J . Chem. Phys. 70,4995-5005. 41. Richards, E. G., Flessel, C. P. & Fresco, J. R. (1963) Biopolymers 1, 431-446. 42. Wits, J. & Luzzati, V. ( 1965) J. Mol. Biol. 11, 620630. 43. Archer, B. G., Craney, C. Z. & Krakauer, H. (1972) Biopolymers 11, 781-809. 44. Stannard, B. S. & Felsenfeld, G. ( 1975) Biopolymers 14, 299-307. 45. Manning, G. S. (1976) Biopolymers 15, 2385-2390. 46. Kwak, J. C. T. & Hayes, R. C. (1975) J . Phys. Chem. 79, 265-269. 47. Katchalsky, A. (1971) Pure Appl. Chem. 26, 327374. 48. Eisenberg, H. (1958) J . Polym. Sci. 30,47-66. 49. Shavit, N. (1973) Zsr. J. Chem. 11, 235-242. 50. Nelson, R. E. & Ander, P. (1971) J. Phys. Chem. 75, 1691-1697. 51. Gregor, H. P. & Gold, D. H. (1957) J . Phys. Chem. 61, 1347-1352. 52. Devore, D. J. & Manning, G. S. (1974) J. Phys. Chem. 78, 1242-1244. 53. Nagasawa, M., Noda, J., Takahashi, T. & Shimamoto, N. (1972) J . Phys. Chem. 76, 2286-2294. 54. Joshi, Y. M. & Kwak, J. C. T. (1980) Biophys. Chem. 12, 323-328. 55. Dolar, D., Span, J. & Pretnar, A. (1968) J . Polym. Sci. Part C 16, 3557-3564. 56. Dolar, D., Span, J. & Isakovic, S. (1974) Biophys. Chem. 1, 312-317. 57. Auer, H. E. & Alexandrowicz, Z. (1969) Biopolymers 8,1-20. 58. Akhrem, A. A., Andrianov, V. T., Vlasov, A. P., Korolyev, N. I. & Kuznetsov, I. A. (1985) Mol. Biol. (MOSCOW) 19,623-628. Received November 11, 1988 Accepted July 3, 1990
