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Rate of unwinding of DNA
J. Mal. Rial. (1963) 6, 39-45
Rate of Unwinding of DNA MARSHALL l!'IXMAN
Institute af Theoretical Science and Department af Chemistry, University af Oregan, Eugene, Oregon, U.S.A. (Received 25 September 1962) The mean time of unwinding of a DNA molecule is calculated. The unwinding is assumed to begin at one end and to proceed under the combined forces of diffusion and a driving torque of arbitrary magnitude. The standard deviation of the unwinding time is calculated approximately. If the free energy decrease on unwinding one turn, in units of kT, times the number of turns in the molecule is a number much greater than unity, diffusion plays a negligible role in the unwinding.
Longuet-Higgins & Zimm (1960) have proposed a mechanism for the unwinding of a double-stranded helix and calculated the time of unwinding. They suppose that both ends ofthe rod- like helix unwind at the same rate and that the two unwound portions form random coils which are attached at their midpoints to the helix. The rotation of the coils about their midpoints is the process which causes the unwinding, and this process is assumed to be driven by a torque large compared to kT. The source of this assumed torque is the increase in rotational entropy that results when sections of the chain are transferred from the helix to the coil. As they point out, their discussion differs mainly from Kuhn's (1957) in the postulate of a driving force which is large compared to kT, and a general discussion would include both the forced unwinding and the pure diffusion as special cases. One of the objects of this note will be to compute the mean time of unwinding for It driving force of arbitrary magnitude, including zero. With less generality, the standard deviation of this time is also computed. A slight but significant change in the model used by Longuet-Higgins & Zimm will be made. Instead of the assumption that unwinding proceeds from both ends, it will be assumed that at time zero the unwinding begins at one end. The probability per unit time that unwinding is initiated is assumed small enough that the event of simultaneous initiation at both ends can be neglected. More precisely, it is assumed that the probability of a second initiation during the unwinding process is small. The consequences of this assumption will be briefly discussed later. For the present it will be noted that in the absence of opposite torques at the two ends of the helical rod the rod, as well as the coil, will be free to rotate in the solution, in contrast to the model of Longuet-Higgins & Zimm. It is only the relative motion of the rod and the coil that are of interest in a determination of the rate of unwinding. The present picture is, accordingly, a combination of the ideas of Longuet-Higgins & Zimm on the one hand and Levinthal & Crane (1956) on the other. See also Delbruck & Stent (1957). It will be recalled that Levinthal & Crane argued that the helical rod, even though not fully extended, would still be free to rotate on its axis much like a 39
40
MARSHALL FIXMAN
speedometer cable. Whatever the relevance of the. present picture to biological problems, it would seem to be the most natural picture of an in vitro solution of DNA. In an intermediate state of unwinding, at a time t after the unwinding has started, the rod will have rotated through an angle 81 and the coil through an angle 82 , The excess 8 = 82 - 81 is the number of radians of the original two-stranded configuration which have actually untwisted and gone into the coil form. Let g( 8, t) d8 be the probability that 8 lies in d8. An equation of conservation 8g/at
(1)
= -8(g8)/88
relates g to the drift velocity 8. If torques T1 and T z act on the rod and the coil respectively, and the rotational friction constants are /31 and /3z' the drift velocities are given by the following balance between the viscous resistance, the torque and the rotational force of diffusion. (2)
The /3i are functions of 8, although not of 81 and 82separately. The functional dependence will be exhibited in detail later but it is noted here that /31is proportional to the length of the rod and /32 is proportional to the cube of the random coil "radius" in the approximation used. Accordingly, /31 is linear in 8 and /32 is proportional to ()3/2. The torque T i is given by T i = -8V(8)j88i
where V(8) is the free energy of the molecule, rod plus coil, and consequently T2 is the free energy decrease on unwinding one radian of rod and is reasonably taken independent of 8. Let (3)
then equations (1) and (2) give
8 = 82 - 81 = kT(/311+/3z1)(a -
8 ln gj88).
(4)
Equations (1), (2) and (4) together give _ 8g =
8t
~
88
[ayg - / 08g]
(5)
where y( 8) is a diffusion constant y
=kT(/311+/3z1).
(6)
The problem is the solution of equation (5) subject to initial and boundary conditions. These are that g( 8, t) satisfy g(8,0) = 8(8)
g(N,t) = 0;
(7)
N = max 8
(ayg-y8gj88)(J=O
= 0.
(8) (9)
Equation (7) expresses the condition that unwinding starts at time zero; 8(8) is a delta function. In equation (8), N is the total number of radians of twist in the original rod, and equation (8) states that molecules pass into the region 8 ~ N but do not return; that is 8 = N is a "sink". Finally, equation (9) requires the flux of molecules through 8 = 0 to vanish, since the possibility of negative 8 is excluded.
RATE OF UNWINDING OF DNA
41
In equation (7), the delta function should be shifted slightly off the origin, but the cumbersome notation required to indicate this will be suppressed. Although an approximate solution of equation (5) can be given, based on the assumption of large a and a slowly varying y, the main object can be reached with no approximation. This object is to determine the mean time i of unwinding, and the standard deviation of this time. Let O(t) = LNg( 6, t) d6
(10)
be the probability that the molecule has not completed its unwinding at time t. Then -dO/dt is the probability that the unwinding is completed at t, per unit time. Consequently (t n
dO >= - f.ooo tn-dt. dt
(11)
In particular
i = LooO(t)dt = J:V(6)d6
(12)
where v( 8) =
Loog( 6, t) dt
and
>= 2 LootG(t)dt =
(t 2
(13)
21
N
w(8)d8
(14)
where w(8)
= Lootg(6,t)dt.
(15)
It is obvious that conditions (8) and (9) also apply to v(6) and w(8). Equations for v and w may be obtained from equation (5). Integration of equation (5) over t gives g(8,0)
a[aYV- Y88av] = 8(6) = 86
.
(16)
Multiplication of equation (5) by t and integration over t gives
8W]
8 [ v = 08 ayw-y 06 .
(17)
Oh] = -e [ayh-y-
(18)
Both equations have the form f(8)
86
88
where y andf are known and h is the unknown function which satisfies equations (8) and (9). Equation (18) integrates to h = e al1 {K2 - f:y-le-al1 [f:f(8)d6+K1 ] d8}
where K 1 and K 2 are constants of integration. Equation (19) gives for any 6 yah+yh' = ffd6+Kl
(19)
MARSHALL FIXMAN
42
and equation (9) therefore requires (20)
Equation (8) requires (21) and therefore (22)
Let
F(8) == y-I
lOfdO.
(23)
The computation of land ( t2 ) requires
LNh(O) d O = a-I IoNF(O) (l-e- aO) d 8,
(24)
as follows from equations (22) and (23). Equations (18) to (24) will now be applied to the computation of land (t 2 ) . First put f( 8) = 8(8) and h = v. Equations (12), (23) and (24) give
l=a-llN~(l-e-aO)d8; ~==y-I.
(25)
The practical evaluation of ( t2 ) cannot be made quite so neatly, but a good approximation is available under the modest condition aN'}> 1. As a preliminary, equation (22) gives
L
v = eao
N
e- ao~dO
= a-l ~+a-2(~' _e-a(N-O) ~'(N» + .. In equations (18) to (24) , put h =
W,
f
(26)
= v/kT. Then
F(O) = ~ S:VdO =
a-l~ lO~do+a-2p+ '"
(27)
and
I~wdO = a-I I:F(O)d8+a-Se- aNF'(N)+ ...
~ a-I I:{a-l~f:~do+a-2P}dO.
(28)
Now (29)
and therefore
( t2) = 2 f: wdO =
a-2[I:~dOr +2a-s f: pd8+ ....
(30)
From equation (25) (31)
RATE OF UNWINDING OF DNA
43
Consequently
or (32)
Even without numerical calculation it can be seen that the neglect of diffusion by Longuet-Higgins & Zimm and Levinthal & Crane has a minor effect on the mean time ofunwinding, I, unless a is extremely small. For example, if the free energy decrease per base pair is only 0·01 leT (a ~ 0·01), equation (25) will be closely approximated by
i=
a-I
LNPdB
if N> 103, as is certainly applicable. This does not mean that a is small; it is surprisingly large in view of the extreme slowness of unbiased diffusion. The friction constants to be used in (33)
are (34) for a cylinder of length L o{1- (B/N)} and radius R 1 in a medium of viscosity Zimm's result
7],
and (35)
for a coil of molecular weight M, where No is Avogadro's number. With a molecular weight of 660/base pair, 10 pairs/twist, a pitch of 38 A/twist, R 1 = 10 A, 7] = 0·04 poise and T = 300 0 K as in previous estimates, M=1050xB
PI = 3·04 x 1O-22(N - B)erg sec P2 = 6·24 X 10-22 ()3/2 erg sec.
(36) (37)
It is evident that P2 > PI for quite small values of B/N; if N = 103, P2 > PI for B/N> 0,06, and the rod will begin rotating faster than the coil. To an adequate approximation, the rotation of the coil can be neglected entirely. Thus if aN}:> I,
i~
a-I
JoNPdB = 3·6 x 10-9 a-I N2
(38)
if P2 is taken to be infinite. If equation (37) is used for P2' a numerical integration gives ionly 20% smaller than equation (38) for N = 103 , or a molecular weight of 106 , and the error decreases rapidly with increasing N. Equation (38) is the result of
44
MARSHALL FIXMAN
Levinthal & Crane, and gives a much smaller t that the result of Longuet-Higgins & Zimm. There is a small error in their calculation. Because unwinding takes place at both ends, they double the rate of unwinding at one end prior to the computation of the time i for total unwinding. What should have been done was to halve the total mass to be unwound in i. Consequently their i was too large by a factor of 23/ 2• The latter estimate a = 17·5 and find, for N = 1()4, t-;;:, 1 sec, but equation (38) gives t= 0-02 sec.
L..----------~-t2
FIG. 1. Schematic graph of the mean unwinding time at the second end of the helix is initiated.
t ver8U8 the tiIUe t. at which unwinding
For unbiased unwinding, a = 0 and equations (25) and (36) give
t=
ioN {3fJdfJ = 1-2 x 10-9 N3
(39)
or t = 20 minutes for N = 1()4. If for some reason only the coil could rotate, equation (37) would give i = 4 X 10-9 N7/2 or some five days. Finally, for aN~ I, equations (32) and (36) give a =
)1/2 = 4
8 (3aN
X
10-3
for a = 17-5 and N = 1()4. Thus a closely approaches the square root of the ratio of unwinding times for the biased and unbiased unwinding (equations (38) and (39». This will seem very reasonable after brief reflection, though an intuitive judgement will probably not supply the square root. Regarding the possibility of initiation at both ends, it seems probable that the calculation of Longuet-Higgins & Zimm is substantially correct if the two initiations take place during a period in which only a few turns have unwound, and the unwinding will be slower than has been calculated here. On the other hand, if the second initiation at end number two occurs toward the end of the unwinding, the opposing torques will only slightly slow the rotation of the rod, because of the large inertia of the coil which has already formed at end number one; and the formation of the eoil will speed up the unwinding. A sketch of t against tz' the time of the second initiation, is given in Fig. I, where to is the time computed for initiation at one end. Intermediate values of t z will of course produce a large dispersion in i.
RATE OF UNWINDING OF DNA
45
A numerical calculation of i from equation (25) is given in Fig. 2 for various values of a and N. Equations (36) and (37) were used for the two friction constants. I07r----r---~--__r---..,
N (radians)
FIG. 2. Unwinding time versus size of helix, where a is the free energy decrease on unwinding one radian, in units of kt,
This work was supported in part by the National Science Foundation and a PHS research grant GM 09153 from the Division of General Medical Sciences, Public Health Service. The author is an Alfred P. Sloan Research Fellow. REFERENCES Delbriick, M. & Stent, G. (1957). In The Chemical Basis of Heredity, ed. by W. McElroy & B. Glass, p. 699. Baltimore: Johns Hopkins Press. Kuhn, W. (1957). Experimenta, 13, 301. Levinthal, C. & Crane, H. R. (1956). Proc. Nat. Acad. s«, Wash. 42, 436. Longuet-Higgins, H. C. & Zimm, B. H. (1960). J. Mol. Biol. 2, 1.
Rate of Unwinding of DNA MARSHALL l!'IXMAN
Institute af Theoretical Science and Department af Chemistry, University af Oregan, Eugene, Oregon, U.S.A. (Received 25 September 1962) The mean time of unwinding of a DNA molecule is calculated. The unwinding is assumed to begin at one end and to proceed under the combined forces of diffusion and a driving torque of arbitrary magnitude. The standard deviation of the unwinding time is calculated approximately. If the free energy decrease on unwinding one turn, in units of kT, times the number of turns in the molecule is a number much greater than unity, diffusion plays a negligible role in the unwinding.
Longuet-Higgins & Zimm (1960) have proposed a mechanism for the unwinding of a double-stranded helix and calculated the time of unwinding. They suppose that both ends ofthe rod- like helix unwind at the same rate and that the two unwound portions form random coils which are attached at their midpoints to the helix. The rotation of the coils about their midpoints is the process which causes the unwinding, and this process is assumed to be driven by a torque large compared to kT. The source of this assumed torque is the increase in rotational entropy that results when sections of the chain are transferred from the helix to the coil. As they point out, their discussion differs mainly from Kuhn's (1957) in the postulate of a driving force which is large compared to kT, and a general discussion would include both the forced unwinding and the pure diffusion as special cases. One of the objects of this note will be to compute the mean time of unwinding for It driving force of arbitrary magnitude, including zero. With less generality, the standard deviation of this time is also computed. A slight but significant change in the model used by Longuet-Higgins & Zimm will be made. Instead of the assumption that unwinding proceeds from both ends, it will be assumed that at time zero the unwinding begins at one end. The probability per unit time that unwinding is initiated is assumed small enough that the event of simultaneous initiation at both ends can be neglected. More precisely, it is assumed that the probability of a second initiation during the unwinding process is small. The consequences of this assumption will be briefly discussed later. For the present it will be noted that in the absence of opposite torques at the two ends of the helical rod the rod, as well as the coil, will be free to rotate in the solution, in contrast to the model of Longuet-Higgins & Zimm. It is only the relative motion of the rod and the coil that are of interest in a determination of the rate of unwinding. The present picture is, accordingly, a combination of the ideas of Longuet-Higgins & Zimm on the one hand and Levinthal & Crane (1956) on the other. See also Delbruck & Stent (1957). It will be recalled that Levinthal & Crane argued that the helical rod, even though not fully extended, would still be free to rotate on its axis much like a 39
40
MARSHALL FIXMAN
speedometer cable. Whatever the relevance of the. present picture to biological problems, it would seem to be the most natural picture of an in vitro solution of DNA. In an intermediate state of unwinding, at a time t after the unwinding has started, the rod will have rotated through an angle 81 and the coil through an angle 82 , The excess 8 = 82 - 81 is the number of radians of the original two-stranded configuration which have actually untwisted and gone into the coil form. Let g( 8, t) d8 be the probability that 8 lies in d8. An equation of conservation 8g/at
(1)
= -8(g8)/88
relates g to the drift velocity 8. If torques T1 and T z act on the rod and the coil respectively, and the rotational friction constants are /31 and /3z' the drift velocities are given by the following balance between the viscous resistance, the torque and the rotational force of diffusion. (2)
The /3i are functions of 8, although not of 81 and 82separately. The functional dependence will be exhibited in detail later but it is noted here that /31is proportional to the length of the rod and /32 is proportional to the cube of the random coil "radius" in the approximation used. Accordingly, /31 is linear in 8 and /32 is proportional to ()3/2. The torque T i is given by T i = -8V(8)j88i
where V(8) is the free energy of the molecule, rod plus coil, and consequently T2 is the free energy decrease on unwinding one radian of rod and is reasonably taken independent of 8. Let (3)
then equations (1) and (2) give
8 = 82 - 81 = kT(/311+/3z1)(a -
8 ln gj88).
(4)
Equations (1), (2) and (4) together give _ 8g =
8t
~
88
[ayg - / 08g]
(5)
where y( 8) is a diffusion constant y
=kT(/311+/3z1).
(6)
The problem is the solution of equation (5) subject to initial and boundary conditions. These are that g( 8, t) satisfy g(8,0) = 8(8)
g(N,t) = 0;
(7)
N = max 8
(ayg-y8gj88)(J=O
= 0.
(8) (9)
Equation (7) expresses the condition that unwinding starts at time zero; 8(8) is a delta function. In equation (8), N is the total number of radians of twist in the original rod, and equation (8) states that molecules pass into the region 8 ~ N but do not return; that is 8 = N is a "sink". Finally, equation (9) requires the flux of molecules through 8 = 0 to vanish, since the possibility of negative 8 is excluded.
RATE OF UNWINDING OF DNA
41
In equation (7), the delta function should be shifted slightly off the origin, but the cumbersome notation required to indicate this will be suppressed. Although an approximate solution of equation (5) can be given, based on the assumption of large a and a slowly varying y, the main object can be reached with no approximation. This object is to determine the mean time i of unwinding, and the standard deviation of this time. Let O(t) = LNg( 6, t) d6
(10)
be the probability that the molecule has not completed its unwinding at time t. Then -dO/dt is the probability that the unwinding is completed at t, per unit time. Consequently (t n
dO >= - f.ooo tn-dt. dt
(11)
In particular
i = LooO(t)dt = J:V(6)d6
(12)
where v( 8) =
Loog( 6, t) dt
and
>= 2 LootG(t)dt =
(t 2
(13)
21
N
w(8)d8
(14)
where w(8)
= Lootg(6,t)dt.
(15)
It is obvious that conditions (8) and (9) also apply to v(6) and w(8). Equations for v and w may be obtained from equation (5). Integration of equation (5) over t gives g(8,0)
a[aYV- Y88av] = 8(6) = 86
.
(16)
Multiplication of equation (5) by t and integration over t gives
8W]
8 [ v = 08 ayw-y 06 .
(17)
Oh] = -e [ayh-y-
(18)
Both equations have the form f(8)
86
88
where y andf are known and h is the unknown function which satisfies equations (8) and (9). Equation (18) integrates to h = e al1 {K2 - f:y-le-al1 [f:f(8)d6+K1 ] d8}
where K 1 and K 2 are constants of integration. Equation (19) gives for any 6 yah+yh' = ffd6+Kl
(19)
MARSHALL FIXMAN
42
and equation (9) therefore requires (20)
Equation (8) requires (21) and therefore (22)
Let
F(8) == y-I
lOfdO.
(23)
The computation of land ( t2 ) requires
LNh(O) d O = a-I IoNF(O) (l-e- aO) d 8,
(24)
as follows from equations (22) and (23). Equations (18) to (24) will now be applied to the computation of land (t 2 ) . First put f( 8) = 8(8) and h = v. Equations (12), (23) and (24) give
l=a-llN~(l-e-aO)d8; ~==y-I.
(25)
The practical evaluation of ( t2 ) cannot be made quite so neatly, but a good approximation is available under the modest condition aN'}> 1. As a preliminary, equation (22) gives
L
v = eao
N
e- ao~dO
= a-l ~+a-2(~' _e-a(N-O) ~'(N» + .. In equations (18) to (24) , put h =
W,
f
(26)
= v/kT. Then
F(O) = ~ S:VdO =
a-l~ lO~do+a-2p+ '"
(27)
and
I~wdO = a-I I:F(O)d8+a-Se- aNF'(N)+ ...
~ a-I I:{a-l~f:~do+a-2P}dO.
(28)
Now (29)
and therefore
( t2) = 2 f: wdO =
a-2[I:~dOr +2a-s f: pd8+ ....
(30)
From equation (25) (31)
RATE OF UNWINDING OF DNA
43
Consequently
or (32)
Even without numerical calculation it can be seen that the neglect of diffusion by Longuet-Higgins & Zimm and Levinthal & Crane has a minor effect on the mean time ofunwinding, I, unless a is extremely small. For example, if the free energy decrease per base pair is only 0·01 leT (a ~ 0·01), equation (25) will be closely approximated by
i=
a-I
LNPdB
if N> 103, as is certainly applicable. This does not mean that a is small; it is surprisingly large in view of the extreme slowness of unbiased diffusion. The friction constants to be used in (33)
are (34) for a cylinder of length L o{1- (B/N)} and radius R 1 in a medium of viscosity Zimm's result
7],
and (35)
for a coil of molecular weight M, where No is Avogadro's number. With a molecular weight of 660/base pair, 10 pairs/twist, a pitch of 38 A/twist, R 1 = 10 A, 7] = 0·04 poise and T = 300 0 K as in previous estimates, M=1050xB
PI = 3·04 x 1O-22(N - B)erg sec P2 = 6·24 X 10-22 ()3/2 erg sec.
(36) (37)
It is evident that P2 > PI for quite small values of B/N; if N = 103, P2 > PI for B/N> 0,06, and the rod will begin rotating faster than the coil. To an adequate approximation, the rotation of the coil can be neglected entirely. Thus if aN}:> I,
i~
a-I
JoNPdB = 3·6 x 10-9 a-I N2
(38)
if P2 is taken to be infinite. If equation (37) is used for P2' a numerical integration gives ionly 20% smaller than equation (38) for N = 103 , or a molecular weight of 106 , and the error decreases rapidly with increasing N. Equation (38) is the result of
44
MARSHALL FIXMAN
Levinthal & Crane, and gives a much smaller t that the result of Longuet-Higgins & Zimm. There is a small error in their calculation. Because unwinding takes place at both ends, they double the rate of unwinding at one end prior to the computation of the time i for total unwinding. What should have been done was to halve the total mass to be unwound in i. Consequently their i was too large by a factor of 23/ 2• The latter estimate a = 17·5 and find, for N = 1()4, t-;;:, 1 sec, but equation (38) gives t= 0-02 sec.
L..----------~-t2
FIG. 1. Schematic graph of the mean unwinding time at the second end of the helix is initiated.
t ver8U8 the tiIUe t. at which unwinding
For unbiased unwinding, a = 0 and equations (25) and (36) give
t=
ioN {3fJdfJ = 1-2 x 10-9 N3
(39)
or t = 20 minutes for N = 1()4. If for some reason only the coil could rotate, equation (37) would give i = 4 X 10-9 N7/2 or some five days. Finally, for aN~ I, equations (32) and (36) give a =
)1/2 = 4
8 (3aN
X
10-3
for a = 17-5 and N = 1()4. Thus a closely approaches the square root of the ratio of unwinding times for the biased and unbiased unwinding (equations (38) and (39». This will seem very reasonable after brief reflection, though an intuitive judgement will probably not supply the square root. Regarding the possibility of initiation at both ends, it seems probable that the calculation of Longuet-Higgins & Zimm is substantially correct if the two initiations take place during a period in which only a few turns have unwound, and the unwinding will be slower than has been calculated here. On the other hand, if the second initiation at end number two occurs toward the end of the unwinding, the opposing torques will only slightly slow the rotation of the rod, because of the large inertia of the coil which has already formed at end number one; and the formation of the eoil will speed up the unwinding. A sketch of t against tz' the time of the second initiation, is given in Fig. I, where to is the time computed for initiation at one end. Intermediate values of t z will of course produce a large dispersion in i.
RATE OF UNWINDING OF DNA
45
A numerical calculation of i from equation (25) is given in Fig. 2 for various values of a and N. Equations (36) and (37) were used for the two friction constants. I07r----r---~--__r---..,
N (radians)
FIG. 2. Unwinding time versus size of helix, where a is the free energy decrease on unwinding one radian, in units of kt,
This work was supported in part by the National Science Foundation and a PHS research grant GM 09153 from the Division of General Medical Sciences, Public Health Service. The author is an Alfred P. Sloan Research Fellow. REFERENCES Delbriick, M. & Stent, G. (1957). In The Chemical Basis of Heredity, ed. by W. McElroy & B. Glass, p. 699. Baltimore: Johns Hopkins Press. Kuhn, W. (1957). Experimenta, 13, 301. Levinthal, C. & Crane, H. R. (1956). Proc. Nat. Acad. s«, Wash. 42, 436. Longuet-Higgins, H. C. & Zimm, B. H. (1960). J. Mol. Biol. 2, 1.