Erratum: “The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling” [J. Chem. Phys. 101, 5186 (1994)]

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Erratum: “The Kirchhoff elastic rod, the nonlinear Schrödinger equation,
and DNA supercoiling” [J. Chem. Phys. 101, 5186 (1994)]
Y. Shi, J. E. Hearst, T. C. Bishop, and H. R. Halvorson
Citation: J. Chem. Phys. 109, 2959 (1998); doi: 10.1063/1.476848
View online: http://dx.doi.org/10.1063/1.476848
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ERRATA
Erratum: ‘‘The Kirchhoff elastic rod, the nonlinear Schro
¨dinger equation,
and DNA supercoiling’’ †J. Chem. Phys. 101, 5186 „1994…‡
Y. Shi, J. E. Hearst,a) and T. C. Bishop
Department of Chemistry, University of California, Berkeley, California 94720-1460
H. R. Halvorson
Molecular Biology Research Program, Henry Ford Hospital, Detroit, Michigan 48202
@S0021-9606~98!00331-6#
Errors have been detected in our paper,1none of which
affect the mathematical results or conclusions. We apologize
for any inconvenience caused. There are errors in Eqs. ~3.9b!
and ~4.24!. In addition, the original authors are grateful to H.
Halvorson for bringing to our attention an error in our nu-
merical implementation of Eq. ~5.14a!in generating Sec.
VD~Numerical examples!. The errors in no way affect the
mathematical validity of the derivations, although some nu-
merical results have changed significantly.
Below are the corrected Eqs. ~3.9b!and ~4.24!, as well
as a corrected Sec. V D. In the text of the latter, only the
values of nin paragraph ~5!just preceding Sec. VI ~Conclu-
sions!have changed. In the table and figures, all of the val-
ues have been affected. Our conclusions still stand. In par-
ticular, the errors have not undermined our conclusion that
Eq. ~3.10!describes the observed folding of DNA into 11 nm
and 30 nm fibers in vivo. Recent examples of the use of these
and more advanced equations in describing the 11 nm and
the 30 nm fibers are presented in Refs. 2, 3, and 4 at the end
of this Erratum. The conclusions and mathematical imple-
mentations in these references are not affected by the correc-
tions described here.
b5E1~
v
3
~0!!21~22l!
v
3
~0!~Q/l!1~12l!~Q/l!2,~3.9b!
T~
k
,
r
!51
N2
r
S
1
k
F
JQ1
S
E21
2Q2
D
k
221
2
k
4
G
,2Q
k
˙,
kk
˙
2P
k
˙,21
k
F
JP1
S
N21
2PQ
D
k
2
G
,
S
E21
2
k
2
D
P2NQ
N
r
k
˙,
S
J
k
2Q
k
2
D
N
r
,
S
1
2
k
22E
D
N
r
D
.~4.24!
TABLE I. Numerical values calculated follow the procedure outlined in Sec. V D.
m
˜
i
n
˜
i
i(
m
i)(
n
i)nDLk Tw/DLk Wr/DLk UTotal /UCircle UTwist /UTotal UBend /UTotal
1 0.499752 0.937809 226 225 0.220696 0.779304 1.051157 0.046262 0.953738
~0.412852!~0.937809!
2 0.562521 0.984847 227 225 0.109145 0.890855 1.121877 0.010601 0.989399
~0.464706!~0.956318!
3 0.577892 0.999228 228 225 0.025301 0.974699 1.202923 0.000531 0.999469
~0.477405!~0.961110!
JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 7 15 AUGUST 1998
LETTERS TO THE EDITOR
The Letters to the Editor section is divided into three categories entitled Notes, Comments, and Errata. Letters to the Editor are
limited to one and three-fourths journal pages as described in the Announcement in the 1 July 1998 issue.
29590021-9606/98/109(7)/2959/3/$15.00 © 1998 American Institute of Physics
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V. TOROIDAL HELIX
D. Numerical examples
In this subsection we shall show briefly some numerical
examples of the toroidal helix. This numerical calculation
has been carried out using the software program
Mathematica5which has built in all the elliptic functions and
elliptic integrals.
~1!We choose l5C/A51, and we assume that the
DNA duplex helix contains 500 duplex turns for which the
total length Lis equal to 1768 nm ~50.34 nm/basepair
310.4 basepair/turn 3500 turn!and the radius of the DNA
duplex helix is 1 nm. We also assume that the linking num-
ber deficiency
s
5DLk/Lk0520.05. Thus DLk5500
3(20.05)5225.
~2!There are four possible sign combinations for
$
sign(J),sign(Q)
%
, namely, ~6,6!and ~6,7!. For
each combination of
$
sign(J),sign(Q)
%
, since
u
n
u
.0,
we find numerically from ~5.14a!that sign(l) is determined.
For simplicity we consider here only the first two cases;
i.e.,
~a!sign(J)5sign(Q)5sign(l)51, which will result in
Wr.0, DTw.0, and n.DLk5Wr1DTw.0;
~b!the mirror image of ~a!, i.e., sign(J)5sign(Q)
5sign(l)521, which will result in Wr,0, DTw,0, and
n,DLk5Wr1DTw,0; Because we have already set
DLk5225, only Case ~b!needs to be considered.
~3!For Case ~b!we find out in the
m
˜
2
n
˜
plane that curve
DLk5225 intersects with curves n5226,227,228,...,at
points
$
m
˜
i,
n
˜
i
%
, which are tabulated in Table I.
~4!We then calculate and tabulate in Table I the follow-
ing quantities for Case ~b!:UTotal /UCircle ,UTwist /UTotal ,
FIG. 1. x–yplot ~top!and
r
–zplot ~bottom!for Case ~b!with
DLk5225 and n5226. The units of
$
x,y,z,
r
%
are nanometers.
The formulas for
$
r
,
f
,z
%
~with x5
r
cos
f
,y5
r
sin
f
) are given
in ~4.19!, whereas the parameters in ~4.19!are given
by
m
50.4128522253359228563,
n
50.9378086691008355865,
w520.0526363802842685069, E50.0039768655823549271,
Q520.0196419884347520515, N50.0012196983004668965, and
P50.0196419884347520515.
FIG. 2. x–yplot ~top!and
r
–zplot ~bottom!for Case ~b!with
DLk5225 and n5227. The units of
$
x,y,z,
r
%
are nanometers.
The formulas for
$
r
,
f
,z
%
~with x5
r
cos
f
,y5
r
sin
f
) are given
in ~4.19!, whereas the parameters in ~4.19!are given
by
m
50.4647064549310418084,
n
50.9563175595835276121,
w520.0558460231662935138, E50.0044031225340830020,
Q520.0097138825351320032, N50.0016083689287922024, and
P50.0354819195407423767.
2960 J. Chem. Phys., Vol. 109, No. 7, 15 August 1998 Letters to the Editor
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UBend /UTotal ,Tw/DLk,Wr/DLk. The energy UCircle is the
total elastic energy of the thin rod whose axis is a circle with
radius 2
p
/Land twist DTw5DLk5225. It is given ex-
plicitly by ~5.17!.
~5!We make three plots ~Figs. 1, 2, and 3!for n
5226, n5227, and n5228 to show the basic features of
the toroidal helices generated by the elliptic function solu-
tions.
We notice immediately, from Figs. 1–3, a distinctive
feature: the cross section of the torus on which the DNA is
wound is nearly, but not precisely, a circle.
a!Author to whom correspondence should be addressed.
1Y. Shi and J. E. Hearst, J. Chem. Phys. 101, 5186 ~1994!.
2Y. Shi, A. Borovik, and J. E. Hearst, J. Chem. Phys. 103, 3166 ~1995!.
3J. E. Hearst and Y. Shi, Nonlinear Sci. Today 5,1~1995!~http://
www.springer-ny.com/nst!.
4T. C. Bishop and J. E. Hearst, J. Phys. Chem. ~in press!.
5Mathematica is a trademark of Wolfram Research, Inc., 100 Trade Center
Drive, Champaign, Illinois 61820-7237.
FIG. 3. x–yplot ~top!and
r
–zplot ~bottom!for Case ~b!with
DLk5225 and n5228. The units of
$
x,y,z,
r
%
are nanometers.
The formulas for
$
r
,
f
,z
%
~with x5
r
cos
f
,y5
r
sin
f
) are given
in ~4.19!, whereas the parameters in ~4.19!are given
by
m
50.4774045841894475342,
n
50.9611103691951271582,
w520.0582353482409647673, E50.0047692646660506139,
Q520.0022517708616683947, N50.0019246021965750458, and
P50.0496116251388530552.
2961J. Chem. Phys., Vol. 109, No. 7, 15 August 1998 Letters to the Editor
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