End-to-end attraction of duplex DNA
Recent experiments [Nakata, M. et al., End-to-end stacking and liquid crystal condensation of 6 to 20 basepair DNA duplexes. Science 2007; 318:1276-1279] have demonstrated spontaneous end-to-end association of short duplex DNA fragments into long rod-like structures. By means of extensive all-atom molecular dynamic simulations, we characterized end-to-end interactions of duplex DNA, quantitatively describing the forces, free energy and kinetics of the end-to-end association process. We found short DNA duplexes to spontaneously aggregate end-to-end when axially aligned in a small volume of monovalent electrolyte. It was observed that electrostatic repulsion of 5'-phosphoryl groups promoted the formation of aggregates in a conformation similar to the B-form DNA double helix. Application of an external force revealed that rupture of the end-to-end assembly occurs by the shearing of the terminal base pairs. The standard binding free energy and the kinetic rates of end-to-end association and dissociation processes were estimated using two complementary methods: umbrella sampling simulations of two DNA fragments and direct observation of the aggregation process in a system containing 458 DNA fragments. We found the end-to-end force to be short range, attractive, hydrophobic and only weakly dependent on the ion concentration. The relation between the stacking free energy and end-to-end attraction is discussed as well as possible roles of the end-to-end interaction in biological and nanotechnological systems.
End-to-end attraction of duplex DNA
, Binquan Luan
and Aleksei Aksimentiev
Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, IL 61801,
Computational Biology Center, IBM Research, 1101 Kitchawan Road, Yorktown Heights, NY,
Received August 26, 2011; Revised October 28, 2011; Accepted November 21, 2011
Recent experiments [Nakata, M. et al., End-to-end
stacking and liquid crystal condensation of 6 to 20
basepair DNA duplexes. Science 2007; 318:1276–
1279] have demonstrated spontaneous end-to-end
association of short duplex DNA fragments
into long rod-like structures. By means of exten-
sive all-atom molecular dynamic simulations,
we characterized end-to-end interactions of duplex
DNA, quantitatively describing the forces, free
energy and kinetics of the end-to-end association
process. We found short DNA duplexes to spontan-
eously aggregate end-to-end when axially aligned
in a small volume of monovalent electrolyte. It
was observed that electrostatic repulsion of
-phosphoryl groups promoted the formation of
aggregates in a conformation similar to the B-form
DNA double helix. Application of an external force
revealed that rupture of the end-to-end assembly
occurs by the shearing of the terminal base pairs.
The standard binding free energy and the kinetic
rates of end-to-end association and dissociation
processes were estimated using two complemen-
tary methods: umbrella sampling simulations of
two DNA fragments and direct observation of the
aggregation process in a system containing 458
DNA fragments. We found the end-to-end force to
be short range, attractive, hydrophobic and only
weakly dependent on the ion concentration. The
relation between the stacking free energy and
end-to-end attraction is discussed as well as
possible roles of the end-to-end interaction in
biological and nanotechnological systems.
Self-assembly properties of nucleic acids are vital to the
basic functions of a biological cell and have been exten-
sively exploited in biotechnology. DNA hybridization—
self-assembly of complementary sequence single-stranded
DNA (ssDNA) into a double helix—is a central biotech-
nological process (1), used, among others, in platforms for
DNA detection (2), programmable assembly of DNA
nanostructures (3,4), directional transport of cargo (5),
molecular computing (6) and nanofabrication (7,8).
Another process of outstanding importance is DNA con-
densation, where counterions transform electrostatic
repulsion between naked DNA molecules into attraction,
facilitating packaging of double-stranded DNA (dsDNA)
in cell nuclei and viral capsids (9,10).
Recently, an entirely different type of DNA self-
assembly was discovered: spontaneous end-to-end aggre-
gation of short duplex DNA fragments into rod-like struc-
tures (11). When water was evaporated from solution
containing a high concentration of short (6–20 bp) DNA
fragments, liquid crystal phases were observed. Since the
DNA fragments were nearly as wide as they were long, the
observation of axial ordering could only be explained if
the fragments formed rod-like supramolecules, suggesting
end-to-end aggregation. Further experimental evidence of
end-to-end association was obtained from the analysis of
small angle X-ray scattering data from a system contain-
ing short DNA fragments and a divalent electrolyte
(12,13). The second virial coefﬁcient extracted from
these data was shown to be positive for DNA fragments
capped with a short hairpin (indicating overall repulsion)
and negative for DNA fragments without such caps
(indicating overall attraction). It was concluded that
end-to-end attraction was large enough to overcome elec-
trostatic repulsion in a divalent electrolyte. While
side-by-side forces between long DNA molecules has
been the subject of many experimental (14–16) and theor-
etical (17,18) studies, little is known about the conditions
and microscopic mechanism of DNA association end-to-
end. Furthermore, the effects of end-to-end attraction of
duplex DNA in biological and technological processes are
Whereas traditional single molecule experiments have
provided extensive information about DNA hybridization
and side-by-side interactions (14,19), applying these tools
to study end-to-end assembly is extremely difﬁcult, as a
*To whom correspondence should be addressed. Tel: +1 217 333 6495; Fax: +1 217 333 9819; Email: email@example.com
3812–3821 Nucleic Acids Research, 2012, Vol. 40, No. 9 Published online 12 January 2012
ß The Author(s) 2012. Published by Oxford University Press.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/
by-nc/3.0), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
DNA duplex cross section is just 5 nm
and the effective
concentration of DNA ends in a solution amenable to
single-molecule studies is typically small. All-atom
molecular dynamics (MD) method is well suited for the
study of systems that share the length scale of short
dsDNA molecules and can be used to probe the atomic
origin of intermolecular forces (20). Here, we use the MD
method to characterize end-to-end association of duplex
DNA in unprecedented detail, elucidating the microscopic
mechanism of spontaneous association, its free energy
costs and the kinetic rates. At the end of the article, we
discuss the relationship between our ﬁndings and the per-
tinent experimental observations.
MATERIALS AND METHODS
General simulation methods
All MD simulations were performed using the program
NAMD (21), the parmbsc0 reﬁnement of the AMBER-
parm99 force ﬁeld (22,23), the TIP3P model of water
(24), standard parameters for ions (25), periodic
boundary conditions, and particle-mesh Ewald (PME)
full electrostatics with a PME grid density of 1A
grid point. Except where speciﬁed, van der Waals and
short-range electrostatic energies were calculated using a
smooth (10–12 A
) cutoff, and integration was performed
using 1–2–4 fs multiple timestepping (21). The temperature
was held constant using a Langevin thermostat (21) applied
to all non-hydrogen atoms; the Langevin damping constant
was set to 0.1 /ps. For simulations in the NPT ensemble,
constant pressure was maintained at 1 bar using the
-Hoover Langevin piston pressure control (26).
Each simulation reported in this study used one of the
following three system types: elongated along the z-axis to
minimize the amount of solvent around two DNA frag-
ments (24 000 atoms, Figure 1a); isotropic to allow two
DNA fragments to tumble freely (56 000 atoms, Figure
2a); and large and isotropic to allow unbiased interaction
between 458 DNA fragments (1.4 M atoms, Figure 4a).
The DNA sequence was poly(dAdT) in all systems.
Counterions were added to each system to neutralize the
DNA charge prior to the addition of a number of ions
corresponding to the reported molarity (100 mM, except
where speciﬁed) of NaCl electrolyte. Steric clashes that
were introduced during the assembly of each system
were removed from each system through minimization
using a conjugate gradient method (27). Equilibration
was performed in the NPT ensemble, and subsequent pro-
duction simulations were performed in the NVT ensemble,
except where speciﬁed.
Collapse of aligned dsDNA
Thirty-six systems were built using the anisotropic unit
cell. The dimensions of each system were chosen to
provide a minimum distance of 2 nm between the
surfaces of the DNA fragments across the periodic
boundary, which should accommodate a majority of
screening counterions for NaCl solutions of 100 mM or
greater concentration (Debye length 1 nm). Steric
clashes were removed through 3000 minimization steps.
Each system was subsequently equilibrated for 65 ps
with the DNA backbone atoms harmonically restrained
to their initial positions. Axial alignment of the DNA frag-
ments was enforced by harmonically restraining each
phosphorous atom of the DNA to the surface of an
radius cylinder (with spring constants of 139 pN/
nm per atom). The DNA fragments could translate
along the axis of the cylinder and rotate azimuthally.
The starting conformation was characterized by a 20.5 A
end-to-end separation, which we deﬁne as the distance
between the centers of mass of the nearest terminal base
pairs, taking the periodic boundary condition into
account. The relative azimuthal angle f of the terminal
base pairs was deﬁned as the angle between the projections
of the vectors connecting the O5
atoms of the
terminal base pairs into the plane normal to the common
DNA axis. For two consecutive base pairs in a B-DNA
, depending on the sequence.
Stability of the end-to-end complex
Systems were built by placing collapsed end-to-end DNA
assemblies in an isotropic volume of 100 mM NaCl elec-
trolyte (Figure 2a). Steric clashes with the solvent were
removed through minimization having the DNA
backbone restrained. Subsequent simulation was per-
formed in the NPT ensemble.
Mechanics of end-to-end dissociation
The coordinates of the system containing
-phosphorylated DNA fragments in a 100 mM electro-
lyte were taken after 140 ns of simulation described in the
section ‘Stability of the end-to-end complex’ to provide
the initial conformation for simulations of rupture of the
DNA fragments. The NVT ensemble was used during
these simulations; Langevin thermostat was applied to
water oxygens and ions. A harmonic spring of k
pN/nm was used to produce the rupture, by increasing its
rest length at a rate of 0.4 or 0.2 A
/ns. The work per-
formed at both rates were in good agreement.
Simulations of the end-to-end fragments having
complementary single-stranded overhangs are described
in Section 1.4 of Supplementary Data.
Potential of mean force of axially aligned DNA duplexes
Umbrella sampling simulations were performed using the
anisotropic systems (Figure 1a) and two simulation proto-
cols different by the method used to set up initial systems
and the alignment restraints. Both protocols enforced the
end-to-end distance r using a harmonic spring of
= 4000 pN/nm for 3.5 < r < 12 A
= 1000 pN/nm for 13 < r < 19 A
The ﬁrst protocol was used to provide the estimate of the
potential of mean force (PMF) (Figure 3). The initial con-
formation for each simulation was obtained by placing the
DNA fragments a speciﬁed distance r apart at one of the
four f = 0, 90, 180 and 270
(four simulations for each r).
The systems were equilibrated for 2 ns before data accu-
mulation during production simulations lasting 16 ns. In
the second protocol, which we used to compute the
Nucleic Acids Research, 2012, Vol. 40, No. 9 3813
relative binding free energies (Table 1), the initial con-
formations were generated iteratively by shifting the
minimum of the restraining potential in steps and
followed by 0.5-ns equilibration, starting from the ﬁnal
frames obtained in the simulations described in the
section ‘Collapse of aligned dsDNA’. Subsequently, each
system was equilibrated for at least 2.5 ns before accumu-
lation of data during production simulations lasting
7.5–15 ns. Axial alignment was maintained as described
in the section ‘‘Collapse of aligned dsDNA’’, using
= 13.9 and 139 pN/nm for the ﬁrst and second proto-
cols, respectively. In the second protocol, a torque
pointing along the common DNA axis was distributed
among the phosphorous atoms of each DNA molecule
to restrain f about 20
, with a spring
constant of 219.4 pN nm/rad
, which roughly corresponds
to an 8
root mean squared ﬂuctuation.
Spontaneous assembly of long end-to-end aggregates
The system (depicted in Figure 4a) was assembled through
the sequential placement of 458 DNA fragments in a cubic
volume (250 A
on each side) of 100 mM NaCl electrolyte.
To place a DNA fragment, trial positions and orientations
were randomly selected until the DNA coordinates did not
clash with any previously placed fragments. During the
ﬁrst 50 ns of equilibration, the system shrank to its equi-
librium size of 238 A
on each side. To improve computa-
tional efﬁciency, a 7–8 A
cutoff was used along with 2–2–6
fs time stepping scheme.
Collapse of aligned dsDNA
Spontaneous end-to-end association of duplex DNA was
observed in the simulations of two (dAdT)
constrained to diffuse along a common axis in a volume of
100 mM NaCl electrolyte. Figure 1a illustrates the initial
state of a typical simulation system. Figure 1b plots the
distance between the DNA fragments versus time for two
simulation systems differed by the termination of the
-ends. The DNA fragments were observed to
freely diffuse along the common axis until the
end-to-end distance fell below &8A
, whereupon the frag-
ments rapidly collapsed into an end-to-end bound
complex. The relative azimuthal orientation of the DNA
fragments f continued to change after the collapse,
switching between several metastable orientations
(Figure 1c). We deﬁne the relative azimuthal angle f as
the angle between the projections of the vectors connect-
ing the O5
atoms of the terminal base pairs into
the plane normal to the common DNA axis (see ‘Materials
and Methods’ section).
In the ﬁnal conformation adopted by the blunt-ended
fragments, the 5
direction of the backbone was dis-
continuous at the end-to-end junction. In the case of the
-phosphorylated fragments the 5
direction of the
backbone was continuous at the end-to-end junction as
though the backbone of a continuous 20 bp B-DNA had
To determine the statistical signiﬁcance of the above
observations, we performed 17 additional simulations
for each system type, different by the relative azimuthal
orientation of the DNA fragments at the onset of the
= i 20
, where i =1,..., 17. In all
simulations, we observed collapse of the DNA fragments
into an end-to-end bound complex. Figure 1d plots the
relative azimuthal orientation at the time of collapse
against the time to collapse. The collapse of blunt-ended
fragments occurs irrespective of their azimuthal orienta-
tion, whereas formation of a 5
end assembly did not occur with f between 90 and 230
Figure 1. Collapse of aligned dsDNA. (a) Simulation system containing axially aligned duplex DNA. Each DNA fragment is free to move along and
rotate about the common axis. The DNA duplexes (blue and green) are shown in van der Waals representation; sodium and chloride ions are shown
as yellow and cyan spheres; water is not depicted. An animation illustrating spontaneous end-to-end collapse of duplex DNA is available in
Supplementary Data. (b and c) The end-to-end distance (b) and the relative azimuthal angle f (c) of two duplex DNA in representative simulations
of the end-to-end collapse. Data from the same pair of simulations are plotted in (b) and (c). (d and e) Scatter plot showing the relative azimuthal
angle f at the time of collapse (d), and at the end of simulation (e). One data point is shown for each of 36 simulations of blunt-ended (black circles)
-phosphorylated (red squares) dsDNA fragments in 100 mM NaCl electrolyte.
3814 Nucleic Acids Research, 2012, Vol. 40, No. 9
After collapse, f continued to evolve, reaching the
states characterized in Figure 1e. The clustering of f
values around 20, 36 and 180
indicates the three
preferred binding states. At f&36
, the conformation
of the bound complex is similar to that of a continuous
B-form DNA. At f&20
, the backbones of the
terminal base pairs overlap slightly (Figure 2b). At
, the 5
-ends of the fragments are in direct
contact (Figure 2c). The conformations of the
end-to-end assemblies with f = 20 and 36
differing primarily in the order in which the 5
-termini are lapped and in their base stacking
geometry. Thus, relative to the coordinates of two con-
secutive base pairs in a canonical B-DNA helix, the
terminal base pairs forming an end-to-end junction have
a time-averaged root mean squared deviation (RMSD) of
2.1 and 0.9 A
for the f = 20 and 36
respectively. For reference, base pairs in the middle of
one of the DNA fragments had an RMSD of 0.7 A
The preference for these three orientations suggests a
hydrophobic origin of the attractive force, as such con-
formations reduce exposure of the hydrophobic bases to
water. The relative orientations of the blunt-ended DNA
were nearly equally split between the 20 and 180
More than 50% of the 5
formed the 20
state, 35% formed either the 36
state (three systems each) and 12% formed the
state with f&100
. We attribute such preferential align-
ment of the 5
-phosphorylated fragments to the electro-
static repulsion between the terminal phosphate groups,
which apparently renders the 180
ally less favorable than the 20
one. The free energy
difference between these bound states is discussed below.
Stability of the end-to-end complex
DNA fragments initially forming a bound state were
simulated in the absence of any restraints using the iso-
tropic system shown in Figure 2a. Three systems were
constructed: one containing 5
bound with f = 20
(Figure 2b) and two containing
blunt-ended DNA fragments bound at f = 180
(Figure 2c) and f = 20
. All three systems contained
100 mM NaCl electrolyte.
The plot of the end-to-end distance, Figure 2d, indicates
that all three assemblies remained bound for the entire
duration of the simulations (>200 ns). The standard devi-
ation of the end-to-end distance in the 5
system was 0.68 A
, twice less than that of the blunt-ended
systems. The greater stability of the 5
complex may be due to hydrogen bonds that were
observed between the 5
-terminal phosphate of one
DNA fragment and the 3
-terminal hydroxyl of the other
fragment. The plot of the relative azimuthal angle reveals
that the 5
-phosphorylated system maintained the
same stable conformation at f&20
, depicted in
Figure 2b. Starting from a similar conformation, the
blunt-ended complex underwent two sudden rotations at
70 and 140 ns that brought f from 20
; the relative orientation continued to evolve after
In the simulation of the 180
blunt-ended complex, the
terminal base pair of one of the fragments ruptured after
33 ns. During the next few nanoseconds, Watson–Crick
pairs within that fragment stochastically broke and
re-annealed, propagating the unpaired base toward the
opposite end of the DNA fragment, and slipping the
entire DNA strand with respect to the other by 1 bp.
Figure 2. Stability of the end-to-end DNA assembly. (a) Simulation system containing two spatially unrestrained dsDNA fragments in 100 mM
NaCl electrolyte. Both fragments are free to rotate and move about the simulation box. The system is drawn as in Figure 1a. An animation
illustrating a typical MD trajectory is available in Supplementary Data. (b and c) End-to-end junction of 5
-phosphorylated dsDNA with f=20
(b) and blunt-ended dsDNA with f=180
(c). (d and e) End-to-end distance (d) and relative azimuthal angle, f, (e) of the DNA fragments in three
unrestrained MD trajectories.
Nucleic Acids Research, 2012, Vol. 40, No. 9 3815
Despite the slippage, the DNA fragments remained stably
Mechanics of end-to-end dissociation
The lifetime of a bound complex sharply decreases when
an external force is applied to disrupt it (28). Thus, under
a constant force of 150 pN directed along the common
axis of the DNA fragments, the assembly remained
intact during a 50-ns simulation but dissociated within
5 ns under a 200 pN force (see Section 1.1 in
Supplementary Data for simulation details).
To determine the dissociation pathway, the rupture
force and the mechanical work required to dissociate the
assemblies, the DNA fragments were subjected to the
force of a harmonic spring whose equilibrium-extension
length was increased at a constant rate. Spring-driven
rupture of this sort has been used extensively in the
study of proteins (29,30). Rupture was induced by
pulling apart the fragments either along (axial stretching)
or perpendicular to (transverse shearing) their common
symmetry axis, applying the force either to the nearest
ends or to the centers of mass (CoM) of the fragments.
At least four simulations were performed for each
protocol (14 in total). Animations in Supplementary
Data illustrate typical simulation trajectories. In all
cases, rupture was observed to occur by sliding of one
terminal base pair relative to the other and was preceded
by stretching of the duplex in the case of the CoM axial
pulling. Although the three rupture protocols yielded dif-
ferent dependencies of the force on the separation distance
(Supplementary Figures S1 and S2), the average work
performed was 9.4±1.5 kcal/mol, independent of the
rupture protocol. The typical rupture forces varied
between 100 and 200 pN and were considerably lower
for CoM pulling. Inclusion of short overhangs of compli-
mentary sequence ssDNA at the ends of the fragments
increased the work required to rupture the end-to-end
assemblies (for details, see Section 1.4 and Figure S3 in
PMF of axially aligned DNA duplexes
To improve our estimates of the force and free energy of
the end-to-end interaction, 100 variants of the system
containing two-DNA fragments in 100 mM NaCl
electrolyte, Figure 1a, were simulated with a constant
end-to-end distance enforced by a harmonic spring poten-
tial. Each simulation explored a unique combination of
the end-to-end distance and the azimuthal angle and
lasted 18 ns (aggregate simulation time was 1.86 ms). The
DNA fragments were kept aligned by weak harmonic re-
straints (see ‘Materials and Methods’ section) that allowed
the terminal base pairs to shear.
Figure 3 shows the dependence of the effective
end-to-end force on the end-to-end distance and the
PMF reconstructed from this set of simulations by the
weighted histogram analysis method (31,32). The PMF
can be thought of as the change of free energy along a
chosen coordinate. The force sharply increases with the
end-to-end distance between 3.5 A
—the distance between
consecutive base pairs in a DNA helix—and 6.5 A
separation allowing water molecules to penetrate the
volume between the ends of the fragments. The force
rapidly decreases as the end-to-end distance exceeds
and becomes slightly repulsive after 13 A
pN). Thus, the end-to-end force has a large but very
short-range attractive component caused by the hydro-
phobic effect and a much smaller long-range repulsive
component that originates from screened electrostatic
interactions between the DNA fragments (19).
A variation of the above protocol (described in Section
2 of Supplementary Data) was used to calculate the
PMF for DNA fragments different by their terminal
chemistry and relative azimuthal orientation and for
several concentrations of the surrounding electrolyte.
Table 1 lists the change in the depth of the PMF
minima ((min[PMF] – PMF(1)) relative to its value
Figure 3. Representative dependence of the effective force (red) and the
free energy (black) of two axially aligned DNA fragments on the
end-to-end distance. The data result from 100 independent simulations
of two 5
-phosphorylated fragments, the end-to-end distance of which
was maintained at a speciﬁed value by a harmonic spring. Additional
restraints were applied to maintain the axial alignment. The strength of
the restraints was found to affect the values but not the general shape
of both curves (see text). The image in the background illustrates the
simulation method. The DNA fragments were immersed in 100 mM
NaCl electrolyte, and were free to rotate about their helical axes.
Table 1. Relative free energy change, G, upon formation of the
chemistry ion concentration
f G (kcal/mol)
Phosphoryl 0.1 36
Phosphoryl 1.0 36
Phosphoryl 1.0 20
Phosphoryl 1.0 180
Hydroxyl 1.0 36
Hydroxyl 1.0 180
Here, the free energy change G is approximated by the minimum of
the end-to-end PMF obtained from umbrella sampling simulations
(Supplementary Figure S4). The values of G are given relative to
the value measured for the system containing 5
ments end-joined in the 36
orientation in 100 mM NaCl. In each
simulation, the relative azimuthal orientation of the DNA fragments
was enforced using harmonic restraints (see ‘Material and Methods’
section). The application of such restraints introduced a bias to the
estimates of G. As all simulations employed the same restraints,
this bias was assumed to cancel out in the calculation of G.
3816 Nucleic Acids Research, 2012, Vol. 40, No. 9
for the 5
-phosphorylated fragments with f =36
100 mM NaCl. These calculations (detailed in Supple-
mentary Data) demonstrate that increasing the electrolyte
concentration from 0.1 to 1 M has negligible effect on the
PMF. Among the three most likely orientations that
-phosphorylated fragments form, the f = 20
has the lowest free energy, in good agreement with the
f = 22
angle frequently observed in the crystal struc-
tures of poly(dAdT) oligonucleotides (33). The 5
phorylated fragments exhibit deeper minima than the
blunt-ended fragments. For the latter, the conformation
of f = 180
is preferred over f =36
. These variations in
the depth of the PMF are consistent with the occupancy of
bound states observed in our simulations of spontaneous
collapse of aligned DNA fragments (Figure 1e).
Standard binding free energy of end-to-end association
We computed the standard free energy of binding G
using a variation of the method described previously (34).
For a system of two DNA fragments, G
fraction of time the fragments form a bound state, which
can be, in principle, observed directly in an all-atom MD
simulation. However, the dissociation rate of the
end-to-end assembly was found to exceed 200 ns
(Figure 2d) and therefore using such a brute force
approach was not possible. Equivalently, G
obtained from the logarithm of the equilibrium binding
constant, which is the ratio of the kinetic rates of end
joining and rupture (k
For our calculations of G
, we considered a process
consisting of the following four steps (Supplementary
Figure S5). First, we evaluated the free energy cost of
enforcing axial alignment restraints on a pair of inﬁnitely
separated DNA fragments. Second, we computed the cost
of bringing a pair of DNA fragments from inﬁnity to the
maximum-separation state considered in our simulations
(CoM–CoM distance of 52 A
). Third, we determined the
free energy cost of forming the end-to-end complex of two
axially aligned DNA fragments. Finally, we evaluated the
cost of releasing the axial alignment restraints from
the end-to-end assembly. The sum of these terms yielded
the free energy change upon formation of the end-to-end
DNA complex G
= 6.3 ± 1 kcal/mol for a DNA con-
centration of 1 M in 120 mM NaCl. Since a pair of DNA
ends has only one binding conﬁguration, we can express
the standard binding free energy in terms of DNA ends, so
¼5:4 kcal/mol. A complete description of the
methods used in this section is provided in Section 3.1 in
The above value for G
represents our best effort to
quantify the strength of the end-to-end interaction. In
may depend on the DNA sequence.
Although the question of sequence dependence is
enticing, we defer investigations of the sequence depend-
ence of the end-to-end interaction to future studies.
The rate of dissociation of an end-to-end assembly, k
can be estimated from the PMF and diffusion coefﬁcient,
D, by computing the mean ﬁrst passage time under the
assumption that reannealing does not occur (35,36) (see
Section 3.2 in Supplementary Data for details). From the
simulations performed in the section ‘Collapse of aligned
dsDNA’, we estimate D 25 A
/ns. Due to uncertainty in
the location of the barrier peak, we obtain the range
170 000–860 000 ns, with a best estimate of
480 000 ns. The use of axial alignment restraints in
our calculations of the PMF limits pathways ordinarily
available to rupturing DNA, so these values likely repre-
sent an upper bound for k
Assuming that end-to-end binding is limited by transla-
tional diffusion, the upper bound estimate for k
7 (ns M)
, where R
is the CoM
distance between a pair of DNA fragments at which
binding is expected to occur. The true value of k
should be smaller because the DNA fragments must be
axially aligned for binding to occur. Furthermore,
long-range electrostatic repulsion may reduce the value
. Our estimates of G
suggest a range
of 0.03–0.16 ns
Spontaneous assembly of long end-to-end aggregates
Our results until this point described the interaction of two
DNA fragments in isolation. To simulate multifragment
aggregation, 458 DNA fragments, each 10 bp in length,
were placed in a cube of 100 mM NaCl solution (23.8 nm
on each side) to form the system shown in Figure 4a.
During a 260-ns MD simulation, the DNA fragments
diffused about their initial positions and interacted with
their neighbors to form aggregates up to 11 DNA frag-
ments (110 bp) in length. The DNA fragments that formed
the longest 10 aggregates are shown at the beginning,
Figure 4b, and the end, Figure 4c, of the simulation.
The number of aggregates of a given length at three dif-
ferent instances of the MD trajectory is shown in
Figure 4d. The plot reveals rapid growth of end-to-end
aggregates and roughly exponential distribution of the
lengths of the aggregates. The number of aggregates did
not reach a steady state by the end of the simulation
(Supplementary Figure S6).
We model the process of end-to-end aggregation using a
simpliﬁed reversible step-growth polymerization model,
which has an analytical time-dependent solution that
relates the mean aggregation number, hNi to G
function of DNA concentration, c (37) (see Section 4 in
Supplementary Data). The DNA concentration in our
multifragment system was 56 mM. Using our prediction
= 6.3 kcal/mol, we obtain hNi = 39 for the
mean aggregation number at equilibrium for this system.
The above simulation partially mimics the experimental
assay of Nakata et al. (11), which involved very dense
ﬂuids of short DNA fragments. For 10 bp DNA frag-
ments, Nakata et al. found a transition from an isotropic
phase to a nematic liquid crystal phase at c = 875 mg/ml
(150 mM). Through a combination of calibration experi-
ments and theory, the authors estimated hNi = 9 at the
isotropic–nematic phase transition. Under the model of
reversible step growth polymerization, hNi =9 at
c = 150 mM corresponds to G
= 3.87 kcal/mol. In
contrast, using our prediction of G
= 6.3 kcal/mol
yields hNi = 64 at a DNA concentration of 150 mM.
Nucleic Acids Research, 2012, Vol. 40, No. 9 3817
In the reversible step growth polymerization model, the
total number of associated molecules is described by a
second-order kinetic equation. Since unbinding of DNA
fragments is negligible in our system, k
can be extracted
by ﬁtting NðtÞ¼N
=ð1 þ tc
Þ to the data, where N(t)is
the total number of end-to-end bound DNA fragments at
time t, N
= N(0) and c
is the initial DNA concentration.
In our simulation of spontaneous aggregation, the associ-
ation rate reduced from k
= 0.37 ns/M observed within
the ﬁrst 50 ns to 0.069 ns/M for the 60–250-ns interval
(Supplementary Figure S6). The change in association
rate occurred as longer end-to-end aggregates formed
(>3 duplex fragments) and the rotational and translation-
al diffusive motions of shorter aggregates slowed.
The aggregation simulation provides a lower bound
estimate for the dissociation rate k
. During the course
of the simulation, 307 DNA ends were bound for 217 ns
on average and unambiguous unbinding was observed for
only one end-to-end associated complex. A few events
were observed in which partial unbinding occurred via
rupture of the Watson–Crick base pair of terminal nucleo-
tides, as well as one instance where a bound DNA
fragment was transferred to an unbound fragment; we
neglect these events for the subsequent analysis. A
Poisson distribution yields the probability that exactly
one end-to-end bound DNA pair would rupture during
the simulation given a value of k
. For k
<1900 ns, it
is likely that more than one rupture would have occurred.
>59600 ns, it is likely that no rupture would have
occurred. Thus, the probability that exactly 1 rupture
occurred is >10% for k
¼ 19 000–596 000 ns. The
greatest probability of exactly 1 rupture occurring is
37% for k
¼ 67 000 ns. The ranges estimated for k
in between 4.4 and 7.6 kcal/mol,
with a most probable range of 5.2–6.2 kcal/mol
= 0.069–0.37 ns/M and k
¼ 67; 000 ns]. Alternative
statistical analysis of the results yields a similar range for
and is described in Section 4 of Supplementary Data.
DISCUSSION AND CONCLUSIONS
Stability of a dsDNA molecule can be conveniently
described as a sum of base stacking and base pairing
energies contributed by individual nucleotides that
constitute the molecule. It is tempting to conceptually
equate the base stacking interactions within a continuous
molecule with the base stacking interactions that drive the
assembly of two disjoint DNA duplexes. Thus, the uniﬁed
nearest neighbor parameters, which can predict the energy
for DNA hybridization based on DNA melting data,
= 16.94+2 6.94 = 3.06 kcal/mol (38)
for the association of two 10 bp poly(dAdT) DNA frag-
ments into a continuous 20-bp molecule. Such a simple
calculation may be, however, ﬂawed as additional con-
formational ﬂexibility afforded by the lack of phospho-
diester bonds at the end-to-end interface should allow
the base stacking geometry to be optimized, magnifying
the base stacking contribution to the free energy.
Accordingly, the average interaction energy between
adjacent base pairs with f = 20
was measured to be
2 kcal/mol lower in bases forming the end-to-end
junction than in bases in the middle of one of the DNA
fragments (Supplementary Figure S7). This ﬁnding is in
good agreement with a survey of crystallographic struc-
tures that found DNA fragments formed of AT base pairs
to stack with f = 22
(33). We note that the stacking
geometry may depend on the sequence of the DNA
The base stacking free energy has been experimentally
quantiﬁed by introducing a dangling nucleotide or a nick
(a cut in the backbone of one strand) to a DNA molecule
and observing the change in melting temperature (39–41),
and by observing the mobility of a nicked DNA molecule
relative to intact DNA and DNA with a gap (42,43). These
experiments provide estimates for the base stacking free
energy between 0.65 and 2.0 kcal/mol for stacks
formed by thymine and adenine. However, the extraction
of the free energy values is indirect with these methods.
Here, we provide the ﬁrst direct estimate of the standard
binding free energy of end-to-end association of DNA
Figure 4. Spontaneous aggregation of duplex DNA into long rod-like
structures. (a) A system containing 458 duplex DNA fragments placed
at random. NaCl solution is shown as a semi-transparent molecular
surface. ( b and c) Initial (b) and ﬁnal (c) conformations of the DNA
fragments that composed the longest 10 aggregates at the end of a
260-ns MD simulation. DNA fragments forming each aggregate are
shown in a different color. (d) The instantaneous number of aggregates,
N(s), formed by s DNA fragments in the MD simulation. The lines
show the equilibrium distribution of the aggregate length according to
the reverse step-growth polymerization model (37) for speciﬁed values
. A movie of the simulation trajectory is provided in
3818 Nucleic Acids Research, 2012, Vol. 40, No. 9
fragments, obtaining G
= 6.3 ± 1 kcal/mol. The
simulations reported here are similar to the liquid crystal
condensation experiments of the Clark and Bellini groups
(11), which estimated 3.8 kcal/mol for the end-to-end
free energy, in reasonable agreement with our estimate.
The small-angle X-ray scattering experiments of the
Pollack group demonstrated that the end-to-end inter-
action dominates over electrostatic repulsion in a
divalent electrolyte (12,13), which, unfortunately, is not
sufﬁcient to estimate the standard free energy of
end-to-end binding. We note that the value of G
obtained in this study is larger than values reported in
experiments. Having employed multiple methods, we are
conﬁdent that the range of values obtained for G
curately reﬂects the standard binding free energy within
the limits of the molecular force ﬁeld used in our study.
Nevertheless, we cannot rule out the possibility that the
present MD force ﬁeld somewhat exaggerates the inter-
actions driving end-to-end self-assembly of duplex DNA.
It is interesting to note that hydrophobic interactions
between DNA bases and inorganic materials can be sig-
niﬁcantly stronger than the stacking energies observed in
biochemical assays. AFM experiments indicate that
ssDNA adheres to graphite with free energies of 4.9
and 6.8 kcal/mol per nucleotide for cytosine and
guanine, respectively (44). For comparison, a computa-
tional investigation of hydrophobic interactions revealed
a free energy of 55 kcal/mol for the adhesion of a pair of
11 12 A
graphene sheets, or about 10 times the energy
per unit surface area (45). Interactions of similar strength
were found to promote DNA–fullerene and ssDNA–
carbon nanotube association (46,47).
Given the relatively large free energy of the end-to-end
interaction, we pose the following question: why has
end-to-end association only recently been observed? In
most biological and nanotechnological settings, the concen-
tration of DNA ends is too low for end-to-end association
to be statistically signiﬁcant, and the lifetime of end-to-end
interactions (70 ms) is somewhat shorter than the temporal
resolution of many experimental techniques. To illustrate
this point, we plot in Figure 5 the fraction of bound DNA
ends versus the concentration of free DNA ends in chemical
equilibrium. The concentration of DNA ends is relatively
high in DNA cyclization assays that measure the fraction of
DNA molecules bent into a circle. The J-factor, which rep-
resents the concentration of one end in the proximity of the
other, has a maximum value of 10
mM (48), which is
two orders of magnitude too small to promote cyclization of
a signiﬁcant fraction of blunt-ended DNA. The introduc-
tion of sticky ends increases the interaction energy by
a few kcal/mol so that the DNA cyclization can be observed.
The large standard binding energy of end-to-end asso-
ciation implies that end-to-end interactions will be import-
ant in systems containing a high concentration of DNA
ends. For instance, a remarkable new method of creating
patterned, self-assembled structures out of DNA—termed
DNA origami (3)—introduces many nicks along a path of
DNA, which may enhance the stability of the resultant
pattern. We speculate that broadened awareness of
end-to-end association will inﬂuence the development of
nanotechnologies where the end-to-end interaction can be
used advantageously, such as with DNA origami, or can
pose a limitation, such as for DNA microarrays.
In cells, double-stranded DNA breakage, which poses a
mortal threat, results in nearby blunt or sticky DNA ends.
During non-homologous end-joining—the dominant
repair pathway for such breaks in multicellular eukaryotes
(49,50)—the ruptured DNA ends are held together by
proteins such as the Ku heterodimer or DNA-PK, or by
nucleosome interactions until damaged DNA can be
removed and ligation of the DNA backbone occurs (49).
Since DNA attracts end-to-end, it is not necessary that
this complex, whose microscopic structure is not yet
known, hold the DNA ends in strict alignment. It is suf-
ﬁcient for the ends to be held proximally so that the effect-
ive concentration of DNA ends is large enough to
promote end-to-end association (Figure 5, purple), where-
upon ligation may occur. The free energy of dsDNA
end-to-end association found in this study suggests that
placing DNA ends in a sphere of 3-nm radius will produce
an end-to-end associated state 95% of the time.
Supplementary Data are available at NAR Online:
Supplementary Methods, Results and Figures 1–7,
Supplementary Animations 1–6 and Supplementary
The authors C.M. and A.A. thank Lois Pollack for useful
Figure 5. The effect of end-to-end attraction in different DNA systems.
The binding free energy (black) and the fraction of bound DNA ends
(red) are plotted against the reference concentration of DNA ends.
Background images schematically illustrate four DNA systems in
which the end-to-end attraction may or may not play a role. From
top left to bottom right: the maximum local concentration of DNA
ends [i.e. the J-factor (51)] is about two orders of magnitude too low to
induce an observable fraction of blunt-ended DNA circles of any length
(orange); translation and rotational conﬁnement of DNA ends at
dsDNA breakage [e.g. by the protein Ku (PDB:1JEY)] will promote
binding of the DNA ends, which likely aids repair of DNA during
non-homologous end joining (49) (purple); the structure factor
obtained from small-angle X-ray scattering experiments of short
DNA duplexes in a divalent electrolyte reveals end-to-end attraction
(12) (blue); at very high DNA concentrations, long DNA aggregates
form and align in liquid crystal phases (11) (green).
Nucleic Acids Research, 2012, Vol. 40, No. 9 3819
Center for the Physics of Living Cells through a grant
from the National Science Foundation (PHY-0822613).
The supercomputer time was provided at TeraGrid
(MCA05S028). Funding for open access charge: NSF.
Conﬂict of interest statement. None declared.
1. Bath,J. and Turberﬁeld,A.J. (2007) DNA nanomachines.
Nat. Nanotech., 2, 275–284.
2. Pihlak,A., Baure
n,G., Hersoug,E., Lo
nnerberg,P., Metsis,A. and
Linnarsson,S. (2008) Rapid genome sequencing with short
universal tiling probes. Nat. Biotech., 26, 676–684.
3. Rothemund,P. (2006) Folding DNA to create nanoscale shapes
and patterns. Nature, 440, 297–302.
4. Maune,H.T., Han,S.-P., Barish,R.D., Bockrath,M., Iii,W.A.G.,
Rothemund,P.W.K. and Winfree,E. (2010) Self-assembly of
carbon nanotubes into two-dimensional geometries using DNA
origami templates. Nat. Nanotech., 5, 61–66.
5. Shin,J.-S. and Pierce,N.A. (2004) A synthetic DNA walker for
molecular transport. J. Am. Chem. Soc., 126, 10834–10835.
6. Benenson,Y., Paz-Elizur,T., Adar,R., Keinan,E., Livneh,Z. and
Shapiro,E. (2001) Programmable and autonomous
computing machine made of biomolecules. Nature, 414,
7. Nykypanchuk,D., Maye,M.M., van der Lelie,D. and Gang,O.
(2008) DNA-guided crystallization of colloidal nanoparticles.
Nature, 451, 549–552.
8. Park,S.Y., Lytton-Jean,A.K.R., Lee,B., Weigand,S., Schatz,G.C.
and Mirkin,C.A. (2008) DNA-programmable nanoparticle
crystallization. Nature, 451, 553–556.
9. Gan,H.H. and Schlick,T. (2010) Chromatin ionic atmosphere
analyzed by a mesoscale electrostatic approach. Biophys. J., 99,
10. Fuller,D.N., Rickgauer,J.P., Jardine,P.J., Grimes,S.,
Anderson,D.L. and Smith,D.E. (2007) Ionic effects on viral DNA
packaging and portal motor function in bacteriophage u29.
Proc. Natl Acad. Sci. USA, 104, 11245–11250.
11. Nakata,M., Zanchetta,G., Chapman,B., Jones,C., Cross,J.,
Pindak,R., Bellini,T. and Clark,N. (2007) End-to-end stacking
and liquid crystal condensation of 6 to 20 base pair DNA
duplexes. Science, 318, 1276.
12. Qiu,X., Andresen,K., Kwok,L., Lamb,J., Park,H. and Pollack,L.
(2007) Inter-DNA attraction mediated by divalent counterions.
Phys. Rev. Lett., 99, 38104.
13. Li,L., Pabit,S.A., Lamb,J.S., Park,H.Y. and Pollack,L. (2008)
Closing the lid on DNA end-to-end stacking interactions.
Appl. Phys. Lett., 92, 223901–1–223901–3.
14. Rau,D.C., Lee,B. and Parsegian,V.A. (1984) Measurement of the
repulsive force between polyelectrolyte molecules in ionic solution:
hydration forces between parallel DNA double helices. Proc. Natl
Acad. Sci. USA, 81, 2621–2625.
15. Bloomﬁeld,V. (1996) DNA condensation. Curr. Op. Struct. Biol.,
16. Wong,G.C.L. and Pollack,L. (2010) Electrostatics of strongly
charged biological polymers: ion-mediated interactions and
self-organization in nucleic acids and proteins. Ann. Rev. Phys.
17. Grosberg,A.Y., Nguyen,T.T. and Shklovskii,B.I. (2002)
Colloquium: the physics of charge inversion in chemical and
biological systems. Rev. Mod. Phys., 74, 329–345.
18. Kornyshev,A.A. (2010) Physics of DNA: unravelling hidden
abilities encoded in the structure of ‘the most important molecule’
. Phys. Chem. Chem. Phys., 12, 12352–12378.
19. Maffeo,C., Scho
pﬂin,R., Brutzer,H., Stehr,R., Aksimentiev,A.,
Wedemann,G. and Seidel,R. (2010) DNA–DNA interactions in
tight supercoils are described by a small effective charge density.
Phys. Rev. Lett., 105, 158101.
20. MacKerell,A. Jr and Nilsson,L. (2008) Molecular dynamics
simulations of nucleic acid-protein complexes. Curr. Op. Struct.
Biol., 18, 194–199.
21. Phillips,J.C., Braun,R., Wang,W., Gumbart,J., Tajkhorshid,E.,
Villa,E., Chipot,C., Skeel,R.D., Kale,L. and Schulten,K. (2005)
Scalable molecular dynamics with NAMD. J. Comp. Chem., 26,
22. Perez,A., Marchan,I., Svozil,D., Sponer,J., Cheatham,T.E.,
Laughton,C.A. and Orozco,M. (2007) Reﬁnement of the AMBER
force ﬁeld for nucleic acids: improving the description of a/g
conformers. Biophys. J., 92, 3817–3829.
23. Cheatham,T.E.I., Cieplak,P. and Kollman,P.A. (1999)
A modiﬁed version of the Cornell et al. force ﬁeld with improved
sugar pucker phase and helical repeat. J. Biomol. Struct. Dyn.,
24. Jorgensen,W.L., Chandrasekhar,J., Madura,J.D., Impey,R.W. and
Klein,M.L. (1983) Comparison of simple potential functions for
simulating liquid water. J. Chem. Phys., 79, 926–935.
25. Beglov,D. and Roux,B. (1994) Finite representation of an inﬁnite
bulk system: solvent boundary potential for computer simulations.
J. Chem. Phys., 100, 9050–9063.
26. Martyna,G.J., Tobias,D.J. and Klein,M.L. (1994) Constant
pressure molecular dynamics algorithms. J. Chem. Phys., 101,
27. Payne,M.C., Teter,M.P., Allan,D.C., Arias,T.A. and
Joannopoulos,J.D. (1992) Iterative minimization techniques for
ab initio total-energy calculations: molecular dynamics and
conjugate gradients. Rev. Mod. Phys., 64, 1045–1097.
28. Dudko,O., Hummer,G. and Szabo,A. (2006) Intrinsic rates and
activation free energies from single-molecule pulling experiments.
Phys. Rev. Lett., 96, 108101.
29. Galera-Prat,A., Go
mez-Sicilia,A., Oberhauser,A.F., Cieplak,M.
zquez,M. (2010) Understanding biology by
stretching proteins: recent progress. Curr. Op. Struct. Biol., 20,
30. Isralewitz,B., Baudry,J., Gullingsrud,J., Kosztin,D. and
Schulten,K. (2001) Steered molecular dynamics investigations of
protein function. J. Mol. Graph. Model., 19, 13–25.
31. Torrie,G. and Valleau,J. (1977) Nonphysical sampling
distributions in Monte Carlo free-energy estimation: umbrella
sampling. J. Comput. Phys., 23, 187–199.
32. Roux,B. (1995) The calculation of the potential of mean force
using computer simulations. Comput. Phys. Commun., 91 ,
33. Campos,L., Valls,N., Urpı
,L., Gouyette,C., Sanmartı
Richter,M., Alechaga,E., Santaolalla,A., Baldini,R., Creixell,M.
et al. (2006) Overview of the structure of all-AT oligonucleotides:
organization in helices and packing interactions. Biophys. J., 91,
34. Woo,H.-J. and Roux,B. (2005) Calculation of absolute protein–
ligand binding free energy from computer simulations.
Proc. Natl Acad. Sci. USA, 102, 6825–6830.
35. Zwanzig,R. (2001) Nonequilibrium Statistical Mechanics. Oxford
University Press, New York.
36. Wereszczynski,J. and Andricioaei,I. (2010) Free energy
calculations reveal rotating-ratchet mechanism for DNA supercoil
relaxation by topoisomerase IB and its inhibition. Biophys. J., 99,
37. Kumar,A. and Khanna,A. (1990) Analytical solution of reversible
step growth polymerization in semibatch reactors. J. Appl.
Polymer Sci., 41, 2077–2094.
38. SantaLucia,J. Jr (1998) A uniﬁed view of polymer, dumbbell,
and oligonucleotide DNA nearest-neighbor thermodynamics.
Proc. Natl Acad. Sci. USA, 95, 1460–1465.
39. Bommarito,S., Peyret,N. and SantaLucia,J. Jr (2000)
Thermodynamic parameters for DNA sequences with dangling
ends. Nucl. Acids Res., 28, 1929–1934.
40. Vasiliskov,V.A., Prokopenko,D.V. and Mirzabekov,A.D. (2001)
Parallel multiplex thermodynamic analysis of coaxial base
stacking in DNA duplexes by oligodeoxyribonucleotide
microchips. Nucl. Acids Res., 29, 2303–2313.
41. Pyshnyi,D.V., Goldberg,E.L. and Ivanova,E.M. (2003) Efﬁciency
of coaxial stacking depends on the DNA duplex structure.
J. Biomol. Struct. Dyn., 21, 459–468.
3820 Nucleic Acids Research, 2012, Vol. 40, No. 9
42. Protozanova,E., Yakovchuk,P. and Frank-Kamenetskii,M.D.
(2004) Stacked–unstacked equilibrium at the nick site of DNA.
J. Mol. Biol., 342, 775–785.
43. Yakovchuk,P., Protozanova,E. and Frank-Kamenetskii,M.D.
(2006) Base-stacking and base-pairing contributions into
thermal stability of the DNA double helix. Nucl. Acids Res., 34,
44. Manohar,S., Mantz,A.R., Bancroft,K.E., Hui,C.-Y., Jagota,A.
and Vezenov,D.V. (2008) Peeling single-stranded DNA from
graphite surface to determine oligonucleotide binding energy by
force spectroscopy. Nano Lett., 8, 4365–4372.
45. Choudhury,N. and Pettitt,B.M. (2005) On the mechanism of
hydrophobic association of nanoscopic solutes. J. Am. Chem.
Soc., 127, 3556–3567.
46. Zhao,X. (2008) Interaction of C
derivatives and ssDNA from
simulations. J. Phys. Chem. C, 112, 8898–8906.
47. Johnson,R.R., Johnson,A.T.C. and Klein,M.L. (2008) Probing
the structure of DNA–carbon nanotube hybrids with molecular
dynamics. Nano Lett., 8, 69–75.
48. Shore,D., Langowski,J. and Baldwin,R.L. (1981) DNA ﬂexibility
studied by covalent closure of short fragments into circles.
Proc. Natl Acad. Sci. USA, 78, 4833–4837.
49. Lieber,M.R., Ma,Y., Pannicke,U. and Schwarz,K. (2003)
Mechanism and regulation of human non-homologous DNA
end-joining. Nat. Rev. Mol. Cell Biol., 4, 712–720.
50. Weterings,E. and van Gent,D.C. (2004) The mechanism of non-
homologous end-joining: a synopsis of synapsis. DNA Repair, 3,
51. Crothers,D.M., Drak,J., Kahn,J.D. and Levene,S.D. (1992) DNA
bending, ﬂexibility, and helical repeat by cyclization kinetics.
Meth. Enzym., 212, 3–29.
52. Isralewitz,B., Izrailev,S. and Schulten,K. (1997) Binding pathway
of retinal to bacterio-opsin: a prediction by molecular dynamics
simulations. Biophys. J., 73 , 2972–2979.
53. Jarzynski,C. (1997) Nonequilibrium equality for free energy
differences. Phys. Rev. Lett., 78, 2690–2693.
54. Stofer,E., Chipot,C. and Lavery,R. (1999) Free energy
calculations of watson-crick base pairing in aqueous solution.
J. Am. Chem. Soc., 121, 9503–9508.
55. Torrie,G. and Valleau,J. (1977) Nonphysical sampling
distributions in Monte Carlo free-energy estimation: umbrella
sampling. J. Comput. Phys., 23, 187–199.
56. Roux,B. (1995) The calculation of the potential of mean force
using computer simulations. Comput. Phys. Commun., 91,
57. Woo,H.-J. and Roux,B. (2005) Calculation of absolute protein–
ligand binding free energy from computer simulations.
Proc. Natl Acad. Sci. USA, 102, 6825–6830.
58. Singh,U.C., Brown,F.K., Bash,P.A. and Kollman,P.A. (1987)
An approach to the application of free energy perturbation
methods using molecular dynamics: applications to the
transformations of CH
OH ! CH
! alanine, and alanine ! phenylalanine in aqueous solution and
in the gas phase. J. Am. Chem.
Soc., 109, 1607–1614.
59. Cordova,A., Deserno,M., Gelbart,W.M., Gelbart,W.M. and
Ben-Shaul,A. (2003) Osmotic shock and the strength of viral
capsids. Biophys. J., 85, 70–74.
60. Hansen,P., Podgornik,R. and Parsegian,V. (2001) Osmotic
properties of DNA: critical evaluation of counterion condensation
theory. Phys. Rev. E, 64, 1–4.
61. Kumar,A. and Khanna,A. (1990) Analytical solution of reversible
step growth polymerization in semibatch reactors. J. Appl.
Polymer Sci., 41, 2077–2094.
Nucleic Acids Research, 2012, Vol. 40, No. 9 3821