Molecular dynamics simulation of multivalent ion mediated DNA attraction
Liang Dai,1 Yuguang Mu,2 Lars Nordenskiöld,2 and Johan R. C. van der Maarel1
1National University of Singapore, Department of Physics, 2 Science Drive 3, Singapore 117542
2Nanyang Technological University, School of Biological Sciences, 60 Nanyang Drive, Singapore, 63755
(January 8, 2008)
All atom molecular dynamics simulations with explicit water were done to study the interaction between
two parallel double-stranded DNA molecules in the presence of the multivalent counterions putrescine (2+),
spermidine (3+), spermine (4+) and cobalt hexamine (3+). The inter-DNA interaction potential is obtained
with the umbrella sampling technique. The attractive force is rationalized in terms of the formation of ion
bridges, i.e. multivalent ions which are simultaneously bound to the two opposing DNA molecules. The life-
time of the ion bridges is short on the order of a few nanoseconds.
PACS numbers: 82.35.Rs, 87.14.Gg, 82.39.Pj
Condensation of DNA induced by multivalent ions has
been studied for many years . Among the condensing
agents, the polyamines constitute an important class. The
polyamine-induced condensates of DNA (150 base pairs)
have been shown to be liquid-crystalline with inter-helical
spacing in the range 2.8-3.2 nm. Polyamines also induce
the collapse of single DNA’s into toroids . Condensa-
tion has been reported for other polyelectrolytes as well,
including actin and microtubules . Although much
work has been done to elucidate the mechanisms involved
in stabilizing the condensed state, the detailed structural
arrangement of the condensing agents is still unclear .
Multivalent ion-induced condensation cannot be ex-
plained with mean-field theory that always predicts a re-
pulsive interaction between like-charged polyelectrolytes.
Recent advances in the physics of strongly interacting
systems go beyond the classical framework and it is now
well established that dynamic correlation of cations
shared by different polyanions gives rise to an attractive
force  and the idea of a strongly correlated 2D liquid of
adsorbed ions, similar to a Wigner crystal, has been pro-
posed [6,7]. In theoretical modeling, DNA is usually
treated as a uniformly charged cylinder, the counterions
as point or spherical charges, and water as a continuous
dielectric medium [8,9,10]. These approximations are
appropriate for interactions over larger distances exceed-
ing the atomic scale, but in dense systems, such as in
DNA condensates, a molecular description is necessary
for an understanding of the condensation phenomenon.
This can now be achieved with the molecular dynamics
(MD) computer simulation method [11,12].
Polyamines are associated with the compaction of DNA
and play a role in the metabolism in eukaryotic cells .
Putrescine (Pu), spermidine (Sd), and spermine (Sm) are
linear polyamines with two cationic nitrogen charges lo-
cated at the terminal ends. Sd and Sm are tri- and tetrava-
lent, respectively, with one or two more nitrogen charges
along the contour. We investigated the interaction be-
tween two parallel double-stranded DNA duplexes with
MD simulations and umbrella sampling . To further
investigate the effects of charge and ligand structure, we
have also done simulations with trivalent cobalt hexamine
(Co). The simulations show an attractive force, which can
be understood in terms of the formation of transient ion
bridges, i.e. counterions which are simultaneously and
temporarily bound to the two opposing DNA’s. To the
best of our knowledge, this is the first validation of multi-
valent ion induced DNA attraction with an atomic model
including a molecular description of the solvent water.
All simulations were for salt free systems using a rec-
tangular cell, which contains one or two identical DNA
decamers in the B-form of 2 nm outer diameter (see Fig.
1). A randomly selected sequence of 10 base-pairs
(G5AAGAGGCTA3-C3TTCTCCGAT5) was chosen.
The DNA charge is neutralized with 10 di, 7 tri or 5 tetra-
valent counterions (excess cationic charge was compen-
sated with chloride). The 3’ end of each strand is con-
nected to the periodic image of the 5’ end along the Z-
axis (periodic boundary condition). This setup mimics an
infinite array of parallel ordered DNA in fibers or liquid
crystals. Note that the periodicity along the longitudinal
FIG. 1. (a): Top view of the simulation box with two parallel
DNA decamers and ten Sm counterions in the initial configura-
tion. The box has a transverse dimension of 7×7 nm2 and 3.4 nm
height. (b): Snapshot illustrating ion bridge formation.
axis matches the helical twist of the duplex with 10 base-
pairs per turn. Furthermore, the connectivity of the de-
camers set by the boundary condition inhibits bending
fluctuations with wave lengths exceeding the 3.4 nm lon-
gitudinal repeat distance of the simulation box. A snap-
shot of the transverse cross-section is shown in Fig. 1.
The AMBER (v. 98) force field was used to model the
DNA molecule, while partial charges, bond lengths, and
bond angles of the counterions were derived employing
the AMBER strategy . The simulation box contains
4956 water molecules described with the simple point
charge (SPC) model . Electrostatic interactions were
treated by the particle mesh Ewald method and the tem-
perature was controlled around 300 K with Berendsen
coupling. The GROMACS software  with a fixed box
volume and a time step of 2 fs was used. Each MD run
lasted more than 20 ns.
DNA molecules form a side-by-side complex if they
are attracting and unconstrained. In equilibrium the sepa-
ration is small with the two duplexes almost touching
each other. To study the interaction at larger separations,
we have applied
( ( , ))2
with two springs. As shown in
Fig. 2, these two springs pull the two duplexes in opposite
directions. We have used a spring constant k = 1000 kJ
mol-1 nm-2 and
( , )
is the deviation of the pull group
with respect to a reference point (
springs have the same spring constant, the total system
experiences no net force. To obtain the interaction energy
as a function of the distance
mass of the duplexes the contribution from Pext has to be
subtracted from the interaction energy. In practice, we
have assigned a weighing factor
sampling point. We then obtained the weighed probability
from the fractional time the
duplexes are separated by a distance
interaction energy follows from
In order to study the DNA-counterion interaction with-
out the influence of other DNA molecules, we have first
done a simulation of a single DNA duplex with Sm coun-
terions. The DNA molecule was positioned in the center
of the box and the counterions were randomly distributed.
In the first few nanoseconds, all Sm ions diffused towards
an external potential
. Since the two
between the centers of
P k T to every
and the true
the duplex and then remained territorially bound in the
following few tens of nanoseconds with at least one of
their four cationic nitrogen charges close to a phosphate
moiety. In agreement with earlier results , the interac-
tion was observed to be unspecific with territorial binding
whereby the Sm’s remain mobile and dynamic. A typical
lifetime of a configuration in which a Sm ion is in close
contact with the duplex is a few nanoseconds.
Next, we have done a simulation of two DNA duplexes
and ten Sm counterions. The initial configuration was
generated using the final state of the single DNA mole-
cule simulation with all counterions territorially bound to
DNA (see Fig. 1). The inter-helical distance was initially
set to 3.8 nm. This distance does not allow a simultaneous
contact of one Sm molecule with the two duplexes (the
contour length of Sm is 1.6 nm). Due to the periodic
boundary conditions and the fact that the top and bottom
base-pairs of each DNA decamer are connected, the du-
plexes can hardly bend and they remain parallel.
The fluctuating inter-helical distance
tion of two DNA duplexes with Sm and without springs is
displayed in Fig. 3. Initially, the two duplexes exhibited
no correlated lateral motion. However, after 12 ns the
duplexes formed a side-by-side complex and from then
onwards they moved coherently with an inter-duplex
separation of about 2.4 nm. Close inspection of the con-
figurations revealed the details of the attraction (an exam-
ple is displayed in Fig. 1). A Sm is usually territorially
bound to one duplex. Since Sm is a linear tetravalent
polyamine with a positive charge at each end, there is a
dangling end jutting outwards in the surrounding medium.
This dangling end can now be territorially bound to the
other duplex and form an ion bridge. Note that the bridge
is only temporarily formed; there is a continuous rear-
rangement of the bridging Sm. We surmise that the for-
mation of these transient ion bridges results in a net at-
traction. Simulations were also done for Pu, Sd and Co.
The result for the trivalent Sd is qualitatively similar.
Control simulations with sodium counterions only, con-
firmed the absence of attraction and resulted in equilib-
rium separations of 5 nm.
To obtain sufficient sampling for larger separations it is
necessary to apply the umbrella sampling technique. Con-
in a simula-
Center of mass
Reference point (x,y) of spring 1
FIG. 2. Illustration of the cross section of the simulation box
and how the two external springs pull the two DNA duplexes in
opposite directions in the transverse plane.
Inter-duplex distance (nm)
Simulation time (ns)
Simulation time (ns)
FIG. 3. Fluctuation in inter-duplex spacing of Sm-DNA. (a):
simulation without springs; (b): as in (a) but with two springs
centered at (±x, ±y) = (0.95, 0.95) nm.
tinuous potential curves are accordingly obtained and
shown in Fig. 4. For the ligands of valence three or
greater, i.e. for Co, Sd, and Sm-DNA, the potential exhib-
its a broad and pronounced minimum at 2.1, 2.3 and 2.4
nm, respectively (results of Co-DNA are not shown). The
positions of the minima agree with the inter-duplex sepa-
ration in the side-by-side complex obtained in the simula-
tions without external forces and are related to the struc-
ture of the ligands. The depth of the potential takes the
values -16 (Co), -9 (Sm), and -6 (Sd)
ing valence and smaller ligand size, the interaction poten-
tial becomes more attractive. For very short separations
the potential is always repulsive due to electrostatic and
hard-core interactions. For larger separations, beyond the
minimum, the potential is attractive and monotonously
increases until it levels off for
interaction is significantly shorter than half the length of
the diagonal of the simulation box (5 nm), so that possible
effects of the periodic boundary conditions are insignifi-
cant. Note that the multivalent ion mediated interaction
energy is an order of magnitude larger than the value
based on screened electrostatics and a helical distribution
of adsorbed monovalent counterions .
The interaction in Pu-DNA is also attractive with a po-
tential depth less than 2
k T . This weak attraction is con-
sistent with the experimental observation that Pu cannot
induce condensation  and the experimentally observed
weak DNA attraction in the presence of divalent magne-
sium . In a simulation of two DNA’s with sodium
counterions we have checked that the potential is always
repulsive. One should bear in mind that our simulations
refer to salt-free systems with counterions only. We have
checked that with the addition of monovalent salt (NaCl)
the potential generally becomes less attractive and the
minimum shifts to a larger inter-duplex distance. Fur-
thermore, we have only considered a pair interaction. In a
DNA condensate or liquid crystal one DNA molecule
interacts with multiple DNA molecules and it is not a
priori clear that the interactions are pair-wise additive
. For Sm-DNA in the absence of monovalent cations
the experimental value of the inter-duplex distance is 2.8
nm . This indicates the pair treatment of the interaction
as a major cause for the shorter equilibrium inter-duplex
distances as compared to the experimental values.
The inter-duplex force can be obtained from the deriva-
tive of the potential with respect to the separation. As an
illustrative example, we have smoothed the data pertain-
ing to Sm-DNA with the help of an arbitrary sixth order
polynomial; the resulting force is shown in Fig. 4. The
force can also be estimated in another way. The duplexes
diffuse under the actions of the attractive force and the
forces exerted by the springs. At the mean separation
, these forces are balanced. If the springs are
stretched by an amount
Δ , the attractive force is ap-
κΔ . The inter-duplex direction is not al-
k T . With increas-
nm. The range of
ways co-linear with the directions of the springs but the
deviations from co-linearity are always quite small and
the resulting forces are consequently good first-order ap-
proximations. Good agreement with the curve as obtained
from the derivative of the inter-duplex potential is ob-
tained (see Fig. 4).
As can already be gauged from the potential curve, for
separations less than, say, 2.4 nm the inter-duplex force is
repulsive. With increasing separation, the force becomes
attractive, shows a maximum at around 2.5 nm, and even-
tually dwindles for distances larger than 3 nm. Note that
the force is of short range on the order of the size of Sm
(in a separate MD simulation the mean end-to-end dis-
tance of Sm was found to be 1.2 nm). For the trivalent
ligands qualitatively similar results are obtained. These
results comply with the notion that the attractive force is
mediated through the formation of transient ion bridges
between the interacting duplexes.
We now focus on the dynamics of the ligands and their
spatial arrangement with respect to the duplexes to reveal
the underlying molecular mechanism for the like-charge
attraction. To characterize the dynamics we have moni-
tored the time-dependence of the closest distance between
any atom of the counterion and any atom of a duplex. The
picture of ligand binding and dynamics in the simulations
with two interacting duplexes is qualitatively the same as
in the above described simulations with one DNA mole-
cule. The minimum distance of closest approach is around
0.2 nm. We consider a ligand to be territorially bound to
the duplex with one of its four charges close to a phos-
phate moiety if this minimum distance is less than 0.25
nm. The binding is highly dynamic; during the MD run, a
counterion often attaches and detaches itself to the duplex
and the lifetime of a bound configuration is typically no
more than a few nanoseconds.
A ligand may be simultaneously bound to the two op-
posing duplexes, forming a temporary ion bridge. To es-
tablish a link between the attractive force and the forma-
tion of ion bridges, we have determined the number of
bridging counterions by counting those which are within
0.25 nm from the two opposing duplexes at the same
Inter−duplex distance (nm)
Inter−duplex distance (nm)
FIG. 4. (a): Interaction potential versus inter-duplex distance
for Pu, Sd, and Sm-DNA. ∇ : no springs. With springs (±x, ±y)
= (0.90, 0.90), +; (0.95, 0.95), ? ; (1.00, 1.00), ∗; (1.05, 1.05),
○ nm. (b): Force-separation curve for Sm-DNA. The dots are
obtained from the spring extensions as described in the text.
4 Download full-text
time. An example of the time-evolution of the total num-
ber of bridging Sm’s is displayed in Fig. 5. Due to the
continuous association and dissociation, the total number
of bridges fluctuates about an average value
a lifetime on the order of a few nanoseconds. Irrespective
structure or charge, three or four ligands can be accom-
modated in the space between duplexes of 10 base-pairs
length. At a particular inter-duplex distance the number of
ion bridges does not depend on ligand charge and struc-
ture. The average number of bridges decreases with in-
creasing mean inter-duplex separation. For separations
larger than 3 nm, the size of the ligands becomes too
small for the formation of an ion bridge.
The range of the attractive force is understood in terms
of ion bridges. Due to simultaneous binding to the counte-
rions, the opposing and parallel duplexes are effectively
connected and exert a force on each other. The depth of
the potential is however determined by ligand charge and
structure and a simple relationship with the number of
bridges seems to be lacking. Furthermore, the ion bridges
are transient with a short lifetime on the order of a few
nanoseconds and characterized by a continuous rear-
rangement of the binding sites (i.e., phosphate moieties on
DNA and nitrogens of the polyamines). The typical life-
time of an ion bridge is hence of the same order of magni-
tude as the lifetime of a contact formed by a ligand and a
single DNA molecule. Long range and persistent two-
dimensional ordering of the associated counterions at the
surface of the DNA molecule was not observed. In this
respect our simulations do not support the notion of a
strongly correlated 2D liquid of adsorbed ions.
Our simulated system differs from the situation in ex-
perimental studies in a number of aspects. First, the axes
of interacting DNA duplexes are often skewed, because in
a densely packed system the molecules undulate with a
wavelength (deflection length) less than the persistence
length . We do not expect that the multivalent ion
mediated interaction in a skewed configuration is qualita-
tively different from the one in a parallel configuration,
because of the absence of long range position correlation
among the adsorbed ions. Second, we only considered a
pair interaction, whereas in a dense phase one DNA
molecule interacts with multiple DNA molecules. Third,
most experimental systems are not salt-free. Despite these
obvious limitations, our simulations provide insight in the
multivalent-ion induced attraction of DNA at the molecu-
lar level which is difficult, if not impossible, to obtain
with alternative theoretical approaches or from experi-
ments. The present study represents the first demonstra-
tion of the experimentally established counterion induced
attraction using a full atomic model. This implies that this
effect is now confirmed in a theoretical description be-
yond the dielectric continuum approximation of the sol-
The support of research grants RG65/06 from Nanyang
Technological University (to YM), T206B3207 from the
MOE (to LN) and R144000145 from National University
of Singapore (to JvdM) is acknowledged.
 V.A. Bloomfield, Curr. Opin. Struc. Biol. 6, 334, (1996).
 E. Raspaud et al., Biophys. J. 74, 381 (1998), ibid 88, 392
 J.X. Tang et al., Phys. Chem. Chem. Phys. 100, 796
 G.C.L. Wong, Curr. Opin. Coll. & Interface Sci. 11, 310
 F. Oosawa, Biopolymers 6, 1633 (1968).
 I. Rouzina and V.A. Bloomfield, J. Phys. Chem., 100,
 B. I. Shklovskii, Phys. Rev. Lett. 82, 3268 (1999).
 L. Guldbrand, L.G. Nilsson, and L. Nordenskiöld, J.
Chem. Phys. 85, 6686 (1986).
 A.P. Lyubartsev and L. Nordenskiöld, J. Phys. Chem. 99,
 B.Y. Ha and A.J. Liu, Phys. Rev. Lett. 79, 1289 (1997).
 J.C. Butler et al., Phys. Rev. Lett. 91, 028301 (2003).
 S.Y. Ponomarev, K.M. Thayer, and D.L. Beveridge, Proc.
Natl. Acad. Sci. USA 101, 14771 (2004).
 S.S. Cohen, A guide to the polyamines (Oxford University
 B. Roux, Comp. Phys. Commun. 91, 275 (1995).
 T.E. Cheatham, P. Cieplak, and P.A. Kollman, J. Biomol.
Struc. Dyn. 16, 845 (1999).
 H.J.C. Berendsen et al., Intermolecular Forces (Reidel
 E. Lindahl, B. Hess, and D. van der Spoel, J. Mol.
Modeling 7, 306 (2001).
 N. Korolev et al., J. Mol. Biol. 308, 907 (2001).
 A.A. Kornyshev and S. Leikin, Phys. Rev. Lett. 82. 4138
 X. Qiu et al., Phys. Rev. Lett. 99, 038104 (2007).
 R. Podgornik and V.A. Parsegian, Phys. Rev. Lett. 80,
 T. Odijk, Macromolecules 16, 340 (1983).
05 10 1520
Simulation time (ns)
Inter-duplex distance (nm)
22.22.4 2.6 2.833.2
FIG. 5. (a): Time-evolution of the number of ion bridges in the
Sm-DNA simulation with (±x, ±y) = (0.90, 0.90) nm (rebinned
in 0.1 ns intervals). (b): Average number of ion bridges vs.
inter-duplex distance.: ○ , Co-DNA; ? , Sm-DNA; ∇ , Sd-
DNA; ? , Pu-DNA.