Molecular dynamics simulation of multivalent-ion mediated attraction between DNA molecules.
ABSTRACT 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 lifetime of the ion bridges is short on the order of a few nanoseconds.
- SourceAvailable from: Xiangyun Qiu[show abstract] [hide abstract]
ABSTRACT: Can nonspecifically bound divalent counterions induce attraction between DNA strands? Here, we present experimental evidence demonstrating attraction between short DNA strands mediated by Mg2+ ions. Solution small angle x-ray scattering data collected as a function of DNA concentration enable model independent extraction of the second virial coefficient. As the [Mg2+] increases, this coefficient turns from positive to negative reflecting the transition from repulsive to attractive inter-DNA interaction. This surprising observation is corroborated by independent light scattering experiments. The dependence of the observed attraction on experimental parameters including DNA length provides valuable clues to its origin.Physical Review Letters 08/2007; 99(3):038104. · 7.94 Impact Factor
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ABSTRACT: We have examined some subtle parameter modifications to the Cornell et al. force field, which has proven quite successful in reproducing nucleic acid properties, but whose C2'-endo sugar pucker phase and helical repeat for B DNA appear to be somewhat underestimated. Encouragingly, the addition of a single V2 term involving the atoms C(sp3)-O-(sp3)-C(sp3)-N(sp2), which can be nicely rationalized because of the anomeric effect (lone pairs on oxygen are preferentially oriented relative to the electron withdrawing N), brings the sugar pucker phase of C2'-endo sugars to near perfect agreement with ab initio calculations (W near 162 degrees). Secondly, the use of high level ab initio calculations on entire nucleosides (in contrast to smaller model systems necessitated in 1994-95 by computer limitations) lets one improve the chi torsional potential for nucleic acids. Finally, the O(sp3)-C(sp3)- C(sp3)-O(sp3) V2 torsional potential has been empirically adjusted to reproduce the ab initio calculated relative energy of C2'-endo and C3'-endo nucleosides. These modifications are tested in molecular dynamics simulations of mononucleosides (to assess sugar pucker percentages) and double helices of DNA and RNA (to assess helical and sequence specific structural properties). In both areas, the modified force field leads to improved agreement with experimental data.Journal of biomolecular structure & dynamics 03/1999; 16(4):845-62. · 4.99 Impact Factor
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ABSTRACT: Conditions of double-stranded DNA precipitation by the polyamines spermidine and spermine have been determined experimentally and compared to theoretical predictions. The influence of the concentrations of DNA and added monovalent salt, and of the DNA length has been investigated in a systematic manner. Three regimes of DNA concentrations are observed. We clarify the dependence of these regimes on the monovalent salt concentration and on the DNA length. Our observations make possible a rationalization of the experimental results reported in the literature. A comparison of the precipitation conditions of different kinds of polyelectrolytes suggests a general process. Our experimental data are compared to the "ion-bridging" model based on short-range electrostatic attractions. By starting from the spinodal equation, predicted by this model, and using the limiting form of Manning's fractions of condensed counterions, analytical expressions of the precipitation conditions have been found in the three regimes. Experimental and theoretical results are in good agreement.Biophysical Journal 02/1998; 74(1):381-93. · 3.67 Impact Factor
Molecular Dynamics Simulation of Multivalent-Ion Mediated Attraction
between DNA Molecules
Liang Dai,1Yuguang Mu,2Lars Nordenskio ¨ld,2and Johan R.C. van der Maarel1
1Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
2School of Biological Sciences, Nanyang Technological University, 60 Nanyang Drive, Singapore 637551
(Received 20 September 2007; published 18 March 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 lifetime of the ion bridges is short on the order of a few nanoseconds.
DOI: 10.1103/PhysRevLett.100.118301 PACS numbers: 82.35.Rs, 87.14.G?, 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 interhelical
spacing in the range 2.8–3.2 nm. Polyamines also induce
the collapse of single DNA’s into toroids . Condensation
has been reported for other polyelectrolytes as well, in-
cluding actin and microtubules . Although much work
has been done to elucidate the mechanisms involved in
stabilizing the condensed state, the detailed structural ar-
rangement 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 proposed [6,7].
In theoretical modeling, DNA is usually treated as a uni-
formly charged cylinder, the counterions as point or spheri-
cal charges, and water as a continuous dielectric medium
[8–10]. These approximations are appropriate for interac-
tions over larger distances exceeding the atomic scale, but
in dense systems, suchas in DNAcondensates, a molecular
description is necessary for an understanding of the con-
densation phenomenon. This can now be achieved with the
molecular dynamics (MD) computer simulation method
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 between
two parallel double-stranded DNA duplexes with MD
simulations and umbrella sampling . To further inves-
tigate 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
multivalent-ion induced DNA attraction with an atomic
model including a molecular description of the solvent
Allsimulations wereforsalt-free systemsusinga rectan-
gular cell, which contains one or two identical DNA dec-
amers in the B-form of 2 nm outer diameter (see Fig. 1).
A sequence of10 base-pairs
C3TTCTCCGAT5) was randomly chosen. The DNA
charge is neutralized with 10 di-, 7 tri-, or 5 tetravalent
counterions (excess cationic charge was compensated with
chloride). The 3’ end of each strand is connected 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
DNA decamers and ten Sm counterions in the initial configura-
tion. The box has a transverse dimension of 7 ? 7 nm2and
3.4 nm height. (b) Snapshot illustrating ion bridge formation.
(a) Top view of the simulation box with two parallel
PRL 100, 118301 (2008)
21 MARCH 2008
© 2008 The American Physical Society
the periodicity along the longitudinal axis matches the
helical twist of the duplex with 10 base pairs per turn.
Furthermore, the connectivity of the decamers set by the
boundary condition inhibits bending fluctuations with
wavelengths exceeding the 3.4 nm longitudinal repeat
distance ofthesimulation box.Asnapshotofthetransverse
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 . Electro-
static interactions were treated by the particle mesh Ewald
method and the temperature 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 separation
is small with the two duplexes almost touching each other.
To study the interaction at larger separations, we have
applied an external potential Pext? 1=2k???x;y??2with
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 kJmol?1nm?2and ??x;y? is the de-
viation of the pull group with respect to a reference point
(?x, ?y). Since the two springs have the same spring
constant, the total system experiences no net force. To
obtain the interaction energy as a function of the distance
Dintbetween the centers of mass of the duplexes the
contribution from Pexthas to be subtracted from the inter-
action energy. In practice, we have assigned a weighing
factor exp?Pext=kBT? to every sampling point. We then
?weighed?Dint? from the fractional time the duplexes are
separated by a distance Dintand the true interaction energy
follows from F ? ?kT ln?weighed.
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 the
duplex and then remained territorially bound in the fol-
lowing 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 interaction 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 molecule
simulation with all counterions territorially bound to DNA
(see Fig. 1). The interhelical distance was initially set to
3.8 nm. This distance does not allow a simultaneous con-
tact of one Smmoleculewith the two duplexes (the contour
length of Sm is 1.6 nm). Because of the periodic boundary
conditions and the fact that the top and bottom base-pairs
of each DNA decamer are connected, the duplexes can
hardly bend and they remain parallel.
The fluctuating interhelical distance Dintin a simulation
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 interduplex sepa-
ration of about 2.4 nm. Close inspection of the configura-
tions revealed the details of the attraction (an example is
displayed in Fig. 1). A Sm is usually territorially bound to
one duplex. Since Sm is a linear tetravalent polyaminewith
a positivecharge 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 rearrangement of the bridg-
ing Sm. We surmise that the formation of these transient
ion bridges results in a net attraction. Simulations were
also done for Pu, Sd and Co. The result for the trivalent Sd
Center of mass
Reference point (x,y) of spring 1
and how the two external springs pull the two DNA duplexes in
opposite directions in the transverse plane.
Illustration of the cross section of the simulation box
Inter-duplex distance (nm)
Simulation time (ns)
Simulation time (ns)
(a) simulation without springs; (b) as in (a) but with two springs
centered at ??x;?y? ? ?0:95;0:95? nm.
Fluctuation in interduplex spacing of Sm-DNA.
PRL 100, 118301 (2008)
21 MARCH 2008
is qualitatively similar. Control simulations with sodium
counterions only, confirmed the absence of attraction and
resulted in equilibrium separations of 5 nm.
To obtain sufficient sampling for larger separations it is
necessary to apply the umbrella sampling technique.
Continuous 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 exhibits 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 interduplex separa-
tion in the side-by-side complex obtained in the simula-
tions without external forces and are related to the structure
of the ligands. The depth of the potential takes the values
?16 (Co), ?9 (Sm), and ?6 (Sd) kBT. With increasing
valence and smaller ligand size, the interaction potential
becomes more attractive. For very short separations the
potential is always repulsive due to electrostatic and hard-
core interactions. For larger separations, beyond the mini-
mum, the potential is attractive and monotonously in-
creases until it levels off for Dint> 3 nm. The range of
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
potential depth less than 2kBT. This weak attraction is
consistent with the experimental observation that Pu can-
not induce condensation  and the experimentally ob-
served weak DNA attraction in the presence of divalent
magnesium . In a simulation of two DNA’s with so-
dium counterions we have checked that the potential is
always repulsive. One should bear in mind that our simu-
lations 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 interduplex distance.
Furthermore, 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 pairwise additive
. For Sm-DNA in the absence of monovalent cations
theexperimental value oftheinterduplexdistance is2.8nm
. This indicates the pair treatment of the interaction as a
major cause for the shorter equilibrium interduplex dis-
tances as compared to the experimental values.
The interduplex 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 pertaining
to Sm-DNAwith the help of an arbitrary sixth order poly-
nomial; 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?Dint, these
forces are balanced. If the springs are stretched by an
amount ?x, the attractive force is approximately ??x.
The interduplex direction is not always colinear with the
directions of the springs but the deviations from colinearity
are always quite small and the resulting forces are con-
sequently good first-order approximations. Good agree-
ment with the curve as obtained from the derivative of
the interduplex potential is obtained (see Fig. 4).
As can already be gauged from the potential curve, for
separations less than, say, 2.4 nm the interduplex 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 distance 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
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 boundto the
duplex with one of its four charges close to a phosphate
moiety if this minimum distance is less than 0.25 nm. The
binding is highly dynamic; during the MD run, a counter-
ion 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-
Interduplex distance (nm)
Interduplex distance (nm)
Pu, Sd, and Sm-DNA. 5: no springs. With springs ??x;?y? ?
?0:90;0:90?, +; (0.95, 0.95), 4; (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.
(a) Interaction potential versus interduplex distance for
PRL 100, 118301 (2008)
21 MARCH 2008
tablish a link between the attractive force andthe formation
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 time. An
example of the time-evolution of the total number of
bridging Sm’s is displayed in Fig. 5. Because of the con-
tinuous association and dissociation, the total number of
bridges fluctuates about an average value ?Nbridgewith 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 interduplex distance the number of
ion bridges doesnot depend on ligand charge and structure.
The average number of bridges decreases with increasing
mean interduplex 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. Because of simultaneous binding to the
counterions, the opposing and parallel duplexes are effec-
tively 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 rearrange-
ment of the binding sites (i.e., phosphate moieties on DNA
and nitrogens ofthe polyamines). The typical lifetime of an
ion bridge is hence of the same order of magnitude as the
lifetime of a contact formed by a ligand and a single DNA
molecule. Long range and persistent two-dimensional or-
dering of the associated counterions at the surface of the
DNA molecule was not observed. In this respect our simu-
lations 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 mole-
cule interacts with multiple DNA molecules. Third, most
experimental systems are not salt-free. Despite these ob-
vious 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 experiments.
The present study represents the first demonstration of
the experimentally established counterion induced attrac-
tion using a full atomic model. This implies that this effect
is now confirmed in a theoretical description beyond the
dielectric continuum approximation of the solvent.
The support of research Grant No. RG65/06 from
No. R144000145 from National University of Singapore
(to J.v.d.M.) is acknowledged.
 V.A. Bloomfield, Curr. Opin. Struct. Biol. 6, 334 (1996).
 E. Raspaud et al., Biophys. J. 74, 381 (1998); 88, 392
 J.X. Tang et al., Ber. Bunsenges. Phys. Chem. 100, 796
 G.C.L. Wong, Curr. Opin. Colloid Interface Sci. 11, 310
 F. Oosawa, Biopolymers 6, 1633 (1968).
 I. Rouzina and V.A. Bloomfield, J. Phys. Chem. 100, 9977
 B.I. Shklovskii, Phys. Rev. Lett. 82, 3268 (1999).
 L. Guldbrand, L.G. Nilsson, and L. Nordenskio ¨ld,
J. Chem. Phys. 85, 6686 (1986).
 A.P. Lyubartsev and L. Nordenskio ¨ld, 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. U.S.A. 101, 14771 (2004).
 S.S. Cohen, A Guide to the Polyamines (Oxford
University, New York, 1998).
 B. Roux, Comput. Phys. Commun. 91, 275 (1995).
 T.E. Cheatham, P. Cieplak, and P.A. Kollman, J. Biomol.
Struct. Dyn. 16, 845 (1999).
 H.J.C. Berendsen et al., Intermolecular Forces (Reidel,
 E. Lindahl, B. Hess, and D. van der Spoel, J. Mol. Model.
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, 1340 (1983).
0510 15 20
Simulation time (ns)
Interduplex distance (nm)
22.2 2.4 2.6 2.83 3.2
Sm-DNA simulation with ??x;?y? ? ?0:90;0:90? nm (rebinned
in 0.1 ns intervals). (b) Average number of ion bridges vs
interduplex distance.: ?, Co-DNA; 4, Sm-DNA; 5, Sd-DNA;
(a) Time-evolution of the number of ion bridges in the
PRL 100, 118301 (2008)
21 MARCH 2008