Informatics and Mathematical Modeling and Department of Physics,
The Technical University of Denmark, 2800 Lyngby, Denmark
104 Y.B. Gaididei et al.
facile in vitro formation of DNA circles shorter than 30 nm (100 base pairs), opened a new
exciting area of research where an interaction of charge carriers with the bending degrees
of freedom of a closed molecular chain is crucial.
Research in solitonic properties of the chains with bending has been initiated in recent
years [7–16]. In particular, it was shown that the bending of the chain could manifest
itself as an effective trap for nonlinear excitations [7, 8, 10, 11, 15, 16] and that the
energy of excitations decreases when the curvature of the bending increases [10, 11].
A phenomenological model for describing the conformational dynamics of biopolymers
via nonlinearity-induced buckling and collapse instability was proposed in [17]. Buckling
instability of semiflexible polyelectrolytes with intrachain attractions due to counterion
correlations was studied in [18].
Quite recently, a simple, generic model for electron–curvature interactions on closed
molecular chains was proposed [19]. It was shown that the presence of charge modifies
(softens or hardens) the local chain stiffness. In the mean field approach, it was found that,
due to the interaction between electrons and the bending degrees of freedom, the circular
shape of the chain may become unstable and the chain takes the shape of an ellipse or, in
general, of a polygon.
In this paper, which we dedicate to the memory of Alwyn C. Scott, we study the charge-
induced conformational transformations of closed molecular chains in the presence of
thermal fluctuations. The paper is organized as follows: In Section 2, we describe a model
and present an analytical solution in the mean-field approach. In Section 3, we compare our
analytical results to results obtained directly by numerical simulations. Section 4 presents
some concluding remarks.
2 The model
We consider a polymer chain consisting of L units (for DNA, each unit is a base pair) labeled
by an index l, and located at the points �rl =(xl, yl), l = 1, . . . , L. We are interested in the
case when the chain is closed, and so we impose the periodicity condition on the coordinates
�rl
�rl = �rl+L. (1)
To describe the chain flexibility, we use a discrete wormlike chain model. In the
framework of this model of chain flexibility, the bending energy of the chain has the form
Ub = k2
∑
l
κ2
l , (2)
where
κl ≡ |�τl − �τl−1| = 2 sinαl
2(3)
determines the curvature of the chain at the point l. Here,
�τl = 1
a(�rl+1 − �rl
)(4)
is the tangent vector at point l of the chain and αl is the angle between the tangent vectors
�τl and �τl−1, a is the equilibrium distance between units (in what follows, we assume a = 1),
and k is the elastic modulus of the bending rigidity (spring constant) of the chain.
Langevin dynamics of conformational transformations 105
In addition to the bending angle, there is a degree of freedom at each segment l, l + 1
describing the change of the distance |�rl − �rl+1| between units. We take the corresponding
stretching energy in the form
Us = σ
2
∑
l
(|�rl − �rl+1| − a)2
(5)
where σ is an elastic modulus of the stretching rigidity of the chain.
We assume that there is a small amount of mobile carriers (electrons, holes in the case of
DNA, protons in the case of hydrogen-bonded systems) on the chain. We take the simplest
theoretical model for the carriers, a nearest neighbor tight-binding Hamiltonian of the form
Hel = E0
∑
l
|ψl|2 + J∑
l
∣∣∣ψl − ψl+1
∣∣∣2
, (6)
where ψl is the complex amplitude that characterizes the probability of finding a carrier at a
site l, E0 is the on-site electron (hole) energy that is determined by the affinity (ionization)
potential of the base, and J describes carrier hopping between adjacent sites. We are
interested in the role of bending degrees of freedom and their coupling with charge carriers.
To derive an explicit form of the electron–conformation interaction, we will assume that
each lth chain unit possesses a quadrupole moment, which is characterized by the tensor
Qα β
l (α, β = x, y, z). Then, the interaction between an electron and chain units takes the
form
Hint =∑
l,l′Vl l′ |ψl|2, (7)
where the matrix element Vl l′ describes a charge–quadrupole interaction between an
electron that occupies the site l in the chain and the group at the site l′
Vl l′ =∑
αβ
e Qα β
l′(�rl − �rl′
)α
(�rl − �rl′)β
|�rl − �rl′ |5 . (8)
We will assume that the subunits are axially symmetric about the z axis. Therefore, the
quadrupole moment tensor Qαβ
l is diagonal and has the form Qxx = Qyy = −Qzz/2 ≡ Q . In
this case, the matrix element of the charge–quadrupole interaction (8) takes the form
Vl l′ = e Q|�rl − �rl′ |3 . (9)
The nearest neighbors l ± 1 do not contribute to the charge–curvature interaction, and
one should take into account the next-nearest neighbor coupling. Taking into account that
the distance between the next-nearest neighbors can be expressed as follows:
|�rl − �rl±2| = a√
2 + 2 �τl · �τl±1 = a√
4 − κ2
l±1(10)
in the small-curvature limit, we obtain the charge–curvature interaction in the form
Hint = 1
2
∑
nχ |ψl|2
(κ2
l+1+ κ2
l−1
)(11)
106 Y.B. Gaididei et al.
where
χ = 3
32
eQa3
(12)
is the coupling constant and we have omitted a constant term.
The total Hamiltonian of the system can be presented as the sum
H = Ub + Us + Hel + Hel−conf. (13)
The quantity
ν ≡ 1
L∑
l
|ψl|2 (14)
gives the total density of charge carriers, which can move along the chain and participate
in the formation of the conformational state of the system. We will neglect the interaction
between electrons (holes). Combining (2) and (11), we notice that the effective bending
rigidity changes close to the points where the electron (hole) is localized. For positive values
of the coupling constant χ , there is a local softening of the chain, while for χ negative, there
is a local hardening of the chain.
The dynamics of the filament is described by the Langevin equations
ηddt
xl = − ∂ H∂ xl
+ Xl(t),
ηddt
yl = − ∂ H∂ yl
+ Yl(t), (15)
iddt
ψl = − ∂ H∂ ψ∗
l(16)
with the Hamiltonian H being defined by (13). Thus, the conformational dynamics is con-
sidered in an overdamped regime when inertia terms (M d2
dt2 xl, M d2
dt2 yl with M being the
mass of the unit) are neglected. The coefficient η gives the friction. The Gaussian white
noise (Xl(t), Yl(t)) is
⟨Xl(t) Xl′
(t′)⟩ = ⟨
Yl(t) Yl′(t′)⟩ = 2 Dδl l′ δ
(t − t′
),
⟨Xl(t) Yl′
(t′)⟩ = 0, (17)
where the standard deviation D, in accordance with the fluctuation–dissipation theorem, is
proportional to the temperature T: D = η T.
2.1 Zero-temperature limit
Some understanding of the dependence of conformational properties of the system on the
strength of the charge–bending interaction χ and bending rigidity κ is obtained from the
continuum equations for the model in the zero-temperature limit [19].
Assuming that the characteristic size of the excitation is much larger than the lattice
spacing, one can replace ψl by a function ψ(s) of the arc length s, which is the continuum
analogue of n. We assume that the chain is inextensible (σ → ∞):
|�rl − �rl+1| = 1. (18)
Langevin dynamics of conformational transformations 107
The inextensibility constraint (18) reads, in the continuum limit,
|∂s�r|2 = 1. (19)
The inextensibility constraint (19) is automatically taken into account by choosing the
parametrization
∂s x(s) = sin θ(s), ∂s y(s) = cos θ(s), (20)
where the angle θ(s) satisfies the conditions
θ(s + L) = 2π + θ(s), (21)
L∫
0
cos θ(s) ds =L∫
0
sin θ(s) ds = 0, (22)
which follow from (1). In the framework of the parametrization (20), the shape of the chain
is determined by the equations
x(s) =s∫
0
sin θ(s′) ds′, y(s) =s∫
0
cos θ(s′) ds′, (23)
the chain curvature is given by
κ(s) ≡ |∂2
s �r(s)| = ∂sθ, (24)
and the continuum version of the Hamiltonian (13) takes the form
E =L∫
0
{J (∂sψ)2 +
(k2
− χ ψ2
)(∂sθ)2
}ds. (25)
The Euler–Lagrange equations for the problem of minimizing E , given by (25) under the
constraint
1
L
L∫
0
ψ2 ds = ν, (26)
become
∂2
s ψ + χ
J(∂sθ)2 ψ − λψ = 0, (27)
∂s
((k2
− χ ψ2
)∂sθ
)= 0, (28)
where λ is the Lagrange multiplier. Solving (27) and (28) under the periodic boundary
conditions
ϕ(s) = ϕ
(s + L
n
), n = 2, 3, . . . , (29)
one finds that there are two kinds of solutions to (27) and (28) [19].
108 Y.B. Gaididei et al.
– Circular chain.
Charge is uniformly distributed along the chain
ψ = √ν (30)
and the curvature of the chain is constant
κ(s) ≡ ∂sθ = 2π
L. (31)
This case corresponds to a circular chain
x = 2π
Lsin
2π sL
, y = −2π
Lcos
2π sL
. (32)
– Polygonally deformed chain.
Solving (27) and (28) under the periodic boundary conditions (29), one finds that the
n-gon structure
θ(s) ≈ 2π
Ls + χ ν
k2
n
√
2
(ν
νn− 1
)sin
(2nπ
Ls)
(33)
appears when the charge density exceeds the threshold value νn:
ν > νn ≡ Jk8 χ2
n2. (34)
Inserting (33) into the closure condition (22), we find that it is satisfied for n ≥ 2.
Equations (23) and (33) describe a polygon: for n = 2, it is an elliptically deformed
chain, while for n = 3, it has a triangular shape (see Fig. 1). The polygon structure
is a result of the self-consistent interaction between electrons and bending degrees of
freedom: extrema of the curvature and of the charge density correlate: in the case of
the softening electron–curvature interaction (χ > 0) maxima, of curvature and charge
density coincide, while, in the case of the hardening interaction (χ < 0), the minima of
the curvature coincide with the maxima of the charge density.
Fig. 1 The analytically
calculated shape of the chain:
the ellipse-like state (n = 2),
the triangular state (n = 3)
Langevin dynamics of conformational transformations 109
Near the threshold (34), the energy difference between the n-gon structure and the
circular chain is given by the expression
En − Ecirc ≈ −16π2
Lχ2
k(ν − νn)
2 . (35)
We note that when the charge density is above the critical value the deformed structure
with spatially inhomogeneous charge distribution is energetically more favorable than
the circular system with a uniformly distributed charge. The state with elliptically
deformed chain (n = 2) is the energetically most preferable.
3 Numerical studies
To check how robust the results obtained are in the zero-temperature limit with respect to
thermal fluctuations, we carried out the dynamical simulations of (15), (16), and (17). We
took as our starting configurations systems involving the electric charge density of the same
magnitude (ψl) at all points. Initially, all the lattice points were placed at symmetric points
on a circle of an appropriate radius. We performed such simulations for several values of the
charge density ν and the standard deviation D. In this paper, we concentrated our attention
on the case when the charge carriers have a positive charge (e.g., holes in DNA) and their
presence leads to a local hardening of the chain (χ > 0). In what follows, we have chosen
the damping coefficient η and the bending rigidity k equal to unity.
To analyze the shape of the chain, it is convenient to use the radius-of-gyration tensor I
[20, 21] with elements
Ixx = 1
L∑
l
(xl(t) − xc(t))2, Iyy = 1
L∑
l
(yl(t) − yc(t))2,
Ixy = 1
L∑
l
(xl(t) − xc(t)) (yl(t) − yc(t)) (36)
where
(xc, yc) = 1
L∑
l
(xl, yl) (37)
is the center-of-mass coordinate. The square roots of the two eigenvalues Rα = √Iα of the
tensor I give the two principal radii of the system. They give the sizes of the filament along
the major and minor axes. As is seen from (36), the eigenvalues have the form
I1,2 = 1
2
{(Ixx + Iyy
) ±√(
Ixx − Iyy)2 + 4 I2
xy
}. (38)
To characterize the shape of the conformation, it is convenient to introduce the quantity A =I1 − I2, which is called the “aspherity” [21]. It characterizes the shape’s overall deviation
from circular symmetry.
Typical final shapes of the filament for the two different values of noise intensity D =0.001 and D = 0.1 are shown in Figs. 2 and 5. They show that the ellipse-like shape is
rather robust. The charge density and the curvature distributions are in full agreement with
the results of the mean-field approach [19]: the ellipse-like shape is the final state of the
shape evolution and the curvature is minimal in the places where the charge density is
110 Y.B. Gaididei et al.
Fig. 2 The top panel shows the
shape of the chain (t = 600)
and the bottom panel shows the
charge distribution (solid line)
and curvature variation (dashedline) along the chain in the case
