1. STATISTICAL MECHANICS OF DNA LOOPING
FELIX X.-F. YE
1,2
, NATHAN BAKER
1
, PANOS STINIS
1
, HONG QIAN
2 1
PNNL,2
UNIVERSITY OF WASHINGTON
INTRODUCTION
The formation of DNA loops is an important part
in the biological processes, including gene expres-
sion, genetic recombination, DNA replication and
repair. Some of binding sites are located many
thousands of base pair apart so it is required for
a DNA molecule
to form a loop.
In this presen-
tation, we are
interested in the
thermodynamic
cost of DNA
looping in terms
of the statistical
distribution of
polymer con-
formations in
different models.
Lisa G. DeFazio et al. EMBO J. 2002; 21:3192-3200
Figure 1: Each end of DNA molecules
is bound to a single protein. The one
in the lower right has been circularized
with what two proteins bound to the
ends of the DNA.
J FACTOR
Figure 2: Intra- and Intermolecular synapsis reactions. The cyclization
reaction is also a part of two step mechanism.
The J factor is a quantification of the free energy
cost of cyclization in terms of the ratio of respec-
tive equilibrium constants Kc and Kd for two dif-
ferent reactions.
J = 8π2 k1/k−1
kd
= 8π2 W(0)
NAv
= 8π2 Zc/Z
NAv
(1)
where W(0) is the probability density for end-
to-end distance ree evaluated at 0, Zc and Z are
the canonical ensemble partition functions for the
loop chain and the chain without any constraints.
FREELY JOINTED MODEL
Consider the DNA
molecule is homoge-
neous and the dynamics
is the local jump process.
The probability density
W(0) can be solved
easily. The J factor is
J = 8π2 (3/2πNl2
)3/2
NAv
(2)
Figure 3: A DNA molecule con-
sists of N links, each of length
l and every bond vector rn =
Rn − Rn−1 are independent of
each other.
ROUSE MODEL
We have the stochas-
tic differential equation
of motion for a freely
draining polymer with
both end are free in
dilute solution. It is
the mechanically over-
damping spring model.
Figure 4: A DNA molecule con-
sists of N beads and each bead is
connected with a homogeneous
spring with spring constant k.
The position of beads is a 3D vector and we can
study each component separately for simplicity.
dR
dt
=
1
ζ
AR +
2kbT
ζ
dW(t)
dt
(3)
where ζ the frictional coefficient, W(t) is the
Wiener process and the matrix A is
A = k






−1 1 0 . . . 0 0
1 −2 1 . . . 0 0
. . . . . . . . . . . . . . . . . .
0 0 . . . 1 −2 1
0 0 . . . 0 1 −1






(4)
The corresponding forward Fokker-Planck equa-
tion for P(R, t) is
∂P
∂t
= · (Dr P −
1
ζ
ARP) (5)
The diffusion constant Dr = kbTI/ζ, that is
fluctuation-dissipation relation. The stationary
distribution is the equilibrium Boltzmann distri-
bution, Ps(R) ∝ exp(1
2 RT
AR/kbT).
ROUSE MODEL CONT.
If we consider 3D model, the dimension of R ex-
tends to 3(N + 1). The J factor is
J ∗ NAv = 8π2
(
k
2πkbT
N + 1
N2
)3/2
(6)
If we choose the spring constant k = 3kbT
l2 , then
the J factor will be
J ∗ NAv = 8π2
(
3
2πl2
N + 1
N2
)3/2
(7)
It is consistent with the result in freely joint model
as N → +∞.
ZIMM MODEL
For long polymer, the hydrodynamic interaction
must be included. The stochastic differential
equation will be
dR
dt
= HAR + 2kbTH
dW(t)
dt
(8)
where this mean hydrodynamic interaction ma-
trix H is
H =
1
ζ







1 1√
2
. . . 1√
N
1 1 . . . 1√
N−1
1√
2
1 . . . 1√
N−2
. . . . . . . . . . . . . . .
1√
N
1√
N−1
1√
N−2
. . .







(9)
The corresponding forward Fokker-Planck equa-
tion for P(R, t) is
∂P
∂t
= · (−HARP + Dz P) (10)
The diffusion constant
Dz = kbTH. The
stationary distribution is
the same as the Rouse
model. So the J factor for
Zimm model is the same
as Rouse model.
Figure 5: Non-local interactions
are included. The further two
beads apart, the weaker the in-
teractions are.
REACTION RATE CONSTANT
Although all J factors
are the same for these
three models, the re-
action rate constants
are different. It is
believed the hydrody-
namic interaction ac-
celerates the reaction
rate.
Figure 6: The capture radius
of DNA looping here is α. The
looping time tL is the mean first
passage time for the length of ree
smaller than the capture radius.
In the normal model analysis,
ree = −4
p:odd
Xp(t) (11)
where Xp is the normal coordinate and each is
governed by decoupled linear stochastic differen-
tial equation. The reaction rate constant k1 is the
reciprocal of the first passage time.
It is now an 1D diffusion model and the passage
time is given by SSS theory
t(x0) =
x0
a
1
DeePs(ree)
dree
L
x
Ps(r )dr (12)
It can be further simplified by Kramer rate theory,
tL ≈
1
DeePs(α)
(13)
INHOMOGENEOUS BOND
The DNA sequence is
not homogeneous so
it is necessary to con-
sider this effect. If
one of the springs has
far greater spring con-
stant k than others
k. It is possible to
exhibit different time
scale behaviors.
Figure 8: In the simplest case, only
one bond is inhomogeneous.
REFERENCES
[1] M. Doi and S. Edwards, The Theory of Polymer Dynamics, The Clarendon Press, Oxford Uni-
versity Press, New York, 1986, 331 pp.
[2] R. Afra and B. Todd, Kinetics of loop formation in worm-like chain polymers The Journal of
Chemical Physics, 138, 174908 (2013)