Dynamics of a polyampholyte hooked around an obstacle
H. Schiessel,* I. M. Sokolov,
²
and A. Blumen
Theoretical Polymer Physics, Freiburg University, Rheinstrasse 12, 79104 Freiburg, Germany
Received 12 December 1996
We consider polyampholytes PAs, which are polymers carrying positive and negative charges. The PAs
are submitted to electric fields and collide with single obstacles. Field separation of PAs depends drastically on
u
, the time to unhook. Our analysis shows
u
to be very sensitive to the charge distribution of the
chains: Unhooking is diffusional for regular, multiblock PAs, with
u
depending on the length of the blocks;
for random charge distributions
u
increases exponentially with the PAs’ length and unhooking is subdiffusive.
S1063-651X9751409-7
PACS numbers: 87.15.-v, 05.40.+j, 82.45.+z
Gel-electrophoresis GE is a widely used technique for
size separation of polyelectrolytes PEs of different length
such as DNA fragments 1. During GE the polymer gets
temporarily hooked around gel fibers, and the release time
determines the overall process. In a more general fashion one
investigates the time
u
needed by polymers to disentangle
from single obstacles say posts2–7.
A related problem is the behavior of polyampholytes
PAs, which are heteropolymers that carry both positive and
negative charges, in gels under external electrical fields. In a
series of works 8–12 we have investigated the behavior of
PAs in dilute solutions and found that the charge distribution
CD leads to a great variety of static and dynamical laws
9,12 which furthermore depend on the intramolecular elec-
trostatic coupling 11,12, on the solvent’s quality 11,12,
on the chain’s extensibility 10cf. also the paper of
Winkler and Reineker 13, and on the hydrodynamically
mediated monomer interactions 12. A study of the PAs’
motion using the biased reptation framework 14 revealed
the paramount role played by the detailed distribution of
positive and negative charges along the chain on the overall
PA’s mobility in a network. We hence focus here on how
u
, the time to unhook from a single obstacle, depends on
the PA’s particular CD. As we proceed to show, we indeed
find
u
to depend drastically on the CD.
Before discussing PAs we recall the findings for PEs en-
tangled to a fiber. Just after the collision the PE coil gets
unraveled by the external field and shows in the simplest
case only two arms, left and right of the fiber, consisting of
monomers 1 to m and m +1 to N , respectively. In general,
we denote by q
n
the charge on the n th monomer, so that for
PE, evidently, q
n
=q for all n . The total charge on the left
arm is Q
m
=mq , and on the right arm Q
tot
-Q
m
=(N-m)q,
where Q
tot
=Q
N
is the total charge of the PE. Hence, the
tangential force F ( m ) acting along the chain from right to
left is F ( m ) =Q
m
E -( Q
tot
-Q
m
)E=2qEm-qEN. Assuming
the chain to be free-draining i.e., the friction to be propor-
tional to N and to be inextensible and neglecting the Brown-
ian motion and the friction between PE and fiber, one has
N bm
˙
t =2 qEm t -qEN . 1
In Eq. 1 m ( t ) denotes the number of the monomer which is
in contact with the fiber at time t , b is the monomer length,
and the friction constant per monomer. The solution of Eq.
1 with initial condition m (0) =m is
m t =
m -
N
2
exp
2 qE
N b
t
+
N
2
. 2
From Eq. 2 the time
u
is given implicitly by m (
u
) =0 for
m N /2 and by m (
u
) =N for m N /2. Thus, say for m
N /2, one has
u
=( N b /2qE )lnN/(N-2m). Hence, as-
suming m =m (0) to be equally distributed we find by aver-
aging over m
u
=
b
2 qE
N . 3
We note that other models, as long as the chain is inexten-
sible, lead to the same dependence on the parameters 3–7.
For Gaussian chains, however, one has
u
N
2
5,6. Con-
sider, namely, a bead-spring model, whose harmonic springs
have equal spring constants K . The tension acting on each
arm increases from the free end to the hook, where the force
is of the order of qEN . Thus the extension l of the springs
near the hook is of the order l qEN / K ; taking l as the
typical segment length, i.e., assuming b l , Eq. 3 takes
now the form
u
N
2
/ K which is independent of E 5,6.
Furthermore, the same power-law dependences of
u
on N
also hold in the non-draining case, despite the complex PE
behavior discussed in Ref. 15. Hence, for PEs
u
is a power
law of N ; now, since the standard deviation
of
u
is of the
same order than
u
itself for the inextensible case, discussed
above, one has
=&
u
, collisions with individual ob-
stacles are not a powerful means to separate different PE
according to length see the discussion in Ref. 4.
As shown, a PE gets unhooked due to the difference in the
forces acting on its two arms. As we proceed to show, the
unhooking scenario of PAs is generally different, since in
many cases a PA requires thermal activations to disentangle
from a fiber. This leads to an exponential N dependence of
*Present address: Materials Department, University of California
at Santa Barbara, Santa Barbara, CA 93106.
²
Also at P. N. Lebedev Physical Institute of the Academy of Sci-
ences of Russia, Leninski Prospect 53, Moscow 117924, Russia.
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