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Anomalous electrophoresis, self-trapping and “freezing”
of partially charged polyelectrolytes
Gary Slater
To cite this version:
Gary Slater. Anomalous electrophoresis, self-trapping and “freezing” of partially charged polyelec- trolytes. Journal de Physique II, EDP Sciences, 1992, 2 (5), pp.1149-1158. �10.1051/jp2:1992193�.
�jpa-00247699�
J. Phys. II France 2 (1992) l149-l158 MAY1992, PAGE l149
Classification
Physics Abstracts
32.20C 87.80 05.40
Anomalous electrophoresis, self-trapping and « freezing » of
partially charged polyelectrolytes
Gary W. Slater
Ottawa-Carleton Institute for Physics, University of Ottawa, Physics Department, 150 Louis Pasteur, Ottawa, Ontario KIN 6N5, Canada
(Received 21August1991, revised 13 December 1991, accepted 13 January 1991)
Rdsumd. Selon le modble de reptation biais6e de l'61ectrophorbse en phase gel, l'orientation des mol£cules d'ADN dons la direction du champ 61ectrique est due h la pr6sence de charges 61ectriques pr~s des bouts des chaines. Cette orientation limite la taille des mot£cules qui peuvent dtre ainsi s6par£es et contribue h l'apparition du ph£nom~ne d'auto-pi£geage. Dons cet article, je pr6sente un modble off la diffusion biais£e dons le tube de reptation est d6coup16e de l'orientation
mo16culaire produite par les extr£mit6s des chaines. Lorsque les extr£mit6s ne sont pas charg£es,
le p16geage domine : la mobilit6 d6croit d'abord comme I/In (E), off E est l'intensit6 du champ
£lectrique appliqu6, avant de d6croitre comme e~~ pour des champs 61ev6s, ce qui indique que la mo16cule est presque totalement « ge16e » sur place. Des changements dans la distribution des
charges 61ectriques le long d'un poly61ectrolyte l1n6aire peuvent donc avoir des effets dramatiques.
Abstract. According to the biased reptation model of gel electrophoresis, the alignment of the DNA molecules in the field direction is due to the presence of electric charges near the ends of the chains. This alignment limits the size of DNA molecules that can be separated, and contributes to the phenomenon of self-trapping. Here, I present a model where the biased diffusion inside the
reptation tube is decoupled from the tube orientation which is produced by the end-segments.
When the ends are uncharged, self-trapping is found to dominate the dynamics : the mobility first decreases with the electric field E as I/In (E) before it vanishes as e~~, indicating that the molecule is
« frozen
» in the gel. This shows that variations in the charge distribution along a chain
polyelectrolyte can have dramatic effects.
Introduction.
The biased reptation model (BRM) was first proposed to explain why the electrophoretic mobility M of DNA molecules scales like I/M, where M is the molecular size [1, 2]. The model also colTectly predicted that molecular alignment is responsible for the co-migration of larger
(Mm40kbp) DNA molecules [3], and that DNA self-trapping (and band inversion) can occasionally occur [4]. More recently, the BRM was used to explain the scaling laws observed in the low-frequency regime of Orthogonal Field Altemation Gel Electrophoresis (OFAGE) [5]. However, the BRM fails to explain the Field-Inversion Gel Electrophoresis (FIGE) effect
[6] and the transient effects [3] which follow from sudden changes in the inten-
l150 JOURNAL DE PHYSIQUE II N° 5
sity/polarity/direction of the applied electric field E. This failure has been interpreted as
evidence in favour of the existence of intra molecular modes whereby the DNA assumes conformations inconsistent with the presence of the reptation tube. Computer simulations [7]
and experimental observations [8] have shown that these non-reptative modes are indeed
responsible for the effects that the BRM cannot explain. Therefore, the BRM should be considered as a first order model where the relevant parameters assume « renormalized » values that take into account some of the effects of the non-reptative modes it is when the
experimental conditions are in resonance with these modes that the BRM fails. The great advantage of the BRM is its simplicity : where computer simulations give only qualitative
information or quantitative results based on a few points, the BRM often gives detailed
analytical results.
In this article, I describe a modified BRM where the two biases (the biased motion inside the tube and the preferential orientation of the new tube segments in the field direction) are decoupled. In the original BRM, each segment of the primitive chain had the same charge
q, and hence these biases were coupled. Here, however, the two (identical) end-segments
have charges qo with q~m qqom 0. This calculation was prompted by recent experiments which show that changing the properties of the ends of a reptative molecule can lead to
dramatic effects even though the same change would show negligible effects in free solution [9] : when a streptavidin molecule was added to one end of the DNA, the larger DNA molecules were found to be completely trapped inside the gel and showed zero mobility. The calculations below study the effect of modifying the charge qo on the head and tail of a
reptating polyelectrolyte. This can be seen as a partial neutralization of the charges on the ends of a DNA molecule, or as a partially charged polyelectrolyte [10].
The model.
For simplicity, I consider a (primitive) molecule with N + 2 segments of length a (the pore size) and friction coefficient ii ; the middle N segments ii
= I,..., N) have a charge
q while the two end segments (I
= 0, N + I) have a charge qo. With a field E
= Ei, the
molecule has curvilinear a velocity [1, 2]
N+i
vi = jj q~ E fi~/(N + 2) ii (I)
=o
where 6~ is the unit vector parallel to the direction of the I-th tube segment (see Fig. la). If h is the end-to-end vector of the middle chain, then (with ~b
= qjq)
qE ihla + ~fi trio + 6N+1)1
~~ " ~~
IN + 2 ii
Finally, the electrophoretic velocity v~ of the molecule is related to its velocity
vi by v~
= vi. (HjL), where H~ is the end-to-end distance of the molecule in the field direction (H
=
h + uo + u~
~ i) and L
= (N + 2) a is the contour length of the tube [I]. The
average electrophoretic mobility is thus
When ~b
= I, equation (3) reduces to the standard BRM result for a chain of N + 2 identical segments [1, 2]. The average is to be computed carefully since it can lead to subtle self-
trapping effects [2].
N° 5 ANOMALOUS ELECTROPHORESIS OF POLYELECTROLYTES list
(~~
Field Direction
i U~
o U~
Fig. I. (a) A partially charged polyelectrolyte reptating in a tube. The two end segments have a
charge qo while the N middle-segments have a charge q. Each segment is characterized by a unit vector
6, and a length a. H and h represent the end-to-end vectors of the entire molecule and of the middle N segments respectively. The angles e, are defined between the vectors fi, and the field axis (x). (b) The polyelectrolyte is considered to be fairly stiff on the length scale of a gel pore (= a). Therefore, conformation (such as this one) where an end-segment is pointing back into the tube
are neglected. Such conformations would largely decrease the effect of changing the charge
qo on the end-segments.
The mobility in equation (3) is not proportional to either (hj) or (Hj), contrary to the case of a uniformly charged polyelectrolyte therefore, M cannot be calculated analytically in general, although computer simulations similar to those in reference [2] can easily be carried out. Here, I will consider the case N » I and 0 w ~b w I since it allows for certain analytical
results to be obtained.
The electric forces acting on the charges qo on the ends of the molecule orient the new tube sections created during reptation ; the average projections of the tube sections are [2]
(ii~ I)
= (cos o)
= coth so I/so m sj3 (4a)
((ii~ I)~)
= (cos~ o)
= 2(cos o)/so m +
~ ~~
(4b)
3 45
where so = qo Ea/2 kT is the scaled electric field and o is the angle between a tube segment (ii~) and the field axis (see Fig. la). The approximate expressions hold for so ~ l. As noted
before, it is the charge q~ that makes the tube align in the field direction.
Results.
The average in equation (3) must take into account the entropy as well as the lifetime of the various tube conformations [2]. As suggested by D6jardin [11], a satisfactory description of the first factor is given by the distribution
I(H~ L (cos o) )~
p(H~) oz exp
~ ~
(4c)
2 La (cos o) (cos o)
because the tube is roughly the trajectory of a biased random walk with projections (4a, b).
The lifetime r(H~) of a conformation is given by the time needed to create a tube section of
length a [2] :
r (H~)
=
~~~ ~
r~ (4d)
152 JOURNAL DE PHYSIQUE II N° 5
with
31Hx)
= ~((~~ (4e)
where r~ =
a~/2 D
oz N is the Brownian time and D is the curvilinear diffusion coefficient of the chain inside its tube (D
= kB T/ (N + 2) ii). The proper weight W (H~) of a conformation in equation (3) is thus W(H~) my (H~)p(H~) [I Ii. When NW I we have H~=h~ (the H~ = h~ = 0 case will be discussed later) and equation (3) becomes
~ j~j)
~ ii N ~a~ ~~~~
~o ~o
= ~~~ hjdh~ W(h~)/ W(h~) dh~ (5b)
N a ii
co co
where the conformational weight W is given by
tanh 8 (h~) (h~ Na (cos o) )~
W(h~)
m exp
~
(5c)
8(h~) 2 Na~
~r
with 8(h~) m sh~la, ~r~
=
(cos~ o) (coso)~ and SW = so. Combining equations we get
M m
~( ln j~ d8 ~i~j~~ e~~~fl~~j1(5d)
ii N s
-co
where a
=
(cos o)/s~r~=
~b w I and I/p
=
2 NW ~s~
=
~ ~~
(the approximate values hold 3
~
for low field intensities). Although the partition function in brackets cannot be computed exactly [11], good approximations can be obtained in three important limits.
In the limit where p » I ~a, I-e- for low fields and/or small molecules such that
NE 2 « 1, only small 8's are relevant and one can use a series expansion for tarn (&) to obtain
M = i f
+ <C°S °>~ ~ l
" ~
+ O IN 8~) 16a)
~ ~2
~
~ ii 3 N ~ 9 ~~ ~~~~ ~ ~~~~
When ~b
= I, one recovers the BRM result for a fully charged polyelectrolyte [11].
However, the second term becomes negative for ~b~ ~ 2/3. This means that a mere 20 9b reduction of the charge on the ends of the molecule changes the qualitative nature of the
problem : increasing the field intensity then decreases the mobility.
In the p « a w I limit, I-e- for large molecules with NE ~ » l and NEl » I, the argument of the integral has a sharp maximum for I
m ) » I. With the proper series expansion about p
I, one obtains :
M m
~ (cos o)~ " ~ ~ "~ O IN ~) (7a)
ii N N~(cos o)~
q s~
~b~ l 2
ii 9 3 N N~s~
~b~
~ ~~~~
N° 5 ANOMALOUS ELECTROPHORESIS OF POLYELECTROLYTES l153
The first two terms are similar to those found by D6jardin [11] and by Doi et al. [12], except for the ~b. This equation implies the existence of a plateau mobility for large molecules (N » I/s(), as well as a minimum mobility (for N
m N *
=
3/s) due to self trapping [4, 11, 12]. Note however that only so(= SW) appears in this expression : this means that reducing
the charge on the end segments largely reduces the plateau mobility and increases the maximum size (= N * ) that can be separated.
Finally, for the intermediate case where am pm I, which colTesponds to having
NEl « I but NE ~» l, the argument of the integral has a maximum for I « I but tails-off slowly for
~ l. One must separate the integral into parts the first part covers the range [- S, + S], with S of order unity, where series expansions of the exponential functions can be used ; and the second part is between ± S and ± oJ, where the tanh (&) can be dropped and the e~~
can be expanded into a series. Calculations give (for S
= 4/3) :
~ ~i ~i~
l.79 +
~~o~~)~~~~~~~N~
~~~~~~~/N~~ e~
+
~~~~
'i
3 N lu(Ns~) ~~~~
Remarkably, the mobility is predicted to decrease as I/In (E) in this intermediate size limit I/~b~ » NE ~ » l (if ~b # 0) or for all molecular sizes such that NE ~ » l (if ~b = 0). Moreover,
the mobility is found to decrease as I/N In (N), which means that it decreases slightly faster than I/N between the small size regime (where M oz I/N see Eq. (6b)) and the large size regime (where M oz N °; see Eq. (7b)). Figure 2a shows a schematic of these behaviours.
The reason why a small value of qo has such a large effect is evident from equation (5c). The
maximum lifetime r(h~) is obtained for conformations with h~ m 0 : this stability of the U-
shaped, zero-velocity conformations is the so-called self-trapping effect [4, 11, 12]. When qo is diminished, the probability p(h~=0) increases sharply and the total weight
W
=
p(0) r(0) of the self-trapping states increases. The logarithmic colTection in equation (8) is due to the alignment process of molecules with small end-to-end distances h~ [13]. The
@ (b) q~=
N~
~-
~
~ ~j_
~-
~f lm = q/2 #f
"° o,ooi
~ ~~ og( £
og (N)
Fig. 2. (a) Schematic log-log diagram showing how the mobility ~1 varies with molecular size
~ l12
N for a given field intensity s. MThen qo~ j)
q, the curves are lower, even in the
I/N regime (see Eq. (fib)), and a new I/[N In (N)] regime appears. The latter replaces the asymptotic plateau when qo =
0. (b) Schematic log-log diagram showing how the velocity ~ls varies with the field
intensity s. Note the exponential decrease for s ~ l when qo = 0.
l154 JOURNAL DE PHYSIQUE II N° 5
motion of such molecules is shown schematically in figure 3a : the molecules hop quickly
between self-trapping states, but stay for long periods of time in the latter. Note that the
logarithmic regime does not exist for ~b
= I, although self-trapping still exists.
Since the h~ m 0 states dominate the dynamics of large molecules when qo = 0, one has to be careful with the approximations used to obtain equation (5d), in particular the fact that
H~ was replaced by h~ in the equations. A simple argument will show that
« high » field intensities is ~ l) can actually « freeze » the molecule, I.e. the mobility becomes negligible.
Figure 4 shows two different conformations with h~
= a. In conformation V, a single jump
of length a towards the + end will increase h~ and the drift towards the + end will accelerate.
In the BRM, the probability of creating a new tube section of length a in the ± direction and in
a time r(h~) is given by [2]
P+ (hx)
=
~
l~
a~~~~
(9)
Since
= shja m s ~ l in the V-conformation, the electric forces strongly favour the + end.
In the W-conformations, however, the situation for s ~ l is quite different. When a molecule
assumes a conformation like W~, with h~ = a and ~ l, the motion along the tube axis is
strongly biased towards the + end and one gets conformation W_. In the latter conformation,
h~ m a, 8
~ l, and the motion is towards the end : for the middle part of the chain, the
W+ conformation is thus recovered. Since 8
~ l in both W_ and W~, the molecule will
direction of migration
~
a)
b--~~
~b)
~
4 ~~~~~
lesc » ~drifl
Fig. 3. (a) Motion of a polyelectrolyte affected by electrophoretic self-trapping- When the end-to- end distance is approximately zero in the direction of migration, the molecule is in a flat « pancake »
conformation which has a long lifetime r
= To. After the molecule escapes from this zero-velocity state, it migrates in an isotropic random-walk conformation until it «falls» into another « pancake » conformation. Because these conformations have long lifetimes and do not contribute to the
displacement of the molecule, they act as traps. Electrophoretic migration is thus a repetition of the
trapping-detrapping cycle schematically shown here. (b) When qo = 0 (uncharged molecular ends), the
cycle includes molecular « freezing » as well as trapping. The escape time from the frozen conformations is much larger than the drift time between these events. Frozen conformations have molecular ends in a
W-like state (see Fig. 4).
N° 5 ANOMALOUS ELECTROPHORESIS OF POLYELECTROLYTES l155
1."~~@~
~..."fi W+
~=;$
~
~~ ~~
,,..:÷"./
~"~/Q~_ w
Fig. 4. Molecular conformations for a qo= 0 molecule. The fat segments represent the two
uncharged segments and the arrows indicate the direction of the field-induced motion. (a) In this V-like conformation, the bias favours motion toward the + end of the tube. This will increase the end-to-end distance and make a further + jump more probable. (b) In the W~ conformation, the electric forces favour a jump towards the + end, which leads to conformation W_. In the latter case, the end-to-end
distance h~ has a different sign and the forces favour a jump towards the end. Conformation
W~ is thus recovered, with the last segment of the end possibly pointing in a different direction. The
W~ = W_ cycle repeats itself many times and thus freezes the chain in space.
simply oscillate between these two conformations, without any net velocity, until a highly improbable move against the direction of the field-induced bias frees the chain from this
frozen bi-stable state.
The escape time can be estimated as follows. The probability of escape is given by
equation (9) with
= s ~ l :
Pesc "
~~ "
~~ ~~ (lo)
which means that an average e~~ trials will be necessary before the chain escapes. Each trial lasts for a duration r(h~) m rj[
m rjs therefore, we have
~~SC ~' fi~) m ~~~X~ e~~
Pesc s
~0IN jj j
For fields s
~ l, de-trapping thus becomes exceedingly difficult.
After the molecule has escaped from a W-trap, it drifts for a time r~~~~ in the field direction until it finds itself in another W-conformation (see Fig. 3b). The net mobility is thus given by
~ l + ~j~ddft ~~~~
where ji is the mobility obtained from equation (5d), I.e. it is the mobility during the drift
periods. The drift time r~~~ can be estimated as follows. Not all conformations with
[h~[ ma are W conformations for which the freezing described above can take place.
However, the fraction p~ of all [h~[ ma conformations that fall into this category is
independent of both s and N. If we consider that a new conformation exists after each jump
of length a, and that those conformations satisfy the distribution function (4c), with
(cos o) = 0 and (cos~ o)
=
1/3, the drift time is given by
~~~~
Pw ~/[~~~
a)
~~~~~
