Table Of ContentDNA electrophoresis in designed channels
Takahiro Sakaue∗
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Wepresentasimpledescriptionontheelectrophoreticdynamicsofpolyelectrolytesgoingthrough
designedchannelswithnarrowconstrictionsofslitgeometry. Byanalyzingrheologicalbehavioursof
thestuckchain,whichiscoupledtotheeffectofsolventflow,threecriticalelectricfields(permeation
field E(per) ∼ N−1, deformation field E(def) ∼ N−3/5 and injection field E(inj) ≃ N0, with N
6
polymerizationindex)areclarified. BetweenE(per)andE(inj),thechainmigrationisdictatedbythe
0
0 drivenactivationprocess. Inparticular,atE >E(def),thestuckchainattheslitentranceisstrongly
2 deformed, which enhances the rate of the permeation. From these observations, electrophoretic
mobilityatagivenelectricfieldisdeduced,whichshowsnon-monotonicdependenceonN. Forlong
n
enough chains, mobility increases with N, in good agreement with experiments. Anabrupt change
a
intheelectrophoreticflowatathresholdelectricfieldisformallyregardedasanonequilibriumphase
J
transition.
8
PACSnumbers: 83.50.-v,36.20.Ey,87.14.Gg
]
t
f
o
I. INTRODUCTION the polymer dynamics in restrictive environments, the
s
. consequences of which seem not to have been explored
t
a The fractionation of DNA molecules depending on well.
m their length is a required step in many sectors of bio- In this article, we investigate the electrophoretic dy-
- logicalsciences. Astandardtechniqueforthis purposeis namics of a polyelectrolyte chain in an artificial chan-
d
a gel electrophoresis, in which negatively charged DNA nel, where two domains of slit constituting of different
n
molecules migrate driven by electric field inside the ran- depths arealternatelyarranged(Fig. 1), withdc electric
o
c domenvironmentof a gel[1, 2]. During the drivendiffu- field. The choice of this particular geometry is moti-
[ sion, DNAs have to negotiate with the geometrical con- vated by recent experimental studies, which suggest the
straints imposed by the gel, resulting in lower mobility potentiality for the DNA separationover the wide range
2
for longer chains. It is well known, however, that there of chain length [5, 6]. A remarkable feature in this sys-
v
3 is an upper limit for the length, beyond which mobility tem is that longer DNAs migrate faster in contrast to
0 of DNAs become length-independent. usual situations in gel electrophoresis, which triggered
4 Many efforts have been carried out to improve on this further experimental and several computer simulation
9 restriction. Among them, artificially designed devices studies[15, 16, 17, 18]. Here we present a simple scaling
0 with micro-patterns have been getting a lot of attention argument on the electrophoretic dynamics, which takes
5
[3, 4, 5, 6]. While in gels the environment for the tar- the chain rheological behaviours and hydrodynamics in-
0
/ get molecule is random and there is an upper bound for teractions into account simultaneously, in the hope of
at the statistical “mesh” size set by the mechanical weak- providing lucid descriptions. An abrupt change of the
m ness of dilute gels (for example, the maximum mesh size dynamical state at a certain threshold electric field, as
in agarose gels is ∼ 0.3µm), the geometry of the device wasobservedinexperiments,isinterpretedasanonequi-
-
d with much larger size constriction can be designed in a librium phase transition by taking an appropriate limit.
n desired way. Another advantage is that the microfabri- Inaddition,webrieflydiscussthedynamicaldiagramob-
o cateddeviceiseasytobuildinintegrativeseparationsys- served in the gel electrophoresis, in particular the tran-
c tems,whichwouldberequiredinthefieldofbiomolecules sition from entropic trapping to reptation regimes upon
:
v analysis in the future. increasing the electric field.
i
X Cleary, the progress in this area relies on the under- In Sec. II, we briefly remind of the basis of the dy-
standing of dynamics of polymers in such restrictive en- namics of polyelectrolytesunder the electric field. Then,
r
a vironments. While the dynamics without or with weak in Sec. III and IV, we analyze the permeation process
external bias has been extensively studied[9, 10, 11, 12], of polyelectrolytes into the narrow channel. In Sec. V,
little isknownaboutthe dynamics involvingoutofequi- we discuss the electrophoretic mobility, in particular, its
librium, large deformation caused by the external field. dependenceonthechainlengthandtheelectricfield. We
Indeed, longpolymersaresosoftthatthey couldbe eas- predict the four distinct regimes for the electrophoretic
ily deformed under weak external forces. Examples are dynamics; with the increase in the electric field, one en-
a chain under stretching[13] and a tetherd chain under counters in order, (i) unbiased or weakly biased dynam-
hydrodynamic flow[14], etc.. The same also applies for ics, (ii) biased dynamics, (iii) biased dynamics with the
significant deformation of the stuck chain, and (iv) free
migration without the trapping. If one relies on the “lo-
cal friction” picture (neglect of the hydrodynamics), the
∗e-mail address: [email protected] regime II disappears, which is discussed in Appendix A.
2
Tohighlighttheeffectofconfinementgeometries,wedis- friction coefficient for the chain in its steady-state con-
cuss in Appendix B the electrophoretic dynamics in a formation. In the situation of our interest, the polymer
cylindrical geometry. We also address the connection to maybeblockedattheentranceofthenarrowslit. Itmay
the flow-injectionproblem,in which polymers aredriven be deformed there, but does not show any net motion,
bythesolventflowinsteadoftheelectricfield(Appendix while the counterions are still driven without any obsta-
C). cle, which induces the solvent flow. Taking these points
in mind, in what follows, we consider the dynamics of
polyelectrolytes in designed channels by focusing on the
elementary step of the channel device, which consists of
one narrow slit facing large enough space (Fig. 1).
III. UNBIASED DYNAMICS
FIG. 1: A schematic picture of the devise. It consists of a
series of wide and narrow slits, whose depth (length) are D˜ Wefirstconsidertheunbiasedorweaklybiaseddynam-
(L ),D (L ), respectively. ics, where the effect of electric field is very small or even
D˜ D
negligible. The dynamics of polymers in restrictive envi-
ronments depends on the relative size ratio between the
polymer and the typical size of the confinement, which
may be called “pore”. The situation of our present in-
terest corresponds to the pore size comparable to, or a
II. DYNAMICS OF POLYELECTROLYTES
WITH GEOMETRICAL TRAPS little smaller than the polymer size in a bulk. Then,
it is known that the free motion (i.e., without external
force) of polymers is described by the so-called entropic
Consider a polyelectrolyte chain with N segments of
trapping process [9, 10, 11, 12]. The physics involved
length a, each of which carrying electric charge of qe
in this process is essentially the same as size-exclusion
(e: elementary charge). For strongly charged polyelec-
chromatography,which states that there is a free energy
trolytes such as DNA, this charge should be regardedas
cost∆F (N)for the polymer to be squeezedin anarrow
therenormalizedoneduetothecounterioncondensation. 0
space. Itshouldbe notedthateveninthe situationwith
As usual for electrophoresis buffers, the ionic strength
non-adsorbing wall in good solvent, the origin of con-
of the solution is rather high such that the bulk chain
finement free energy ∆F (N) comes from the segmental
size is given by that of a neutral chain in good solvent, 0
interactions or the reduction in the conformational en-
R ≃ aN3/5. It also affects the dynamics through the
tropy depending on the architecture of the polymer and
screeningofhydrodynamiccouplingonscaleslargerthan
thegeometryoftheconfinement,whichisreflectedinthe
Debye length, which leads to the so-called local friction
scaling form of ∆F [19]. Since the case of our interest,
picture, i.e., Rouse dynamics, for the free electrophore- 0
i.e.,alinearpolymerconfinedinaslit,correspondstothe
sis. As a consequence, polyelectrolytes in the free so-
entropicconfinement,theterminologyofentropicbarrier
lution are treated as free draining coils and the elec-
would be appropriate. In this case, the scaling form of
trophoreticmobilityµ =qe/ηa(ηbeingthesolventvis-
el the confinement free energy is [19, 20]
cosity) is neither dependent on the chain length nor the
conformation[1]. However, as was pointed out in ref.[7], ∆F (N) a 5
forpolyelectrolytesinteractingwithanimmobileobstacle 0 ∼ 3 N (1)
k T D
(moregenerally,forpolyelectrolytesunderthesimultane- B
(cid:16) (cid:17)
ous action of electric fields and non-electric forces), this This free energy barrier leads to the exponential slowing
localfrictionpicturebreaksdown. Insuchsituations,one down of the chain mobility with the increase in chain
hastodealwiththe electrohydrodynamics,whichiscou- length and/or the decrease in the pore size:
pled with the deformation behaviours of the chain. Al-
though this does not allow, in general, a simple analytic µ ∼exp −∆F(N) (2)
0
treatment, it is possible to linearize the coupled electro- k T
(cid:18) B (cid:19)
hydrodynamic equations in the case relevant to typical
electrophoresis experiments (low Raynolds number situ-
ations). This scheme allows the superposition of electric IV. STRONGLY BIASED DYNAMICS
and non-electric forces, which leads to a hydrodynamic-
electric equivalence[1, 7, 8]. It says that, for example, If the motion of the chain is biased by the external
the deformation of an end-anchored polyelectrolyte un- force,one has to modify the abovediscussionfor the un-
der the uniform electric field E is equivalent to that of biased dynamics. There are several attempts to this di-
an end-anchored polymer in a uniform solvent flow at rection including the injection of linear chains [21] and
speed v = µ E. And the force to immobilize that poly- branchedchains[22]into anarrowcapillarybyhydrody-
el
electrolyte (stall force) f = γµ E, with γ being the namicflow,biasedelectrophoreticmobilityofDNAusing
stall el
3
a slit channel [5, 6, 15, 16, 17, 18]. In these studies, a ofblobsinaslit,the primarilyinjectedblobwillproceed
commonstrategyis that one noticesevents occurringin- straight toward the direction of the electric field, which
side the channel only, which we call “inside approach”. indeed corresponds to the most energy-saving strategy.
The validity of the inside approach can be justified in In other words, the nucleated hernia would be more or
some cases [21, 22]. However, the events occurring out- less straight. This means that the essential part of the
side of the channel become important depending on the permeation process is described as the one-dimensional
dissipationmechanismand/orthespacedimensionofthe problem (Fig. 2). This notion is consistent with the
channel (Appendix A, B). Indeed, we will show that the video images of ref. [5].
systemunderconsiderationfallsuponthelattercase. For Letusdenotethelengthofasequenceofblobsalready
thesakeofconvenience,nevertheless,wefirstpresentthe permeated as y. The free energy in the corresponding
inside approach. The modification due to the outside state is
events is necessary at high electric field, which will be
y y
done later. F (y)= ΠD2−qeE dy (3)
in in
a
Z0 (cid:16) (cid:17)
The first term in the integrand comes from the confine-
ment effect, where Π ≃ k T/(D3) is the osmotic pres-
B
sure. The second term represents the force imposed by
the electric field inside the slit E . Equation (3) indi-
in
cates the presence of a free energy barrier at y =y∗ :
in
∆F k T a
in ≃ B (4)
k T qeE D2
B in
(cid:16) (cid:17)
and
k T a
y∗ ≃ B (5)
in qeE D
in
(cid:16) (cid:17)
The mainresultofthe inside approachwith the external
force is that the free energy barrier does not depend on
the chain length N, which is in a remarkable contrastto
eq. (1).
FIG. 2: A schematic picture of the elementary unit of the B. Modification at High Electric Field: Single
channel device ((a)side and (b)top views) and a process of Chain Rheology
polyelectrolyte permeation by electric field, where the chain
section outside the narrow channel is assumed to be unper- The mainassumptionofthe inside approachisthe ne-
turbed (“inside approach”). In (b) and FIG. 3, the shaded
glectofthechainportionstilloutsideofthenarrowchan-
part designates the narrow region.
nel. However, this assumption is not always valid in the
case of our problem, since as we shall show below, the
chain facing the entropic barrier, i.e., the chain stuck at
theentranceofanarrowregion,isdeformedtoanotable
degree under the influence of the rather high external
A. Inside Approach
field. Then, one has to deal with the single chain rheol-
ogy;howthechainisdeformedundertheinfluenceofthe
We now consider the situation, where a large poly- external force. This case with high external force is in-
electrolyte chain is facing the entrance of the narrowslit deed interesting in the context of the application, which
(Fig. 2). Then the process of chain permeation may be requires the shorter separation time.
induced by the electric field. In the early stage of the Let us first clarify when the deformation of the stuck
permeation, the insertion of a single blob of size D is re- chain becomes evident. In experiments, the dielectric
alized easily. The corresponding number of monomers is constant of the channel is usually much lower than that
denoted as g : D =ag3/5. of the water solution. Then, the electric flux lines inside
D D
But for the chain to be further permeated, a sufficient the channel interior scarcely leak out. From the conser-
electrostatic force is required to overcome the osmotic vation of electric flux and the geometrical symmetry of
force due to the confinement. If the moderate range of the channel, the electric field at distance r from the en-
the electric field is applied, this permeation process is trance is evaluated as E(r) ≃ E D/r. In the light of a
in
formulated as a barrier crossing problem [5]. Although hydrodynamic-electricequivalence,wecalculatethe stall
there is a two-dimensional freedom for the arrangement forceforthestuckchain. Undertheactionoftheelectric
4
field, the chain is pressed against the wall, and its shape where f is the force acting on each blob;
ξ
is distorted to non-spherical form. If the electric field is
small enough, it would be plausible to assume that the f ≃qeE D/a (9)
ξ in
averagechain shape is non-spherical(reflecting the sym-
metry of the electric field), but the overall size is still From these, we obtain
given by R ≃ aN3/5 (unpertubed size). Assuming that
k T a
the stuck chain is not essentially unperturbed (overall ξ ≃ B (10)
chainsize is givenby R=aN3/5), the electrostatic force qeEin D
(cid:16) (cid:17)
acting on the whole chain is evaluated as
Therefore, the length of the stuck chain along the slit
qeE D entrance is
f ≃γ µ E¯ = in (6)
0 el
a
2
where γ0 ≃ ηR is the friction coefficient for an unper- Rk ≃aN qekEiTnD 3 (11)
turbed chain, and E¯ ≃ RE(r)r/R2dr is the average (cid:18) B (cid:19)
0
electric field acting on the stuck chain. If this force IftheblobsizeξissmallerthantheslitsizeD,thereisno
R
is smaller than kBT/R, the assumption of unperturbed entropicbarrierarisingfromthedifferenceintheosmotic
chain conformation is valid. In the opposite case, how- pressure; the chain will be injected smoothly inside the
ever, the chain is significantly deformed due to the con- slit. The critical field for the injection is, thus, obtained
vergent electric field. Equating kBT/R with eq. (6), we from eq. (10) with ξ =D as
have the chain length dependent threshold electric field
(in a narrow slit): k T a R
E(inj) ≃ B ≃E(def) (12)
qe D2 D
E(def) ≃ kBTN−53 (7) (cid:16) (cid:17) (cid:18) (cid:19)
qeD
At electric field above E(inj), the chain trapping at the
slit entrance is irrelevant, therefore, the local friction as-
Equation(7)indicatesthatthelongerchainismoreeasily
sumption becomes valid for the description of the elec-
deformed.
trophoretic dynamics of polyelectrolytes. Then the elec-
trophoretic mobility is independent on both the chain
length and the electric field. Note that the same phe-
nomenon for the polyelectrolyte without hydrodynamic
interactions corresponds to the observation in gel elec-
trophoresisathighelectricfield(seeAppendixA,B)[10].
Equation (12) provides a simple scaling estimate for the
critical field of the abrupt transition on flow behaviours.
For quantitative analyses, however, more details regard-
ing the prefactor are required, which shall be discussed
below.
We consider cases with the electric field slightly lower
than the critical injection field given by eq. (12); E =
in
FIG. 3: A stuck chain at the entrance of the narrow channel αE(inj) (α<1). Thentherestillremainsasmallfreeen-
under strong electric field (E(def) < E <E(inj)). A chain is
ergy barrierand an activation process of the permeation
strongly deformed by the convergent field, and its conforma-
will be governed by the following free energy, which is
tion is represented as a parallel alignment of blobs of size ξ.
a modified version of eq. (3) with the effect of outside
(top view corresponding to FIG. 2 (b))
events.
y y
F(y)= ∆ΠD2−qeE dy (13)
theAtslEitinen>traEn(cdee,f)a,ntdhethcehaminoliesciunltaernsshelaypperbesesceodmaegsauinnsit- Z0 (cid:16) ina(cid:17)
axialsomethinglikeacigar. Theresultingdeformedcon- where ∆Π is the difference of osmotic pressures inside
formationis describedas a parallelalignmentofblobs of and outside of the slit and is given by
sizeξ alongtheslitentry(Fig. 3). Insideblobs,theforce
duetoelectricfieldisnotsignificant,thus,theblobsizeξ 1 1
∆Π≃k T − (14)
isrelatedtothenumberofmonomersinsideitg asinthe B D3 ξ3
bulk, i.e., ξ =ag3/5. Accordingly, blos size ξ is obtained (cid:18) (cid:19)
by the following equation Free energy eq. (13) takes a maximum value at
ξ ≃ kBT (8) y∗ ≃D(1−α3) (15)
fξ α
5
and the corresponding free energy barrier is At electric field below E(def), the chain deformation
becomes irrelevant. The free energy barrier is, then, ob-
∆F (1−α3)2
≃ . (16) tainedthroughthe inside approach(eq. 4) andthe tran-
kBT α sition rate is given by κ=τ1−1 ≃(τ˜1D/R)−1;
Nthoetethtrheashtotlhdi,sEdesc=ripEt(idoenf)i,sthvaelibdloabtsEizienb>ecoEm(deesf)e.quAatl κ=t−1w(2)(α, D)N35 exp −∆Fin (23)
in D k T
to the unperturbed coil ξ ≃ R ≃ R. At E < E(def) (cid:18) B (cid:19)
k in
the inside approachis valid. (E(per) <Ein <E(def))
An important consequence of the chain deformation
forE >E(def) is thatthere areN/g blobs stuckatthe with
in
slit entry. Every blob has an equal chance to climb the a
w(2)(α, D)≃ α23 (24)
barrier toward a nucleation. The transition rate for this D
activation process is, therefore, estimated as (cid:16) (cid:17)
Whentheelectricfieldisfurtherdecreased,thecritical
N nucleationsizeofthehernia,y∗ (eq. (5))becomeslonger
κ≃τ−1 (17) in
1 g and equal to the length of the completely confined chain
L=Na5/3/D2/3 at E =E(per)
where τ ≃ τ˜D/ξ is the mean escape time from the
1 1
free energy trap, which is initiated from any single blob; 2 5
τ˜ ∼ exp(∆F/k T) with ∆F given by eq. (16). The E(per) ≃ kBT D 3 N−1 ≃E(inj) D 3 (25)
1 B qeD a R
factorD/ξ in τ1 reflectsthe factthat the largerblobhas (cid:18) (cid:19) (cid:18) (cid:19)
larger contact area with the slit entrance[5]. The escape
AtelectricfieldbelowE =E(per),thefreeenergybarrier
time is calculated by analysing the stochastic process of
isnolonger∆F ,butgivenbyF (L),whichapproaches
the barrier crossing. From eq. (13), the Fokker-Plank in in
the confinement free energy ∆F (eq. (1)) in the limit
equation for this process is 0
of zero electric field. The effect of electric field is, then,
∂P(y, t) ∂ 1 ∂F(y) ∂ very small, and the transition rate is given by
= + k T P(y, t) (18)
B
∂t ∂yγ(y) ∂y ∂y
(cid:20) (cid:21) κ=t−1w(1)(α, D, N)N53 exp −Fin(L) (E <E(per)()26)
where, γ(y) ≃ ηy is y-dependent friction coefficient. D (cid:18) kBT (cid:19)
Evaluation for τ˜ is given as a mean first passage time
1
with
from y ≃ 0 to y >> y∗[23]. By assuming the bound-
0
aries at y = 0 and at y = y0 >> y∗ to be reflective and w(1)(α, D, N)
absorbing, respectively, we obtain
a L D 2 1 D 2 −1
τ˜ ≃ y0dy y exp F(y) ydy′exp −FkB(yT′) (19) ≃ D (D−αL −(cid:18)D−αL(cid:19) + α +(cid:18)αL(cid:19) )(27)
1 k T yn′D(y′) o
Z0 (cid:26) B (cid:27)Z0 To obtain eq. (26), one has to take notice that the lo-
where D(y) ≃ kBT/γ(y) is the (y-dependent) diffusion cation of the free energy maximum is no longer givenby
coefficientofthepartlyconfinedchain[24]. Followingthe y = y∗ (eq. (5)), but given by y = L (the length of the
in
standardprocedures(utilizingthefactthattheintegrand completely confined chain). In contrast to other regimes
of the first integral in eq. (19) is sharply peaked at y = for higher electric field, the transition rate becomes ex-
y∗, while that of the latter is very small around there), ponentially small with increase in the chain length (Sec.
it is approximated as III).
Hereafter, we refer to the migration dynamics at E <
∆F
τ˜1 ≃tDα−32 exp (20) E(per), E(per) <E <E(def) and E(def) <E <E(inj) as
k T
(cid:18) B (cid:19) regime I, regime II and regime III, respectively.
where t = ηD3/(k T) is the characteristic time scale
D B
corresponding to the length scale of D. Substituting eq.
V. ELECTROPHORETIC MOBILITY
(16), (20) into eq. (17), we obtain
∆F After having clarified the physics involved in the ele-
κ=t−1w(3)(α, D)Nexp − (21)
D k T mentary step of the device, it is now possible to discuss
(cid:18) B (cid:19)
themobilityofthechaingoingthroughanartificialchan-
(E(def) <E <E(inj))
in nel consisting of a succession of geometrical traps. The
mobility of the chain could be expressed as [5]
with
w(3)(α, D)≃ Da 53 α163 (22) µµel = t0+t0κ−1 (28)
(cid:16) (cid:17)
6
Thetimelapseforachaintotravelanelementaryunitis
t = (µ E )−1(L +L D˜/D), which is a controllable
0 el in D D˜
quantity from the device design (see Fig. 1). Using eq.
(21), (23) and (26), the explicit formula for the mobility
is obtained;
µ t
≃ 0 (E(def) <E <E(inj)) (29)
µ t +AN−1
el 0
µ t
≃ 0 (E(per) <E <E(def))(30)
µel t0+BN−53
µ t
≃ 0 (31)
µel t0+(tD/w(1)(N))N−53 exp Fin(a5k/3BNT/D2/3)
(cid:16) (E <E(per)(cid:17))
where,forsimplicity,weshowtheexplicitdependenceon
N only.
Figure 4 shows dependences of mobility on applied
electric field for various chain lengths with different de-
viceparameters(Thebottomapproximatelycorresponds
to the experimentalcondition in ref. ([5])). With the in-
crease on field strength, a crossover is observed from a
stuck state to a freely migrating state. Two trends are
perceivable here: (i) If the chain is longer than a cer-
tain threshold N∗, a sharp crossover occurs around the
well-definedfield strength E∗, which is almostlengthin-
FIG. 4: Electrophoretic mobility of apolyelectrolyte in a de-
dependent. One can recognize, however, a clear trend signedchannelasafunctionofelectricfieldwiththefollowing
that the crossover takes place at slightly lower field and parameters: (Top) D = 8 a, t = 5×10−9 (sec), t = 0.2
D 0
becomes sharper with the increase in the chain length. (sec) and N = 200, N = 500, 2×103 and 104, (Bottom)
(ii)Forsufficiently short(butstilllargerthanD)chains: D=2 a, tD =10−6 (sec), t0 =5 (sec) and N =25, N =50,
N(D) = (D/a)5/3 < N < N∗, on the other hand, the 250 and 103. Around the transition region of α shown here,
crossover becomes less sharp and is realized at much thedynamicalbehaviourscorrespondtoregimeIIandregime
III for the chain with (Top) N = 2×103, 500, (Bottom) 50
weaker field.
and (Top) N = 104, (Bottom) N = 103, N = 250, respec-
Therefore,thedependence ofthecrossoverfieldonthe
tively. Forthe chain with (Top) N =200, (Bottom) N =25,
chainlengthisnon-monotonic. Asisclearfromouranal-
thetransition is realized in theregime I.
ysis,thesecondpoint,i.e.,largercrossoverfieldforlonger
chains, is a consequence of the weakly biased entropic
trapping process (regime I), and the observation of the
opposite dependence signals a strongly biased dynamics resultsobtainedfromthelocalfrictionpicture(Appendix
(regimes II and III). While the effect of chain length is A) are also shown in Fig. 6. Note that in these figures,
exponentialintheregimeI,itshowsupthroughaprefac- the right graphsapproximately correspondto the exper-
torinregimesIIandIII.Thethresholdlengthseparating imental condition in ref. ([5]).
above two opposite tendencies is given by The electrophoretic behaviours can be classified into
the following four regimes. (0)There is no effect of the
5
D 3 geometrical trap for very short chain (N < N(D) =
N∗ =N(per)(E =E∗)= α∗−1 (32)
a (D/a)5/3), then, the mobility is the same as that in the
(cid:18) (cid:19)
free draining limit. (I)For short chains (N(D) < N <
where α∗ = E∗/E(inj) and the definition of N(per) is N(per)), the process is described by weakly biased en-
introduced below. tropic trapping (regime I). The mobility decreases ex-
Of great practical interest is how the electrophoretic ponentially with the chain length (eq. (31)). Here,
mobility depends on chain length at a fixed dc electric N(per) ≃(k T/qeD)(D/a)2/3E−1 fromeq. (25). (II)For
B
field. An appreciable dependence on the chain length is chainswithmoderatelength(N(per) <N <N(def)), the
expected if one tunes the electric field in place (slightly permeationis drivenbythe electricfield, butthe confor-
smaller than E(inj), see Fig. 4). This is exemplified in mation of the stuck chain is still in equilibrium (regime
Fig. 5 for two different situations. For comparison, the II). The mobility increaseswith the chainlength accord-
7
FIG.5: Lengthdependenceoftheelectrophoreticmobilityof
polyelectrolytes at a given dcelectric field with thefollowing
parameters: (Left)D=8a,tD =5×10−9(sec),t0=0.2(sec) FIG. 7: A dynamical diagram of the electrophoresis in
andα=0.068,(Right)D=2a,tD =10−6 (sec),t0 =5(sec) (N, Ein)plane(Logarithmicscale). Linesseparatingregimes
and α = 0.075. Corresponding characteristic chain lengths are: E(per) ≃E(inj)(D/R)5/3 for the borderbetween (I) and
are (Left) N(D) = 32, N(per) ≃ 470,N(def) ≃ 2800,Nc ≃ (II) and E(def) ≃ E(inj)(D/R) for the border between (II)
1200,(Right)N(D) ≃3,N(per) ≃40,N(def) ≃240,Nc ≃270. and(III).Thedashedcurvebetweenwhiteandgrayareasdes-
ignatesacrossoverfromastucktoaflowingstate,whichisob-
tained from κt =1 (see eq. (28)). The gray area designates
0
the regimes, in which the effect of the geometrical constric-
tion is not apparent because of either too short chain length
(left part) or sufficient strong electric field (upper part). At
N >> N∗, this dashed curve is given by E∗ ∼ [lnN]−1 (see
eq. (33)) and this crossover exhibits a feature of a nonequi-
librium phase transition in thelimit of infinitechain length.
tionalbehavioursofthe stuckchain,whichseparatesthe
regimeIIandIII.Weexpectthatasufficientlylongchain
ismarkedlydeformedattheentrance,whichenhancesthe
rate of the permeation.
FIG. 6: Length dependence of the electrophoretic mobility On the other hand, if one neglects the effect of
of polyelectrolytes at a given dc electric field calculated ac- hydrodynamic interactions (i.e., local friction picture),
cording to the local friction model. The parameters are the N(per) = N(def) is expected; the regime (II) does not
same as Fig. 5. Corresponding characteristic chain lengths exist. Therefore, the deformation of the stuck chain is
are (Left) N(D) = 32, N(per) = N(def) ≃ 470,N ≃ 2600,
c expected to be evident even for chains with moderate
(Right) N(D) ≃ 3, N(per) =N(def) ≃ 40,N ≃ 170. Around
c length. The computer simulation studies so far should
N =N(def), thedynamicalregime isshifted from thealmost
be compared with this model[15, 16, 17, 18].
unbiased entropic trap tothe biased forced penetration.
VI. DISCUSSION
ing to eq. (30). Here, N(def) ≃ (k T/qeDE)5/3 from
B
eq. (7). (III)For longer chains (N > N(def)), the chain A. Comparison with Experiments
distortionattheentrancebecomesapparent(regimeIII),
which leads to more enhanced mobility for longer chains First, let us compare our results with the experiment.
(eq. (29)). In ref. [5], electrophoresis of DNA was carried out using
As a brief summary, a diagram of the electrophoretic the narrow slit D = 90 nm. Then the sharp crossover
dynamics is shown in Fig. 7. It is worth mentioning a from the trapped state to the freely migrating state was
similarity between this diagram and the experimentally found around the electric field E∗ ≃30 (V/cm), where
ave
observed dynamical diagram of the gel electrophoresis E is the average electric field over the channel. From
ave
[10] (see also Appendix A, B). the device parametersusedin the experiment, the corre-
ThechainwithN >N(per) isinthedrivenpermeation sponding electric field in the narrow slit is found to be
regime, and the longer chains migrate faster. This is a E∗ ≃ 50 (V/cm). The sharpness of the transition (de-
consequenceof the increasedattempt frequency to climb fined as the ratio of the range of the electric field ∆E
the free energy barrier. How this attempt frequency de- between zero and the asymptotic mobility to E∗) is also
pends on the chain length is a matter of the conforma- found to be ∆E/E∗ ≃ 0.6. Putting values correspond-
8
ing to DNA (segment length a ≃ 60 nm and effective entrance for larger chains[5]. For long enough chains
chargeper segmentq ≃85)into eq. (12), we obtainthat (N(def) <N),orunderthestrongelectricfield(E(def) <
E(inj) ≃20 (V/cm), and ∆E/E∗ ≃0.5 is obtained from E), this effect is further enhanced by the electric field
Fig. 4(Bottom). Thus,wefindagoodagreementforthe induced chain distortion at the geometrical trap, which
sharpnessofthetransition. Ontheotherhand,wesome- generates more blobs for longer chains as steady state
what underestimate the threshold field, since the experi- conformations(regime III). An increase in blob numbers
mentallyobservedvalueisslightlysmallerthanE(inj) by increases the frequency for the trial to climb the free en-
a factor of α∗ ∼ 0.1 (see Fig. 4): Note that the precise ergy barrier as is clearly seen in eq. (21). If one neglects
value of α∗ depends on t , thus, the device parameters the effect caused by the solvent flow, the regime II dis-
0
such as D, L and L . However, given the overall ac- appears. Then, the chain stuck at the slit entrance and
D D˜
curacy of scaling laws and the assumption involved in driven by the electric field is inevitably distorted. This
the simplified model (no leakage of the electric flux at explainsthereasonwhythestuckchaindeformationwas
the dielectric boundary, neglect of the effect of electro- observed in Monte Carlo simulations[15], while it is not
osmotic flow), this discrepancy in the threshold field is apparent in real experiments using moderate length of
not surprising. DNA[5].
For very long chains of N >> N = A/t (or N =
c 0 c
(B/t )5/3 if (B/t )5/3 < N(def)), however, the chain
0 0
B. Long Chain Limit: Nonequilibrium Phase trapping becomes irrelevant (t >> κ−1), and the mo-
0
Transition bility approachesto the free draining limit. This bounds
the range for efficient separation in a single run of elec-
It may be intriguing to examine dynamic behaviours trophoresis at a given set of parameters. However, we
at the asymptotic limit of the long chain length. As we nowknowthetrendonhowthisappropriatelengthrange
shall argue now, a sharp crossover of the electrophoretic depends on controllable parameterssuch as size of chan-
flow exhibits features of“phase transition”. We consider nels(D,LD,LD˜)andelectricfield. Thesamealsoapplies
thesituation,wherethelengthofthedeviceunitL , L forthemigratingspeedofchains,whichisanotherimpor-
D D˜
islargeenoughcomparedtothechainlength. (Although tant factor for the practical purpose. In addition, such
this is, of course, not the case for very long chains, one microfabricated devices could be easily assembled into
can take the limit, in which the device size is made ar- practical integrative systems. Therefore, by construct-
bitrary large in proportion to the chain length.) In this ing multiple channels with designed different conditions,
limit,thethresholdfieldseparatingastuckandaflowing whichoperateinsuccessionorinparallel,itmaybe pos-
state becomes sible to achieve an efficient separation over a wide range
of chain length. This is, in fact, of advantage to use de-
E∗
α∗ = ∼[lnN]−1 (33) signed channels for separations.
E(inj)
and the electrophoretic mobility is expressed as
VII. FINAL REMARKS
µ 1
∼ (34)
µ 1+const. Nǫ Our present analysis is based on the steady state rhe-
el
ological behaviours of the chain. This approach is moti-
where ǫ ≃ (α∗ −α)/α is a reduced electric field. The vated by the microscopic images of the experiment[5, 6],
crossover becomes shaper and sharper with the increase in which the transient behaviours (the relaxation after
in N and becomes singular in the limit of infinite chain reentering the thick region, the stretching under high
length as announced. Therefore, this dynamical state electric field gradient, etc.) seem not to be important
change (from the state with zero to some finite current) very well. However, if the period of the device is made
caused by the external field is formally regarded as a shorter, successive injection and ejection events start to
nonequilibrium phase transition. correlate, and the transient dynamics of the chain be-
As another possible choice of the limit, one may con- comesrelevant[15,17,18]. Also,recentexperimentaland
sider increasing the chain length for a fixed device size. simulation studies have observed the transition between
In such a case, a chain interacts with many elementary two distinct states with different migration speeds using
device units simultaneously, and some unscripted new relativelydeepchannels,inwhichDNAsmaybetrapped
features should appear [18]. deepinthe wellandcannotescapefromit[17,18]. Such
featuresarenotstudiedhere,butshouldbeincorporated
in the future analysis.
C. Practical Aspects In summary, we have shown that the deformation of
the stuck chain and the effect of the flow are important
A unique property of the present system is that there ingredients for the DNA electrophoresis in the designed
is a regime where longer chains migrate faster. This is channel under consideration. Such rheological proper-
a consequence of the larger contact area with the slit ties in a single chain level are expected to manifest their
9
importance in many other situations in the dynamics of independent electrophoretic mobility. This corresponds
polymers in restrictive environments. totheobservationingelelectrophoresisthattheentropic
trapping effect disappears at electric field higher than a
certainthresholdvalue[10]. (Notethatthetopologicalre-
Acknowledgments strictionimposedbygelswithratherlargeporesizemay
be better modelled by the cylindrical pore geometry, for
I am grateful to E. Rapha¨el, discussion with whom whichwefindthesamescalingformoftheinjectionfield,
motivated the present work. I am also indebted to T. see Appendix B.)
Ohta for useful discussion and his critical reading of the The activation process at α(=E /E(inj))<1 can be
in (R)
manuscript. This research was supported by JSPS Re-
analyzed similarly. Now the free energy (eq. (13)) for
search Fellowships for Young Scientists (No. 01263).
the permeation process is modified as
y y
APPENDIX A: ELECTROPHORESIS BASED ON F(R)(y)= ∆Π(R)D2−qeEingDD dy (A6)
THE LOCAL FRICTION ASSUMPTION Z0 (cid:16) (cid:17)
with
Inthis appendix,webrieflydiscussthe electrophoretic
dynamics based on the local friction picture, where the 1 1
∆Π ≃k T − (A7)
role of the solvent flow is neglected. Although this does (R) B D3 ξ3
notcorrespondtotherealexperiments,itprovidesuseful (R)!
information to analyze the result from computer simula-
This produces a free energy barrier
tions performed so far. In addition, this argument may
berelevanttothegelelectrophoresis,wherethehydrody- ∆F (1−α9/5)2
namics interactions are screened at the scale of the pore (R) ≃ (A8)
k T α
size (see also Appendix B). B
Then, the rheology of the stuck chain is modified as and its location
follows. Instead of eq. (6), the electric force acting on
the stuck chain is y∗ ≃D(1−α95) (A9)
f ≃ qeEinDN = qeEinDR32 (A1) (R) α
(R) R a35 By taking into account that the diffusion coefficient of
the partly confined chain is now given by D (y) ≃
Here and in the followings, the subscript (R) indicates (R)
(k T/η)(a/D)2/3y−1, the transitionrateis calculatedas
the local friction (Rouse) dynamics. One finds for the B
deformation field
∆F
κ =t−1w(3)(α, D)Nexp − (R) (A10)
E(def) ≃ kBTN−1 (A2) (R) D (R) (cid:18) kBT (cid:19)
(R) qeD
E(def) <E <E(inj)
(R) in (R)
AtelectricfieldaboveE(def),thestuckchainisdeformed,
(R) with
anditsconformationisaparallelalignmentofblobofsize
ξ(R) (cf. Fig. 3). a 7
w(3)(α, D)≃ 3 α1190 (A11)
(R) D
3
ξ ≃a kBT 5 (A3) (cid:16) (cid:17)
(R) (cid:18)qeEinD(cid:19) Note that this description is valid at Ein > E((Rde)f)
(def)
The length of the stuck chain along the entrance if (regime III). At the threshold, Ein = E(R) , or
equivalently α ≃ (D/a)5/3N−1, we have y∗ ≃
2 (R)
R ≃aN qeEinD 5 (A4) (a5/3/D2/3)N = L, which is the length of the com-
(R)k k T
(cid:18) B (cid:19) pletely confined chain. Therefore, at this point, we
TheconditionforthesmoothchaininjectionisD ≥ξ , have a crossover to the almost unbiased entropic bar-
(R)
rier regime (regime I) and the regime II does not exist
which leads to the critical injection filed;
(E(per) ≃E(def)). The transition rate in regime I is cal-
(R) (R)
E(inj) ≃ kBT a35 ≃E(def) R 5/3 (A5) culated as
(R) qe D83! (R) (cid:18)D(cid:19) F (L)
κ=t−1w(1)(α, D, N)N3/5exp − in(R) (A12)
AtEin >E((Rin)j),thetrappingattheslitentranceisirrel- D (R) (cid:18) kBT (cid:19)
evant, which results in the length and also electric field Ein <E((Rde)f)
10
where F describesthe free energy ofthe permeation where R(def) is the overall size of the deformed stuck
in(R)
process based on the inside approach, and chainandγ isthefrictioncoefficientoftheoutmost
R(def)
blob.
w(1)(α, D, N)
(R)
a 5/3 L D 2 1 D 2 −1 ξ(R(def))≃ kBTa 12 R(def) ≃ E(def) 21 R(def(B) 4)
≃ − + + (A13) qeE D2 E
(cid:16)D(cid:17) (D−αL (cid:18)D−αL(cid:19) α (cid:18)αL(cid:19) ) (cid:18) in (cid:19) (cid:18) in (cid:19)
The dependence of the electrophoretic mobility on the The force acting on the inner blobs (r < R(def)) in a
chain length is represented in Fig. 6. radialdirection is the sum of drag forces acting on outer
blobs;
APPENDIX B: CHANNELS WITH R(def) γ(r)µ E(r)
CYLINDRICAL PORES f(r)≃ el dr (B5)
ξ(r)
Zr
qeE D2 1 1
If the channeliscomposedofthe alternatingwide and ≃ in − (B6)
a r R(def)
narrow cylindrical pores, instead of deep and shallow (cid:18) (cid:19)
slits,theelectrophoreticdynamicsofthechainisaltered.
By balancing this force with k T/ξ(r), the blob size at
In this geometry, the convergent field near the entrance B
r ≥D is expressed as
of narrow constriction is E(r) ≃ E D2/r2. The stall
in
force for the unperturbed stuck chain is evaluated as
k Ta rR(def)
ξ(r)≃ B (B7)
f ≃γ µ E¯ = qeEinD2 (B1) qeEinD2R(def)−r
0 el
aR
At r < D, the blob does not further shrink any more;
where γ ≃ ηR is the friction coefficient for an unper-
0 ξ(r) = ξ(D) at r < D. The size of the deformed chain
turbed chain, and E¯ ≃ RE(r)r2/R3dr is the average
0 R(def) isdeterminedfromthe segmentnumberconserva-
electric field acting on the stuck chain. By comparing
R tion;
this force with k T/R, the deformation field is obtained
B
E(def) ≃ kBTa (B2) R(def) g(r) d3r ≃N (B8)
qeD2 ξ(r)3
Z0
In contrast to the slit geometry (eq. (7)), E(def) does
where g(r)=(ξ(r)/a)5/3 is the number of segments in a
not depend on the chain length.
blob at r.
At the electric field above E(def), the chain is highly
Since D <<R(def), the blob size at the entrance (r =
compressed toward the narrow pore and the resultant
D) is readily approximated as
conformationcanbe viewedasadensestackingofblobs,
whosesizeshrinkswithapproachingtotheporeentrance
k Ta
i.e., an inhomogeneous semidilute solution (Fig. 8). The ξ(D)≃ B (B9)
qeE D
in
R(def) We expect the smooth injection whenξ(D)≤D. There-
fore, we find that the injection field E(inj) is sparsely
ξ(r)
D larger than the deformation field, but no dramatic dif-
ference is expected; E(inj) ≃E(def). Thus, the injection
r
field is almost the same as that in a slit geometry (eq.
(12)). It means that once the stuck chain is deformed at
the entrance, there is almost no free energy barrier left,
and the smooth injection takes place (the range for the
regime III is very narrow), therefore, the use of the in-
E side approach (Sec. IVA) is justified in this case. The
field E(per), below which the process is described by the
weakly biased entropic trapping, is the same as that in
FIG.8: Astuckchainattheentranceofthecylindricalpore. the slit geometry (eq. (25)).
The colour strength represents thesegment density. The case with localfriction picture can be workedout
similarly. For a chain obeying local friction (Rouse) dy-
namics, we find that
size of outmost blobs is obtained from
ξ(R(def))≃ kBT (B3) E(def) ≃ kBTaN−25 (B10)
γ µ E(R(def)) (R) qeD2
R(def) el
