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ELECTROPHORESIS. Hwa A. Lim, Jaan Noolandi and Gary W. Slater ... 1985; Slater, 1985). ..... Watson, J.D., N.H. Hopkins, J.W. Roberts, J.A. Steitz and A.M. ...
Math1 Comput. Modelling, Printed in Great Britain

Vol. 14, pp. 494-499,

GENERALIZED ELECTROPHORESIS Hwa A. Lim,

Jaan

TUBE

Xerox Research L5K 2Ll

MODEL

Noolandi

Supercomputer Computations Florida 32306-4052

08957177/90 $3.00 + 0.00 Pergamon Press plc

1990

OF

and Gary Research

BIASED

W.

REPTATION

FOR

DNA-GEL

Slater

Institute,

Florida State University,

Centre of Canada, 2660 Speakman

Drive,

Tallahassee,

Ontario,

MiBBiBBaUga,

Canada

Abstract. We present a generalized tube model for the theoretical analysis of the reptational motion of DNA in gel electrophoresis. The generalized model incorporates effects of molecular fluctuations and is capable of explaining some main features observed in electrophoresis experiments involving periodic inversions of the external electric field. These features may be attributed to a delicate interplay between the internal motion and the center of mass motion of the DNA molecules under pulsed field conditions. Keywords. Chromosomes, computer namics, reptation, tube model

INTRODUCTION

AND

simulation,

DNA, electrophoresis,

gel, molecular

dy-

- lo3 bp have been and are separated routinely; but for larger DNA, there is a problem of resolution of DNA of neighboring sizes, e.g. 2.0 x lo6 bp from 2.1 x lo6 bp. The technical difficulty is associated with band inversion or compression of DNA during migration (Alberts, 1988).

MOTIVATIONS

In two recent papers (Lim, 1989a; Noolandi, 1987,1989), we provide a generalization of the biased-reptation model of DNA (Bean, 1983; Lerman, 1982; Lumpkin, 1982, 1985; Slater, 1985). The biased-reptation model is based on the formalism developed by de Gennes 1971, 1985), and Doi and Edwards (1978a, 1978b, 1986 I to describe the dynamics of a polymer chain in a highly constrained environment. A salient feature of the biased-reptation model of DNA-gel electrophoresis is to regard the gel fibres obstructing the lateral motions of the DNA molecule as forming a tube as shown in Fig. 1. The DNA molecule then slides longitudinally along the tube. To the first approximation, the intra-molecular motions are neglected and the DNA molecule is replaced by segments of rigid rod as shown in Fig. lc. But recently, it has become clear that for the more advanced separation technique (e.g. field inversion gel electrophoresis (FIGE]) (Hostock, 1988; Holzwarth, 1987, 1989), the internal modes of the DNA molecule play a vital role in the molecular mechanism responsible for the separation and the rigid rod tube model cannot explain many of the features (Noolandi, 1989).

Hence, without a strong theoretical basis for the understanding of the electrophoretic behavior of DNA, one is left with a large technological gap. A better understanding of the underlying mechanisms responsible for the behavior of DNA mobilities under various experimental protocols will certainly lead to improvement in DNA-gel electrophoresis equipment design, and most importantly, to the separation of human chromosomal DNA in a reasonable runtime.

GENERALIZED REPTATION

BIASED MODEL

In the generalized tube model of biased reptation for DNA-gel electrophoresis (Lim, 1989a; Noolandi, 1989), the set of connected, flexible entropic springs may be visualized as N beads initially placed in the middle of N

A refinement thus is to generalize the tube model to incorporate intramolecular motions (Lim, 1989a; Noolandi, 1989). This can be achieved by replacing the DNA molecule as a set of connected, flexible entropic springs, rather than rigid rods. Fig. Id depicts such an example. This constitutes the subject of this report. Further refinehernias (Schoenherr, 1989) or wrappings around gel fibres (Hood, 1989), will not be considered in this report.

consecutive pores (see Fig. 1). If we denote the curvilinear position of bead i at time t along the tube axis as s(i, t), then the governing equation of motion is (Lim, 1989a; Noolandi, 1989)

Other motivations for this study arise from the drive towards the ultimate goal of separating human chromosoma1 DNA (Alberts, 1988). The current size record of chromosomal DNA separable by DNA-gel electrophoresis technique is - 9.0 x lo6 bp in a runtime of approximately a week (Turmel, 1989). However, the amount of human chromosome, - 1.4 x lo* bp Alberts, 1988; Watson, 1987), is at least lo-fold or lOO-I old greater than the maximum size separable by the application of current gelelectrophoretic technique. Therefore, very little progress is likely to be made in separating DNA molecules any larger than that contained in chromosome bands (- lo6 bp). Extensions of the technique to isolating individual chromosome appear to be extremely difficult (Alberts, 1988). In fact, some very large DNA molecules, for un-

where 7 is the coefficient

a +i,t)

=FJ(Ari)

- F,(AT~-~)

(I)

+ qE cos ty;, t) + 7](i, t),

of friction, 0(i,t) is the included angle between the tube axis and the field direction, u(;,t) is the curvilinear random force acting on i satisfying (q(;, t)n(j, t’)) = 2ykgT&jb(t - t’) and Fs (Ai) = Fs (Is (; + 1, t) - s (i, t) I) is the anharmonic entropic spring force given by (Tanner, 1985) bead

B(r) b is the b-‘kgTB_’

Kuhn

= cothz

length.

% is exerted

-

$.

An additional

0 to prevent the chain from collapsing 1989a; Noolandi, 1989).

known reasons, appear unable to enter the running gel and remain embedded in the gel plug. DNA molecules of 494

entropic

force

on each of the two end beads inside the tube (Lim,

. . .c

495

Proc. 7th Int. Conf. on Mathematical and Computer Modelling

.

.

l

.

.

reptation

= 02, model

(4) is

recovered

l

l

. . .

.

the usual biased (Lim, 1989a).

l

l

. .

[AT(~)]’ = ((AT)‘)

UN .

(a)

*a

.

l

“1. .

.

;@p$

0.



.

.

.

.

(d) .

l

;>>

:

Figure 1. (a) Schematics

S+

picture of a DNA

molecule in a gel medium. Solid circles represent gel fibre cross-sections.

The average

pore size is a; (b) In the Rouse model, the

Figure 2. (a) A schematic picture of a tube

DNA molecule is replaced by a bead-spring chain; (c) In the usual reptational the gel obstacles

limiting

in a biased reptational

model,

the lateral

model.

The first

and the last segments are marked r+ and

mo-

r_, respectively;

tion of the DNA molecule are replaced by

(b) According to the nor-

a tube and the DNA molecule is replaced

mal rules of reptation,

by a primitive

ment leaves the tube and goes in the S+

segments;

chain of freely-jointed

rigid

(which is chosen randomly),

(d) In the generalized reptation

model, the rigid segments

are further

placed by springs to incorporate

when the r_+ sega new tube

section is created, and the trailing- segment -

re-

is destroved; -

the intra-

.\,(c) Similarlv. “I

when the r_

segment leaves the tube and goes in the S-

molecular modes of motion.

direction (which is also chosen randomly), a corresponding An alternative

will be to replace the b-‘/cBTB-l

i

0 boundary entropic force by a Boltzmann form (Deutsch, 1989a, 1989b; Duke, 1989; Lim, 1989b). a is the average pore size satrsfying the condition (Lim, 1989a) lb = ((AT)‘) 2 = max(Ar)

=

o2, >> b.

new tube section is cre-

ated and the leading segment is destroyed.

RESULTS (3o) (3b)

The evolution of the two end beads follows the normal rules of reptational motion, i.e., new tube sections are created in random directions when the leading or the trailing (first or N) bead leaves the tube and old ones are destroyed. Fig. 2 illustrates how this is achieved. The tube thus guides the overall motion of the DNA molecule, which is allowed to fluctuate longitudinally inside the tube. In the limit of

To enhance visualization, the computer simulation code is combined with a graphics interface to permit interactive experiments. Fig. 3 presents some representative snapshots taken at various times in a movie recorded during a computer simulation of the migration of a DNA molecule in a DNA-gel electrophoresis experiment. It is seen that in the absence of an electric field, the DNA molecule is a random coil under thermal agitations as in Fig. 3s. The length is cx JL where L is the length of the DNA molecule (see Fig. la). The external electric field E (left to right) is then turned on at t = 0. Un-

496

Proc.

7th Int. Conf. on Mathematical

and Computer

Modelling

dec the influence E, the two free arms of the DNA begin to migrate in the field direction (Figs. 3b-3c). In this particular simulation, the motion of the middle section of the DNA molecule is hindered by a gel obstacle. As time elapses, the longer arms pulls on the shorter arms and lengthens at the expense of the shorter arm (Figs. itself 3d-3e). At t x 100, the DNA molecule disengages completely from the obstructing gel fibce and begins to migrate as a whole in the gel medium (Figs. 3f-3g). The leading segments ace seen to be more compressed than the trailing segments, which is apparent in Figs. 4e-4f, which show average the spring length at various times t. This is easily explained. During the migration, the leading bead has to seek out an obstacle-free region so that it can ceptate in the field direction while the trailing beads ace pulled along. The lead to compressions.

hesitations

Fig.

physically

5 contains

of interest:

several

the

end-to-end

average

distance

tube

(hz)

=

of the

leading

observable

length

beads

parameters

y,

the

dwjzl

as a func-

tion of the time. This observation is in good agreement with the observations of recent fluorescence linear dichcoism experiments (Holzwacth, 1987, 1989) (insect of Fig. 5) in which the initial overshoot and the decrease in tube orientation when the E is reversed have been observed, and that the chain has moved over a distance appcoximately equal to half its contour length at time t = Tag. The latter is shown in the upper ordinate-axis in Fig. 5, which shows that

the curvilinear

position

(13

(f ,t)

I)Na

of the center bead of the chain-a measure of the number of tube renewals undergone by the chain-is of the order that

of 0.5 at the ratio

analytical The

t = rmax. In w

calculatiods

variation

particular, -

(Kckmec,

it is also observed

6

as predicted

1988;

of the electcophocetic

Slatec,

velocity

by

1986). v with pulse

duration t , in FIGE simulation runs, is shown in Fig. 6. A gca B ual increase of the velocity occucs when the pulse duration tp < y to t, > Tdisrwhere T&s is the tube disengagement (or renewal) time, is in agreement with the experimental observation (Bostock, 1988). The curve also shows a marked minimum for a critical pulse duration tp = T* =: Arrev. Since Arcev is proportional to the molecular size, this sudden decrease of the velocity permits the separation of larger molecules that cannot be separated in continuous fields. This sudden deIt is believed that if the crease shown here is - 15-20%. entropic force exerted on the two end beads to prevent them from collapsing inside the tube is of some exponential form (Deutsch, 1989a, 1989b, Duke, 1989) rather the linear form used here [see also paragraph preceding Eq. (3a)], the sudden decrease will be more pronounced. This is currently under intensive investigation (Lim, 198913). The mechanisms involved in the decrease of the tube length upon field reversal ace apparently responsible for the increased separation power in FIGE. To better understand the mechanisms, we study the intca-tube distribution of the DNA molecule during migration (Figs. 3 and 4).

The

average

spring

length

~3(‘+“t~-s(i’t)l)

as

for various sizes a function of the spring index $ij of the DNA molecule is shown in Fig. 7. Most springs ace slightly overstretched, and there is net asymmetry in the distribution: leading springs ace compressed; springs in the middle ace very overstretched and the trailing springs ace slightly stretched. Moreover, this asymmetry is mote pronounced the larger the DNA molecule (Noolandi, 1989). When the field is reversed, this asym-

Figure

3. Typical

of a DNA

molecular

migrating

conformations

in a gel medium.

The

solid lines ace tube

sections.

sence of an electric

field, the conformation

is a random

coil under

thermal

(b-c)

When

electric

field

plied.

the two free arms

orient

in the ders

the

(a) In the ab-

field direction. the

motion

of the

molecule;

lengthens

at the expense

until “V

the

DNA itself

to migrate

ap-

themselves

A gel obstacle

the DNA

enzaees

agitations; is first

(d-e)

molecule

middle The

section longer

of the shorter finally;

from the obstacle

hin-

(f-g)

of

arms arm dis-

and begins

as a whole in the gel medium.

497

Proc. 7th Int. Conf. on Mathematical and Computer Modelling N=30.8=2.0 Ensemble=iOOO Chains

~~f~~

I

c d

___--_-__-_-__~-----~-----d .j

‘.

‘.

-I- 0.4

.g

0

0

Figure

e g

3 (e)....!+~

4 2

zB1 “.. . . . . .

/

1

5. Relative

average

+@Yi+@(

upper

age end-to-end

distance

#$

curve)

= j-1

of the time (in units

T = &)foraN

10

Figure

4. The average

spring

index

is N

spring

i at various

= 30 beads

20

i

middle.

length

&@

lation,

versus

times t. The molecular

or 29 springs,

the scaled

0 = 2.0, i = 0.25, and the results an ensemble

The applied

30

of 2000 chains.

field is

are averaged

In this particular

the size

over

each spring

arms

and

are therefore

(b) At t = 40, most the pulling

slightly

springs

overstretched;

are overstretched

force of the growing

arms;

(d) At t = 100, the smaller

i’s is retracting tube;

back

inside

the initial

under

i.e., th,

.___~

force on

arm (small random-walk

(e) At f ~~ 150, the i m: 1 end of the chain

has just

left the initial

t, steady

state

is attained.

end of the chain at the trailing

random-walk

is slightly

The spring compressed,

end are slightly

stretched.

= 30beadchain. electric

The upper

field is shown abscissa-axis

curvilinear

position

ter bead

of the chain.

the orientation

q

give the of the cen-

The insert

of A DNA during

field inversion

sequence

in the

of Ref.

show a short 11. f

is the orientation fluorescence-detected

function linear

measured

by

dichroism.

metry has to be reversed before the steady state can be achieved, a process that take r’ z AT,,, and is responsible for the different velocity of the molecule just after the field is switched.

(c) At t = 78,

the springs reach t,heir maximum extension, ___~_. ~~~_. ~~~_ -~~ ~. spring force just counterbalance the electric the two arms;

of

simu-

corresponds to w 1000 bp, 1.00 z ___ In terms of these, ~1 z 0.02 seconds. 0.3 V/cm. ______ (a) At t = 10, a few springs at both the frees ends

form

and the aver-

(lower curve)

as a function

0

tube length

tube and __~_

the

in the leading while those

CONCLUSION We have reported a generalized tube model of biased reptation for DNA-gel electrophoresis (Noolandi, 1989). The model incorporates effects of intra-molecular fluctuations. Even though the model allows for only longitudinal fluctuations along the tube and does not allow lateral hernias), its predictions are in good fluctuations (e,g. agreement with experiment observations in FIGE (Bostock, 1988; Holzwarth, 1987, 1989). Phenomena like lateral fluctuations (Schoenherr, 1989) or wrappings around a gel fibre (Hood, 1989) are expected to be important in electrophoresis experiments like cross field electrophoresis (Mathew, 1988), in high fields (Shaffer, 1989) or for long large DNA molecules. Non-homogeneous gel may also increase the effects of fluctuations. Investigations of

Proc. 7th Int. Conf. on Mathematical and Computer Modelling

498

of this model with those of the original biased reptation model. It further predict8 the observations of two dimensional simulations (Deutsch, 1989a, 1989b) and experimental observation (Smith, 1989) of bunching effects of the leading edge of the migrating DNA molecule. Acknowledgements

100

IO

3

PULSE DURATION

$17

loo0

(dimensionless)

Figure 6. The average center-of-mass

The authors thank the Supercomputer Computations Research Institute staff and Ms. D.E. Burnette for assistance. HAL is partially supported by the US Department of Energy under the contract number DE-FCOS85ER250000 and the Technological Research and Development Authority under the contract number TRDA 103. JN and GWS are supported by contribution agreement number CA949-6-0007Y of the Biotechnology Research Institute’s external research program of the National Reaearch Council of Canada.

ve-

locity of a DNA molecule in FIGE as a REFERENCES

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5. During the forward pulse duration of tp, E = +l and during the reverse pulse duration of 0.4t,,

E = -1.

There is a marked

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I

’ 0.3

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W-1)

average

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’ 0.7

I 0.8

I 0.9

1

1.0

length

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