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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