Scaling of megabase DNA undergoing gel electrophoresis

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Scaling of megabase DNA undergoing gel electrophoresis

S. Obukhov, M. Rubinstein

To cite this version:

S. Obukhov, M. Rubinstein. Scaling of megabase DNA undergoing gel electrophoresis. Journal de

Physique II, EDP Sciences, 1993, 3 (10), pp.1455-1459. �10.1051/jp2:1993212�. �jpa-00247918�

Classification Physics Abstracts

5.20 36.20 C 82.45

Short Communication

Scaling of megabase DNA undergoing gel electrophoresis

S. P. Obukhov (~) and M. Rubinstein (~)

(~) Department of

Physics, University

of Florida, Gainesville, Florida 32611, U-S-A- and Landau Institute for Theoretical

Physics,

Moscow, Russia

(~)

Corporate

Research Laboratories, Eastman Kodak

Company

Rochester, New York 14650- 02l IO, U-S- A.

(Received 8 December 1992, revised 29 Jane 1993, accepted 22 July 1993)

Abstract. We present a

simple description

of

configuration

and

dynamical properties

of

megabase

DNA

undergoing gel electrophoresis.

The molecule _moves in the field in the

shape

of a

self-similar tree-like structure. The

electrophoretic mobility

of DNA in this

regime

is

independent

of its molecular

weight.

1. Introduction.

In the absence of extemal field _a

high

molecular

weight polymer

is confined

by

the strands of the

gel

to a tube

(of

diameter al

Ill

and _moves

through

_a

gel along

this tube

by

_a

reptation

mechanism

[2].

The presence ^of an electric field E introduces _a bias in the motion of

charged polymer.

In

relatively

weak fields

reptation

with _a bias is still _a dominant mode of

polymer

motion

[3].

As electric field E becomes stronger, ^{the chain} can no

longer

move

exclusively along

its linear _contour. If the electric energy of the

entanglement

strand with

charge

_{q is}

comparable

to the thermal energy kT

qaEmkT, (1)

the walls of the tube _{are no}

longer

able to confine the

polymer.

The strands of _a chain leak _out of the tube in the

shape

of hemias under the influence of strong ^{electric field. For}

single-

stranded DNA in I iii agarose this condition

(Eq.(I)) corresponds

to electric fields

Em lo

V/cm [4].

These hernias have been observed both in computer simulations

[5-8]

and in

experiments [9- III-

It _was verified

[5-1II

that in

strong

^{fields chain}

spends

_a lot of time in _a hooked

configuration

with _a

polymer

section between

neighboring

_« hooks _»

pulled by

the field into the

shape

of _a hemia

(Fig. I).

1456 JOURNAL DE

PHYSIQUE

II N° 10

Fig.

I. Hooked

configuration

of _a

polymer

chain. The strands of _a

gel acting

as « hooks _{» are} denoted by filled circles. The direction of electric field is indicated

by

an arrow.

The number of hernias increases with molecular

weight.

When this number is

large,

it is reasonable _to discuss _a

typical

chain

configuration

with many hernias. The remarkable tree- like _structure of these hemias _was observed in _a computer simulation of Duke and

Viovy [12]

for

ultra-high polymer

molecular

weights.

The

snapshot pictures

shown in

[12]

_suggest the

possibility

of the self-similar _structure of _a chain. We

investigate

this

possibility

in the present paper.

Below _we

provide

_a

simple description

of the fractal

shape

and

dynamics

of _a very

high

molecular

weight polymer moving through

a

gel

in _a strong ^{external field. In}

particular,

_we

will show that

I) polymer mobility

is

independent

of its molecular

weight

and iii the

structure of DNA

undergoing gel electrophoresis

is _a self-similar _tree.

2. Structure and its evolution.

One

simple

limit of DNA

gel electrophoresis corresponds

to weak fields with biased

reptation

models

describing

chain motion. It will be shown below that the

polymer dynamics

is also

simple

in the

opposite

^limit of very strong ^{electric fields. The}

key

_new element

controlling

the

structure and

simplifying

the

dynamics

at

high

fields is the

high

tension in the chain. Due to

this

high

tension the

equations

of motion for _most of the very

high

molecular

weight

chain in strong ^{electric field} are deterministic. The

only

parts ^{of the}

polymer

where stochastic motion takes

place

_are low tension sections _at chain ends and

tips

of hemias where stored elastic

energy is less than kT.

The schematic sketch of _a

typical

structure of _a

moving

linear

[13] polymer

with many hernias is shown in

figure

2. It is _a

oversimplified sketch,

the

displacements perpendicular

to the electric field _are

unimportant

and _are not shown in scale. This structure consists of _a

hierarchy

^{of hemias of various sizes. Hernias down the field} _grow at the expense of hernias up the field. But while these hemias grow,

they

encounter strands of _a

gel

and

split, creating

new

small hemias. Some of these small hernias grow, while others

disappear.

At each moment of time

polymer configuration

consists of _a

leading tip

Y and _a

hierarchy

of hemias

(side brunches)

a/ tY of different sizes. As chain _moves down the field, the details of its

configuration

are

constantly

modified, but _we argue below that _a

steady

state statistical

description

of the

moving

chain is still

possible.

It is

important

to note

that,

_{at any}

given

moment, the tension in the

tips

of all hernias a/ tY in

figure

2 is _zero. There is _no forces

acting

on the

leading tip

3' from hemias a/ and tY. Therefore, hernias i@ and if _screen the

leading tip 9',

which has moved the furthest down the field, from the rest of the

polymer. Similarly,

hernias a/ and ti

effectively

separate

the section i@Yif from the remainder of the chain. If _we ask Maxwell Demon _{to cut} the

polymer

_at

points

a/ and

tY,

the

velocity

of the

leading tip

^Y would not be

immediately

E

n

T

Fig.

2. Self-similar branched _structure of _a DNA

undergoing

gel

electrophoresis.

The scale

perpendicular

to the field direction is

exaggerated.

The

constraining

strands of _a gel (« hooks ») _are represented

by

solid circles.

affected, because there is _no interaction between the

leading tip

and the remote branches of the

tree. In

particular,

the average

velocity

^of the

tip

should _not

depend

_on the number of hemias

separating

it from the rest of the

polymer.

This _means that the

mobility

of the

polymer

is

independent

of its size. This conclusion is in agreement ^{with the numerical} simulations of Duke and

Viovy [12].

The tree-like _structure of the chain

(Fig. 2)

can be subdivided into _a main trunk and side branches. The side branches of the _tree go

through

different stages ^{of their evolution.}

They

_are born

by hemia-splitting

of the

leading tip

_upon

encountering

_a strand of _a

gel, they

_grow, then retract and

disappear.

^Most of the side branches

disappear

soon after their formation and

only

very few live

long enough

to grow

large.

The

splitting

of these

long-lived

side branches does not lead to any stable sub-branches. Therefore the side hemias _are

practically

linear.

Despite

the fact that each

particular

branch is either

growing

_or

shrinking,

the average

steady

state

picture

of the whole _structure does _not

change.

3.

Self-similarity

of the

moving polymer.

The

self-similarity

of the structure follows from the dimensional

analysis

of the

equation

of motion of the section a/dJi@

(see Fig. 2),

vJva~ =

CE(L~~ LJvg)/(Lg~

+

LJvg) (21

Here L~V~ and _{L~~W are}

lengths

of strands ala and

dJi#,

E is the

strength

of electric field and

c is _a

proportionality

constant that

depends

_on the effective

polymer charge density

and monomeric friction coefficient. The _{numerator on} the

right-hand

side of this

equation

^is

proportional

to the _net force

acting

on the

fragment (the

difference of forces

acting

on sections ala and dJi@). The denominator is

proportional

to the friction coefficient of the section

1458 JOURNAL DE PHYSIQUE II N° 10

a/dJi@

(the

total

length

of this

section).

Similar

equations

_can be written for the velocities of branches of all sizes. In all these

equations

velocities _are related _to the ratio of strand

lengths (e.g.,

in

Eq. (2)

_v~g~

depends only

_on

L~vg/Lg~).

Therefore _mass transfer around _a hook is

scale-invariant and the size of _a hemia is

proportional

to its lifetime.

We

already

know that the average

velocity

of the

leading tip

^is

independent

of

polymer

mass. Therefore the lifetime and size of _a side hernia _are

proportional

to its distance from the

leading tip

Y. This

implies

that the structure of the

polymer

is

statistically

self-similar. For

example,

the branch size ala

(and

its characteristic lifetime

t~gg)

^is

proportional

to the distance from the

leading tip

Y to the hook dJ of that branch. This

scaling

follows from the result that the _mass transfer around all hooks is

approximately

the _same and

proportional

to the transport

velocity

of the

polymer.

The whole _structure is thus

statistically

invariant under the affine transformation of all distances R

=

(Rjj, Ri )

measured from the

tip

Y

R(

=

ARj

and R

[

= A

~'~Ri

,

(3)

where Rjj and

Ri

_{are components} of _vector R

parallel

and

perpendicular

to the direction of the electric field.

It _can be shown that the

splittings

of side hemias do not

produce

_any stable sub-hemias

II 4].

The structure of these side hemias is

practically

linear and their _mass

proportional

to their size.

Therefore,

the overall linear size of the

polymer

is

proportional

to its _mass. The total number of hernias is

proportional

_to the

logarithm

of

polymer

mass.

4. Conclusions.

We derived the

independence

of the

electrophoretic mobility

of _a

polymer

on its _mass. This is in

agreement

^with

experiments

and computer simulations

II 2].

Our result of the statistical self-

similarity

of

polymer steady

state structure

(Eq. (2))

_seems to be in agreement ^with recent simulations

[15]

but still awaits

experimental

verification.

Our main

assumption

is that there is _a tree-like

typical configuration

of _a

high

molecular

weight polymer moving through

_a

gel

in _a strong ^electric field. This _tree consists of _a well- defined

leading tip,

main trunk and side branches. New side branches _are

generated by

the

splitting

of the

leading tip.

We did not consider

special

events when _a

particular

branch overtakes the

leading tip

and wins,

forcing

the old

tip

to

disappear.

We assumed these _events to be _rare and _{not to} affect the

steady

state distribution. We

hope

that further results for the Duke-

Viovy

model

[12]

will

clarify

the relevance of these effects. In addition we are

currently studying

the new computer ^model

[14]

which

exploits

the almost deterministic motion of

polymer.

In this model the

dynamics

is controlled

by equation (2)

and randomness is introduced

only during tip splitting.

It is remarkable

that,

under _very

general conditions,

_a

moving polymer

chain

dynamically develops statistically

self-similar

shapes.

This _can be viewed _{as one more}

example

of the

phenomenon

of

self-organized criticality recently

^discovered

by

P. Bak _et al.

[16].

Acknowledgements.

We

acknowledge

the

illuminating

discussions and critical remarks of

Viovy

and Duke.

References [Ii Edwards S. F., Proc. Phys. Soc. London 92 (1967) 9.

[2] de Gennes P. G., J. Chem.

Phys.

55 (1971) 572.

[3] Lerrnan L. S., Frish H. L.,

Biopolymers

21(1982) 995

Lumpkin

O. J., Zimm B. H., Biopolymers 21 (1982) 2315

Slater G. W., Noolandi, J. Phys. Rev. Lett. 55 (1985) 1579.

[4] The distribution of pore sizes in _a

gel

is

quite polydisperse.

Hemias _can leak out of larger _{pores even}

at weaker fields.

[5] Deutsch J. M., Science 240 (1988) 922.

[6] Deutsch J. M., J. Chem.

Phys.

90 (1989) 7436.

[7] Deutsch J. M., Madden T. L., J. Chem.

Phys.

90 (1989) 2476.

[8] Madden T. L., Deutsch J. M., J. Chem.

Phys.

94 (1991) 1584.

[9] Smith S. B.,

Aldridge

P. K., Callis J. B., Science 243 (1989) 203.

[10] Schwartz D. S. and Koval M., Nature 338 (1989) 520.

[I Ii Gurrieri S., Rizzarelli E., Beach D., Bustamante C., Biochemistry 29 (1990) 3396.

[12] Duke T. A. J.,

Viovy

J. L.,

Phys.

Rev. Lett. 68

(1992)

542.

[13] There should be _no essential difference between

gel electrophoresis

of linear and

ring

DNA in strong fields.

[14] Boris D., Rubinstein M., Obukhov S. P., to be

published.

[15] Duke T. A. J.,

Viovy

J. L.,

private

communications.

[16] Bak P,,

Tang

C, and Wiesenfeld K., Phys. Rev. Lett. 59 (1987) 381.