Repton model of gel electrophoresis and diffusion

(1)

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Repton model of gel electrophoresis and diffusion

B. Widom, J. Viovy, A. Defontaines

To cite this version:

B. Widom, J. Viovy, A. Defontaines. Repton model of gel electrophoresis and diffusion. Journal de

Physique I, EDP Sciences, 1991, 1 (12), pp.1759-1784. �10.1051/jp1:1991239�. �jpa-00246450�

(2)

Classification

Physics

Abstracts

05.40 66.10 82.45

Repton model of gel electrophoresis and diffusion

B. Widom

(*),

J. L.

Viovy

and A. D. Defontaines

Laboratoire de

Physicochimie Thdodque,

ESPCI, 10 _rue

Vauquelin,

F-75231Pads Cedex 05, France

(Received

3 June 1991, revised13

August, accepted14 August)

Abstract. We

analyze

the repton model of Rubinstein _as

adapted by

Duke _as _a model for the

gel electrophoresis

of DNA. Parameters in the model _are the number Nor reptons in the chain, a

length

_a, _a

microscopic

transition

frequency

_w, and the

product

eE of the electric field E

(assumed constant)

and the

charge

e per repton.

Fornlally

exact formulas _are derived for the dimensionless

diffusion coefficient

Dla~w

and drift

velocity Vlaw,

the latter _as _a function of the field.

Calculation of Vlaw

requires

the

eigenvector

associated with the leading eigenvalue of _a 3~ x 3~ ' matrix. For short chains exact results _are obtained

analytically

_:

Vlaw

for all eE for I « N _« 4, and

Dla~

_w for I _« N _« 5. For large N _we deduce that

Dla~

w vanishes

proportionally

to

I/N~,

the standard de Gennes reptation result, but _we have not evaluated the coefficient

analytically.

We have determined

Dla~w

for N up ^to 150

by

simulation and verified the I

/N~

law.

1. Indoducfion and outline.

Rubinstein

[I]

has introduced _a model that he calls the repton model for the diffusion of _a

polymer

chain in _a dense medium. It is _a lattice model that

incorporates

and expresses de Gennes'

reptation

mechanism

[2]

in its purest

form, viz.,

_as the diffusion of stored

length along

the chain's _own contour. The chain is

represented by

_a _sequence of reptons

(beads)

connected

by

links. A number 0 _or I is associated with each

link, according

to whether the associated beads

belong

to the _same site _or _to

adjacent

_ones. This model was

generalized by

Duke

[3]

to _a chain of three-state links

(0, 1, 1),

_as a model for the

gel electrophoresis

of DNA. To avoid _any

confusion,

_we refer in the

following

to the

original

model

Ii

_as the _« two-

state repton model _» and to Duke's

generalization [3]

as the _« three-state

repton

^model ».

This paper is concemed ~vith the latter.

For _a

description

of the

physical

model in the real two- or three-dimensional _space of the

diffusing

chain _we refer to the

original

_papers

[1, 3J;

here _we shall describe

only

the

equivalent

one-dimensional model into which that _one _maps. The

mapping

_was demonstrated

by

Rubinstein

[1].

(*)

Permanent address: Department of

Chemistry,

Baker

Laboratory,

Comell

University,

Ithaca, New York 14853, U-S-A-

(3)

Figure

I shows _a connected chain of N _« reptons », each

representing

_some

(large)

number of base

pairs

of _a

charged

DNA chain. The discrete variable _x~

= .,

2,

1,

0,

1,

2,

(I

=

I,.., N) gives

the location of the I-th

repton,

hence of the I-th element of the DNA

chain,

in _some direction of space that is

conveniently

chosen to be that of _an extemal electric field E that acts _on the

charges

of the chain. When E

=

0,

_so that the chain diffuses but has _no net drift

velocity,

this direction is

arbitrary.

The chain

connectivity requires

that Ji~ t x, =

0 _or _± I for all I

=

I,

,

N -1.

6 5

~

-i

2 3 4 5 7 8 9 II 13

repton number

Fig.

I.

-Repton

chain with N =15 in _a

representation

with _a

single spatial

coordinate _x in the direction of _an

extemally applied

electric field E

(after

Duke, Ref. [3]). The arrows represent allowed

moves.

Neighboring

_reptons that have _a _common value of their _x coordinate represent, via the

mapping,

_a cluster of successive

reptons occupying

_a common cell in the

original

two- or

three-dimensional _space. In the

example

of

figure I,

taken from Duke

[3], _reptons

2 and 3 _are

a cluster of two,

reptons 8, 9,

10 _are _a cluster of

three,

and

14,

15 _are another cluster of _two.

The

reptation

mechanism in the

original

_space then maps into

dynamical

_move rules in the space of

figure

I. An interior repton I

(I

₌

2,

...,

N

I)

is allowed to _move

only

if it is the end

repton

^of a

cluster,

and then it is allowed to _move

only

in the direction of that _one of its two

neighbors

that does not

belong

to the cluster. This is shown in

figure

I

by

the _arrows,

I

_or

(,

on the interior reptons

2, 3, 8, 10,

and 14. An end

repton (I

= I _or

N~

^is

always

allowed _to

move. If its coordinate _x is not the _same _as that of its

neighbor,

_so that it is not part of _a

cluster,

it is allowed to _move

only

in the direction of its

neighbor,

_as

exemplified by _repton

I in

figure

I. If the end repton ^is at the _same _x _as its

neighbor,

_so that it is part of _a

cluster,

it is

allowed _to _move either

I

or

(,

_as

exemplified by repton15

in

figure

I. A _move

I

_or

( by

_a

repton changes

its _x coordinate

by

+ I _or

I, respectively.

Define

B

=

e~~W~~~

(I.I)

where E is the constant extemal electric

field,

_e the

charge

_on each repton, ^and a a

single

length parameter

^that may be taken to be both the

length

_per _repton and the linear dimension of _a cell

(gel pore)

in the

original

_space _; and let _w be _a constant transition

probability

_per unit

(4)

time. We then

assign

wB and wB _as the transition

probabilities

_per unit time for _an allowed

move

I (direction

of

increasing x)

_or

( (direction

of

decreasing x), respectively.

This is _a little different from the

assignments

made

by

Rubinstein

[I]

and Duke

[3],

^who ⁱⁿ

assigning

transition rates to arrows

distinguish

between _arrows that _are _on interior and _on end

reptons,

and between the two arrows on an end

repton (when

it carries

two).

Such differences in the

assignment

of transition rates affect the

quantitative

but not the

qualitative

behavior of the model. We choose to have

only

the transition rates wB _or wB for every

I

_or

(

_arrow without further distinction _so _as _not _to have _any _more

parameters

^than

N,

_a, _w, ^and eE.

The

configuration

of the

repton

^chain ⁱⁿ ^the one-dimensional

representation

of

figure

I

(one spatial dimension, x) changes

in time

according

to the rules

just specified. Any

_one such

change

of

configuration, I-e-,

_any allowed _move

by

_a

single

_repton,

changes

the _mean

position

x of the

reptons,

^and so, as time progresses, we see a drift of the chain _as _a whole in the _x direction. This _may be viewed _as the

projection

onto _one dimension of the drift of the

original

chain in its

original, lligher-dimensional

_space. The drift

velocity

and diffusion coefficient of the chain in the _x direction in the

representation

of

figure

I _are the _same _as that of the real

chain in the

original

_space. Because the _moves _are

stochastic,

the transition rates

being

transition

probabilities

_per unit

time,

_we should think of the chain _as _one member of _an ensemble of many such

chains,

all of the _same

length

N.

Since x;~t -x; for each I

=

I,...,N-

I may have _any of the three values 0 _or

±

I,

the

repton

^chain ^has

3'~~~ possible

distinct internal

configurations irrespective

of its location _as _a whole in the _x direction. We index those internal

configurations by

_y. We choose any one of the _x; _to define the location of the chain in the _x direction. We call that Ji

simply

_x. Then _x and _y

together

define both the location and intemal

configuration

of the

chain.

Let

p~,~(t)

be the

probability

that at time t the chain will be _at _x in

configuration

_y. In section 2 _we write and

formally

solve the

equation

that determines

p~,~(t)

for any

given p~,~(0).

^The ^solution ^involves ^the

eigenvalues A~(q) (n =0,

1,

2,

..,

3'~~~ -1)

^of a

3'~~~ x3'~-~

matrix of transition

probabilities

the elements of which

depend

_on _a

(dimensionless)

_wave number q as well _as _on B.

We remark in section 2 that _one of the

eigenvalues,

which _we label _n

=

0,

vanishes at q =

0,

and that the

field-dependent

drift

velocity

V and diffusion coefficient 5~ of the chain _are

obtained from the coefficients in the

expansion

of

Ao(q)

in powers of

qla,

A

o(q)

= I

Vqla s~q ia~

₊

_(1.2)

V and 5~ _are

respectively

odd and _even in eE,

V(eE)

= V'

(0 )

eE ₊ V

"'(0 ) (eE)~

₊

5~(eE)

=

5~(°)

₊

5~"(°)(eE)~

₊

_(l.3)

Here

5~(0)

=

D, (1.4)

the

ordinary phenomenological

diffusion coefficient in the absence of _an external field.

Since each repton carries the

charge

e the total force exerted

by

the field E _on the chain is NeE.

Hence, by

the Nemst-Einstein relation

[4],

^the coefficient

V'(0)

in the first of

equations (1.3)

and the diffusion coefficient D in

(1.4)

must be related

by

V'(0)/N

=

D/kT. (1.5)

(5)

This is _a necessary relation between

a~Aolaq

aE and

a~Aolaq~

at E

= 0 that should in

principle

be derivable _as _a property ^{of the} ^matrix ^of ^wuich

Ao(q)

is the

leading eigenvalue.

As _a

purely

mathematical

problem

associated with this model it is _a

challenging

_one and has not been solved for

general N, although

we later

(Sect. 4) verify

it for N

= 1,

2,

and 3. In sections

3, 4,

and 5 _we _use

(1.5)

to obtain D from

V(eE)

_even for N

~

3,

_on the

grounds

that _as _a

physical,

as distinct from

mathematical, principle (1.5)

is unassailable and _we _are confident that it is satisfied

by

this model for all N.

There is another _as

yet

^unsolved mathematical

puzzle

in tills

model,

to which _we _are

again

confident of the _answer _on

physical grounds. Physically,

the drift

velocity V(eE) (hence

also

D)

must be

independent

of which x;

(I

=

I,

,

A§

is chosen to be the variable _x that

specifies

the location of the chain _as _a whole in the _x direction. The _same is

probably

true, more

generally,

of all the

eigenvalues A~(q).

The

question

arises

only

for

q#0;

^when

q=0

the matrix

A(q)

of which the

A~(q)

_are the

eigenvalues (Sect. 2)

is

manifestly

independent

of that

choice,

but

explicitly depends

on it when q # 0.

Also,

the

question only

comes up for N _m 3 _; for N

= I there is

only

_one _x;, while for N

=

2, although

there _are two,

they

_are

equivalent by

symmetry. ^In section 4 _we

verify

that

V(eE)

for N

=

3 is indeed

independent

of which _x, is chosen to be _x. In section 3 _we

exploit

the

certainty

that

V(eE)

is

independent

of that choice for any ^N

(although

it remains _unproven

mathematically)

to

simplify

the

general

formula for

V(eE) by removing

from it reference _to the

arbitrary

choice of marker.

In section 3 _we obtain

formally

exact formulas for the drift

velocity

V and diffusion coefficient D based _on

(1.2)-(1.5)

and other results in section 2.

Then,

in section

4,

_we

explicitly

calculate the dimensionless

V(eE)law

for N

=1,

2,

3,

and

4,

and

Dla~w

for N

=1, 2, 3, 4,

and 5.

We deduce in section 5 that

Dla~w

vanishes

proportionally

to

I/N~

_as N

- ~x~, which is characteristic of

reptation [2],

^but we have _not _evaluated the coefficient. In section

6,

_we present ^the ^results ^of simulations for N up ^to 150 _we make _some connections with earlier

work,

and

verify

the

I/N~

_law. _In section 7 _we summarize _our results and refer

briefly

to

current work _on

scaling properties

of the model for

large

N and small E.

2,

p~,~(t);

identification of Vand D.

In section I _we chose _an

arbitrary

_one of the repton coordinates x; and called it _x, _we indexed the

3'~

internal

configurations

of the chain

by

_y, and _we defined p~,

~(t

to be the

probability

that the chain is at x in internal

configuration

_y at time _t. The _move rules and transition rates described in section I may be

represented symbolically by time-independent probabilities

_per

unit time W~,

~, _~ _~ of the transitions _{x, y}

-

x', y'.

Then

dp~,~(t)/dt

=

it _W~,~,~,,~, p~,,~,(t) W~,,~,,~,~ p~,~(t)j (2.1)

From the rules of section I _we know that W~, _~,.~

~

for

given

_y and

y' depends

_on _x and x'

only

via the difference x' _x and vanishes if

tiis'difference

_is _not ₀

or ± I. We then define 1k~7~~,

lf~i),

^and

lf~,§~ by

w(~

) ~'_ _~

= j

y y '

w(f)

_{~'_}

~ _~ ~

w,

~

y y'

~ ~~ "~

lljfy~,

^x'- x #

0

,

otherwise

(2.2)

(6)

By

the

dynamical

rules in section I many of the

Ik~T~),

W)I),

^and _lk~f~~ ^vanish. ^The

only lljT~~

^that ^do ^not ^vanish ^are ^those ^for ^which ^the ^transition y -

y'

ⁱⁿ the internal

configuration

occurs

by

_an allowed _move

( by

the marker

repton (the _repton

whose _x coordinate _we have

adopted

as the _x variable for the

chain),

and then that

non-vanishing llj§~

is wB-

[with

B defined in

(I.I)]

the

only lljf~)

^that ^do not vanish _are those for which the transition

y -

y'

_occurs

by

_an allowed _move

I by

the marker repton, and then that

non-vanishing

ll~§~

^is

^wB;

^and ^the

^only W)I)

^that ^do ^not ^vanish are those for which the transition

y -

y'

_occurs

by

_an allowed _move

by

_some

_repton

other than the marker

repton,

and then that

non-vanislling lk~i)

^is ^wB ^if ^the

repton

move that _causes the transition is

I

_and

wB-'

if it is

(.

The solution of

(2.I)

is then of the form

J'x,

y(t)

~

£ j~ ^Cn(q) ^fj~~(q)

^~~~~~^~~~~~

^dq ^(2.3)

where the

A~(q) (n

=

0,

1,

2,.., 3'~~

_l

are the

eigenvalues

of the

3'~

_x

3'~~

transition-rate matrix

A(q)

with elements

A~~,(q)

^defined

^by

~YY' ~~iY'~~~~

~

~~~

_~

_~~i~

~ ~~ ~Y' # Y~

Ayy =

Z _(Hl~y~

₊

_Wlfl

₊

Wlty~), ^(2.4)

y,<~y>

where the

fj~~(q)

_are the y-components of the associated

right eigenvectors fl~~(q),

~(~) f~~~(~)

~

n(~) f~~~(~)

,

(2.5)

and where the functions

c~(q)

_are determined

by

the initial _p~

~(0). Formally,

if the left

eigenvectors

of

A(q)

_are called

g~~)(q),

g~~~

(q)

A

(q)

"

An (q ) g~~>(q )

>

(2.6)

and if _we choose the normalization

g~(q) f~(q)

= ,

(2.7)

then

Cn(~)

_~

(2 ")~ z z

~~ ^~~~

~j~~(~)Px, y(0) (2.8)

We _note from

(2.4)

that the

diagonal

elements A~~ ^of

A(q)

_are

independent

_{of q and} _are

related to the

off-diagonal

elements _at _q

= 0

by

A~~ =

£ A~,~(0) ^(2.9)

v>(#y>

It then follows from

(2.5)

and

(2.6)

that when q = 0 there is

necessarily

_an

eigenvalue A(0)

=0 and the associated left

eigenvector g(0)

is _a constant;

I-e-,

its

components

g~(0)

are

independent

_{of y. We} index this

eigenvalue by

_n

= 0 _:

Ao(0)

= 0.

(2.10)

At all other _q this

Ao(q)

has _a

negative

real

part,

as do all the other

A~(q)

at all q ; if any

(7)

A~(q)

had _a

positive

real

part p~,~(t)

would increase without bound _as _t

- ~x~, which cannot be.

The

long-time

behavior of

p~,~(t)

is dominated

by

the _n

= 0 term in

(2.3),

and

by

_q _near

0,

because of

(2.10). Thus,

Px,y(t) j~ ^co(q) ^f)°~(q) ^e'~~~

^~°~~~

^dq ^(t

^- ^~x~

^), ^{(2.i i)}

with the

major

contribution

coming

from the

neighborhood

of _q

= 0. Then if

Ao(q)

is

expanded

_as in

(1.2),

_we find that after

long times,

@Px,y/°t

^V

@Px,y/@(aX)

⁺ ^5~

@~Px,y/@(ax)~

⁺

(2.12)

(Recall

that _x is

dimensionless;

it is _ax that _measures distance in the

x-direction.)

We

compare this with the

phenomenological equation aplat

=

Vapla(ax)+

5~

a~pla(ax)~

appropriate

to the

long-time, small-gradient limit,

with V and 5~ the

(field-dependent)

drift

velocity

and diffusion

coefficient,

and thus make the identifications

anticipated

in section

I, Equations (1.2)-(1.4).

We refer _to the

long-time

limit in

(2. II)

_as the _«

steady

state », and _we then _see that the

eigenvector #°)(0)

associated with the

leading eigenvalue o(0)

at q = 0 is

proportional

to the

steady-state

distribution _among the internal

configurations

_y ^of ^the

drifting

and

diffusing

chains. Its

components f)°)(0)

_may then be taken to be

positive

real numbers. We remarked above that the components

g)°)(0)

of the associated left

eigenvector

_are

independent

of _y. If

the _common value of these

components

is chosen to be

I,

and if _we continue to

adopt

the

normalization

(2.7),

then the

steady-state

distribution

f)°)(0)

would be normalized to

unity, g)°)(0)

m I

,

£ f)°)(0)

=

(2.13)

y

When the extemal field

vanishes,

so that B

=

I,

the

probability

_per unit time of any transition and its inverse _are

equal.

The

steady-state

distribution _among internal _states _y is then the

equilibrium

distribution in which all _states _are

equally populated.

With the

normalization

(2.13),

and

recalling

that there _are

3'~~

_internal _states,

lim

f)°)(0)

w

1/3'~

,

(2.14)

B _~ l

independently

of _y. In this _same E=0 limit the

eigenvalues A~(0)

at

q=0

for

n # 0 are all

negative

real numbers.

They

_are the

negatives

of the relaxation _rates

(reciprocal

relaxation

times)

in the modes _n

= 1,

2,..., 3'~

l

by

^which the distribution _among internal states relaxes _to the

equilibrium

distribution

(2.14)

when there is _no extemal field.

When E # 0

(B

# I

),

the

f)°)(0)

do

depend

on y ; the

steady-state

distribution is _not _a uniform _one. Not

only

is such _a

steady-state

distribution different from the uniform

equilibrium distribution,

but it is also

partly

maintained

by cyclic

_processes,

which,

at

equilibrium,

would be ruled out

by

detailed balance. In

figure

2a is _an

example

with N

=

3,

in which three of the nine

configurations ~labeled

_y

=

0,

1, ^and

3, following

the nomenclature _we shall introduce in Sect.

4)

_are

shown,

with _arrows

I

_and

( indicating

the

moves allowed

by

the rules of section I. If eE

~

0,

_say, _so that B

~ l

by (I.I),

the transitions y = 3

- - 0

- 3 _are all _« downhill _»

(transition

rate wB

~

w)

while _y

=

0

- - 3

- 0 _are

all _«

uphill

_»

(transition

rate wB~ _<

w),

_as ^shown

schematically

ⁱⁿ

figure

2b. That each of these

configurations

is both

uphill

and downhill from itself is _a

paradoxical

circumstance that

(8)

~ _~~~ @

§q~ _Y~' ^wB~' ~ ~ ^wi'

~~~§ ^Y~3 f ~@

wB"

(a) (b)

Fig.

2.

(a)

Three of the nine

configurations

of _a repton chain with N

= 3, and

(b)

the transition rates

connecting

those

configurations

in

pairs.

cannot _occur at

equilibrium

but _can and does _occur in the

steady

state. For the

steady-state

distribution to be an

equilibrium

_one the

product

of the rates of the transitions

going

clockwise around _any

cycle

must

equal

the

product

of the counterclockwise rates,

by

detailed

balance. In the present

example, by

contrast, those two

products

_are

w~B~

_and

w~B~~,

which _are

equal only

when E

= 0

(B

=

I

).

When

llji)

in

(2.2)

and

(2.4)

is wB

(or wB~~)

then

lljf~)

^is

^wB~~ ^(or ^wB)

^and

lk~i)

₌

lf~()

₌ _11~7~) ₌

W)))

₌

0,

whereas when the transitions y ++

y'

do not involve the

marker repton ^then

lljj

= W)T~) =

ll~)

⁾

=

lf~f~)

=

0 while

ll~(I

=

wB

(or

wB~ _~) and

ll~i)

= wB

(or wB). Thus,

_except when B

=

I,

the matrix

A(q ),

the elements of which _are

given

in

(2.4),

is not Hermitian for

general

_q _nor

symmetric

at q = 0.

Because,

_as _we _saw, there _are unbalanced

cycles

in the transitions among

configurations

when B #

I,

the matrix

A(0)

is not

only

not

symmetric

when B # I but is then not even

symmetrizable I-e-,

there is

no vector f

[and,

in

particular, f1°)(0)

is not such _a

vector]

such that

f~A~,~(0)m

f~,A~~,(0);

which is another way of

stating

the failure of detailed balance. Since

A(0)

for _{B # I} is neither

symmetric

nor

symmetrizable

_any of the

eigenvalues A~(0)

_may be

complex [except

for

Ao(0),

which

by (2.10)

is

always 0].

When

~(0)

is

complex

the relaxation to the

steady

state via mode _n is

oscillatory

in time

although

the oscillations _are

exponentially damped

since the real part ^of

A~(0)

remains

negative. Complex A~(0)

for B # I _are illustrated for N

= 3 in section 4.

The _sum

lljj _{)+ W)(I}

₊

_W)))

is the transition

probability

_per unit time of the transition

y'

- y

irrespective

of whether that transition _occurs

by

_move of the marker repton or

by

move

of any of the N I non-marker

reptons.

^That sum is wB if the transition

y'

- y occurs when

some repton moves

I,

it is wB if the transition _occurs when _some repton moves

(,

and it is 0 if _no

single

allowed _move

by

_any

repton

can

bring

about that transition.

By (2.4)

the matrix

A(0)

is then

independent

of the

arbitrary

choice of

marker,

and _so then also _are all the

eigenvalues A~(0).

But that is not true of

A(q)

when _q # 0 : some of the

off-diagonal

elements _are wB~ _~, _some _are wB e~~, some are

wB,

and _some _are wB _e

~'«,

and which _ones

are in which

positions

in the matrix

depends

_on the

arbitrary

choice of marker.

Nevertheless,

since that choice is

arbitrary

_no

physical quantity

_can

depend

_on it. In

particular,

the drift

velocity V(eE),

hence also the diffusion coefficient

D,

must be

independent

of that choice.

As _we remarked in section

I,

_we shall later

(Sect. 4) verify

that

independence

for

N

=

3,

but elsewhere

(starting

ⁱⁿ Sect.

3)

_we

just

_assume

it,

_even

though

it has not yet ^been proven

mathematically

for N

~ 3.

(9)

3. Formulas for V and D.

It is apparent ^from the

expansion (1.2)

and from the identification of the coefficient of

iqla

in that

expansion

_as the drift

velocity

V

(Sect. 2)

that V for

general

eE may be obtained from

A(q) by

first-order

perturbation theory

with

A(0)

treated _as the

unperturbed

matrix.

There _are two distinct routes to the diffusion coefficient D. One is via

(1.2)-(1.4),

from which

it is _seen that D may be obtained from

A(q) (at

E=

0) by

first- and second-order

perturbation theory [terms

O

(q)

in the

perturbation

to be treated in second order and those

O(q~)

_to _first

_order],

_with

_{A(0) (at}

_E

=

0)

the

unperturbed

matrix. The second _route _to D is via

(1.5),

_once

V(eE)

has first been found

by

first-order

perturbation theory

_as described

above.

By

this route,

then, differentiating V(eE)

to obtain

V'(0)

is

equivalent

to

going

to the next order of

perturbation theory

at E

= 0.

From

(2.4),

the

y'y

element of the

perturbation

is

Ay~

y(q) _Ay~y(0)

" H§~y~ Wj~y~)

iq

_(11~~7y~ ₊

Wjfy~) q~

₊ O

(q~) (3.1)

(The configuration changes

if _any

repton

_moves,

including

the marker

repton,

so

necessarily lJ§p

⁾ ₌

W)(

⁾ ₌ ⁰ ^for any

y.)

Then with the

right eigenvector #°)(0)

normalized

by (2.13),

and

recalling (2.10), Ao(q)

"

iq z z ( _Hj~y~ _Wjt~~) /j°~(0)

₊ O

(q~) (3.2j

We also recall that if the transition _y

-

y'

results from the allowed _move

( by

the marker repton then

lljT~)

= wB and

lJjt~)

₌ ⁰ ; if that transition results from the allowed _move

I by

the marker repton then

lJ~t~)

= wB and

lljT~)

=

0 and

llj,j)

=

lljf~)

₌ ⁰ ^otherwise.

Therefore

Ao(q)

=

iqw B~

~if)°~(o)

B

~( fj°)(o)j

⁺ ^o

^(q2), ^(3.3)

<->

where

£

means sum over all

configurations

_y from which _a _move

I by

the marker repton ^is

~ i+

allowed and

£

means sum over all

configurations

_y from which _a _move

I by

^the marker

y

repton ^is allowed. Then from

(1.2),

Vlaw

₌ B

£ f)°~(0)

B~

jj f)°)(0) (3.4)

The formula

(3.4)

for the drift

velocity

V is exact for all

E,

but it still makes reference _to _an

arbitrarily

chosen marker

repton.

^We

know, however,

^that it is

really independent

of that

choice

(as

we illustrate for N =3 in Sect.

4). Hence,

_we _may choose each

repton

I

=

I,

...,

N in turn to be the marker repton, ^write

(3.4)

for each such

choice,

add all the

results,

and then divide

by N,

thus

obtaining

for

Vlaw

a formula that _no

longer

refers to any

marker _:

Viaw

= N

z (Br~

B

s~) /j°~(0)

,

(3.5)

(10)

where r~ ^is the total number of

I

_arrows _on all of the

reptons

^of the chain in

configuration

_y

and s~ ^is ^the ^number ^of

(

arrows.

(For example,

in the

configuration

_y illustrated for

N

= 15 in

Fig. I,

_r~

=

5 and s~ =

3.)

Equation (3.5)

for

Vlaw

is exact for the model. It

yields

the drift

velocity

V in fields E of

arbitrary strength.

It

requires knowing

the

fight eigenvector #°~(0)

associated with the

eigenvalue Ao(0)

= 0 of the transition-rate matrix

A(0).

From

(2.4) (with

the known values of W)7~),

etc.), (2.5)

and

(2.10),

and with the _same r~, s~ ^notation ^as ⁱⁿ

(3.5),

the

equations

for the components

f)°~(0)

_are

B

£ f)?~(0)

+ B

£ fj?~(0)

₌

(Br~

+ B ^'

s~) fj°~(0

,

(3.6)

~Y.

I)

I

<y'£)1

where the notation is meant to indicate that the first _sum is _over all

configurations y'

from which the

configuration

_y _may be achieved

by

an allowed move

I by

_any repton ^of the

chain,

and likewise for the second _sum with

(

_instead _of

I.

The condition _on the first _sum could

equally

well have been written _y-

y'(, _meaning

a sum over all

configurations y'

achievable from the

configuration

_y

by

_an allowed _move

( by

_any repton ^{of the} chain. There

are therefore _as _many terms in that _sum _as there _are

(

_arrows _on all the

reptons

^{of the} ^chain ⁱⁿ

configuration

_y _;

viz.,

_s~ and there _are likewise r~ ^terms ⁱⁿ ^the second _sum.

Note, then,

that

on the left of

(3.6)

the factor B

multiplies

_s~ terms and the factor

B~~ multiplies

r~ terms, whereas _on the

right

it is the

opposite.

For _use in

(3.5),

the solution

f)°~(0)

of

(3.6)

is to be normalized _as in

(2.13).

The first route to the diffusion coefficient D _as described earlier is via

(1.2)-(1.4).

This

requires

the terms O

(q~)

in

(3.2),

at E

=

0.

They

_are obtained

by taking

account of the terms

O(q)

and

O(q~)

_in

_(3.I)

_in _second- _and first-order

perturbation theory, respectively.

The

O(q)

term in

Ao(q) (Eq. (3.2))

vanishes _at E

= 0 because

f)°)(0)

is

independent

of _y when E

=

0, by (2.14),

and because

£ £

_W)T~)

=

£ £

_W)f~~,

by

symmetry.

(Thus,

the drift

velocity

vanishes _at E

=

0,

_as it

must.)

Taking

account of

(2.13)

and

(2.14),

_we _see that the contribution _to the terms O

(q~)

in

Ao(q)

that _come from the _terms

O(q~)

in

(3.I) by

first-order

perturbation theory

is

~^~

i~- I ~'Y~

~

~~~~

_~^~

$~ _'

Y Y

where M is the number of distinct

configurations

_y in which the marker

repton

^carries an

I

arrow

(=

the number in which it carries _a

(

_arrow,

by symmetry).

The contribution _to the

terms

O(q~)

in

Ao(q)

that _come from the terms

O(q )

ⁱⁿ

(3.I) by

second-order

perturbation

theory

is

~2 ~2

~

z j-

A

~(0)i~ iv f~~~(0)i~ (at

E

=

0

,

(3.8)

3 n<#o>

where the components v~ of the _vector _v _are

v~ ⁼ ^I if in

configuration

_y the marker

repton

carries _an

I

_arrow but not _a

(

_arrow,

v~ = I if it carries _a

(

_arrow _but _not

an

I

_arrow, and

v = 0 otherwise

(either

_no _arrow _or both _an

I

and _a

( arrow)

_; and with the

eigenvectors

/~(0)

_for

n # 0 normalized

by

j#~~(0)

^~

=

l

(n

# 0

) (3.9)

(11)

[This

is in contrast with

(2.13),

which is the _more convenient normalization for _n

=

0,

when

the

components f)°)(0)

_are

proportional

to the

steady-state populations

and _may thus be taken real and

positive.

Since E

= 0

here,

the matrix

A(0)

is

symmetric,

_so _except for

arbitrary

normalization _constants the left

eigenvectors g~~)(0)

_are

just

the

right eigenvectors

#~)(0) transposed.

For convenience _we take

g)~~(0)

=

f)~~(0) here,

^for n #

0,

and then

(2.7)

becomes

(3.9).J

The _sum of

(3.7)

and

(3.8)

is the

O(q~)

_term in

(3.2)

at E

= 0.

Therefore, by (1.2)-(1.4),

the dimensionless diffusion coefficient

Dla~

_w is

f

=

/ (M

^w

£ [-

A

~(0) J-

_[v

~~)(0)J~)

,

(3.10)

a w 3

~,~o~ ^E=o

with M and _v defined beneath

(3.7)

and

(3.8), respectively,

and with

#~)(0)

normalized

by (3.9). By

_a convention for

numbering

the states y that _we shall

adopt

in the next

section,

the

components

_{v~ of} v will be odd in _y while the

components f)~~(0)

^{will be} ^either even or odd

depending

_on n. The summation _over _n in

(3.10)

_may then be restricted to those _n for which

f)~)(0)

is odd in _y. There is still in

(3.10)

_a

seeming dependence

_on the choice of marker repton

(via

_v and

M~,

but _as _we shall _see

explicitly

for N

=

3 in the next

section,

and _as _we know from

general principles

for all

N,

that

dependence

is

only

_apparent, not real.

The second route to D is via

(3.5)

and

(1.5). Suppose

for small E

(I.e.,

for B _near

I)

the

fj°~(0),

which _at E

=

0 _are

given by (2.14),

have the

expansion

f)°~(0)

₌

/ _II

+

ay(B

I

)

₊ O

I(B

I

)~l (3.I1)

Then from

(3.5), (1.5)

and

(I.I),

~~~~

_~

~

~~2~~N

-1

I

~~Y ~

~Y ~~Y ~Y~ ~Y~

(~.l~)

Y

floote

from the definition of s~ ^and r~ ^beneath

(3.5)

that

£ _r~

=

£

_s~,

by symmetry.]

It follows from

(3.6)

and

(3.I I),

and

by recalling

that there _are s~ ^terms ⁱⁿ ^the ^first ^sum ⁱⁿ

(3.6)

and r~ ^terms ⁱⁿ ^the

second,

that the

a~ satisfy

£

_a~,

= 2

(r~ s~)

+

(r~

+

s~) a~ (3.13)

y>

<y> -y>

where the notation

(y'++ y)

_means that the _sum is _over all

configurations y'

that _may be obtained from _or that

yield

the

configuration

_y

by

_any

single

move

I

or

( by

any one repton ^of the chain. The number of terms in that _sum is _r~ ₊

s~,

^the ^total ^number ^of

I

_and

(

arrows on

the

repton

^chain ⁱⁿ

configuration

_y. Therefore if

a~

îs âny ^solution ôf

(3.13)

_so is q~ ⁺

A,

where A is any

quantity independent

of _y. This

non-uniqueness

of the solution of the

inhomogeneous

linear

equations (3.13)

is _a result of the

vanishing

of the determinant of coefficients of the a~. ^To ^determine ^the a~

uniquely requires adding

_a condition to

(3.13),

_as

follows.

The

configurations

_y _occur in

mirror-image pairs,

_y and

f,

reflected in what in

figure

^I

would be _a horizontal

mirror,

_so that

I

_arrows _in

y become

(

arrows in

fl

and vice _versa. The

configuration

in which all

reptons

are at the _same _x, _so that _xi

= x~ = = x~, is its _own

image;

in the notation to be

adopted

in the next section that

configuration

is called y = 0. Recall that

f)°)(0)

with the normalization

(2.13)

is the

steady-state

distribution in the

(12)

field E. Then

by

symmetry,

f)°)(0)

in the field E must be the _same _as

f~°~(0)

in the field

E;

_so from

(I.I)

and

(3.ll),

a~

=

ay,

ao = 0.

(3.14)

With these conditions the

3'~ equations (3.13)

_may be reduced to

(3'~

l

equations

2

for that smaller number of

a~'s.

^This ^is

again

_a set of

inhomogeneous

linear

equations,

but

now with _a

non-vanishing

determinant of coefficients. The

unique

solution of

these, together

with

(3.14), yield

all

3'~~

_of _the

a~.

Equation (3.10)

with

(3.9),

and

equation (3.12)

with

(3.13)

and

(3.14),

_are two altemative but

equivalent

formulas for the diffusion coefficient D. Their

equivalence

is asserted

by

the

Nemst-Einstein relation

(1.5).

In the next section _we evaluate D

by

both formulas for N

=

1, 2,

and 3 and

verify

their

equivalence [as

well _as

verifying

for N

=

3 that

(3.10)

is

independent

of the choice of

marker],

and _we evaluate D

by

the second formula for N

=

4 and 5.

4. N ₌

t, 2, 3, 4,

S.

Here _we

illustrate,

for small

N,

the

application

of the

principles

and formulas of sections 1-3.

N

= i.

Here there is

only

_one intemal state y, call it _y

=

0,

and then

f)°~(0)

= 1,

by (2.13),

and ao =

0, by (3.14).

There _are _no _terms in the _sum _over _n in

(3.10).

There is _now

only

_one

repton

ⁱⁿ ^the chain of

figure

I and it carries both _an

I

_and _a

(

_arrow. _Then _ro=

so = I in

(3.5)

and

(3.12),

and M

= I in

(3.10).

Then from

(3.5),

Vlaw

= B B

(4.I)

and from either

(3.10)

_or

(3.12)

Dla~

_w

=

(4.2)

We have thus verified the

equivalence

of

(3.10)

and

(3.12) [I.e.,

the Nemst-Einstein relation

(1.5)]

for N= I. The drift

velocity (4.I)

_as _a function of the dimensionless field

eEa/kT [=

2 In

B, by (I.I)]

is shown _as the _curve labeled N

= I in

figure

3.

N

= 2.

There _are _now

3'~~~

= 3

configurations

_y. We show them with their attached _arrows in

figure 4,

where

they

_are numbered _y

=

-1, 0,

1. This follows _a

numbering

convention _we

adopt

hereafter. The

configuration

in which all reptons are at the _same _x _we call y =

0,

and

configurations

that _are mirror

images by

reflection in that _one,

I-e-, by

reflection in

a horizontal

mirror,

_are numbered _y and _y. Such _a

mirror-image pair

_was called _y and

f

ⁱⁿ ^the text above

(3.14).

Whichever of the two

reptons

^is ^chosen ^to ^be ^the

marker,

the matrix

A(q)

of

(2.4),

_at

q =

0,

is

I 0

A

(0)

= w

(B

₊ B ~) l 2

(4.3)

0

(13)

o.7 ~~' N=2

O.6

O.5 Vlaw

O.4

O.3

o. 2

O.

-I 2 3 4 5 6 7 8 9 IO

-°.' eEa/kT

Fig.

3.- Drift

velocity

V in units of _aw _as _a function of the field E in units of

kTlea

for N

= 1, 2, 3, and 4. V is _an odd function of eE.

~-i

~

o

f~

Fig.

4. The three

configurations

_y

= 1, 0, for N

=

2.

with

eigenvalues Ao(o)

=

o,

i

(o)

= w

(B

₊ B

i), _~(o)

=

3 _w

(B

₊ B

-1) (4.4)

and

corresponding eigenvectors

1/3 Ill Ill

f1°)(0)

=

1/3j

,

f~~~(0)

=

0

,

f~~~(0)

=

2/ Qi _(4.5)

1/3 1/ / _1/ Qi

Here

f1°~(0)

is normalized _as in

(2.13)

while

fl~)(0)

and

fl~~(0)

are normalized _as in

(3.9).

In

figure

4 _we _see _{r_}

j ⁼ s_ _j = rt = sj =

I,

_ro

= so =

2,

M

=

2

(with

either choice of marker

repton),

and _v

= ±

[1, 0,

J, the

(inconsequential)

choice of

sign depending

on which of the two reptons ^is ^chosen as marker.

(14)

From

(3.5)

_we then find the drift

velocity, Vlaw

=

~

(B

B

~). (4.6)

This is

plotted

_as the _curve N

=

2 in

figure

3.

The

only

odd

eigenvector

in

(4.5)

is

fl~~(0),

_so in the _sum in

(3.10)

the

only non-vanishing

term is _n

=

I. We then calculate

Dla~

w =

1/3 (4.7)

Either from

(3.13), (3.14),

and

figure 4,

_or from

(3.ll)

and

(4.5),

_we see that a_j =

ao = at =

0 here _so from

(3.12)

we calculate

Dla~w

=

1/3,

in

agreement

^with

(4.7), again verifying

the Nernst-Einstein law for this model.

N = 3.

The nine

configurations

_y

=

4,

,

4 of the repton ^chain ^with N

=

3 _are

pictured

in

figure

5.

The matrix

A(q)

of

(2.4),

at q =

0,

is

(B+B~')

B 0 B~' _o ₀ ₀ ₀ ₀

B~'

-2(B+B~')

B B B~' ₀ _o 0 0

0 B~' _-28 0 0 B~' ₀ ₀ 0

B B~' ₀

-2(B+B~')

_B ₀ B~' ₀ 0

A(0)

= W 0 B 0 B~^' 2(B ₊ B~ ') B 0 B~ ^' 0

0 0 B 0 B~' -2(B+B~') ₀ _B B~'

0 0 0 B 0 0 -28~' _B ₀

0 0 0 0 B B~' B~'

-2(B+B~')

_B

0 0 0 0 0 B 0 B~'

(B+B~')

(4.8)

With the index _y _as in

figure 5,

in

(4.8)

it _runs from 4 _to 4

reading

left to

right

and top to bottom.

The

leading eigenvalue

of

A(0),

and the associated

right eigenvector,

the latter normalized

as in

(2.13),

_are

(B+B~~)~

~2

~ ~~

~-2 1+B+28

28~+1+B~~

Ao(0)

=

0, f~°~(0

= ~ ~ ~ ~

(B

₊ B )~

(4.9)

2(B

+58 +6+58~ +B~

~~~

_~~

~_~

28~+B~+1 28~+1+B~~

(B+B~~)~

The

eight remaining eigenvalues,

evaluated at E

=

0

(B

=

I

),

_are, in order of