Experiments have been carried out in the earth's gravitational field with a =

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A METHOD FOR SEPARATING ACCORDING TO MASS A MIXTURE OF MACROMOLECULES OR SMALL PARTICLES SUSPENDED IN A FLUID,

I. THEOR Y

BY HOWARD C. BERG*AND EDWARD M. PURCELL LYMANLABORATORYOFPHYSICS, HARVARD UNIVERSITY

Communicated July 20, 1967

Wehave developed a method for separating a mixture of macromolecules or small particles suspended in a fluid. The separation is according to mass, or more pre-cisely, according to what we shall call the effective mass, the mass of a particle minus the mass of the fluid it displaces. We expose the fluid to a steady acceleration, allowing the suspended particles to distribute themselves in equilibrium "atmos-pheres" whose scale heights,

h,,

depend on the effective masses, me, through the familiar relation

meahs = kT. (1)

Here kisBoltzmann's constant, Tis the absolute temperature, and a is the accelera-tion. Experimentshave been carried out in the earth's gravitational field with a = g, the acceleration of gravity, and in centrifugal fields with values of a as large as

12,000g.1 The particles are suspended initially at one end of a layer of fluid which islatermade to flowinadirection perpendicular to the accelerationfield. The flow velocity variesacrossthe thickness of the layer, so that a particle is transported at aratewhich depends on its height in its atmosphere. Theaveragevelocity of par-ticles of a given effective mass depends on the scale height of their distribution. That alone does notsuffice to separate particles ofagiven effective mass, foratany instant particles of the same massarebornealong at different velocities, since they arefound at different heights in their own atmosphere. However, ifthe time for diffusion through anatmospheric height is very short compared to theduration of the experiment, every particle samples the fluid velocity distribution within its atmosphere many times. The dispersion in the total distances of travel is then small forparticles of the same effective mass, anda separation of particlesof dif-ferent effective mass becomesfeasible.

The same kind of statistical averaging is exploited in all forms of partition

chromatography. The transfer between two phases is, for a single molecule, a randomevent. Itisonly because the transferoccurssomanytimes in thehistory

of eachmolecule thatwe find in paper chromatography, for example, well-defined spots rather than superposed streaks extending from the origin. In this respect ourmethod could be viewed as a form ofchromatography inwhich there is a con-tinuum of carrier velocities through the fluid, so that the average speed of a sus-pended particle dependson its relative affinity for the top or bottom of thefluid, anaffinity determined by thebarometric law.

The Mean Distance of Travel. To describe the process more

precisely,

let y be the coordinate in the direction opposite to that of the

acceleration, ranging

from y = 0 atthebottom of thefluidtoy = hoatthe top. Weassumethat the accelera-tion isnearly constant over thedepth of the fluid, that interactions between par-ticles can be neglected, that adsorption does not occur at the boundaries of the

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PHYSICS: BERG AND PURCELL

fluid, and that the size of a particle is negligible compared to its scale height.

Particlesofeffective massme are distributed in an equilibrium atmosphere of density proportional to

exp(-y/h,).

Theirmean height, y, is

Y =

h,

-

ho/[exp(ho/h)

- 1].

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Forscale heights small compared toho, y h,; for scaleheights large compared to

ho, y

ho/2.

The fluid is made to flow in the x-direction, perpendicular to y, with a velocity profilevz(y),which we assume to be independent of x and the time t. Examples are theparabolic profiles associated with viscous flow with a fixed boundary at y = 0

andafree boundary aty = ho = ha, or withafixed boundary at y = 0 and a fixed

boundaryaty = ho = 2ha:

vX(y)

=

3va(y/ha

-

y2/2ha2),

(3)

where Va is the averagevelocityof thefluid.2 As a result of the flow, particles of scale height hsaretransported in the x-direction at a mean velocity vgiven by

gho

= J

v(y)

exp

(-y/h,)dY/J

exp

(-y/h8)dy.

(4)

If the particles are clustered about x = 0 at t =

0,

thentheir mean x-coordinate at timetis

x= vt. (5)

The Dispersion in the Total Distances of Travel.-Consider the behavior of an ensemble ofidentical, independent particles. The velocity vO atwhichaparticle is

transportedattime t2, which dependsonits heightatthatinstant, issubstantially

independent of the velocity at

ti,

provided that the interval t2-

tj

is long enough to permit diffusion over a distance about equal to

y.

We will define a "diffusion

time" T by the relation

T = y2/2D, (6)

where D is the diffusionconstantfor the

particle

in thefluid. This definitionmakes T exactlyequaltothe time forone-dimensional diffusionthroughthe rms distance

y.

Weassumethat Tis much less than theduration of flowt. In thewhole timet each particle will have experiencedsomething like t/r independent samples of the

velocityfield

vx(y).

We may expect this to reduce the variation fromparticle to

particleindistancetravelled-whichotherwisewouldbeaboutaslargeasx

itself-by afactorroughly equalto

(T/t)11'.

At thesametime, an independent contribu-tiontothe variation in finalx-positionswill havearisensimplyfrom diffusion in the x-direction.

We will call the x-distribution ofparticles of a

given

kind a band and define as

the mean-square bandwidth thequantity

(x

-_ )2,wherexis the

position

ofa par-ticleattime t,and the average, denotedbythebar,isovertheensemble of

particles.

From what has beensaid, weexpectthe mean-squareband widthtobe

given

by

a

relation ofthe form

(X-X)2 = f2X2T/t + 2Dt,

Voi,.'587 1967 96.3

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where f is a number of order unity whosevalue dependson the form of thevelocity

profile and the ratio of scale height to fluid depth. An accurate treatment, to be describedbelow, will confirm this and will yield values of the factor f for particular cases.

TheResolving Power.-For high resolution, we want the root-mean-square band width (x

-)2

/ to besmall compared to the banddisplacement x. From equa-tion (7) the ratio of these quantities is

(X ,

*)

// = (flT/t + 2Dt/ 2)'/2 (8)

Inpractice, x is limited by the length of the apparatus. If all the quantities except the flow time t are fixed, the expression in equation (8) has its minimum value when the two terms are equal, or, usingequation (6), when

t =fxy/2D. (9)

For this most favorable condition,

(X -

/)2'2/g

-

(2fy/j)1/2

(10)

The ability to separate particles of different effective mass depends not only on the relative band width but on the variation of band displacement with effective mass, that is, on (me/2)62/bme. If the scaleheightissmall comparedto thedepth ofthefluid, and if thevelocity profile

vx(y)

hasthe constant gradient to be expected nearthe fixed boundary aty = 0, the 1 per centincrease in

h,

associatedwith a 1 per cent decrease in me will causea 1 percentincrease inv. In this case, the one with whichweshall bemostconcerned,

(re/t)b.i/bme

= -1. Theresolvingpower for effectivemassesisthenwhollydeterminedby therelative band width.

Equation (10) shows that a small ratio of scale height to apparatus length is desirable. Inpractice, useful resolutionscanbeobtained withflow times consider-ably smaller than theoptimum specified by equation (9). In this case the second termin equation (8), thesimplediffusion term, isrelatively unimportant.

Calculationofthe Width-Factorf.-To analyzethe transport processwerepresent it by a discrete random walk with draft. This is easy to visualize and leads di-rectly to formulas suitable for machine calculation. In the

y-direction

we mark off Mdiscreteequidistant levels,the lowestaty = ho/2M, thehighest aty =

he-ho/2M1.

The distances betweenadjacentlevels is thus

s = ho/M. (11)

M issufficientlylarge thats<< y. With each level k = 1, 2,.. ., Mweassociate a

local drift velocity

Uk =

vx(ks

-

s/2).

(12)

Time is divided into N equal intervals of duration t, << r. A

particle

atx in the kth level moves asfollows in the nth interval. Its coordinatex is increased by

in =

Ukts

+ as, (13)

where a is a random variable with

equiprobable

values ± 1. The first term ex-pressesthe drift duetothe fluid

velocity;

the

second,

the diffusionin the

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tion. At the same time, diffusion in the

y-direction

transfers the particle to level k + 1 or to level k - 1 with probabilities

W+

and

W-,

respectively, with

W+

+ W- = 1. Theseprobabilities must be biased in deference to the acceleration field:

W+IW-

= r = exp (-meas/kT). (14)

Aspecial rule is needed for the bottom and top levels. A particle at k = 1 remains therewith probability W-; a particle at k = M remains there with probability

W+.

Exceptfor the discretenessof the steps, this model accurately represents our system, provided that the step length s and the interval

t,

are related to the actual diffusionconstant D by

t= s2/2D. (15)

LetPk, k = 1, 2,..., M, be theprobability thatagiven particle will be found in the kth level. This isjust the Boltzmann weighting,

M

Pk= (1-r)rkl/(1 - rM),

E

= 1. (16)

k=1

Let

Qklm,

k = 1,2,.. .,M, 1 = 1, 2,..., M, m = 1, 2,..., co (orm -1, -2, ....

- co),bethejoint probability that a particle now in level k occupies level 1, m steps later(or-m steps earlier). Under our assumptions the

Qii,,m

arecompletely deter-minedby thePk, and

Qklm

=

Qklm..

The Pk describe the distribution which is stationary against diffusion in the

y-direction,

as wehave specified it. Clearly

liM (Qklm -

PI)

= 0. (17)

Consider now an ensembleof identicalparticles starting atx = 0. After along timet =

Nt,

aparticle will have reached someposition

N

x =

E

in (18)

n=1

Averaging over the independent histories of theparticles of the ensemble, we have

x = Nt = at, (19)

where aisgivenby

M

U= PkUk- (20)

k=1

In order to find (x-)2, which is the same asx2 -x2, weneed to computex2. Now

N N

x2 = Adn2 + E E tt, (21)

n=1 j=1 i~j

The average overthe ensemble of the firstsum is

VOL.58, 1967 865

t:2 U2t2

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PHYSICS: BERG AND PURCELL with N u= E

PkUk

*

(23)

k=1 N _

The average of the second sum of equation (21) is E

E (>t,.

If aparticleis

j=1 ipj

in the kth level during the jth interval, it will be in the lth level during the [j + (i - j)]th or ith interval with probability Qk

,,-j.

Therefore,

M M

tote = tv2 +d PkUk E Qk. -P)-jU(

k=1 1=1

M M M

= t's2 E Pk~lk E(Qkl~i-j PO)8I + EPluI

k=1 =11

M M

=

42aU2

+ ts2 A) PkUk E (Qkl.i-j -

POlU

I (24)

k=l 1el

The first term ofequation (24) contributes to the sum an amount N(N

-)ts2a2

If jdoes not lie toonearthebeginning or the end of the sequence of N intervals,we mayleti-j runfrom 1 to o and from -1 to -oo for each j, in whichcasethe

second termofequation (24) contributes anamount

M OD M

2NtS2

Ad PkUk E E

(Qklm

-

PO)uI

k=1 m=1 1=1

Theerrorintroduced by this approximation is small provided that thenumber of intervals N is very large compared to the number of steps required for Qkl,m to settle down to itsasymptotic limit. Westipulate that N is large, i.e., that there is anmo << N such that the quantities (Qkl,m-

PI)

areall negligible for m > mo. This is practically equivalent to the condition t/T»> 1, a condition which will be met in allthecasesofinterest tous.

Putting this all together, with the help of equations (19) and (22), we find

-t2 - 2t2 M as M

X2 - x2 = NS2 + N

(U2

-

a2)

+ k

PkUk

m=

(Qk

rn -

P1)uI.

(25)

The termNs2canbewritten, using the relation (15), as2DNtI orsimply 2Dt. Itis the secondterminequation(7),which arises fromsimple diffusioninthex-direction and needs no further comment. The second term in equation (25) vanishes for N-a 0. It is an artifact of the discrete random walk model and should be dis-carded. Thethirdtermdoesnot vanish asN increases, for the number of signifi-canttermsinthesumovermincreasesastheinterval

t,

and steplengths are short-ened with increasing N. In fact, the number of intervals required for the joint probabilitiesQk

Im

torelax most of the waytotheirasymptotic valuesP1isabout thenumber of intervals neededtotravel y by arandomwalk, that is, (Y/S)2 inter-vals. Hence thethirdterminequation (25)will be of order ofmagnitude

(t2/N)a2-(y2/S2) which, sinceNs2 = 2Dt, y2 = 2DT, and x = at, isindeed the same as the

expression C2r/t which appears in equation (7). Comparing equation (7) and

equation (25) weobtain thefollowingformula forf2:

f2

= 2 (-) >3 Pk (u) >3 (Qm -P

(I))

(26) PROC. N. A. S. 866

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We have computed f for various relative scale heights for the single and double fixed-boundary parabolic velocity profiles of equation (3). An arbitrary profile could be handled just as easily. Thearray

Qkl,m,

1 =

1,.

. .

,M,

isgenerated from the array Qklml- by applying the transition probabilities

W+

and W_. Here, too, a more general behavior-for instance, a temporary adsorption of the particle at thewall-could be accommodated by a minor change in the program. The elemen-tary step s wastaken as

0.2y.

In thecase of very small scale-heights, h,<< ho, the ladder of levels was arbitrarily terminated at M = 40, corresponding to approxi-mately eight scale-heights. The sum over m was extended until convergence to betterthan0.001 in the value of f seemed to be achieved.

The results of these calculations are given in Figure 1. For convenience ofscale,

1.4;I 1.3 I-WALL 1.2 2L 2WL 1.c -va 0.7-a s p0.6 .01 .02 .04.06 .10 .20 .40 60 1.0 2.0 4.06.0 10 h /ha

FIG. 1.-Thewidth-factorf, plotted asf/2, the ratio of the average particle height to the scaleheight,5/h8,andthe ratio of the average particle velocity tothe average fluid velocity,0/V.,as afunction of the relative scale heighth./h.. The "1-wall" curves are for the single fixed-boundaryparabolic profile, equation (3), withha = h0,and the "2-wall" curves are-for the doublefixed-boundary parabolic proffle, equation (3), withha = ho/2, wherehiis thefluid depth.

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theordinate is f/2 rather than f, and the abscissa, ha/ha, is logarithmic. The "1-wall" curves refer to the single fixed-boundary profile of equation (3), for which ha = ho. The "2-wall" curves refer to the double fixed-boundary profile, for which ha = h0/2. Weinclude the corresponding values of the ratio of the average particle height to the scale height, y/h8, and the average particle velocity to the average fluid velocity,

tD/v,,

which were derived from the analytical expressions, equations (2) and (4).

TheShape of the Bands.-We conclude that if the flow is stopped at a time t >>

T,

the ensemble of identical particles will be distributed in a band around x = Dt,

withvariance (x -x)2 given by equation (7). Moreover, the distribution will be normal. This is assured for the discrete random walk model by a theorem about finiteMarkov chains.3 With each of the M levels we can associate two states of a finite Markov chain of 2M states, assigning one of these states a displacement Ukts +

s,

theotheradisplacement Ukts- s. The theorem then implies that the sum of thedisplacements, equation (18), obeys the central limit theorem. Thus as t in-creases, theprobabilityF(x)dx of finding a particle in dx at time t will approach

F(x)dx =

exp[-(x

- g)2/2q2]dx (27)

with x = Ntand o2 = fC2,/t + 2Dt. Wehave tacitly assumed that the initial dis-tribution of particles hasawidthwhich is negligiblecompared to a-.

In some experiments the observation is made not by stopping the flow at time t andexaminingthedistribution in x, but by detecting theparticles as soon asthey leave the apparatus at x = L. Theresulting distribution in arrival times is that encountered in the well-known "first-passage" problem for diffusion with drift. IfF(x)dx isgiven by equation (27), then theprobability G(t)dt that a particle first arrives at x = Lwithindtwill be4

G(t)dt = A exp[-(t - t)2/2E2t]dt (28)

with I = L/vand 62 = [a(t)

]2/L-D.

Themean emergencetime is

1,

and the second momentof the distribution G(t) taken about I has exactly the samerelative value asthe secondmomentofF(x) about x. Thereare minor differences inshape; the maximum ofG(t) occurs not at t = Ibutalittleearly, at t I - 362/2.

Summary.-Itispossibleinprincipletoseparatethe components ofamixture of smallparticles suspended in afluid and to measurethe effective massand the dif-fusion constant of each in a single experiment. The individual components form discrete bands. Theposition ofaband dependson theeffective massof its

parti-cles, and its widthdepends aswellon theirdiffusionconstant in the fluid. In the second and third parts of this

report' experiments

will be described which

largely

confirm these expectations. One set has been done in the earth's

gravitational

field with polystyrene latex spheres, and the other has been done in

centrifugal

fields up to12,000gwith the E. colibacteriophageR17.

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VOL. 58, 1967 PHYSICS: BERG AND PURCELL 869 We are grateful to Costas Papaliolios forhis comments on the manuscript. Thework was begunwhile one of us(H.C.B.)wassupportedby a Harvard JuniorFellowship.

* Present address: Department of Biochemistry and Molecular Biology,Harvard University. 1Berg, H. C., and E. M. Purcell, thesePROCEEDINGS,inpress.

2Landau, L. D., and E. M. Lifshitz, FluidMechanics (Reading: Addison-WesleyPublishing Company, 1959),p. 56.

3Feller, W., An Introduction to Probability Theory and Its Applications (New York: JohnWiley and Sons, 1957), 2d ed., p. 374.