905

THE PHYSICAL PROPERTIES OF CYTOPLASM. A STUDY BY
MEANS OF THE MAGNETIC PARTICLE METHOD. PART II.
THEORETICAL TREATMENT

F. H.C. CRICK
Strangeways Research Laboratory, Cambridge

Received February 9, 1950

A. INTRODUCTION

In Part I (1) a method was described for measuring some of the physical properties
of the cytoplasm of chick cells in tissue culture by means of magnetic particles. The
cells were allowed to phagocytose these particles, which were then acted on hy
magnetic fields, their movements being observed simultaneously under high magni-
fication.

In this paper the theoretical basis for the experimental methods used has been set
out. The results are mainly standard pieces of magnetism and hydrodynamics, but
as they are scattered about in the literature it was thought worth while to bring them
all together in one place.

The paper has been written for workers who may wish to use the method them-
selves, or who wish to examine its foundations critically. For those only interested
in the results an extended summary of the theoretical conclusions has already been
given in Part I. An elementary knowledge of magnetism and hydrodynamics is
therefore assumed. There are occasional remarks from a more advanced standpoint,
but they are not crucial to the main results.

The experimental methods have been set out in part I. It will suffice here to state
the general theoretical problems for which we require solutions. There are three
main cases. They are

(1) Twisting: the permanent magnet case.

In this case the magnetic particle is turned into a little permanent magnet by
applying a large magnetic field momentarily. It is subsequently twisted by a much
smaller field applied in a direction roughly perpendicular to its permanent magnetic
moment.

(2) Twisting: the soft iron case.

In this case the material is considered to have no hysteresis. A magnetic field is
applied at a small angle to the length of the particle, which is thus twisted.
506 IP) H.C. Crick

(3) Dragging.

In this case a large ficld, with a large field gradient is applied. The magnetic
particle is magnetically saturated, and after being twisted into line, is dragged by
the field gradient.

We wish to calculate the velocity (or the angular velocity) of the particle in terms
of its size, shape, and magnetic properties, and the physical properties and bounda-
ries of the medium in which the particle is embedded.

We Lackle the problems in the following order. We first show that we can neglect
the effects of inertia, and by elementary arguments find how the forces vary with
scale. We then consider the behaviour of the medium, discussing the effects of the
shape of the particle, of boundaries, and of non-newtonian and elastic behaviour.
Next we give the formulae for the magnetic forces on the particle, and then in sec-
tion F we show how all the factors can be combined to evaluate the velocity (or
angular velocity) of the particle for the three main cases.

l‘inally we give some brief theoretical notes on the production of large field gra-
dients and a note on some comparative numerical values for the stress.

B. GENERAL CONSIDERATIONS
1. Inertia

Inertia will delay the approach to the steady state and will alter the final
velocity distribution. We shall show that both of these effects can be neg-
lecled in our experiments mainly because the particles are so small.

There are two problems that can be considered separately. Firstly, the
inertia of the fluid, secondly the inertia of the particle.

For the inertia of the fluid the retevant characteristic of the motion is the
ratio of the inertia forces to the viscous lorces (Reynold’s number). It is
given by

cow
a

where o = reciprocal of a characteristic time
@ =: density of the fluid
4 = viscosity. of the fluid
a= a characteristic length

(see for example the case of an oscillating sphere, (7), paragraph 354). This
formula applies strictly only to short particies.
If the above parameter is << 1, the inertia of the liquid is negligible. An-
. . . 1 . cow
other use of the parameter is that the value of (*) defined by putting »
equal to unity, gives the order of the time required to approach the steady state,
We shall not be considering particles bigger than 10 @ in diameter, so we

The phystcal properties of cytoplasm. Part IT. 507

may puta = 5 x10°' em. If we take as a lower bound for 7 the value for
water (0.0L poise), since biological fluids can scarcely be less viscous, and

40
This is naturally very much smaller than anything we have measured. We
should note in passing that if the margin were not so great a more exact
treatment would be advisable. Our formula gives the time for the particle

1' 1 :
put @ = 1, we obtain for the upper bound of (‘) the value ~~ milliseconds.

to reach a good fraction of its final velocity, but in certain cases the later
stages of the asymptotic approach to the final velocity may take much Jonger
than the earlier stages.

These results only apply strictly to the case of an infinite fluid. We can
give an argument which suggests that the effect of adding fixed boundaries
will usually be to decrease the time to approach the steady state. Consider
lwo cases: firstly a particle in an infinite fluid, secondly the same particle
with fixed boundaries added to the fluid. Let the forces applied to the part-
icles be such that the same steady velocity is attained in the two cases. We
will assume that as a rough measure of the time to approach equilibrium
we may take the ratio of the kinetic energy of the fluid to the rate of dissi-
pation of energy, both for the steady state. The effect of fixed boundaries
is to increase the resistance and therefore the rate of dissipation of energy.
The boundaries also reduce the amount of fluid in motion and over most of
the volume! decrease the fluid’s velocity. The total kinetic energy is thus
likely to be reduced. Therefore the ratio referred to above will be decreased.

To estimate the effect of the inertia of the particle we consider the case
of the dragging of an iron sphere. The ratio of the inertia to the viscous
forees is

da ap" dp
3S dt
OGaHyav
where p= velocity
eo = density of sphere

. . eg 1 :
It we define the characteristic lime — by the equation
so

dvi op

" ())

* This assumes that the boundaries do not force the flow into a very restricted channel, in
which case the velocity would be increased considerably.
508 Plat. C. Crick

this ratio becomes
200’ a
9 4
which apart from the numerical factor is the same form as before, except
that 9’ is now the density of the iron. It can easily be shown that exactly the
same type of parameter is involved in the case of rotation. Thus the effects
of the inertia of the particle are negligible.

The case for non-newtonian fluids, whose “viscosity” varies with shear
is not quite so clear cut. However the margin in our experiments is so big
that we can simply consider the extreme case where the inner parts of the
liquid move effectively as a solid, and the outer parts as a newtonian liquid.
This is clearly similar to the movements of a body of increased radius in
water. In our experiments the radius of the body is bounded by the size
of the cell, so that we again get a very small value for the time to reach equi-
librium.

The case for the visco-elastic medium is given on page 519.

The conclusion is the same. Thus for all possible cases in our experiments
the steady state is reached in a time very much smaller than anything we
can measure.

2. Scale

We shall next show, by simple dimensional arguments, how the forces
involved in dragging and twisting change with scale. We only consider a

. . dH .
range of seale over which the magnetic factors CB. pean be considered
dx

constant.

(a) Dragging
As before let @ == characteristic length of particle
p= characteristic velocity of particle
y= viscosily of (newlonian) liquid
o = density of fluid

The density of the particle is clearly not involved in the steady state condi-
lion, We restrict ourselves to a range of scale over which the density of the
fluid can also be ignored, for the reasons given above.
The magnetic field will produce a force per unit volume given by
dit
I

dx

The physical properties of cyloplasm. Part 1. D09
where J = magnetic induction of the particle
dil . oa . . :
dn = magnetic field gradient producing the force.

A force per unit volume has the dimensions M L-? T-?. The only combina-
. : 2 v
lion of a, v and 7 which will give this is ("2).
a

Whence we obtain

aa (“2)
p~—.7{22).
7] dx

Thus, as the scale is reduced, the velocity decreases as the square of the

characteristic length. The time for the body to traverse its own length in-
creases linearly with the reciprocal of the characteristic length.

(b) Twisting

Here the magnetic couple per unit volume depends on

(1 H) f(0)

where @ is an angle.
This has the dimensions M L~! T7? and from @ (the typical angular ve-
locity), 47 and a we can only form the combination 4a.

lH
Thus an £68)

Therefore the angular velocity does not vary wilh seale.

Note that if the liquid has boundaries they, too, must be sealed for the
above results to apply.

If the liquid is non-newtonian comparisons can only be made belween
conditions under which the shear is the same. For dragging this occurs
when the time for the particles to go their own lengths is the same; for twisting,
when the angular velocities are the same. The interesling result for Uwisting,
that the angular velocily is independent of scale is therefore also true for
the non-newlonian case.

Finally note that the magnetic conditions have nof been scaled. S valing
the magnets producing the field makes no difference to the value of the field,
hut does alter the field gradient. This reservation is therefore Important
in dragging but not in twisting. [Lis -asy to see that if we do seale the magnets
for the dragging case the lime for a particle to be dragged its own length is
independent of seale for both the newtonian and the non-newtonian cases.
d10 EF. H.C. Crick

Cc. THE FORCES ON A BODY IN A VISCOUS FLUID
t. Variation with shape

‘The variation wilh shape is naturally more complicated than the variation
with scale, and has only been worked out for special cases, usually in an
infinite fluid. Itis possible to obtain the result for the general ellipsoid, but
we shall only qnote those for ovary cllipsoids of revolution, as we require
them merely to give some idea of the general behaviour. We consider in
this section the formulae for a body immersed in an infinite newtonian lig-
uid, leaving to the two following sections the consideration of boundaries
and of non-newtonian behaviour.

We shall use the following notation for the ovary ellipsoid:

major axis =a

minor axes = b = c
b2
eccentricity, e, given by 1—e? = a

We denote
2(1 — e?) l+e
aaa “3 —_—— (3 log le —e by QX-
(1 — e?) i+e 1
— ( “yg 3 log T—e _— ee by Bo-
and
b? t+e
— log -——— b .
e 08 1—e y Xo
(All logs are natural logs).
(a) Twisting
For a sphere: couple = 8a%y40o
where w» = angular velocity
(7, para 334)

For an ovary cllipsoid of revolution:

We shall only consider the case of rotation about a minor axis. This has
been solved by Edwards (2), but owing to an algebraical slip towards the
end he omits the factor 2/3. The correct result, in his notation, is, for the
general ellipsoid,

2
b? 4 “~@
32 une + 3

couple = —s PBC

The physical properties of cytoplasm. Part II. o11

This being adapted to our notation, and restricted to the case of the ovary
ellipsoid, becomes

» 2b?

at
32y20 "3

rouple = ef Po 2,

couple 5 Gay +B fy ab

We express this as couple = k-8ayq@a?b and evaluate k numerically for
different values of a/b.
Some values are given in ‘Table [.

 

 

 

TABLE [|
a
b 1.0 2.0 3.0 4.0 5.0 10.0 | 20.0
k 1.0 | 0.84 0.91 1,00 1.10 1.60 2.50

 

 

 

 

 

 

 

 

In the limit (a/b) > 00, k > 3 (5) ___} .
5\b (tog 2a 1)
b
(b) Dragging
For a sphere:
the force is 627av with the usual notation
(7, para 337)
For an ovary ellipsoid of revolution
(i) in the direction of its major axis, a

we have force = 627 Rv

where R= I
3 x9 + a a?
(7, paras 339 and 114)
This reduces to
_ 8 ae
3 1 l+e 2
(1 + =) log -— —_

R

For the special case where the ellipsoid is very long, so that a = b

Ro 4. a

(2 log * -- 7)
ol2 PL. C. Crick

Numerical values are given in Table I below.
(ii) in the direction of its minor axis, b

8 ab?

we have R= 50 ie pol?

(7, paras 339 and 114)

whence we obtain

16 ae
B38 Tg) tog LE 2) 4 2
e? oe ) e
and when a > b
roo. 4.
3 2a
(2 log b + 1)

Numerical values are given in Table IL.

TaBLe ITI

Values of a/R for the dragging of an ovary ellipsoid.

 

0.3 | 0.4

 

0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 10

 

bia | 0.0 | O01 | 0.2

 

 

Lane

 

 

t
2.799) 2.267 Lvt4 1609 1467 1.314 L0H 1.091

 

a/R (along major axis) | oo 3.701

 

1.606 1.451 1.326 1.224 1.138! 1.063 1tert

 

a/R (along winor axis) | co leoi7 2.1081 1.815

The numerical values are taken from Gans (4), where the numerical results
for a planclary cHipsoid are also given.
2. The effect of boundaries

(a) Boundaries are, in general, more important in dragging than in twisting.
This is not surprising when we remember that the viscosity of a VISCOUS

. . 1 .
fluid through which a sphere is dragged falls off as 7a Jong way from the

sphere (in contrast to the case where the inertia is important and viscosity
negligible), while the angular velocity in the Muid round a rotating sphere

. . 1
falls off as 4:
r

r

The physical properties of cytoplasm. Part Il. d13

The following examples illustrate this point. In all cases a is the inner ra-
dius, b the outer radius.
(i) translation of a sphere in a fixed cylindrical tube

force = banav(t + 2.1045 +]

(ii) rotation of a sphere in a fixed spherical shell
couple = 8 aa 1
I muy aw i— ails

(iii) translation of a cylinder in a fixed cylinder

Force per unit length = 2anv (seeing)

(iv) rotation of a cylinder in a fixed cylinder

couple per unit length = 4aya°7w (<7) .

_(b) Dragging

For the translation of a sphere along the axis of a fixed cylindrical tube
the solution with the higher terms included is

 

force = Oanav a
1—2.104(") + 2.09 (“ *_o.93(“)
a0") © 2.00 (“) —0.99(%)
where a = radius of sphere
b = radius of tube.

This formula is for the case when Reynold’s number is infinitesimal. (3.)
Thus when b = 3a the formula gives a resistance of 2+ 7 times that for
an infinile fluid, and for 6 = 4a, a factor of about 2-0, so thal increases
of this sort are very probable in a small cell. Neighbouring inclusions mas
well have quite a large effect.
For bodies of a shape not greatly different from a sphere, a good approx-
a” the value for the

.

imation is to use the above formula taking for “oquiv-
alent sphere’? in the infinile Muid case. This approximation can only be
very rough for the case of a very elongated body, or of a wall very close to

a body.
Dl FL. C. Crick

(c) Twisting

The effect of boundaries on the couple exerted on a compact body of re-
volution rotating about its axis of symmetry is fairly casily grasped, and is
small wiless the boundaries are near the equatorial belt of the body.

The effect of the boundaries on less restricted bodies (including our par-
ticles) has not been worked out, but below we try to give a rough bound.

(i) Bodies of revolution.

These have been treated by Jeffery (6). He gives a general treatment which
can be described as follows: for any body of revolution rotating in an in-
finite fluid about its axis of symmetry we can find a family of surfaces each
of which rotates with constant angular velocity. We can then always ob-
tain the solution for the body rotating within one of these surfaces, regarded
as a fixed wall, by superimposing a uniform counter-rotation on the whole
system to bring the “wall” to rest. This explains the form and intimate
relauionship between the fall-off of angular velocity and the increase of
couple due to a houndary in the two simple cases of rotation quoted above.

Jeffery gives the formulae for the case of an ellipsoid of revolution. A
typical result, for a planetary ellipsoid with an axial ratio of 2.24, sur-
rounded by a confocal planetary ellipsoidal shell such that the spacing at
the equator is 20 per cent of the body’s equatorial radius, shows that the
shell increases the viscous couple by a factor of 1.9.

Jeffery has also solved another illuminating case; that of a sphere rotating
close lo a fixed plane perpendicular to the axis of rotation. The results show
that the plane has to be extremely close to the pole to have any consider-
able effect c.g. ala distance of 2 per cent of the radius it inereases the re-
sistance by only £7 per cent. This is because the major part of the viscous
couple comes from the equatorial bell, where both the arm of the couple
and the velocity are big.

We may thus conclude that unless the boundary approaches close to the
body at points far from the axis of rotation, the increase in couple is unlikels
to be big.

(ii) Other’ bodies.

In general our particles are not bodies of revolution, and even if they were

we could nol casily distinguish their different angular positions under the

microscope, Apparently no case has been solved which helps us here.
We propose to estimate the couple on an ovary ellipsoid of revolution

rotating inside a fixed spherical shell, radius d (d not too close to a) as fol-

lows: we suspect that such a shell would not increase the couple on the el-

The physical properties of cytoplasm. Part 11. 515.

lipsoid by more than it would increase that of a sphere of radius a. This
implies that the couple would be increased by no more than

3
neglecting terms higher than (<r)
Now we have shown (page 51 1) that the couple on the ellipsoid in an infinite
fluid is approximately given by

8aynat bw

so that the total couple could be written as
2 ae
Sanadbha (1 pS]

where p is probably between (a/b) and 1. However this estimate is little
more than a guess.

For a comparatively short particle the boundaries are not likely to cause
a large variation in the couple e.g. for d one half greater than a, and for a
particle of axial ratio 2:1, the couple is probably not more than doubled.

Finally we consider how a fixed obstruction at one end of the particle
affects its behaviour. A rough estimate can be made by comparing this case
with that of a particle of twice the length, and acted upon by double the
magnelic couple, with obstructions near its middle. It is clear that in this
second case the motion would hardly be altered at all by the obstructions,
as they are so near the axis of rotation. The viscous couple on the seeond
particle in a newtonian fluid. is very roughly + fimes that of the first (sce
page 512). Therefore the angular velocity will be halved.

Thus in an infinite newtonian liquid an obstruction al one end inereases
the resistance so thal about twice the couple needs to be applied to give the
sume angular velocity. It also, of course, produces a small translation of the
particle.

Stated in this way it seems probable that the result would also apply to
most non-newtonian liquids.

Note that it is not essential for the obstruction to touch the particle, as a
very close approach may produce sufficient resistance to reduce the angular
velocity appreciably.
ot6 BOW C. Crick
3. Non-newlonian behaviour

So far we have only considered newtonian liquids. This is sufficient for
calibration liquids, but may not be for the cytoplasm.

Consider first the behaviour of a liquid, whose “viscosity” varies with
shear, in a concentric cylinder viscometer (neglecting end effects). Ht can
be shown that what we measure is the average value of the fluidity (’\. In

0

svinbols

Sb
[as
Ne = Sa
‘b
ds
Sa
where 4, = experimental value of the viscosity
s = shearing stress
Sq == Shearing stress at inner cylinder

8, == shearing stress at outer cylinder

That is, if we substitute the experimental values of angular velocity, ete in
the usual formula for the viscosity, we obtain the above value of He Wis
assumed that the ‘viscosity’ is a function of the instantaneous value of the
shear only.

This solution is possible because the surfaces in the fluid which retain
their shape (i.e. move as if solid) are surfaces of constant stress, and one
‘an therefore give the stress distribution across the annulus irrespeclive of

I .
the curve of (;) against s. This condition also applies to the only other case

which has been worked out, namely that of the flow through a capillary.
although the actual formula given above does not. ,

We thus see, as is intuitively obvious, that the effect is to slur over the de-
tailed variations in the curve of (; against s, and to give an average value.

The details are usually got by making b only slightly greater than a, so that
the average is taken over only a very small portion of the curve on cach we-
fasion, and by repeating at different rates of shear lo cover a wide range.
We are clearly unable to do anything of this sort in our experiments. Even
for a sphere in a non-newlonian liquid the problem is of a different order of

The physical properties of cytoplasm. Part 11. 517

difficulty and does not appear to have been solved. The shear stress varies
with latitude from zero to a maximum and itis not at all clear how it distri-
butes itself in the non-newtonian case. ‘The problem of the ellipsoid is even
more hopeless.

However it appears extremely plausible that if we substitute the expert-
mental values in the formula, the value of 9 derived will correspond to some
average value; that is, to some point on the curve of 7 against s between the
maximum value of s and some minimum, probably zero. Moreover it seems
very likely that the apparent change of 7 with shear will be less than the
maximum change anywhere within this region. Sinee we cannol rely on
our experimental arrangements to measure small differences accurately,
we conclude that this method will only show up large changes of “viscosity”
with shear, and may conceal small changes.

We note at this point a feature which may be expected in the behaviour
of a non-spherical particle in a non-newtonian fluid of the type which be-
comes very viscous at low shearing stresses. Since the shear near the axis
is much less than that near the ends of the particle, the fluid may behave
almost as a solid at points near the axis, and also at points far from the par-
ticle, so that the flow may take place over a rather restricted region.

D. The forces on a body in a jelly

We can apply almost all the previous formulae to the case of a particle
in an isotropic elastic medium. Since we are only dealing with feeble jel-
lies we may take Poisson's ratio equal to $. For small strains all the algebra-
ical results for velocity in a viscous newtonian fluid apply to the deflection
in a hookian elastic medium, providing we substitute n, the rigidity modulus,
for 4 the viscosity.

For example, the couple on a sphere rotating in a viscous liquid, which is

8an@a
nH = Viscosily
wo = angular velocity
a = radius of sphere

enables us to write down the couple on a sphere embedded in an clastic
medium as (for small angles)
8anad 0

n = rigidity modulus
0 = angular deflection

36— 503704
518 FP... Crick

‘ s
$ Stress cress

: applied removed

Deflection

    
 

   

 

x
Th we ne snnnwcenewsneh.s

Time
Fig. 1.

To clarify our terms consider next a simple visco-elastic medium, which
behaves as shown in figure 1 when a constant stress is suddenly applied
and later suddenly removed. In this figure the elastic properties are re-
presented by AB and DE, the viscous damping by the exponential rise AC
and fall DG, and the pseudo-viscous yield occurring along CD, by EF. The
relaxation time is HA.

If we have a particle embedded in a simple visco-elastic medium whose
relaxation lime is constant with stress, we can obtain the relaxation time ex-
perimentaly without knowing any of the details of the particle, boundaries,
ete. by simply dividing the elastic deflection (for small strains) by the cor-
responding pseudo-viscous yicld rate for any given applied couple. This
follows from the similarity in form of the viscous and clastic coefficients
mentioned above. If the relaxation time is a funetion of the stress, there is
no simple solution for the general case.

We have so far neglected inertia. We now take it into account and show
that the period of free oscillation is very short, and the damping high, again
mainky beeause the particles are so small.

We first consider a particle in a simple clastic medium without Viscosity.
Due to the inertia it will be capable of free osciltations of both translation
and rotation, ICean easily be shown that, neglecting for simplicily the inertia
of the medium, the dimensionless parameter for both cases is of the form

oo’ a
oR

where — is the order of the period of free oscillation, and eis the density
a

of the particle.

The physical properties of cytoplasm. Part 11. 519

Putting ! = 1 millisecond, a = 5 yw, and @’ = 4, we see that for the pa-
a

rameter to equal unity, m must be 1 dyne/em?. This is extremely feeble (for
anormal gelatin gel n = 103 to 10° dynes/em?). For a stiffer medium the
time is of course shorter.

It is clear that for a substance as feeble as this the viscous damping would
in practice be important. We will therefore consider a particle in a medium
with both viscous and elastic properties, and find the condition for critical
damping. We assume that the times involved are so short that we can neg-
lect any pscudo-viscous yield.

If we work through a particular case, such as a sphere undergoing rotary
oscillations about its axis, and neglect for simplicity the inertia of the me-
dium, we obtain the condition for critical damping as

2
natn
— ea

oe’ = density of sphere.

We could have derived this, without the constant, in a rough and ready
Manner by equating the values of ¢? which make the two previous dimen-
sionless parameters (page 506 and page 518) equal to unity.

Pulling a = 3 x, 7 = 0-01 poise, and ge’ = 4, we obtain n = 1,100 dynes/
em? ‘This is not high, but it is rather higher than our estimates. Moreover as
already observed it is highly unlikely that the “viscosity” coefficient in bio-
logical materials is as low as that of water.

As regards the effects of boundaries, a closer examination shows. that
since the ratio of the viscous to clastic forees is independent of them, the
condition for the damping remaining critical or greater when boundaries
are added, reduces to the condilion that the ratio of the damping forces to
the inertia forces shall not deerease, which we have already shown (page
007) lo be probable in’ most cases. ,

We thus conclude that the time-period of free oscillation of our particles
is less than a milli-second (probably much less) and that the damping is
critical or greater unless the rigidity modulus is high and the viscosity low,
which is not the case in our experiments.

We have only considered the case of small strains. Large strains may
well increase the apparent value of the rigidity modulus, calculated using
the simple theory. [tis unlikely however in our cases to increase il suffi-
ciently to alter our general conclusions.
520 PLO C. Crick

We have dealt with this point about critical damping because it is some-
limes suggested that a resonance method should be used. Apart from the
experimental difficulties due to the very short time periods, it is clear that
the extremely low *'Q”’ of the system would make this unprofitable.

It has also been suggested that the restoring force on a particle could be
increased by magnetic means, though this would mask any elastic effect
due to the medium, The simple theory shows that in the case of a sphere
this is equivalent to the medium having a rigidity modulus of

i: dynes/em?,

A more exact treatment would be required if the method were seriously con-
templated. It is thus not impossible that oscillations could be produced. The
envelope of these oscillations is determined by the ratio of the viscous to
the inertia forces, and this could be used to measure the former. As has
been shown this involves making measurements in a time probably of the
order of microseconds. This is not impossible, but it is certainly not easv.
The only advantage of such a method is that it is not necessary to know the
magnetic forces accurately, although il is essential to know the exact formnu-
lae for the viscous and the inertia forces. These could if necessary be checked
by calibration experiments in a known liquid. We have not pursued this
approach further.

E. THE FORCES ON A BODY IN A MAGNETIC FIELD

1. Twisting: the permanent magnet case

Assuming that the particle is magnetically homogeneous we nole that
the magnetic condition is independent of scale, and therefore the forces
per unit volume are also independent of scale, over a range where H and ‘Le
can be considered constant,

The variation with shape is more complex. The ellipsoid is the only bods
for which the magnetic conditions are constant throughout the volume fer
a uniform applied ficld. As in the hydrodynamic cases, we will consider
only ellipsoids of revolution.

We proceed as follows: we first calculate a factor depending on the shayu
of the ellipsoid. Using this we find from the B/H curve of the material th:
relevant value of B for an ellipsoid permanently magnetised along its ma-

The physical properties of cytoplasm. Parl IT. 521

jor axis. From this we easily obtain the magnetic moment (7) of the ellip-
soid. The couple due to a small applied field, A, perpendicular to the length
of the ellipsoid, is then Afh.

The factors which are calculated for ellipsoids are ‘“‘demagnetising fac-
tors.” These express the amount by which the magnetisation of the ellip-
soid produces a reversed magnetic field acting upon itself, and thus tending
to demagnetise itself, and are defined by the equation

H’=DI
where H’ = the demagnetising field produced
I = the intensity of magnetisation
D = demagnetlisation co-efficient.

D will in general depend upon direction, and will have three different values
corresponding to the three axes of the general ellipsoid. The behaviour for
other directions can be found by compounding J and H vectorially.

We shall, as usual, only quote the formulae for an ovary ellipsoid of re-
volution. They are

1 1 ite . :
D, = an(5— 1) (5. log ime t) for the major axis.

1 1 — e? l+e .
D, = 22|-5 —- -5-q— log j-— -} for the minor axes.
e 2e 1—e
(The logs are natural logs)
. . . 2
where a = major axis, b = ¢ = minor axes and 1-e? =-,-W hen az b,
a

these formulae become

D,sL 2a

be 2b
D, 24 x7, (log -1):

a
: : da
For the particular case of a sphere D, = D, =.
. 4a tn
hat is == fh
that 1s D, ~ Dg 3

((8). The notation has been altered.)

. a . . . : da
Thus if we consider the case of Bb fixed and a increasing, we sce that (,
1
a2 Pal. C. Crick

D)

2

INcereases rapidly, while ( ) tends to the value 2. We give a few values
in Table TH.

TABLE ITI

 

 

 

 

 

 

 

 

 

 

 

 

; | og 20 | 3.0 | 40 5.0 | 10.0} 20.0
i |
4m |
3.00) 5.71 | 917) 132] 17.9] 49.5} 148
Dy
4 |
p, | 200 | 242) 224) 216) 212) 204] 202

 

 

 

 

 

 

As is well known there is a simple graphical construction to find the work-
ing point on the B/H curve for a body with a given Din a (parallel) external
field H,. The curve of (B-H) against H is plotted, and a line is drawn from

. ws 4 . :
the point H, on the H axis with slope — (a) The working point is the point

where this line cuts the curve. This construction applies for any value of
the applied field JZ, For a permanent magnet we usually have H, = 0;
in this case we are working in the top left-hand quadrant of the (B-H)
against H curve.

Once we have found the working value of (B-H) the magnetic moment
(M) of the ellipsoid is simply

M = (B— HF) Vv
4x

where V = volume of the eHipsoid.

2. Twisting: the soft iron case

We consider the case of an ellipsoid of soft iron ina magnetic field inclined
ata small angle to its major axis. By soft iron we mean here a material with
no hysteresis. We do not restrict ourselves to the case of constant perme-
ability, and will in fact consider a material which becomes magnetically
saturated. ,

We calculate the case of an ovary ellipsoid of revolution (major axis =
a, minor axes = b = c) where the applied field, H, makes and angle §
with the major axis, and the intensity of magnetisation, J, makes an angle
a with the major axis. In general «+ 0. Lis constant throughout the body

The physical properties of eyloplasm. Part 1. 23
hoth in magnitude and direction. We solve by splitting into components
We obtain

Icos a= ere cos @—- D, Icos «) along the major axis
x

I sin « = Mt H sin @— D, Isin «) along a minor axis.
cr

We shall only consider the cases where the angles are small. We therefore
put sin 8 = 0 and cos 0 = J, elec, and eliminating I we obtain

roa a

(u—1) (52)

6 : Dy}
me 1

——__.... -+ wee
(u—1) (4x
[to
Now to this approximation the couple (C) experienced by the particle is
C=IVH (6—a)

where V = volume of particle.
That is

C=I1VH (1-4)

which we can write

1 + — i
i) 63
D, D,
C=1VH6 Ant Ml
w= * (a)
D,

This is the expression we require.
It follows that

(i) if (u—1) > (5 (the suffix 2 referring to the broadways-on case) the
term in the bracket is effectively constant as H varies, and

C varies as IH
as we should expect.
O24 FI. C. Crick

(ii) if we have a material of low permeability, or one which is saturated
4a
D,
varies from 3 fer a sphere to 2 for a long ellipsoid (page 522). Writing the
expression for the couple as

so that (—1) has become low, we may have (u—1) < ( ) This latter factor

1 1
i4n\ ~~ any
C=IVH(n,—1)0 tn) Lo)

4x
D,
we see that C varies as I H (u—1) approximately
that is C varies as I? approximately.
Thus if the material is saturated the couple does not increase with the

field indefinitely, but tends to a limit.
(iii) for short ellipsoids the couple is less than might be expected on simple

theory by the factor
dD,
( - 2)

1+. Dy} .

(uw — 1)
which is always less than 1, and moreover becomes zero for a sphere, for
which Dy == Dg.

A moment's thought shows that this latter point is obvious. If a soft iron
sphere is subjected to a slowly rotating magnetic field, the magnetism rotates,
not the sphere. This is in fact the clue to all the effects. As the magnetic
material saturates with increasing field, for example, it becomes easier for
the magnetism to rolate. For magnetite, where (g—1) can be small these
effeels may be quite important.

3. Dragging

As we are concerned with obtaining the maximum drag, we will only
give the case where the particle is in a magnetic field large enough to saturate
it. The magnetic moment (M) will normally be in the direction of the ap-
plied field. The foree on the particle in the x direction, Fy, is given by

The physical properties of cytoplasm. Part II. 525

0H, OH
F, a Mee + Mi Gy + M,

OH,
Oz
where M,, M,, M, are the vectorial components of M

and H, is the x component of the applied field H.
There are similar expression for F, and F,.

HA,
The force thus depends on terms of the form (22) rather than of the

Ox
0 Hz . . .
form { Hx Da which occur in certain other cases.

The force on the particle is not necessarily along its length. For example,
if the particle is in a magnetic field which lies in the y direction (H,=H,=0),
so that it, too, points in the y direction (M, = M, = 0), there will never-

H.
theless be a force on the particle in the x direction if (2) is not zero.

F. THE VARIATION IN RATE OF MOVEMENT WITH SHAPE

We can now combine the results of the previous sections.

1, Twisting: the permanent magnet case

We assume that an ovary ellipsoid is magnetised parallel to its major axis
so that it becomes a permanent magnet, of magnetic moment M, and that it
is then acted on by a small magnetic field (fh) perpendicular to its length.
This will produce a couple Mh and if the ellipsoid is immersed in a new-
tonian liquid it will rotate with an angular velocity @. The problem we wish
to solve is, how big is @ and how does it vary with shape?

To obtain the value of w for any particular case we merely have to work
out the magnetic and the viscous couples from the formulae given in the
previous sections and equate them. However it is useful to get a qualita-
tive idea of how ow changes with shape (we know that it is independent
af scale) so we shall suppose that b is kept constant and a alowed to in-
crease, The nature of the variation depends on the nature of the magnetic
material.

We take the extreme case first. If the ellipsoid is long, so that the slope of

: . dn yep at
the (B- A) against H curve is much less than (37) then (B-JZ) will effectively
be constant, and M will only increase due to the increase in volume, that is
proportional to ab? The viscous couple, however, increases at a rate be-
026 LH, C. Crick

tween a®b and a Thus in this range the angular velocity decreases rather

Fast tl (

aster than .
(aib)

On the other hand, if the ellipsoid is short, and the magnetic material such

thal the slope of the (B- H) against H curve is much greater than (;;} then

dD,
ab ( p)

katb

the angular velocity varies as

where & is tabulated on page S11.

- a Pdad by. : .
We thus evaluate (;. D, ea) for various values of a/b.

TasBLe IV

 

2.0 | 3.0

4.0 | 5.0

 

 

1.13 1.12 1.10 1.09

 

|

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(! 4m 1 ')
tea  £.00

30D, k al ff

It can be seen that the variation with shape is not very great. Eventually
the angular velocity will fall off, but by this time the approximation used
is unlikely to be still valid. Thus fora real (8-H) against H curve the angu-
lar velocity will eventually decrease with increasing (a/b). It may be roughiv
constant over a range for (a/b) small, but this depends on the shape of the
curve. The exact values can be calculated for any given curve from the for-
mulae given.

The above results apply strictly to the special ellipsoids chosen, It seems
reasonable to assume that in the region where the shape of the ellipsoid is
making a large difference the approximation for a body of arbitrary shape
will not be as good as for ranges where the ellipsoid’s shape is having litt
effect on the angular velocity. However it is not easy to put a figure to the
usefulness of the approximation,

We have not pursued this further, as the problem is complicated and we
have in any case in our actual experiments taken an average value. If greater
accuracy is required the solutions for the viscous forces and the demagnet-
ising co-elficients for the general ellipsoid are available, and might give a
better idea of the effects of irregular shape.

The physical properties of cyloplasm. Part II. O27

2. Twisting: the soft iron case
The qualitative results for the corresponding problem in the soft iron case
can easily be seen. For very long ovary ellipsoids the angular velocity will

1 . . > .
fall off rather faster than (cais)) as in the previous case. For almost spherical

ones it will again be small. Somewhere in between there will be a maximum,
depending on the properties of the material and the size of the applied tietd.
The exact values can be calculated for particular cases from the formulac
given.

It seems probable that for actual particles of irregular shape we shall get
similar effects to those calculated for the ellipsoid. That is, for very short
particles we shall get smaller couples than might be expected on the simple
theory, and for larger applied fields the couple tending to a maximum
instead of increasing indefinitely. In the case of any particular material the
evaluation of a few cases for the ellipsoid should give a good idea of the
general behaviour, though the reduction in couple due to shortness is likely
lo be less important for irregular bodies.

The treatment will not apply to materials which show hysteresis.

3, Dragging

We will only consider the cases of an ovary ellipsoid of revolution being
moved either parallel or perpendicular to its length. Other directions can
be solved by compounding vectorially. We consider the relevant field gra-
dient as fixed, and investigate how the velocity of movement depends on the
dimensions. For our case the magnetic force, for a given value of (B-H)
at saturation, depends only on the volume, not on the shape. We have al-
realy shown (page 509) that the effect of size is to make the velocity vary as
the square of the characteristic length, so that it only remains to investigate
shape variations. As before the formulae will give an exact solution for any
chosen case.

(a) dragging parallel to the major axis.

The formula for the viscous resistance and a selection of values are given
on page 512. These show that for a fixed b, the drag increases with a, initially

4
rather slowly, say as Va, and gradually increases to rather slower than a.

Taking Va as a typical value, the velocity of the particle will roughly be pro-
portional to
ba

Va be
928 F. H.C. Crick

(b) dragging parallel to a minor axis.

The formula and a selection of the values are given on page 512. These
show that if b is fixed and a increased, the drag increases initially a litle
faster than the previous case, so that we may take Va as typical, giving

ba

velocity ~ --—--
Va

What is wanted in fact in both cases is a good estimate of the volume of
the particle, plus an approximate estimate of a and b. This conclusion is
likely to stand for particles having irregular shapes.

G. PRODUCING FIELD GRADIENTS
We first note that since

0 He 4 0Hy + OH; =)
Ox dy Oz

OH, _, :
we cannot get a large value of or without either one or both of the other

two being large, and of opposite sign. This implies that the lines of force
cannot be parallel in such a region. They must either diverge or be bent.

It can be shown that a magnetically saturated particle can never be im
true stable equilibrium under the influences of magnetic forces alone. Tha
follows simply by regarding the particle as having a (fixed) surface dism-
bution of magnetic poles, and applying the appropriate analogue of Eam-
shaw’s Theorem (5, 374 and 167). The particle will in fact be either in wt-
stable equilibrium or be moving towards one of the magnets producing tbr
field.

We next wish to show, quile generally, that a very large ficld gradient vn
only be produced (leaving aside electric currents for the moment) by having
ferromagnetic material near the particle. This is perhaps obvious on di:
mensional grounds. A magnet of a given shape and of a given material will
produce the same field at corresponding points, irrespective of seale. Thus,
clearly, the smaller the magnet, the greater the field gradient. As there is en
upper limit lo the size of (B-H) for magnetic materials, there must come 4
time when the gradient can only be increased by making everything smaile

We can illustrate this by calculating the result for an ideal polepicce ah

The physical properties of cytoplasm. Part II. 529

the shape of a truncated cone, semi-angle «, with the particle at the apex of
the cone, which we will take as the origin. We will assume that the direction
of magnetisation is everywhere parallel to the axis of the cone. The solution
of this problem, which is quite straightforward, gives the field gradicnt alt

the origin as
, OH 6alI
mo =-——— sin? « cos? «
Ox xz=0 Xp . /

where I = intensity of magnetisation of the pole-piece, which is as-
sumed to be uniform.
9 = distance of pole-piece from the particle at the origin.
There are three points to notice about this answer. Firstly that the expres-

‘: . . 3
sion has a maximum with respect to « at cos a = —==. Secondly that we can

V15

wrile this maximum (putting 421 = B-H) as

(B— H) 18V3 .
— ———= (B, H refer to the polepiece
% 50V5 ( polepiece)

 

so that the field gradient at the origin is of the form

(B— H)

Vo

where p is a constant a bit less than 1. This form of result is very general.
Thirdly we note that if we had not continued the pole to infinity, but stop-
ped it at the point 2,, we should have obtained

1 1

p(B—H) (5 -=

Xt
shich shows that as long as 2, is several times wg, the result is not sensitive
Wits exact value. This obviously follows from the fact that we are inle-

. . . BH .
@ating an expression of the form ree through a volume. Thus distant
F

entributions have hardly any effect, because of the upper limit to (RoI)

his not necessary, however, to produce the magnetic gradient dire: any
sith the primary magnet. We can produce a large uniform field, and con-
veer the gradient near a small body of soft iron placed in this field. For
simplicity we will consider the case where this body is a sphere. This is
530 F. H.C. Crick

extremely casy, as for points outside it behaves exactly as a doublet of

strength

located at its center (a = radius of sphere). Let the particle be a distance
r from the centre of the sphere (r > a).

If we consider the case where the applied field is parallel to the line join-
ing the particle and the sphere, the force is an attraction given by

_ 3
|F[ = “= (3) 2M
r r

CV refers to the particle, B and H to the soft iron sphere.)
For the case where the field is perpendicular to the joining line, we have

a repulsion of
_. 3
jr) = Bo (‘) 3M
r r

 

It is thus possible to control the direction of the force to some extent by al-
tering the direction of the applied field. It should be noted that the force

falls off as g> so that it will vary rapidly with the position of the particle.
r , .

To sum up, the maximum gradient will usually be of the form

BH
pF)

Xp

where (B-H) is the value for the iron in the immediate vicinity, av is the
distance of the nearest iron [rom the particle, and p is a constant depending
in a complicated way on the configuration, but approaching a value of the
order of 1 in well-designed cases. The more distant parts of the magnetic
cireuit do not affect the gradient directly, but only in so far as they deter-
mine (B-H) in the iron near the particle.

We must consider briefly the possibility of producing high field gradicnts
by cleetric currents in air-cored coils. We first observe that we require a
sustained force for our purposes; a short pulse is in general not sufficient.
The limitation is therefore the steady heating effect: cither the small rise in
temperature which the culture will tolerate, which would be important for
coils close to, or the rise in temperature of the coil itself for larger, more

The physical properties of cytoplasm. Part 11, 531

distant coils. We will not give a general treatment, but will give one simple
example. Consider the conical polepiece of page 529. Instead of a cone of
magnetic material, imagine that this cone is a former upon which a coil is
wound, What is the current through such a coil which would produce the
same field-gradient as the magnet did? By considering the equivalent mag-
netic shell we arrive at the simple answer that

nt = Iwhere n = number of turns per cm
i = current in coil (e.m.u.)
J = magnetic intensity of the iron.

Now we can easily make TJ = 103, so that if we had 1 turn per mm (n =
10) i must be 10?, or 10? amps. This will clearly give an enormous amount
of heat. The margin in the calculation is so big that more precise considera-
tions would not be appropriate. Briefly we note that the heating effect al-
lows (ni) to increase as 127, where 1 is a characteristic length, so that air-
core coils can only compete with magnets if they are both very large, which,
as we have shown is the case which produces low field gradients. Thus,
in general, magnets are much better than air-cored coils for our purpose.

H. SOME NUMERICAL VALUES FOR THE STRESS

Although we have argued that the method is a very poor one for finding how
the “viscosity’’ of a non-newtonian liquid varies with stress, it is clear that
if the range of stresses is very wide indeed we may expect quite considerable
differences in behaviour. It is therefore useful to compare the maximum
values of the stresses due to twisting and dragging magnetically, and due to
gravity. To simplify matters, since we are only concerned with orders of
magnitude, we will consider the case of a sphere, taking its radius (r) as 1 ye.

1. twisting of a sphere magnetically.
The maximum stress in this case is

BH
~_ -- dynes/cem?,
4a

For B = 225 and H = 45 oersteds we get

~ 400 dynes /em?.
mye . en
532 Io. C. Crick
2. dragging a sphere magnetically.
The maximum: stress is

B
12a

1H
(‘ =) dynes /em?
dx] ~

. - 1H
for B = 1500 and “ix = 10* oersteds/em (say)

and r= 107+ «em
we get
~ 40 dynes/em?,
3. dragging a sphere due to gravity,

For the general case (as in a centrifuge) where the centrifugal acee-
leration is ng we haye the maximum: stress equal to

n
(e — @0) “2 T
where @ = density of particle
Oo = density of liquid.
There are two cases of interest.
(a) for magnetic particles under gravity.
Taking @=4 o=1 n=1 r=10-4 cm
we get

1
= -—~ dynes 2
io ‘ ynes/em

(b) for natural inclusions of the cell, in a centrifuge.
Take, arbitrarily, (@ — gy) = 0-1.
We obtain for the maximum stress

n
a, dynes/em?
300° /
for an acceleration of ng.
The point we wish to bring out is not merely that the stresses produced
during magnelic twisting are rather bigger than in magnetic dragging, bul
that both are enormously bigger than the effect of gravity. Moreover, these

The physical properties of cytoplasm. Part II. 533

stresses are only equaled when centrifuging natural inclusions of the same
size by centrifugal fields of the order of 10° times gravity.

Finally we must emphasize that these results only apply for particles of
the chosen size, as can be scen from the factor r in the later expressions.

SUMMARY!

1. The paper gives the theory of the magnetic particle method, in which
some of the mechanical properties of a fluid are estimated by observing the
movements of magnetic particles in it due to applied fields.

2. For the very small particles likely to be used in biological systems the
inertia can be neglected.

3. The effect of scale is derived for particles of irregular shape in a new-
tonian liquid.

4. Exact formulae are quoted, for the three cases most often encountered,
for an ovary ellipsoid of revolution in an infinite newtonian liquid. Refe-
rences to the general ellipsoid are given.

5. The effects of the irregular shape of a particle, of boundaries, and of
non-newtonian and elastic behaviour of the fluid are discussed qualita-
tively.

6. Some theoretical notes are given on producing large gradients of mag-
netic field.

7. Some comparative numerical values of the stresses in certain biological
applications are evaluated.

ACKNOWLEDGEMENTS

The author wishes to thank Dr. Honor B. Fell for the hospitality of the Strange-
ways Research Laboratory, the Medical Research Council for a Studentship, and Mr.
G. Kreisel for many helpful and characteristic suggestions on presentation,

REFERENCES

. Crick, F. H. C., and Huaues, A. F. W., Exp. Cell Res. 1, 37 (1950).

. Epwarps, Quart. J. Maths, XXVI, 70 (1893).

Faxkn, O., quoted in G. W. Oseen’s Hydrodynamik, p. 198, Leipzig, 1927.

Gans, Silzungsb. K. B. Acad., Miinchen, 191 (1911).

Jeans, J., Electricity and Magnetisin. 5th Ed. Camb. Univ. Press, 1925.

. Jerreny, G. B., Proc. London Math. Soc. (2) XIV, 327 (1915).

. Lams, H., Hydrodynamics, Gth Ed. Camb. Univ. Press, 1932.

. MAXxwe Lt, J. C., A treatise on Electricity and Magnetism, 3rd Ed. Vol II, Oxford Univ. Press,
1892.

ONS akwNe

+ A more extended summary has been given in Part I (1, p. 79).
37 — 503704