. 5
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denote the magnitude of the block™s displacement in the x direction as a function of z.

friction, will slow down the ¬‚ow and contribute to the bulk property
known as viscosity. It is intuitively obvious that diffusion and friction
in a liquid cannot be independent of each other: the stronger the fric-
tion the slower the diffusion. In addition, the higher the temperature,
the more thermal energy the molecule has and the more intense its
diffusion. Indeed, under very general conditions we have D = kT/f,
which is known as the Einstein--Smoluchowski relation (Berg, 1993).
Since F = fv, the faster an object moves, the stronger the friction it
To illustrate the molecular basis of friction in liquids, imagine
pulling a plate (or any object) of area A through a liquid with a con-
stant force F x in the x direction (see Fig. 1.5). For it to move, the plate
has to displace the liquid molecules it encounters. The state of the
liquid is thus perturbed. This perturbation in the state of the liquid
is called shear and leads to friction, i.e., viscous drag, acting on the
plate (Howard, 2001). A measure of the shear, or rather the rate of
shearing, is the modi¬cation of the liquid™s velocity in the vicinity of
the plate. For forces that are not too strong, the shear rate is propor-
tional to F x . The proportionality constant between F x and the shear
rate is the viscosity ·, obviously a property of the liquid (for the pre-
cise de¬nition of viscosity see Box 1.1). The customary unit of · is the
Pa s (pascal second). The viscosity of water is 0.001 Pa s. The viscosity
of the cytoplasm varies strongly with cell type (for an overview, see
Valberg and Feldman, 1987) and even with location within the cell
(Bausch et al., 1999; Yamada et al., 2000; Tseng et al., 2002).

Box 1.1 De¬nition of viscosity

Consider Fig. 1.5, where a force of magnitude F x is applied along a layer of a liquid
in the x direction. For simplicity, we ¬rst assume that the velocity of the resulting
¬‚ow v x depends only on a single variable, z, and increases linearly in the z direction,
and that the liquid is at rest in the plane z = 0 (no-slip boundary condition). This
is illustrated by the blue arrows, which represent the magnitude of the velocity
as a function of z, and trace a straight line. Under these conditions, the de¬ning
equation for the viscocity · is

vx (z)
F x = ·A . (B1.1a)
Here A is the area of the liquid layer acted upon by F x and vx and z are
small changes in the corresponding quantities. Note that because of the assumed
linearity of vx on z, the ratio vx / z is constant and gives the slope of the line
traced by the arrowheads.
More generally, the velocity pro¬le is not linear (e.g., for ¬‚ow in a pipe it is
parabolic), in which case vx / z itself depends on z and should be evaluated
in the limit when the changes in both vx and z become in¬nitesimal. This pro-
cedure de¬nes the derivative of vx with respect to z, dvx /dz (or for a function
f (x), d f /dx). A derivative thus represents the rate of change of one quantity
with respect to another. In the future, when it will create no confusion, derivatives
will be written as ratios of ¬nite differences as in the above equation.
Even more generally vx might depend on other variables, in which case the
notation for the ordinary derivative, d, is replaced by the notation for a partial
derivative, ‚. Thus, in the most general case, Eq. (B1.1a) becomes

‚vx (x,y , z, t)
F x = ·A , (B1.1b)
‚z z=h

‚vx ‚ ‚x ‚ ‚x
= =
‚z ‚z ‚t ‚t ‚z

(the order of multiple differentiation can be interchanged) is the shear rate ex-
pressed in terms of the liquid™s velocity pro¬le, the derivative being evaluated at
the location of the plate in Fig. 1.5 (assumed to be at height h); ‚ x/‚z represents
the shear.

The shear rate caused by the moving object is dif¬cult to calculate
(it is usually found from measurement); for this we need the velocity
of the perturbed liquid as a function of position, which requires the
solution of the complicated Navier--Stokes equation (this is discussed
in more detail in Chapter 8). Such calculations can be carried out
for simple cases. Thus, for a sphere of radius r moving in a liquid of
viscosity ·, with constant velocity v, the frictional drag is given by
the Stokes formula (see, for example, Hobbie, 1997)

F = 6π ·r v. (1.2)

As a consequence, for a sphere of radius r the friction coef¬cient
(de¬ned earlier by F = f v) and the diffusion coef¬cient (related to
f through the Einstein--Smoluchowski relationship D = kT / f , see
above) are given respectively by fsphere = 6π ·r and D sphere = kT /
(6π·r ). These expressions imply that f and D are strongly shape-
dependent. For example, for a long cylindrical molecule of length L
and diameter d the friction coef¬cient depends on whether the mo-
tion is lengthways (parallel to L ) or sideways (perpendicular to L ). In
the limit of large aspect ratio, L /d 1, the corresponding friction
coef¬cients are (Berg, 1993; Howard, 2001)

4π ·d 8π ·d
f|| = , f⊥ = . (1.3)
2L 1 2L 1
’ +
ln ln
d 2 d 2

Viscous transport of cells
Everyone is familiar with two extreme behaviors of moving bodies:
there are ˜˜inertial” objects that when set in motion tend to remain
in motion (as described by Newton™s ¬rst law) and ˜˜frictional” objects
that won™t move unless you continue to push them along. Since in
this book we will often be interested in the motion of individual
cells in the embryo, we will again switch scales from molecular to
the cellular (as we did for diffusion) and see what our analysis of
viscosity can tell us about cell motion. Here we will use one of the fa-
vorite tools of physicists, ˜˜dimensional analysis,” to show the relative
contribution of inertial and viscous behaviors to the movement of
cells. Dimensional analysis allows us to compare the magnitudes of
the different factors contributing to a complex process after having
rendered them nondimensional.
According to Newton™s second law, mass times acceleration = the
sum of all forces acting on a body. Let us assume that a cell moving
through a tissue experiences other forces, collectively denoted by F ,
along with the frictional forces. According to Newton,

d2 x dx
=F ’f .
m (1.4)
dt 2 dt

Here m is the mass of the cell, x and t denote distance and time
respectively, and in the expression for the frictional force the velocity
is denoted as the derivative of distance. The minus sign in the last
term expresses the fact that the direction of the frictional force is
opposite to the direction of motion.
Let us introduce nondimensional (hence, unit-less) quantities
s = x/L , „ = t/T . Here L and T are some typical values of the dis-
tance and time. In terms of these parameters (and with a slight re-
arrangement of the various factors), Eq. 1.4 becomes

m d2 s T ds
= F’ . (1.5)
f T d„ fL d„

Each term in Eq. 1.5 is now dimensionless and we can thus com-
pare their magnitudes (note that d2 s/d„ 2 and ds/d„ are themselves
dimensionless). We can take L to be the typical linear size of a cell,
so that m = ρ L 3 , ρ being the density. Thus the coef¬cient of the di-
mensionless acceleration is ρvL 2 / f , where we have introduced the
typical velocity v = L /T . Since f /L ≈ · (see above in connection with
Eq. 1.2), the ratio of the inertial term (proportional to the mass) and
the frictional term contains the expression ρvL /·. We can now plug
in known values of these factors. The typical size L of a cell is of order
10 micrometers and the cellular density is of order that of water. A
characteristic time T could be identi¬ed with the early-embryo cell
cycle time. For the sea urchin embryo, which we will discuss in Chap-
ters 4 and 5, T is about an hour (≈103 s), thus v = L /T ≈ 10’2 µm/s
(1 µm = 10’6 m). Using these values we obtain ρV L /· ≈ 10’7 , a very
small number (remember, this result is independent of the units of
measurement). This analysis shows that when dealing with physical
motion in the early embryo, inertial effects can safely be neglected;
see also the example in Box 1.2.

Box 1.2 Inertial versus frictional motion; coasting
of a bacterium

Let us consider the motion of a bacterium in the viscous intracellular environment.
A bacterium is propelled into motion by a rotary motor in its tail. The typical speed
of such motion is 25 µm/s. We now ask the question, how long will the bacterium
coast once its rotary motor stops working?
The bacterium will keep moving due to its inertia. To see how far this inertial
motion will take it, we have to solve the appropriate equation of motion, which
states that mass m times acceleration equals the sum of all forces acting on the
bacterium. Once the motor is turned off the only force acting is the viscous force,
fv. Thus

m (B1.2a)

The solution of this equation is v (t) = v (0)e’t/„ , where „ = m/ f and v (0) is the
speed of the bacterium at the moment when its motor turned off. Approximating
the bacterium by a spherical particle of radius r = 1 µm and density ρ that of
water, we obtain m = (4/3)πr 3 ρ ≈ 4 — 10’15 kg. Using Stoke™s law (see Eq. 1.2)
we have f = 6π ·r ≈ 20 nN s/m. The total distance the bacterium coasts is speed
times time and is approximately v (0)„ = v (0)m/ f ≈ 5pm = 5 — 10’12 m, a mi-
nuscule distance even on the scale of the bacterium. (Since the speed varies with
time, the mathematically accurate way of obtaining the distance is to integrate the
speed with respect to time from zero to in¬nity, which would lead to a value of the
same order of magnitude.) This example illustrates that inertial effects indeed can
be neglected when the motion of cells is considered. The ratio of inertial forces
to viscous forces is known as the Reynolds number. For a fascinating discussion of
“life at low Reynolds number” see Purcell (1977).

Elasticity and viscoelasticity
A viscous material will readily change its shape (deform) when a
force is applied to it. The following experiment showed that cells are
not constructed entirely of viscous materials. Ligand-coated magnetic
beads were attached to transmembrane proteins on the surfaces of
cells linked to the cytoskeleton. These beads, and thus the cytoplasm,
were subjected to a twisting force by an applied magnetic ¬eld. It
was found that wild-type cells (i.e., genetically normal cells) exhib-
ited higher stiffness and greater stiffening response to applied stress
than cells that were genetically de¬cient in the cytoskeletal protein
vimentin or wild-type cells in which vimentin ¬laments were chemi-
cally disrupted (Wang and Stamenovic, 2000). The properties of stiff-
ness and stiffening measured in these experiments were directly re-
lated to the elastic modulus (or Young™s modulus) of the cytoplasm.
Cells and tissues can easily be deformed by external forces and
to some extent without sustaining any damage, because they have
elastic properties (just push with your ¬nger on your stomach). The
prototype elastic device is the spring. The force needed to compress
or extend a spring is

F = k x, (1.6)

where k is the spring constant or stiffness and x is the deviation of
the spring from its equilibrium length. Equation 1.6 is Hooke™s law,
which expresses the fact that for elastic bodies the deformation force
is proportional to the elongation (or in more general terms to the
magnitude of the deformation caused). Hooke™s law often is written
in a slightly different form, namely

=E . (1.7)
Equation 1.7 states that if an elastic linear body (a rod, for exam-
ple) of original length L and cross-sectional area A is extended by
L , the stress (F /A) needed to achieve this is proportional to the
strain (i.e., the relative deformation, L /L ). The parameter E is the
Young™s modulus of the rod™s material; its unit is the pascal, Pa. Even
though Eqs. 1.6 and 1.7 are de¬ned for a linear body, one can de¬ne
an elastic modulus for any biological material. Its value would be
determined, for example, by simply stretching a piece of such mate-
rial with known force in some direction and measuring the original
length and the deformation in the same direction. The cross-sectional
area then is measured perpendicular to the direction of the force. (For
nonisotropic materials, the stiffness varies with direction). Any ma-
terial if stretched or compressed with suf¬ciently moderate force (in
the ˜˜Hookean regime”) will obey Eq. 1.6. Davidson et al. (1999) listed
the stiffness of a number of cells and tissues. Comparing Eqs. 1.6 and
1.7 one can de¬ne an effective spring constant for any material in
terms of its Young™s modulus using the relation k = E A/L .

Another type of deformation to which cells and tissues are often
exposed to is shear. It has been discussed in connection with viscosity
(Box 1), but it can also be de¬ned for any elastic material. If the upper
face of an elastic rectangular body is moved with a force F (acting
parallel to that face) relative to its lower face (see Fig. 1.5) then
F x
=G . (1.8)
A z
Equation (1.8) expresses the fact that in the elastic regime the shear
stress F /A is proportional to the shear; the constant of proportion-
ality, G , is the shear modulus in Pa. Note that in the case of viscous
liquids shear stress can be maintained only if it causes the shear to


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