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= D 2.
‚t ‚x
For the meaning of the symbol ‚ see Box 1.1 below.)

Diffusion inside the cell
So far we have assumed that the diffusing particles execute their mo-
tion in a homogeneous liquid environment, free of any forces or ob-
stacles other than those due to random collisions with other particles.
The inside of the cell is far from being homogeneous: it represents
a crowded environment (Ellis and Minton, 2003) with a myriad of
organelles. The cytoskeleton, the interconnected network of cytoplas-
mic protein ¬bers (actin, microtubules, and intermediate ¬laments),
is a dynamical structure in fertilized eggs, changing markedly in
organization in different regions and at different stages (Capco and
McGaughey, 1986; see Fig. 1.1). These objects and structures are likely
to hinder the motion of any molecule (Ellis and Minton, 2003). It is
not surprising, therefore, that simple diffusion is increasingly seen
to be not an entirely adequate mechanism for extended intracellular
transport (Ellis, 2001; Hall and Minton, 2003; Medalia et al., 2002); at
best it can act on a submicrometer scale (Goulian and Simon, 2000;
Shav-Tal et al., 2004).
Modern experimental techniques make it possible to follow the
motion of individual molecules inside the cell. Goulian and Simon
(2000), for example, tracked single proteins for close to 250 millisec-
onds in the cytoplasm and nucleoplasm of mammalian cells. They
found that even on this small time scale the observed motion cannot
be modeled by simple diffusion with a unique diffusion constant.
To explain the experimental ¬ndings a broad distribution of diffu-
sion coef¬cients, or ˜˜anomalous” diffusion, had to be assumed. The
latter notion refers to ¬tting data using the more general expression
xt2 = 2D t ± , with ± different from 1 (cf. Eq. 1.1). (Anomalous diffusion
typically takes place in heterogeneous materials with complex struc-
ture. In special cases the exponent ± can be determined theoretically.
For a review on the subject see Ben-Avraham and Havlin, 2000). Other
authors (reviewed by Agutter and Wheatley, 2000) claim outright that
diffusion is an incorrect description of intracellular molecular trans-
port. Such views are based on the supposition that the intracellular
milieu represents a ˜˜dynamical gel” rather than a ¬‚uid medium (Pol-
lack, 2001). Even though the major component of the cell interior is
1 THE CELL: FUNDAMENTAL UNIT OF DEVELOPMENTAL SYSTEMS 13


water, intracellular ¬laments and organelles render its architecture
highly structured. The transport of molecules is more likely to take
place along structural elements rather than in the intervening cyto-
plasm or nucleoplasm (von Hippel and Berg, 1989; Kabata et al., 1993;
Agutter and Wheatley, 2000; Kalodimos et al., 2004).
Shafrir and coworkers (2000) presented a model of intracellular
transport in which the relevant structural elements are the ¬laments
of the cytoskeleton. Since diffusion is assumed to take place along
these linear elements it is effectively one-dimensional. Using numer-
ical techniques, the authors demonstrated that for realistic ¬lament
densities and diffusion coef¬cients this constrained transport mecha-
nism is no slower than free diffusion. In comparison with free diffu-
sion, the proposed mechanism has however distinct advantages. The
¬laments provide guiding tracks and thus transport becomes more
focused. Because the cytoskeleton avoids organelles, movement is less
hindered. As many of the cell™s proteins are bound to the cytoskeletal
mesh ( Janmey, 1998; Forgacs et al., 2004), the network may provide
sites of concentrated enzyme activity for metabolic transformation
during transport. Diffusive transport along cytoskeletal components
should not be confused with molecular motor-driven motion. In the
former case no extra source of energy is needed (other than ther-
mal energy), whereas in the latter case a constant supply of energy,
provided by ATP hydrolysis, is required.
In summary, contrary to common notions simple diffusion across
the crowded intracellular environment cannot be the principal mech-
anism for distributing molecules within the cell. Classical diffusion as
described earlier may be relevant on small length scales, especially for
small molecules such as Ca2+ or cyclic AMP (estimated to be around
20 nm by Agutter and Wheatley, 2000). For larger distances, various
transport mechanisms utilizing cellular structural components or el-
ements (e.g., cytoskeletal ¬laments, DNA) take over. While extended
free diffusion inside the cell is unlikely, conditions in the extracellu-
lar space are more favorable for it (Lander et al., 2002). As we will see
in Chapter 7, in combination with biochemical processes diffusion is
an important mechanism in setting up molecular gradients that give
rise to speci¬c tissue patterns in the embryo.

Diffusion in the presence of external forces
Although free diffusion is not a realistic large-scale transport process
within the cell, it does occur on a scale that is larger than most
biomolecules (but small relative to the dimensions of the cell). This
will be suf¬cient to transport molecules through pores in the plasma
membrane or nuclear envelope or short distances in the cytoplasm.
In many cases this diffusion will be subject to external forces, which
can accelerate or reverse the direction of passive ¬‚ow. The presence of
external forces will modify the diffusive process discussed so far and,
signi¬cantly, permit insight into another basic physical parameter of
cellular and embryonic materials -- the viscosity.
14 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


Let us consider particles with concentration c(x, t), which in addi-
tion to random collisions experience a constant attractive or repulsive
force F in the positive x direction (say due to electrostatic interac-
tions). When F = 0, there will be a net diffusion current, jD (x, t) =
’D ‚c(x, t)/‚ x, along the concentration gradient, although for each
individual particle in the ¬‚ow x = 0. (The minus sign indicates that
the diffusion current is directed from higher concentration to lower
concentration.) The diffusion current is zero for uniform concentra-
tion. For F = 0 there will be an additional current. Force causes ac-
celeration (equal to force/mass), but as a result of the numerous col-
lisions experienced by the diffusing particles they quickly reach a
terminal velocity called the drift velocity, vd . Thus, even for constant
c the motion is biased in the direction of vd (see Fig. 1.3). (The total
particle current is now jT (x, t) = ’D ‚c(x, t)/‚ x + vd c(x, t).) The drift
velocity vd is de¬ned as F / f , f being the friction coef¬cient, a charac-
teristic property of the medium in which the motion takes place. If
F acts opposite to the diffusion current, it may reverse the direction
of motion: ¬‚ow may proceed against the concentration gradient.
A speci¬c case of diffusion in the presence of an external force is
of particular interest to biologists. If a diffusing molecule is electri-
cally charged then its motion will also be in¬‚uenced by any electrical
potential difference in its environment (e.g., a membrane potential).
The overall mass transport of a collection of such molecules will be
due to the combination of the concentration gradient and the elec-
trical gradient, which is termed the electrochemical gradient. (See
Chapter 9 for a description of the role of electrochemical gradients
during fertilization).
The term ˜˜diffusion” turns up in a number of contexts in cell and
developmental biology and it is important to understand how its dif-
ferent uses relate to the concepts described above. A cell biologist will
use ˜˜facilitated diffusion” to refer to free diffusion under conditions
in which speci¬c channels permit the selective passage through a
barrier (usually a membrane) of molecules for which this would not
otherwise be possible. Molecules with certain shapes, for example,
can be facilitated in their diffusion by pores with a complementary
structure. Such facilitation, of course, is not capable of causing mass
transport against a concentration gradient. For more details on diffu-
sion in biological systems the reader may consult the excellent book
by Berg (1993).

“Diffusion” of cells and chemotaxis
Locomoting cells, in the absence of any chemical gradient, typically
execute random, amoeboid motion without preferred direction. As
will be seen later, to interpret some aspects of such motion it is
useful to introduce an effective diffusion coef¬cient. It is impor-
tant, however, to keep in mind that the randomness here is not
due to thermal ¬‚uctuations but is the result of inherent cellular mo-
tion powered by metabolic energy. If such ˜˜diffusive” motion of cells
such as slime mold amoebae or bacteria occurs in the presence of a
1 THE CELL: FUNDAMENTAL UNIT OF DEVELOPMENTAL SYSTEMS 15


chemical gradient, the gradient can be considered as the source of an
external force that biases the direction of cell ¬‚ow. This phenomenon
is called ˜˜chemotaxis” and, while it is a unique property of living
systems, the source of the biased motion can be interpreted in phys-
ical terms.



Osmosis
A phenomenon closely related to diffusion, with important biological
implications, is osmosis, the selective movement of molecules across
semi-permeable membranes. An example of such a membrane is the
lipid bilayer surrounding the eukaryotic cell. It is permeable to water
but not to numerous organic and inorganic molecules needed for the
cell™s survival. (These must be transported through special pores com-
posed of proteins embedded in the lipid bilayer, as discussed above.
Here we will ignore this facilitated transport.)
Consider Fig. 1.4, which shows a container with two compartments
(L and R) separated by a semi-permeable membrane, permissive for the
solvent (e.g., water), but restrictive for the solute (e.g., sugar). The mov-
able walls in L and R act as pistons: if they are attached to appropriate
gauges the pressure inside the two compartments can be measured.
At equilibrium one ¬nds that the two pressures are not the same:
pL > pR . The reason for this is as follows. The pressure is due to the
bombardment of the container walls by molecules executing thermal
motion. Since the solvent can freely diffuse across the membrane,
it will do so until the average number of collisions per unit time
of its molecules with the movable container walls (i.e., the partial




PR
PL


R
L R
L


Fig. 1.4 Physical origin of osmotic pressure. The two compartments, L and R, are
separated by a semi-permeable wall, represented by the broken line. The two walls at
the ends of the compartments are movable (and can be thought of as the stretchable
membranes of “cells” L and R) and attached to springs, which measure the pressure in L
and R. The larger, pink particles (the solute) cannot pass through the wall in the middle,
but the smaller blue particles (the solvent) can. At equilibrium the pressure exerted by
the solvent on the movable walls in L and R is the same. If the two compartments
contain the larger particles at different concentrations (in the ¬gure their concentration
in R is zero), the pressures they exert on the movable walls are not equal: their
difference is called the osmotic pressure. In particular, if the two springs are made of the
same material, the one attached to L will be more compressed, corresponding to the
membrane of “cell” L being more stretched.
16 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


pressure due to the solvent) is the same in the two compartments.
The extra pressure in L, pos = pL ’ pR (acting, in particular, on the
semi-permeable membrane) is the osmotic pressure and is due to the
imbalance in the concentration of the solute in L and R.
The biological signi¬cance of osmosis is obvious. Osmotic pres-
sure is clearly an important determinant of cell shape. As long as
the overall concentration of organic and inorganic solutes inside the
cell remains higher than outside, the cell membrane is stretched (be-
cause water enters the cell in an effort to balance the concentration
difference, thus increasing the volume of the cell). The membrane
can tolerate increases in osmotic pressure only within limits, beyond
which it bursts. To avoid this happening, cells have evolved active
transport mechanisms: they are able to pump molecules across the
lipid bilayer through channels.


Viscosity
Viscosity of cytoplasm
There are numerous biological transport mechanisms other than
those discussed so far. For instance, microscopic observation of the
interior of certain cells has revealed the phenomenon of ˜˜cytoplasmic
streaming.” Streaming is an example of convection. Unlike diffusion,
which can take place in a stationary medium, convection is always
associated with the bulk movement of matter (e.g., ¬‚owing water,
¬‚owing blood). Cytoplasmic streaming is seen in certain regions of
locomoting cells such as amoeba, where it contributes to the cell™s re-
shaping and movement, as well as in axons -- the extended processes
of nerve cells -- where it is employed to move molecules and vesicles
to the axon™s end or terminus. Molecules or organelles present in the
streaming cytoplasm are transported just as an unpowered boat would
be in a river. Convective ¬‚ow is maintained by a pressure difference
(as in blood ¬‚ow), whereas diffusion current is due to the difference
in concentration. Yet another transport mechanism, less important
in biological systems, is conduction, which is not associated with any
net mass transport. A typical example is heat conductance, which
is possible due to the collisions between atoms performing localized
thermal motion around their equilibrium positions.
When there is a relative velocity difference between a liquid and a
body immersed in the liquid (either because the body moves through
the liquid or because the liquid moves past the body), the body ex-
periences a drag force. The type of drag to which organelles and cy-
toskeletal ¬bers moving through the cell™s cytoplasm are subject is
viscous drag.
An ideal gas (whose molecules collide elastically with each other
but do not otherwise interact) will ¬‚ow without generating any inter-
nal resistance. For any other ¬‚uid, including cytoplasm, interactions
among the molecular constituents, collectively leading to internal
1 THE CELL: FUNDAMENTAL UNIT OF DEVELOPMENTAL SYSTEMS 17



z

Fx
A




x

Fig. 1.5 Illustration of the phenomena described by Eqs. B1.1a and 1.8. For Eq. B1.1a
consider the ¬gure as showing a plate of area A pulled through a liquid in the x direction
with a shearing force F x . As a result of internal friction, as the plate moves the ¬‚uid
particles will also be displaced. The horizontal blue arrows show schematically the
magnitude of the ¬‚uid™s z component of velocity in the vicinity of the plate. Only the ¬‚uid
below the plate is shown. For Eq. 1.8 the object shown in the ¬gure is to be considered
as an originally rectangular solid block with its lower surface immobilized. The shearing
force F x now acts along the top surface of the block. The horizontal blue arrows

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