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Plasma membrane
Rough endoplasmic reticulum
Mitochondrion

Fig. 1.1 Schematic representation of an animal cell. Most of the features shown are
characteristic of all eukaryotic cells, including those of protists (e.g., protozoa and
cellular slime molds), fungi, and plants. Animal cells lack the rigid cell walls found in fungi
and plants, but contain specializations in their plasma membranes that permit them to
bind to other cells or to extracellular matrices, hydrated materials containing proteins
and polysaccharides that surround or adjoin one or more surfaces of certain cell types.
These specializations are depicted in Fig. 4.1.



Modern eukaryotic cells, whether free-living or part of multicel-
lular organisms (˜˜metazoa”), have evolved far beyond the ancient
communities of prokaryotes in which they originated. Cellular sys-
tems with well-integrated subsystems are at a premium in evolution,
as are organisms with ¬‚exible responses to environmental changes.
The result of millions of years of natural selection for such properties
is that a change in any one of a cell™s subsystems (by genetic muta-
tion or environmental perturbation) will have repercussions in other
subsystems that tend either to restore the original functional state
or to bring the subsystem in question to another appropriate state.
The physicist™s aim of isolating relevant variables to account for the
system™s behavior would seem to be all but impossible under these
circumstances.
Despite these complications, no one questions that all cellular
processes are subject to physical laws. For a cell to function properly,
the values of its physical parameters must be such that the govern-
ing physical laws serve the survival and reproduction of the cell. If
the osmotic pressure is too high inside the cell, it may lyse. If the
voltage across the membrane is not appropriate, a voltage-gated ion
channel will not function. If the viscosity of the cytoplasm is too
8 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


large, diffusion across it slows down and processes that are normally
coordinated by diffusing signals become incoherent.
One way in which a cell sets the values of its physical parameters
is by regulated expression of the cell™s genes. Genes specify proteins,
which themselves control or in¬‚uence every aspect of the cell™s life,
including the values of its physical parameters. The embryo begins as
a fertilized egg, or zygote, a single cell that contains the full comple-
ment of genes needed to construct the organism. The construction
of the organism will necessitate changes in various physical param-
eters at particular stages of development and typically this will be
accomplished by new gene expression. These changes will be set off
by signals that may originate from outside the embryo or by interac-
tions between the different parts of the embryo itself. As cells reach
their de¬nitive differentiated states later in development their appro-
priate functioning will require stability of their physical properties. If
these change in an unfavorable direction then other signals are gen-
erated and transmitted to the nucleus to produce further proteins
that reverse or otherwise correct the altered physical state.
In the remainder of this chapter we will consider certain trans-
port and mechanical properties that are essential to an individual
cell™s existence. Because these properties have physical analogues at
the multicellular level they will also ¬gure in later discussions. We
will begin with the ˜˜default” physical description, the simplest char-
acterization of what might be taking place in the interior of the cell.
We will then see how this compares with measurements in real bio-
logical systems. Where there is a disparity between the theoretical
and actual behaviors we will examine ways in which the physical
description can be modi¬ed to accommodate biological reality.


Diffusion
Free diffusion
The interior of the zygote, like most other cells, represents a crowded
aqueous environment (Ellis and Minton, 2003) with many thousands
of molecules in constant motion in the presence of complex struc-
tures such as organelles. Metabolism, the transformation of small
molecules within the zygote, requires nutrients to be brought in
and waste molecules to be carried out. In the course of intracellu-
lar signaling, information is passed on from one molecule to an-
other and ¬nally delivered to various destinations. How do biologi-
cal molecules move inside the cell? Unless they are bound to a sur-
face these molecules typically bump into each other and collide with
water molecules and thus are constantly changing their direction of
motion. Under such conditions how can they reach their destinations
and carry out their tasks with high ¬delity?
Motion requires energy. In general it can be active or passive.
Of the two, only active motion has a preferred direction. Molecu-
lar motors moving along cytoskeletal ¬laments use the chemical en-
ergy of adenosine triphosphate (ATP) and transform it into kinetic
1 THE CELL: FUNDAMENTAL UNIT OF DEVELOPMENTAL SYSTEMS 9


energy. They shuttle from the plus to the minus end of microtubules
(˜˜plus-directed motor”) or in the opposite direction (˜˜minus-directed
motor”).
As an example of passive motion we may consider a molecule in-
side the cell, kicked around by other molecules, just as a stationary
billiard ball would be by a moving ball, and triggering a cascade of
collisions. In the case of a billiard ball, motion is generated in the ¬rst
place by a cue; cytoplasmic molecules are not set in motion by any
such external device. Rather, they have a constant supply of energy
that causes them to be in constant motion. The energy needed for this
motion is provided by the environment and is referred to as ˜˜ther-
mal energy.” In fact, any object will exchange kinetic energy with its
environment, and the energy content of the environment is directly
proportional to its absolute temperature T (measured in kelvins, K).
For everyday objects (e.g., a billiard ball weighing 500 g) at typical
temperatures (e.g., room temperature, 298 K = 25 —¦ C) the effect of
thermal energy transfer on the object™s motion is negligible. A cyto-
plasmic protein weighing on the order of 10’19 g, however, is subject
to extensive buffeting by the thermal energy of its environment.
For a particle allowed to move only along the x axis the kinetic
energy E kin imparted by thermal motion is given by E kin = m v2 /2 = x
kB T /2, where m is the mass of the particle, vx is its velocity, and kB
is Boltzmann™s constant. The symbol denotes an average over an
ensemble of identical particles (Fig. 1.2). Averaging is necessary since
one is dealing with a distribution of velocities rather than a single
well-de¬ned velocity. To understand better the meaning of , ima-
gine following the motion of a particle fueled exclusively by ther-
mal energy, for a speci¬ed time t (at which it is found at some
point xt ) and measuring its velocity at this moment. If the particle
is part of a liquid or gas it moves in the presence of obstacles (i.e.,
other particles) and thus its motion is irregular. Therefore, repeating
this experiment N times will typically yield N different values for v2 . x
The average over these N values (i.e., the ensemble average) gives us
the interpretation of v2 (Fig. 1.2). For a G-actin molecule at 37 —¦ C,
x
1/2
= 7.8 m/s. (For a bil-
thermal energy would provide a velocity v2 x
liard ball at room temperature the velocity would be a billion times
smaller.) In the absence of obstacles, this velocity would allow an
actin monomer to traverse a typical cell of 10 µm diameter in about
1 microsecond.
The interior of a cell represents a crowded environment. A mole-
cule starting its journey at the cell membrane would not get very
far before bumping into other molecules. Collisions render the mo-
tion random or diffusive. When discussing diffusion (referred to as
Brownian motion in the case of a single particle), a reasonable ques-
tion to ask is, what distance would a molecule cover on the average
in a given time? For one-dimensional motion the answer is (see, for
example, Berg, 1993 or Rudnick and Gaspari, 2004)


xt2 = 2D t, (1.1)
10 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO



6

5
4

3

2




DISPLACEMENT
1

0

’1

’2

’3

’4

’5

’6
0 2 4 6 8 10 12

TIME

Fig. 1.2 The meaning of the ensemble average . The ¬gure can be interpreted as
showing either the diffusive trajectories of four random walkers in one dimension,
allowed to make discrete steps of unit length in either direction with equal probability
along the vertical axis, or four different trajectories of the same walker. The trajectories
are shown up to 10 time steps. The walkers make one step in one (discrete) unit of time.
(Displacement and time are measured in arbitrary units.) The average distance after 10
time steps is x (t = 10) = 1 (4 + 2 + 0 ’ 2) = 1, the average squared displacement is
4
x 2 (t = 10) = 1 (16 + 4 + 0 + 4) = 6. The mean ¬rst passage time of arrival at x = 2 is
4
T1 (x = 2) = 1 (2 + 2 + 4 + 8) = 4. These values are independent of how the
4
trajectories are interpreted. Note the difference between T1 and t. The latter is a ¬xed
quantity (10 in the ¬gure), whereas the former is a statistical quantity.




assuming that at t = 0 the particle was at x = 0. (Diffusion in three di-
mensions can be decomposed into one-dimensional diffusions along
the main coordinate axis, each giving the same contribution to the
three-dimensional analogue of Eq. 1.1. Thus, when x 2 above is re-
placed by r 2 = x 2 + y 2 + z2 , r being the length of the radius vector,
the factor 2 on the right-hand side of Eq. 1.1 changes to 6.) Here D
is the diffusion coef¬cient, whose value depends on the molecule™s
mass, shape and on properties of the medium in which it diffuses
(temperature and viscosity). The average is calculated as above for the
case of v. The diffusion coef¬cient of a G-actin molecule in water at
37—¦ C is approximately 102 µm2 /s, which allows its average displace-
ment to span the 10 µm distance across a typical cell in about 1
second. Comparing with the earlier result, obtained assuming unim-
peded translocation with the thermal velocity, we see that collisions
slow down the motion a million-fold.
There are several remarks to be made about Eq. 1.1. The most stri-
king observation is that distance is not proportional to time: there
1 THE CELL: FUNDAMENTAL UNIT OF DEVELOPMENTAL SYSTEMS 11


is no well-de¬ned velocity in the sense of the distance traversed by
a single particle per unit time. A ˜˜diffusion velocity” of sorts could
be de¬ned by vD = xt2 1/2 /t = (2D /t)1/2 . This ˜˜velocity” is large for
small t and gradually diminishes with time. Because of the statistical
nature of these properties, Eq. 1.1 does not tell us where we will ¬nd
the molecule at time t. On average it will be a distance (2D t)1/2 from
the origin (in three dimensions the average particle will be located
at the surface of a sphere of radius (6D t)1/2 ). But it is also possible that
the molecule has reached this distance earlier than t (or will reach
it later, see Fig. 1.2). We may be interested to know the time T 1
at which a given molecule will, on average, arrive at a well-de¬ned
target site (e.g., the cell nucleus) for the ¬rst time. This ˜˜¬rst passage
time” is in many instances a more appropriate or useful quantity than
xt2 (Redner, 2001). For diffusion in one dimension, the ¬rst passage
time for a particle to arrive at a site a distance L from the origin
is T 1 = L 2 /(2D ) (Shafrir et al., 2000). Even though this expression
resembles the expression t = L 2 /(2D ) (Eq. 1.1), owing to the mean-
ing of averaging, T 1 and t are entirely different quantities (see also
Fig. 1.2). In particular, t denotes real time measured by a clock,
whereas T 1 cannot be measured, only calculated.
Equation 1.1 relates to the Brownian motion of a single molecule.
For any practical purpose, molecules in a cell are represented by their
concentration and one needs to deal with the simultaneous random
motion of many particles. Brownian motion in this case ensures that
even if the molecules are initially con¬ned to a small region of space,
they will eventually spread out symmetrically (in the absence of any
force) towards regions of lower concentration (see Fig. 1.3). In one


F




B
A C

Fig. 1.3 The simultaneous Brownian motion of many particles. Particles con¬ned
initially to a small region of space (A) diffuse symmetrically outward in the absence of
forces (B) or, when an external force is present, preferentially in the direction of the
force (C).
12 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


dimension, if all the particles are initially at the origin with con-
centration c 0 then after time t their concentration at x is
2
c 0 e’x /4D t
c(x, t) = .
(4π D t)1/2
(The mathematically more sophisticated reader will recognize this
expression as being the solution of the one-dimensional diffusion
equation
‚c ‚ 2c

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