Untangling Complex Systems, page 54
ate filaments, but the protein subunits of these structures vary. Some intermediate filaments
are associated with specific cell types. For example, neurofilaments are found specifically in
neurons, desmin filaments are found specifically in muscle cells, and keratins are found spe-
cifically in epithelial cells. (III) Microtubules are the largest of the three types of filaments.
Their diameter is about 25 nm. A microtubule is a hollow, rigid tube made of a protein called
tubulin. Microtubules are ever-changing, with reactions continually adding and subtracting
tubulin dimers at both ends of the filament. The rates of change, at either end, are not bal-
anced; one end grows more rapidly and is called the plus end, whereas the other end is known
as the minus end. In cells, the minus ends of microtubules are anchored in structures called
microtubule-organizing centers (MTOCs). The primary MTOC in a cell is called the centro-
some, and it is usually located adjacent to the nucleus. Microtubules are involved in the sepa-
ration of chromosomes during cell division, in the transport within the cell and the formation
of specialized structures called cilia and flagella (the latter is true only in eukaryotic cells).
If the morphogenetic process, which we are focusing on, is slow enough that morphogens cover the
required distances by diffusion, then the Brownian motion is fast enough to account for the spatial
patterns that can be observed within cells and tissues. However, when the development of patterns
is too fast than the diffusion, other processes are necessary for pattern formation. One is the phe-
nomenon of traveling chemical waves that will be described in paragraphs 9.6 and 9.7. Two others
are mechanical processes within cells and tissues that act faster over long distances than Brownian
motion, and they are presented right here. They are two types of advection that require chemical
energy resources (Horward et al. 2011).
The first is transport driven by motor proteins (read Box 9.4 of this chapter to know more about
motor proteins). Typically, a motor protein proceeds with a speed ( v) of 1 μm s−1. A useful way to
compare the relative importance of advection and diffusion over the distance L is to calculate the
Péclet number ( Pe). The Péclet number is the ratio of the diffusion time driven by thermal energy to
the advection time driven by motor proteins.
τ dif
vL
Pe =
=
[9.41]
τ adv
D
When Pe > 1, the advection is faster than diffusion, whereas when Pe < 1 the opposite is true.
The Péclet number is less than 1 when we consider short distances. For example, if the diffusion
coefficient of a protein is 5 μm2 s−1, Pe is less than 1 as long as L < 5 μm. On the other hand,
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Untangling Complex Systems
BOX 9.4 MOTOR PROTEINS
The myriad of chemical reactions that occur within a cell cannot be described simply as the
result of a complex series of parallel and consecutive chemical transformations brought about by
diffusion and random collisions of chemical species embedded in different cellular compartments.
Cells are polar structures, and their internal composition and structure are not homogeneous neither
isotropic. Moreover, most of the essential cellular functions, like the maintenance of a transmem-
brane electric potential, cell division, and translocation of organelles, all require directional move-
ment and vectorial transport of chemical species. To overcome the randomizing effect of Brownian
motion and to carry out directional processes, cells have macromolecules that behave like tiny
machines. These are the so-called molecular motors: the energy to drive their motion ultimately
comes from chemical reactions taking place in the catalytic site(s) of the motors.
Molecular motors dominate the randomness of single molecular events and generate uni-
directional processes in the cell. Cells have hundreds of different types of molecular motors
(Bustamante et al. 2001). Some of them operate cyclically: the steps of the mechanical cycle
are coupled to the steps of a chemical cycle that fuels the molecular motor. The chemical
steps typically involve the two most common energy repositories, the nucleotides ATP or
GTP (in myosin, kinesin, and helicases) or translocation of an ion across an electrochemi-
cal gradient (in F ATPase and the bacterial flagellar motor). A parameter that measures the
0
mechanochemical coupling is the ratio χ:
v
χ =
[B4.1]
L vr
In [B4.1], v is the mean velocity of the motor, L its step size and vr the mean reaction rate.
If the mechanical motion and the chemical reaction are tightly coupled, χ~1 because the product
L vr will equal v . On the other hand, if many chemical steps do not lead to mechanical motion,
χ will be less than 1. Other molecular motors are “one-shot”: they unleash previously stored elas-
tic potential energy and then disassemble (like in spasmoneme and actin polymerization).
A molecular motor moving on a molecular route and interacting with all the surrounding
molecules has many degrees of freedom. Many of them are in thermal equilibrium with the
microenvironment and define the so-called bath variables. Bath variables are not involved
directly in the directional movement of the motor, but they affect its motion as fluctuating sto-
chastic forces, as sources of friction, and as entropic contributions to the thermodynamics of
molecular movement. The rest of the variables that are involved in the unidirectional motion of
the motor are called system variables. Since a molecular motor must be fueled by a chemical
reaction, at least one of the system variables measures the progress of the chemical reaction
and is called reaction coordinate. All the other system variables are mechanical variables. The simplest case is when the motor has only one reaction coordinate (x ) and one mechanical vari-1
able ( x ). If it were a macroscopic motor, its motion would be determined only by the shape of
2
the energy potential surface. But we are considering microscopic molecular motors, and they
suffer the random thermal forces. The motion of a molecular motor is stochastic and only statis-
tically biased by the potential. In fact, the inertial forces due to the potential are negligible com-
pared to friction forces. The molecular motors always operate at low Reynold’s number. The
Reynold’s number is the ratio of inertial ( F ) to viscous forces ( F ) on a body moving in a fluid:
in
v
Fin
dvl
Re =
=
[B4.2]
Fv
η
( Continued)
The Emergence of Order in Space
267
BOX 9.4 (Continued) MOTOR PROTEINS
In equation [B4.2], d is the density of the fluid, v is the velocity of the motor having length
l and η is the viscosity of the fluid. Macroscopic motors usually operate at high Reynold’s
numbers, where inertial forces dominate viscous forces. For example, a car 4 m long, mov-
ing in the air (with d = 1.225 Kg/m3 and η = 1.81 × 10−5 Pa*s) at the velocity of 50 Km/h, has
Re = 3.8 × 106. On the other hand, all microscopic motors, either unicellular organisms like
protists (they use flagella or cilia to move), or organelles or molecular motors within cells,
move in an environment characterized by a low Reynold’s number. For example, a protein
10 nm long moving at a velocity of 1 μm/s within the water (having d = 1000 Kg/m3 and
η = 0.001 Pa*s) has Re = 1 × 10−8! It is as if our car moved through tar; it is like swimming
through a very viscous liquid. Locomotion at low Re requires the continuous expenditure
of work. If work stops, the motion stops immediately. Locomotion at low Re drags along a
large added mass of fluid by viscosity, and it is more energetically demanding than motion
at high Re.
For molecular motors, the random forces are rapid compared with the time scale of the
movement of the motor. Therefore, the random forces loose correlation between steps of the
walk and the motion is Markovian. It is “memoryless” motion: its next step depends on its
present state but not on the motion’s full history. Therefore, the movement of the molecular
motor over the potential energy surface in the mechanochemical space is a uniform motion
described by the Langevin equations:
dx
∂ V x , x
1
( 1 2)
−γ1
−
+ 1 f( t) + F 1
B ( t ) = 0
[B4.3]
dt
∂ 1
x
dx
∂ V x , x
2
( 1 2)
−γ2
−
+ f 2( t) + FB 2( t) = 0
[B4.4]
dt
∂ 2
x
where the subscripts 1 and 2 refer to the reaction and mechanical coordinates, respectively.
V ( x
)
1, x 2 is the energy potential of the motor; f 1( t ) and f 2 ( t ) represent the external mechanical forces acting on the motor and exerted by a laser trap, an atomic force microscope, or
the forces due to the load that the motor is driving. The other two terms of equations [B4.3]
and [B4.4] derive from the surrounding fluid, with γ and being the friction coefficients,
1
γ 2
and FB 1( t) and FB 2( t) being the Brownian random bath forces. Because of the random forces, individual trajectories obtained for solving these equations are meaningless; only statistical
distributions over many trajectories are useful. The final solution of the Langevin equations
will be a probability distribution w ( x
)
1, x 2 , t . Since the total probability is constant, it will
obey a continuity equation without a source term:
∂ w
2
+ ∇
w
J
J = ∂ +
∂ i = 0
[B4.5]
t
t
∑
∂
∂
∂
x
i=
i
1
where J represents the probability current (with [ L]/[ t] as dimensions). In our case of a biased diffusion process, J is the sum of two contributions:
kT w
∂
f
J
i
i = −
+ w
[B4.6]
γ i x
∂ i γ i
( Continued)
268
Untangling Complex Systems
BOX 9.4 (Continued) MOTOR PROTEINS
The first term represents the diffusion, whereas the second is the advection driven by the
forces f including that due to the potential and external forces, but excluding the stochastic
i
forces, which are included in the first term.
Inserting equation [B4.6] into [B4.5], we obtain the Smoluchowski equation:
∂
w
w
2 k T 2
1
+ ∇ J = ∂ +
B
fi w 0
[B4.7]
t
t
∑− ∂ + ∂ =
∂
∂
2
γ i ∂ xi
γ i ∂
x
i=1
i
Molecular motors convert chemical energy into mechanical motion by two general
mechanisms: the power stroke and the Brownian ratchet. In the power stroke mechanism,
the chemical reaction and the mechanical movement are tightly coupled. For example, the
chemical step could be the binding of a substrate to the site of the protein, and the move-
ment, the corresponding conformational change of the macromolecule. In the Brownian
ratchet mechanism, the chemical process is not coupled to the mechanical movement. The
motor is driven by thermal fluctuations, but the chemical reaction rectifies or selects the
forward fluctuations. The path of a Brownian ratchet drawn in a free-energy surface having
one mechanical coordinate and one chemical coordinate will be zigzag. It will consist of
pairs of transitions at right angles: one nearly parallel to the chemical axis and one nearly
parallel to the mechanical axis. The chemical step is irreversible with its Δ G large and
negative. The free energy is expended to favor forward motion, rather than doing mechani-
cal work on the protein directly. A Brownian ratchet is efficient when each fuel molecule
burned results in exactly one step forward.
Pe is larger than 1, when L > 5 μm. It means that the fastest way to cross an entire eukaryotic cell that has a diameter of 10 μm is through advection. In fact, it will take 10 s, whereas the Brownian
motion would take the double: (λ2 D) = 20s. When the advection moves the motor proteins in
a direction that is the opposite with respect to the diffusion, a new length scale emerges that is
λ = D v
/ .
The second type of advection, which is faster than diffusion, is movement due to mechanical
stress. Forces exerted by motor proteins locally on the cytoskeleton create an active stress gradient
and lead to a velocity gradient. If there is friction with the surroundings, then the velocity gradient
will have the characteristic length λ = η / γ where η is the viscosity and γ is the friction coefficient.
Such velocity gradient leads to rapid movement of material, at the speed of sound, called cytoplas-
mic streaming (Quinlan 2016). Morphogens embedded in this material are moved along with it, like
objects transported by a flowing river. Within a cell, such bulk-material transport can be driven by
flows of cytoplasm. But bulk-material flow also occurs at the tissue scale, resulting from changes in
cell position, orientation, and shape.
The mechanochemical patterning is responsible for cell polarity (Goehring and Grill 2013). Cell
polarity is the asymmetric organization of a cell. Cell polarity is crucial for certain cellular func-
tions, such as cell migration, directional cell growth, and asymmetric cell division. It is also relevant
in the formation of tissues. For example, epithelial cells are examples of polarized cells that feature
distinct apical and basal plasma membrane domains (see Figure 9.14). The apical-basal polarity
drives the opposing surfaces of the cell to acquire distinct functions and chemical components.
The Emergence of Order in Space
269
Planar polarity
Apical
Basal
FIGURE 9.14 Polarity of epithelial cells.
There is also planar cell polarity that aligns cells and cellular structures such as hairs and bristles
within the epithelial plane. Cell polarity is so fundamental that it is ubiquitous among living beings:
it is present not only in animals and plants, but also in fungi, prokaryotes, protozoa, and even
archaebacteria.
9.6 CHEMICAL WAVES
Spontaneously, we are inclined to think of diffusion as a process that tends to homogenously spread
chemicals in space. However, in this chapter we are discovering that this is not always the case. For
example, in paragraph 9.2 we have learned that a periodic structure emerges from a homogeneous
system when diffusion is coupled to an autocatalytic reaction. In paragraph 9.3, we have discov-
ered how diffusion combined with activator-inhibitor dynamics can give rise to stationary patterns
having several shapes. In this paragraph, we are going to learn that, when diffusion works in the
presence of an excitable system or an oscillatory reaction carried out in an unstirred reactor, amazing
phenomena, called chemical waves, emerge (Figure 9.15). The most common type of chemical wave
is the single propagating front that is a relatively sharp boundary between reacted and unreacted
(a)
(b)
(c)
FIGURE 9.15 Examples of chemical waves generated in a chemistry laboratory by using the BZ reaction: a
propagating wave front in (a); a target pattern in (b), and a spiral pattern in (c).
270
Untangling Complex Systems
chemicals and that can be circular like that shown in picture (a) of Figure 9.15. In some systems, the medium restores its original state, relatively quickly, after the passage of the front. Such phenomenon is referred to as a pulse. Even more astonishing are the target patterns in which pulses are emit-
ted periodically from the same point, named as “pacemaker” or “leading center” (see picture (b) in
Figure 9.15). If an expanding pulse is broken at a point, it begins to curl up at the broken ends and produces oppositely rotating spiral patterns (see picture (c) in Figure 9.15). The images of chemical waves shown in Figure 9.15 have been collected in a chemistry laboratory by exploiting the BZ reaction. But in nature there are many other examples of chemical waves. Some of them are described
