Biomedical Engineering Reference
In-Depth Information
8.5 Hydrodynamic Interactions in Other Biological
Systems
Hydrodynamic interactions also play an important role in other biological systems.
As an example, we discuss the properties of a collection of molecular motors exhibit-
ing rotational motion such as ATPsynthase. The major difference from the ciliar
arrays is that the motors are mobile. Thus, here hydrodynamic interactions also
lead to collective motion and an effective repulsion (or attraction) between rotating
motors. The analysis of this system will be presented in Section 8.5.1. In Section
8.5.2 the implications for possible technological applications will be discussed.
8.5.1 Collective Effects in Arrays of Rotatory Motors
Some molecular motors exhibit rotational motion. One of them is ATPsynthase,
which is a truly astonishing molecular machine. Its function is to synthesize ATP.
In doing so, a large enzymatic protruding portion
F
1
is performing a rotation in a
“reaction chamber” consisting of
α
and
β
subunits (see Figure 8.8). The
F
1
-part is
attached to a membrane-embedded, proton-conducting portion
F
0
[19]. It is believed
that protons passing through the transmembrane carrier generate the rotation of the
F
1
part [72, 73, 74]. When protons flow through
F
0
, ATP is synthesized in
F
1
.The
motor is reversible and an excess of ATP provokes a rotation in the opposite direction
and a reverse flux of protons.
In an ingenious experiment, Kinosita et al. have visualized the rotational motion
of ATPsynthase. By attaching a fluorescently labeled actin filament to the
F
1
-part
(see Figure 8.8) the rotation can be directly observed under an optical microscope
[20]. With this technique, it is also possible to directly measure the torque
τ
trans-
fered by the motor to the surrounding liquid. It turns out that 2
πτ
3
ΔG
,where
ΔG
=20
kT
is the energy difference between ATP and ADP. Because every rotation
consumes three ATP this little machine is astonishingly ecient.
In the chloroplasts of plants and in human mitochondria these motors are densely
packed. Typically, a membrane of area 1
μ
m
2
contains around 1,000 motors. Thus,
the mean distance between them is roughly 30nm. All motors rotate and the induced
movement in the cell liquid influences the neighboring motors. Because the motors
are mobile in the cell membrane, this induced velocity field gives rise to a repulsive
hydrodynamic interaction as shown below (see Figure 8.9). We will now analyze the
consequences of this interaction for the collective behavior in an array of motors and
the associated order phenomena [42, 43].
To discuss these effects, we focus here on the most simple description of the
rotating motor where it is modeled as a rotating sphere of radius
R
.Furthermore,
we neglect the surface viscosity of the embedding membrane. For a more general
analysis, see [42, 43]. Then, the rotation of motor
i
at position
x
i
induces a velocity
field
R
3
r
3
ω
×
r
=
τ
8
πηr
2
e
ϕ
,
v
i
=
(8.37)
where
τ
=8
πηR
3
ω
is the torque transfered by the rotating sphere on the surrounding
liquid. The velocity field is in the direction of
e
ϕ
≡
e
z
×
r
/
(
r
sin
θ
), where in spherical
coordinates
r
=
r
(sin
θ
cos
ϕ,
sin
θ
sin
ϕ,
cos
θ
)and
e
z
=(0
,
0
,
1).
