Chemistry Reference
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parallel (
µ
//
,
E
//
) to its long axis. Consequently, a microtubule oriented under an angle
with E (as defined in Figure 1.15(b)), will have velocity components parallel (
v
y
) and
perpendicular (
v
x
) to the electric field:
1
=
sin 2
θ
E
x
//
⊥
2
(1.12)
2
=
sin
θ
+
+
E
//
⊥
⊥
E O
F
where
µ
EOF
is the mobility of the electro-osmotic flow in our channels. We determine
orientation-dependent velocity for a large number of microtubules. In Figure 1.15(c,d) we
show binned values of measured
v
x
and
v
y
for microtubules at
E
= 4 kV/m. As expected
from Equation 1.12, microtubules that are oriented under an angle with
E
move
perpendicular to
E
in the positive
x
-direction if
< 90
and in the negative x-direction
otherwise (Figure 1.15(c)). Moreover, microtubules that are oriented parallel to
E
(
= 90
)
move faster than microtubules that are oriented perpendicular to E (
= 0
) (Figure
1.15(d)), which is expected if
µ
//
≥
µ
⊥
(Equation 1.12 and References 122 and 129). The
red lines in Figure 1.15(c,d) are fits of Equation 1.12 to the data. The fitted amplitude A =
(
µ
//
-
µ
⊥
)
E
and offset B = (
µ
⊥
+
µ
EOF
)
E
yield information about the different mobility
components.
We measured orientation-dependent velocities for different electric fields and
display the fitted A and B as a function of
E
in the insets of Figure 1.15(c,d). From the
linear fit through the data we derive the values (
µ
//
-
µ
⊥
) = -(4.42 ± 0.12) 10
-9
m
2
/Vs, and
(
µ
⊥
+
µ
EOF
) = -(8.75 ± 0.04)10
-9
m
2
/Vs. In order to determine the values of
µ
//
and
µ
⊥
, we
need to measure the value of the electro-osmotic flow mobility. We do this by a current-
monitoring method
122,130
and we find
µ
EOF
= (1.28 ± 0.01)10
-8
m
2
/Vs. This allows us to
calculate
µ
//
= -(2.59 ± 0.02)10
-8
m
2
/Vs and
µ
⊥
= -(2.15 ± 0.01)10
-8
m
2
/Vs.
The measured mobility anisotropy
µ
⊥
/
µ
//
= 0.83 ± 0.01 is clearly different from
the well-known factor 0.5 in Stokes-drag coefficients for long cylinders. The reason is that
in purely hydrodynamic motion, in which a particle is driven by an external force, the fluid
disturbance around a particle is long-range, decaying inversely proportional to the
characteristic length scale of the particle. However, in electrophoresis the external electric
force acts on the charged particle itself, but as well on the counter ions around the particle.
As a result the fluid disturbance around the particle is much shorter range and decays
inversely to the cube of the characteristic length scale of the particle.
122,131
The force on the counterions has important implications for the interpretation of
electrophoresis experiments in terms of the effective charge. In previous reports of the
electrophoretic mobility of microtubules, their motion was interpreted as a balance
between the electric force on the particle and the hydrodynamic Stokes-drag
coefficient.
117,132
However, because in hydrodynamic motion the fluid is sheared over a
much larger distance than in electrophoresis, this interpretation seriously underestimates
the restraining force, which leads to a similarly large underestimation of the effective
charge.
Instead, we determine the effective charge of a microtubule by calculation of the
-potential and using the Grahame-equation that relates the -potential to effective surface-
charge density. For cylinders, the mobility
µ
//
is directly proportional to the -potential via
µ
//
=
,
129
where and are the solution's dielectric constant and viscosity, respectively.
This yields
= -32.6 ± 0.3 mV, which corresponds to an effective surface-charge density
of -36.7 ± 0.4 mC/m
2
. Using the surface area of a microtubule, we calculate an effective
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