Department of Physics, Wake Forest University, Winston-Salem, North
Carolina 27109 USA
The purpose of this paper is to deduce
whether the maximum force, steplike movement, and rate of ATP
consumption of kinesin, as measured in buffer, are sufficient for the
task of fast transport of vesicles in cells. Our results show that
moving a 200-nm vesicle in viscoelastic COS7 cytoplasm, with the same
steps as observed for kinesin-driven beads in buffer, required a
maximum force of 16 pN and work per step of 1 ± 0.7 ATP, if the
drag force was assumed to decrease to zero between steps. In buffer,
kinesin can develop a force of 6-7 pN while consuming 1 ATP/step,
comparable to the required values. As an alternative to assuming that
the force vanishes between steps, the measured COS7 viscoelasticity was
extrapolated to zero frequency by a numerical fit. The force required
to move the bead then exceeded 75 pN at all times and peaked briefly to
92 pN, well beyond the measured capabilities of a single kinesin in
buffer. The work per step increased to 7 ± 5 ATP, greatly
exceeding the energy available to a single motor.
 |
INTRODUCTION |
The motor protein kinesin transports
organelles within cells. Especially within neurons, where transport
distances along axons can be large, it has long been conjectured that
the work of transport presents a significant energy cost to the cell.
Although considerable progress has recently been made in understanding
the force, velocity, and energy coupling as kinesin drags a latex bead
along microtubules in solution, no quantitative connections have been
made to the forces and work of fast transport in cells.
A single kinesin molecule in buffer can generate a steady force
of no more than 7.5 pN while dragging an attached bead up the potential
well of an optical trap (Svoboda and Block, 1994
; Kojima et al., 1997;
Visscher et al., 1999). When the constraining force is less than the
stall force, a single kinesin can drag the bead at an average velocity
of 800 nm/s in buffer. If the position of the latex bead is measured
more carefully, it is found that kinesin moves along the microtubule in
abrupt 8-nm steps, with each step coupled to the hydrolysis of 1 ATP.
To achieve the observed time-averaged bead velocity of 800 nm/s, a
single kinesin must carry out 100 such steps per second. Each step
takes a mere 50 µs in buffer (Nishiyama et al., 2001). Thus, a
kinesin motor and its vesicle load in buffer are stationary 99.5% of
the time, but when they move, their instantaneous velocity briefly exceeds 100,000 nm/s. Work is done only during the brief intervals when
the bead moves.
In cells, the time-averaged velocity of fast axonal transport is
800-4500 nm/s (Howard 2001). However, kinesin molecules pulling a
vesicle in a cell face a very different load. In the buffer-filled trap, the load is almost purely elastic, whereas in a cell, the load is
almost purely viscous. The effect of viscous load has been explored in
gliding assays (Hunt et al., 1994). The mobility decreases from 1.2 µm/s in buffer to 0.2-0.5 µm/s in an increasingly viscous mixture
of dextran, Ficoll, and trypsin inhibitor. The limiting average value
of the force generated by kinesin in the viscous medium is 4.0-5.2 pN.
Because the position of the microtubule was determined at intervals of
0.1-1.0 s, the individual steps were not resolved.
Recently, high-resolution optical tracking of the Brownian motion of
intracellular particles has enabled researchers to measure, for the
first time, the complex viscoelastic modulus G*(
) within a living cell over a broad frequency range (Yamada et al., 2000). The
modulus was determined for 0.5 
30,000 rad/s in a
kidney epithelial cell line, COS7; the tracked particles were
endogenous lipid droplets with radius 130-250 nm. The method used to
determine G* from Brownian motion has been extensively
tested (Mason et al., 2000).
The purpose of this paper is to calculate the drag force and work
required to move a spherical vesicle within a cell from the
measurements of G* in COS7 and the measurements of kinesin motion in an optical trap. It is assumed that kinesin moves in cytoplasm with the same quick steps as in an optical trap.
| |
FORCE AND WORK IN A NEWTONIAN FLUID |
In a Newtonian fluid at low Reynolds number, the drag force on a
sphere moving with steady velocity is given by the well-known Stokes'
formula, F = 6
a
v, where a is the
radius of the sphere and is the viscosity of the fluid. The work
done by an external force in moving the sphere a distance L
is then given by
|
(1)
|
where T is the time required to travel distance
L. For a sphere with a = 100 nm moving at a
constant 800 nm/s in water ( = 0.001 Pa·s) the work per 8-nm
displacement is 0.012 pN·nm. In vitro, kinesin consumes 1 ATP per
step (Schnitzer and Block, 1997; Coy et al., 1999). Because the
hydrolysis of 1 ATP releases approximately 100 pN·nm of energy
(Howard, 2001), only a negligible fraction of the available energy is
required to drag the sphere at constant velocity in buffer. To
understand the drag force and work required to move organelles within
cells, we need to modify these results, because in cells is many
orders of magnitude larger than in water. In addition, cytoplasm is
shear thinning and viscoelastic.
| |
TIME-DEPENDENT VELOCITY AND FORCE FOR KINESIN IN AN OPTICAL TRAP |
Latex spheres in buffer moved by single kinesin motors advance
with quick, 8-nm jumps rather than steadily (Svoboda et al., 1993;
Coppin et al., 1996; Visscher et al., 1999; Nishiyama et al., 2001).
The individual steps follow the relation
|
(2)
|
with 
= 47 µs and L = 8.7 ± 0.7 nm
for cargo spheres of radius a = 100 nm (Nishiyama et
al., 2001). The velocity corresponding to this displacement is
|
(3)
|
Graphs of x(t) and v(t) from Eqs. 2 and 3,
and the corresponding force for different values of
x0, are shown in Fig. 1,
A-C.
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FIGURE 1
Dynamics of kinesin during a single step in an optical
trap, according to the experimental results of Nishiyama et al. (2001).
(A) x(t) according to Eq. 2. (B)
v(t). (C) The force that kinesin must exert on
the bead for offset x0 = 0, 40, and 80 nm
from the center of the trap.
|
|
The peak velocity v0 is 1.7 × 105 nm/s, about 200 times the average speed; nevertheless,
the Reynolds number is less than 10
6. In a Newtonian
fluid, W is proportional to v2 (Eq. 1), so quick steps are expected to require more energy than a steady
pull over the same distance and same time interval. Indeed, for the
same bead in water, the work done against viscous drag by a quick step
is 1.25 pN·nm. This is about 100 times the work required to move the
same sphere with steady v, but still less than 2% of the
energy available from 1 ATP.
Nishiyama et al. (2001) observed that the 8-nm step consists of two
sequential steps of 4 nm, with time constants of <25 and 40 µs, in
quick succession. However, the time interval between the two substeps
could not be determined with certainty. We therefore choose to use in
our modeling their better-defined parameters for a single 8-nm step.
The calculation of the drag force requires values for
(), the Fourier transform of the bead velocity
v(t). To eliminate the infinite acceleration of the sphere
at t = 0 in Eq. 3, we replaced the leading edge
discontinuity with a linear velocity ramp over 12.5 µs, a time
interval consistent with the finite bandwidth of Yanagida's
measurements. The resultant () extends from
= 0 to = 4 × 105 s1
(Fig. 2). Viscoelastic data are needed
over the same range.
| |
VISCOELASTICITY OF CYTOPLASM |
Stokes' Law and Eq. 1 are inappropriate for the analysis of the
drag force and work required to move vesicles in cells because cytoplasm is shear-thinning and elastic, so energy can be stored and
recovered from the medium during each step. For small stresses and
strains, the time-dependent stress 
(t) is a convolution
of the shear modulus G(t) with the rate of strain
(t) (Tschoegl, 1989)
|
(4)
|
If the fluid is subjected to periodic strain excitation, 
= 0ei
t, the stress is also periodic:
(t) 
() = G*()(), where G*() is the complex modulus. It is customary to split
G* into real and imaginary parts: G* = G' + iG".
Alternatively, it is often useful to relate the stress to the rate of
strain rather than the strain, so that
|
(5)
|
where *() is the complex viscosity (Tschoegl, 1989). The
complex viscosity can also be written in terms of a viscous part '
and an elastic part ": * = ' 
i". Absorption
of energy occurs through G" and '; storage of energy
occurs through G' and ".
It can be shown that *() is the Fourier transform of the shear
modulus (Tschoegl, 1989),
|
(6)
|
This leads to the relations ' = G"/ and
" = G'/.
Recent determinations of G' and G" by laser
tracking microrheology are particularly promising for the current
problem because the frequency range of the measurements is large, the
probe is endogenous, and the probe is comparable in size to many
vesicles. Laser tracking microrheology has been used to track spherical lipid storage granules of radius 130-250 nm in a kidney epithelial cell line, COS7 (Yamada et al., 2000). Figure
3 shows the real and imaginary parts of
G* over the frequency range 0.5-3 × 104
radians/s in the lamellar region of the COS7 cells (Yamada et al.,
2000). The figure shows that G' and G" are
comparable in magnitude to one another, a common behavior in
concentrated high-polymer solutions. Although G' and
G" increase with , the rate of increase is sublinear, so
that both ' = G"/ and " = G'/
decrease markedly at high frequencies. This shear-thinning aspect of
G" reduces the drag work for short, quick pulses; this
effect on W is opposite from that caused by the
v2 factor in Eq. 1.
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FIGURE 3
The storage modulus G' and loss modulus
G" for COS7 cells and for a buffer-filled optical trap. The
COS7 data were measured in the lamellar region of the cells, using
laser tracking microrheology (Yamada et al., 2000). The trap is
characterized as a harmonic potential with spring constant k = 0.071 dyne/cm (Nishiyama et al., 2001), which results in a
frequency-independent storage modulus. Buffer in the trap provides a
frequency-dependent absorption modulus with a slope of 1.0.
|
|
| |
EFFECTIVE VISCOELASTIC PARAMETERS FOR AN OPTICAL TRAP |
The load faced by kinesin in a buffer-filled optical trap differs
from the load in a viscoelastic medium. By evaluating the forces on a
bead executing harmonic oscillation in a water-filled trap with
potential energy V = 1/2kx2,
one finds that the trap alone can be represented by G' = (k/6a), G" = 0. There is no way to include the nonoscillatory
force arising from offset of the oscillation midpoint from x = 0 in this formulation.
The values of G' for a 200-nm-diameter sphere in Yanagida's
optical trap (k = 0.071 dyne/cm (Nishiyama et al.,
2001)) and G" for water ( = 0.001 Pa·s) are shown
in Fig. 3 for comparison to the viscoelastic moduli for COS7 cells.
Note that the absorptive modulus G" in the buffer is many
orders of magnitude smaller than the absorptive modulus of cytoplasm at
low frequencies. Also, the elastic modulus G' for the trap
is less than the elastic modulus of COS7 cytoplasm for all 
10 rad/s.
| |
FORCE IN A VISCOELASTIC FLUID |
Starting from a very general model for the linear stress-strain
relations in a viscoelastic fluid, Thomas and Walters (1965) used
Laplace transform methods to obtain an expression for the drag force on
a sphere moving in a viscoelastic medium. Their analysis was developed
to describe the motions of a sphere falling under the influence of
gravity, as in a falling-ball viscometer. Hwang et al. (1969)
reformulated the Thomas-Walters solution, using Fourier transform
methods, to analyze periodic motion in their magnetic rheometer, in
which a spherical bead immersed in mucus is driven by a time-dependent
sinusoidal force. Their analysis shows that 
(),
the Fourier transform of F(t), is given by a Stokes-like
relation,
|
(7)
|
Because one frequently knows G*() rather than
(), it is helpful to substitute Eq. 6 into Eq. 7
to obtain
|
(8)
|
Eq. 8 specifies how to obtain () from
measurable quantities.
| |
EVALUATION OF  ( ) FOR A SPHERE IN CYTOPLASM |
G*() is known for 0.5 < < 30,000 rad/s for the cytoplasm of COS7 cells (Fig. 3). We assumed that kinesin
moves in cytoplasm with the same quick 8-nm steps as in the optical
trap, and that these steps repeat every 0.01 s. This repetition
time ensures that the average velocity is 8/0.01 = 800 nm/s, which
is typical of fast axonal transport. The force scales linearly with the
radius a of the sphere being dragged. We assume a = 100 nm, the radius in Yanagida's experiments. Many endogenous
vesicles are of comparable size.
The specification of ( = 0) also poses a
problem because G*()/ has not been measured at
= 0. This region is important because any net processive
movement requires a nonzero () at = 0. The problem is not unique to the current situation, and methods for
dealing with it are known (Mason, 2000). If cytoplasm behaves like most
viscoelastic fluids, and lacks a yield point, the contribution to
W could be small compared to the contributions by other frequencies.
For our purposes, we have calculated force versus time curves under two
conditions. Case I extrapolates G*/ to = 0 from 1 = 628 and 2 = 1256 rad/s, the
two smallest nonzero values in our 4096-point array. This gives
' = 2.8 × 107 pN·s/nm2 at = 0. With this assumption, the force on the sphere becomes negligible
between pulses. Case II extrapolates measured values of
G'/ and G"/ from 0.5 1.0 rad/s to = 0. This gives ' = 5.1 × 105 pN·s/nm2 at = 0. As will be
shown below, this adds a "DC offset" to F(t) at all
times during processive movement of the sphere.
| |
CALCULATION OF F(t) |
The force F(t) was calculated by numerical Fourier
inversion of (). The elastic and viscous
components of the force and the total force are shown in Fig.
4 for the case in which G"/ was extended to zero by method I. The viscous component was always positive and showed a roughly Lorentzian shape (Fig. 4 A),
with a peak of 18 pN and a full width at half maximum of 70 µs, about twice the full width at half maximum of the Yanagida velocity function
(Fig. 1). The elastic component of the drag force was negative for
t < 0 and positive for t > 0, with
peak magnitude roughly 20% of the absorptive peak. For a continuous
series of such velocity pulses, as in processive motion, the elastic
energy stored in the cytoplasm at the end of one step was returned to the sphere at the beginning of the next pulse. The resultant total force was initially negative, rose briefly to 16 pN, then dropped to
zero after 5 ms (Fig. 4 B). The cycle then repeated itself.
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FIGURE 4
The forces required to drag a 100-nm-radius sphere in
COS7 cytoplasm, with case I extrapolation of viscosity to = 0. The sphere is assumed to move with the softened Yanagida function (Eq. A1). (A) Elastic and viscous forces. (B) Total
force.
|
|
The force was also calculated by method II, which assigns a much
greater value to '( = 0). The larger value of '( = 0) caused a DC offset of 77 pN in the viscous part of F(t)
and increased the maximum total force to 93 pN (Fig.
5).
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FIGURE 5
The forces required to drag a 100-nm-radius sphere in
COS7 cytoplasm, with case II extrapolation to a much larger value of
'( = 0). The sphere is assumed to move with the softened
Yanagida function. (A) Elastic and viscous components of the
force. (B) The total force.
|
|
In single-molecule studies of kinesin in buffer, in which kinesin drags
a sphere against the force of an optical trap, kinesin becomes detached
from the microtubule if the optical force exceeds 6-7.5 pN (Svoboda
and Block, 1994; Meyhoefer and Howard, 1995; Kojima et al., 1997;
Crevel et al., 1999; Visscher et al., 1999).
| |
WORK IN CYTOPLASM |
If we assumed that the velocity of the sphere was the same in
buffer as in cytoplasm and that '( = 0) was small (case I), the work of dragging a sphere 8 nm evaluated to W = 100 ± 70 pN·nm. The stated uncertainty arises from uncertainties in
the measured values of G" in COS7 cells (Yamada et al.,
2000). An energy of 100 pN·nm is released during the hydrolysis of
one ATP under typical cellular conditions (Howard, 2001). This suggests
that kinesin could function in the cell with the same pulse-like steps
that it exhibits in buffer. However, when the larger case II value of
'( = 0) was used, the work increased to 700 ± 500 pN·nm, or 7 ± 5 ATPs.
In all of the above calculations, the time interval between steps was
fixed at 0.01 s. The results did not change significantly if the
time intervals obeyed the distribution expected for a random process,
as observed for single kinesin molecules in a trap (Schnitzer and
Block, 1997).
| |
DISCUSSION |
Our calculations show that the pulsed forces and velocity that
kinesin exhibits while pulling a bead up the potential gradient of an
optical trap are similar in magnitude to the forces required to move a
small spherical vesicle with similar pulse-like velocity in cytoplasm,
if F goes to zero between pulses (case I). Although the
maximum force required in the case I extrapolation scenario for '(0)
is 16 pN in cytoplasm, 2-3 times the maximum steady force that kinesin
can exert in a trap, this force is needed only 1% of the time in
cytoplasm. In case II, the force required exceeds the maximum available
force generated by 1 kinesin in buffer by a factor of 10 at all times.
More than one kinesin would thus be required to move 200-nm vesicles in
cells if case II is appropriate.
The chemical energy available from the hydrolysis of 1 ATP in a typical
cellular environment is given by (Alberty and Goldberg, 1992; Howard,
2001)
Because 
G0 depends on pH, magnesium
concentration, and temperature, and the concentrations of ADP,
Pi, and ATP can vary from cell to cell, G is
uncertain by at least ±20% for vertebrate cells. It is encouraging
that, at least for the case I extrapolation of cytoplasmic viscosity,
the work required to move a sphere one step, 100 ± 70 pN·nm,
was comparable to the available chemical energy. For larger spheres,
the work increases linearly with radius.
There are a number of assumptions that increase the uncertainty of the
calculated values of F(t) and W:
| 1. |
The cargo particle was assumed to be a rigid 100-nm-radius sphere, but in cells, vesicles are deformable and have a range of radii and shapes. For example, in hippocampal neurons, cargo is packaged in deformable cylinders of length 10 µm and diameter <0.5 µm (Kaether et al., 2000).
|
| 2. |
During single kinesin experiments in an optical trap, and in our calculations, the microtubules are stationary. However, microtubules in cells may move or deform when subjected to forces by bound kinesin.
|
| 3. |
The values of G' and G" vary among different types of cells, and within different regions or times within the same cell (Sato et al., 1984; Valberg and Feldman, 1987; Bausch et al., 1999; Yamada et al., 2000). Moreover, optical and electron microscopy show that actin fibers and microtubules are not randomly oriented, so G' and G" should really be tensors rather than scalars. Movement parallel to the fibers will take less force than perpendicular travel. Similarly, the existence of a nearby stationary cell wall will increase the drag force (Jones et al., 1994).
|
| 4. |
It was assumed that only one kinesin moves the model vesicle.
|
Experiments are needed to reduce these uncertainties and to
understand how motor proteins couple their internal machinery to
intracellular vesicle transport. Optical microscopic methods using a
quad cell have the spatial and temporal resolution to determine whether
kinesin-driven vesicles move with the same quick steps as beads in
buffer. Fast-tracking of green fluorescent protein-labeled organelles (Kaether et al., 2000) in cells with a microscope and line-scan camera may also be able to reveal the presence of individual transport steps. The load-dependence of kinesin stepping could be
studied microscopically in an optical trap if the buffer normally surrounding the trapped bead were replaced by a viscoelastic polymer solution. It may be possible to determine the drag forces on vesicles in cells with optical or magnetic tweezers. Motion of microtubules under load might be observable if the microtubules are fluorescently tagged or are attached to an atomic force microscope tip. Finally, it
may be feasible to count the number of kinesin molecules pulling a
vesicle by labeling the kinesin with green fluorescent protein.
Our model can provide additional insight into the effect of slowing
down individual steps. This could occur because of load dependence
within the kinesin motor or because of elasticity within the kinesin
molecule (Svoboda and Block, 1994; Kojima et al., 1997). We modeled the
possible overall effect of this on F(t) and W by
increasing all time constants in the assumed velocity of the sphere by
a factor of 10, while keeping step size, step rate, and viscoelasticity
unchanged. Using case I extrapolation to = 0, the maximum
force on the bead decreased from 16 pN (Fig. 4 B) to 3 pN,
about half the maximum force kinesin produces while working against the
constant force of an optical trap. The drag work generated by the
slower steps also decreased dramatically, from 100 to 20 pN·nm per
step. If the viscosity were extrapolated to = 0 by the case II
method, lengthening the velocity pulse had only minor effects on the
maximum force, and the work decreased from 700 to 600 pN·nm.
It has been shown experimentally that the average velocity of kinesin
decreases approximately linearly with the magnitude of opposing force
from an optical trap (Svoboda and Block, 1994; Coppin et al., 1997;
Kojima et al., 1997; Visscher et al., 1999). These observations have
been modeled by introducing force-dependent rate constants into a
Michaelis-Menten description of the rate of ATP utilization (Schnitzer
et al., 2000). The breaking of noncovalent bonds, such as those between
kinesin and microtubules, probably depends critically on how fast the
bond-breaking force is applied (Evans, 2001). Bonds under faster
loading withstand larger forces but have shorter lifetimes.
The present study quantifies the drag force and work per step for
intracellular transport by kinesin, assuming kinesin behaves within
cells as it does within an optical trap. A single kinesin motor might
be unable to move 200-nm vesicles through COS7 cytoplasm if it moves
with the same quick jumps as in an optical trap. However, if each step
occurred more slowly, the measured maximum force and energy from a
single kinesin could be adequate.
D.B.H. is grateful for the support of a Graduate Dean's Fellowship
at Wake Forest University.
Address reprint requests to G. Holzwarth, Department of Physics, Wake
Forest University, PO Box 7507, Winston-Salem, NC 27109. Tel.:
336-758-5533; Fax: 336-758-6142; E-mail: gholz{at}wfu.edu.
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ABSTRACT
INTRODUCTION
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