One-dimensional models are presented for the macroscopic
intracellular transport of vesicles and organelles by molecular motors on a network of aligned intracellular filaments. A motor-coated vesicle
or organelle is described as a diffusing particle binding intermittently to filaments, when it is transported at the motor velocity. Two models are treated in detail: 1) a unidirectional model,
where only one kind of motor is operative and all filaments have the
same polarity; and 2) a bidirectional model, in which filaments of both
polarities exist (for example, a randomly polarized actin network for
myosin motors) and/or particles have plus-end and minus-end motors
operating on unipolar filaments (kinesin and dynein on microtubules).
The unidirectional model provides net particle transport in the absence
of a concentration gradient. A symmetric bidirectional model, with
equal mixtures of filament polarities or plus-end and minus-end motors
of the same characteristics, provides rapid transport down a
concentration gradient and enhanced dispersion of particles from a
point source by motor-assisted diffusion. Both models are studied in
detail as a function of the diffusion constant and motor velocity of
bound particles, and their rates of binding to and detachment from
filaments. These models can form the basis of more realistic models for
particle transport in axons, melanophores, and the dendritic arms of
melanocytes, in which networks of actin filaments and microtubules
coexist and motors for both types of filament are implicated.
 |
INTRODUCTION |
The aim of this paper is to provide a simple
macroscopic theory of intracellular transport of cell organelles and
vesicles, here termed "particles." Numerous experimental studies
have established that these particles are equipped with bound motor
proteins, which move them along microtubules and actin filaments
(reviewed by Kelleher and Titus, 1998
; Langford,
1995; Lambert et al., 1999). For example,
anterograde transport of particles along microtubules in nerve axons is
mediated by the motor protein kinesin (Vale et al.,
1985a, b). In this
system the motion of particles is not continuous, but saltatory
(Adams and Bray, 1983; Allen et al., 1982; Rebhun, 1963; Weiss et al.,
1986): particles are transported for distances of typically
~10 µm at a more or less steady velocity of ~1 µm · s
1, but there are pauses lasting for upward of 1 s
in which a given particle is apparently undergoing Brownian motion and
has presumably detached from the microtubule, or is stuck. There is
apparently no published theoretical treatment of the kinetic motion of
particles moving under the combined action of diffusion and motor
transport, and no treatment at all for bidirectional motor transport.
As a first step we have developed a "reaction-diffusion-transport" model: using simple kinetics to describe the interaction of particles with microtubules or actin filaments, and allowing free diffusion of
unattached particles and steady motion of attached particles, net
movement is described by partial differential equations which we have
solved for a number of boundary conditions.
Unidirectional motor transport along a single filament system is the
simplest case found in nature. However, motor transport along
microtubules has been shown in some cases to be bidirectional, that is,
particles can be transported in either direction, and individual
particles sometimes appear to switch direction at random (Cooper
and Smith, 1974). Bidirectional motion occurs (Schnapp et al., 1985) either because microtubules of both polarity are present, or because of the presence on the same particle of two motor
proteins (kinesin and dynein) with opposite polarity (Schnapp and Reese, 1989; Schroer et al., 1989). At first
sight bidirectional motor action would seem to be an ineffective
mechanism for net transport, but in the presence of a concentration
gradient it could nevertheless accelerate the rate of material
transport compared with diffusion. There is an analogy with the process
of "facilitated diffusion," in which the diffusion of a solute is
aided by binding to a protein (e.g., O2 to myoglobin), thus
increasing the amount in solution (Wittenberg, 1966;
Wittenberg et al., 1975; Wyman, 1966).
Facilitated diffusion has also been reported for the
faster-than-diffusion movement by which DNA-binding proteins find their
target sequence, by hopping or sliding along the DNA (Hannon et
al., 1986). We compare the results for unidirectional and
bidirectional transport for a number of boundary conditions.
A further (and perhaps more general) complication in transport studies
is the coexistence of a myosin-mediated transport system (Bridgman, 1999; Tabb et al., 1998;
Wu et al., 1997) which transports particles along the
actin cytoskeletal network (Schliwa et al., 1981).
Actin-based transport is the sole system in the leading edge and
filopodia of nerve growth cones (Evans and Bridgman, 1995; Cramer, 1997 and refs. therein) and at the
tips of melanocyte dendrites (Wu et al., 1998), but
elsewhere actin-based transport coexists with microtubular transport.
The actin cytoskeleton can be considered to be bidirectional because in
general it consists of a network of cross-linked randomly polarized
filaments (although there is at least one unidirectional exception in
the case of Nitella; Sheetz and Spudich,
1983). Bidirectional particle transport on the actin network
has been observed by depolymerizing the microtubules in axons
(Bridgman, 1999; Morris and Hollenbeck,
1995), melanocytes (Wu et al., 1998), and
melanophores (Rodionov et al., 1998; Rogers and
Gelfand, 1998). We have not attempted to include actin-based and microtubule-based transport in the same model, but we show that
bidirectional motor transport may reduce to a type of diffusion: in the
one-dimensional case in which a particle detaches and re-attaches many
times from the filament system in the period of observation, bulk
movement is equivalent to diffusion with a modified diffusion constant.
This is readily accommodated in our model.
Melanin-producing cells are a particularly attractive prospect for
quantitative analysis and theoretical modeling. In the melanophores of
fish and frogs, rapid darkening of the skin is achieved by the
dispersion of pigment granules from a band near the nucleus to the
cytoplasm, with pigment granules being retained in the cytoplasm by a
myosin-actin filament system. The distribution of pigment can be
reversed, presumably via control of the functionally active motor
protein type by a signaling pathway. In mammals, the melanocyte is
responsible for producing pigment granules, the melanosomes, which are
transported down to the ends of dendritic processes, where they are
engulfed by keratinocytes, and thus lend the skin its coloration
(Jimbow and Sugiyama, 1998). Transport is again mixed:
the bidirectional microtubular system transports melanosomes to the
tips of dendrites, where they are captured by an actin system
(Wu et al., 1998). The motor for actin-based melanosome
transport is myosin V, for which motor speeds of 0.3-0.4 µm/s have
been observed (Cheney et al., 1993; Evans et al.,
1998; Wolenski et al., 1995; Mehta et
al., 1999).
With transport in axons and dendrites particularly in mind, we have
found one-dimensional solutions of a reaction-diffusion-transport model
that give the flux of particles and their spatial distribution in
various situations:
| 1. |
Steady-state transport of particles from the cell body along an axon or dendrite ("arm") of finite length, when the concentrations of free particles at each end of the arm are held constant. The results include several experimentally relevant boundary conditions at the tip of the arm, for example when the arm is closed and stationary or growing longitudinally, or when particles are trapped by a cold block. The problem of loading onto microtubules is considered;
|
| 2. |
The rise time for transporting a step increase in the concentration of particles at one end (the cell body) is also calculated as a function of the length of the arm and fitted to a formula based on random walks;
|
| 3. |
Dispersion of particles from their starting position within a long arm, after injection or pulse-labeling at the midpoint of a long arm, corresponding to diffusion along an infinite tube. The results also apply to particles injected or pulse-labeled at a particular location.
|
Some of these situations have analogs in the classical theories of
diffusion or heat conduction (Carslaw and Jaeger, 1959) but, as indicated above, the phenomena are generally more complex. For
example, diffusion of free particles and motor transport cannot be
considered as separate pathways, except when attachments to filaments
are irreversible. Although the literature suggests that cellular
organelles may not diffuse readily in cytoplasm, it is important to be
able to predict the contribution of free-particle diffusion for a given
value of the diffusion constant D. Free diffusion adds
significantly to motor transport over short distances when particles
bind weakly to filaments.
In the Discussion section, the ability of these unidirectional and
bidirectional models to describe specific cellular transport systems is
assessed after reviewing the experimental literature, and ways are
suggested of overcoming some obvious deficiencies of the models. For
example, motor transport is treated phenomenologically by assuming a
steady motor velocity v, which should be viewed as a
constitutive coefficient for a law of active transport (flux 
density of bound particles) analogous to the diffusion constant for
Fick's law of free diffusion (flux density gradient of free particles). Because the models work with particle densities, they predict only the macroscopic behavior of a large number of particles viewed as a continuous fluid moving in the cytoplasm. However, the
densities as functions of position can also be interpreted as
probability distributions for the location of a single particle. The
meanings of mathematical symbols used in this paper are defined in
Table 1.
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TABLE 1
Glossary of mathematical symbols. Alternative formulas
apply to unidirectional and bidirectional models, respectively
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A THEORY OF MOTOR-ASSISTED TRANSPORT |
General equations
If attention is restricted to a single filament system
(microtubules or actin), a macroscopic transport theory of particles can be formulated in terms of the laws of diffusion and kinetics. For
simplicity, all motions are restricted to one space dimension, but
generalizations to particle motions in three dimensions and two- or
three-dimensional filament networks are straightforward.
The basic assumptions are 1) a "particle" consists of a complex
between an organelle or vesicle and motor proteins (permanently attached to the surface membrane); 2) particles either diffuse freely
in solution or move on a filament at a steady velocity v
(the "motor velocity"), which may depend on the number of motors on
the particle; 3) binding to and detachment from filaments are kinetic
processes specified by first-order rate constants, which include
factors as appropriate for lateral diffusion and the density of motor
proteins and filaments; and 4) in the general case of bidirectional
transport, binding is followed by motion in either direction, as a
result of the presence of filaments and/or motors with both polarities.
For convenience it is assumed that it is the polarity of the filaments
that determines the direction in which particles are transported.
The one-dimensional case describes transport between two planar
boundaries, say at x = 0 and x = L, all
particle concentrations varying only along the x-axis (Fig.
1 A). A fraction of the space between the boundary planes is homogeneously occupied with filaments oriented along the x-axis. The remaining space allows
diffusion of unbound particles in the x-direction.
"Outward" filaments transport particles toward the right-hand end
(x = L) and "inward" filaments toward the left-hand
end (x = 0). All filaments are assumed to span the
intervening space. This model may be interpreted as a simplified
description of axial transport in an axon or cellular dendrite
("arm") between the cell body and the tip of the arm (Fig.
1 B), which motivates various boundary conditions at each end of the arm (discussed in the following sections). For convenience we use the terms related to the cell biology ("arm," "cell
body," "tip") in most of what follows.
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FIGURE 1
(A) Geometry of the one-dimensional
bidirectional model. Diffusion and transport occur in a medium between
planes x = 0 and x = L at fixed
temperature and pressure. A fixed fraction of this medium is filled
with a homogeneous mixture of right-directed ("outward") and
left-directed ("inward") filaments in known proportions, on which
particles (not shown) are moved right or left by motor transport at
velocities v+, v. The remaining
space allows diffusion of unbound particles in the
x-direction, while lateral diffusion is assumed to have
homogenized any lateral concentration gradients of free particles.
First-order rate constants k+, k
determine binding to outward and inward filaments. The medium is open
at x = 0 and x = L to reservoirs of
free particles at concentrations n, ñ. Outward
filaments project into the reservoir at x = 0 and
inward filaments into the reservoir at x = L by
distances lpu,  pu,
along which "loading" of particles onto the projecting filaments
occurs. (B) A cartoon of bidirectional particle transport in
a cell "arm" (axon or dendrite), equivalent to A. The
cell body and the tip of the arm act as reservoirs. Particle fluxes
(number/second/unit area) in the arm are assumed to be axial,
homogeneous throughout the arm, and equal to those obtained in
A.
|
|
We first derive particle equations of motion for the most general
bidirectional transport model. Let v+ > 0
and v < 0 be the motor velocities in the
direction of increasing x for particles traveling on outward
and inward filaments, respectively. Let k+ and
k be the corresponding first-order rate constants for binding to filaments, and
k'+ and
k' the rate constants for detachment.
The final parameter is the diffusion constant D of the free particle.
Let no(x, t) be the number density
(per unit volume) of free particles at distance x along the
arm at time t, and n±(x, t) the
densities on right- and left-directed filaments.
no(x, t) and
n±(x, t) satisfy reaction-diffusion-transport
equations
|
(1a)
|
|
(1b)
|
The particle flux J(x, t) (the number per second per
unit area normal to filaments at position x) arises from
diffusion of free particles and convection of bound ones, so
|
(2a)
|
where
|
(2b)
|
Because 
(no + n+ + n)/t = 
J/x,
J is a constant of the motion under steady-state conditions. The
sign of bound-state fluxes is determined by the polarity of the
filament, while the diffusion flux can be of either sign; thus
particles can be exchanged between the ends of an arm even when the net flux is zero.
Motor-assisted transport can be understood in terms of mean lifetimes
off, ± and mean path lengths
loff, l± for free and
bound particles, where loff2 = Doff and l± = |v±|±, so
|
(3)
|
The average speed vD = loff/off = 
of free
diffusion over the lifetime of the "off" state is also useful. As
an example, values for a 1-µm diameter particle moving on
microtubules might be v± = ±1 µm/s, k± = 1 s1,
l± = 10 µm, and D = 0.1
µm2/s (Table 2), giving
loff = 0.224 µm and
vD = 0.447 µm/s. Binding rates reflect
the density of filaments and intracellular structures may reduce the
apparent value of the diffusion constant; thus this estimate for
vD may be an upper limit.
Dispersion and drift
Consider a sequence of many particle displacements, each
initiated by binding to a randomly selected filament which determines the direction of motion and terminated by detachment. If free diffusion
is absent and periods of detachment are negligibly small, these random
walks define a form of facilitated diffusion with known mean bound path
lengths l±. However, this effect is generally
accompanied by convection of particles at the drift velocity
|
(4a)
|
When 
0, motor-assisted diffusion occurs
in a frame of reference moving with this velocity, with an effective
diffusion constant
|
(4b)
|
the equilibrium average of free and bound contributions with
binding constants K± for filament systems of
opposite polarity. This formula is exact in the limit of many
attachment cycles, even for unidirectional transport
(K = 0) where all displacements are in
the same direction. In this case a spread of displacements about the
mean arises from variable attachment times on filaments.
Boundary conditions, loading
To describe particle transport in a cell arm (for example, an
axon or dendrite), solutions of Eqs. 1 require appropriate boundary conditions. In the first instance, let the boundaries at x = 0 and L be open to reservoirs of free particles at
fixed concentrations n and ñ, respectively.
Throughout, the reservoir in x < 0 is identified as
the cell body, which is assumed to be large enough that n is
constant. At the tip of the arm, the situation is more complicated and
is dealt with below. If filaments in the arm do not protrude into these
reservoirs, the boundary concentrations for bound particles must be
zero. However, outward filaments are known to extend back into the cell
body, for example under the plasma membrane (Wu et al.,
1998). In that case, outward filaments emerging from the cell
body are already "loaded" with particles, and the boundary value
for n+(x, t) at x = 0 may be
written as 
n, where will be called the "degree of loading."
The tip of a cell arm is, in some cases, closed rather than open to a
particle reservoir, though a store of particles in the tip can be
achieved by the presence of an auxiliary filament system (Wu et
al., 1998). Moreover, outward transport of particles at the tip
is often associated with its physical growth, which is compatible with
a closed but moving boundary. For the time being, we choose to work
with fixed concentrations of free and minus-directed particles at the
tip end x = L, giving boundary conditions
|
(5)
|
The degree of loading 
in the tip may be smaller than
or even zero. Under steady-state conditions, predictions obtained with these boundary conditions may readily be transferred to a closed
tip, whether stationary or moving.
The loading coefficients may be calculated kinetically in terms of the
"pick-up" lengths defined in Fig. 1. For outward filaments at
x = 0 extending back into the reservoir by a distance
lpu, solving the steady-state reaction-transport
equation v+dn+(x)/dx = k+n k'+n+(x)
for lpu < x < 0 and
n+(lpu) = 0 gives
n+(x) = K+n{1 exp(k'+(x + lpu)/v+}. Hence
= K+{1 exp(k'+lpu/v+)} < K+.
Solution of equations, scaling
Solutions of Eqs. 1 are first sought for the case of
unidirectional motor transport where all particles have only one kind of active motor protein and the filaments are unipolar
(k+ = k, k = 0).
Bidirectional motor transport, in which filaments of both polarities
exist or different motors of opposite polarity exist on the same
particle, is studied here only for the symmetrical case
k± = k, k'± = k',
and v± = ±v. Algebraic solutions simplify
considerably when the arm is longer than the diffusion length
loff, which is expected and assumed throughout. For the bidirectional case, it is convenient to make separate predictions for the case when particles bind irreversibly to filaments until motor action takes them to the end. In both cases the predicted behavior is a function of the four basic parameters D, v, k,
k' plus loading parameters and the length of the arm.
The required amount of computation is eased by using scaling
relationships that follow from the existence of scaled dimensionless solutions. These may be obtained by choosing v/k and
1/k as units of length and time, which leads to a
dimensionless detachment rate 1/K = k'/k and a
dimensionless diffusion constant Dk/v2. In
this way, scaling laws for the concentrations
|
(6a)
|
|
(6b)
|
in which the four basic parameters are displayed can be derived
from Eqs. 1, where 
> 0 is an arbitrary scaling factor and i = 0, ±. Thus the number of independent parameters is
reduced from four to two, say the detachment rate k' and
motor velocity v, while D and k can be
held fixed. This procedure is adopted throughout the paper, setting
D = 0.1 µm2/s and k = 1
s1. Equation 6a shows that the effects of reducing the
diffusion constant by a factor of 2 < 1 are
equivalent to those obtained by raising the motor velocity and position
x along the arm by a factor of 1/, so computed solutions should be available for more than one motor velocity. Similarly, the
effect of reducing the binding rate by a factor of is equivalent to
keeping the equilibrium constant K unchanged, raising the
motor velocity by a factor of 1/2, and reducing the
position coordinate by 1/2 (Eq. 6b). Similar results
follow for the net outward flux J at the tip of a cell arm.
In terms of the mobility J/n,
|
(7a)
|
|
(7b)
|
so results for a range of arm lengths are required to access the
effects of variations in D or k in terms of known
effects of variations in v and k'.
| |
UNIDIRECTIONAL TRANSPORT |
Unidirectional transport occurs when the filament system is
unipolar and all active particle motors have the same polarity. The
mathematical description of this model is equivalent to theories of
sedimentation or electrophoresis for a unimolecular reaction (Cann, 1970; Gilbert and Jenkins, 1959;
van Holde, 1962). These theories often ignore free
diffusion and focus on finding localized propagating solutions
generated by nonlinear reaction kinetics. The binding of particles to
filaments is a simple bimolecular reaction, for which the binding rate
is a product of the concentrations of free particle and free binding
sites and nonlinear in the above sense. For organelle transport, it can
safely be assumed that particle concentrations are dilute, leading to
Eqs. 1, which are linear in the concentrations. For these equations,
stable traveling-wave solutions for a group of particles are not expected.
A unidirectional model follows from Eqs. 1 by setting
k = 0, which is true when inward
filaments are absent. The polarity subscript for rate constants for
outward filaments is now omitted. Steady-state solutions are sought
first, then transient solutions resulting from a step increase in
particle concentration in the cell body, or a localized pulse injection
of particles within a dendritic arm.
Steady-state solutions
Solutions of the steady-state form of the two remaining equations
of (1), namely
|
(8a)
|
|
(8b)
|
can be obtained by noticing that the flux J = Ddno(x)/dx + n+(x)v is
independent of x (a first integral), giving the single
differential equation
|
(9)
|
for the bound concentration profile. The general solution can be
written as
|
(10a)
|
|
(10b)
|
where
|
(11)
|
and no(x) is obtained from Eq. 8b. The constants of integration A, B, and J
follow by applying the first three boundary conditions of Eq. 5 for
fixed free-particle concentrations n, ñ at
x = 0 and L, respectively (Fig.
1 A).
Irreversible attachment
When k' = 0, then
q± = ±q, where 1/q = loff = 
is the mean path
length on filaments. Assuming that L > loff, the boundary conditions yield A = kn/qv, B = kñ/qv, and a net outward flux
|
(12)
|
where vD is a diffusional velocity, or
diffusive displacement over the mean binding time
off = 1/k. In contrast to transport by
free diffusion, this flux is independent of the length of the arm.
Because exp(qL) 
1, particles cannot diffuse freely
down the whole arm without binding, and the flux is independent of particle concentration ñ in the tip, even although
such particles may diffuse back into the arm and bind to filaments.
When the tip is closed and stationary, there is an accumulation of
particles in the tip and steady-state conditions do not apply. If the
tip is closed but extending at velocity u, then J = ñu and Eq. 12 determines the tip concentration
ñ = n(vD + v)/u,
which will be higher than the cell-body concentration n if
the arm is growing slowly.
Equation 12 expresses the outward steady-state flux in terms of the
concentration n of free particles in the body, but this flux
is conserved along the arm. Away from the cell-body end x = 0, all particles have bound to filaments and the flux is entirely due to motor transport. Thus the concentration of such particles is
n(vD + v)/v, since
multiplication by v yields the predicted flux. This interior
concentration is generally not equal to n; this can be
understood as follows. With no loading in the cell body ( = 0),
the flux in the entrance to the arm where particles have not yet bound
is entirely diffusional and proportional to the velocity
vD, which is usually slower than the motor speed v; as particles bind to filaments and are transported more
rapidly at speed v, their lineal density is decreased if a
steady state prevails. The disparity between the effective mobilities
(flux per unit particle density) in the entrance and the interior of the arm is reduced when particles are loaded onto filaments in the cell
body ( > 0), but it should be remembered that the cell body
then contains bound particles and the total density of such particles
is (1 + )n. Such loading creates a parallel
transport path in which particles remain bound throughout the entire
outward journey, with a flux equal to (n)v.
The particle flux in the presence of motor filaments is generally much
higher than from diffusion alone, JD = nD/L by Fick's law. The degree of facilitation
|
(13)
|
is much greater than unity even with no loading ( = 0),
except when the arm is shorter than loff, which
is under 1 µm for a 1-µm-diameter particle (Table 2).
Multiple attachments
With a finite detachment rate k', the expression for
the flux is more complicated but differs little from Eq. 12 unless
detachment is so rapid that the mean path lengths
loff and lon 
v/k' are similar. Assuming |q±L| > 1, the complete expression for the net flux is
|
(14)
|
where K = k/k' and 
Kv/vD = lon/loff is the mean-path
ratio, which is large if the motor speed is high or particles remain bound for long periods. The corresponding facilitation factor is
|
(15)
|
These results are very similar to Eqs. 12 and 13, which are
recovered when k' 
0. However, the formulae differ in
detail. When K, loading in the cell body is weak
and particles must diffuse into the arm before binding; the flux is
limited by the motor velocity v for slow motors (v
vD) and by the diffusional velocity
vD in the opposite limit of fast motors. When
= K, cell-body loading is optimal and the flux
arises entirely from particles that bind before entering the arm (Fig.
2). These differences arise because, with
reversible detachments, pathways into the arm by diffusion and
cell-body loading are not independent.
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FIGURE 2
Unidirectional transport, steady state. The
steady-state outward flux J in a cell arm per unit
concentration n of free particles in the cell body as a
function of motor velocity from Eq. 14, different loading parameters
in the cell body as shown and K = 2. With no
cell-body loading, the limiting flux of J/n at high motor
speeds is the diffusional velocity vD =  , here equal to 0.316 µm/s. With optimal
loading ( = 2), J/n is twice the motor velocity,
reflecting the fact that the concentration of bound particles in the
body is 2n.
|
|
Equations 10 show that the concentration profiles for free and bound
particles within the arm are basically flat except at boundary layers
of widths 1/q+ and 1/q
at the ends. The absence of concentration gradients in the central zone
shows that transport in this unidirectional model is clearly convective rather than diffusive, even though the flux is limited by diffusion into the arm when cell-body loading is ineffective. The concentrations of free and bound particles in the central zone are
no = J/Kv, n+ = J/v, showing that in this zone reaction-equilibrium is established
with n+/no = K,
though not in the boundary layers. The flux J can therefore
be interpreted in terms of the total concentration of particles
no + n+ in the
central zone, moving at the mean speed
|
(16)
|
for particles with a duty ratio K/(K + 1) (the
bound steady-state fraction). Equation 15 shows that free diffusion in
the free periods increases the central concentrations.
Transient solutions
The rise of flux in the tip
If the arm is initially free of particles, and particles are
suddenly introduced at concentration n in the cell body
(x < 0), there will be a time delay before particles
arrive at the tip. If particles arriving in the tip region are
prevented from diffusing back into the arm and rebinding, for example
by imposing a cold block or sink, the tip response is measured by the
net outward flux at the end of the arm. If the tip is closed and
stationary, the response is measured by the concentration of particles
in the tip. Although the latter may be closer to in vivo conditions, the tip concentration is sensitive to the value of , which is raised by mechanisms for storing particles in the tip region, so
numerical calculations were made for the rise of flux in the presence
of a sink at x = L. What behavior is expected?
In the unidirectional model, particles binding in the cell body will
travel down an arm of length L in time L/v when
no detachments occur (L lon v/k'). Initially free particles experience an extra delay of
order of the binding time 1/k, which will be partially offset if they can diffuse into the arm. In the opposite limit L
lon, the rise time for flux at the tip should be
approximately L/, where is the
mean displacement velocity (Eq. 16). These estimates ignore diffusion
of free particles, which operate between pauses and should therefore
speed up the rise of flux somewhat for short arms and weak binding
(K < 1). Diffusion down the entire arm contributes
negligibly to transport in long arms, since the rise time is of order
L2/2D, which is greater than
L/ for L > 2D/, typically
under 1 µm for microtubule motors with ~ 1
µm/s.
Fig. 3 shows the rise of flux at the end
of a 20-µm arm, calculated for various rates of detachment that span
the limiting cases described above. The time for the flux to rise to
50% of its final value is qualitatively described by the empirical
formula 0.5 
L/. The computed
rise time increases linearly with the length of the arm except for very
short arms and rapid detachment, where the flux rise is more rapid.
This difference is due to free diffusion because it disappears when
calculations are made with D = 0. There is also a
spread of arrival times arising from pauses, which is most significant
if K < 1, when particles are mostly paused. The length
of each pause is controlled kinetically and obeys a Poisson
distribution with a mean pause time of 1/k. Conversely, when
K > 1, a distribution of excursion times for bound
particles is expected, but only if the arm is long enough to allow many attachment cycles; this condition was not fulfilled in calculations presented in the figure. Thus the computed rise times can be simply understood, but the dispersion of arrival times reflected in the shape
of the flux-time curve requires a deeper analysis. Dispersive aspects
of motor-assisted transport are considered next in relation to a
different experimental protocol.
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FIGURE 3
Unidirectional transport, stepwise increase of
concentration in the cell body. (A) Rise of flux with time
in the tip region of a 20-µm arm, initially without particles, after
introducing unit concentration of particles in the body at time zero,
for the unidirectional model for different rates of detachment k' = 5 (black line), 0.5 (red line), 0.05 (green line), and 0.005 s1 (blue
line). The motor speed is 1 µm/s, for which the full-transit
time on filaments is 20 s, and k = 1
s1, D = 0.1 µm2/s. Fluxes
are normalized to their steady-state values calculated at long times,
which agreed with the values predicted by Eq. 15. Numerical
calculations were made by direct integration of Eqs. 1, using upwind
differencing on the convective term (Press et al., 1992)
and a smaller time step for the diffusion component. (B)
Rise times to half the maximum flux as a function of arm length for the
same set of detachment rates, plotted logarithmically (the
green and blue curves overlap). Except at the
highest rate of detachment and the second highest rate for the shortest
arm, the results fit a linear law, as expected from the empirical
formula given in the text.
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Dispersion from a point distribution
Distributions p(x, t) of particle displacements
x as a function of time t can be studied
experimentally by tracking particles from their initial positions
within the arm, or by injecting particles into the arm at one point.
The form of these distributions may depend on whether the particles are
initially free or bound, but the effects of initial conditions are
removed after several cycles of attachment. For the unidirectional
model, Fig. 4 shows the spatial
distribution of particles for various motor speeds at a fixed time
t = 5 s after injection of free particles at
x = 0. The initial delta-function distribution is
translated by motor action, and broadened by motor action and free
diffusion before binding and during subsequent pauses. The figures show
that single transits occur, producing a sharp right-hand edge in the
distribution of displacements at x = vt in Fig.
4 B when D = 0, although in Fig.
4 A this edge is broadened by free diffusion. The most
probable displacement for each motor speed is close to the mean
displacement t, where /v = 2/3
as K = 2.
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FIGURE 4
Unidirectional transport, dispersion. Computed
distributions of particle displacement x in the middle of an
infinite arm 5 s after starting at x = 0 with all
particles detached from filaments. The two cases are (A)
with free diffusion (D = 0.1 µm2/s), and
(B) without free diffusion. The curves correspond to
different motor speeds v = 0.1 to 1.0 µm/s as shown. The
binding constant K was set at 2.0 and other parameters as in
Table 2.
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The asymptotic form of these distributions at large times was not
achieved in Fig. 4, but can be obtained analytically by Fourier-transform methods. The expected form after many attachment cycles is the classical diffusion law
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(17)
|
about the asymptotic mean displacement x(t) ~ t, which also appears in theories of electrophoresis
(Cann, 1970). The effective diffusion constant is
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(18)
|
reflecting diffusion of free particles in solution and a Poisson
distribution of bound periods. This result is also obtained from Eq. 4b
by setting K = 0. The second term
contains the variance of this Poisson distribution, proportional to
r(1 r), where r = K/(K + 1) is
the bound fraction or duty ratio in attachment equilibrium. Diffusion
is enhanced if Don > D, where
Don = v2/(K + 1)(k + k') (v )2/k',
as expected from Eq. 4. When the duty ratio tends to unity at fixed
k, Don becomes small and particle motions
approximate to uniform translation at the motor speed v.
The distribution (Eq. 17) was confirmed computationally by plotting a
time-scaled distribution against a time-scaled displacement from the
mean (Fig. 5), which asymptotes to the
exponential factor in (17). At intermediate times, a truncated form of
this distribution may appear because a significant fraction of
displacements arise from full-transit events (those in which particles
attach in the cell body and are transported to the tip without
detachment) rather than multiple attachments, as can be seen in Fig.
5B, where free diffusion is absent.
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FIGURE 5
Unidirectional transport, dispersion at long times. The
approach to a Gaussian distribution of scaled deviations y = (x t)/t1/2 from the mean displacement
for particles spreading from a point distribution, as in Fig. 4 with
v = 1 µm/s. The indicated distribution slowly
approaches the function
(4 Deff)1/2
exp(y2/4Deff) from Eq. 17 (results
shown are for t = 5, 20, 100, and 1000 s). The
standard deviation of the last curve (0.341 µm) is close to the value
2Deff = 0.363 from Eq. 18 with D = 0.1 µm2/s, k = 1 s1,
k' = 0.5 s1. The first curve shows the
truncation effect seen in Fig. 4 and associated with single
excursions.
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Apart from this truncation effect, it turns out that the persistence of
the effects of initial conditions, such as the proportion of particles
initially bound, is felt only for the time 1/(k + k')
required to bring free and bound particles into reaction equilibrium. No further change in the form of this distribution occurs over the
cycling time c = 1/k + 1/k', which
is much larger than the equilibration time if K
1. Thus the initial equilibration of free and bound particles is
all that matters, and subsequent attachment cycles merely produce
dispersion about the average velocity according to
Eq. 17. This feature is peculiar to the unidirectional model; very
different behavior is found with bidirectional models.
The persistence of initial conditions is also reflected in the
time-dependence of low-order moments
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(19)
|
of the distribution, in particular the mean displacement
and variance S(t) = 
2. For the models of this paper,
these functions can be calculated exactly, from the appropriate
differential equations (Appendix A) or by Fourier methods. The former
method is more efficient. For the unidirectional model,
|
(20a)
|
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(20b)
|
where p is the initial fraction of bound particles and
D* is given by Eq. 18. The constants A, B,
C are given in Eqs. A6. There is an initial temporal phase
reflecting the bound fraction that persists for the equilibration time,
followed by a second phase of diffusion about the mean, which lasts
indefinitely (Fig. 6). Endogenous
particles are expected to be in kinetic equilibrium with their
filaments (p = K/(K + 1)), in which case
there is no transient in the mean displacement and the variance-time
curve approaches linearity with a single exponential function
(B = C = 0); the predicted behavior for injected
particles (p = 0) is more complex. These predictions
could be tested by fitting experimental moment-time curves obtained
from an ensemble of tracked-particle distributions; the same method has
been used for bead assays of kinesin motility (Svoboda et al.,
1994).
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FIGURE 6
Unidirectional transport, time-dependence of the mean
displacement , and variance
S(t) in the unidirectional model, calculated from Eqs. 20 with k' = 5 s1 (A) and 0.05 s1 (B), and values of D, v, and
k in Table 2. The initial transients are functions of the
initial particle state, either free (p = 0), bound
(p = 1), or an equilibrium mixture (p = K/(K + 1)). Memory of the initial state persists over a
time of order 1/(k + k') = 0.17 s (A)
or 0.95 s (B).
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SYMMETRIC BIDIRECTIONAL TRANSPORT |
The reaction-diffusion-transport equations (1) define a general
bidirectional transport model. Here we consider only the symmetric case, taking k+ = k k,
k'+ = k' k' and v+ = v v. This
symmetric model describes particles with only one type of motor moving
on a bipolar filament network with an equal mixture of polarities, for
example myosin-V on F-actin. The same model could also be used for a
unipolar filament network if particles possess two kinds of motors with
opposite polarity but the same motor speed, and the same attachment and
detachment rates, which may be approximated by kinesin and dynein
motors on microtubules. The relevance of these models is further
considered in the Discussion section, but we attempt to address both
systems by presenting computation results for a range of motor speeds and detachment rates. The binding rate and diffusion constant are
usually fixed in the following examples at 1 s1 and 0.1 µm2/s, but the scaling laws (Eqs. 7 and 8) are structured
in such a way that predictions for lower values of both these
quantities can also be obtained.
As before, steady-state transport properties are investigated first,
followed by transient responses and dispersive behavior.
Steady-state solutions
Irreversible attachment
Bound particles are likely to proceed down the arm in a single
pass when L lon, which is possible with
microtubule motors in short arms (under 10 µm). Here we consider the
limiting case k' = 0. The steady-state solutions of the
symmetrized form of Eqs. 1 are
|
(21)
|
where 1/q = loff = 
and qL 1. From Eq. 2, the
net outward flux is
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(22)
|
The diffusion velocity vD loff/off is now equal to
, but only half of the particles entering the
arm bind to filaments directed into the arm as required; the remainder
are returned by motor action to their starting points.
With a sink at the tip (ñ = 0), Eq. 22 has the
same structure as Eq. 13 for unidirectional transport and the
discussion underneath applies in equal measure. When the tip
concentration ñ is not held fixed and the arm is
closed, the "no-flux" condition at the outer end is achieved when
the concentration of free particles has risen to its steady-state value
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(23)
|
for which J = 0. Under these conditions, particles
are exchanged by motor action on both filaments, at a rate
Jex obtained from bound-state fluxes in the
central zone of the arm away from boundary layers, as
|
(24)
|
This rate of exchange equals the net rate of outward transport
with a sink at the tip. The way in which transport is shared between
free diffusion and motor action in the loading zones is shown in Fig.
7 for both types of boundary conditions
at the tip.
TOP
ABSTRACT
INTRODUCTION
