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Proc. R. Soc. A (2012) 468, 3384–3397
doi:10.1098/rspa.2012.0061
Published online 23 May 2012
An exact solution describing slow axonal
transport of cytoskeletal elements:
the effect of a finite half-life
BY A. V. KUZNETSOV*
Department of Mechanical and Aerospace Engineering, North Carolina State
University, Campus Box 7910, Raleigh, NC 27695-7910, USA
This paper presents an exact solution for a two kinetic state model of slow axonal
transport that is based on the stop-and-go hypothesis. The model accounts for two
populations of cytoskeletal elements (CEs): pausing and running. The model also
accounts for a finite half-life of CEs involved in slow axonal transport. It is assumed that
initially CEs are injected into the axon such that their concentration forms a rectangular
pulse; initially all CEs are assumed to be in the pausing state. Kinetic processes quickly
redistribute CEs between the pausing and running states. After less than a minute,
equilibrium is established, forming two pulses, representing concentrations of pausing
and running CEs, respectively. As these pulses propagate, their shape changes and they
turn to bell-shaped waves. The amplitude of the waves decreases, and the waves spread
out as they propagate down the axon. The rate of the amplitude decrease is larger for
CEs with a shorter half-life, but even if CE half-life is infinitely long, some decrease of
the waves’ amplitudes is observed. The velocity of the waves’ propagation is found to be
independent of the CE half-life and is in good agreement with published experimental
data for slow axonal transport of neurofilaments.
Keywords: slow axonal transport; stop-and-go hypothesis; neurons; molecular motors;
exact solution
1. Introduction
Neurons are very polarized cells that have two types of long processes, axons
and dendrites (dendrites receive signals while axons transmit signals); the total
volume of the processes can exceed the volume of the neuron body (soma) by a
factor of 1000. The fact that most chemical synthesis occurs in the neuron soma
and that axons are very long (in a human body they can be up to 1 m in length)
present a problem for axonal growth and maintenance. Indeed, various organelles
and cytoskeletal elements (CEs) need to be transported significant distances in
the anterograde direction; various chemical signals and lysosomal vesicles also
need to be transported retrogradely (back to the neuron soma; Goldstein & Yang
2000; Alberts et al. 2008).
*[email protected]
Received 1 February 2012
Accepted 23 April 2012
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This journal is © 2012 The Royal Society
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Exact solution for slow axonal transport
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Nature solved this problem by employing molecular motors, kinesin and dynein,
which run on microtubules (MTs), to propel various cargos. In axons, kinesin
motors run in the anterograde direction, whereas dynein motors run in the
retrograde direction. The fast anterograde and retrograde transport in axons that
occurs with average velocities ranging from 1 to 5 mm s−1 is easily explained by
the action of these motors because these are typical velocities at which kinesin
and dynein motors walk on MTs (Gallant 2000; Gross 2004; Welte 2004; Pilling
et al. 2006; Ally et al. 2009). Explaining slow axonal transport (characteristic
velocities are 0.002–0.01 mm s−1 for slow component A and 0.02–0.09 mm s−1 for
slow component B) is more difficult because there are no motors that move at
these velocities (Vallee & Bloom 1991; Yabe et al. 1999, 2000; Shah et al. 2000;
Xia et al. 2003; Roy et al. 2007).
Brown and colleagues (Brown 2000; Brown et al. 2005; Craciun et al. 2005)
put forward the stop-and-go hypothesis, according to which CEs involved in slow
axonal transport are propelled by the same molecular motors, kinesin and dynein,
but now the periods of rapid movement are followed by short on-track and long
off-track pauses. Recently, Jung & Brown (2009) developed several models of
various complexities based on this hypothesis. Some extensions of these models
that included accounting for diffusivity of CEs, and numerical and perturbation
solutions of Jung–Brown equations were reported in Kuznetsov et al. (2009a,b,
2010a,b, 2011a,b) and Kuznetsov (2011).
The linearity of equations developed in Jung & Brown (2009) suggests using
analytical techniques to attack this problem. On the other hand, kinetic terms
describing CE transition rates between the pausing and running states present a
significant difficulty on that path. In the appendix to their paper, Jung & Brown
(2009) used the Fourier transform to analyse equations of their model, but have
not presented any explicit solutions of their equations, which would require finding
the inverse Fourier transform. The present paper continues the investigation
initiated in Kuznetsov (2012) and presents an exact solution describing the
propagation of a CE concentration wave in an axon. It is demonstrated that the
obtained solution can be used to compute the wave propagation and spreading.
Unlike Kuznetsov (2012), the solution presented here accounts for a finite half-life
of CEs and is also obtained for a modified, more natural initial condition.
2. Governing equations
A basic two-state equation model developed in Jung & Brown (2009), given by
equation (2.7) in their paper, is considered. This model is extended here by
accounting for a finite half-life of CEs. It is assumed that the half-life of CEs
in the pausing and running states is the same.
Figure 1a displays a neuron, an axon, the injection point of CEs and
the coordinate system. Figure 1b depicts a kinetic diagram showing two CE
populations, pausing and running, and the kinetic processes between them.
It is assumed that the kinetic processes are described by first-order reactions.
Figure 1c illustrates the initial condition. It is assumed that all injected CEs are
initially in the pausing state, and that they initially form a rectangular pulse of
width xc∗ . The origination point of the coordinate system is placed at the left-hand
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3386
A. V. Kuznetsov
(a)
presynaptic terminal
CE wave
x* = 0
x*
axon
point of CE injection
neuron body
(b)
n*0
*
pausing CEs, half-life T 1/2
g *10
g *01
n*1
*
running CEs, half-life T 1/2
(c)
n*0(x*, 0)
n*0in
x*c
direction of CE wave propagation
x*
x* = 0
Figure 1. (a) Sketch of a neuron with an axon; also, an injection point of cytoskeletal elements
(CEs) and a coordinate system in the axon; (b) a diagram showing running and pausing CEs and
kinetic processes between them; both populations of CEs are subject to degradation; it is assumed
∗ , is the same for both populations of CEs; (c) initial condition; it is
that the half-life of CEs, T1/2
assumed that initially all CEs are in the pausing state.
side of the initial pulse. Under these assumptions, the governing equations are
vn0∗
= −g∗01 n0∗ + g∗10 n1∗ − g∗deg r n0∗
vt ∗
and
(2.1)
∗
vn1∗
∗ vn1
=
−v
− g∗10 n1∗ + g∗01 n0∗ − g∗deg r n1∗ ,
(2.2)
vt ∗
vx ∗
where n0∗ is the number density of CEs in the pausing state, n1∗ is the number
density of CEs in the running state, g∗01 is a first-order rate constant describing
the probability of transition from the pausing to the running state, g∗10 is a firstorder rate constant describing the probability of transition from the running to the
pausing state (figure 1b), v ∗ is the net average velocity of CEs in the motor-driven
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Exact solution for slow axonal transport
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state (calculated excluding pauses), x ∗ is the linear coordinate along the axon and
t* is the time. Asterisks denote dimensional variables. Also, in equations (2.1)
and (2.2), g∗deg r is the rate of CE degradation (it is assumed that CEs can degrade
in either pausing or running state). g∗deg r can be evaluated as
g∗deg r =
ln 2
∗ ,
T1/2
(2.3)
∗
∗
where T1/2
is the half-life of particular CEs. For T1/2
→ ∞, equations (2.1) and
(2.2) collapse to equation (2.7) of Jung & Brown (2009). The degradation rates
in the pausing and running states are assumed to be the same. This assumption
is justified as follows. Because the CE haft-life is typically quite large, the
degradation rate is much smaller than the rates of transition between the pausing
and running states. This means that during its half-life, a typical CE will switch
many times between the pausing and running states, which implies that the
solution is determined by the average CE degradation rate calculated accounting
for the CE residence time in each of the kinetic states.
When CEs are injected, they are not linked to kinesin motors; therefore, they
are all initially assumed to be in the pausing state. It is also assumed that initially
∗
.
CEs form a pulse confined between 0 ≤ x ∗ ≤ xc∗ with a uniform amplitude of n0in
Mathematically, this initial condition can be described as follows:
and
∗
(H [x ∗ ] − H [x ∗ − xc∗ ])
n0∗ (x ∗ , 0) = n0in
(2.4)
n1∗ (x ∗ , 0) = 0,
(2.5)
where H is the Heaviside step function. It should be noted that for the initial
condition described by equations (2.4) and (2.5), there are no CEs in the domain
x ∗ < 0 at any time because the motors are assumed to move CEs only in the
anterograde direction (figure 1c), and governing equations (2.1) and (2.2) do not
contain any terms describing diffusion of CEs.
Jung & Brown (2009) also presented more complicated models that included
reversals from anterograde to retrograde motion and vice versa, but the most
important features of the stop-and-go hypothesis are captured by equations (2.1)
and (2.2). Indeed, the stop-and-go hypothesis postulates that the slow average
velocity of CEs can be explained by the fact that they spend most of their time
in the pausing state, and only a small portion of their time in the running state,
when they move at the velocity of fast axonal transport.
The total number density of CEs (in the pausing and running states), which
is the parameter accessible to experiments, is
n ∗ (x ∗ , t ∗ ) = n0∗ (x ∗ , t ∗ ) + n1∗ (x ∗ , t ∗ ).
(2.6)
Because CEs in the motor-driven state are the only CEs that can move, the total
flux of CEs is given by
(2.7)
j ∗ = v ∗ n1∗ .
At x ∗ = 0, a no CE-flux condition is imposed. Equations (2.1) and (2.2) are solved
subject to the following boundary condition:
n1∗ (0, t ∗ ) = 0.
Proc. R. Soc. A (2012)
(2.8)
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3388
A. V. Kuznetsov
The dimensionless forms of equations (2.1) and (2.2) are
vn0
= −n0 + g10 n1 − gdeg r n0
vt
(2.9)
vn1
vn1
=−
− g10 n1 + n0 − gdeg r n1 ,
vt
vx
(2.10)
and
where
n0 =
and gdeg r =
n0∗
∗ ,
n0in
g∗deg r
g∗01
n1 =
=
n1∗
∗ ,
n0in
n=
n∗
,
n0∗in
t = t ∗ g∗01 ,
x=
x ∗ g∗01
,
v∗
g10 =
ln 2
.
T1/2
g∗10
g∗01
(2.11)
∗
g∗01 is the dimensionless half-life of the CEs.
In equation (2.11), T1/2 = T1/2
The dimensionless forms of initial conditions (2.4), (2.5) are
n0 (x, 0) = H [x] − H [x − xc ]
(2.12)
n1 (x, 0) = 0,
(2.13)
xc∗ g∗01
.
v∗
(2.14)
and
where
xc =
The subsidiary equations are
s n̄0 − H [x] + H [x − xc ] = −n̄0 + g10 n̄1 − gdeg r n̄0
(2.15)
and
s n̄1 = −
vn̄1
− g10 n̄1 + n̄0 − gdeg r n̄1 .
vx
(2.16)
Equations (2.15) and (2.16) are solved subject to the following boundary
condition: n̄1 (0, s) = 0, which stems from equation (2.8). The solutions of the
subsidiary equations are
n̄0 = −
(s + g10 + gdeg r )(−1 + H [x − xc ])
(s + gdeg r )(1 + s + g10 + gdeg r )
−e
−sx
g10 exp[−x(g10 + gdeg r )]
g10 x
exp
(s + gdeg r )(1 + s + gdeg r )(1 + s + g10 + gdeg r )
1 + s + gdeg r
g10 H [x − xc ] exp[−g10 (x − xc ) − gdeg r (x − xc )]
(s + gdeg r )(1 + s + gdeg r )(1 + s + g10 + gdeg r )
g10 (x − xc )
× exp
1 + s + gdeg r
+ e−s(x−xc )
Proc. R. Soc. A (2012)
(2.17)
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Exact solution for slow axonal transport
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and
n̄1 =
1 − H [x − xc ]
(s + gdeg r )(1 + s + g10 + gdeg r )
− e−sx
exp[−x(g10 + gdeg r )]
g10 x
exp
(s + gdeg r )(1 + s + g10 + gdeg r )
1 + s + gdeg r
H [x − xc ] exp[−g10 (x − xc ) − gdeg r (x − xc )]
(s + gdeg r )(1 + s + g10 + gdeg r )
g10 (x − xc )
.
× exp
1 + s + gdeg r
+ e−s(x−xc )
(2.18)
In calculating the inverse Laplace transforms of the right-hand sides of equations
(2.17) and (2.18), use is made of the fact that the Laplace transform of the
convolution of two functions is equal to the product of the Laplace transforms
of these functions (Abramowitz & Stegun 1965; Carslaw & Jaeger 1959) as well
as of the property of the inverse Laplace transform that L−1 {e −as F (s)} = f (t −
a)H [t − a]. The following solutions for the CE concentrations are obtained:
n0 (x, t) = −
+
(exp[−t(1 + g10 + gdeg r )] + g10 exp[−tgdeg r ])(−1 + H [x − xc ])
1 + g10
⎧
exp[−x(g10 + gdeg r ) − (t − x)(1 + g10 + gdeg r )]
⎪
⎪
⎨ ×(−1 + exp[g (t − x)](1 + g − g exp[t − x]))
10
⎪
⎪
⎩
10
10
1 + g10
t−x
1
exp[−x(g10 + gdeg r ) − (t − x − t)(1 + g10 + gdeg r )]
1 + g10 0
× (−1 + exp[g10 (t − x − t)](1 + g10 − g10 exp[t − x − t]))
⎫
⎪
⎪
√
√
g10 x I1 [2 g10 tx] ⎬
× exp[−t(1 + gdeg r )]
dt H [t − x]
√
⎪
t
⎪
⎭
⎧
⎨ 1
exp[−(x − xc )(g10 + gdeg r ) − (t − x + xc )
+
⎩ 1 + g10
+
× (1 + g10 + gdeg r )]
× {1 + exp[g10 (t − x + xc )](−1 + (−1 + exp[t − x + xc ])g10 )}
t−x+xc
1
+
exp[−(x − xc )(g10 + gdeg r )
1 + g10 0
− (t − x + xc − t)(1 + g10 + gdeg r )]
× {1 + exp[g10 (t − x + xc − t)]
× (−1 + (−1 + exp[t − x + xc − t])g10 )}
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3390
A. V. Kuznetsov
⎫
⎬
g10 (x − xc ) I1 2 g10 t(x − xc )
× exp[−t(1 + gdeg r )]
dt
√
⎭
t
× H [x − xc ]H [t − x + xc ]
(2.19)
and
exp[−t(1 + g10 + 2gdeg r )](exp[tgdeg r ]
− exp[t(1 + g10 + gdeg r )])(−1 + H [x − xc ])
n1 (x, t) =
1 + g10
⎧
exp[−x(g10 + gdeg r ) − (t − x)(1 + g10 + 2gdeg r )]
⎪
⎪
⎨ ×(exp[(t − x)g ] − exp[(t − x)(1 + g + g )])
deg r
10
deg r
+
⎪
1
+
g
10
⎪
⎩
1
+
1 + g10
t−x
exp[−x(g10 + gdeg r ) − (t − x − t)(1 + g10 + 2gdeg r )]
0
× (exp[(t − x − t)gdeg r ] − exp[(t − x − t)(1 + g10 + gdeg r )])
⎫
⎪
⎪
√
√
xg10 I1 [2 g10 tx] ⎬
× exp[−t(1 + gdeg r )]
dt
H [t − x]
√
⎪
t
⎪
⎭
⎧
exp[−(x − xc )(g10 + gdeg r ) − (t − x + xc )
⎪
⎪
⎨ ×(1 + g + g )](−1 + exp[(t − x + x )(1 + g )])
10
deg r
c
10
+
⎪
1 + g10
⎪
⎩
1
+
1 + g10
t−x+xc
exp[−(x − xc )(g10 + gdeg r )
0
− (t − x + xc − t)(1 + g10 + gdeg r )]
× (−1 + exp[(t − x + xc − t)(1 + g10 )])
× exp[−t(1 + gdeg r )]
⎫
⎪
⎪
⎬
g10 (x − xc )I1 [2 g10 t(x − xc )]
dt
√
⎪
t
⎪
⎭
× H [x − xc ]H [t − x + xc ],
(2.20)
where I1 (h) is the modified Bessel function of the first kind of order 1.
3. Results and discussion
For figures 2–5, parameter values typical for neurofilaments (NFs) are used.
Different estimates for the half-life of NFs are found in the literature. Nixon &
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3391
Exact solution for slow axonal transport
(a)
(b)
1.0
0.10
0.8
0.08
n*1/n0*in
n*0 /n0*in
n
0.6
0.06
0.4
0.04
0.2
0.02
0
2
4
6
x* (mm)
8
10
0
2
4
6
x* (mm)
8
10
(c)
*
1.0
T 1/2 = 20 days; t*= 0
*
T 1/2 = 200 days; t*= 0
0.8
*
T 1/2 = 2000 days; t*= 0
n*/n0*in
*
T 1/2 = 20 days; t*= 1 min
0.6
*
T 1/2 = 200 days; t*= 1 min
*
T 1/2 = 2000 days; t*= 1 min
0.4
0.2
0
2
4
6
x* (mm)
8
10
Figure 2. (a) Number density of pausing CEs. (b) Number density of CEs propelled by kinesin
motors. (c) Total number density of CEs (pausing and running). t ∗ = 0 and 1 min (60 s).
Logvinenko (1986) estimated the half-life of NF proteins to be approximately 20
days; however, data presented in Millecamps et al. (2007) suggest that the halflife of NFs in long peripheral axons with a dense NF network can exceed several
months. In order to show the effect of the NF half-life on the solution, the results
∗
: 20, 200 and 2000 days. The latter value
are presented for different values of T1/2
effectively simulates the case when NFs do not degrade.
Trivedi et al. (2007) estimated the net average velocity of NFs in the motordriven state (calculated excluding pauses), v ∗ , to be 0.2 m ms−1 . The value of
0.093 s−1 , reported in Jung & Brown (2009) for the rat superior cervical ganglion
neuron, is used for g∗10 . According to Trivedi et al. (2007), on average NFs
spend only 3 per cent of their time in the running state and 97 per cent
of their time in the pausing state (doing either short or long pauses; during
long pauses, they completely disengage from MTs). Hence, g∗01 /g∗10 = 0.03, which
leads to g∗01 = 0.0028 s−1 . The average velocity of NFs (including pauses) can be
estimated as (see Jung & Brown 2009) v̄ ∗ = v ∗ g∗01 /(g∗01 + g∗10 ) = 0.0058 mm s−1 ,
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3392
A. V. Kuznetsov
(a)
(b)
1.0
0.10
0.08
n*1/n0*in
n*0 /n0*in
0.8
0.6
0.06
0.4
0.04
0.2
0.02
0
2
4
6
x* (mm)
8
10
0
(c)
2
4
6
x* (mm)
8
10
*
T 1/2 = 20 days; t*= 1 day
1.0
*
T 1/2 = 200 days; t*= 1 day
n*/n0*in
0.8
*
T 1/2 = 2000 days; t*= 1 day
*
T 1/2 = 20 days; t*= 2 weeks
0.6
*
T 1/2 = 200 days; t*= 2 weeks
0.4
*
T 1/2 = 2000 days; t*= 2 weeks
0.2
2
0
4
6
x* (mm)
8
10
Figure 3. Similar to figure 2, but now for t ∗ = 1 day (8.64 × 104 s) and 2 weeks (1.21 × 106 s).
which is approximately equal to 0.5 mm per day, a typical velocity of NFs in slow
axonal transport. Computations are performed for the initial pulse width, xc∗ ,
of 1 mm.
Figure 2 shows the initial stage of transport when kinetic processes (see the
kinetic diagram displayed in figure 1b) quickly redistribute CEs, which are
initially assumed to be in the pausing state, between pausing and running states.
Computations are presented at two times, t ∗ = 0 and 1 min. Figure 2a displays
the number density of pausing CEs, figure 2b displays the number density of CEs
propelled by kinesin motors and figure 2c displays the total number density of
CEs (pausing and running). At the initial stage of the process, the CE degradation
can be neglected and the number densities of CEs in kinetic equilibrium can be
calculated as follows:
∗
n0eq
∗
n0in
=
g10
(1 + g10 )
and
∗
n1eq
∗
n0in
=
1
.
(1 + g10 )
(3.1)
∗
∗
∗
∗
Using g10 = g∗10 /g∗01 = 33.2 leads to n0eq
/n0in
= 0.97 and n1eq
/n0in
= 0.03.
Figure 2a,b thus shows that the equilibrium values of n0 and n1 are reached
within the first minute. This figure also shows that there is no visible motion
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3393
Exact solution for slow axonal transport
(b)
1.0
0.10
0.8
0.08
n*1/n0*in
n*0 /n0*in
(a)
0.6
0.06
0.4
0.04
0.2
0.02
0
10
12
14
16
x* (mm)
18
0
10
20
12
14
16
x* (mm)
18
20
*
T 1/2 = 20 days; t*= 4 weeks
(c)
*
T 1/2 = 200 days; t*= 4 weeks
1.0
*
T 1/2 = 2000 days; t*= 4 weeks
n*/n0*in
0.8
0.6
0.4
0.2
0
10
12
14
16
x* (mm)
18
20
Figure 4. Similar to figure 2, but now for t ∗ = 4 weeks (2.42 × 106 s).
of the waves in this timeframe, which is because the time scale for establishing
the kinetic equilibrium is much shorter than the time scale at which motion of
the concentration waves is noticeable. Figure 2c shows no change in the total
concentrations of CEs in the first minute. This means that the only process that
occurs during the first minute is related to redistribution of CEs between the
pausing and running states, but the total concentration of CEs is not affected
by this.
Figure 3 is similar to figure 2, but is computed at much larger times, t ∗ =
1 day and two weeks. Two interesting features can be observed in figure 3. (i)
CEs are injected in the axon such that their concentration at t ∗ = 0 forms a
rectangular-shaped pulse. After the kinetic processes redistributed CEs between
the pausing and running states, both running and pausing CE concentrations
form rectangular-shaped pulses (see the curves corresponding to t ∗ = 1 min in
figure 2a,b). As time progresses however, the shapes of the waves change from
rectangular to bell-like. In figure 3 at t ∗ = 1 day, this change has just started,
and concentrations of CEs in the central regions of the waves remain constant.
However, at t ∗ = 2 weeks, the bell-shaped waves are already formed. (ii) For t ∗ = 2
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3394
A. V. Kuznetsov
(b)
1.0
0.10
0.8
0.08
n*1/n0*in
n*0 /n0*in
(a)
0.6
0.06
0.4
0.04
0.2
0.02
0
16
18
20
22
x* (mm)
0
24
(c)
18
20
x* (mm)
22
24
*
T 1/2 = 20 days; t*= 6 weeks
1.0
*
T 1/2 = 200 days; t*= 6 weeks
*
T 1/2 = 2000 days; t*= 6 weeks
0.8
n*/n0*in
16
0.6
0.4
0.2
0
16
18
20
x* (mm)
22
24
Figure 5. Similar to figure 2, but now for t ∗ = 6 weeks (3.63 × 106 s).
weeks, the effect of CE degradation is already very significant for the CEs with
∗
= 200 and 2000
the shortest half-life (20 days). The curves corresponding to T1/2
days are still close.
Figure 4 is similar to figure 2, but is computed for t ∗ = 4 weeks. By this time,
the waves have advanced by approximately 15 mm. It should be noted that the
average rate of wave propagation can be changed by changing the values of kinetic
constants g∗01 and g∗10 . These constants determine the ratio of the residence times
of a CE in the pausing and motor-driven states. Because a CE moves only when it
is in the motor-driven (running) state, increasing its residence time in the pausing
state would decrease the average velocity of the wave, and vice versa. Another
interesting feature in figure 4 is that degradation of CEs does not influence the
velocity of the wave. This is because CE degradation does not influence the ratio
of the CE residence times in the pausing and running states; it simply decreases
the number of CEs in both states, hence decreasing the wave amplitude but not
its velocity.
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Exact solution for slow axonal transport
3395
Figure 5 is similar to figure 2, but is computed for t ∗ = 6 weeks. By that time,
the waves have advanced by approximately 22 mm. The effect of NF degradation
∗
= 20 days. The behaviour of the total
is quite significant, especially for T1/2
concentration displayed in figures 2c–5c is qualitatively similar to that displayed
in fig. 6 of Jung & Brown (2009), which depicts profiles measured by experiments
with radiolabelled NFs (these profiles are based on experimental data reported
in Xu & Tung 2000). The velocity of the wave propagation is the same, and both
experimentally and numerically obtained waves exhibit an amplitude decrease
and spreading as they propagate down the axon, but experimental profiles exhibit
faster spreading than those displayed in figures 2c–5c. This is because equations
solved here are based on the simplest model developed in Jung & Brown (2009),
which does not account for the possibility of switching the direction of transport
(there is no retrograde running state in equations (2.1) and (2.2)). Physically, a
switch to retrograde motion can happen if a pausing CE is picked up by a dynein
motor. The presence of CEs moving in both directions, in addition to pausing
CEs, would result in faster spreading of the waves. This is accounted for in the
four- and six-kinetic states models developed in Jung & Brown (2009), but these
models are too complicated to solve analytically.
4. Conclusions
An exact solution for a model of slow axonal transport that accounts for CE
degradation is obtained. It is shown that the process can be divided into two
stages. During the first stage, which lasts less than a minute, equilibrium between
pausing and running CEs is established. Since this time is too short for the CEs
to move a significant distance, the initial shape of the pulse does not change
during this time. During the second stage that can last several weeks, the CE
concentration waves move anterogradely. As the waves move, their amplitude
decreases and the waves spread out. The velocity of the waves is independent of
the rate of CE degradation, but the rate of amplitude decrease depends on that,
although the decrease occurs even if CEs do not degrade (in the latter case, the
amplitude decreases slowly; computations show that in six weeks it decreased by
only about 10%).
The shape of the initial pulse changes slowly; there is very little change during
the first day, but in two weeks the concentration wave becomes bell-shaped.
Because CEs do not move in the pausing state, the velocity of the wave depends on
the ratio between the residence times in the pausing and running states, and the
latter depends on the values of constants determining the rates of CE transition
between these two kinetic states.
The obtained results are in a quantitative agreement with published
experimental results concerning the velocity of the wave and in a qualitative
agreement concerning the rate of amplitude decay and the rate of wave spreading.
To improve the agreement for the last two items, the model needs to be extended
to include the possibility for a CE to change the direction of its motion (due to
attachment to a dynein motor). Such four- and six- kinetic state models have been
developed in Jung & Brown (2009), but they are too complicated for attempting
an analytical solution.
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3396
A. V. Kuznetsov
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