## Abstract

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.

## 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).

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 μm 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 μm s^{−1} for slow component A and 0.02–0.09 μm 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.* (2009*a*,*b*, 2010*a*,*b*, 2011*a*,*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 1*a* displays a neuron, an axon, the injection point of CEs and the coordinate system. Figure 1*b* 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 1*c* 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 *x**_{c}. The origination point of the coordinate system is placed at the left-hand side of the initial pulse. Under these assumptions, the governing equations are
2.1and
2.2where is the number density of CEs in the pausing state, is the number density of CEs in the running state, is a first-order rate constant describing the probability of transition from the pausing to the running state, is a first-order 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 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), is the rate of CE degradation (it is assumed that CEs can degrade in either pausing or running state). can be evaluated as
2.3where is the half-life of particular CEs. For , 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**≤*x**_{c} with a uniform amplitude of *n**_{0in}. Mathematically, this initial condition can be described as follows:
2.4and
2.5where *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 1*c*), 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
2.6Because CEs in the motor-driven state are the only CEs that can move, the total flux of CEs is given by
2.7At *x**=0, a no CE-flux condition is imposed. Equations (2.1) and (2.2) are solved subject to the following boundary condition:
2.8The dimensionless forms of equations (2.1) and (2.2) are
2.9and
2.10where
2.11In equation (2.11), is the dimensionless half-life of the CEs.

The dimensionless forms of initial conditions (2.4), (2.5) are
2.12and
2.13where
2.14The subsidiary equations are
2.15and
2.16Equations (2.15) and (2.16) are solved subject to the following boundary condition: , which stems from equation (2.8). The solutions of the subsidiary equations are
2.17and
2.18In 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:
2.19and
2.20where *I*_{1}(*η*) 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 & 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 half-life 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 are presented for different values of and 2000 days. The latter value effectively simulates the case when NFs do not degrade.

Trivedi *et al.* (2007) estimated the net average velocity of NFs in the motor-driven state (calculated excluding pauses), *v**, to be 0.2 μ ms^{−1}. The value of 0.093 s^{−1}, reported in Jung & Brown (2009) for the rat superior cervical ganglion neuron, is used for . 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, , which leads to . The average velocity of NFs (including pauses) can be estimated as (see Jung & Brown 2009) , 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, *x**_{c}, of 1 mm.

Figure 2 shows the initial stage of transport when kinetic processes (see the kinetic diagram displayed in figure 1*b*) 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 2*a* displays the number density of pausing CEs, figure 2*b* displays the number density of CEs propelled by kinesin motors and figure 2*c* 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:
3.1Using leads to and . Figure 2*a*,*b* thus shows that the equilibrium values of *n*_{0} and *n*_{1} are reached within the first minute. This figure also shows that there is no visible motion 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 2*c* 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 in figure 2*a*,*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 weeks, the effect of CE degradation is already very significant for the CEs with the shortest half-life (20 days). The curves corresponding to and 2000 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 and . 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.

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 is quite significant, especially for days. The behaviour of the total concentration displayed in figures 2*c*–5*c* 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 2*c*–5*c*. 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.

- Received February 1, 2012.
- Accepted April 23, 2012.

- This journal is © 2012 The Royal Society