Modelling transport and mean age of dense core vesicles in large axonal arbours

Abstract

A model simulating the transport of dense core vesicles (DCVs) in type II axonal terminals of Drosophila motoneurons has been developed. The morphology of type II terminals is characterized by the large number of en passant boutons. The lack of both scaled-up DCV transport and scaled-down DCV capture in boutons results in a less efficient supply of DCVs to distal boutons. Furthermore, the large number of boutons that DCVs pass as they move anterogradely until they reach the most distal bouton may lead to the capture of a majority of DCVs before they turn around in the most distal bouton to move in the retrograde direction. This may lead to a reduced retrograde flux of DCVs and a lack of DCV circulation in type II terminals. The developed model simulates DCV concentrations in boutons, DCV fluxes between the boutons, age density distributions of DCVs and the mean age of DCVs in various boutons. Unlike published experimental observations, our model predicts DCV circulation in type II terminals after these terminals are filled to saturation. This disagreement is likely because experimentally observed terminals were not at steady state, but rather were accumulating DCVs for later release. Our estimates show that the number of DCVs in the transiting state is much smaller than that in the resident state. DCVs travelling in the axon, rather than DCVs transiting in the terminal, may provide a reserve of DCVs for replenishing boutons after a release. The techniques for modelling transport of DCVs developed in our paper can be used to model the transport of other organelles in axons.

1. Introduction

Investigating axons with large arbours is important for understanding Parkinson's disease (PD) [1]. Loss of dopaminergic neurons in the substantia nigra is implicated in PD pathogenesis. The extensive axonal arborization of these neurons may put them at an increased risk, possibly due to the resulting tight energy budget [2] and the potential inability to supply organelles to synaptic boutons by a single axon sufficiently, especially to distally located boutons [3,4].

In order to understand the transport of organelles in large arbours, simpler animal models can be examined. Drosophila melanogaster is a popular model for basic studies of the nervous system [5]. Varicosities located along the axon terminals are called en passant boutons (hereafter referred to as boutons); they are active terminal sites that release neurotransmitters. Three types of boutons with different morphologies are identified in Drosophila: I, II and III. Type I boutons are further divided into Ib and Is boutons, which represent big and small boutons, respectively [5–7].

Neuropeptides, a type of signal molecules that transmit signals across the synaptic cleft, are synthesized in the soma and transported toward the synapses in the dense core vesicles (DCVs) by means of fast axonal transport [8,9]. Tao et al. [3] investigated DCV transport in axons with type II endings, which are characterized by the greatest number of boutons when compared with the other Drosophila motoneurons. Here, we investigate how the issue of potentially insufficient supply with organelles of distal boutons in large axonal arbours is solved in motoneurons with type II endings [10].

Wong et al. [11] suggested that neuropeptide delivery to boutons is based on the sporadic capture of DCVs from the circulation that these authors discovered in type Ib terminals. However, Bulgari et al. [12] reported that large neuropeptide content in type III boutons is due to increased DCV capture rather than delivery, which results in fewer transiting DCVs and a lack of DCV circulation in type III terminals. Furthermore, Tao et al. [3] reported an inefficient supply of DCVs to distal boutons and a lack of DCV circulation in type II terminals. In this paper, we attempt to better understand the lack of similarity between type Ib and type II terminals.

The models simulating DCV transport in type Ib and type III axon terminals were developed in [13–15]. Kuznetsov & Kuznetsov [16] investigated the effects of reversibility of DCV capture in boutons. In the present paper, we extend our model to simulate DCV transport in type II terminals, which contain a much larger number of boutons. Additionally, our model is now capable of evaluating the mean age of DCVs and their age density in various boutons. We compared the mean ages of DCVs in proximal and distal boutons as predicted by the model with those observed experimentally and reported in Tao et al. [3]. Our DCV age simulations assumed that, after being captured into a resident state, the captured DCVs are eventually destroyed in a bouton. The lack of agreement with the experimental results of [3] suggests that DCVs captured into the resident state in boutons are not destroyed in boutons, but rather are re-released from the resident state back to the transiting pool.

2. Material and models

(a). Governing equations

We assumed that DCVs are synthesized in the soma, transported in the axon and then enter one of the terminal branches where they can be captured in boutons (figure 1). We assumed that an axon has three identical branches (figure 1a). We followed [11] and numbered the boutons 1 to 26 from the most distal to the most proximal (figure 1a).

Figure 1.

(a) A schematic diagram showing a neuron with an axon whose terminal consists of three branches. Each branch contains 26 boutons. We numbered the boutons following the convention adopted in [11]: bouton 1 is the most distal bouton while bouton 26 is the most proximal bouton. We also show 27 compartments simulated in the model (26 compartments representing the boutons and a single compartment representing the axon), as well as the sizes of these compartments. (b) A magnified portion of the terminal showing boutons 20, 19, 18 and 17, fluxes between the compartments occupied by these boutons, as well as resident and transiting DCVs in the terminal. The rates of transition of anterogradely and retrogradely moving DCVs into the resident state are also shown. (Online version in colour.)

There are two pools of DCVs in this system [11]: resident DCVs in boutons and transiting DCVs that travel in the axon and terminal (table 1). We assume that the number of resident DCVs in the whole region containing boutons greatly exceeds the number of transiting DCVs. This is supported by the following estimate. The number of DCVs in the transiting state can be estimated as $j_{\text{ax}\to 26}(0)L_{26}/v$, where $j_{\text{ax}\to 26}(0)$ is the flux from the axon to the most proximal bouton at t = 0, L₂₆ is the length of a compartment occupied by the most proximal bouton (10 µm) and $v$ is the average DCV velocity, which is estimated to be approximately 1 µm s⁻¹ [11,17–19]. This estimates that the number of DCVs transiting in the terminal is approximately 0.33 vesicles, which is about 1% of the number of vesicles in the resident state (approx. 34 DCVs reside in the most proximal type II bouton, [3]).

Table 1.

Various pools of DCVs in the axon and terminal.

type of DCVs	location	possible fate(s)
resident	boutons	resident DCVs can re-enter the transiting pool (this scenario is simulated by δ = 1) or be destroyed in boutons (simulated by δ = 0). If 0 < δ < 1, a portion of DCVs re-enter the transiting pool after spending some time in boutons and some are destroyed in boutons
transiting	axon and terminal	the axon contains only transiting DCVs. In a terminal, transiting DCVs can travel from bouton to bouton, either anterogradely or retrogradely, and can be captured into the resident state in boutons. The capture rate is determined by the capture efficiency of boutons $\varpi$. For δ = 1, the capture is reversible, and after spending some time in the resident state in boutons (determined by the half-residence time of DCVs in boutons, T1/2) the captured DCVs return back to the transiting pool. For δ = 0, the capture is irreversible, and after spending some time in boutons (determined by the half-life of DCVs in boutons, also denoted as T1/2), the DCVs are destroyed

The variables used in our model are summarized in table 2.

Table 2.

Model variables.

symbol	definition	units
ja → b(t)	flux of DCVs from compartment ‘a’ to compartment ‘b’ (figure 1b)	vesicles s−1
nax(t)	average concentration of DCVs in the axon (axon is modelled as a single compartment)	vesicles µm−1
ni(t)	concentrations of resident DCVs in bouton i (i = 1, … ,26)	vesicles µm−1
t	time	s

As DCV transport in an axon is essentially one dimensional, we characterized the DCV concentration by its linear number density, which we defined as the number of DCVs per unit length of the axon. The concentration of DCVs residing in a bouton can change either by DCV capture into the resident state (as DCVs pass the bouton) or by DCV release back to the transiting state (alternatively by DCV destruction in boutons). DCV capture is shown in figure 1b by block arrows. We used a multi-compartment model [20–22] to develop the governing equations in 27 compartments: the axon and 26 boutons (figure 1). The number of DCVs in a compartment is the conserved property.

We first state the conservation of DCVs in the resident state in the most proximal bouton:

$$L_{26}\frac{\text{d}{n}_{26}}{\text{d}t}=min[h_{26}^a(n_{\text{sat0,26}}-n_{26}),j_{\text{ax}\to 26}]+min[h_{26}^r(n_{\text{sat0,26}}-n_{26}),j_{25\to 26}]-L_{26}\frac{n_{26}\ln (2)}{T_{1/2}}.$$ 2.1

The meanings of different terms in equations (2.1)–(2.3) are explained in table 3.

Table 3.

Significance of various terms in the conservation equations for DCVs in the resident state in boutons. Applies to equations (2.1)–(2.3).

physical process	terms that simulate the process	notes
accumulation of transiting DCVs into the resident state	the term on the left-hand side of equations (2.1)–(2.3)
DCV capture into the resident state as DCVs pass the bouton anterogradely or retrogradely	the first and second terms on the right-hand side of equations (2.1) and (2.2). In equation (2.3), DCV capture is described by a single term only (the first term on the right-hand side) because DCVs pass bouton 1 only once	if the number of DCVs in the transiting state is sufficiently large, the rate of DCV capture by a bouton is assumed to be proportional to nsat0,i − ni (the difference between the steady-state DCV concentration in a bouton for infinite DCV half-residence time, or infinite half-life, and the current DCV concentration in the bouton). The DCV concentration for infinite DCV half-residence time, or infinite half-life, is used to allow DCV capture to continue even at steady state if T1/2 is finite. This is necessary to compensate for the DCV re-release from boutons or their destruction. The conservation equations are not affected by how DCVs leave the resident state, whether they are released back to the transiting pool or destroyed in boutons. The min function simulates the fact that the rate of DCV capture into the resident state cannot exceed the flux of DCV into the bouton. Note that retrograde fluxes between the boutons will initiate with a 300 s delay (equations (2.8)–(2.10)) that is required for DCVs to change anterograde to retrograde motors in bouton 1 (E. S. Levitan 2017, personal communication). Hence, the capture of retrogradely moving DCVs, described by the second term on the right-hand side of equations (2.1) and (2.2), will initiate 300 s after the DCVs start entering the terminal
release back into the transiting state (or destruction in the bouton)	the last term on the right-hand side of equations (2.1)–(2.3)

The conservation of DCVs in the resident state in boutons 25 through 2 is stated as follows:

$$\begin{array}{r}{L}_i\frac{\text{d}{n}_i}{\text{d}t}=min[h_i^a(n_{\text{sat}0,i}-n_i),j_{i+1\to i}]+min[h_i^r(n_{\text{sat}0,i}-n_i),j_{i-1\to i}]-L_i\frac{n_i\ln (2)}{T_{1/2}}(i=25,24,\dots ,2).\end{array}$$ 2.2

In the most distal bouton (bouton 1), the requirement of DCV conservation in the resident state produces the following equation:

$$L_1\frac{\text{d}{n}_1}{\text{d}t}=min[h_1(n_{\text{sat}0,1}-n_1),j_{2\to 1}]-L_1\frac{n_1\ln (2)}{T_{1/2}}.$$ 2.3

In addition to the conservation of resident DCVs in boutons, a statement can be made based on the conservation of DCVs travelling in the axon, which results in the following equation:

$$L_{\text{ax}}\frac{\text{d}{n}_{\text{ax}}}{\text{d}t}=j_{\text{soma}\to \text{ax}}+3j_{26\to \text{ax}}-3j_{\text{ax}\to 26}-L_{\text{ax}}\frac{n_{\text{ax}}\ln (2)}{T_{1/2,\text{ax}}}.$$ 2.4a

Our model does not simulate DCV processing in the soma (processes such as the production of DCVs and destruction of returning DCVs in the somatic lysosomes). Therefore, $j_{\text{soma}\to \text{ax}}$ in equation (2.4a) should be interpreted as the net DCV flux from the soma into the axon, calculated as the DCV production rate in the soma minus the DCV destruction rate in the somatic lysosomes.

We also consider the situation when the flux of DCVs from the axon to the most proximal bouton remains constant. In this case, equation (2.4a) must be replaced with the following equation:

$$\frac{\text{d}{n}_{\text{ax}}}{\text{d}t}=0.$$ 2.4b

Equations (2.1)–(2.4) include DCV fluxes from the axon to the most proximal bouton, $j_{\text{ax}\to 26}$, and back to the axon, $j_{26\to \text{ax}}$, as well as anterograde and retrograde fluxes between the boutons (figure 1b). These fluxes need to be modelled. We assumed that the DCV flux from the axon to the most distal bouton is proportional to the average DCV concentration in the axon:

$$j_{\text{ax}\to 26}=h_{\text{ax}}{n}_{\text{ax}}.$$ 2.5

Here, we follow our previous paper [16], where we developed the method of simulating the effect of re-entry of resident DCVs into the transiting pool. The portion of DCVs that escape from the captured state in boutons back into the transiting pool was characterized by parameter δ. The case when all captured DCVs eventually re-enter the transiting pool is simulated by δ = 1 and the case when all captured DCVs are eventually destroyed in boutons is simulated by δ = 0. The effect of adopting one of these hypotheses should be carefully analysed. Indeed, the hypothesis that DCVs captured into the resident state in boutons are eventually re-released to the transiting pool (δ = 1) is supported by the fact that organelles like DCVs are usually destroyed in lysosomes, which are abundant in the soma but not in the terminals. On the other hand, the hypothesis that the released DCVs return to the soma for degradation requires a non-zero retrograde flux from the terminal back to the axon, which seemingly contradicts Tao et al. [3], who reported almost no retrograde flux in type II terminals.

Equations for the DCV fluxes between the boutons, $j_{26\to 25},\dots ,j_{2\to 1}$ (figure 1b), are written by stating the conservation of DCVs in the transiting state. The anterograde flux between the most proximal bouton (bouton 26) and bouton 25 is

$$j_{26\to 25}=j_{\text{ax}\to 26}-min[h_{26}^a(n_{\text{sat}0,26}-n_{26}),j_{\text{ax}\to 26}]+\delta \epsilon L_{26}\frac{n_{26}\ln (2)}{T_{1/2}}.$$ 2.6

The meaning of different terms on the right-hand side of equations (2.6)–(2.10) is explained in table 4.

Table 4.

Significance of various terms in the equations for DCV fluxes between the boutons. Applies to equations (2.6)–(2.10).

physical process	terms that simulate the process	notes
difference between the flux of DCVs entering a bouton and the DCV capture rate into the resident state in the bouton	the first and second terms on the right-hand side of equations (2.6)–(2.10)
rate of DCV release from the resident state back to the transiting state	the last term on the right-hand side of equations (2.6)–(2.10)	if δ = 0, then all DCVs are destroyed in boutons and none are released back to the transiting pool. On the other hand, if δ = 1, then all DCVs are released to the transiting pool after spending some time in the resident state. Parameter ε describes how the DCVs that are released from the transiting state are split between the anterogradely and retrogradely moving pools (ε is the portion of DCVs that join the anterograde pool and (1 − ε) is the portion that join the retrograde pool). The multiplier H[t − t1] in equations (2.8)–(2.10) accounts for the 300 s delay that it takes for the retrograde flux to start

Anterograde fluxes between boutons 25 through 2 are modelled by the following equations:

$$j_{i\to i-1}=j_{i+1\to i}-min[h_i^a(n_{\text{sat}0,i}-n_i),j_{i+1\to i}]+\delta \epsilon L_i\frac{n_i\ln (2)}{T_{1/2}}(i=25,24,\dots ,2).$$ 2.7

The retrograde flux from bouton 1 into bouton 2 is

$$j_{1\to 2}=H[t-t_1]\{j_{2\to 1}-min[h_1(n_{\text{sat}0,1}-n_1),j_{2\to 1}]\}+\delta L_1\frac{n_1\ln (2)}{T_{1/2}},$$ 2.8

where H is the Heaviside step function.

Note that the last term in equation (2.8) does not contain $\epsilon$. This is because DCVs released from the resident state in this bouton can only join the pool of retrogradely moving vesicles.

Retrograde fluxes between boutons 2 through 25 are modelled by the following equations:

$$j_{i\to i+1}=H[t-t_1]\{j_{i-1\to i}-min[h_i^r(n_{\text{sat}0,i}-n_i),j_{i-1\to i}]\}+\delta (1-\epsilon )L_i\frac{n_i\ln (2)}{T_{1/2}}(i=2,3,\dots ,25).$$ 2.9

Finally, the retrograde flux from bouton 26 back to the axon is modelled by the following equation:

$$j_{26\to \text{ax}}=H[t-t_1]\{j_{25\to 26}-min[h_{26}^r(n_{\text{sat}0,26}-n_{26}),j_{25\to 26}]\}+\delta (1-\epsilon )L_{26}\frac{n_{26}\ln (2)}{T_{1/2}}.$$ 2.10

In equations (2.5)–(2.10), fluxes have units of vesicles s⁻¹. Equations (2.1)–(2.4) describe a system of 27 first-order ordinary differential equations that require 27 initial conditions. In Drosophila, release of DCVs by exocytosis, which is associated with moulting behaviour, can be massive, as much as approximately 90% of the content (E. S. Levitan 2017, personal communication). Since is it not our goal to simulate the specific experiments reported in Tao et al. [3] but rather to gain a fundamental understanding of DCV transport in large axonal arbours, for simplicity we simulate the process of refilling the terminal after a compete release, which means that, initially, the terminal is empty,

$$n_1(0)=0,\dots ,n_{26}(0)=0,\hspace{0.17em}{n}_{ax}(0)=n_{\text{sat},\text{ax}}.$$ 2.11

The last equation in (2.11) assumes that, initially, the axon is filled to saturation. Since the terminal is initially empty, DCVs enter the terminal at a large rate earlier on in the simulation. If the rate of DCV synthesis in the soma is small, there may be an initial decrease in the concentration of transiting DCVs in the axon. The axonal DCV concentration will recover when the boutons in the terminal are filled to (or close to) saturation, and thus less DCVs are needed in the terminal.

(b). Estimation of values of parameters involved in the model

We used two methods to estimate the values of model parameters. First, we estimated the parameters whose values we could find in published literature or assume on physical grounds. These are $j_{\text{ax}\to 26}$, L₁, … , L₂₆, L_ax, t₁, T_1/2, $T_{1/2,\text{ax}}$ and $\varpi$. We summarized the values of these parameters in table 5. We then estimated the values of parameters which we were unable to find in the literature ($n_{\text{sat},\text{ax}}$, $n_{\text{sat},1}$, … , $n_{\text{sat},26}$, h₁, … , h₂₆, h_in, $n_{\text{sat}0,1}$, … , $n_{\text{sat}0,26}$, $n_{\text{sat, ax}}$ and $j_{\text{soma}}$) by stating DCV conservation at the initial moment (t = 0) and at steady state. We report the values of these parameters in table 6. (In what follows, the words ‘saturated’ and ‘steady state’ are used interchangeably.)

Table 5.

Model parameters estimated based on values found in the literature or assumed on physical grounds.

symbol	definition	units	estimated value(s) or range	reference(s)
a	parameter characterizing a decrease in the number of DCVs in the saturated state from the most proximal to the most distal bouton, as defined in equation (2.12)		1.1a
jax→26	flux from the axon to the most proximal bouton	vesicles s−1	0.0333b	[3,11]
L1, … ,L26	lengths of compartments occupied by boutons 1, 2, … , 26 (defined in figure 1a)	µm	10c	(E. S. Levitan 2018, personal communication)
Lax	length of the axon, figure 1a	µm	500	(E. S. Levitan 2018, personal communication)
t1	time required for DCVs to change direction in the most distal bouton, if they are not captured	s	300	[11], (E. S. Levitan 2017, personal communication)
T1/2	half-life or half-residence time of captured DCVs	s	2.16 × 104	[23]
T1/2,ax	half-life of DCVs in the axon	s	(1 … 100) × T1/2d	(E. S. Levitan 2017, personal communication), [23]
ε	parameter simulating how DCVs released from the resident state in boutons are split between the anterograde and retrograde transiting pools. ε is the portion of released DCVs that join the anterograde pool while (1 − ε) is the portion of released DCVs that join the retrograde pool		0.5e
δ	parameter determining the fate of DCVs captured into the resident state in boutons. δ = 0 simulates the situation when all DCVs are eventually destroyed in boutons while δ = 1 simulates the situation when DCVs, after spending some time in the resident state, are released back to the transiting pool		0–1
$\varpi$	capture efficiency defined as the percentage of DCVs captured in a bouton when DCVs pass the bouton. In boutons 2, … ,26 capture occurs twice, when DCVs pass the bouton in anterograde and in retrograde directions		0.1f	[3]

^aTao et al. [3] reported a decrease in the DCV concentration in distal boutons compared with proximal boutons in type II terminals.

^bAccording to Tao et al. [3], the anterograde flux into a terminal with type II boutons is half of that into a terminal with type Ib boutons (approx. 4 vesicles min⁻¹ [11]), that is, approximately 2 vesicles min⁻¹.

^cThe length of a compartment occupied by a bouton equals the spacing between two adjacent boutons (figure 1), which is approximately 10 µm according to E. S. Levitan (2018, personal communication).

^dDCVs are transported over large distances in the axons (up to 1 m in humans). This means that they must somehow be protected from degradation in the axon, and thus the DCV half-life in the axon is probably larger than the DCV half-life or half-residence time in boutons. A possible physical mechanism explaining such protection is the scarcity of organelle degradation machinery in axons (E. S. Levitan 2017, personal communication). We investigated the effect of DCV half-life in the axon on DCV transport, specifically cases when T_1/2,ax varies in the range between (T_1/2,100 × T_1/2).

^eThere are no experimental data that indicate how the DCVs are split. For computations presented in this paper, we assumed that ε = 0.5. In future research, a sensitivity analysis similar to that reported in [24,25], with respect to parameter ε, as well as to other parameters, should be performed.

^fFor type II boutons, we assumed that $\varpi =0.1$, the same value when DCVs pass the bouton in anterograde or retrograde directions, for all 26 boutons. This estimate is based on data reported in [3]. Data presented in fig. 6E of [3] may suggest that retrograde capture is slightly greater. However, since, in a terminal with type II boutons, the retrograde DCV flux is small compared with the anterograde flux, this difference would not affect our model.

Table 6.

Model parameters estimated based on DCV balances at the initial state and at steady state as well as on values reported in table 5 (see §2b for the details on how the estimates were made).

symbol	definition	units	estimated value(s)
h1, h2, … ,h26	mass transfer coefficients characterizing the rates of capture of DCVs into the resident state in boutons 1, … ,26, respectively (for boutons 2, … ,26, we assumed that $h_i^a=h_i^r=h_i$)	µm s−1	data are summarized in electronic supplementary material, table S1
hax	mass transfer coefficient characterizing the rate at which DCVs leave the axon and enter the most proximal bouton	µm s−1	0.0980
jsoma	the rate of DCV synthesis in the soma minus the rate of DCV destruction in the somatic lysosomes	vesicles s−1	calculated by equation (2.20)
nsat,1,…, nsat,26	saturated (steady state) concentrations of DCVs in boutons 1, … , 26, defined by equation (2.12)	vesicles µm−1	3.4/a26−i
nsat0,1, … ,nsat0,26	saturated concentrations of DCVs in boutons 1, … , 26, respectively, at infinite DCV half-life or at infinite DCV residence time	vesicles µm−1	data are summarized in electronic supplementary material, table S2
nsat,ax	saturated concentration of DCVs in the axon	vesicles µm−1	0.34

(i). Saturated DCV concentrations in boutons, $n_{\mathrm{sat},i}$ (i = 1, … ,26)

Type II terminals, studied in [3], have approximately 80 boutons per muscle, usually distributed on three or four branches (E. S. Levitan 2018, personal communication). We thus estimated that there are approximately 26 boutons per single branch. Based on Tao et al. [3], we estimated that, in the saturated state, there are approximately 34 DCVs in the most proximal type II bouton. Also, Tao et al. [3] reported a decreased DCV content in distal boutons; for example, neuropeptide release was 50% lower for distal boutons than for proximal boutons. To model this, we assumed that the number of DCVs in the saturated state decreases in more distal boutons as $34\text{/}{a}^{26-i}$, where a = 1.1 and i is the number of a bouton (figure 1). Since spacing between boutons, L_i, is approximately 10 µm (E. S. Levitan 2018, personal communication), the saturated concentration of DCVs in type II boutons is

$$n_{\text{sat},i}=\frac{34\text{/}{L}_i}{a^{26-i}}=\frac{3.4}{a^{26-i}}\text{vesicles}\hspace{0.17em}\text{μ}{\text{m}}^{-1}\hspace{0.17em}(i=1,\dots ,26).$$ 2.12

Equation (2.12) postulates a set capacity for vesicles (e.g. akin to parking spaces) that limits accumulation. This allows boutons with an excess supply of vesicles to fill to a set amount and allow more vesicles to continue travelling distally.

One of the goals of our research is to estimate whether it is realistic to assume that a saturated state can be reached in type II terminals. The reason why it may not be reached is because, in the process of transitioning from being a larva to a fly, a Drosophila larva retracts the neurons, destroys the muscles, builds new muscles and reinnervates (E. S. Levitan 2018, personal communication). This may happen before transport processes in type II terminals reach steady state.

(ii). Average saturated DCV concentration in the axon, $n_{\mathrm{sat},\mathrm{ax}}$

We assumed, following [14], that the average DCV concentration in the axon is 10% of the saturated DCV concentrations in boutons. This leads to the following estimate:

$$n_{\text{sat, ax}}=0.1\times 3.4\hspace{0.17em}\text{vesicles}\hspace{0.17em}\text{μ}{\text{m}}^{-1}=\text{0}\text{.34}\hspace{0.17em}\text{vesicles}\hspace{0.17em}\text{μ}{\text{m}}^{-1}.$$ 2.13

(iii). Mass transfer coefficient characterizing the rate at which DCVs enter the most proximal bouton from the axon, h_ax

We assumed that the flux from the axon into the terminal branch is proportional to the average DCV concentration in the axon:

$$j_{\text{ax}\to 26}=h_{\text{ax}}{n}_{\text{sat, ax}}.$$ 2.14

Since the flux into a type II terminal branch is approximately 2 vesicles min⁻¹ (see footnote ‘b’ after table 5), by solving equation (2.14) for h_ax, we obtained that

$$h_{\text{ax}}=0.0980\hspace{0.17em}\text{μ}\text{m}\hspace{0.17em}{\text{s}}^{-1}.$$ 2.15

(iv). Mass transfer coefficients characterizing DCV capture into the resident state in boutons, h₁, h₂, … , h₂₆; and saturated concentrations of DCVs in boutons at infinite DCV half-life or half-residence time, $n_{\mathrm{sat}0,1}$, $n_{\mathrm{sat}0,2}$, … , $n_{\mathrm{sat}0,26}$

We assumed that the mass transfer coefficients characterizing DCV capture when DCVs pass a bouton in anterograde and retrograde directions are equal,

$$h_i^a=h_i^r=h_i(i=2,\dots ,26).$$ 2.16

We also assumed that, initially, there are no resident DCVs in the boutons. We then wrote equations simulating the reduction of the DCV flux after DCVs pass boutons 26, 25, … , and 1, respectively, at t = 0. Capture efficiency, $\varpi$, characterizes capture initiation [3]. Therefore, after passing each bouton, the DCV flux is initially reduced by 10%, and we can state the following:

$$\varpi j_{\text{ax}\to 26}=h_{26}(n_{\text{sat}0,26}-0),$$ 2.17a

$$\varpi j_{\text{ax}\to 26}{(1-\varpi )}^{26-i}=h_i(n_{\text{sat}0,i}-0)(i=25,\dots ,2)$$ 2.17b

$$\text{and}\varpi j_{\text{ax}\to 26}{(1-\varpi )}^{25}=h_1(n_{\text{sat}0,1}-0).$$ 2.17c

On the other hand, at steady state, the rate at which DCVs are captured must be equal to the rate at which they are destroyed (or re-enter the transiting pool). It should be noted that steady state may never be reached in type II terminals in Drosophila because larva may transition into a fly before it is reached, but the model must still be able to simulate the steady-state situation. This leads to the following equation:

$$2h_i(n_{\text{sat}0,i}-n_{\text{sat},i})=L_i\frac{n_{\text{sat},i}\ln (2)}{T_{1/2}}(i=\hspace{0.17em}2,\dots ,26).$$ 2.18

A factor of two on the left-hand side of equation (2.18) appears because DCVs have two chances to be captured in boutons 2, … 26: they can be captured as they travel anterogradely or retrogradely through a bouton.

In a similar equation describing DCV conservation at steady state in bouton 1, the factor of 2 on the left-hand side is absent because DCVs pass this bouton only once:

$$h_1(n_{\text{sat}0,1}-n_{\text{sat},1})=L_1\frac{n_{\text{sat},1}\ln (2)}{T_{1/2}}.$$ 2.19

Equations (2.17)–(2.19) were solved by using Matlab's (Matlab R2018b, MathWorks, Natick, MA, USA) Solve solver, and the results are summarized in electronic supplementary material, tables S1 and S2.

(c). The net rate of DCV production in the soma

Following E. S. Levitan (2018, personal communication), we assumed that an axon splits into three branches. Wong et al. [11] argued that there is no active address system for directing DCVs to a particular bouton. Also, the results reported in [3] suggest the depletion of distal boutons of DCVs, which would not be likely if an actively controlled DCV delivery system existed. Therefore, we assumed that the DCV transport is passively regulated and neglected any possible feedback effects on DCV transport.

At steady state, the net rate of DCV production in the soma, j_soma (defined as the rate of DCV synthesis minus the rate of DCV destruction in somatic lysosomes), must be equal to the rate of DCV destruction in the axon and three branches,

$$j_{\text{soma}}=3\sum _{i=1}^{26}{L}_i{n}_{\text{sat},i}(1-\delta )\frac{\ln (2)}{T_{1/2}}+L_{\text{ax}}{n}_{\text{sat, ax}}\frac{\ln (2)}{T_{\text{1/2, ax}}}.$$ 2.20

According to our model, the DCV flux from the soma into the axon is heavily dependent on the fate of DCVs in boutons. If δ = 0, then all DCVs captured in boutons are eventually destroyed. To maintain steady state, this scenario requires more DCVs to enter the axon than the scenario in which DCVs captured in boutons are re-released and re-enter the transiting pool (δ = 1).

(d). Age distribution of DCVs in boutons and mean age of DCVs in boutons

In order to investigate the DCV age distribution in boutons, we followed [26,27]. The compartmental system is displayed in figure 2. Governing equations (2.1)–(2.3) were recast as:

$$\frac{\text{d}}{\text{d}t}\mathbf{\text{n}}(t)=\text{B}(\mathbf{\text{n}}(t),t)\mathbf{\text{n}}(t)+\mathbf{\text{u}}(t).$$ 2.21

Matrix B for the case displayed in figure 2 is a diagonal matrix with the same elements on the main diagonal,

$$b_{ii}=-\frac{\ln (2)}{T_{1/2}}(i=26,25,\dots ,1).$$ 2.22

Figure 2.

Schematic diagram showing transiting DCVs in the terminal and DCVs in the resident states in boutons. This compartmental representation is used for the analysis of DCV age distribution and average DCV age in the resident state in boutons. Capture of DCVs from the transiting state, destruction of DCVs in the resident state and re-release of DCVs from the resident to the transiting state are shown by arrows. DCVs in the transiting state are assumed to have zero age. (Online version in colour.)

The last element of vector u is

$$u_{26}=\frac{\{min[h_{26}^a(n_{\text{sat0,26}}-n_{26}),j_{\text{ax}\to 26}]+min[h_{26}^r(n_{\text{sat0,26}}-n_{26}),j_{25\to 26}]\}}{L_{26}}.$$ 2.23

The other elements of vector u are

$$u_i=\frac{\{min[h_i^a(n_{\text{sat}0,i}-n_i),j_{i+1\to i}]+min[h_i^r(n_{\text{sat}0,i}-n_i),j_{i-1\to i}]\}}{L_i}(i=25,24,\dots ,2)$$ 2.24

and

$$u_1=\frac{\{min[h_1(n_{\text{sat}0,1}-n_1),j_{2\to 1}]\}}{L_1}.$$ 2.25

The fact that matrix B is diagonal means that the analysed compartmental system does not simulate the direct transfer of DCVs between the boutons. Indeed, the model displayed in figure 2 treats the transiting pool as a reservoir of DCVs all of which have the age of zero. For δ = 0, this assumption is valid as there is no transfer of DCVs between different boutons. All the DCVs that are captured into a resident state in a bouton are eventually destroyed in that state. There is also a sufficient supply of transiting DCVs from the axon (see the analysis in electronic supplementary material, section S2.6), such that DCV capture into the resident state is controlled by bouton capture kinetics rather than by DCV supply (electronic supplementary material, equations (S1) and (S2)). However, for δ > 0, a captured DCV can re-enter the transiting state and be subsequently re-captured into the resident state in one of the boutons located downstream (figure 2). For δ > 0, treatment of the transiting pool as a reservoir of DCVs with zero age is not valid.

In order to exactly account for the age accumulated by a DCV that re-entered the transiting pool after residing in a bouton for some time, a two-concentration model, which would simulate DCV concentrations not only in the resident but also in the transiting states, needs to be developed. The difficulty of developing such a model lies in the fact that it would require the introduction of a large number of additional coefficients, which would describe transitions between transiting and resident states. In the current formulation of the model, the fluxes between the boutons contain both DCVs that have not yet been captured (their age is zero) and DCVs that have already spent some time residing in one of the boutons. Attributing previously captured DCVs to vector u (see equations (2.23)–(2.25)) results in resetting their ages to zero. Therefore, for δ > 0, the presented analysis of age distribution and mean age of DCVs is an approximation.

It should also be noted that we neglected the time it takes DCVs to travel between the boutons. A portion of DCVs that re-entered the axon ($j_{26\to \text{ax}}$, see equation (2.10)) can turn again, by changing the retrograde to anterograde motors, and re-enter the terminal. This situation needs to be modelled because, if returning DCVs previously resided in boutons, their re-entry will affect the DCV age distribution. In type II terminals, the effect of DCV return to the terminal is expected to be minor [3].

The state transition matrix, $\mathrm{\Phi }$, was determined by solving the following matrix equation [26]:

$$\frac{\text{d}}{\text{d}t}\mathrm{\Phi }(t,t_0)=\text{B}(\mathbf{\text{n}}(t),t)\mathrm{\Phi }(t,t_0),$$ 2.26

with the initial condition

$$\mathrm{\Phi }(t_0,t_0)=\text{I}\text{, }$$ 2.27

where I is an identity matrix. Note that $\mathrm{\Phi }$ depends on two time variables.

Initially, the terminal did not contain any DCVs (equation (2.11)); hence, the age density of DCVs at t  =  0 was zero. To proceed, we need an assumption concerning the age of DCVs entering the terminal. If the DCVs had just been synthesized in the soma, they could be treated as new; however, if they had already recirculated in the terminal many times, they could be old. Tao et al. [3] found no DCV circulation in large axonal arbours so we assumed that all DCVs entering the terminal were new; therefore, their age was set to zero. This assumption neglects the transiting time of DCVs from the soma to the terminal; therefore, the calculated DCV age should be interpreted as the age of DCVs after they enter the terminal.

The age density of DCVs that entered the terminal after t = 0 is then calculated as:

$$\mathbf{\text{p}}(a,t)=1_{[0,t-t_0)}(a)\mathrm{\Phi }(t,t-a)\mathbf{\text{u}}(t-a),$$ 2.28

where $1_{[0,t-t_0)}$ is the indicator function that is equal to 1 if $0\le a<t-t_0$; otherwise, $1_{[0,t-t_0)}$ is equal to 0.

The mean age of DCVs in boutons, which changes as time progresses, was calculated as [28]

$${\overline{a}}_i(t)=\frac{{\int }_0^{\mathrm{\infty }}a{p}_i(a,t)\text{d}a}{{\int }_0^{\mathrm{\infty }}{p}_i(a,t)\text{d}a}(i=1,\dots ,26).$$ 2.29

${\overline{a}}_i(t)$ (i = 1, … ,26) were obtained by solving the following mean age system [26,28]:

$$\frac{\text{d}}{\text{d}t}\overline{\mathbf{\text{a}}}(t)=\text{G}(\mathbf{\text{n}}(t),t),$$ 2.30

with the initial condition

$$\overline{\mathbf{\text{a}}}(0)=0,$$ 2.31

where $\overline{\mathbf{\text{a}}}=({\overline{a}}_1,\dots ,{\overline{a}}_{26})$.

In our case, G is the diagonal matrix, which is defined as follows:

$$g_{ii}(t)=1-\frac{{\overline{a}}_i(t)u_i}{n_i(t)}(i=1,\dots ,26).$$ 2.32

(e). Numerical solution

We used Matlab's ode45 solver (Matlab R2017b, MathWorks, Natick, MA, USA) to solve equations (2.1)–(2.11) numerically. We set the error tolerance parameters, RelTol and AbsTol, to 10⁻⁶ and 10⁻⁸, respectively. We checked that the solution was not affected by a further decrease of RelTol and AbsTol; see electronic supplementary material, figure S1, which shows a comparison of $n_{26}(t)$, $n_{13}(t)$ and $n_1(t)$ computed with a standard accuracy, RelTol = 10⁻⁶ and AbsTol = 10⁻⁸, with those computed with an increased accuracy, RelTol = 10⁻⁸ and AbsTol = 10⁻¹⁰. The computational results with standard and increased accuracy are virtually identical.

The position of a turn in the curve was determined by finding a location of large curvature. The latter was estimated by the curvature of a circle drawn through three adjacent points.

3. Results

Figures displaying estimated values of mass transfer coefficients characterizing the rates of DCV capture in boutons (electronic supplementary material, figure S2a) and saturated concentrations of DCVs assuming that DCVs have an infinite half-life (or infinite half-resident time) in boutons (electronic supplementary material, figure S2b) are given in the electronic supplementary material.

(a). Comparison of assumed steady-state concentrations with numerical results

We checked the solution by computing steady-state concentrations in the resident state in boutons, $n_{\text{sat},i}$, and compared them with the values assumed in equation (2.12) in the process of estimating the model parameters (figure 3a). It should be noted that values of $n_{\text{sat},i}$ are not explicitly involved in the governing equations (2.1)–(2.10). A similar comparison is shown for the steady-state concentration of transiting DCVs in the axon, $n_{\text{sat, ax}}$ (figure 3a). By plotting concentrations of resident DCVs in boutons at various times (t = 0.2, 1 and 5 h), we established that the most proximal boutons are filled first and the distal boutons are filled later in the process (figure 3b).

Figure 3.

(a) Saturated DCV concentrations in the resident state in various boutons and in the transiting state in the axon. Estimated values of these concentrations, calculated using equation (2.12) and equation (2.13), are compared with numerically obtained values of saturated concentrations (obtained at t → ∞). (b) Concentrations of captured DCVs in various boutons at three times: t = 0.2, 1, and 5 h for the case when $j_{\text{ax}\to 26}$ is kept constant (at 2 DCVs min⁻¹). (Online version in colour.)

It should be noted that Tao et al. [3] did not show the monotonic tapering-off of bouton vesicle content with distance that is output by the model. Rather, the drop-off in organelle content appeared suddenly at the furthest ends of the arbour. The model should be amended in the future to improve the agreement with the experiments in this regard.

(b). The case when the DCV flux from the axon to the most proximal bouton remains constant

(i). DCV concentrations in boutons

Figure 4 shows how boutons reach their maximum capacity in terms of DCV accumulation. If the DCV flux from the axon to the most proximal bouton is kept constant, it takes about 8 h for the DCV concentration in boutons to reach steady state (figure 4; electronic supplementary material, table S3). An exception to this is bouton 1, where it takes about 16 h to reach steady state (figure 4b; electronic supplementary material, table S3). It takes longer to fill bouton 1 because, in all other boutons, DCV capture occurs twice, first when the DCVs pass the bouton moving anterogradely and second when moving retrogradely (figure 1b). However, DCVs pass bouton 1 only once, which explains why fewer DCVs are captured and why it takes longer to fill bouton 1.

Figure 4.

The build-up toward steady state: concentrations of captured DCVs in various boutons. (a) Boutons 26 through 14. (b) Boutons 13 through 1. The case when $j_{\text{ax}\to 26}$ is kept constant (at 2 DCVs min⁻¹). Results are independent of $\delta$. (Online version in colour.)

(ii). DCV fluxes: the case when DCVs captured into the resident state in boutons escape and re-enter the transiting pool (δ = 1)

Anterograde (figure 5a) and retrograde (figure 5b) fluxes between the axon and the most proximal bouton and between various boutons, at the initial state (t = 0) and at steady state ($t\to \mathrm{\infty }$), show that, initially, the anterograde DCV flux decays from the most proximal bouton to the most distal bouton (figure 5a). This is because fewer and fewer DCVs are left in the transiting pool as some of them are captured by boutons while travelling anterogradely. At the initial state, all retrograde fluxes are equal to zero (figure 4b) because it takes 300 s to change anterograde to retrograde motors in bouton 1 (most distal); hence, retrograde fluxes cannot begin before 300 s.

Figure 5.

Fluxes between the axon and the most proximal bouton and between various boutons at the initial state and at steady state. (a) Anterograde fluxes. (b) Retrograde fluxes. The case when $j_{\text{ax}\to 26}$ is kept constant (at 2 DCVs min⁻¹), δ = 1 (which refers to the case when all captured DCVs eventually re-enter the transiting pool). (Online version in colour.)

At steady state, all anterograde and retrograde fluxes are equal to the flux of DCVs from the axon to the most proximal bouton, 2 DCVs min⁻¹ (figure 5), because, at δ = 1, all captured DCVs re-enter the transiting pool, after spending some time in the resident state. At steady state, the rate of DCV capture is equal to the rate of DCV return to the transiting state; hence, the DCV flux is not decreasing from bouton to bouton.

It takes the anterograde and retrograde fluxes about 8 h to reach the same steady-state value, $j_{\text{ax}\to 26}$ (figures 6 and 7; electronic supplementary material, table S4). It should be noted that retrograde fluxes, especially in proximal boutons, do not begin immediately, but with some delay (up to about 1 h) which is required to fill the boutons, so that the retrograde component of the flux does not get completely depleted of DCVs before it reaches these boutons.

Figure 6.

The build-up toward steady state: the flux from the axon to the most proximal bouton and anterograde fluxes between various boutons. (a) Fluxes ax → 26 through 1 → 14. (b) Fluxes 14 → 13 through 2 → 1. The case when $j_{\text{ax}\to 26}$ is kept constant (at 2 DCVs min⁻¹), δ = 1 (which refers to the case when all captured DCVs eventually re-enter the transiting pool). (Online version in colour.)

Figure 7.

The build-up toward steady state: retrograde fluxes between various boutons and the flux from the most proximal bouton to the axon. (a) Fluxes 1 → 2 through 13 → 14. (b) Fluxes 14 → 15 through 26 → ax. The case when $j_{\text{ax}\to 26}$ is kept constant (at 2 DCVs min⁻¹), δ = 1 (which refers to the case when all captured DCVs eventually re-enter the transiting pool). (Online version in colour.)

(iii). Distribution of DCV age in the terminal

At steady state, the age density changes from 0 to approximately 0.4 vesicles µm⁻¹ h⁻¹. The new DCVs prevail in boutons as older DCVs leave the resident state because they are destroyed in boutons (for δ = 0) (figure 8a). Our model also makes it possible to calculate the mean age of DCVs in various boutons. At steady state, the mean DCV age in all boutons is approximately 8.66 h (figure 8b; electronic supplementary material, table S5).

Figure 8.

(a) Age density of DCVs in various boutons at steady state. (b) Mean age of resident DCVs in various boutons versus time. The case when $j_{\text{ax}\to 26}$ is kept constant (at 2 DCVs min⁻¹). The presented age of DCV analysis applies to the situation characterized by δ = 0 (which refers to the case when all captured DCVs are eventually destroyed in boutons). (Online version in colour.)

To investigate possible simplifications of our model, we used the fact that anterograde fluxes remain positive at all times (figure 6), while the retrograde fluxes become positive in less than 1 h after the process of filling the terminal begins (figure 7). This fact enabled us to linearize matrix B in equation (2.21). The analysis is presented in section S2.5 of the electronic supplementary material. The numerical results for the linearized case, displayed in electronic supplementary material, figure S25, are practically identical to the results presented in figure 8.

Apparently, the case of δ = 0 does not correctly simulate the experimental findings reported in Tao et al. [3], who investigated the age of DCVs by marking the DCVs with a photoconvertible construct. Their construct switches from green to red fluorescence over a period of hours. The results reported in [3] indicate that DCVs residing in distal boutons are older than those residing in proximal boutons. The fact that our model does not capture this observation (electronic supplementary material, table S5) suggests that DCVs are not destroyed in the resident state in boutons, but rather, after spending some time in the resident state, are returned to the transiting pool. As older DCVs are returned to the transiting state, for δ = 1, the average age of DCVs is expected to increase from proximal to distal boutons.

(c). DCV fluxes: the case when DCVs captured into the resident state are destroyed in boutons (δ = 0)

Anterograde fluxes decay toward more distal boutons. This applies to the fluxes at the initial state and at steady state, but, at steady state, the decay is slower (electronic supplementary material, figure S3a). This is because, at steady state, the boutons are filled to saturation, and DCV capture is only needed to replace the DCVs destroyed in the resident state (the rate of their destruction is controlled by the half-life of DCVs in the resident state, T_1/2). Retrograde fluxes decay from more distal to more proximal boutons (electronic supplementary material, figure S3b). This is because, as DCVs move retrogradely from more distal to more proximal boutons, their capture continues, even at steady state.

It takes again about 8 h for the anterograde (electronic supplementary material, figure S4) and retrograde (electronic supplementary material, figure S5) fluxes to reach steady state, as for the case with no DCV destruction (electronic supplementary material, table S4). However, fluxes in the next bouton reach a smaller value than in the previous bouton, because, even at steady state, there is now some DCV destruction in boutons which must be compensated by DCV capture from the transiting pool.

(d). Implications of the obtained results on DCV circulation in the terminal

It should be noted that, unlike [3], our model predicts that DCV circulation in type II terminals would develop if time allowed for DCV transport to reach steady state. This circulation is stronger if DCVs are re-released to the transiting pool after spending some time in boutons (δ = 1). In this case, at steady state, the anterograde flux of DCVs entering the terminal equals the retrograde DCV flux leaving the terminal (figure 5). The circulation is slightly weaker if DCVs are destroyed in boutons (δ = 0, electronic supplementary material, figure S3). Our explanation for this disagreement with results of [3] is that the terminals observed in [3] were not at steady state but rather were accumulating DCVs for later release (E. S. Levitan 2017, personal communication). Our results indicate that, if the DCV flux from the axon to the most proximal bouton, $j_{\text{ax}\to 26}$, remains constant (at 2 vesicles min⁻¹) during the process of filling the terminal, it takes about 8 h for the DCV fluxes to reach steady state. In the electronic supplementary material, figures S6–S24 and the discussion of these figures, by simulating the DCV concentration in the axon, we investigate the situation when $j_{\text{ax}\to 26}$ depends on time, for different values of the DCV half-life in the axon. We show that a variation of $j_{\text{ax}\to 26}$ may result in a much longer time required for DCV fluxes in the axon to reach the steady state, which may exceed the time that the Drosophila third instar larval stage lasts (approx. 48 h); the third instar was used in experiments of [3].

Noteworthy, we used slightly different criteria for vesicle circulation from [3,11]. For example, Wong et al. [11] emphasized that vesicles in type Ib terminals tend to accumulate at the distal bouton early on as part of vesicle circulation. In our definition, circulation is present when a large portion of DCVs return from the most proximal bouton to the axon; in other words, when $j_{26\to \mathrm{ax}}/j_{\mathrm{ax}\to 26}$ is much larger than 0. This occurs because, when a large portion of DCVs return to the axon, they can switch the direction of their motion to anterograde and re-enter the circulation [11]. For example, for the case displayed in figure 5, $j_{26\to \mathrm{ax}}\text{/}{j}_{\text{ax}\to 26}$ at steady state is equal to unity.

4. Discussion and future directions

Wong et al. [11] and Bulgari et al. [12] suggested that the circulation of transiting DCVs in the terminal forms a DCV pool that can be tapped into when DCV reserves in boutons need to be replenished. Surprisingly, experiments with type III [12] and type II [3] terminals found a lack of such DCV circulation. Our results suggest that DCV circulation in a type II terminal may develop if the terminal is filled to saturation, especially if the hypothesis that all DCVs captured in the resident state in boutons are eventually released back to the transiting pool is adopted. This hypothesis is supported by the scarcity of organelle degradation machinery in terminals. The time it takes for the DCV fluxes in the axon to reach steady state depends on the model of DCV transport in the axon and can be long, even exceeding the duration of Drosophila's third instar larval stage (animals in this stage were used in the DCV transport experiments). Our estimates show that the number of DCVs in the transiting state may be about 1% of the number of DCVs in the resident state. Thus, the transiting state may not provide a sufficient reserve of DCVs for replenishing boutons after a release. Instead, the DCVs travelling in the axon may play the role of a reserve source for replenishing DCV stores in boutons after a release.

Our investigation of DCV transport in the axon, presented in the electronic supplementary material, suggests that the rate at which DCVs enter the axon from the soma depends on the DCV half-life in the axon. If the DCV half-life is small, a large number of DCVs must enter the axon such that some would reach the terminal. On the other hand, if the DCVs in the axon are protected from degradation (their half-life is large), then DCV transport in the axon is accomplished with minimal losses, and the rate at which DCVs enter the axon from the soma is small.

Supporting DCV circulation, the presence of which is established at least in type Ib terminals [11], requires a large amount of energy to support the synthesis of DCVs that are present in the circulation. Thus, it appears that the DCV transport system is not optimized in terms of energy efficiency (to minimize the need for DCV synthesis in the soma), but rather in terms of the ability to perform a complicated task (in this case, DCV delivery) in a robust way.

A one-compartment model of the axon probably overpredicts the drop of the DCV concentration in the axon during the initial stages of filling the terminal. A more detailed model of the axon (for example, simulating the axon as consisting of several smaller sub-compartments rather than one large compartment) could be developed. However, such a model would require more mass transport coefficients to simulate DCV transport between axonal sub-compartments.

Data accessibility

Additional data accompanying this paper are available in the electronic supplementary material.

Authors' contributions

I.A.K. and A.V.K. contributed equally to the performing of computational work and article preparation.

Competing interests

We declare we have no competing interests.

Funding

A.V.K. acknowledges funding from the National Science Foundation (award CBET-1642262).