Comparison of active transport in neuronal axons and dendrites

Mathematical Biosciences 228 (2010) 195–202

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Comparison of active transport in neuronal axons and dendrites A.V. Kuznetsov * Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA

a r t i c l e

i n f o

Article history: Received 23 May 2010 Received in revised form 5 October 2010 Accepted 8 October 2010 Available online 16 October 2010 Keywords: Mathematical modeling Molecular motors Fast axonal transport Neurons Axons and dendrites Intracellular organelles

a b s t r a c t This paper presents a theoretical study, based on modified Smith–Simmons equations, that compares transport of intracellular organelles in two different neurite outgrowths, dendrites and axons. It is demonstrated that the difference in microtubule polarity orientations in dendrites and axons has significant implications on motor-assisted transport in these neurite outgrowths. The developed approach presents a qualitative theoretical basis for understanding important questions such as why axons exhibit almost an unlimited grows potential in vitro while dendrites remain relatively short. It is shown that the difference in a microtubule polarity arrangement between axons and dendrites may be a regulatory mechanism for limiting dendritic growth. Other biological implications of the developed theory as well as other possible reasons for the difference in microtubule structure between axons and dendrites are discussed. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Neurons are highly polarized cells that have two types of long processes, axons that transmit signals and dendrites that receive signals. Axons in a human body can reach up to one meter in length. Since most organelles needed for the growth and maintenance of axons and dendrites are synthesized in the neuron body, organelles must be constantly delivered from the cell body toward distal parts of axons and dendrites. Diffusion is not a sufficiently fast process for transporting large intracellular organelles. For example, it would take diffusion 1.5 h to move mitochondria (particles with an effective diameter of 0.6 lm) to a distance of 10 lm; it would take molecular motors transporting mitochondria at a speed of 0.5 lm/s only 20 s to do the same job [1]. Therefore, transport of large intracellular organelles is mostly accomplished by active transport that relies on molecular motors that attach themselves to microtubules (MTs) and pull various intracellular cargos as they walk on MTs. Active transport toward MT plus-ends is powered by kinesin-family molecular motors while that toward MT minus-ends is accomplished by dynein-family molecular motors [2–4]. Axons and dendrites have different polarity orientations of their MTs. The MT polarity orientation in axons is uniform; the plus end of each MT is directed away from the neuron soma toward the axon terminal (synapse). The MT polarity orientation in dendrites is non-uniform; approximately half of MTs have their plus-ends directed toward the synapse and half toward the cell soma [5,6]. There is obviously something fundamental about different MT ⇑ Tel.: +1 919 515 5292. E-mail address: [email protected] 0025-5564/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2010.10.003

arrangements in axons and dendrites. During development most neurons become polarized when one neurite starts growing at a fast rate and becomes an axon and the other neurites become dendrites [7]. Hayashi et al. [8] and Takahashi et al. [5] described a culture system in which dendrites of rat cortical neurons can be converted to axons. They discovered that the minus-end-distal MTs present in the original dendrites disappear within 24 h of the beginning of conversion. The minus-end-distal MTs are somehow removed from the dendrite that is being converted to an axon and a uniform arrangement of MTs is established. Why dendrites are designed with a mix of positively and negatively oriented MTs remains unclear. One possible explanation is that the distinct organization of the MT structure in axons and dendrites may provide spatial cues governing axon/dendrite specification and maintenance [3]; there are also other theories that are based on experiments involving direct mapping of MT tracks [9]. The purpose of this research is to develop a mathematical model of organelle transport in neurite outgrowths and apply it to simulating fast motor-assisted transport in axons and dendrites with a goal of gaining fundamental understanding into similarities and differences in organelle transport in these two types of neuronal processes. Continuum-based models of organelle transport have been recently investigated in [10–15]. 2. Governing equations Fig. 1(a) shows a coordinate system in an axon and Fig. 1(b) shows that in a dendrite. For the purpose of comparison, both segments have the same dimensionless length, L. In an axon all MTs have uniform orientation with their plus-ends distal to neuron

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A.V. Kuznetsov / Mathematical Biosciences 228 (2010) 195–202

There is a possibility that multiple differently oriented motors can be attached to a single organelle. This situation is not considered in the current model. This imposes certain restrictions on the resolution of the model; however, the model, being a continuum model, has an inherently limited resolution; if one wishes to obtain a more detailed information, one has to resort to Monte Carlo-type models. However, the developed model is well suited for obtaining information about averaged organelle concentrations and fluxes that provides an overall picture of organelle transport in an axon or dendrite. Equations governing transport of organelles in a segment of an axon displayed in Fig. 1(a) are

  @n0þ @ 2 n0þ 0 ð1Þ ð2Þ  ð2 þ k0 Þn0þ þ k0þ n0 þ kþ nþ þ nþ ; ¼ D0þ 2 @t @x @n0 @ 2 n0  ð2k þ k0þ Þn0 þ k0 n0þ ¼ D0 @t @x2  0  ð1Þ þ k n þ nð2Þ ; ð1Þ

Fig. 1. (a) Coordinate system for an axon, (b) coordinate system for a dendrite, (c) kinetic diagram showing various organelle populations and kinetic processes between them.

soma while in dendrites the polarity orientation of MTs is non-uniform with half of the MTs having their plus-ends distal and half their minus-ends distal [5]. The purpose of this research is to investigate consequences of these different MT arrangements on organelle transport in axons and dendrites. The model developed here is based on modified Smith–Simmons equations governing molecular-motor-assisted transport in neurons [16]. The Smith–Simmons model [16] assumes three kinetic states for organelles: free organelles, which can be transported by diffusion; organelles transported on MTs by plus-end-directed molecular motors, and organelles transported on MTs by minusend-directed molecular motors. In this paper, the Smith–Simmons model is extended by allowing for four kinetic states of organelles (see Fig. 1(c)). This division is motivated by the kinetic diagrams suggested in Jung and Brown [17] (it should be noted that the diagrams in Jung and Brown [17] were developed for slow rather than fast axonal transport; the diagram presented in Fig. 1(c) is adjusted for fast axonal transport). All organelles are divided into the plus-end-directed and the minus-end-directed organelles (the plus-end-directed organelles are attached to kinesin molecular motors while the minus-end-directed organelles are attached to dynein molecular motors). The plus-end-directed organelles are divided into the free organelles (suspended in the cytosol and ~ 0þ , and the transported by diffusion), whose number density is n organelles pulled by molecular motors running on MTs, whose ~ þ . The same is done with the minus-endnumber density is n ~ 0 , directed organelles: they are divided into the free organelles, n ~ . and the organelles transported on MTs by molecular motors, n The reason why the free (off-track) organelles are split into two ~ 0þ , and the free groups (the free plus-end-directed organelles, n ~ 0 ) is the fact that switching the minus-end-directed organelles, n type of a molecular motor (or molecular motors, if an organelle is pulled by more than one motor) requires an elaborate molecular mechanism and therefore occurs much less frequently than the attachment of free organelles to MTs.

ð2aÞ

ð1Þ

@nþ @n 0 ð1Þ ¼ n0þ  kþ nþ  þ ; @t @x ð2Þ

ð1aÞ

ð3aÞ

ð2Þ

@nþ @n 0 ð2Þ ¼ n0þ  kþ nþ  þ ; @t @x

ð4aÞ

@nð1Þ @nð1Þ 0   ¼ k n0  k nð1Þ ;   v @t @x

ð5aÞ

@nð2Þ @nð2Þ 0   ¼ k n0  k nð2Þ :   v @t @x

ð6aÞ

Eqs. (3a), (4a) and Eqs. (5a), (6a) can be added up to obtain only two ð1Þ ð2Þ ð1Þ equations for the quantities nþ þ nþ and n þ nð2Þ  (for the axon the two groups of MTs are undistinguishable, see Fig. 1(a)). The reason why the model for the axon is formulated in the way given by Eqs. (1a)–(6a) is to have a symmetric formulation to the dendrite case. Equations governing transport of organelles in a segment of a dendrite displayed in Fig. 1(b) are

  @n0þ @ 2 n0þ 0 ð1Þ ð2Þ  ð2 þ k Þn þ k n þ k n þ n ¼ D0þ ; 0 0þ 0þ 0 þ þ þ @t @x2 @n0 @ 2 n0  ð2k þ k0þ Þn0 þ k0 n0þ ¼ D0 @t @x2  0  ð1Þ þ k n þ nð2Þ ; ð1Þ

ð2bÞ

ð1Þ

@nþ @n 0 ð1Þ ¼ n0þ  kþ nþ  þ ; @t @x ð2Þ

ð1bÞ

ð3bÞ

ð2Þ

@nþ @n 0 ð2Þ ¼ n0þ  kþ nþ þ þ ; @t @x

ð4bÞ

@nð1Þ @nð1Þ 0   ¼ k n0  k nð1Þ ;   v @t @x

ð5bÞ

@nð2Þ @nð2Þ 0   ¼ k n0  k nð2Þ :  þ v @t @x

ð6bÞ

In Eqs. (1a), (2a), (3a), (4a), (5a), (6a) and (1b), (2b), (3b), (4b), (5b), (6b) superscripts (1) and (2) on n+ and n refer to two groups of MTs in the segments. These groups of MTs have the same orientation in an axon and opposite orientations in a dendrite (compare Fig. 1(a) and (b)). Note that the signs before the last terms on the right-hand sides of Eqs. (4b) and (6b) have been reversed compared to those in

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A.V. Kuznetsov / Mathematical Biosciences 228 (2010) 195–202 Table 1 Parameters used in the model. e 0þ D e 0 D

Diffusivity of free (off-track) organelles with plus-end-directed motors attached to them

~þ k ~0 k

First order rate constant for binding to MTs for plus-end-oriented organelles

~ k 0þ ~ k

First order rate constant describing the probability for a free minus-end-oriented organelle to become a free plus-end oriented organelle

~0 k  ~ k

First order rate constant for detachment from MTs for minus-end-oriented organelles

e L ~þ n ~ 0þ n ~ n ~ 0 n ~t v~  v~ þ ~ x

Length of the segment of the axon or dendrite in which transport of organelles is modeled

Diffusivity of free (off-track) organelles with minus-end-directed motors attached to them First order rate constant for detachment from MTs for plus-end-oriented organelles

þ

First order rate constant for binding to MTs for minus-end-oriented organelles First order rate constant describing the probability for a free plus-end-oriented organelle to become a free minus-end oriented organelle

0

Number density of organelles transported on MTs by plus-end-directed motors Number density of free organelles with plus-end-directed motors attached to them Number density of organelles transported on MTs by minus-end-directed motors Number density of free organelles with minus-end-directed motors attached to them Time Average velocity of organelles moving toward the MT minus end Average velocity of organelles moving toward the MT plus end Linear coordinate along the MT tracks

Eqs. (4a) and (6a) to account for the change of polarity of 50% of MTs in a dendrite. Dimensionless variables in Eqs. (1a), (2a), (3a), (4a), (5a), (6a) and (1b), (2b), (3b), (4b), (5b), (6b) are defined as follows:

D0 ¼

~ e 0 k D þ ; 2 ~ vþ

v~ v ¼ ~ ; vþ

x¼

k ¼ ~þ ~xk ; ~ vþ

~ k  ; ~þ k

0

k ¼

~þ ; t ¼ ~t k

~0 k  ; ~þ k

k0 ¼

~ 0 n0 ¼ n

~ k 0 ~þ k

v~ 3þ ~3 k þ

;

L¼

~ e Lk þ ; ~ vþ

~ n ¼ n

v~ 3þ ~3 k þ

ð7Þ

ð8Þ

Parameters used in the model are summarized in Table 1. Within the continuum framework utilized in this research intracellular transport of organelles can be described by a timeindependent quiescent solution. Therefore, transient terms in Eqs. (1a), (2a), (3a), (4a), (5a), (6a) and (1b), (2b), (3b), (4b), (5b), (6b) are neglected and the resulting steady-state equations are solved subject to the following boundary conditions. For an axon, at the left-hand side boundary of the simulated segment, x = 0, the boundary conditions are:

n0þ ¼ N 0þ ;

n0 ¼ N0 ;

ð1Þ

nþ ¼ r0 N0þ ;

ð2Þ

nþ ¼ r0 N0þ ;

ð9aÞ

n0 ¼ N0 ;

ð1Þ

nþ ¼ r0 N0þ ;

nð2Þ  ¼ r0 N 0 :

ð9bÞ

Note that at x = 0 the boundary condition given by the last equation ð2Þ in (9b) imposes the end-condition on nð2Þ  (rather than on nþ , compare that with the last equation in (9a)); this is because in the case of a dendrite the polarity of 50% of MTs has been reversed. In Eqs. (9a), (9b) the new dimensionless parameters are defined as follows:

N0þ

~3 e 0þ v þ ; ¼N ~3 k þ

N0

~3 e 0 v þ ; ¼N ~3 k þ

ð10Þ

e 0þ and N e 0 are constant number densities of free particles where N with kinesin/dynein motors attached to them, respectively, maintained at x = 0 (at the neuron soma) and r0 is the degree of loading of organelles at x = 0. For an axon, at the right-hand side boundary of the segment, x = L, the boundary conditions are

n0þ ¼ N Lþ ;

n0 ¼ NL ;

nð1Þ 

while for a dendrite those are

n0 ¼ NL ;

nð1Þ ¼ rL NL ;

ð2Þ

nþ ¼ rL NLþ :

ð11bÞ

Note that at x = L the boundary condition given by the last equation ð2Þ in (11b) imposes the end-condition on nþ (rather than on nð2Þ  , compare that with the last equation in (11a)); again, this is because in the case of a dendrite the polarity of 50% of MTs has been reversed. In Eqs. (11a), (11b) the new dimensionless parameters are defined as follows:

~3 e Lþ v þ ; NLþ ¼ N ~3 k þ

~3 e L v þ ; NL ¼ N ~3 k þ

ð12Þ

e Lþ and N e L are constant number densities of free particles where N with kinesin/dynein motors attached to them, respectively, maintained at x = L and rL is the degree of loading of organelles at x = L. In simulations, N0+ and N0 are are assumed high (both equal to 0.1) since components are being synthesized at the cell soma while NL+ and NL are assumed low (both equal to 0.01) since components are being utilized at the growth cone of a neurite. The total dimensionless flux of organelles is the sum of fluxes of organelles due to diffusion and motor-driven transport:

j ¼ jdiff þ jmotor ;

ð13Þ

where the diffusion flux is calculated by the same equation for both axon and dendrite:

while for a dendrite those are

n0þ ¼ N 0þ ;

n0þ ¼ NLþ ;

¼ rL NL ;

nð2Þ 

¼ rL N L ;

ð11aÞ

jdiff ¼ D0þ

dn0þ dn0  D0 ; dx dx

ð14Þ

while the motor-driven flux is calculated by different equations. In an axon, the flux of organelles by motor-driven transport is ð1Þ

ð2Þ

jmotor ¼ nþ þ nþ þ v  nð1Þ þ v  nð2Þ  ;

ð15aÞ

while in a dendrite that is calculated as: ð1Þ

ð2Þ

jmotor ¼ nþ  nþ þ v  nð1Þ  v  nð2Þ  :

ð15bÞ

To evaluate the efficiency of motor-assisted transport in an axon and dendrite, the flux of organelles in the segment by pure diffusion (that would be happening if all MTs in the axon or dendrite were depolymerized) is also presented. This is calculated as:

jdiff ¼ D0þ

NLþ  N0þ NL  N0  D0 : L L

ð16Þ

The concentration of free organelles with kinesin motors attached in the case of pure diffusion is calculated as:

N0þ  NLþ x; L

ð17Þ

while the concentration of those with dynein motors attached is calculated as:

n0 ¼ N0 

N0  NL x: L

ð18Þ

Distributions of n0+(x) and n0(x) for the case of pure diffusion (those given by Eqs. (17) and (18)) are also presented in figures below for comparison purposes. 3. Results and discussion According to Carter and Cross [18] and Vale et al. [19], kinesin-1 (conventional kinesin) walks to the MT plus-end with the average velocity of 1 lm/s while according to King and Schroer [20] and Toba et al. [21] cytoplasmic dynein walks to the MT minus-end with approximately the same average velocity of 1 lm/s. These values are for the case when cargo is moved by a single molecular motor; several motors pulling in the same direction can move cargo at a much faster rate; maximum velocities of organelles in neurons are reported to be 3.5–5 lm/s [22,23]. The average attachment rate to MTs for kinesin-1 is estimated as 5 s1 [24,25] while for cytoplasmic dynein it is estimated as 1.5 s1 [18,19]. The average detachment rate from MTs for kinesin-1 is estimated as 1 s1 [26,19] while for cytoplasmic dynein it is estimated as 0.25 s1 [20,27]. On the other hand, Smith and Simmons [16] used the value of 1 s1 for all attachment/detachment rates in a standard set of primary parameters for numerical work. There is not much data on kinetic constants for switching the type of molecular motor that the organelle is attached to. Jung and Brown [17] estimated the rate of transition of free cytoskeketal elements with minus-end-directed motors attached to those with plus-end-directed motors attached as 1.4  105 s1 and backwards as 4.2  106 s1, but these estimates are made for slow axonal transport and the rates will probably be larger for fast axonal transport. Smith and Simmons [16] used Einstein–Stokes relation to determine the diffusivity of a 1 lm sphere in water and obtained the value of diffusivity of 0.4 lm2/s; then they rounded down this value to 0.1 lm2/s to allow for an irregular surface topology and a larger cytoplasmic viscosity. Thus there is quite a large spread in the values of parameters found in the literature. Based on the analysis of the above data, the following values of dimensionless parameters have been selected: D0þ ¼ D0 ¼ 0:5; k0þ ¼ k0 ¼ 0 0:05; k ¼ 0:5; L ¼ 15; N 0þ ¼ N 0 ¼ 0:1; N Lþ ¼ N L ¼ 0:01; v  ¼ 1; r0 ¼ 0:1, and rL = 0.1. Although these parameter values fall into biologically reasonable ranges, the presented simulation results should not be viewed as modeling transport of any particular organelle species in a particular type of an axon or dendrite, but rather as an attempt to understand possible reasons and implications of the difference in the MT structure of axons and dendrites by means of numerical modeling. Segments of an axon and dendrite that originate from the neuron soma and have the same length are considered, and it is assumed that concentrations of free organelles at the ends of these segments are the same for both the dendrite and axon. Two values of k are used: 1 and 0.1. k = 1 corresponds to a symmetric situation (see kinetic diagram in Fig. 1(c)) when free organelles with plus-end-directed motors are as likely to attach to MTs as free organelles with minus-end-directed motors. k = 0.1 corresponds to the case when free organelles with minus-end-directed motors are much less likely to attach to MTs than organelles with plusend-directed motors; this creates an asymmetry in molecular-motor-assisted transport. There are indications that such asymmetry can exist in real neurons. For example, a recent paper by Müller ~ =k ~þ (which is equal et al. [28] utilized the value 0.3 for the ratio k

to the dimensionless parameter k utilized in this research) for their model of motor coordination in bidirectional transport. There is also experimental evidence that there are sophisticated control mechanisms of transport processes in neurons (some of them can be simulated through variation of parameter values in the developed model). For example, the direction of bidirectional motor-driven transport (when an organelle is simultaneously attached to several molecular motors which may pull it in opposite directions) can be changed by adding certain factors. Indeed, at a certain time in Drosophila development, when levels of the transacting factor Halo (which is a small very basic protein) are low, the net transport of lipid droplets is directed toward the minus-end while when levels of Halo are high, it is directed toward the plus-end [29,30]. Numerical solutions of Eqs. (1a), (2a), (3a), (4a), (5a), (6a), subject to boundary conditions (9a) and (11a) for the axon and Eqs. (1b), (2b), (3b), (4b), (5b), (6b) subject to boundary conditions (9b) and (11b) for the dendrite are obtained using an ODE solver NDSolve included in Mathematica 7.0 package. Fig. 2(a) displays

(a) 0.1 n0+ axon k−=1 n0+ dendrite k−=1 n0+ axon k−=0.1 n0+ dendrite k−=0.1 n0+ pure diffusion

0.08

0.06

n0+

n0þ ¼ N0þ 

A.V. Kuznetsov / Mathematical Biosciences 228 (2010) 195–202

0.04

0.02

0

0

5

x

10

15

(b) 0.1 n0− axon k−=1 n0− dendrite k−=1 n0− axon k−=0.1 n0− dendrite k−=0.1 n0− pure diffusion

0.08

0.06

n0−

198

0.04

0.02

0

0

5

10

15

x Fig. 2. (a) Distributions of the number densities of free organelles with plus-enddirected motors attached to them, n0+(x), (b) distributions of the number densities of free organelles with minus-end-directed motors attached to them, n0(x), for an axon, dendrite, and by pure diffusion (with no motor-driven component).

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A.V. Kuznetsov / Mathematical Biosciences 228 (2010) 195–202

(a)

0.14 (1)

(2)

0.1 0.08

(2)

(1) n(1) + +n+ , n− +n−

(2)

n+ +n+ axon k−=1 +n(2) axon k−=1 n(1) − −

0.12

characterizes net motor-driven organelle flux

0.06 0.04 0.02 0

0

5

x

10

15

(b) 0.14 n(1) +n(2) dendrite k−=1 + − n(1) +n(2) dendrite k−=1 − +

0.12

(1)

(2) n+ +n−(2), n(1) − +n+

distributions of the number densities of free organelles with plusend-directed motors attached to them, n0+(x). The distribution that corresponds to the case of pure diffusion (with no motor-driven component) is depicted by a straight line that connects given concentrations at the segment boundaries. n0+ concentrations in an axon are on average larger than those in a dendrite; this applies to both k = 1 and k = 0.1 situations. In an axon, there is a large region where n0+ is almost independent of x; n0+ exhibits a strong dependence on x only in the two boundary layers, one near the neuron soma and the other at the end of the neurite segment (if the length of the simulated segment corresponds to the total length of the neurite the latter boundary layer is real, otherwise it is an artifact of imposing a fixed concentration boundary condition at the end of the segment). However, in a dendrite n0+ changes gradually with x, there is no region where n0+ is independent of x. Fig. 2(b) displays distributions of the number densities of free organelles with minus-end-directed motors attached to them, n0(x). Again, the distribution that corresponds to the case of pure diffusion is depicted by a straight line that connects given concentrations at the neuron soma and at the end of the neurite segment. For the case of k = 1 in an axon, n0 has a region of independence of x (in fact, for this case over most of the axon length n0 is close to zero). In a dendrite, n0 concentrations are on average larger than for the corresponding axon cases (for the same value of k). Figs. 3(a) and (b) are computed for k = 1. Fig. 3(a) displays distributions of the number densities of organelles transported on MTs anterogradely and retrogradely in an axon. In an axon the concentration of anterogradely transported organelles is calculated as ð1Þ ð2Þ nþ ðxÞ þ nþ ðxÞ and the concentration of retrogradely transported ð2Þ organelles is calculated as nð1Þ  ðxÞ þ n ðxÞ since all MTs have the same polarity. Remarkably, the number density of anterogradely transported organelles (solid line) is parallel to that of retrogradely transported organelles (dashed line) over most of the axon length (except in the diffusion boundary layers at the ends of the axon segment). If the average velocities of kinesin and dynein motors are assumed to be the same (as it is done in this research), the distance between the solid and dashed lines represents the net motordriven organelle transport toward the end of the axon segment (anterograde less retrograde motor-driven flux). Over most of the segment length the lines have a very small slope; in fact they are pretty close to horizontal lines. This means that anterograde and retrograde motor-driven fluxes decay very slowly with x, which indicates a high efficiency of motor-driven transport in axons and explains why axons have an almost unlimited growth potential in vitro. Indeed, this means that once organelles are loaded on MTs at the neuron soma, they are continuously transported anterogradely to the end of the segment (if any organelles detach from MTs during this process about the same amount of organelles is loaded back to MTs, so that the concentration of organelles transported anterogradely on MTs remains approximately constant). Also, the distance between the solid and dashed lines is quite large, which indicates that the net motor driven flux in an axon is quite large. Fig. 3(b) shows concentrations of anterogradely and retrogradely motor-transported organelles in a dendrite. In a dendrite half of MTs have their plus-ends-out and half have their minusends-out, so the concentration of anterogradely transported ð1Þ organelles is nþ ðxÞ þ nð2Þ  ðxÞ and the concentration of retrogradely ð2Þ transported organelles is nð1Þ  ðxÞ þ nþ ðxÞ. The difference with Fig. 3(a) is remarkable: the slope of the lines is much larger. This means that both anterograde and retrograde fluxes decay strongly with the distance from the soma. This explains why dendrites cannot grow long. In general, motor-driven transport in dendrites resembles a counter-flow heat exchanger: in such a heat exchanger most heat from a hot pipe is transferred to a cold pipe before the fluid reaches the end of the heat exchanger. Similarly, in a dendrite

0.1 0.08 0.06 0.04 0.02 0

0

5

x

10

15

Fig. 3. (a) Distributions of the number densities of organelles transported on MTs anterogradely and retrogradely in an axon (a) and dendrite (b). (k = 1). In an axon ð1Þ ð2Þ the concentration of anterogradely transported organelles is nþ ðxÞ þ nþ ðxÞ and the ð2Þ concentration of retrogradely transported organelles is nð1Þ ðxÞ þ n ðxÞ. In a dendrite   ð1Þ the concentration of anterogradely transported organelles is nþ ðxÞ þ nð2Þ  ðxÞ and the ð2Þ concentration of retrogradely transported organelles is nð1Þ  ðxÞ þ nþ ðxÞ.

transport of organelles between the two groups of MTs with opposite polarities is strong, and a large portion of anterogradely transported organelles are lost before they reach the growth cone of a dendrite. Also, the distance between the lines corresponding to anterograde and retrograde transport is much smaller; this indicates that the net flux to the end of the dendrite segment is much smaller than that in an axon. Figs. 4(a) and (b) are similar to Figs. 3(a) and (b), but they are computed for k = 0.1. For an axon (Fig. 4(a)) the distance between the lines is even larger than that in Fig. 3(a), which indicates that the net flux to the end of the segment is larger. This is as expected because decreasing the value of k decreases the rate of minusend-transport on MTs. However, in Fig. 4(b) the distance between the lines is smaller than in Fig. 3(b), which means that reducing k from 1 to 0.1 reduces the net flux toward the end of the dendrite segment. This seems surprising; one would expect that reducing

200

A.V. Kuznetsov / Mathematical Biosciences 228 (2010) 195–202

(a)

(a) 0.14 0.12

(2)

0.1

jdiff axon k−=1 jdiff dendrite k−=1 jdiff axon k−=0.1 jdiff dendrite k−=0.1 jdiff pure diffusion

0.1

j diff

0.08

(2)

(1) n(1) + +n+ , n− +n−

0.15

n(1) +n(2) axon k−=0.1 + + (2) n(1) +n axon k−=0.1 − −

0.06

0.05 0.04 0.02 0

0 0

5

x

10

15

0

5

x

10

15

(b)

(b) 0.14 (1)

(1)

0.05

0.1

j motor

n+ +n−(2), n−(1)+n+(2)

(2)

n+ +n− dendrite k−=0.1 +n(2) dendrite k−=0.1 n(1) − +

0.12

0.08 0.06

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the rate of attachment of organelles with minus-end motors to MTs would reduce the rate of retrograde transport, thus increasing the rate of net transport toward the end of the dendrite segment (as it happens in the axon). However, just the opposite happens, and this is explained by the fact that a dendrite has half of plus-end-out and half of minus-end-out MTs, so decreasing the rate of attachment of organelles with minus-end-directed motors does not do anything to improve the dendrite’s performance with respect to net motor-driven organelle transport. Fig. 5(a) displays distributions of the flux of organelles due to diffusion. The horizontal line in Fig. 5(a) corresponds to pure diffusion transport, which would be the case if all MTs were depolymerized in an axon or dendrite. Fig. 5(a) shows that over most of the axon or dendrite length (except in the boundary layers near the soma and the end of the segment) diffusion flux is relatively small and would be insufficient to provide enough organelle flux. Notably, the diffusion flux for all cases over most of the axon or dendrite length is smaller than that for the case of pure diffusion; this is because a large number of organelles that otherwise would be involved in diffusion transport are involved in motor-driven trans-

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Fig. 5. (a) Distributions of the flux of organelles due to diffusion, jdiff(x), for an axon, dendrite, and by pure diffusion (b) distributions of the flux of organelles due to motor-driven transport, jmotor(x), for an axon and dendrite.

port. Fig. 5(b) displays distributions of the net flux of organelles due to motor-driven transport, jmotor(x). In an axon jmotor is large, which indicates a high efficiency of motor-driven transport in axons; it is a little larger for the k = 0.1 case than for the k = 1 case because reducing k reduces the retrograde motor-driven flux in an axon thus increasing the net flux toward the axon growth cone. In a dendrite the net motor-driven flux is significantly smaller in an axon; it is lager for the k = 1 case and smaller for the k = 0.1 case. This is explained by a mixed polarity of MTs in a dendrite; because of it decreasing k does not selectively decrease retrograde motordriven transport. Notably, the net motor-driven flux is almost independent of x in axons and dendrites. In an axon this is as expected since both anterograde and retrograde motor-driven fluxes are almost independent of x (see Fig. 3(a)). The reason why this is also true for a dendrite follows from Fig. 3(b): although both anterograde and retrograde motor-driven fluxes decay with x, they decay at the same rate, and the net motor-driven flux thus remains constant. Coming back to the conversion of dendrites to axons described in Takahashi et al. [5] and Hayashi et al. [8], it is

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Fig. 6. (a) Distributions of the total number densities of organelles, nt(x), for an axon and dendrite, (b) total flux of organelles versus the length of the axon or dendrite, j(L), for an axon, dendrite, and by pure diffusion.

clear now why this conversion is accompanied by changing the MT structure from a mixed MT polarity arrangement in a dendrite to a uniform MT polarity arrangement in an axon. As a dendrite that is being converted to an axon grows, the MT arrangement with a mixed polarity orientation simply cannot provide enough organelle flux to the growth cone of the growing axon; therefore, the neurite has to change its MT polarity arrangement. This is in agreement with the explanation offered in Takahashi et al. [5], who observed that the regeneration of an axon from the dendritic tip required 5 h longer after plating than did the regeneration of axons from the cell body; the delay is explained by the reorganization of the dendritic cytoskeleton needed to support organelle flux necessary for axonal elongation. This also suggests that the difference in an MT polarity arrangement between axons and dendrites may be a regulatory mechanism for limiting dendritic growth. Fig. 6(a) displays distributions of the total number densities of organelles, nt(x), which is a quantity accessible to experiments. In an axon, nt(x) is much more horizontal than in a dendrite, which indicates that most of organelle flux in an axon occurs by motor-driven

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transport. In a dendrite, nt(x) exhibits a larger variation over the dendrite length and has almost a constant slope, which suggests that in a dendrite diffusion transport plays a more important (although still fairly insignificant) role than in an axon. Fig. 6(b) displays the total flux of organelles versus the length of the axon or dendrite, j(L). In an axon, j is much larger than that by pure diffusion (with no motor-assisted component). j for k = 0.1 is a little larger than for k = 1, which is as expected since in an axon retrograde motor-driven transport is significantly reduced by reducing k. In a dendrite the total flux of organelles is still larger than that by pure diffusion, but it is significantly smaller than that in an axon. Decreasing k does not increase the total organelle flux in a dendrite because dendrites have mixed MT polarities. Most importantly, in a dendrite j decreases fast when x is increased, meaning that dendrites can support sufficient organelle transport only if they are short. Unlike that, in an axon j is almost independent of x, meaning that axons can grow to significant lengths without reducing organelle transport to their growth cones. A very slight decrease of j that happens in an axon is mostly due to the decrease of a diffusion component of the total organelle flux. Predictions of Fig. 6(b) agree well with the experimental observation that dendrites are much slower growing than axons and reach much shorter lengths [31]. The slower growing occurs because dendrites have a much smaller organelle flux to the growth cone. Also, Fig. 6(b) shows that the total flux of organelles decays fast as the length of the dendrite increases; this conclusion applies to both k = 1 and k = 0.1 cases. Thus the developed model predicts that the MT structure in dendrites can provide sufficient organelle flux necessary for dendritic growth only at small dendrite lengths. On the other hand, by looking at Fig. 6(b), one can easily understand why it is found in experiments that axons have an effectively unlimited growth potential in vitro [32]. Indeed, the total flux of organelles decays very slowly with L (and this decay is mostly due to the decay of the diffusion component of the total flux). The motordriven flux in axons is independent of x over most of the axon’s length (see Fig. 5(b)); the uniformly polarized MT structure of an axon can thus transport large organelle fluxes in very long axons. The obtained results throw some light on the question why dendrites have a mix of plus-end-out and minus-end-out MTs. The mix of MT polarities in a dendrite may be beneficial for flexibility of active unidirectional transport, providing the opportunity to change the direction of transport without changing the type of a molecular motor or without controlling and coordinating the work of molecular motors, in the case when motors with opposing polarities are attached to an organelle. This may provide necessary flexibility for transport of certain cargo in dendrites, especially if the cargo is small and it is impossible to dock motors of different polarity to the same cargo or if controlling the motors presents a problem. It also may be beneficial if, say, only kinesin motors are readily available at a certain location. It is then possible to use MTs whose ends are oriented toward the end of the dendrite for transport deeper into the dendrite and to use MTs whose ends are oriented toward the neuron soma for transport toward the cell center. Another possible reason for different MT orientations in dendrites is that such an arrangement makes it possible to avoid traffic jams. For example, if while moving down the dendrite the cargo encounters a blockage region, it can switch to an MT pointing into the opposite direction and back off this region without switching the type of a molecular motor. This is similar to one of possible explanations of bidirectional transport given in Gross [30].

4. Conclusions This paper compared transport of intracellular vesicles in segments of axons and dendrites of the same length with the goal of understanding the implications of different MT polarizations in

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these two neurite outgrowths on transport of organelles. Motordriven transport of organelles in axons is highly efficient: anterograde and retrograde motor-driven fluxes decay very slowly with the distance from the soma, which indicates a high efficiency of motor-driven transport in axons and explains why axons have almost unlimited growth potential. Indeed, this means that once organelles are loaded on MTs at the neuron soma, they are continuously transported anterogradely to the end of the axon. In a dendrite, the picture is drastically different: both anterograde and retrograde fluxes decay strongly with the distance from the soma. This is caused by organelle transport between two groups of MTs with opposite polarities in a dendrite; as a result of this interaction a large portion of anterogradely transported organelles are lost before they reach the growth cone of a dendrite. As a result, the net flux of organelles to the end of the dendrite segment is much smaller than that in an axon. It is also demonstrated that the diffusion flux for all cases over most of the axon or dendrite length is smaller than that for the case of pure diffusion. This is explained by the fact that a large number of organelles that otherwise would be involved in diffusion are involved in motor-driven transport. The obtained computational results demonstrate that the MT arrangement in dendrites can provide sufficient organelle flux necessary for dendrite growth only at small dendrite lengths while axons have an effectively unlimited growth potential. This is because the total flux of organelles in axons decreases very slowly with the axon length while it decreases very fast in dendrites. Concerning the reasons for different MT polarity orientations in axons and dendrites, the presented modeling supports the following possible explanations. One benefit of having a mix of positively and negatively oriented MTs in dendrites is the flexibility of using a particular molecular motor that is available at a certain location (if kinesin is available, then the plus-end-out MTs can be used to transport organelles into the dendrite and minus-end-out MTs can be used to transport organelles out of the dendrite). Another benefit of the mixed MT orientation in dendrites is the possibility to avoid traffic jams. Indeed, if an organelle moving on a plusend-out MT encounters a region with a traffic jam, it can back off this region without switching the molecular motor by using an MT with the opposing orientation. On the other hand, since axons require a much larger organelle flux, they have to have all MTs oriented with their plus-ends out. Acknowledgements The author gratefully acknowledges critical comments of the anonymous reviewer. References [1] S.L. Mironov, Spontaneous and evoked neuronal activities regulate movements of single neuronal mitochondria, Synapse 59 (2006) 403. [2] R.B. Vallee, G.S. Bloom, Mechanisms of fast and slow axonal-transport, Annual Review of Neuroscience 14 (1991) 59. [3] E.L.F. Holzbaur, Axonal transport and neurodegenerative disease, in: P. St.George-Hyslop et al. (Eds.), Intracellular Traffic and Neurodegenerative Disorders, Springer, Berlin, 2009, p. 27. [4] M. Linial, The secrets of a functional synapse-from a computational and experimental viewpoint, BMC Bioinformatics 7 (2006) S6.

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