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A coupled model of fast axonal transport of organelles and slow axonal transport of tau protein a
b
I.A. Kuznetsov & A.V. Kuznetsov a
Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD 21218-2694, USA b
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA Published online: 28 May 2014.
Click for updates To cite this article: I.A. Kuznetsov & A.V. Kuznetsov (2015) A coupled model of fast axonal transport of organelles and slow axonal transport of tau protein, Computer Methods in Biomechanics and Biomedical Engineering, 18:13, 1485-1494, DOI: 10.1080/10255842.2014.920830 To link to this article: http://dx.doi.org/10.1080/10255842.2014.920830
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Computer Methods in Biomechanics and Biomedical Engineering, 2015 Vol. 18, No. 13, 1485–1494, http://dx.doi.org/10.1080/10255842.2014.920830
A coupled model of fast axonal transport of organelles and slow axonal transport of tau protein I.A. Kuznetsova and A.V. Kuznetsovb* a
Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD 21218-2694, USA; bDepartment of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA
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(Received 31 December 2013; accepted 30 April 2014) We have developed a model that accounts for the effect of a non-uniform distribution of tau protein along the axon length on fast axonal transport of intracellular organelles. The tau distribution is simulated by using a slow axonal transport model; the numerically predicted tau distributions along the axon length were validated by comparing them with experimentally measured tau distributions reported in the literature. We then developed a fast axonal transport model for organelles that accounts for the reduction of kinesin attachment rate to microtubules by tau. We investigated organelle transport for two situations: (1) a uniform tau distribution and (2) a non-uniform tau distribution predicted by the slow axonal transport model. We found that non-uniform tau distributions observed in healthy axons (an increase in tau concentration towards the axon tip) result in a significant enhancement of organelle transport towards the synapse compared with the uniform tau distribution with the same average amount of tau. This suggests that tau may play the role of being an enhancer of organelle transport. Keywords: tau protein; organelle transport; fast axonal transport; slow axonal transport
1.
Introduction
Tau protein can affect cellular functions in many ways. The pathological role of tau in Alzheimer’s disease is well known (Grundke-Iqbal et al. 1986; Wood et al. 1986). Indeed, intraneuronal tangles composed mostly of hyperphosphorylated tau protein are one of the defining features of Alzheimer’s disease (Selkoe 2013); it is possible that tau oligomers are even more neurotoxic than tau tangles (Lasagna-Reeves et al. 2011). Originally, it was believed that, in healthy axons, the major function of tau is microtubule (MT) stabilisation, but later research has suggested the function of tau to be much more complex (Fanara et al. 2010; Morris et al. 2011). It is now believed that tau plays a complicated role in regulating organelle traffic in axons (Morris et al. 2011). Understanding the role of tau as a regulator of intracellular traffic is quite controversial and requires further research. In vitro studies suggest that tau retards kinesin-driven transport along MTs (Ebneth et al. 1998; Stamer et al. 2002; Dixit et al. 2008; Xu et al. 2013). This could be due to tau occupying binding sites along MTs, leaving less binding sites available to kinesin. It is also possible that tau regulates the number of kinesin motors that transport an organelle (Vershinin et al. 2007). On the other hand, other research (Morfini et al. 2007; Yuan et al. 2008) seems to indicate that tau does not affect kinesin motility. A possible explanation of this contradiction was suggested by McVicker et al. (2011) who pointed out that tau’s effect on kinesin may depend on the mode of tau’s interaction with MTs.
*Corresponding author. Email: [email protected] q 2014 Taylor & Francis
In this paper, we attempt to model how tau affects fast axonal transport of intracellular organelles. We rely on experimental research by Dixit et al. (2008) who reported that tau regulates organelle transport by reducing the rate at which kinesin-1 binds to MTs. We neglected the effect of tau on cytoplasmic dynein because a much higher tau concentration is needed to affect dynein (Dixit et al. 2008). We also neglected the effect of tau on kinesin velocity and kinesin detachment rate from MTs because, according to experimental data reported by Seitz et al. (2002), once kinesin motors are attached to MTs, their speed and runlength are not affected by the presence of tau. To complete the model describing the effect of tau on fast axonal transport, the distribution of tau itself needs to be modelled. In developing a model of tau transport, we followed experimental reports indicating that tau is transported in axons at a rate consistent with slow axonal transport (Utton et al. 2002; Saha et al. 2004). Data reported in Utton et al. (2005) suggest similarity between tau transport and transport of neurofilaments (NFs). Similar to NFs, particles containing tau exhibit rapid and bi-directional movements as well as extended periods of pausing. There is even similarity between parameters describing tau transport and transport of NFs. Indeed, tau spends 73% of the time in the pausing state while NFs spend , 80% of the time in the pausing state (Utton et al. 2005). This allowed us to use the slow axonal transport model, which was originally developed in Jung and Brown (2009) and Li et al. (2012) for transport of NFs, in order to simulate tau distribution in the axon.
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I.A. Kuznetsov and A.V. Kuznetsov
2. A model of slow axonal transport of tau 2.1 Equations governing slow axonal transport of tau A sketch of the problem is displayed in Figure 1. To quantify tau protein transport, we characterised tau concentrations in various kinetic states by their linear number density, which is the number of tau protein molecules residing in a particular kinetic state per unit length of the axon (mm21). We considered tau concentrations in the six kinetic states displayed in Figure 2(a). To obtain dimensionless concentrations, we scaled all concentrations with respect to the concentration of off-track tau molecules at the tip of the axon, c*x¼L :
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ca ¼
c*a c*x¼L
;
cr ¼
c* cr0 ¼ *r0 ; cx¼L
c*r c*x¼L
cap ¼
;
ca0 ¼
c*ap c*x¼L
;
c*a0 ; c*x¼L
crp ¼
c*rp c*x¼L
ð1Þ
Figure 1. Sketch of a neuron with an axon. The coordinate system in the axon is also displayed.
v*r are the instantaneous velocities of kinesin and dynein motors when they transport tau protein, respectively (mm s21) (the signs in Equations (2) and (3) are chosen such that both v*a and v*r are positive); g*10 is a first-order rate constant describing the probability of tau transition from the running (anterograde or retrograde) to the
:
We described tau transport by a modified version of Jung –Brown’s model (Jung and Brown 2009) originally developed to simulate transport of NFs. The modifications include incorporating the effects of tau diffusion and degradation (Kuznetsov IA and Kuznetsov AV 2014a). It is assumed that only off-track tau can be transported by diffusion. As tau degradation mainly occurs in proteasomes (Poppek et al. 2006), we also assumed that only offtrack tau can be degraded. Indeed, in order to be degraded, tau must first enter the proteasome’s proteolytic chamber (Kierszenbaum 2000), and in order to do that, tau must be detached from MTs. The equations governing tau transport are dca 2v*a * 2 g*10 ca þ g*01 ca0 ¼ 0; ð2Þ dx dcr v*r * 2 g*10 cr þ g*01 cr0 ¼ 0; ð3Þ dx 2 g*01 þ g*ar þ g*off ca0 þ g*10 ca þ g*ra cr0 þ g*on cap ¼ 0; ð4Þ * 2 g01 þ g*ra þ g*off cr0 þ g*10 cr þ g*ar ca0 þ g*on crp ¼ 0; ð5Þ d2 cap þ g*off ca0 2 g*on þ g*ar cap þ g*ra crp 2 * dx cap ln ð2Þ 2 ¼ 0; T *1=2
ð6Þ
d2 crp þ g*off cr0 2 g*on þ g*ra crp þ g*ar cap 2 * dx crp ln ð2Þ 2 ¼ 0; T *1=2
ð7Þ
D*ap
D*rp
where D*ap and D*rp are the diffusivities of tau protein in the off-track state associated with anterograde (kinesin) and retrograde (dynein) motors, respectively (mm2 s21); v*a and
Figure 2. (a) Kinetic diagram used for the model of slow axonal transport of tau proteins. Six kinetic states for tau proteins and kinetic processes between these states are displayed. The diagram is based on the kinetic scheme of slow axonal transport suggested in Figure 4 of Jung and Brown (2009) and Figure 1 of Li et al. (2012). (b) Kinetic diagram used for the model of fast axonal transport of organelles.
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Computer Methods in Biomechanics and Biomedical Engineering corresponding pausing state (s21 ); g*01 is a first-order rate constant describing the probability of tau transition from the pausing (anterograde or retrograde) to the corresponding running state (s21 ); g*ar is a first-order rate constant describing the probability of tau transition from the anterograde (pausing or off-track) to the corresponding retrograde state (s21 ); g*ra is a first-order rate constant describing the probability of tau transition from the retrograde (pausing or off-track) to the corresponding anterograde state (s21 ); g*on is a first-order rate constant describing the probability of tau transition from the offtrack (anterograde or retrograde) to the corresponding pausing state (s21 ); g*off is a first-order rate constant describing the probability of tau transition from the pausing (anterograde or retrograde) to the corresponding off-track state (s21 ); and x * is the linear coordinate along the axon (mm). Equations (2) –(7) were solved subject to the following boundary conditions: At x * ¼ 0 :
cap ¼ cx¼0 ;
crp ¼ cx¼0 ;
ca ¼ cx¼0 ; At x * ¼ L * :
cap ¼ 1;
crp ¼ 1;
cr ¼ 0;
ð8Þ ð9Þ
where cx¼0 ¼ c*x¼0 =c*x¼L in which c*x¼0 is the concentration of off-track tau molecules at the axon hillock. The last condition in Equation (9), cr ¼ 0, reflects the fact that tau cannot reverse its direction and move back into the axon. This is supported by experimental observations reflecting that tau is trapped at the distal axon by interaction with membrane proteins (Weissmann et al. 2009; Gauthier-Kemper et al. 2011).
Table 1. Parameter D*ap D*rp T *1=2 v*a v*r g*10 g*01 g*ar g*ra g*off g*on a
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The total dimensionless concentration of tau is defined as ctot ¼ ca þ cr þ ca0 þ cr0 þ cap þ crp :
ð10Þ
2.2 Estimation of parameter values for the tau transport model The ranges of parameters and the values we used in our computations are summarised in Table 1.
2.3
Results for the tau distribution
All numerical solutions were obtained using the BVP4C solver that is a part of Matlab 7 (MathWorks, Natick, MA, USA). The BVP4C solver’s default settings were used. We checked that decreasing the values of parameters that control the error tolerance properties (RelTol and AbsTol) did not affect the solutions. This insured that the obtained solutions were accurate. In Figure 3, distributions of the total tau concentration obtained by numerically solving Equations (2) – (7) (and then using Equation (10)) are compared with experimentally measured tau distributions. The experimental tau distributions were taken from figure 7(B),(D) of Black et al. (1996) and then normalised by using the condition ctot jx¼0 ¼ 1. The numerical solutions correctly capture the overall increase in tau concentration towards the axon tip. The slow axonal transport model given by Equations (2) –(7) accounts for two mechanisms of tau transport: (1) by diffusion; this mode of transport does not require energy input but can only occur from a location with a high concentration of free tau towards a location with a low concentration of free tau; (2) by motor-driven transport;
Estimated values of parameters involved in the tau transport model. Dimension 2 21
mm s mm2 s21 s mm s21 mm s21 s21 s21 s21 s21 s21 s21
Value or range 3 3 2:16 £ 105 0.3 –1 0.3 –1 9:30 £ 1022 4:10 £ 1022 3:10 £ 1025 6:90 £ 1025 4:45 £ 1023 2:75 £ 1024
Reference Konzack et al. (2007) Konzack et al. (2007) Poppek et al. (2006) Konzack et al. (2007) Konzack et al. (2007) Li et al. (2012) Li et al. (2012) Li et al. (2012) Li et al. (2012) Li et al. (2012) Li et al. (2012)
Value used in computations 3 3 2:16 £ 105 0.51a 0.52a 9:30 £ 1022 4:10 £ 1022 3:10 £ 1025 6:90 £ 1025 4:45 £ 1026 b 2:75 £ 1024
Here we used values reported in Jung and Brown (2009) for slow NF transport (since NFs are propelled by the same fast transport motors that propel tau proteins). These values are within the range of instantaneous tau velocities reported in Konzack et al. (2007). b The set of kinetic constants given in Li et al. (2012) was optimised to simulate the transport of NFs. Some of these constants had to be modified to model tau transport. We found that in order to match the tau profile reported in Black et al. (1996), the value of g*off had to be reduced. Physically, this means that tau proteins transition from the pausing to the corresponding off-track state less frequently than NFs.
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I.A. Kuznetsov and A.V. Kuznetsov calculated the average tau concentration:
(a) 3.5 numerical data from Fig. 7B of Black et al.
3
ctot; av ¼
2.5
1 L*
ctot 1.5
* cdev tot ðx Þ ¼
1
0
50
100
150
200
250
300
ðL *
350
x *, µm 0
(b) 3.5
ctot ðx * Þ 2 ctot; av : ctot ðL * Þ 2 ctot; av
ð12Þ
2.5 2 1.5 1 0.5 0
100
200
300
400
* * cdev tot ðx Þ dx ¼ 0
ð13Þ
* cdev tot ðL Þ ¼ 1:
ð14Þ
and
numerical . data from Fig. 7D of Black et al.
3
ctot
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ð11Þ
* cdev tot ðx Þ defined by Equation (12) satisfies two properties, which are given as follows:
0.5
0
ctot ðx * Þ dx * :
0
The renormalised tau distribution was computed as follows:
2
0
ðL *
500
600
x *, µm
Figure 3. (a) The total concentration of tau protein for a 350-mmlong axon. Experimental data from figure 7(B) of Black et al. (1996) are also shown. (b) The total concentration of tau protein for a 600mm-long axon. Experimental data from figure 7(D) of Black et al. (1996) are also shown. These experimental data were normalised such that at the axon hillock, x * ¼ 0, ctot was equal to 1.
this mode requires energy input but can occur in any direction; the direction depends on whether transport is driven by kinesin or dynein motors. The complexity of the model, which includes two modes of transport, explains why numerical results can correctly reproduce the complicated tau distribution that includes an initial slight decrease in tau concentration and then a sharp increase closer to the axon tip (Figure 3(b)).
3. A model of fast axonal transport of organelles and vesicles 3.1 Equations governing fast axonal transport of organelles and vesicles After ctot (defined by Equation (10)) was computed, we renormalised it by the following procedure. First we
* cdev tot ðx Þ thus characterises the dimensionless deviation of tau concentration, ctot ðx * Þ, from its average value, * ctot; av . cdev tot ðx Þ is normalised such that its maximum deviation from the average value (at the axon tip) is equal * to 1 and the average value of cdev tot ðx Þ is equal to 0 (see Figure 4(a),(b)). Dixit et al. (2008) reported that the attachment rate of kinesin to MTs is reduced by tau. We assumed that this is the only effect that tau has on organelle transport. Also, relying on Dixit et al. (2008), we neglected the effect of tau on dynein. In developing our model, we used the simplest linear correlation to characterise the dependence of the kinetic constant describing the rate of kinesin attachment to MTs, k*þ , on the dimensionless tau concentration, * cdev tot ðx Þ (Kuznetsov IA and Kuznetsov AV 2014b): ð15Þ k*þ ¼ k*þ0 1 2 a cdev tot ;
where k*þ0 is a parameter that characterises the rate of kinesin attachment to MTs for the uniform (cdev tot ¼ 0) tau distribution (s21) and a is a parameter that characterises by how much the increase in tau concentration reduces the probability of attachment of kinesin motors to MTs (0 # a # 1). The Smith –Simmons model describing fast axonal transport of vesicles and organelles (Smith and Simmons 2001) includes three organelle concentrations (Figure 2 (b)). These are the concentration of free organelles, n*0 ; the concentration of MT-bound organelles driven anterogradely by kinesin motors, n*þ ; and the concentration of MTbound organelles driven retrogradely by dynein motors, n*2 . Similar to the concentration of tau protein, we characterised all organelle concentrations by their linear number densities (mm21 ). To obtain dimensionless concentrations, all concentrations were rescaled with
Computer Methods in Biomechanics and Biomedical Engineering (a)
where D*0 is the diffusivity of free organelles (mm2 s21), v*þ is the velocity of organelles transported by kinesin motors (mm s21), v*2 is the velocity of organelles transported by dynein motors (mm s21), k*2 is the kinetic constant describing the rate of dynein attachment to MTs (s21), k0 *þ is the kinetic constant describing the rate of kinesin detachment from MTs (s21) and k0 *2 is the kinetic constant describing the rate of dynein detachment from MTs (s21). Equations (17) – (19) were solved subject to the following boundary conditions:
1 uniform non−uniform (numerical)
0.8 0.6
cdev tot
0.4 0.2 0 −0.2
n0 ð0Þ ¼ 1;
−0.4 0
50
100
150
200
250
300
NL ¼
uniform non−uniform (numerical)
0.8 0.6
0.2 0 −0.2
N *L ; N *0
ntot ðxÞ ¼ n0 ðxÞ þ nþ ðxÞ þ n2 ðxÞ: 0
100
200
300
400
500
ð21Þ
Figure 4. Deviation of tau concentration from its average value, cdev tot , (a) for a 350-mm-long axon and (b) for a 600-mm-long axon. cdev tot is used in modelling the effect of tau on kinesin attachment rate (see Equation (15)).
The total organelle flux, which includes the diffusion-, kinesin- and dynein-driven components, respectively, is calculated as j*tot ¼ 2D*0
respect to the concentration of free organelles at x * ¼ 0, N *0 : n* n0 ¼ 0* ; N0
n* nþ ¼ þ* ; N0
n* n2 ¼ 2* : N0
d2 n0 0* 0* * ¼ 2k*þ0 1 2 a cdev tot n0 2 k2 n0 þ k þ nþ þ k 2 n2 ; *2 dx ð17Þ dnþ 0* ¼ k*þ0 1 2 a cdev tot n0 2 k þ nþ ; dx * 2v*2
dn2 ¼ k*2 n0 2 k0 *2 n2 ; dx *
ð18Þ ð19Þ
›n*0 þ v*þ n*þ 2 v*2 n*2 : ›x *
ð23Þ
The total dimensionless organelle flux is then found as jtot ¼
ð16Þ
We modified the Smith – Simmons equations (Smith and Simmons 2001) by using Equation (15) for k*þ . We assumed that all other kinetic constants (k*2 , k0 *þ and k0 *2 ) are independent of tau concentration. For a steady-state situation, the governing equations are
v*þ
ð22Þ
600
x *, µm
2D*0
ð20Þ
where N *L is the concentration of free organelles at x * ¼ L *; s0 and sL are the degrees of loading of free organelles on MTs at the axon hillock and at the axon tip, respectively, and N *L is the concentration of free organelles at x * ¼ L *. The total dimensionless organelle concentration is then found as
0.4
−0.4
nþ ð0Þ ¼ s0 ;
In Equation (20),
1
cdev tot
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n0 ðL * Þ ¼ N L ;
n2 ðL * Þ ¼ sL N L :
350
x *, µm (b)
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j*tot : v*þ N *0
ð24Þ
3.2 Estimation of parameter values for the tau transport model Estimated ranges of parameters for the model of organelle transport and the values we used in our computations are summarised in Table 2.
3.3
Results for the organelle transport
To evaluate the effect of tau on fast axonal transport of organelles, we used tau distributions computed in Section 2.3 (displayed by solid lines in Figure 3(a),(b)). These computed tau distributions were then renormalised by using Equation (12); the renormalised distributions
1490 Table 2. Notation
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a D*0 k*þ0
I.A. Kuznetsov and A.V. Kuznetsov Estimated values of parameters involved in the fast axonal transport model of organelles and vesicles. Dimension 2 21
mm s s21
Value or range b
0.1 – 0.4 5
k*2
s21
1.5
k0 *þ
s21
1
k0*2
s21
0.25
L* NL v*þ
mm
,103 to 106
mm s21
1
v*2
mm s21
1
s0 sL
0 – 10 0 – 10
Reference Smith and Simmons (2001) Mu¨ller et al. (2010), Beeg et al. (2008), Leduc et al. (2010) Mu¨ller et al. (2010), King and Schroer (2000), Reck-Peterson et al. (2006) Mu¨ller et al. (2010), Schnitzer et al. (2000), Vale et al. (1996) Mu¨ller et al. (2010), King and Schroer (2000), Reck-Peterson et al. (2006) Debanne et al. (2011) Vale et al. (1996), Carter and Cross (2005) King and Schroer (2000), Toba et al. (2006) Smith and Simmons (2001) Smith and Simmons (2001)
Value used in computations 0.02, 0.04, 0.06a 0.1 1c 1c 1c 1c 350, 600d 0.5e 1 1 0.1 0.1
a These values were chosen to demonstrate the effect of parameter a, which describes a decrease in kinesin attachment rate to MTs by tau protein. The increase in a increases organelle concentration in the axon. This is because when less kinesin-driven organelles attach to MTs, fewer of them are transported towards the synapse, and more accumulate in the axon. We have chosen the value of a to satisfy two conditions: (1) there is no unrealistically high accumulation of organelles in the axon and (2) there is a visible effect of tau on organelle transport (if a ¼ 0, tau does not affect organelle transport, see Equation (15)). Condition (2) is consistent with the results reported in Dixit et al. (2008) which indicate that tau does have effect on fast axonal transport because it inhibits kinesin more strongly than dynein. b This range is based on the estimate reported in Smith and Simmons (2001) for a vesicle with 1-mm diameter. c We followed Smith and Simmons (2001) who used the same value of 1 s21 for all attachment/detachment rates in a standard set of primary parameters for numerical work. d We simulated axons with lengths of 350 and 600 mm because these are the axon lengths for which experimentally measured tau distributions are reported in Black et al. (1996). e Synthesis of most organelles occurs in the neuron soma; the organelles are then transported towards the axon synapse by fast axonal transport (Goldstein and Yang 2000). Based on this model, we assumed that the concentration of free organelles at the base of the axon (x * ¼ 0) is two times larger than that at the axon tip (x * ¼ L * ).
(showing the deviation from the average concentration of tau) are displayed by solid lines in Figure 4(a),(b). The renormalised distributions were used in Equation (15) to quantify the reduction of kinesin attachment rate to MTs by tau. Figures 5 and 6 show the results for a 350-mm-long axon. In Figure 5, we show organelle concentrations in the free (Figure 5(a)), kinesin-driven (Figure 5(b)) and dynein-driven (Figure 5(c)) kinetic states. We consider a uniform tau distribution [corresponding to ctot ðx * Þ ¼ ctot; av * or cdev tot ðx Þ ¼ 0] and a numerically computed tau distribution, shown by a solid line in Figure 3(a). Both uniform and non-uniform tau distributions are for the same total number of tau molecules in the axon (Figure 4(a)), but for the non-uniform tau distribution, there are fewer tau molecules in the proximal axon and more tau molecules in the distal axon. We characterise the strength of the effect of tau on kinesin attachment rate to MTs by specifying three values for parameter a: 0.02, 0.04 and 0.06. A larger value of a corresponds to stronger inhibition of kinesin attachment to MTs by tau. For a uniform tau distribution, the organelle concentrations are almost linear, except for sharp variations very close to the axon base
(x * ¼ 0) and axon tip (x * ¼ L * ). These sharp variations are due to diffusion boundary layers close to the axon base and axon tip; in the rest of the axonal domain, organelle transport is mostly motor driven. The effect of a non-uniform tau distribution in Figure 5 is to increase organelle concentrations in the axon in all three kinetic states. The increase is larger for larger values of a; larger a corresponds to a stronger reduction of kinesin attachment to MTs by tau. The qualitative similarity between n0 , nþ and n2 displayed in Figure 5(a) – (c), respectively, is due to the fact that all kinetic constants that control transitions between various organelle populations (Figure 2(a)) were set to 1 (Table 2). The total organelle concentration, ntot , is the sum of n0 , nþ and n2 . ntot is approximately three times larger than any of these concentrations, but the trends are qualitatively the same (Figure 6(a)). The flux of organelles towards the axon tip (Figure 6 (b)) is independent of x * . This is a consequence of the steady-state formulation of the problem. Indeed, in the steady-state situation, organelles cannot accumulate in any control volume inside the axon; hence, organelle fluxes in and out of the control volume must be the same.
Computer Methods in Biomechanics and Biomedical Engineering (a)
(a) 1.5 uniform a = 0.02 a = 0.04 a = 0.06
4 uniform a = 0.02 a = 0.04 a = 0.06
3.5 3
1
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ntot
n0
2.5 2 1.5
0.5
1 0.5 0
0
50
100
150
200
250
300
0
350
0
50
100
x *, µm (b)
uniform a = 0.02 a = 0.04 a = 0.06
3
250
300
350
150 200 * x , µm
250
300
350
x 10−3 uniform a = 0.02 a = 0.04 a = 0.06
2.5
1
jtot
n+
2 1.5
0.5 1 0.5 0
0
50
100
150
200
250
300
350
0
x *, µm (c) 1.5
1
50
100
concentration (see the solid line in Figure 3(a)), depends on parameter a. The more the kinesin attachment rate is reduced by tau (larger a), the larger the flux towards the axon tip becomes (Figure 6(b)). Table 3 shows by how much the organelle flux is increased compared with a
0.5
0
0
Figure 6. Fast axonal transport model. (a) Dimensionless total organelle concentration and (b) dimensionless total organelle flux. Results for a 350-mm-long axon.
uniform a = 0.02 a = 0.04 a = 0.06
n–
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(b) 1.5
150 200 x *, µm
Table 3. Effect of non-uniformity of tau distribution on organelle flux towards the axon tip. 0
50
100
150
200
250
300
350
x *, µm
Figure 5. Fast axonal transport model. Dimensionless concentrations of (a) free organelles, (b) anterogradely moving organelles transported by kinesin motors and (c) retrogradely moving organelles transported by dynein motors. Results for a 350-mm-long axon.
The flux for the uniform tau distribution is the smallest (Figure 6(b)). The flux for the non-uniform tau distribution, exhibiting proximal-to-distal increase in tau
Increase compared Parameter characterising with the uniform tau distribution, the effect of 100ðjtot 2 jtot; uniform Þ= tau on kinesin Axon length, attachment rate, a jtot; uniform (%) L * (mm) 350 350 350 600 600 600
0.02 0.04 0.06 0.02 0.04 0.06
35 77 124 42 93 147
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I.A. Kuznetsov and A.V. Kuznetsov (a)
(a) 1.5 uniform a = 0.02 a = 0.04 a = 0.06
4 uniform a = 0.02 a = 0.04 a = 0.06
3.5 3
1 n0
ntot
2.5 2 1.5
0.5
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Figure 8. Fast axonal transport model. (a) Dimensionless total organelle concentration and (b) dimensionless total organelle flux. Results for a 600-mm-long axon.
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0
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x *, µm
Figure 7. Fast axonal transport model. Dimensionless concentrations of (a) free organelles, (b) anterogradely moving organelles transported by kinesin motors and (c) retrogradely moving organelles transported by dynein motors. Results for a 600-mm-long axon.
uniform tau distribution. For a ¼ 0.02, 0.04 and 0.06, the increase is by 35%, 77% and 124%, respectively, which indicates that the organelle flux is highly sensitive to the
effects of tau. This also suggests that the reason for the non-uniform tau distribution (the proximal-to-distal increase in tau concentration displayed in Figure 3(a)) may be to increase the flux of organelles towards the axon tip. Thus, tau may not just be a regulator, but also an enhancer of organelle transport. Figures 7 and 8 show the results for a 600-mm-long axon. The trends are similar to those displayed in Figures 5 and 6 for a 350-mm-long axon. A notable difference is in the effect of parameter a on the organelle flux towards the axon tip for a non-uniform tau distribution (Figure 8(b)). The enhancements of the organelle flux for different values of parameter a are summarised in Table 3. For a ¼ 0.02, 0.04 and 0.06, the increase in the flux is by 42%, 93% and 147%, respectively. It is noteworthy that the percentage increase in the flux for a non-uniform tau distribution is larger for a longer axon. This is because the effect of tau accumulates over the axon length. This begs an interesting question. As axons may be much longer than 600 mm in
Computer Methods in Biomechanics and Biomedical Engineering
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length, does this mean that kinesin attachment rate to MTs is reduced differently for axons with different lengths?
4. Conclusions We investigated the effect of tau on fast axonal transport of organelles towards the axon synapse by coupling a model of fast axonal transport and a model of tau protein transport by the slow axonal transport mechanism. To the best of our knowledge, this is the first attempt to investigate the interaction of fast and slow axonal transport mechanisms. Our results indicate that the proximal-to-distal increase in tau concentration, observed in healthy axons, enhances organelle transport towards the synapse, compared with a uniform tau distribution. The enhancement depends on how strongly tau reduces the kinesin attachment rate to MTs. The greater the effect of tau on kinesin attachment rate, the larger the enhancement of organelle transport towards the synapse. This suggests a potential role of tau as an enhancer of fast axonal transport. We also investigated the effect of axon length on the enhancement of organelle transport towards the synapse. We found that the enhancement is larger for longer axons. This suggests that the effect of tau on kinesin attachments rate may accumulate along the axon length, causing stronger enhancement for longer axons.
Acknowledgements AVK gratefully acknowledges the support of the Alexander von Humboldt Foundation (Germany) through the Humboldt Research Award.
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Schnitzer MJ, Visscher K, Block SM. 2000. Force production by single kinesin motors. Nat Cell Biol. 2:718 – 723. Seitz A, Kojima H, Oiwa K, Mandelkow EM, Song YH, Mandelkow E. 2002. Single-molecule investigation of the interference between kinesin, tau and MAP2c. Embo J. 21:4896–4905. Selkoe DJ. 2013. Snapshot: pathology of Alzheimer’s disease. Cell. 18:468 – 468.e1 Smith DA, Simmons RM. 2001. Models of motor-assisted transport of intracellular particles. Biophys J. 80:45– 68. Stamer K, Vogel R, Thies E, Mandelkow E, Mandelkow E. 2002. Tau blocks traffic of organelles, neurofilaments, and APP vesicles in neurons and enhances oxidative stress. J Cell Biol. 156:1051 – 1063. Toba S, Watanabe TM, Yamaguchi-Okimoto L, Toyoshima YY, Higuchi H. 2006. Overlapping hand-over-hand mechanism of single molecular motility of cytoplasmic dynein. Proc Nat Acad Sci USA. 103:5741– 5745. Utton M, Connell J, Asuni A, van Slegtenhorst M, Hutton M, de Silva R, Lees A, Miller C, Anderton B. 2002. The slow axonal transport of the microtubule-associated protein tau and the transport rates of different isoforms and mutants in cultured neurons. J Neurosci. 22:6394– 6400.
Utton M, Noble W, Hill J, Anderton B, Hanger D. 2005. Molecular motors implicated in the axonal transport of tau and alpha-synuclein. J Cell Sci. 118:4645 –4654. Vale RD, Funatsu T, Pierce DW, Romberg L, Harada Y, Yanagida T. 1996. Direct observation of single kinesin molecules moving along microtubules. Nature. 380: 451– 453. Vershinin M, Carter BC, Razafsky DS, King SJ, Gross SP. 2007. Multiple-motor based transport and its regulation by tau. Proc Nat Acad Sci USA. 104:87 – 92. Weissmann C, Reyher H, Gauthier A, Steinhoff H, Junge W, Brandt R. 2009. Microtubule binding and trapping at the tip of neurites regulate tau motion in living neurons. Traffic. 10:1655 – 1668. Wood JG, Mirra SS, Pollock NJ, Binder LI. 1986. Neurofibrillary tangles of Alzheimer-disease share antigenic determinants with the axonal microtubule-associated protein tau (tau). Proc Nat Acad Sci USA. 83:4040 – 4043. Xu J, King SJ, Lapierre-Landry M, Nemec B. 2013. Interplay between velocity and travel distance of kinesinbased transport in the presence of tau. Biophys J. 105: L23– L25. Yuan A, Kumar A, Peterhoff C, Duff K, Nixon RA. 2008. Axonal transport rates in vivo are unaffected by tau deletion or overexpression in mice. J Neurosci. 28:1682– 1687.
Computer Methods in Biomechanics and Biomedical Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcmb20
A coupled model of fast axonal transport of organelles and slow axonal transport of tau protein a
b
I.A. Kuznetsov & A.V. Kuznetsov a
Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD 21218-2694, USA b
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA Published online: 28 May 2014.
Click for updates To cite this article: I.A. Kuznetsov & A.V. Kuznetsov (2015) A coupled model of fast axonal transport of organelles and slow axonal transport of tau protein, Computer Methods in Biomechanics and Biomedical Engineering, 18:13, 1485-1494, DOI: 10.1080/10255842.2014.920830 To link to this article: http://dx.doi.org/10.1080/10255842.2014.920830
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Computer Methods in Biomechanics and Biomedical Engineering, 2015 Vol. 18, No. 13, 1485–1494, http://dx.doi.org/10.1080/10255842.2014.920830
A coupled model of fast axonal transport of organelles and slow axonal transport of tau protein I.A. Kuznetsova and A.V. Kuznetsovb* a
Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD 21218-2694, USA; bDepartment of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA
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(Received 31 December 2013; accepted 30 April 2014) We have developed a model that accounts for the effect of a non-uniform distribution of tau protein along the axon length on fast axonal transport of intracellular organelles. The tau distribution is simulated by using a slow axonal transport model; the numerically predicted tau distributions along the axon length were validated by comparing them with experimentally measured tau distributions reported in the literature. We then developed a fast axonal transport model for organelles that accounts for the reduction of kinesin attachment rate to microtubules by tau. We investigated organelle transport for two situations: (1) a uniform tau distribution and (2) a non-uniform tau distribution predicted by the slow axonal transport model. We found that non-uniform tau distributions observed in healthy axons (an increase in tau concentration towards the axon tip) result in a significant enhancement of organelle transport towards the synapse compared with the uniform tau distribution with the same average amount of tau. This suggests that tau may play the role of being an enhancer of organelle transport. Keywords: tau protein; organelle transport; fast axonal transport; slow axonal transport
1.
Introduction
Tau protein can affect cellular functions in many ways. The pathological role of tau in Alzheimer’s disease is well known (Grundke-Iqbal et al. 1986; Wood et al. 1986). Indeed, intraneuronal tangles composed mostly of hyperphosphorylated tau protein are one of the defining features of Alzheimer’s disease (Selkoe 2013); it is possible that tau oligomers are even more neurotoxic than tau tangles (Lasagna-Reeves et al. 2011). Originally, it was believed that, in healthy axons, the major function of tau is microtubule (MT) stabilisation, but later research has suggested the function of tau to be much more complex (Fanara et al. 2010; Morris et al. 2011). It is now believed that tau plays a complicated role in regulating organelle traffic in axons (Morris et al. 2011). Understanding the role of tau as a regulator of intracellular traffic is quite controversial and requires further research. In vitro studies suggest that tau retards kinesin-driven transport along MTs (Ebneth et al. 1998; Stamer et al. 2002; Dixit et al. 2008; Xu et al. 2013). This could be due to tau occupying binding sites along MTs, leaving less binding sites available to kinesin. It is also possible that tau regulates the number of kinesin motors that transport an organelle (Vershinin et al. 2007). On the other hand, other research (Morfini et al. 2007; Yuan et al. 2008) seems to indicate that tau does not affect kinesin motility. A possible explanation of this contradiction was suggested by McVicker et al. (2011) who pointed out that tau’s effect on kinesin may depend on the mode of tau’s interaction with MTs.
*Corresponding author. Email: [email protected] q 2014 Taylor & Francis
In this paper, we attempt to model how tau affects fast axonal transport of intracellular organelles. We rely on experimental research by Dixit et al. (2008) who reported that tau regulates organelle transport by reducing the rate at which kinesin-1 binds to MTs. We neglected the effect of tau on cytoplasmic dynein because a much higher tau concentration is needed to affect dynein (Dixit et al. 2008). We also neglected the effect of tau on kinesin velocity and kinesin detachment rate from MTs because, according to experimental data reported by Seitz et al. (2002), once kinesin motors are attached to MTs, their speed and runlength are not affected by the presence of tau. To complete the model describing the effect of tau on fast axonal transport, the distribution of tau itself needs to be modelled. In developing a model of tau transport, we followed experimental reports indicating that tau is transported in axons at a rate consistent with slow axonal transport (Utton et al. 2002; Saha et al. 2004). Data reported in Utton et al. (2005) suggest similarity between tau transport and transport of neurofilaments (NFs). Similar to NFs, particles containing tau exhibit rapid and bi-directional movements as well as extended periods of pausing. There is even similarity between parameters describing tau transport and transport of NFs. Indeed, tau spends 73% of the time in the pausing state while NFs spend , 80% of the time in the pausing state (Utton et al. 2005). This allowed us to use the slow axonal transport model, which was originally developed in Jung and Brown (2009) and Li et al. (2012) for transport of NFs, in order to simulate tau distribution in the axon.
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2. A model of slow axonal transport of tau 2.1 Equations governing slow axonal transport of tau A sketch of the problem is displayed in Figure 1. To quantify tau protein transport, we characterised tau concentrations in various kinetic states by their linear number density, which is the number of tau protein molecules residing in a particular kinetic state per unit length of the axon (mm21). We considered tau concentrations in the six kinetic states displayed in Figure 2(a). To obtain dimensionless concentrations, we scaled all concentrations with respect to the concentration of off-track tau molecules at the tip of the axon, c*x¼L :
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ca ¼
c*a c*x¼L
;
cr ¼
c* cr0 ¼ *r0 ; cx¼L
c*r c*x¼L
cap ¼
;
ca0 ¼
c*ap c*x¼L
;
c*a0 ; c*x¼L
crp ¼
c*rp c*x¼L
ð1Þ
Figure 1. Sketch of a neuron with an axon. The coordinate system in the axon is also displayed.
v*r are the instantaneous velocities of kinesin and dynein motors when they transport tau protein, respectively (mm s21) (the signs in Equations (2) and (3) are chosen such that both v*a and v*r are positive); g*10 is a first-order rate constant describing the probability of tau transition from the running (anterograde or retrograde) to the
:
We described tau transport by a modified version of Jung –Brown’s model (Jung and Brown 2009) originally developed to simulate transport of NFs. The modifications include incorporating the effects of tau diffusion and degradation (Kuznetsov IA and Kuznetsov AV 2014a). It is assumed that only off-track tau can be transported by diffusion. As tau degradation mainly occurs in proteasomes (Poppek et al. 2006), we also assumed that only offtrack tau can be degraded. Indeed, in order to be degraded, tau must first enter the proteasome’s proteolytic chamber (Kierszenbaum 2000), and in order to do that, tau must be detached from MTs. The equations governing tau transport are dca 2v*a * 2 g*10 ca þ g*01 ca0 ¼ 0; ð2Þ dx dcr v*r * 2 g*10 cr þ g*01 cr0 ¼ 0; ð3Þ dx 2 g*01 þ g*ar þ g*off ca0 þ g*10 ca þ g*ra cr0 þ g*on cap ¼ 0; ð4Þ * 2 g01 þ g*ra þ g*off cr0 þ g*10 cr þ g*ar ca0 þ g*on crp ¼ 0; ð5Þ d2 cap þ g*off ca0 2 g*on þ g*ar cap þ g*ra crp 2 * dx cap ln ð2Þ 2 ¼ 0; T *1=2
ð6Þ
d2 crp þ g*off cr0 2 g*on þ g*ra crp þ g*ar cap 2 * dx crp ln ð2Þ 2 ¼ 0; T *1=2
ð7Þ
D*ap
D*rp
where D*ap and D*rp are the diffusivities of tau protein in the off-track state associated with anterograde (kinesin) and retrograde (dynein) motors, respectively (mm2 s21); v*a and
Figure 2. (a) Kinetic diagram used for the model of slow axonal transport of tau proteins. Six kinetic states for tau proteins and kinetic processes between these states are displayed. The diagram is based on the kinetic scheme of slow axonal transport suggested in Figure 4 of Jung and Brown (2009) and Figure 1 of Li et al. (2012). (b) Kinetic diagram used for the model of fast axonal transport of organelles.
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Computer Methods in Biomechanics and Biomedical Engineering corresponding pausing state (s21 ); g*01 is a first-order rate constant describing the probability of tau transition from the pausing (anterograde or retrograde) to the corresponding running state (s21 ); g*ar is a first-order rate constant describing the probability of tau transition from the anterograde (pausing or off-track) to the corresponding retrograde state (s21 ); g*ra is a first-order rate constant describing the probability of tau transition from the retrograde (pausing or off-track) to the corresponding anterograde state (s21 ); g*on is a first-order rate constant describing the probability of tau transition from the offtrack (anterograde or retrograde) to the corresponding pausing state (s21 ); g*off is a first-order rate constant describing the probability of tau transition from the pausing (anterograde or retrograde) to the corresponding off-track state (s21 ); and x * is the linear coordinate along the axon (mm). Equations (2) –(7) were solved subject to the following boundary conditions: At x * ¼ 0 :
cap ¼ cx¼0 ;
crp ¼ cx¼0 ;
ca ¼ cx¼0 ; At x * ¼ L * :
cap ¼ 1;
crp ¼ 1;
cr ¼ 0;
ð8Þ ð9Þ
where cx¼0 ¼ c*x¼0 =c*x¼L in which c*x¼0 is the concentration of off-track tau molecules at the axon hillock. The last condition in Equation (9), cr ¼ 0, reflects the fact that tau cannot reverse its direction and move back into the axon. This is supported by experimental observations reflecting that tau is trapped at the distal axon by interaction with membrane proteins (Weissmann et al. 2009; Gauthier-Kemper et al. 2011).
Table 1. Parameter D*ap D*rp T *1=2 v*a v*r g*10 g*01 g*ar g*ra g*off g*on a
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The total dimensionless concentration of tau is defined as ctot ¼ ca þ cr þ ca0 þ cr0 þ cap þ crp :
ð10Þ
2.2 Estimation of parameter values for the tau transport model The ranges of parameters and the values we used in our computations are summarised in Table 1.
2.3
Results for the tau distribution
All numerical solutions were obtained using the BVP4C solver that is a part of Matlab 7 (MathWorks, Natick, MA, USA). The BVP4C solver’s default settings were used. We checked that decreasing the values of parameters that control the error tolerance properties (RelTol and AbsTol) did not affect the solutions. This insured that the obtained solutions were accurate. In Figure 3, distributions of the total tau concentration obtained by numerically solving Equations (2) – (7) (and then using Equation (10)) are compared with experimentally measured tau distributions. The experimental tau distributions were taken from figure 7(B),(D) of Black et al. (1996) and then normalised by using the condition ctot jx¼0 ¼ 1. The numerical solutions correctly capture the overall increase in tau concentration towards the axon tip. The slow axonal transport model given by Equations (2) –(7) accounts for two mechanisms of tau transport: (1) by diffusion; this mode of transport does not require energy input but can only occur from a location with a high concentration of free tau towards a location with a low concentration of free tau; (2) by motor-driven transport;
Estimated values of parameters involved in the tau transport model. Dimension 2 21
mm s mm2 s21 s mm s21 mm s21 s21 s21 s21 s21 s21 s21
Value or range 3 3 2:16 £ 105 0.3 –1 0.3 –1 9:30 £ 1022 4:10 £ 1022 3:10 £ 1025 6:90 £ 1025 4:45 £ 1023 2:75 £ 1024
Reference Konzack et al. (2007) Konzack et al. (2007) Poppek et al. (2006) Konzack et al. (2007) Konzack et al. (2007) Li et al. (2012) Li et al. (2012) Li et al. (2012) Li et al. (2012) Li et al. (2012) Li et al. (2012)
Value used in computations 3 3 2:16 £ 105 0.51a 0.52a 9:30 £ 1022 4:10 £ 1022 3:10 £ 1025 6:90 £ 1025 4:45 £ 1026 b 2:75 £ 1024
Here we used values reported in Jung and Brown (2009) for slow NF transport (since NFs are propelled by the same fast transport motors that propel tau proteins). These values are within the range of instantaneous tau velocities reported in Konzack et al. (2007). b The set of kinetic constants given in Li et al. (2012) was optimised to simulate the transport of NFs. Some of these constants had to be modified to model tau transport. We found that in order to match the tau profile reported in Black et al. (1996), the value of g*off had to be reduced. Physically, this means that tau proteins transition from the pausing to the corresponding off-track state less frequently than NFs.
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I.A. Kuznetsov and A.V. Kuznetsov calculated the average tau concentration:
(a) 3.5 numerical data from Fig. 7B of Black et al.
3
ctot; av ¼
2.5
1 L*
ctot 1.5
* cdev tot ðx Þ ¼
1
0
50
100
150
200
250
300
ðL *
350
x *, µm 0
(b) 3.5
ctot ðx * Þ 2 ctot; av : ctot ðL * Þ 2 ctot; av
ð12Þ
2.5 2 1.5 1 0.5 0
100
200
300
400
* * cdev tot ðx Þ dx ¼ 0
ð13Þ
* cdev tot ðL Þ ¼ 1:
ð14Þ
and
numerical . data from Fig. 7D of Black et al.
3
ctot
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ð11Þ
* cdev tot ðx Þ defined by Equation (12) satisfies two properties, which are given as follows:
0.5
0
ctot ðx * Þ dx * :
0
The renormalised tau distribution was computed as follows:
2
0
ðL *
500
600
x *, µm
Figure 3. (a) The total concentration of tau protein for a 350-mmlong axon. Experimental data from figure 7(B) of Black et al. (1996) are also shown. (b) The total concentration of tau protein for a 600mm-long axon. Experimental data from figure 7(D) of Black et al. (1996) are also shown. These experimental data were normalised such that at the axon hillock, x * ¼ 0, ctot was equal to 1.
this mode requires energy input but can occur in any direction; the direction depends on whether transport is driven by kinesin or dynein motors. The complexity of the model, which includes two modes of transport, explains why numerical results can correctly reproduce the complicated tau distribution that includes an initial slight decrease in tau concentration and then a sharp increase closer to the axon tip (Figure 3(b)).
3. A model of fast axonal transport of organelles and vesicles 3.1 Equations governing fast axonal transport of organelles and vesicles After ctot (defined by Equation (10)) was computed, we renormalised it by the following procedure. First we
* cdev tot ðx Þ thus characterises the dimensionless deviation of tau concentration, ctot ðx * Þ, from its average value, * ctot; av . cdev tot ðx Þ is normalised such that its maximum deviation from the average value (at the axon tip) is equal * to 1 and the average value of cdev tot ðx Þ is equal to 0 (see Figure 4(a),(b)). Dixit et al. (2008) reported that the attachment rate of kinesin to MTs is reduced by tau. We assumed that this is the only effect that tau has on organelle transport. Also, relying on Dixit et al. (2008), we neglected the effect of tau on dynein. In developing our model, we used the simplest linear correlation to characterise the dependence of the kinetic constant describing the rate of kinesin attachment to MTs, k*þ , on the dimensionless tau concentration, * cdev tot ðx Þ (Kuznetsov IA and Kuznetsov AV 2014b): ð15Þ k*þ ¼ k*þ0 1 2 a cdev tot ;
where k*þ0 is a parameter that characterises the rate of kinesin attachment to MTs for the uniform (cdev tot ¼ 0) tau distribution (s21) and a is a parameter that characterises by how much the increase in tau concentration reduces the probability of attachment of kinesin motors to MTs (0 # a # 1). The Smith –Simmons model describing fast axonal transport of vesicles and organelles (Smith and Simmons 2001) includes three organelle concentrations (Figure 2 (b)). These are the concentration of free organelles, n*0 ; the concentration of MT-bound organelles driven anterogradely by kinesin motors, n*þ ; and the concentration of MTbound organelles driven retrogradely by dynein motors, n*2 . Similar to the concentration of tau protein, we characterised all organelle concentrations by their linear number densities (mm21 ). To obtain dimensionless concentrations, all concentrations were rescaled with
Computer Methods in Biomechanics and Biomedical Engineering (a)
where D*0 is the diffusivity of free organelles (mm2 s21), v*þ is the velocity of organelles transported by kinesin motors (mm s21), v*2 is the velocity of organelles transported by dynein motors (mm s21), k*2 is the kinetic constant describing the rate of dynein attachment to MTs (s21), k0 *þ is the kinetic constant describing the rate of kinesin detachment from MTs (s21) and k0 *2 is the kinetic constant describing the rate of dynein detachment from MTs (s21). Equations (17) – (19) were solved subject to the following boundary conditions:
1 uniform non−uniform (numerical)
0.8 0.6
cdev tot
0.4 0.2 0 −0.2
n0 ð0Þ ¼ 1;
−0.4 0
50
100
150
200
250
300
NL ¼
uniform non−uniform (numerical)
0.8 0.6
0.2 0 −0.2
N *L ; N *0
ntot ðxÞ ¼ n0 ðxÞ þ nþ ðxÞ þ n2 ðxÞ: 0
100
200
300
400
500
ð21Þ
Figure 4. Deviation of tau concentration from its average value, cdev tot , (a) for a 350-mm-long axon and (b) for a 600-mm-long axon. cdev tot is used in modelling the effect of tau on kinesin attachment rate (see Equation (15)).
The total organelle flux, which includes the diffusion-, kinesin- and dynein-driven components, respectively, is calculated as j*tot ¼ 2D*0
respect to the concentration of free organelles at x * ¼ 0, N *0 : n* n0 ¼ 0* ; N0
n* nþ ¼ þ* ; N0
n* n2 ¼ 2* : N0
d2 n0 0* 0* * ¼ 2k*þ0 1 2 a cdev tot n0 2 k2 n0 þ k þ nþ þ k 2 n2 ; *2 dx ð17Þ dnþ 0* ¼ k*þ0 1 2 a cdev tot n0 2 k þ nþ ; dx * 2v*2
dn2 ¼ k*2 n0 2 k0 *2 n2 ; dx *
ð18Þ ð19Þ
›n*0 þ v*þ n*þ 2 v*2 n*2 : ›x *
ð23Þ
The total dimensionless organelle flux is then found as jtot ¼
ð16Þ
We modified the Smith – Simmons equations (Smith and Simmons 2001) by using Equation (15) for k*þ . We assumed that all other kinetic constants (k*2 , k0 *þ and k0 *2 ) are independent of tau concentration. For a steady-state situation, the governing equations are
v*þ
ð22Þ
600
x *, µm
2D*0
ð20Þ
where N *L is the concentration of free organelles at x * ¼ L *; s0 and sL are the degrees of loading of free organelles on MTs at the axon hillock and at the axon tip, respectively, and N *L is the concentration of free organelles at x * ¼ L *. The total dimensionless organelle concentration is then found as
0.4
−0.4
nþ ð0Þ ¼ s0 ;
In Equation (20),
1
cdev tot
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n0 ðL * Þ ¼ N L ;
n2 ðL * Þ ¼ sL N L :
350
x *, µm (b)
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j*tot : v*þ N *0
ð24Þ
3.2 Estimation of parameter values for the tau transport model Estimated ranges of parameters for the model of organelle transport and the values we used in our computations are summarised in Table 2.
3.3
Results for the organelle transport
To evaluate the effect of tau on fast axonal transport of organelles, we used tau distributions computed in Section 2.3 (displayed by solid lines in Figure 3(a),(b)). These computed tau distributions were then renormalised by using Equation (12); the renormalised distributions
1490 Table 2. Notation
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a D*0 k*þ0
I.A. Kuznetsov and A.V. Kuznetsov Estimated values of parameters involved in the fast axonal transport model of organelles and vesicles. Dimension 2 21
mm s s21
Value or range b
0.1 – 0.4 5
k*2
s21
1.5
k0 *þ
s21
1
k0*2
s21
0.25
L* NL v*þ
mm
,103 to 106
mm s21
1
v*2
mm s21
1
s0 sL
0 – 10 0 – 10
Reference Smith and Simmons (2001) Mu¨ller et al. (2010), Beeg et al. (2008), Leduc et al. (2010) Mu¨ller et al. (2010), King and Schroer (2000), Reck-Peterson et al. (2006) Mu¨ller et al. (2010), Schnitzer et al. (2000), Vale et al. (1996) Mu¨ller et al. (2010), King and Schroer (2000), Reck-Peterson et al. (2006) Debanne et al. (2011) Vale et al. (1996), Carter and Cross (2005) King and Schroer (2000), Toba et al. (2006) Smith and Simmons (2001) Smith and Simmons (2001)
Value used in computations 0.02, 0.04, 0.06a 0.1 1c 1c 1c 1c 350, 600d 0.5e 1 1 0.1 0.1
a These values were chosen to demonstrate the effect of parameter a, which describes a decrease in kinesin attachment rate to MTs by tau protein. The increase in a increases organelle concentration in the axon. This is because when less kinesin-driven organelles attach to MTs, fewer of them are transported towards the synapse, and more accumulate in the axon. We have chosen the value of a to satisfy two conditions: (1) there is no unrealistically high accumulation of organelles in the axon and (2) there is a visible effect of tau on organelle transport (if a ¼ 0, tau does not affect organelle transport, see Equation (15)). Condition (2) is consistent with the results reported in Dixit et al. (2008) which indicate that tau does have effect on fast axonal transport because it inhibits kinesin more strongly than dynein. b This range is based on the estimate reported in Smith and Simmons (2001) for a vesicle with 1-mm diameter. c We followed Smith and Simmons (2001) who used the same value of 1 s21 for all attachment/detachment rates in a standard set of primary parameters for numerical work. d We simulated axons with lengths of 350 and 600 mm because these are the axon lengths for which experimentally measured tau distributions are reported in Black et al. (1996). e Synthesis of most organelles occurs in the neuron soma; the organelles are then transported towards the axon synapse by fast axonal transport (Goldstein and Yang 2000). Based on this model, we assumed that the concentration of free organelles at the base of the axon (x * ¼ 0) is two times larger than that at the axon tip (x * ¼ L * ).
(showing the deviation from the average concentration of tau) are displayed by solid lines in Figure 4(a),(b). The renormalised distributions were used in Equation (15) to quantify the reduction of kinesin attachment rate to MTs by tau. Figures 5 and 6 show the results for a 350-mm-long axon. In Figure 5, we show organelle concentrations in the free (Figure 5(a)), kinesin-driven (Figure 5(b)) and dynein-driven (Figure 5(c)) kinetic states. We consider a uniform tau distribution [corresponding to ctot ðx * Þ ¼ ctot; av * or cdev tot ðx Þ ¼ 0] and a numerically computed tau distribution, shown by a solid line in Figure 3(a). Both uniform and non-uniform tau distributions are for the same total number of tau molecules in the axon (Figure 4(a)), but for the non-uniform tau distribution, there are fewer tau molecules in the proximal axon and more tau molecules in the distal axon. We characterise the strength of the effect of tau on kinesin attachment rate to MTs by specifying three values for parameter a: 0.02, 0.04 and 0.06. A larger value of a corresponds to stronger inhibition of kinesin attachment to MTs by tau. For a uniform tau distribution, the organelle concentrations are almost linear, except for sharp variations very close to the axon base
(x * ¼ 0) and axon tip (x * ¼ L * ). These sharp variations are due to diffusion boundary layers close to the axon base and axon tip; in the rest of the axonal domain, organelle transport is mostly motor driven. The effect of a non-uniform tau distribution in Figure 5 is to increase organelle concentrations in the axon in all three kinetic states. The increase is larger for larger values of a; larger a corresponds to a stronger reduction of kinesin attachment to MTs by tau. The qualitative similarity between n0 , nþ and n2 displayed in Figure 5(a) – (c), respectively, is due to the fact that all kinetic constants that control transitions between various organelle populations (Figure 2(a)) were set to 1 (Table 2). The total organelle concentration, ntot , is the sum of n0 , nþ and n2 . ntot is approximately three times larger than any of these concentrations, but the trends are qualitatively the same (Figure 6(a)). The flux of organelles towards the axon tip (Figure 6 (b)) is independent of x * . This is a consequence of the steady-state formulation of the problem. Indeed, in the steady-state situation, organelles cannot accumulate in any control volume inside the axon; hence, organelle fluxes in and out of the control volume must be the same.
Computer Methods in Biomechanics and Biomedical Engineering (a)
(a) 1.5 uniform a = 0.02 a = 0.04 a = 0.06
4 uniform a = 0.02 a = 0.04 a = 0.06
3.5 3
1
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ntot
n0
2.5 2 1.5
0.5
1 0.5 0
0
50
100
150
200
250
300
0
350
0
50
100
x *, µm (b)
uniform a = 0.02 a = 0.04 a = 0.06
3
250
300
350
150 200 * x , µm
250
300
350
x 10−3 uniform a = 0.02 a = 0.04 a = 0.06
2.5
1
jtot
n+
2 1.5
0.5 1 0.5 0
0
50
100
150
200
250
300
350
0
x *, µm (c) 1.5
1
50
100
concentration (see the solid line in Figure 3(a)), depends on parameter a. The more the kinesin attachment rate is reduced by tau (larger a), the larger the flux towards the axon tip becomes (Figure 6(b)). Table 3 shows by how much the organelle flux is increased compared with a
0.5
0
0
Figure 6. Fast axonal transport model. (a) Dimensionless total organelle concentration and (b) dimensionless total organelle flux. Results for a 350-mm-long axon.
uniform a = 0.02 a = 0.04 a = 0.06
n–
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(b) 1.5
150 200 x *, µm
Table 3. Effect of non-uniformity of tau distribution on organelle flux towards the axon tip. 0
50
100
150
200
250
300
350
x *, µm
Figure 5. Fast axonal transport model. Dimensionless concentrations of (a) free organelles, (b) anterogradely moving organelles transported by kinesin motors and (c) retrogradely moving organelles transported by dynein motors. Results for a 350-mm-long axon.
The flux for the uniform tau distribution is the smallest (Figure 6(b)). The flux for the non-uniform tau distribution, exhibiting proximal-to-distal increase in tau
Increase compared Parameter characterising with the uniform tau distribution, the effect of 100ðjtot 2 jtot; uniform Þ= tau on kinesin Axon length, attachment rate, a jtot; uniform (%) L * (mm) 350 350 350 600 600 600
0.02 0.04 0.06 0.02 0.04 0.06
35 77 124 42 93 147
1492
I.A. Kuznetsov and A.V. Kuznetsov (a)
(a) 1.5 uniform a = 0.02 a = 0.04 a = 0.06
4 uniform a = 0.02 a = 0.04 a = 0.06
3.5 3
1 n0
ntot
2.5 2 1.5
0.5
1 0.5 0
0
100
200
300 * x , µm
400
500
0
100
200
300
400
500
600
400
500
600
x *, µm
(b) 1.5
(b)
uniform a = 0.02 a = 0.04 a = 0.06
3
x 10−3 uniform a = 0.02 a = 0.04 a = 0.06
2.5
1
jtot
n+
2 1.5
0.5 1 0.5 0
0
100
200
300
400
500
600
x *, µm
0
0
100
200
300 x *, µm
(c) 1.5
Figure 8. Fast axonal transport model. (a) Dimensionless total organelle concentration and (b) dimensionless total organelle flux. Results for a 600-mm-long axon.
uniform a = 0.02 a = 0.04 a = 0.06 1 n–
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0
600
0.5
0
0
100
200
300
400
500
600
x *, µm
Figure 7. Fast axonal transport model. Dimensionless concentrations of (a) free organelles, (b) anterogradely moving organelles transported by kinesin motors and (c) retrogradely moving organelles transported by dynein motors. Results for a 600-mm-long axon.
uniform tau distribution. For a ¼ 0.02, 0.04 and 0.06, the increase is by 35%, 77% and 124%, respectively, which indicates that the organelle flux is highly sensitive to the
effects of tau. This also suggests that the reason for the non-uniform tau distribution (the proximal-to-distal increase in tau concentration displayed in Figure 3(a)) may be to increase the flux of organelles towards the axon tip. Thus, tau may not just be a regulator, but also an enhancer of organelle transport. Figures 7 and 8 show the results for a 600-mm-long axon. The trends are similar to those displayed in Figures 5 and 6 for a 350-mm-long axon. A notable difference is in the effect of parameter a on the organelle flux towards the axon tip for a non-uniform tau distribution (Figure 8(b)). The enhancements of the organelle flux for different values of parameter a are summarised in Table 3. For a ¼ 0.02, 0.04 and 0.06, the increase in the flux is by 42%, 93% and 147%, respectively. It is noteworthy that the percentage increase in the flux for a non-uniform tau distribution is larger for a longer axon. This is because the effect of tau accumulates over the axon length. This begs an interesting question. As axons may be much longer than 600 mm in
Computer Methods in Biomechanics and Biomedical Engineering
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length, does this mean that kinesin attachment rate to MTs is reduced differently for axons with different lengths?
4. Conclusions We investigated the effect of tau on fast axonal transport of organelles towards the axon synapse by coupling a model of fast axonal transport and a model of tau protein transport by the slow axonal transport mechanism. To the best of our knowledge, this is the first attempt to investigate the interaction of fast and slow axonal transport mechanisms. Our results indicate that the proximal-to-distal increase in tau concentration, observed in healthy axons, enhances organelle transport towards the synapse, compared with a uniform tau distribution. The enhancement depends on how strongly tau reduces the kinesin attachment rate to MTs. The greater the effect of tau on kinesin attachment rate, the larger the enhancement of organelle transport towards the synapse. This suggests a potential role of tau as an enhancer of fast axonal transport. We also investigated the effect of axon length on the enhancement of organelle transport towards the synapse. We found that the enhancement is larger for longer axons. This suggests that the effect of tau on kinesin attachments rate may accumulate along the axon length, causing stronger enhancement for longer axons.
Acknowledgements AVK gratefully acknowledges the support of the Alexander von Humboldt Foundation (Germany) through the Humboldt Research Award.
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