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Modeling of retrograde nanoparticle transport in axons and dendrites
International Communications in Heat and Mass Transfer 38 (2011) 543–547
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Modeling of retrograde nanoparticle transport in axons and dendrites☆ A.V. Kuznetsov Dept. 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
Available online 22 March 2011 Keywords: Nanoparticle transport Neurons Axons and dendrites Nanomedicine Molecular motors Fast axonal transport
a b s t r a c t This paper presents a pioneering modeling study on nanoparticle internalization and transport in neurons. The model developed in this paper is based on recent experimental results that indicate that after entering a neurite by endocytosis, nanoparticles are transported toward the neuron soma in endocytic vesicles by retrograde molecular-motor-driven transport. Experimental results also indicate that nanoparticles enter axons at axon terminals while in dendrites they enter through the entire plasma membrane. The model equations developed in this paper are based on these experimental observations. The analytical solution of these equations is obtained; the solution predicts the distribution of the concentration of nanoparticles associated with free nanoparticle-loaded vesicles (NLVs) (not transported on microtubules (MTs)) as well as the distribution of the concentration of nanoparticles associated with NLVs transported on MTs by dynein motors. The fluxes of nanoparticles by diffusion and motor-driven transport as well as the total (combined) flux of nanoparticles are also predicted. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Nanotechnology has numerous emerging applications in medicine. Diagnostic, therapeutic, and targeting agents can be incorporated within different types of nanoparticles, such as quantum dots, liposomes, and viruses [1]. One of the advantages of nanoparticles is that, if properly designed, they are capable of penetrating various tissue barriers, such as the blood-brain barrier [2,3]. The blood-brain barrier presents the major obstacle for treatment of many brain disorders because it is difficult for many therapeutic and diagnostic agents to cross this barrier in adequate amounts. Patients with neurodegenerative conditions, such as Alzheimer's, Parkinson's, and Huntington's diseases, may benefit from novel drugs, if these drugs can be delivered to specific areas of the brain. This paper is motivated by the recent research of Wong et al. [4] who investigated the utilization of layered double hydroxide nanoparticles for the delivery of small interfering RNAs (siRNAs) [5] to neuron bodies. Such siRNAs are capable of destroying specific messenger RNAs. This makes siRNAs suitable for treatment of such disorders as Huntington's disease, which is linked to the production of abnormal proteins. The advantages of using nanoparticles as delivery vehicles for siRNAs include the fact that they can be administered intravenously [6]; also, layered double hydroxide nanoparticles exhibit low cytotoxicity, are biocompatible, and much more efficiently internalized by cortical neurons than by many non-neuronal cell lines.
☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.03.015
Nanoparticles most likely enter neurons by clathrin-dependent endocytosis [7]. Once they entered neurons, nanoparticles are transported toward the neuron body (where they are released into the cytoplasm) by retrograde transport in endocytic vesicles through association with dynein molecular motors [4,8,9]. Neurons have two types of long processes, axons and dendrites; axons transmit signals and dendrites receive signals. Interestingly, there is evidence that nanoparticles enter axons at axon terminals while in dendrites nanoparticles enter through the entire plasma membrane [4,10,11]. This paper develops, based on the above experimental observations, models of nanoparticle transport from their place of entry to the neuron body. Two models are developed, for an axon and a dendrite. Transport of nanoparticles in these two neuronal processes is then compared. 2. Governing equations Fig. 1a and b shows schematic diagrams of the problem. In an axon (Fig. 1a) nanoparticles enter through the axon synapse only, where the nanoparticle concentration is assumed to be the highest. In a dendrite (Fig. 1b) nanoparticles enter along the whole length of the dendrite. After they have entered, nanoparticles are transported in nanoparticleloaded vesicles (NLVs) retrogradely toward the neuron soma, where nanoparticle concentration is assumed to be the lowest. It is assumed that after releasing their cargo in the cell body, nanoparticles are directed to lysosomes for degradation (which provides a sink for nanoparticles in the soma); this assumption allows solving the problem in a steady-state formulation. Equations describing nanoparticle transport are based on the model of molecular motor-assisted transport of intracellular organelles
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A.V. Kuznetsov / International Communications in Heat and Mass Transfer 38 (2011) 543–547
Nomenclature D D˜ j jdiff jmotor k˜ k′
dimensionless diffusivity of free NLVs (not transported ˜
on MTs), Dν̃2k̃ diffusivity of free NLVs (not transported on MTs), (μm2/s) total dimensionless flux of nanoparticles (by diffusion and motor-driven transport) dimensionless flux of nanoparticles due to diffusion dimensionless flux of nanoparticles due to motordriven transport first order rate constant characterizing the rate of NLVs binding to MTs (1/s) dimensionless first order rate constant characterizing the rate of NLVs detachment from MTs,
k˜ ′ L
first order rate constant characterizing the rate of NLVs detachment from MTs (1/s) dimensionless length of an axon or dendrite (see Fig. 1a, b),
L˜ n ñ nMT
ñMT N0 Ñ0 NL ÑL Q
k˜′ k˜
L˜ k˜ v˜
length of an axon or dendrite (μm) dimensionless number density of nanoparticles asso3 ciated with free NLVs (not transported on MTs), ñ ṽ 3 k˜
number density of nanoparticles associated with free NLVs (not transported on MTs), (1/μm3) dimensionless number density of nanoparticles associated with NLVs transported retrogradely by dynein 3 motors, ñ MT ν̃3 k˜
number density of nanoparticles associated with NLVs transported retrogradely by dynein motors (1/μm3) dimensionless number density of nanoparticles asso3 ciated with free NLVs at x = 0, N˜ 0 ṽ 3
developed in Smith and Simmons [12]. Two populations of nanoparticles are considered. The first population is comprised of nanoparticles associated with free NLVs; their concentration is denoted by ñ, this population can be transported by diffusion. The second population is comprised of nanoparticles associated with NLVs that are transported on MTs retrogradely by dynein motors; their concentration is denoted by ñMT. NLVs can switch between the two populations; free NLVs can attach to MTs, the rate of this process is characterized by the first order ˜ NLVs associated with MTs can detach from MTs and rate constant k. become freely suspended NLVs; the rate of this process is characterized by the first order rate constant k˜ ′ . It should be noted that in axons all MTs have the same polarity; the plus-ends of all MTs are directed away from the neuron soma toward the axon terminal (synapse). Therefore, cytoplasmic dynein, which is the major retrograde motor in axons, always pulls organelles toward the neuron body. However, the MT polarity orientation in dendrites is non-uniform [13,14]; one would assume that dynein would then pull endocytic vesicles in both directions, depending on the orientation of a particular MT. However, [10] found that transport of endocytosed material in peripheral regions of dendrites was always unidirectional, except in a small 20 to 30 μm-thick segment closest to the cell body where some vesicles changed direction. Even in that region approximately 80% of vesicles moved into the cell body without change of direction. Based on these experimental observations, in the model it is assumed that NLVs are transported on MTs in the retrograde direction only, and no change of direction occurs. This assumption significantly simplifies the problem formulation and makes it possible to obtain the exact analytical solution. Equations governing transport of nanoparticles in an axon (Fig. 1a) are D
k˜
number density of nanoparticles associated with free NLVs at x = 0 (1/μm3) dimensionless number density of nanoparticles asso3 ciated with free NLVs at x = L, NL̃ ṽ 3 k˜ number density of nanoparticles associated with free NLVs at x = L (1/μm3) dimensionless volumetric rate of nanoparticle inter-
∂2 n −n + k′ nMT = 0 ∂x2
ð1Þ
∂nMT =0 ∂x
ð2Þ
n−k′ nMT +
Dimensionless variables in Eqs. (1) and (2) are defined as follows: D=
3
˜ Q
ṽ x x̃
nalization in a dendrite, ṽ4 Q˜ k˜ volumetric rate of nanoparticle internalization in a dendrite (in a dendrite internalization occurs over the entire plasma membrane but is represented as a volumetric source based on the assumption that nanoparticle concentration in a dendrite cross-section is uniform) (1/μm3s) average velocity of NLVs transported on MTs retrogradely (μm/s) x̃ ̃ dimensionless linear coordinate along the MT tracks, ṽk linear coordinate along the MT tracks (see Fig. 1a and b) (μm)
D˜ k˜ ṽ
2
; k′ =
x̃ ̃ ṽ 3 ṽ3 k˜′ ; x = k; n = ñ 3 ; nMT = ñ MT 3 ṽ k̃ k̃ k̃
ð3Þ
where D˜ is the dimensionless diffusivity of free NLVs (not transported on MTs), ṽ is the average velocity of NLVs transported on MTs retrogradely, and x̃ is the linear coordinate along the MT tracks (see Fig. 1a and b). Equations governing transport of nanoparticles in a dendrite (Fig. 1b) are
D
∂2 n −n + k′ nMT + Q = 0 ∂x2
n−k′ nMT +
∂nMT =0 ∂x
ð4Þ ð5Þ
In Eq. (4) Greek symbols degree of loading at x = L σL
Abbreviations MT microtubule NLV nanoparticle-loaded vesicle
Q =
ṽ3 ̃ 4Q k̃
ð6Þ
where Q˜ is the volumetric rate of nanoparticle internalization in a dendrite (assumed constant). Based on the data presented in Refs. [4,10], in a dendrite internalization occurs by endocytosis (most likely by clathrin-dependent uptake) over the entire length of the dendrite. This uptake is represented by a volumetric source based on the assumption that nanoparticle concentration in a dendrite cross-section is uniform.
A.V. Kuznetsov / International Communications in Heat and Mass Transfer 38 (2011) 543–547
545
Fig. 1. Schematic diagram of the problem for (a) an axon and (b) a dendrite.
Once n is obtained, nMT is found by integrating Eq. (2), as
Eqs. (1) and (2) for the axon or Eqs. (4) and (5) for the dendrite must be solved subject to the following boundary conditions: At x = 0 : n = N0 At x = L : n = NL ; nMT = σL NL
ð7Þ ð8a; bÞ
nMT = −
c0 c1 c2 expðμ 1 xÞ + expðμ 2 xÞ + c3 expðk′ xÞ ð14Þ + k′ k′ −μ 1 k′ −μ 2
where the integration constant c3 is found by substituting Eq. (14) into boundary condition (8b).
where 4. Solution for a dendrite ̃ ̃ ṽ3 ṽ3 N0 = N˜ 0 3 ; NL = Ñ L 3 ; L = L k ṽ ̃ ̃ k k
ð9Þ
Eliminating nMT from Eqs. (4) and (5) and integrating with respect to x results in:
In Eqs. (8) and (9) σL is the degree of loading at x = L, Ñ0 is the number density of nanoparticles associated with free NLVs at x = 0, ÑL is the number density of nanoparticles associated with free NLVs at x = L, and L˜ is the length of the axon or dendrite (see Fig. 1a and b).
D
3. Solution for an axon
n = −d0 + k′ Q Dk′ −x + d1 expðλ1 xÞ + d2 expðλ2 xÞ
Eliminating nMT from Eqs. (1) and (2) and integrating with respect to x results in:
where λ1 and λ2 are the roots of
D
d2 n dn −Dk′ −n = c0 dx dx2
ð10Þ
2 Dλ −Dk′ λ−1 = 0
ð12Þ
ð17Þ
ð13Þ
Eqs. (7), (8a), and (13) give a system of three linear equations for c0, c1, and c2.
d n −Q + ð1−k′ σL ÞNL = D dx2
ð18Þ
Eqs. (7), (8a), and (18) give a system of three linear equations for d0, d1, and d2. Once n is obtained, nMT is found by integrating Eq. (5), as nMT = −
2
At x = L :
ð16Þ
2
At x = L :
Constants c0, c1, and c2 in Eq. (11) are found from boundary conditions given by Eqs. (7) and (8a) and the following boundary condition that is obtained by eliminating nMT(L) from Eq. (8b): d n ð1−k′ σL ÞNL = D dx2
The general solution of Eq. (15) is
ð11Þ
where μ1 and μ2 are the roots of 2 Dμ −Dk′ μ−1 = 0
ð15Þ
Constants d0, d1, and d2 in Eq. (16) are found from boundary conditions given by Eqs. (7) and (8a) and the following boundary condition that is obtained by eliminating nMT(L) from Eq. (8b):
The general solution of Eq. (10) is n = −c0 + c1 expðμ 1 xÞ + c2 expðμ 2 xÞ
d2 n dn −Dk′ −n−k′ Qx = d0 dx dx2
+
d0 1 d1 expðμ 1 xÞ +Q − + Dk′ −x + k′ k′ −μ 1 k′
ð19Þ
d2 expðμ 2 xÞ + d3 expðk′ xÞ k′ −μ 2
where the integration constant d3 is found by substituting Eq. (19) into boundary condition (8b).
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A.V. Kuznetsov / International Communications in Heat and Mass Transfer 38 (2011) 543–547
The dimensionless diffusion-driven flux of nanoparticles is calculated as jdiff = −D
dn dx
nanoparticle transport in these two processes; the analytical solutions obtained above make it possible to calculate nanoparticle distributions in segments of any length. The results presented in Figs. 2 and 3 are
ð20Þ
while the motor-driven flux is calculated as jmotor = −nMT
ð21Þ
The total flux of nanoparticles is jtotal = −D
dn −nMT dx
ð22Þ
5. Results and discussion For the purpose of comparison, in numerical results reported in Figs. 2 and 3 both neurites (axon and dendrite) are assumed to have the same dimensionless length, L, although axons are usually much longer than dendrites. The equality of the lengths of segments of the axon and dendrite is assumed only to ease the comparison between
Fig. 2. (a) Distributions of the number density of nanoparticles associated with free NLVs (not actively transported by retrograde motors), n(x); (b) Distributions of the number density of nanoparticles associated with NLVs transported on MTs by dynein motors, nMT(x), in an axon and dendrite.
Fig. 3. (a) Distributions of the flux of nanoparticles due to diffusion, jdiff (x); (b) distributions of the flux of nanoparticles due to motor-driven retrograde transport, jmotor(x); (c) distributions of the total flux of nanoparticles (due to motor-driven and diffusion transport mechanisms), jtotal(x), in an axon and dendrite.
A.V. Kuznetsov / International Communications in Heat and Mass Transfer 38 (2011) 543–547
obtained for the following typical values of dimensionless parameters: D = 0.5, k′ = 0.5, L = 20, N0 = 0.01, NL = 0.2, σL = 0.1, and Q = 0.01. Fig. 2a displays distributions of the number density of nanoparticles associated with free NLVs (not actively transported by retrograde motors), n(x), in an axon and dendrite. Since nanoparticles enter the dendrite over the entire dendritic plasma membrane, the concentration of nanoparticles associated with free NLVs is increasing when x is decreasing, as more and more nanoparticles enter the dendrite, see Fig. 1b (the lateral area of a segment of a dendrite between x and L is 2πR(L − x), where R is the radius of a dendrite; this quantity becomes larger when x becomes smaller). Fig. 2b displays distributions of the number density of nanoparticles associated with NLVs transported on MTs retrogradely, nMT(x). In an axon nMT is independent of x over most of the axon length, except in the boundary layers near x = 0 and x = L. In a dendrite, nMT is increasing when x is decreasing, which is again explained by the nanoparticle's uptake by the lateral surface of the dendrite. Fig. 3a displays distributions of the flux of nanoparticles due to diffusion, jdiff(x). jdiff(x) is significant (and negative, directed toward the axon soma) only in the boundary layers near x =0 and x = L. In the middle region of neurites jdiff(x) is slightly negative in an axon and slightly positive in a dendrite, the latter is explained by the negative slope of n(x) in a dendrite, see Fig. 2a. Fig. 3b displays distributions of the flux of nanoparticles due to motor-driven retrograde transport, jmotor(x). In an axon jmotor(x) is almost constant over of the axon length. In a dendrite the magnitude of jmotor(x) increases quickly when x is decreasing (as one gets closer to the neuron soma); this is again explained by the fact that for smaller x the number of nanoparticles becomes larger due to nanoparticle internalization over the lateral area of a dendrite. The negative sign of the flux simply means that the flux is directed toward the neuron soma. Fig. 3c displays the total flux of nanoparticles, jtotal(x), due to diffusion and motor-assisted transport. In an axon jtotal is constant, which is explained by the steady-state formulation of the problem (in this formulation no nanoparticle accumulation in a control volume is possible; therefore, the rate at which nanoparticles enter the control volume must equal the rate at which nanoparticles leave the control volume). In a dendrite, the absolute value of jtotal becomes larger when x is decreasing; this is because nanoparticles enter through the lateral area of a dendrite. jtotal at x =L is slightly different in an axon and dendrite,
547
although the boundary conditions are the same (see Eqs. (7) and (8)). This is an interesting property of the reaction–diffusion system considered in this paper; it has been confirmed by analytical investigation of the solutions for the dendrite and axon, which demonstrated that this is due to the dependence of the diffusion component of the total flux, jdiff (L), on Q. It should be noted that in a purely diffusion system (if one drops Eqs. (2) and (5) and sets nMT to zero) jdiff (L) would not be affected by Q. References [1] R.K. Jain, T. Stylianopoulos, Delivering nanomedicine to solid tumors, Nature Reviews Clinical Oncology 7 (2010) 653–664. [2] R.B. Huang, S. Mocherla, M.J. Heslinga, P. Charoenphol, O. Eniola-Adefeso, Dynamic and cellular interactions of nanoparticles in vascular-targeted drug delivery (review), Molecular Membrane Biology 27 (2010) 312–327. [3] S. Singh, Nanomedicine-nanoscale drugs and delivery systems, Journal of Nanoscience and Nanotechnology 10 (2010) 7906–7918. [4] Y. Wong, K. Markham, Z.P. Xu, M. Chen, G.Q. Lu, P.F. Bartlett, H.M. Cooper, Efficient delivery of siRNA to cortical neurons using layered double hydroxide nanoparticles, Biomaterials 31 (2010) 8770–8779. [5] D.M. Dykxhoorn, J. Lieberman, Knocking down disease with siRNAs, Cell 126 (2006) 231–235. [6] M.E. Davis, J.E. Zuckerman, C.H.J. Choi, D. Seligson, A. Tolcher, C.A. Alabi, Y. Yen, J.D. Heidel, A. Ribas, Evidence of RNAi in humans from systemically administered siRNA via targeted nanoparticles, Nature 464 (2010) 1067–1070. [7] A. Sorkin, M. von Zastrow, Endocytosis and signalling: intertwining molecular networks, Nature Reviews. Molecular Cell Biology 10 (2009) 609–622. [8] M. Busse, A. Kraegeloh, D. Stevens, Modeling the effects of nanoparticles on neuronal cells: from ionic channels to network dynamics, Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE, Anonymous, 2010, pp. 3816–3819. [9] M. Praetorius, C. Brunner, B. Lehnert, C. Klingmann, H. Schmidt, H. Staecker, B. Schick, Transsynaptic delivery of nanoparticles to the central auditory nervous system, Acta Oto-Laryngologica 127 (2007) 486–490. [10] R.G. Parton, K. Simons, C.G. Dotti, Axonal and dendritic endocytic pathways in cultured neurons, The Journal of Cell Biology 119 (1992) 123–137. [11] R.G. Parton, C.G. Dotti, Cell biology of neuronal endocytosis, Journal of Neuroscience Research 36 (1993) 1–9. [12] D.A. Smith, R.M. Simmons, Models of motor-assisted transport of intracellular particles, Biophysical Journal 80 (2001) 45–68. [13] D. Takahashi, W. Yu, P.W. Baas, R. Kawai-Hirai, K. Hayashi, Rearrangement of microtubule polarity orientation during conversion of dendrites to axons in cultured pyramidal neurons, Cell Motility and the Cytoskeleton 64 (2007) 347–359. [14] A.C. Kwan, D.A. Dombeck, W.W. Webb, Polarized microtubule arrays in apical dendrites and axons, Proceedings of the National Academy of Sciences of the United States of America 105 (2008) 11370–11375.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Modeling of retrograde nanoparticle transport in axons and dendrites☆ A.V. Kuznetsov Dept. 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
Available online 22 March 2011 Keywords: Nanoparticle transport Neurons Axons and dendrites Nanomedicine Molecular motors Fast axonal transport
a b s t r a c t This paper presents a pioneering modeling study on nanoparticle internalization and transport in neurons. The model developed in this paper is based on recent experimental results that indicate that after entering a neurite by endocytosis, nanoparticles are transported toward the neuron soma in endocytic vesicles by retrograde molecular-motor-driven transport. Experimental results also indicate that nanoparticles enter axons at axon terminals while in dendrites they enter through the entire plasma membrane. The model equations developed in this paper are based on these experimental observations. The analytical solution of these equations is obtained; the solution predicts the distribution of the concentration of nanoparticles associated with free nanoparticle-loaded vesicles (NLVs) (not transported on microtubules (MTs)) as well as the distribution of the concentration of nanoparticles associated with NLVs transported on MTs by dynein motors. The fluxes of nanoparticles by diffusion and motor-driven transport as well as the total (combined) flux of nanoparticles are also predicted. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Nanotechnology has numerous emerging applications in medicine. Diagnostic, therapeutic, and targeting agents can be incorporated within different types of nanoparticles, such as quantum dots, liposomes, and viruses [1]. One of the advantages of nanoparticles is that, if properly designed, they are capable of penetrating various tissue barriers, such as the blood-brain barrier [2,3]. The blood-brain barrier presents the major obstacle for treatment of many brain disorders because it is difficult for many therapeutic and diagnostic agents to cross this barrier in adequate amounts. Patients with neurodegenerative conditions, such as Alzheimer's, Parkinson's, and Huntington's diseases, may benefit from novel drugs, if these drugs can be delivered to specific areas of the brain. This paper is motivated by the recent research of Wong et al. [4] who investigated the utilization of layered double hydroxide nanoparticles for the delivery of small interfering RNAs (siRNAs) [5] to neuron bodies. Such siRNAs are capable of destroying specific messenger RNAs. This makes siRNAs suitable for treatment of such disorders as Huntington's disease, which is linked to the production of abnormal proteins. The advantages of using nanoparticles as delivery vehicles for siRNAs include the fact that they can be administered intravenously [6]; also, layered double hydroxide nanoparticles exhibit low cytotoxicity, are biocompatible, and much more efficiently internalized by cortical neurons than by many non-neuronal cell lines.
☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.03.015
Nanoparticles most likely enter neurons by clathrin-dependent endocytosis [7]. Once they entered neurons, nanoparticles are transported toward the neuron body (where they are released into the cytoplasm) by retrograde transport in endocytic vesicles through association with dynein molecular motors [4,8,9]. Neurons have two types of long processes, axons and dendrites; axons transmit signals and dendrites receive signals. Interestingly, there is evidence that nanoparticles enter axons at axon terminals while in dendrites nanoparticles enter through the entire plasma membrane [4,10,11]. This paper develops, based on the above experimental observations, models of nanoparticle transport from their place of entry to the neuron body. Two models are developed, for an axon and a dendrite. Transport of nanoparticles in these two neuronal processes is then compared. 2. Governing equations Fig. 1a and b shows schematic diagrams of the problem. In an axon (Fig. 1a) nanoparticles enter through the axon synapse only, where the nanoparticle concentration is assumed to be the highest. In a dendrite (Fig. 1b) nanoparticles enter along the whole length of the dendrite. After they have entered, nanoparticles are transported in nanoparticleloaded vesicles (NLVs) retrogradely toward the neuron soma, where nanoparticle concentration is assumed to be the lowest. It is assumed that after releasing their cargo in the cell body, nanoparticles are directed to lysosomes for degradation (which provides a sink for nanoparticles in the soma); this assumption allows solving the problem in a steady-state formulation. Equations describing nanoparticle transport are based on the model of molecular motor-assisted transport of intracellular organelles
544
A.V. Kuznetsov / International Communications in Heat and Mass Transfer 38 (2011) 543–547
Nomenclature D D˜ j jdiff jmotor k˜ k′
dimensionless diffusivity of free NLVs (not transported ˜
on MTs), Dν̃2k̃ diffusivity of free NLVs (not transported on MTs), (μm2/s) total dimensionless flux of nanoparticles (by diffusion and motor-driven transport) dimensionless flux of nanoparticles due to diffusion dimensionless flux of nanoparticles due to motordriven transport first order rate constant characterizing the rate of NLVs binding to MTs (1/s) dimensionless first order rate constant characterizing the rate of NLVs detachment from MTs,
k˜ ′ L
first order rate constant characterizing the rate of NLVs detachment from MTs (1/s) dimensionless length of an axon or dendrite (see Fig. 1a, b),
L˜ n ñ nMT
ñMT N0 Ñ0 NL ÑL Q
k˜′ k˜
L˜ k˜ v˜
length of an axon or dendrite (μm) dimensionless number density of nanoparticles asso3 ciated with free NLVs (not transported on MTs), ñ ṽ 3 k˜
number density of nanoparticles associated with free NLVs (not transported on MTs), (1/μm3) dimensionless number density of nanoparticles associated with NLVs transported retrogradely by dynein 3 motors, ñ MT ν̃3 k˜
number density of nanoparticles associated with NLVs transported retrogradely by dynein motors (1/μm3) dimensionless number density of nanoparticles asso3 ciated with free NLVs at x = 0, N˜ 0 ṽ 3
developed in Smith and Simmons [12]. Two populations of nanoparticles are considered. The first population is comprised of nanoparticles associated with free NLVs; their concentration is denoted by ñ, this population can be transported by diffusion. The second population is comprised of nanoparticles associated with NLVs that are transported on MTs retrogradely by dynein motors; their concentration is denoted by ñMT. NLVs can switch between the two populations; free NLVs can attach to MTs, the rate of this process is characterized by the first order ˜ NLVs associated with MTs can detach from MTs and rate constant k. become freely suspended NLVs; the rate of this process is characterized by the first order rate constant k˜ ′ . It should be noted that in axons all MTs have the same polarity; the plus-ends of all MTs are directed away from the neuron soma toward the axon terminal (synapse). Therefore, cytoplasmic dynein, which is the major retrograde motor in axons, always pulls organelles toward the neuron body. However, the MT polarity orientation in dendrites is non-uniform [13,14]; one would assume that dynein would then pull endocytic vesicles in both directions, depending on the orientation of a particular MT. However, [10] found that transport of endocytosed material in peripheral regions of dendrites was always unidirectional, except in a small 20 to 30 μm-thick segment closest to the cell body where some vesicles changed direction. Even in that region approximately 80% of vesicles moved into the cell body without change of direction. Based on these experimental observations, in the model it is assumed that NLVs are transported on MTs in the retrograde direction only, and no change of direction occurs. This assumption significantly simplifies the problem formulation and makes it possible to obtain the exact analytical solution. Equations governing transport of nanoparticles in an axon (Fig. 1a) are D
k˜
number density of nanoparticles associated with free NLVs at x = 0 (1/μm3) dimensionless number density of nanoparticles asso3 ciated with free NLVs at x = L, NL̃ ṽ 3 k˜ number density of nanoparticles associated with free NLVs at x = L (1/μm3) dimensionless volumetric rate of nanoparticle inter-
∂2 n −n + k′ nMT = 0 ∂x2
ð1Þ
∂nMT =0 ∂x
ð2Þ
n−k′ nMT +
Dimensionless variables in Eqs. (1) and (2) are defined as follows: D=
3
˜ Q
ṽ x x̃
nalization in a dendrite, ṽ4 Q˜ k˜ volumetric rate of nanoparticle internalization in a dendrite (in a dendrite internalization occurs over the entire plasma membrane but is represented as a volumetric source based on the assumption that nanoparticle concentration in a dendrite cross-section is uniform) (1/μm3s) average velocity of NLVs transported on MTs retrogradely (μm/s) x̃ ̃ dimensionless linear coordinate along the MT tracks, ṽk linear coordinate along the MT tracks (see Fig. 1a and b) (μm)
D˜ k˜ ṽ
2
; k′ =
x̃ ̃ ṽ 3 ṽ3 k˜′ ; x = k; n = ñ 3 ; nMT = ñ MT 3 ṽ k̃ k̃ k̃
ð3Þ
where D˜ is the dimensionless diffusivity of free NLVs (not transported on MTs), ṽ is the average velocity of NLVs transported on MTs retrogradely, and x̃ is the linear coordinate along the MT tracks (see Fig. 1a and b). Equations governing transport of nanoparticles in a dendrite (Fig. 1b) are
D
∂2 n −n + k′ nMT + Q = 0 ∂x2
n−k′ nMT +
∂nMT =0 ∂x
ð4Þ ð5Þ
In Eq. (4) Greek symbols degree of loading at x = L σL
Abbreviations MT microtubule NLV nanoparticle-loaded vesicle
Q =
ṽ3 ̃ 4Q k̃
ð6Þ
where Q˜ is the volumetric rate of nanoparticle internalization in a dendrite (assumed constant). Based on the data presented in Refs. [4,10], in a dendrite internalization occurs by endocytosis (most likely by clathrin-dependent uptake) over the entire length of the dendrite. This uptake is represented by a volumetric source based on the assumption that nanoparticle concentration in a dendrite cross-section is uniform.
A.V. Kuznetsov / International Communications in Heat and Mass Transfer 38 (2011) 543–547
545
Fig. 1. Schematic diagram of the problem for (a) an axon and (b) a dendrite.
Once n is obtained, nMT is found by integrating Eq. (2), as
Eqs. (1) and (2) for the axon or Eqs. (4) and (5) for the dendrite must be solved subject to the following boundary conditions: At x = 0 : n = N0 At x = L : n = NL ; nMT = σL NL
ð7Þ ð8a; bÞ
nMT = −
c0 c1 c2 expðμ 1 xÞ + expðμ 2 xÞ + c3 expðk′ xÞ ð14Þ + k′ k′ −μ 1 k′ −μ 2
where the integration constant c3 is found by substituting Eq. (14) into boundary condition (8b).
where 4. Solution for a dendrite ̃ ̃ ṽ3 ṽ3 N0 = N˜ 0 3 ; NL = Ñ L 3 ; L = L k ṽ ̃ ̃ k k
ð9Þ
Eliminating nMT from Eqs. (4) and (5) and integrating with respect to x results in:
In Eqs. (8) and (9) σL is the degree of loading at x = L, Ñ0 is the number density of nanoparticles associated with free NLVs at x = 0, ÑL is the number density of nanoparticles associated with free NLVs at x = L, and L˜ is the length of the axon or dendrite (see Fig. 1a and b).
D
3. Solution for an axon
n = −d0 + k′ Q Dk′ −x + d1 expðλ1 xÞ + d2 expðλ2 xÞ
Eliminating nMT from Eqs. (1) and (2) and integrating with respect to x results in:
where λ1 and λ2 are the roots of
D
d2 n dn −Dk′ −n = c0 dx dx2
ð10Þ
2 Dλ −Dk′ λ−1 = 0
ð12Þ
ð17Þ
ð13Þ
Eqs. (7), (8a), and (13) give a system of three linear equations for c0, c1, and c2.
d n −Q + ð1−k′ σL ÞNL = D dx2
ð18Þ
Eqs. (7), (8a), and (18) give a system of three linear equations for d0, d1, and d2. Once n is obtained, nMT is found by integrating Eq. (5), as nMT = −
2
At x = L :
ð16Þ
2
At x = L :
Constants c0, c1, and c2 in Eq. (11) are found from boundary conditions given by Eqs. (7) and (8a) and the following boundary condition that is obtained by eliminating nMT(L) from Eq. (8b): d n ð1−k′ σL ÞNL = D dx2
The general solution of Eq. (15) is
ð11Þ
where μ1 and μ2 are the roots of 2 Dμ −Dk′ μ−1 = 0
ð15Þ
Constants d0, d1, and d2 in Eq. (16) are found from boundary conditions given by Eqs. (7) and (8a) and the following boundary condition that is obtained by eliminating nMT(L) from Eq. (8b):
The general solution of Eq. (10) is n = −c0 + c1 expðμ 1 xÞ + c2 expðμ 2 xÞ
d2 n dn −Dk′ −n−k′ Qx = d0 dx dx2
+
d0 1 d1 expðμ 1 xÞ +Q − + Dk′ −x + k′ k′ −μ 1 k′
ð19Þ
d2 expðμ 2 xÞ + d3 expðk′ xÞ k′ −μ 2
where the integration constant d3 is found by substituting Eq. (19) into boundary condition (8b).
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A.V. Kuznetsov / International Communications in Heat and Mass Transfer 38 (2011) 543–547
The dimensionless diffusion-driven flux of nanoparticles is calculated as jdiff = −D
dn dx
nanoparticle transport in these two processes; the analytical solutions obtained above make it possible to calculate nanoparticle distributions in segments of any length. The results presented in Figs. 2 and 3 are
ð20Þ
while the motor-driven flux is calculated as jmotor = −nMT
ð21Þ
The total flux of nanoparticles is jtotal = −D
dn −nMT dx
ð22Þ
5. Results and discussion For the purpose of comparison, in numerical results reported in Figs. 2 and 3 both neurites (axon and dendrite) are assumed to have the same dimensionless length, L, although axons are usually much longer than dendrites. The equality of the lengths of segments of the axon and dendrite is assumed only to ease the comparison between
Fig. 2. (a) Distributions of the number density of nanoparticles associated with free NLVs (not actively transported by retrograde motors), n(x); (b) Distributions of the number density of nanoparticles associated with NLVs transported on MTs by dynein motors, nMT(x), in an axon and dendrite.
Fig. 3. (a) Distributions of the flux of nanoparticles due to diffusion, jdiff (x); (b) distributions of the flux of nanoparticles due to motor-driven retrograde transport, jmotor(x); (c) distributions of the total flux of nanoparticles (due to motor-driven and diffusion transport mechanisms), jtotal(x), in an axon and dendrite.
A.V. Kuznetsov / International Communications in Heat and Mass Transfer 38 (2011) 543–547
obtained for the following typical values of dimensionless parameters: D = 0.5, k′ = 0.5, L = 20, N0 = 0.01, NL = 0.2, σL = 0.1, and Q = 0.01. Fig. 2a displays distributions of the number density of nanoparticles associated with free NLVs (not actively transported by retrograde motors), n(x), in an axon and dendrite. Since nanoparticles enter the dendrite over the entire dendritic plasma membrane, the concentration of nanoparticles associated with free NLVs is increasing when x is decreasing, as more and more nanoparticles enter the dendrite, see Fig. 1b (the lateral area of a segment of a dendrite between x and L is 2πR(L − x), where R is the radius of a dendrite; this quantity becomes larger when x becomes smaller). Fig. 2b displays distributions of the number density of nanoparticles associated with NLVs transported on MTs retrogradely, nMT(x). In an axon nMT is independent of x over most of the axon length, except in the boundary layers near x = 0 and x = L. In a dendrite, nMT is increasing when x is decreasing, which is again explained by the nanoparticle's uptake by the lateral surface of the dendrite. Fig. 3a displays distributions of the flux of nanoparticles due to diffusion, jdiff(x). jdiff(x) is significant (and negative, directed toward the axon soma) only in the boundary layers near x =0 and x = L. In the middle region of neurites jdiff(x) is slightly negative in an axon and slightly positive in a dendrite, the latter is explained by the negative slope of n(x) in a dendrite, see Fig. 2a. Fig. 3b displays distributions of the flux of nanoparticles due to motor-driven retrograde transport, jmotor(x). In an axon jmotor(x) is almost constant over of the axon length. In a dendrite the magnitude of jmotor(x) increases quickly when x is decreasing (as one gets closer to the neuron soma); this is again explained by the fact that for smaller x the number of nanoparticles becomes larger due to nanoparticle internalization over the lateral area of a dendrite. The negative sign of the flux simply means that the flux is directed toward the neuron soma. Fig. 3c displays the total flux of nanoparticles, jtotal(x), due to diffusion and motor-assisted transport. In an axon jtotal is constant, which is explained by the steady-state formulation of the problem (in this formulation no nanoparticle accumulation in a control volume is possible; therefore, the rate at which nanoparticles enter the control volume must equal the rate at which nanoparticles leave the control volume). In a dendrite, the absolute value of jtotal becomes larger when x is decreasing; this is because nanoparticles enter through the lateral area of a dendrite. jtotal at x =L is slightly different in an axon and dendrite,
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