Model setup.

<p>(A) The model geometry. The computational domain <i>D</i> represents a cross-section of an axon. The black circle is the domain boundary representing the axon membrane. Small grey dots are neurofilaments (NF), large black dots are microtubules (MT) and cyan filled disks are organelles (Org). (B) The relation of <i>D</i> to the whole axon. Thin grey lines are neurofilaments, thick black lines are microtubules, the cyan body is an organelle, red triangles represent molecular motors that move microtubules and organelles along microtubule tracks. (C) The shape of organelles considered in this model. The cross-sectional radius of an organelle in <i>D</i> depends on its position relative to <i>D</i>.</p

The effect of microtubule zippering by moving organelles.

<p>The mean of PDMT is plotted over time. The maximum number of microtubules that a single organelle can interact with simultaneously (<i>m</i>_<i>max</i>) is set to be 1, 2, 4, 8, and 16 for the blue, green, red, cyan, purple, and yellow curves respectively. Each curve represents the average over 5 realizations of the model and the error bars are the standard deviation. All other parameters are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.g004" target="_blank">Fig 4</a>.</p

A Stochastic Multiscale Model That Explains the Segregation of Axonal Microtubules and Neurofilaments in Neurological Diseases

<div><p>The organization of the axonal cytoskeleton is a key determinant of the normal function of an axon, which is a long thin projection of a neuron. Under normal conditions two axonal cytoskeletal polymers, microtubules and neurofilaments, align longitudinally in axons and are interspersed in axonal cross-sections. However, in many neurotoxic and neurodegenerative disorders, microtubules and neurofilaments segregate apart from each other, with microtubules and membranous organelles clustered centrally and neurofilaments displaced to the periphery. This striking segregation precedes the abnormal and excessive neurofilament accumulation in these diseases, which in turn leads to focal axonal swellings. While neurofilament accumulation suggests an impairment of neurofilament transport along axons, the underlying mechanism of their segregation from microtubules remains poorly understood for over 30 years. To address this question, we developed a stochastic multiscale model for the cross-sectional distribution of microtubules and neurofilaments in axons. The model describes microtubules, neurofilaments and organelles as interacting particles in a 2D cross-section, and is built upon molecular processes that occur on a time scale of seconds or shorter. It incorporates the longitudinal transport of neurofilaments and organelles through this domain by allowing stochastic arrival and departure of these cargoes, and integrates the dynamic interactions of these cargoes with microtubules mediated by molecular motors. Simulations of the model demonstrate that organelles can pull nearby microtubules together, and in the absence of neurofilament transport, this mechanism gradually segregates microtubules from neurofilaments on a time scale of hours, similar to that observed in toxic neuropathies. This suggests that the microtubule-neurofilament segregation can be a consequence of the selective impairment of neurofilament transport. The model generates the experimentally testable prediction that the rate and extent of segregation will be dependent on the sizes of the moving organelles as well as the density of their traffic.</p></div

Dependence of microtubule-neurofilament segregation on konN.

<p>Each curve plots the mean of the PDMT over time averaged over five realizations, and the error bars are the standard deviations. The rate </p><p></p><p></p><p></p><p><mi>k</mi></p><p><mi>o</mi><mi>n</mi></p><mi>N</mi><p></p><p></p><p></p><p></p> is reduced to 50% (blue), 20% (green), and 0% (red) of the value in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.g004" target="_blank">Fig 4</a> at <i>t</i> = 1 h. All other parameters are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.g004" target="_blank">Fig 4</a>.<p></p

IDPN-induced segregation of microtubules and neurofilaments in axonal cross-sections.

<p>We show here drawings that are based loosely on the electron micrographs in Fig 3 in the reference Papasozomenos et al. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.ref050" target="_blank">50</a>]. Due to copyright restrictions we are not able to show the actual micrographs. The authors administered IDPN in physiological saline to adult male rats by intraperitoneal injection (2 mg/g body weight). Control injections consisted of physiological saline alone. For the micrographs on which these drawings are based, the animal was sacrificed after 2 weeks and the nerve was fixed and examined in cross-section by electron microscopy. Full experimental details can be found in the original article. (A) Drawing of a control axon in cross-section showing that the microtubules (large black dots), neurofilaments (small black dots) and membranous organelles (irregularly shaped grey blobs) are normally interspersed throughout the axonal cross-section. (B) Drawing of an axon in cross-section after IDPN treatment showing that the microtubules and organelles form a central core surrounded by a peripheral rim of neurofilaments. Note that the central core of microtubules and organelles contains very few neurofilaments and the outer rim of neurofilaments contains very few microtubules and organelles. The dark grey area outside of the axon is the myelin sheath. The scale bar is 1 <i>μ</i>m.</p

Reversible segregation of microtubules and neurofilaments in a single realization of the model.

<p>Neurofilament transport is blocked starting at <i>t</i> = 1 h and restored at <i>t</i> = 13 h. (A-F) Snapshots of the positions of microtubules, neurofilaments and organelles at <i>t</i> = 1 h, 3 h, 6 h, 12 h, 14 h, 20 h. All panels are from a single realization of the model. Large black dots are microtubules; small grey dots are free neurofilaments; small purple dots are neurofilaments engaged with microtubules; large cyan circles are organelles. (A) Microtubules and neurofilaments form a mixture under normal conditions. (B-D) Blockage of neurofilament transport leads to gradual segregation of microtubules and neurofilaments. (E,F) Restoration of neurofilament transport causes remixing of microtubules and neurofilaments. Parameters used: <i>n</i>^<i>M</i> = 56, <i>n</i>^<i>N</i> = 361. Neurofilament on-rate </p><p></p><p></p><p></p><p><mi>k</mi></p><p><mi>o</mi><mi>n</mi></p><mi>N</mi><p></p><p></p><p></p><p></p> equals 0 between <i>t</i> = 1 h and 13 h. All other parameters are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.t001" target="_blank">Table 1</a>.<p></p

Simulated neurofilament distributions with different densities and repulsion magnitudes.

<p>Left: snapshots of neurofilament positions after randomizing for 25 sec. Middle: the radial distribution functions (RDF, <i>g</i>(<i>r</i>)). Right: the bars are histograms of the occupancy probability distribution (OPD, <i>p</i>_<i>n</i>) using randomly chosen circular windows with a radius 60 nm, and the black curves are their Gaussian fits. Middle and right plots represent averages over 50 time frames between <i>t</i> = 25 sec and <i>t</i> = 30 sec. In all cases, <i>ε</i> is short for <i>ε</i>^<i>NN</i>. (A) <i>n</i>^<i>F</i> = 200, <i>ε</i>^<i>NN</i> = 0.5 pN; (B) <i>n</i>^<i>F</i> = 400, <i>ε</i>^<i>NN</i> = 0.5 pN; (C) <i>n</i>^<i>F</i> = 400, <i>ε</i>^<i>NN</i> = 0.25 pN. Other parameters are the same as specified in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.t001" target="_blank">Table 1</a>.</p

Statistics of the pairwise distances between microtubules (PDMT).

<p>(A, B) IDPN treatment started at t = 1 hour and stopped at t = 13 hours. (C, D) Control. (A, C) Distribution of the PDMT; data plotted for every 20 min. The pseudo color key represents the number of microtubule pairs. (B, D) Mean of the PDMT; data plotted for every min. Parameters used are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.g004" target="_blank">Fig 4</a>.</p

Segregation proceeds by the coalescence of microtubule islands.

<p>(A-C) Snapshots of the segregation process in a single realization of the model. Large black dots are microtubules, small grey dots are neurofilaments, and cyan circles are organelles. (A) Microtubules form three clusters by <i>t</i> = 4 h. (B) These clusters remain separated for several hours until two of them merge around <i>t</i> = 8 h. (C) Finally, all microtubules form a single big cluster near the center of the domain. The dimension of the organelles: <i>b</i> = 70 nm, <i>a</i>/<i>b</i> = 10. The flux rate of the organelles: 0.21 /s. All other parameters are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.g004" target="_blank">Fig 4</a>. (D) A drawing based loosely on the electron micrograph in Fig 9A of Zhu et al. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.ref073" target="_blank">73</a>], showing a cross-section of an L5 ventral nerve root axon from an animal that was exposed to IDPN. Due to copyright restrictions we are not able to show the actual micrographs. The authors administered IDPN in physiological saline to adult mice by intraperitoneal injection (1.5 mg/g body weight) and supplemented with 0.02% IDPN in the drinking water. For the micrograph on which this drawing is based, the animal was sacrificed 1 week after the first injection. Full experimental details can be found in the original article. Note the presence of multiple microtubule clusters, which resembles the intermediate stages of segregation in the simulations. Microtubules are represented by the large black dots, neurofilaments by the small black dots, and membranous organelles by the irregularly shaped grey blobs. The dark grey area outside of the axon is the myelin sheath. The scale bar is 0.4 <i>μ</i>m.</p

Model parameter values.

<p>E.E.: estimated from experiments; see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004406#pcbi.1004406.s001" target="_blank">S1 Text</a> for detailed information.</p><p>Model parameter values.</p