A dynamical system model of neurofilament transport in axons

Deep learning of material transport in complex neurite networks

Neurons exhibit complex geometry in their branched networks of neurites which is essential to the function of individual neuron but also brings challenges to transport a wide variety of essential materials throughout their neurite networks for their survival and function. While numerical methods like isogeometric analysis (IGA) have been used for modeling the material transport process via solving partial differential equations (PDEs), they require long computation time and huge computation resources to ensure accurate geometry representation and solution, thus limit their biomedical application. Here we present a graph neural network (GNN)-based deep learning model to learn the IGA-based material transport simulation and provide fast material concentration prediction within neurite networks of any topology. Given input boundary conditions and geometry configurations, the well-trained model can predict the dynamical concentration change during the transport process with an average error less than 10% and [Formula: see text] times faster compared to IGA simulations. The effectiveness of the proposed model is demonstrated within several complex neurite networks.

A possible mechanism for neurofilament slowing down in myelinated axon: Phosphorylation-induced variation of NF kinetics

Neurofilaments(NFs) are the most abundant intermediate filaments that make up the inner volume of axon, with possible phosphorylation on their side arms, and their slow axonal transport by molecular motors along microtubule tracks in a “stop-and-go” manner with rapid, intermittent and bidirectional motion. The kinetics of NFs and morphology of axon are dramatically different between myelinate internode and unmyelinated node of Ranvier. The NFs in the node transport as 7.6 times faster as in the internode, and the distribution of NFs population in the internode is 7.6 folds as much as in the node of Ranvier. We hypothesize that the phosphorylation of NFs could reduce the on-track rate and slow down their transport velocity in the internode. By modifying the ‘6-state’ model with (a) an extra phosphorylation kinetics to each six state and (b) construction a new ‘8-state’ model in which NFs at off-track can be phosphorylated and have smaller on-track rate, our model and simulation demonstrate that the phosphorylation-induced decrease of on-track rate could slow down the NFs average velocity and increase the axonal caliber. The degree of phosphorylation may indicate the extent of velocity reduction. The Continuity equation used in our paper predicts that the ratio of NFs population is inverse proportional to the ratios of average velocity of NFs between node of Ranvier and internode. We speculate that the myelination of axon could increase the level of phosphorylation of NF side arms, and decrease the possibility of NFs to get on-track of microtubules, therefore slow down their transport velocity. In summary, our work provides a potential mechanism for understanding the phosphorylation kinetics of NFs in regulating their transport and morphology of axon in myelinated axons, and the different kinetics of NFs between node and internode.

Statistical Laws of Protein Motion in Neuronal Dendritic Trees

Across their dendritic trees, neurons distribute thousands of protein species that are necessary for maintaining synaptic function and plasticity and that need to be produced continuously and trafficked to their final destination. As each dendritic branchpoint splits the protein flow, increasing branchpoints decreases the total protein number downstream. Consequently, a neuron needs to produce more proteins to maintain a minimal protein number at distal synapses. Combining in vitro experiments and a theoretical framework, we show that proteins that diffuse within the cell plasma membrane are, on average, 35% more effective at reaching downstream locations than proteins that diffuse in the cytoplasm. This advantage emerges from a bias for forward motion at branchpoints when proteins diffuse within the plasma membrane. Using 3D electron microscopy (EM) data, we show that pyramidal branching statistics and the diffusion lengths of common proteins fall into a region that minimizes the overall protein need.

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Surface proteins are more efficient at reaching distal sites than soluble proteins

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Daughter radius optimization reduces the number of proteins needed to populate dendrites

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Ratios of daughter radii at branchpoints are cell type specific

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Highly diffusive proteins incur a smaller extra cost for non-optimized radii

Intra- and intercellular transport of substances: Models and mechanisms

Non-equilibrium-statistical models of intracellular transport are built. The most significant features of these models are microscopic reversibility and the explicit considerations of the driving forces of the process - the ATP-ADP chemical potential difference. In this paper, water transport using contractile vacuoles, the transport and assembly of microtubules and microfilaments, the protein distribution within a cell, the transport of neurotransmitters from the synaptic cleft and the transport of substances between cells using plasmodesmata are discussed. Endocytosis and phagocytosis models are considered, and transport tasks and information transfer mechanisms inside the cell are explored. Based on an analysis of chloroplast movement, it was concluded that they have a complicated method of influencing each other in the course of their movements. The role of quantum effects in sorting and control transport mechanisms is also discussed. It is likely that quantum effects play a large role in these processes, otherwise reliable molecular recognition would be impossible, which would lead to very low intracellular transport efficiency.

Local Acceleration of Neurofilament Transport at Nodes of Ranvier

Myelinated axons are constricted at nodes of Ranvier. These constrictions are important physiologically because they increase the speed of saltatory nerve conduction, but they also represent potential bottlenecks for the movement of axonally transported cargoes. One type of cargo are neurofilaments, which are abundant space-filling cytoskeletal polymers that function to increase axon caliber. Neurofilaments move bidirectionally along axons, alternating between rapid movements and prolonged pauses. Strikingly, axon constriction at nodes is accompanied by a reduction in neurofilament number that can be as much as 10-fold in the largest axons. To investigate how neurofilaments navigate these constrictions, we developed a transgenic mouse strain that expresses a photoactivatable fluorescent neurofilament protein in neurons. We used the pulse-escape fluorescence photoactivation technique to analyze neurofilament transport in mature myelinated axons of tibial nerves from male and female mice of this strain ex vivo Fluorescent neurofilaments departed the activated region more rapidly in nodes than in flanking internodes, indicating that neurofilament transport is faster in nodes. By computational modeling, we showed that this nodal acceleration can be explained largely by a local increase in the duty cycle of neurofilament transport (i.e., the proportion of the time that the neurofilaments spend moving). We propose that this transient acceleration functions to maintain a constant neurofilament flux across nodal constrictions, much as the current increases where a river narrows its banks. In this way, neurofilaments are prevented from piling up in the flanking internodes, ensuring a stable neurofilament distribution and uniform axonal morphology across these physiologically important axonal domains.SIGNIFICANCE STATEMENT Myelinated axons are constricted at nodes of Ranvier, resulting in a marked local decrease in neurofilament number. These constrictions are important physiologically because they increase the efficiency of saltatory nerve conduction, but they also represent potential bottlenecks for the axonal transport of neurofilaments, which move along axons in a rapid intermittent manner. Imaging of neurofilament transport in mature myelinated axons ex vivo reveals that neurofilament polymers navigate these nodal axonal constrictions by accelerating transiently, much as the current increases where a river narrows its banks. This local acceleration is necessary to ensure a stable axonal morphology across nodal constrictions, which may explain the vulnerability of nodes of Ranvier to neurofilament accumulations in animal models of neurotoxic neuropathies and neurodegenerative diseases.

A stochastic model that explains axonal organelle pileups induced by a reduction of molecular motors

Nerve cells are critically dependent on the transport of intracellular cargoes, which are moved by motor proteins along microtubule tracks. Impairments in this movement are thought to explain the focal accumulations of axonal cargoes and axonal swellings observed in many neurodegenerative diseases. In some cases, these diseases are caused by mutations that impair motor protein function, and genetic depletion of functional molecular motors has been shown to lead to cargo accumulations in axons. The evolution of these accumulations has been compared to the formation of traffic jams on a highway, but this idea remains largely untested. In this paper, we investigated the underlying mechanism of local axonal cargo accumulation induced by a global reduction of functional molecular motors in axons. We hypothesized that (i) a reduction in motor number leads to a reduction in the number of active motors on each cargo which in turn leads to less persistent movement, more frequent stops and thus shorter runs; (ii) as cargoes stop more frequently, they impede the passage of other cargoes, leading to local 'traffic jams'; and (iii) collisions between moving and stopping cargoes can push stopping cargoes further away from their microtubule tracks, preventing them from reattaching and leading to the evolution of local cargo accumulations. We used a lattice-based stochastic model to test whether this mechanism can lead to the cargo accumulation patterns observed in experiments. Simulation results of the model support the hypothesis and identify key questions that must be tested experimentally.

Assembly and turnover of neurofilaments in growing axonal neurites

Neurofilaments (NFs) are thought to provide stability to the axon. We examined NF dynamics within axonal neurites of NB2a/d1 neuroblastoma by transient transfection with green fluorescent protein-tagged NF-heavy (GFP-H) under the control of a tetracycline-inducible promoter. Immunofluorescent and biochemical analyses demonstrated that GFP-H expressed early during neurite outgrowth associated with a population of centrally-situated, highly-phosphorylated crosslinked NFs along the length of axonal neurites (‘bundled NFs’). By contrast, GFP-H expressed after considerable neurite outgrowth displayed markedly reduced association with bundled NFs and was instead more evenly distributed throughout the axon. This differential localization was maintained for up to 2 weeks in culture. Once considerable neurite outgrowth had progressed, GFP that had previously associated with the NF bundle during early expression was irreversibly depleted by photobleaching. Cessation of expression allowed monitoring of NF turnover. GFP-H associated bundled NFs underwent slower decay than GFP-H associated with surrounding, less-phosphorylated NFs. Notably, GFP associated with bundled NFs underwent similar decay rates within the core and edges of this bundle. These results are consistent with previous demonstration of a resident NF population within axonal neurites, but suggest that this population is more dynamic than previously considered.

Summary: Immunofluorescent and radiolabel analyses demonstrate that neurofilaments establish a resident population within growing axonal neurites that undergoes exchange with a surrounding, transporting pool.

Kymograph analysis with high temporal resolution reveals new features of neurofilament transport kinetics

We have used kymograph analysis combined with edge detection and an automated computational algorithm to analyze the axonal transport kinetics of neurofilament polymers in cultured neurons at 30 ms temporal resolution. We generated 301 kymographs from 136 movies and analyzed 726 filaments ranging from 0.6 to 42 µm in length, representing ∼37,000 distinct moving and pausing events. We found that the movement is even more intermittent than previously reported and that the filaments undergo frequent, often transient, reversals which suggest that they can engage simultaneously with both anterograde and retrograde motors. Average anterograde and retrograde bout velocities (0.9 and 1.2 µm s^-1 , respectively) were faster than previously reported, with maximum sustained bout velocities of up to 6.6 and 7.8 µm s^-1 , respectively. Average run lengths (∼1.1 µm) and run times (∼1.4 s) were in the range reported for molecular motor processivity in vitro, suggesting that the runs could represent the individual processive bouts of the neurofilament motors. Notably, we found no decrease in run velocity, run length or run time with increasing filament length, which suggests that either the drag on the moving filaments is negligible or that longer filaments recruit more motors.

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

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.

The shape and function of axons is dependent on a dynamic system of microscopic intracellular protein polymers (microtubules, neurofilaments and microfilaments) that comprise the axonal cytoskeleton. Neurofilaments are cargoes of intracellular transport that move along microtubule tracks, and they accumulate abnormally in axons in many neurotoxic and neurodegenerative disorders. Intriguingly, it has been reported that neurofilaments and microtubules, which are normally interspersed in axonal cross-sections, often segregate apart from each other in these disorders, which is something that is never observed in healthy axons. Here we describe a stochastic multiscale computational model that explains the mechanism of this striking segregation and offers insights into the mechanism of neurofilament accumulation in disease.

Drag of the cytosol as a transport mechanism in neurons

Axonal transport is typically divided into two components, which can be distinguished by their mean velocity. The fast component includes steady trafficking of different organelles and vesicles actively transported by motor proteins. The slow component comprises nonmembranous materials that undergo infrequent bidirectional motion. The underlying mechanism of slow axonal transport has been under debate during the past three decades. We propose a simple displacement mechanism that may be central for the distribution of molecules not carried by vesicles. It relies on the cytoplasmic drag induced by organelle movement and readily accounts for key experimental observations pertaining to slow-component transport. The induced cytoplasmic drag is predicted to depend mainly on the distribution of microtubules in the axon and the organelle transport rate.

Axonal transport cargo motor count versus average transport velocity: is fast versus slow transport really single versus multiple motor transport?

Cargos have been observed exhibiting a "stop-and-go" behavior (i.e. cargo "pause"), and it has generally been assumed that these multi-second pauses can be attributed to equally long pauses of cargo-bound motors during motor procession. We contend that a careful examination of the isolated microtubule experimental record does not support motor pauses. Rather, we believe that the data suggests that motor cargo complexes encounter an obstruction that prevents procession, eventually detach and reattach, with this obstructed-detach-reattach sequence being observed in axon as a "pause." Based on this, along with our quantitative evidence-based contention that slow and fast axonal transport are actually single and multi-motor transport, we have developed a cargo level motor model capable of exhibiting the full range of slow to fast transport solely by changing the number of motors involved. This computational model derived using first-order kinetics is suitable for both kinesin and dynein and includes load-dependence as well as provision for motors encountering obstacles to procession. The model makes the following specific predictions: average distance from binding to obstruction is about 10 μm; average motor maximum velocity is at least 6 μm/s in axon; a minimum of 10 motors is required for the fastest fast transport while only one motor is required for slow transport; individual in-vivo cargo-attached motors may spend as little as 5% of their time processing along a microtubule with the remainder being spent either obstructed or unbound to a microtubule; and at least in the case of neurofilament transport, kinesin and dynein are largely not being in a "tug-of-war" competition.

Deciphering the axonal transport kinetics of neurofilaments using the fluorescence photoactivation pulse-escape method

Neurofilaments are transported along axons stochastically in a stop-and-go manner, cycling between brief bouts of rapid movement and pauses that can vary from seconds to hours in length. Presently the only way to analyze neurofilament pausing experimentally on both long and short time scales is the pulse-escape method. In this method, fluorescence photoactivation is used to mark a population of axonal neurofilaments and then the loss of fluorescence from the activated region due to neurofilament movement is monitored by time-lapse imaging. Here we develop a mathematical description of the pulse-escape kinetics in terms of the rate constants of a tested mathematical model and we show how this model can be used to characterize neurofilament transport kinetics from fluorescence photoactivation pulse-escape experiments. This combined experimental and computational approach is a powerful tool for the analysis of the moving and pausing behavior of neurofilaments in axons.

Analytical comparison between Nixon-Logvinenko's and Jung-Brown's theories of slow neurofilament transport in axons

This paper develops analytical solutions describing slow neurofilament (NF) transport in axons. The obtained solutions are based on two theories of NF transport: Nixon-Logvinenko's theory that postulates that most NFs are incorporated into a stationary cross-linked network and only a small pool is slowly transported and Jung-Brown's theory that postulates a single dynamic pool of NFs that are transported according to the stop-and-go hypothesis. The simplest two-kinetic state version of the model developed by Jung and Brown was compared with the theory developed by Nixon and Logvinenko. The model for Nixon-Logvinenko's theory included stationary, pausing, and running NF populations while the model used for Jung-Brown's theory only included pausing and running NF populations. Distributions of NF concentrations resulting from Nixon-Logvinenko's and Jung-Brown's theories were compared. In previous publications, Brown and colleagues successfully incorporated slowing of NF transport into their model by assuming that some kinetic constants depend on the distance from the axon hillock. In this paper we defined the average rate of NF transport as the rate of motion of the center of mass of radiolabeled NFs. We have shown that for this definition, if all kinetic rates are assumed constant, Jung-Brown's theory predicts a constant average rate of NF transport. We also demonstrated that Nixon-Logvinenko's theory predicts slowing of NF transport even if all kinetic rates are assumed constant, and the obtained slowing agrees well with published experimental data.

The discontinuous nature of neurofilament transport accommodates both establishment and repair of the axonal neurofilament array

Neurofilaments (NFs) provide structural support to axons. Timely and regional deposition of NFs is essential during axonogenesis, since progressive stabilization of proximal axons is essential to support continued pathfinding of distal axonal regions. NFs undergo short bursts of microtubule-mediated axonal transport interspersed by prolonged pauses. We demonstrate herein that it is this unique "on-off" method of axonal transport, coupled with the ability of NFs to form cation-dependent, phosphomediated lateral associations that allow neurons to mediate the orderly transition from exploratory process to stabilized axon following synaptogenesis. We further demonstrate how this transport method provides for NF maintenance following maturation and encompasses the potential for regeneration.

A critical reevaluation of the stationary axonal cytoskeleton hypothesis

Neurofilaments are transported along axons in a rapid intermittent and bidirectional manner but there is a long-standing controversy about whether this applies to all axonal neurofilaments. Some have proposed that only a small proportion of axonal neurofilaments are mobile and that most are deposited into a persistently stationary and extensively cross-linked cytoskeleton that remains fixed in place for many months without movement, turning over very slowly. In contrast, others have proposed that this hypothesis is based on a misinterpretation of the experimental data and that, in fact, all axonal neurofilaments move. These contrary perspectives have distinct implications for our understanding of how neurofilaments are organized and reorganized in axons both in health and disease. Here, we discuss the history and substance of this controversy. We show that the published data on the kinetics and distribution of neurofilaments along axons favor a simple "stop and go" transport model in which axons contain a single population of neurofilaments that all move in a stochastic, bidirectional and intermittent manner. Based on these considerations, we propose a dynamic view of the neuronal cytoskeleton in which all neurofilaments cycle repeatedly between moving and pausing states throughout their journey along the axon. The filaments move infrequently, but the average pause duration is on the order of hours rather than weeks or months. Against this fluid backdrop, the action of molecular motors on neurofilaments can have dramatic effects on neurofilament organization that would not be possible if the neurofilaments were extensively cross-linked into a truly stationary network.

An exact solution describing slow axonal transport of cytoskeletal elements: the effect of a finite half-life

This paper presents an exact solution for a two kinetic state model of slow axonal transport that is based on the stop-and-go hypothesis. The model accounts for two populations of cytoskeletal elements (CEs): pausing and running. The model also accounts for a finite half-life of CEs involved in slow axonal transport. It is assumed that initially CEs are injected into the axon such that their concentration forms a rectangular pulse; initially all CEs are assumed to be in the pausing state. Kinetic processes quickly redistribute CEs between the pausing and running states. After less than a minute, equilibrium is established, forming two pulses, representing concentrations of pausing and running CEs, respectively. As these pulses propagate, their shape changes and they turn to bell-shaped waves. The amplitude of the waves decreases, and the waves spread out as they propagate down the axon. The rate of the amplitude decrease is larger for CEs with a shorter half-life, but even if CE half-life is infinitely long, some decrease of the waves' amplitudes is observed. The velocity of the waves' propagation is found to be independent of the CE half-life and is in good agreement with published experimental data for slow axonal transport of neurofilaments.

MODELING TRAFFIC JAMS IN SLOW AXONAL TRANSPORT

The purpose of this paper is to develop a model capable of simulating traffic jams in slow axonal transport. Slowing of slow axonal transport is an early sign of some neurodegenerative diseases. Axonal swellings observed near the end stage of such diseases may be an indication of traffic jams developing in axons that cause the slowing down of slow axonal transport. Traffic jams may result from misregulation of microtubule-associated proteins caused by an imbalance in intracellular signaling or by mutations of these proteins. This misregulation leads to a decay of microtubule tracks in axons, effectively reducing the number of "railway tracks" available for molecular-motor-assisted transport of intracellular organelles. In this paper, the decay of microtubule tracks is modeled by a reduction of the number density of microtubules in the central part of the axon. Simulation results indicate that the model predicts the build-up of the bell-shaped concentration wave, as the wave approaches the bottleneck (blockage) region. This increase in concentration will likely plug the bottleneck region resulting in a traffic jam that would hinder the slow axonal transport.

An exact solution of transient equations describing slow axonal transport

An exact analytical solution of equations describing slow axonal transport of cytoskeletal elements (CEs) injected in an axon is presented. The equations modelling slow axonal transport are based on the stop-and-go hypothesis. The simplest model implementing this hypothesis postulates that CEs switch between pausing and running kinetic states, and that the probabilities of CE transition between these two states are described by first-order rate constants. It is assumed that initially CEs are injected such that they form a uniform pulse of a given width. All injected CEs are initially attributed to the pausing state. It is shown that within 30 s kinetic processes redistribute CEs between pausing and running states; after that the process occurs under quasi-equilibrium conditions. The parameter accessible to experiments is the total concentration of CEs (pausing plus running). As the initial rectangular-shaped pulse moves, it changes its shape to become a bell-shaped wave that spreads out as it propagates. The wave's amplitude is decreasing during the wave's propagation. It is also shown that the system forgets its initial condition, meaning that if one starts with pulses of different widths, after sometime they converge to the same bell-shaped wave.

Axonal transport of neurofilaments: a single population of intermittently moving polymers

Studies on mouse optic nerve have led to the controversial proposal that only a small proportion of neurofilaments are transported in axons and that the majority are deposited into a persistently stationary and extensively cross-linked cytoskeletal network that remains fixed in place for months without movement. We have used computational modeling to address this issue, taking advantage of the wealth of published kinetic and morphometric data available for neurofilaments in the mouse visual system. We show that the transport kinetics and distribution of neurofilaments in mouse optic nerve can all be explained fully by a "stop-and-go" model of neurofilament transport, in which axons contain a single population of neurofilaments that all move stochastically in a rapid, intermittent, and bidirectional manner. Importantly, we find that the transport kinetics are not consistent with deposition of neurofilaments into a persistently stationary phase, and that deposition models cannot account for the observed distribution of neurofilaments along mouse optic nerve axons. Finally, we show that the apparent existence of a stationary neurofilament network in mouse optic nerve is most likely an experimental artifact due to contamination of the neurofilament transport kinetics with cytosolic proteins that move at faster rates. Thus, there is no evidence for the deposition of axonally transported neurofilaments into a persistently stationary neurofilament network in optic nerve axons. We conclude that all of the neurofilaments move and that they do so with a single broad and continuous distribution of average rates that is dictated by their intermittent and stochastic motile behavior.

Cargo distributions differentiate pathological axonal transport impairments

Axonal transport is an essential process in neurons, analogous to shipping goods, by which energetic and cellular building supplies are carried downstream (anterogradely) and wastes are carried upstream (retrogradely) by molecular motors, which act as cargo porters. Impairments in axonal transport have been linked to devastating and often lethal neurodegenerative diseases, such as Amyotrophic Lateral Sclerosis, Huntington's, and Alzheimer's. Axonal transport impairment types include a decrease in available motors for cargo transport (motor depletion), the presence of defective or non-functional motors (motor dilution), and the presence of increased or larger cargos (protein aggregation). An impediment to potential treatment identification has been the inability to determine what type(s) of axonal transport impairment candidates that could be present in a given disease. In this study, we utilize a computational model and common axonal transport experimental metrics to reveal the axonal transport impairment general characteristics or "signatures" that result from three general defect types of motor depletion, motor dilution, and protein aggregation. Our results not only provide a means to discern these general impairments types, they also reveal key dynamic and emergent features of axonal transport, which potentially underlie multiple impairment types. The identified characteristics, as well as the analytical method, can be used to help elucidate the axonal transport impairments observed in experimental and clinical data. For example, using the model-predicted defect signatures, we identify the defect candidates, which are most likely to be responsible for the axonal transport impairments in the G93A SOD1 mouse model of ALS.

Predictive models for pressure-driven fluid infusions into brain parenchyma

Direct infusions into brain parenchyma of biological therapeutics for serious brain diseases have been, and are being, considered. However, individual brains, as well as distinct cytoarchitectural regions within brains, vary in their response to fluid flow and pressure. Further, the tissue responds dynamically to these stimuli, requiring a nonlinear treatment of equations that would describe fluid flow and drug transport in brain. We here report in detail on an individual-specific model and a comparison of its prediction with simulations for living porcine brains. Two critical features we introduced into our model-absent from previous ones, but requirements for any useful simulation-are the infusion-induced interstitial expansion and the backflow. These are significant determinants of the flow. Another feature of our treatment is the use of cross-property relations to obtain individual-specific parameters that are coefficients in the equations. The quantitative results are at least encouraging, showing a high fraction of overlap between the computed and measured volumes of distribution of a tracer molecule and are potentially clinically useful. Several improvements are called for; principally a treatment of the interstitial expansion more fundamentally based on poroelasticity and a better delineation of the diffusion tensor of a particle confined to the interstitial spaces.

Mathematical modeling of axonal formation. Part I: Geometry

A stochastic model is proposed for the position of the tip of an axon. Parameters in the model are determined from laboratory data. The first step is the reduction of inherent error in the laboratory data, followed by estimating parameters and fitting a mathematical model to this data. Several axonogenesis aspects have been investigated, particularly how positive axon elongation and growth cone kinematics are coupled processes but require very different theoretical descriptions. Preliminary results have been obtained through a series of experiments aimed at isolating the response of axons to controlled gradient exposures to guidance cues and the effects of ethanol and similar substances. We show results based on the following tasks; (A) development of a novel filtering strategy to obtain data sets truly representative of the axon trail formation; (B) creation of a coarse graining method which establishes (C) an optimal parameter estimation technique, and (D) derivation of a mathematical model which is stochastic in nature, parameterized by arc length. The framework and the resulting model allow for the comparison of experimental and theoretical mean square displacement (MSD) of the developing axon. Current results are focused on uncovering the geometric characteristics of the axons and MSD through analytical solutions and numerical simulations parameterized by arc length, thus ignoring the temporal growth processes. Future developments will capture the dynamic growth cone and how it behaves as a function of time. Qualitative and quantitative predictions of the model at specific length scales capture the experimental behavior well.

A synergic simulation-optimization approach for analyzing biomolecular dynamics in living organisms

A synergic duo simulation-optimization approach was developed and implemented to study protein-substrate dynamics and binding kinetics in living organisms. The forward problem is a system of several coupled nonlinear partial differential equations which, with a given set of kinetics and diffusion parameters, can provide not only the commonly used bleached area-averaged time series in fluorescence microscopy experiments but more informative full biomolecular/drug space-time series and can be successfully used to study dynamics of both Dirac and Gaussian fluorescence-labeled biomacromolecules in vivo. The incomplete Cholesky preconditioner was coupled with the finite difference discretization scheme and an adaptive time-stepping strategy to solve the forward problem. The proposed approach was validated with analytical as well as reference solutions and used to simulate dynamics of GFP-tagged glucocorticoid receptor (GFP-GR) in mouse cancer cell during a fluorescence recovery after photobleaching experiment. Model analysis indicates that the commonly practiced bleach spot-averaged time series is not an efficient approach to extract physiological information from the fluorescence microscopy protocols. It was recommended that experimental biophysicists should use full space-time series, resulting from experimental protocols, to study dynamics of biomacromolecules and drugs in living organisms. It was also concluded that in parameterization of biological mass transfer processes, setting the norm of the gradient of the penalty function at the solution to zero is not an efficient stopping rule to end the inverse algorithm. Theoreticians should use multi-criteria stopping rules to quantify model parameters by optimization.

Myosin Va increases the efficiency of neurofilament transport by decreasing the duration of long-term pauses

We investigated the axonal transport of neurofilaments in cultured neurons from two different strains of dilute lethal mice, which lack myosin Va. To analyze the motile behavior, we tracked the movement of green fluorescent protein (GFP)-tagged neurofilaments through naturally occurring gaps in the axonal neurofilament array of cultured superior cervical ganglion neurons from DLS/LeJ dilute lethal mice. Compared with wild-type controls, we observed no statistically significant difference in velocity or frequency of movement. To analyze the pausing behavior, we used a fluorescence photoactivation pulse-escape technique to measure the rate of departure of PAGFP (photoactivatable GFP)-tagged neurofilaments from photoactivated axonal segments in cultured dorsal root ganglion neurons from DLS/LeJ and dl20J dilute lethal mice. Compared with wild-type controls, we observed a 48% increase in the mean time for neurofilaments to depart the activated regions in neurons from DLS/LeJ mice (p < 0.001) and a 169% increase in neurons from dl20J mice (p < 0.0001). These data indicate that neurofilaments pause for more prolonged periods in the absence of myosin Va. We hypothesize that myosin Va is a short-range motor for neurofilaments and that it can function to enhance the efficiency of neurofilament transport in axons by delivering neurofilaments to their microtubule tracks, thereby reducing the duration of prolonged off-track pauses.

Organizational dynamics, functions, and pathobiological dysfunctions of neurofilaments

Neurofilament phosphorylation has long been considered to regulate their axonal transport rate, and in doing so it provides stability to mature axons. We evaluate the collective evidence to date regarding how neurofilament C-terminal phosphorylation may regulate axonal transport. We present a few suggestions for further experimentation in this area, and expand upon previous models for axonal NF dynamics. We present evidence that the NFs that display extended residence along axons are critically dependent upon the surrounding microtubules, and that simultaneous interaction with multiple microtubule motors provides the architectural force that regulates their distribution. Finally, we address how C-terminal phosphorylation is regionally and temporally regulated by a balance of kinase and phosphatase activities, and how misregulation of this balance might contribute to motor neuron disease.

A quantitative examination of the role of cargo-exerted forces in axonal transport

Axonal transport, via molecular motors kinesin and dynein, is a critical process in supplying the necessary constituents to maintain normal neuronal function. In this study, we predict the role of cooperativity by motors of the same polarity across the entire spectrum of physiological axonal transport. That is, we examined how the number of motors, either kinesin or dynein, working together to move a cargo, results in the experimentally determined velocity profiles seen in fast and slow anterograde and retrograde transport. We quantified the physiological forces exerted on a motor by a cargo as a function of cargo size, transport velocity, and transport type. Our results show that the force exerted by our base case neurofilament (D(NF)=10 nm, L(NF)=1.6 microm) is approximately 1.25 pN at 600 nm/s; additionally, the force exerted by our base case organelle (D(org)=1 microm) at 1000 nm/s is approximately 5.7 pN. Our results indicate that while a single motor can independently carry an average cargo, cooperativity is required to produce the experimental velocity profiles for fast transport. However, no cooperativity is required to produce the slow transport velocity profiles; thus, a single dynein or kinesin can carry the average neurofilament retrogradely or anterogradely, respectively. The potential role cooperativity may play in the hypothesized mechanisms of motoneuron transport diseases such as amyotrophic lateral sclerosis (ALS) is discussed.

Regulation of neurofilament dynamics by phosphorylation

Neurofilament (NF) phosphorylation has long been considered to regulate axonal transport rate and in doing so to provide stability to mature axons. Studies utilizing mice in which the C-terminal region of NF subunits (which contains the vast majority of phosphorylation sites) has been deleted has prompted an ongoing challenge to this hypothesis. We evaluate the collective evidence to date for and against a role for NF C-terminal phosphorylation in regulation of axonal transport and in providing structural support for axons, including some novel studies from our laboratory. We present a few suggestions for further experimentation in this area, and expand upon previous models for axonal NF dynamics. Finally, we address how C-terminal phosphorylation is regionally and temporally regulated by a balance of kinase and phosphatase activities, and how misregulation of this balance can contribute to motor neuron disease.

Effect of spastic paraplegia mutations in KIF5A kinesin on transport activity

Hereditary spastic paraplegia (HSP) is a neurodegenerative disease caused by motoneuron degeneration. It is linked to at least 30 loci, among them SPG10, which causes dominant forms and originates in point mutations in the neuronal Kinesin-1 gene (KIF5A). Here, we investigate the motility of KIF5A and four HSP mutants. All mutations are single amino-acid exchanges and located in kinesin's motor or neck domain. The mutation in the neck (A361V) did not change the gliding properties in vitro, the others either reduced microtubule affinity or gliding velocity or both. In laser-trapping assays, none of the mutants moved more than a few steps along microtubules. Motility assays with mixtures of homodimeric wild-type, homodimeric mutant and heterodimeric wild-type/mutant motors revealed that only one mutant (N256S) reduces the gliding velocity at ratios present in heterozygous patients, whereas the others (K253N, R280C) do not. Attached to quantum dots as artificial cargo, mixtures involving N256S mutants produced slower cargo populations lagging behind in transport, whereas mixtures with the other mutants led to populations of quantum dots that rarely bound to microtubules. These differences indicate that the dominant inheritance of SPG10 is caused by two different mechanisms that both reduce the gross cargo flux, leading to deficient supply of the synapse.

Role of phosphorylation on the structural dynamics and function of types III and IV intermediate filaments

Phosphorylation of types III and IV intermediate filaments (IFs) is known to regulate their organization and function. Phosphorylation of the amino-terminal head domain sites on types III and IV IF proteins plays a key role in the assembly/disassembly of IF subunits into 10 nm filaments, and influences the phosphorylation of sites on the carboxyl-terminal tail domain. These phosphorylation events are largely under the control of second messenger-dependent protein kinases and provide the cells a mechanism to reorganize the IFs in response to the changes in second messenger levels. In mitotic cells, Cdk1, Rho kinase, PAK1 and Aurora-B kinase are believed to regulate vimentin and glial fibrillary acidic protein phosphorylation in a spatio-temporal manner. In neurons, the carboxyl-terminal tail domains of the NF-M and NF-H subunits of heteropolymeric neurofilaments (NFs) are highly phosphorylated by proline-directed protein kinases. The phosphorylation of carboxyl-terminal tail domains of NFs has been suspected to play roles in forming cross-bridges between NFs and microtubules, slowing axonal transport and promoting their integration into cytoskeleton lattice and, in doing so, to control axonal caliber and stabilize the axon. The role of IF phosphorylation in disease pathobiology is discussed.

Traffic jams: dynamic models for neurofilament accumulation in motor neuron disease

The dynamics of axonal transport are often colloquially described using highway traffic as a model system. Examination of the physics of traffic patterns, with emphasis on traffic jams and accidents, provides unique and perhaps counterintuitive insight into the aberrant accumulation of neurofilaments that accompanies amyotrophic lateral sclerosis/motor neuron disease.

Neurofilaments switch between distinct mobile and stationary states during their transport along axons

We have developed a novel pulse-escape fluorescence photoactivation technique to investigate the long-term pausing behavior of axonal neurofilaments. Cultured sympathetic neurons expressing a photoactivatable green fluorescent neurofilament fusion protein were illuminated with violet light in a short segment of axon to create a pulse of fluorescent neurofilaments. Neurofilaments departed from the photoactivated regions at rapid velocities, but the overall loss of fluorescence was slow because many of the neurofilaments paused for long periods of time before moving. The frequency of neurofilament departure was more rapid initially and slower at later times, resulting in biphasic decay kinetics. By computational simulation of the kinetics, we show that the neurofilaments switched between two distinct states: a mobile state characterized by intermittent movements and short pauses (average = 30 s) and a stationary state characterized by remarkably long pauses (average = 60 min). On average, the neurofilaments spent 92% of their time in the stationary state. Combining short and long pauses, they paused for 97% of the time, resulting in an average transport rate of 0.5 mm/d. We speculate that the relative proportion of the time that neurofilaments spend in the stationary state may be a principal determinant of their transport rate and distribution along axons, and a potential target of mechanisms that lead to abnormal neurofilament accumulations in disease.