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

Coordination, cooperation, competition, crowding and congestion of molecular motors: Theoretical models and computer simulations

Cytoskeletal motor proteins are biological nanomachines that convert chemical energy into mechanical work to carry out various functions such as cell division, cell motility, cargo transport, muscle contraction, beating of cilia and flagella, and ciliogenesis. Most of these processes are driven by the collective operation of several motors in the crowded viscous intracellular environment. Imaging and manipulation of the motors with powerful experimental probes have been complemented by mathematical analysis and computer simulations of the corresponding theoretical models. In this article, we illustrate some of the key theoretical approaches used to understand how coordination, cooperation and competition of multiple motors in the crowded intra-cellular environment drive the processes that are essential for biological function of a cell. In spite of the focus on theory, experimentalists will also find this article as an useful summary of the progress made so far in understanding multiple motor systems.

Single-particle fluorescence tracking combined with TrackMate assay reveals highly heterogeneous and discontinuous lysosomal transport in freely orientated axons

Axonal transport plays a significant role in the establishment of neuronal polarity, axon growth, and synapse formation during neuronal development. The axon of a naturally growing neuron is a highly complex and multifurcated structure with a large number of bends and branches. Nowadays, the study of dynamic axonal transport in morphologically complex neurons is greatly limited by the technological barrier. Here, a sparse gene transfection strategy was developed to locate fluorescent mCherry in the lysosome of primary neurons, thus enabling us to track the lysosome-based axonal transport with a single-particle resolution. Thereby, several axonal transport models were observed, including the forward or backward transport model, stop-and-go model, repeated back-and-forth transport model, and cross-branch transport model. Then, the accurate single-particle velocity quantification by TrackMate revealed a highly heterogeneous and discontinuous transportation process of lysosome-based axonal transport in freely orientated axons. And, multiple physical factors, such as the axonal structure and the size of particles, were disclosed to affect the velocity of particle transporting in freely orientated axons. The combined single-particle fluorescence tracking and TrackMate assay can be served as a facile tool for evaluating axonal transport in neuronal development and axonal transport-related diseases.

High-throughput quantitative analysis of axonal transport in cultured neurons from SOD1H46R ALS mice by using a microfluidic device

Amyotrophic lateral sclerosis (ALS) is a neurodegenerative disease characterized by selective loss of motor neurons. We have previously shown that autophagosome-like vesicular structures are progressively accumulated in the spinal axons of an ALS mouse model, overexpressing human Cu/Zn superoxide dismutase (SOD1) mutant, prior to the onset of motor symptoms. This suggests that axonal transport perturbation can be an early sign of neuronal dysfunction. However, the exact causal relationship between axonal transport deficits and neurodegeneration is not fully understood. To clarify whether axonal transport of organelles even in neurons at early developmental stages was affected by overexpression of mutant SOD1, we conducted a microfluidic device-based high-throughput quantitative analysis of the axonal transport of acidic vesicles and mitochondria in primary cultured cortical neurons established from SOD1^H46R transgenic mice. Compared to wild-type (WT), a significantly increased number of motile acidic vesicles, i.e., autophagosomes and/or late-endosomes, was observed in the axons of SOD1^H46R neurons. By contrast, mitochondria moving along the axons were significantly decreased in SOD1^H46R compared to WT. Since such phenotypes, where the axonal transport of these organelles is differently affected by mutant SOD1 expression, emerge before axonal degeneration, axonal transport deficits could dysregulate axon homeostasis, thereby ultimately accelerating neurodegeneration.

CaV β controls the endocytic turnover of CaV 1.2 L-type calcium channel

Membrane depolarization activates the multisubunit Ca_V 1.2 L-type calcium channel initiating various excitation coupling responses. Intracellular trafficking into and out of the plasma membrane regulates the channel's surface expression and stability, and thus, the strength of Ca_V 1.2-mediated Ca²⁺ signals. The mechanisms regulating the residency time of the channel at the cell membrane are unclear. Here, we coexpressed the channel core complex Ca_V 1.2α₁ pore-forming and auxiliary Ca_V β subunits and analyzed their trafficking dynamics from single-particle-tracking trajectories. Speed histograms obtained for each subunit were best fitted to a sum of diffusive and directed motion terms. The same mean speed for the highest-mobility state underlying directed motion was found for all subunits. The frequency of this component increased by covalent linkage of Ca_V β to Ca_V 1.2α₁ suggesting that high-speed transport occurs in association with Ca_V β. Selective tracking of Ca_V 1.2α₁ along the postendocytic pathway failed to show the highly mobile state, implying Ca_V β-independent retrograde transport. Retrograde speeds of Ca_V 1.2α₁ are compatible with myosin VI-mediated backward transport. Moreover, residency time at the cell surface was significantly prolonged when Ca_V 1.2α₁ was covalently linked to Ca_V β. Thus, Ca_V β promotes fast transport speed along anterograde trafficking and acts as a molecular switch controlling the endocytic turnover of L-type calcium channels.

MR Imaging of Peripheral Nerves Using Targeted Application of Contrast Agents: An Experimental Proof-of-Concept Study

Introduction: Current imaging modalities for peripheral nerves display the nerve's structure but not its function. Based on a nerve's capacity for axonal transport, it may be visualized by targeted application of a contrast agent and assessing the distribution through radiological imaging, thus revealing a nerve's continuity. This concept has not been explored, however, may potentially guide the treatment of peripheral nerve injuries. In this experimental proof-of-concept study, we tested imaging through MRI after administering gadolinium-based contrast agents which were then retrogradely transported. Methods: We synthesized MRI contrast agents consisting of paramagnetic agents and various axonal transport facilitators (HSA-DTPA-Gd, chitosan-DTPA-Gd or PLA/HSA-DTPA-Gd). First, we measured their relaxivity values in vitro to assess their radiological suitability. Subsequently, the sciatic nerve of 24 rats was cut and labeled with one of the contrast agents to achieve retrograde distribution along the nerve. One week after surgery, the spinal cords and sciatic nerves were harvested to visualize the distribution of the respective contrast agent using 7T MRI. In vivo MRI measurements were performed using 9.4 T MRI on the 1st, 3rd, and the 7th day after surgery. Following radiological imaging, the concentration of gadolinium in the harvested samples was analyzed using inductively coupled mass spectrometry (ICP-MS). Results: All contrast agents demonstrated high relaxivity values, varying between 12.1 and 116.0 mM^-1s^-1. HSA-DTPA-Gd and PLA/HSA-DTPA-Gd application resulted in signal enhancement in the vertebral canal and in the sciatic nerve in ex vivo MRI. In vivo measurements revealed significant signal enhancement in the sciatic nerve on the 3rd and 7th day after HSA-DTPA-Gd and chitosan-DTPA-Gd (p < 0.05) application. Chemical evaluation showed high gadolinium concentration in the sciatic nerve for HSA-DTPA-Gd (5.218 ± 0.860 ng/mg) and chitosan-DTPA-Gd (4.291 ± 1.290 ng/mg). Discussion: In this study a novel imaging approach for the evaluation of a peripheral nerve's integrity was implemented. The findings provide radiological and chemical evidence of successful contrast agent uptake along the sciatic nerve and its distribution within the spinal canal in rats. This novel concept may assist in the diagnostic process of peripheral nerve injuries in the future.

Modeling tau transport in the axon initial segment

By assuming that tau protein can be in seven kinetic states, we developed a model of tau protein transport in the axon and in the axon initial segment (AIS). Two separate sets of kinetic constants were determined, one in the axon and the other in the AIS. This was done by fitting the model predictions in the axon with experimental results and by fitting the model predictions in the AIS with the assumed linear increase of the total tau concentration in the AIS. The calibrated model was used to make predictions about tau transport in the axon and in the AIS. To the best of our knowledge, this is the first paper that presents a mathematical model of tau transport in the AIS. Our modeling results suggest that binding of free tau to microtubules creates a negative gradient of free tau in the AIS. This leads to diffusion-driven tau transport from the soma into the AIS. The model further suggests that slow axonal transport and diffusion-driven transport of tau work together in the AIS, moving tau anterogradely. Our numerical results predict an interplay between these two mechanisms: as the distance from the soma increases, the diffusion-driven transport decreases, while motor-driven transport becomes larger. Thus, the machinery in the AIS works as a pump, moving tau into the axon.

Bidirectional sliding of two parallel microtubules generated by multiple identical motors

It is often assumed in biophysical studies that when multiple identical molecular motors interact with two parallel microtubules, the microtubules will be crosslinked and locked together. The aim of this study is to examine this assumption mathematically. We model the forces and movements generated by motors with a time-continuous Markov process and find that, counter-intuitively, a tug-of-war results from opposing actions of identical motors bound to different microtubules. The model shows that many motors bound to the same microtubule generate a great force applied to a smaller number of motors bound to another microtubule, which increases detachment rate for the motors in minority, stabilizing the directional sliding. However, stochastic effects cause occasional changes of the sliding direction, which has a profound effect on the character of the long-term microtubule motility, making it effectively diffusion-like. Here, we estimate the time between the rare events of switching direction and use them to estimate the effective diffusion coefficient for the microtubule pair. Our main result is that parallel microtubules interacting with multiple identical motors are not locked together, but rather slide bidirectionally. We find explicit formulae for the time between directional switching for various motor numbers.

Investigating sensitivity coefficients characterizing the response of a model of tau protein transport in an axon to model parameters

Evaluating the sensitivity of biological models to various model parameters is a critical step towards advancing our understanding of biological systems. In this paper, we investigated sensitivity coefficients for a model simulating transport of tau protein along the axon. This is an important problem due to the relevance of tau transport and agglomeration to Alzheimer's disease and other tauopathies, such as some forms of parkinsonism. The sensitivity coefficients that we obtained characterize how strongly three observables (the tau concentration, average tau velocity, and the percentage of tau bound to microtubules) depend on model parameters. The fact that the observables strongly depend on a parameter characterizing tau transition from the retrograde to the anterograde kinetic states suggests the importance of motor-driven transport of tau. The observables are sensitive to kinetic constants characterizing tau concentration in the free (cytosolic) state only at small distances from the soma. Cytosolic tau can only be transported by diffusion, suggesting that diffusion-driven transport of tau only plays a role in the proximal axon. Our analysis also shows the location in the axon in which an observable has the greatest sensitivity to a certain parameter. For most parameters, this location is in the proximal axon. This could be useful for designing an experiment aimed at determining the value of this parameter. We also analyzed sensitivity of the average tau velocity, the total tau concentration, and the percentage of microtubule-bound tau to cytosolic diffusivity of tau and diffusivity of bound tau along the MT lattice. The model predicts that at small distances from the soma the effect of these two diffusion processes is comparable.

Simulating the effect of formation of amyloid plaques on aggregation of tau protein

In this paper, we develop a mathematical model that enables the investigation of the production and intracellular transport of amyloid precursor protein (APP) and tau protein in a neuron. We also investigate the aggregation of APP fragments into amyloid-β (Aβ) as well as tau aggregation into tau oligomers and neurofibrillary tangles. Using the developed model, we investigate how Aβ aggregation can influence tau transport and aggregation in both the soma and the axon. We couple the Aβ and tau agglomeration processes by assuming that the value of the kinetic constant that describes the autocatalytic growth (self-replication) reaction step of tau aggregation is proportional to the Aβ concentration. The model predicts that APP and tau are distributed differently in the axon. While APP has a uniform distribution along the axon, tau's concentration first decreases and then increases towards the synapse. Aβ is uniformly produced along the axon while misfolded tau protein is mostly produced in the proximal axon. The number of Aβ and tau polymers originating from the axon is much smaller than the number of Aβ and tau polymers originating from the soma. The rate of production of misfolded tau polymers depends on how strongly their production is facilitated by Aβ.

Slow and fast grouping of cargo velocities in axonal transport due to single versus multi-motor transport

We have recently been exploring the idea that axonal transport velocity is "track and motor limited." That is, microtubule length as well as microtubule-associated obstructions interact with the number of motors attached to a specific cargo to determine average cargo velocities. We assert that "slow" and "fast" transport as they are commonly referred to in the literature are really single- versus multi-motor transport along interrupted and obstructed track. To this end, we have recently developed a cargo-level motor model that appears to readily reproduce fast and slow transport simply by altering the number of motors. In the work presented here, we explore the ramifications of this model across a wide range of cargo sizes and motor-motor interactions. We find that categorization of cargo transport into "slow" and "fast" might be a natural consequence of track and motor limited transport as cargo load versus average velocity distribution produced by this model are clearly bi-modal with a curved (roughly square-root) relationship between number of motors and cargo load being the best at reproducing experimental data.

Rotation of endosomes demonstrates coordination of molecular motors during axonal transport

Long-distance axonal transport is critical to the maintenance and function of neurons. Robust transport is ensured by the coordinated activities of multiple molecular motors acting in a team. Conventional live-cell imaging techniques used in axonal transport studies detect this activity by visualizing the translational dynamics of a cargo. However, translational measurements are insensitive to torques induced by motor activities. By using gold nanorods and multichannel polarization microscopy, we simultaneously measure the rotational and translational dynamics for thousands of axonally transported endosomes. We find that the rotational dynamics of an endosome provide complementary information regarding molecular motor activities to the conventionally tracked translational dynamics. Rotational dynamics correlate with translational dynamics, particularly in cases of increased rotation after switches between kinesin- and dynein-mediated transport. Furthermore, unambiguous measurement of nanorod angle shows that endosome-contained nanorods align with the orientation of microtubules, suggesting a direct mechanical linkage between the ligand-receptor complex and the microtubule motors.

How the formation of amyloid plaques and neurofibrillary tangles may be related: a mathematical modelling study

We develop a mathematical model that enables us to investigate possible mechanisms by which two primary markers of Alzheimer's disease (AD), extracellular amyloid plaques and intracellular tangles, may be related. Our model investigates the possibility that the decay of anterograde axonal transport of amyloid precursor protein (APP), caused by toxic tau aggregates, leads to decreased APP transport towards the synapse and APP accumulation in the soma. The developed model thus couples three processes: (i) slow axonal transport of tau, (ii) tau misfolding and agglomeration, which we simulated by using the Finke-Watzky model and (iii) fast axonal transport of APP. Because the timescale for tau agglomeration is much larger than that for tau transport, we suggest using the quasi-steady-state approximation for formulating and solving the governing equations for these three processes. Our results suggest that misfolded tau most likely accumulates in the beginning of the axon. The analysis of APP transport suggests that APP will also likely accumulate in the beginning of the axon, causing an increased APP concentration in this region, which could be interpreted as a 'traffic jam'. The APP flux towards the synapse is significantly reduced by tau misfolding, but not due to the APP traffic jam, which can be viewed as a symptom, but rather due to the reduced affinity of kinesin-1 motors to APP-transporting vesicles.

Simulating tubulin-associated unit transport in an axon: using bootstrapping for estimating confidence intervals of best-fit parameter values obtained from indirect experimental data

In this paper, we first develop a model of axonal transport of tubulin-associated unit (tau) protein. We determine the minimum number of parameters necessary to reproduce published experimental results, reducing the number of parameters from 18 in the full model to eight in the simplified model. We then address the following questions: Is it possible to estimate parameter values for this model using the very limited amount of published experimental data? Furthermore, is it possible to estimate confidence intervals for the determined parameters? The idea that is explored in this paper is based on using bootstrapping. Model parameters were estimated by minimizing the objective function that simulates the discrepancy between the model predictions and experimental data. Residuals were then identified by calculating the differences between the experimental data and model predictions. New, surrogate 'experimental' data were generated by randomly resampling residuals. By finding sets of best-fit parameters for a large number of surrogate data the histograms for the model parameters were produced. These histograms were then used to estimate confidence intervals for the model parameters, by using the percentile bootstrap. Once the model was calibrated, we applied it to analysing some features of tau transport that are not accessible to current experimental techniques.

Utilization of the bootstrap method for determining confidence intervals of parameters for a model of MAP1B protein transport in axons

This paper develops a model of axonal transport of MAP1B protein. The problem of determining parameter values for the proposed model utilizing limited available experimental data is addressed. We used a global minimum search algorithm to find parameter values that minimize the discrepancy between model predictions and published experimental results. By analyzing the best fit parameter values it was established that some processes can be dropped from the model without losing accuracy, thus a simplified version of the model was formulated. In particular, our analysis suggests that reversals in MAP1B transport are infrequent and can be neglected. The detachment of anterogradely-biased MAP1B from microtubules (MTs) and the attachment of retrogradely-biased MAP1B to MTs can also be neglected. An analytical solution for the simplified model was obtained. Confidence intervals for the determined parameters were found by bootstrapping model residuals. The resultant analysis heavily constrained the values of some parameters while showing that some could vary without significantly impacting model error. For example, our analysis suggests that, above a certain threshold value, the kinetic constant determining the rate of MAP1B transition from the retrograde pausing state to the off-track state has little impact on model error. On the contrary, the kinetic constant describing MAP1B transition from a pausing to a running state has great impact on model error.

Seeking homeostasis: temporal trends in respiration, oxidation, and calcium in SOD1 G93A Amyotrophic Lateral Sclerosis mice

Impairments in mitochondria, oxidative regulation, and calcium homeostasis have been well documented in numerous Amyotrophic Lateral Sclerosis (ALS) experimental models, especially in the superoxide dismutase 1 glycine 93 to alanine (SOD1 G93A) transgenic mouse. However, the timing of these deficiencies has been debatable. In a systematic review of 45 articles, we examine experimental measurements of cellular respiration, mitochondrial mechanisms, oxidative markers, and calcium regulation. We evaluate the quantitative magnitude and statistical temporal trend of these aggregated assessments in high transgene copy SOD1 G93A mice compared to wild type mice. Analysis of overall trends reveals cellular respiration, intracellular adenosine triphosphate, and corresponding mitochondrial elements (Cox, cytochrome c, complex I, enzyme activity) are depressed for the entire lifespan of the SOD1 G93A mouse. Oxidant markers (H2O2, 8OH2'dG, MDA) are initially similar to wild type but are double that of wild type by the time of symptom onset despite early post-natal elevation of protective heat shock proteins. All aspects of calcium regulation show early disturbances, although a notable and likely compensatory convergence to near wild type levels appears to occur between 40 and 80 days (pre-onset), followed by a post-onset elevation in intracellular calcium. The identified temporal trends and compensatory fluctuations provide evidence that the "cause" of ALS may lay within failed homeostatic regulation, itself, rather than any one particular perturbing event or cellular mechanism. We discuss the vulnerabilities of motoneurons to regulatory instability and possible hypotheses regarding failed regulation and its potential treatment in ALS.

State of the field: An informatics-based systematic review of the SOD1-G93A amyotrophic lateral sclerosis transgenic mouse model

Numerous sub-cellular through system-level disturbances have been identified in over 1300 articles examining the superoxide dismutase-1 guanine 93 to alanine (SOD1-G93A) transgenic mouse amyotrophic lateral sclerosis (ALS) pathophysiology. Manual assessment of such a broad literature base is daunting. We performed a comprehensive informatics-based systematic review or 'field analysis' to agnostically compute and map the current state of the field. Text mining of recaptured articles was used to quantify published data topic breadth and frequency. We constructed a nine-category pathophysiological function-based ontology to systematically organize and quantify the field's primary data. Results demonstrated that the distribution of primary research belonging to each category is: systemic measures an motor function, 59%; inflammation, 46%; cellular energetics, 37%; proteomics, 31%; neural excitability, 22%; apoptosis, 20%; oxidative stress, 18%; aberrant cellular chemistry, 14%; axonal transport, 10%. We constructed a SOD1-G93A field map that visually illustrates and categorizes the 85% most frequently assessed sub-topics. Finally, we present the literature-cited significance of frequently published terms and uncover thinly investigated areas. In conclusion, most articles individually examine at least two categories, which is indicative of the numerous underlying pathophysiological interrelationships. An essential future path is examination of cross-category pathophysiological interrelationships and their co-correspondence to homeostatic regulation and disease progression.