Mathematical modeling and parameter estimation of axonal cargo transport

Why slow axonal transport is bidirectional - can axonal transport of tau protein rely only on motor-driven anterograde transport?

Slow axonal transport (SAT) moves multiple proteins from the soma, where they are synthesized, to the axon terminal. Due to the great lengths of axons, SAT almost exclusively relies on active transport, which is driven by molecular motors. The puzzling feature of slow axonal transport is its bidirectionality. Although the net direction of SAT is anterograde, from the soma to the terminal, experiments show that it also contains a retrograde component. One of the proteins transported by SAT is the microtubule-associated protein tau. To better understand why the retrograde component in tau transport is needed, we used the perturbation technique to analyze how the full tau SAT model can be simplified for the specific case when retrograde motor-driven transport and diffusion-driven transport of tau are negligible and tau is driven only by anterograde (kinesin) motors. The solution of the simplified equations shows that without retrograde transport the tau concentration along the axon length stays almost uniform (decreases very slightly), which is inconsistent with the experimenal tau concentration at the outlet boundary (at the axon tip). Thus kinesin-driven transport alone is not enough to explain the empirically observed distribution of tau, and the retrograde motor-driven component in SAT is needed.

Mathematical models of neuronal growth

The establishment of a functioning neuronal network is a crucial step in neural development. During this process, neurons extend neurites-axons and dendrites-to meet other neurons and interconnect. Therefore, these neurites need to migrate, grow, branch and find the correct path to their target by processing sensory cues from their environment. These processes rely on many coupled biophysical effects including elasticity, viscosity, growth, active forces, chemical signaling, adhesion and cellular transport. Mathematical models offer a direct way to test hypotheses and understand the underlying mechanisms responsible for neuron development. Here, we critically review the main models of neurite growth and morphogenesis from a mathematical viewpoint. We present different models for growth, guidance and morphogenesis, with a particular emphasis on mechanics and mechanisms, and on simple mathematical models that can be partially treated analytically.

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.

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.

Bidirectional actin transport is influenced by microtubule and actin stability

Local and long-distance transport of cytoskeletal proteins is vital to neuronal maintenance and growth. Though recent progress has provided insight into the movement of microtubules and neurofilaments, mechanisms underlying the movement of actin remain elusive, in large part due to rapid transitions between its filament states and its diverse cellular localization and function. In this work, we integrated live imaging of rat sensory neurons, image processing, multiple regression analysis, and mathematical modeling to perform the first quantitative, high-resolution investigation of GFP-actin identity and movement in individual axons. Our data revealed that filamentous actin densities arise along the length of the axon and move short but significant distances bidirectionally, with a net anterograde bias. We directly tested the role of actin and microtubules in this movement. We also confirmed a role for actin densities in extension of axonal filopodia, and demonstrated intermittent correlation of actin and mitochondrial movement. Our results support a novel mechanism underlying slow component axonal transport, in which the stability of both microtubule and actin cytoskeletal components influence the mobility of filamentous actin.

Ribosomal trafficking is reduced in Schwann cells following induction of myelination

Local synthesis of proteins within the Schwann cell periphery is extremely important for efficient process extension and myelination, when cells undergo dramatic changes in polarity and geometry. Still, it is unclear how ribosomal distributions are developed and maintained within Schwann cell projections to sustain local translation. In this multi-disciplinary study, we expressed a plasmid encoding a fluorescently labeled ribosomal subunit (L4-GFP) in cultured primary rat Schwann cells. This enabled the generation of high-resolution, quantitative data on ribosomal distributions and trafficking dynamics within Schwann cells during early stages of myelination, induced by ascorbic acid treatment. Ribosomes were distributed throughout Schwann cell projections, with ~2-3 bright clusters along each projection. Clusters emerged within 1 day of culture and were maintained throughout early stages of myelination. Three days after induction of myelination, net ribosomal movement remained anterograde (directed away from the Schwann cell body), but ribosomal velocity decreased to about half the levels of the untreated group. Statistical and modeling analysis provided additional insight into key factors underlying ribosomal trafficking. Multiple regression analysis indicated that net transport at early time points was dependent on anterograde velocity, but shifted to dependence on anterograde duration at later time points. A simple, data-driven rate kinetics model suggested that the observed decrease in net ribosomal movement was primarily dictated by an increased conversion of anterograde particles to stationary particles, rather than changes in other directional parameters. These results reveal the strength of a combined experimental and theoretical approach in examining protein localization and transport, and provide evidence of an early establishment of ribosomal populations within Schwann cell projections with a reduction in trafficking following initiation of myelination.

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.

A one-dimensional moving-boundary model for tubulin-driven axonal growth

A one-dimensional continuum-mechanical model of axonal elongation due to assembly of tubulin dimers in the growth cone is presented. The conservation of mass leads to a coupled system of three differential equations. A partial differential equation models the dynamic and the spatial behaviour of the concentration of tubulin that is transported along the axon from the soma to the growth cone. Two ordinary differential equations describe the time-variation of the concentration of free tubulin in the growth cone and the speed of elongation. All steady-state solutions of the model are categorized. Given a set of the biological parameter values, it is shown how one easily can infer whether there exist zero, one or two steady-state solutions and directly determine the possible steady-state lengths of the axon. Explicit expressions are given for each stationary concentration distribution. It is thereby easy to examine the influence of each biological parameter on a steady state. Numerical simulations indicate that when there exist two steady states, the one with shorter axon length is unstable and the longer is stable. Another result is that, for nominal parameter values extracted from the literature, in a large portion of a fully grown axon the concentration of free tubulin is lower than both concentrations in the soma and in the growth cone.

From the cell membrane to the nucleus: unearthing transport mechanisms for dynein

Mutations in the motor protein cytoplasmic dynein have been found to cause Charcot-Marie-Tooth disease, spinal muscular atrophy, and severe intellectual disabilities in humans. In mouse models, neurodegeneration is observed. We sought to develop a novel model which could incorporate the effects of mutations on distance travelled and velocity. A mechanical model for the dynein mediated transport of endosomes is derived from first principles and solved numerically. The effects of variations in model parameter values are analysed to find those that have a significant impact on velocity and distance travelled. The model successfully describes the processivity of dynein and matches qualitatively the velocity profiles observed in experiments.

An integro-partial differential equation for modeling biofluids flow in fractured biomaterials

A novel mathematical model in the framework of a nonlinear integro-partial differential equation governing biofluids flow in fractured biomaterials is proposed, solved, verified, and evaluated. A semi-analytical solution is derived for the equation, verified by a mass-lumped Galerkin finite element method (FEM), and calibrated with two in vitro experimental datasets. The solution process uses separation of variables and results in explicit expression involving complete and incomplete beta functions. The proposed semi-analytical model shows reasonable agreements with the finite element simulator as well as with two in vitro experimental time series and can be successfully used to simulate biofluids (e.g. water, blood, oil, etc.) flow in natural and synthetic porous biomaterials.

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.