Efficient simulations of tubulin-driven axonal growth

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman--Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g.\ its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.Comment: Authors' accepted version, (post refereeing). The final publication (in Journal of Computational Neuroscience) is available at Springer via http://dx.doi.org/10.1007/s10827-016-0604-

Intracellular transport driven by cytoskeletal motors: General mechanisms and defects

Cells are strongly out-of-equilibrium systems driven by continuous energy supply. They carry out many vital functions requiring active transport of various ingredients and organelles, some being small, others being large. The cytoskeleton, composed of three types of filaments, determines the shape of the cell and plays a role in cell motion. It also serves as a road network for the so-called cytoskeletal motors. These molecules can attach to a cytoskeletal filament, perform directed motion, possibly carrying along some cargo, and then detach. It is a central issue to understand how intracellular transport driven by molecular motors is regulated, in particular because its breakdown is one of the signatures of some neuronal diseases like the Alzheimer. We give a survey of the current knowledge on microtubule based intracellular transport. We first review some biological facts obtained from experiments, and present some modeling attempts based on cellular automata. We start with background knowledge on the original and variants of the TASEP (Totally Asymmetric Simple Exclusion Process), before turning to more application oriented models. After addressing microtubule based transport in general, with a focus on in vitro experiments, and on cooperative effects in the transportation of large cargos by multiple motors, we concentrate on axonal transport, because of its relevance for neuronal diseases. It is a challenge to understand how this transport is organized, given that it takes place in a confined environment and that several types of motors moving in opposite directions are involved. We review several features that could contribute to the efficiency of this transport, including the role of motor-motor interactions and of the dynamics of the underlying microtubule network. Finally, we discuss some still open questions.Comment: 74 pages, 43 figure

Molecular and Cellular Approaches Toward Understanding Dynein-Driven Motility

Active transport is integral to organelle localization and their distribution within the cell. Kinesins, myosins and dynein are the molecular motors that drive this long range transport on the actin and microtubule cytoskeleton. Although several families of kinesins and myosins have evolved, there is only one form of cytoplasmic dynein driving active retrograde transport in cells. While dynactin is an essential co-factor for most cellular functions of dynein, the mechanistic basis for this evolutionarily well conserved interaction remains unclear. Here, I use single molecule approaches with purified dynein to reconstitute processes in vitro, and implement an optogenetic tool in neurons to further dissect regulatory mechanisms of dynein-driven transport in cells. I demonstrate for the first time, at the single molecule level, that dynactin functions as a tether to enhance the initial recruitment of dynein onto microtubules but also acts as a brake to slow the motor. I then extend this work in neurons to understand regulation of the dynein motor at the cellular level. Neurons are particulary dependent on long-range transport as organelles and macromolecules must be efficiently moved over the extended length of the axon and further, have mechanisms in place for the compartment-specific regulation of trafficking in axons and dendrites. I use a light-inducible dimerization tool to recruit motor proteins or motor adaptors to organelles in real time to examine downstream effects of organelle motility and compartment-specific regulation of motors. I find that while dynein works efficiently in both axons and dendrites, kinesins are differentially regulated in a compartment-specific manner. I further demonstrate that dynein-driven motility in neurons is largely governed by microtubule orientation and requires microtubule dynamics for efficient navigation in axons and dendrites. Together, this work sheds light on the molecular and cellular mechanisms of dynein function both in vitro and in vivo using a combination of approaches. My findings converge to a model wherein dynactin enhances the recruitment of dynein onto microtubule plus ends, leading to efficient minus-end directed motility of dynein. This becomes especially critical in neuronal growth cones and dendrites owing to the large number of highly dynamic microtubules in these compartments

Spatial and Temporal Sensing Limits of Microtubule Polarization in Neuronal Growth Cones by Intracellular Gradients and Forces

Neuronal growth cones are the most sensitive amongst eukaryotic cells in responding to directional chemical cues. Although a dynamic microtubule cytoskeleton has been shown to be essential for growth cone turning, the precise nature of coupling of the spatial cue with microtubule polarization is less understood. Here we present a computational model of microtubule polarization in a turning neuronal growth cone (GC). We explore the limits of directional cues in modifying the spatial polarization of microtubules by testing the role of microtubule dynamics, gradients of regulators and retrograde forces along filopodia. We analyze the steady state and transition behavior of microtubules on being presented with a directional stimulus. The model makes novel predictions about the minimal angular spread of the chemical signal at the growth cone and the fastest polarization times. A regulatory reaction-diffusion network based on the cyclic phosphorylation-dephosphorylation of a regulator predicts that the receptor signal magnitude can generate the maximal polarization of microtubules and not feedback loops or amplifications in the network. Using both the phenomenological and network models we have demonstrated some of the physical limits within which the MT polarization system works in turning neuron.Comment: 7 figures and supplementary materia

Mathematical modelling and numerical simulation of the morphological development of neurons

BACKGROUND: The morphological development of neurons is a very complex process involving both genetic and environmental components. Mathematical modelling and numerical simulation are valuable tools in helping us unravel particular aspects of how individual neurons grow their characteristic morphologies and eventually form appropriate networks with each other. METHODS: A variety of mathematical models that consider (1) neurite initiation (2) neurite elongation (3) axon pathfinding, and (4) neurite branching and dendritic shape formation are reviewed. The different mathematical techniques employed are also described. RESULTS: Some comparison of modelling results with experimental data is made. A critique of different modelling techniques is given, leading to a proposal for a unified modelling environment for models of neuronal development. CONCLUSION: A unified mathematical and numerical simulation framework should lead to an expansion of work on models of neuronal development, as has occurred with compartmental models of neuronal electrical activity

The Role Of Kinesin-2 In Navigating Microtubule Obstacles: Implications For The Regulation Of Axonal Transport

Neurons are specialized cells that transmit information through electrical and chemical signals using structural processes known as dendrites and axons. Dendrites receive information for the cell to interpret while the exceedingly long axon transmits the processed information to its target destination. To ensure the neuron properly carries out its extracellular functions, the orchestration of intracellular cargo (e.g. mitochondria) is critical. This is especially true in the axon, which can be up to a meter in length. There are many challenges involved in the spatial and temporal regulation of cargo over such vast cellular distances. In order to accomplish cargo transport between the cell body and axon terminus the neuron has developed an efficient process to overcome this challenge called axonal transport. Axonal transport utilizes a system of molecular motors coupled to cargo, creating a multi-motor complex, which walks along a set of tracks to position the cargo at the right time and place. One class of molecular motors, called kinesin, are used to traffic cargo away from the cell body and walk along microtubule tracks. These motors work in teams to navigate a complex microtubule landscape that is rich in microtubule-associated proteins (MAPs). One MAP abundantly found within the axon is called Tau and is implicated in a variety of neurodegenerative disorders (e.g. Alzheimer\u27s disease). Much attention has been focused on the kinesin-1 motor while investigating the axonal transport process. However, kinesin-2 plays an equally important role and is not as well characterized as kinesin-1. Previously, it has been demonstrated, in vitro, that Tau disrupts kinesin-1 transport, even below physiological concentrations, however, in vivo evidence suggests the contrary. Given this discrepancy, there are likely other cellular systems in place to provide the necessary navigation of Tau obstacles. One solution may involve multi-motor complexes using two kinesin family members attached to cargo, as both kinesin-1 and kinesin-2 have been observed coupled to cargo. In order to peel away the complex layers of kinesin-1 and kinesin-2 coupled cargo inside the axon, single-molecule imaging techniques were employed to observe the individual behavior of both kinesin-1 and kinesin-2, in vitro. Further, using a combination of genetic engineering, single-molecule analysis and mathematical modeling has helped elucidate differences between these two motors. Kinesin-2 was found to be insensitive to Tau obstacles, unlike kinesin-1, which is in part due to a longer region of the motor called the neck-linker. This region connects the motor domain, which interfaces with the microtubule track, to the coiled-coil stock, which interfaces with the cargo. When the neck-linker lengths were swapped between the motors their behavior in the presence of Tau also switched, and kinesin-2 became sensitive to Tau. To understand kinesin-2\u27s mechanism of navigating Tau obstacles, we looked at the lateral stepping characteristics of both motors. We observed kinesin-2\u27s lateral stepping frequency to be 2-4 fold higher than kinesin-1 and independent of the microtubule obstacle concentration. Thus, kinein-2\u27s longer neck-linker allows a more agile walk along the microtubule surface to navigate obstacles more efficiently than kinesin-1. In a multi-motor complex containing both motors, kinesin-2 is more efficient at maneuvering around MAPs while kinesin-1, which has previously been demonstrated to sustain a higher stall force, is more efficient at towing cargo. This work demonstrates how teams of directionally similar motors may work together to position cargo during axonal transport

Stochastic models of intracellular transport

The interior of a living cell is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an over-damped Brownian particle. On the other hand, active transport requires chemical energy, usually in the form of ATP hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review we present a wide range of analytical methods and models of intracellular transport. In the case of diffusive transport, we consider narrow escape problems, diffusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion. In the case of active transport, we consider Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasi-steady-state reduction methods, and mean field approximations. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, protein-DNA interactions, virus trafficking, and the self–organization of subcellular structures

Analyses and Applications of the Peaceman--Rachford and Douglas--Rachford Splitting Schemes

Splitting methods are widely used as temporal discretizations of evolution equations. Such methods usually constitute competitive choices whenever a vector field can be split into a sum of two or more parts that each generates a flow easier to compute or approximate than the flow of the sum. In the research presented in this Licentiate thesis we consider dissipative evolution equations with vector fields given by unbounded operators. Dynamical systems that fit into this framework can for example be found among Hamiltonian systems and parabolic and hyperbolic partial differential equations (PDEs). The goal of the presented research is to perform convergence analyses for the lternating direction implicit (ADI) methods in the setting of dissipative operators. In this context these methods are known to possess excellent stability properties. Additionally, they generate easily computable numerical flows and are ideal choices for studying convergence to stationary solutions, a property related to their favorable local error structure. In this thesis we consider the Peaceman--Rachford and Douglas--Rachford schemes, which were the first ADI methods to be constructed and to this day are the most representative members of the ADI method class. We perform convergence studies for the Peaceman--Rachford and Douglas--Rachford schemes when applied to semilinear, dissipative evolution equations, that is, when the vector fields are given by the sum of a linear and a nonlinear dissipative operator. Optimal convergence orders are proven when the solution is sufficiently regular. With less regularity present we are still able to prove convergence, however of suboptimal order or without order. In contrast to previous convergence order analyses we do not assume Lipschitz continuity of the nonlinear operator. In the context of linear, dissipative evolution equations we consider full space-time discretizations. We assume that the full discretization is given by combining one of the two aforementioned ADI methods with a general, converging spatial discretization method. In this setting we prove optimal, simultaneous space-time convergence orders. Advection-diffusion-reaction models, encountered in physics, chemistry, and biology are important examples of dissipative evolution equations. In this thesis we present such a model describing the growth of axons in nerve cells. The model consists of a parabolic PDE, which has a non-trivial coupling to nonlinear ordinary differential equations via a moving boundary, which is part of the solution. Since additionally the biological model parameters imply a wide range of scales, both in time and space, the application of a numerical method is involved. We make an argument for a discretization consisting of a splitting which is integrated by the Peaceman--Rachford scheme. The choice is motivate by the results of some numerical experiments

Doctor of Philosophy

dissertationThe efficient transport of particles throughout a cell plays a fundamental role in several cellular processes. Broadly speaking, intracellular transport can be divided into two categories: passive and active transport. Whereas passive transport generally occurs via diffusive processes, active transport requires cellular energy through adenosine triphosphate (ATP). Many active transport processes are driven by molecular motors such as kinesin and dynein, which carry cargo and travel along the microtubules of a cell to deliver specific material to specific locations. Breakdown of molecular motor delivery is correlated with the onset of several diseases, such as Alzheimer's and Parkinson's. We mathematically model two fundamental cellular processes. In the first part, we introduce a possible biophysical mechanism by which cells attain uniformity in vesicle density throughout their body. We do this by modeling bulk motor density dynamics using partial differential equations derived from microscopic descriptions of individual motor-cargo complex dynamics. We then consider the cases where delivery of cargo to cellular targets is (i) irreversible and (ii) reversible. This problem is studied on the semi-infinite interval, disk, and spherical domains. We also consider the case where exclusion effects come into play. In all cases, we find that allowing for reversibility in cargo delivery to cellular targets allows for more uniform vesicle distribution. In the second part, we see how active transport by molecular motors allows for length control and sensing in flagella and axons, respectively. For the flagellum, we model length control using a doubly stochastic Poisson model. For axons, we model bulk motor dynamics by partial differential equations, and show how spatial information may be encoded in the frequency of an oscillating chemical signal being carried by dynein motors. Furthermore, we discuss how frequency-encoded signals may be decoded by cells, and how these mechanisms break down in the face of noise