Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth

Biased Random Walk Model of Neuronal Dynamics on Substrates with Periodic Geometrical Patterns

Neuronal networks are complex systems of interconnected neurons responsible for transmitting and processing information throughout the nervous system. The building blocks of neuronal networks consist of individual neurons, specialized cells that receive, process, and transmit electrical and chemical signals throughout the body. The formation of neuronal networks in the developing nervous system is a process of fundamental importance for understanding brain activity, including perception, memory, and cognition. To form networks, neuronal cells extend long processes called axons, which navigate toward other target neurons guided by both intrinsic and extrinsic factors, including genetic programming, chemical signaling, intercellular interactions, and mechanical and geometrical cues. Despite important recent advances, the basic mechanisms underlying collective neuron behavior and the formation of functional neuronal networks are not entirely understood. In this paper, we present a combined experimental and theoretical analysis of neuronal growth on surfaces with micropatterned periodic geometrical features. We demonstrate that the extension of axons on these surfaces is described by a biased random walk model, in which the surface geometry imparts a constant drift term to the axon, and the stochastic cues produce a random walk around the average growth direction. We show that the model predicts key parameters that describe axonal dynamics: diffusion (cell motility) coefficient, average growth velocity, and axonal mean squared length, and we compare these parameters with the results of experimental measurements. Our findings indicate that neuronal growth is governed by a contact-guidance mechanism, in which the axons respond to external geometrical cues by aligning their motion along the surface micropatterns. These results have a significant impact on developing novel neural network models, as well as biomimetic substrates, to stimulate nerve regeneration and repair after injury.

Biomechanical models and mechanisms of cellular morphogenesis and cerebral cortical expansion and folding

Morphogenesis of the nervous system involves a highly complex spatio-temporal pattern of physical forces (mainly tension and pressure) acting on cells and tissues that are pliable but have an intricately organized cytoskeletal infrastructure. This review begins by covering basic principles of biomechanics and the core cytoskeletal toolkit used to regulate the shapes of cells and tissues during embryogenesis and neural development. It illustrates how the principle of 'tensegrity' provides a useful conceptual framework for understanding how cells dynamically respond to forces that are generated internally or applied externally. The latter part of the review builds on this foundation in considering the development of mammalian cerebral cortex. The main focus is on cortical expansion and folding - processes that take place over an extended period of prenatal and postnatal development. Cortical expansion and folding are likely to involve many complementary mechanisms, some related to regulating cell proliferation and migration and others related to specific types and patterns of mechanical tension and pressure. Three distinct multi-mechanism models are evaluated in relation to a set of 18 key experimental observations and findings. The Composite Tension Plus (CT+) model is introduced as an updated version of a previous multi-component Differential Expansion Sandwich Plus (DES+) model (Van Essen, 2020); the new CT+ model includes 10 distinct mechanisms and has the greatest explanatory power among published models to date. Much needs to be done in order to validate specific mechanistic components and to assess their relative importance in different species, and important directions for future research are suggested.

Stochastic partial differential equations driven by space-time fractional noises

In this paper, we study a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises. Our method consists in studying at first the non-local SPDEs and thereafter showing the convergence of the family of these equations. The limit gives the solution of the SPDE.

Cell Migration in 1D and 2D Nanofiber Microenvironments

Understanding how cells migrate in fibrous environments is important in wound healing, immune function, and cancer progression. A key question is how fiber orientation and network geometry influence cell movement. Here we describe a quantitative, modeling-based approach toward identifying the mechanisms by which cells migrate in fibrous geometries having well controlled orientation. Specifically, U251 glioblastoma cells were seeded onto non-electrospinning Spinneret based tunable engineering parameters fiber substrates that consist of networks of suspended 400 nm diameter nanofibers. Cells were classified based on the local fiber geometry and cell migration dynamics observed by light microscopy. Cells were found in three distinct geometries: adhering two a single fiber, adhering to two parallel fibers, and adhering to a network of orthogonal fibers. Cells adhering to a single fiber or two parallel fibers can only move in one dimension along the fiber axis, whereas cells on a network of orthogonal fibers can move in two dimensions. We found that cells move faster and more persistently in 1D geometries than in 2D, with cell migration being faster on parallel fibers than on single fibers. To explain these behaviors mechanistically, we simulated cell migration in the three different geometries using a motor-clutch based model for cell traction forces. Using nearly identical parameter sets for each of the three cases, we found that the simulated cells naturally replicated the reduced migration in 2D relative to 1D geometries. In addition, the modestly faster 1D migration on parallel fibers relative to single fibers was captured using a correspondingly modest increase in the number of clutches to reflect increased surface area of adhesion on parallel fibers. Overall, the integrated modeling and experimental analysis shows that cell migration in response to varying fibrous geometries can be explained by a simple mechanical readout of geometry via a motor-clutch mechanism.

Bio-Nano-Magnetic Materials for Localized Mechanochemical Stimulation of Cell Growth and Death

Magnetic nanoparticles are promising new tools for therapeutic applications, such as magnetic nanoparticle hyperthermia therapy and targeted drug delivery. Recent in vitro studies have demonstrated that a force application with magnetic tweezers can also affect cell fate, suggesting a therapeutic potential for magnetically modulated mechanical stimulation. The magnetic properties of nanoparticles that induce physical responses and the subtle responses that result from mechanically induced membrane damage and/or intracellular signaling are evaluated. Magnetic particles with various physical, geometric, and magnetic properties and specific functionalization can now be used to apply mechanical force to specific regions of cells, which permit the modulation of cellular behavior through the use of spatially and time controlled magnetic fields. On one hand, mechanochemical stimulation has been used to direct the outgrowth on neuronal growth cones, indicating a therapeutic potential for neural repair. On the other hand, it has been used to kill cancer cells that preferentially express specific receptors. Advances made in the synthesis and characterization of magnetic nanomaterials and a better understanding of cellular mechanotransduction mechanisms may support the translation of mechanochemical stimulation into the clinic as an emerging therapeutic approach.

The dynamics of growth cone morphology

BACKGROUND

Normal brain function depends on the development of appropriate patterns of neural connections. A critical role in guiding axons to their targets during neural development is played by neuronal growth cones. These have a complex and rapidly changing morphology; however, a quantitative understanding of this morphology, its dynamics and how these are related to growth cone movement, is lacking.

RESULTS

Here we use eigenshape analysis (principal components analysis in shape space) to uncover the set of five to six basic shape modes that capture the most variance in growth cone form. By analysing how the projections of growth cones onto these principal modes evolve in time, we found that growth cone shape oscillates with a mean period of 30 min. The variability of oscillation periods and strengths between different growth cones was correlated with their forward movement, such that growth cones with strong, fast shape oscillations tended to extend faster. A simple computational model of growth cone shape dynamics based on dynamic microtubule instability was able to reproduce quantitatively both the mean and variance of oscillation periods seen experimentally, suggesting that the principal driver of growth cone shape oscillations may be intrinsic periodicity in cytoskeletal rearrangements.

CONCLUSIONS

Intrinsically driven shape oscillations are an important component of growth cone shape dynamics. More generally, eigenshape analysis has the potential to provide new quantitative information about differences in growth cone behaviour in different conditions.

Low piconewton towing of CNS axons against diffusing and surface-bound repellents requires the inhibition of motor protein-associated pathways

Growth cones, dynamic structures at axon tips, integrate chemical and physical stimuli and translate them into coordinated axon behaviour, e.g., elongation or turning. External force application to growth cones directs and enhances axon elongation in vitro; however, direct mechanical stimulation is rarely combined with chemotactic stimulation. We describe a microfluidic device that exposes isolated cortical axons to gradients of diffusing and substrate-bound molecules, and permits the simultaneous application of piconewton (pN) forces to multiple individual growth cones via magnetic tweezers. Axons treated with Y-27632, a RhoA kinase inhibitor, were successfully towed against Semaphorin 3A gradients, which repel untreated axons, with less than 12 pN acting on a small number of neural cell adhesion molecules. Treatment with Y-27632 or monastrol, a kinesin-5 inhibitor, promoted axon towing on substrates coated with chondroitin sulfate proteoglycans, potent axon repellents. Thus, modulating key molecular pathways that regulate contractile stress generation in axons counteracts the effects of repellent molecules and promotes tension-induced growth. The demonstration of parallel towing of axons towards inhibitory environments with minute forces suggests that mechanochemical stimulation may be a promising therapeutic approach for the repair of the damaged central nervous system, where regenerating axons face repellent factors over-expressed in the glial scar.

Neuronal growth as diffusion in an effective potential

Current understanding of neuronal growth is mostly qualitative, as the staggering number of physical and chemical guidance cues involved prohibit a fully quantitative description of axonal dynamics. We report on a general approach that describes axonal growth in vitro, on poly-D-lysine-coated glass substrates, as diffusion in an effective external potential, representing the collective contribution of all causal influences on the growth cone. We use this approach to obtain effective growth rules that reveal an emergent regulatory mechanism for axonal pathfinding on these substrates.

Avalanche-like behavior in ciliary import

Cilia and flagella are microtubule-based organelles that protrude from the cell body. Ciliary assembly requires intraflagellar transport (IFT), a motile system that delivers cargo from the cell body to the flagellar tip for assembly. The process controlling injections of IFT proteins into the flagellar compartment is, therefore, crucial to ciliogenesis. Extensive biochemical and genetic analyses have determined the molecular machinery of IFT, but these studies do not explain what regulates IFT injection rate. Here, we provide evidence that IFT injections result from avalanche-like releases of accumulated IFT material at the flagellar base and that the key regulated feature of length control is the recruitment of IFT material to the flagellar base. We used total internal reflection fluorescence microscopy of IFT proteins in live cells to quantify the size and frequency of injections over time. The injection dynamics reveal a power-law tailed distribution of injection event sizes and a negative correlation between injection size and frequency, as well as rich behaviors such as quasiperiodicity, bursting, and long-memory effects tied to the size of the localized load of IFT material awaiting injection at the flagellar base, collectively indicating that IFT injection dynamics result from avalanche-like behavior. Computational models based on avalanching recapitulate observed IFT dynamics, and we further show that the flagellar Ras-related nuclear protein (Ran) guanosine 5'-triphosphate (GTP) gradient can in theory act as a flagellar length sensor to regulate this localized accumulation of IFT. These results demonstrate that a self-organizing, physical mechanism can control a biochemically complex intracellular transport pathway.

Surface microstructures on planar substrates and textile fibers guide neurite outgrowth: a scaffold solution to push limits of critical nerve defect regeneration?

The treatment of critical size peripheral nerve defects represents one of the most serious problems in neurosurgery. If the gap size exceeds a certain limit, healing can't be achieved. Connection mismatching may further reduce the clinical success. The present study investigates how far specific surface structures support neurite outgrowth and by that may represent one possibility to push distance limits that can be bridged. For this purpose, growth cone displacement of fluorescent embryonic chicken spinal cord neurons was monitored using time-lapse video. In a first series of experiments, parallel patterns of polyimide ridges of different geometry were created on planar silicon oxide surfaces. These channel-like structures were evaluated with and without amorphous hydrogenated carbon (a-C:H) coating. In a next step, structured and unstructured textile fibers were investigated. All planar surface materials (polyimide, silicon oxide and a-C:H) proved to be biocompatible, i.e. had no adverse effect on nerve cultures and supported neurite outgrowth. Mean growth cone migration velocity measured on 5 minute base was marginally affected by surface structuring. However, surface structure variability, i.e. ridge height, width and inter-ridge spacing, significantly enhanced the resulting net velocity by guiding the growth cone movement. Ridge height and inter-ridge distance affected the frequency of neurites crossing over ridges. Of the evaluated dimensions ridge height, width, and inter-ridge distance of respectively 3, 10, and 10 µm maximally supported net axon growth. Comparable artificial grooves, fabricated onto the surface of PET fibers by using an excimer laser, showed similar positive effects. Our data may help to further optimize surface characteristics of artificial nerve conduits and bioelectronic interfaces.

EXISTENCE OF STRONG SOLUTIONS FOR NEURONAL NETWORK DYNAMICS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

We study the existence of strong solutions for a class of stochastic differential equations in an infinite dimensional space. Our investigation is specially motivated by the stochastic version of a common model of potential spread in a dendritic tree. We do not assume the noise in the junction points to be Markovian. In fact, we allow for long-range dependence in time of the stochastic perturbation. This leads to an abstract formulation in terms of a stochastic diffusion with dynamic boundary conditions, featuring fractional Brownian motion. We prove results on existence, uniqueness and asymptotics of weak and strong solutions to such a stochastic differential equation.

Engineered in vitro/in silico models to examine neurite target preference

Research on spinal cord injury (SCI) repair focuses on developing mechanisms to allow neurites to grow past an injury site. In this article, we observe that numerous divergent paths (i.e., spinal roots) are present along the spinal column, and hence guidance strategies must be devised to ensure that regrowing neurites reach viable targets. Therefore, we have engineered an in vitro micropatterned model in which cultured E7 dorsal root ganglia (DRG) explants may enter alternate pathways (?roots?) along a branching micropattern. Alongside this in vitro model, we have developed an in silico simulation that we validate by comparison with independent experiments. We find in both in silico and in vitro models that the probability of a neurite entering a given root decreases exponentially with respect to the number of roots away from the DRG; consequently, the likelihood of neurites reaching a distant root can be vanishingly small. This result represents a starting point for future strategies to optimize the likelihood that neurites will reach appropriate targets in the regenerating nervous system, and provides a new computational tool to evaluate the feasibility and expected success of neurite guidance in complex geometries.

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.

Theoretical models of neural circuit development

Proper wiring up of the nervous system is critical to the development of organisms capable of complex and adaptable behaviors. Besides the many experimental advances in determining the cellular and molecular machinery that carries out this remarkable task precisely and robustly, theoretical approaches have also proven to be useful tools in analyzing this machinery. A quantitative understanding of these processes can allow us to make predictions, test hypotheses, and appraise established concepts in a new light. Three areas that have been fruitful in this regard are axon guidance, retinotectal mapping, and activity-dependent development. This chapter reviews some of the contributions made by mathematical modeling in these areas, illustrated by important examples of models in each section. For axon guidance, we discuss models of how growth cones respond to their environment, and how this environment can place constraints on growth cone behavior. Retinotectal mapping looks at computational models for how topography can be generated in populations of neurons based on molecular gradients and other mechanisms such as competition. In activity-dependent development, we discuss theoretical approaches largely based on Hebbian synaptic plasticity rules, and how they can generate maps in the visual cortex very similar to those seen in vivo. We show how theoretical approaches have substantially contributed to the advancement of developmental neuroscience, and discuss future directions for mathematical modeling in the field.

Computational model provides insight into the distinct responses of neurons to chemical and topographical cues

Neuronal cell polarization (i.e., establishment of an axon) and axon guidance are mediated and controlled by mechanical and chemical signals from the environment. Unfortunately, an integrated approach to study cell-substrate interactions in a unified framework incorporating structural and chemical effects of the substrate has been lacking. In this paper, we present a new model combining experimental and computational methods to better understand the distinct behavior of E18 hippocampal neurons in response to topographical vs. immobilized chemical cues. We present results from our coarse-grain physiological computational model that correctly describes previously observed phenomena and predicts behavior that was subsequently tested through new experiments. The model differentiates topographical from chemical cues via a difference in cue spacing in these two substrates. Using the feature size spacing for topographical cues and a minimum step size, governed by the physics of filopodia protrusion, for chemical cues, the model successfully mimics the trend observed in experimental polarization probability for four different topographical feature sizes and constant chemical cue spacing. Our results not only show good agreement with experiments, but also provide novel suggestions for development of substrates for finer control of neuronal cell polarization.

Tensile force-dependent neurite elicitation via anti-beta1 integrin antibody-coated magnetic beads

Previous work using glass microneedles to apply calibrated, localized force to neurons showed that tensile force is a sufficient signal for neurite initiation and elongation. However, previous studies did not examine the kinetics or probability of neurite initiation as a function of force or the rate of force application. Here we report the use of a new technique-magnetic bead force application-to systematically investigate the role of force in these phenomena with better ease of use and control over force than glass microneedles. Force-induced neurite initiation from embryonic chick forebrain neurons appeared to be a first-order random process whose rate increased with increasing force, and required the presence of peripheral microtubules. In addition, the probability of initiation was more than twofold lower for neurons exposed to rapid initial force ramps (450 pN/s) than for neurons exposed to slower ramps (1.5 and 11 pN/s). We observed a low force threshold for elongation (15-100 pN), which was not previously detected in chick forebrain neurites elongated by glass microneedles. Finally, neurites subjected to constant force elongated at variable instantaneous rates, and switched abruptly between elongation and retraction, similar to spontaneous, growth-cone-mediated outgrowth and microtubule dynamic instability.

Neurotrophins and the dynamic regulation of the neuronal cytoskeleton

The morphology of neuronal axons and dendrites is dependent on the dynamics of the cytoskeleton. An understanding of neurodevelopment and adult neuroplasticity must therefore include a detailed description of the intrinsic and extrinsic mechanisms that regulate the organization and dynamics of actin filaments and microtubules. In this paper we review recent advances in the understanding of the dynamic regulation of neuronal morphology by interactions among cytoskeletal components and the regulation of the cytoskeleton by neurotrophins.

Autocorrelation function and power spectrum of two-state random processes used in neurite guidance

During development neurons extend and retract cytoskeletal structures, chiefly microtubules and filopodia, to process informational cues from the extracellular environment and thereby guide growth cone migration toward an appropriate synaptic partner. This cytoskeleton-based exploration is achieved by stochastic switching, with microtubules and filopodia alternating between growing and shortening phases apparently at random. If stabilizing signals are not detected during the growth phase, then the structures switch to a shortening state, from which they can again return to a growth phase, and so forth. A useful means of characterizing these stochastic processes in a model-independent way is by autocorrelation and spectral analysis. Previously, we compared experiment to theory by performing Monte Carlo simulations and computing the autocorrelation function and power spectrum from the simulated dynamics, an approach that is computationally intensive and requires recalculation whenever model parameters are changed. Here we present analytical expressions for the autocorrelation function and power spectrum, which compactly characterize microtubule and filopodial dynamics based on the stochastic, two-state model. The model assumes that the phase times are of variable duration and gamma-distributed, consistent with experimental evidence for microtubules assembled in vitro from purified tubulin. The analytical expressions permit the precise quantitative characterization of changes in microtubule and filopodial searching behavior corresponding to changes in the shape of the gamma distribution.