Continuum model for tubulin-driven neurite elongation

Modeling neuron growth using isogeometric collocation based phase field method

We present a new computational framework of neuron growth based on the phase field method and develop an open-source software package called “NeuronGrowth_IGAcollocation”. Neurons consist of a cell body, dendrites, and axons. Axons and dendrites are long processes extending from the cell body and enabling information transfer to and from other neurons. There is high variation in neuron morphology based on their location and function, thus increasing the complexity in mathematical modeling of neuron growth. In this paper, we propose a novel phase field model with isogeometric collocation to simulate different stages of neuron growth by considering the effect of tubulin. The stages modeled include lamellipodia formation, initial neurite outgrowth, axon differentiation, and dendrite formation considering the effect of intracellular transport of tubulin on neurite outgrowth. Through comparison with experimental observations, we can demonstrate qualitatively and quantitatively similar reproduction of neuron morphologies at different stages of growth and allow extension towards the formation of neurite networks.

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.

Dendrite and Axon Specific Geometrical Transformation in Neurite Development

We propose a model of neurite growth to explain the differences in dendrite and axon specific neurite development. The model implements basic molecular kinetics, e.g., building protein synthesis and transport to the growth cone, and includes explicit dependence of the building kinetics on the geometry of the neurite. The basic assumption was that the radius of the neurite decreases with length. We found that the neurite dynamics crucially depended on the relationship between the rate of active transport and the rate of morphological changes. If these rates were in the balance, then the neurite displayed axon specific development with a constant elongation speed. For dendrite specific growth, the maximal length was rapidly saturated by degradation of building protein structures or limited by proximal part expansion reaching the characteristic cell size.

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.

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.

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.

Dynamics of notch activity in a model of interacting signaling pathways

Networks of interacting signaling pathways are formulated with systems of reaction-diffusion (RD) equations. We show that weak interactions between signaling pathways have negligible effects on formation of spatial patterns of signaling molecules. In particular, a weak interaction between Retinoic Acid (RA) and Notch signaling pathways does not change dynamics of Notch activity in the spatial domain. Conversely, large interactions of signaling pathways can influence effects of each signaling pathway. When the RD system is largely perturbed by RA-Notch interactions, new spatial patterns of Notch activity are obtained. Moreover, analysis of the perturbed Homogeneous System (HS) indicates that the system admits bifurcating periodic orbits near a Hopf bifurcation point. Starting from a neighborhood of the Hopf bifurcation, oscillatory standing waves of Notch activity are numerically observed. This is of particular interest since recent laboratory experiments confirm oscillatory dynamics of Notch activity.

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.

Study and simulation of reaction-diffusion systems affected by interacting signaling pathways

Possible effects of interaction (cross-talk) between signaling pathways is studied in a system of Reaction-Diffusion (RD) equations. Furthermore, the relevance of spontaneous neurite symmetry breaking and Turing instability has been examined through numerical simulations. The interaction between Retinoic Acid (RA) and Notch signaling pathways is considered as a perturbation to RD system of axon-forming potential for N2a neuroblastoma cells. The present work suggests that large increases to the level of RA-Notch interaction can possibly have substantial impacts on neurite outgrowth and on the process of axon formation. This can be observed by the numerical study of the homogeneous system showing that in the absence of RA-Notch interaction the unperturbed homogeneous system may exhibit different saddle-node bifurcations that are robust under small perturbations by low levels of RA-Notch interactions, while large increases in the level of RA-Notch interaction result in a number of transitions of saddle-node bifurcations into Hopf bifurcations. It is speculated that near a Hopf bifurcation, the regulations between the positive and negative feedbacks change in such a way that spontaneous symmetry breaking takes place only when transport of activated Notch protein takes place at a faster rate.

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.

Dynamics of outgrowth in a continuum model of neurite elongation

Neurite outgrowth (dendrites and axons) should be a stable, but easily regulated process to enable a neuron to make its appropriate network connections during development. We explore the dynamics of outgrowth in a mathematical continuum model of neurite elongation. The model describes the construction of the internal microtubule cytoskeleton, which results from the production and transport of tubulin dimers and their assembly into microtubules at the growing neurite tip. Tubulin is assumed to be largely synthesised in the cell body from where it is transported by active mechanisms and by diffusion along the neurite. It is argued that this construction process is a fundamental limiting factor in neurite elongation. In the model, elongation is highly stable when tubulin transport is dominated by either active transport or diffusion, but oscillations in length may occur when both active transport and diffusion contribute. Autoregulation of tubulin production can eliminate these oscillations. In all cases a stable steady-state length is reached, provided there is intrinsic decay of tubulin. Small changes in growth parameters, such as the tubulin production rate, can lead to large changes in length. Thus cytoskeleton construction can be both stable and easily regulated, as seems necessary for neurite outgrowth during nervous system development.

Transport limited effects in a model of dendritic branching

A variety of stochastic models of dendritic growth in developing neurons have been formulated previously. Such models indicate that the probability of a new branch forming in a growing tree may be modulated by factors such as the number of terminals in the tree and their centrifugal order. However, these models cannot identify any underlying biophysical mechanisms that may cause such dependencies. Here, we explore a new model in which branching depends on the concentration of a branch-determining substance in each terminal segment. The substance is produced in the cell body and is transported by active transport and diffusion to the terminals. The model reveals that transport-limited effects may give rise to the same modulation of branching as indicated by the stochastic models. Different limitations arise if transport is dominated by active transport or by diffusion.