Instabilities in Cellular Dendritic Morphogenesis

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.

Shape, connectedness and dynamics in neuronal networks

The morphology of neurons is directly related to several aspects of the nervous system, including its connectedness, health, development, evolution, dynamics and, ultimately, behavior. Such interplays of the neuronal morphology can be understood within the more general shape-function paradigm. The current article reviews, in an introductory way, some key issues regarding the role of neuronal morphology in the nervous system, with emphasis on works developed in the authors' group. The following topics are addressed: (a) characterization of neuronal shape; (b) stochastic synthesis of neurons and neuronal systems; (c) characterization of the connectivity of neuronal networks by using complex networks concepts; and (d) investigations of influences of neuronal shape on network dynamics. The presented concepts and methods are useful also for several other multiple object systems, such as protein-protein interaction, tissues, aggregates and polymers.

Mechanical control of bacterial cell shape

In bacteria, cytoskeletal filament bundles such as MreB control the cell morphology and determine whether the cell takes on a spherical or a rod-like shape. Here we use a theoretical model to describe the interplay of cell wall growth, mechanics, and cytoskeletal filaments in shaping the bacterial cell. We predict that growing cells without MreB exhibit an instability that favors rounded cells. MreB can mechanically reinforce the cell wall and prevent the onset of instability. We propose that the overall bacterial shape is determined by a dynamic turnover of cell wall material that is controlled by mechanical stresses in the wall. The model affirms that morphological transformations with and without MreB are reversible, and quantitatively describes the growth of irregular shapes and cells undergoing division. The theory also suggests a unique coupling between mechanics and chemistry that can control organismal shapes in general.

Interplay between long- and short-range interactions drives neuritogenesis on stiff surfaces

Substrate factors such as surface energy distribution can affect cell functions, such as neuronal differentiation of PC12 cells. However, the surface effects that trigger such cell responses need to be clarified and analyzed. Here we show that the total surface tension is not a critical parameter. Self-assembled monolayers of alkylsiloxanes on glass were used as culture substrates. By changing the nanoscale structure and ordering of the monolayer, we designed surfaces with a range of dispersive (γ(d) ) and nondispersive (γ(nd) ) potentials, but with a similar value for total free-energy (50 ≤ γ(d) + γ(nd) ≤ 55 mN m⁻¹). When seeded on surfaces displaying γ(d) /γ(nd) ≤ 3.7, PC12 cells underwent low level of neuritogenesis. On surfaces exhibiting γ(d) /γ(nd) ≥ 5.4, neurite outgrowth was greatly enhanced and apparent by only 24 h of culture in absence of nerve growth-factor treatment. These data indicate how the spatial distribution of surface potentials may control neuritogenesis, thus providing a new criterion to address nerve regeneration issues on rigid biocompatible surfaces.

Quantifying neurite growth mediated by interactions among secretory vesicles, microtubules, and actin networks

Neurite growth is a fundamental process of neuronal development, which requires both membrane expansions by exocytosis and cytoskeletal dynamics. However, the specific contribution of these processes has not been yet assessed quantitatively. To study and quantify the growth process, we construct a biophysical model in which we relate the overall neurite outgrowth rate to the vesicle dynamics. By considering the complex motion of vesicles in the cell soma, we demonstrate from biophysical consideration that the main step of finding the neurite initiation site relies mainly on a two-dimensional diffusion/sequestration/fusion at the cell surface and we obtain a novel formula for the flux of vesicles at the neurite base. In the absence of microtubules, we show that a nascent neurite initiated by vesicular delivery can only reach a small length. By adding the microtubule dynamics to the secretory pathway and using stochastic analysis and simulations, we study the complex dynamics of neurite growth. Within this model, depending on the coupling parameter between the microtubules and the neurite, we find different regimes of growth, which describe dendritic and axonal growth. To validate one aspect of our model, we demonstrate that the experimental flux of TI-VAMP but not Synaptobrevin 2 vesicles contributes to the neurite growth. We conclude that although vesicles can be generated randomly in the cell body, the search for the neurite position using the microtubule network and diffusion is quite fast. Furthermore, when the TI-VAMP vesicular flow is large enough, the interactions between the microtubule bundle and the neurite control the growth process. In addition, all of these processes intimately cooperate to mediate the various modes of neurite growth: the model predicts three different growing modes including, in addition to the stable axonal growth and the stochastic dendritic growth, a fast oscillatory regime. Finally our study demonstrates that cytoskeletal dynamics is necessary to generate long protrusion, while vesicular delivery alone can only generate small neurite.

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.

Curvature induced periodic attractor on growth interface

We experimentally address the long-time dynamics of an artificially curved growth interface in directional solidification. Repetitive cell nucleations are found to appear in a disordered way but to eventually organize themselves coherently, at long times. This behavior is recovered by simulation of a nonlinear advection-diffusion model for the phase dynamics. The existence of a periodic attractor is shown by deriving a Liapunov functional for the cellular pattern organization on time ranges that include the singular events of cell nucleation.

Self-organized dynamics on a curved growth interface

We experimentally address long-time dynamics of an artificially curved growth interface in directional solidification. Repetitive cell nucleations are found to appear in a disordered way, but to eventually organize themselves in a coherent way, for long times. This behavior is recovered by simulation of a nonlinear advection-diffusion model for phase dynamics. The existence of a periodic attractor is supported by the derivation of a Lyapunov functional for this model.

Quantitative analysis of concentration gradient and ionic currents associated with hyphal tip growth in fungi

It has been shown previously that the nutrient gradient generated by a tip growing elongated cell induces an ionic current entering the cell tip and looping back in the extracellular medium [L. Limozin, B. Denet and P. Pelcé, Phys. Rev. Lett. 78, 4881 (1997)]. We apply this mechanism to the case of hyphae of fungi, using realistic cell geometries, symport kinetics, proton pump permeabilities, and buffer concentrations. We show that this mechanism contributes to a noticeable part of the external current intensity, related inner electrical field and pH gradient, in agreement with experimental measurements. This provides a good example in biological cells of interaction between shape and field, a common property of growing nonliving systems, such as crystalline dendrites or electrodeposition.

Predicted acetylcholine layer around embryonic motoneurones

The secretion of acetylcholine (ACh) from cultures of dissociated rat purified embryonic motoneurones into the medium was measured at different times using a chemiluminescent assay. From these results it is proposed that a motoneurone is able to generate an extracellular ACh layer surrounding its membrane. This layer is expected to modulate the ACh concentration at the membrane in a range where ACh receptors can be sensitive, and thus to affect the response of the motoneurone. The influences of both membrane-bound and soluble-secreted acetylcholinesterase (AChE) were simulated and a local ACh membrane efflux was deduced. Since ACh usually exerts an inhibitory effect on neurite outgrowth, a possible autocrine influence of the ACh layer in motoneurone morphogenesis is discussed.

Diffusion-regulated control of cellular dendritic morphogenesis

Highly branched dendritic shapes are distinguishing characteristics of neurons and certain other cell types, but the physical mechanisms responsible for their formation are not well understood. Here, we model the growth of cells under the control of diffusible growth-regulating factors (morphogens such as calcium ion) whose local internal concentration results from influx and active extrusion across the cell membrane. Nonlinearities in voltage-dependent ionic permeabilities enhance unstable growth, so that branching dendritic outgrowths results from self-sustaining internal morphogen gradients. Simulations display complex patterns of branching growth, influenced by membrane conductance, galvanotropism and chemotropism. This self-organizing pattern formation is in agreement with the development of real neurons under corresponding conditions.