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

Large-scale analysis of neurite growth dynamics on micropatterned substrates

2011 September 15During both development and regeneration of the nervous system, neurons display complex growth dynamics, and several neurites compete to become the neuron's single axon. Numerous mathematical and biophysical models have been proposed to explain this competition, which remain experimentally unverified. Large-scale, precise, and repeatable measurements of neurite dynamics have been difficult to perform, since neurons have varying numbers of neurites, which themselves have complex morphologies. To overcome these challenges using a minimal number of primary neurons, we generated repeatable neuronal morphologies on a large scale using laser-patterned micron-wide stripes of adhesive proteins on an otherwise highly non-adherent substrate. By analyzing thousands of quantitative time-lapse measurements of highly reproducible neurite growth dynamics, we show that total neurite growth accelerates until neurons polarize, that immature neurites compete even at very short lengths, and that neuronal polarity exhibits a distinct transition as neurites grow. Proposed neurite growth models agree only partially with our experimental observations. We further show that simple yet specific modifications can significantly improve these models, but still do not fully predict the complex neurite growth behavior. Our high-content analysis puts significant and nontrivial constraints on possible mechanistic models of neurite growth and specification. The methodology presented here could also be employed in large-scale chemical and target-based screens on a variety of complex and subtle phenotypes for therapeutic discoveries using minimal numbers of primary neurons.United States. Dept. of Defense. (National Defense Science and Engineering Graduate Fellowship

A diffusion-based neurite length-sensing mechanism involved in neuronal symmetry breaking

Shootin1, one of the earliest markers of neuronal symmetry breaking, accumulates in the neurite tips of polarizing neurons in a neurite length-dependent manner. Thus, neurons sense their neurites' length and translate this spatial information into a molecular signal, shootin1 concentration.Quantitative live cell imaging of shootin1 dynamics combined with mathematical modeling analyses reveals that its anterograde transport and retrograde diffusion in neurite shafts account for the neurite length-dependent accumulation of shootin1.The neurite length-dependent shootin1 accumulation and shootin1-induced neurite outgrowth constitute a positive feedback loop that amplifies stochastic shootin1 signals in neurite tips.Quantitative mathematical modeling shows that the above positive feedback loop, together with shootin1 upregulation, constitutes a core mechanism for neuronal symmetry breaking

Biologically Plausible Models of Neurite Outgrowth

The growth of a neuronal dendritic tree depends on the neuron’s internal state and the environment within which it is situated. Different types of neuron develop dendritic trees with specific characteristics, such as the average number of terminal branches and the average length of terminal and intermediate segments. A key aspect of the growth process is the construction of the microtubule cytoskeleton within the dendritic tree. Neurite elongation requires assembly of microtubules from free tubulin at the growth cone. The stability of microtubule bundles is an important factor in determining how likely it is for a growth cone to split to form new daughter branches. Microtubule assembly rates and bundle stability are controlled by microtubule-associated proteins, principally MAP2 in dendrites. Extending previous work (Hely et al, J. Theor. Biol. 210:375-384, 2001) I have developed a mathematical model of neurite outgrowth in which elongation and branching rates are determined by the phosphorylation state of MAP2 at the tips of each terminal branch. Tubulin and MAP2 are produced in the cell body and transported along the neurite by a combination of diffusion and active transport. Microtubule (dis)assembly at neurite tips is a function of tubulin concentration. The rate of assembly depends on the amount of unphosphorylated MAP2 bound to the microtubules and linking them together. Phosphorylation of MAP2 destroys its linking capability and destabilises the microtubule bundles. Each terminal has a probability of branching that depends on the phosphorylation of MAP2 which, in turn, is a function of calcium concentration. Results from this model show that changes in the (de)phosphorylation rates of MAP2 affect the topology of the final dendritic tree. Higher phosphorylation promotes branching and results in trees with many short terminal branches and relatively long intermediate segments. Reducing phosphorylation promotes elongation and inhibits branching

MAPping out distribution routes for kinesin couriers

In the crowded environment of eukaryotic cells, diffusion is an inefficient distribution mechanism for cellular components. Long-distance active transport is required and is performed by molecular motors including kinesins. Furthermore, in highly polarized, compartmentalized and plastic cells such as neurons, regulatory mechanisms are required to ensure appropriate spatio-temporal delivery of neuronal components. The kinesin machinery has diversified into a large number of kinesin motor proteins as well as adaptor proteins that are associated with subsets of cargo. However, many mechanisms contribute to the correct delivery of these cargos to their target domains. One mechanism is through motor recognition of subdomain-specific microtubule (MT) tracks, sign-posted by different tubulin isoforms, tubulin post-translational modifications (PTMs), tubulin GTPase activity and MT associated proteins (MAPs). With neurons as a model system, a critical review of these regulatory mechanisms is presented here, with particular focus on the emerging contribution of compartmentalised MAPs. Overall, we conclude that – especially for axonal cargo – alterations to the MT track can influence transport, although in vivo, it is likely that multiple track-based effects act synergistically to ensure accurate cargo distribution

Dysregulation of microtubule stability impairs morphofunctional connectivity in primary neuronal networks

Functionally related neurons assemble into connected networks that process and transmit electrochemical information. To do this in a coordinated manner, the number and strength of synaptic connections is tightly regulated. Synapse function relies on the microtubule (MT) cytoskeleton, the dynamics of which are in turn controlled by a plethora of MT-associated proteins, including the MT-stabilizing protein Tau. Although mutations in the Tau-encodingMAPT gene underlie a set of neurodegenerative disorders, termed tauopathies, the exact contribution of MT dynamics and the perturbation thereof to neuronal network connectivity has not yet been scrutinized. Therefore, we investigated the impact of targeted perturbations of MT stability on morphological (e.g., neurite- and synapse density) and functional (e.g., synchronous calcium bursting) correlates of connectivity in networks of primary hippocampal neurons. We found that treatment with MT-stabilizing or -destabilizing compounds impaired morphofunctional connectivity in a reversible manner. We also discovered that overexpression of MAPT induced significant connectivity defects, which were accompanied by alterations in MT dynamics and increased resistance to pharmacological MT depolymerization. Overexpression of a MAPT variant harboring the P301L point mutation in the MT-binding domain did far less, directly linking neuronal connectivity with Tau's MT binding affinity. Our results show that MT stability is a vulnerable node in tauopathies and that its precise pharmacological tuning may positively affect neuronal network connectivity. However, a critical balance in MT turnover causes it to be a difficult therapeutic target with a narrow operating window

Distinct roles of c-Jun N-terminal kinase isoforms in neurite initiation and elongation during axonal regeneration

c-Jun N-terminal kinases (JNKs) (comprising JNK1-3 isoforms) are members of the MAPK (mitogen-activated protein kinase) family, activated in response to various stimuli including growth factors and inflammatory cytokines. Their activation is facilitated by scaffold proteins, notably JNK-interacting protein-1 (JIP1). Originally considered to be mediators of neuronal degeneration in response to stress and injury, recent studies support a role of JNKs in early stages of neurite outgrowth, including adult axonal regeneration. However, the function of individual JNK isoforms, and their potential effector molecules, remained unknown. Here, we analyzed the role of JNK signaling during axonal regeneration from adult mouse dorsal root ganglion (DRG) neurons, combining pharmacological JNK inhibition and mice deficient for each JNK isoform and for JIP1. We demonstrate that neuritogenesis is delayed by lack of JNK2 and JNK3, but not JNK1. JNK signaling is further required for sustained neurite elongation, as pharmacological JNK inhibition resulted in massive neurite retraction. This function relies on JNK1 and JNK2. Neurite regeneration of jip1(-/-) DRG neurons is affected at both initiation and extension stages. Interestingly, activated JNKs (phospho-JNKs), as well as JIP1, are also present in the cytoplasm of sprouting or regenerating axons, suggesting a local action on cytoskeleton proteins. Indeed, we have shown that JNK1 and JNK2 regulate the phosphorylation state of microtubule-associated protein MAP1B, whose role in axonal regeneration was previously characterized. Moreover, lack of MAP1B prevents neurite retraction induced by JNK inhibition. Thus, signaling by individual JNKs is differentially implicated in the reorganization of the cytoskeleton, and neurite regeneration

A Framework for Modeling the Growth and Development of Neurons and Networks

The development of neural tissue is a complex organizing process, in which it is difficult to grasp how the various localized interactions between dividing cells leads relentlessly to global network organization. Simulation is a useful tool for exploring such complex processes because it permits rigorous analysis of observed global behavior in terms of the mechanistic axioms declared in the simulated model. We describe a novel simulation tool, CX3D, for modeling the development of large realistic neural networks such as the neocortex, in a physical 3D space. In CX3D, as in biology, neurons arise by the replication and migration of precursors, which mature into cells able to extend axons and dendrites. Individual neurons are discretized into spherical (for the soma) and cylindrical (for neurites) elements that have appropriate mechanical properties. The growth functions of each neuron are encapsulated in set of pre-defined modules that are automatically distributed across its segments during growth. The extracellular space is also discretized, and allows for the diffusion of extracellular signaling molecules, as well as the physical interactions of the many developing neurons. We demonstrate the utility of CX3D by simulating three interesting developmental processes: neocortical lamination based on mechanical properties of tissues; a growth model of a neocortical pyramidal cell based on layer-specific guidance cues; and the formation of a neural network in vitro by employing neurite fasciculation. We also provide some examples in which previous models from the literature are re-implemented in CX3D. Our results suggest that CX3D is a powerful tool for understanding neural development