Competition for tubulin between growing neurites during development

Calibration of stochastic, agent-based neuron growth models with approximate Bayesian computation

Understanding how genetically encoded rules drive and guide complex neuronal growth processes is essential to comprehending the brain's architecture, and agent-based models (ABMs) offer a powerful simulation approach to further develop this understanding. However, accurately calibrating these models remains a challenge. Here, we present a novel application of Approximate Bayesian Computation (ABC) to address this issue. ABMs are based on parametrized stochastic rules that describe the time evolution of small components-the so-called agents-discretizing the system, leading to stochastic simulations that require appropriate treatment. Mathematically, the calibration defines a stochastic inverse problem. We propose to address it in a Bayesian setting using ABC. We facilitate the repeated comparison between data and simulations by quantifying the morphological information of single neurons with so-called morphometrics and resort to statistical distances to measure discrepancies between populations thereof. We conduct experiments on synthetic as well as experimental data. We find that ABC utilizing Sequential Monte Carlo sampling and the Wasserstein distance finds accurate posterior parameter distributions for representative ABMs. We further demonstrate that these ABMs capture specific features of pyramidal cells of the hippocampus (CA1). Overall, this work establishes a robust framework for calibrating agent-based neuronal growth models and opens the door for future investigations using Bayesian techniques for model building, verification, and adequacy assessment.

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.

A generative growth model for thalamocortical axonal branching in primary visual cortex

Axonal morphology displays large variability and complexity, yet the canonical regularities of the cortex suggest that such wiring is based on the repeated initiation of a small set of genetically encoded rules. Extracting underlying developmental principles can hence shed light on what genetically encoded instructions must be available during cortical development. Within a generative model, we investigate growth rules for axonal branching patterns in cat area 17, originating from the lateral geniculate nucleus of the thalamus. This target area of synaptic connections is characterized by extensive ramifications and a high bouton density, characteristics thought to preserve the spatial resolution of receptive fields and to enable connections for the ocular dominance columns. We compare individual and global statistics, such as a newly introduced length-weighted asymmetry index and the global segment-length distribution, of generated and biological branching patterns as the benchmark for growth rules. We show that the proposed model surpasses the statistical accuracy of the Galton-Watson model, which is the most commonly employed model for biological growth processes. In contrast to the Galton-Watson model, our model can recreate the log-normal segment-length distribution of the experimental dataset and is considerably more accurate in recreating individual axonal morphologies. To provide a biophysical interpretation for statistical quantifications of the axonal branching patterns, the generative model is ported into the physically accurate simulation framework of Cx3D. In this 3D simulation environment we demonstrate how the proposed growth process can be formulated as an interactive process between genetic growth rules and chemical cues in the local environment.

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.

Developmental time windows for axon growth influence neuronal network topology

Early brain connectivity development consists of multiple stages: birth of neurons, their migration and the subsequent growth of axons and dendrites. Each stage occurs within a certain period of time depending on types of neurons and cortical layers. Forming synapses between neurons either by growing axons starting at similar times for all neurons (much-overlapped time windows) or at different time points (less-overlapped) may affect the topological and spatial properties of neuronal networks. Here, we explore the extreme cases of axon formation during early development, either starting at the same time for all neurons (parallel, i.e., maximally overlapped time windows) or occurring for each neuron separately one neuron after another (serial, i.e., no overlaps in time windows). For both cases, the number of potential and established synapses remained comparable. Topological and spatial properties, however, differed: Neurons that started axon growth early on in serial growth achieved higher out-degrees, higher local efficiency and longer axon lengths while neurons demonstrated more homogeneous connectivity patterns for parallel growth. Second, connection probability decreased more rapidly with distance between neurons for parallel growth than for serial growth. Third, bidirectional connections were more numerous for parallel growth. Finally, we tested our predictions with C. elegans data. Together, this indicates that time windows for axon growth influence the topological and spatial properties of neuronal networks opening up the possibility to a posteriori estimate developmental mechanisms based on network properties of a developed network.

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.

Competitive dynamics during resource-driven neurite outgrowth

Neurons form networks by growing out neurites that synaptically connect to other neurons. During this process, neurites develop complex branched trees. Interestingly, the outgrowth of neurite branches is often accompanied by the simultaneous withdrawal of other branches belonging to the same tree. This apparent competitive outgrowth between branches of the same neuron is relevant for the formation of synaptic connectivity, but the underlying mechanisms are unknown. An essential component of neurites is the cytoskeleton of microtubules, long polymers of tubulin dimers running throughout the entire neurite. To investigate whether competition between neurites can emerge from the dynamics of a resource such as tubulin, we developed a multi-compartmental model of neurite growth. In the model, tubulin is produced in the soma and transported by diffusion and active transport to the growth cones at the tip of the neurites, where it is assembled into microtubules to elongate the neurite. Just as in experimental studies, we find that the outgrowth of a neurite branch can lead to the simultaneous retraction of its neighboring branches. We show that these competitive interactions occur in simple neurite morphologies as well as in complex neurite arborizations and that in developing neurons competition for a growth resource such as tubulin can account for the differential outgrowth of neurite branches. The model predicts that competition between neurite branches decreases with path distance between growth cones, increases with path distance from growth cone to soma, and decreases with a higher rate of active transport. Together, our results suggest that competition between outgrowing neurites can already emerge from relatively simple and basic dynamics of a growth resource. Our findings point to the need to test the model predictions and to determine, by monitoring tubulin concentrations in outgrowing neurons, whether tubulin is the resource for which neurites compete.

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.

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.

Cell morphology and size must be properly controlled to ensure cellular function. Although there has been significant progress in understanding the molecular signals that change cell morphology, the manner in which cells monitor their size and length to regulate their morphology is poorly understood. Cultured hippocampal neurons polarize by forming a single long axon and multiple short dendrites (Craig and Banker, 1994; Arimura and Kaibuchi, 2007), and symmetry breaking is the initial step of this process. This symmetry-breaking step reproduces even when the neuronal axon is transected; the longest neurite usually grows rapidly to become an axon after transection, regardless of whether it is the axonal stump or another neurite (Goslin and Banker, 1989). Elongation of an immature neurite by mechanical tension also leads to its axonal specification (Lamoureux et al, 2002). These results suggest that cultured hippocampal neurons can sense neurite length, identify the longest one, and induce its subsequent axonogenesis for symmetry breaking. However, little is known about the mechanism for this process.

Shootin1 is one of the earliest markers of neuronal symmetry breaking (Toriyama et al, 2006). During the symmetry-breaking step, it undergoes a stochastic accumulation in neurite tips, and eventually accumulates predominantly in a single neurite that subsequently grows to become an axon. In this study, we demonstrated that shootin1 accumulates in neurite tips in a neurite length-dependent manner, regardless of whether it is the axonal stump or another neurite (Figure 3A, C–F). Thus, morphological information (neurite length) is translated into a molecular signal (shootin1 concentration in neurite tips).

We previously reported that shootin1 is transported from the cell body to neurite tips as discrete boluses and diffuses back to the cell body (Toriyama et al, 2006). The boluses containing variable amounts of shootin1 traveled repeatedly but irregularly along neurites, and their arrival caused large stochastic fluctuations in shootin1 concentration in the neurite tips. To understand the mechanism of length-dependent shootin1 accumulation, we performed quantitative live cell imaging of the anterograde transport and retrograde diffusion of shootin1 and fitted the obtained data into mathematical models of the anterograde transport and retrograde diffusion. The parameters of these two models were derived entirely from quantitative experimental data, without any adjustment. Shootin1 concentration at neurite tips, calculated by integrating the two models, was neurite length dependent (Figure 3B) and showed good agreement with the experimental data (Figure 3A). These results suggest that the neurite length-dependent accumulation of shootin1 is quantitatively explained by its anterograde transport and retrograde diffusion.

This length-dependent shootin1 accumulation constitutes a positive feedback interaction with the previously reported shootin1-induced neurite outgrowth (Shimada et al, 2008). To analyze the functional role of this feedback loop, we quantified shootin1 upregulation (Toriyama et al, 2006) and shootin1-induced neurite outgrowth, and integrated them, together with the above model of length-dependent shootin1 accumulation, into a model neuron (Figure 7A). Furthermore, the parameters of the model components were chosen to give the best fit to the quantitative experimental data without any adjustment. Integrating the three components into a model neuron resulted in spontaneous symmetry breaking (Figure 7B and C). Furthermore, there are a total of 15 agreements between the model predictions and the experimental data, including the neurite length-dependent axon specification and regeneration (Goslin and Banker, 1989; Lamoureux et al, 2002). These data suggest that the three components in our model—namely, diffusion-based neurite length sensing system, shootin1-induced neurite outgrowth and shootin1 upregulation—are sufficient to induce neuronal symmetry breaking.

Bolus-like transport of shootin1 caused large stochastic fluctuations in shootin1 concentration in neurite tips. Interestingly, the generation of continuous shootin1 transport in our model neuron impaired the symmetry-breaking process (Figure 7D). This is consistent with theoretical models in which feedback amplification of fluctuations in signaling can give rise to robust patterns (Turing, 1952; Meinhardt and Gierer, 2000; Kondo, 2002), and underscores the importance of the stochastic fluctuating signals in spontaneous neuronal symmetry breaking.

The combination of quantitative experimentation and mathematical modeling is regarded as a powerful strategy for attaining a profound understanding of biological systems (Hodgkin and Huxley, 1952b; Lewis, 2008; Ferrell, 2009). By focusing on a simple system involving one of the earliest markers of neuronal symmetry breaking, shootin1, we were able to evaluate here the core components of neuronal symmetry breaking on the basis of quantitative experimental data. The present model may thus provide a core mechanism of neuronal symmetry breaking, to which other possible mechanisms can be added to increase the model's complexity in future studies.

Although there has been significant progress in understanding the molecular signals that change cell morphology, mechanisms that cells use to monitor their size and length to regulate their morphology remain elusive. Previous studies suggest that polarizing cultured hippocampal neurons can sense neurite length, identify the longest neurite, and induce its subsequent outgrowth for axonogenesis. We observed that shootin1, a key regulator of axon outgrowth and neuronal polarization, accumulates in neurite tips in a neurite length-dependent manner; here, the property of cell length is translated into shootin1 signals. Quantitative live cell imaging combined with modeling analyses revealed that intraneuritic anterograde transport and retrograde diffusion of shootin1 account for its neurite length-dependent accumulation. Our quantitative model further explains that the length-dependent shootin1 accumulation, together with shootin1-dependent neurite outgrowth, constitutes a positive feedback loop that amplifies stochastic fluctuations of shootin1 signals, thereby generating an asymmetric signal for axon specification and neuronal symmetry breaking.

Large-scale analysis of neurite growth dynamics on micropatterned substrates

During 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.

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.

Spatial embedding of neuronal trees modeled by diffusive growth

The relative importance of the intrinsic and extrinsic factors determining the variety of geometric shapes exhibited by dendritic trees remains unclear. This question was addressed by developing a model of the growth of dendritic trees based on diffusion-limited aggregation process. The model reproduces diverse neuronal shapes (i.e., granule cells, Purkinje cells, the basal and apical dendrites of pyramidal cells, and the axonal trees of interneurons) by changing only the size of the growth area, the time span of pruning, and the spatial concentration of 'neurotrophic particles'. Moreover, the presented model shows how competition between neurons can affect the shape of the dendritic trees. The model reveals that the creation of complex (but reproducible) dendrite-like trees does not require precise guidance or an intrinsic plan of the dendrite geometry. Instead, basic environmental factors and the simple rules of diffusive growth adequately account for the spatial embedding of different types of dendrites observed in the cortex. An example demonstrating the broad applicability of the algorithm to model diverse types of tree structures is also presented.

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.

Deterministic and Stochastic Elements of Axonal Guidance

An enormous literature has been developed on investigations of the growth and guidance of axons during development and after injury. In this review, we provide a guide to this literature as a resource for biomedical investigators. We first review briefly the molecular biology that is known to regulate migration of the growth cone and branching of axonal arbors. We then outline some important fundamental considerations that are important to the modeling of the phenomenology of these guidance effects and of what is known of their underlying internal mechanisms. We conclude by providing some thoughts on the outlook for future biomedical modeling in the field.

Statistical determinants of dendritic morphology in hippocampal pyramidal neurons: A hidden Markov model

Dendritic structure is traditionally characterized by distributions and interrelations of morphometric parameters, such as Sholl-like plots of the number of branches versus dendritic path distance. However, how much of a given morphology is effectively captured by any statistical description is generally unknown. In this work, we assemble a small number of standard geometrical parameters measured from experimental data in a simple stochastic algorithm to describe the dendrograms of hippocampal pyramidal cells. The model, consistent with the hidden Markov framework, is feedforward, local, and causal. It relies on two "hidden" local variables: the expected number of terminal tips in a given subtree, and the current path distance from the soma. The algorithm generates dendrograms that statistically reproduce all morphological essentials of dendrites observed in real neurons, including the distributions of branching and termination points, branch lengths, membrane area, topological asymmetry, and (assuming passive membrane parameters within physiological range) electrotonic characteristics. Thus, this algorithm and the small number of its morphometric parameters constitute a remarkably complete description of the dendrograms of hippocampal pyramidal cells. Specifically, it is found that CA3 and CA1 basal dendrites and CA3 apical dendrites can each be described as homogeneous morphological classes. In contrast, the accurate generation of CA1 apical dendrites necessitates the separate sampling of two types of branches, main and oblique, suggesting their derivations from different developmental mechanisms (terminal and interstitial growth, respectively). We further offer a plausible biophysical interpretation of the model hidden variables, relating them to microtubules and other intracellular resources.

Competition in neurite outgrowth and the development of nerve connections

During the development of the nervous system, neurons form their characteristic morphologies and become assembled into synaptically connected networks. In both neuronal morphogenesis and the development of nerve connections, competition plays an important role. Although the notion of competition is commonly used in neurobiology, there is little understanding of the nature of the competitive process and the underlying molecular and cellular mechanisms. In this chapter, we review a model of competition between outgrowing neurites, as well as various models of competition that have been proposed for the refinement of connections that takes place in the development of the neuromuscular and visual systems. We describe in detail a model that links competition in the development of nerve connections with the underlying actions and biochemistry of neurotrophic factors.