Mathematical Modeling of Axonal Formation Part I: Geometry

Pearson, Yanthe E.; Castronovo, Emilio; Lindsley, Tara A.; Drew, Donald A.

doi:10.1007/s11538-011-9648-2

Cited by 16 publications

(20 citation statements)

References 40 publications

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“…These studies show that physical stimuli (gradients of various molecular species, stiffness of the growth substrate, traction forces generated during axonal extension etc.) play a key role in the wiring of the nervous system [3][4][5][6][7][8][9]. However, our current understanding of neuronal growth is mostly qualitative, the vast complexity of the parameter space still prohibiting fully quantitative predictions of outcomes from given initial conditions such as: geometry of the neuronal circuit, type of biochemical cues on the growth substrate, topography or mechanical properties of the substrate.…”

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confidence: 99%

Neuronal growth as diffusion in an effective potential

et al. 2013

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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-Dlysine 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.

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confidence: 99%

Neuronal growth as diffusion in an effective potential

et al. 2013

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“…Similar models have been adopted for axon growth in the literature for symmetric surfaces. 18,19 To interpret the observed alignment of the axonal growth along the asymmetric surfaces, we derived the expected distribution of the tangent angles from a biased random walk model. The motion of the growth cone is described by the following Langevin equation, m _ v ¼ Àav þ F þ nðtÞ, where m is an effective mass, t is the time, a is a Stokes drag coefficient, F is a constant force, and nðtÞ is a random force which has zero mean hni ¼ 0 and is Markovian hnðtÞ Á nðt 0 Þi ¼ m 2 Cdðt À t 0 Þ, where C represents the strength of the noise and d is the Dirac delta function.…”

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confidence: 99%

Neuronal alignment on asymmetric textured surfaces

et al. 2012

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Axonal growth and the formation of synaptic connections are key steps in the development of the nervous system. Here, we present experimental and theoretical results on axonal growth and interconnectivity in order to elucidate some of the basic rules that neuronal cells use for functional connections with one another. We demonstrate that a unidirectional nanotextured surface can bias axonal growth. We perform a systematic investigation of neuronal processes on asymmetric surfaces and quantify the role that biomechanical surface cues play in neuronal growth. These results represent an important step towards engineering directed axonal growth for neuro-regeneration studies. These surfaces also provide physical guidance, and chemical support for neuronal cell adherence, axonal extension, network formation, and function. The axons, and in particular their dynamic unit known as the growth cone are able to detect and respond to environmental signals such as functionalization of surfaces with extracellular matrix proteins, biomolecules released by neighboring neurons at extremely low concentrations (molecular level), substrate stiffness and topographical and geometrical cues. 6 Over the past decade, there has been rapid progress in our understanding of the role played by chemical signaling and surface-based biochemical guidance on the growth cone dynamics and axonal elongation. For example, it is known that axonal navigation to their target depends on the precise arrangement of extracellular proteins on the growth surfaces. 2,6,7 It is also now recognized that mechanical interactions between neurons and their environment are playing an essential role in neuronal growth and development. 5,8 However, the neuronal response to mechanical and topographical stimuli, and the details of cell-surface interactions such as adhesion forces and traction stress generated during growth are currently poorly understood. 9,10Directional surfaces composed of asymmetric structures are widely used in nature for wet and dry adhesion.11 Inspired by these surfaces, Demirel et al. synthesized an asymmetric textured surface 12 and reported an engineered nanotextured surface deriving its anisotropic adhesive wetting directly from its asymmetric nanoscale roughness. 13 In an earlier study, Demirel et al. studied the fibroblast adhesion and removal on directional nanofilms, 14 using a fluidic shear stress to remove cells from a microfluidic channel. It has been shown that cells were removed with lower shear stresses when the flow was in the direction of nanorod tilt, compared to flow against the tilt.14 Adhesion and retraction under asymmetric mechanical cues demonstrated unique properties. 15Cell polarization (i.e., response to external cues such as chemical gradients and mechanical deformation) has been studied extensively on textured surfaces to understand cell fate. 16 However, unidirectional polarization in response to surface mechanical cues has not been demonstrated earlier.Here, we report axonal extension and network formation on asymmetri...

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“…Successive steps were anticorrelated (Fig 6.11H), which was not accounted for in a previous model [123]. This helps the paths remain relatively straight: if successive steps were positively correlated, the paths would become more bent over time.…”

Section: Ditionsmentioning

confidence: 88%

“…A common way to achieve correlation between successive steps is through a non-uniform turning angle distribution. Pearson et al [123] used a correlated random walk model to describe the trajectories of rat pyramidal neurons. The instantaneous turning angle θ is a function of the arc length s from the axon's initiation point.…”

Section: Random Walk Models Of Growth Cone Trajectoriesmentioning

confidence: 99%

“…Subsequently, various phenomenological models have been built that differ as to how they treat stochasticity, and mechanisms for directional preference, namely turning or growth rate modulation. One set attempted to fit the dynamics of growth cone movement to a random walk with drift [123][124][125]129]. Li et al simulated trajectories by assuming the turning angle of the growth cone is in proportion to the angle between the neurite and the resultant filopodial tension [131].…”

Section: Introductionmentioning

confidence: 99%

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Mathematical modelling of axon guidance in chemical gradients

Nguyen¹

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Correct wiring is crucial to the proper functioning of the nervous system. Key signals guiding growth cones, the motile tips of developing axons, to their targets are molecular gradients. The receptors on the membrane of the growth cone bind stochastically with guidance molecules in the environment, giving an estimate of the direction of the gradient. The growth cone then executes biased random movements to navigate toward the chemoattractant source or away from the chemorepellent source. This thesis addresses two questions: 1) how the positioning and diffusion of receptors on the growth cone membrane affect the accuracy of gradient estimation, 2) how trajectories are influenced by chemical gradients.Membrane receptors can diffuse freely on the membrane, smearing out the directional information about the gradient. It was unknown how the positioning and diffusion of receptors affect the estimate of the gradient. To address question (1) above, we utilise an ideal-observer approach, assuming that the growth cone can perform maximum likelihood estimation of the gradient based on the stochastic binding patterns with ligand molecules. The performance of gradient sensing is measured by the Fisher Information between the binding pattern and the gradient direction. The quality of gradient sensing decreases with higher diffusion constant of the receptors and improves with higher concentration. We then extend to a two-dimensional model of an elliptical growth cone with a general prior distribution. With a random uniform distribution of receptors, the shape of the growth cone can introduce bias in the gradient estimate. This bias can be corrected by a non-uniform distribution of receptors with higher density near the minor axis of the growth cone.Besides stochastic gradient sensing, growth cones also trace highly stochastic trajectories, and it is unclear how molecular gradients bias their movement. We then introduce a mathematical model of a correlated random walk based on persistence, bias and noise to describe growth cone trajectories, constrained directly by measurements of the detailed statistics of growth cone movements in both attractive and repulsive gradients in a microfluidic device. This model explains the long-standing mystery that average axon turning angles in gradients in vitro plateau very rapidly with time at relatively small values of 10-20• . This work introduces the most accurate predictive model of growth cone trajectories to date, and calls into question the ability of molecular gradients alone to provide reliable guidance cues for growing axons.ii Declaration by AuthorThis thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

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Mathematical Modeling of Axonal Formation Part I: Geometry

Cited by 16 publications

References 40 publications

Neuronal growth as diffusion in an effective potential

Neuronal growth as diffusion in an effective potential

Neuronal alignment on asymmetric textured surfaces

Mathematical modelling of axon guidance in chemical gradients

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