Research Article: Anomalous diffusion for neuronal growth on surfaces with controlled geometries

Date Published: May 6, 2019

Publisher: Public Library of Science

Author(s): Ilya Yurchenko, Joao Marcos Vensi Basso, Vladyslav Serhiiovych Syrotenko, Cristian Staii, Keng-Hwee Chiam.

http://doi.org/10.1371/journal.pone.0216181

Abstract

Geometrical cues are known to play a very important role in neuronal growth and the formation of neuronal networks. Here, we present a detailed analysis of axonal growth and dynamics for neuronal cells cultured on patterned polydimethylsiloxane surfaces. We use fluorescence microscopy to image neurons, quantify their dynamics, and demonstrate that the substrate geometrical patterns cause strong directional alignment of axons. We quantify axonal growth and report a general stochastic approach that quantitatively describes the motion of growth cones. The growth cone dynamics is described by Langevin and Fokker-Planck equations with both deterministic and stochastic contributions. We show that the deterministic terms contain both the angular and speed dependence of axonal growth, and that these two contributions can be separated. Growth alignment is determined by surface geometry, and it is quantified by the deterministic part of the Langevin equation. We combine experimental data with theoretical analysis to measure the key parameters of the growth cone motion: speed and angular distributions, correlation functions, diffusion coefficients, characteristics speeds and damping coefficients. We demonstrate that axonal dynamics displays a cross-over from Brownian motion (Ornstein-Uhlenbeck process) at earlier times to anomalous dynamics (superdiffusion) at later times. The superdiffusive regime is characterized by non-Gaussian speed distributions and power law dependence of the axonal mean square length and the velocity correlation functions. These results demonstrate the importance of geometrical cues in guiding axonal growth, and could lead to new methods for bioengineering novel substrates for controlling neuronal growth and regeneration.

Partial Text

Neuronal cells are the primary working units of the nervous system. A single neuron is a very specialized cell that develops two types of processes during growth: a long axon and several shorter dendrites (Fig 1A). These processes extend (grow) and make connections with other neurons thus wiring up the nervous system. Once the neuronal network is formed, a neuron can send electrical signals to other neurons through functional connections (synapses) made between axons and dendrites. During the development of the nervous system axons actively navigate over large distances (~ 10–100 cell diameters in length) to find target dendrites from other neurons and to form neural circuits [1, 2]. Axonal motion is controlled by the growth cone, a dynamic unit located at the leading edge of the axon. The growth cone is sensitive to a great number of external stimuli including biochemical, electrical, mechanical and geometrical cues [1–4].

The experimental data for speed distributions (Fig 3 and S2 Fig), as well as velocity autocorrelation functions and mean square length (Fig 4 and S3 Fig) for axons grown on micro-patterned PDMS surfaces show a gradual cross over between normal diffusion (described by the OU process), observed for low/intermediate time scales (t < 48 hrs), and superdiffusion observed for larger time scales (t ≥ 48 hrs). In contrast, experiments performed on flat PDMS substrates show normal diffusion at all times. The OU process, which is inspired by the study of the Brownian motion, represents the simplest stochastic model used for describing cellular motility. It has been successfully used for modeling the dynamics of many types of cells including endothelial cells [25], human granulocytes [21], fibroblasts and human keratinocytes [23], as well as cortical neurons [12, 19, 20]. The values we have obtained for the diffusion coefficient D = (19±2)μm2/hr and the characteristic rms speed for growth cones on PDMS surfaces (Eq 10) are comparable with the corresponding values reported for human peritoneal mesothelial cells [29], one order of magnitude smaller than the values reported for human keratinocytes [23], and for endothelial cells [25], and about two orders of magnitude smaller than the corresponding values reported for glioma cells [30]. These results are consistent with the relatively slower dynamics expected for growth cones as they move to form connections and to wire up the nervous system [1, 2]. In this paper, we have used stochastic analysis to model neuronal growth on micro-patterned PDMS substrates coated with PDL. We have shown that the experimental data for small and intermediate time scales are well-described by Ornstein-Uhlenbeck (OU) processes (linear Langevin equations with white noise). On the other hand, growth measured at longer time scales displays superdiffusive dynamics, characterized by non-Gaussian speed distributions, and power-law behavior of velocity autocorrelation function and of the axonal mean square length.   Source: http://doi.org/10.1371/journal.pone.0216181

 

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