Anomalous diffusion for neuronal growth on surfaces with controlled geometries

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.

Neuronal Growth and Formation of Neuron Networks on Directional Surfaces

“…The cells used in this work are cortical neurons obtained from embryonic day 18 rats. For cell dissociation and culture, we have used established protocols presented in our previous work [ 10 , 15 , 16 , 19 , 20 , 30 , 31 , 32 ]. In our previous work, we have performed immunostaining experiments that show high neuron cell purity in these cultures [ 30 ].…”

“…These studies have shown that neurons grown on substrates with periodic geometrical features develop different growth patterns when compared to neurons grown on surfaces lacking a periodic geometry. Such differences include populations of axons that are significantly longer and that tend to align their growth with preferred spatial directions [ 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 ].…”

“…In our previous work, we have shown that axonal growth on surfaces with controlled geometries arises as the result of an interplay between deterministic and stochastic components of growth cone motility [ 10 , 15 , 16 , 19 , 20 ]. Deterministic influences include, for example, the presence of preferred directions of growth along specific geometric patterns on substrates, while stochastic components come from the effects of polymerization of cytoskeletal elements (actin filaments and microtubules), neuron signaling, low concentration biomolecule detection, biochemical reactions within the neuron, and the formation of lamellipodia and filopodia [ 1 , 2 , 7 , 21 ].…”

“…We have also reported that neurons cultured on poly-D-lysine coated polydimethylsiloxane (PDMS) substrates with periodic parallel ridge micropatterns of spatial periodicity d (henceforth referred to as the pattern spatial period) grow axons parallel to the surface patterns. We have studied axonal growth as a function of time on these micropatterned surfaces and found that axonal alignment increases as a function of time [ 16 ]. The axonal dynamics are described by non-linear Langevin equations, involving quadratic velocity terms and non-zero coefficients for the angular orientation of the growing axon [ 20 ].…”

“…We have also measured the angular distributions and the coefficients of diffusion and angular drift on these substrates [ 10 ]. Our results provide a detailed analysis of axonal growth on substrates with different geometrical patterns by measuring speed and acceleration distributions as a function of substrate geometry [ 20 ], axonal alignment as a function of time [ 16 ], as well as axonal angular distributions, angular drift, and diffusion coefficients [ 10 , 16 , 20 ]. However, a comprehensive model of the growth dynamics on these substrates, which takes into account the mechanisms of axonal alignment and cell-surface interactions, is still missing.…”