A mathematical model explains saturating axon guidance responses to molecular gradients
- The University of Queensland, Australia
- University College London, United Kingdom
Figures
Figure 1
Model set-up and the noiseless case.
(A) The axon starts growing from the soma (black segment) at initiation angle ϕ(0). At each time point, the bearing is θ(t), and the bearing change between t and t + 1 is Δθ(). ϕ() is the angle …
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Figure 1—source code 1
The code to simulate the trajectories based on Equation 1 in the noiseless case.
- https://doi.org/10.7554/eLife.12248.005
Figure 2
Model results with noise.
(A) Long-term behavior of growth cones: Simulation of 9 axons with fixed growth rate and noise in bearing changes (ξ ∼ N(0, π/4) radians) starting at ϕ(0) = 90° subject to the gradient direction Ψ = …
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Figure 2—source code 1
The code to simulate the trajectories based on Equation 1 in the noisy case.
- https://doi.org/10.7554/eLife.12248.007
Figure 3
The microfluidic assay.
(A) The design of the chamber: the two solutions were pumped into the inlets and mix in the mixing channels before flowing into the growth chamber where the cells are plated. The mixing channels …
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Figure 3—source data 1
The average brightness intensity and noise in the microfluidic chamber at 0 and 20 hr.
The average and noise were estimated from 5 min interval timelapse imaging over an 1-hr period.
- https://doi.org/10.7554/eLife.12248.009
Figure 4
Turning in microfluidic gradients.
(A) Images of a representative axon initially almost perpendicular to the gradient at the beginning and end of the measurement after 80 min. Scale bar 20 μm. The red dots are the positions of the …
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Figure 4—source data 1
The turning angles at different time points from the start of the experiment.
- https://doi.org/10.7554/eLife.12248.011
Figure 5
The gradient did not affect branch extension and retraction rates.
(A) Histogram of the number of cells with different numbers of branches after 5 hr of growth. The number (mean ± std) of branches per neuron in the control condition was 4.2 ± 1.8 (n=324 cells) and …
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Figure 5—source data 1
The number of branches per cell after 5 hr of growth in the control and NGF gradient (Columns A,B).
In the gradient, we counted the number of branches pointing up and down the gradient (Columns C,D). We measured the time intervals between successive branching events in the same cell in the control and NGF gradient over 5 hr (Columns E-F). For branches that retracted in the 5 hr imaging time, we measured their lifetimes in the control and NGF gradient (Columns G,H).
- https://doi.org/10.7554/eLife.12248.013
Figure 6
Flow did not affect step statistics.
(A) Axons growing in different directions were grouped into four quadrants. (B) Growth cones’ step sizes in different quadrants. n values refer to the number of steps in each quadrant. There was no …
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Figure 6—source data 1
We divided the axons into four quadrants as explained in Figure 6 and measured the bearing changes and stepsizes in each quadrant.
This file contains the step sizes (Sheet 1) and and bearing changes (Sheet 2) in the control condition (Column A) and in each quadrant of the NGF gradient condition (Columns B-E).
- https://doi.org/10.7554/eLife.12248.015
Figure 7
Axons were dragged by growth cones.
(A–C) Timelapse images of three example growth cones. Red arrows point to the putative anchor points and green arrows point to the growth cones. Time is shown in hours and minutes. (D) We measured …
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Figure 7—source data 1
We measured the angle of the growth cone from its putative anchor point (Column A) and compared with the angle of the most distal 20 μm segment of the axon (Column B) 1 hr after the start of the experiment.
- https://doi.org/10.7554/eLife.12248.017
Figure 8
Trajectories of 300 axons growing over 80 min in the control condition, ordered by the initial angle.
The red segments indicate the initial direction of the axon and the blue segments show the traces of the growth cones’ trajectories. Scale bar = 100 μm.
Figure 9
Trajectories of 300 axons growing over 80 min in the NGF gradient, ordered by the initial angle.
Only axons in the box were selected for turning angle measurements as they were almost perpendicular to the gradient, hence most affected by it. Scale bar = 100 μm.
Figure 10
Trajectories of 300 axons growing over 80 min in the NGF gradient with 70 nM KT5720 added, ordered by the initial angle.
Only axons in the box were selected for turning angle measurements. Scale bar = 100 μm.
Figure 11
Trajectories were straight with step sizes and bearing changes independent of each other.
(A) Distribution of straightness indices of all paths with mean straightness of 0.72. (B) There was no correlation between bearing change and step size (R2 = 0.1, p = 0.7). (C) The distribution of …
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Figure 11—source data 1
From our 5-min interval tracings, we measured the bearing changes and step sizes in the control condition (Columns A, B), the mean step sizes of all the growth cones (Column C) and the step sizes in the NGF gradient and NGF gradient + KT5720 conditions (Columns D,E).
- https://doi.org/10.7554/eLife.12248.025
Figure 12
Model captured key statistics of trajectories.
(A) The evolution of simulated turning angles (mean ± std) of n=5000 growth cones over time in the attractive gradient condition. (B) Simulated turning angles after 16 steps (80 min) had mean 9.8° …
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Figure 12—source code 1
The code to simulate the trajectories based on Equation 1 with the step sizes and bearing changes described in Section Turning angles over time were well predicted by the model.
- https://doi.org/10.7554/eLife.12248.027
Figure 13
Simulated trajectories from three conditions.
(A) control, (B) NGF gradient, (C) NGF gradient + KT5720.
Figure 14
More variability with more anchor points.
(A–C) Trajectories of growth cones with probability of putting down a new anchor r= 0.01, 0.05, 0.1 at each timestep and the same parameters as Figure 2A (a = 1, b = 0.1, T = 150 timesteps). The …
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Figure 14—source code 1
The code to simulate the trajectories based on Equation 1 with normally distributed noise in bearing changes described in Section Multiple anchor points achieved sharp turns but also increased variability.
In the regular anchoring case, the growth cone position after every 1/r steps becomes a new anchor point. In the probabilistic anchoring case, each growth cone position has a probability of r to become a new anchor point.
growth-cone-tracker-5min. Growth cone tracking code. The code tracks the position of the growth cone centre every 5 mins from timelapse AVI files.
extract-GC-positions. Growth cone position extraction code. The code to extract the position of the growth cone from the tracings.
- https://doi.org/10.7554/eLife.12248.030
Videos
Video 1
Timelapse images of an example growth cone.
One-minute interval phase contrast timelapse imaging of a growth cone in a microfluidic chamber.
Video 2
Timelapse images of an example growth cone.
One-minute interval phase contrast timelapse imaging of a growth cone in a microfluidic chamber.
Video 3
Timelapse images of an example growth cone.
One-minute interval phase contrast timelapse imaging of a growth cone in a microfluidic chamber.
Tables
Table 1
Summary of model parameters (GC: growth cone).
| Symbol | Meaning |
| GC’s current bearing | |
| GC’s overall angle | |
| Gradient direction | |
| Bearing change | |
| Turning angle after 80 min | |
| Persistence strength | |
| Bias strength | |
| Noise in bearing change | |
| Standard deviation of | |
| Step size every 5 min | |
| Distance from origin to GC | |
| Straightness index | |
| Anchoring rate |
