How to assemble a scale-invariant gradient
- Department of Physics, Brandeis University, United States
Figures
Figure 1
The polar transport model of gradient formation.
Transport of proteins (red) in the cell is a combination of diffusion and polar transport. Diffusion occurs throughout the cytoplasm. When diffusing proteins encounter motor proteins (black arrows) moving along polar filaments (green) that contour the cell surface, they can be taken up by the motors and delivered to the cell’s anterior pole. The anterior pole acts as a source, while the filaments along the surface of the cell serve as a sink of diffusing proteins. The result of this combined polar transport and diffusion is a protein gradient that extends along the polar () axis of the cell, with proteins accumulating toward the anterior pole.
Figure 2
Concentration gradient in a spherical cell.
(A) Sample steady-state configuration obtained from the direct simulation of the polar transport model in a spherical cell with 1000 proteins (red). Direction of polar transport on the surface of the cell is toward the anterior pole, where the accumulation of proteins is observed. (B) Concentration of proteins along the polar, z-axis normalized by the average cytoplasmic concentration, for spherical cells of different radii. (C) Concentration profiles from B scaled by the cell radius. (D) Comparison between the steady-state solution of the polar transport equations (Equation 3), compared to the protein concentration obtained from simulations, for a spherical cell with radius . For all plots, the diffusion constant and the transport speed along the cortex is .
Figure 3
Protein gradient in a cylindrical cell.
(A) Configuration of the proteins obtained from simulation of the polar transport model with 1000 proteins (red circles). Directed transport on the surface of the cell is along positive z-direction. (B) Protein concentration along the z axis, normalized by the average cytoplasmic concentration, for cylindrical cells of different lengths and fixed radius, . (C) Comparison between the steady-state solution of the polar transport equations and simulations. The analytic solution is an infinite series, and we compare the first term of this series to the results of simulations. The inset compares simulation results to the sum of the first 40 terms of the analytic solutions. For all plots, the diffusion constant and the surface transport speed, .
Figure 4
Concentration gradient in a spheroidal cell.
(A) Configuration of the proteins obtained from simulation of the polar transport model in a spheroidal cell of radius and major axis , and 1000 proteins. Direction of transport on the surface of the cell is along the positive z direction. (B) Protein concentration normalized by the average cytoplasmic concentration for spheroidal cells of different radii, with a fixed aspect ratio . The inset shows concentration profiles where the distance from the pole is scaled by a. (C) Comparison between the analytic solution and simulations. (D) Concentration profiles for spheroids with the same minor axis () and a varying major axis (). The dashed line represents exponential decay with a decay constant as we computed above, for the case of a cylindrical cell. For all plots, the diffusion constant and the transport speed along the cortex is .
Figure 5
Effect of imperfect capture on the protein gradient.
(A) Protein concentration gradients (normalized by the average cytoplasmic concentration ) in spherical cells of different radii, computed by simulations of the polar transport model with 1000 particles. We use reactive boundary conditions with m/s. (B) Comparison of the simulated protein gradient in a spherical cell, m, with reactive boundary conditions ( m/s), to the analytical solution for the concentration profile (see Appendix C). (C) Analytic results for concentration gradients for different cell radii plotted against the distance from the pole scaled by the radius; m/s. The scaling observed for fully absorbing boundary conditions (see Figure 2C) is absent. (D) Same as C, but now with m/s. Scaling is approximately restored for most distances away from the pole. For all plots, the diffusion constant and the transport speed along the cortex is .
Figure 6
Effect of imperfect transport of proteins on the protein gradient.
(A) Concentration gradient obtained from simulation with 1000 proteins for various values of with a spherical cell of radius . All other parameters are the same as in Figure 2. The analytic curve represents the gradient obtained from the polar transport model for a spherical cell. (B) Concentration gradients in a scaled coordinate for different values of with . For all plots, the diffusion constant and the transport speed along the cortex is..
Appendix 1—figure 1
Surface concentration gradient for a spherical cell.
Concentration profiles on the surface of spherical cells of different radii, obtained from simulations of the polar transport model of gradient formations. Inset shows the same data, just with the distance from the pole scaled by the cell radius. Parameters used in the simulation: protein diffusion constant , transport speed along the cortex , and is the release point.
Author response image 1
Concentration gradient for cells with the source at different distances from the pole (ε).
Concentration profiles with differing source points. We start very close to the pole and move further away. The radius of the sphere is , the diffusion constant and the transport speed along the cortex is .
Author response image 2
Concentration gradient in XY plane for a spherical cell.
Concentration Profiles in the XY plane for three different distances from the pole. The radius of the sphere is , the diffusion constant and the transport speed along the cortex is .
Author response image 3
Surface concentration gradient for a spherical cell.
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How to assemble a scale-invariant gradient
eLife 11:e71365.
https://doi.org/10.7554/eLife.71365
