Computational and Systems Biology

Balancing reaction-diffusion network for cell polarization pattern with stability and asymmetry

Yixuan Chen
Guoye Guan
Lei-Han Tang author has email address
Chao Tang author has email address

South Bay Interdisciplinary Science Center, Songshan Lake Materials Laboratory, Guangdong 523808, China
Department of Physics, Sichuan University, Sichuan 610065, China
School of Physics, Peking University, Beijing 100871, China
Center for Quantitative Biology, Peking University, Beijing 100871, China
Department of Physics, Hong Kong Baptist University, Hong Kong 999077, China
Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Hong Kong 999077, China
State Key Laboratory of Environmental and Biological Analysis, Hong Kong Baptist University, Hong Kong 999077, China
Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China

https://doi.org/10.7554/eLife.96421.1

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Figures and data

The single modification on the antagonistic 2-node network causes the collapse of the cell polarization pattern.
(a) Basic network with the interface region shaded by grey. (b) Two subtypes of single-sided self-regulation. (c) Four subtypes of single-sided additional regulation. (d) Two subtypes of unequal system parameters, exemplified by unequal inhibition intensity and unequal cytoplasmic concentration. For each network, the corresponding concentration distribution of [A_m] and [P_m] at t = 0, 100, 200, 300, 400, and 500 are shown beneath with a color scheme listed in the bottom left corner. Note that within a network, normal arrows and blunt arrows symbolize activation and inhibition respectively.

The combination of two opposite modifications recovers the stability of the cell polarization pattern.
The basic network and the ones added with a single modification are shown in the 1^st and 2^nd columns respectively; the three combinatorial networks composed of any two of the three single modifications are shown in the 3^rd column. For each network, the corresponding concentration distribution of [A_m] and [P_m] at t = 0, 100, 200, 300, 400, and 500 are shown beneath with a color scheme listed on the right. Here, the value assignments on the modifications in the 3^rd column are as follows: = 0.012 and = 0.01 for 1^st row, = 0.012 and = 1.24 for 2^nd row, and = 0.012 and = 2 for 3^rd row. Note that within a network, normal arrows and blunt arrows symbolize activation and inhibition respectively.

The combination of two opposite modifications recovers the stability of the cell polarization pattern.
The basic network and the ones added with a single modification are shown in the 1^st and 2^nd columns respectively; the three combinatorial networks composed of any two of the three single modifications are shown in the 3^rd column. For each network, the corresponding concentration distribution of [A_m] and [P_m] at t = 0, 100, 200, 300, 400, and 500 are shown beneath with a color scheme listed on the right. Here, the value assignments on the modifications in the 3^rd column are as follows: = 0.012 and = 0.01 for 1^st row, = 0.012 and = 1.24 for 2^nd row, and = 0.012 and = 2 for 3^rd row. Note that within a network, normal arrows and blunt arrows symbolize activation and inhibition respectively.

The detailed balance between system parameters is needed for maintaining pattern stability.
(a) The phase diagram between and in the network modified by self-activation (quantified by ) and additional inhibition (quantified by ) on [A]. The representative parameter assignment for each phase are marked with ① (i.e., = 0.015 and = 0 with a homogeneous state dominated by [A]), ② (i.e., = 0.015 and = 0.0135 with a stable polarized state), and ③ (i.e., = 0.02 and = 0 with a homogeneous state dominated by [P]). The corresponding concentration distribution of [A_m] and [P_m] at t = 0,100,200, 300,400, and 500 are shown around the phase diagram with a color scheme listed on top. (b) The phase diagram between responsive concentration k₁, basal on-rate γ, basal off-rate α, cytoplasmic concentration [X_c], and inhibition intensity k₂. For each phase diagram in (a)(b), the final state dominated by [A] or [P] or stable polarized is colored in orange, green, and gray, respectively. Note that within a network, normal arrows and blunt arrows symbolize activation and inhibition respectively.

The detailed balance between system parameters is needed for maintaining pattern stability.
(a) The phase diagram between and in the network modified by self-activation (quantified by ) and additional inhibition (quantified by ) on [A]. The representative parameter assignment for each phase are marked with ① (i.e., = 0.015 and = 0 with a homogeneous state dominated by [A]), ② (i.e., = 0.015 and = 0.0135 with a stable polarized state), and ③ (i.e., = 0.02 and = 0 with a homogeneous state dominated by [P]). The corresponding concentration distribution of [A_m] and [P_m] at t = 0,100,200, 300,400, and 500 are shown around the phase diagram with a color scheme listed on top. (b) The phase diagram between responsive concentration k₁, basal on-rate γ, basal off-rate α, cytoplasmic concentration [X_c], and inhibition intensity k₂. For each phase diagram in (a)(b), the final state dominated by [A] or [P] or stable polarized is colored in orange, green, and gray, respectively. Note that within a network, normal arrows and blunt arrows symbolize activation and inhibition respectively.

Adopting parameter sets corresponding to opposite interface velocities on two sides of the interface, the stability of the polarity pattern recovers, with a regulable zero-velocity solution of the interface localization.
(a) Spatially uniform parameters of a symmetric 2-node network generate a symmetric pattern. (b) Using the parameter combination with the posteriorshifting interface on the left and anterior-shifting interface on the right, a stable polarity pattern can also be obtained by increasing to 1.5 at x < 0 and decreasing γ_A to 0.01 at x > 0. (cd) The stable interface localization can be optionally adjusted by setting the change position of the step-up function. (c) As in (b), but changing the step position to x = 0.1. (d) As in (b), but changing the step position to x = –0.1. For each parameter set, the corresponding concentration distribution of [A_m] and [P_m] at t = 0,100,200, 300,400, and 500 are shown beneath with a color scheme listed in the right bottom corner.

Adopting parameter sets corresponding to opposite interface velocities on two sides of the interface, the stability of the polarity pattern recovers, with a regulable zero-velocity solution of the interface localization.
(a) Spatially uniform parameters of a symmetric 2-node network generate a symmetric pattern. (b) Using the parameter combination with the posteriorshifting interface on the left and anterior-shifting interface on the right, a stable polarity pattern can also be obtained by increasing to 1.5 at x < 0 and decreasing γ_A to 0.01 at x > 0. (cd) The stable interface localization can be optionally adjusted by setting the change position of the step-up function. (c) As in (b), but changing the step position to x = 0.1. (d) As in (b), but changing the step position to x = –0.1. For each parameter set, the corresponding concentration distribution of [A_m] and [P_m] at t = 0,100,200, 300,400, and 500 are shown beneath with a color scheme listed in the right bottom corner.

The molecular interaction network in C. elegans zygote and its natural advantages in terms of pattern stability, viable parameter sets, balanced network configuration, and parameter robustness.
(a) The schematic diagram of the network is composed of five molecules or molecular complexes, each of which has a polarized concentration distribution on the cell membrane shown beneath. Note that within the network, normal arrows and blunt arrows symbolize activation and inhibition respectively. (b-d) The structure of 4-Node, LGL-1⁻, and WT networks (1^st row). The final concentration distribution averaged over all established viable parameter sets for each molecule, shown by a solid line (2^nd row). For each position, MEAN ± STD (i.e., standard deviation) calculated with all viable parameter sets is shown by shadow. The moving velocity of the pattern (3^rd row). For each subfigure in 3^rd row, a unique color represents the simulation of a viable parameter set, and = 10^–4 is marked by a dashed line. (e) The viable parameter sets of WT and 4-Node networks. (f) The detailed balance between [A]~[C] mutual activation and [A]~[L] mutual inhibition. The contour map of the interface velocity with different parameter combinations of and represents the moving trend of the pattern. (g) The averaged pattern error in a perturbed condition compared to the original 4-Node, LGL-1⁻, and WT networks.

The molecular interaction network in C. elegans zygote and its natural advantages in terms of pattern stability, viable parameter sets, balanced network configuration, and parameter robustness.
(a) The schematic diagram of the network is composed of five molecules or molecular complexes, each of which has a polarized concentration distribution on the cell membrane shown beneath. Note that within the network, normal arrows and blunt arrows symbolize activation and inhibition respectively. (b-d) The structure of 4-Node, LGL-1⁻, and WT networks (1^st row). The final concentration distribution averaged over all established viable parameter sets for each molecule, shown by a solid line (2^nd row). For each position, MEAN ± STD (i.e., standard deviation) calculated with all viable parameter sets is shown by shadow. The moving velocity of the pattern (3^rd row). For each subfigure in 3^rd row, a unique color represents the simulation of a viable parameter set, and = 10^–4 is marked by a dashed line. (e) The viable parameter sets of WT and 4-Node networks. (f) The detailed balance between [A]~[C] mutual activation and [A]~[L] mutual inhibition. The contour map of the interface velocity with different parameter combinations of and represents the moving trend of the pattern. (g) The averaged pattern error in a perturbed condition compared to the original 4-Node, LGL-1⁻, and WT networks.

The control of the interface velocity and position by adjusting parameters in a multidimensional system.
(a) The parameter space and the linear relationship between interface velocity and parameters. The discrete parameter space of WT is fitted by a blue curved surface to represent its null surface. The benchmark point P* and its neighborhood are marked by an orange cube (top). Centering on the benchmark point P*(γ = 0.039, k₁ = 1.55, q₂ = 0.05), the relationship between the velocity interface and parameters is shown by slice planes orthogonal to the γ-axis at the values 0.034, 0.036, 0.038, 0.04, 0.042, and 0.044. (b-d) As in Fig. 4, the control of the interface position by spatially inhomogeneous parameters can be applied to the realistic C. elegans network. (b) Using P*(γ* = 0.039, = 1.55, = 0.05) as a representative, spatially uniform parameters generate a stable polarity pattern. (c) A stable polarity pattern with its interface around x = 0 can be obtained by increasing to 0.12 at x < 0 and increasing γ_P, γ_L, and γ_H to 0.06 at x > 0. (d) As in (c) but changing the step position to x = –0.1, the interface stabilizes around x = –0.1. (e) As in (c), but changing the step position to x = 0.1.

The control of the interface velocity and position by adjusting parameters in a multidimensional system.
(a) The parameter space and the linear relationship between interface velocity and parameters. The discrete parameter space of WT is fitted by a blue curved surface to represent its null surface. The benchmark point P* and its neighborhood are marked by an orange cube (top). Centering on the benchmark point P*(γ = 0.039, k₁ = 1.55, q₂ = 0.05), the relationship between the velocity interface and parameters is shown by slice planes orthogonal to the γ-axis at the values 0.034, 0.036, 0.038, 0.04, 0.042, and 0.044. (b-d) As in Fig. 4, the control of the interface position by spatially inhomogeneous parameters can be applied to the realistic C. elegans network. (b) Using P*(γ* = 0.039, = 1.55, = 0.05) as a representative, spatially uniform parameters generate a stable polarity pattern. (c) A stable polarity pattern with its interface around x = 0 can be obtained by increasing to 0.12 at x < 0 and increasing γ_P, γ_L, and γ_H to 0.06 at x > 0. (d) As in (c) but changing the step position to x = –0.1, the interface stabilizes around x = –0.1. (e) As in (c), but changing the step position to x = 0.1.

The initial state at t = 0 (shown on the left) and stable state at t = 500 (shown in the middle) of the cell polarization pattern generated by the simple 2-node network and C. elegans 5-node network (shown on the right).
Note that the concentration distribution on the cell membrane (c) is averaged over all established viable parameter sets for each molecule (i.e., 122 sets for the simple 2-node network and 602 sets for the C. elegans 5-node network).

The in silico perturbation experiments on [L] based on the C. elegans modeling framework.
(a) The final concentration distribution of [A] and [C] on the cell membrane generated by C. elegans mutant networks with depletion on [P] (2^nd column), [L] (3^rd column), or both (4^th column), in comparison with the one generated by C. elegans wild-type network (1^st column). (b) The pattern error in C. elegans mutant networks with depletion on [P], [L], or both (shown from left to right), compared to the pattern in C. elegans wild-type network. (c) The final concentration distribution of [A] and [C] on the cell membrane generated by C. elegans mutant networks with depletion on [P] (2^nd column), depletion on [P] but overexpression of [L] ([LC] from 1 to 1.5) (3^rd column), and depletion on both [P] and [H] but overexpression of [L] ([LC] from 1 to 1.5) (4^th column), in comparison with the one generated by C. elegans wild-type network (1^st column). (d) The pattern error in C. elegans mutant networks with depletion on [P] or both [P] and [H] but overexpression of [L] ([LC] from 1 to 1.5), compared to the pattern in C. elegans wild-type network. For (a)(c), the final concentration distribution on the cell membrane generated by the C. elegans wild-type network (1^st column) is illustrated with dashed lines in the C. elegans mutant networks (2^nd ~ 4^th column) as a comparative reference. Note that within a network, normal arrows and blunt arrows symbolize activation and inhibition respectively. For (b)(d), the pattern error is shown by a bar chart with the average of all viable parameter sets, while the values from each viable parameter set are plotted with gray dots then the ones from the same viable parameter set but in different networks are connected with gray lines.

The 34 possible additional feed loops between LGL-1 (abbr., [L]) and PAR-3/PAR-6/PKC-3 (abbr., [A]) or PAR-1/PAR-2 (abbr., [P]).
Note that within a network, normal arrows and blunt arrows symbolize activation and inhibition respectively, while dashed lines mean no regulation.

(a) The progress bar showing the running progress. (b) The output subfolders in the folder “PolarSim”.

The results of Example 1 (Simple Antagonistic 2-Node Network) and Example 2 (C. elegans Wild-Type 5-Node Network) of PolarSim.
(a) The flow chart for computing Example 1: ① input parameters; ② click “Run”; ③ the simulation is completed with a progress bar shown and the files are saved in the folder “Output 2-Node”; the file “Pattern_500.mat” is used to show the data format in the right, where the first part stores the name and location of the nodes (molecules) while the second part stores the concentration distribution of each node (molecule) on the cell membrane; ④ input the pathway of the outputted pattern file; ⑤ click “Plot”; ⑥ the simulation is completed with a figure shown and the position of the transition plane given in the box “Output: Transition Plane”; ⑦ input the pathway of the two outputted files “Pattern_*.mat” at different time points into “Import: Start Time” and “Import: End Time”, where “*” denotes the start time and end time respectively; ⑧ click “Run”; ⑨ the simulation is completed with the interface velocity given in the box “Output: Interface Velocity”. (b) The same as (a) but for the C. elegans Wild-Type 5-Node Network.

The results of Example 3 (LGL-1 Mutant 4-Node Network) of PolarSim.
(a) The transition plane is shown at t = 500, with a result of XT = 0.207071. The interface velocity is calculated between t = 300 and t = 500, with the value v_I = 0.000429293 representing an unstable pattern (top). The figure is plotted at t = 500 (bottom). (b) The transition plane and the figure are shown at t = 1000 and the interface velocity is calculated between t = 800 and t = 1000. The pattern collapses to a homogeneous state with [A] and [C] invading the posterior domain at t = 1000, and thereby the transition plane doesn’t exist and the interface can’t be calculated.

The effects of cell size (length) on the cell polarization pattern.
(a-e) The pattern of Simple Antagonistic 2-Node Network at t = 500. From left to right, the cell lengths are 0.1, 0.2, 0.3, 0.4 and 0.5, respectively. (f-j) The same as (a-e) but for the C. elegans Wild-Type 5- Node Network.

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