The single modification on the antagonistic 2-node network causes the collapse of the cell polarization pattern.

(a) Basic network with the transition plane marked by grey dashed line.(b) Two subtypes of single-sided self-regulation. (c) Four subtypes of single-sided additional regulation. The corresponding spatial concentration distribution of [Lm] at t = 0, 100, 200, 300, 400, and 500, and the subtle difference between left and right panels are detailed in Fig. S3. (d) Two subtypes of unequal system parameters, exemplified by unequal inhibition intensity and unequal cytoplasmic concentration. For each network, the corresponding spatial concentration distribution of [Am] and [Pm] 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 1st and 2nd columns respectively; the three combinatorial networks composed of any two of the three single modifications are shown in the 3rd column. For each network, the corresponding concentration distribution of [Am] and [Pm] 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 3rd column are as follows: and for 1st row, and for 2nd row, and and for 3rd row. Note that within a network, normal arrows and blunt arrows symbolize activation and inhibition respectively.

The 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., and with a homogeneous state dominated by [A]), ② (i.e., and with a stable polarized state), and ③ (i.e., and with a homogeneous state dominated by [P]). The corresponding concentration distribution of [Am] and [Pm] 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 ϒ, basal off-rate α, cytoplasmic concentration [Xc], 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 zero-velocity interface position of the polarity pattern is shifted and regulatable.

(a) Spatially uniform parameters (2nd column) of a symmetric 2-node network generate a symmetric pattern (1st column), from the same simulation in Fig. 1a. (b) Using the parameter combination with the posterior-shifting interface on the left and anterior-shifting interface on the right, a stable polarity pattern (1st column) can be remained by increasing to 1.5 at x < 0 and decreasing ϒA to 0.01 at x> 0 (2nd column). (c-d) The stable interface position can be optionally adjusted by setting the change position of the step-up function (2nd column). (c) As in (b), but changing the step position to x = −0.1, the interface stabilizes around x = −0.1 (1st column). (d) As in (b), but changing the step position to x = 0.1, the interface stabilizes around x = 0.1 (1st column). For each parameter set, the corresponding spatial concentration distribution of [Am] and [Pm] at t = 0, 100, 200, 300, 400, and 500 are shown beneath with a color scheme listed on the right. Note that all the interface positions in (b-d) are marked by the vertical dashed line in gray.

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 composed of five molecular species, each of which has a polarized spatial concentration distribution on the cell membrane shown beneath. The left nodes PAR-3/PAR-6/PKC-3 (i.e. [A]) and CDC-42 (i.e. [C]) exhibit high concentration in the anterior pole and low concentration in the posterior pole, schemed by the purple line beneath; the right nodes PAR-1/PAR-2 (i.e. [P]), LGL-1 (i.e. [L]), and CHIN-1 (i.e. [H]) exhibit low concentration in the anterior pole and high concentration in the posterior pole, schemed by the cyan line 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 (1st row). The final spatial concentration distribution (t = 500) averaged over all established viable parameter sets for each molecular species (i.e., 62 among 262,701 sets, ~0.024%, for the 4-Node network; 62 among 5,151 sets, ~1.2%, for the LGL-1- network; 602 among 262,701 sets, ~0.229%, for the WT network), shown by a solid line (2nd 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 (v) over evolution time (t) (3rd row). For each subfigure in 3rd row, each line with a unique color represents the simulation of a unique viable parameter set, and v = 10−4 is marked by a dashed line. For (a-d), the legend for the relationship between molecular species and corresponding color is placed in the bottom right corner. (e) The viable parameter sets of WT (blue points; 602 viable solutions among 262,701 sets, ~0.229%) and 4-Node networks (red points; 62 viable solutions among 5,151 sets, ~1.20%). (f) The interplay between [A]~[C] mutual activation and [A]~[L] mutual inhibition on the interface velocity. The contour map of the interface velocity vI with different parameter combinations of and represents its moving trend. The darker the red is, the faster the interface is moving toward the posterior pole; the darker the blue is, the faster the interface is moving toward the anterior pole; the closer the color is to white, the more stable the interface is. (g) The averaged pattern error in a perturbed condition compared to the original 4-Node, LGL-1-, and WT networks. The Gaussian noise is simultaneously exerted on all the parameters ϒ, α, k1, k2, q1 and q2, where standard deviation σ denotes the noise amplification. For a specific noise level, the pattern error is averaged over 1000 independent simulations.

The control of the interface velocity and position by adjusting parameters in a multi-dimensional system.

(a) The parameter space (1st row) and the linear relationship (2nd row) between interface velocity and parameters. The discrete viable parameter space of WT network is fitted by a blue curved surface to represent its null surface with little or no pattern interface moving (1st row). The benchmark point and its neighborhood are marked by an orange cube. Centering on the benchmark point Ω∗, the relationship between the velocity interface and parameters in the region marked by the orange cube is shown by slice planes orthogonal to the ϒ-axis at the values 0.034, 0.036, 0.038, 0.04, 0.042, and 0.044 (2nd row). The darker the red is, the faster the interface is moving toward the posterior pole; the darker the blue is, the faster the interface is moving toward the anterior pole; the closer the color is to white, the more stable the interface is. (b-d) As in Fig. 4, the control of the interface position by spatially inhomogeneous parameters corresponding to opposite interface velocities on two sides of the interface can be applied to the C. elegans 5-node network. (b) Using Ω∗ as a representative, spatially uniform parameters (2nd column) generate a stable polarity pattern (1st column). (c) On top of (b), a stable polarity pattern (1st column) with its interface around x = 0 can be obtained by increasing to 0.12 at x < 0 and increasing ϒP, ϒL, and ϒH to 0.06 at x > 0 (2nd column). (d) As in (c) but changing the step position to x = −0.1 (2nd column), the interface stabilizes around x = −0.1 (1st column). (e) As in (c), but changing the step position to x = 0.1 (2nd column), the interface stabilizes around x = 0.1 (1st column). For each parameter set, the corresponding spatial concentration distribution of [Am], [Cm], [Pm], [Lm], and [Hm] at t = 0, 100, 200, 300, 400, and 500 are shown beneath with a color scheme listed on the right. Note that all the interface positions in (b-e) are marked by the vertical dashed line in gray.