Figure 1. Overview of integrating in silico and wet lab approaches in pattern formation. The utilization of genetic circuits in bacteria or mammalian cells can yield patterns analogous to those obtained through computational simulations.

Figure 2. Illustrates the simulation of cell growth and differentiation in a 10 × 100 area over 500 unit times. In this example, the concentration of morphogen u spreads from left to right along the x-axis with the initial concentration of 1, the diffusion coefficient Du is 0.1; the threshold β is 10 times that of α (α = 0.0001). Periodic boundary condition is used. Top panel: The simulation result of morphogen concentration distribution. - White represents the maximum concentration. - Black represents a concentration of 0. This panel shows how the morphogen concentration forms a gradient, with the highest concentration on the left side (white) and decreasing concentration towards the right side (black). Bottom panel: The simulation result of cell differentiation based on morphogen concentration. - Yellow represents fully differentiated cells (fate 3). - Dark green represents undifferentiated cells (fate 1). - Light green represents semi-differentiated cells (fate 2). This panel demonstrates how cells differentiate according to the morphogen concentration. Cells near the high-concentration region (left side) differentiate into fate 3 (yellow), while cells in the middle region with intermediate concentration differentiate into fate 2 (light green). Cells in the low-concentration region (right side) remain undifferentiated (dark green). This simulation provides a visual representation of how morphogen gradients direct cell differentiation and pattern formation in a developing tissue. It highlights the importance of understanding the role of morphogens in the organization and development of multicellular organisms.

Figure 3. The impact of varying morphogen diffusion coefficients and cell differentiation thresholds on the resulting pattern of cell differentiation. In all the simulations, threshold β is assumed to be 10α. The initial and boundary condition is the same as the simulation shown in Figure 2.

Figure 4. Simulation results of bacterial movement represented by diffusion equation. The initial state assumes that cells grow uniformly in a 10 × 100 area and run for 300 time units; D0 = 0.001, β1 = 0.1, β2 = 1, K1 = K2 = 0.01. Periodic boundary condition is used. For activating effect simulation (Positive effect), select dv>0 and use Dv (u)activating to calculate the diffusion coefficient of each grid; for repressing effect simulation (Negative effect), select dv < 0 and use Dv (u)repressing to calculate the diffusion coefficient of each grid. For chemical gradients, white indicates high concentration and black for low concentration; for cell concentrations, yellow indicates relatively high concentration and green indicates relatively low concentration.

Figure 5. Simulation results of the concentration gradient formed by the spontaneous diffusion of chemical u with Du = 0.1. The initial state involves adding chemical u to a 1-unit-width strip in the center and running the simulation for 70 time units. Other conditions are same as the simulation in Figure 4.

Figure 6. Simulation results of the reaction-diffusion model. Two 100 × 100 random matrices were generated as the initial states of u (left panel) and v (right panel), and the simulation was run for 500 time units. Du = 0.001, Dv = 0.005, du = dv = 0.01, gu = gv = 0.02. Periodic boundary condition is used. White and yellow indicate high levels, while black and green represent low levels.

Figure 7. Simulation results of lateral inhibition, run for 200 time units; β = 0.1, threshold = 4. One cell was seeded at the center of the area in the initial state. In each running step of the time unit, every cell with fewer than 6 neighbors have the potential to grow a new cell in the empty neighboring space. The new cell will inherit the expression level of signal u from its parent. Different inhibitory rates (α = 0.0001, 0.001, 0.01, 0.1, 0.5) were simulated respectively, and the threshold is set at 1.0.

Figure 8. A diagram of CDX2, BRA, SOX17, and SOX2 gradients and germ layer differentiation in 2D cultured hESCs [2]. In this experiment, 2D cultured hESC on micropatterned coverslip was treated with BMP4 for 24 h. The fluorescence intensity of CDX2, BRA, and SOC2 showed germ layers differentiation (adapted from Warmflash et al. [2,38]).

Figure 9. Sequential patterning of E. coli with a quorum-sensing circuit of AHL sensing and motility control [4]. (A) The motility control circuit based on AHL quorum-sensing. (B) The establishment of a stripe pattern while the cell population expands. (C) The control strain without the quorum-sensing circuit cannot form a pattern (adapted from Liu et al. [4]).

Figure 10. The lateral activation circuit with synNotch bioparts [5,62]. (A) Direct cell–cell contact. The notch core will release the effector (shown as Gal4) when GFP binds to its receptor, then the effector will activate the expression of BFP. Only receiver cells in direct contact with sender cells will be activated for BFP expression. (B) Cell–cell contact mediated by artificial GFP gradient (refer to section 4.1). The notch core will release the effector (shown as Gal4) only when free GFP binds to the anchor (shown as LaG2) of neighboring cells, which leads to the expression of mCherry. The activation of mCherry expression can be formed by a soluble GFP gradient (adapted from Morsut et al. [5] and Toda et al. [62]).

Figure 11. The structure of RsgI, and the model of RsgI self-cleavage and signal transduction [6]. The binding of substrate to CBM/GH triggers the self-cleavage of PD-β1 in RsgI, which leads to the cleavage of NTD and release of SigI which affects the expression of the downstream target gene (adapted from Chen et al. [6]).

Figure 12. The mechanism of MinDE gradient formation [70]. MinD and MinE achieve dynamic association and dissociation through ATP phosphorylation and dimerization, thereby causing oscillations (adapted from Ramm et al. [70]).

Figure 13. The artificial auxin quorum sensing circuit in mammalian cell. The concentration of auxin or auxin precursor regulates the survival of cell by binding to AID and osTIR1, which causes the degradation of BlastR via ubiquitination (adapted from Ma et al. [8]).

Figure 14. The structure and mechanism of synthetic biomolecular condensates. The structure of synIDP-DBD-DD binds to target sequence of plasmids while driving the phase transition process, thereby causing the encapsulation of the plasmids in the protein condensate (adapted from Dai et al. [9]).