Fig 1.
Schematic diagram of the research scheme.
Top: The branching model was decoupled to obtain the activator-inhibitor model. Bottom: The S-Y parameter space of the activator-inhibitor model was calculated for the Turing instability. A crescent-shaped Turing region is presented as a result. The SY-curve for the differentiation trajectory of a cell in the branching system was plotted in the S-Y plane. Each state (S, Y) of cell differentiation is represented as a point on the SY-curve. To acquire the Turing pattern underlying the branching pattern, a point (e.g., point p1) on the trajectory within the Turing region was taken as the values of Sand Y, and simulation of the activator-inhibitor model was performed.
Fig 2.
Turing spot patterns underlying the branching patterns.
(A) For the tip bifurcation pattern (Aa), the underlying Turing pattern is a spot distribution (Ab), which appears at the point of the cell differentiation trajectory (Ac, green curve) within the crescent-shaped Turing region (Ac, gray shadow region). (B) Same as (A) but for the side branching pattern. The Turing pattern underlying the side branching pattern is also a spot distribution. (Ca) The growth of tip bifurcation in the tip bifurcation pattern (Aa) in 3D form. (Cb) The tip bifurcation pattern (Aa) in 3D form. (Cc) The underlying spot pattern (Ab) in 3D form. (Da) The growth of side branching in the side branching pattern (Ba) in 3D form. (Db) The side branching pattern (Ba) in 3D form. (Dc) The underlying spot pattern (Bb) in 3D form. Parameters: c = 0.002, μ = 0.16, ρA = 0.03, DA = 0.02, v = 0.04, ρH = 0.0001, DH = 0.3, c0 = 0.02, γ = 0.02, Ds = 0.06, d = 0.008, e = 0.1, f = 10, (Aa) ε = 1.0, (Ba) ε = 0.06, (Ab) S = 0.352, Y = 0.248, (Bb) S = 0.679, Y = 0.510. In the 2D patterns, black color indicates a high concentration, while gray indicates a low concentration. In the 3D patterns, red indicates a high concentration, while blue indicates a low concentration.
Fig 3.
Comparison of the concentration gradients in different Turing patterns.
(A) Spot pattern. (B) Stripe pattern. (C) Hole pattern. (D-F) The Turing patterns shown in 3D form corresponding to patterns A-C. The spot patterns exhibit the highest concentration gradient among the Turing patterns. The Turing patterns were generated by the activator-inhibitor model[4] with a saturation of activator production.
Fig 4.
Sparse spot patterns underlying different tip bifurcation patterns.
(A-C) The tip bifurcation patterns with increasing spatial interval between bifurcation events. (D-F) The underlying spot patterns have a sparse distribution, with increasing number of spots, which correspond to the tip bifurcation patterns A-C. Parameters: c = 0.002, μ = 0.16, ρA = 0.03, DA = 0.02, v = 0.04, ρH = 0.0001, DH = 0.3, c0 = 0.02, γ = 0.02, DS = 0.06, d = 0.008, e = 0.1, f = 10, (A-C) ε = 1.5/1.0/0.7, (D-F) S = 0.320, Y = 0.185; S = 0.352, Y = 0.248; S = 0.395, Y = 0.313. In 2D patterns, black color indicates high concentration while gray indicates low.
Fig 5.
Dense spot patterns underlying different side branching patterns.
(A-C) The side branching patterns. (D-F) The underlying spot patterns have a dense distribution that corresponds to the side branching patterns A-C. The spatial interval has a small decrease in the side branching patterns with the production of more branches as the number of spots slightly increases in the underlying spot patterns. Parameters: c = 0.002, μ = 0.16, ρA = 0.03, DA = 0.02, v = 0.04, ρH = 0.0001, DH = 0.3, c0 = 0.02, γ = 0.02, DS = 0.06, d = 0.008, e = 0.1, f = 10, (A-C) ε = 0.1/0.06/0.045, (D-F) S = 0.614, Y = 0.478; S = 0.679, Y = 0.510; S = 0.716, Y = 0.530. In the 2D patterns, black indicates a high concentration while gray indicates a low concentration.
Fig 6.
Dispersion relations for the spot patterns underlying the branching patterns in Figs 4 and 5.
(A) Dispersion relations for the spot patterns underlying the branching patterns in Figs 4 and 5. λ represents the eigenvalue with the largest real part for a given wavenumber k. k1-6 are the critical wavenumbers at which the maximum value of λ occurs at. k1-3 are the wavenumbers of the spot patterns in Fig 4D–4F, while k4-6 are the wavenumbers of the spot patterns in Fig 5D–5F. (B) Comparison of the wavelength (2π/k) of the spot patterns underlying the branching patterns in Figs 4 and 5. Patterns 1–3 represent the spot patterns in Fig 4D–4F, while patterns 4–6 represent the spot patterns in Fig 5D–5F.
Fig 7.
Five Turing regions obtained by varying ρH.
The Turing region C is the same region shown in Fig 2 for ρH = 0.0001. Turing regions A (ρH = 0.00005) and B (ρH = 0.00007) were obtained by decreasing ρH. Turing regions D (ρH = 0.00013) and E (ρH = 0.00015) were obtained by decreasing ρH.
Fig 8.
Effect of the wavelength on the branching patterns in the Turing region for ρH = 0.00005.
(A) The cell differentiation trajectories of the branching patterns cross the Turing region. Points 1–7 correspond to the position of the underlying Turing patterns. (B) Branching patterns (top) and underlying spot patterns (bottom). Patterns 1–7 correspond to points 1–7 in A. (C) Dispersion relations for patterns, with wavenumber k1-7 corresponding to patterns 1–7 in B. (D) Comparison of the wavelength (2π/wavenumber) from patterns 1–7 in B. Parameters: c = 0.002, μ = 0.16, ρA = 0.03, DA = 0.02, v = 0.04, ρH = 0.00005, DH = 0.3, c0 = 0.02, DS = 0.06, (B1-7) ε = 3.0/2.0/1.0/0.5/0.1/0.06/0.045. In the 2D patterns, black indicates a high concentration, and gray indicates a low concentration.
Fig 9.
Effect of the wavelength on the branching patterns in the Turing region for ρH = 0.00007.
(A) The cell differentiation trajectories of the branching patterns cross the Turing region. Points 1–7 correspond to the position of the underlying Turing patterns. (B) Branching patterns (top) and underlying spot patterns (bottom). Patterns 1–7 correspond to points 1–7 in A. (C) Dispersion relations for patterns, with wavenumber k1-6 corresponding to patterns 1–7 in B. (D) Comparison of the wavelength (2π/wavenumber) from patterns 1–7 in B. Parameters: c = 0.002, μ = 0.16, ρA = 0.03, DA = 0.02, v = 0.04, ρH = 0.00007, DH = 0.3, c0 = 0.02, DS = 0.06, (B1-7) ε = 1.5/1.2/0.9/0.45/0.1/0.06/0.045. In the 2D patterns, black indicates a high concentration, and gray indicates a low concentration.
Fig 10.
Effect of the wavelength on the branching patterns in the Turing region for ρH = 0.00013.
(A) The cell differentiation trajectories of the branching patterns cross the Turing region. Points 1–6 correspond to the position of the underlying Turing patterns. (B) Branching patterns (top) and underlying spot patterns (bottom). Patterns 1–6 correspond to points 1–6 in A. (C) Dispersion relations for patterns, with wavenumber k1-6 corresponding to patterns 1–6 in B. (D) Comparison of the wavelength (1π/wavenumber) from patterns 1–6 in B. Parameters: c = 0.002, μ = 0.16, ρA = 0.03, DA = 0.02, v = 0.04, ρH = 0.00013, DH = 0.3, c0 = 0.02, DS = 0.06, (B1-6) ε = 1.1/0.85/0.7/0.07/0.06/0.045. In the 2D patterns, black indicates a high concentration, and gray indicates a low concentration.
Fig 11.
Effect of the wavelength on the branching patterns in the Turing region for ρH = 0.00015.
(A) The cell differentiation trajectories of the branching patterns cross the Turing region. Points 1–6 correspond to the position of the underlying Turing patterns. (B) Branching patterns (top) and underlying spot patterns (bottom). Patterns 1–6 correspond to points 1–6 in A. (C) Dispersion relations for the patterns, with wavenumber k1-6 corresponding to patterns 1–6 in B. (D) Comparison of the wavelength (2π/wavenumber) from patterns 1–6 in B. Parameters: c = 0.002, μ = 0.16, ρA = 0.03, DA = 0.02, v = 0.04, ρH = 0.00015, DH = 0.3, c0 = 0.02, DS = 0.06, (B1-6) ε = 0.9/0.8/0.65/0.07/0.06/0.045. In the 2D patterns, black indicates a high concentration, and gray indicates a low concentration.