Fig 1.
Schematic illustration of the N-cell model.
Each cell has a common catalytic network, while the nutrient X0 transported from the medium is transformed to catalytic components for cellular growth. The nutrient X0 is supplied to the medium from the exterior. The reaction from Xi to Xj is catalyzed by another component, Xl. Some components (orange circles) are diffusible and exchanged via the medium, while others (green squares) are not. In the medium, n cells coexist and interact with each other (1≤n≤N).
Table 1.
The parameters and variables of the N-cell model.
Fig 2.
Typical behavior of the N-cell model in category (a).
(A) Chemical compositions of N cells surviving at time t = 5×105. The concentration is plotted with a color code, with the vertical axis indicating the cell index m, and the horizontal axis indicating the chemical index i, while the top band designates the composition of an “isolated” cell. Cells differentiate into two types with distinct compositions. (B) The time series of
in interacting N cells surviving at time t = 4×104, overlaid for all cells shown as different colors. Each line represents the concentration of the chemical X1 in a different cell, plotted with different colors over 100 cells. (C) The time series of the number of cell divisions per cell for interacting (red) and isolated (blue) cells. Interacting cells grow faster than isolated cells; i.e., Rμ > 1.
Fig 3.
Behavior of simplified networks.
(A) Network 1. (B) Network 2. In (A) and (B), each number represents the chemical index, with orange-circle and green-square nodes representing diffusible and non-diffusible chemicals, respectively. The difference between networks 1 and 2 lies only in the diffusibility of X5. The dashed arrows denote the diffusive fluxes of chemicals, and the thick arrows indicate catalytic reactions. The chemical at the arrowtail is transformed to the chemical at the arrowhead, catalyzed by the chemical labeled on the edge; e.g., the left arrow from X0 in (A) denotes the catalytic reaction X0 + αX1 → X3 + αX1. When the cells differentiate, type-1 (type-2) cells mainly produce the chemicals X1 and X3 (X2 and X4), and receive X4 (X3) from type-2 (type-1) cells, which are illustrated in the lower panel with color. (C) and (D) are examples of the behavior of network 1 for (C, Vmed, D) = (0.15, 100, 1). (C) Chemical concentrations in N cells surviving at time t = 105 are plotted according to the color code shown in the sidebar, with the vertical axis as the cell index m and the horizontal axis as the chemical index i. The top band designates an “isolated” cell. (D) The time series of the number of cell divisions per cell for interacting (red) and isolated (blue) cells.
Fig 4.
The behavior of the r1cell and r2cell models with network 1.
(A) A phase diagram (D = 1). The blue area designates phase (I) in which cells cannot exhibit differentiation, and reach a single fixed point in both the r1cell and r2cell models. In phases (II) and (III), cells always differentiate. In phase (II), shown in cream color, cells exhibit pitch-fork bifurcation from a homogeneous state. In phase (III), “oscillation-death” occurs in the dynamics of the chemical compositions of the two cells. In phase (IV), depicted by orange, if the initial difference between the compositions of the two cells is large enough, “oscillation-death” occurs; otherwise, the r2cell model exhibits non-differentiated, synchronized oscillation. (B) The time series of concentrations of X1 in phase (II) for (C, V, D) = (0.27, 0.1, 1). (C) The time series in phase (III) for (C, V, D) = (0.1, 0.8, 0.1). For both (B) and (C), the time series for the r1cell (r2cell) model is displayed on the upper (lower) column.
Fig 5.
Growth enhancement with network 1 (D = 1).
(A) Dependence of Rμ on parameters C and V. Cells can differentiate given the parameters to the left of the green line. (B) Dependence of Rμ on V (C = 0.1). Rμ is the ratio of the growth rate of differentiated cells to that of isolated cells. (C) Dependence of Rp on V (C = 0.1). Rp is the degree of increase in the production of X3-X4 conferred by differentiation.
Fig 6.
The stability of collective growth.
(A) Network 1 in the r2cell model for (C, V, D) = (0.1, 1, 0.1), representing unstable collective growth. (B) Network 2 in the r2cell model for (C, V, D) = (0.15, 3, 1), representing balanced collective growth. In (A) and (B), the horizontal axis denotes the volume ratio of the two cells, r(1)≡v(1)/(v(1) + v(2)), and each black line displays the growth rate in the r1cell model μ(iso). Blue circles and orange triangles denote the growth rate of type-1 μ(1) and type-2 μ(2), respectively. Green rectangles indicate the total growth rate of the cell aggregate μ(tot)≡r(1)μ(1) + r(2)μ(2). (C) and (D) show the cell number distribution in the N-cell model (N = 100). Blue, red, and gray bars designate the numbers of type-1, type-2, and non-differentiated cells, respectively. (C) Case for network 1 with (C, Vmed, D) = (0.01, 100, 0.1). (D) Case for network 2 with (C, Vmed, D) = (0.3, 50, 0.1).