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Fig 10. Approximating the dynamics of trait distribution.(A) Schematic diagrams of the calculation for effective fitness. In a pure population, each cell has an elimination rate (solid green line) and cellular fitness (dashed purple line) determined according to its own mechanical trait, e.g., the coefficient of line tension (ΛW and ΛG, W and G indicate the white cells and gray cells, respectively). In contrast, in a mixed population, the elimination rate of each cell depends on the mechanical properties of adjacent cells. Suppose that each cell makes contacts with cells of the same or different kind with equal probability. Then, when the effective coefficient of line tension of an edge is modeled by max(ΛW, ΛG) (see text) and the inequality ΛW < ΛG is assumed, the effective value is Λ˜W=(3ΛW+3ΛG)/6 for the white cells and Λ˜G=ΛG for the gray cells. Using these effective trait values, the fitness of each cell can be calculated as ϕCell(Λ˜W) and ϕCell(Λ˜G), where ϕCell is the function of cellular fitness obtained in Fig 3A. (B) Time evolution of the spatial correlation of mechanical cell traits. We used Moran’s I for the index of spatial correlation. For higher heritability, the correlation was higher, meaning that cells with the same trait (i.e., descendant cells) tended to easily form clusters, for which the approximation (i) works well. In contrast, for lower heritability, the correlation was lower, and cells with different traits would be well mixed, for which approximation (ii) works well. (C) Time evolution of the total number of eliminated cells estimated by the proposed approximation method in a case with much larger tissue size. When the number of cells within a tissue reaches 106, the total number of eliminated cells through tissue growth was 4×104 in the case of h2 = 1 (obtained by approximation (i)), while 26×104 for h2 = 0 (obtained by approximation (ii)). This demonstrates that cell competition through MCE, especially in earlier phases of development, is able to considerably reduce energy loss and improve tissue growth efficiency in the presence of high heritability. (D) Frequency distribution of a mechanical trait could evolve so that it has an intermediate peak value when tissue fluidity affects both cell elimination and proliferation rates (right). (Middle) Specifically, the cell proliferation rate was given as a monotonously increasing function of Λ (solid red line; 1/(1+Exp[-10Λ])). The blue dashed line shows the energy efficiency (1-m/μ), and h2 = 0.87 was adopted. In actuality, mechanical cell traits are expected to be homogenized through tissue growth under different tradeoffs. Image published in: Lee SW and Morishita Y (2017) © 2017 Lee, Morishita. This image is reproduced with permission of the journal and the copyright holder. This is an open-access article distributed under the terms of the Creative Commons Attribution license Larger Image Printer Friendly View |