Lee SW and Morishita Y (2017), Possible roles of mechanical cell elimination i...

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PLoS Comput Biol 2017 Jul 13;137:e1005651. doi: 10.1371/journal.pcbi.1005651.

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Possible roles of mechanical cell elimination intrinsic to growing tissues from the perspective of tissue growth efficiency and homeostasis.

Lee SW , Morishita Y .

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Cell competition is a phenomenon originally described as the competition between cell populations with different genetic backgrounds; losing cells with lower fitness are eliminated. With the progress in identification of related molecules, some reports described the relevance of cell mechanics during elimination. Furthermore, recent live imaging studies have shown that even in tissues composed of genetically identical cells, a non-negligible number of cells are eliminated during growth. Thus, mechanical cell elimination (MCE) as a consequence of mechanical cellular interactions is an unavoidable event in growing tissues and a commonly observed phenomenon. Here, we studied MCE in a genetically-homogeneous tissue from the perspective of tissue growth efficiency and homeostasis. First, we propose two quantitative measures, cell and tissue fitness, to evaluate cellular competitiveness and tissue growth efficiency, respectively. By mechanical tissue simulation in a pure population where all cells have the same mechanical traits, we clarified the dependence of cell elimination rate or cell fitness on different mechanical/growth parameters. In particular, we found that geometrical (specifically, cell size) and mechanical (stress magnitude) heterogeneities are common determinants of the elimination rate. Based on these results, we propose possible mechanical feedback mechanisms that could improve tissue growth efficiency and density/stress homeostasis. Moreover, when cells with different mechanical traits are mixed (e.g., in the presence of phenotypic variation), we show that MCE could drive a drastic shift in cell trait distribution, thereby improving tissue growth efficiency through the selection of cellular traits, i.e. intra-tissue "evolution". Along with the improvement of growth efficiency, cell density, stress state, and phenotype (mechanical traits) were also shown to be homogenized through growth. More theoretically, we propose a mathematical model that approximates cell competition dynamics, by which the time evolution of tissue fitness and cellular trait distribution can be predicted without directly simulating a cell-based mechanical model.

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	Fig 1. A classification of cell competition phenomena.We classified the experimental observations of cell competition based on two criteria. One is whether the elimination of cells is based on genetic differences or not; i.e. the focal cell population is either genetically homogeneous or heterogeneous. The other is the mechanical relevance; i.e. cell elimination is a mechanically-driven or non-mechanical event. As described in the text, the latter criterion is not completely exclusive and thus we included cases in which the mechanical relevance is unclear into the “non-mechanical” category. In this study we mainly focus on the case in the upper-left, i.e. mechanical cell elimination from a genetically-homogeneous growing tissue.
	Fig 2. Settings for mechanical tissue simulation using a vertex dynamics model.(A) A schematic diagram of the vertex dynamics model. In this model, each cell is represented as a polygon formed by linking several vertices. Each vertex moves in a manner that decreases the energy U of the system. U is composed of three terms: area constraint, line tension and perimeter contractility. The mechanical traits of each cell are represented by two kinds of parameters Λαβ and Γα. Λαβ is the coefficient for tension acting on a cell edge between cell α and cell β (blue line). The other parameter Γα is the contractility of the apical surface (red dashed line). (B) The rule of cell division orientation. A cell divides with an axis through its center (left). The orientation of the axis is a random variable obeying the von Mises distribution f(θ;κ) around the shortest axis θ obtained by elliptical approximation of the cell. The randomness can be regulated by a single parameter κ (right). (C) Cell rearrangement (T1-process) and elimination (T2-process). As a consequence of push-pull dynamics between cells in a growing tissue, the spatial rearrangement and elimination of cells occur. The rearrangement occurs when the edge length is less than the T1-threshold θT1 (left). Elimination is implemented simply by removing the cell whose area is less than the T2-threshold θT2, which is called a T2-process (right).
	Fig 3. Cell elimination rate and fitness in a pure population.(A) Dependence of the cell elimination rate ε (green circles) and the cellular fitness ϕCell (= ϕTissue) (purple crosses) on mechanical/growth parameters in simulations, which can be well approximated by Gaussian-type functions (solid green and dashed purple lines), ε = Exp[-α(ζ-β)2]+γ, where ζ represents any one of the mechanical/growth parameters (Λ, Γ, θT1, κ and μ). Parameters: (α, β, γ) = (245, 0.24, 0) for Λ, (1800, 0.076, 0) for Γ, (12.5, 0.465, 0.0875) for θT1, (0.095, 5.2, 0.039) for κ, (0.0016, 24.3, -0.31) for μ. (B) Time evolution of the elimination rate ε in a case with the reference parameter set (see the red circles in Fig 3A; Λ = 0.14, Γ = 0.04, θT1 = 0.1, κ = 0 and μ = 3.47×10−3); the red line is the average at each time point over 10 trials shown by the gray lines. The black line is the temporal average of the red line. Since the variation in elimination rate is not large during tissue growth when the number of cells ranges from N = 250 to N = 20,000, we adopt its temporal average as the typical value of the elimination rate for each parameter set.
	Fig 4. Cell size variance as the common geometrical determinant of the elimination rate.When the cellular mechanical/growth parameters change, geometrical quantities (specifically, variation in cell size, cell rearrangement frequency, and cell shape anisotropy) and elimination rate change. (A) For example, when the tissue fluidity (that is determined by Λ, Γ, and θT1) increases (or decreases), both the variation in cell size and elimination rate decrease (or increase; top panel). Similarly, for changes in the other parameters, κ (which determines the randomness of division orientation) and μ (growth rate), both cell size variation and the elimination rate change with the same tendency. In this manner, the responses of cell size variation and elimination rate to changes in any parameter are highly and positively correlated. In contrast, the responses in cell rearrangement frequency (B) and cell shape anisotropy (C) to parameter change does not necessarily show the same tendency as the response in cell elimination; for instance, for changes in tissue fluidity and growth rate, the responses of rearrangement frequency and elimination rate are negatively correlated, whereas for changes in division orientation they are positively correlated. Thus, among these three geometrical quantities, only cell size variation had a response consistent with the response in elimination rate, and we can conclude that cell size variance is the common geometrical determinant of the cell elimination rate. Note that cell shape anisotropy was calculated as 1-(length of the shortest axis/length of the longest axis) after approximating each cell by an ellipse. All simulations were performed using a pure population. Symbols: blue crosses (Λ), cyan pluses (Γ), green triangles (θT1), orange diamonds (κ), and red circles (μ).
	Fig 5. Relationship between spatial heterogeneity in cell density and MCE.(A) A schematic diagram of the calculation for regional dependence in cell density. The whole tissue was divided into three regions, and for each region the temporal averages in cell density, cell size variance, and cell elimination rate were calculated. (B) The relationships between the temporal averages in cell density and cell size variance (left), between the temporal averages in cell size variance and cell elimination rate (middle), and between the temporal averages in cell density and T1-frequency (right). When the growth rate is not high (e.g., for the reference value), all three values were nearly the same for all three regions. In contrast, in cases with a higher growth rate, those values were strongly dependent on region. More central regions have higher density, which inhibits smooth cell rearrangement (i.e. cell density and T1-frequency were negatively correlated) and thus leads to an increase in cell size variance and elimination. Cell density and size variance had a clear positive correlation, as did size variance and elimination rate. (C) A schematic diagram of the calculation for change in cell size variance around the elimination point before and after its occurrence. (D) Cell size variance clearly decreased due to MCE, demonstrating that recovery in the homogeneity of cell density (i.e., density homeostasis) is one possible role for MCE.
	Fig 6. Spatial heterogeneity in stress magnitude as the mechanical cause of cell elimination.(A) The stress within a tissue was evaluated by two types of stress tensors that are discrete versions of Cauchy’s stress and defined by using the forces acting on the vertices that compose each polygonal cell. The forces acting on vertex i of cell α are composed of the pressure inside the cell, Pα, and tension at the two edges linked to the vertex, Tij,α, Tik,α. The edge tension involving cell α was assumed to be half of the tension acting on the focal edge, Tij, and the remaining half was allotted to the other cell β that shares the edge, i.e., Tij = Tij,α+Tij,β = 2Tij,α. The stress tensor σ(A)α given by Eq (25) was calculated using the force vector Fi, positional vector ri from cell center, and normal vector njk (right upper). The stress tensor σ(B)α given by Eq (27) was calculated using the pressure Pα, identity matrix I, tension Tij,α, and positional vector rij from focal vertex i to the adjacent one j (right lower; also see the Models section). (B) Stress magnitude, stress anisotropy and its orientation. The calculated tensors were characterized by the two scalars, stress magnitude σ1+σ2 and stress anisotropy σ1-σ2, where σ1 and σ2 (σ1>σ2) are the principal stresses. The left panel shows an example of stress distribution. The color indicates stress magnitude: blue for tensile states and red for compressed states. The line inside each cell indicates the degree of stress anisotropy and its orientation. Longer lines show more anisotropic states. The orientation is the direction of the maximum principal stress. (C) Correlations between the stress state and cellular geometry. Relationship between stress magnitude and cell area (top). Relationship between stress anisotropy and cell shape anisotropy (middle). Relationship between directions of stress anisotropy and cell shape anisotropy (bottom). Each graph is composed of 10,000 data points calculated from cells randomly selected at a specific time in one simulation run. Parameter values: the reference set, Λ = 0.14, Γ = 0.04, θT1 = 0.1, κ = 0 and μ = 3.47×10−3. Correlation coefficient: ρ≈0.94 (top), ρ≈0.97 (middle), ρ≈0.99 (bottom). (D) Good correlation between the mean change in local stress magnitude and stress heterogeneity over the tissue (left) or the elimination rate (right). All simulations were performed using a pure population. Symbols: blue crosses (Λ), cyan pluses (Γ), orange diamonds (κ), and red circles (μ).
	Fig 7. Summary of the results obtained with a pure population in the 2nd-4th subsections.During tissue growth, cell divisions induce surrounding tissue compression, and this results in an increase in the spatial heterogeneity of stress magnitude and cell size. As the cell size variance increases, more cells are eliminated from the tissue. In turn, cell elimination releases the compression due to cell division. From the viewpoint of energy efficiency, this process should be reduced and more newly-born cells should contribute to tissue growth; this can be achieved by higher tissue fluidity, division along the shortest axis, and lower growth rate.
	Fig 8. Possible mechanical feedback mechanisms that could improve tissue growth efficiency and homeostasis.(A) Time evolution of the elimination rate ε with and without mechanical feedback mechanisms. All feedback mechanisms outlined in the text significantly reduced the elimination rate ε. (B) Time evolution of tissue size with and without mechanical feedback mechanisms. Although the density-dependent growth regulation (red line) leads to a drastic decrease in the elimination rate, it takes much more time to attain a certain tissue size (e.g. 20,000 cells) compared to cases without feedback (black line), meaning that it was not necessarily efficient in regard to developmental speed. In contrast, the other feedback mechanisms, including feedback regulation of tissue fluidity given by Eqs (6) and (7) (cyan and blue lines), and that of cell division orientation (green line), could improve tissue growth efficiency by reducing the elimination rate and by maintaining a normal growth speed. (C) (Left) Incompatibility between the reductions in cell size variance and stress magnitude under a feedback model of tissue fluidity (Γ) given by Eq (6). When the feedback strength c is positive and larger, the cell size/density variance and elimination rate decreased but the variance in stress magnitude increased (see also (D) for an example of the simulation results). In contrast, when c is negative and smaller (i.e., \|c\| is larger), the opposite is true; although the cell size/density variance and elimination rate increased, the stress state was homogenized. (Right) For comparison, the relationship between the elimination rate and variance in cell size or stress magnitude is shown in the cases without feedback. The variances and elimination rate are lower for higher tissue fluidity, and vice versa. Graph symbols: red circles indicate the variance in cell size; blue diamonds indicate the variance in stress magnitude. Parameter values: (left) c ranges from -0.004 to 0.004. (right) Γ ranges from 0.02 to 0.05. (D) Examples of the spatial distribution of cell size (upper) and stress magnitude (lower) in the presence and absence of mechanical feedback; from the left: without feedback, with density-dependent growth regulation, feedback to tissue fluidity, and feedback to cell division orientation. All feedbacks could reduce the variance in cell size and density, but in the case of the feedback to tissue fluidity, there was incompatibility between the reductions in variances of cell size and stress magnitude. For the other two feedback mechanisms, the both were compatible.
	Fig 9. Improvement in tissue growth efficiency and homeostasis over time with cell competition through MCE in a mixed population.(A) A simulation result for the time evolution of frequency distribution for a mechanical cell trait. In the case of perfect inheritance of the trait (i.e., h2 = 1), the frequency distribution drastically changes (left). At the same time, tissue-level fitness and density homeostasis are improved through tissue growth (right). In other words, intra-tissue evolution can occur. In contrast, in the case of no inheritance (i.e., h2 = 0), the frequency distribution of a cellular trait, the tissue fitness, and density homeostasis never changes. (B) Schematic diagrams of a model for inheritance of a mechanical trait. The trait of a daughter cell is inherited from its parent (blue) with probability q and is uniformly and randomly chosen from all the possibilities (red; among a discrete value of N-kinds) with probability 1-q (thus, q = 1 for perfect inheritance and q = 0 for no inheritance). (C) Growth curves for different values of heritability. The black solid lines are the curves obtained by simulations of vertex dynamics model. The green dashed lines show the growth curves obtained by the approximation (i), which is effective in cases with much higher heritability where each cell population of each trait can be regarded to grow independently (see Eq (9)). The purple dashed lines show the growth curves by the mean field approach (approximation (ii)), which is effective in a case with lower heritability (see Eqs (8)–(13)).
	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.

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