Patterns of proliferative activity in the colonic crypt determine crypt stability and rates of somatic evolution

Abstract

Epithelial cells in the colon are arranged in cylindrical structures called crypts in which cellular proliferation and migration are tightly regulated. We hypothesized that the proliferation patterns of cells may determine the stability of crypts as well as the rates of somatic evolution towards colorectal tumorigenesis. Here, we propose a linear process model of colonic epithelial cells that explicitly takes into account the proliferation kinetics of cells as a function of cell position within the crypt. Our results indicate that proliferation kinetics has significant influence on the speed of cell movement, kinetics of mutation propagation, and sensitivity of the system to selective effects of mutated cells. We found that, of all proliferation curves tested, those with mitotic activities concentrated near the stem cell, including the actual proliferation kinetics determined in in vivo labeling experiments, have a greater ability of delaying the rate of mutation accumulation in colonic stem cells compared to hypothetical proliferation curves with mitotic activities focused near the top of the crypt column. Our model can be used to investigate the dynamics of proliferation and mutation accumulation in spatially arranged tissues.

Figure 1. Schematics of the linear process and proliferation kinetic curves.

A: We designed a linear process model to describe the essential features of cell movement in a colonic crypt. During each time step, a cell at position i is selected to divide. During mitosis, a mutation may occur with probability u, giving rise to one mutated and one wild type daughter cell, with equal probability of occupying either position i or i+1. Cell division pushes all cells to the right of position i upwards in the crypt by one position. The last cell is shed into the lumen of the colon. B: Cell death may occur after each round of cell division. The number of dying cells follows a Poisson distribution with mean λ; the positions of the dying cells are selected according to a uniform distribution. Dead cells are replaced by replenishing cell divisions. Dying cells at position j can only be replaced by cells of a similar (j+1≤k≤j+δ) or less (k<j) differentiated stage. Position k is selected according to the proliferation kinetics curve. If multiple cells die simultaneously, the replenishing cell divisions occur sequentially, in the order of j₍₁₎, j₍₂₎ … j_(m), where j₍₁₎, j₍₂₎ … j_(m) are ordered death position. The m positions for the replenishing cell divisions are selected according to the reweighted kinetic curve. C: Proliferation kinetic curves as a function of cell positions. The black curve represents the measured labeling index for normal human colon using bromodeoxyuridine (BRDU) . The colored curves represent the five kinetic curves under investigation. See the Methods section for details. Note that curve 1 is in good agreement with the measured labeling index.

Figure 2. The rate of cellular movement in the crypt column.

A: The left panel shows the distribution of the number of mitoses needed to push a cell at position i out of the crypt column for each proliferation curve. Colors match the corresponding proliferation curves in Fig. 1C. The horizontal axis represents the position of the cell in the crypt. Box plots indicate the distribution of the number of mitoses, assuming no cell death. The size of each bar indicates the amount of variation in the number of cell divisions. The right panel provides a zoomed-in view focusing on curves 1, 2 and 3. B: The effects of cell death on cellular movement in the crypt for three selected representative positions (2, 12, and 22) in the crypt column under the assumption of a uniform death selection function. As the death rate, λ, increases, the rate of cell movement increases, as shown by the decreasing number of mitoses needed to push a cell out of the crypt column. C: The panel shows the effects of cell death on the mitotic stress of the stem cell assuming a uniform death selection function. The average number of times the stem cell is selected for divisions is displayed as a function of cell death for the proliferation kinetic curves. Without cell death, the number of times the stem cell is selected for cell division is identical for all curves. As λ increases, the mitotic stress on the stem cell increases. The magnitude of increase depends on the shape of proliferation kinetic curve. Dots represent results from simulations, whereas the lines are exact results based on the terms inside the parentheses in Eq. 8. All graphs are generated based on 1,000 simulations for each kinetic curve under each scenario. All cells are assumed to have identical relative fitness values.

Figure 3. The single mutation model.

A: Schematic representation of the single mutation model. B: An example of the dynamics of somatic evolution in the crypt column. The curves show the proportions of cells in the crypt column. The colors correspond to cell types in panel A. The gray shaded region indicates the stem cell is mutated to APC ^+/−. For illustration purposes, u₀ = 0.01. C: The number of mitoses needed for a wild type crypt column to transition to an APC ^+/− state for various proliferation kinetic curves in the absence of cell death. As expected, the numbers of cell division needed to reach mutant fixation are the same for all curves. The mutation rate is u ₀ = 10⁻⁷ per cell division. Box plots are color coded, corresponding to the curves in Fig. 1C. D: The number of mitoses needed for fixation of APC ^+/− cells for various proliferation kinetic curves, measured from the time at which the stem cell accumulates the APC ^+/− mutation assuming no cell death. The gray area corresponds to the gray shaded interval in panel B. E: Acceleration of mutation accumulation due to cell death. As the death rate, λ, increases, fewer cell divisions are required for a mutated stem cell to arise. The comparison between panels C–E highlights the importance of proliferation kinetics of non-stem cells in the presence of cell death. F: Effects of fitness differences and proliferation curves on the rate of somatic evolution. The range of relative fitness spans from 0.5 to 2.0. G: The left panel shows the effects of fitness differences and different proliferation curves on the rate of APC ^+/− fixation, starting from an APC ^+/− stem cell in the absence of cell death. The panel on the right provides a zoomed-in view on curves 1, 2 and 3.

Figure 4. The two mutation model.

A: Schematic representation of the mutations leading to inactivation of both APC alleles, at rates u ₀ and u₁, respectively. B: A representative example of the dynamics of somatic evolution in the linear process. Each color-coded curve represents the proportion of each cell type in the crypt column, with colors corresponding to those in panel A. The gray shaded region represents the time interval during which the stem cell is in the APC ^+/− state. The yellow shaded region represents the time interval during which the stem cell is in the APC ^−/− state. C–F: Conditional probability for losing the second APC allele before an APC ^+/− cell at a particular position along the crypt column is “flushed” out of the crypt in the absence of cell death. At each position between 2 and 80, 1,000 simulation runs are generated in the absence of cell death to determine the probabilities of APC ^+/− cells gaining new mutations before being “flushed” out of the crypt column. Four different rates of inactivating the second APC allele, u ₁, were investigated and are shown on the top of the four sub panels. The bottom two panels provide zoomed-in views.

Figure 5. The effects of chromosomal instability and tunneling.

A: Schematic representation of the mutations leading to the inactivation of both APC alleles incorporating chromosomal instability. B: The upper panel displays a representative example of linear somatic evolution dynamics. Each color-coded curves represent the proportion of each cell type in the crypt column, with colors corresponding to those in panel A. Dashed and solid lines correspond to cells with and within CIN respectively. The gray and yellow shaded regions represent the time interval during which the stem cell is in the APC ^+/− state and APC ^−/− state respectively, regardless of CIN status. The lower panel provides a zoomed-in view. Notice that the dashed red curve does not reach 1, which signifies tunneling. This representative simulation run is performed using the uniform proliferation kinetics in the absence of cell death and fitness differences. The mutation rate is inflated to u ₀ = u₁ = u₂ = 10⁻³ and u₃ = 0.01 for computational speed. C: The tunneling probability as a function of u₃ and the cell death rate, λ, for the five proliferation curves under uniform death selection. To reduce the extent of complexity, tunneling rate is simulated using a three-state system consisting of APC ^+/−, APC ^+/− CIN and APC ^−/− CIN cells instead of the six-state system as illustrated in Panel A. Each simulation run starts with a crypt column seeded with an APC ^+/− cell at the stem position. The number of simulation runs is set at 1,000. Stem cell death is allowed. D: Concordance between simulated tunneling rates and analytical rates for linear systems of length N = 10, 20, …100 with equal proliferation probability at each position, using mutation rate u ₃ = 0.001, 0.01, 0.1 and 1.0. All simulations were performed for 1,000 runs. E: Analytical tunneling rate for the five proliferation curves at different mutation rate.