Crypt reconstruction and analysis. (*A*) Crypt reconstruction. The arrangement of 5,462 crypts presumed to reside on the intestinal surface imaged in is reconstructed by Metropolis chain simulation of a distribution over all possible crypt arrangements. This posterior distribution is informed by the known crypt arrangement from a *ROSA11* mouse (expanded version of ), which constitutes a prior distribution of arrangements, and by the binary image (), according to techniques from Bayesian image reconstruction (see *Materials and Methods*). The techniques give higher likelihood to arrangements in which all of the pixels within a crypt have the same color. Crypt centers from one realization of the posterior distribution are plotted; crypts are colored according to a majority rule of contained pixels. Crypt neighbors are determined by Delaunay triangulation, and the connecting edges are drawn in red if the adjacent crypts are heterotypic. This reconstruction contains 16,902 neighboring crypt pairs, 14% of which are heterotypic. If, for example, polyclonal tumors form by the interaction of two neighboring crypts in a chimeric intestinal surface like , then we estimate that 14% of such tumors would be heterotypic. This is the crypt pair phenotype index (). (*B*) Crypt reconstruction detail. Shown is a higher magnification of the area marked in *A*.(*C*) Crypt neighborhood system. One crypt from *B* is highlighted in red and its neighbors of various orders are indicated by different colors. For instance, this crypt has 6 nearest neighbors (green) and 13 second-order neighbors (yellow). (*D*) Crypt neighborhood statistics. Crypt reconstructions as in *A* were obtained for 17 representative images, and neighborhoods were identified for all crypts. Plotted for various neighborhood orders is the average number of neighbors for each crypt. To avoid boundary problems, the average is computed over all interior crypts in all images, where a crypt is interior if it resides in the middle 80% of the image in both coordinates. On average, crypts are within a few steps of many other crypts. (*E*) Heterotypic crypt neighborhoods. Averaging as in *D*, we compute for each white crypt the average proportion of white crypts in its *n*th-order neighborhood (white circles) and for each blue crypt the average proportion of white crypts in its *n*th order neighborhood (blue circles). White crypts tend to lie near white crypts, and similarly for blue, but the patch sizes are such that two crypts seven or eight steps apart have independent colors. (*F*) Heterotypic tumor fraction. Averaging as in *D* and *E* we compute the probability ψ that a tumor is heterotypic, using various levels of crypt interaction. Shown in red is ψ when a tumor is formed from all crypts within an *n*th order neighborhood of some initiated crypt. Less than complete involvement is indicated by the other curves. For instance, the blue curve shows ψ when each curve within an *n*th-order neighborhood of an initiated crypt tosses a fair coin to decide whether or not to participate in the tumor. Importantly, the observed heterotypic fraction (22%) can be explained by intercryptal interactions between first- or second-order neighbors for a wide range of participation rates.

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