Crypt fusion as a homeostatic mechanism in the human colon

Objective The crypt population in the human intestine is dynamic: crypts can divide to produce two new daughter crypts through a process termed crypt fission, but whether this is balanced by a second process to remove crypts, as recently shown in mouse models, is uncertain. We examined whether crypt fusion (the process of two neighbouring crypts fusing into a single daughter crypt) occurs in the human colon. Design We used somatic alterations in the gene cytochrome c oxidase (CCO) as lineage tracing markers to assess the clonality of bifurcating colon crypts (n=309 bifurcating crypts from 13 patients). Mathematical modelling was used to determine whether the existence of crypt fusion can explain the experimental data, and how the process of fusion influences the rate of crypt fission. Results In 55% (21/38) of bifurcating crypts in which clonality could be assessed, we observed perfect segregation of clonal lineages to the respective crypt arms. Mathematical modelling showed that this frequency of perfect segregation could not be explained by fission alone (p<10−20). With the rates of fission and fusion taken to be approximately equal, we then used the distribution of CCO-deficient patch size to estimate the rate of crypt fission, finding a value of around 0.011 divisions/crypt/year. Conclusions We have provided the evidence that human colonic crypts undergo fusion, a potential homeostatic process to regulate total crypt number. The existence of crypt fusion in the human colon adds a new facet to our understanding of the highly dynamic and plastic phenotype of the colonic epithelium.

Therefore, under the null hypothesis, the probability of 21 (Nseg) or more of the 309 bifurcating crypts (Nbifurcate) to be perfectly segregated is then given by the binomial distribution:

Probability of perfect segregation in a crypt with an odd number of stem cells
Above, the probability of a crypt undergoing fission perfectly segregating all CCO-and CCO+ cells to each arm of the bifurcating crypt was discussed for the case of an even number of stem cells.
Considering the case where the number of stem cells is odd is straightforward. Instead of each arm of the crypt receiving S/2 stem cells, one arm receives (S+1)/2 and the other (S-1)/2. The number of possible bifurcation planes is then S. In analogue to the even case, the probability of a crypt undergoing fission giving rise to perfectly segregated arms can be estimated as: Substituting the distribution of partially labelled crypts as described above, we estimate ! to be the same as in the even case: The rest of the calculation proceeds as previously.

Crypt fission and fusion rate
Crypt fission allows the expansion of patches of clonal tissue within the colon. Previous studies 2, 3 have attempted to infer the rate of crypt fission from the distribution of patch sizes, identifying clonal patches using neutral mutational markers (such as CCO deficiency, as used in this study). These studies assumed that clonal patches can only grow via crypt fission, however the existence of crypt fusion suggests that clonal patches can also shrink.
We modelled the processes of crypt fission and fusion as a linear birth-death process. The rate at which a patch transitions from containing 8 labelled crypts to containing 8 + 1 labelled crypts is the fission rate per crypt, n Eoppoqr , multiplied by the patch size 8. The death rate is more complicated to calculate. If a CCO-crypt fuses with a neighbouring CCO+, the patch size will decrease by 1 half of the time (e.g. for the 50% of cases where the CCO+ clone 'wins'). If instead a CCO-crypt fuses with another CCO-crypt within the patch, the patch size will always decrease by 1. The rate at which a patch decreases in size depends on the size of the patch. For simplicity we shall consider the case where the crypts are organised into a regular structure with 4 nearest neighbours, in line with the observed average coordination in the colon. An isolated CCO-crypt has 4 unlabelled neighbours, so that loss will occur at rate s tubvwx k . Each CCO-crypt in a patch of size 2 has 3 unlabelled neighbours and 1 labelled neighbour, so each crypt will transition to CCO+ at a rate . In this manner, the rate at which larger patches shrink can be calculated relatively simply. However, 95% of the patches observed contained 4 or fewer crypts. Therefore, to allow the application of an analytically-tractable linear birth-death model, we calculate an effective crypt loss rate by finding the weighted mean of the loss rate for patches of size 4 or less. For our data, we find an effective loss rate of 0.544 n EÅpoqr , hence we model the rate at which a patch of size 8 transitions to a patch of size 8 − 1 as 0.544 n EÅpoqr 8.
$ ≠ É ∧ 8 ≥ 1: Prior to fitting the patch size distributions, we corrected our data for the possibility of spontaneous, independent CCO-mutations in adjacent crypts, leading to the misclassification of two individual clones as a patch of size 2. If we estimate the probability that a crypt becomes labelled, ! = Çî, as the number of labelled patches divided by the total number of crypts, we can estimate the number of patches of size 2 that are, in fact, separate clones as 2! k (1 − !) õ N +,+() (where we are still assuming that the crypts are organized into a regular structure with 4 nearest neighbours). Similar corrections could be performed for patches of size 3 and above, however ! ≪ 1 so the number of misclassified size 3 patches was assumed to be negligible.
To fit to the data, we performed a maximum likelihood estimate on a patient-by-patient basis (Supplementary Figure 2). The log-likelihood of observing Npatches patches, each of size mi, is: The log-likelihood is purely a function of ç, and ç($, É) is a many-to-one function, hence we cannot separately determine n Eoppoqr and n EÅpoqr from our data. Instead, the fission rate was fixed to be equal to the fusion rate, as discussed in the main text. The log-likelihood was maximised numerically using the 'Nelder-Mead' method and the confidence intervals on each fit were estimated from the Fisher information matrix.

The duration of crypt fission/fusion
If we assume that the rate of crypt fission per crypt is constant over time, then the probability that a given crypt undergoes fission in a time-interval ∆7 is simply n Eoppoqr ∆7. Let the duration of crypt fission be î, then if we observe a crypt undergoing fission at a given time 7, that bifurcation must have begun later than 7 − î. Hence, the probability that a crypt is undergoing crypt fission at a given time is n Eoppoqr î. Due to the inability to accurately evaluate Type I bifurcations as fission or fusion events, we must estimate the number of bifurcations arising from crypt fission as half the total number of bifurcations (A •oEÅ ¶ßCå® ). If we model the number of crypt-fission bifurcations as a binomial process with probability n Eoppoqr î, then we can estimate the duration of crypt fission as î = a ©vtu™´¨≠c ks tvbbvwx a ≠w≠¨AE .

The spatial distribution of bifurcation events
Visual Where 5 o∑ is the pairwise distance between the : åB and ∫ åB points, ¥ is the mean density of points (estimated as ¥ µ = r ª ), ; is a search radius defining the scale of spatial interactions, ∂ is a Heaviside step function and ∏ o∑ is an edge correction factor (∏ o∑ is the fraction of the circumference of a circle of radius 5 o∑ centred on the point that lies within the sampling region. For a spatially homogeneous Poisson process ∞ ± (;) should approximately equal ;. To determine whether deviations from ; are due to spatial clustering or consistent with random noise, a set of 999 simulations containing / randomly scattered points within Ø were generated, and for each simulation Ripley's L function (∞ poà (;)) was calculated. For each simulation, the maximum difference between the simulation and the theoretical value of ∞, É = max(|∞ poà (;) − ;|), was recorded. To construct a global simulation envelope of significance â = 0.01, the 10 th largest É value was found. If any value of ∞ ± (;) has a larger deviation from ; than this critical value, then the spatial distribution is significantly different from a spatially homogeneous Poisson process.
For a significance of 0.05, samples 4, 6, 7a, 7b, 8 and 10 exhibited evidence of spatial clustering (Supplementary Figure 4). This analysis is robust to our choice of â, for a stringent significance of â = 0.001, only sample 7b is no longer statistically significant. Note that all of these patients are FAP or AFAP, and each sample contained far larger numbers of bifurcations than the IBD or diseasefree samples. This analysis cannot distinguish between underlying spatial heterogeneity of the fission/fusion rates within the sample, or local interactions between fission/fusion events.