Inheritance entropy quantifies epigenetic regulation of cell-cycle exit in human bone marrow stromal cells

TL;DR

The study tackles colony-level heterogeneity in human bone marrow stromal cells (BMSC) by testing whether non-genetic, hereditary factors regulate cell-cycle exit. It introduces inheritance entropy, a topology-based metric derived from inactivity imbalances in single-cell lineage trees, and applies it to 32 clonal colonies with a null model based on generation-wise scrambling. The results show strong inheritance signals in 21 of 28 testable colonies, with an inactivity mutation lag averaging $3.1$ generations, supporting an epigenetic mechanism that precedes the appearance of inactivity. The findings link lineage topology to epigenetic regulation of colony behavior, offering a potential explanation for inter-colony heterogeneity and informing strategies to modulate BMSC potency for skeletal regeneration therapies. Mathematical constructs such as $I_m=|N_m^{\mathrm{left}}-N_m^{\mathrm{right}}|$, $w_m=\frac{I_m}{\sum_n I_n}$, and $S=-\sum_m w_m \log w_m$ underpin the analysis, with $S_{biological}$ typically lower than scrambled controls, and the null-hypothesis testing yielding $P<0.05$ in a majority of cases.

Abstract

Human bone marrow stromal cells (BMSC) include skeletal stem cells with ground-breaking therapeutic potential. However, BMSC colonies have very heterogeneous in vivo behaviour, due to their different potency; this unpredictability is the greatest hurdle to the development of skeletal regeneration therapies. Colony-level heterogeneity urges a fundamental question: how is it possible that one colony as a collective unit behaves differently from another one? If cell-to-cell variability were just an uncorrelated random process, a million cells in a transplant-bound colony would be enough to yield statistical homogeneity, hence washing out any colony-level traits. A possible answer is that the differences between two originating cells are transmitted to their progenies and collectively persist through an hereditary mechanism. But non-genetic inheritance remains an elusive notion, both at the experimental and at the theoretical level. Here, we prove that heterogeneity in the lineage topology of BMSC clonal colonies is determined by heritable traits that regulate cell-cycle exit. The cornerstone of this result is the definition of a novel entropy of the colony, which measures the hereditary ramifications in the distribution of inactive cells across different branches of the proliferation tree. We measure the entropy in 32 clonal colonies, obtained from single-cell lineage tracing experiments, and show that in the greatest majority of clones this entropy is decisively smaller than that of the corresponding non-hereditary lineage. This result indicates that hereditary epigenetic factors play a major role in determining cycle exit of bone marrow stromal cells.

Figures (8)

• Figure 1: Lineage trees of four samples of BMSC clonal colonies. Red segments represent active cells and black discs represent mitosis; the length of each segment is fixed, not related to the division time. Inactive cells (which are all G$_0$ in these samples) are represented as black segments ending in a cap. The first cell of the colony is not represented, as we do not record its birth, but only its division (the central disc of the tree). Ovals separate different generations. Cells at the last generation (the $k_7$ leaves), are tracked up to their birth, not up to their division, hence they are neither labeled as active nor inactive and we represent them in grey.
• Figure 2: Illustration of how the inactivity imbalance is calculated, here using a portion of a real lineage (20230503_13). White dotted segments represent $k_7$ leaves that are missing because of the presence of some inactive cells upstream in that branch. For each mitosis/node, the inactivity imbalance is defined as the modulus of the difference between the number of missing $k_7$ leaves in the right branch and the number of missing $k_7$ leaves in the left branch.
• Figure 3: The distribution of the inactivity imbalance across the different mitosis is very different in biological vs. randomly scrambled trees. (a) For lineage 20230503_13, the inactivity imbalance, $I_m$, is graphically represented as a purple vertical brushstroke on each mitosis/node (null imbalances are not marked). In this lineage most of the imbalance is concentrated on just two nodes; of these two mitosis, the one connecting generation 1 to generation 2 (imbalance 22) is most likely that at which there has been a mutation in the probability of emergence of inactive cells. This large heterogeneity in the distribution of the imbalances across the lineage is the clearest symptom of inheritance. The entropy $S$ measures exactly this heterogeneity; in this lineage $S_\text{biological} = 2.18$, which is rather low compared to: (b) One instance out of the $10^6$ randomly scrambled versions of lineage 20230503_13; the number of inactive cells and their generation is the same as in the biological lineage, but their radial positions across the tree have been randomly reshuffled, hence severing all potential hereditary relationships. In this tree the imbalance is more homogeneously distributed across the nodes than in the biological tree, thus giving a much lower inheritance signal: the entropy is indeed significantly larger than the biological one, $S_\mathrm{scrambled} = 3.17$. If we repeat the scrambling $10^6$ times, in only 7 cases we find $S_\mathrm{scrambled} \leq S_\text{biological}$, proving that the evidence of inheritance is very strong in this clone.
• Figure 4: Two examples of how the hereditary structure of the inactive cells is erased by the scrambling procedure and how this impacts on the inheritance entropy. Top: Lineage 20241203_03 (left) has a strong concentration of inactive cells, and therefore of missing cells, in the west wing of the lineage; in this tree the imbalance is generated at the very first mitosis (the central node). Instead, a randomly scrambled version of the biological tree (center) has inactive cells distributed evenly across all branches. The entropy of the biological tree is $S_\text{biological}=0.81$, while the entropy of the corresponding scrambled tree is $S_\mathrm{scrambled}=2.11$. When we produce $10^6$ scrambled versions of 20241203_03, we very rarely find $S_\mathrm{scrambled} \leq S_\text{biological}$; this means that the probability that the entropy of the biological case is so small by pure chance is extremely low. The distribution of the randomly reshuffled entropies is shown as a violin-plot on the right: it is evident that the biological value of the entropy is way below the bulk of the distribution of the reshuffled entropies. The P-value is simply the integral of this distribution below $S_\text{biological}$, which gives P$=9.0 \times 10^{-5}$, making the inheritance test for colony 20241203_03 highly significant. Bottom: A similar situation holds for lineage 20230503_13, which also gives a very strong inheritance signal.
• Figure 5: Result of the inheritance test for all colonies in the dataset for which the test can be run (i.e. for lineages with more than one inactive cell). For each colony, the violin plot represents the probability distribution of the entropy for the set of $10^6$ randomly scrambled trees, while the black line is the value of the entropy of the non-scrambled biological tree. The P-value is the total area of the violin plot which lies below the black line. Green: significant result of the inheritance test (P-value $< 0.05$). Gray: non-significant result of the inheritance test (P-value $\geq 0.05$).