Integrating mathematical modeling with high throughput imaging explains how polyploid populations behave in nutrient-sparse environments
Gregory J. Kimmel, Mark Dane, Laura M. Heiser, Philipp M. Altrock and Noemi Andor
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Why does a cancer cell move? Cell motility facilitates infiltration and invasion and is the first response of cells to sub-optimal growth conditions, such as deficiency in oxygen- or essentials nutrients. The identities of the growth-limiting resources that ultimately drive a cell's fate decision vary in space and time. To investigate some of that variability, we integrate experimental data of microenvironment diversity and RNA-seq analysis of a primary adherent breast cancer cell line with a mathematical model of invasion of a polyploid population. To quantify response to microenvironment diversity, cells were previously exposed to hepatocyte growth factor (HGF) in combination with 48 extracellular matrices (ECMs), followed by multi-color imaging (1). This was achieved with a technique called MEMA profiling. Two considerations were key to capture this experimental setting into a mathematical model. First, binding to the ECM can cause soluble factors (like HGF) to act and signal as solid phase ligands. Proteolytic degradation of these ECMs then creates haptotactic gradients. Thus, depending on the ECM, both, chemo- and haptotaxis can occur. Second, maximal growth-stimulating effects for HGF have been reported at concentrations twice as high as concentrations that maximize migration. This suggests a shift from proliferation to migration as resources get depleted.
Both assumptions were incorporated in a partial differential equations (PDE) model of growth dynamics in polyploid populations of various subpopulation compositions. At the core of the model lies the assumption that chemotactic/haptotactic response to an energy gradient is a function of the cell’s energetic needs. This trade-off implies that, in energy-poor environments, heterogeneous populations will segregate spatially, with more energy-demanding cells leading the front of tumor growth and invasion. In contrast, for an energy-rich environment we expect the cells to grow in a similar way as they will have no need to search for places of higher energy density. We calibrated the model to recapitulate spatial growth patterns measured for the HCC1954 ductal breast carcinoma cell line in 48 different ECM environments. The inferred parameter space revealed three clusters, with different ECMs segregating mainly into different clusters. The two largest clusters differed mostly in their chemotactic/haptotactic- and energy diffusion coefficients; while the small cluster stood out by a high sensitivity to low energy and fast chemotactic/haptotactic response. The cells’ energy consumption rate was negatively correlated with RNA-seq derived expression of the corresponding ECM. A potential explanation for this negative correlation is that our model does not account for the possibility that cells can replace the ECM they degrade. The slower the rate of this replacement is, the higher the consumption rate appears to be.
Figure 1. Imbalance between nutrient efficacy and chemotactic superiority accelerates invasion of polyploid populations: subpopulation will segregate and only one type dominates at the invasion front.
When calibrating our model to a given ECM environment, growth patterns of a substantial fraction of replicates of that ECM could not be explained by fixed choices of energy sensitivity. A potential explanation for this is variable cell compositions across experimental replicates. The observation of a bimodal distribution in the DNA content of S-phase cells strengthened the hypothesis that HCC1954 is a polyploid cell line. One event that could have led to clones of variable ploidies coexisting in this cell line is WGD. Selection pressures for WGD suggests that it mitigates the accumulation of deleterious somatic alterations (2). However, there may be an energetic cost the cells pay for this robustness. In line with this, we find that high-ploidy breast cancer cell lines are resistant to cytotoxic drugs, but tend to be more sensitive to inhibitors of mTOR, EGFR, and MAPK signaling pathways. We used our model to investigate how variable energetic drawbacks of additional DNA content, affects competitiveness of high-ploidy cells as members of a polyploid population. We find that long-term coexistence of low- and high-ploidy clones occurs when sensitivity of the latter to energy scarcity is well-offset by their chemotactic ability to populate new terrain (3). Only when the high-ploidy clone has an intermediate motility, does it persistently coexist with the low-ploidy clone, both at the center and edge of the dish. Delaying chemotactic response of highly chemotactic cells could slow down invasion by maximizing competition within a polyploid population. If, on the other hand, chemotactic response of high-ploidy cells is already at an intermediate level, our simulation suggests that further reduction may accelerate invasion of low-ploidy cells.
In contrast to cell lines, WGD events in primary tumors are mostly clonal, not subclonal. Clones carrying a doubled genome often sweep over the population, such that by the time the tumor is detected, the diploid ancestor no longer exists. A related scenario is advanced, therapy-exposed tumors shown to revert to genomic stability, potentially bringing a WGD population back to a genomic state that more closely resembles its diploid ancestral state (4). The model presented here can investigate how dynamics between the two subpopulations unfold in both of these scenarios—early, shortly after the WGD or late, after therapy exposure. This would characterize what circumstances prevent the WGD carrying clone from becoming dominant or from retaining its dominance and could help explain WGD incidence in primary and recurrent tumors.
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