Tunable Regulatory Networks As Models of Chromosomal Copy Number Variations Across Cancers

Chromosomal copy number variations (CNVs) are among the most prevalent mutations in tumor cells. Recent analysis of clinical data (Discher et al., 2025) shows surprising simplicity in the statistics of chromosome copy number variation across cancer types: the variance $\sigma^2_{\text{CNV}}$ in the fraction of chromosomes that exhibit copy number variation is related to the mean fraction $\mu_{\text{CNV}}$ by the relation $\sigma^2_{\text{CNV}}=\alpha \mu_{\text{CNV}}(1-\mu_{\text{CNV}})$ with a prefactor $\alpha \approx 0.34$. We model cancer regulatory networks as a network of chemical reactions with rate constants that can be tuned to minimize a prescribed cost function, given as the difference between desired and actual concentrations of certain regulated species. We tune the reaction constants using evolutionary dynamics, with the condition that once each reaction rate drops below a threshold it cannot change. This condition reflects the reality that once the number of copies of a genome sequence drops to zero, the sequence cannot be recovered by mutation. In the limit of large tumor size, we find that regulatory network models with heterogeneous reaction pathways yield results with exactly the form found in clinical data, with a prefactor of $\alpha=0.33$. The evolution trajectory of tumors in the space of reaction constants is fractal with dimension 1.5, indicating persistence in the evolution direction as the tumor descends in cost. These results sharply contrast with predictions from neutral genetic drift models such as Wright-Fisher, highlighting the role of strong selection in CNV statistics. Our findings suggest that the statistics of chromosome copy number variation encode signatures of optimization in cancer evolution.