Cusp bifurcation in a metastatic regulatory network.

Understanding the potential for cancers to metastasize is still relatively unknown. While many predictive methods may use deep learning or stochastic processes, we highlight a long standing mathematical concept that may be useful for modeling metastatic breast cancer systems. Ordinary differential equations (ODEs) can model cell state transitions by considering the pertinent environmental variables as well as the paths systems take over time. Bifurcation theory is a branch of dynamical systems which studies changes in the behavior of an ODE system while one or more parameters are varied. Many studies have applied concepts in one-parameter bifurcation theory to model biological network dynamics, and cell division. However, studies of two-parameter bifurcations are much more rare. Two-parameter bifurcations have not been studied in metastatic systems. Here we show how a specific two-parameter bifurcation phenomenon called a cusp bifurcation separates two qualitatively different metastatic cell state transitions modalities and propose a new perspective on defining such transitions based on mathematical theory. We hope the observations and verification methods detailed here may help in the understanding of metastatic potential from a basic biological perspective and in clinical settings.

Amino Acid metabolism conflicts with protein diversity.

The 20 protein-coding amino acids are found in proteomes with different relative abundances. The most abundant amino acid, leucine, is nearly an order of magnitude more prevalent than the least abundant amino acid, cysteine. Amino acid metabolic costs differ similarly, constraining their incorporation into proteins. On the other hand, a diverse set of protein sequences is necessary to build functional proteomes. Here, we present a simple model for a cost-diversity trade-off postulating that natural proteomes minimize amino acid metabolic flux while maximizing sequence entropy. The model explains the relative abundances of amino acids across a diverse set of proteomes. We found that the data are remarkably well explained when the cost function accounts for amino acid chemical decay. More than 100 organisms reach comparable solutions to the trade-off by different combinations of proteome cost and sequence diversity. Quantifying the interplay between proteome size and entropy shows that proteomes can get optimally large and diverse.