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From the perfect radial symmetries of radiolarian mineral skeletons to the broken symmetry of homochirality, the logic of Nature's regularities has fascinated scientists for centuries. Some of Nature's symmetries are clearly visible in morphology and physical structure, whereas others are hidden in the network of interactions among system components. Just as visible symmetries and asymmetries contribute to the system's beauty, might hidden symmetries contribute to the system's functional harmony? And if so, how? Here we demonstrate that the interaction networks of biological systems from cell signaling to cancer display a form of dynamical reflection symmetry that serves to expand their dynamical complexity. The expansion proceeds according to precise rules regarding the lengths of dynamical cycles, made possible by a peculiar imbalance between excitation and inhibition. To probe the conditions under which this reflection symmetry evolves, we use a multi-objective genetic algorithm to produce networks with long dynamical cycles. We find that local structural motifs that break the reflection symmetry are deleted in the evolutionary process, providing evidence for symmetry's causal role in dynamical complexity. Finally, we identify symmetries in real continuous-valued scRNA-seq data of gene expression in drug-resistant cancer cells. Broadly, our work reveals a class of hidden symmetries in biology, present in the network of interactions among system components, and it underscores their functional importance. Mathematically simple and computational inexpensive, our approach is applicable to the types of biological data commonly acquired in modern experiments.