Mathematics at the eve of a historic transition in biology

Figure 1: Geometric modeling of a protein surface in the Eulerian representation, which is crucial for the understanding of the protein structure and interactions, and bridges the gap between biomolecular structure data and mathematical models such as Poisson-Boltzmann equation or Poisson-NernstPlanck equations. Image credit: Rundong Zhao. Biology concerns the structure, function, development and evolution of living organisms. It underwent a dramatic transformation from macroscopic to microscopic (i.e.,“molecular”) in the 1960s and assumed an omics dimension around the dawn of the millennium. Understanding the rules of life is the major mission of biological sciences in the 21st century. The technological advances in the past few decades have fueled the exponential growth of biological data. For example, the Protein Data Bank (https://www.rcsb.org/pdb/home/home.do) has archived more than one hundred thirty thousand threedimensional (3D) biomolecular structures and Genbank (https://www.ncbi.nlm.nih.gov/genbank/) has recorded more than 200 million sequences. The accumulation of various biological data in turn has paved the way for biology to undertake another historic transition from being qualitative, phenomenological and descriptive to being quantitative, analytical and predictive. Such a transition provides both unprecedented opportunities and grand challenges for mathematicians. One of major challenges in biology is the understanding of structure-function relationships in biomolecules, such as proteins, DNA, RNA, and their interacting complexes. Such an understanding is the holy grail of biophysics and has a profound impact to biology, biotechnology, bioengineering and biomedicine. Mathematical apparatuses, including simplicial geometry, differential geometry, differential topology, algebraic topology, geometric topology, knot theory, tiling theory, spectral graph theory and topological graph, are essential for deciphering biomolecular structurefunction relationships [1, 2, 3, 4]. In general, geometric modeling is paramount for the conceptualization of biomolecules and their interactions, which is vital to the understanding of intricate biomolecules. Geometric modeling also bridges the gap between biological data and mathematical models involving topology, graph theory and partial differential equations (PDEs) [5, 6] (see Figure 1). Topology dramatically simplifies biological complexity and renders insightful high level abstraction to large biological data [7, 8, 9, 10, 11] (see Figure 2). Graph theory is able to go beyond topological connectivity and incorporates harmonic analysis and optimization