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Abstract

Proteins are the workhorse molecules of the cell and perform their biological functions by binding to other molecules through physical contact. Protein function is then regulated through coupling of bindings on the protein (allosteric regulation). Just as the genetic code provides the blueprint for protein synthesis, the coupling is thought to provide the basis for protein communication and interaction. However, it is not yet fully understood how binding of a molecule at one site affects binding of another molecule at another distal site on a protein, even more than $60$ years after its discovery in 1961. In this paper, I propose a simple mathematical model of protein interactions, using a ``quantized'' version of differential geometry, i.e., the discrete differential geometry of $n$-simplices. The model is based on the concept of electron delocalization, one of the main features of quantum chemistry, Allosteric regulation then follows tautologically from the definition of interactions. No prior knowledge of conventional discrete differential geometry, protein science, or quantum chemistry is required. I hope this paper will provide a starting point for many mathematicians to study chemistry and molecular biology.