Algebraic Geometry Tools in Systems Biology

Abstract

Many models in the sciences and engineering are ex- pressed as solution sets to systems of polynomial equa- tions, that is, as affine algebraic varieties. This is a basic no- tion in algebraic geometry, a vibrant area of mathematics which is particularly good at counting (solutions, tangen- cies, obstructions, etc.), giving structure to interesting sets (varieties with special properties, moduli spaces, etc.) and, principally, understanding structure. Starting in the 1980s with the development of computer algebra systems, and increasingly over the last years, ideas and methods from algebraic geometry are being applied to a great number of new areas (both in mathematics and in other disciplines including biology, computer science, physics, chemistry, etc.). The aim of this note is to give a glimpse of how methods and concepts from algebraic geometry (in particular, from computational and real algebraic geometry) can be used to analyze standard models in molecular biology. These models occur in systems and synthetic biology, which fo- cus on understanding the design principles of living sys- tems. The past ten years have experienced an intense ac- tivity in the field and a rapidly growing literature. In turn, this application has challenged the current theory, mainly in the realm of real algebraic geometry.