Bio-medical symbolic modeling with algebraic patches

Abstract

The mathematical description of geometry is of paramount importance in the modeling of systems of all kinds. Standard geometric modeling describes curved objects through parametric functions as the image of compact domains. Alternatively, as in algebraic geometry, one may describe curved geometry as the zero-set of polynomials. Geometric modeling of biological systems highlight certain persistent and open problems more effectively addressed using algebraic geometry. In this thesis a framework for computer-based geometric representation based on algebraic geometry is introduced. Several algebraic representation schemes known as A-splines and A-patches are detailed as a description of geometry, by using a piecewise continuous gluing of algebraic curves and surfaces. The application of Asplines and A-patches to biological modeling is discussed in the context of protein molecular interface modeling. The rationale is to present an algebraic representation of bio-modeling under an unified point of view. This framework provides a suitable background for the main contribution of this thesis: the formulation and implementation of algorithms for Boolean operations (union, intersection, difference, etc.) on the algebra of curved polyhedra whose boundary is triangulated with A-patches. Boolean operations on curved geometry are yet an open obstinate research problem and its exact solution is only definable within the domain of algebraic geometry. The exact formulation is here used as basis for a geometrically approximate yet topologically accurate solution, closed in the geometric domain of A-patches. The prototype implementation has been applied to pairs of molecular models of ligand proteins in docking configuration. To date, the computational use of algebraic geometry is still experimental and is far from being a major component of current systems. This thesis shows an evidence that representation techniques derived from algebraic geometry have strong potential in bio-medical modeling, still needing much further research and engineering.