This thesis develops a mathematical model of the biological procedure SELEX (Systematic Evolution of Ligands by EXponential Enrichment). The procedure is an in vitro method for identifying nucleic acid (NA) molecules that have an ability to bind tightly and specifically to target species of interest, such as small organic molecules, peptides or proteins.

We explore two main algorithms: multiple target (positive) SELEX and alternate SELEX. The schemes are considered as discrete time dynamical systems, and the limiting (steady-state) behaviors of the processes are characterized by the initial parameters of each system: concentration of total targets, concentration of a pool of nucleic acids and fractional distributions of NA and target molecules. To gain a better understanding of our systems, we also construct (simplex-based) geometric structures of the limiting states, in terms of chemical thermodynamics.

We find that the dynamical system defined by the multiple target (positive) SELEX process is globally asymptotically stable, when and only when a certain family of chemical potentials at infinite target dilution has at most one critical point. That is, the SELEX iteration scheme converges to a unique subset of nucleic acids and does not depend on the distribution of nucleic acids present in the pool, but does depend upon the total nucleic acid concentration, the initial fractional distribution of the targets, and the overall limiting equilibrium association constant.

We also present a mathematical model for “alternate SELEX”. The goal here is to minimize an enrichment of non-specifically binding nucleic acids against multiple targets by alternating two processes: positive selection and negative selection, which in combination result in the selection of nucleic acids with high “selectivity” and “specificity”.