A Computational Study of Alternate SELEX

Systematic evolution of ligands by exponential enrichment (SELEX) is a procedure for identifying nucleic acid (NA) molecules with affinities for specific target species, such as proteins, peptides, or small organic molecules. Here, we extend the work in Seo et al. (Bull Math Biol 72:1623–1665, 2010) (multiple-target SELEX or positive SELEX) and examine an alternate SELEX process with multiple targets by incorporating negative selection into a positive SELEX protocol. The alternate SELEX process is done iteratively by alternating several positive selection rounds with several negative selection rounds. At the end of each positive selection round, NAs are eluted from the bound product and amplified by polymerase chain reaction (PCR) to increase the size of the pool of NA species that bind preferentially to the given positive target vector. The enriched population of NAs is then exposed to the negative targets (undesired targets). The free NA species (instead of the bound NA species being eluted) are retained and amplified by PCR (negative selection). The goal is to minimize an enrichment of nonspecifically binding NAs against multiple targets. While positive selection alone results in a pool of NAs that bind tightly to a given target vector, negative selection results in the subset of the NAs that bind best to the nontarget vectors that are also present. By alternating the two processes, we eventually obtain a refined population of nucleic acids that bind to the desired target(s) with high “selectivity” and “specificity.” In the present paper, we give formulations of the negative and alternate selection processes and define their efficiencies in a meaningful way. We study the asymptotic behavior of alternate SELEX system as a discrete-time dynamical system. To do this, we use the chemical potential to examine how alternate SELEX leads to the selection of NAs with more specific interactions when the ratio of the number of positive selection rounds to the number of negative selection rounds is fixed. Alternate SELEX is said to be globally asymptotically stable if, given the initial target vector and a fixed ratio, the distribution of the limiting NA fractions does not depend on the relative concentrations of the NAs in the initial pool (provided that all of the NA species are initially present in the initial pool). We state conditions on the matrix of NA—target affinities that determine when the alternate SELEX process is globally asymptotically stable in this sense and illustrate these results computationally.