A Mathematical Analysis of Multiple-Target Selex

SELEX (Systematic Evolution of Ligands by Exponential Enrichment) is a procedure by which a mixture of nucleic acids can be fractionated with the goal of identifying those with specific biochemical activities. One combines the mixture with a specific target molecule and then separates the target-NA complex from the resulting reactions. The target-NA complex is separated from the unbound NA by mechanical means (such as by filtration), the NA is eluted from the complex, amplified by PCR (polymerase chain reaction), and the process repeated. After several rounds, one should be left with the nucleic acids that best bind to the target. The problem was first formulated mathematically in Irvine et al. (J. Mol. Biol. 222:739–761, 1991). In Levine and Nilsen-Hamilton (Comput. Biol. Chem. 31:11–25, 2007), a mathematical analysis of the process was given. In Vant-Hull et al. (J. Mol. Biol. 278:579–597, 1998), multiple target SELEX was considered. It was assumed that each target has a single nucleic acid binding site that permits occupation by no more than one nucleic acid. Here, we revisit Vant-Hull et al. (J. Mol. Biol. 278:579–597, 1998) using the same assumptions. The iteration scheme is shown to be convergent and a simplified algorithm is given. Our interest here is in the behavior of the multiple target SELEX process as a discrete “time” dynamical system. Our goal is to characterize the limiting states and their dependence on the initial distribution of nucleic acid and target fraction components. (In multiple target SELEX, we vary the target component fractions, but not their concentrations, as fixed and the initial pool of nucleic acids as a variable starting condition.) Given N nucleic acids and a target consisting of M subtarget component species, there is an M×N matrix of affinities, the (i,j) entry corresponding to the affinity of the jth nucleic acid for the ith subtarget. We give a structure condition on this matrix that is equivalent to the following statement: For any initial pool of nucleic acids such that all N species are represented, the dynamical system defined by the multiple target SELEX process will converge to a unique subset of nucleic acids, each of whose concentrations depend only upon the total nucleic acid concentration, the initial fractional target distribution (both of which are assumed to be the same from round to round), and the overall limiting association constant. (The overall association constant is the equilibrium constant for the system of MN reactions when viewed as a composite single reaction.) This condition is equivalent to the statement that every member of a certain family of chemical potentials at infinite target dilution can have at most one critical point. (The condition replaces the statement for single target SELEX that the dynamical system generated via the process always converges to a pool that contains only the nucleic acid that binds best to the target.) This suggests that the effectiveness of multiple target SELEX as a separation procedure may not be as useful as single target SELEX unless the thermodynamic properties of these chemical potentials are well understood.