Discrete Dynamical Systems in Multiple Target and Alternate SELEX

Abstract

Dynamical systems are often used to model biochemical and biological processes. In Seo et al. (2010, 2014) we studied two mathematical models of the iterative biochemical procedure known as SELEX (Systematic Evolution of Ligands by EXponential Enrichment): multiple target SELEX and alternate SELEX. It is the purpose of this paper to revisit the mathematics of these processes in the language of dynamical systems on compact manifolds but for a dynamical system on a manifold with compact closure. From the experimentalist's point of view, multiple target SELEX provides a way of obtaining the best binding ligands to a pool of several fixed targets, whereas alternate SELEX provides a way to specify which of the best binding ligands also bind best to a specified subtarget. Because these procedures are iterative, it is natural to investigate them in the context of the theory of discrete dynamical systems. Although the iterative schemes are nonautonomous, they have the same limiting properties as two closely related autonomous iteration schemes, called simplified multiple target SELEX and simplified alternate SELEX. The iteration scheme defined by simplified multiple target SELEX (simplified MTS) is not defined by the gradient of a potential function as in the standard theory (Akins, 1993). However, there is associated with this scheme, a related function, called the efficiency. From its structure, we show that the basic sets for simplified MTS are the sets of extreme points of this function and only occur on the boundary of the compact manifold. Their union, together with the repeller manifold, constitutes the set of fixed points for the dynamics. We discuss the attracting properties of the basic sets for simplified MTS and multiple target SELEX (or positive SELEX). They can be ordered by their ability to attract the flows, from the strongest attracting set to the repeller manifold. Under the hypothesis that as the SELEX scheme evolves, fewer and fewer nucleic acids can bind with greater efficiency than the overall efficiency for the given round, we prove that simplified MTS possesses a set of global attractors with highest possible overall efficiency. We show that positive SELEX has the same basic sets and that the same attracting properties as simplified MTS hold when the total target concentration decreases neither too quickly nor too slowly as a function of iteration number (Levine et al., 2007). We introduce an iteration scheme for negative SELEX, in which a subtarget is removed and, instead as in positive SELEX, where the bound target is retained and amplified by PCR (polymerase chain reaction) at each step, the free nucleic acids are retained and amplified by PCR. Simplified alternate SELEX defines a scheme in which each iteration consists of several iterations of simplified MTS followed by several iterations of negative SELEX. The number of simplified MTS iterations need not be the same as the number of iterations of negative SELEX, but these numbers are fixed for all iterations of simplified alternate SELEX. We examine the convergence properties of alternate SELEX and introduce the notion of limiting ultimate specificity as a consequence of alternating between positive and negative SELEX iterations.