A finite characterization of perfect equilibria

Govindan and Klumpp [7] provided a characterization of perfect equilibria using Lexicographic Probability Systems (LPSs). Their characterization was essentially finite in that they showed that there exists a finite bound on the number of levels in the LPS, but they did not compute it explicitly. In this note, we draw on two recent developments in Real Algebraic Geometry to obtain a formula for this bound.

Calculating Positive Invariant Sets: A Quantifier Elimination Approach

A Lyapunov-based approach for calculating positive invariant sets in an automatic manner is presented. This is done using real algebraic geometry techniques, which are summed up under the term quantifier elimination (QE). Using available tools, the approach presented yields an algorithmizable procedure whose conservatism only depends on the initial choice for the Lyapunov candidate function. The performance of the approach is illustrated on a variant of the Rössler system and on the Lorenz-Haken system.

TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

This paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

An analytical approach to bistable biological circuit discrimination using real algebraic geometry

Biomolecular circuits with two distinct and stable steady states have been identified as essential components in a wide range of biological networks, with a variety of mechanisms and topologies giving rise to their important bistable property. Understanding the differences between circuit implementations is an important question, particularly for the synthetic biologist faced with determining which bistable circuit design out of many is best for their specific application. In this work we explore the applicability of Sturm's theorem—a tool from nineteenth-century real algebraic geometry—to comparing ‘functionally equivalent’ bistable circuits without the need for numerical simulation. We first consider two genetic toggle variants and two different positive feedback circuits, and show how specific topological properties present in each type of circuit can serve to increase the size of the regions of parameter space in which they function as switches. We then demonstrate that a single competitive monomeric activator added to a purely monomeric (and otherwise monostable) mutual repressor circuit is sufficient for bistability. Finally, we compare our approach with the Routh–Hurwitz method and derive consistent, yet more powerful, parametric conditions. The predictive power and ease of use of Sturm's theorem demonstrated in this work suggest that algebraic geometric techniques may be underused in biomolecular circuit analysis.