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

Abstract

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 19th-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 suggests that algebraic geometric techniques may be underutilized in biomolecular circuit analysis.

Ultrasensitive molecular controllers for quasi-integral feedback

“…Given an input-output map k u (ȳ, q, r) in polynomial form, these inequalities could be used to find bounds on the reaction rate constants that will satisfy the specifications, and to find whether the specifications are realistic in practice for the chosen implementation of the controller. The inequalities could be examined analytically using methods such as the Routh-Hurwitz or Sturm theorem [27,28] for simple controller networks, or characterized numerically in more complex controllers.…”

An algebraic approach to parameter optimization in biomolecular bistable systems

“…Polynomial Px(λ) = det(λI − J(x)) = λ n + a n−1 λ n−1 + · · · + a 1 λ + a 0 is the characteristic polynomial of the Jacobian evaluated atx. In many practical cases, the characteristic polynomials of bistable networks have a unique coefficient sign pattern that we summarize in the definition below [6].…”

“…The number of equilibria of system (1) can be in many cases established by finding the roots of a master (polynomial) equilibrium condition [6]. The number of positive roots of a polynomial can be established using methods such as Routh table [7] or Sturm's theorem [6] on the master equilibrium condition of the system. These methods can provide parametric conditions for the presence of a desired number of positive roots, i.e.…”

“…Specifically we focus on a class of bistable systems for which the bistability property can be related solely to the number of equilibria of the system and to the sign of the coefficients of the characteristic polynomial. Equilibrium polynomials can be analyzed using Sturm's theorem [6] or a Routh table [7], obtaining a set of inequalities that guarantee a desired number of positive roots. Bistability conditions (global) can thus be expressed as nonlinear constraints to be enforced in the optimization procedure.…”

Robustness and parameter geography in post-translational modification systems

“…Many kinds of approaches have been taken to quantitatively assess robustness in this 69 way, such as by parametric sensitivity [37][38][39][40], or by estimating volume and 70 shape [41][42][43][44][45][46][47]. Algebraic methods can sometimes provide an analytical description of 71 parametric regions [46,[48][49][50][51], but these methods tend to scale poorly with the 72 complexity of the system. For systems arising from networks of biochemical reactions, 73 methods also exist which give parametric conditions under which bistability 74 occurs [52-60] and some of these apply to PTM systems [55,[59][60][61][62][63][64].…”