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 nineteenth-century real algebraic geometryto 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.

Stability and Robustness of Unbalanced Genetic Toggle Switches in the Presence of Scarce Resources

“…Comprising the repressor proteins Y and Z , the behavior of the toggle switch [ 60 ] evolves according to where and are the dissociation and decay rate constants of Y , respectively, with Hill coefficient , and , , and are defined similarly. In this paper, we consider corresponding to the most commonly considered case of repressors binding as dimers [ 50 , 52 , 69 , 70 , 71 , 72 , 73 ], but our analysis can be easily extended to other cases as well [ 20 , 49 , 56 , 59 , 74 , 75 ]. Assuming that protein decay is primarily determined by cell growth, in this manuscript we consider where is the cell growth rate.…”

Robustness and parameter geography in post-translational modification systems

“…Many kinds of approaches have been taken to quantitatively assess robustness in this way, such as by parametric sensitivity [37][38][39][40], or by estimating volume and shape [41][42][43][44][45][46][47]. Algebraic methods can sometimes provide an analytical description of parametric regions [46,[48][49][50][51], but these methods tend to scale poorly with the complexity of the system. For systems arising from networks of biochemical reactions, methods also exist which give parametric conditions under which bistability occurs [52][53][54][55][56][57][58], or does not occur [59][60][61][62], and some of these apply to PTM systems [54,57,58,[63][64][65][66].…”

Numerical algebraic geometry for model selection and its application to the life sciences

“…We demonstrate model compatibility (algorithm 1) on an example from receptor-mediated programmed cell death, which is initiated by the activation of death receptors upon the detection of extracellular death ligands [36][37][38][39]. We consider, in particular, the 'cluster' model of [12], which was inspired by crystallographic data [40] and describes the recruitment of receptors by ligands into local self-activating clusters capable of bistability. The cluster model is a system of three degree-four polynomials in the form of (2.2) in three variables (representing various receptor states) and six rate parameters, supplemented by ligand and receptor conservations (see the electronic supplementary material).…”