Robustness and parameter geography in post-translational modification systems

Abstract

<jats:title>Abstract</jats:title><jats:p>Biological systems are acknowledged to be robust to perturbations but a rigorous understanding of this has been elusive. In a mathematical model, perturbations often exert their effect through parameters, so sizes and shapes of parametric regions offer an integrated global estimate of robustness. Here, we explore this “parameter geography” for bistability in post-translational modification (PTM) systems. We use the previously developed “linear framework” for timescale separation to describe the steady-states of a two-site PTM system as the solutions of two polynomial equations in two variables, with eight non-dimensional parameters. Importantly, this approach allows us to accommodate enzyme mechanisms of arbitrary complexity beyond the conventional Michaelis-Menten scheme, which unrealistically forbids product rebinding. We further use the numerical algebraic geometry tools Bertini, Paramotopy, and alphaCertified to statistically assess the solutions to these equations at ∼10<jats:sup>9</jats:sup> parameter points in total. Subject to sampling limitations, we find no bistability when substrate amount is below a threshold relative to enzyme amounts. As substrate increases, the bistable region acquires 8-dimensional volume which increases in an apparently monotonic and sigmoidal manner towards saturation. The region remains connected but not convex, albeit with a high visibility ratio. Surprisingly, the saturating bistable region occupies a much smaller proportion of the sampling domain under mechanistic assumptions more realistic than the Michaelis-Menten scheme. We find that bistability is compromised by product rebinding and that unrealistic assumptions on enzyme mechanisms have obscured its parametric rarity. The apparent monotonic increase in volume of the bistable region remains perplexing because the region itself does not grow monotonically: parameter points can move back and forth between monostability and bistability. We suggest mathematical conjectures and questions arising from these findings. Advances in theory and software now permit insights into parameter geography to be uncovered by high-dimensional, data-centric analysis.</jats:p><jats:sec><jats:title>Author Summary</jats:title><jats:p>Biological organisms are often said to have robust properties but it is difficult to understand how such robustness arises from molecular interactions. Here, we use a mathematical model to study how the molecular mechanism of protein modification exhibits the property of multiple internal states, which has been suggested to underlie memory and decision making. The robustness of this property is revealed by the size and shape, or “geography,” of the parametric region in which the property holds. We use advances in reducing model complexity and in rapidly solving the underlying equations, to extensively sample parameter points in an 8-dimensional space. We find that under realistic molecular assumptions the size of the region is surprisingly small, suggesting that generating multiple internal states with such a mechanism is much harder than expected. While the shape of the region appears straightforward, we find surprising complexity in how the region grows with increasing amounts of the modified substrate. Our approach uses statistical analysis of data generated from a model, rather than from experiments, but leads to precise mathematical conjectures about parameter geography and biological robustness.</jats:p></jats:sec>

Reformulating non-equilibrium steady-states and generalised Hopfield discrimination

“…Independently of these developments, a graph-theoretic approach to analysing biochemical systems under timescale separation, the "linear framework", was introduced in systems biology [14,26,25,38,39]. This was applied to bulk populations of biochemical entities, such as posttranslational modification systems [8,28], but the same mathematics can be used to analyse individual stochastic entities, such as genes [9,3,36,29,37]. In this context, as in the approaches described above, the linear framework provides a treatment of continuous-time, finite state Markov processes based on directed graphs with labelled edges.…”

Machine learning the real discriminant locus

“…Given a compact subset of the parameter space R k , one approach to generate sample points is to randomly, e.g., uniformly, select a parameter value and use a parameter homotopy to count the number of real solutions. Such an approach has been applied to a variety of problems, e.g., [7,40,56]. With the aim of approximating the real discriminant locus, i.e., the classification boundaries, the following method uses domain knowledge regarding the classical discriminant locus to provide sample points near the boundaries to guide the learning of the boundaries.…”

Algebraic Geometry Tools in Systems Biology

“…A different line of attack is presented for instance in [14], like in some other previous papers, where tools from numerical algebraic geometry are used to describe the "geography" of the space of parameters in particular enzymatic systems. Robustness of multistationarity is measured in terms of the size and the shape of the region where the property holds, with an extensive sampling of parameter points in an 8-dimensional space.…”