Characterizing the Relationship between Steady State and Response Using Analytical Expressions for the Steady States of Mass Action Models

The steady states of cells affect their response to perturbation. Indeed, diagnostic markers for predicting the response to therapeutic perturbation are often based on steady state measurements. In spite of this, no method exists to systematically characterize the relationship between steady state and response. Mathematical models are established tools for studying cellular responses, but characterizing their relationship to the steady state requires that it have a parametric, or analytical, expression. For some models, this expression can be derived by the King-Altman method. However, King-Altman requires that no substrate act as an enzyme, and is therefore not applicable to most models of signal transduction. For this reason we developed py-substitution, a simple but general method for deriving analytical expressions for the steady states of mass action models. Where the King-Altman method is applicable, we show that py-substitution yields an equivalent expression, and at comparable efficiency. We use py-substitution to study the relationship between steady state and sensitivity to the anti-cancer drug candidate, dulanermin (recombinant human TRAIL). First, we use py-substitution to derive an analytical expression for the steady state of a published model of TRAIL-induced apoptosis. Next, we show that the amount of TRAIL required for cell death is sensitive to the steady state concentrations of procaspase 8 and its negative regulator, Bar, but not the other procaspase molecules. This suggests that activation of caspase 8 is a critical point in the death decision process. Finally, we show that changes in the threshold at which TRAIL results in cell death is not always equivalent to changes in the time of death, as is commonly assumed. Our work demonstrates that an analytical expression is a powerful tool for identifying steady state determinants of the cellular response to perturbation. All code is available at http://signalingsystems.ucsd.edu/models-and-code/ or as supplementary material accompanying this paper. Author Summary: Diagnostic markers are derived from steady state measurements, but are used to predict the cellular response to therapy. To develop new and better diagnostics, we would like to systematically characterize the relationship between steady state and the response to a given therapeutic. Mathematical models have powerfully complemented empirical studies in this regard, but it remains challenging to employ these models to characterize the effects of steady state. To do so requires a mathematical expression for the steady state, for which no universal method has been developed. Here, we present a method for deriving a mathematical expression for the steady state of a common class of models, those that obey the Law of Mass Action. We show that our method is easy to use and scales well to large models. We then use our method to characterize the relationship between steady state and the sensitivity to the anti-cancer drug candidate, dulanermin. We find that sensitivity to the drug is strongly affected by the concentration of the signaling molecule, procaspase 8, and its inhibitor, Bar. Our work thus demonstrates the utility of analytical studies of the steady state and its relationship to drug sensitivity.