Analytical solution of steady-state equations for chemical reaction networks with bilinear rate laws

An integrated modelling approach for targeted degradation: insights on optimization, data requirements and PKPD predictions from semi- or fully-mechanistic models and exact steady state solutions

The value of an integrated mathematical modelling approach for protein degraders which combines the benefits of traditional turnover models and fully mechanistic models is presented. Firstly, we show how exact solutions of the mechanistic models of monovalent and bivalent degraders can provide insight on the role of each system parameter in driving the pharmacological response. We show how on/off binding rates and degradation rates are related to potency and maximal effect of monovalent degraders, and how such relationship can be used to suggest a compound optimization strategy. Even convoluted exact steady state solutions for bivalent degraders provide insight on the type of observations required to ensure the predictive capacity of a mechanistic approach. Specifically for PROTACs, the structure of the exact steady state solution suggests that the total remaining target at steady state, which is easily accessible experimentally, is insufficient to reconstruct the state of the whole system at equilibrium and observations on different species (such as binary/ternary complexes) are necessary. Secondly, global sensitivity analysis of fully mechanistic models for PROTACs suggests that both target and ligase baselines (actually, their ratio) are the major sources of variability in the response of non-cooperative systems, which speaks to the importance of characterizing their distribution in the target patient population. Finally, we propose a pragmatic modelling approach which incorporates the insights generated with fully mechanistic models into simpler turnover models to improve their predictive ability, hence enabling acceleration of drug discovery programs and increased probability of success in the clinic.

Tailored parameter optimization methods for ordinary differential equation models with steady-state constraints

BACKGROUND

Ordinary differential equation (ODE) models are widely used to describe (bio-)chemical and biological processes. To enhance the predictive power of these models, their unknown parameters are estimated from experimental data. These experimental data are mostly collected in perturbation experiments, in which the processes are pushed out of steady state by applying a stimulus. The information that the initial condition is a steady state of the unperturbed process provides valuable information, as it restricts the dynamics of the process and thereby the parameters. However, implementing steady-state constraints in the optimization often results in convergence problems.

RESULTS

In this manuscript, we propose two new methods for solving optimization problems with steady-state constraints. The first method exploits ideas from optimization algorithms on manifolds and introduces a retraction operator, essentially reducing the dimension of the optimization problem. The second method is based on the continuous analogue of the optimization problem. This continuous analogue is an ODE whose equilibrium points are the optima of the constrained optimization problem. This equivalence enables the use of adaptive numerical methods for solving optimization problems with steady-state constraints. Both methods are tailored to the problem structure and exploit the local geometry of the steady-state manifold and its stability properties. A parameterization of the steady-state manifold is not required. The efficiency and reliability of the proposed methods is evaluated using one toy example and two applications. The first application example uses published data while the second uses a novel dataset for Raf/MEK/ERK signaling. The proposed methods demonstrated better convergence properties than state-of-the-art methods employed in systems and computational biology. Furthermore, the average computation time per converged start is significantly lower. In addition to the theoretical results, the analysis of the dataset for Raf/MEK/ERK signaling provides novel biological insights regarding the existence of feedback regulation.

CONCLUSION

Many optimization problems considered in systems and computational biology are subject to steady-state constraints. While most optimization methods have convergence problems if these steady-state constraints are highly nonlinear, the methods presented recover the convergence properties of optimizers which can exploit an analytical expression for the parameter-dependent steady state. This renders them an excellent alternative to methods which are currently employed in systems and computational biology.

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization.

A technique for determining the signs of sensitivities of steady states in chemical reaction networks

This paper studies the direction of change of steady states to parameter perturbations in chemical reaction networks, and, in particular, to changes in conserved quantities. Theoretical considerations lead to the formulation of a computational procedure that provides a set of possible signs of such sensitivities. The procedure is purely algebraic and combinatorial, only using information on stoichiometry, and is independent of the values of kinetic constants. Three examples of important intracellular signal transduction models are worked out as an illustration. In these examples, the set of signs found is minimal, but there is no general guarantee that the set found will always be minimal in other examples. The paper also briefly discusses the relationship of the sign problem to the question of uniqueness of steady states in stoichiometry classes.

The impact of high density receptor clusters on VEGF signaling

Vascular endothelial growth factor (VEGF) signaling is involved in the process of blood vessel development and maintenance. Signaling is initiated by binding of the bivalent VEGF ligand to the membrane-bound receptors (VEGFR), which in turn stimulates receptor dimerization. Herein, we discuss experimental evidence that VEGF receptors localize in caveloae and other regions of the plasma membrane, and for other receptors, it has been shown that receptor clustering has an impact on dimerization and thus also on signaling. Overall, receptor clustering is part of a complex ecosystem of interactions and how receptor clustering impacts dimerization is not well understood. To address these questions, we have formulated the simplest possible model. We have postulated the existence of a single high affinity region in the cell membrane, which acts as a transient trap for receptors. We have defined an ODE model by introducing high- and low-density receptor variables and introduce the corresponding reactions from a realistic model of VEGF signal initiation. Finally, we use the model to investigate the relation between the degree of VEGFR concentration, ligand availability, and signaling. In conclusion, our simulation results provide a deeper understanding of the role of receptor clustering in cell signaling.