[Paper Review] ODE and Random Boolean networks in application to modelling of 6-mercaptopurine metabolism

Anastasia I. Lavrova, Eugene B. Postnikov+2 more|arXiv (Cornell University)|Oct 31, 2016

Gene Regulatory Network Analysis19 references3 citations

TL;DR

This study proposes a hybrid Boolean-ODE model to analyze 6-mercaptopurine (6-MP) metabolism, showing that ATP concentration acts as a key switch between therapeutic and toxic metabolic pathways. By integrating ODE-based kinetic modeling with probabilistic Boolean networks, the approach captures smooth transitions in pathway activation while preserving computational simplicity, revealing that an intermediate ATP level (0.7 μmol/ml) optimizes therapeutic efficacy and minimizes toxicity.

ABSTRACT

We consider two approaches to modelling of cell metabolism of 6-mercaptopurine, which is one of the important chemotherapy drugs used for treating of acute lymphocytic leukemia: kinetic ordinary differential equations and random Boolean networks, and analyse their interplay with respect to taking into account ATP concentration as a key parameter of switching between different pathways. It is shown that Boolean networks, which allow for avoiding complexity of general kinetic modelling, preserve an opportunity to the reproduction of the principal switching mechanism. To keep a detailed quantitative measure of the control parameter, a combined Boolean-ODE method is proposed.

Motivation & Objective

• To identify the key regulatory parameter controlling the switch between therapeutic and toxic pathways in 6-mercaptopurine metabolism.
• To overcome limitations of purely ODE-based models (high parameter sensitivity) and purely Boolean models (oversimplification) by developing a hybrid approach.
• To investigate the role of ATP concentration in modulating metabolic fluxes and toxicity in 6-MP treatment.
• To demonstrate that probabilistic Boolean networks can accurately reproduce key dynamic switching behaviors observed in ODE models, even without detailed kinetic parameters.
• To propose a scalable framework for modeling complex metabolic networks by integrating continuous dynamics with discrete logical rules.

Proposed method

• Development of a simplified ODE system describing 6-MP metabolic transformations, with ATP concentration as a central regulatory variable.
• Formulation of a probabilistic Boolean network where each network instance corresponds to a distinct metabolic pathway, with pathway selection governed by ATP concentration.
• Use of non-stationary, continual ATP levels as a dynamic control parameter in the Boolean network, enabling smooth transitions between pathways.
• Calibration of ODE model using kinetic constants from the literature and validation against known metabolic behavior, including TIMP and TGMP dynamics.
• Application of Hill-type kinetics in ODE model to simulate enzyme saturation and non-linear responses, particularly for TPMT and other key enzymes.
• Integration of ODE and Boolean models by mapping ODE-derived dynamic behavior (e.g., TIMP accumulation) onto Boolean state transitions, preserving key regulatory logic.

Experimental results

Research questions

• RQ1 What is the role of ATP concentration in switching between the therapeutic (TGMP-producing) and toxic (TIMP/TITP-accumulating) pathways in 6-MP metabolism?
• RQ2 Can probabilistic Boolean networks accurately reproduce the dynamic switching behavior observed in detailed ODE models without requiring full kinetic parameter sets?
• RQ3 How does intermediate ATP concentration (0.7 μmol/ml) affect the balance between therapeutic metabolite production and toxic intermediate accumulation?
• RQ4 To what extent can a hybrid ODE-Boolean model preserve quantitative dynamics while reducing model complexity?
• RQ5 Can non-stationary ATP levels be effectively used as a continuous control parameter in a discrete Boolean framework to model smooth metabolic transitions?

Key findings

• An ATP concentration of 0.7 μmol/ml maximizes therapeutic efficacy and minimizes toxicity by enabling complete conversion of 6-MP to TGMP while limiting accumulation of toxic intermediates.
• At low ATP (0.1 μmol/ml), TIMP and TXMP concentrations remain elevated, indicating incomplete metabolism and increased risk of toxicity due to inhibition of de novo purine synthesis.
• High ATP levels (above 0.7 μmol/ml) inhibit glycolysis, leading to glucose accumulation and potential cardiomyopathy, suggesting a narrow therapeutic window for ATP.
• The ODE model shows that ATP acts as a key regulator, with its concentration directly influencing flux distribution between the TGMP and catabolic pathways.
• The probabilistic Boolean network successfully reproduces the ODE-based switching behavior, demonstrating that discrete logic models can capture complex, continuous dynamics when guided by a non-stationary control parameter like ATP.
• The hybrid Boolean-ODE approach provides a scalable and computationally efficient alternative to full kinetic models, preserving essential regulatory mechanisms without requiring extensive kinetic parameterization.