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A Bilevel Integer Programming Approach for the Synchronous Attractor Control Problem

Kyungduk Moon, Kangbok Lee, Loïc Paulevé

Abstract

Boolean networks are dynamical models of disease development in which the activation levels of genes are represented by binary variables. Given a Boolean network, controls represent mutations or medical treatments that fix the activation levels of selected genes so that all states in every attractor (i.e., long-term recurrent states) satisfy a desired phenotype. Our goal is to enumerate all minimal controls, identifying critical gene subsets in disease development and therapy. This problem has an inherent bilevel integer programming structure and is computationally challenging. We propose an infeasibility-based Benders decomposition, a logic-based Benders framework for bilevel integer programs with multiple subproblems. In our application, each subproblem finds a forbidden attractor of a given length and yields a problem-specific feasibility cut. We also propose an auxiliary IP called subspace separation that finds a Boolean subspace that includes multiple forbidden attractors and thereby strengthens the cut. Numerical experiments show that the resulting algorithms are much more scalable than state-of-the-art methods and that subspace separation substantially improves performance.

A Bilevel Integer Programming Approach for the Synchronous Attractor Control Problem

Abstract

Boolean networks are dynamical models of disease development in which the activation levels of genes are represented by binary variables. Given a Boolean network, controls represent mutations or medical treatments that fix the activation levels of selected genes so that all states in every attractor (i.e., long-term recurrent states) satisfy a desired phenotype. Our goal is to enumerate all minimal controls, identifying critical gene subsets in disease development and therapy. This problem has an inherent bilevel integer programming structure and is computationally challenging. We propose an infeasibility-based Benders decomposition, a logic-based Benders framework for bilevel integer programs with multiple subproblems. In our application, each subproblem finds a forbidden attractor of a given length and yields a problem-specific feasibility cut. We also propose an auxiliary IP called subspace separation that finds a Boolean subspace that includes multiple forbidden attractors and thereby strengthens the cut. Numerical experiments show that the resulting algorithms are much more scalable than state-of-the-art methods and that subspace separation substantially improves performance.

Paper Structure

This paper contains 39 sections, 6 theorems, 18 equations, 8 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Precision Medicine and Boolean Networks
  3. The Synchronous Attractor Control Problem (SACP)
  4. Bilevel Integer Programming and Logic-Based Benders Decomposition
  5. Bilevel IP Model for the Synchronous Attractor Control Problem
  6. Overview
  7. Mathematical model
  8. The lower-level problem $\mathcal{L}^{T_{\max}}(\mathbf{d})$
  9. Transition condition.
  10. Attractor length selection.
  11. Phenotype.
  12. The upper-level problem $\mathcal{U}(\lambda,\mathcal{D})$
  13. An Infeasibility-Based Benders Decomposition for SACP($T_{\max}$)
  14. The Infeasibility-Based Benders Decomposition for SACP($T_{\max}$)
  15. The Attractor Cut
  16. ...and 24 more sections

Key Result

Theorem 4

Let $(\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{w}},\hat{p})$ be a feasible solution of $\mathcal{S}^{T}(\hat{\mathbf{d}})$ for some $T\in\llbracket T_{\max} \rrbracket$ and feasible control $\hat{\mathbf{d}}$. Let $\alpha_j$, $\beta_j$, and $k_j$ be the binary indicators defined in def:binary- Then, for a control $\mathbf{d}$ satisfying eq:ulp-exclusivity, $\hat{\mathbf{x}}$ is an attractor

Figures (8)

  • Figure 1: A Boolean network example and state transition graphs (STGs)
  • Figure 2: Overview of the Benders cuts
  • Figure 3: Cumulative number of minimal controls over time for $T_{\max}=5$
  • Figure EC.1: A counterexample showing that the trap space cut may not be stronger than the attractor cut.
  • Figure EC.2: Cumulative number of minimal controls over time for $T_{\max}=1$
  • ...and 3 more figures

Theorems & Definitions (9)

  • Definition 1: Minimal control
  • Definition 2: Infeasibility-based Benders decomposition (IBBD)
  • Definition 3: Binary indicators
  • Theorem 4: Characterization of feasible controls for a synchronous attractor
  • Theorem 5: The attractor cut
  • Theorem 6: Trap spaces remaining under a control
  • Theorem 7: The trap space cut
  • Theorem 8
  • Theorem 9: The strength of a trap space cut