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Approximate Constrained Lumping of Chemical Reaction Networks

Alexander Leguizamon-Robayo, Antonio Jiménez-Pastor, Micro Tribastone, Max Tschaikowski, Andrea Vandin

TL;DR

This work proposes approximate constrained lumping, which allows for a relaxation of exactness within a given tolerance parameter $\varepsilon$ while still working in polynomial time, and provides a heuristic algorithm to find the smallest $\varepsilon$ for a given maximum allowable size of the lumped system.

Abstract

Gaining insights from realistic dynamical models of biochemical systems can be challenging given their large number of state variables. Model reduction techniques can mitigate this by decreasing complexity by mapping the model onto a lower-dimensional state space. Exact constrained lumping identifies reductions as linear combinations of the original state variables in systems of nonlinear ordinary differential equations, preserving specific user-defined output variables without error. However, exact reductions can be too stringent in practice, as model parameters are often uncertain or imprecise -- a particularly relevant problem for biochemical systems. We propose approximate constrained lumping. It allows for a relaxation of exactness within a given tolerance parameter $\varepsilon$, while still working in polynomial time. We prove that the accuracy, i.e., the difference between the output variables in the original and reduced model, is in the order of $\varepsilon$. Furthermore, we provide a heuristic algorithm to find the smallest $\varepsilon$ for a given maximum allowable size of the lumped system. Our method is applied to several models from the literature, resulting in coarser aggregations than exact lumping while still capturing the dynamics of the original system accurately.

Approximate Constrained Lumping of Chemical Reaction Networks

TL;DR

This work proposes approximate constrained lumping, which allows for a relaxation of exactness within a given tolerance parameter while still working in polynomial time, and provides a heuristic algorithm to find the smallest for a given maximum allowable size of the lumped system.

Abstract

Gaining insights from realistic dynamical models of biochemical systems can be challenging given their large number of state variables. Model reduction techniques can mitigate this by decreasing complexity by mapping the model onto a lower-dimensional state space. Exact constrained lumping identifies reductions as linear combinations of the original state variables in systems of nonlinear ordinary differential equations, preserving specific user-defined output variables without error. However, exact reductions can be too stringent in practice, as model parameters are often uncertain or imprecise -- a particularly relevant problem for biochemical systems. We propose approximate constrained lumping. It allows for a relaxation of exactness within a given tolerance parameter , while still working in polynomial time. We prove that the accuracy, i.e., the difference between the output variables in the original and reduced model, is in the order of . Furthermore, we provide a heuristic algorithm to find the smallest for a given maximum allowable size of the lumped system. Our method is applied to several models from the literature, resulting in coarser aggregations than exact lumping while still capturing the dynamics of the original system accurately.

Paper Structure

This paper contains 10 sections, 12 theorems, 46 equations, 8 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constrained Approximated Lumping
  4. Definition and error estimation
  5. Lumping Algorithm
  6. Heuristic search of lumping tolerance
  7. Evaluation
  8. Reduction power
  9. Scalability analysis
  10. Conclusion

Key Result

theorem 1

Given a system of $m$ ODEs of the form given by Equation eq:ode and a matrix $L\in \mathbb{R}^{l\times m}$ with rank $l$, the following are equivalent.

Figures (8)

  • Figure 1: Evolution of observable S2P in a FceRI-like network of a cell-surface receptor borisov_domain-oriented_2008 using different models. The simulation using the original model with 24 variables is displayed in blue.
  • Figure 2: Points projected to $\mathop{\mathrm{rowsp}}\limits(L)$leguizamon-robayo_approximate_2023.
  • Figure 3: Example \ref{['ex:applump']}: evolution of $\mathop{\mathrm{dev}}\nolimits_{L}(f(x(t)))$.
  • Figure 4: Reduced system and error computation for Example \ref{['ex:appredsys']} using the matrix $L$ of Example \ref{['ex:firstode']} on Equation \ref{['eq:examplepert']}.
  • Figure 5: Decomposition of $rJ_i$ into $\mathop{\mathrm{rowsp}}\limits(L)$ and $\mathop{\mathrm{rowsp}}\limits(L) ^{\perp}$leguizamon-robayo_approximate_2023.
  • ...and 3 more figures

Theorems & Definitions (32)

  • definition 1
  • definition 2
  • definition 3
  • theorem 1: Characterization of Exact Lumping tomlin_effect_1997
  • theorem 2: jimenez_clue_2022
  • definition 4
  • definition 5
  • remark 1
  • definition 6
  • definition 7
  • ...and 22 more