Automatic partitioning for the low-rank integration of stochastic Boolean reaction networks

TL;DR

This paper tackles the computational bottleneck of stochastic Boolean reaction networks by combining dynamical low-rank approximation with an automatic, entropy-guided partitioning scheme. Partitions are generated with Kernighan-Lin to minimize pathway cuts, then selected via a rule based on information entropy to minimize loss of pathway information, improving accuracy for a given rank. The approach is demonstrated on multiple biochemical networks, showing substantial memory savings and accuracy gains, especially when using hierarchical tree tensor networks. The method offers a practical preprocessing step that can adapt to model inference tasks and potentially extend to probabilistic Boolean networks and general chemical master equations, enabling scalable stochastic simulations.

Abstract

Boolean reaction networks are an important tool in biochemistry for studying mechanisms in the biological cell. However, the stochastic formulation of such networks requires the solution of a master equation which inherently suffers from the curse of dimensionality. In the past, the dynamical low-rank (DLR) approximation has been repeatedly used to solve high-dimensional reaction networks by separating the network into smaller partitions. However, the partitioning of these networks was so far only done by hand. In this paper, we present a heuristic, automatic partitioning scheme based on two ingredients: the Kernighan-Lin algorithm and information entropy. Our approach is computationally inexpensive and can be easily incorporated as a preprocessing step into the existing simulation workflow. We test our scheme by partitioning Boolean reaction networks on a single level and also in a hierarchical fashion with tree tensor networks. The resulting accuracy of the scheme is superior to both partitionings chosen by human experts and those found by simply minimizing the number of reaction pathways between partitions.

Figures (13)

• Figure 1: An example for a Boolean reaction network with five species. The corresponding graph is shown in Figure \ref{['fig:example-graph']}.
• Figure 2: (a) Graphical representation of Equation (\ref{['eq:DLR-approximation']}) and (b) reaction graph for the five-dimensional example problem of Section \ref{['sec:boolean-reaction-system']} with two partitions. The time dependency of the low-rank factors and the coefficient matrix has been omitted.
• Figure 3: (a) Graphical representation of Equation (\ref{['eq:DLR-approximation']}), with low-rank factors being decomposed according to Equation (\ref{['eq:TTN-X']}) and (b) reaction graph for the example problem of Section \ref{['sec:boolean-reaction-system']} with two subpartitions for each of the two partitions. For sake of simplicity, the time dependency of the low-rank factors, the coefficient matrix and the connection tensors have been omitted.
• Figure 4: Reaction graph of the mTOR pathway example consisting of $22$ species.
• Figure 5: Time-dependent error of the DLR approximation ($r=16$) in the infinity norm for nine one-level partitionings of the mTOR pathway example. The maximum error over time for each partitioning is highlighted with a cross.