Beyond Boolean networks: new tools for the steady state analysis of multivalued networks

TL;DR

This work extends Boolean regulatory networks to multivalued networks by employing MV-algebras on $X_m$ to capture multi-level gene actions. It develops a constructive representation of multivalued update functions using $\odot$ and $\mathrm{neg}$, introduces the $\odot$-$\mathrm{neg}$ network framework, and proves a fixed-point bijection with the original system via a projection, enabling efficient computation through reductions and lattice-point counting in rational polytopes. The authors present algorithmic tools (including Algorithm 1 and Algorithm 2) and demonstrate their approach on a Mammalian Cell Cycle motif and a denitrification model in Pseudomonas aeruginosa, obtaining fixed points consistent with established qualitative models and enabling larger-scale analyses. The methodology blends multivalued logic, combinatorial geometry, and algebraic techniques to provide scalable, interpretable steady-state analysis for complex biological networks, accompanied by a public implementation.

Abstract

Boolean networks can be viewed as functions on the set of binary strings of a given length, described via logical rules. They were introduced as dynamic models into biology, in particular as logical models of intracellular regulatory networks involving genes, proteins, and metabolites. Since genes can have several modes of action depending on their expression levels, binary variables are often not sufficiently rich, requiring the use of multivalued networks instead. In this paper, we explore the multivalued generalization of Boolean networks by writing the standard $(\wedge, \vee, \lnot)$ operations on $\{0, 1\}$ in terms of the operations $(\odot, \oplus, \neg)$ on $\big\{0,\frac{1}{m}, \frac{2}{m}, \dots, \frac{m-1}{m}, 1\big\}$ from multivalued logic. We recall the basic theory of this mathematical framework, and give a novel algorithm for computing the fixed points that in many cases has essentially the same complexity as in the binary case. Our approach provides a biologically intuitive representation of the network. Furthermore, it uses tools to compute lattice points in rational polytopes, tapping a rich area of algebraic combinatorics as a source for combinatorial algorithms for network analysis. An implementation of the algorithm is provided.

Figures (3)

• Figure 1: (a) The nodes take the values $\{0,\tfrac{1}{2},1\}$. The repression shown in the table is naturally described by the function $f_1=x_1 \ominus x_2 = x_1\odot\mathrm{neg}(x_2) = \max \{0, x_1-x_2\}$. In (b) and (c) the nodes take the values $\{0,1,2\}$ and we consider the translated table. In (b), the polynomial $f_1\in\mathbb{F}_3[x_1,x_2]$ summarizes the values in the table over the field $\mathbb F_3$. In (c), the table is instead expressed by the logical function $f_1$, where the notation means $\&$$\leftrightarrow$ AND, $|$$\leftrightarrow$ OR, $x_j{:}i$$\leftrightarrow$$x_j=i$ and $!x_j$$\leftrightarrow$$x_j=0$.
• Figure 3: Screenshot of the app Aye
• Figure 4: $NirQ$ update rule. The table on the left is the multivalued non-smooth version of S3 Table in ABL. On the right: the update rule approximation shown in ABL; the update rule obtained by interpolation from the table according to Theorem \ref{['th:logic']}; and the expression obtained by considering $Dnr$ as an activator and $NarXL$ as a mild activator. The last two expressions coincide with the values in the table.

Theorems & Definitions (45)

• Definition 1
• Example 2
• Remark 3
• Proposition 4
• Remark 5
• Definition 6
• Remark 7
• Remark 8
• Example 9
• Lemma 10