Petri net on expanded networks computes the dynamics of finite multilevel models

Abstract

We discuss connections between different types of models of gene-regulatory network dynamics that include Boolean, finite multilevel and monotone Boolean models. We recast the methodology of describing attractors of Boolean and finite multilevel dynamics using motifs of the expanded network $eRN$ in terms of Petri net dynamics on $eRN$. We show that the Petri net can be used to recover dynamics of the finite multilevel dynamics on the state transition graph. We conclude with new insights that the expanded network viewpoint offers for the robustness of equilibria and complex attractors.

1. Introduction

Modelling the dynamic behaviour of complex systems is a challenging problem in many areas of science. Apart from the inherent difficulty of describing the dynamics of a fully specified high-dimensional dynamical system, the modelling challenges are also severe. In many problems, and this is particularly true in biology, the nonlinearities are often not known and/or their parametrization is only approximate. In addition, one is interested in dynamics for either all or a representative collection of initial conditions. There are no mathematical tools that are able to describe the dynamics of complex systems for all initial conditions across all parameters.

One potential simplification is offered by a concept of a network of interactions, for example, ecological interactions, or, most prominently, gene regulatory interactions. A regulatory network is a directed signed graph where nodes represent genes or proteins and signed directed edges indicate the effect that the presence of one gene/protein has on the abundance of another. The key simplifications in comparison to an arbitrary complex system are (i) the effect of one gene on another is monotone; (ii) the graph is usually far from a complete graph.

There are at least two broad categories of models that are used to model network dynamics. The first are Boolean and closely related finite multilevel models. In Boolean models, each node is either active or inactive, and thus the state variable corresponding to this node assumes values $1$ or $0$, respectively [1,2]. In finite multilevel models, the state variable can assume any of a discrete set ${\displaystyle \{0,\dots ,k\},k\ge 1}$ of values [3]. One way to bridge the binary Boolean and multilevel models is to use the Boolean models, where the state has values in a Boolean algebra with $2^p$ elements with $p>1$ [4]. A subset of Boolean models that are compatible with the network structure are monotone Boolean models [5–7]. Large Boolean models of cellular networks have been constructed and successfully interrogated [8–11]. Most of the analysis focuses on understanding the number and types of steady states (equilibria), since these potentially represent different phenotypes that the network can support and that can arise from different initial conditions. Unfortunately, finding steady states is equivalent to the satisfiability problem, which is nondeterministic polynomial time (NP)-complete [12–14]. Despite this fact, many algorithms have been employed [15–17] to address this problem. A recent review [18] considers several special classes of models (i.e. maximum in-degree is bounded, the Boolean functions are limited to AND or OR [19] or each Boolean function is given as a nested canalysing function [20]) where the time complexity of attractor finding algorithms is $o(2^n)$.

One powerful approach to find all the attractors of finite multilevel models has been proposed by Albert and collaborators [21,22]. Their approach captures the logical structure of the finite multilevel maps in an expanded network (EN). The analysis of an EN finds motifs that correspond to partial fixed points as well as those that correspond to more complex attractors. The advantage of this approach is that it can be computationally feasible for larger networks since it does not need to analyse the entire state space of the Boolean system, whose size grows exponentially with the size of the network. We describe this approach in more detail in §2.5.

The question of parametrization of monotone Boolean functions (MBFs), i.e. the description of all possible MBFs that are compatible with the network structure, and how the choice of MBF affects dynamics have not been discussed until relatively recently [6,7,23,24].

An alternative class of network models are ordinary differential equationbeen discussed until relatively recently (ODE) models, where the nonlinearities are often chosen to be Hill functions. These models are difficult to analyse in their full generality, since their dynamics depend on parameters and initial conditions. Often, the only recourse is numerical simulation of dynamics at a sample of initial conditions and parameters.

Fortunately, a further simplification of the ODE-type model allows the description of dynamics across initial data and parameters. Switching systems are systems of ODEs where each nonlinearity is a piecewise constant function. Studying switching systems as models of network [25–32] dynamics revealed that the parameter space for these models is decomposed into a finite number of domains, which can be organized by proximity into a parameter graph $\mathsf{P}\mathsf{G}$ [33–35]. For each parameter domain, one can compute a state transition graph (STG), which represents the phase space of the ODE switching system. Finally, from the STG, one can compute the set of strongly connected path components and organize them in a Morse graph that captures all attractors and the global organization of recurrent dynamics of the STG. This approach is called Dynamical Signatures Generated by Regulatory Networks (DSGRN) [33–35], and the code is publicly available [36,37]. The finiteness of the parameter graph $\mathsf{P}\mathsf{G}$ allows, in principle, the description of all dynamics that the network is capable of over the entire range of parameters. Importantly, switching models are a bridge between traditional ODE models that use Hill function nonlinearities and finite multilevel models, since

• —the switching nonlinearities are limits of Hill functions [38,39]; and
• —each parameter domain of the switching system corresponds to a finite multilevel system [7,40,41].

It has been recognized for a long time [28,29,32,42] that the most realistic assumption for the ODE models is that the concentration thresholds at which a gene $x$ activates other downstream genes are different. This naturally leads to consideration of finite multilevel models where each gene assumes a set of discrete states that may be greater than two [24,42]. This raises natural questions on (i) what is the relationship between monotone Boolean, finite multilevel and ODE models for a given regulatory network, and (ii) what is the relationship between the expanded network analysis of multilevel models and dynamical system analysis of the STG.

1.1. Overview of the results

In this paper, we first describe a natural class of finite multilevel models that arise from the assumption that the underlying biological processes are continuous. We then delineate the relationship between these multilevel models and the more restricted class of traditional Boolean models. Our main insight is that while in Boolean models the state of each vertex $v_i$ is either ON or OFF, in finite multilevel models, the state $k\in \{0,\dots ,m_i\}$ is the number of downstream nodes that $v_i$ activates (see equation (2.5)). The set of all multilevel functions compatible with the regulatory network is organized in a parameter graph $\mathsf{P}\mathsf{G}$ (§2.2). The justification for this name comes from the fact that this graph also describes the partition of the space of continuous parameters of the ODE switching system that models the same network [7,40,41].

In the second part of the paper, we apply the expanded network framework [21,22] to finite multilevel maps. Our goal is to expand the focus on finding attractors to finding the entire Morse graph $\mathsf{M}\mathsf{G}$ of the STG. To this end, by a slight modification of the original expanded network, we build a bipartite expanded network $eRN$ and define a Petri net update on $eRN$ that mimics the dynamics of the STG [43]. We then provide a new characterization of fixed points (theorem 3.10) and complex attractors (theorem 3.15) using the Petri net and relate these to the approach of Gan & Albert [21] that uses $eRN$ motifs (theorem 3.17). This connection allows us to define the concepts of core oscillator and core of the fixed point (§3.5) that can be related to the robustness of fixed points and oscillations with respect to different choices of multilevel models, which are related by proximity within the parameter graph $PG$.

The exponential growth of the STG size with respect to the size of the network limits the applicability of finite multilevel models to large networks. Specialized algorithms have been developed for the existence of fixed points and the reachability problem that asks for the existence of a path from an initial to a final state in the STG. These include computational tree logic [42,44,45] and symbolic model checking [46]. An extension of this approach [24] allows searching for all finite multilevel functions that support a fixed finite path in STG. Compared to the exponential growth of STG size, the expanded network description often grows more slowly, so the method of computing the Morse graph from the expanded network presented in this paper may be beneficial for studying the dynamics of large networks.

In summary, this work builds close connections between monotone Boolean, finite multilevel discrete and ODE switching system models of network dynamics and shows that the expanded network approach with a Petri net framework can be used to compute the key dynamical signatures of network dynamics.

Remark 1.1. We will consistently use the language of finite multilevel function to denote discrete functions with $k\ge 1$ values and reserve the name multivalued map for maps that associate more than one value to some elements in their domains. The multivalued maps arise as asynchronous update functions of Boolean and finite multilevel functions, as explained in the next section.

2. Background

A regulatory network $RN=(V,E,\delta )$ is a directed graph with nodes $V$, directed edges $E$ and an edge sign function $\delta :E\to \{-\mathrm{1,1}\}$. We denote an edge from node $v_i$ to node $v_j$ without indicating its sign by $v_i⊸v_j$. The edge $v_i⊸v_j$ is activating if ${\delta }_i^j=1$ and repressing if ${\delta }_i^j=-1$. Graphically, an activating edge of a regulatory network edge is denoted by $v_i\to v_j$ and a repressing edge by $v_i⊣v_j$. The sources and targets of a node $v_i$ are given by

$$\mathbf{S}(v_i):=\left\{v_k\in V|v_k⊸v_i\in E\right\}\text{ and }\mathbf{T}(v_i):=\left\{v_j\in V|v_i⊸v_j\in E\right\},$$

respectively.

2.1. Discrete models on regulatory networks

Our goal is to describe a class of finite multilevel models that are compatible with the structure of the regulatory network. Compatibility with the structure will mean respecting the signs of the edges, and this will be enforced by considering only monotone update functions. We impose additional restrictions on the set of functions: we require that each finite multilevel model corresponds to a system of ODEs that is compatible with the network and that this model is as simple as possible.

A traditional construction of a finite state model is a Boolean [25,27,47,48] model that associates to each vertex $v_i$ a Boolean variable $b_i\in \mathbb{B}$ where $\mathbb{B}=\{\mathrm{0,1}\}$ as well as a Boolean-valued update function $h_i:{\mathbb{B}}^{|\mathbf{S}(v_i)|}\to \mathbb{B}$. The interpretation of the function $h_i$ is the following: given an input word $d\in {\mathbb{B}}^{|\mathbf{S}(v_i)|}$, if $h_i(d)=1$, the node $v_i$ is turned ON, and when $h_i(d)=0$, the node $v_i$ is turned OFF. The key assumption of the Boolean model is that when $v_i$ is on, it activates all $v_j\in \mathbf{T}(v_i)$, or, equivalently, all edges from $v_i$ to $v_j\in \mathbf{T}(v_i)$. We use the word activate in a manner that is independent of the sign of the edge $v_i⊸v_j$. When edge $v_i\to v_j$ is activated, the input to $v_j$ is $1$, and when edge $v_i⊣v_j$ is activated, the input to $v_j$ is $0$.

By listing all potential MBFs at each vertex (definition 2.1), one can consider a collection of MBFs $h=(h_1,\dots ,h_n)$, one for each node in $V$, as a model of network dynamics. Listing all such collections describes a set of Boolean models compatible with the network $RN$.

We and others proposed [33,42,49,50] a larger class of finite multilevel models compatible with the network structure by focusing on edges rather than on vertices of the network $RN$. We consider all possible ways the collection of input edges $\{v_j⊸v_i|v_j\in \mathbf{S}(v_i)$} can activate a subset of output edges $\{v_i⊸v_k|v_k\in \mathbf{T}(v_i)$}. This contrasts with the Boolean model, where either all of the edges are activated at the same time or none of them are. Our requirement that this map represents an ODE implies that this activation pattern must be representable as a response to varying levels of a single variable $x_i$ associated to node $v_i$. In the ODE model of the network dynamics, $x_i$ represents a concentration. As the value of $x_i$ passes the threshold associated with the edge $v_i⊸v_j$ with $v_j\in \mathbf{T}(v_i)$, the node $v_i$ activates the corresponding edge. In principle, there can be more than one threshold associated to edge $v_i⊸v_j$. The simplest model satisfying these conditions has a single threshold for each edge. The crucial constraint imposed by this view of activation comes from the fact that these thresholds are linearly ordered. This, together with the assumption of monotonicity of the response, implies that if an input activates the $k$th threshold, it will also activate the lower thresholds $1,\dots ,k-1.$ Since there are $|\mathbf{T}(v_i)|$ thresholds of variable $x_i$ associated to $v_i$, we assume that the activation levels of node $v_i$ attain a discrete set of values $x_i\in X_i=\{0,1,\dots ,t_i\}$ where $t_i:=|\mathbf{T}(v_i)|$ represents the number of thresholds that the node $v_i$ can activate. We use this approach to describe the set of parameters associated with a regulatory network $RN=(V,E,\delta )$ [51,52], each of which uniquely defines a finite multilevel discrete function.

An order parameter for node $v_i$ is a bijection ${\theta }_i:\mathbf{T}(v_i)\to \{1,\dots ,t_i\}$, which defines an ordering of the out edges of $v_i$ and thus the ordering of thresholds that $x_i$ can activate. The set of order parameters for $v_i$ is denoted by $\mathrm{\Theta }(v_i)$. The set of all order parameters is given by $\mathrm{\Theta }:={\prod }_{v_i\in V}\mathrm{\Theta }(v_i)$.

Next, we define logic parameters for $RN=(V,E,\delta )$. It is useful to treat the set $\mathbb{B}$ as a Boolean lattice with natural order $0\prec 1$, i.e. $\mathbb{B}:=(\{\mathrm{0,1}\};0\prec 1\})$. Let ${\mathbb{B}}^n$ be a lattice of Boolean $n$ vectors with order induced component-wise by $\prec$.

Definition 2.1. A function $f:{\mathbb{B}}^n\to \mathbb{B}$ is an increasing MBF if $b^1\prec b^2$ implies $f(b^1)⪯f(b^2)$.

Definition 2.2. A logic parameter is a collection of increasing MBFs

$$f_i=(f_i^1,\dots ,f_i^{t_i})$$

satisfying the ordering condition

$$\begin{array}{lrr}& f_i^1(b)⪰\cdots ⪰f_i^{t_i}(b)& \end{array}$$ (2.1)

for all $b\in {\mathbb{B}}^{|\mathbf{S}(v_i)|}$.

The purpose of the function $f_i^{{\theta }_i(v_j)}$ is to describe when an input $b$ activates the edge $v_i⊸v_j$:

• —$f_i^{{\theta }_i(v_j)}(b)=1$ denotes the fact that $b$ activates the edge $v_i⊸v_j$ associated with the threshold ${\theta }_i(v_j)$;
• —$f_i^{{\theta }_i(v_j)}(b)=0$ $b$ does not activate the edge $v_i⊸v_j$.

Note that the function $f_i$ has a domain and a range

$$\begin{array}{lcr}& f_i:{\mathbb{B}}^{|\mathbf{S}(v_i)|}\to {\mathbb{B}}^{|\mathbf{T}(v_i)|}.& \end{array}$$ (2.2)

A collection $f:=(f_i{)}_{v_i\in V}$ is a logic parameter. The set of all logic parameters for node $v_i$ is denoted $\mathcal{L}(v_i)$, while the set of all logic parameters is $\mathcal{L}:={\prod }_{v_i\in V}\mathcal{L}(v_i)$.

The set of parameters is the $\mathcal{P}:=\mathcal{L}\times \mathrm{\Theta }$. We call $\mathcal{P}(v_i):=\mathcal{L}(v_i)\times \mathrm{\Theta }(v_i)$ the set of parameters for node $v_i$, and note that all parameters are given by the product $\mathcal{P}={\prod }_{v_i\in V}\mathcal{P}(v_i)$. The following section endows the set $\mathcal{P}$ with the structure of a graph by defining adjacency between elements of $\mathcal{P}$.

For illustration, consider the network in figure 1a. In figure 1b, we describe a particular parameter $p$. Since node $v_1$ has two out edges $|\mathbf{T}(v_1)|=2$, we list one of the two possible choices of the order parameter ${\theta }_{11}<{\theta }_{21}$. Since $|\mathbf{T}(v_2)|=1$, there is only one choice of threshold order for node $v_2$. The logic parameter choice at $v_2$ respects the fact that the edge $v_1⊣v_2$ is negative. The pair of logical functions $f_1=(f_1^1,f_1^2)$ completes the description of the parameter $p$.

Figure 1.

(a) A regulatory network RN. (b) A choice of network parameter. (c) Circles and arrows depict the STG of the RN in (a) at the parameter in (b). Axes represent continuous concentrations associated to nodes $v_1$ and $v_2$ divided by thresholds ${\theta }_{11},{\theta }_{12}$ and ${\theta }_{21}$ (dashed lines). Resulting domains are in one-to-one correspondence with the states of STG. The state $(\mathrm{0,1})$ represents a fixed point FP. An FC through states $(\mathrm{1,0})$, $(\mathrm{1,1})$, $(\mathrm{2,1})$ and $(\mathrm{2,0})$ is also marked. (d) Morse graph of the STG in (c).

2.2. The parameter graph

The parameters are related by a proximity relationship, which we now define. Two parameters for a node $v_i$, $(f_i,{\theta }_i),({\hat{f}}_i,{\hat{\theta }}_i)\in \mathcal{P}(v_i)$ are adjacent if exactly one of the following conditions is satisfied:

• —Order adjacency: $f_i={\hat{f}}_i$ and the values of the order parameters ${\theta }_i$ and ${\hat{\theta }}_i$ are exchanged on a single pair of neighbouring entries on which the logic parameters agree, i.e. there exists $v_{j_1},v_{j_2}\in \mathbf{T}(v_i)$ such that ${\theta }_i(v_{j_1})+1={\theta }_i(v_{j_2})$ and ${\hat{\theta }}_i(v_{j_1})-1={\hat{\theta }}_i(v_{j_2})$ with ${\theta }_i(v_j)={\hat{\theta }}_i(v_j)$ for all $j\notin \{j_1,j_2\}$.
• —Logical adjacency: ${\theta }_i={\hat{\theta }}_i$ and the Boolean functions $f_i$ and ${\hat{f}}_i$ differ in a single input, i.e. there exists exactly one $b\in {\mathbb{B}}^{|\mathbf{S}(v_i)|}$ such that $f_i(b)\ne {\hat{f}}_i(b)$.

The factor graph for node $v_i$ is the undirected graph $\mathsf{P}\mathsf{G}(v_i):=(\mathcal{P}(v_i),\mathcal{E}(v_i))$ whose nodes are parameter nodes for $v_i$ and whose edges are given by adjacency.

The parameter graph $\mathsf{P}\mathsf{G}:=(\mathcal{P},\mathcal{E})$ is the Cartesian product

$$\mathsf{P}\mathsf{G}:=\prod _{v_i\in V}\mathsf{P}\mathsf{G}(v_i).$$

That is, there is an edge $(p^1,p^2)\in \mathcal{E}$ if and only if there is a unique $v_i\in V$ such that $(p_i^1,p_i^2)\in \mathcal{E}(v_i)$ and $p_i^1=p_i^2$ otherwise. Note that each factor graph $\mathsf{P}\mathsf{G}(v_i)$ has $|\mathbf{T}(v_i)|$ isomorphic subgraphs, one for each choice of order parameter ${\theta }_i$. Each of these sub-factor graphs [53] is given by the collection of all Boolean functions $f_i$ with edges defined by logical adjacency. Different sub-factor graphs are then joined by edges defined by order adjacency.

It follows that the size and structure of the parameter graph $\mathsf{P}\mathsf{G}$ depend only on the number of input and output edges for each $v_i\in V$, but not on the sign structure of edges given by $\delta =({\delta }_i^j{)}_{v_i⊸v_j}$. The sign structure $\delta$ affects dynamics at each $p\in \mathsf{P}\mathsf{G}$, which we describe next.

2.3. Dynamics

For any parameter $p=(f,\theta )\in \mathsf{P}\mathsf{G}$, the finite multilevel dynamics occur on the state space

$$\mathcal{D}:=\prod _{v_i\in V}{X}_i,X_i:=\{\mathrm{0,1},\dots ,t_i\}.$$

We call $d\in \mathcal{D}$ a state of the $RN$ and construct the finite multilevel update function $g:\mathcal{D}\to \mathcal{D}$ from the information contained in $(f,\theta )$ in three steps.

First, the state $d=(d_i{)}_{v_i\in V}$ gives rise to an input to a node $v_j\in V$ via the input map

$${\displaystyle B^j:\mathcal{D}\to {\mathbb{B}}^{|\mathbf{S}(v_j)|},B^j:=(B_i^j{)}_{v_i\in \mathbf{S}(v_j)},}$$ (2.3)

where

$${\displaystyle B_i^j(d):=\{\begin{array}{ll}0,& \text{ if }{d}_i<{\theta }_i\left(v_j\right)\text{ and }{\delta }_i^j=1\text{ or }{d}_i\ge {\theta }_i\left(v_j\right)\text{ and }{\delta }_i^j=-1\\ 1,& \text{ if }{d}_i\ge {\theta }_i\left(v_j\right)\text{ and }{\delta }_i^j=1\text{ or }{d}_i<{\theta }_i\left(v_j\right)\text{ and }{\delta }_i^j=-1.\end{array}}$$

Note that for an activating edge $v_i\to v_j$, if $d_i$ is below the threshold ${\theta }_i(v_j)$, then the input from $v_i$ to $v_j$ is $B_i^j(d)=0$, while if $d_i$ is above the threshold ${\theta }_i(v_j)$, then the input from $v_i$ to $v_j$ is $B_i^j(d)=1$. This assignment is reversed if the edge $v_i⊣v_j$ is repressing. Given $n=|V|$, the range of the map $B:=(B^1,B^2,\dots ,B^n)$ is ${\mathbb{B}}^{|E|}$ with

$$B:\mathcal{D}\to {\mathbb{B}}^{|E|}$$

since every edge in $E$ is in $\mathbf{S}(v_i)$ for exactly one node $v_i\in V$. Therefore, $B$ describes the assignment of $0$ and $1$ to every edge in the network for any $d\in \mathcal{D}$. The function $B$ depends on the order parameter $\theta$ and the sign structure function $\delta$, but not on the logic parameter $f$. It is only through this function $B$ that the sign structure $\delta =({\delta }_i^j{)}_{v_i⊸v_j}$ and the threshold order ${\theta }_i$ affect the dynamics.

In the second step, we take into consideration the logic parameter $f$ from parameter $p=(f,\theta )\in \mathsf{P}\mathsf{G}$. We apply the logic parameter map $f_i$ (see equation (2.2)) associated with the vertex $v_i$ to the output of the function $B^i$ and set

$$\begin{array}{lcr}& g_i(d):=\sum _{j\in \mathbf{T}(v_i)}{f}_i^j(B^i(d)).& \end{array}$$ (2.4)

The construction of the finite multilevel update function $g$ is summarized by the commutative diagram

where $g:=(g_1,g_2,\dots ,g_n)$.

We remark that the class of finite multilevel functions $g$ that arise from this construction is a strict subset of all finite multilevel functions on $\mathcal{D}$, since they have to be filtered through the collection of MBFs $f$.

Definition 2.3. We define three types of dynamics for the network $RN$ on the set $\mathcal{D}$ for every parameter $(f,\theta )\in \mathcal{P}$. These three types of dynamics are all derived from the function $g$.

Synchronous update: The map $g:\mathcal{D}\to \mathcal{D}$ is a synchronous update.

Asynchronous update: The asynchronous update (aU) ${\mathcal{G}}_0:\mathcal{D}\rightrightarrows \mathcal{D}$ is a multivalued map given by

$${\displaystyle \overline{d}\in {\mathcal{G}}_0(d)\iff \overline{d}=(d_1,\dots ,g_i(d),\dots d_n)\text{ for single index }i.}$$

Nearest neighbour asynchronous update: The nearest neighbor asynchronous update (nnU) $\mathcal{G}:\mathcal{D}\rightrightarrows \mathcal{D}$ is a multivalued map generated by and defined by

• —If $g(d)=d$, then $\mathcal{G}(d)=\{d\}$.
• —For any $v_i$ and $\eta \in \{-\mathrm{1,1}\}$ satisfying $\eta g_i(d)>\eta d_i$, the state
  $${\overline{d}}_i=d_i+\eta ,{\overline{d}}_j=d_j\text{for}j\ne i$$

  satisfies $\overline{d}\in \mathcal{G}(d)$.

The maps $g$, ${\mathcal{G}}_0$ and $\mathcal{G}$ depend on the choice of network and threshold order through the order parameter $\theta$. We explicitly include these dependencies as arguments if needed.

Note that for the Boolean models, the asynchronous and nearest neighbor asynchronous updates are the same since every variable has only two levels, $0$ and $1$. For Boolean models, our definition of synchronous and asynchronous updates coincides with the concepts of synchronous and asynchronous update of the Boolean maps [17,54]. We carefully distinguish the concepts of aU and nnU since only the latter update can capture the behaviour of solutions of a system of differential equations, since ODE solutions are continuous and can only travel between neighbouring domains. Such transitions correspond to continuous changes in a variable $x_i$ associated with node $v_i$. For this reason, we will focus on the nnU update $\mathcal{G}$. The multivalued map $\mathcal{G}$ can be represented as a state transition graph $\mathsf{S}\mathsf{T}\mathsf{G}(\mathcal{D})$ with vertices given by the states $\mathcal{D}$ and with directed edge $d\to \overline{d}$ in $\mathsf{S}\mathsf{T}\mathsf{G}(\mathcal{D})$ if and only if $\overline{d}\in \mathcal{G}(d)$.

We interpret the multivalued map $\mathcal{G}$ as a set-valued map. There are at least two alternative interpretations of $\mathcal{G}$. One is to add transition probabilities to each edge of STG and regard the resulting system as a probabilistic Boolean network [55]. Alternatively, one $\mathcal{G}$ can be regarded as a collection of possible deterministic transitions. In this case, for any finite sequence of specified order of coordinate updates, one obtains a deterministic map [56]. It is easy to see that trajectories for any such map are a subset of those generated by $\mathcal{G}.$

In figure 1c, we show the nnU update of the multilevel function $g$ that corresponds to the parameter $p$ described in figure 1b. In this case, the underlying space $\mathcal{D}=\{\mathrm{0,1,2}\}\times \{\mathrm{0,1}\}$ of the STG is represented by circles that lie within rectangular domains in $R_2^{+}$. The partition of $R_2^{+}$ into rectangular squares anticipates the correspondence between the STG and the switching ODE system: nodes $v_1$ and $v_2$ represent continuous concentration values, and these values are divided into intervals by the activation thresholds.

2.4. The Morse graph

The recurrent dynamics of $\mathcal{G}(\cdot ;p)$ are encoded by a Morse graph $\mathsf{M}\mathsf{G}(p)$. The Morse graph $\mathsf{M}\mathsf{G}(p)=(\mathsf{S}\mathsf{C}\mathsf{C},A)$ is a directed graph with nodes $\mathsf{S}\mathsf{C}\mathsf{C}$ consisting of strongly connected components (SCCs) of $\mathsf{S}\mathsf{T}\mathsf{G}(\mathcal{D},p)$. We say that $M_2\in \mathsf{S}\mathsf{C}\mathsf{C}$ can be reached from $M_1\in \mathsf{S}\mathsf{C}\mathsf{C}$ if there is a directed path in $\mathsf{S}\mathsf{T}\mathsf{G}(\mathcal{D},p)$ starting at some node in $M_2$ and ending in a node in $M_1$. The Morse graph is the Hasse diagram on $\mathsf{S}\mathsf{C}\mathsf{C}$ of this reachability relation. We label each SCC $s\in \mathsf{S}\mathsf{C}\mathsf{C}$ according to the following:

• —If $s\in \mathsf{S}\mathsf{C}\mathsf{C}$ consists of a single recurrent state, $s=\{x\}$, then $x$ is a fixed point of $\mathcal{G}$, and we label $s$ by $\mathsf{F}\mathsf{P}(x)$. Note that by definition 2.3, this is equivalent to $g(x)=x$.
• —If $s\in \mathsf{S}\mathsf{C}\mathsf{C}$ is not an $\mathsf{F}\mathsf{P}$, then we label $s$ as a partial cycle ($\mathsf{P}\mathsf{C}$) or a full cycle ($\mathsf{F}\mathsf{C}$). The SCC $s$ is a $\mathsf{P}\mathsf{C}$ if $s$ is constant in at least one coordinate: there is a node $u\in V$ and an integer $k$ such that $x\in s$ implies $x_u=k$. If $s$ is not an $\mathsf{F}\mathsf{P}$ or a $\mathsf{P}\mathsf{C}$, then $s$ is an $\mathsf{F}\mathsf{C}$.
• —If $s\in \mathsf{S}\mathsf{C}\mathsf{C}$ has no out edges in $\mathsf{M}\mathsf{G}(p)$, then $s$ is stable. Otherwise, $s$ is unstable.

The Morse graph of the STG in figure 1c is shown in figure 1d. In this case, there is no order relationship between the two nodes representing a fixed point and the FC.

2.5. The expanded network

The expanded network model [21] is a graph theoretical approach to finding the attractors of either a Boolean or finite multilevel model with fixed update functions without explicitly constructing the STG. We briefly describe the construction of the expanded network in [21] that applies to a general finite multivalued map. Later, we provide a more precise mathematical description of this construction and show how it relates to a finite multivalued map $g:\mathcal{D}\to \mathcal{D}$ that we defined in (2.4).

Given $RN=(V,E,\delta )$, the expanded network approach defines new nodes and new edges. First, it represents each state $k\in \{0,\dots ,m_i\}$ of $v_i\in V$ as a Boolean virtual node $v_i^k$. The ON state of $v_i^k$ means that the regulatory network node $v_i$ is in state $k$. We denote the set of virtual nodes for $v_i$ by

$$Q_i:=\{v_i^0,\dots ,v_i^{m_i}\}.$$

We call $Q_i$ the $i$th set of sister nodes. Since a network node $v_i$ can be only in one state $k\in \{0,\dots ,m_i\}$, only one of the sister nodes can be in the ON state (i.e. $v_i^k=1$) at any given time.

By a temporary abuse of notation, which will be resolved in §2.6, we identify the nodes $v_i^j\in {\prod }_{i=1}^n{Q}_i$ by their Boolean values, with $v_i^j=1$ indicating that $v_i^j$ is ON. A regulatory function ${\ell }_i^k:{\prod }_{v_j\in \mathbf{S}(v_i)}^n{Q}_j\to \mathbb{B}$ is a Boolean function that governs the update of the value of the node $v_i^k$. Following [21], we assume that any given input $q\in \prod Q_i$ can activate at most one node in each sister node set $Q_i$, i.e.

$$\begin{array}{lcr}& {\ell }_i^k(q)=1⟹{\ell }_i^j(q)=0\text{for all}j\ne k.& \end{array}$$ (2.6)

The inputs of ${\ell }_i^k$ are Boolean values of the virtual nodes, expressed in a special disjunctive normal form called Blake canonical form (BCF) [57]. In this form, ${\ell }_i^k$ is a disjunction of conjunctive clauses indicating which virtual nodes in their ON state turn ON the node $v_i^k$, written as follows:

$$v_i^k\leftarrow (v_1^{m_1}\wedge v_2^{m_2}\wedge \cdots \wedge v_n^{m_n})\vee (v_1^{w_1}\wedge v_2^{w_2}\wedge \cdots \wedge v_n^{w_n})\vee \cdots \vee (v_1^{r_1}\wedge v_2^{r_2}\wedge \cdots \wedge v_n^{r_n}).$$

We call this the logical representation of the function ${\ell }_i^k$. For example, consider the two-node RN in figure 1, where $v_1$ has three states $\{\mathrm{0,1,2}\}$ while $v_2$ has two states $\{\mathrm{0,1}\}$. Note that at the parameter $p$ described in figure 1b, $v_1$ can assume state $0$ if either both $v_1$ and $v_2$ are zero, or $v_1$ is zero and $v_2$ is one. Therefore, the regulatory function ${\ell }_1^0$ is given by

$$v_1^0\leftarrow (v_1^0\wedge v_2^0)\vee (v_1^0\wedge v_2^1).$$

This clause indicates that $v_1^0$ is ON when both $v_1^0$ and $v_2^0$ are ON, or when both $v_1^0$ and $v_2^1$ are ON. Notice that all states of node $v_2$ are present on the right-hand side in conjunction with $v_1^0$, implying that the state of $v_2$ does not determine the state of $v_1^0$ when $v_1^0$ is ON. In BCF, these conjunctive clauses are merged, and the function becomes

$$v_1^0\leftarrow v_1^0.$$

The benefit of writing the regulatory function in BCF is the following: information about which inputs affect the network dynamics is preserved while simultaneously removing redundant information.

We represent every conjunctive clause of the BCF of ${\ell }_i^k$, for all $i,k$ as a composite node $w$ in the expanded network. Let $V^c$ be the set of all composite nodes. Additionally, let

$$V^v:=\underset{v_i\in V}{⨆}{Q}_i$$

be the set of all virtual nodes. We are now set up to define the expanded network.

Definition 2.4. Given a regulatory network $RN=(V,E,\delta )$ and a regulatory function ${\ell }_i^k:{\prod }_{v_j\in \mathbf{S}(v_i)}^n{Q}_j\to \mathbb{B}$ for each $i,k$, the expanded network $eRN=(eV,eE)$ of $RN$ has nodes

$$eV:=V^v\cup V^c,$$

with edges constructed as follows:

• (1) for each composite node $w=v_{i_1}^{k_1}\wedge v_{i_2}^{k_2}\wedge \cdots \wedge v_{i_m}^{k_m}\in V^c$, add edges $(v_{i_j}^{k_j},w)\in eE$ for all $j=1,\dots ,m$, and

• (2) for each virtual node $v_i^k\in V^v$, add an edge from each clause in ${\ell }_i^k$ to $v_i^k$.

In our running example of the network in figure 1 at parameter $p$, the update functions are listed in figure 2a. The second entry $v_1^1\leftarrow (v_1^1\wedge v_2^0)\vee (v_1^2\wedge v_2^0)$ will be represented by three nodes: $v_1^1$, $w_1$ with label $(v_1^1\wedge v_2^0)$ and $w_2$ with label $(v_1^2\wedge v_2^0)$. Then there are edges from virtual nodes to the composite node $w_1$ given by

Figure 2.

(a) Regulatory functions, depicted in logical notation, for the RN and parameter shown in figure 1. Notice how the regulatory function for ${\displaystyle v_1^0}$ had conjunctive clauses that could be merged using BCF. (b) The expanded network representation of the regulatory functions in (a), where white circular nodes are virtual nodes and rectangular grey nodes are the composite nodes. (c) The bipartite graph representation of the regulatory functions in (a). Note that the expanded network and bipartite graph are equivalent.

$${\displaystyle v_1^1\to w,v_2^0\to w_1,\text{ as well as }{w}_1\to v_1^1,}$$

and the second composite node $w_2$ connected in an analogous way.

In the first entry $v_1^0\leftarrow v_1^0$, the first copy of $v_1^0$ is the virtual node, but the second copy is a composite node $w$ with label $v_1^0$. In a bipartite graph, we will keep these nodes separate and connect $v_1^0\to w$ and $w\to v_1^0$. Anytime there is a self-loop in the expanded network like $v_1^0\to v_1^0$ in figure 2b, we realize that the node $v_1^0$ plays the role of both a composite node and a virtual node. Therefore, in the bipartite description, we separate the virtual node $v_1^0$ from the composite node $v_1^0$. Observe that the expanded graph is a directed graph with unsigned edges. We are reusing the notation $\to$ to represent the directed edges of the expanded graph, where, unlike in the regulatory networks, this notation does not indicate edge sign.

Remark 2.5. Nodes in regulatory networks may or may not have self-loops. It has been suggested that biological networks contain a significant number of self-loops that support auto-regulation [58,59]. However, not all nodes exhibit auto-regulation; in fact, it is estimated that only 10% of yeast gene-encoding regulators are auto-regulated [60]. To maintain generality, our approach allows any number of nodes without self-loops, including the case where none are present.

2.6. Mathematical description of the expanded network

This expanded network construction applies to any finite multilevel function [21]. While the mathematical description below of the expanded network is also valid in this generality, our goal is to relate it to the regulatory network dynamics described by the finite multilevel function $g:\mathcal{D}\to \mathcal{D}$. Therefore, we will apply the expanded network construction in this context. To do so, we construct regulatory functions ${\ell }_i^k$ from $g$. These regulatory functions are constructed as a Booleanization of $g$, by first constructing a map $\phi :\mathcal{D}\to Z$, where $Z$ is the set of all possible values of the set of nodes ${\prod }_{i=1}^n{Q}_i$, and then defining a map $L:Z\to Z$ by

$$\begin{array}{lcr}& L=\phi \circ g\circ {\phi }^{-1},& \end{array}$$ (2.7)

so that the following diagram commutes:

Lastly, we use the map $L$ to construct regulatory functions ${\ell }_i^k$ that are induced by $g$. To represent the multilevel function $g:\mathcal{D}\to \mathcal{D}$, we define the set of virtual nodes as

$${\displaystyle Q_i=\{v_i^0,\dots ,v_i^{t_i}\},}$$

where $t_i=|\mathbf{T}(v_i)|$. To construct $Z$ as a Booleanization of the space $\mathcal{D}$, consider a state $d\in D$ and note that the $i$th component of the state $d$ assumes values $d_i\in X_i=\{0,\dots ,t_i\}$. We associate to each $d_i$ the vector $y_i:=(y_i^0,\dots ,y_i^{t_i})\in {\mathbb{B}}^{|Q_i|}$ such that $y_i^k=1$ if $k=d_i$ and $y_i^k=0$ otherwise. Note that the vector $y_i$ uniquely represents $d_i$.

Let ${\phi }_i:X_i\to Y_i$ be the map defined by

$${\phi }_i(k)=(y_i^0,\dots ,y_i^{t_i})\text{ where }{y}_i^k=1\text{ and }{y}_i^j=0\text{ for all }j\ne k.$$

Finally,

$$Z_i:=\mathrm{range}\left({\phi }_i\right)=\{(y_i^0,\dots ,y_i^{t_i})\in Y_i|y_i^j=1\text{ for exactly one }j\}.$$

The collection of Boolean vectors $(y_1,\dots ,y_n)\in Z:={\mathrm{\Pi }}_{i=1}^n{Z}_i$ represents a Booleanization of the state space $\mathcal{D}$.

To connect this to the virtual nodes of the expanded network, note that the non-zero entry $y_i^k$ of the vector $y\in Z_i$ identifies the state $k\in X_i$ that $v_i\in V$ is in and, therefore, uniquely identifies the node $v_i^k\in Q_i$ as being in state $1$. As a result, there is a bijection between the sets $Z_i$ and $Q_i$ for every $i$, i.e.

$$\begin{array}{lcr}& Z:={\mathrm{\Pi }}_{i=1}^n{Z}_i\cong {\mathrm{\Pi }}_{i=1}^n{Q}_i.& \end{array}$$ (2.9)

Hence, $Z$ is also the Booleanization of the collections of virtual nodes $(v_1^{i_1},\dots ,v_n^{i_n})$, where $v_j^{i_j}\in Q_j$. Let $\phi :\mathcal{D}\to Z$ be defined by $\phi =({\phi }_1,\dots ,{\phi }_n)$. Given the multilevel function $g:\mathcal{D}\to \mathcal{D}$ and using the function $\phi :\mathcal{D}\to Z$, we define $L:Z\to Z$ by

$$L=\phi \circ g\circ {\phi }^{-1}.$$

We note that since $\phi$ is a bijection, maps $g$ and $L$ are conjugate.

The second step of the Booleanization construction turns the components $L_i:Z\to Z_i$ of the map $L=(L_i{)}_{i=1}^n$ into a collection of Boolean-valued maps ${\ell }_i^k:Z\to \mathbb{B}$ that will facilitate their decomposition into Blake normal form and construction of composite nodes in the expanded network. We first decompose $Z$ into disjoint sets based on the value of the function $L_i$. Let

$${\displaystyle U_i^k:=\left\{z\in Z\mid L_i(z)=e_i^k\right\},\text{ for }k=1,\dots ,t_i,}$$

where $e_i^k=(0,\dots ,1,\dots 0)\in {\mathbb{B}}^{|X_i|}$ with value $1$ in position $k$. Note that the sets $U_i^k$ are a decomposition of the set $Z$ for every $i=1,\dots ,n$. Each set $U_i^k$ is a truth set of a function with a Boolean range

$${\ell }_i^k:Z\to \mathbb{B},$$

given by

$${\displaystyle \begin{array}{r}{\ell }_i^k(z)=\{\begin{array}{ll}1& \text{ if }z\in U_i^k\\ 0& \text{ if }z\notin U_i^k.\end{array}\end{array}}$$ (2.10)

Therefore, every function $L_i$ can be represented as a collection of Boolean functions $({\ell }_i^1,\dots ,{\ell }_i^{t_i})$, which by (2.10) satisfy condition (2.6).

This construction finishes the Booleanization of the function $g:\mathcal{D}\to \mathcal{D}$, where we replace states in $\mathcal{D}$ by a set of Boolean states $Z\subset Y$ and represent function $g$ first by $L=(L_i{)}_{i=1}^n$ and then represent each component $L_i$ as a series of regulatory functions $({\ell }_i^k)$. This Booleanization can be directly translated into the expanded network description in [21]. First, each $z\in Z$ (and hence each state $d\in \mathcal{D}$) can be represented as a collection of virtual nodes

$$(v_1^{i_1},v_2^{i_2},\dots ,v_n^{i_n}),$$

one for each $i\in \{1,\dots ,n\}$.

Second, the decomposition of regulatory functions ${\ell }_i^k$ in Blake normal form facilitates its representation as a collection of composite nodes $V^c$, one for each conjunctive clause in the normal form.

Definition 2.6. For any composite node $w$, we define two related subsets of $V^v$. Let $\alpha (w)\subset V^v$ be the set of virtual nodes that appear in the conjunctive clause that correspond to $w$. For $C\subset V^c$ with $C=\{w_1,\dots ,w_k\}$ let

$$\alpha (C):=\bigcup _{i=1}^k\alpha (w_i).$$

We call $\alpha (w)$ the support of $w$ . Let $\omega (w)\subset V^v$ be the set of virtual nodes $v\in V^v$ such that the edge $(w\to v)\in eE$. We set

$$\omega (C):=\bigcup _{i=1}^k\omega (w_i).$$

For example, consider the composite node $v_1^1\wedge v_2^1$ in the expanded network example from figure 2. The support of $v_1^1\wedge v_2^1$ is $\alpha (v_1^1\wedge v_2^1)=\{v_1^1,v_2^1\}$, and the downset of $v_1^1\wedge v_2^1$ is given by $\omega (v_1^1\wedge v_2^1)=\{v_1^2\}$. We note that the expanded network in figure 2 matches the STG dynamics in figure 1c, exhibited by the network in figure 1a at the DSGRN parameter in figure 1b. The Morse graph representation of the SCCs of the dynamics in figure 1d should therefore be represented in the expanded network in figure 2b,c. As the expanded network is a static structure, we proceed to define a dynamical system in terms of Petri nets that would mimic dynamics on the STG in figure 1c.

3. Dynamics on the expanded network $eRN$

A Petri net [61] is a model developed originally for the simulation of chemical reactions. It is a bipartite directed graph with edges connecting two disjoint sets of nodes called places and transitions.

Definition 3.1. A net is a tuple $N=(P,T,F)$ where

• —$P$ and $T$ are disjoint finite sets of places and transitions, respectively, and
• —$F\subset (P\times T)\cup (T\times P)$ is a set of (directed) arcs (or flow relations).

The expanded network $eRN$ is a net with $P=V^v$, $T=V^c$ and $F=eE$, the set of directed edges. However, we now modify the basic concept of a Petri net to a stratified Petri net where the vertices $V^v$ admit decomposition into disjoint sets $Q_i$. We use the stratification to specify a stratified Petri net transition that, as we will show, is equivalent to an asynchronous update of the function $g$.

Definition 3.2. A stratified net is a net $N=(P,T,F)$ where the set $P$ is stratified into disjoint sets $P={⨆}_{i=1}^n{P}_i$.

The expanded network $eRN$ is a stratified net with $V^v={⨆}_{i=1}^n{Q}_i$.

Definition 3.3. Given a net $N=(P,T,F)$, a complete node configuration is a set $S\subset V^v$ containing exactly one sister node from each $Q_i$, i.e. $|S\cap Q_i|=1$ for all $i=1,\dots ,n$. Let $CN$ be the set of all complete node configurations. By using (2.9), we note that

$$\begin{array}{lcr}& CN:=\{S\subset V^v||S\cap Q_i|=1\forall i=1,\dots ,n\}\cong Z.& \end{array}$$ (3.1)

A complete clause configuration is a set $C=\{c_1,\dots ,c_k\}\subset V^c$ such that the support of $C$ is a complete node configuration, i.e.

$$\alpha (C)\in CN.$$

Let $CC$ be the collection of all complete clause configurations for $N$, i.e.

$$CC:=\{C\subset V^c|\alpha (C)\in CN\}.$$

The set

$$CN=\{\{v_1^0,v_2^0\},\{v_1^0,v_2^1\},\{v_1^1,v_2^0\},\{v_1^1,v_2^1\},\{v_1^2,v_2^0\},\{v_1^2,v_2^1\}\}$$

is the set of all complete node configurations for the expanded graph in figure 2, and the set of all complete clause configurations for this expanded network is

$$\begin{array}{lrlr}& CC=& \{\{v_1^1\wedge v_2^0\},\{v_1^1\wedge v_2^1\},\{v_1^2\wedge v_2^0\},\{v_1^2\wedge v_2^1\},\{v_1^1,v_1^1\wedge v_2^0\},& \\ & & \{v_1^1,v_1^1\wedge v_2^1\},\{v_1^2,v_1^2\wedge v_2^0\},\{v_1^2,v_1^2\wedge v_2^1\}\}.& \end{array}$$ (3.2)

In figure 3, the set $C=\{v_1^2\wedge v_3^0,v_2^1\}$ is an example of a complete clause configuration for the expanded network in figure 3c, but not a complete clause configuration for figure 3b,d. Note that in figure 3, we do not draw the complete bipartite graph as in figure 2, but any time there is an edge of the form $v_i^k\to v_j^{\ell }$, the node $v_i^k$ is also a composite node.

Figure 3.

(a) Regulatory network for the expanded networks shown in (b–d) representing dynamics at different parameters. (b) Full cycle: grey nodes are the core oscillation. (c) Stable motif ${\displaystyle \left\{v_1^2,v_2^0\right\}}$ (blue nodes) stabilizing multiple FPs: FP(2,0,1) given by ${\displaystyle \left\{v_1^2,v_2^0,v_3^1\right\}}$ (blue and green nodes) and FP(2,0,2) given by ${\displaystyle \left\{v_1^2,v_2^0,v_3^2\right\}}$ (blue and red nodes). (d) Stable and oscillating motif: the stable motif ${\displaystyle \left\{v_3^2\right\}}$ (blue node) stabilizes nodes ${\displaystyle \left\{v_1^0,v_2^0\right\}}$ (grey striped), giving FP(0,0,2), while the PC (grey and grey striped nodes) has value ${\displaystyle \left\{v_3^0\right\}}$ fixed and oscillates between ${\displaystyle \left\{v_2^1,v_1^1,v_2^0,v_1^0\right\}}$.

Usual simulation of Petri net dynamics consists of an initial placing of finite tokens on place nodes. A new token is placed on a place node $x$, if there is a directed edge from a transition node $e$ to $x$ and there are tokens on all place nodes $y_i$ that have directed edges $y_i\to e.$ Usually, these tokens are removed and a new one is added to $x$. In chemistry, the removed tokens represent chemicals being consumed in a reaction, and new tokens represent the chemicals produced by the reaction. In this context, a place node can contain any number of tokens. However, there are many variations on modelling Petri net dynamics, and we define three below that will be useful for our purposes. To define our update functions on a Petri net $N=(P,T,F)$, we will use the following terms:

• (1) we say a transition $t\in T$ is enabled if each $p_i\in \{p_j\in P|(p_j,t)\in F\}$ has a token,

• (2) we say an enabled transition $t\in T$ is executed when at least one $p\in P$ is tokened with the property that $(t,p)\in F$. Note that this execution protocol is slightly different from the usual protocol, which for every enabled transition $t$ puts a token on all $p\in P$ that satisfy $(t,p)\in F$.

Definition 3.4. We define three update algorithms on a stratified net $N=(V^v,V^c,E)$ that map a cohort $S_1\in CN$ to another cohort $S_2\in CN$. As any transition in a regular Petri net, these protocols may fire in state $s\in CN$ if that transition is enabled (i.e. there are sufficient tokens in all of its input places).

Synchronous update: Define ${\displaystyle S_2=L(S_1)}$, where S₂ is the state where all transitions enabled by S₁ have been executed simultaneously. That is, consider the sets

$$W=\{w_j\in V^c|\text{if }(s_i,w_j)\in E\text{ then }{s}_i\in S_1\}\text{ and }U=\{v_k\in V^v|(w_j,v_k)\in E\text{ and }{w}_j\in W\}.$$

We place tokens on all $v_k\in U$. For each $i=1,\dots ,n$, if $v_k\in Q_i\cap U\ne \varnothing$, we remove the token from $s_i\in Q_i\cap S_1$. We set $S_2:=U$.

Asynchronous update: We define the multivalued map ${\mathcal{F}}_0$ by setting $S_2\in {\mathcal{F}}_0(S_1)$ if S₂ is obtained from S₁ by executing any single transition in W enabled by S₁. That is, S₂ is obtained by placing a token on ${\displaystyle v_i\in U\cap Q_i}$, removing the token from ${\displaystyle s_i\in Q_i}$ and keeping other tokens on s_j for ${\displaystyle j\ne i}$.

Nearest neighbor asynchronous update: For every ${\displaystyle v_i^j\in Q_i\cap U}$ that is produced by the transition ${\mathcal{F}}_0$, we replace the token on ${\displaystyle s_i=v_i^k\in Q_i}$ by a new token on ${\displaystyle v_i^{k+1}}$ when j > k, and by a new token on ${\displaystyle v_i^{k-1}}$when ${\displaystyle j<k}$. Tokens remain in place if j = k. Let $\mathcal{F}:CN\rightrightarrows CN$ be a collection of all such nearest neighbour asynchronous transitions.

By construction, the three Petri net updates $L,{\mathcal{F}}_0,\mathcal{F}$ match $g,{\mathcal{G}}_0,\mathcal{G}$ from definition 2.3, respectively. Therefore, we have the following theorem.

Theorem 3.5. The following diagrams commute:

Note that the first diagram is identical to diagram (2.8) in view of isomorphism ( 3.1 ).

3.1. Fixed points of the state transition graph in the Petri net representation of $eRN$

The goal of this subsection is to describe an algorithm that computes the fixed points of an STG using Petri net dynamics on the expanded network $eRN$.

Recall from definition 2.3 that a domain $d^{\ast }\in \mathcal{D}$ represents a fixed point at parameter $p=(g,\theta )$ if and only if $g(d^{\ast })=d^{\ast }$, i.e. for $d^{\ast }=(d_1^{\ast },d_2^{\ast },\dots ,d_n^{\ast })$, we have

$$\begin{array}{lcr}& g_i(d^{\ast })=d_i^{\ast }\forall i\in \{1,2,\dots ,n\}.& \end{array}$$ (3.3)

In the general Petri net that models chemical reactions, the number of tokens on each node is important as it signifies the capacity of the system to execute the same reaction multiple times. In our application, tokens only signify whether a particular subset of $V^v$ or $V^c$ can support such a transition. Therefore, from now on, we replace the language of tokens by the language of subsets that these tokens occupy.

We first characterize a pair of sets $(S,C)\subset V^v\times V^c$ that corresponds to a fixed point of $g:\mathcal{D}\to \mathcal{D}$.

Definition 3.6. Given sets $S\subset V^v$ and $C\subset V^c$, we say that S supports C if $\alpha (C)\subset S$, and we say that C supports S if $S\subset \omega (C)$. We denote the relationship of set $A$ supporting set $B$ by writing $A\Rrightarrow B$.

To illustrate, the set $S=\{v_1^2,v_2^1\}$ supports the set $C=\{v_1^2\wedge v_2^1\}$ in the expanded graph from figure 2 since $\alpha (C)=\{v_1^2,v_2^1\}=S$; however, since $\omega (C)=\{v_1^2\}$, we see that $C$ does not support $S$. Our first main result relates a fixed point in an STG to a particular bipartite subgraph structure in the expanded network.

Theorem 3.7. A domain $d^{\ast }\in \mathcal{D}$ is a fixed point at parameter $p=(g,\mathrm{\Theta })$ if and only if given the set

$$S:=\{v_1^{d_1^{\ast }},v_2^{d_2^{\ast }},\dots ,v_n^{d_n^{\ast }}\}$$

there is a set of composite nodes $C\subset V^c$ in the expanded network $eRN$ such that

• (1) $S$ supports $C$, and

• (2) $C$ supports $S$.

We call a pair of sets $(S,C)\subset V^v\times V^c$ a fixed point pair.

Proof. $(\Rightarrow )$ Suppose $d^{\ast }\in \mathcal{D}$ is a fixed point. We want to show that there is a set of composite nodes $C\subset V^c$ in the expanded network $eRN$ such that given

$$S:=\{v_1^{d_1^{\ast }},v_2^{d_2^{\ast }},\dots ,v_n^{d_n^{\ast }}\}$$

we have $S\Rrightarrow C$ and $C\Rrightarrow S$.

Since (3.3) holds, the commutativity of the first diagram in theorem 3.5 and (2.10) implies that

$${\ell }_i^{d_i^{\ast }}(\phi (d^{\ast }))=1.$$

It follows that in the BCF of the function ${\ell }_i^{d_i^{\ast }}$, there is at least one composite node $c_i\in V^c$ such that $\alpha (c_i)\subset S$, i.e. $S\Rrightarrow c_i$. In addition, the superscript $d_i^{\ast }$ shows that the composite node $c_i$ supports the virtual node $v_i^{d_i^{\ast }}\in V^v$. Then the set

$$C:=\{c_i|i=1,\dots ,n\}$$

is the desired set, since our first observation shows that $S\Rrightarrow C$ and our second observation shows that $C\Rrightarrow S$.

$(\Leftarrow )$ Suppose there is a set of composite nodes $C\subset V^c$ in the expanded network $eRN$ such that for a set

$$\begin{array}{lcr}& S:=\{v_1^{d_1^{\ast }},v_2^{d_2^{\ast }},\dots ,v_n^{d_n^{\ast }}\}& \end{array}$$ (3.4)

we have $S\Rrightarrow C$ and $C\Rrightarrow S$. We want to show that the domain $d^{\ast }=(d_1^{\ast },d_2^{\ast },\dots ,d_n^{\ast })$ is a fixed point. Since $C\Rrightarrow S$, then for all $v_i^{d_i^{\ast }}$, there exists $c\in C$ such that $(c,v_i^{d_i^{\ast }})\in eE$. Since $S\Rrightarrow C$, then for all $c\in C$, there is a collection of $s\in S$ that verify $C$. In other words, each $c\in C$ has the form

$$c=\underset{k\in B}{\bigwedge }{v}_k^{d_k^{\ast }}$$

for some $B\subset S$. Since $c$ corresponds to a conjunctive clause in BCF for some $d_j^{\ast }$ for some $j$, we have ${\ell }_j^{d_j^{\ast }}(\phi (d^{\ast }))=1$. The first commutative diagram in theorem 3.5 now verifies $g_i(d^{\ast })=d_i^{\ast }$. Since this holds for any $i=1,\dots ,n$, we have $g(d^{\ast })=d^{\ast }$.∎

We illustrate the concept of fixed point pairs in figure 3c,d. Consider the set $S=\{v_1^2,v_2^0,v_3^1\}$ from the expanded network in figure 3c and the set $C=\{v_1^2,v_2^0,v_1^2\wedge v_3^1\}$. Notice that $S\Rrightarrow C$ since $\alpha (C)=\{v_1^2,v_2^0,v_3^1\}\subset S$, and $C\Rrightarrow S$ since $(v_1^2,v_2^0)$, $(v_2^0,v_1^2)$, and $(v_1^2\wedge v_3^1,v_3^2)$ are edges in the expanded network. Thus, $(S,C)$ is a fixed point pair representing a fixed point in domain $(\mathrm{2,0,1})$. To see an example of a fixed point in an STG represented as a fixed point pair in the expanded network, see figure 2 with the fixed point pair $S=\{v_1^0,v_2^1\}$ and $C=\{v_1^0\}$ representing a fixed point in domain $(\mathrm{0,1})$. Compare this to the STG shown in figure 1, which, as expected, also has a fixed point in domain $(\mathrm{0,1})$.

3.2. Construction of fixed point pairs $(S,C)$

We now describe an algorithm that constructs fixed point pairs $(S,C)$. Given an expanded network $eRN$, recall from definition 3.3 that any $S\in CN$ has the property that $|S\cap Q_i|=1$ for all $i=1,\dots ,n$. Additionally, recall that $CC$ consists of all $C\subset V^c$ such that $\alpha (C)\in CN$, which implies $|\alpha (C)\cap Q_i|=1$. In addition to the set of complete clause configurations $CC$, we will also consider a larger set of all clause configurations $AC$ with $CC\subset AC$.

Definition 3.8. Let $AC$ be the collection of all subsets $C\subset V^c$ such that $\alpha (C)\subset V^v$ satisfies

$|\alpha (C)\cap Q_i|\le 1$. Note that this relaxes the completeness assumption $|\alpha (C)\cap Q_i|=1$ and therefore $CC\subset AC$. Recalling the set of complete clause configurations $CC$ in (3.2), we have

$$AC=CC\cup \{\{v_1^0\},\{v_1^1\},\{v_1^2\}\}.$$

We define a multivalued map $\psi :AC\rightrightarrows AC$ by the following procedure. Take any $C\in AC$ and consider $\alpha (C)\subset V^v$. Then the multivalued image of $C$ is the set

$${\displaystyle \psi (C):=\{C_i\},}$$

where $C_i\in AC$ is any subset that supports the set $\alpha (C)$. That is, for any $s\in \alpha (C)$ and any $i$, there is a $c\in C_i$ such that $(c,s)\in eE$.

Definition 3.9. We say that the set $C_0\in AC$ is focal if there is an element $C_1\in \psi (C_0)$ such that $C_1\subset C_0$. We say that $C_0\in AC$ is completely focal if $C_0$ is focal, and there is $S_0\subset \omega (C_1)$ with $S_0\Rrightarrow C_0$ and $S_0\in CN$.

Given the expanded network in figure 3, consider $C=\{v_1^2,v_2^0,v_1^2\wedge v_3^2\}\in AC$ and note that $C\in \psi (C)$ since $v_2^0\to v_1^2$, $v_1^2\to v_2^0$, and $v_1^2\wedge v_3^2\to v_3^2$ are edges in the expanded network (i.e. for each $s\in \alpha (C)=\{v_1^2,v_2^0,v_3^1\}$, there is a $c\in C$ such that $(c,s)\in eE$). Since $C\subset C$, then $C$ is focal, and since the set $\omega (C)=\{v_1^2,v_2^0,v_3^1\}\in CN$ supports $C$, then $C$ is also completely focal. The following theorem characterizes fixed points of the function $g$ in terms of the map $\psi$.

Theorem 3.10.

• (1) For any completely focal set $C_0$, the pair $(S_0,C_1)\subset V^v\times V^c$ is a fixed point pair.

• (2) For any fixed point pair $(S,C)$, there is a completely focal set $C_0$ such that $C_1=C$ and $S_0=S$.

Proof. (1) Let $C_0$ be completely focal. Then there exists a set $C_1\subset C_0$ such that $C_1\Rrightarrow \alpha (C_0)$ and a set $S_0\subset \omega (C_1)$ such that $S_0\Rrightarrow C_0$. By theorem 3.7, to show that $(S_0,C_1)$ is a fixed point pair, we need to show that $S_0\Rrightarrow C_1$ and $C_1\Rrightarrow S_0$. Note that $C_1\Rrightarrow S_0$ by construction of $S_0$, and since $C_1\subset C_0$ and $S_0\Rrightarrow C_0$, then $S_0$ must also support $C_1$. Thus, $(S_0,C_1)$ is a fixed point pair.

(2) Suppose that $(S,C)$ is a fixed point pair. We want to show that there exists a completely focal set $C_0$ such that $C_1=C$ and $S_0=S$. By theorem 3.7, we know that $C\Rrightarrow S$ and $S\Rrightarrow C$.

First, we show that $C\in AC$. Notice that since $S\Rrightarrow C$, we have $\alpha (C)\subset S$. At the same time, since $S\in CN$, then $|\alpha (C)\cap Q_i|\le 1$ for all $i$. This implies $C\in AC$.

Choose $C_1:=C$ and $C_0:=C$. Then $C_1\subset C_0$ with $C_1\Rrightarrow S$ and $S\Rrightarrow C_0$, showing $C_0$ is focal. To show $C_0$ is completely focal, observe that since $C\Rrightarrow S$, then $S\subset \omega (C)$, concluding that $S_0:=S$ satisfies $S_0\subset \omega (C_0)$ with $S_0\Rrightarrow C_0$ and $S_0\in CN$. Therefore, $C_0$ is completely focal with $C_1=C$ and $S_0=S$.∎

Note that we use the set $\omega (C_1)$ instead of $\alpha (C_0)$ when finding a set $S_0$. To see that this is necessary, consider the example of the fixed point pair $S=\{v_1^0,v_2^0\}$ and $C=\{v_1^0\}$ from the expanded network in figure 2. Using $C_1=C_0=C$, we see that $\alpha (C)=\{v_1^0\}$, while $\omega (C)=\{v_1^0,v_2^0\}$. Notice that there is no set $C\in AC$ for which the set $S$ is equal to $\alpha (C)$.

3.3. Non-equilibrium recurrent dynamics

Another dynamical behaviour that is important in the study of dynamics is recurrent dynamics. Fixed points constitute trivially recurrent dynamics, so the interest here is in non-trivial recurrent dynamics. In an STG, this is represented as an SCC, i.e. where each vertex is reachable from any other vertex that contains more than one state $d\in \mathcal{D}$. While fixed point dynamics are independent of the considered updates $g,{\mathcal{G}}_0$ and $\mathcal{G}$, non-trivial SCCs are not [21]. Thus, what constitutes an SCC will depend on the type of update implemented. As explained in §2.4, in [33,34,62], two types of non-trivial SCC were considered: FC and a PC. The FC is an SCC where there exists a closed path within the SCC where every component changes at least once (and hence at least twice). The PC is a non-trivial SCC where there is a subset of components that remain fixed on any path within the SCC.

We now provide a characterization of these sets in the expanded network $eRN$ using the map $\psi$. Since the presence of a non-trivial SCC depends on a choice of update between $g$, ${\mathcal{G}}_0$ or $\mathcal{G}$, the characterization will involve the corresponding Petri net updates $L,{\mathcal{F}}_0$ and $\mathcal{F}$.

Definition 3.11. We say that $C_0\in AC$ is $k$-periodic if there exists a sequence of sets

$$C_1\in \psi (C_0),C_2\in \psi (C_1),\dots ,C_k\in \psi (C_{k-1}),$$

such that the following conditions hold:

• (1) at least two sets $C_i$ and $C_j$ are distinct,

• (2) $C_k\subset C_0$,

• (3) for each $j=1,\dots ,k$, there exists $S_{j-1}\subset \omega (C_j)$ such that $S_j\in CN$ and $S_j$ supports $C_j$, and

• (4) for each $j=1,\dots ,k$, there exists at least one $\ell \in \{1,\dots ,n\}$ such that

  $${\displaystyle S_j\cap Q_{\ell }\ne S_{j-1}\cap Q_{\ell }.}$$ (3.5)

Note that for $k=1$, this definition coincides with the definition of a completely focal set, definition 3.9.

As an example, consider the set $C_0=\{v_1^0,v_2^0\}$ from the expanded network in figure 3b, and the sequence

$$C_1=\{v_1^2,v_2^0\},C_2=\{v_1^2,v_2^1\},C_3=\{v_1^0,v_2^1\},C_4=\{v_1^0,v_2^0\}$$

satisfying $C_j\in \psi (C_{j-1})$ for $j=\mathrm{1,2,3,4}$ and $C_4=C_0$. Additionally, we set $S_j=\omega (C_{j+1})$ for each $j$ and get

$$S_0=\{v_1^0,v_2^0,v_3^1\},S_1=\{v_1^2,v_2^0,v_3^1\},S_2=\{v_1^2,v_2^1,v_3^0\},S_3=\{v_1^0,v_2^1,v_3^0\},$$

which satisfy

$$C_4\Rrightarrow S_3\Rrightarrow C_3\Rrightarrow S_2\Rrightarrow C_2\Rrightarrow S_1\Rrightarrow C_1\Rrightarrow S_0\Rrightarrow C_0.$$

Figure 4a shows a sequence of these sets in detail. Figure 4b shows another example of a 4-periodic set in the expanded network shown in figure 3d.¹

Figure 4.

(a) A depiction of the 4-periodic set ${\displaystyle C_0=\left\{v_1^0,v_2^0\right\}}$ from figure 3b and the sequence of sets ${\displaystyle C_4\Rrightarrow S_3\Rrightarrow \dots \Rrightarrow S_0\Rrightarrow C_0}$ satisfying the conditions in definition 3.11. Each set is indicated as a column of nodes below the set label, and the edges between sets correspond to edges in the expanded network. Note that there are two changes between the sets S₂ and S₁, as well as between S₀ and S₃, but this supports periodic paths under nearest neighbour asynchronous update. (b) A depiction of the 4-periodic set ${\displaystyle C_0=\left\{v_1^0\wedge v_3^0,v_2^0\right\}}$ from figure 3d corresponding to a PC. Notice how in (a) that ${\displaystyle \alpha (C_j)\subset S_j}$ for all j, while in (b), we have ${\displaystyle \alpha (C_j)=S_j}$ for all j.

Definition 3.12. We say the set $C_0\in AC$ is $k$-periodic under synchronous update $L$ if it is $k$-periodic.

Notice in figure 4a that between sets $S_2$ and $S_1$, as well as between sets $S_0$ and $S_3$, the changes involve two different sister nodes. For a synchronous update map $L$, the sequence

$$S_0\to S_3\to S_2\to S_1\to S_0$$

can be directly translated to a periodic path through the corresponding nodes in the STG. However, this is not a valid set of nodes and transitions for an asynchronous update map. To obtain a valid asynchronous path in the STG from the collection of sets $S_0\to S_3\to S_2\to S_1\to S_0$, additional sets need to be added between $S_2$ and $S_1$ as well as between $S_0$ and $S_3$ in a way that the difference between consecutive sets may only be in a single sister node. For example, given the sets $S_{1.5}=\{v_1^2,v_2^0,v_3^0\}$ and $S_{3.5}=\{v_1^0,v_2^0,v_3^0\}$, the sequence

$$S_0\to S_{3.5}\to S_3\to S_2\to S_{1.5}\to S_1\to S_0$$

represents a periodic path in the STG under the asynchronous update map. Notice that replacing $S_{1.5}$ with $\{v_1^2,v_2^1,v_3^1\}$ or $S_{3.5}$$\{v_1^0,v_2^1,v_3^1\}$ also produces valid paths. The following definition and corollary formalize this idea.

Definition 3.13. Given a $k$-periodic set $C_0\in AC$, let $C:=\{C_j{\}}_{j=0}^k$ be a collection of transition sets generated by $C_0$. Define $\mathcal{S}:=\{{\mathcal{S}}_q,{\mathcal{S}}_{q-1},\dots ,{\mathcal{S}}_0\}$ with $q>k$ as an ordered collection, where for each ${\mathcal{S}}_i\in \mathcal{S}$, we have ${\mathcal{S}}_i\in CN$, and there exists exactly one $\ell \in \{1,\dots ,n\}$ where (3.5) holds. We say that $\mathcal{S}$ is a path under asynchronous update ${\mathcal{F}}_0$ consistent with $C$ if there is a partition of the collection $\mathcal{S}$ into $k+1$ subsets

$$U_k,U_{k-1},\dots ,U_0,U_j\subset \mathcal{S},$$

each of which contains consecutive sets in $\mathcal{S}$, such that each transition ${\mathcal{S}}_i\to {\mathcal{S}}_{i-1}$ with ${\mathcal{S}}_i\in U_j$ is enabled by an asynchronous Petri net update of the set $C_j$.

We say that a $k$-periodic set $C_0\in AC$ supports a periodic path under asynchronous update if there is a path under asynchronous update ${\mathcal{F}}^0$ consistent with $C$ with ${\mathcal{S}}_q={\mathcal{S}}_0$.

There is an analogous definition for when $C_0\in AC$ supports a periodic path under nearest neighbour asynchronous update $\mathcal{F}$. The only difference is that the difference between consecutive sets in collection $S$ satisfies the additional requirement that for each $j=\{1,\dots ,k\}$, when there exists some $v_{\ell }^a\in {\mathcal{S}}_j\cap Q_{\ell }$ and $v_{\ell }^b\in {\mathcal{S}}_{j-1}\cap Q_{\ell }$ such that $v_{\ell }^a\ne v_{\ell }^b$ then $a=b\pm 1$.

Returning to the figure 4a example, define $S_{1.5}=\{v_1^2,v_2^0,v_3^0\}$ and $S_{3.5}=\{v_1^0,v_2^0,v_3^0\}$. Then the set

$$\mathcal{S}:=\{S_0,S_{3.5},S_3,S_2,S_{1.5},S_1,S_0\}$$

with the partition

$$U_4:=\{S_0\},U_3:=\{S_{3.5},S_3\},U_2:=\{S_2\},U_1:=\{S_{1.5},S_1\},U_0:=\{S_0\}$$

is a periodic nearest neighbour path consistent with $C=\{C_4,C_3,C_2,C_1,C_0\}$. To see this, note that transition $v_1^2\in C_1$ (upper arrow) enables transition from $S_2$ to $S_{1.5}$ and transition $v_1^2\in C_1$ (lower arrow) enables transition from $S_{1.5}$ to $S_1$. Similarly, transition $v_1^0\in C_4$ (upper arrow) enables transition from $S_0$ to $S_{3.5}$ and transition $v_1^0\in C_4$ (lower arrow) enables transition from $S_{3.5}$ to $S_3$. Again, notice that replacing $S_{1.5}$ with $\{v_1^2,v_2^1,v_3^1\}$ or $S_{3.5}$ with $\{v_1^0,v_2^1,v_3^1\}$ is also valid.

Corollary 3.14. Given a $k$-periodic set $C_0$, any nearest neighbour periodic path $S$ consistent with the collection of updates $C$ corresponds to a periodic path in STG.

Proof. Since all states in the collection of sets $S_j\in S$ satisfy $S_j\in CN$, each $S_j$ corresponds to exactly one $x\in \mathcal{D}$. Then a periodic path under nearest neighbour asynchronous update corresponds to a closed path in the set $\mathcal{D}$, where each transition is between two nodes $x_i,x_j$ that are neighbours in $\mathcal{D}$.∎

The following theorem characterizes the periodic behaviour in terms of the map $\psi$.

Theorem 3.15. For any $k$-periodic set $C_0$, the pair $(S_0,C_k)\subset V^v\times V^c$ is a fixed point pair for a fixed point of $g^k$. The periodic path in the STG is encoded in sets of vertices $S_{k-1},S_{k-2},\dots ,S_1,S_0$, which are all subsets of complete node configurations $CN$.

Proof. Direct consequence of theorem 3.10 applied to $g^k$ instead of $g$.

Definition 3.11 does not distinguish between periodic sets that would produce FCs versus PCs. Whether a periodic set supports an FC or a PC can be determined by examining the sequence of sets $S_j,j=0,\dots ,k-1$. In particular, if

$$|Q_{\ell }\cap \bigcup _{j=1}^k{S}_j|\ge 2$$

for all $\ell =1,\dots ,n$, then the periodic set supports an FC; otherwise, it supports a PC. See figures 2b, 3b and 5b for examples of expanded networks with FC oscillating motifs and figure 3d for an expanded network with a PC oscillating motif. The sequences of sets $S_j,j=0,\dots ,3$ of the 4-periodic set describing the FC in figure 3b and PC in figure 3d are described in detail in figure 4.

Figure 5.

(a) Regulatory network RN consisting of a negative cycle. (b) The expanded network that admits an oscillatory motif consisting of all nodes. The two sets of token distributions represent two distinct 8-periodic sets ${\displaystyle C_0=\left\{v_1^0,v_2^0,v_3^0,v_4^0\right\}}$ (blue/upper) and ${\displaystyle C_0^{\prime }=\left\{v_1^0,v_2^0,v_3^1,v_4^0\right\}}$ (green/lower). ${\displaystyle C_0}$ corresponds to a stable FC in the STG, while ${\displaystyle C_0^{\prime }}$ corresponds to an unstable FC. (c) Depiction of the two FCs in STG, stable (left) and unstable (right). Each node label a, b, c, d corresponds to domain in STG and a collection of virtual nodes ${\displaystyle v_1^a,v_2^b,v_3^c,v_4^d}$. The coloured nodes indicate the domain represented by the 8-periodic sets ${\displaystyle C_0}$ and ${\displaystyle C_0^{\prime }}$ in (b) for each cycle.

3.4. Correspondence between motifs and fixed points

In this section, we describe the connection between motifs of the expanded network $eRN$ and the Morse nodes in the Morse graph $\mathsf{M}\mathsf{G}$ of the STG. The motif approach to describing attractors of Boolean and discrete models was introduced in the work of Albert and co-workers [21,22].

Definition 3.16. Consider an expanded network $eRN$.

• (1) A motif $M$ is an SCC in $eRN$ that satisfies the following condition: if a composite node is in the SCC, then all of its input nodes must also be in the SCC.

• (2) A stable motif is the smallest motif $M$ such that $M$ contains no sister node pairs, i.e. $|M\cap Q_i|\le 1$ for all $i=1,\dots ,n$.

• (3) An oscillating motif is the largest motif in $eRN$ such that at least one virtual node in the motif has at least two sister nodes in the motif, i.e. $|M\cap Q_i|>1$ for at least one $i\in \{1,\dots ,n\}$.

Smallest and largest refer to the partial order induced by inclusion.

We separate condition (1) since it guarantees that the subset of nodes stabilize regardless of the state of the rest of the system. Conditions (2) and (3) then specify the type of the motif. There is a one-to-one correspondence between a stable motif and a partial fixed point of the system, defined as a state in which a subset of nodes stabilize regardless of the state of the rest of the system [21].

The following theorem describes the connection between completely focal sets (definition 3.9), periodic (definition 3.11) sets and motifs.

Theorem 3.17.

• (1) Given a completely focal set $C_0$, the bipartite graph on vertices $C_1\cup S_0$ contains at least one stable motif.

• (2) Given a $k$-periodic set $C_0$ with $k>1$, the bipartite graph on vertices ${\bigcup }_{i=1}^k{C}_i\cup S_{i-1}$ contains at least one oscillating motif.

Proof. Let $eRN=(eV,eE)$ be an expanded network. (1) Consider a completely focal set $C_0$ in $eRN$. We want to show that $C_1\cup S_0$ contains an SCC $M$ such that every composite node in $M$ also has all of its inputs in $M$ (i.e. $M$ is a motif). Recall that since $C_1\in \psi (C_0)$, then $C_1\Rrightarrow \alpha (C_0)$ and $\alpha (C_0)\Rrightarrow C_0$ by construction. Additionally, since $C_1\subset C_0$, then $\alpha (C_1)\subset \alpha (C_0)$, which implies

$$C_1\Rrightarrow \alpha (C_1)\Rrightarrow C_1.$$

Recall that a graph is strongly connected if it has a path from $u$ to $v$ and a path from $v$ to $u$ for every pair of vertices $u,v$. Thus, the graph on vertices $C_1\cup \alpha (C_1)$ must decompose into $m\ge 1$ maximal strongly connected subgraphs.

Suppose that $M_i$ for $i=1,\dots ,m$ are the disjoint maximal SCCs. Observe that each $M_i$ is a motif. Indeed, if $s\in C_1\cap M_j$, then by construction all its inputs are in $M_j\cap \alpha (C_1)$ since $M_j$ is a maximal SCC. Since $C_1\subset AC$, we have $|M_j\cap Q_i|\le 1$ for all $j=1,\dots ,m$. The proof of part (a) is complete by noting that each $M_j$ must contain at least one minimal SCC; these minimal components in each $M_j$ are stable motifs.

(2) Suppose that $C_0$ is a $k$-periodic set in $eRN$ with $k>1$. Then there exists a sequence of distinct sets

$$C_1\in \psi (C_0),C_2\in \psi (C_1),\dots ,C_k\in \psi (C_{k-1}),$$

such that $C_k\subset C_0$. Note that $C_k$ may be equal to $C_0$. Specifically, we have

$$C_k\Rrightarrow \alpha (C_{k-1})\Rrightarrow \dots \Rrightarrow C_2\Rrightarrow \alpha (C_1)\Rrightarrow C_1\Rrightarrow \alpha (C_0)\Rrightarrow C_k.$$

Using a similar argument to (1), we know that the subgraph of the bipartite graph on the vertices ${\bigcup }_{j=1}^k{C}_j\cup \alpha (C_j)$ decomposes into $m\ge 1$ maximal SCCs $M_i,i=1,\dots ,m$, which are also motifs. Therefore, we need to show that at least one motif $M_i$ has $|M_i\cap Q_{\ell }|>2$ for some $\ell =1,\dots ,n$, and, therefore, it is an oscillating motif.

Assume $S_j=\alpha (C_j)$ for all $j$, if $|M_i\cap Q_{\ell }|\le 1$ for all $\ell$, then we know

$$S_j\cap Q_{\ell }\cap M_i=S_{j-1}\cap Q_{\ell }\cap M_i$$

for all $\ell$ and $j$, which implies that

$$S_j\cap M_i=S_{j-1}\cap M_i$$

for all $j$. If this is true for all motifs $i$, then $S_j=S_{j-1}$ for all $j$, contradicting that $C_0$ is $k$-periodic. This is because, for each $j$, we need $S_j\cap Q_{\ell }\ne S_{j-1}\cap Q_{\ell }$ for at least one $\ell$ by definition 3.11.

Now assume that $\alpha (C_j)\subset S_j$ for at least one $j$ and that $|M_i\cap Q_{\ell }|\le 1$ for all $\ell$ and $i$. Then the same argument as given above shows that $\alpha (C_j)=\alpha (C_{j-1})$ for all $j$. Since $C_0$ is $k$-periodic, then for some $\ell$ it must be true that

$$(S_j\setminus \alpha (C_j))\cap Q_{\ell }\ne (S_{j-1}\setminus \alpha (C_{j-1}))\cap Q_{\ell }.$$

Specifically, this implies that there are sister nodes $v_{\ell }^a,v_{\ell }^b\in Q_{\ell }$ with $a\ne b$ such that there exist composition nodes $c_1,c_2\in {\bigcup }_{i=1}^k{C}_i$ where $(c_1\to v_{\ell }^a),(c_2\to v_{\ell }^b)\in eE.$ Since the composition nodes are derived from BCF, any complete configuration of virtual nodes $q\in \prod Q_i$ (each of which by diagram (2.8) and equation (2.9) represent a state in $\mathcal{D}$) such that $q\cap \alpha (c)=\alpha (c)$ will validate the logical expression represented by the composition node $c$. Therefore, such a configuration of virtual nodes $q$ enables execution of the composite node $c$.

Consider the sets of all such configurations

$${\mathcal{Q}}_1:=\{q\in \prod Q_i|q\cap \alpha (c_1)=\alpha (c_1)\}\text{ and }{\mathcal{Q}}_2:=\{q\in \prod Q_i|q\cap \alpha (c_2)=\alpha (c_2)\}.$$

Note that ${\mathcal{Q}}_1$ contains all complete virtual node configurations that turn on $v_{\ell }^a$, and ${\mathcal{Q}}_2$ contains all complete virtual node configurations that turn on $v_{\ell }^b$.

Importantly, ${\mathcal{Q}}_1\cap {\mathcal{Q}}_2\ne \varnothing$ because if $\alpha (c_1)\cap Q_i\ne \varnothing$ and $\alpha (c_2)\cap Q_i\ne \varnothing$, then $\alpha (c_1)\cap Q_i=\alpha (c_2)\cap Q_i$ since $\alpha (C_j)=\alpha (C_{j-1})$ for all $j$.

We conclude that there exists a complete virtual node configuration $q\in {\mathcal{Q}}_1\cap {\mathcal{Q}}_2$ such that the regulatory functions (see definition 2.4) for the sister nodes $v_{\ell }^a$ and $v_{\ell }^b$ are active, i.e.

$${\ell }_{\ell }^a(q)={\ell }_{\ell }^b(q)=1.$$

This is a contradiction since we know by equation (2.6) that only one node in $Q_{\ell }$ can be active for any given $q$. This contradiction completes the proof.∎

As described in [21], the stable motifs of the expanded network are self-sustaining subnetworks representing partial fixed points of a system. Their existence indicates the presence of attractors, but does not, in general, fully specify a fixed point in the STG. A complete fixed point can be determined through methods such as combining compatible stable motifs or analysing downstream nodes stabilized by the motifs. By contrast, the Petri net framework directly identifies fixed points via completely focal sets. There are also advantages to the Petri net method when considering oscillatory dynamics. As noted in [21], there can be several periodic paths in the STG represented by a single oscillatory motif in the expanded network. In figure 5, we use the characterization of $k$-periodic sets to construct two different periodic paths in the STG forming two different Morse sets $FC$ in the STG, which correspond to the same periodic motif.

Consider the four node regulatory network in figure 5a consisting of a single negative loop. In figure 5b, we present an expanded network for a parametrization of the network in figure 5a, where each node represents both a virtual node and corresponding composite node. This expanded network has a single oscillatory motif consisting of all nodes corresponding to two FCs in the STG, one stable and the other unstable. We show two different initial distributions of tokens representing different periodic sets $C_0=\{v_1^0,v_2^0,v_3^0,v_4^0\}$ (blue upper tokens) and $C_0^{\prime }=\{v_1^0,v_2^0,v_3^1,v_4^0\}$ (green lower tokens). The Petri net update starting at the blue token distribution, which corresponds to the state $(1000)$ in the STG, leads to sequence of states depicted in figure 5c (left). This sequence of states is 8-periodic and represents a stable Morse set FC in STG. On the other hand, starting from green token distribution in figure 5b, the Petri net update produces a periodic sequence of states depicted in figure 6c (right). This corresponds to unstable Morse set FC in STG. This example illustrates that the same oscillating motif in the expanded network can correspond to two different Morse sets in STG, and how they can be recovered using the $k$-periodic sets.

3.5. Core of strongly connected components

Theorem 3.10 allows us to relate the size of the set $C_1\subset C_0$ for a completely focal set $C_0$ to robustness of the corresponding fixed point.

As shown in theorem 3.10, for every completely focal set $C_0$, there is a fixed point pair $(S_0,C_1)\in V^v\times V^c$ where $S_0\in CN$ and $C_1\in AC$. This means that for every $v_i^k\in S_0$, there is at least one edge $c_1\to v_i^k$ from $c_1\in C_1$, which is supported by nodes in the set $S_0$. However, there can be multiple composite nodes $U(v_i^k):=\{c_1,\dots ,c_j\}$ that support $v_i^k$, each of which corresponds to a term in BCF of ${\ell }_i^k$. Let $\mathcal{U}:={\bigcup }_{v\in S_0}U(v)$ be the set of all composite nodes that support $S_0$ and are validated by nodes in $S_0$. Then the set of those $U\in \mathcal{U}$ where $(S_0,U)$ is a fixed point pair is a partially ordered set $\mathcal{L}$ under the operation of union of sets. The set $U$ can be any sub-collection of composite nodes in $\mathcal{U}$ that are sufficient to support all $v_i^k\in S_0$. Clearly, the set $C_1\in \mathcal{L}$ is one of such sets. Therefore, there is a minimal element $C_{\min }$ in $\mathcal{L}$ and a maximal element $C_{\max }$, either one of which need not be unique. Assume that $C_{\min }\ne C_{\max }$. Since $C_{\min }$ already supports every $s\in S_0$, the largest set $C_{\max }\in \mathcal{L}$ provides redundant support for at least one $v_i^k\in S_0$. Such redundant support implies that any perturbation that abolishes this support edge does not disrupt the existence of equilibrium. Therefore, a large redundant support set $C_{\max }\setminus C_{\min }$ contributes to robustness of the equilibrium.

We call the minimal element $C_{\min }$ the core support of the equilibrium. If this set is very small, it increases robustness of the equilibrium, since all perturbations that do not hit the set of composite nodes $C_{\min }$ will not affect the existence of the equilibrium. This robustness is independent of the robustness coming from the size of the set $C_{\max }\setminus C_{\min }$.

Similar comments apply to periodic sets $C_0$. The definition of a core oscillator is analogous: it is the smallest set $C_{min}\in AC$ that supports the set $S_0$ and $(S_0,C_{min})$ is a fixed point pair for $g^k$. Figure 3b shows an example of a core oscillator (grey nodes) consisting of sets

$$\{v_1^0,v_2^0\},\{v_1^0,v_2^1\},\{v_1^2,v_2^1\},\{v_1^2,v_2^0\}.$$

This core oscillator drives the oscillation in nodes $v_3^0,v_3^1$.

The idea of the core oscillator is important in systems biology, where a primary goal is the description of oscillatory behaviour like the cell cycle [63,64], circadian rhythm [65] and other cellular oscillations [66]. As many gene products and proteins oscillate with the cell cycle, it is important to distinguish those that drive the rhythm and those that are being driven by the oscillation. The former comprises the core oscillator of the oscillatory network. The approach described in this paper can be used to find a core oscillator of a gene network that exhibits oscillations.

4. Future extensions

The first step in the analysis of a dynamical system is usually finding the set of attractors, since these describe the range of long-term behaviour of most of the solutions in the system. For nonlinear systems, this is challenging since non-equilibrium attractors are difficult to find and characterize.

Since the boundaries of basins of attraction contain other recurrent dynamics, in order to understand the global organization of the dynamics, i.e. which basins are near each other and how they are organized, one has to understand the lattice of attractors of the system or its dual construction, a Morse decomposition [67–69]. The structure of the Morse decomposition is captured for an STG in a Morse graph. A Morse graph $MG$ for $\mathcal{F}:\mathcal{D}\to \mathcal{D}$ consists of Morse sets, which represent SCCs of $\mathcal{F}$ dynamics and directed edges that capture reachability, i.e. a Morse set $M_j$ is connected to $M_k$ if there is a path in STG from $M_j$ to $M_k.$ Attractors are Morse sets with no out edges.

Since reachability under $\mathcal{F}:\mathcal{D}\to \mathcal{D}$ is equivalent to reachability under $\mathcal{G}:CN\to CN$, the Morse graph $\mathsf{M}\mathsf{G}$ of STG can be computed by first computing Morse nodes labelled $\mathsf{F}\mathsf{P}$, $\mathsf{F}\mathsf{C}$ and $\mathsf{P}\mathsf{C}$ (see §2.4) and then using $\mathcal{G}$ to establish reachability between them. We leave the details of the reachability construction and verification for future work.

5. Discussion

Modelling dynamics of gene-regulatory networks provides some particular challenges. There are no universal laws that specify what form the interaction nonlinearities should take. Since experimental access to individual genes is very difficult in the first place, lack of universally accepted nonlinearities means that parameters that are fitted to one model cannot be used in a different model. Therefore, selecting a new model requires either refitting the parameters or the new function to the data. Since experimental data often have a high level of experimental noise, there is a need for dynamical system analysis that describes dynamics in broad strokes, but is valid for a broad range of nonlinearities, parameters and initial data.

These demands lead to discrete time models that include Boolean, finite multilevel and Petri nets and to continuous time models that allow for finite and coarse descriptions of dynamics (such as switching system models). Even when considering regular ODE models, the emphasis is on employing simplifications like separate time scales and other approximations that allow the description of dynamics on a coarse scale [70].

In this paper, we describe two types of Boolean models of network dynamics. In the classical Boolean model, the activity of network nodes is represented by binary or non-binary [4] Boolean values, and the state of each node is updated by a Boolean function of its inputs.

We also propose a different Boolean model where the interaction network is interpreted as describing which input edges activate which output edges of the same node. In this perspective, the activity level is assigned to edges, rather than to nodes. The state of each out edge of a vertex $v_j$ is updated by a Boolean function of the states of the in-edges of $v_j$. We show in this paper that this type of Boolean model naturally leads to a finite multilevel model where the node assumes one more value than the number of its out edges.

We have shown previously [7,40,41] that these same finite multilevel models arise as update maps of STGs for switching ODE network models. Furthermore, the entire parameter space of a switching system network model is decomposed into a finite number of parameter domains, each of which gives rise to an update function that is a finite multilevel function.

The dynamics of the STG, which is an asynchronous update of the (multilevel) Boolean map, can be characterized by computing its SCCs and paths between them. The Hasse diagram of the reachability structure between the SCC is the Morse graph, which is a succinct summary of the dynamics [33,50]. The Morse nodes with no out edges are the SCC that are attractors of the dynamics.

Computation of attractors and SCCs is still difficult for large networks since the size of the STG grows exponentially with the size of the network. Many algorithms have been proposed and used to find both the fixed point attractors and more complex attractors. One of the most intriguing methods encodes the logical structure of the update function into the expanded network and shows how particular motifs in the expanded network can be used to find both fixed points and complex attractors.

We extend this method and show that a Petri net update on the expanded network recapitulates dynamics on the STG. We define a condition on Petri nets that leads to the detection of SCCs of the STG, and hence the Morse nodes can be recovered from the expanded network. Since the expanded network is, in general, smaller than the STG, this provides tangible benefits for computation of Morse graphs. Finally, we connect motifs of the expanded network to Morse nodes of the STG.

By providing the connections between the key modelling modalities of network dynamics, we hope that the paper will stimulate further discussion and research in this area.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

The source code for this method (used to compute these examples) can be found on Zenodo [71].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

E.A.: formal analysis, investigation, visualization, writing—original draft; E.Q.: methodology, writing—review and editing; T.G.: conceptualization, formal analysis, investigation, supervision, writing—original draft.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interests

We declare we have no competing interests.

Funding

T.G. was partially supported by NSF grant DMS-1951510. E.A. was partially supported by National Science Foundation DMS-1748883, as well as the SMART Scholarship funded by OUSD/R&E (The Under Secretary of Defense-Research and Engineering) and the National Defense Education Program (NDEP)/BA-1, Basic Research.