Characterization of monotone Boolean models supporting fixed points and multistability in balanced networks

Sarah Adigwe; Harshavardhan BV; Mohit Kumar Jolly; Tomáš Gedeon

doi:10.1371/journal.pcsy.0000103

Abstract

Gene regulatory networks (GRN) control the expression levels of proteins in cells, and understanding their dynamics is key to potentially controlling disease processes. Steady states of GRNs are interpreted as cellular phenotypes, and the first step in understanding GRN dynamics is describing the collection of steady states the network can support in different conditions. We consider a collection of all monotone Boolean function models compatible with a given GRN, and ask which steady states are supported by most models. We find that for balanced networks, there is an explicit hierarchy in the prevalence of individual steady states, as well as the prevalence of bistability and multistability. The key insight that we use is that monotone Boolean models supporting a given equilibrium are a product of prime ideals and prime filters of the lattices of monotone Boolean functions. To illustrate our result, we show that in the EMT network associated with cancer metastasis, the most common equilibria correspond to epithelial (E) and mesenchymal (M) states, and the bistability between them is the most common bistability among all network-compatible monotone Boolean models.

Author summary

Cells adjust their behavior in response to external inputs via networks of genes that regulate each other’s expression until they arrive at a new steady state. Each network of genes can behave in different ways that depend on internal and external cellular conditions. In this paper, we consider, for a given network, an entire collection of a particular type of models (monotone Boolean models) that represent all different ways that the network can behave. Then, for any given state a network can potentially be in, we describe all monotone Boolean models that have that state as a steady state. We consider those states that are supported by more models to be more likely to represent the states that the network will be in. We apply our approach to the EMT network, which is important in cancer metastasis. We show that the most common steady states correspond to epithelial and mesenchymal states, and that the bistability between these two states is the most common bistability. This confirms the experimental results that these are the most common states of the EMT network.

Citation: Adigwe S, BV H, Jolly MK, Gedeon T (2026) Characterization of monotone Boolean models supporting fixed points and multistability in balanced networks. PLOS Complex Syst 3(5): e0000103. https://doi.org/10.1371/journal.pcsy.0000103

Editor: Vera Pancaldi, INSERM U1037, FRANCE

Received: October 17, 2025; Accepted: March 21, 2026; Published: May 7, 2026

Copyright: © 2026 Adigwe et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The code used to generate the Figures and Tables are in the following public depositories: https://doi.org/10.5281/zenodo.18682834 https://github.com/Harshavardhan-BV/Monotone-Bool-Net.

Funding: This work was supported by the Prime Minister’s Research Fellowship (PMRF), (to HBV), Param Hansa Philanthropies (to MKJ) and the National Science Foundation (DMS-1951510 to TG). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

1 Introduction

Unicellular organisms and cells in multicellular organisms solve complicated control problems related to the allocation of resources, division, and many others quickly and efficiently. Most of these tasks involve activation, deactivation, or expression of new proteins. The main conceptual model of how the chemical signal results in these changes takes the form of a network of directed interactions, where the presence of one signal results in increased or decreased activity/abundance of a downstream signal.

Many regulatory networks have been constructed [1–8] for organisms and tasks ranging from Escherichia coli to humans. An important question with obvious biological implications is to understand the range of dynamics a given network can support. In particular, a first step is to describe the steady states the network can support, as these are interpreted as different phenotypes that the network can support.

Experimentally, this is an untractable question, since the answer depends on both the external conditions and the internal state of the cell. Using mathematical models to answer this question is difficult, as the answer may depend on the choice of a model, its parameterization, and the choice of an initial condition. Unknown parameters can be addressed using parameter sampling methods [9] or using combinatorial parameterization of all ODE models with steep nonlinearities [10,11], but both of these approaches quickly approach their limits when dealing with large networks. An alternative modeling approach of using a Boolean network model [12–15] has the (apparent) advantage of not needing any parameters, but the choice of update functions is a form of parameterization. Often, these models analyze a single, or only a few, Boolean models compatible with the network, raising the question of whether some potential dynamics have been missed.

We propose a different approach [16] by constructing a finite collection of all monotone Boolean models (MBM) compatible with the network [17]. This includes monotone Boolean models whose influence graph is a strict subgraph of the directed graph of the network.These are constructed as collections of monotone Boolean functions [18] (MBF) that update the state of each node based on its inputs. The monotonicity of the Boolean function reflects the network activation vs. deactivation edges. Monotone Boolean functions can be constructed by induction, but their size grows rapidly [19,20]. In this approach, the network dynamics is a collection of all dynamics supported by all monotone Boolean models, which is weighted in importance by its prevalence. That is, if a particular steady state is supported by 80% of all MBMs, then it is more important than a steady state that is only supported by 10% of all MBMs.

Since the number of MBMs for a given network is finite, the prevalence of all dynamics is, in principle, computable. However, the number of MBM grows very quickly, and even computing steady states for a single MBM is NP-complete [21–23].

In this paper, we concentrate on balanced strongly connected networks, which are networks that have no negative loops (cycles) and there is a directed path between any two nodes. We show that for such networks, the MBMs that support particular equilibria have a structure of products of down- and up-sets in lattices of MBFs. As a result of this characterization, we are able to completely describe the prevalence of each Boolean steady state, bistability, and multistability for any such network. This leads to a general computational procedure that we illustrate on monotone Boolean functions with k = 1,2,3 inputs, MBF(k), and several monotone Boolean models comprised of such MBFs.

We start by showing that there is a change of variables that results in a one-to-one correspondence between monotone Boolean models for any balanced network [24,25] and monotone Boolean models compatible with networks with all activating edges (positive networks). We then analyze positive networks and use the change of variables to obtain results for the original balanced networks. Balanced directed signed graphs (networks) are closely related to balanced undirected signed graphs. Balanced signed graphs are important in many areas of science [26–28]. In fact, in strongly connected signed directed graphs, every undirected cycle is positive if every directed cycle is [29]. Therefore, for balanced directed networks, considered in this paper, the underlying undirected signed graph is also balanced.

After illustrating our approach on a toggle switch network, we consider networks with a 2-team network structure, which were recently described [30,31]. Here, edges connecting the nodes from the same team are activating, but those connecting nodes from opposite teams are repressing. We show that such networks are balanced and that the most prevalent phenotypes are those where one team is active, while the other is not, and the most prevalent bistability is between these two phenotypes.

We then study two EMT networks: one a six-node network [1] and one with 15 nodes and 59 edges [2]. We show that the most prevalent states supported by both networks are the epithelial and mesenchymal states, and that bistability between them is the most prevalent bistability. This work also allows studying the prevalence of other steady states apart from mesenchymal and epithelial states, which are linked with the discussion about so-called intermediate steady states, which have biological significance [32].

The paper is organized as follows: After introducing basic definitions, we discuss balanced networks in section balanced networks. The main result of this section is that all monotone Boolean models of a balanced network are in one-to-one correspondence with monotone Boolean models of a positive network. In section Steady states for positive MBMs, we characterize MBMs that support a particular steady state, and in sections Bistability and Multistability, those that support bistability and multistability, respectively. We illustrate our results throughout the text on several networks, including toggle switch, toggle switch with self-loops, 2-team network, and two EMT network models with 6 and 15 nodes. In the Methods section, we use lattice theory to describe the sets of MBF, which we use to construct MBMs that support steady states. We close the paper with a discussion.

2 Results

Definition 2.1. A regulatory network is a directed graph G = (V,E) with nodes V with |V| = N, directed edges E, and an edge sign function . We denote an edge from node v_i to node v_j without indicating its sign by or . The edge is activating if and repressing if . Graphically, an activating edge of a regulatory network edge is denoted by and a repressing edge by . The sources and targets of a node v_i are given by

respectively. In addition, we assume that the directed graph is strongly connected and so there is a directed path between any two nodes.

Definition 2.2. A loop in network RN is a set of edges with . A loop is positive (negative) if the product of the signs of edges

is 1 (–1).

Definition 2.3. A regulatory network is balanced if it has no negative loops.

Definition 2.4. A Boolean function is increasing with respect to input j if for any input

It is strictly increasing with respect to j if there is at least one input where the inequality is strict.

A Boolean function is decreasing with respect to input j if for any collection of

It is strictly decreasing with respect to j if there is at least one input where the inequality is strict.

Definition 2.5. A Boolean function is monotone if it is increasing or decreasing with respect to each input . In this case, f is called a monotone Boolean function.

A set of increasing monotone Boolean functions will be denoted by MBF(k).

In Fig 1a, we show three increasing monotone Boolean functions MBF(1) with a single input denoted X. These are the zero function 0, the identity function Id, and the one function 1. We represent these three functions as nodes, connected by edges if their truth values differ on a single input string. In Fig 1b we list six increasing monotone Boolean functions MBF(2) with two inputs denoted X and Y. Their names are in the top row of the table; function X copies the values of the input X, while the function Y copies the input Y. Similarly, we represent these functions in the form of a graph where again, nodes are connected by edges if their truth values differ on a single input string.

Download:

Fig 1. (a) There are three monotone Boolean functions with single input MBF(1) and (b) six monotone Boolean functions with 2 inputs MBF(2).

https://doi.org/10.1371/journal.pcsy.0000103.g001

As we will see in section 3.1, the sets MBF(k) for any k form a mathematical structure called lattice. The language and results from lattice theory will help in understanding which Boolean models support which steady states.

We now discuss monotone Boolean models of network dynamics where the update function f_i at the node v_i is a monotone Boolean function.

Definition 2.6. A Boolean model , is monotone if for every i the function f_i is monotone.

A Boolean model is positive if for every i the function f_i is monotone increasing.

Definition 2.7. An influence graph of a monotone Boolean model is a directed graph with nodes for each variable and a directed edge if, and only if, f_i is strictly increasing or strictly decreasing with respect to j.

Since our definition of monotonicity is not strict, for a given regulatory network RN with directed graph G = (V,E) we consider a monotone Boolean model f to be a valid model of network dynamics if its influence network is a subnetwork of the directed graph G. Since the vertex set of I_f and G is the same, this is equivalent to the inclusion of the edges . The inclusion of the functions f with makes MBF(k) a lattice. This, in turn, allows a compact description of sets of functions that support a particular fixed point.

2.1 Balanced networks

Consider a monotone Boolean model , where each . Apply a change of variables where for each i either or . Let be the indices i where .

Then for any

Then

On the other hand, applying the negation operation to the function f_i, we get

We have the following Theorem.

Theorem 2.8. Consider a balanced network RN and a monotone Boolean model compatible with RN. Then there is a change of variables with , and a positive Boolean model such that the following diagram commutes

Proof can be found in section 3.1.

Note that the commutative diagram above implies that any trajectory under f, say

exists if, and only if there is a corresponding trajectory under g

In particular, x is a fixed point under f, if, and only if is a fixed point under g

This shows that every balanced network has the same dynamics as a positive network, where all edges are positive. Since an MBM for a network with all positive edges has all functions f_i that are increasing MBFs, it is sufficient to study steady states and multistability for positive networks.

We illustrate the change of variables in Fig 2.

Download:

Fig 2. Schematic representation of the change of variables to obtain a positive network.

The cyan nodes represent the nodes that have ¬ in . (a): Toggle switch (left: without self-regulation, right: with self-activation), (b): 2-team network with 3 nodes, and (c): EMT network.

https://doi.org/10.1371/journal.pcsy.0000103.g002

In Fig 2a is a toggle switch [33], one of the basic network examples. Using Theorem 2.8, note that the map results in both g_B(A) and g_A(B) to be increasing. Thus transforms f to a positive Boolean model g (Fig 2a). The change of variables is not unique; it is easy to see that also transforms f to a positive Boolean model g. Similarly, a toggle switch with self-activations can be transformed into a positive Boolean model using the same transformation.

In Fig 2b we consider a toy example of a 2-team network, studied in [34]. We define a 2-team network as a network where the set of nodes V can be decomposed into two disjoint sets A and B, , in such a way that the nodes of the same team activate each other while inhibiting the nodes of the opposite team. In other words, if, and only if, , or , and , if, and only if , or .

Here node A belongs to one team, whereas B and C belong to the other team. It is easy to see that transformation , or transformation transforms f into a positive Boolean model g (Fig 2b).

Finally, we consider the biologically important EMT network which governs epithelial-to-mesenchymal transition (EMT) [1,35], where we do not consider the input and output nodes from the original network and exclude the self-inhibition present on SNAI1. Then = (TGF, ¬miR200, SNAI1, ¬OVOL2, ZEB1, ¬miR34a) transforms f into a positive model g (Fig 2c).

2.2 Steady states for positive MBMs

We are now ready to address the main question posed in this paper: How many MBMs compatible with a balanced network support a particular steady state?

Because of Theorem 2.8, it is sufficient to address this question only for positive monotone Boolean models. To this end, we define the following sets

Definition 2.9. Consider the set MBF(k), a set of monotone Boolean functions . For any input let

In other words, U(b) (L(b)) is the set of functions that evaluate to 1 (0). As an illustration, we show these intervals for the set MBF(2) of positive monotone Boolean functions (see Fig 1b) and Table 1.

Download:

Table 1. Sets of functions in MBF(2) that evaluate to 0 (left) and 1 (right). The third column in each part lists the number of functions in the corresponding set.

https://doi.org/10.1371/journal.pcsy.0000103.t001

We are ready for the main result.

Theorem 2.10. Consider a positive network RN and an arbitrary Boolean state of the network RN.

Then e is a steady state of any monotone Boolean model where

where

where e_S(i) are the values of the input nodes of v_i evaluated at the state e.

Proof. Vector is an equilibrium under a monotone Boolean model where if and only if

That is, f_i evaluated on the values of e restricted to sources of node i evaluates to the value e_i. Since is the set of functions that evaluate to 1 on input e_S(i) and is the set of functions that evaluate to 0 on input e_S(i), the result follows.

In the section 3.1 we show that the sets U are ideals (down-sets) and filters (up-sets) in lattices. In addition, we show there the following characterization of the largest set U and the largest set L.

Lemma 2.11.

The largest set is the set where is the vector of ones of length k, with
the largest set is the set where is the vector of zeros of length k, with .

Proof can be found in section 3.1.

The next corollary of Lemma 2.11 characterizes the equilibria that are supported by most monotone Boolean models.

Theorem 2.12. Consider a positive network RN with N nodes. Then

Every state in is an equilibrium for some monotone Boolean model.
For positive network RN with N among all the states, the states and are most prevalent, i.e., supported by most MBMs. The set of all MB models compatible with RN is

where .

The sets of MBMs that supports has the form

which has the size

(1)

The sets of MBMs that supports has the form

which also has the size (1).
Proof can be found in section 3.1.

Remark 2.13. The first conclusion of Theorem 2.12 crucially depends on our choice to consider non-strict definition of monotonicity, since the class of monotone Boolean functions always includes constant functions and any fixed point is supported by at least a set of constant functions with the values matching components of the fixed point.

We are now ready to apply the theory to the examples in Fig 2. In the toggle switch, the change of variables allows us to look at the equivalent problem of equilibria in a positive toggle switch where both edges are positive. Since each node has a single input, the set of MBMs f has the form f = (f₁,f₂) with . Therefore there are 3 × 3 = 9 monotone Boolean models f = (f₁,f₂), see Fig 1a for MBF(1) and Fig 3a for the set of models. It is easy to see that

Download:

Fig 3. Parameter graphs (f₁, f₂) and the steady states supported by (a):

and (b):

.

The states supported are denoted in each square. The red dashed line, blue dash-dotted line, and the teal bold line delineate the set of models that support the steady states (00), (11), and (01), respectively. The filled colors represent the type of exact multistability supported by the corresponding Boolean model f = (f₁, f₂).

https://doi.org/10.1371/journal.pcsy.0000103.g003

Therefore

steady state (00) is supported by which has 4 models;
steady state (10) is supported by which has 1 model;
steady state (01) is supported by which has 1 model;
steady state (11) is supported by which has 4 models.

Applying the change of variable function of the form to the equilibria set in this list, we have the following result for the toggle switch

steady state (10) is supported by 4 models;
steady state (00) is supported by 1 model;
steady state (11) is supported by 1 model;
steady state (01) is supported by 4 models.

For the toggle switch with self-activation, the same change of variables as in the toggle switch changes the model to g = (g₁, g₂) where each has two inputs, see Fig 1b The set of MBMs has 6 × 6 = 36 models, see Fig 3b for the set of models and Table 1 for the list of intervals . The dominant steady states are the same as in the toggle switch network, but they are supported by a larger number of models:

steady state (00) is supported by which has 5 × 5 = 25 models;
steady state (11) is supported by which also has 5 × 5 = 25 models.

After applying , the most prevalent steady states in the toggle switch with self-activation are

steady state (10) supported by 25 models;
steady state (01) supported by 25 models.

For the 2-team network in Fig 2b, any MBM f = (f₁, f₂, f₃) can be changed to g = (g₁, g₂, g₃) with . Again, all states are supported by some models, but the most prevalent steady states are

steady state (000) is supported by which has 5³ = 125 models;
steady state (111) is supported by which also has 5³ = 125 models.

Applying , , gives the most common steady states as

(100) supported by 125 models;
(011) supported by 125 models.

Finally, for the EMT network in Fig 2c the change of variables

transforms f into a positive model g (Fig 2c). The collection of MBMs for the positive EMT (P-EMT) network is

where the numbers correspond to the number of inputs of the corresponding nodes. The collection MBF(3) is shown in Fig 6 and has 20 functions. Therefore, there is a total of

Download:

Fig 4. Schematic representation of the change of variables to obtain a positive network of EMT network from [2].

The cyan nodes represent the nodes that have ¬ in .

https://doi.org/10.1371/journal.pcsy.0000103.g004

Download:

Fig 5. (a) Poset P; (b) Lattice of up-sets of U(P); (c) The set of monotone Boolean functions with two inputs MBF(2) is isomorphic to U(P) via the map

).

(d) Poset P; (e) Lattice of down-sets of J(P); (f) The set of monotone Boolean functions with two inputs MBF^a(2) with anti-order is isomorphic to J(P) via the map .

https://doi.org/10.1371/journal.pcsy.0000103.g005

Download:

Fig 6. Lattice of monotone Boolean functions MBF(3).

Nodes aligned horizontally have the same size of truth set, ranging from 0 at the bottom to 8 = 2³ on the top. Each node has DNF description of the function, where 0 and 1 are constant functions and . Down-sets of meet-irreducible nodes (red) are the sets L(*); while up-sets of join-irreducible elements (dashed) are the sets U(*). Both collections are isomorphic to the poset .

https://doi.org/10.1371/journal.pcsy.0000103.g006

Within this collection of MBMs, the most common steady states are

(111111) which is supported by models in 5 × 5 × 5 × 2 × 19 × 5 = 23750 models;
(000000) which is also supported by 23750 models.

Applying the change of variables , it follows that the most common steady states in the original EMT network are states

each of which is supported by 23750 out of 77760 MBMs compatible with the EMT network. These two states represent the mesenchymal and epithelial state, respectively, with the mesenchymal state characterized by TGF, SNAI1, and ZEB1 high, whereas the epithelial state is characterized by miR200, OVOL2, and miR34a high.

Similarly, Theorem 2.10 can be used to calculate the number of MBMs supporting the other states for the positive network, as given in Table 2. These states correspond to the intermediate states characterized by mixed expression of the canonical epithelial and mesenchymal markers.

Download:

Table 2. Number of models supporting the top 12 most common steady states. P-EMT-state represents the state in the positive EMT network and EMT state represents the state in the original network after transformation. Calculations we done with code included in depository [36].

https://doi.org/10.1371/journal.pcsy.0000103.t002

Consider now a larger EMT network in Fig 4, described in [2], with 15 nodes and 59 edges (excluding the input/output nodes and self-inhibitions). The description of sets becomes computationally challenging due to nodes like ZEB1 and ZEB2 having many inputs (12 and 10, respectively). However, leveraging Theorem 2.12 allows identification of the most common states by transforming the most common states and of the positive network using . It can be seen that this network is balanced, and there exists given by:

Therefore the most common states in the network in Fig 4 are (1) TGF, ZEB1, ZEB2, SNAI1, SNAI2, FOXC2, TWIST1, TWIST2, GSC high and miR141, miR100, miR34a, miR200a, miR200b, miR200c low, which corresponds to the mesenchymal state and (2) the state where the first group of nodes are low and the second group is high, which corresponds to the epithelial state. This mathematical result supports the biological insight that the main role of this model is to modulate the transition between these two states. At the same time, the fact that the most prevalent states in the MB models are indeed the epithelial and mesenchymal states supports the idea that looking for the most prevalent states in MB models of a network may reveal the biological significance of the network.

2.3 Bistability

As we have shown in Theorem 2.12, the most common equilibria in any balanced network have the same abundance. A natural question arises whether the bistability between these two equilibria is also the most common form of bistability. Note that in our work, we will refer to the presence of two or more steady states by the term bistability. We will use the term exact bistability for the presence of exactly two steady states. Specifically, when we say that a model supports bistability between states d and e, we mean that d and e are among the steady states, but other steady states beyond d and e may also exist.

The following theorem extends Theorem 2.10 by characterizing the MBMs that support a bistability between two steady states in the set of positive MBMs.

Theorem 2.14. Consider a positive network RN and two arbitrary Boolean states . Then the bistability between steady states d and e is supported by any positive monotone Boolean model where

where

and

where are the values of the input nodes of v_i evaluated at the state d, (e), respectively.

There are several important consequences of this description. If i-th components of d and e agree, d_i = e_i then

However, if then the intersection may be empty. Since d and e must differ in at least one component, not all pairs of states d,e can be bistable states for an MBM.

For instance, in the positive toggle switch, the bistability between d = (00) and e = (11) is supported by the set of MBMs f = (f₁, f₂) with

These intersections are non-empty and contain a single model

see Fig 3a. This corresponds to a single MBM supporting bistability between (01) and (10) in the toggle switch. However, bistability between (00) and (10) in is supported by models f = (f₁, f₂) with

where . Applying the map , we conclude there are no MBMs for the toggle switch supporting bistability between (10) and (00). In fact, no other pair of states except (01) and (10) can be bistable in a toggle switch, see Fig 3a.

In the case of a toggle switch with self-activation, a larger set of models offers bistability between the states (00) and (11) in the positive network, see Fig 3b. These are the set of MBMs

Hence, bistability between (01) and (10) is supported by 16 models for the toggle switch with self-activation. However, unlike the toggle switch, bistability between the other pairs of steady states is also supported. The number of models that support these bistabilities in the positive toggle switch are

(00) and (01): 6 models;
(00) and (10): 6 models;
(01) and (11): 6 models;
(10) and (11): 6 models;
(01) and (10): 1 model.

We note that while there is only one monotone Boolean model where both functions are strictly increasing with respect to all its inputs (Fig 3a), there are 4 such models

While f supports bistability and one could reasonably argue that only this model represents the true dynamics of the toggle switch, this conclusion is harder to sustain in the larger model. Here, two of the models support bistability and two support tristability; the context provided by the non-strict monotone Boolean models is essential for understanding how these results fit together.

Lemma 2.15. Consider a positive network RN with N vertices. Then the most common bistability supported by MBMs with

is between the steady states and .

It is supported by

MBMs.

Proof can be found in section 3.1.

For 2-team network example, Lemma 2.15 shows that the bistability between (100) and (011) is supported by 4³ = 64 MBMs, which is 64/216 = 0.296; or almost 30% of all models.

For the 6-node EMT network, the bistability between epithelial and mesenchymal states is supported by 4 × 4 × 4 × 1 × 18 × 4 = 4608 MBMs. The resulting prevalence is 4608/77760 = 0.059, or about 6%.

2.4 Multistability

Mirroring our comment about bistability, we will say that an MB model supports q–multistability if it has n ≥ q steady states. We will use the term exact q-multistablity for MBM that supports exactly q-steady states.

In order to determine the collection of MBMs that support exact q–multistability, it is sufficient to know MBMs that support q–multistability for all n ≥ q. More precisely, let m_n(E) be the set of MB models that support n–multistability between states , for some n ≥ q, and let em_q(A) be the set of MBMs that support exact q–multistability between states . Then

The extension of Theorems 2.10 and 2.14 to multistability is straightforward.

Theorem 2.16. Consider a positive network RN and q Boolean states . Then the multistability between these steady states is supported by any positive monotone Boolean model where

where

where are the values of the input nodes of v_i evaluated at the state d^j, respectively.

The conditions under which these products are non-empty are subject to ongoing investigation.

As an example, note that in Fig 3b there is one set of functions (f₁, f₂) that supports multistability between all 4 states (00, 10, 01, 11), 4 models that support tristability between (00, 01, 11) and 4 models that support tristability between (00, 10, 11).

The question of what is the highest multistability supported by the network is related to the number of positive loops in the network, and only partial solutions and estimates are known [37].

3 Methods

3.1 Lattice theory

Definition 3.1 ([38]). A partially ordered set P (or poset) is a set P together with a binary relation denoted ≤, satisfying the following three axioms:

1. For all , (reflexivity).
2. If and , then s = t (antisymmetry).
3. If and , then (transitivity).

We say that two elements s and t of P are comparable if or ; otherwise s and t are incomparable, denoted .

Definition 3.2. A subset D of a poset P is a down-set or order ideal if

We denote the down-sets in P by J(P). Dually, a subset U of P is an up-set or order filter if

We denote the up-sets of P by U(P). Since we will only deal with order ideals and order filters, we will shorten these terms to ideal and filter, respectively.

Definition 3.3. Let P be a poset with two binary operations ∨, called join, and ∧, called meet. P is a lattice if every pair of elements of P has both a meet and a join.

In this paper, we will only consider finite posets and lattices. In a finite lattice, all ideals (filters) are principal [39] and thus have the form

Definition 3.4. Let be a lattice.

1. A nonzero element is join-irreducible if j is not the join of two smaller elements, that is, if

2. Dually, a nonunit element is meet-irreducible if m is not the meet of two larger elements, that is, if

A subset is a proper subset of B.

Definition 3.5. ([39]). Let be a lattice.

A proper ideal J is prime if

Dually, a proper filter F is prime if

Theorem 3.6 (Theorem 3.37, [39]). The properties of being a prime ideal and being a prime filter are complementary, that is, I is a prime ideal if and only if I^c is a prime filter.

Definition 3.7 ([39]). If a lattice has a smallest element (which we call 0), and is another lattice then the ideal kernel of a lattice homomorphism is the set

This is easily seen to be an ideal of .

Given a lattice , choosing to be the Boolean lattice , we arrive at the concept of the indicator function of a subset . The indicator function of S is the function defined by setting f_S(x) = 0 if and only if . In general, an indicator function need not be a lattice homomorphism. However, the indicator function of a prime ideal P is a lattice epimorphism.

Definition 3.8 ([38]). Lattice is a distributive lattice if

holds for any elements

The fundamental Theorem for finite distributive lattices is due to Birkhoff [40].

Theorem 3.9 (Birkhoff Theorem). Any finite distributive lattice is isomorphic to the lattice of down-sets of its join-irreducible elements.

This representation is crucial for understanding the relationship between join-irreducibles and prime ideals.

Lemma 3.10. Down set of join-irreducible element in finite distributive lattices are prime ideals.

Proof. Consider an ideal . If , then by join-irreducibility of v, either or , and therefore that element is also in . This directly satisfies the definition of a prime ideal. □

Theorem 3.11 (Theorem 3.41, [39]). In a lattice , the prime ideals in are precisely the kernels of the indicator epimorphisms.

By duality Theorem 3.6, the prime filters are precisely sets U which map to 1 under an indicator epimorphism.

3.2 Lattice of increasing monotone Boolean functions

Consider the poset of Boolean vectors under the order induced by 0 < 1 on each component.

The set of increasing monotone Boolean functions MBF(k) is a set of order preserving maps

Definition 3.12 (Truth set and zero set). The truth set T(f) of a Boolean function is

while the zero set is

Note that is a poset with order given by

Lemma 3.13. The poset is a distributive lattice with operations given by union and intersection of the associated truth sets

Let taking

The map G is a lattice isomorphism between and where we made the orders explicit.

We also define an anti-order on MBF(k) by using kernels of the maps f

It is straightforward to see that

Consider a map taking and so

Here J(P) is the set of order ideals in P. It is easy to see that the map F is an isomorphism and thus .

We have an immediate result.

Lemma 3.14. is a lattice with operations

where the operations are operations in lattice

We will use notation MBF(k) for and notation MBF^a for lattice .

Let S=2^P be a set of all subsets of finite poset P. Let

(2)

which assigns to its complement Q^c in S. Then the map applied to poset of all possible inputs to maps T(f) to and to T(f).

Corollary 3.15. The map induces an isomorphism between lattices and which maps meet-irreducible elements to join-irreducible elements and join-irreducible elements to meet-irreducible elements. The isomorphism takes each to itself but exchanges the lattice operations.

Proof. Since for any two functions the union is the complement of and intersection is the complement of by de Morgan laws, the result follows from Lemma 3.14.

Recall that in Definition 2.9 we set

Theorem 3.16. We have the following characterizations of the sets :

For each x the set L(x) is a prime ideal in MBF(k).
There is a unique meet-irreducible element such that L(x) is a down-set of g in MBF(k), i.e., .
For each x the set U(x) is a prime filter in MBF(k)
There is a unique join-irreducible element such that U(x) is an up-set of g in MBF(k), i.e., .

Proof. For illustration of the arguments below, please see Fig 5 for MBF(2). The set MBF(3) is shown in Fig 6.

We start by describing all epimorphisms . First, for every let

be a bounded lattice homomorphism which maps all up-sets of P containing x to 1 and all other up-sets to 0. Note that this set of epimorphisms is in one-to-one correspondence with .

We now show that every lattice epimorphism has the form u_x for some . Consider any lattice epimorphism . Then the elements such that f(y)=0 that are mapped to 0 must have a unique maximal element , which is the join of the elements y that are mapped to 0. This follows from the homomorphism property requiring that if f(u)=0 and f(v) = 0, then also . Importantly, must be meet-irreducible, since it cannot be the meet of any elements u,v with f(u)=f(v) = 1.

Since and , the isomorphism from Corollary 3.15 between lattices

induces isomorphism between U(P) and J(P), By Lemma 3.14 the meet irreducible elements in U(P) are mapped to join-irreducible elements in J(P). Since the isomorphism between and is induced by , it follows from (2) that is a join irreducible element of J(P).

We illustrate this map on the example MBF(2) in Fig 5. The set U(P) is depicted in panel Fig 5b and the red nodes correspond to meet-irreducible elements of U(P). Note that , , and and the images are join-irreducible nodes of the lattice J(P) in Fig 5e.

By Lemma 3.10 down-sets of join-irreducible elements in J(P) are prime ideals. Therefore for each there exists such that

(3)

To summarize our construction so far, for each lattice epimorphism we associate a meet-irreducible element , to which we assign a join-irreducible element , which has a form of (3) form some . We claim that . First, we note that the kernel of the epimorphism f are all the up-sets that are subsets of the largest up-set . By construction, the set is the complement of the up-set in S=2^P and so it is disjoint from any up-sets in U(P) that are below . Furthermore, this complement is a prime ideal generated by . Therefore, is the largest element in P which does not belong to the upper set . As a consequence, if contains , then f(y)=1, but if , then f(y) =0. This shows that .

Therefore, every lattice epimorphism f has the form f = u_x for some x.

Consider for some . Then the set of up-sets in P that do not contain x is a down-set in MBF(k) that consists of all functions whose truth set does not contain x, and therefore f(x)=0. By definition of the sets L(x), these are precisely the functions that belong to the set L(x). This down-set in MBF(k) is mapped by u_x to 0 and therefore is a kernel of a homomorphism u_x. By Theorem 3.11 L(x) is the kernel of indicator epimorphism and therefore prime ideal in MBF(k). Furthermore there is unique maximal element which is meet -irreducible and such that

This proves statements 1 and 2 of the Theorem.

We illustrate this in Fig 5c. Down-sets of meet-irreducible elements (in red) are At the same time , , and .

By Theorem 3.6 the complements of prime ideals are prime filters. This shows U(x) are upsets of join-irreducible elements which proves 3 and 4.

To illustrate, note that in Fig 5f the down-sets of meet-irreducible elements in MBF^a(2) (in red), which are upsets of join-irreducibles in MBF(2), are

3.3 Structure of MBF models that support fixed points

In this subsection, we use the lattice theory language to describe the sets .

Proof of Lemma 2.11.

Proof. By Theorem 3.16, each set U(x) is a prime filter in MBF(k). Since the logical function in k arguments X_i given by

has unique predecessor , m is joint-irreducible element in MBF(k). Consequently, is a prime filter and it contains all functions in MBF(k), except the function 0. Therefore, this is the maximal filter with size . Finally, since 0 is the only function that evaluates to 0 on input , it follows that .

Analogously, by Theorem 3.16 each set L(x) is a prime ideal in MBF(k). The logical function in k arguments X_i given by

has unique successor , M is meet-irreducible element in MBF(k) and is a prime ideal that contains all functions in MBF(k), except the function 1. Therefore, this is the maximal ideal with size . Finally, since 1 is the only function that evaluates to 1 on input , it follows that .

Proof of Theorem 2.12

Proof. Any state is supported by the monotone Boolean model where f_i is a constant function f_i = b_i. Therefore, the set of models supporting b is non-empty.

The rest of the Theorem follows by realizing that the equilibrium supported by most MB models is one where for each component i, the monotone Boolean function f_i will be from the largest set , or . These sets were characterized in Lemma 2.11.

Proof of Lemma 2.15

Proof. By Theorem 2.14 the models that support bistability between and satisfy

Since and it follows that

The size of this set is

By the complementarity property in Theorem 3.6 that for any k and any the sets U(b) and L(b) satisfy

Note, however, that if and then in general, see (Table 3).

Download:

Table 3. Prime ideals (left) and prime filters (right) in MBF(3). Both sets are posets isomorphic to

by Birkhoff Theorem.

https://doi.org/10.1371/journal.pcsy.0000103.t003

We conclude with a result that is a direct consequence of Corollary 3.15. For illustration, see Table 3 for the table of all sets U(b), L(b) in MBF(3), which is depicted in Fig 6.

Lemma 3.17. For any the sets satisfy

where ¬ is negation.

There are additional symmetries on the lattice MBF(k), induced by permutations on the set of coordinates for vectors . Each such permutation p induces isomorphism . Since each function is uniquely determined by its truth set , and the lattice operations are defined in terms of union and intersections of truth sets, the isomorphism induces an automorphism

(4)

where .

Consider collection of prime ideals and collection of prime filters . Each of these collections has 2^k elements, but the existence of this automorphism implies that these sets only have k + 1 different sizes.

Theorem 3.18. Consider collection of prime ideals and collection of prime filters . Then there is a sequence of k + 1 integers

with , such that

Proof. The fact that follows from Lemma 2.11.

Consider any permutation on the set of coordinates for vectors and the induced automorphism in (4).

It is easy to check that preserves the property of join- and meet-irreducibility and therefore induces an automorphism on the set . Since the map preserves the number of 1s in the string, it follows that the size . Therefore, the size of |U(b)| only depends on the number of entries 1 in the string b.

Finally, since MBF(k) is a collection of increasing monotone Boolean functions, then in implies and thus |U(b)| < |U(c)|. This finishes the proof for the set . The result for the set follows now from Lemma 3.17.

To illustrate the Theorem recall from Table 1 that for MBF(2) the sizes of the sets U(b), L(b) are

from Table 3 the sizes are

Proof of Theorem 2.8

Proof. We order nodes . If there is no j such that f_j is decreasing in x₁, then all edges with source x₁ in the influence network are positive. We set . Otherwise, if there is an f_j which is decreasing in x₁ we set

and for any f_k that depends on x₁ we have

(5)

However, since the update function for the node x₁ is f₁(x) if we change variable x₁ to we also need to change the sign of function f₁. In other words,

Setting it follows that

(6)

Consider now the influence network of a new model with state . By (5) all influence edges of the original network f that start in x₁ have changed signs in the influence network of ; and by (6) all influence edges that terminate in x₁ also changed signs.

Importantly, in any loop that passes through node x₁, there were two changes. Hence, the sign of the loop did not change, and we eliminated the negative edge from x₁ to x_j from the influence network. We conclude that the sign of every loop in the influence network of f is the same as in the influence network of .

We now consider the sources of node x₁ and ask if g₁ is decreasing in any of its inputs. If so, we list these inputs when i < s and apply transformation in this order. This eliminates all negative edges until g₁ is increasing in all its inputs. Note that if g₁ is increasing in x_j no change is applied to x_j. We then move to sources of node and eliminate all negative inputs to function ; then do the same until all elements of I₁ are exhausted. If at any time all inputs are positive, we take the next node from the initial ordered list that has not been considered in the previous steps.

Since there are no negative loops in the network RN when a previously considered node appears in the list of sources, it only can appear as a positive input, since the corresponding loop that has just formed will have all positive edges. As a result, no change will be required for such a node and every node in the list will be only handled once. Since there is a finite number of nodes in the network and we will only need to consider each of them for the change only once, this process will terminate. Since every loop in the network RN is positive, the change of variables will yield g with positive influence network.

4 Discussion

This paper presents a first step towards a mathematical understanding of the relationship between network structure and the dynamics it supports. The answer clearly depends on a class of models that is selected to represent the network dynamics; answering this question in the class of all differential equation models compatible with the network runs immediately into problems with basic concepts: What does it mean to have the same dynamics when both the class of models and the state space of every single model are uncountable?

We investigate this question within the class of monotone Boolean models where the set of models is finite (but large) and the state space is a finite set of Boolean vectors . In spite of this finiteness, Boolean models are very popular in systems biology as they combine the finiteness with sufficient expressiveness to capture qualitative dynamic behavior of regulatory networks.

We make two additional simplifications. First, we seek to describe the relationship between network structure and steady states, rather than more general dynamic behavior. Second, because we focus on steady states, we only consider balanced networks that do not contain negative loops. It is known [41] that negative loops are associated with the existence of attractors that consist of more than one state (for instance, complex attractors), and, in fact, the existence of such attractors requires negative loops in a network.

In networks with negative loops, some of the MBMs will not support any equilibria, only complex attractors, automatically decreasing the number of MBMs supporting a given equilibrium. This is consistent with results from the toggle triad, where the prevalence/stability of single high states is lower than the corresponding prevalence in two-team networks (balanced) networks [42]. This assumption, which excludes networks with negative loops, has been made for convenience, in order to describe the most common steady states and bistability. However, the description of models supporting steady states and multistability by up-sets and down-sets is applicable to any network structure. The work on understanding the connection between negative loops and complex attractors is the subject of our current work, and we believe that the approach outlined in this paper can be extended to general networks that contain negative loops.

Our results are consistent with work on the prevalence of steady states for the EMT network. Paper [43] found that positive loops are enhanced in the EMT network, which is consistent with the high prevalence of bistability in ODE simulations of EMT models. Two EMT models that are studied in this paper are balanced, and therefore, the E and M states are the most prevalent steady states across all MBM models for these networks. Further study of the robustness of this bistability was examined in [44]. This paper studied 13 different EMT networks and showed that while these wild-type networks show the highest prevalence of E and M states in sampled ODE simulations, small perturbations of the network structure (i.e., change of edge sign) produced networks where the prevalence of E and M states decreased, and the prevalence of intermediate states increased. While our results do not directly address robustness under network perturbations, our results imply that among all networks with the same number of nodes and edges, the network that has all positive edges will have the highest prevalence of the constant and states. The removal of self-inhibition on SNAI1 simplifies our analysis of the EMT networks studied in this paper to a balanced network. Thus, E and M states are the most prevalent; any perturbation of this network that changes the signs of the edges will result in a network with decreased prevalence of E and M states and increased prevalence of the intermediate states. Unlike self-activation on ZEB, which has been experimentally demonstrated to maintain the M phenotype [45], the role of self-inhibition on SNAI1 has not been explored experimentally and is considered a noise buffer in ODE models [46]. While this exclusion may limit the biological accuracy of our model, it should have minimal impact on predictions of the Boolean models without noise considered here.

The realization that the sets of monotone Boolean functions that evaluate to either 0 or 1 as prime ideals or filters in the lattice of monotone Boolean functions with k inputs, allows us to prove general results that are valid for all k. This is significant since the size of MBF(k) grows rapidly with k and it is only known for k ≤ 9, but a construction of meet- and join-irreducible elements in MBF(k) may be possible by inductive procedure from those elements in MBF(k − 1) in a way similar to the construction of MBF(k) from MBF(k − 1) [17]. This in turn would enable explicit construction of monotone Boolean models with certain multistability properties, or, alternatively, prove that a particular network does not support multistability greater than some upper bound q.

We note that we allow monotone Boolean functions to have redundant inputs that do not affect the output of the function. Apart from the fact that inclusion of such functions allows monotone Boolean functions to form a lattice, such functions represent the parameter regimes where a gene can be constitutively activated or suppressed in certain tissue, or under some external condition.

Our approach can be compared to [47], which searches for all (locally) monotone Boolean models that satisfy a set of constraints imposed by partially observed data. These constraints include fixed point behavior, but also other attractors, as well as reachability and non-reachability between pairs of states. In both of our approaches, a model is considered valid if its influence graph is a subset of the given network graph. Chevalier et al. [47] encode the set of constraints into Answer Set Programming, a declarative programming used in difficult, often NP-hard, search problems, and then search for all models that satisfy them. Our approach considers only constraints given by the desired fixed points, but uses the explicit structure of the set of monotone Boolean models to compute the models with the desired behavior explicitly. In the future, we plan to extend our work from balanced to arbitrary networks, hoping to include more general attractors that arise in models of networks as a constraint.

Monotone Boolean function models are closely linked with ODE models with steep nonlinearities, which in the limit become Glass models with piecewise constant nonlinearities. Each such ODE model generates a finite state transition graph (STG) that encodes its dynamics, and there is only a finite collection of STGs that are compatible with the given network. This collection is organized in a DSGRN database, which takes the form of a graph [10,11]. Since MB network models are a subset of the DSGRN collection, the work presented here should describe multistability in ODE network models with steep nonlinearities. The precise nature of this connection is a subject of current investigation.

Reaction network theory takes a different approach to modeling network dynamics using mass action kinetics. The delineation of conditions that guarantee (or preclude) multistability has been a subject of many papers, see for instance [48–50].

The underlying philosophy of our approach, which aims to describe the full suite of network dynamics, is that important dynamics will occur with higher frequency within this set. The importance of particular network dynamics is weighted by the prevalence of MBMs that support it. That is, we view a steady state that is supported by 80% of all MBMs as more important than a steady state that is only supported by 10% of all MBMs. It is certainly possible that biological control external to the network can keep the network in a (set of) MBMs which supports the dynamics with 10% prevalence, but in the absence of such information, we will assume that the more prevalent dynamics is more important.

Acknowledgments

Part of this work has been done while the last author visited Department of Bioengineering, Indian Institute of Science, Bengaluru, India. Its hospitality is gratefully acknowledged.

References

1. Hong T, Watanabe K, Ta CH, Villarreal-Ponce A, Nie Q, Dai X. An Ovol2-Zeb1 Mutual Inhibitory Circuit Governs Bidirectional and Multi-step Transition between Epithelial and Mesenchymal States. PLOS Computational Biology. 2015;11(11):e1004569.
- View Article
- Google Scholar
2. Huang B, Lu M, Jia D, Ben-Jacob E, Levine H, Onuchic JN. Interrogating the topological robustness of gene regulatory circuits by randomization. PLoS Comput Biol. 2017;13(3):e1005456. pmid:28362798
- View Article
- PubMed/NCBI
- Google Scholar
3. Ríos O, Frias S, Rodríguez A, Kofman S, Merchant H, Torres L, et al. A Boolean network model of human gonadal sex determination. Theor Biol Med Model. 2015;12:26. pmid:26573569
- View Article
- PubMed/NCBI
- Google Scholar
4. Chang R, Shoemaker R, Wang W. Systematic search for recipes to generate induced pluripotent stem cells. PLoS Comput Biol. 2011;7(12):e1002300. pmid:22215993
- View Article
- PubMed/NCBI
- Google Scholar
5. Udyavar AR, Wooten DJ, Hoeksema M, Bansal M, Califano A, Estrada L, et al. Novel Hybrid Phenotype Revealed in Small Cell Lung Cancer by a Transcription Factor Network Model That Can Explain Tumor Heterogeneity. Cancer Res. 2017;77(5):1063–74. pmid:27932399
- View Article
- PubMed/NCBI
- Google Scholar
6. Martínez-Antonio A, Janga SC, Thieffry D. Functional organisation of Escherichia coli transcriptional regulatory network. J Mol Biol. 2008;381(1):238–47. pmid:18599074
- View Article
- PubMed/NCBI
- Google Scholar
7. Zhang R, Shah MV, Yang J, Nyland SB, Liu X, Yun JK, et al. Network model of survival signaling in large granular lymphocyte leukemia. Proc Natl Acad Sci U S A. 2008;105(42):16308–13. pmid:18852469
- View Article
- PubMed/NCBI
- Google Scholar
8. Abou-Jaoudé W, Traynard P, Monteiro PT, Saez-Rodriguez J, Helikar T, Thieffry D. Logical Modeling and Dynamical Analysis of Cellular Networks. Frontiers in Genetics. 2016;7.
- View Article
- Google Scholar
9. Huang B, Jia D, Feng J, Levine H, Onuchic JN, Lu M. RACIPE: a computational tool for modeling gene regulatory circuits using randomization. BMC Syst Biol. 2018;12(1):74. pmid:29914482
- View Article
- PubMed/NCBI
- Google Scholar
10. Gedeon T. Multi-parameter exploration of dynamics of regulatory networks. Biosystems. 2020;190:104113. pmid:32057819
- View Article
- PubMed/NCBI
- Google Scholar
11. Cummins B, Gedeon T, Harker S, Mischaikow K, Mok K. Combinatorial representation of parameter space for switching networks. SIAM J Appl Dyn Syst. 2016;15(4):2176–212. pmid:30774565
- View Article
- PubMed/NCBI
- Google Scholar
12. Font-Clos F, Zapperi S, La Porta CAM. Topography of epithelial-mesenchymal plasticity. Proc Natl Acad Sci U S A. 2018;115(23):5902–7. pmid:29784817
- View Article
- PubMed/NCBI
- Google Scholar
13. Chaouiya C, Naldi A, Thieffry D. Logical modelling of gene regulatory networks with GINsim. Methods Mol Biol. 2012;804:463–79. pmid:22144167
- View Article
- PubMed/NCBI
- Google Scholar
14. Stoll G, Caron B, Viara E, Dugourd A, Zinovyev A, Naldi A, et al. MaBoSS 2.0: an environment for stochastic Boolean modeling. Bioinformatics. 2017;33(14):2226–8.
- View Article
- Google Scholar
15. Albert R, Thakar J. Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions. Wiley Interdiscip Rev Syst Biol Med. 2014;6(5):353–69. pmid:25269159
- View Article
- PubMed/NCBI
- Google Scholar
16. Gedeon T. Network topology and interaction logic determine states it supports. NPJ Syst Biol Appl. 2024;10(1):98. pmid:39198512
- View Article
- PubMed/NCBI
- Google Scholar
17. Gedeon T. Lattice structures that parameterize regulatory network dynamics. Math Biosci. 2024;374:109225. pmid:38866065
- View Article
- PubMed/NCBI
- Google Scholar
18. Korshunov AD. Monotone Boolean functions. Russ Math Surv. 2003;58(5):929–1001.
- View Article
- Google Scholar
19. Fidytek R, Mostowski AW, Somla R, Szepietowski A. Algorithms counting monotone Boolean functions. Information Processing Letters. 2001;79(5):203–9.
- View Article
- Google Scholar
20. Jäkel C. A computation of the ninth Dedekind number. Journal of Computational Algebra. 2023;6–7:100006.
- View Article
- Google Scholar
21. Mori T, Akutsu T. Attractor detection and enumeration algorithms for Boolean networks. Comput Struct Biotechnol J. 2022;20:2512–20. pmid:35685366
- View Article
- PubMed/NCBI
- Google Scholar
22. TAMURA T, AKUTSU T. Detecting a Singleton Attractor in a Boolean Network Utilizing SAT Algorithms. IEICE Trans Fundamentals. 2009;E92-A(2):493–501.
- View Article
- Google Scholar
23. Dubrova E, Teslenko M, Ming L. Finding Attractors in Synchronous Multiple-Valued Networks Using SAT-Based Bounded Model Checking. In: 2010 40th IEEE International Symposium on Multiple-Valued Logic, 2010. 144–9. https://doi.org/10.1109/ismvl.2010.35
24. Beineke LW, Harary F. Consistency in marked digraphs. Journal of Mathematical Psychology. 1978 Dec;18(3):260-9. Available from: https://linkinghub.elsevier.com/retrieve/pii/0022249678900548
- View Article
- Google Scholar
25. Harary F, Palmer EM. Graphical Enumeration. Elsevier; 2014. Google-Books-ID: ZrvSBQAAQBAJ.
26. Harary F. On the notion of balance of a signed graph. Michigan Math J. 1953;2(2).
- View Article
- Google Scholar
27. Shi G, Altafini C, Baras JS. Dynamics over Signed Networks. SIAM Rev. 2019;61(2):229–57.
- View Article
- Google Scholar
28. Diaz-Diaz F, Candellone E, Gonzalez-Casado MA, Fraxanet E, Vendeville A, Ferri I, et al. Signed Networks: theory, methods, and applications. arXiv; 2025. ArXiv:2511.17247 [physics]. Available from: http://arxiv.org/abs/2511.17247
29. Harary F, Norman RZ, Cartwright D. Structural Models: An Introduction to the Theory of Directed Graphs. Wiley; 1965. Available from: https://books.google.com/books?id=DbY9AAAAIAAJ
30. Hari K, Harlapur P, Saxena A, Haldar K, Girish A, Malpani T, et al. Low dimensionality of phenotypic space as an emergent property of coordinated teams in biological regulatory networks. iScience. 2025;28(2):111730. pmid:39898023
- View Article
- PubMed/NCBI
- Google Scholar
31. Hari K, Ullanat V, Balasubramanian A, Gopalan A, Jolly MK. Landscape of epithelial-mesenchymal plasticity as an emergent property of coordinated teams in regulatory networks. Elife. 2022;11:e76535. pmid:36269057
- View Article
- PubMed/NCBI
- Google Scholar
32. Pastushenko I, Brisebarre A, Sifrim A, Fioramonti M, Revenco T, Boumahdi S, et al. Identification of the tumour transition states occurring during EMT. Nature. 2018;556(7702):463–8. pmid:29670281
- View Article
- PubMed/NCBI
- Google Scholar
33. Gardner TS, Cantor CR, Collins JJ. Construction of a genetic toggle switch in Escherichia coli. Nature. 2000;403(6767):339–42. pmid:10659857
- View Article
- PubMed/NCBI
- Google Scholar
34. Hebbar A, Moger A, Hari K, Jolly MK. Robustness in phenotypic plasticity and heterogeneity patterns enabled by EMT networks. Biophys J. 2022;121(19):3600–15. pmid:35859419
- View Article
- PubMed/NCBI
- Google Scholar
35. Xin Y, Cummins B, Gedeon T. Multistability in the epithelial-mesenchymal transition network. BMC Bioinformatics. 2020;21(1):71. pmid:32093616
- View Article
- PubMed/NCBI
- Google Scholar
36. Adigwe S, BV H, Gedeon T, Jolly MK. Characterization of monotone Boolean models supporting fixed points and multistability in balanced networks. Zenodo; 2026. Available from:
- View Article
- Google Scholar
37. Aracena J, Richard A, Salinas L. Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks. SIAM J Discrete Math. 2017;31(3):1702–25.
- View Article
- Google Scholar
38. Stanley RP. Enumerative Combinatorics. vol. 1 of Cambridge Studies in Advanced Mathematics. 2nd ed. Cambridge: Cambridge University Press; 2011. Available from: https://www.cambridge.org/core/books/enumerative-combinatorics/3155CDE1D973D49F873BDE2EAF8D7651. https://doi.org/10.1017/CBO9781139058520
39. Lattices RS. Lattices and Ordered Sets. New York, NY: Springer. 2008. https://doi.org/10.1007/978-0-387-78901-9
40. Birkhoff G. On the combination of subalgebras. Math Proc Camb Phil Soc. 1933;29(4):441–64.
- View Article
- Google Scholar
41. Snoussi EH. Necessary Conditions for Multistationarity and Stable Periodicity. J Biol Syst. 1998;06(01):3–9.
- View Article
- Google Scholar
42. Duddu AS, Sahoo S, Hati S, Jhunjhunwala S, Jolly MK. Multi-stability in cellular differentiation enabled by a network of three mutually repressing master regulators. J R Soc Interface. 2020;17(170):20200631. pmid:32993428
- View Article
- PubMed/NCBI
- Google Scholar
43. Nordick B, Hong T. Identification, visualization, statistical analysis and mathematical modeling of high-feedback loops in gene regulatory networks. BMC Bioinformatics. 2021;22(1):481. pmid:34607562
- View Article
- PubMed/NCBI
- Google Scholar
44. Rashid M, Hari K, Thampi J, Santhosh NK, Jolly MK. Network topology metrics explaining enrichment of hybrid epithelial/mesenchymal phenotypes in metastasis. PLoS Comput Biol. 2022;18(11):e1010687. pmid:36346808
- View Article
- PubMed/NCBI
- Google Scholar
45. Preca BT, Bajdak K, Mock K, Sundararajan V, Pfannstiel J, Maurer J, et al. A self-enforcing CD44s/ZEB1 feedback loop maintains EMT and stemness properties in cancer cells. International Journal of Cancer. 2015 Dec;137(11):2566-77. Publisher: John Wiley & Sons, Ltd. Available from: https://onlinelibrary.wiley.com/doi/full/10.1002/ijc.29642
- View Article
- Google Scholar
46. Lu M, Jolly MK, Levine H, Onuchic JN, Ben-Jacob E. MicroRNA-based regulation of epithelial-hybrid-mesenchymal fate determination. Proc Natl Acad Sci U S A. 2013;110(45):18144–9. pmid:24154725
- View Article
- PubMed/NCBI
- Google Scholar
47. Chevalier S, Froidevaux C, Paulevé L, Zinovyev A. Synthesis of Boolean Networks from Biological Dynamical Constraints using Answer-Set Programming. In: 2019 IEEE 31st International Conference on Tools with Artificial Intelligence (ICTAI), 2019. 34–41. http://doi.org/10.1109/ictai.2019.00014
48. Dickenstein A, Millán MP, Shiu A, Tang X. Multistationarity in Structured Reaction Networks. Bull Math Biol. 2019;81(5):1527–81. pmid:30788691
- View Article
- PubMed/NCBI
- Google Scholar
49. Sadeghimanesh A, England M. Polynomial superlevel set representation of the multistationarity region of chemical reaction networks. BMC Bioinformatics. 2022;23(1):391. pmid:36167486
- View Article
- PubMed/NCBI
- Google Scholar
50. Conradi C, Feliu E, Mincheva M, Wiuf C. Identifying parameter regions for multistationarity. PLoS Comput Biol. 2017;13(10):e1005751. pmid:28972969
- View Article
- PubMed/NCBI
- Google Scholar

[ref1] 1. Hong T, Watanabe K, Ta CH, Villarreal-Ponce A, Nie Q, Dai X. An Ovol2-Zeb1 Mutual Inhibitory Circuit Governs Bidirectional and Multi-step Transition between Epithelial and Mesenchymal States. PLOS Computational Biology. 2015;11(11):e1004569.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Huang B, Lu M, Jia D, Ben-Jacob E, Levine H, Onuchic JN. Interrogating the topological robustness of gene regulatory circuits by randomization. PLoS Comput Biol. 2017;13(3):e1005456. pmid:28362798
View Article
PubMed/NCBI
Google Scholar

[5] View Article

[6] PubMed/NCBI

[7] Google Scholar

[ref3] 3. Ríos O, Frias S, Rodríguez A, Kofman S, Merchant H, Torres L, et al. A Boolean network model of human gonadal sex determination. Theor Biol Med Model. 2015;12:26. pmid:26573569
View Article
PubMed/NCBI
Google Scholar

[9] View Article

[10] PubMed/NCBI

[11] Google Scholar

[ref4] 4. Chang R, Shoemaker R, Wang W. Systematic search for recipes to generate induced pluripotent stem cells. PLoS Comput Biol. 2011;7(12):e1002300. pmid:22215993
View Article
PubMed/NCBI
Google Scholar

[13] View Article

[14] PubMed/NCBI

[15] Google Scholar

[ref5] 5. Udyavar AR, Wooten DJ, Hoeksema M, Bansal M, Califano A, Estrada L, et al. Novel Hybrid Phenotype Revealed in Small Cell Lung Cancer by a Transcription Factor Network Model That Can Explain Tumor Heterogeneity. Cancer Res. 2017;77(5):1063–74. pmid:27932399
View Article
PubMed/NCBI
Google Scholar

[17] View Article

[18] PubMed/NCBI

[19] Google Scholar

[ref6] 6. Martínez-Antonio A, Janga SC, Thieffry D. Functional organisation of Escherichia coli transcriptional regulatory network. J Mol Biol. 2008;381(1):238–47. pmid:18599074
View Article
PubMed/NCBI
Google Scholar

[21] View Article

[22] PubMed/NCBI

[23] Google Scholar

[ref7] 7. Zhang R, Shah MV, Yang J, Nyland SB, Liu X, Yun JK, et al. Network model of survival signaling in large granular lymphocyte leukemia. Proc Natl Acad Sci U S A. 2008;105(42):16308–13. pmid:18852469
View Article
PubMed/NCBI
Google Scholar

[25] View Article

[26] PubMed/NCBI

[27] Google Scholar

[ref8] 8. Abou-Jaoudé W, Traynard P, Monteiro PT, Saez-Rodriguez J, Helikar T, Thieffry D. Logical Modeling and Dynamical Analysis of Cellular Networks. Frontiers in Genetics. 2016;7.
View Article
Google Scholar

[29] View Article

[30] Google Scholar

[ref9] 9. Huang B, Jia D, Feng J, Levine H, Onuchic JN, Lu M. RACIPE: a computational tool for modeling gene regulatory circuits using randomization. BMC Syst Biol. 2018;12(1):74. pmid:29914482
View Article
PubMed/NCBI
Google Scholar

[32] View Article

[33] PubMed/NCBI

[34] Google Scholar

[ref10] 10. Gedeon T. Multi-parameter exploration of dynamics of regulatory networks. Biosystems. 2020;190:104113. pmid:32057819
View Article
PubMed/NCBI
Google Scholar

[36] View Article

[37] PubMed/NCBI

[38] Google Scholar

[ref11] 11. Cummins B, Gedeon T, Harker S, Mischaikow K, Mok K. Combinatorial representation of parameter space for switching networks. SIAM J Appl Dyn Syst. 2016;15(4):2176–212. pmid:30774565
View Article
PubMed/NCBI
Google Scholar

[40] View Article

[41] PubMed/NCBI

[42] Google Scholar

[ref12] 12. Font-Clos F, Zapperi S, La Porta CAM. Topography of epithelial-mesenchymal plasticity. Proc Natl Acad Sci U S A. 2018;115(23):5902–7. pmid:29784817
View Article
PubMed/NCBI
Google Scholar

[44] View Article

[45] PubMed/NCBI

[46] Google Scholar

[ref13] 13. Chaouiya C, Naldi A, Thieffry D. Logical modelling of gene regulatory networks with GINsim. Methods Mol Biol. 2012;804:463–79. pmid:22144167
View Article
PubMed/NCBI
Google Scholar

[48] View Article

[49] PubMed/NCBI

[50] Google Scholar

[ref14] 14. Stoll G, Caron B, Viara E, Dugourd A, Zinovyev A, Naldi A, et al. MaBoSS 2.0: an environment for stochastic Boolean modeling. Bioinformatics. 2017;33(14):2226–8.
View Article
Google Scholar

[52] View Article

[53] Google Scholar

[ref15] 15. Albert R, Thakar J. Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions. Wiley Interdiscip Rev Syst Biol Med. 2014;6(5):353–69. pmid:25269159
View Article
PubMed/NCBI
Google Scholar

[55] View Article

[56] PubMed/NCBI

[57] Google Scholar

[ref16] 16. Gedeon T. Network topology and interaction logic determine states it supports. NPJ Syst Biol Appl. 2024;10(1):98. pmid:39198512
View Article
PubMed/NCBI
Google Scholar

[59] View Article

[60] PubMed/NCBI

[61] Google Scholar

[ref17] 17. Gedeon T. Lattice structures that parameterize regulatory network dynamics. Math Biosci. 2024;374:109225. pmid:38866065
View Article
PubMed/NCBI
Google Scholar

[63] View Article

[64] PubMed/NCBI

[65] Google Scholar

[ref18] 18. Korshunov AD. Monotone Boolean functions. Russ Math Surv. 2003;58(5):929–1001.
View Article
Google Scholar

[67] View Article

[68] Google Scholar

[ref19] 19. Fidytek R, Mostowski AW, Somla R, Szepietowski A. Algorithms counting monotone Boolean functions. Information Processing Letters. 2001;79(5):203–9.
View Article
Google Scholar

[70] View Article

[71] Google Scholar

[ref20] 20. Jäkel C. A computation of the ninth Dedekind number. Journal of Computational Algebra. 2023;6–7:100006.
View Article
Google Scholar

[73] View Article

[74] Google Scholar

[ref21] 21. Mori T, Akutsu T. Attractor detection and enumeration algorithms for Boolean networks. Comput Struct Biotechnol J. 2022;20:2512–20. pmid:35685366
View Article
PubMed/NCBI
Google Scholar

[76] View Article

[77] PubMed/NCBI

[78] Google Scholar

[ref22] 22. TAMURA T, AKUTSU T. Detecting a Singleton Attractor in a Boolean Network Utilizing SAT Algorithms. IEICE Trans Fundamentals. 2009;E92-A(2):493–501.
View Article
Google Scholar

[80] View Article

[81] Google Scholar

[ref23] 23. Dubrova E, Teslenko M, Ming L. Finding Attractors in Synchronous Multiple-Valued Networks Using SAT-Based Bounded Model Checking. In: 2010 40th IEEE International Symposium on Multiple-Valued Logic, 2010. 144–9. https://doi.org/10.1109/ismvl.2010.35

[ref24] 24. Beineke LW, Harary F. Consistency in marked digraphs. Journal of Mathematical Psychology. 1978 Dec;18(3):260-9. Available from: https://linkinghub.elsevier.com/retrieve/pii/0022249678900548
View Article
Google Scholar

[84] View Article

[85] Google Scholar

[ref25] 25. Harary F, Palmer EM. Graphical Enumeration. Elsevier; 2014. Google-Books-ID: ZrvSBQAAQBAJ.

[ref26] 26. Harary F. On the notion of balance of a signed graph. Michigan Math J. 1953;2(2).
View Article
Google Scholar

[88] View Article

[89] Google Scholar

[ref27] 27. Shi G, Altafini C, Baras JS. Dynamics over Signed Networks. SIAM Rev. 2019;61(2):229–57.
View Article
Google Scholar

[91] View Article

[92] Google Scholar

[ref28] 28. Diaz-Diaz F, Candellone E, Gonzalez-Casado MA, Fraxanet E, Vendeville A, Ferri I, et al. Signed Networks: theory, methods, and applications. arXiv; 2025. ArXiv:2511.17247 [physics]. Available from: http://arxiv.org/abs/2511.17247

[ref29] 29. Harary F, Norman RZ, Cartwright D. Structural Models: An Introduction to the Theory of Directed Graphs. Wiley; 1965. Available from: https://books.google.com/books?id=DbY9AAAAIAAJ

[ref30] 30. Hari K, Harlapur P, Saxena A, Haldar K, Girish A, Malpani T, et al. Low dimensionality of phenotypic space as an emergent property of coordinated teams in biological regulatory networks. iScience. 2025;28(2):111730. pmid:39898023
View Article
PubMed/NCBI
Google Scholar

[96] View Article

[97] PubMed/NCBI

[98] Google Scholar

[ref31] 31. Hari K, Ullanat V, Balasubramanian A, Gopalan A, Jolly MK. Landscape of epithelial-mesenchymal plasticity as an emergent property of coordinated teams in regulatory networks. Elife. 2022;11:e76535. pmid:36269057
View Article
PubMed/NCBI
Google Scholar

[100] View Article

[101] PubMed/NCBI

[102] Google Scholar

[ref32] 32. Pastushenko I, Brisebarre A, Sifrim A, Fioramonti M, Revenco T, Boumahdi S, et al. Identification of the tumour transition states occurring during EMT. Nature. 2018;556(7702):463–8. pmid:29670281
View Article
PubMed/NCBI
Google Scholar

[104] View Article

[105] PubMed/NCBI

[106] Google Scholar

[ref33] 33. Gardner TS, Cantor CR, Collins JJ. Construction of a genetic toggle switch in Escherichia coli. Nature. 2000;403(6767):339–42. pmid:10659857
View Article
PubMed/NCBI
Google Scholar

[108] View Article

[109] PubMed/NCBI

[110] Google Scholar

[ref34] 34. Hebbar A, Moger A, Hari K, Jolly MK. Robustness in phenotypic plasticity and heterogeneity patterns enabled by EMT networks. Biophys J. 2022;121(19):3600–15. pmid:35859419
View Article
PubMed/NCBI
Google Scholar

[112] View Article

[113] PubMed/NCBI

[114] Google Scholar

[ref35] 35. Xin Y, Cummins B, Gedeon T. Multistability in the epithelial-mesenchymal transition network. BMC Bioinformatics. 2020;21(1):71. pmid:32093616
View Article
PubMed/NCBI
Google Scholar

[116] View Article

[117] PubMed/NCBI

[118] Google Scholar

[ref36] 36. Adigwe S, BV H, Gedeon T, Jolly MK. Characterization of monotone Boolean models supporting fixed points and multistability in balanced networks. Zenodo; 2026. Available from:
View Article
Google Scholar

[120] View Article

[121] Google Scholar

[ref37] 37. Aracena J, Richard A, Salinas L. Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks. SIAM J Discrete Math. 2017;31(3):1702–25.
View Article
Google Scholar

[123] View Article

[124] Google Scholar

[ref38] 38. Stanley RP. Enumerative Combinatorics. vol. 1 of Cambridge Studies in Advanced Mathematics. 2nd ed. Cambridge: Cambridge University Press; 2011. Available from: https://www.cambridge.org/core/books/enumerative-combinatorics/3155CDE1D973D49F873BDE2EAF8D7651. https://doi.org/10.1017/CBO9781139058520

[ref39] 39. Lattices RS. Lattices and Ordered Sets. New York, NY: Springer. 2008. https://doi.org/10.1007/978-0-387-78901-9

[ref40] 40. Birkhoff G. On the combination of subalgebras. Math Proc Camb Phil Soc. 1933;29(4):441–64.
View Article
Google Scholar

[128] View Article

[129] Google Scholar

[ref41] 41. Snoussi EH. Necessary Conditions for Multistationarity and Stable Periodicity. J Biol Syst. 1998;06(01):3–9.
View Article
Google Scholar

[131] View Article

[132] Google Scholar

[ref42] 42. Duddu AS, Sahoo S, Hati S, Jhunjhunwala S, Jolly MK. Multi-stability in cellular differentiation enabled by a network of three mutually repressing master regulators. J R Soc Interface. 2020;17(170):20200631. pmid:32993428
View Article
PubMed/NCBI
Google Scholar

[134] View Article

[135] PubMed/NCBI

[136] Google Scholar

[ref43] 43. Nordick B, Hong T. Identification, visualization, statistical analysis and mathematical modeling of high-feedback loops in gene regulatory networks. BMC Bioinformatics. 2021;22(1):481. pmid:34607562
View Article
PubMed/NCBI
Google Scholar

[138] View Article

[139] PubMed/NCBI

[140] Google Scholar

[ref44] 44. Rashid M, Hari K, Thampi J, Santhosh NK, Jolly MK. Network topology metrics explaining enrichment of hybrid epithelial/mesenchymal phenotypes in metastasis. PLoS Comput Biol. 2022;18(11):e1010687. pmid:36346808
View Article
PubMed/NCBI
Google Scholar

[142] View Article

[143] PubMed/NCBI

[144] Google Scholar

[ref45] 45. Preca BT, Bajdak K, Mock K, Sundararajan V, Pfannstiel J, Maurer J, et al. A self-enforcing CD44s/ZEB1 feedback loop maintains EMT and stemness properties in cancer cells. International Journal of Cancer. 2015 Dec;137(11):2566-77. Publisher: John Wiley & Sons, Ltd. Available from: https://onlinelibrary.wiley.com/doi/full/10.1002/ijc.29642
View Article
Google Scholar

[146] View Article

[147] Google Scholar

[ref46] 46. Lu M, Jolly MK, Levine H, Onuchic JN, Ben-Jacob E. MicroRNA-based regulation of epithelial-hybrid-mesenchymal fate determination. Proc Natl Acad Sci U S A. 2013;110(45):18144–9. pmid:24154725
View Article
PubMed/NCBI
Google Scholar

[149] View Article

[150] PubMed/NCBI

[151] Google Scholar

[ref47] 47. Chevalier S, Froidevaux C, Paulevé L, Zinovyev A. Synthesis of Boolean Networks from Biological Dynamical Constraints using Answer-Set Programming. In: 2019 IEEE 31st International Conference on Tools with Artificial Intelligence (ICTAI), 2019. 34–41. http://doi.org/10.1109/ictai.2019.00014

[ref48] 48. Dickenstein A, Millán MP, Shiu A, Tang X. Multistationarity in Structured Reaction Networks. Bull Math Biol. 2019;81(5):1527–81. pmid:30788691
View Article
PubMed/NCBI
Google Scholar

[154] View Article

[155] PubMed/NCBI

[156] Google Scholar

[ref49] 49. Sadeghimanesh A, England M. Polynomial superlevel set representation of the multistationarity region of chemical reaction networks. BMC Bioinformatics. 2022;23(1):391. pmid:36167486
View Article
PubMed/NCBI
Google Scholar

[158] View Article

[159] PubMed/NCBI

[160] Google Scholar

[ref50] 50. Conradi C, Feliu E, Mincheva M, Wiuf C. Identifying parameter regions for multistationarity. PLoS Comput Biol. 2017;13(10):e1005751. pmid:28972969
View Article
PubMed/NCBI
Google Scholar

[162] View Article

[163] PubMed/NCBI

[164] Google Scholar