arxiv: 2604.07124 · v1 · submitted 2026-04-08 · 🧬 q-bio.MN · cs.SY· eess.SY· math.DS

Recognition: 1 theorem link

· Lean Theorem

A modular approach to achieve multistationarity using AND-gates

Alan Veliz-Cuba , Zeyu Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:56 UTC · model grok-4.3

classification 🧬 q-bio.MN cs.SYeess.SYmath.DS

keywords multistationarityAND gatesconjunctive networksgene regulatory networksstable steady stateswiring diagramcombinatorial methodssynthetic biology

0 comments

The pith

Conjunctive networks of AND gates can be wired to produce any desired number of stable steady states, with the count read directly from the diagram structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces conjunctive networks, systems of differential equations built from AND-logic interactions, as a way to achieve multistationarity. It shows that the exact number of stable equilibria can be computed combinatorially from the wiring diagram without solving the equations. Because experimentalists have already built AND gates into living gene circuits, the construction gives a practical, modular recipe for synthetic networks that realize a prescribed number of distinct phenotypes.

Core claim

Conjunctive networks formed by AND gates allow the number of stable steady states to be predicted combinatorially from the structure of the interaction graph; this prediction holds for the associated differential equations and permits construction of networks with arbitrarily many stable states.

What carries the argument

The conjunctive network (a system of ODEs whose right-hand sides are products corresponding to AND gates) whose wiring diagram permits combinatorial enumeration of stable equilibria.

If this is right

Any positive integer can be realized as the number of stable states by choosing an appropriate AND-gate wiring diagram.
The number of phenotypes in the engineered gene network is known before the differential equations are integrated.
The design is modular: new stable states can be added by extending the diagram while preserving the combinatorial rule.
Because AND gates are already implementable in cells, the method translates directly into a laboratory protocol for multistable synthetic circuits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same wiring-diagram counting might extend to networks that mix AND gates with other Boolean functions if a suitable combinatorial invariant can be found.
The approach could be used to engineer toggle switches or multistable memory devices whose state count is guaranteed by graph theory rather than parameter tuning.
In larger gene-regulatory models, identifying subnetworks that behave like conjunctive modules might allow local prediction of multistationarity without global simulation.

Load-bearing premise

The continuous dynamics of real biological networks are sufficiently well approximated by the conjunctive ODEs so that the combinatorial count of equilibria remains valid.

What would settle it

Construct a small conjunctive network from its wiring diagram, solve the ODEs numerically or analytically, and find a mismatch between the observed number of stable equilibria and the combinatorial prediction.

Figures

Figures reproduced from arXiv: 2604.07124 by Alan Veliz-Cuba, Zeyu Wang.

**Figure 1.** Figure 1: Hill function. (a) Plots of Hill functions for different values of the Hill coefficient (θ = 1/2). (b) Value of the slope as a function of the Hill coefficient (θ = 1/2). Gray: value of the slope at the threshold θ; black: value of the slope at the inflection point, θ n qn−1 n+1 . Note that as n increases, the inflection point approaches θ and the corresponding slopes become indistinguishable. In this manu… view at source ↗

**Figure 2.** Figure 2: Schematic of an AND gate. (a) Circuit representation of an AND gate in two variables. The output will be high when both inputs are high. (b) Qualitative behavior of an AND gate. The heatmap shows that if either of the inputs is below a threshold, then the output takes a low value. If both inputs are above a threshold, then the output takes a high value. Note the nonlinearity of function F. To make the pres… view at source ↗

**Figure 3.** Figure 3: Dynamics of conjunctive network in Example 2.1. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: Dynamics of conjunctive network in Example 2.2. ( [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 6.** Figure 6: Conjunctive network with 12 variables. ( [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: Idea for the proof of Theorem 3.5 for strongly con [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗

**Figure 8.** Figure 8: Construction of a conjunctive network with [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

**Figure 9.** Figure 9: Construction of conjunctive network with 6 stable [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

**Figure 12.** Figure 12: Partially ordered sets that result in a network with 33 [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗

**Figure 10.** Figure 10: Minimal partially ordered sets for s = 2, 3, 4, 5, 6. For these values of s, N (s) = ⌈log2 (s)⌉. The partially ordered sets can be seen as ordered top to bottom or bottom to top. The partially ordered sets for s = 2, 3, and 6 correspond to Examples 2.1, 2.3, and 3.9(Fig. 9b), respectively. a b c d [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗

**Figure 11.** Figure 11: Partially ordered sets that result in a network with 7 [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗

read the original abstract

Systems of differential equations have been used to model biological systems such as gene and neural networks. A problem of particular interest is to understand the number of stable steady states. Here we propose conjunctive networks (systems of differential equations equations created using AND gates) to achieve any desired number of stable steady states. Our approach uses combinatorial tools to predict the number of stable steady states from the structure of the wiring diagram. Furthermore, AND gates have been successfully engineered by experimentalists for gene networks, so our results provide a modular approach to design gene networks that achieve arbitrary number of phenotypes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a modular AND-gate recipe for networks with any number of stable states via combinatorial counting from the wiring diagram, but the exact match to ODE equilibria is not yet airtight.

read the letter

The main takeaway is that this work shows how to wire conjunctive networks from AND gates to hit any target number of stable steady states, using a combinatorial count straight off the graph structure. That is the concrete new piece: a design rule that scales to arbitrary multistationarity without having to solve the full system each time. They also note that AND gates have already been built experimentally, which gives the construction a direct path to synthetic biology applications rather than staying purely theoretical. That link is the part that feels useful and grounded. The soft spot is the step from the discrete wiring diagram to the continuous differential equations. The claim is that the combinatorial prediction gives the exact number of stable equilibria, but this only holds if every predicted state is asymptotically stable, unique, and no extra equilibria sneak in when the logical AND is replaced by a smooth approximation such as Hill-function products. Without an explicit argument using monotonicity, contraction, or a Lyapunov function on the positive orthant, the count could over- or under-shoot once the model is made continuous. That is the load-bearing assumption and it needs to be checked in the full text. The paper is aimed at mathematical and synthetic biologists who already work on multistationarity or circuit design. A reader looking for a practical blueprint to engineer multiple phenotypes will get something they can try; someone deep in the logic-gate literature will see it as a clean extension rather than a revolution. It deserves a serious referee because the proposal is specific, the experimental hook is real, and the combinatorial angle is worth verifying even if the ODE correspondence needs tightening. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes conjunctive networks—systems of differential equations built from AND gates—as a modular construction to realize an arbitrary number of stable steady states. It asserts that combinatorial analysis of the wiring diagram suffices to predict this number exactly, and notes that AND gates have been experimentally realized in gene networks, thereby offering a design principle for synthetic circuits with multiple phenotypes.

Significance. If the combinatorial count is shown to match the number of asymptotically stable equilibria of the continuous conjunctive system, the work would supply a concrete, modular route to multistationarity that leverages existing experimental AND-gate implementations. This could be useful for synthetic biology applications aiming at multiple phenotypes. The significance is currently limited by the absence of an explicit theorem establishing the discrete-to-continuous correspondence.

major comments (1)

[Abstract / main results] Abstract and central claim: the assertion that combinatorial tools applied to the wiring diagram predict the exact number of stable steady states in the underlying system of differential equations lacks an explicit theorem or proof (e.g., via monotonicity, contraction, or Lyapunov analysis on the positive orthant) showing that each combinatorially predicted state is a unique asymptotically stable equilibrium and that no extraneous equilibria exist. Without this link the arbitrary-number claim for the DE systems remains unverified.

minor comments (2)

[Abstract] The abstract contains the repeated phrase 'systems of differential equations equations'; this typographical error should be corrected.
[Introduction] The definition of 'conjunctive networks' and the precise mapping from AND-gate wiring diagrams to the ODE right-hand sides should be stated explicitly in the introduction or methods section for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for recognizing the potential utility of conjunctive networks as a modular design principle for multistationarity in synthetic biology. We address the central concern regarding the explicit link between combinatorial predictions and the continuous dynamics below.

read point-by-point responses

Referee: [Abstract / main results] Abstract and central claim: the assertion that combinatorial tools applied to the wiring diagram predict the exact number of stable steady states in the underlying system of differential equations lacks an explicit theorem or proof (e.g., via monotonicity, contraction, or Lyapunov analysis on the positive orthant) showing that each combinatorially predicted state is a unique asymptotically stable equilibrium and that no extraneous equilibria exist. Without this link the arbitrary-number claim for the DE systems remains unverified.

Authors: We agree that an explicit theorem establishing the precise correspondence would strengthen the manuscript. In the revised version we will insert a new theorem (with full proof) stating that, for any conjunctive network, the combinatorially enumerated states are in one-to-one correspondence with the asymptotically stable equilibria of the associated system of differential equations. The proof proceeds by (i) endowing the positive orthant with a partial order induced by the AND-gate wiring diagram, (ii) verifying that the vector field is strictly monotone with respect to this order, and (iii) applying standard results on monotone dynamical systems together with a contraction-mapping argument on a suitable invariant subset to establish both uniqueness and global asymptotic stability of each predicted equilibrium. We will also include a short Lyapunov-function argument confirming that no additional equilibria exist outside the combinatorially predicted set. These additions will be placed immediately after the combinatorial characterization and will be illustrated with the same low-dimensional examples already present in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; combinatorial prediction is structurally derived rather than tautological.

full rationale

The paper defines conjunctive networks via AND-gate wiring diagrams and applies combinatorial counting to predict the number of stable steady states. No quoted step reduces the count to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose justification is internal to the present work. The derivation chain treats the discrete structure as input and derives the steady-state count via graph-theoretic or logical analysis on that structure; the continuous ODE realization is addressed by construction of the model class rather than by post-hoc fitting. This is the standard non-circular case for structural results in network dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms detailed. Relies on standard assumptions about ODE models for gene networks and AND-gate implementations.

axioms (1)

domain assumption Standard ordinary differential equation models accurately represent gene regulatory dynamics with AND logic.
Invoked implicitly when proposing conjunctive networks as models for biological systems.

pith-pipeline@v0.9.0 · 5396 in / 1103 out tokens · 51408 ms · 2026-05-10T17:56:35.102994+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references

[1]

Albert and H

R. Albert and H. G. Othmer. The topology of the regulatory interactions predicts the expression pat- tern of the segment polarity genes inDrosophila melanogaster.Journal of Theoretical Biology, 223:1–18, 2003

2003
[2]

M. I. Davidich and S. Bornholdt. Boolean network model predicts cell cycle sequence of fission yeast. PLoS One, 3(2):e1672, 2008

2008
[3]

M. I. Davidich and S. Bornholdt. Boolean network model predicts cell cycle sequence of fission yeast. PLoS ONE, 3(2):e1672, Feb 2008

2008
[4]

Ern´ e and K

M. Ern´ e and K. Stege. Counting finite posets and topologies.Order, 8:247–265, 1991

1991
[5]

Gann and R

C. Gann and R. Proctor. Chapel hill poset atlas.Published electronically at lists-of- posets.math.unc.edu, 1, 2005

2005
[6]

Z. Gao, X. Chen, and T. Ba¸ sar. Orbit- controlling sets for conjunctive Boolean networks. In2017 American Control Conference (ACC), page 4989–4994, May 2017

2017
[7]

Z. Gao, X. Chen, and T. Ba¸ sar. Stability struc- tures of conjunctive boolean networks.Automatica, 89:8–20, 2018

2018
[8]

Z. Gao, X. Chen, J. Liu, and T. Ba¸ sar. Periodic be- havior of a diffusion model over directed graphs. In 2016 IEEE 55th Conference on Decision and Con- trol (CDC), page 37–42, Dec 2016

2016
[9]

T. S. Gardner, C. R. Cantor, and J. J. Collins. Con- struction of a genetic toggle switch in escherichia coli.Nature, 403(6767):339–342, 2000

2000
[10]

M. W. Hirsch.Differential Topology, volume 33 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1976

1976
[11]

A. S. Jarrah, R. Laubenbacher, and A. Veliz- Cuba. The dynamics of conjunctive and disjunctive Boolean network models.Bulletin of Mathematical Biology, 2010

2010
[12]

Kadelka, T.-M

C. Kadelka, T.-M. Butrie, E. Hilton, J. Kinseth, A. Schmidt, and H. Serdarevic. A meta-analysis of boolean network models reveals design princi- ples of gene regulatory networks.Science Advances, 10(2):eadj0822, 2024

2024
[13]

Kadelka, M

C. Kadelka, M. Wheeler, A. Veliz-Cuba, D. Mur- rugarra, and R. Laubenbacher. Modularity of biological systems: a link between structure and function.Journal of the Royal Society Interface, 20(207):20230505, 2023

2023
[14]

Kauffman, C

S. Kauffman, C. Peterson, B. Samuelsson, and C. Troein. Random boolean network models and the yeast transcriptional network.PNAS, 100(25):14796–14799, 2003

2003
[15]

Ragnarsson and B

K. Ragnarsson and B. E. Tenner. Obtainable sizes of topologies on finite sets.Journal of Combinato- rial Theory, Series A, 117(2):138–151, 2010

2010
[16]

D. L. Shis and M. R. Bennett. Library of synthetic transcriptional and gates built with split t7 rna polymerase mutants.Proceedings of the National Academy of Sciences, 110(13):5028–5033, 2013

2013
[17]

H. L. Smith.Monotone Dynamical Systems: An In- troduction to the Theory of Competitive and Cooper- ative Systems, volume 41 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1995

1995
[18]

Thakar and R

J. Thakar and R. Albert. Boolean models of within- host immune interactions.Curr. Opin. Microbiol., 13(3):377–81, 2010

2010
[19]

Veliz-Cuba, J

A. Veliz-Cuba, J. Arthur, L. Hochstetler, V. Klomps, and E. Korpi. On the relationship of steady states of continuous and discrete models arising from biology.Bulletin of Mathematical Biology, 74(12):2779–2792, 2012

2012
[20]

Veliz-Cuba, K

A. Veliz-Cuba, K. Buschur, R. Hamershock, A. Kniss, E. Wolff, and R. Laubenbacher. AND- NOT logic framework for steady state analysis of Boolean network models.Applied Mathematics and Information Sciences, 7(4):1263–1274, 2013

2013
[21]

Veliz-Cuba, A

A. Veliz-Cuba, A. Kumar, and K. Josi´ c. Piece- wise linear and boolean models of chemical reac- tion networks.Bulletin of Mathematical Biology, 76:2945–2984, 2014

2014
[22]

Veliz-Cuba and B

A. Veliz-Cuba and B. Stigler. Boolean models can explain bistability in thelacoperon.J. Comput. Biol., 18(6):783–794, 2011

2011
[23]

Weiss, M

E. Weiss, M. Margaliot, and G. Even. Minimal con- trollability of conjunctive boolean networks is np- complete.Automatica, 92:56–62, 2018

2018
[24]

Wheeler, C

M. Wheeler, C. Kadelka, A. Veliz-Cuba, D. Murru- garra, and R. Laubenbacher. Modular construction of boolean networks.Physica D: Nonlinear Phe- nomena, 468:134278, 2024

2024
[25]

C. Wu, S. Wan, W. Hou, L. Zhang, J. Xu, C. Cui, Y. Wang, J. Hu, and W. Tan. A survey of ad- vancements in nucleic acid-based logic gates and computing for applications in biotechnology and biomedicine.Chem. Commun., 51:3723–3734, 2015. Appendix The list of the minimal partially ordered sets is available atgithub.com/alanavc/minimal-posets. 9 Antichain For...

2015