# Emergent Criticality in Coupled Boolean Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Methods

#### 2.1. Boolean Networks

#### 2.2. Dynamic Control Kernel

#### 2.3. Coupled Boolean Networks

#### 2.4. Detecting Cell Types

## 3. Results

**Case 1**.- If ${h}_{0}=0$, the CK is not affected by unpinned dynamics. Thus, the Hamiltonian reduces to the standard r-layer Ising model under the influence of an external field.
**Case 2**.**Case 3**.- For ${h}_{0}\approx J$, the r Ising layers are coupled through the dynamics of each BN. This is a nontrivial, bidirectional, and time-dependent nonlinear coupling.

#### 3.1. Spontaneous Cell Differentiation

#### 3.2. Self-Tuned Cell Differentiation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Table of parameters, definitions, and parameter values utilized in the model development and simulations. There are two control parameters: the intercellular intrinsic noise (T) and the tendency of the system to retain its original BN dynamics (${h}_{0}$). Both parameters are measured in J units.

Parameter Notation | Definition | Parameter Values |
---|---|---|

k | BN connectivity | 2 |

p | Boolean function bias | $0.5$ |

q | perturbation probability | $0.1,0.02$ |

r | control kernel size | $1,2$ |

J | interaction strength/energy unit | 1 |

h | external field constant | 0 |

$\alpha $ | relaxation coefficient | $0.8$ |

$\mathcal{T}$ | simulation time (MC steps) | ${10}^{4}$ |

${t}_{eq}$ | equilibration time (MC steps) | ${10}^{4}$ |

## References

- Kærn, M.; Elston, T.C.; Blake, W.J.; Collins, J.J. Stochasticity in gene expression: From theories to phenotypes. Nat. Rev. Genet.
**2005**, 6, 451–464. [Google Scholar] [CrossRef] - Huang, S. Reprogramming cell fates: Reconciling rarity with robustness. BioEssays
**2009**, 31, 546–560. [Google Scholar] [CrossRef] - Hu, M.; Krause, D.; Greaves, M.; Sharkis, S.; Dexter, M.; Heyworth, C.; Enver, T. Multilineage gene expression precedes commitment in the hemopoietic system. Genes Dev.
**1997**, 11, 774–785. [Google Scholar] [CrossRef][Green Version] - Garcia-Ojalvo, J.; Martinez Arias, A. Towards a statistical mechanics of cell fate decisions. Curr. Opin. Genet. Dev.
**2012**, 22, 619–626. [Google Scholar] [CrossRef] - Macarthur, B.D.; Lemischka, I.R. Statistical Mechanics of Pluripotency. Cell
**2013**, 154, 484–489. [Google Scholar] [CrossRef] [PubMed][Green Version] - Silva, J.; Smith, A. Capturing Pluripotency. Cell
**2008**, 132, 532–536. [Google Scholar] [CrossRef][Green Version] - Goldenfeld, N. Lectures on Phase Transitions and the Renormalization Group; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Ehrenfest, P. Phasenumwandlungen im ueblichen und erweiterten Sinn, classifiziert nach den entsprechenden Singularitaeten des thermodynamischen Potentiales; NV Noord-Hollandsche Uitgevers Maatschappij: Amsterdam, The Netherlands, 1933. [Google Scholar]
- Jaeger, G. The Ehrenfest classification of phase transitions: Introduction and evolution. Arch. Hist. Exact Sci.
**1998**, 53, 51–81. [Google Scholar] [CrossRef] - Pujadas, E.; Feinberg, A.P. Regulated Noise in the Epigenetic Landscape of Development and Disease. Cell
**2012**, 148, 1123–1131. [Google Scholar] [CrossRef][Green Version] - MacArthur, B.D.; Ma, A.; Lemischka, I.R. Systems biology of stem cell fate and cellular reprogramming. Nat. Rev. Mol. Cell Biol.
**2009**, 10, 672–681. [Google Scholar] [CrossRef][Green Version] - Okamoto, K.; Germond, A.; Fujita, H.; Furusawa, C.; Okada, Y.; Watanabe, T.M. Single cell analysis reveals a biophysical aspect of collective cell-state transition in embryonic stem cell differentiation. Sci. Rep.
**2018**, 8, 1–13. [Google Scholar] [CrossRef] [PubMed] - Shmulevich, I.; Dougherty, E.R.; Kim, S.; Zhang, W. Probabilistic Boolean networks: A rule-based uncertainty model for gene regulatory networks. Bioinformatics
**2002**, 18, 261–274. [Google Scholar] [CrossRef][Green Version] - Kauffman, S.; Peterson, C.; Samuelsson, B.; Troein, C. Random Boolean network models and the yeast transcriptional network. Proc. Natl. Acad. Sci. USA
**2003**, 100, 14796–14799. [Google Scholar] [CrossRef][Green Version] - Serra, R.; Villani, M.; Graudenzi, A.; Kauffman, S.A. Why a simple model of genetic regulatory networks describes the distribution of avalanches in gene expression data. J. Theor. Biol.
**2007**, 246, 449–460. [Google Scholar] [CrossRef] - Albert, R.; Othmer, H.G. The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. J. Theor. Biol.
**2003**, 223, 1–18. [Google Scholar] [CrossRef] - Huang, S.; Ingber, D.E. Shape-dependent control of cell growth, differentiation, and apoptosis: Switching between attractors in cell regulatory networks. Exp. Cell Res.
**2000**, 261, 91–103. [Google Scholar] [CrossRef][Green Version] - Saez-Rodriguez, J.; Simeoni, L.; Lindquist, J.A.; Hemenway, R.; Bommhardt, U.; Arndt, B.; Haus, U.U.; Weismantel, R.; Gilles, E.D.; Klamt, S.; et al. A logical model provides insights into T cell receptor signaling. PLoS Comput. Biol.
**2007**, 3, e163. [Google Scholar] [CrossRef] - Derrida, B.; Pomeau, Y. Random networks of automata: A simple annealed approximation. EPL (Europhys. Lett.)
**1986**, 1, 45. [Google Scholar] [CrossRef] - Luque, B.; Solé, R.V. Lyapunov exponents in random Boolean networks. Phys. A Stat. Mech. Its Appl.
**2000**, 284, 33–45. [Google Scholar] [CrossRef][Green Version] - Kauffman, S.A. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol.
**1969**, 22, 437–467. [Google Scholar] [CrossRef] - Kauffman, S. The large scale structure and dynamics of gene control circuits: An ensemble approach. J. Theor. Biol.
**1974**, 44, 167–190. [Google Scholar] [CrossRef] - Bornholdt, S.; Kauffman, S. Ensembles, dynamics, and cell types: Revisiting the statistical mechanics perspective on cellular regulation. J. Theor. Biol.
**2019**, 467, 15–22. [Google Scholar] [CrossRef][Green Version] - Shmulevich, I.; Gluhovsky, I.; Hashimoto, R.F.; Dougherty, E.R.; Zhang, W. Steady-state analysis of genetic regulatory networks modelled by probabilistic Boolean networks. Comp. Funct. Genom.
**2003**, 4, 601–608. [Google Scholar] [CrossRef][Green Version] - Ching, W.K.; Zhang, S.; Ng, M.K.; Akutsu, T. An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks. Bioinformatics
**2007**, 23, 1511–1518. [Google Scholar] [CrossRef] - Kang, C.; Aguilar, B.; Shmulevich, I. Emergence of diversity in homogeneous coupled Boolean networks. Phys. Rev. E
**2018**, 97, 052415. [Google Scholar] [CrossRef] - Brun, M.; Dougherty, E.R.; Shmulevich, I. Steady-state probabilities for attractors in probabilistic Boolean networks. Signal Process.
**2005**, 85, 1993–2013. [Google Scholar] [CrossRef] - Elowitz, M.B.; Levine, A.J.; Siggia, E.D.; Swain, P.S. Stochastic gene expression in a single cell. Science
**2002**, 297, 1183–1186. [Google Scholar] [CrossRef][Green Version] - Raser, J.M.; O’Shea, E.K. Control of stochasticity in eukaryotic gene expression. Science
**2004**, 304, 1811–1814. [Google Scholar] [CrossRef][Green Version] - Chen, Y.; Golding, I.; Sawai, S.; Guo, L.; Cox, E.C. Population fitness and the regulation of Escherichia coli genes by bacterial viruses. PLoS Biol.
**2005**, 3, e229. [Google Scholar] [CrossRef] - Lei, X.; Tian, W.; Zhu, H.; Chen, T.; Ao, P. Biological sources of intrinsic and extrinsic noise in cI expression of lysogenic phage lambda. Sci. Rep.
**2015**, 5, 1–12. [Google Scholar] [CrossRef][Green Version] - Cheng, D.; Qi, H. Controllability and observability of Boolean control networks. Automatica
**2009**, 45, 1659–1667. [Google Scholar] [CrossRef] - Serra, R.; Villani, M.; Semeria, A. Genetic network models and statistical properties of gene expression data in knock-out experiments. J. Theor. Biol.
**2004**, 227, 149–157. [Google Scholar] [CrossRef] - Kim, J.; Park, S.M.; Cho, K.H. Discovery of a kernel for controlling biomolecular regulatory networks. Sci. Rep.
**2013**, 3, 1–9. [Google Scholar] [CrossRef][Green Version] - Joo, J.I.; Zhou, J.X.; Huang, S.; Cho, K.H. Determining relative dynamic stability of cell states using boolean network model. Sci. Rep.
**2018**, 8, 1–14. [Google Scholar] [CrossRef] - Villani, M.; Serra, R.; Ingrami, P.; Kauffman, S.A. Coupled random Boolean network forming an artificial tissue. In Proceedings of the International Conference on Cellular Automata, Springer, Berlin/Heidelberg, Germany; 2006; pp. 548–556. [Google Scholar]
- Serra, R.; Villani, M.; Damiani, C.; Graudenzi, A.; Colacci, A.; Kauffman, S.A. Interacting random boolean networks. In Proceedings of the ECCS07: European Conference on Complex Systems; Citeseer: University Park, PA, USA, 2007; pp. 1–15. [Google Scholar]
- Damiani, C.; Serra, R.; Villani, M.; Kauffman, S.A.; Colacci, A. Cell—Cell interaction and diversity of emergent behaviours. IET Syst. Biol.
**2011**, 5, 137–144. [Google Scholar] [CrossRef] - Flann, N.S.; Mohamadlou, H.; Podgorski, G.J. Kolmogorov complexity of epithelial pattern formation: The role of regulatory network configuration. Biosystems
**2013**, 112, 131–138. [Google Scholar] [CrossRef] - Kim, H.; Pineda, O.K.; Gershenson, C. A multilayer structure facilitates the production of antifragile systems in boolean network models. Complexity
**2019**, 2019, 2783217. [Google Scholar] [CrossRef] - Lee, D.D.; Seung, H.S. Learning the parts of objects by non-negative matrix factorization. Nature
**1999**, 401, 788–791. [Google Scholar] [CrossRef] - Liu, L.; Michowski, W.; Kolodziejczyk, A.; Sicinski, P. The cell cycle in stem cell proliferation, pluripotency and differentiation. Nat. Cell Biol.
**2019**, 21, 1060–1067. [Google Scholar] [CrossRef] - Waddington, C.H. Organisers and Genes; Cambridge University Press: Cambridge, UK, 1940. [Google Scholar]
- Wang, J.; Xu, L.; Wang, E.; Huang, S. The potential landscape of genetic circuits imposes the arrow of time in stem cell differentiation. Biophys. J.
**2010**, 99, 29–39. [Google Scholar] [CrossRef][Green Version] - Zhou, J.X.; Aliyu, M.; Aurell, E.; Huang, S. Quasi-potential landscape in complex multi-stable systems. J. R. Soc. Interface
**2012**, 9, 3539–3553. [Google Scholar] [CrossRef] - McElroy, M.; Voulgarakis, N.K. 2023; in preparation.
- Tsuchiya, M.; Giuliani, A.; Hashimoto, M.; Erenpreisa, J.; Yoshikawa, K. Emergent Self-Organized Criticality in gene expression dynamics: Temporal development of global phase transition revealed in a cancer cell line. PLoS ONE
**2015**, 10, e0128565. [Google Scholar] [CrossRef] [PubMed][Green Version] - Martello, G.; Smith, A. The nature of embryonic stem cells. Annu. Rev. Cell Dev. Biol.
**2014**, 30, 647–675. [Google Scholar] [CrossRef] - Hackett, J.A.; Surani, M.A. Regulatory principles of pluripotency: From the ground state up. Cell Stem Cell
**2014**, 15, 416–430. [Google Scholar] [CrossRef] [PubMed][Green Version]

**Figure 1.**A 6-gene network with $k=2$, and the bias of $p=0.5$. (

**a**) Wiring diagram of the network. (

**b**) Truth table of Boolean functions. (

**c**) State transition diagram. (

**d**) Steady-state distribution, ${g}_{0}\left(s\right)$, with perturbation $q=0.1$.

**Figure 2.**(

**Left**) Pinned BN with $r=1$; ${x}_{1}$ is either 0 or 1. The two resulting steady-state distributions ${g}_{1}^{\left(1\right)}\left(s\right)$ and ${g}_{2}^{\left(1\right)}\left(s\right)$ have partitioned state spaces and preserve attractor points. (

**Right**) Pinned BN with $r=2$; ${x}_{1}$ and ${x}_{2}$ are pinned to either 0 or 1. The four resulting steady-state distributions ${g}_{1}^{\left(2\right)}\left(s\right)$, ${g}_{2}^{\left(2\right)}\left(s\right)$, ${g}_{3}^{\left(2\right)}\left(s\right)$, and ${g}_{4}^{\left(2\right)}\left(s\right)$, once again, have partitioned state spaces and preserve attractor points.

**Figure 3.**Stochastic pinning of the BN: (

**a**) transition diagram for a two-state, ${\mathbf{W}}_{1}=\{1,2\}$, stochastic pinning process; (

**b**) first 100 sampled values of the stochastic process ${\eta}_{t}$; and (

**c**) steady-state distribution of the stochastic CK, $g\left(s\right)$, obtained after $\mathcal{T}={10}^{5}$ simulation steps.

**Figure 4.**Differentiation process for $r=1$ and a $50\times 50$ tissue size. First row: spontaneous differentiation process as a function of ${h}_{0}$ for (

**a**) $T=0.25$, (

**b**) $T=1$, and (

**c**) $T=1.25$. Second row: spontaneous differentiation process as a function of T for (

**d**) ${h}_{0}=0$, (

**e**) ${h}_{0}=0.5$, and (

**f**) ${h}_{0}=1$. The arrow approximately indicates the differentiation starting point.

**Figure 5.**Differentiation process for $r=2$ and on a $50\times 50$ tissue. First row: spontaneous differentiation process as a function of ${h}_{0}$ for (

**a**) $T=0.25$, (

**b**) $T=1$, and (

**c**) $T=1.25$. Second row: spontaneous differentiation process as a function of T for (

**d**) ${h}_{0}=0$, (

**e**) ${h}_{0}=0.5$, and (

**f**) ${h}_{0}=1$. The arrow and dashed arrow denote approximate starting points of two different differentiation processes.

**Figure 6.**Cell-to-cell variability for $r=2$ with ${h}_{0}=0.5$ on a $50\times 50$ tissue. First row ((

**a**,

**c**,

**e**) shows time evolutions of the cell fractions, and the second row ((

**b**,

**d**,

**f**) shows representative snapshots of the tissue states at time $t=67$ respectively. Temperature is $T=3$ for the first column ((

**a**,

**b**)), $T=1.85$ for the second column ((

**c**,

**d**)), and $T=1$ for the third column ((

**e**,

**f**)). The time unit is ${10}^{2}$ MC steps.

**Figure 7.**Simulations of a $32\times 32$ Ising model with a self-tuning feedback equation (Equation (8)). Here, $h=0$, ${h}_{0}=0$, and $\alpha =0.8$. (

**a**) Magnetization trajectories show two systems are driven to ${M}_{t}\approx 0$ immediately upon initialization and eventually self-tune to two different homogeneous cell types (${M}_{t}\approx -1$ or ${M}_{t}\approx 1$). (

**b**) The simulations show that the temperature drop slows as it descends below the critical temperature (${T}_{c}$) and eventually reaches a steady temperature. Combining (

**a**) and (

**b**), we see that as the feedback temperature reaches the Ising critical temperature (${T}_{c}$), the magnetizations begin to diverge and tissue differentiates homogeneously to one of the two possible cell types. The time unit is ${10}^{2}$ MC steps with ${N}_{MC}=10$.

**Figure 8.**One hundred independent tissue simulations of a $32\times 32$ Ising Hamiltonian (Equation (5) with the temperature–magnetization feedback mechanism (Equation (8)) are shown: (

**a**) At time $t=1$, where the initial temperature is high ($T=2.8$), all cells are of cell type ${\mathbf{g}}_{2}^{\left(1\right)}$. (

**b**) At time $t=1\times {10}^{5}$, where the tissues are ${M}_{t}\approx 0$, there is a mixture of ${\mathbf{g}}_{1}^{\left(1\right)}$ and ${\mathbf{g}}_{2}^{\left(1\right)}$ cell types from Figure 2 (left), and the average cell state at the colony-level remains unimodal, centered between the distributions ${g}_{1}^{\left(1\right)}\left(s\right)$ and ${g}_{2}^{\left(1\right)}\left(s\right)$. (

**c**) At time $t=6\times {10}^{5}$, the tissues decide on the fate of the cells with a drop in temperature to critical and subcritical points, and hence, ${M}_{t}\approx -1$ or ${M}_{t}\approx 1$. This results in a split, bimodal distributions of gene states at the cellular and colony level.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kang, C.; McElroy, M.; Voulgarakis, N.K. Emergent Criticality in Coupled Boolean Networks. *Entropy* **2023**, *25*, 235.
https://doi.org/10.3390/e25020235

**AMA Style**

Kang C, McElroy M, Voulgarakis NK. Emergent Criticality in Coupled Boolean Networks. *Entropy*. 2023; 25(2):235.
https://doi.org/10.3390/e25020235

**Chicago/Turabian Style**

Kang, Chris, Madelynn McElroy, and Nikolaos K. Voulgarakis. 2023. "Emergent Criticality in Coupled Boolean Networks" *Entropy* 25, no. 2: 235.
https://doi.org/10.3390/e25020235