# Tipping the Balance: A Criticality Perspective

## Abstract

**:**

## 1. Introduction

## 2. Model of Tumour Heterogeneity

#### 2.1. Similarity with Population Genetics Model

#### 2.2. Critical-Point Transition to Bimodality

#### 2.3. Critical Exponents

## 3. Nonequilibrium Model with λ ≠ 0

#### Cancer as a Phase Transition

## 4. Quantitative Signatures of the Onset of Dominance

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Steady state PDF, ${p}_{S}\left(q\right)$, versus $q$ for $\lambda =0.$ The noise intensity ${\sigma}^{2}$ has values 1.0 $\left(a\right)$, 2.0 $\left(b\right)$, and 4.0 $\left(c\right)$. A purely noise-induced transition from unimodality to bimodality occurs when the noise intensity exceeds the critical value ${\sigma}_{c}^{2}=2$.

**Figure 2.**Parametric plots of bifurcation curves (22) and (23) in the $\left(\lambda -{\sigma}^{2}\right)$ plane. The curves separate a region of bimodality from the regions of unimodality.

**Figure 3.**Hysteresis plots for ${q}_{m}$ and $1-{q}_{m}$ versus λ with ${\sigma}^{2}=4$. The two plots correspond to the two phenotypically distinct subpopulations.

**Figure 4.**Cumulative distribution function (CDF) of $q$ versus λ for the cases $\left(d\right){F}_{S}\left(0.5\right)$ and $\left(e\right)1-{F}_{S}\left(0.5\right)$ with ${\sigma}^{2}=8.0.$.

**Figure 5.**Evolution of the steady state PDF ${P}_{S}\left(q\right)$ versus $q$ as the parameter decreases from positive to negative values with ${\sigma}^{2}=4.0.$ The same set of plots represent the evolution of ${P}_{S}\left(1-q\right)$ versus $q$ but with the sign of λ changed.

**Figure 7.**The variance of the steady state probability distribution, given in (9), versus λ for ${\sigma}^{2}=4$.

**Figure 8.**The plot of ${D}_{JS}\left(P\left|\right|Q\right)$ versus λ. For the P-distribution, ${\sigma}^{2}=4,\lambda \ne 0$. For the Q-distribution, ${\sigma}^{2}=2,\lambda =0$, i.e., the distribution is the critical distribution.

**Figure 9.**The plot of ${D}_{JS}\left(P\left|\right|Q\right)$ versus λ. For both the distributions $P$ and $Q$, ${\sigma}^{2}=4$ but the λ values are opposite in sign.

**Figure 10.**The plot of $CPE$ (28) versus λ with ${\sigma}^{2}=4.0$. The entropic measure has the highest value at λ = 0.

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Bose, I.
Tipping the Balance: A Criticality Perspective. *Entropy* **2022**, *24*, 405.
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Bose I.
Tipping the Balance: A Criticality Perspective. *Entropy*. 2022; 24(3):405.
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**Chicago/Turabian Style**

Bose, Indrani.
2022. "Tipping the Balance: A Criticality Perspective" *Entropy* 24, no. 3: 405.
https://doi.org/10.3390/e24030405