# Some Aspects of Nonlinearity and Self-Organization In Biosystems on Examples of Localized Excitations in the DNA Molecule and Generalized Fisher–KPP Model

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## Abstract

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## 1. Introduction

## 2. Soliton Excitations in Molecular Chains

#### 2.1. The Concept of Solitons in Molecules

#### 2.2. Kink Dynamics asExternal Factors for the Sine-Gordon DNA Model

#### 2.3. Localized Energy Distributions in the Framework of The Peyrard–Bishop Model

#### 2.4. Summary

## 3. Pattern Formation in Cell Populations

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Relative kink velocity $v\left(\tau \right)$ (the solid curve) and the trend velocity ${v}_{\mathrm{trend}}\left(\tau \right)$ (the open diamonds curve) for $\lambda =0.1$: (

**a**) ${v}_{0}=0.2$, $\Omega =0.8$ and $\lambda =0.1$; (

**b**) ${v}_{0}=0.6$, $\Omega =10$.

**Figure 2.**Trend velocity ${v}_{\mathrm{trend}}\left(\tau \right)$ (the dotted curve) and the average relative velocity $\langle v\left(\tau \right)\rangle $ (the open diamonds curve) for ${v}_{0}=0.6$, $\Omega =10$ and $\lambda =0.1$.

**Figure 3.**Root-mean-square value of the kink momentum for $\alpha =0.1$, $\tilde{D}=0.01$ and ${t}_{0}=0$.

**Figure 4.**Average value of the kink momentum for $\alpha =0.1$, ${P}_{\gamma}=4$, and ${t}_{0}=0$. (

**a**) The solid curve is for ${P}_{0}=0.1$, $\mu =0.01$; the dashed curve is for ${P}_{0}=0.1$, $\mu =0$. (

**b**) The solid curve is for ${P}_{0}=-0.1$, $\mu =0.01$; the dashed curve is for ${P}_{0}=-0.1$, $\mu =0$.

**Figure 5.**Graph of the function ${\rho}_{0}(t,s)$ for $a=1$, $a>\kappa {\lambda}_{0}{v}_{0}{\beta}_{00}$.

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Shapovalov, A.V.; Obukhov, V.V.
Some Aspects of Nonlinearity and Self-Organization In Biosystems on Examples of Localized Excitations in the DNA Molecule and Generalized Fisher–KPP Model. *Symmetry* **2018**, *10*, 53.
https://doi.org/10.3390/sym10030053

**AMA Style**

Shapovalov AV, Obukhov VV.
Some Aspects of Nonlinearity and Self-Organization In Biosystems on Examples of Localized Excitations in the DNA Molecule and Generalized Fisher–KPP Model. *Symmetry*. 2018; 10(3):53.
https://doi.org/10.3390/sym10030053

**Chicago/Turabian Style**

Shapovalov, A. V., and V. V. Obukhov.
2018. "Some Aspects of Nonlinearity and Self-Organization In Biosystems on Examples of Localized Excitations in the DNA Molecule and Generalized Fisher–KPP Model" *Symmetry* 10, no. 3: 53.
https://doi.org/10.3390/sym10030053