# Exploring Evolutionary Fitness in Biological Systems Using Machine Learning Methods

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Generic Framework to Estimate Evolutionary Fitness

- $\rho (v,t)=0$ signifies the absence of v in the system at time $t;$
- $\rho (v,t)>0$ signifies the presence of v in the system at time $t;$
- $\rho (v,t)$ is a continuous function of v in $V;$
- $\rho (v,t)$ is a continuous function of time;
- approaching zero by $\rho (v,t)$ over time means the extinction of the strategy $v;$
- if $\rho (v,{t}_{0})=0$ at some time ${t}_{0}$, then $\rho (v,t)=0$ for all $t>{t}_{0};$ and
- $\rho (v,t)$ is uniformly bounded by a constant, i.e., $\rho (v,t)<C$.

#### 2.2. Predicting Patterns of Optimal DVM via Machine Learning

## 3. Results

#### 3.1. Revealing Fitness in a Non-Structured Population

#### 3.2. Revealing Evolutionary Fitness in a Structured Population

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- Input box. Input two vectors v and w of vertical locations of zooplankton measured every 3 h (determining two different strategies of DVM). The components of these vectors are denoted by ${x}_{i}\left(v\right)$, ${x}_{i}\left(w\right)$ with $i=1,2,\dots ,8$.
- Evaluation of values of the functions E, S, G as well as the velocity ${x}^{\prime}$ for both strategies v and w at the considered discrete points ${x}_{i}\left(v\right)$, ${x}_{i}\left(w\right)$ with $i=1,2,\dots ,8$. We calculate the values of $E\left({x}_{i}\left(v\right)\right)$, $S\left({x}_{i}\left(v\right)\right)$, $G\left({x}_{i}\left(v\right)\right)$, ${\left({x}_{i}^{\prime}\left(v\right)\right)}^{2}$, $E\left({x}_{i}\left(w\right)\right)$, $S\left({x}_{i}\left(w\right)\right)$, $G\left({x}_{i}\left(w\right)\right)$, ${\left({x}_{i}^{\prime}\left(w\right)\right)}^{2}$.
- Computation of the key parameters ${M}_{i}$ via summation of components obtained in step 2 with appropriate signs: ${M}_{1}={\sum}_{1}^{8}E\left({x}_{i}\left(v\right)\right)$; ${M}_{2}=-{\sum}_{1}^{8}{\left({x}_{i}^{\prime}\left(v\right)\right)}^{2}$; ${M}_{3}=-{\sum}_{1}^{8}S\left({x}_{i}\left(v\right)\right)$; ${M}_{4}=-{\sum}_{1}^{8}G\left({x}_{i}\left(v\right)\right)$ (for w the corresponding expressions will be similar).
- Calculation of the difference between the key parameters ${M}_{i}(v,w)={M}_{i}\left(v\right)-{M}_{i}\left(w\right)$, $i=1,2,3,4$.
- Computation of convolution of the differences ${M}_{i}(v,w)$ with the corresponding weighting coefficients as $\alpha {M}_{1}(v,w)+\gamma {M}_{2}(v,w)+\beta {M}_{3}(v,w)+\delta {M}_{4}(v,w)$.
- Implementation of a sigmoid function to the convolution found in step 5. Comparison with a fixed threshold value.
- Output box: interpretation of the obtained result as comparison of strategies by concluding if $v\succ w$ or $w\succ v$.

**Figure A1.**Flowchart of the AI procedure used in the current study to reveal the ranking order of two different strategies of zooplankton DVM.

## Appendix B

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**Figure 1.**Revealing evolutionary fitness in an unstructured (single-stage) zooplankton DVM model by separating pair strategies with the different ranking order. (

**a**) Two-dimensional cross section $({M}_{1}\left(v\right)-{M}_{1}\left(w\right)$, ${M}_{2}\left(v\right)-{M}_{2}\left(w\right))$ of the four-dimensional parameter space $\{{M}_{i}\left(v\right)-{M}_{i}\left(w\right)\}$, $i=1,2,3,4$. (

**b**) Three-dimensional cross section $({M}_{1}\left(v\right)-{M}_{1}\left(w\right)$, ${M}_{2}\left(v\right)-{M}_{2}\left(w\right)$, ${M}_{3}\left(v\right)-{M}_{3}\left(w\right))$ of the four-dimensional parameter space. In both figures, each dot represents the difference ${M}_{i}\left(v\right)-{M}_{i}\left(w\right)$ (or equivalently, ${M}_{i}\left(w\right)-{M}_{i}\left(v\right))$ and corresponds to the pair strategies $(v,w)$. In both figures, dark dots denote pairs with $w\succ v$ and bright dots denote pairs where $w\prec v$. The considered key parameters denote food availability $\left({M}_{1}\right),$ predator pressure $\left({M}_{2}\right),$ and the metabolic cost of migration $\left({M}_{3}\right)$. The strategies are obtained from empirical data on vertical trajectories of DVM of Calanus euxinus (adult stage) in the northeastern Black Sea in the summer period (see Section 2 for detail).

**Figure 2.**Optimal trajectories of DVM of zooplankton predicted by an unstructured (single-stage) population model: (

**a**) $E,{S}_{x},$ and G in ${M}_{i}$ are modeled by the simple linear and quadratic parameterization; (

**b**) $E,{S}_{x},$ and G are modeled by hyperbolic functions. Blue dots show the empirical data of vertical migration of Calanus euxinus in the northeastern Black Sea in summer (adult stage) [42]. For details on the construction of trajectories and parameterization of $E,{S}_{x},$ and G see the text. The depth x in the graphs is measured in meters.

**Figure 3.**Optimal trajectories of DVM of zooplankton predicted by a two-stage model in the case where the functions E, ${S}_{x}$, and G are the given by linear and quadratic functions (

**a**) and in the case where E, ${S}_{x}$, and G are given by hyperbolic functions (

**b**). In both figures, the dots show the empirical data of vertical migration of Calanus euxinus in the northeastern Black Sea in summer from [42]: blue and red color denote, respectively, juveniles and adults. The blue and red lines denote the optimal trajectories for juveniles and adults, respectively, predicted by the optimization of fitness (6). For details on the construction of trajectories, see the text. The depth x in the graphs is measured in meters.

**Table 1.**Comparison of the coefficients of evolutionary fitness function from the analytical form and the computational method using recognition of ranking of pairs.

Coefficient | Analytical Model | Recognition of Pairs |
---|---|---|

${h}_{3}$ | 1.00 | 1.00 |

${h}_{1}$ | 0.6478 | 0.6332 |

${h}_{2}$ | −1.3472 | −1.3692 |

${h}_{4}$ | −1.4809 | −2.4853 |

${h}_{33}$ | −2.1204 | −2.0251 |

${h}_{34}$ | 0.1417 | 0.1436 |

${h}_{23}$ | −0.1417 | −0.1589 |

${h}_{13}$ | 4.1073 | 3.8467 |

${h}_{44}$ | 4.2301 | 4.0135 |

${h}_{24}$ | −4.2301 | −4.7115 |

${h}_{14}$ | −3.9474 | −3.3190 |

${h}_{22}$ | 4.2301 | 4.8529 |

${h}_{12}$ | 3.9474 | −3.9102 |

${h}_{11}$ | −4.7742 | −5.4857 |

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Kuzenkov, O.; Morozov, A.; Kuzenkova, G.
Exploring Evolutionary Fitness in Biological Systems Using Machine Learning Methods. *Entropy* **2021**, *23*, 35.
https://doi.org/10.3390/e23010035

**AMA Style**

Kuzenkov O, Morozov A, Kuzenkova G.
Exploring Evolutionary Fitness in Biological Systems Using Machine Learning Methods. *Entropy*. 2021; 23(1):35.
https://doi.org/10.3390/e23010035

**Chicago/Turabian Style**

Kuzenkov, Oleg, Andrew Morozov, and Galina Kuzenkova.
2021. "Exploring Evolutionary Fitness in Biological Systems Using Machine Learning Methods" *Entropy* 23, no. 1: 35.
https://doi.org/10.3390/e23010035