# Poincaré Maps and Aperiodic Oscillations in Leukemic Cell Proliferation Reveal Chaotic Dynamics

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## Abstract

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## 1. Introduction

#### 1.1. Acute Lymphoblastic Leukemia of Childhood

#### 1.1.1. Phenotypic and Molecular Characteristics of Leukemia

#### 1.1.2. Leukemogenesis

#### 1.1.3. Non-Linear Dynamics in Leukemia Proliferation

#### 1.2. The Dynamics of Cell Proliferation in Leukemia. An In Vitro Model

#### 1.3. The Mathematical Model and Analysis

#### 1.3.1. A General Description of the Growth Model

#### 1.3.2. Dynamic Systems

_{k}(t), for k = 1, 2, …, N, which have as their only independent variable time t and are components of the vector:

_{1}(t) and x

_{2}(t) are the populations of two different species, which grow, interact and compete. Their state can be described by a series of equation systems:

#### 1.3.3. Lyapunov Exponent

_{0}is the difference at time 0 (that would be Δ

_{1}− Δ

_{0}), x

_{n}is the population at time t and x

_{n+}

_{1}is the population at time t + 1. The sensitivity of the system to its initial conditions can be quantified by the introduction of the Lyapunov exponent λ. The exponent calculates the sensitivity to initial conditions by actually comparing the trajectories of the same system for times t and t + 1. Assuming the Napierian logarithm:

λ_{i} < 0, i = 1,2,…,n: Fixed point. System reaches equilibrium |

λ_{1} = 0 and λ_{i} < 0, i = 2,…,n: Periodic. System oscillates periodically |

λ_{1} = 0, λ_{2} = 0 and λ_{i} < 0, i = 3,…,n: Torus. |

At least one λ_{i} > 0, i = 1,2,…,n: System is chaotic. |

_{1}) for the orbit {x

_{1},x

_{2},…,x

_{n}} is defined as:

#### 1.3.4. Poincaré Maps

_{f}of the function f that intersects the plane at two points A and B. If A intersects the plane for the k

^{th}time, and B intersects the plane for the k

^{th+1}time, then it can be proven that there is a graph such as:

_{t}, with a period T, then it can be proven that φ(t + T, x

_{0}) = φ(t, x

_{0}). If a transversal cross-section (Σ) is taken to the function’s flow then a Poincaré map P(x), is described as P(x): V $\in $ Σ→Σ, which correlates the vector $\overline{x}$, in space V with a point P($\overline{x}$) for each intersection [33]. Therefore, when the function’s flow crosses the transversal plane for the first time, it moves on and crosses the plane in a second point, then on a third, a fourth and so on. This process is mapped through the operator P such as:

_{0}) = x

_{0}, this is called an equilibrium point [33]. This was the solution invented by Poincaré, to solve the three-body problem. Poincaré “invented” a plane, perpendicular to the trajectories of the bodies, which would describe the orbits on a two-dimensional plane. The first advantage of his conception was that it reduced the dimensions of the problem by 1. Thus, if three bodies (or planets) would follow Newtonian dynamics then their mapping would be manifested by single points on a plane (which is actually the phase-space of the trajectories). On the contrary, trajectories that do not follow predictable dynamics would be represented by the mapping of infinitesimally large number of points on a plane. Therefore, complex trajectories would also manifest complex Poincaré maps. From his observations, Poincaré found that there are trajectories that change their behavior due to small changes in their initial conditions, thus introducing chaos [33].

#### 1.4. Aim and Objectives

## 2. Materials and Methods

#### 2.1. The Cellular System

_{2}and ~100% humidity. Cells were cultured in 75 cm

^{2}flasks in total medium volume of 25 mL. Cells were seeded at an initial concentration of 20 cells/μL and ~200 cells/μL and were fed at regular intervals thereafter. Medium changes took place by centrifugation at 1000 rpm for 10 min, the supernatant was discarded and the remaining cells were rediluted in 25 mL media and were allowed to grow.

#### 2.2. Cell Population Measurements

^{6}spheres/μL. Initially, 1ml of the sphere solution was obtained and a 1:10 dilution was performed, in order to obtain a concentration of 1100 spheres/μL. Further on, a series-dilution was performed ranging from 1100 spheres/μL down to ~9 spheres/μL. The theoretical (Figure 2A) and experimental (Figure 2B) concentrations were measured and recorded. Since the Coulter counter was not able to represent a value <100 particles/μL, we have evaluated all dilutions using the histograms provided by the manufacturer. In particular, the histograms for all dilutions from 1100 spheres/μL to 9 spheres/μL are presented in Figure 2C–J. The reason for this type of evaluation, was to examine whether the instrument could detect much diluted sphere solutions and thus diluted cell populations.

#### 2.3. Cell Viability Assays

#### 2.4. Microscopic Evaluation of the Growth Model

#### 2.5. Data Analysis

^{®}for preprocessing. Data were treated as time-series and in particular, we have studied the cell populations with respect to the cell population per volume unit (μL), to the total cell population, as well as with respect to the extrapolation of the population to a single volume. Further on, the phase-space of the time-series has been used and the Poincaré mapping was estimated. In addition, we have calculated the Lyapunov exponents for the time-series trajectories.

## 3. Results

#### 3.1. Time-Series

_{0}≅ 20 cells/μL (Figure 5A) and n

_{0}≅ 200 cells/μL (Figure 5B). In addition, we have superimposed the two time-series and it appeared that the two curves progressed similarly, but with slight divergence (Figure 5C). The two experiments manifested small typical errors, indicating that there was a good repeatability in the experimental setups, which we also examined by regressing cell proliferation data with known growth models (Figure S1).

#### 3.2. Cell Viability

#### 3.3. Cell Cycle

#### 3.4. The Phase-Space of Cell Proliferation

_{0}= 20 cells/μL (Figure 9A) and x

_{0}= 200 cells/μL (Figure 9B). Thus, this was an indication that cell proliferation followed a chaotic orbit.

#### 3.5. Lyapunov Exponents

#### 3.6. The Phase-Space of Cell Proliferation II

#### 3.7. The Phase-Space of Cell Proliferation III

_{0}> 0 such as $x(t+{t}_{0})=x(t)\forall t\in \mathbb{R}$, $\forall $ $t\in \mathbb{R}$). This is an equilibrium point where S $\equiv $ S′. On the contrary, in our experimental setup we did not observe such an evidence of a global non-chaotic behavior. The system converged in the first convolution of the trajectory (Figure 12A,B) as well as after cell culture media supplementation (Figure 12E). In the other cases the trajectory (Figure 12C,D,F–K) diverged from equilibrium indicating that a Poincaré cross-section followed chaotic dynamics.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

^{2}values are presented for each respective curve (using the same coloring scheme). Although regressions were performed, they still concerned an approximation of the growth curves, and therefore our intention was to investigate the details of the manifested non-linear dynamics with more specialized tools, Table S1: Cell Proliferation Data, Video S1: Animated Figure 11.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Examples of Poincaré maps. A trajectory intersects a plane at points x and P(x), where it is x = P(x) (

**A**) (reproduced from https://en.wikipedia.org/wiki/Poincar%C3%A9_map, accessed on 24 May 2021). The phase portrait of a Hamiltonian function (

**B**), with the respective Poincaré map (

**C**), as well as the Poincaré map of a Duffing system (

**D**) are presented.

**Figure 2.**The calibration measurements with the Coulter counter, investigating the sensitivity of the instrument. In particular, the theoretical (

**A**) and experimental (

**B**) regressions of the dilution series are presented. Further on, the actual coulter-counter measurements are presented for sphere concentrations of 1100 spheres/μL (

**C**), 550 spheres/μL (

**D**), 275 spheres/μL (

**E**), 137 spheres/μL (

**F**), 69 spheres/μL (

**G**), 34 spheres/μL (

**H**), 17 spheres/μL (

**I**) and 9 spheres/μL (

**J**).

**Figure 3.**The experimental setup of the growth model [43].

**Figure 4.**CCRF-CEM morphology during the experimental setup. Image acquisition was performed with an inverted microscope and a polarized light filter (magnification ×100). The micrometer presented is a Nikon micrometer, type MBM 11100 (stage micrometer type A), of 1mm total length and graduations of 0.01 mm (=100 μm). The micrometer is in the same magnification (×100) as the microscopy image [43].

**Figure 5.**The time-series presentation of the cell population CCRF-CEM with respect to cell concentration (cells/μL) and in particular, for initial cell population of ~20 cells/μL (

**A**), ~200 cells/μL (

**B**). Further on, the two time-series are presented together in one diagram for comparison (

**C**) (n

_{0}: the initial population).

**Figure 6.**The time-series presentation of the cell population CCRF-CEM with respect to total cell concentration for initial cell populations of ~20 cells/μL (

**A**) and ~200 cells/μL (

**B**) (n

_{0}: the initial population. The arrows indicate the time-points of cell culture media replacement).

**Figure 7.**The time-series presentation of the cell population’s viability (

**A**) and total cell death (

**B**) of the CCRF-CEM cells.

**Figure 8.**The cell population’s cell cycle. The cell cycle of our cellular system manifested oscillatory patterns for G

_{1}(

**A**), G

_{2}(

**B**) and S-phase (

**C**). In addition, a scatter plot of the three cell cycle phases in all combinations showed a discrete separation of the three cell cycle phases (

**D**).

**Figure 10.**Calculations of the Lyapunov exponent (λ) for initial population 20 cells/μL and 200 cells/μL. We have calculated the Δ

_{0}as Δ

_{0}= |x

_{0}− x

_{0′}|, where x

_{0}= 20/max and x

_{0′}= 200/max. The maximum population was considered to be the max population reached for each experiment in cells/μL.

**Figure 11.**The phase-space steps of the experiment for x

_{0}= 20 cells/μL. The red arrows indicate the course of the trajectory step-by-step. The x-axis corresponds to the population of cells at time t and the y-axis corresponds to the population of cells at time t + 1. Each of the subfigures (

**A**–

**W**) presents the phase-space recurrences from the first (

**A**) to the 23rd (

**W**).

**Figure 12.**The phase-space of the experiment for x

_{0}= 20 cells/μL. The red arrows indicate the course of the trajectory step-by-step. In addition, in all subfigures the diagonal is drawn, which shows the Poincaré map crossings. The x-axis corresponds to the population of cells at time t and the y-axis corresponds to the population of cells at time t + 1. The system converged in the first convolution of the trajectory (

**A**,

**B**) as well as after cell culture media supplementation (

**E**). In the other cases (

**C**,

**D**,

**F**–

**K**) diverged from equilibrium.

**Figure 13.**The Poincaré map cross-sections. We have presented all sections of the trajectory on the diagonal (

**A**), along with the transmission points of S→S′ (

**B**), as well as the respective points with their mapping from S onto S′ (

**C**). Finally, the 3D representation of the cross-sections along with the cell populations is presented (

**D**). The x-axis corresponds to the population of cells at time t and the y-axis corresponds to the population of cells at time t + 1.

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**MDPI and ACS Style**

Adamopoulos, K.; Koutsouris, D.; Zaravinos, A.; Lambrou, G.I.
Poincaré Maps and Aperiodic Oscillations in Leukemic Cell Proliferation Reveal Chaotic Dynamics. *Cells* **2021**, *10*, 3584.
https://doi.org/10.3390/cells10123584

**AMA Style**

Adamopoulos K, Koutsouris D, Zaravinos A, Lambrou GI.
Poincaré Maps and Aperiodic Oscillations in Leukemic Cell Proliferation Reveal Chaotic Dynamics. *Cells*. 2021; 10(12):3584.
https://doi.org/10.3390/cells10123584

**Chicago/Turabian Style**

Adamopoulos, Konstantinos, Dimitis Koutsouris, Apostolos Zaravinos, and George I. Lambrou.
2021. "Poincaré Maps and Aperiodic Oscillations in Leukemic Cell Proliferation Reveal Chaotic Dynamics" *Cells* 10, no. 12: 3584.
https://doi.org/10.3390/cells10123584