Alternating regimes of motion in a model with cell-cell interactions

Nara Guisoni, Karina I. Mazzitello, and Luis Diambra

Phys. Rev. E 101, 062408 – Published 11 June 2020

Abstract

Cellular movement is a complex dynamic process, resulting from the interaction of multiple elements at the intra- and extracellular levels. This epiphenomenon presents a variety of behaviors, which can include normal and anomalous diffusion or collective migration. In some cases, cells can get neighborhood information through chemical or mechanical cues. A unified understanding about how such information can influence the dynamics of cell movement is still lacking. In order to improve our comprehension of cell migration we have considered a cellular Potts model where cells move actively in the direction of a driving field. The intensity of this driving field is constant, while its orientation can evolve according to two alternative dynamics based on the Ornstein-Uhlenbeck process. In one case, the next orientation of the driving field depends on the previous direction of the field. In the other case, the direction update considers the mean orientation performed by the cell in previous steps. Thus, the latter update rule mimics the ability of cells to perceive the environment, avoiding obstacles and thus increasing the cellular displacement. Different cell densities are considered to reveal the effect of cell-cell interactions. Our results indicate that both dynamics introduce temporal and spatial correlations in cell velocity in a friction-coefficient and cell-density-dependent manner. Furthermore, we observe alternating regimes in the mean-square displacement, with normal and anomalous diffusion. The crossovers between diffusive and directed motion regimes are strongly affected by both the driving field dynamics and cell-cell interactions. In this sense, when cell polarization update grants information about the previous cellular displacement, the duration of the diffusive regime decreases, particularly in high-density cultures.

Received 30 December 2019
Accepted 26 May 2020

DOI:https://doi.org/10.1103/PhysRevE.101.062408

Physics Subject Headings (PhySH)

Anomalous diffusion Brownian motion Cell locomotion Cell migration Diffusion Random walks

Monte Carlo methods

Statistical Physics & Thermodynamics

Authors & Affiliations

Nara Guisoni ^1,*, Karina I. Mazzitello ^2,†, and Luis Diambra ^3,‡

¹Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Universidad Nacional de La Plata, CONICET, 1900 La Plata, Buenos Aires, Argentina
²Instituto de Investigaciones Científicas y Tecnológicas en Electrónica, Universidad Nacional de Mar del Plata, CONICET, B7608 Mar del Plata, Buenos Aires, Argentina
³Centro Regional de Estudios Genómicos, Universidad Nacional de La Plata, CONICET, 1900 La Plata, Buenos Aires, Argentina

^*naraguisoni@gmail.com
^†kmazzite@gmail.com
^‡ldiambra@gmail.com

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Vol. 101, Iss. 6 — June 2020

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Figure 1
A log-log plot of the mean-square displacement versus time (distance in units of average cell diameter $d$ and time in units of $τ$ ). The direction of the driving field is actualized according to the Ornstein-Uhlenbeck process (OU naive) and the Ornstein-Uhlenbeck process with feedback (OU $+$ feedback) for (a) $λ = 0.01$ and (b) $λ = 0.10$ , with $ρ = 0.2$ (dashed line) and $ρ = 0.9$ (solid line). Lines with slopes equal to 2 (ballistic) and 1 (random walk) are shown for comparison purposes only. The MSD is calculated between the starting point at $t = 0$ and the actual positions at time $t$ and averaged over all $Q$ cells of the simulation. More details are given in the text.
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Figure 2
Rescaled MSD of the curves of the inset, for results obtained from OU naive actualization model at (a) low and (b) high cell density. Scaling functions are defined as $γ = 1 / ln [MSD (t_{c 2})] = 1 / ln (a λ^{- h})$ and $χ = 1 / ln (t_{c 2} / t_{c 1})$ , with constant values (a) $a = 3.0 \times 10^{- 5}$ , $h = 6.4$ , $t_{c 1} = 9$ (units of $τ$ ), $t_{c 2} = 4.04 \times 10^{- 4} λ^{- h}$ (units of $τ$ ), $c = 0.015$ , and $\bar{β} = 1.8$ and (b) $a = 3.5 \times 10^{- 5}$ , $h = 5.6$ , $t_{c 1} = 8$ (units of $τ$ ), $t_{c 2} = 1.26 \times 10^{- 3} λ^{- h}$ (units of $τ$ ), $c = 0.0086$ , and $\bar{β} = 1.6$ . The insets correspond to the log-log plot of the MSD versus time (distance in units of average cell diameter $d$ and time in units of $τ$ ), for results obtained from the OU naive actualization model at (a) low and (b) high cell density. The curves correspond to different values of the friction coefficient: $λ = 0.01, 0.05, 0.07, 0.08, 0.09, 0.10$ (violet, orange, blue, green, red, and black, respectively).
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Figure 3
Logarithmic derivative of the MSD $β (t)$ versus time (in units of $τ$ ), calculated from data shown in Fig. 1 and additional data. The direction of the driving field is actualized according to (a) the OU naive process and (b) the OU process with feedback for different values of $λ = 0.01, 0.05, 0.10$ (dark, red, and blue, respectively) and $ρ = 0.2, 0.9$ (dashed and solid lines, respectively). We used a time-sliding window, the size of which depends on time. Successive measurements of $β$ are made in the time interval $[⌊ 1 . 775^{k} ⌋, ⌊ 1 . 775^{(k + 4)} ⌋]$ in time units of MCSs, with $k = 1, 1.2, 1.4, ..., 16$ , since the MSD data are considered until $t = 10^{5}$ MCSs. Here $⌊ \dots ⌋$ indicates the greatest integer less than or equal to the argument.
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Figure 4
Semilogarithmic plot of the velocity ACF function $C (t)$ versus time (in units of $τ$ ). The direction of the driving field is actualized according to the OU naive process and the OU process with feedback, for different densities $ρ = 0.2, 0.9$ (dashed and solid lines, respectively) and friction coefficients (a) $λ = 0.01$ and (b) $λ = 0.05$ . Here $C (t)$ was averaged over all $Q$ cells of the simulation and over $t_{0}$ for each simulation sample. We consider three samples for $ρ = 0.2$ and one for $ρ = 0.9$ , each with $6 \times 10^{4}$ MCSs. More details are given in the text.
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Figure 5
Spatial correlation function of the velocities $C (r)$ as a function of the distance $r$ between cell pairs (in units of average cell diameter $d$ ). The direction of the driving field is actualized according to the OU naive process and the OU process with feedback, for different densities $ρ = 0.2, 0.9$ (dashed and solid lines, respectively) and friction coefficients (a) $λ = 0.01$ and (b) $λ = 0.10$ . Here $C (r)$ is averaged over all $Q$ cells of the simulation and over time until $t = 10^{5}$ MCSs. More details are given in the text.
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Figure 6
(a) Average distance traveled by a cell during the time interval that the driving field is operating, $\bar{D}$ (in units of average cell diameter $d$ ), for densities $ρ = 0.2$ and 0.9. The direction of the driving field is actualized according to the OU naive process and the OU process with feedback (black and red, respectively) with friction coefficients $λ = 0.01$ and 0.10 (different color patterns, as indicated). Histograms of the direction of the cell movement $α$ are shown for different conditions: (b) $ρ = 0.2$ and $λ = 0.10$ , (c) $ρ = 0.9$ and $λ = 0.10$ , (d) $ρ = 0.2$ and $λ = 0.01$ , and (e) $ρ = 0.9$ and $λ = 0.01$ . The update of the direction of the driving field is according to labels in (a). Here $\bar{D}$ is averaged over all $Q$ cells in the simulations until $10^{5}$ MCSs (the first 500 MCSs are disregarded). In addition, $α$ was calculated for simulations until $5 \times 10^{4}$ MCSs for $ρ = 0.2$ and $10^{4}$ MCSs for $ρ = 0.9$ , in order to have similar quantity of data, that is, $Q$ times the number of time steps. Error bars are smaller than the line thickness.
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