# Matrix Stiffness Modulates Mechanical Interactions and Promotes Contact between Motile Cells

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Observations Motivate Model for Cell Elastic Interactions

## 3. Materials and Methods

#### 3.1. Model for Two-Cell Interactions

#### 3.2. Dimensionless Parameters Quantifying Cell Motion and Interactions

^{2}/minute to 50 $\mathsf{\mu}$m

^{2}/minute. Time scales are estimated from experiments as well and 250 s in real time correspond to a dimensionless time duration of unity.

#### 3.3. Numerical Solution and Tracking Cell Trajectories

## 4. Results

#### 4.1. Cell–Cell Contact Frequency Is Controlled by Matrix Elastic Interactions

#### 4.2. Cell Motility Characteristics Depend on Elastic Interactions

#### 4.3. Elastic Interactions Lead to Effective Capture of Motile Cell

## 5. Discussion and Future Extensions to Other Forms of Interactions

#### 5.1. Anisotropic Cell-Cell Elastic Interactions

#### 5.2. Extensions to Near-Contact Biochemical or Bond Interactions

## 6. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MSD | Mean Square Displacement |

## Appendix A Model for a Moving Cell Interacting with a Stationary Cell Via Substrate Elasticity

## References

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**Figure 1.**Traction forces that are exerted by motile cells on soft substrates can be modeled as force dipoles. (

**a**) Schematic of an animal cell, e.g., an isolated fibroblast in culture [17] adhered to a compliant substrate through focal adhesions comprising of integrins and a host of other mechanosensitive adhesion proteins. Mechanical forces are actively generated by molecular motors of the myosin-II family that put the actin cytoskeleton under tension. Stress fibers are bundles of crosslinked actin filaments with a periodic (sarcomeric) organization of myosin [44] that often span the length of the cell and are anchored at the focal adhesions to the extracellular substrate or matrix (ECM). The contractile forces are transmitted at these sites from the stress fibers to the underlying substrate, which can be strongly deformed if soft. Such deformations are long range (they extend up to a few cell lengths away from the cell) and they can be measured by Traction Force Microscopy (TMF). This is a common index of cell–substrate mechanical interactions. (

**b**) (i) A simplified top view of the same cell showing the alignment of the stress fibers and, therefore, of the contractile forces that are generated by them. In order to model the effects of the cell on the substrate, we use classical linear elasticity theory with the stress distributions effectively modeled as a contractile force dipole, a pair of equal and opposite forces separated by some distance acting along the dipole axis (marked). (ii) This leads to a very simplified mechanical model of the cell in terms as a contractile force dipole that is exerted on the substrate along its average axis of orientation defined by the alignment of its stress fibers.

**Figure 2.**Schematic of the cell–cell mechanical interactions model: (

**A**) Two cells A and B cultured on the surface of thick elastic substrate can sense each other and interact at long range (when the inter-cell distance r is longer than typical cell sizes, depicted here by dashed red circles) through mechanical deformations of the underlying substrate; here the contractile stresses set up in the substrate yield deformations as indicated by green arrows. The cells are restricted to move on the surface of the substrate. (

**B**) We study with our computational model how a motile cell (M, Cell A, pink) moves in the presence of a fixed central cell (Cell B, yellow). This two cell system on a substrate (schematic shown as a top view) also mimics scenarios where a motile cell may encounter an elastic impurity or obstacle on the medium. Shown in blue circles are contours of constant elastic potential (in simplified form) that determine the inter-cell elastic force that is experienced by the motile cell B as a result of the elastic deformations of the medium by both cells A and B. Also shown (in black) is a representative simulated trajectory of the motile cell which starts outside the area of influence of the stationary cell.

**Figure 3.**The number of cell–cell contact events measured in a fixed interval of time depends strongly on the elastic interaction parameter. A contact event is identified as cell A coming within a prescribed contact radius of cell B with cell A initialized randomly in a certain area around cell B. Thus the number of contact is be interpreted as the average number of contacts of the two cells. The number of simulation runs conducted were 50 for each combination of ${D}_{T}$ and $\alpha $. The dashed curves are guides to the eye illustrating the trends seen with increasing values of $\alpha $. Diffusion is the major factor in governing the number of contacts for low values of $\alpha $. For higher $\alpha $, the attractive potential increases the probability of the cell to stay near the contact radius and controls the number of contacts. The trajectories for highlighted data points (1)–(4) are shown on the right. The box plots show the distribution of contact numbers. The lower and upper bounds of the box are the first and the third quartiles respectively, while the line in middle is the median. The lower and upper limits of the dashed lines are the minimum and maximum number of contacts observed for cells for each combination of $\alpha $ and ${D}_{T}$. The simulation was run for a total time of $T=1000$ and updates in the cell position were made every $\delta t=0.001$.

**Figure 4.**The number of cell–cell contact events in a fixed interval of time ($T=1000$) plotted here as a function of the scaled effective diffusivity, ${D}_{T}$, which represents the random motility of cell B. Here, we show how the number of cell–cell contact varies for three different elastic interaction strength values, $\alpha $, corresponding to substrates with three different stiffness. The highlighted points numbered from (1)–(4), show representative cell trajectories over long times and highlight how varying $\alpha $ and ${D}_{T}$ can yield states where the cells are in close proximity most of the time (low ${D}_{T}$, high $\alpha $) or states where cells interact rarely (high ${D}_{T}$, low $\alpha $). The interpretation of the box plots is the same as in Figure 2. The simulation was run for a total time of $T=1000$ and updates in cell position were made every $\delta t=0.001$.

**Figure 5.**Mean square displacement (MSD) as a function of the delay time interval $\tau $ (calculated from Equation (9)), for the motile cell A is shown. Here we explore the variation in the MSD for various values of substrate-mediated elastic interactions, $\alpha $. The diffusivity ${D}_{T}$ is held constant for these simulations with ${D}_{T}=2$. Other diffusivities were explored (results not shown). At low elastic interaction strengths, $\alpha $, corresponding to stiff substrates, the cell shows a purely diffusive trajectory, whereas at higher values of $\alpha $, the motile cell is captured by the strong attractive interaction from the stationary cell, resulting in a flattening of the MSD (blue curve). At an intermediate interaction regime (green curve), the motile cell makes repeated contact with the fixed cell, but it is never fully captured.

**Figure 6.**Capture statistics of motile cell. (

**A**) Probability that cell B is inside contact radius as a function of time. (

**B**,

**C**) The dependence of steady state capture probability, ${P}_{ss}$, i.e., the fraction of cells captured within the contact radius after a long time interval, on simulation parameters. (

**B**) shows the dependence on diffusivity, ${D}_{T}$ at different values of the elastic interaction parameter, $\alpha $, whereas (

**C**) shows the dependence on $\alpha $ for different values of ${D}_{T}$. (

**D**) The steady state capture probability, ${P}_{ss}$, data can be collapsed into a single master curve, when plotted vs. the key parameter, $alpha/{D}_{T}$, the strength of the elastic interactions relative to the diffusivity. This is expected since our model steady state is a thermal equilibrium with the effective temperature set by the noisy cell motility, ${D}_{T}$, and the competition between attractive interactions and noise dictates the number of cells (cell trajectories) captured vs. the number that escape.

**Figure 7.**Dipolar cell orientation and trajectory. The equilibrium orientation of contractile cells fixed in position, but free to reorient, and that are uniformly distributed in a square box of size $10\sigma $, are depicted by two arrows (red) pointing towards each other. Each cell is influenced by the central stationary cell B (green) and not by each other. Two possible trajectories of cell A (blue and black) are recorded for ${D}_{T}=0.1,\phantom{\rule{0.222222em}{0ex}}\alpha =40$ for total time $T=500$ with time steps of $dt=0.001$. The cells did not have any self propulsion or rotational diffusion. The Poisson’s ratio $\nu $ of the substrate was considered to be 0.3 for this simulation.

Quantity | Interpretation | Experimental Values |
---|---|---|

$\sigma $ | Cell size | 10–100 $\mathsf{\mu}$m |

T | Temperature | 25 °C |

${D}_{0}$ | Thermal Diffusivity | 25 $\mathsf{\mu}$m^{2}/min |

${D}_{\mathrm{eff}}$ | Effective Diffusivity | 3–50 $\mathsf{\mu}$m^{2}/min |

E | Young’s modulus | 0.5–33 kPa |

$\nu $ | Poisson ratio | 0.3–0.5 |

P | Contractility | ${10}^{-14}$ Nm |

Parameter | Interpretation | Definition | Simulation Values |
---|---|---|---|

${D}_{T}$ | Diffusivity | ${D}_{\mathrm{eff}}/{D}_{0}$ | 0.1–10 |

$\alpha $ | Cell-cell interaction | ${P}^{2}\varphi \left(\nu \right)/\left(E{k}_{B}T{\sigma}^{3}\right)$ | 0.1–100 |

${k}_{\mathrm{steric}}$ | Self-avoidance | $k{\sigma}^{2}/{k}_{B}T$ | ${10}^{3}$–${10}^{4}$ |

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**MDPI and ACS Style**

Bose, S.; Dasbiswas, K.; Gopinath, A.
Matrix Stiffness Modulates Mechanical Interactions and Promotes Contact between Motile Cells. *Biomedicines* **2021**, *9*, 428.
https://doi.org/10.3390/biomedicines9040428

**AMA Style**

Bose S, Dasbiswas K, Gopinath A.
Matrix Stiffness Modulates Mechanical Interactions and Promotes Contact between Motile Cells. *Biomedicines*. 2021; 9(4):428.
https://doi.org/10.3390/biomedicines9040428

**Chicago/Turabian Style**

Bose, Subhaya, Kinjal Dasbiswas, and Arvind Gopinath.
2021. "Matrix Stiffness Modulates Mechanical Interactions and Promotes Contact between Motile Cells" *Biomedicines* 9, no. 4: 428.
https://doi.org/10.3390/biomedicines9040428