Figures and data in Bridging the gap between single-cell migration and collective dynamics

Version of Record: January 30, 2020
Accepted Manuscript: December 6, 2019

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Florian Thüroff

Andriy Goychuk

Matthias Reiter

Erwin Frey

(2019)
Bridging the gap between single-cell migration and collective dynamics

eLife 8:e46842.

https://doi.org/10.7554/eLife.46842
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Figures
Tables
Additional files

10 figures, 1 table and 2 additional files

Figures

Figure 1

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Illustration of the computational model with the pertinent simulation steps.

(A) Illustration of a small cell cohort that adheres to a surface ( $(x, y)$ -plane). The polarization field, $ϵ$ , is defined on the contact surface with the adhesion plane. The magnitude of the polarization …

Figure 2 with 2 supplements

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Cell shape and persistence of migration as a function of cell polarizability.

(A) Mean-squared displacement (MSD) for single-cell movements at different maximum cell polarity $Δ ϵ$ (stiffness parameters $κ_{P} = 0.060$ , $κ_{A} = 0.18$ ; average polarization field $ϵ_{0} = 225$ ; signaling radius $R = 5$ ; cell-substrate …

Figure 2—figure supplement 1

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Role of substrate dissipation for cell shape and motility.

(**A–D**) Role of substrate dissipation for cells of varying maximum cell polarity $Δ ϵ$ . The aspect ratio $l_{+} / l_{-}$ (A), the speed $v$ (B), and the persistence time $τ_{p}$ (C) as a function of substrate dissipation $D$ …

Figure 2—video 1

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posterframe for video — Single cell motility and shape for different maximum cell polarities ( $κ_{P} = 0.060$ , $R = 5$ ).

Figure 3 with 1 supplement

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Migratory behavior of single cells as a function of the cell’s signaling radius $R$ at different values for the maximal cytoskeletal polarity $Δ ϵ$ .

(Stiffness parameters $κ_{P} = 0.060$ , $κ_{A} = 0.18$ ; average polarization field $ϵ_{0} = 225$ ; cell-substrate dissipation $D = 0$ ; cell-substrate adhesion penalty $φ = 0$ ; cytoskeletal update rate $μ = 0.1$ ; $100$ independent simulations for each set …

Figure 3—video 1

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Figure 4 with 6 supplements

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Phases of collective motion.

( $4$ -cell systems; confinement radius $r_{0} = 30.6$ ; area stiffness $κ_{A} = 0.18$ ; average polarization field $ϵ_{0} = 225$ ; signaling radius $R = 5$ ; cytoskeletal update rate $μ = 0.1$ ; cell-cell adhesion $B = 0$ ; cell-cell dissipation $Δ B = 12$ ; …

Figure 4—figure supplement 1

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Collective motion for varying number of cells at low polarizability.

( $N$ -cell systems; confinement radius $r_{0} = \sqrt{234 N}$ ; stiffness parameters $κ_{P} = 0.060$ , $κ_{A} = 0.18$ ; average polarization field $ϵ_{0} = 225$ ; maximum cell polarity $Δ ϵ = 28$ ; signaling radius $R = 5$ ; cytoskeletal update rate $μ = 0.1$ ; cell-cell adhesion $B = 0$ ;…

Figure 4—figure supplement 2

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Collective motion for varying number of cells at intermediate polarizability.

( $N$ -cell systems; confinement radius $r_{0} = \sqrt{234 N}$ ; stiffness parameters $κ_{P} = 0.060$ , $κ_{A} = 0.18$ ; average polarization field $ϵ_{0} = 225$ ; maximum cell polarity $Δ ϵ = 50$ ; signaling radius $R = 5$ ; cytoskeletal update rate $μ = 0.1$ ; cell-cell adhesion $B = 0$ ;…

Figure 4—figure supplement 3

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Collective motion for varying number of cells at high polarizability.

( $N$ -cell systems; confinement radius $r_{0} = \sqrt{234 N}$ ; stiffness parameters $κ_{P} = 0.060$ , $κ_{A} = 0.18$ ; average polarization field $ϵ_{0} = 225$ ; maximum cell polarity $Δ ϵ = 70$ ; signaling radius $R = 5$ ; cytoskeletal update rate $μ = 0.1$ ; cell-cell adhesion $B = 0$ ;…

Figure 4—video 1

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Figure 4—video 2

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Figure 4—video 3

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Figure 5 with 3 supplements

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Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at $x = \pm 175$ for two different parameter settings.

(Stiffness parameters $κ_{P} = 0.12$ , $κ_{A} = 0.18$ ; average polarization field $ϵ_{0} = 35$ ; signaling radius $R = 2$ ; cytoskeletal update rate $μ = 0.1$ ; cell-cell adhesion $B = 12$ ; cell-cell dissipation $Δ B = 0$ ; cell-substrate dissipation $D = 0$ ; cell-sub…

Figure 5—figure supplement 1

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Monolayer expansion depends on dissipation and cell polarizability.

Cell monolayer expansion depends on the cell-cell dissipation, cell-substrate dissipation, and maximum cell polarity. (Initially a $2500$ -cell system; stiffness parameters $κ_{P} = 0.12$ , $κ_{A} = 0.18$ ; average polarization …

Figure 5—video 1

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Figure 5—video 2

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Figure 6 with 2 supplements

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Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at $x = \pm 175$ for two different parameter settings that produce rough tissue fronts.

(Initially a $2500$ -cell system; stiffness parameters $κ_{P} = 0.10$ , $κ_{A} = 0.18$ ; average polarization field $ϵ_{0} = 35$ ; maximum cell polarity $Δ ϵ = 20$ ; signaling radius $R = 5$ ; cytoskeletal update rate $μ = 0.1$ ; cell-cell adhesion $B = 5$ ; cell-cell …

Figure 6—video 1

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Figure 6—video 2

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Appendix 1—figure 1

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Illustration of the various sets defining a cell and its environment.

Grid sites occupied by cell $α$ , i.e. its domain $𝒟^{(α)}$ , are indicated in red colors. The cell’s membrane sites, $ℬ^{(α)}$ , are indicated by the lighter red color, the cell’s immediate neighborhood, $𝒩^{(α)}$ , is …

Appendix 1—figure 2

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Distribution of regulatory factors on the basis of accepted elementary events.

For ease of reference, grid rows have been numbered from $1$ to $10$ . *Left* (A): Solid black lines indicate cells’ membrane positions *after* acceptance of the respective elementary event; colors indicate …

Appendix 1—figure 3

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Cell-cell adhesion.

(A) Adhesive energy contribution in a cyclic process, where a protrusion of source cell $α$ against target cell $β$ is followed by the inverse retraction event. Both events involve a third party cell $γ$ , leading to net energy dissipation after the cyclic process has been completed. *Protrusion*: (i) Three pre-existing cell-cell contacts between $β$ and $γ$ are torn apart (red dashed contacts); (ii) three new contacts between $α$ and $γ$ are formed; (iii) the contact length between source cell $α$ and target cell $β$ increases by one unit of length. This implies $Δ ℋ_{adh} (𝒯_{pro}) = ℓ (3 B_{β, γ}^{'} - 3 B_{α, γ} - B_{α, β})$ . *Retraction*: (i) Three pre-existing cell-cell contacts between $α$ and $γ$ are torn apart (red dashed contacts); (ii) three new contacts between $β$ and $γ$ are formed; (iii) the contact length between source cell $α$ and target cell $β$ decreases by one unit of length. This implies $Δ ℋ_{adh} (𝒯_{ret}) = ℓ (3 B_{α, γ}^{'} - 3 B_{β, γ} + B_{α, β})$ . Altogether, this leads to $Δ ℋ_{adh}^{(cycl)} = Δ ℋ_{adh} (𝒯_{pro}) + Δ ℋ_{adh} (𝒯_{ret}) = ℓ (3 (Δ B)_{α, γ} + 3 (Δ B)_{β, γ}) \geq 0$ , i.e. a (non-negative) dissipative contribution, whose magnitude depends on the dissipation matrix $(Δ B)_{α, β} = B_{α, β}^{'} - B_{α, β} \geq 0$ . (B) Shear viscosity due to cell-cell adhesion. Consider two rows of adhesive cells sliding past each other as indicated in the figure (left row of cells moves up by one grid site; colors indicate different cells). The associated adhesion energy change (per cell) reads $Δ ℋ_{adh} / n_{c} = 2 (B^{'} - B) \geq 0$ , where $n_{c}$ denotes the number of cells sliding past each other, and where we assumed cells of like type, i.e. $B_{α, β} \equiv B$ and $B_{α, β}^{'} \equiv B^{'}$ ( $α \neq β$ ). The condition $B^{'} > B$ , Equation S15e, thus implies positive friction associated with cellular shear flows, whose magnitude is proportional to the number of cells sliding past each other. Note that this shear viscosity vanishes for $B^{'} = B$ , i.e. for zero dissipation matrix.

Appendix 2—figure 1

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Role of area stiffness $κ_{A}$ for cell size and motility.

(A) The cell area increases linearly with $1 / κ_{A}$ . The aspect ratio (B), speed (C) and persistence (D) of the cell decrease with increasing cell size. In the simulations, the area elasticity was varied …

Tables

Table 1

Source and parameter files used for each figure.

All source and parameter files are found in Source data 1.

Figure	Simulation code	Processing code	Parameters
Figure 2	CPM_NoDivision	TrajectoryAnalysisSingle	single_Q
Figure 2—figure supplement 1 (A-D)	CPM_NoDivision	TrajectoryAnalysisSingle	single_DQ
Figure 2—figure supplement 1 (E-H)	CPM_NoDivision	TrajectoryAnalysisSingle	single_DM
Figure 3	CPM_NoDivision	TrajectoryAnalysisSingle	single_R
Figure 4	CPM_NoDivision	TrajectoryAnalysisCircularPattern	rotation_Q
Figure 4—figure supplement 1	CPM_NoDivision	TrajectoryAnalysisCircularPattern	rotation_N_R1
Figure 4—figure supplement 2	CPM_NoDivision	TrajectoryAnalysisCircularPattern	rotation_N_R2
Figure 4—figure supplement 3	CPM_NoDivision	TrajectoryAnalysisCircularPattern	rotation_N_R3
Figure 5 (A-D)	CPM_Division		wound_nodiv
Figure 5 (E-H)	CPM_Division		wound_div
Figure 5—figure supplement 1 (A-B)	CPM_Division_Supplement	FrontAnalysis	wound_div_A
Figure 5—figure supplement 1 (C-D)	CPM_Division_Supplement	FrontAnalysis	wound_div_D
Figure 5—figure supplement 1 (E, F)	CPM_Division_Supplement	FrontAnalysis	wound_div_Q
Figure 6 (A-D)	CPM_Division		wound_div_fing_1.0
Figure 6 (E-H)	CPM_Division		wound_div_fing_1.1
Appendix 2—figure 1	CPM_NoDivision	TrajectoryAnalysisSingle	single_A

Additional files

Source code 1 Simulation code, processing code and parameter files.: https://cdn.elifesciences.org/articles/46842/elife-46842-code1-v2.tar
Download elife-46842-code1-v2.tar
Transparent reporting form: https://cdn.elifesciences.org/articles/46842/elife-46842-transrepform-v2.pdf
Download elife-46842-transrepform-v2.pdf

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Figures

Illustration of the computational model with the pertinent simulation steps.

Cell shape and persistence of migration as a function of cell polarizability.

Role of substrate dissipation for cell shape and motility.

Single cell motility and shape for different maximum cell polarities ( $κ_{P} = 0.060$ , $R = 5$ ).

Migratory behavior of single cells as a function of the cell’s signaling radius $R$ at different values for the maximal cytoskeletal polarity $Δ ϵ$ .

Single cell motility and shape for different signaling radii ( $Δ ϵ = 60$ , $κ_{P} = 0.060$ ).

Phases of collective motion.

Collective motion for varying number of cells at low polarizability.

Collective motion for varying number of cells at intermediate polarizability.

Collective motion for varying number of cells at high polarizability.

Collective rotations of $4$ cells in the $ℛ_{1}$ -phase ( $Δ ϵ = 28$ ; $Δ ϵ / κ_{P} = 467$ ).

Collective rotations of $4$ cells in the $ℛ_{2}$ -phase ( $Δ ϵ = 50$ ; $Δ ϵ / κ_{P} = 833$ ).

Collective rotations of $4$ cells in the $ℛ_{3}$ -phase ( $Δ ϵ = 70$ ; $Δ ϵ / κ_{P} = 1167$ ).

Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at $x = \pm 175$ for two different parameter settings.

Monolayer expansion depends on dissipation and cell polarizability.

Motility-dominated tissue dynamics.

Proliferation-dominated tissue dynamics.

Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at $x = \pm 175$ for two different parameter settings that produce rough tissue fronts.

Weak monolayer roughening (fingering) in motility-dominated tissue with quick proliferation.

Strong monolayer roughening in motility-dominated tissue with slow proliferation.

Illustration of the various sets defining a cell and its environment.

Distribution of regulatory factors on the basis of accepted elementary events.

Cell-cell adhesion.

Role of area stiffness $κ_{A}$ for cell size and motility.

Tables

Source and parameter files used for each figure.

Additional files

Source code 1

Transparent reporting form

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Illustration of the computational model with the pertinent simulation steps.

Cell shape and persistence of migration as a function of cell polarizability.

Role of substrate dissipation for cell shape and motility.

Single cell motility and shape for different maximum cell polarities (κP=0.060<math id="inf131"><mrow><msub><mi>κ</mi><mi>P</mi></msub><mo>=</mo><mn>0.060</mn></mrow></math>, R=5<math id="inf132"><mrow><mi>R</mi><mo>=</mo><mn>5</mn></mrow></math>).

Migratory behavior of single cells as a function of the cell’s signaling radius R<math id="inf137"><mi>R</mi></math> at different values for the maximal cytoskeletal polarity Δ⁢ϵ<math id="inf138"><mrow><mi mathvariant="normal">Δ</mi><mo>⁢</mo><mi>ϵ</mi></mrow></math>.

Phases of collective motion.

Collective motion for varying number of cells at low polarizability.

Collective motion for varying number of cells at intermediate polarizability.

Collective motion for varying number of cells at high polarizability.

Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at x=±175<math id="inf303"><mrow><mi>x</mi><mo>=</mo><mrow><mo>±</mo><mn>175</mn></mrow></mrow></math> for two different parameter settings.

Monolayer expansion depends on dissipation and cell polarizability.

Motility-dominated tissue dynamics.

Proliferation-dominated tissue dynamics.

Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at x=±175<math id="inf357"><mrow><mi>x</mi><mo>=</mo><mrow><mo>±</mo><mn>175</mn></mrow></mrow></math> for two different parameter settings that produce rough tissue fronts.

Weak monolayer roughening (fingering) in motility-dominated tissue with quick proliferation.

Strong monolayer roughening in motility-dominated tissue with slow proliferation.

Illustration of the various sets defining a cell and its environment.

Distribution of regulatory factors on the basis of accepted elementary events.

Cell-cell adhesion.

Role of area stiffness κA<math id="inf852"><msub><mi>κ</mi><mi>A</mi></msub></math> for cell size and motility.

Source and parameter files used for each figure.

Source code 1

Transparent reporting form

Single cell motility and shape for different maximum cell polarities ( $κ_{P} = 0.060$ , $R = 5$ ).

Migratory behavior of single cells as a function of the cell’s signaling radius $R$ at different values for the maximal cytoskeletal polarity $Δ ϵ$ .

Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at $x = \pm 175$ for two different parameter settings.

Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at $x = \pm 175$ for two different parameter settings that produce rough tissue fronts.

Role of area stiffness $κ_{A}$ for cell size and motility.