Bridging the gap between single-cell migration and collective dynamics

  1. Florian Thüroff
  2. Andriy Goychuk
  3. Matthias Reiter
  4. Erwin Frey  Is a corresponding author
  1. Ludwig-Maximilians-Universität München, Germany
10 figures, 1 table and 2 additional files

Figures

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
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 Δϵ (stiffness parameters κP= 0.060, κA= 0.18; average polarization field ϵ0= 225; signaling radius R= 5; cell-substrate …

Figure 2—figure supplement 1
Role of substrate dissipation for cell shape and motility.

(A–D) Role of substrate dissipation for cells of varying maximum cell polarity Δϵ. The aspect ratio l+/l (A), the speed v (B), and the persistence time τp (C) as a function of substrate dissipation D

Figure 2—video 1
Single cell motility and shape for different maximum cell polarities (κP=0.060, R=5).
Figure 3 with 1 supplement
Migratory behavior of single cells as a function of the cell’s signaling radius R at different values for the maximal cytoskeletal polarity Δϵ.

(Stiffness parameters κP=0.060κA=0.18; average polarization field ϵ0=225; cell-substrate dissipation D=0; cell-substrate adhesion penalty φ=0; cytoskeletal update rate μ=0.1100 independent simulations for each set …

Figure 3—video 1
Single cell motility and shape for different signaling radii (Δϵ=60, κP=0.060).
Figure 4 with 6 supplements
Phases of collective motion.

(4-cell systems; confinement radius r0=30.6; area stiffness κA=0.18; average polarization field ϵ0=225; signaling radius R=5; cytoskeletal update rate μ=0.1; cell-cell adhesion B=0; cell-cell dissipation ΔB=12; …

Figure 4—figure supplement 1
Collective motion for varying number of cells at low polarizability.

(N-cell systems; confinement radius r0=234N; stiffness parameters κP=0.060, κA=0.18; average polarization field ϵ0=225; maximum cell polarity Δϵ=28; signaling radius R=5; cytoskeletal update rate μ=0.1; cell-cell adhesion B=0;…

Figure 4—figure supplement 2
Collective motion for varying number of cells at intermediate polarizability.

(N-cell systems; confinement radius r0=234N; stiffness parameters κP=0.060, κA=0.18; average polarization field ϵ0=225; maximum cell polarity Δϵ=50; signaling radius R=5; cytoskeletal update rate μ=0.1; cell-cell adhesion B=0;…

Figure 4—figure supplement 3
Collective motion for varying number of cells at high polarizability.

(N-cell systems; confinement radius r0=234N; stiffness parameters κP=0.060, κA=0.18; average polarization field ϵ0=225; maximum cell polarity Δϵ=70; signaling radius R=5; cytoskeletal update rate μ=0.1; cell-cell adhesion B=0;…

Figure 4—video 1
Collective rotations of 4 cells in the 1-phase (Δϵ=28Δϵ/κP=467).
Figure 4—video 2
Collective rotations of 4 cells in the 2-phase (Δϵ=50Δϵ/κP=833).
Figure 4—video 3
Collective rotations of 4 cells in the 3-phase (Δϵ=70Δϵ/κP=1167).
Figure 5 with 3 supplements
Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at x=±175 for two different parameter settings.

(Stiffness parameters κP=0.12κA=0.18; average polarization field ϵ0=35; signaling radius R=2; cytoskeletal update rate μ=0.1; cell-cell adhesion B=12; cell-cell dissipation ΔB=0; cell-substrate dissipation D=0; cell-sub…

Figure 5—figure supplement 1
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 κP=0.12, κA=0.18; average polarization …

Figure 5—video 1
Motility-dominated tissue dynamics.
Figure 5—video 2
Proliferation-dominated tissue dynamics.
Figure 6 with 2 supplements
Expansion of a confluent epithelial cell sheet after removal of boundaries positioned at x=±175 for two different parameter settings that produce rough tissue fronts.

(Initially a 2500-cell system; stiffness parameters κP=0.10, κA=0.18; average polarization field ϵ0=35; maximum cell polarity Δϵ=20; signaling radius R=5; cytoskeletal update rate μ=0.1; cell-cell adhesion B=5; cell-cell …

Figure 6—video 1
Weak monolayer roughening (fingering) in motility-dominated tissue with quick proliferation.
Figure 6—video 2
Strong monolayer roughening in motility-dominated tissue with slow proliferation.
Appendix 1—figure 1
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, (α), are indicated by the lighter red color, the cell’s immediate neighborhood, 𝒩(α), is …

Appendix 1—figure 2
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
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 Δadh(𝒯pro)=(3Bβ,γ3Bα,γBα,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 Δadh(𝒯ret)=(3Bα,γ3Bβ,γ+Bα,β). Altogether, this leads to Δadh(cycl)=Δadh(𝒯pro)+Δadh(𝒯 ret)=(3(ΔB)α,γ+3(ΔB)β,γ)0, i.e. a (non-negative) dissipative contribution, whose magnitude depends on the dissipation matrix (ΔB)α,β=Bα,βBα,β0. (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 Δadh/nc=2(BB)0, where nc denotes the number of cells sliding past each other, and where we assumed cells of like type, i.e. Bα,βB and Bα,βB (αβ). 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
Role of area stiffness κA for cell size and motility.

(A) The cell area increases linearly with 1/κA. 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.

FigureSimulation codeProcessing codeParameters
Figure 2CPM_NoDivisionTrajectoryAnalysisSinglesingle_Q
Figure 2—figure supplement 1 (A-D)CPM_NoDivisionTrajectoryAnalysisSinglesingle_DQ
Figure 2—figure supplement 1 (E-H)CPM_NoDivisionTrajectoryAnalysisSinglesingle_DM
Figure 3CPM_NoDivisionTrajectoryAnalysisSinglesingle_R
Figure 4CPM_NoDivisionTrajectoryAnalysisCircularPatternrotation_Q
Figure 4—figure supplement 1CPM_NoDivisionTrajectoryAnalysisCircularPatternrotation_N_R1
Figure 4—figure supplement 2CPM_NoDivisionTrajectoryAnalysisCircularPatternrotation_N_R2
Figure 4—figure supplement 3CPM_NoDivisionTrajectoryAnalysisCircularPatternrotation_N_R3
Figure 5 (A-D)CPM_Divisionwound_nodiv
Figure 5 (E-H)CPM_Divisionwound_div
Figure 5—figure supplement 1 (A-B)CPM_Division_SupplementFrontAnalysiswound_div_A
Figure 5—figure supplement 1 (C-D)CPM_Division_SupplementFrontAnalysiswound_div_D
Figure 5—figure supplement 1 (E, F)CPM_Division_SupplementFrontAnalysiswound_div_Q
Figure 6 (A-D)CPM_Divisionwound_div_fing_1.0
Figure 6 (E-H)CPM_Divisionwound_div_fing_1.1
Appendix 2—figure 1CPM_NoDivisionTrajectoryAnalysisSinglesingle_A

Additional files

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