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Integer topological defects of cell monolayers: Mechanics and flows

Carles Blanch-Mercader, Pau Guillamat, Aurélien Roux, and Karsten Kruse
Phys. Rev. E 103, 012405 – Published 12 January 2021
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  • INTRODUCTION
  • HYDRODYNAMIC DESCRIPTION OF MONOLAYERS…
  • ACTIVE FORCES IN INTEGER TOPOLOGICAL…
  • ASTERS
  • SPIRALS
  • CHARACTERIZATION OF MYOBLAST MONOLAYERS
  • EXTENSIONS
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Monolayers of anisotropic cells exhibit long-ranged orientational order and topological defects. During the development of organisms, orientational order often influences morphogenetic events. However, the linkage between the mechanics of cell monolayers and topological defects remains largely unexplored. This holds specifically at the timescales relevant for tissue morphogenesis. Here, we build on the physics of liquid crystals to determine material parameters of cell monolayers. In particular, we use a hydrodynamical description of an active polar fluid to study the steady-state mechanical patterns at integer topological defects. Our description includes three distinct sources of activity: traction forces accounting for cell-substrate interactions as well as anisotropic and isotropic active nematic stresses accounting for cell-cell interactions. We apply our approach to C2C12 cell monolayers in small circular confinements, which form isolated aster or spiral topological defects. By analyzing the velocity and orientational order fields in spirals as well as the forces and cell number density fields in asters, we determine mechanical parameters of C2C12 cell monolayers. Our work shows how topological defects can be used to fully characterize the mechanical properties of biological active matter.

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    • Received 3 June 2020
    • Revised 11 September 2020
    • Accepted 10 December 2020

    DOI:https://doi.org/10.1103/PhysRevE.103.012405

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Cell mechanicsEmergence of patternsMorphogenesis
    Physics of Living Systems

    Authors & Affiliations

    Carles Blanch-Mercader1,2, Pau Guillamat1, Aurélien Roux1, and Karsten Kruse1,2,3

    • 1Department of Biochemistry, University of Geneva, 1211 Geneva, Switzerland
    • 2Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland
    • 3NCCR Chemical Biology, University of Geneva, 1211 Geneva, Switzerland

    See Also

    Quantifying Material Properties of Cell Monolayers by Analyzing Integer Topological Defects

    Carles Blanch-Mercader, Pau Guillamat, Aurélien Roux, and Karsten Kruse
    Phys. Rev. Lett. 126, 028101 (2021)

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    Vol. 103, Iss. 1 — January 2021

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    • Figure 1
      Figure 1

      Confined C2C12 monolayers. (a) Schematic of the experimental setup. (b) Phase-contract image of a spiral in a circular domain of 100μm radius. (c) Orientational order (left) and velocity fields (right) averaged over N=12 spirals. Colors correspond to S and speeds (see legend); the vertical line separates the two fields. Gray lines: velocity stream lines. (d) Phase-contrast image of an aster in a circular domain of 100μm radius. Scale bar in panels (b) and (d): 50μm.

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    • Figure 2
      Figure 2

      Active forces associated with integer topological defects: asters (a, c, e) and spirals (b, d, f). Active forces generated only by traction forces T0p (a, b), by anisotropic active stresses proportional to ζΔμ (c, d), and by isotropic active stresses proportional to ζ′′Δμ (e, f). Gray lines indicate the polarization field, which points outwards. The angle of the spiral is ψ0=π/3 (b, d, f). Magenta arrows: surface active force density at r/R={1/3,2/3,1}, fa,s in Eq. (27). Green arrows: line active force density, fa,l in Eq. (28). Black circle: boundary at r=R. The shafts of the magenta arrows are scaled by fa,s(r=R) and of the green arrows by Rfa,s(r=R), where fa,s=|fa,s|. Scale bars indicate fa,s(r=R)=Rfa,s(r=R)=1. We assumed T0,ζΔμ,ζ′′Δμ>0.

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    • Figure 3
      Figure 3

      Steady-state profiles for asters. (a) Cell number density B(n−ntot)/n0, Eq. (32), and (b) radial force density fi·r̂, Eq. (39), as a function of the radial distance r for varying values of the dimensionless ratio T0R/ζΔμ as indicated in the legend. We consider ζ′′Δμ=0 (a) and −ζ′′Δμ2−Bntot−n0n0=0 (b). Units are set by ζΔμ=R=1.

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    • Figure 4
      Figure 4

      Steady-state profiles of the orientational order in spirals and with R2≪K/χ. (a) Polarization angle ψ and (b) polar order parameter S. Purple lines: ψ=ψ0 and S=r/R, respectively. Green dots: numerical solution of the dynamic equations. Parameter values are χ=0.1, ν=−1.4, ζ=10−2, T0=0, η=102, and ξ=1 with the units being set by R=K=γ=1. For these parameter values |γνvrθsin(2ψ0)|<2×10−5≪χ.

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    • Figure 5
      Figure 5

      Steady-state azimuthal velocity for flows driven by traction forces and with R2≪K/χ. (a) Regime I with η=50, 100, 200 and ξ=1, (b) Regime II with 100η=0.5, 1, 2 and ξ=10−2, (c) Regime III with η=100 and 10−5ξ=0.5, 1, 2, and (d) Regime III with η=0.01 and 10−3ξ=0.5, 1, 2. Purple lines: Eq. (43). Green dots: numerical solutions of the dynamic equations. Other parameter values are χ=10−1, ν=−1.4, T0=10−2, and ζΔμ=0 with the units being set by R=K=γ=1.

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    • Figure 6
      Figure 6

      Steady-state azimuthal velocity for flows driven by gradients in active stresses and with R2≪K/χ. (a) Regime I with η=50, 100, 200 and ξ=1, (b) Regime II with 104η=0.5, 1, 2 and ξ=10−2, (c) Regime III with η=100 and 10−5ξ=0.5, 1, 2, and (d) Regime III with η=0.01 and 10−3ξ=0.5, 1. Purple lines: (a) Eq. (48), (b) Eq. (49), (c, d) Eqs. (50) and (52). Green dots: numerical solution of the dynamic equations. Other parameter values are χ=10−1, ν=−1.4, T0=0, and ζΔμ=10−2 with the units being set by R=K=γ=1.

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    • Figure 7
      Figure 7

      Half-integer topological defects in C2C12 myoblast monolayers. (a) Schematic representation of the director field for a +1/2 topological defect. (b) Theoretical profile ϕ(θ), Eq. (67), with s=+1/2 for varying ε as indicated in the legend. The ratio of Frank constants is K1/K3={0.25,0.54,1,1.86,4.} for ε={−0.6,−0.3,0,0.3,0.6}. (c) Representative experimental curves ϕe(θ) for varying radial distance r as indicated in the legend. (d) Fitted ratio K1/K3 as a function of the radial coordinate r. Error bars correspond to the standard deviation of all values of ε that lead to E<1.1Emin.

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    • Figure 8
      Figure 8

      Probability density of the polarization angle with respect to the radial direction ψ. The data were obtained from C2C12 monolayers in spiral configurations that were confined to an island of 50μm radius (N=11). The dashed line indicates ψ=90∘.

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    • Figure 9
      Figure 9

      Parameter values leading to an error E<1.1Emin for the error function (70). The cuts of the parameter space are (a) T0 vs ζΔμ, (b) η vs ξ, (c) χ vs ζΔμ, and (d) ζΔμξ/T0η vs ξ. The units are fixed by K=γ=R=1, and ν=−1.2. Gray areas indicate parameter regions that were not analyzed. Green squares: active stress predominant region, dark green star: local minimum. Magenta circles: traction force dominant region, dark magenta star: global minimum.

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    • Figure 10
      Figure 10

      Theoretical fits to experimental data. (a) Polar order parameters S and (b) azimuthal velocity vθ as a function of the radial distance r. Mean theoretical profiles for the active stress predominant parameter region in solid magenta and for the traction force dominant parameter region in dashed green; see Fig. 9 and Table 1. Blue: experimental profiles of spirals that were treated with 10μM mitomycin-C (N=(11,12,5) for confining domain radius (50,100,150)μm). Error bars in theoretical fits correspond to the standard deviation of parameter values that lead to E<1.1Emin and in experimental curves to the standard error of the mean. Profiles for three different confinement radii R=50, 100, and 150μm are shown. The theoretical curves are endowed with physical units such that S(R)=1 and vθ(R)=21.4μm/h for R=50μm.

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    • Figure 11
      Figure 11

      Theoretical fits of steady-state profiles for asters. (a) Cell number density n and (b) radial force density as a function of the radial distance r. Averaged experimental profiles (orange filled circles in (a): N=10 spirals treated with 10μM mitomycin-C 1 day after seeding, purple empty circles in (a): N=9 asters 2 days after seeding, blue cross in (b): N=3 experiments), mean fit in the active-stress predominant (magenta, full lines) and in the traction force dominant parameter region (green, dashed lines) of the cell number density of asters and the radial force density, respectively. The theoretical solutions are Eq. (32) in (a) and Eq. (39) in (b). Parameters are given in Table 2. We used ζ′′Δμ=0. Error bars in theoretical fits correspond to the standard deviation of all parameter values with E<1.1Emin and in experimental curves to the standard error of the mean.

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    • Figure 12
      Figure 12

      Polarization field of C2C12 monolayers. (a) Phase-contrast image for a spiral defect. Yellow segments represent the polarization field obtained from the structure tensor method in Ref. [44]. The yellow segments in the inset correspond to the actual experimental resolution. (b) Fluorescence image of SiR-actin for a spiral defect. In both panels (a) and (b) the circular domain has a radius of 50μm. (c) Histogram of ψBF defined as the angle between the radial direction and the polarization field from phase-contrast images. The average angle is ψBF=76∘±22∘ (mean±std, N=11). Panel (c) is the same as Fig. 8. (d) Histogram of ψSA defined as the angle between the radial direction and the polarization field from fluorescence images of SiR-actin. The average angle is ψSA=82∘±34∘ (mean±std, N=7). For panels (c) and (d), values of the angles positioned at a radial distance from the circular domain center below 48μm as well as with a polar order parameter S>0.4 were considered. Dashed lines in panels (c) and (d) indicate ψ=90∘.

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