Generating active T1 transitions through mechanochemical feedback

  1. Rastko Sknepnek  Is a corresponding author
  2. Ilyas Djafer-Cherif
  3. Manli Chuai
  4. Cornelis Weijer  Is a corresponding author
  5. Silke Henkes  Is a corresponding author
  1. School of Science and Engineering, University of Dundee, United Kingdom
  2. School of Life Sciences, University of Dundee, United Kingdom
  3. School of Mathematics, University of Bristol, United Kingdom
  4. Leiden Institute of Physics, Leiden University, Netherlands
8 figures, 2 tables and 1 additional file

Figures

Figure 1 with 2 supplements
An active junction.

(A) An external pulling force of magnitude Text induces tension T in a cell–cell junction of length l, which consists of passive viscoelastic and active components. The passive component consists of a Maxwell element with stiffness k and viscous relaxation time τv and a harmonic spring of stiffness B and rest length a connected to it in parallel. The active component is due to myosin motors (green and blue dots) with concentration m that act to contract cortical actin filaments (red lines), exerting a force of magnitude βm. Myosin motors bind to the actin cortex with association rate τm-1 and unbind with a tension-dependent dissociation rate τm1F(T). (B) Heatmap plot of the contraction force FC(Text,β). For B=0, the junction contraction rate is l˙=FC/ζ, where ζ is the friction coefficient with the surrounding medium. The mechanochemical feedback loop is contractile in the top-right quadrant where β>βc, Text>T, and FC>0. Negative values of FC correspond to an extending junction. (C) Junction length vs. time for Text=0.5ka, β=2.5ka, τv=τm=10t* (black dot in B) for increasing values of the elastic barrier B. An active T1 corresponds to reaching l=0. Increasing B slows down contractions, until, for BFC/a, the equilibrium length l0 and no T1 is possible. Inset: myosin dynamics for the same set of junctions; the horizontal dashed line indicates meqα=1, T*=0.3ka, k0=2/T*, and m0=0.5. Length is measured in units of a, time in units of t*=ζ/k, and force in units of ka.

Figure 1—figure supplement 1
Key ingredients of the single active junction model.

(A) The junction is modelled as a Maxwell element with stiffness k and viscosity η (and the relaxation timescale τv=η/k) connected in parallel with the active forcing, and an elastic barrier with stiffness B and rest length a. It is coupled in parallel to a dashpot with viscosity ζ modelling dissipation with the surrounding medium. The red arrow indicates the feedback loop between tension and myosin kinetics. (B) Myosin–tension response curve, for F(T)=α+e-k0(T-T*) with α=1, T*=0.3ka, k0=2/T*, m0=0.5. (C) Trajectories in the u-m plane for several values of B, where u=l-l0. G0 (GB) is the fixed point with no (B>0) elastic barrier. Fixed points are located at the intersection of their respective u (black) and m (green) nullclines. Arrows indicate the direction of motion. α=1, T*=0.3ka, k0=2/T*, and m0=0.5. Length is measured in units of a, time in units t*=ζ/k, and force in units of ka.

Figure 1—figure supplement 2
A linear chain of active junctions.

(A) Schematic of a linear chain for viscoelastic junctions with active tension feedback. The chain is formed by connecting in series the active viscoelastic elements shown in Figure 1—figure supplement 1A, with the pulling force of magnitude Text applied to both ends. (B) Time evolution of a linear chain for Text=0.385ka and β=2.5ka with an initial pulse of myosin. The central contraction propagates rapidly outwards, resulting in a contraction and collapse of the chain. Activity and pulling force correspond to a point on the contraction–extension separatrix in Figure 1B, putting the system in a mechanically neutral state. The remaining parameters are the same as in Figure 1C.

Figure 2 with 14 supplements
An active T1 transition event.

(A) Top panel: the mechanical anisotropy in the initial state is produced by applying pulling forces (green arrows) in the horizontal direction to the left and right boundaries. Bottom panel: the final state after the active T1 shows a clear convergence–extension deformation (red arrows). Cells are coloured by type: passive (light grey), buffer (medium grey), and active (dark grey). Junctions are coloured by junctional myosin. (B) Time sequence of the active T1 transition measured from the moment activity and viscoelasticity were switched on. Cells in the top row are coloured by type and junctions are coloured by tension. Cells in the bottom row are coloured by activated myosin mactC, and junctions are coloured by myosin. Parameters: A0=323a2, P0=6a, β=0.8f* (active), β=0.4f* (buffer), β=0 (passive), M=6, T*=0.3f*, k0=2/T*, τv=20t*, τm=100t, α=0.1, f=1, fpull=0.15f*, with nx=15 (ny=11) cells in the horizontal (vertical) direction. Units: length (a), time (t*=ζ/(Γ+k)), force (f*=(Γ+k)a).

Figure 2—figure supplement 1
Schematic representation of the key ingredients in the vertex model with active junctions.

(A) The total force on vertex i can be expressed as a sum of forces due to junctions. For the junction e, Ce,l and Ce,r are the cells to the left and the right, respectively, when looking in the direction of the vector lemeC1(meCr) is the myosin attached to junction e due to the cell to the left (right) of it. The area (pCe,l/r) and perimeter (tCe,l/r) terms are defined in the text above. (B) Example of the redistribution of myosin after a T1 transition for one side of the junction, and one of its inner and outer shoulder junctions.

Figure 2—video 1
Simulation of a passive system with applied horizontal pulling for fpull=0.15f*.

Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). In this video activity is turned off, and there is no difference between those three cell types.

Figure 2—video 2
Low activity system with no T1 transitions.

Activity is set to β=0.4f* for the four central cells and the patch is pulled horizontally with fpull=0.15f*, τm=100t*, and τv=20t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 3
Active T1 transition.

Activity is set to β=0.8f* for the four central cells and the patch is pulled horizontally with fpull=0.15f*, τm=100t*, and τv=20t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 4
System with low pulling force.

Activity is set to β=0.8f* for the four central cells and the patch is pulled horizontally with fpull=0.05f*, τm=100t*, and τv=20t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 5
System with too high pulling force.

Activity is set to β=0.8f* for the four central cells and the patch is pulled horizontally with fpull=0.25f*, τm=100t*, and τv=20t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 6
System with too high activity.

Activity is set to β=1.2f* for the four central cells and the patch is pulled horizontally with fpull=0.15f*, τm=100t*, and τv=20t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 7
Active T1 transition with low τm and τv.

Activity is set to β=1.0f* and the patch is pulled horizontally with fpull=0.15f* with τm=20t* and τv=20t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 8
Active T1 transition with low τm.

Activity is set to β=1.0f* and the patch is pulled horizontally with fpull=0.15f* with τm=20t* and τv=100t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 9
Active T1 transition with low τv.

Activity is set to β=1.0f* and the patch is pulled horizontally with fpull=0.15f* with τm=100t* and τv=20t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 10
Active T1 transition at intermediate values of τm and τv.

Activity is set to β=1.0f* and the patch is pulled horizontally with fpull=0.15f* with τm=100t* and τv=100t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 11
Active T1 transition at high τv.

Activity is set to β=1.0f* and the patch is pulled horizontally with fpull=0.15f* with τm=100t* and τv=500t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 12
Active T1 transition at high τm.

Activity is set to β=1.0f* and the patch is pulled horizontally with fpull=0.15f* with τm=500t* and τv=100t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 2—video 13
Active T1 transition at high τm and τv.

Activity is set to β=1.0f* and the patch is pulled horizontally with fpull=0.15f* with τm=500t* and τv=500t*. Left panel - junction and cell myosin; right panel - junction tension, cells are coloured by type, active (dark grey), buffer (intermediate grey), passive (light grey). Activity of buffer cells is set to β/2.

Figure 3 with 1 supplement
Junction dynamics during the active T1 transition shown in Figure 2.

(A) Definition of central (red/blue – junction that disappears/appears), inner (orange), and outer (blue-green) shoulder junctions through the T1 transition. (B) Central junction: myosin, m (green; two curves for myosin on two sides of the junction), tension, T (yellow), junction length, l (black), and rest length l0 (purple) vs. time. The vertical line indicates the T1 transition, at which point junctional myosin is redistributed according to the rules outlined in Figure 2—figure supplement 1B. (C) Same as in panel (A) but averaged over four shoulder junctions, with variance indicated as shade. (D) Polar histogram of the orientation of the first T1 event measured with respect to the pulling direction, from n=32 simulations. Blue (red) indicates appearing (disappearing) junctions. Parameters are β=0.8f* and fpull=0.15f* and as in Figure 2.

Figure 3—figure supplement 1
Continuous strain tensors through the active T1 transition, for β=0.8f*, fpull=0.15f*, and the other parameters corresponding to Figure 2 and Figure 3.

(A) Statistical strain tensor U^ defined in Equation 20, with the initial undeformed state used as reference configuration. (B) Total strain, that is, integrated V^ tensor defined in Equation 24. (C) Integrated plastic strain tensor P^ defined in Equation 22. One can see that the active T1 proceeds at near constant elastic strain with cells elongated in the pulling direction, that is, εxx>0 and εyy<0. Conversely, there is clear convergence–extension since εxx<0 and εyy>0 in the total strain. Here, the traces are an average over n=32 simulations, and the shading corresponds to the error of the mean.

Existence and timescales of T1 transitions in the vertex model with active junctions as a function of fpull and β, averaged over n=32 simulations with different realisations of the myosin noise.

(A) Probability of a central T1 transition. The red line is the 50% probability contour of any T1 occurring in the simulation. (B) Typical timescale for the T1 transition to occur, measured as the length of the contraction phase. The other parameters are the same as in Figure 2. (C) Magnitude of the convergence–extension deformation as a function of fpull and characterised by measuring εxxtotεyytot induced by the T1 transition.

Robustness of the T1 mechanism as a function of τv and τm for β=1.0f* and fpull=0.15f*.

(A) Probability of a central T1 event, averaged over n=32 simulations with different realisations of the myosin noise. The probability of any T1 event is 1 throughout. (B) Contraction time to collapse for the central T1 as a function of τv, for different values of τm, for points where a central T1 event occurred in at least 25% of simulations. (C) Peak of the total convergence extension strain εxxtotεyytot, showing very weak dependence on viscoelastic and myosin timescales. Shading in panels (B) and (C) indicates the standard error of the mean.

Figure 6 with 2 supplements
Disordered active tissue at time t700t* as a function of the magnitude of the pulling force fpull and activity β.

The region of convergence–extension is at the centre of the diagram, around β0.4-0.6f* and fpull0.1-0.3f*. The remaining parameters are the same as in Figure 2.

Figure 6—video 1
Video of simulations of active contractions in disordered tissues for a range of activities, β, and magnitudes of the pulling force, fpull.

The system is first stretched for t=150t with activity switched off. During the initial stretching period, myosin is allowed to build anisotropy, but it does not feed back onto tension. This results in the tissue extending in the direction of external force and contracting in the direction perpendicular to it. After the tissue had reached mechanical equilibrium, the activity was turned on, with myosin being allowed to exert active contractions. All systems were simulated for an additional 800t. Snapshots used to make the video were recorded once every 10t. Simulations parameters are given in the caption of Figure 6.

Figure 6—video 2
Active T1 transitons in a random patch.

Activity is set to β=0.5f* and the patch is pulled horizontally with fpull=0.2f* with τm=100t and τv=20t. Left panel - junction and cell myosin; right panel - junction tension, and all cells are active.

Figure 7 with 3 supplements
Characterisation of convergence-extension in a random tissue patch.

(A) Snapshot of a random tissue patch for β=0.5f* and fpull=0.2f* at t700t*,that is during the convergence–extension flow. Red arrow indicates that a constant pulling force is applied throughout the entire simulation. (B) Angular histogram of T1 events for same values of β and fpull. Cells and junctions are coloured as in Figure 6. (C) Magnitude of the convergence–extension deformation as a function of fpull characterised by measuring ϵxxtot-ϵyytot induced by the T1 transition. (D) Anisotropy along (xx, solid line) and perpendicular to (yy, dashed line) the direction of the external pulling force for myosin (green), mechanical stress (red), and shape tensor (black) as functions of fpull for β=0.5f* at t700t*. In (C) and (D), each point was averaged over n=33 independent samples and the error bar is smaller than the symbol size.

Figure 7—figure supplement 1
Continuous strain tensors for the fully active tissue at β=0.5f*, fpull=0.2f*, and the other parameters corresponding to Figure 2 and Figure 3.

(A) Statistical strain tensor U^ defined in Equation 20, with the initial undeformed state used as reference configuration. (B) Total strain, that is, integrated V^ tensor defined in Equation 24. (C) Integrated plastic strain tensor P^ defined in Equation 22. Convergence–extension flow proceeds with cells elongated in the pulling direction, that is, εxx>0 and εyy<0 while εxx<0 and εyy>0 in the total strain. Here, the traces are an average over n=5 simulations, and the shading corresponds to the standard deviation.

Figure 7—figure supplement 2
Measuring T1 transitions in the fully active random patch.

A representative example taken from a sample of n=5 simulations for β=0.5f*, fpull=0.2f*. (A) Appearing and disappearing junctions as a function of simulation frame number and angle, before filtering. Inset: raw T1 histogram. (B) Same dataset after filtering procedure.

Figure 7—figure supplement 3
T1 histograms in the parameter range where no convergence–extension occurs.

(A) Passive tissue at β=0 pulled with fpull=0.2f*, T1s are passive and aligned along pulling direction. (B) Active tissue at β=0.5f* with no applied force fpull=0.0f*, here T1s are distributed isotropically. (C) Experimental T1 distribution in the anterior region of the streak, strongly resembling the active isotropic case.

Figure 8 with 3 supplements
Analysis of the tissue flows in the early-stage chick embryo.

(A) Image of a typical early-stage chick embryo prior to the gastrulation (i.e. primitive streak formation). The primitive streak will form along the yellow dashed line. The direction of myosin anisotropy is shown by the green double-headed arrows, and the direction of the tissue flow is indicated by the red arrows. x-axis is chosen to coincide with the long direction of the sickle-shaped active region in the embryo’s posterior (Rozbicki et al., 2015). (A’) Zoom-in of the rectangular region on the posterior side of the embryo; myosin II (green), actin (red), and nuclei (blue). (B) Measured distribution of the orientation of T1 events in a circular patch of diameter ≈ 190 μm tracked over the period of ≈ 6 hr (cf. model distribution in Figure 7B). Blue (red) denotes junctions that appear (disappear); arrows have the same meaning as in (A). (C) Angle (red) and magnitude (green) of tissue shape anisotropy (dots) and tissue flow (crosses) for n=6 rectangular patches along the sickle with corresponding anisotropy and flow patterns shown in (C’). Components of the elastic strain tensor U^ (D) and the total integrated strain tensor V^ (E; definition in Equation 24) as a function of time during first 4 hr of the streak formation for two central regions of the sickle (yellow stripes in C). Details of the analysis are given in ‘Materials and methods’.

Figure 8—figure supplement 1
Region of interest in the anterior of the embryo.

(A) Integrated total strain ε^tot(t) and (B) elastic strain U^(t).

Figure 8—video 1
Image sequence taken at the base of the forming streak showing cell intercalations.

Centres of some cells are labelled with coloured dots for easier identification. Crossed blue and red lines indicate individual intercalation events. Blue lines indicate the direction of junction contraction and red lines the direction of extension of newly formed junctions. White scale bar 25 μm.

Figure 8—video 2
Image sequence taken in the region of epiblast in front of forming streak.

Centres of some cells are labelled with coloured dots for easier identification. Crossed blue and red lines indicate individual intercalation events. Blue lines indicate the direction of junction contraction and red lines the direction of extension of newly formed junctions. Images were taken at 3 min intervals. White scale bar 50 μm.

Tables

Table 1
Values of the parameters in the single-junction model.

Units: length (a), time (t*=ζ/k), force (ka).

Base
ParameterDescription
kSpring constant
aBarrier rest length
ζFriction with substrate
Model
ParameterDescriptionValue range
BBarrier spring constant0-0.2k
TextApplied external tension0-1ka
βMyosin activity0-3ka
τvViscoelastic time10t*
τmMyosin time10t*
m0Myosin reference level0.5
T*Threshold tension0.3ka
k0Slope of m vs. T at T2/T*
αTension-independent myosin dissociation1
Table 2
Values of the parameters used in the vertex model with active junctions.

Units: length (a), time (t*=ζ/(Γ+k)), force (f*=(Γ+k)a).

Base
ParameterDescription
aHexagonal cell edge length
ΓPerimeter modulus
kSpring constant
ζFriction with substrate
Model
ParameterDescriptionValue range
κArea modulus1f*/a3
A0Target cell area33a2/2
P0Target cell perimeter6a
fpullPulling force0.0-0.3f*
βMyosin activity0.0-1.4f*
τvViscoelastic time100-103t*
τmMyosin time101-103t*
T*Threshold tension0.3f*
K0Slope of m vs. T at T2/T*
m0Myosin reference level0.5
MTotal cell myosin6
fVariance of myosin fluctuations1
αTension-independent myosin dissociation0.1

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  1. Rastko Sknepnek
  2. Ilyas Djafer-Cherif
  3. Manli Chuai
  4. Cornelis Weijer
  5. Silke Henkes
(2023)
Generating active T1 transitions through mechanochemical feedback
eLife 12:e79862.
https://doi.org/10.7554/eLife.79862