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Long-range mutual activation establishes Rho and Rac polarity during cell migration.

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  1. [1] § Results › Mechanochemical model of Rho and Rac partitioning in cells ↔ mechanochemical_2D_axisymmetric/output/T=350.0_dt=1.0_ n=40/D=0.3_eta=10000.0_vC=-0.3 S=0.0_M=17.32_a0=1.0 b0=1.0 wrac=6.0 Drac=0.05/rac0=0.01 koff=1.4 aopto = 2.0 da=0.04 db=0.04_sig=200.0_x=2.0 L=16.28 a=1.0 b=1.0 sig0=0.01_tenth=0.025_MCAbth=0.035_rho0=0.001 ten0=0.02/MechanochemicalAxisymmetricVector.jl, lines 1–51 · score 0.58 · mechanochemical model, cortical flow, reproduces, surface, coupling, active

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  1. # Mechanochemical model on an axisymmetric cell surface.
  2. # Simulates coupled Rac/Rho signalling, MCA (Moesin-like cortical actin) dynamics,
  3. # cortical flow, and membrane deformation on a sphere.
  4. #
  5. # Written by Andreu F Gallen (Turlier lab) and Eric Neiva,
  6. # in collaboration with Orion Weiner's lab.
  7. #
  8. # Reference: INSERT DOI
  9. #
  10. # To reproduce pure local-inhibition behaviour, set:
  11. # coup.vCTE = 0, coup.α = 0, coup.β = 0, σₐ⁰ = 0
  13. include("Plots_RhoRacSinglet.jl")
  15. # ── Parameter structs ────────────────────────────────────────────────────────
  17. @kwdef struct MechanicalParams
  18. η::Real # Viscosity
  19. χ::Real # Friction coefficient
  20. χ₀::Real # Basal friction
  21. sigmaₐ⁰::Real # Basal active tension
  22. sigmaρ⁰::Real # Rho-dependent active tension coefficient
  23. S::Real # Prestress / geometric parameter
  24. Λ::Real # Lamé-like parameter
  25. M::Real # Lamé-like parameter
  26. R::Real # Cell radius [μm]
  27. end
  29. @kwdef struct KineticParams
  30. koff::Real; kon::Real; M0::Real; D::Real
  31. dᵃ::Real; dᵇ::Real; λᵇ::Real; λʳᴬ::Real
  32. Drac::Real; Drho::Real; α₀::Real; β₀::Real; wrac::Real
  33. end
  35. @kwdef struct CouplingParams
  36. rac0::Real; rho0::Real; ten0::Real; vCTE::Real
  37. tenth::Real; sig0::Real; MCAbth::Real
  38. α::Real; β::Real; αopto::Real; βopto::Real
  39. end
  41. @kwdef struct SimControl
  42. domain::Tuple
  43. ls::Any # Level-set object
  44. n::Int # Mesh partition
  45. Δt::Real
  46. T::Real
  47. order::Int
  48. output_frequency::Int
  49. γᶜ::Real
  50. τᵈkₒ::Real
  51. end
  54. # ── Helper functions ─────────────────────────────────────────────────────────
  56. function plotting(ylabel_str, data, path, label)
  57. plot(data)
  58. xlabel!("ξ [μm]")
  59. ylabel!(ylabel_str)
  60. savefig(path * "$ylabel_str" * label * ".png")
  61. end
  63. """Mass conservation residual for MCA protein."""
  64. function conservation(sMCAu, sMCAb, Minitial)
  65. return 1.0 * (Minitial - (sMCAu + sMCAb))
  66. end
  68. threshold(x, x₀, xth) = 0.5 * (tanh ∘ (x / x₀ - xth / x₀) + 1)
  69. threshold2(x, x₀, xth) = 0.5 * (tanh.(x / x₀ .- xth / x₀) .+ 1)
  71. sinθ(x) = x[2] / norm(x)
  72. cotθ(x) = x[1] / x[2]
  73. cotθ2(x) = cotθ(x) * cotθ(x)
  74. cotθ3(x) = cotθ(x) * cotθ(x) * cotθ(x)
  75. cscθ2(x) = 1 + cotθ2(x)
  78. # ── Main simulation function ─────────────────────────────────────────────────
  80. """
  81. Run a single mechanochemical axisymmetric simulation.
  83. # Arguments
  84. - `mech` : Mechanical parameters
  85. - `kin` : Kinetic/chemical parameters
  86. - `coup` : Mechanochemical coupling and optogenetics parameters
  87. - `control` : Simulation control (mesh, time stepping, output)
  88. - `name` : Output folder name
  89. - `writesol` : Write VTU output if true
  90. """
  91. function run_mechanochemical_axisymmetric_vector(
  92. mech::MechanicalParams,
  93. kin::KineticParams,
  94. coup::CouplingParams,
  95. control::SimControl;
  96. name::String,
  97. writesol::Bool = true
  98. )
  101. # ── MCA weak forms ───────────────────────────────────────────────────────────
  103. """
  104. Weak forms for MCA bound and unbound species.
  105. Returns (mMCA, aMCAb, bMCAb, aMCAu, bMCAu).
  106. """
  107. function MCA_bound_unbound_weak_forms(Δt, kon, koff, λᵇ, λ, R2, D, nΓ, dΓ)
  109. τ = TensorValue(0.0, -1.0, 1.0, 0.0) ⋅ nΓ # unit tangent vector
  111. # Bound MCA
  112. mMCA(Δt, MCA_b, w) = ∫(((MCA_b * w) / Δt) * y)dΓ
  114. aMCAb(MCA_b, v, w) =
  115. ∫(((MCA_b * w) / Δt) * y)dΓ +
  116. ∫((0.0005 * (∇ᵈ(MCA_b, nΓ) ⋅ ∇ᵈ(w, nΓ))) * y)dΓ + # diffusion for stabilisation
  117. ∫((koff * (MCA_b * w)) * y)dΓ +
  118. ∫((w * ((v ⋅ τ) * (∇ᵈ(MCA_b, nΓ) ⋅ τ))) * y)dΓ +
  119. ∫(w * (MCA_b * ((τ ⋅ ∇ᵈ(v, nΓ)) ⋅ τ)) * y)dΓ
  121. bMCAb(w, MCA_u, MCAb_old, λ) =
  122. ∫((kon * (MCA_u * w) + (MCAb_old * w) / Δt) * y)dΓ +
  123. ∫((λ / (π * R2) * (kon / (kon + koff)) * w) * y)dΓ
  125. # Unbound MCA
  126. aMCAu(MCA_u, x, x_old, w) =
  127. ∫(((MCA_u * w) / Δt) * y)dΓ +
  128. ∫((D * (∇ᵈ(MCA_u, nΓ) ⋅ ∇ᵈ(w, nΓ))) * y)dΓ +
  129. ∫((kon * (MCA_u * w)) * y)dΓ +
  130. ∫((w * (((x - x_old) ⋅ τ / Δt) * (∇ᵈ(MCA_u, nΓ) ⋅ τ) +
  131. MCA_u * ((τ ⋅ ∇ᵈ(x, nΓ) ⋅ τ) - (τ ⋅ ∇ᵈ(x_old, nΓ) ⋅ τ)) / Δt)) * y)dΓ
  133. bMCAu(w, MCA_b, MCAu_old, λ) =
  134. ∫((koff * (MCA_b * w)) * y)dΓ +
  135. ∫(((MCAu_old * w) / Δt) * y)dΓ +
  136. ∫((λ / (π * R2) * (koff / (kon + koff)) * w) * y)dΓ
  138. mMCA, aMCAb, bMCAb, aMCAu, bMCAu
  139. end
  141. activity::Function = unit_activity_axisymmetric
  142. redistance_frequency::Int = 1
  144. # Background mesh
  145. cells = (control.n, div(control.n, 2))
  146. h = (control.domain[2] - control.domain[1]) / control.n
  147. bgmodel = CartesianDiscreteModel(control.domain, cells)
  148. Ω = Triangulation(bgmodel)
  149. degree = control.order < 3 ? 3 : 2 * control.order
  150. R2 = mech.R
  152. # Level-set buffer (updated each time step)
  153. buffer = Ref{Any}((Ωᶜ=nothing, dΩᶜ=nothing, dΓ=nothing, nΓ=nothing,
  154. φ₋=nothing, cp₋=nothing, t=nothing, Vbg=nothing))
  156. function update_buffer!(i, t, dt, v₋₂, mv₋₂)
  157. buffer[].t == t && return true
  159. Ωᶜ = buffer[].Ωᶜ; Vbg = buffer[].Vbg
  161. if buffer[].Ωᶜ === nothing
  162. Vbg = TestFESpace(Ω, ReferenceFE(lagrangian, Float64, control.order))
  163. _φ₋ = interpolate_everywhere(control.ls.φ, Vbg)
  164. else
  165. cp₋₂ = buffer[].cp₋; φ₋₂ = buffer[].φ₋
  166. __φ = get_free_dof_values(φ₋₂.φ)
  167. Ωⱽ = get_triangulation(Vbg)
  168. _ϕ₋ = compute_normal_displacement(cp₋₂, φ₋₂, v₋₂, dt, Ωⱽ)
  169. _φ₋ = FEFunction(Vbg, __φ - _ϕ₋)
  170. end
  172. φ₋ = AlgoimCallLevelSetFunction(_φ₋, ∇(_φ₋))
  173. (i % redistance_frequency == 0) && begin
  174. _φ₋ = compute_distance_fe_function(bgmodel, Vbg, φ₋, control.order, cppdegree=3)
  175. φ₋ = AlgoimCallLevelSetFunction(_φ₋, ∇(_φ₋))
  176. end
  178. cp₋ = compute_closest_point_projections(Vbg, φ₋, control.order,
  179. cppdegree=3, trim=true, limitstol=1.0e-2)
  180. squad = Quadrature(algoim, φ₋, degree, phase=CUT)
  181. s_cell_quad, is_c₋ = CellQuadratureAndActiveMask(bgmodel, squad)
  183. δ₋ = 2.0 * mv₋₂ * dt
  184. _, is_nᶜ = narrow_band_triangulation(Ω, _φ₋, Vbg, is_c₋, δ₋)
  185. Ωᶜ, dΓ = TriangulationAndMeasure(Ω, s_cell_quad, is_nᶜ, is_c₋)
  186. dΩᶜ = Measure(Ωᶜ, 2 * control.order)
  187. nΓ = normal(φ₋, Ω)
  189. buffer[] = (Ωᶜ=Ωᶜ, dΩᶜ=dΩᶜ, dΓ=dΓ, nΓ=nΓ, cp₋=cp₋, φ₋=φ₋, t=t, Vbg=Vbg)
  190. return true
  191. end
  193. N = num_dims(bgmodel)
  194. reffeʷ = ReferenceFE(lagrangian, VectorValue{N,Float64}, control.order - 1)
  195. reffeᵉ = ReferenceFE(lagrangian, Float64, control.order - 1)
  197. function update_all!(i, t, dt, disp, val)
  198. Ωᶜ = buffer[].Ωᶜ; dΩᶜ = buffer[].dΩᶜ
  199. dΓ = buffer[].dΓ; nΓ = buffer[].nΓ
  200. φ = buffer[].φ₋
  202. τ = TensorValue(0.0,-1.0, 1.0, 0.0) ⋅ nΓ # vector tangente
  203. Vʷ = TestFESpace(Ωᶜ, reffeʷ, dirichlet_tags=[5, 8])
  204. UXʷ = TrialFESpace(Vʷ, [p -> VectorValue(0, x₀), p -> -xₗ * τ(p)])
  205. UVʷ = TrialFESpace(Vʷ, [p -> VectorValue(0, v₀), p -> -vₗ * τ(p)])
  207. Vᵉ = TestFESpace(Ωᶜ, reffeᵉ)
  208. Vᴿ = TestFESpace(Ωᶜ, reffeᵉ)
  209. Vˡ = ConstantFESpace(bgmodel)
  211. Uʷ = TrialFESpace(Vʷ)
  212. Uᵉ = TrialFESpace(Vᵉ)
  213. Uᴿ = TrialFESpace(Vᴿ)
  214. Uˡ = TrialFESpace(Vˡ)
  216. Yᵛ = Vʷ
  217. Xᵛ = MultiFieldFESpace([Uʷ, Uˡ])
  218. Yʳ = MultiFieldFESpace([Vᵉ, Vˡ])
  219. Xʳ = MultiFieldFESpace([Uᵉ, Uˡ])
  221. UXʷ, UVʷ, Vʷ, Xᵛ, Yᵛ, Xʳ, Yʳ, Uᵉ, Vᵉ, Vᴿ, Uᴿ, Ωᶜ, dΩᶜ, dΓ, nΓ, φ
  222. end
  224. # Create output directories
  225. pVTU = "./output/" * name * "VTU/"
  226. pPNG = "./output/" * name
  227. mkpath(pVTU)
  228. mkpath(pPNG)
  229. mkpath(pPNG * "Rac_time/")
  230. mkpath(pPNG * "Rho_time/")
  231. mkpath(pPNG * "MCAb_time/")
  232. mkpath(pPNG * "Rac_time_initial/")
  233. mkpath(pPNG * "Rho_time_initial/")
  235. # Save source files alongside results for reproducibility
  236. cp(@__FILE__, pPNG * split(@__FILE__, "/")[end], force=true)
  237. cp("./src/WeakForms.jl", pPNG * "WeakForms.jl", force=true)
  238. cp("./examples/SurfaceViscousFlows/SurfaceViscousFlows.jl",
  239. pPNG * "SurfaceViscousFlows.jl", force=true)
  241. # Time discretisation
  242. t₀ = 0.0
  243. u₀ = VectorValue(0.0, 0.0)
  244. m₀ = 2.0
  245. nΔt = trunc(Int, control.T / control.Δt + 0.5) + 1
  246. tol = 1e-8
  248. # Boundary condition values (updated during time loop)
  249. x₀ = 0.0; xₗ = 0.0
  250. v₀ = 0.0; vₗ = 0.0
  252. update_buffer!(0, t₀, control.Δt, u₀, m₀)
  253. UXʷ, UVʷ, Vʷ, Xᵛ, Yᵛ, Xʳ, Yʳ, Uᵉ, Vᵉ, Vᴿ, Uᴿ, Ωᶜ, dΩᶜ, dΓ, nΓ, φ =
  254. update_all!(0, t₀, control.Δt, u₀, m₀)
  256. τ = TensorValue(0.0, -1.0, 1.0, 0.0) ⋅ nΓ
  258. γʷ = control.γᶜ / h # velocity stabilisation parameter
  259. γᵉ = control.γᶜ / h # concentration stabilisation parameter
  261. # Initial conditions
  262. _υₕ(x) = VectorValue(0.0, 0.0)
  263. υₕ = interpolate_everywhere(_υₕ, UVʷ)
  264. _xₕ(x) = VectorValue(0.0, 0.0)
  265. xₕ = interpolate_everywhere(_xₕ, UXʷ)
  266. xₕ_old = xₕ
  268. Tm = SparseMatrixCSR{0,PetscScalar,PetscInt}
  269. Tv = Vector{PetscScalar}
  270. ps = PETScLinearSolver(mykspsetup)
  272. i = 0
  273. t = t₀
  275. # Arc-length function and optogenetic activation profiles
  276. arclength(x) = R2 * atan(x[2], -x[1])
  278. α₀opto(x) = kin.α₀
  279. β₀opto(x) = kin.β₀
  280. α₀opto2(x) = kin.α₀ + coup.αopto * exp(-0.5 * (arclength(x) - π * R2)^2 / (kin.wrac)^2)
  281. β₀opto2(x) = kin.β₀ + coup.βopto * exp(-0.5 * (arclength(x))^2 / (kin.wrac)^2)
  283. γ₀ = 0.1 / h
  284. γ₀R = 0.1 / h
  285. γ₀M = 0.1 / h
  287. m₀opto(u, v) = ∫(u * v)dΓ
  288. s₀opto(u, v) = ∫(γ₀ * ((nΓ ⋅ ∇(u)) ⊙ (nΓ ⋅ ∇(v))))dΩᶜ
  289. s₀R(u, v) = ∫(γ₀R * ((nΓ ⋅ ∇(u)) ⊙ (nΓ ⋅ ∇(v))))dΩᶜ
  290. s₀MCA(u, v) = ∫(γ₀M * ((nΓ ⋅ ∇(u)) ⊙ (nΓ ⋅ ∇(v))))dΩᶜ
  291. s₀x(υ, μ) = ∫(γʷ * ((nΓ ⋅ ε(υ)) ⊙ (nΓ ⋅ ε(μ))))dΩᶜ
  293. λ = 0.0
  295. A₀opto(u, v) = m₀opto(u, v) + s₀opto(u, v)
  296. bα₀opto(v) = m₀opto(α₀opto, v)
  297. bβ₀opto(v) = m₀opto(β₀opto, v)
  298. bα₀opto2(v) = m₀opto(α₀opto2, v)
  299. bβ₀opto2(v) = m₀opto(β₀opto2, v)
  301. op_α₀ = AffineFEOperator(A₀opto, bα₀opto, Uᴿ, Vᴿ)
  302. op_β₀ = AffineFEOperator(A₀opto, bβ₀opto, Uᴿ, Vᴿ)
  303. α₀v = solve(op_α₀)
  304. β₀v = solve(op_β₀)
  306. # Initial Rac and Rho (coarse equilibration run with larger diffusion)
  307. Rₕ = interpolate_everywhere(0.0, Uᴿ)
  308. ρₕ = interpolate_everywhere(1.0, Uᴿ)
  309. Rₕ_old = Rₕ; ρₕ_old = ρₕ
  311. a_R, b_R, a_ρ, b_ρ = rac_rho_weak_forms2(
  312. control.Δt, 200 * kin.dᵃ, 200 * kin.dᵇ, kin.Drac, kin.Drho,
  313. nΓ, dΓ, coup.α, coup.β)
  315. # MCA initial conditions (equilibrium bound/unbound split)
  316. mMCA, aMCAb, bMCAb, aMCAu, bMCAu = MCA_bound_unbound_weak_forms(
  317. control.Δt, kin.kon, kin.koff, kin.λᵇ, λ, R2, kin.D, nΓ, dΓ)
  319. uh_MCAb = interpolate_everywhere(kin.kon * kin.M0 / (π * R2) / (kin.koff + kin.kon), Uᴿ)
  320. uh_MCAb_old = uh_MCAb
  321. uh_MCAu = interpolate_everywhere(kin.koff * kin.M0 / (π * R2) / (kin.koff + kin.kon), Uᴿ)
  322. uh_MCAu_old = uh_MCAu
  323. sum_uh_MCAu = ∑(∫(uh_MCAu)dΓ)
  324. sum_uh_MCAb = ∑(∫(uh_MCAb)dΓ)
  325. Minitial = sum_uh_MCAu + sum_uh_MCAb
  326. println("Initial total MCA: ", Minitial)
  327. λ = conservation(sum_uh_MCAu, sum_uh_MCAb, Minitial)
  329. I = TensorValue(1.0, 0.0, 0.0, 1.0)
  331. # Membrane tension bilinear / linear forms
  332. N(u) = mech.S * I + mech.Λ * tr(εᶜ(u, nΓ)) * I + mech.M * εᶜ(u, nΓ)
  333. ∂u(u) = mech.R * (τ ⋅ (∇ᶜ(u, nΓ)) ⋅ τ)
  334. mten(u, v) = ∫((u * v) * y)dΓ
  335. mten2(u, v) = ∫(((τ ⋅ (N(u) ⋅ τ)) * v) * y)dΓ
  336. sten(u, v) = ∫(10 * γ₀ * ((nΓ ⋅ ∇(u)) ⊙ (nΓ ⋅ ∇(v))))dΩᶜ
  338. Aten(u, v) = mten(u, v) + sten(u, v)
  339. bten(v) = mten2(xₕ, v)
  340. op_ten = AffineFEOperator(Aten, bten, Uᴿ, Vᴿ)
  341. ten = solve(op_ten)
  343. # Membrane displacement (nonlinear strain bilinear forms)
  344. aᴹ(M, R, x_old, x, w) =
  345. ∫((2 * M * (x ⋅ w / 2 +
  346. R * R * (∇ᶜ(x, nΓ) ⊙ ∇ᶜ(w, nΓ)) +
  347. (cotθ2) * (x ⋅ w))) * sinθ)dΓ +
  348. ∫((2 * M / R / 2 *
  349. ((x_old ⋅ x + R * R * (∇ᶜ(x_old, nΓ) ⊙ ∇ᶜ(x, nΓ))) * (R * ∇ᶜ(w, nΓ) ⋅ τ) +
  350. (cotθ3 * x_old) * (x ⋅ w))) ⋅ τ * sinθ)dΓ
  352. aᴸ(L, R, x_old, x, w) =
  353. ∫(L * (R * R * (∇ᶜ(x, nΓ) ⊙ ∇ᶜ(w, nΓ)) +
  354. (cotθ * (x ⋅ ∇ᶜ(w, nΓ)) + cotθ * (w ⋅ ∇ᶜ(x, nΓ))) ⋅ τ +
  355. (cotθ2) * (x ⋅ w)) * sinθ)dΓ +
  356. ∫(0.5 / R * L *
  357. ((cscθ2) * (x_old ⋅ x) + R * R * ∇ᶜ(x, nΓ) ⊙ ∇ᶜ(x_old, nΓ)) *
  358. (R * ∇ᶜ(w, nΓ) ⋅ τ + cotθ * w) ⋅ τ * sinθ)dΓ
  360. bₓ(MCA_b, v, w) = ∫((mech.χ * (MCA_b) * v ⋅ w) * (mech.R * mech.R) * sinθ)dΓ
  361. m(MCA_b, Δt, x, w) = ∫(((mech.χ * MCA_b) * (x ⋅ w) / Δt) * (mech.R * mech.R) * sinθ)dΓ
  363. mυ(Δt, v, w) = ∫(((v ⋅ w) / Δt) * y)dΓ
  365. # Weak tangentiality penalty
  366. bo = 10.0 / ((2 / 40)^2)
  367. wt(x, w) = ∫(bo * ((x ⋅ nΓ) * (w ⋅ nΓ)))dΓ
  369. # Equilibrate Rac/Rho at t = 0
  370. Arac(rac, w) = a_R(rac, w, υₕ) + s₀R(rac, w)
  371. Brac(w) = b_R(w, ρₕ, α₀v, Rₕ_old, uh_MCAb, coup.MCAbth, coup.rho0)
  372. op_rac = AffineFEOperator(Arac, Brac, Uᴿ, Vᴿ)
  373. Rₕ = solve(op_rac); Rₕ_old = Rₕ
  375. Arho(rho, w) = a_ρ(rho, w, υₕ) + s₀R(rho, w)
  376. Brho(w) = b_ρ(w, Rₕ, β₀v, ρₕ_old, ten, coup.sig0, coup.tenth)
  377. op_rho = AffineFEOperator(Arho, Brho, Uᴿ, Vᴿ)
  378. ρₕ = solve(op_rho); ρₕ_old = ρₕ
  380. # MCA operators at t = 0
  381. AMCAb(MCA_b, w) = aMCAb(MCA_b, υₕ, w) + s₀MCA(MCA_b, w)
  382. BMCAb(w) = bMCAb(w, uh_MCAu, uh_MCAb_old, λ)
  383. AMCAu(MCA_u, w) = aMCAu(MCA_u, xₕ, xₕ_old, w) + s₀MCA(MCA_u, w)
  384. BMCAu(w) = bMCAu(w, uh_MCAb, uh_MCAu_old, λ)
  386. # Extract quadrature points ordered by arc length
  387. xΓ = dΓ.quad.cell_point.values
  388. xΓ = lazy_map(Reindex(xΓ), dΓ.quad.cell_point.ptrs)
  389. alenΓ = lazy_map(Broadcasting(x -> atan(x[2], -x[1])), xΓ)
  390. flat_xΓ = vcat(xΓ...)
  391. flat_alenΓ = vcat(alenΓ...)
  392. num_qpoints = length(flat_xΓ)
  393. perm = sortperm(flat_alenΓ)
  394. flat_alenΓ = R2 * flat_alenΓ[perm]
  396. # Pre-allocate time-series arrays
  397. ract = zeros(nΔt, num_qpoints)
  398. rhot = zeros(nΔt, num_qpoints)
  399. MCAbt = zeros(nΔt, num_qpoints)
  400. σₐt = zeros(nΔt, num_qpoints)
  401. χt = zeros(nΔt, num_qpoints)
  402. vt = zeros(nΔt, num_qpoints)
  403. vnt = zeros(nΔt, num_qpoints)
  404. xt = zeros(nΔt, num_qpoints)
  405. tent = zeros(nΔt, num_qpoints)
  407. # Record initial state
  408. vt[1,:] = vcat(lazy_map(υₕ ⋅ τ, xΓ)...)[perm]
  409. vnt[1,:] = vcat(lazy_map(υₕ ⋅ nΓ, xΓ)...)[perm]
  410. xt[1,:] = vcat(lazy_map(xₕ ⋅ τ, xΓ)...)[perm]
  411. MCAbt[1,:] = vcat(lazy_map(uh_MCAb, xΓ)...)[perm]
  412. tent[1,:] = vcat(lazy_map(ten, xΓ)...)[perm]
  414. # Pre-equilibration loop (coarse diffusion coefficients)
  415. for ti in 1:100
  416. op_rho = AffineFEOperator(Arho, Brho, Uᴿ, Vᴿ)
  417. ρₕ = solve(op_rho); ρₕ_old = ρₕ
  418. op_rac = AffineFEOperator(Arac, Brac, Uᴿ, Vᴿ)
  419. Rₕ = solve(op_rac); Rₕ_old = Rₕ
  421. ractt = vcat(lazy_map(Rₕ, xΓ)...)[perm]
  422. rhott = vcat(lazy_map(ρₕ, xΓ)...)[perm]
  423. plotting("rac", ractt, pPNG * "Rac_time_initial/", "$ti")
  424. plotting("rho", rhott, pPNG * "Rho_time_initial/", "$ti")
  425. end
  427. # Switch to physical diffusion coefficients
  428. msₕ = get_maximum_magnitude_with_dirichlet(υₕ)
  429. a_R, b_R, a_ρ, b_ρ = rac_rho_weak_forms2(
  430. control.Δt, kin.dᵃ, kin.dᵇ, kin.Drac, kin.Drho,
  431. nΓ, dΓ, coup.α, coup.β)
  432. #Arac(rac, w) = a_R(rac, w, υₕ) + s₀R(rac, w)
  434. # ── Main time loop ───────────────────────────────────────────────────────
  436. while t < control.T + tol
  438. # Optogenetic activation window: on after step 50, off after step 150
  439. if i == 50
  440. op_α₀ = AffineFEOperator(A₀opto, bα₀opto2, Uᴿ, Vᴿ)
  441. op_β₀ = AffineFEOperator(A₀opto, bβ₀opto2, Uᴿ, Vᴿ)
  442. α₀v = solve(op_α₀); β₀v = solve(op_β₀)
  443. end
  444. if i == 150
  445. op_α₀ = AffineFEOperator(A₀opto, bα₀opto, Uᴿ, Vᴿ)
  446. op_β₀ = AffineFEOperator(A₀opto, bβ₀opto, Uᴿ, Vᴿ)
  447. α₀v = solve(op_α₀); β₀v = solve(op_β₀)
  448. end
  450. # Update mass conservation Lagrange multiplier
  451. i1 = ∑(∫(uh_MCAb)dΓ); i2 = ∑(∫(uh_MCAu)dΓ)
  452. λ = conservation(i2, i1, Minitial)
  454. @info "Time step $i, time $(trunc(t, digits=4)), Δt = $(control.Δt)"
  456. # Solve velocity
  457. aᵛ, bᵛ = cortical_flow_problem_mechanochemical_axisymmetric_dimensional(
  458. xₕ, xₕ_old, control.Δt, mech.η,
  459. ρₕ, uh_MCAb, dΩᶜ, dΓ, nΓ, γʷ,
  460. mech.χ, mech.χ₀, activity, mech.sigmaₐ⁰, mech.sigmaρ⁰)
  461. op = AffineFEOperator(aᵛ, bᵛ, UVʷ, Yᵛ)
  462. υₕ = solve(op)
  463. υₕtan = to_tangent_vector(υₕ, nΓ)
  465. # Solve membrane displacement
  466. aˣ(x, w) = m(uh_MCAb, control.Δt, x, w) +
  467. aᴸ(mech.Λ, R2, xₕ, x, w) +
  468. aᴹ(mech.M, R2, xₕ, x, w) +
  469. s₀x(x, w) + wt(x, w)
  470. bˣ(w) = m(uh_MCAb, control.Δt, xₕ, w) + bₓ(uh_MCAb, υₕ, w)
  471. op_x = AffineFEOperator(aˣ, bˣ, UXʷ, Vʷ)
  472. xₕ = solve(op_x)
  474. # Update tension
  475. op_ten = AffineFEOperator(Aten, bten, Uᴿ, Vᴿ)
  476. ten = solve(op_ten)
  477. xₕ_old = xₕ
  479. i = i + 1
  480. t = t + control.Δt
  482. writesol && postprocess_all_with_tangent(
  483. φ, dΩᶜ.quad.trian, Rₕ, ρₕ, xₕ, υₕ, υₕtan, uh_MCAb, ten;
  484. i=i, of=control.output_frequency, name=pVTU)
  486. # Update leading-edge boundary condition (velocity and displacement)
  487. Rₕaux = vcat(lazy_map(Rₕ, xΓ)...)[perm]
  488. _ten = vcat(lazy_map(ten, xΓ)...)[perm]
  489. _ten = _ten[end] * threshold2(_ten[end], 0.0001, 0.0)
  490. vₗ = coup.vCTE *
  491. threshold2(Rₕaux[end], coup.rac0, 1.3 * Rₕaux[1]) /
  492. (1 + _ten * _ten / coup.ten0)
  493. xₗ = xₗ * 0.9 + control.Δt * vₗ # slight spatial relaxation at boundary
  495. # Rebuild FE spaces on updated geometry
  496. UXʷ, UVʷ, Vʷ, Xᵛ, Yᵛ, Xʳ, Yʳ, Uᵉ, Vᵉ, Vᴿ, Uᴿ, Ωᶜ, dΩᶜ, dΓ, nΓ, φ =
  497. update_all!(i, t, control.Δt, υₕ, msₕ)
  499. # Solve Rac and Rho
  500. op_rac = AffineFEOperator(Arac, Brac, Uᴿ, Vᴿ)
  501. Rₕ = solve(op_rac); Rₕ_old = Rₕ
  502. op_rho = AffineFEOperator(Arho, Brho, Uᴿ, Vᴿ)
  503. ρₕ = solve(op_rho); ρₕ_old = ρₕ
  505. # Solve MCA bound and unbound
  506. op_MCAb = AffineFEOperator(AMCAb, BMCAb, Uᴿ, Vᴿ)
  507. uh_MCAb = solve(op_MCAb); uh_MCAb_old = uh_MCAb
  508. op_MCAu = AffineFEOperator(AMCAu, BMCAu, Uᴿ, Vᴿ)
  509. uh_MCAu = solve(op_MCAu); uh_MCAu_old = uh_MCAu
  511. # Record time-series data
  512. ract[i,:] = vcat(lazy_map(Rₕ, xΓ)...)[perm]
  513. rhot[i,:] = vcat(lazy_map(ρₕ, xΓ)...)[perm]
  514. MCAbt[i,:] = vcat(lazy_map(uh_MCAb, xΓ)...)[perm]
  515. σₐt[i,:] = mech.sigmaₐ⁰ .+ mech.sigmaρ⁰ * rhot[i,:]
  516. χt[i,:] = mech.χ₀ .+ mech.χ * MCAbt[i,:]
  517. vt[i,:] = vcat(lazy_map(υₕ ⋅ τ, xΓ)...)[perm]
  518. vnt[i,:] = vcat(lazy_map(υₕ ⋅ nΓ, xΓ)...)[perm]
  519. xt[i,:] = vcat(lazy_map(xₕ ⋅ τ, xΓ)...)[perm]
  520. tent[i,:] = vcat(lazy_map(ten, xΓ)...)[perm]
  522. plotting("rac", ract[i,:], pPNG * "Rac_time/", "$i")
  523. plotting("rho", rhot[i,:], pPNG * "Rho_time/", "$i")
  524. plotting("MCAb", MCAbt[i,:], pPNG * "MCAb_time/", "$i")
  525. end
  527. plots_run_singlet(nΔt, vt, vnt, xt * mech.R, MCAbt, ract, rhot, pPNG,
  528. num_qpoints, π * R2, control.Δt, control.T, flat_alenΓ, σₐt, χt, tent)
  529. end

MechanochemicalAxisymmetricVector.jl at commit cd51f0f, under CC-BY-SA-4.0 · at the source

Overview

Authors: Henry De Belly1,2,3, Andreu F. Gallén4, Evelyn Strickland1,2,5, Dorothy C. Estrada1,2, David Sanchez Godinez1,2, Eric Neiva4,6, Patrick J. Zager1,2, Tamas L. Nagy1,2,7, Janis K. Burkhardt8, Hervé Turlier4,9, Orion D. Weiner1,2
ORCID iDs: Henry De Belly, Tamas L. Nagy, Janis K. Burkhardt, Hervé Turlier, Orion D. Weiner
  1. Cardiovascular Research Institute, University of California, San Francisco,San Francisco, CA USA
  2. Department of Biochemistry and Biophysics, University of California, San Francisco,San Francisco, CA USA
  3. Present Address: Children’s Medical Center Research Institute, University of Texas Southwestern Medical Center,Dallas, TX USA
  4. Center for Interdisciplinary Research in Biology (CIRB), Collège de France, CNRS, INSERM, Université PSL,Paris, France
  5. Present Address: Immunology Program, Memorial Sloan Kettering Cancer Center,New York, NY USA
  6. Present Address: Department of Fluid Mechanics, Universitat Politècnica de Catalunya - Barcelona Tech (UPC),Barcelona, Spain
  7. Present Address: Broad Stem Cell Research Center, University of California, Los Angeles,Los Angeles, CA USA
  8. Department of Pathology and Laboratory Medicine, Children’s Hospital of Philadelphia Research Institute and Perelman School of Medicine, University of Pennsylvania,Philadelphia, PA USA
  9. Present Address: Center for Integrative Biology (CBI), University of Toulouse, CNRS,Toulouse, France
Institutions: University of California, San Francisco (United States); The University of Texas Southwestern Medical Center (United States); Université Paris Sciences et Lettres (France); Memorial Sloan Kettering Cancer Center (United States); Universitat Politècnica de Catalunya (Spain); University of California, Los Angeles (United States); Children's Hospital of Philadelphia (United States); Université de Toulouse (France)
Journal: Nature cell biology, volume 28, issue 6, pages 1244-1257
Dates: received 13 April 2025; accepted 17 April 2026; published online 10 June 2026; in print 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1038/s41556-026-01965-1 · PMID 42270977 · PMCID PMC13279283 · OpenAlex W4403067808
Open access: hybrid, a free copy (OpenAlex)
Preprint: doi:10.1101/2024.10.01.616161
Status: code verified
Categories: human (organism), cellular / molecular (subfield)
Methods: Statistics, Evoked potentials, fMRI & imaging
Keywords: Cell polarity, Chemotaxis, Membrane biophysics, Actin
MeSH: Cell Movement*, Cell Polarity*, rac GTP-Binding Proteins*, rho GTP-Binding Proteins*, Actins, Animals, Cell Membrane, Epithelial Cells, Humans, Mechanistic Target of Rapamycin Complex 2, Optogenetics, rac1 GTP-Binding Protein, Signal Transduction (* major topic)
Topic: Protein Kinase Regulation and GTPase Signaling (Molecular Biology, Biochemistry, Genetics and Molecular Biology), according to OpenAlex
Funding: NIGMS (R35GM118167, K99GM154115, R21AI1739938); RCUK | Biotechnology and Biological Sciences Research Council (BBSRC) (2019598); H2020 Marie Skłodowska‐Curie Actions (101150259); American Heart Association (24PRE1198865); EC | Horizon 2020 Framework Programme (EU Framework Programme for Research and Innovation H2020) (101105565, 949267)
Citations: cited by 1 paper (Europe PMC); 89 references in the paper

Abstract

In migrating cells, the GTPase Rac organizes a protrusive front, whereas Rho organizes a contractile back. How these GTPases are positioned at opposite poles remains unclear. We leverage optogenetics, mechanical perturbations, and mathematical modelling to reveal a surprising mechanochemical long-range mutual activation between front and back polarity programmes that complements their well-known local mutual inhibition. Rac-based protrusions elevate membrane tension, stimulating an mTORC2-dependent activation of Rho at the opposite side of the cell. Conversely, Rho-mediated contractility induces cortical-flow-based regulation of phosphoinositide signalling that triggers Rac activation distally. We develop a minimal mechanochemical model to explain how long-range facilitation, together with local inhibition, enables robust Rho and Rac partitioning. Our findings demonstrate how the actin cortex and plasma membrane interact as an integrated mechanochemical system for long-range Rac–Rho patterning. This circuit is required for efficient polarity and migration in primary human T cells and is conserved in epithelial cells, highlighting the generality of this mechanism.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repositories

Its files are read in the Code ↔ Paper reader above, with 1 match between paragraphs and lines of code.

VirtualEmbryo/mechanochemical_polarization

License: CC-BY-SA-4.0
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Commit: cd51f0f91e039929abac2b6ce466ff2edbb11b67, 15 April 2026
Languages: Julia (24)
Size: 466 files, 24 scripts
Software Heritage: not archived
Found in: “Code availability”
Holds: README, license file, environment (mechanochemical_2D_axisymmetric/Manifest.toml, mechanochemical_2D_axisymmetric/Project.toml)
Not found: CITATION.cff, tests, continuous integration, documentation
Tools: Plots.jl (11 files)
Availability: 1 check, the latest on 27 September 2026: the link answers
  • 27 September 2026: the link answers
26 files
At the source: github.com/VirtualEmbryo/mechanochemical_polarization

Zenodo 19591544

License: CC-BY-SA-4.0
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Size: 1 file
Software Heritage: not checked
Found in: “Code availability”
Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: Plots.jl (11 files)
Availability: 1 check, the latest on 27 September 2026: the link answers (HTTP 200)
  • 27 September 2026: the link answers (HTTP 200)
26 files
At the source:

Code availability

Unique code generated for this study can be found on GitHub at https://github.com/VirtualEmbryo/mechanochemical_polarization and Zenodo at 10.5281/zenodo.19591544 (ref. 89).

Reproduced under the paper's license (CC BY), from the paper cited above.

Tracing map

Proposed by the machine: these links were found in the paper and verified at the source, without human review. The map will receive a Zenodo DOI once one of the paper's authors has validated it with their ORCID.

What the map holds:

  • 2 repositories of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
  • 48 scripts, each with its path and the digest of its content;
  • 1 match between paragraphs of the paper and lines of the code (method lexical-v1);
  • neither the text of the paper nor the code itself.

Its JSON (tracing-map.json) is deposited on Zenodo with its DOI once the map is validated.

Data

No dataset and no data link were found in the paper.

Data availability

All data supporting the findings of this study will be made available within the manuscript. All other data supporting the findings of this study can be obtained from the corresponding authors upon reasonable request. Source data are provided with this paper.

Reproduced under the paper's license (CC BY), from the paper cited above.

Versions

The history of this record: each version stored by the harvester or made by a correction of its authors or of the maintainers of its code, and what changed in its facts. The texts of the paper (its abstract, its availability statements) are not part of it; versions that changed only those are not listed.

Version 2, 28 September 2026

  • Publisher: n/a → Nature Portfolio

Version 1, 27 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 11 authors, 4 keywords, 13 MeSH terms, 5 funders, 89 references.

Cite

This paper

De Belly, H., Gallén, A. F., Strickland, E., Estrada, D. C., Godinez, D. S., Neiva, E., Zager, P. J., Nagy, T. L., Burkhardt, J. K., Turlier, H., & Weiner, O. D. (2026). Long-range mutual activation establishes Rho and Rac polarity during cell migration. Nature cell biology, 28(6), 1244-1257. https://doi.org/10.1038/s41556-026-01965-1

BibTeX

@article{debelly2026long,
author = {De Belly, Henry and Gallén, Andreu F. and Strickland, Evelyn and Estrada, Dorothy C. and Godinez, David Sanchez and Neiva, Eric and Zager, Patrick J. and Nagy, Tamas L. and Burkhardt, Janis K. and Turlier, Hervé and Weiner, Orion D.},
title = {{Long-range mutual activation establishes Rho and Rac polarity during cell migration}},
journal = {Nature cell biology},
year = {2026},
month = jun,
volume = {28},
number = {6},
pages = {1244--1257},
publisher = {Nature Portfolio},
issn = {1465-7392},
doi = {10.1038/s41556-026-01965-1},
url = {https://doi.org/10.1038/s41556-026-01965-1},
pmid = {42270977},
pmcid = {PMC13279283}
}

RIS

TY - JOUR
AU - De Belly, Henry
AU - Gallén, Andreu F.
AU - Strickland, Evelyn
AU - Estrada, Dorothy C.
AU - Godinez, David Sanchez
AU - Neiva, Eric
AU - Zager, Patrick J.
AU - Nagy, Tamas L.
AU - Burkhardt, Janis K.
AU - Turlier, Hervé
AU - Weiner, Orion D.
TI - Long-range mutual activation establishes Rho and Rac polarity during cell migration
T2 - Nature cell biology
J2 - Nat Cell Biol
PY - 2026
DA - 2026/06/10
VL - 28
IS - 6
SP - 1244
EP - 1257
SN - 1465-7392
PB - Nature Portfolio
DO - 10.1038/s41556-026-01965-1
UR - https://doi.org/10.1038/s41556-026-01965-1
LA - en
ER -

CSL-JSON

{
"id": "10.1038/s41556-026-01965-1",
"type": "article-journal",
"title": "Long-range mutual activation establishes Rho and Rac polarity during cell migration",
"container-title": "Nature cell biology",
"author": [
{
"family": "De Belly",
"given": "Henry"
},
{
"family": "Gallén",
"given": "Andreu F."
},
{
"family": "Strickland",
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"family": "Estrada",
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},
{
"family": "Godinez",
"given": "David Sanchez"
},
{
"family": "Neiva",
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"given": "Patrick J."
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{
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"given": "Tamas L."
},
{
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"given": "Janis K."
},
{
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},
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"family": "Weiner",
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}
],
"container-title-short": "Nat Cell Biol",
"volume": "28",
"issue": "6",
"page": "1244-1257",
"DOI": "10.1038/s41556-026-01965-1",
"PMID": "42270977",
"PMCID": "PMC13279283",
"ISSN": "1465-7392",
"publisher": "Nature Portfolio",
"URL": "https://doi.org/10.1038/s41556-026-01965-1",
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