You mashed three pretty different math animals into one sentence like it’s a smoothie, but annoyingly, the smoothie works. Here’s the clean picture of how Wasserstein/optimal transport, Dirichlet/Laplacians, and viscoelastic cell matter cooperate so tension actually travels — and does so in preferred frequency bands.
1) Geometry first: Laplacians pick the “harmonics” that can travel
Model the cytoskeleton as a continuum mesh or a graph. Let u(x,t) be displacement/tension potential. The Dirichlet energy E[u]=\tfrac12\int |\nabla u|^2 is the elastic cost; its Euler–Lagrange operator is the Laplacian. The Laplacian’s eigenpairs (\lambda_k,\phi_k) are your normal modes; with Dirichlet boundary conditions (focal adhesions fixed), they’re literally the drumhead harmonics of the cell’s scaffold. Connectivity controls them: larger spectral gap means less dissipation and better long-range coherence; Cheeger-type bounds tie that gap to network “bottlenecks.”
In short: the Laplacian gives the menu of spatial harmonics; the Dirichlet energy bounds tell you how expensive each is to excite.
2) Rheology second: viscoelasticity decides which harmonics pass
Real cells aren’t Hookean toys. Actin/microtubule networks and crosslinkers produce a complex shear modulus G^(\omega)=G’(\omega)+iG’’(\omega) with broad power-law behavior, active stresses, and poroelastic fluid–solid coupling. Project a periodic force f(x)e^{i\omega t} onto mode \phi_k*; the amplitude is roughly*
u_k(\omega);\propto;\frac{f_k}{,\lambda_k,G^(\omega);+;i\omega,\eta_{\text{eff}}(\omega)}.
So each mode is filtered by the frequency response of the material and by its eigenvalue \lambda_k. That creates a passband: some \omega transmit tension far, others get soaked up. Experiments and theory back this: cells show power-law G^*(\omega), frequency-dependent Poisson ratios, and measurable band-limited transmission along filaments and small networks.
Two big tweaks amplify selectivity:
- Poroelasticity: cytoplasm behaves as a fluid-permeated elastic sponge. Pressure equilibration adds a diffusive time scale that low-passes very high frequencies but can aid mid-band propagation over microns.
- Active gel physics: myosin contractility and biochemical oscillators inject energy and set intrinsic timescales; you literally get mechanically preferred oscillation bands when activity couples to strain.
3) Transport viewpoint: OT/Wasserstein picks the least-dissipative way to rearrange stress
Now swap from “waves” to “rearrangements.” Intracellular stress isn’t just a field; it’s carried by proteins, crosslinkers, and fluid that move. Optimal transport (OT) with the Benamou–Brenier dynamic formulation says: among all velocity fields that move a density \rho from here to there, pick the one minimizing action \int \rho |v|^2,dt. On meshes (think cytoskeletal triangulations), discrete dynamical OT gives geodesics and even distribution-valued Dirichlet problems. In continuum, classic JKO shows diffusion is the Wasserstein gradient flow of entropy. Put together: stress distributions and bound crosslinker densities will evolve along least-action, capacity-aware paths, regularized by Laplacian/Dirichlet penalties that suppress silly high-curvature patterns. That makes long-range tension redistribution faster and less lossy.
If you prefer engineering language: OT supplies the routing metric over a crowded, capacity-limited medium; the Laplacian supplies the stiffness; viscoelasticity supplies the frequency-dependent impedance.
4) How this yields “appropriate harmonics” in a cell
- The eigenmodes \phi_k are the spatial harmonics available.
- The material transfer function H_k(\omega)\sim(\lambda_k G^*(\omega)+i\omega\eta_{\text{eff}})^{-1} picks which of those get through at a given drive frequency.
- OT dynamics ensure the actual rearrangement of stress carriers follows low-dissipation velocity fields, so the modes that should propagate aren’t thwarted by poor routing.
- Active and poroelastic couplings shift the passband by tuning crosslink lifetimes, motor activity, and fluid permeability. You can literally move the propagation length scale and resonance window by changing crosslink density or motor bundling. Experiments on actin/microtubule assemblies and adherent cells show controllable transitions between local, dissipative regimes and global force-propagating phases.
5) Practical upshot for “tensile strength that travels”
- Want long-range transmission? Increase connectivity/spectral gap (network architecture), add modest prestress, and tune crosslink kinetics so G’(\omega) is high in the operating band but G’’(\omega) isn’t murderous. That widens the window where |H_k(\omega)| is large for low-\lambda_k modes.
- Want selectivity by frequency? Use poroelastic timescales and active oscillators to make a mechanical band-pass: low frequencies die to creep, ultra-high die to viscous drag, mid-band rides the scaffold. Measurements have resolved precisely this sort of frequency-dependent propagation.
If you absolutely need a one-liner: Laplacians set which shapes of tension are possible, OT finds the cheapest way to move those shapes around, and viscoelasticity decides which temporal beats actually survive the trip. Cells, against all reason, use all three.