02 · Force balance over time
Pulling and pushing alternate through the sequence
Inward contractility is plotted above zero; outward contractility is plotted below. T003 is push-dominant, while T023 carries the strongest inward pull.
3D traction reconstruction · 30 timepoints
Cell–Matrix
Interactions During
Pore Traversal
The cell alternates between front and rear anchoring while exerting intermittent outward pressure near its center. A stabilized inverse model resolves these forces without discarding the large collagen deformations.
- Max Contractility
- 3.22 nN
- Max Deformation
- 1.41 µm
- Max Pushing force
- 1.41 nN
↓ Advance one view at a time
01 · T003
Rear anchoring accompanies swimming-like propulsion
The opening view places the cell within the deformation-colored collagen network. The following views isolate the projected force pattern around the same cell.
- Pulling P95 / P99
- 0.69 / 0.98 µm
- Pushing P95 / P99
- 0.56 / 0.71 µm
- Total force
- 4.38 nN
Pulled inwardPushed outward
Drag to rotate
03 · Physical scale
Nanonewton forces are sufficient
A linear elastic estimate gives the correct scale. Near the cell, a complete 2–3 nN traction field can generate approximately 0.4–0.7 µm of displacement.
For an unbounded linear material, u = F / 4πGr. Using the project’s linearized shear modulus, 2.4 nN produces 0.40 µm at 5 µm from the load and 0.66 µm at 3 µm.
Move across the curves to inspect force, distance, and predicted displacement.
04 · Forward simulation
The measured peak force produces the measured deformation scale
A balanced SAENOpy force dipole uses the peak inward contractility, 3.22 nN. The simulated peak displacement is 1.08 µm, close to the 1.01 µm maximum measured at T023.
- Applied dipole
- 3.22 nN
- Simulated peak
- 1.08 µm
- Measured T023 peak
- 1.01 µm
Displacement arrows are enlarged 5× for visibility; all reported values use the unscaled field.
lowhigh displacement
Drag to rotate · scroll to zoom
05 · Reconstruction method
The reconstruction passes through three regimes
Each regime solves a slightly different objective. The force field is warm-started from the previous solution, but the loss can reset when a penalty changes.
Broad fit
α = 10⁸ · bulk / shell = 1
Fit the measured displacement before preferring forces near the cell surface.
Surface localization
α = 10⁸ · bulk / shell: 10³ → 10⁶
Raise the cost of bulk forces in two warm-started solves.
Force-scale refinement
bulk / shell = 10⁶ · α: 10⁸ → 10⁶
Lower α along the stable branch. Force and torque balance remain active throughout.
One curve, three regimes
Loss falls within each regime. At both boundaries the objective changes, causing a small reset; the previous force field is reused as the next starting point.
minf ‖umodel(f) − umeasured‖² + αΣshell‖fᵢ‖² + rαΣbulk‖fᵢ‖² + wF‖Σfᵢ‖² + wτ‖Σrᵢ × fᵢ‖²
The curve explains the optimization logic; it is not presented as a measured convergence trace.
06 · Production settings
One parameter set is used for all 30 frames
- Surface shell
- 3 µm
- Penalty ratios
- 1 → 10³ → 10⁶
- α path
- 10⁸ → 5·10⁷ → 2·10⁷ → 10⁷ → 5·10⁶ → 2·10⁶ → 10⁶
- Selected α
- 10⁷
- Material
- k = 1,449 Pa · d₀ = 0.0022 · λₛ = 0.032 · dₛ = 0.055
- Solver
- step 0.33 · 40 iterations · relative convergence 0.005
- Balance weights
- wF = 1 · wτ = 1
Beyond slide 04
Supplementary and work-in-progress slides
The main presentation ends here. Continue to the reconstruction details, validation plots, additional 3D frames, and current interpretation.
S1 · Parameter stability
A stable force scale emerges
Whole-series contractility plateaus from α = 107 to 108.
- Total force variation
- 1.152×
- Pulling variation
- 1.118×
- Quality pass
- 100%
S2 · Validation
The localization pattern persists
Pulling is enriched at a cell end in 29 frames; central pushing appears in 18 frames.
S3 · T022
The dominant pull moves to the front
Later in the sequence, the leading end becomes the main anchor. Central outward traction remains detectable while the cell advances through the constriction.
- Front pull enrichment
- 2.08×
- Center push enrichment
- 1.34×
- Total force
- 4.63 nN
Pulled inwardPushed outward
Drag to rotate · scroll to zoom
S4 · T023
Central pushing accompanies strong end traction
The strongest force frame combines pulling at both ends with outward traction near the central internal signal. This is consistent with pressure used to widen a pore.
- End pulling
- front + rear
- Center push enrichment
- 1.28×
- Total force
- 5.86 nN
Pulled inwardPushed outward
Drag to rotate · scroll to zoom
S5 · Interpretation
Anchoring shifts as the cell traverses the pore
Early frames are more often rear-dominant. From T013 onward, front pulling becomes more common before rear dominance returns near the end.
Central pushing is intermittent. The robust conclusion is an alternating end-anchor mechanism with occasional central outward pressure—not a fixed pattern at every timepoint.
Rear anchor
→T000–T012
Central pressure
→intermittent
Front anchor
T013–T026
- End pulling enriched
- 29 / 30
- Central pushing enriched
- 18 / 30
- Front / rear dominant
- 16 / 14
S6 · Method and limits
Stable numerically,
conditional physically
The force scale is reproducible across α and predicts held-out displacement blocks. Absolute nN values still depend on the assumed collagen material law and the displacement reference.
- 01
Broad-domain fit keeps bulk forces visible during reconstruction.
- 02
Penalty continuation transfers support into a 3 µm surface shell.
- 03
Force and torque balance removes non-physical branches.
- 04
Seven-α validation identifies the decade-wide contractility plateau.
Validated reconstruction · α = 107 · production full series