Recipe — Noncontact FM-AFM Hair Bundle Mechanics

Recipe — Noncontact FM-AFM for Hair Bundle Mechanics

How to compute hair-bundle passive stiffness $k_b$ and viscous damping $c_b$ from acoustic-frequency-modulated AFM measurements, with parameter-selection heuristics and pitfalls. Use this when modelling, planning a wet-lab calibration, or interpreting Cartagena-Rivera-style data without re-reading the source paper.

When to use this recipe

• You need a noncontact measurement at physiological acoustic frequencies (kHz) with sub-piconewton stimulus.
• You want stiffness AND damping from one experiment.
• The bundle is intact (no peeled tectorial membrane needed if a $Tect{a}^{-/-}$ background is available).
• Microprobe stiff-stick methods would over-stretch links and underestimate stiffness.

Master equations

Bundle stiffness (Eq. 1):

$$k_b=2k_c^{2/3}\Delta f^{2/3}{\left(\frac{3\mu }{\pi R{f}_{\text{far}}}\right)}^{1/3}\left(\frac{A_b\sin \beta }{\left(h_m^2\right(\frac{h_{\text{far}}}{h_m}-1)\cos \beta {)}^{2/3}}\right)$$

Bundle damping (Eq. 2):

$$c_b=-\frac{3k_b}{A_b\sin \beta }\frac{1}{\pi f_{\text{far}}^2\frac{d\Phi }{df}}$$

Symbol legend (verbatim from [Cartagena-Rivera 2019]):

Symbol	Meaning	Typical value
$k_c$	calibrated cantilever spring constant	0.10–0.17 N·m⁻¹
$\mu$	fluid viscosity	water at 37 °C
$R$	bead radius	5 µm (10-µm sphere)
$A_b$	effective area at bundle top	5–6 µm² (mouse OHC apical, see OHC Bundle Geometry Apical Mouse Cartagena 2019)
$\beta$	sensory-epithelium angle	4°–15° (age-dependent)
$h_m$	minimum gap (sphere ↔ tallest stereocilia)	50 nm
$h_{\text{far}}$	far gap (unperturbed reference)	1 µm (or 500 nm)
$f_{\text{near}},f_{\text{far}}$	resonant frequencies at $\Phi =\pi /2$, near vs. far	30–45 kHz
$\Delta f$	$f_{\text{near}}-f_{\text{far}}$	the measured shift
$d\Phi /df$	slope of phase–frequency curve at $f_{\pi /2}$	—

Decision tree: parameter selection

1. Cantilever — gold-coated triangular silicon-nitride (Bruker MLCT or equivalent), tipless, with a 10-µm borosilicate microsphere attached. Pre-calibrated $k_c\approx 0.12$ N·m⁻¹ from supplier (e.g. Novascan); confirm in-house with the thermal-tune method. Validate the workflow first against a tipless cantilever of known stiffness — the FM-AFM-derived value should match thermal calibration ($P>0.05$).
2. Drive frequency — sweep around resonance in liquid; choose the largest peak with a well-defined phase change (typically 30–45 kHz). Forest-of-peaks behaviour is normal in liquid piezo drive ([Schäffer 2002 J. Appl. Phys.]).
3. Drive amplitude — adjust the piezo so that bundle-side oscillation amplitude at $f_{\pi /2}$ is ≤ 5 nm (assumption: amplitude ≪ $h_m$ ≪ $R$).
4. Approach — engage tapping mode, gently touch the bundle, then retract to $h_{\text{far}}=1$ µm, set phase to $\pi /2$. Record $f_{\text{far}}$.
5. Sweep series — record phase–frequency sweeps at multiple gaps from 1 µm or 500 nm down to 50 nm. Then pull to 4 µm and re-record $f_{\infty }$ (drift check).
6. Fit $f_{\pi /2}$ — second-order polynomial in the vicinity of $\pi /2$ extracts $f_{\pi /2}$ at each gap.
7. Compute $\Delta f$ — $f_{\text{near},\pi /2}-f_{\text{far},\pi /2}$ at the 50-nm gap.
8. Compute $d\Phi /df$ — third-order polynomial fit on the 50-nm sweep. Slope is constant across gaps in practice (this is the validity check for Eq. 2 — see [Cartagena-Rivera 2019, Fig. 5A]).

Force scale (sanity check)

Applied force on the bundle is

$$F=2k_{c}A\frac{\Delta \omega }{\omega }\approx 1\text{–}10\text{pN}$$

i.e. the same order as physiological hearing forces ([Hudspeth 2014 Nat. Rev. Neurosci.]). This is 10×–1000× lower than stiff-microprobe deflection. If your computed $F$ exceeds 100 pN you are no longer in the linear/passive regime — re-tune amplitude.

Physical assumptions you must respect

• $A\ll h_m\ll R$ (5 nm ≪ 50 nm ≪ 5 µm). If you change bead radius, re-derive.
• Sphere oscillation is normal-dominated as long as $\beta \le 15°$. Above that, the in-plane component of $V\sin \beta$ is no longer a small perturbation.
• Drive frequency is well above the bundle’s excitatory band (≈ 4–10 kHz in mouse apex) — you measure passive mechanics, not the active gating compliance.
• Bundle is treated as Kelvin–Voigt (single elastic + single dashpot, in parallel). Sub-component decomposition (gating spring vs. pivot vs. HTC) requires additional perturbations like tip-link cleavage; see Hair Bundle Stiffness Decomposition.

Pitfalls

• Tectorial-membrane peeling damages the bundle and underestimates stiffness/damping at mature ages (P ≥ 9). Use $Tect{a}^{-/-}$ background to keep the bundle intact while detaching the TM constitutively.
• Geometric inputs decay fast with age: $\beta$ shifts ~4× from P10 → P14. Re-measure $A_b$ and $\beta$ for every developmental stage rather than reusing P14 values.
• Reticular-lamina interference: the apical surface itself is 5–10× stiffer than a bundle ([Cartagena-Rivera 2019, fig. S1]). Position the bead above the bundle, not over a supporting cell.
• No SI numbers for HTC counts: the paper does NOT report links per bundle or per-link stiffness. Per-link values that appear elsewhere in the STRC vault (e.g. ~0.16 pN/nm) are external derivations, not measurements (see audit notes in stereocilia-mechanics).

Output you should expect (mouse OHC, apical, P13–P15)

Quantity	$Str{c}^{+/-}/Tect{a}^{-/-}$	$Str{c}^{-/-}/Tect{a}^{-/-}$
Bundle stiffness $k_b$	5.12 ± 0.46 pN/nm	2.05 ± 0.15 pN/nm
Bundle damping $c_b$	10.76 ± 1.2 kPa·s·m⁻¹	2.85 ± 0.3 kPa·s·m⁻¹

(Source: dulon-2019-htc-bundle-mechanics; full developmental series in OHC Bundle Damping Cartagena 2019.)

Connections

• [source] dulon-2019-htc-bundle-mechanics
• [part-of] stereocilia-mechanics
• [see-also] OHC Bundle Damping Cartagena 2019
• [see-also] OHC Bundle Geometry Apical Mouse Cartagena 2019
• [see-also] Hair Bundle Stiffness Decomposition
• [applies] h09-hydrogel — calibration target for HTC-substitute peptide hydrogel
• [applies] h05-calcium-oscillation — required for acoustic-driven OHC mechanical model
• [applies] h01-pharmacochaperone — bundle-mechanics gate on rescue-quality