# Red blood cells rheology measured using AFM. A comparison with other studies

Agathe Nidriche, Magalie Faivre, Taha Benyattou, Claude Verdier

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Agathe Nidriche, Magalie Faivre, Taha Benyattou, Claude Verdier. Red blood cells rheology measured using AFM. A comparison with other studies. Multidisciplinary Biomechanics Journal, 2025, 50th congress of the.., 10.46298/mbj.16234. hal-05191789v2

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Submitted on 23 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
dishes, using a Nanowizard 4XP AFM (Bruker, Berlin) in contact and oscillary modes, using tips (QUEST20, UK) with a 3µm-diameter bead glued at the edge. Cantilever stiffness calibration gave k=0.02 N/m. A typical approach curve was made with a small set-point F₀~50pN, followed by cantilever retraction at velocities of 1µm/s. The analysis was made using Hertz model for a sphere i.e. where E is the Young modulus (Pa), v is the Poisson ratio (~0.5), R the bead radius and δ₀ is the indentation. For microrheology, a similar approach was made, followed by application of small sinusoidal indentations d=50nm at angular frequency ω=2πf (frequency f in Hz) superposed to the initial δ₀. Linearization of Hertz equation leads to G*(ω)=G'+iG''=[(1-ν)(F*/δ*-iωb(0))]/4(Rδ₀)¹/², the dynamic modulus, with b(0) determined as previously (Abidine *et al.*, 2018).

## 3. Results and discussion

### 3.1 Young modulus

The Young's modulus was measured using Hertz equation up to 20pN, corresponding to d₀ < 1µm, to reduce substrate effects. Measurements (N~20) were made for 5 donors and shown in Table 1.

<table><caption>**Table 1.** Young's moduli (Pa) for five donors at different poly-L-lysine concentration C<sub>PLL</sub> (mg/mL).</caption><thead><tr><th>C<sub>PLL</sub></th><th>0.001</th><th>0.01</th><th>0.1</th><th>1</th></tr></thead><tbody><tr><td>E (Pa)</td><td>14±8.7†</td><td>17±7.8†</td><td>20±10†</td><td>40±21†</td></tr></tbody></table>

† medians with p<0.05.

Significant differences were found, and show an increase of the Young's modulus (E) which scales with Poly-L-lysine concentration. The higher the concentration, the higher E, as expected, corresponding to firm adhesion on the glass leading to a higher membrane tension (Hategan *et al.*, 2004).

### 3.2 Viscoelastic data

Further microrheology measurements were carried out in the range [0.5-100]Hz. Fig. 1 shows the typical behavior corresponding to almost constant G' and linearly increasing G''. The slope of G'' is smaller than 1, but overall this corresponds to a viscous effect with viscosity ~ 0.3 Pa.s, probably due to spectrin effects. We observe that a Voigt model (spring and dashpost in parallel) is sufficient to describe well this behavior. The

dependence on the poly-L-lysine concentration (C<sub>PLL</sub>) is clear as the viscous moduli are almost the same (except for 0.001 mg/mL) whereas G' increases with C<sub>PLL</sub>. A soft glassy rheology model such as G*=G₀(iωλ)ⁿ + ηω provides a better fit of the data (Figure 1, solid lines). G₀ takes its values accordingly to those in Table 1, exponent n (~0.1) is close to 0 as expected from previous observations, and η decreases from 0.51 Pa.s to 0.35 Pa.s with increasing C<sub>PLL</sub>.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Elastic (G', circle) and viscous (G'', triangle) moduli (Pa) at different C<sub>PPL</sub>.</figcaption></figure>

To sum-up, these results are in very good agreement with Puig-de-Morales-Marinkovic *et al.*, 2007 but have a different dimension (Pa/nm). Since the torque applied to the bead produces shear, moduli are obtained by dividing their stress by shear deformation, i.e. displacement by RBC height (~2µm). This gives both quantitative and qualitative agreement (within a factor ~5). A bottom correction to take into account substrate effects could further improve results and comparisons with authors.

## 4. Conclusions

In this work, it was shown that viscoelastic data using AFM can give similar results as compared to magnetic twisting cytometry. The elastic shear modulus G' was rather constant in the range considered, while the shear loss modulus G'' increased with a power slope roughly equal to one. Interestingly the value of the loss modulus did not depend so much on the poly-L-lysine coating that was used for surface coverage. But the elastic shear
modulus G' was correlated significantly with the poly-L-lysine content. This means that rheological properties depend strongly on the tension developed during adhesion to the surface governing membrane properties.

Further works are needed to model the shape of RBCs more precisely and account for this effect, and results should also consider possible substrate effects.

## Acknowledgements

We thank Delphine Débarre and Lionel Bureau at LIPhy for stimulating discussions.

## Conflict of Interest Statement

The authors declare no conflict of interest.

## Contributor Roles

AN: Conceptualization, Data curation, Investigation, Formal analysis, Methodology, Writing-review & editing; MF: Methodology, Investigation, Validation;

TB: Funding acquisition, Project administration; CV: Conceptualization, Methodology, Formal analysis, Supervision, Writing-review & editing.

## Funding

This work was partly funded by the CellDance project (ANR-21-CE09-0011).

## References

Evans, E., & Hochmuch, R.M. (1976). A solid-liquid composite model of the red cell membrane. *J Membrane Biol*, *30*, 351-362. https://doi.org/10.1007/BF01869676

Turlier, H., Fedosov, D.A., Audoly, B., Auth, T., Gov, N.S., Sykes, C., Joanny, J-F., Gompper, G., & Betz, T. (2016). Equilibrium physics breakdown reveals the active nature of redbood cell flickering. *Nature Physics*, *12*(5), 513-519. https://doi.org/10.1038/nphys3621

Hategan, A., Sengupta, K., Sackmann, E., & Discher D. (2004). Topological pattern dynamics in passive adhesion of cell membranes. *Biophys J*, *87*, 3547-3560. https://doi.org/10.1529/biophysj.104.041475

Puig-de-Morales-Marinkovic, M., Turner, K.T., Butler, J. P., Fredberg, J.J., & Suresh, S. (2007). Viscoelasticity of the human red blood cell. *Am J Physiol Cell Physiol*, *293*, C597-C605. https://doi.org/10.1152/ajpcell.00562.2006

Abidine, Y., Constantinescu, A., Laurent, V.M., Sundar Rajan, V., Michel, R., Laplaud, V., & Verdier, C. (2018). Mechanosensitivity of cancer cells in contact with soft substrates using AFM. *Biophys J*, *114*, 1165-1175. https://doi.org/10.1016/j.bpj.2018.01.005