Viscoelasticity and Noise Properties Reveal the Formation of Biomemory in Cells

Living cells are neither perfectly elastic nor liquid and return a viscoelastic response to external stimuli. Nanoindentation provides force–distance curves, allowing the investigation of cell mechanical properties, and yet, these curves can differ from point to point on the cell surface, revealing its inhomogeneous character. In the present work, we propose a mathematical method to estimate both viscoelastic and noise properties of cells as these are depicted on the values of the scaling exponents of relaxation function and power spectral density, respectively. The method uses as input the time derivative of the response force in a nanoindentation experiment. Generalized moments method and/or rescaled range analysis is used to study the resulting time series depending on their nonstationary or stationary nature. We conducted experiments in living Ulocladium chartarum spores. We found that spores in the approaching phase present a viscoelastic behavior with the corresponding scaling exponent in the range 0.25–0.52 and in the retracting phase present a liquid-like behavior with exponents in the range 0.67–0.85. This substantial difference of the scaling exponents in the two phases suggests the formation of biomemory as a response of the spores to the indenting AFM mechanical stimulus. The retracting phase may be described as a process driven by bluish noises, while the approaching one is driven by persistent noise.


Section I.
In this section the viscoelastic exponent, β, of a rubber "non-living" material is obtained. We conducted two sets of four AFM-NI experiments at room temperature where Polydimethylsiloxane (PDMS) was used as a sample. In the first set of experiments approach and retraction rates are similar to that have been used in spores ~0.029 µm/sec, while in the second set of experiments, approach and retraction rates are equal to ~ 0.49 µm/sec (~17 times higher than the first control condition). PDMS is widely used in bio-transducers due to its biocompatibility and its mechanical compliance. 1 Viscoelasticity is an inherent property of PDMS, and it changes with loading rates and exposition times on loads. 2 For a PDMS sample, the viscoelastic exponent depends on the chain length and the concentration of the precursor material. 3 It has been also reported that for a scarcely cross-linked PDMS at room temperature the viscoelastic exponent presents a cross over point from low (β=0.

S 3
Right, the differentiation of the recorded deflection signal with respect to time for the signals depicted on the left. For the differentiation, we considered only the points to the right/left of the contact point (CP) for approach/retraction respectively. Color code: red for approach and green for retraction.
Α direct estimate of the scaling exponent can be made by fitting eq.(4) of the main text to the raw data (see Figure S1a and S1b left panels) for the approach phase, and by fitting eq.(6) for the retraction. The experimental curves as well the corresponding fittings for each phase are depicted in Figure S2, and the obtained scaling exponents are listed in Table S1. pretty much the same value for all experiments, (β ~0.16). The findings indicate that for a much higher velocity of penetration, the sample material behaves as a viscoelastic material, which, in the retraction phase, seems to have become more elastic because of the reduction of the scaling exponent by a factor of two.
We apply our method on the recorded FDC and compare the results with the scaling exponents returned by a direct fit of eqs (4) and (6) Table S1, whose connection with the scaling exponent is made through eq. (14), and the values of these exponents are listed in Table S1. In the approach phase, the agreement of the values of β's obtained by fitting eq.(4) and by H GMM is excellent, values in bold in Table S1. For the retraction phase, the β's obtained by GMM have constantly lower values with respect to corresponding values of the approach phase obtained by the same method. We have to notice that the linear fittings (eq.11) displayed in Figure S3  underlines the existence of a multiplicative mechanism of two competitive and independent sources shaping the overall response signal. The first one is assigned to the viscoelastic environment, and the second one is likely attributed to creep, or drift, or feedbacks from electronics. A quantification of them requires a more sophisticated treatment; a fractional Langevin equation with multiplicative noise as well with an additive fractional Gaussian noise could be a way of modeling it. 6 We leave this challenging task for future work.

Section II
Power spectrum scaling exponents obtained either directly by linear regression of (PSD), equation (15) Table S2.