SupplementaryInformation (SI)

Zhou et al., Power-law rheology analysis of cells undergoing micropipette aspiration

SI Figures

Fig. S1. The stress-free projection length (diamonds) is linearly dependent on the pipette size for 2Rp > ~ 7 m at a loading rate of 1/30 cmH2O/s. The solid line represents and the dash line represents. LpSF is significantly larger LpG and the difference,  = LpSF − LpG, reflects the optical pipette edge effect. The hypothetical origin of this effect is illustrated in the right diagram, in which we postulate that the manually detected pipette edge deviates from the real pipette edge by a distance of . Pipette size (2Rp) and total projection length (LpT) are also illustrated here.

Fig. S2. The size effect is dominated by the pipette size (2Rp) but not the pipette-to-cell size ratio (Rp/Rc). Two groups of cells are compared here: small (Rc = 7.1 - 8.4 m) and large (Rc = 9 – 13.8m). (A) r2 as a function of Rp/Rc; (B): r2 as a function of Rp; (C) S/Rp as a function of Rp/Rc; (D): S/Rp as a function of Rp. For both r2 and S/Rp, there is no unique threshold (the vertical dash lines) in Rp/Rc, but a unique threshold exists for Rp.

Fig. S3. Statistical distribution and association of the power-law rheology parameters. (A)−(C) Probability density for the distribution of shear complianceconstant (A), power-law exponent (B) and shear stiffness constant (C) measured for untreated fibroblasts (n = 81). The best fits of normal and log-normal distribution functions are also shown.(D) Positive association between AG and  for untreated and cytoD-treated fibroblasts.

Fig. S4. A general trend for power-law rheology measured with various experimental techniques, cell types and drug treatments. For the data acquired with the current work (MA), the drug treatments are indicated by a label with an arrow. Error bars indicate standard deviation. The data sources are as follow: OMTC (Fabry et al. 2003; Lenormand et al. 2004; Puig-De-Morales et al. 2004; Stamenovic et al. 2004; Trepat et al. 2004; Laudadio et al. 2005), MMTC (Puig-De-Morales et al. 2001), AFM (Alcaraz et al. 2003), OT (IC) (Yanai et al. 2004), OT (Balland et al. 2005), MPM (Desprat et al. 2005), MA (this work). (Abbreviations: OMTC − optical MTC, MMTC − magnetic MTC, OT − optical tweezers, MPM − microplate manipulation, (IC) − intracellular measurements, Col − colchicine. CD − cytochalasin D. For more details including cell types and drug treatments, see Table A.3 in (Zhou 2006).) For OMTC, the conversion factor 6.8m was used(Fabry et al. 2003).
Fig. S5. Predicting the ramp-test deformation based on the creep-test results. The pressure-deformation relationship for ramp tests was computed based on Eq. (S6) (cf. SIText 4) for two loading rates, 1/30 and 1/120 cmH2O/s. The apparent deformability was taken as the average slope of the curves.

SI Text 1. Is the size effect dictated by the pipette size (Rp) or the pipette-to-cell size ratio (Rp/Rc)?

To answer this, we now present Fig. 2.A and 2.B (in raw data) with either Rp or Rp/Rcas x-axis (Fig. S2). We compare two groups of cells, the 40% smallest (Rc = 7.1 - 8.4 m) and the 40% largest (Rc = 9 – 13.8 m); a desirable size parameter should give a unique size threshold for both groups of cells. For both r2 and S/Rp, there is no unique threshold (the vertical dash lines) in Rp/Rc (Fig. S2A and C), but a unique threshold exists for Rp (Fig. S2B and D). This suggests that the pipette diameter itself is a more desirable parameter for quantifying the size effect.

SIText2. Stress-free projection length and pipette edge effect measured with large pipettes

Because the stiffness of suspended animal cells (including NIH 3T3 fibroblasts) at small deformation is in the order of 100 Pa (Wottawah et al. 2005; Roca-Cusachs et al. 2006), comparable to that measured here with relatively large deformation, we assume that that the linearity observed of the P-LpT relations with large pipettes (Fig. 1 C) extends to zero pressure. Therefore, LpSF correspond to the projection length at the undeformed, stress-free state. LpSFis found to depend weakly on cell size but strongly on pipette diameter for 2Rp = 6.8 ~ 10.1 m. The measured LpSFfor 1/30 cmH2O/s is plotted against pipette diameter inFig. S1, together with the geometric prediction (Rc was taken as the average cell radius, 8.8m). The average SF projection lengthmeasured with 1/30 cmH2O/s can be fitted by

which provides a reference for calculating the deformed projection length, especially for the ensuing creep experiments. The observation that LpSF was significantly larger than LpGat a given pipette size indicates that the real pipette entrance may not correspond to the most distinct edge. The distance between the most distinct edge and the real pipette entrance can thus be computed as  = LpSF − LpG, which increases slightly with pipette size but has an average of 1.3 ± 0.4 m for 2Rp = 6.8 ~ 10.1 m (Fig. S1, right diagram). Neglecting the pipette edge effect may lead to severe overestimation of the actual cell deformation in micropipette aspiration.Because it mainly depends on the position of the most distinct edge with respect to the real pipette edge, we assume that the SF projection length do not change substantially with the type of measurements or drug treatments.

In this work, the pipette edge effect is estimated indirectly from ramp experiments. Alternatively, direct approach can be employed to enhance the imaging resolution such that the deformed projection length can be directly and more accurately quantified.
SI Text 3. Functional forms of spring-dashpot models

The creep functions of the three-parameter standard linear solid (SLS) model(Schmid-Schonbein et al. 1981; Sato et al. 1990) and the four-parameter standard linear solid-dashpot (SLS-D)model(Bausch et al. 1998) can be written as

(S1)

and

(S2)

wherek1 and k2 are two elastic shear constants,µ is a viscous constant in series with k2 (not explicit in the equations), is the characteristic creep time and H(t) is the Heaviside function,and 0 is an additional viscous constant.

SI Text 4. Compatibility between creep tests and ramp tests

In the ramp tests, the apparent deformability S/Rpis measured at two loading rates by applying linear curve fitting to the pressure-deformation relationship (cf. Quantification of the apparent elasticity). On the other hand, the apparent deformability can also be theoretically predicted based on the power-law creep function, measured from creep tests. Thus, the compatibility between creep tests and ramp tests can be examined by comparing S/Rp measured with ramp tests versus that predicted from the creep function measured by creep tests.

Based on the half-space model, nominal strainand stress in micropipette aspiration can be defined as

.(S3)

Thus, the creep deformation in micropipette aspiration (Eq.(2))can be expressed by. Using the Boltzmann superposition principle, the evolution of the representative strain in response to certain loading history can be written as

(S4)

In a ramp test, the loading history is

(S5)

where is the increasing rate of the pressure. Substitution of Eqs. (3), (S3) and (S5) into Eq. (S4) results in

(S6)

whichis used to predict the apparent deformability from the measured creep function of cells (Fig. S5).

Given the average power-law parameters ( = 0.3 and AJ = 1.0 (10−2 Pa−1)) measured with the creep tests and the loading rates used in the ramp tests, the relation between pressure and deformation can be calculated for the typical pressure range between 1 and 10 mmH2O (Fig. S5). From the average slope of Lp(t)/Rp versus P(t), the apparent deformability was calculated as 0.0725 (1/mmH2O) for the loading rate of 1/30 cmH2O/s, and 0.110 (1/mmH2O) for 1/120 cmH2O/s(Fig. S5), which compares favorably (within 5% error) with those found with the ramp tests. This further substantiates the applicability of the power-law rheology model.

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