The atomic force microscope (AFM) can detect the mechanical fingerprints of normal
and diseased cells at the single cell level under physiological conditions
1,2
. However, AFM studies of cell mechanics is limited by the "bottom effect" artifact
that arises from the stiff substrates used to culture cells. Because cells adhered
to substrates are very thin
3
, this artifact makes cells appear stiffer than they really are
4
. Here we show an analytical correction that accounts for this artifact when conical
tips are used for AFM measurements of thin samples. Our Bottom Effect Cone Correction
(BECC) corrects the Sneddon's model
5
, which is widely used to measure Young's modulus (E). Comparing the performance of
BECC and Sneddon's model on thin polyacrylamide gels, we find that while Sneddon's
model overestimates E, BECC yields E values that are thickness-independent and similar
to those obtained on thick regions of the gel. Application of BECC to measurements
on live adherent fibroblasts demonstrates a significant improvement on the estimation
of their local mechanical properties.
The pioneering work of Lekka et al. showed that AFM could be used to identify malignant
cancer cells by measuring their reduced Young’s modulus
6
. Following this work, similar studies on different types of cancer cells have emerged
7–9
, along with a better understanding of how various factors (such as the coating of
the cell substrate, force loading rate or culture time) influence the ability to unequivoquely
distinguish a malignant cell from a normal one
10,11
. Altered mechanical phenotypes have also been characterized using AFM for other pathological
conditions and diseases (for a review see Kuznetsova et al.
12
).
It is widely acknowledged that AFM measurements on adherent cells are affected by
artifacts stemming from the large stiffness of the substrates typically used for cell
culture
3
. For that reason, AFM users limit the indentations to 10% of the cell's thickness
13
. Nevertheless, >400 nm indentations are required to avoid errors due to uncertain
determination of the contact point
14
. As a result, measurements are restricted to the central region of the cell
10,13
, likely probing the mechanics of the nucleus rather than the cytoskeleton. A less
restrictive approach would use a more sophisticated model that accounts for the bottom
effect when estimating (E). We previously derived such a model for spherical tips
4
. Nevertheless, sharpened tips are better suited to reach the full potential of AFM
as a high-resolution biomechanical tool, since they allow for simultaneous topographical
and nanomechanical mapping of single cells
2,14
.
To that account, we have now derived BECC, a multiplicative analytical correction
to the commonly used Sneddon’s Model (SM) for conical tips
5
:
F
=
8
E
tan
θ
δ
2
3
π
{
1
+
1.7795
2
tan
θ
π
2
δ
h
+
16
(
1.7795
)
2
tan
2
θ
δ
2
h
2
+
Ο
(
δ
3
h
3
)
}
where F is the applied force, δ is indentation, θ is the half-opening angle of the
cone, h is the height of the sample at that location, and Poisson's ratio was assumed
to be 0.5 (formula derivation can be found in online materials).
To compare the performance of BECC and SM, we used polyacrylamide gels of graded thickness
(<1 µm to hundreds of microns), specifically crafted to resemble the height profile
of an adherent cell (fig1C, suppl. fig1). Polyacrylamide gels are homogenous and isotropic,
which makes them an ideal substrate to test Hertzian-like contact models like SM or
BECC. We find that SM grossly overestimates E up to 100-fold, with values heavily
dependent on gel thickness (Fig1A, Fig2A). Conversely, when we use BECC, computed
values for E are thickness-independent (fig1B) and similar to the values obtained
on thick regions of the same gel (suppl. Fig2A,C). Furthermore, BECC performs equally
well for a wide range of gel stiffness (Fig2B). When we intentionally applied very
large indentations (>85% of gel thickness), the observed E values began to increase,
likely indicating that we had reached the non-linear elasticity regime of the gel
(suppl. Fig. 3).
We also compared our correction to the finite element calculation of Kang et al, who
considered the indentation of a finite thickness soft incompressible elastic layer
bonded to a rigid substrate by a slightly blunted rigid frictionless cone
15
. When we input the parameters used in our experiments, our analytical result and
the finite element result agree within 4% of each other (see online materials).
We then tested the performance of BECC on measurements carried out on adherent fibroblasts
cultured on fibronectin-coated glass surfaces. To avoid remodelling of the cytoskeleton
due to prolonged cell poking or too large applied forces, we limited our indentations
to ~500 nm, using maximal forces of 2.5 nN and being in contact with the cell only
for ~12% of the cycle time. As shown in fig 3, we were able to discern regions with
distinct ranges of stiffness, likely corresponding to stress fibres or the nucleus
(Fig. 3B). The location of those regions, as well as the height profile of the cell,
was in agreement with the cell morphology observed in the phase contrast image that
was recorded simultaneously (Fig. 3C–D). On the contrary, regions of distinct stiffness
were barely evident when using SM (Fig. 3A).
To characterize the mechanical cell phenotype associated with a disease, multiple
locations (usually on thick regions) are probed for each cell, and several cells on
a population are studied. For our cell type and culture conditions, we find that cell
regions up to 4 µm thickness display the largest variability (Fig 4). Therefore, targeting
measurements to these cell regions would maximize the odds of measuring a statistically
significant difference in cell mechanical properties when studying a disease or pharmacological
treatment. We thus recommend a similar preliminary assessment when performing AFM
indentation measurements to distinguish mechanical cell phenotypes. Standard studies
pool together E values obtained from many cells, usually displaying the data in the
form of histograms
8
. It has been suggested that the skewness of the E distribution constitutes a reliable
fingerprint of diseased cell populations
8
. Not surprisingly, we find that when the bottom effect is not corrected, the distribution
of E values becomes artifactually skewed to the right (suppl. Fig. 3A), mainly due
to the overestimated E values that thin areas contribute to the distribution. Thus,
in light of our results, bottom effect artifacts should be ruled out to all certainty
before using skewness as a mechanical hallmark of disease. Another artifact arising
from the bottom effect impacts the determination of the contact point, which is slightly
displaced to the right of the force-indentation curve when using SM (suppl. Fig. 4).
As a result, thin regions appear to be even thinner. This artifact is again corrected
using BECC and should be considered when performing force-volume measurements that
correlate AFM mechanical measurements with cell topography
11,16
.
Both SM and BECC are Hertzian-like models that attempt to characterize the whole mechanical
response of an adherent cell with a single parameter E. A more complete approach would
be to generate a completely new constitutive model that takes into account the true
cell architecture, including the presence of a membrane, a heterogeneous cytoskeleton
and a nucleus. Such a model would then contain multiple parameters for the distinctive
mechanical responses of these three elements
17
. Nevertheless this goal has not been fully achieved yet. Hertzian-like contact models
have been extensively used as an alternative, although they make certain assumptions
on the nature of the probed sample. Namely, they assume the sample is isotropic, homogeneous
and linear elastic. These assumptions, which are not necessarily fulfilled by adherent
cells, constitute the main limitations of applicability of these models. As a result,
BECC can’t, on its own, account for cell viscoelasticity or changes in cell stiffness
along its depth. Nevertheless, researchers have devised clever ways to modify AFM
force-displacement protocols, so that Hertzian-like models can provide additional
information on the cell’s mechanical behaviour. Cell viscoelasticity has been addressed
by superimposing small oscillations to a constant indentation and analysing the results
as a complex elastic modulus
18
. A recent approach based on multi-harmonic analysis yields a much larger throughput,
and allows mapping of the local properties of a cell by using the 0th, 1st and 2nd
harmonic components of the Fourier spectrum of the AFM cantilevers interacting with
a cell’s surface
19
. Mechanical heterogeneity along the cell thickness has been characterized by comparing
the relative E values obtained from shallow and deep indentations
20
. In addition, a similar approach can be used to measure non-linear elasticity
21
. Most importantly, since all these approaches are based on SM, our multiplicative
correction can be readily combined with any of these protocols. For a discussion of
additional potential model extensions, the reader is addressed to the online supplementary
material.
In conclusion, BECC enables non-artifactual nanomechanical mapping of the whole cell
surface using AFM. The correction can also be readily combined with existing protocols
for viscoelasticy, non-linear elasticity and depth-sensing analysis. We thus predict
that the mechanical abnormalities so far measured in diseased cells will be further
evident once larger parts of the cell cytoskeleton are non-artifactually probed, thus
solidifying AFM as a diagnostic tool for malignancy.
Methods
Preparation of polyacrylamide gels
Polyacrylamide gels constitute an elastic and repeatable test material, with small
point-to-point variations in stiffness
22
(coefficient of variation for E is ~30%). Polyacrylamide gels were prepared via photopolymerization
initiated by Irgacure 2959 as described previously
23
. Different final concentrations of acrylamide and bis-acrylamide were diluted in
water to obtain gels of a wide range of stiffness A drop of gel mixture was deposited
on a chemically activated glass slide and the drop was left uncovered. Polymerization
was achieved by exposure to UV light. After polymerization, gels remained firmly attached
to the slide and displayed a hill-like shape. At their edges, gels displayed a smoothly
increasing height profile, with the thinnest areas being less than 1 µm tall.
Cells
Cell measurements were performed in living fibroblasts, cell line NIH-3T3 (CCL-1658,
ATCC). The culture medium consisted of hepes-buffered DMEM (Gibco) with 10% calf serum
(SAFC Biosciences) and 1:100 Penicillin-Streptomycin (Sigma). Measurements were performed
on glass-bottomed petri dishes coated with fibronectin, at 37 °C by heating the stage
of the microscope.
AFM setup
Measurements were performed using a Catalyst AFM (Bruker Corp.) instrument mounted
on the stage of an Axiovert 200 inverted microscope (Zeiss) placed on a vibration-isolation
table (Isostation). A V-shaped gold-coated silicon nitride cantilever with a four-sided
pyramidal tip (MLCT, Bruker Corp.) was used as probe. The spring constant of the cantilever
was 0.047± 0.003 N m−1 as calibrated using the thermal fluctuations method
24
. Detailed descriptions of the measurement protocol and data analysis can be found
in the online methods.
Supplementary Material
1