Annals of Biomedical Engineering, Vol. 33, No. 2, February 2005 (©2005) pp. 131–141
DOI: 10.1007/s10439-0058972-9
Computation of Adherent Cell Elasticity for Critical Cell-Bead
Geometry in Magnetic Twisting Experiments
JACQUES OHAYON and PHILIPPE TRACQUI
Laboratoire TIMC-IMAG, équipe DynaCell, CNRS, UMR 5525, Institut de l’Ingénierie et de l’Information de Santé (In3 S), 38706 La
Tronche cedex France
(Received 26 March 2004; accepted 4 August 2004)
Abstract—Quantification of the cell elastic modulus is a central
issue of micromanipulation techniques used to analyze the mechanical properties of living adherent cells. In magnetic twisting
cytometry (MTC), magnetic beads of radius R, linked to the cell
cytoskeleton through transmembrane receptors, are twisted. The
relationships between imposed external torque and measured resulting bead rotation or translation only provide values of the apparent cell stiffness. Thus, specific correcting coefficients have to
be considered in order to derive the cell elastic modulus. This issue
has been highlighted in previous studies, but general relationships
for handling such corrections are still lacking while they could help
to understand and reduce the large dispersion of the reported values
of cell elastic modulus. This work establishes generalized abacuses
of the correcting coefficients from which the Young’s modulus of
a cell probed by MTC can be derived. Based on a 3D finite element
analysis of an hyperelastic (neo-Hookean) cell, we show that the
dimensionless ratio hu /2R, where hu is the cell height below the
bead, is an essential parameter for quantification of the cell elasticity. This result could partly explain the still intriguing question
of the large variation of measured elastic moduli with probe size.
ei j
h
hu
h∞
p
u
(Aα , Bα )
(Aβ , Bβ )
(Aαi , Bαi )
(Aβi , Bβi )
C
(Cβ , Dβ )
Keywords—Cytomechanical model, Finite element method,
Magnetic twisting cytometry, Correcting coefficients, Cell
stiffness.
(Cβi , Dβi )
Ecell
G
Gθ
NOMENCLATURE
ax , a y
a1
d
(e1 , e2 , e3 )
(er , e , eφ )
eeff
side lengths of the representative cellular
volume
rheological constant of the neo-Hookean
strain energy function
lateral bead translation along the ox axis
Cartesian unit base vectors associated to
coordinates (x, y, z)
spherical unit base vectors associated to
coordinates (r, , φ)
local effective strain field
Gδ
I
I1
R
Re
T
V
W
Address correspondence to Pr. J. Ohayon or Dr. Ph. Tracqui,
Equipe Dynacell, Laboratoire TIMC-IMAG, CNRS, UMR 5525, Institut de l’Ingénierie et de l’Information de Santé (In3 S), 38706
La Tronche cedex, France. Electronic mail: [email protected];
[email protected]
α
components of the deviatoric strain field
cell height
distance between the bead bottom and the
substrate
infinite cell height (approximated in our
computation by a 20-µm-high cell)
Lagrangian multiplier resulting from the
incompressibility constraint
displacement vector
functions of γ used for the fit of the
correcting coefficient α
functions of γ used for the fit of the
correcting coefficient β
coefficients of the polynomial functions
Aα (γ ) and Bα (γ )
coefficients of the polynomial functions
Aβ (γ ) and Bβ (γ )
right Cauchy-Green strain tensor
functions of γ used for the fit of the
correcting coefficient β
coefficients of the polynomial functions
Cβ (γ ) and Dβ (γ )
cell Young modulus used for linear material
shear modulus for linear material
apparent cell stiffness related to the bead
rotation
apparent cell stiffness related to the bead
translation
identity matrix
first invariant of the right Cauchy-Green
strain tensor C
bead radius
radius of a spherical external volume
immersing the rigid bead
external magnetic torque applied
bead volume
strain energy function defining the elastic
properties of the cell medium
correcting coefficient related to the apparent
cell stiffness Gθ
131
C 2005 Biomedical Engineering Society
0090-6964/05/0200-0131/1 132
β
γ
δ
ε
θ
θs
λ
σ
ξ
J. OHAYON and P. TRACQUI
correcting coefficient related to the apparent
cell stiffness Gδ
half-angle of bead immersion or embedding
angle
normalized lateral bead translation along the ox
axis (δ = d/R)
linear lagrangian strain tensor of components εi j
total bead rotation angle
slipping bead rotation angle
extension ratio
Cauchy stress tensor of components σi j
normalized ratio h u /2R
INTRODUCTION
In living tissues, cells are exposed to a variety of mechanical stresses and strains which determine cell shape. In
return, cell deformability is a central feature of the cellular mechanochemical signaling pathways which control a
variety of fundamental cell functions, including spreading,
migration, proliferation and gene expression.6,11,15,20 In this
context, the characterization of cell rheological properties,
and especially the quantification of cell elastic modulus,
are crucial for understanding both physiological and pathological cell behavior, as well as for evaluating the action of
pharmacological drugs which potentially modify cell architecture and thus cell rheological properties.14
Among the different micromanipulation methods used
to analyze cell mechanical response,19 magnetic twisting cytometry (MTC) has been widely used, since the
pioneering work of Wang et al.26,27 to probe mechanical properties of adherent cells on various substrate and
in different experimental situations.5,7,17,18 In the MTC
method, ferromagnetic beads (3.5–5.5-µm diameters) are
coated with arginine–glycine–aspartic acid (RGD) peptide
in order to permit their binding to integrin transmembrane
mechanoreceptors.23 After beads premagnetization using a
short uniform magnetic pulse, a magnetic torque is then created by a uniform magnetic field.17,27 The resulting measured bead rotation or/and lateral bead translation reflect
the mechanical resistance of the cell to the imposed deformation. Recently, this experimental approach also permits to distinguish between cortical and deep cytoskeleton
responses.18
The so obtained experimental data allow the computation
of the apparent cell stiffness, either from the ratio between
the applied magnetic torque and the resulting angular rotation or from the ratio between the applied torque and the
lateral bead displacement. Such quantifications appear quite
appealing since they only involve one additional parameter, the bead radius.8,21 Nevertheless, Mijailovich et al.21
pointed out recently that such apparent cell stiffness may
vary with cell height and bead embedding angle. Thus, in
order to derive the intrinsic value of the cell elastic modulus, they computed a limited set of correcting coefficient
for a given bead radius of 2.5 µm and considering three
specific cell heights [20 µm (semi-infinite medium), 5 and
1 µm]. In this work, we extend and generalize the computation of the cell elastic modulus by establishing analytical
expressions of the correcting coefficients in a whole range
of experimental values taken by the bead embedding angle
γ , cell height h, and bead radius. Additionally the mechanical response of deeply embedded bead (i.e. with an angle
γ in the range 90◦ –180◦ ) is also analysed. As in the study
of Caille et al.,1 we assumed the cell behaves as an hyperelastic (neo-Hookean) material which takes into account
the contribution of the strain tensor nonlinear terms, while
preserving a linear stress-strain relationship. We used a
three-dimensional finite element method to extensively
compute and analyze the cell mechanical response to applied torques for a large set (800 cases) of different cell
geometries. Our goals are (i) to point out the most relevant
cellular geometrical parameters in the computation of the
correcting coefficients, and (ii) to establish a set of generalized abacuses from which the Young’s modulus of a cell
probed by MTC can be computed, as well as critical values
of the cell height below which correcting coefficients must
necessary be considered.
METHODS
Finite Element Formulation of the Cell Model
Cell Geometry
The cell is represented by parallelepipedic volume element of height h with rectangular basal area of size
ax × a y , where ax = 2a y = 36 µm. Considering for example a mean cell height h = 4 µm, we obtained a cellular
volume of 2592 µm3 , similar to the measured volume of
2500 ± 300 µm3 obtained by Schneider et al.24 in dog kidney epithelial cells. A single bead is embedded in the middle
of the upper surface of the cellular volume. Using the symmetrical plane passing across the center of the bead, we
restricted the analysis to half the cellular volume (Fig. 1).
The main parameters defining the cell-bead model geometry are the cell length a y , the cell height h, the bead radius R
and the embedding angle γ (see Fig. 1). Experimental values of the bead embedding angles, measured with alveolar
epithelial cell,17 range from 15◦ to 90◦ : we thus considered this latter value as a first step in our analysis, before
considering larger bead embedding angles up to 180◦ .
Finite Element Model
The cellular medium is assumed to be an homogeneous quasi-incompressible and hyperelastic neo-Hookean
continuum,12 characterized by a strain-energy function already used to model the mechanical response of tissues and
living cells1 and given by
W = a1 (I1 − 3)
(1)
Generalized Elastic Modulus Estimation of Adherent Cell
where a1 is the cellular material constant (in Pa), while I1
is the first invariant of the right Cauchy–Green strain tensor
C [I1 = Trace(C)].11
For this incompressible medium, the cell shear modulus
G is related to the Young’s modulus Ecell by the equation
E cell = 3 G. Furthermore, an additional explicit relationship between Ecell and the material constant a1 can be obtained for uniaxial tension or compression of the medium:
in this case, E cell = 2a1 (2 + λ−3 ), where λ is the extension
ratio in the direction of the uniaxial stress. Notice that for
small extension ratio (λ ∼ 1), this expression reduces to
E cell ∼ 6a1 .
Computations of the bead rotations induced by increasing value of the applied magnetic torque were performed
using a finite element approach (Ansys 8.0 software,
Ansys, Inc., Cannonsburg, PA). The bead surface was
modeled as a rigid shell since its stiffness is far larger than
the cell stiffness.
Boundary Conditions
The boundary conditions were written on half of the cell
volume. The conditions imposed at the cell boundaries are
as follows (Fig. 1):
(i) zero displacement, modeling full cell attachment to
the substrate, was imposed to the cell membrane in
contact with the rigid substrate,
(ii) zero normal displacement condition was imposed
on the cell section belonging to the plane of
symmetry,
(iii) free boundary conditions were assumed for all the
other cell surfaces,
(iv) no-slip condition at the bead-cell interface.
133
RESULTS
In MTC experiments, the bead rotation induced by the
applied magnetic torque deforms the adherent cell. Our finite element computations simulate the cell resistance to
both bead rotation and bead lateral translation (the vertical
translation toward the substrate remains negligible compared to the lateral translation). Thus, cell stiffness can be
computed from the imposed torque and the resulting bead
motion for a given cell-bead geometry, considering either
bead rotation or bead lateral translation as described in the
following paragraph.
Background for the Calculation of Apparent Cell Stiffness
from Bead Rotation or Bead Lateral Translation
In the case of a bead embedded in an infinite linear elastic
and incompressible medium, one can obtain an analytical
relationship between the applied mechanical torque T and
the resulting bead rotation θ, as a function of the medium
shear modulus G and bead radius R (see Appendix):
T = 6V θ G
(2)
where V is the bead volume.
The apparent stiffness modulus of the cell is computed from either bead rotation θ or associated lateral bead
translation d using the following relationships21 :
G θ = T /6V θ
(3)
G δ = T /6V δ
(4)
where δ is the normalized bead translation (δ = d/R).
In order to take into account the differences between
the infinite medium case and the real cell-bead geometry,
FIGURE 1. Finite element mesh of the half cell volume of size ax × ay/2 × h. The arrow indicates the orientation of the bead rotation
and torque applied at the bead center. Free boundary conditions are considered, except when explicitly indicated in the figure.
134
J. OHAYON and P. TRACQUI
several authors9,21 introduced two correcting coefficients
α and β which incorporate the effect of the geometrical
parameters γ , h, and R on the apparent cell stiffness estimations. The correcting coefficients α and β are defined by
G θ = αG
(5)
G δ = βG
(6)
In the case of a bead embedded in an infinite medium (see
Appendix), the correcting coefficient α is equal to 1, but
because no bead translation occurs (δ = 0) the coefficient β
tends to infinity. In real experiments, variations of the bead
radius can significantly modify the mechanical response
of adherent cells for low values of the cell height. It is thus
of importance to study the sensitivity of the correcting
coefficients values to modifications of the geometrical
parameters and to explicit the expression of α and β as
functions of γ , h, and R.
Influence of the Bead Radius and Cell Height on the Cell
Mechanical Response
Spatial Distribution of Effective Strains with Variations of
Bead Radius and Cell Height
Figure 2 highlights the effects of increasing bead radius
and decreasing cell height on the simulated spatial distri-
butions of effective strains within the cell. In all cases, the
same 15◦ bead rotation θ has been imposed, which corresponds to a torque amplitude (in pN µm) ranging from
1.65 Ecell [Fig. 2(A)] to 12.58 Ecell [Fig. 2(D)]. The bead
embedding angle has been kept to a constant value γ taken
as 65◦ . Figure 2 shows that decreasing cell height from 5
to 2 µm has no major effect on the effective strain values
for bead radius of 1.5 µm [Fig. 2(A) and 2(C)]. This is no
longer the case when considering bead radius of 2.5 µm:
the maximum effective strain remains close to the previous
values (56% compared to 48% for 1.5-µm bead radius) for
5-µm-high cell [Fig. 2(B)], but increases sharply to 80%
for 2-µm-high cell [Fig. 2(D)]. This influence of the degree
of bead embedding at low cell heights is more precisely
analyzed in the next paragraph.
Nonlinear Corrections of the Apparent Cell Stiffness with
Variations of Bead Radius and Bead Embedding
As in Mijailovich et al.,21 we first consider the percentage of bead diameter as the geometrical parameter characterizing bead embedding in the cell, while bead radius
is taken as a second geometrical parameter. We then simulated the cell response for 800 different realistic cellbead geometries, each corresponding to a given value of R
(1 µm ≤ R ≤ 5 µm), γ (15◦ ≤ γ ≤ 180◦ ), and h (1 µm ≤
FIGURE 2. 3D color maps showing
the cell deformed shapes and the spatial distributions of the effective strains eeff in the neigh
borhood of the bead (e eff = 2ei j ei j /3, with ei j the components of the deviatoric strain tensor). Four different cell-bead geometries
have been considered for given values γ = 65◦ and ay = 18 µm. For each geometry, a bead rotation of 15◦ (0.262 rad) has been
imposed, giving rise to a lateral bead translation d. Parameters and variables values are respectively (with Ecell in Pascals): (A) h =
5 µm, R = 1.5 µm, d = 0.222 µm, T = 1.651 Ecell pN µm; (B) h = 5 µm, R = 2.5 µm, d = 0.371 µm, T = 7.779 Ecell pN µm; (C) h = 2 µm,
R = 1.5 µm, d = 0.208 µm, T = 1.828 Ecell pN µm; (D) h = 2 µm, R = 2.5 µm, d = 0.316 µm, T = 12.577 Ecell pN µm. The bead has been
removed in order to visualize more clearly the spatial strain distributions in the neighborhood of the cell-bead contact area.
Generalized Elastic Modulus Estimation of Adherent Cell
FIGURE 3. Influence of the bead radius R and of the percentage
of embedding angle on the correcting coefficient α values for
two extreme cell heights h∞ = 20 µm (A) and h = 1 µm (B).
Solid points correspond to iso-values of bead radius when the
percentage of bead diameter is varied and to iso-values of bead
embedding angle when bead radius is varied.
h ≤ 20 µm). The associated correcting coefficient α is then
computed from Eqs. (3) and (5) (Fig. 3). When the bead
is embedded in a semi-infinite medium (approximated here
by a 20-µm-high cell), the coefficient α does not change
with the bead radius and becomes significantly a nonlinear
function of the bead embedding angle for values larger than
90◦ [Fig. 3(A)]. This graph is drastically modified when the
cell height decreases up to 1 µm: the correcting coefficient
α now exhibits a sharp nonlinear variation with the bead
embedding [Fig. 3(B)].
Similar results have been obtained for the correcting coefficient β (results not shown). Taken as a whole, these
results indicate that a complete set of abacus for the functions α(γ , h, R) and β(γ , h, R) must be provided in order to
estimate the intrinsic cell elastic modulus of adherent cell
probed by MTC.
Analytical Expressions of the Correcting Coefficients
α and β
In our simulations, the ratio ξ = h u /2R, i.e. cell height
below the bead over bead diameter (Fig. 1), appears as a key
135
FIGURE 4. Best fits obtained for the two correcting functions
α(hu /2R ) [(A), r 2 = 0.990] and β(hu /2R ) [(B), r 2 = 0.984] for
the fixed value γ = 50◦ of the bead embedding angle. The dimensionless geometrical invariant hu /2R is the ratio of the cell
height below the bead over the bead diameter.
dimensionless geometrical parameter for the quantification
of the cell response. This property is supported by the series
of fits performed, at given embedding angles γ , against the
computed values of α and β when the ratio ξ is increased.
Based on experimental measurements22 of hu , ξ is taken in
the range 0.1 ≤ ξ ≤ 10. A very satisfactory fit (exemplified
in Fig. 4 when γi = 50◦ ) has been obtained when assuming
for each embedding angle γ i hyperbolic-like relationships
of the form
Bα (γi )
α(γi , ξ ) = Aα (γi ) +
(7)
ξ
β(γi , ξ ) = Aβ (γi ) +
Bβ (γi ) + Cβ (γi )ξ 2
Dβ (γi ) + ξ
(8)
In a second step, we then try to find mathematical expressions of the functions Aα (γ ), Bα (γ ), Aβ (γ ), Bβ (γ ), Cβ (γ ),
and Dβ (γ ), from which the correcting coefficients α(γ , ξ )
and β(γ , ξ ) can be continuously computed. We found that
the best fit (Fig. 5) to the simulated cell response for the 800
geometries we considered is obtained when considering the
following expressions:
Aα (γ ) = Aα1 γ + Aα2 γ 2 + Aα3 γ 3
Bα (γ ) = Bα0 + Bα1 γ + Bα2 γ 2 + Bα3 γ 3
(9)
(10)
136
J. OHAYON and P. TRACQUI
FIGURE 5. Best mathematical fits of the six functions Aα (γ) [(A), r 2 = 0.999], Bα (γ) [(B), r 2 = 0.998], Aβ (γ) [(C), r 2 = 0.999 if
15◦ ≤γ≤90◦ and r 2 = 0.998 if 90◦ ≤γ≤180◦ ], Bβ (γ) [(D), r 2 = 0.998 if 15◦ ≤γ≤90◦ and r 2 = 0.999 if 90◦ ≤γ≤180◦ ], Cβ (γ) [(E), r 2 = 0.998]
and Dβ (γ) [(F), r 2 = 0.999] given in Eqs. (9)–(14) and used for the estimation of the two correcting coefficient α and β [see Eqs. (7)
and (8)].
Aβ (γ ) = Aβ0 + Aβ1 γ + Aβ2 γ 2 + Aβ3 γ 3
(11)
Bβ (γ ) = Bβ0 + Bβ1 γ + Bβ2 γ 2 + Bβ3 γ 3
(12)
Cβ (γ ) = Cβ0 + Cβ1 γ + Cβ2 γ
(13)
2
Dβ (γ ) = Dβ0 e Dβ1 γ
(14)
An excellent goodness of fit has been obtained with numerical values of coefficients given in Table 1. Let us remark that
in the limiting case where the bead-cell interface reduces
to one contact point (i.e. when γ → 0◦ and h u → ∞), the
correcting coefficients α and β tend to zero.
(ii) by Laurent et al.17 when considering the approximated
analytical solution α = sin3 (γ )/2 obtained in the case of a
bead embedded in a semi-infinite medium. Figure 6(A) and
6(B) show that, for all considered embedding angles, our
results are in very good agreement with those of Mijailovich
et al.21 for the three cell heights considered in their analysis, namely h ∞ = 20 µm, h = 5 µm, and h = 1 µm. On
the other hand, our results only agree with those of Laurent
et al.17 for bead embedding angles smaller than 20◦ [see
enlargement in Fig. 6(A)]: above this value, the approximation based on the sin3 (γ ) function gives relative error on α
larger than 50%.
Comparison with Previous Results
For the range of experimentally observed embedding
angles22 (i.e. 15◦ ≤ γ ≤ 90◦ ), we compared the values of
the correcting coefficients α and β to those derived—(i) by
Mijailovich et al.21 using finite element simulations and—
Elaboration of a Set of Abacuses for Analyzing Magnetic
Twisting Cytometry Data
We took benefit of the analytical expressions of the correcting coefficients α(γ , ξ ) and β(γ , ξ ) established above to
Generalized Elastic Modulus Estimation of Adherent Cell
137
TABLE 1. Numerical values of the coefficients defining the
mathematical expressions of the interpolation functions Aα (γ),
Bα (γ), Aβ (γ), Bβ (γ), Cβ (γ), and Dβ (γ) given in Eqs. (9)–(14).
Aα1
Aα2
Aα3
Bα0
Bα1
Bα2
Bα3
15◦ ≤ γ ≤ 180◦
−6.497 × 10−2
2.966 × 10−1
−6.240 × 10−2
−2.811 × 10−3
1.133 × 10−2
7.418 × 10−3
−2.466 × 10−3
Aβ0
Aβ1
Aβ2
Aβ3
Bβ0
Bβ1
Bβ2
Bβ3
Cβ0
Cβ1
Cβ2
Dβ0
Dβ1
15◦ ≤ γ ≤ 90◦
0
−7.205 × 10−3
−9.690 × 10−3
2.778 × 10−1
−4.673 × 10−3
2.403 × 10−2
−2.125 × 10−2
3.362 × 10−2
0
0
0
0
0
α
β
90◦ ≤ γ ≤ 180◦
29.688
−43.252
20.229
−2.734
−10.830
16.823
−8.694
1.540
0.593
−0.756
0.242
0.002 × 10−1
2.892
construct the set of abacuses which appear as the most relevant for quantifying the Young’s modulus of a cell probed
by MTC. Moreover, we considered as the main dependent
variable the ratio 1/ξ instead of ξ in order to exhaust the
sensitivity of the correcting coefficients variations in the
region of interest, i.e. when hu tends to small values, as exemplified in Fig. 4. From the mathematical expressions of
α and β given in Eqs. (7) and (8), we derived the critical
value of the ratio ξ = h u /2R for which the slopes of the
curves α(γ i , ξ ) and β(γ i , ξ ) at given embedding angle γ i
become larger than 10%. Figure 7 shows that this threshold
increases with increasing embedding angles: for example,
the influence of hu on the correcting coefficient α must be
considered as soon as hu is lower than ∼R (i.e. ξ ∼ 1/2)
when the embedding angle equals 90◦ [Fig. 7(A)]. Estimation of the cell elastic modulus from the bead translation is
even more sensitive: the variation of the correcting coefficient β with hu must be taken into account for still larger
values of hu , approximately equal to 2R (i.e. ξ ∼ 1) when
γ = 90◦ (Fig. 7B). For values of hu larger than these critical
bead sizes, the correcting coefficient α is almost constant
for fixed γ : its value is then given by the function Aα (γ )
(Eq. (9)).
In practical terms, the use of the abacuses can be
illustrated in two cases of interest. The first one is a
straightforward evaluation of the correcting coefficients α or
β for a given cell-bead geometry (Fig. 8). The cell Young’s
modulus is then computed from Eqs. (3) and (5) or/and Eqs.
(4) and (6), knowing the amplitude of the applied magnetic
torque as well the bead rotation or/and bead translation.
FIGURE 6. Variation of the correcting coefficients α [Fig. 6(A)]
and β [Fig. 6(B)] with increasing values of the embedding angle γ and for three different cell heights chosen for a seek of
comparison with previous works: h = 1 µm (square symbols),
h = 5 µm (triangular symbols) and h∞ = 20 µm (circular symbols). Our results (solid symbols) are compared to the analytical expression proposed by Laurent et al.17 (α = sin3 (γ)/2) in
the semi-infinite medium case (enlargement in graph A) and
to the numerical results of Mijailovich et al.21 (open symbols)
obtained for finite cell height and with the same bead radius
R = 2.25 µm.
However, the measurement of the bead embedding angle
remains a quite involved task10 which could prevent the
use of the above procedure. One can thus originally underline that such a measurement of the embedding angle, as
well as the measurement of the value hu , can be avoided
if both bead rotation and bead lateral translation are measured. From the abacus of Fig. 9(A), one can indeed determine, from the knowledge of the cell height h and the bead
radius R, the intersection point between the iso-(h/2R) values curve [for example h/2R = 0.8, Fig. 9(A)] with the
horizontal line corresponding to the measured θ/δ ratio
[for example θ/δ = 2, Fig. 9(A)]. Only one iso-γ curve
goes through this point [solid point in Fig. 9(A)], which
gives the corresponding embedding angle γ (γ = 65◦ in
this example). In addition, the abscissa of this intersection point gives the 2R/hu ratio which will be used for
the derivation of the correcting coefficients α and β from
the abacus of Fig. 8 [solid points, Fig. 8(A) and 8(B)
respectively].
138
J. OHAYON and P. TRACQUI
FIGURE 7. Critical values of the ratio ξ = hu /2R above which
the slopes of the correcting functions α(ξ) [Fig. 7(A)] and
β(γ i , ξ) [Fig. 7(B)] are equal to or less than 10% for a given
embedding angle γ i . Below these thresholds, the correcting
coefficients α and β vary significantly with decreasing values
of ξ (see Fig. 4).
DISCUSSION
Estimation of the cell stiffness is a key issue for understanding how changes in cytoskeleton tension modulate intracellular mechanochemical signaling, i.e. mechanotransduction cell properties. Moreover, quantification of
cell stiffness is also a prerequisite for analyzing the global
mechanical response of cells cultured on extracellular matrices with different stiffness6 and on which they undergo
a limited or complete spreading.5 On the other hand, cell
elasticity values reported from experimental data in the literature exhibit large dispersions which may be related not
only to differences between the micromanipulation assays
or cell types,19 but also to the method used for estimating the
cell elastic modulus from rough experimental data. In this
work, we revisited such estimation methods when adherent
cells are probed by MTC, with the aim to correct and extend
previous linear models taking into account the geometrical
variations of the cell-bead interface.17,21 Indeed, even if the
need for corrected quantification of cell elastic modulus
has been highlighted by Mijailovich et al.,21 general relationships allowing the derivation of this modulus over the
complete range of experimental parameters values are still
lacking, especially when the influence of the cell height
comes into play.
Considering the cell as hyperelastic (neo-Hookean) material undergoing large strains, our finite element computations first indicate that only two geometrical parameters,
namely the embedding angle γ and the ratio of the under bead height over the bead diameter (i.e. the normalized
geometrical parameter ξ = h u /2R), are relevant for determining the correcting coefficients α and β that allow the
cell elastic modulus to be estimated from measurements
of either bead rotation or lateral bead translation, respectively. Secondly, the consideration of a neo-Hookean cellular material validates our analysis in the domain of large
deformations, i.e. for magnetic bead rotation going up to
∼15◦ , since the correcting coefficients α and β remain
quasi-constant in this rotation range. This is true even for the
lowest h u values we considered, i.e. when the nonlinear geometrical effects are important: for example, the computed
relative variation α/α and β/β of the correcting coefficients are both in the order of 3% when the bead rotation
increases from 1◦ to 15◦ for the cell-bead geometry considered in Fig. 2D, with h u = 2 µm and R = 2.5 µm (ξ ∼ 0.4).
Consequently, our analysis kept only two varying parameters, the bead embedding angle γ and the dimensionless geometrical invariant ξ = h u /2R. We have shown that
the corresponding α(γ , ξ ) and β(γ , ξ ) functions can be
accurately described by mathematical functions. This result can be discussed in the light of a recent work using
magnetocytometry16 and in which the relationship between
the applied force and the bead displacement was reported
to scale with sin3 γ . We indeed found that a good fit to our
computed α and β values can be obtained with analytical
functions α(γ , ξ ) and β(γ , ξ ) of the form (k1 + k2 /ξ ) sin3 γ ,
where k1 and k2 are real constants. However, this is only true
for embedding angles γ lower than 40◦ for the fit of function β(γ , ξ ) and for values of γ lower than 30◦ for the fit
of the function α(γ , ξ ). Above these values, the sin3 γ -type
fitting function is no longer valid.
The critical role of the geometrical invariant ξ = h u /2R
has been more precisely highlighted by computing the
threshold values ξ ∗ (γ ) of the ratio ξ above which α(γ ,
ξ ) and β(γ , ξ ) no longer vary significantly (Fig. 7), i.e.
α(γ , ξ ) ∼ α(γ ) and β(γ , ξ ) ∼ β(γ ) for a given embedding
angle γ . Below this threshold value ξ ∗ (γ ), the varying correcting coefficients α(γ , ξ ) and β(γ , ξ ) can be easily determined thanks to the relationships we established [Eqs. (7)
and (8)].
We also analyzed the influence of deeply embedded
beads and computed the corresponding abacuses for increasing values of γ up to 180◦ . Our computations of the
Generalized Elastic Modulus Estimation of Adherent Cell
139
FIGURE 8. Generalized abacuses establishing the relationships between the correcting coefficients α [Fig. 8(A) and 8(C)] and β
[Fig. 8(B) and 8(D)] when the two geometrical parameters 2R/hu and γ are varied. These abacuses cover the following ranges of
parameter values: 0.1 ≤ 2R/ hu ≤ 10; 25◦ ≤ γ ≤ 180◦ . The iso-embedding angle curves are computed by increment of 5◦ . The two
solid points, corresponding to a specific embedding angle of 65◦ , exemplify the application of theses abacuses (see text for details).
correcting coefficients α(γ , ξ ) and β(γ , ξ ) also agree with
and extend previous results of Mijailovitch et al.21
In this work, we assumed that the bead is firmly attached
to the cell cytoskeleton through integrin trans-membranous
receptors. Thus, a no slipping boundary condition has been
imposed at the cell/bead interface. However, if a partial
slipping θ s of the bead occurs during bead rotation, the
estimation of the shear modulus G would be biased by improper evaluation of the correcting factor α. From Eqs. (3)
and (5), we derived the expression of the relative error δG
on the shear modulus G as δG = θs /(θ − θs ), where θ is
the overall bead rotation. The influence of various degrees
of slipping can be analyzed by this relationship. Typically,
the relative error δG is lower than 10% if the ratio θ /θ s is
larger than 10.
Finally, despite of static conditions assumed for this finite element analysis, our results may be used to characterize the linear viscoelastic properties of the cytoskeletal
medium.4,9,25 Indeed, knowing that the correcting coefficients α(γ , ξ ) and β(γ , ξ ) are only geometry-dependent
and not frequency-dependent, the computed results can be
generalized9 such that G̃ θ ( f ) = α(γ , ξ )G̃( f ) and G̃ δ ( f ) =
β(γ , ξ )G̃( f ), where G̃ θ , G̃ δ , and G̃ are complex moduli
and f is the excitation frequency. The associated spatial
stress distribution could thus be compared to recent experimental data obtained by intracellular stress tomography.13
In conclusion, this work provides a refined analysis of the
experimentally measured cell response to external torques
imposed at the cell apical surface. In particular, consideration of the here reported abacuses is of importance to deal
with variations in parameter hu due to the random localization of bead attachment at the cell surface. Indeed, the
decrease of the cell height when going from the cell body
to flattened lamellipods would lead to drastic variation in
the estimated cell elastic modulus.2 Consideration of such
variations of the correcting coefficients could be still more
crucial when measuring the integrated mechanical response
of cells anchored on gels with varying stiffness.6 In this
case, the substrate stiffness modifies the cell spreading, and
thus the mean cell height, with direct implications on the
140
J. OHAYON and P. TRACQUI
APPENDIX
Rigid Bead Immersed in a Finite (or Infinite) Isotropic
Incompressible Medium
Let us denoted (er , e , eφ ) and (r, , φ) the spherical
unit base vectors and the associated physical coordinates
respectively. An exact solution of a single spherical rigid
bead of radius R embedded (γ = π ) in an elastic finite
spherical concentric medium of radius Re can be found, in
linear elasticity, when the bead is submitted to a small bead
rotation θ in the parallel plane (r, φ) going through the bead
center. In this test problem, relevant for twisting magnetocytometry experiments, we are looking for the relationship
between applied torque T and bead rotation θ . The medium
is assumed to be incompressible and isotropic, and is thus
described by the constitutive law:
σ = − p I + 2Gε
(A.1)
where σ and ε are the stress and strain tensors, I is the identity matrix, p is the Lagrangian multiplier resulting from
the material incompressibility,12 and G is the material shear
modulus, related to the Young’s modulus Ecell by the relationship E cell = 3G.
Moreover, the linearized material incompressibility constraint is given by
∇ u = 0
(A.2)
where u is the unknown displacement vector.
If gravity and inertial forces are neglected, the condi or in terms of the
tion for local equilibrium is ∇·[θ ] = 0,
displacement vector:
FIGURE 9. Generalized abacuses giving the relationship between the ratio of bead rotation over normalized bead translation (θ/δ) when the three geometrical parameters 2R /hu , h/2R,
and γ are varied [Fig. 9(A)]. Notice that the ratio θ/δ can be
derived from the correcting coefficients α and β using the relationship θ/δ = β/α. These abacuses are built for the following
parameter ranges: 0.1 ≤ 2R/hu ≤ 10, 0.2 ≤ h/2R ≤ 10, and
25◦ ≤ γ ≤ 180◦ . The iso-γ curves are computed by increment
of 5◦ , while the iso-h/2R curves [Fig. 9(A)] are computed from
h/2R = 0.2 to 2 with an initial step of 0.1, then with the steps
size specified on the abacus. The solid point corresponds to
the values used to exemplify the application of these abacuses
(see text for more details).
estimated apparent cell stiffness: in the MTC experiments
conducted by Doornaert et al.,5 the apparent cell stiffness
almost doubles with cell spreading, i.e. when maximum cell
height decreases from 22 to 9 µm.
Therefore, although focusing on data analysis obtained
by magnetic twisting cytometry, this study provides a framework in which finite element modeling of cell rheology
would help to decrease the dispersion of cell elastic modulus estimated from experimental cell mechanical response.
More fundamentally, it would improve our knowledge of the
mechanisms of mechanotransduction and modified genes
expression triggered by micromanipulations techniques, including MTC.3
∇ p = µ∇ 2 u
(A.3)
The displacement field u must satisfy the following boundary conditions: (i) a small bead rotation θ with no translation
is imposed to the bead and (ii) zero displacements are imposed on the external surface r = Re . These two conditions
are expressed respectively as:
u (r, , φ) = r sin () θ eφ at r = R
(A.4)
u (r, , φ) = 0
at r = R
(A.5)
The displacement field can be found by using the method
of separation of variables and by looking for a u vector
solution of the form
u (r, , φ) = r sin ()w(r )eφ
(A.6)
where w(r ) becomes the unknown function of the problem.
For a solution field u which satisfies the above Eqs. (A.2)(A.5), one gets the following function w(r ):
3
θ R3
Re
w(r ) = 3
−1
(A.7)
Re − R 3 r 3
So, the relation between the applied external torque T and
the small bead rotation θ is given by:
π 2π
T =
σr φ | R R 3 sin () d dφ
(A.8)
0
0
Generalized Elastic Modulus Estimation of Adherent Cell
or, in the developed form, by
T =
R3 R3
8
π E cell θ 3 e 3
3
Re − R
11
(A.9)
In the particular case of a rigid bead immersed in an infinite
medium, the torque-bead rotation relation is obtained from
the limit lim T and is given by
Re →∞
8
π E cell θ R 3 = 6V Gθ
3
where V is the bead volume.
T =
(A.10)
ACKNOWLEDGMENTS
We gratefully acknowledge Dr Daniel Isabey and Pr
François Gallet for helpful discussions. Ph. Tracqui and
J. Ohayon are supported by a grant “ACI-Bioinformatique”
from the French Centre National de la Recherche Scientifique (CNRS).
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