doi: 10.1371/journal.pone.0069850.
eCollection 2013.
Three-dimensional quantification of cellular traction forces and mechanosensing of thin substrata by fourier traction force microscopy
Affiliations
- PMID: 24023712
- PMCID: PMC3762859
- DOI: 10.1371/journal.pone.0069850
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Three-dimensional quantification of cellular traction forces and mechanosensing of thin substrata by fourier traction force microscopy
PLoS One.
.
Abstract
We introduce a novel three-dimensional (3D) traction force microscopy (TFM) method motivated by the recent discovery that cells adhering on plane surfaces exert both in-plane and out-of-plane traction stresses. We measure the 3D deformation of the substratum on a thin layer near its surface, and input this information into an exact analytical solution of the elastic equilibrium equation. These operations are performed in the Fourier domain with high computational efficiency, allowing to obtain the 3D traction stresses from raw microscopy images virtually in real time. We also characterize the error of previous two-dimensional (2D) TFM methods that neglect the out-of-plane component of the traction stresses. This analysis reveals that, under certain combinations of experimental parameters (cell size, substratums' thickness and Poisson's ratio), the accuracy of 2D TFM methods is minimally affected by neglecting the out-of-plane component of the traction stresses. Finally, we consider the cell's mechanosensing of substratum thickness by 3D traction stresses, finding that, when cells adhere on thin substrata, their out-of-plane traction stresses can reach four times deeper into the substratum than their in-plane traction stresses. It is also found that the substratum stiffness sensed by applying out-of-plane traction stresses may be up to 10 times larger than the stiffness sensed by applying in-plane traction stresses.
Conflict of interest statement
Figures
The three panels at the top of the figure (a–c) show the first horizontal slice of a 
–pixel 
stack, which is focused at the free surface of the substratum (
). (
a
), Tracer beads fluorescence in undeformed conditions used as reference for traction force microscopy. The scale bar is 10 microns long. The axes indicate the reference system for both the substratum deformation and the traction stresses. (
b
), Tracer beads fluorescence when the substratum is deformed by a migrating cell, whose outline is indicated by the green contour. (
c
), Image obtained by merging the fluorescence from tracer beads in undeformed (a) and deformed (b) conditions, which reveals the deformation of the substratum. White speckles indicate perfect match between undeformed and deformed conditions and thus zero local deformation. Blue and orange speckles indicate mismatch between undeformed and deformed conditions and thus non-zero local deformation. Regions of locally large deformation are indicated with yellow arrows. The dashed green line indicates the location of the vertical section shown in panels d–g. (
d
), Three-dimensional illustration of the relative positions of the horizontal plane in panel (x–y, yellow axes) and the vertical plane in panel (s–z, red axes). The black axes indicate the three-dimensional reference system used to express both the substratum deformation and the traction stresses. The three panels at the bottom right corner of the figure (e–g) show vertical slices of the same 
–stack passing through the dashed green line in panel at 
. (
e
), Image obtained by merging the fluorescence from tracer beads in undeformed (f) and deformed (g) conditions. Speckle patterns with orange top and blue bottom are found in locations where the cell is pulling up on the substratum. Conversely, speckle patterns with blue top and orange bottom are found the when cell is pushing down on the substratum. (
f
), Tracer beads fluorescence in deformed condition. (
g
), Tracer beads fluorescence in underformed condition.
–pixel stack, which is focused at the free surface of the substratum (). (
a
), Tracer beads fluorescence in undeformed conditions used as reference for traction force microscopy. The scale bar is 10 microns long. The axes indicate the reference system for both the substratum deformation and the traction stresses. (
b
), Tracer beads fluorescence when the substratum is deformed by a migrating cell, whose outline is indicated by the green contour. (
c
), Image obtained by merging the fluorescence from tracer beads in undeformed (a) and deformed (b) conditions, which reveals the deformation of the substratum. White speckles indicate perfect match between undeformed and deformed conditions and thus zero local deformation. Blue and orange speckles indicate mismatch between undeformed and deformed conditions and thus non-zero local deformation. Regions of locally large deformation are indicated with yellow arrows. The dashed green line indicates the location of the vertical section shown in panels d–g. (
d
), Three-dimensional illustration of the relative positions of the horizontal plane in panel (x–y, yellow axes) and the vertical plane in panel (s–z, red axes). The black axes indicate the three-dimensional reference system used to express both the substratum deformation and the traction stresses. The three panels at the bottom right corner of the figure (e–g) show vertical slices of the same –stack passing through the dashed green line in panel at . (
e
), Image obtained by merging the fluorescence from tracer beads in undeformed (f) and deformed (g) conditions. Speckle patterns with orange top and blue bottom are found in locations where the cell is pulling up on the substratum. Conversely, speckle patterns with blue top and orange bottom are found the when cell is pushing down on the substratum. (
f
), Tracer beads fluorescence in deformed condition. (
g
), Tracer beads fluorescence in underformed condition.
The input data are the measured three-dimensional deformation caused by the cell on the free surface of the substratum (
, red), and it is assumed that the deformation of the substratum is zero at the bottom surface in contact with the glass coverslip (
, blue). We assume that the substratum has linear, homogeneous, isotropic material properties, with Young modulus E and Poisson's ratio σ. Fourier series with spatial periods L and W are used to express the dependence of the variables in the horizontal directions.
, red), and it is assumed that the deformation of the substratum is zero at the bottom surface in contact with the glass coverslip (, blue). We assume that the substratum has linear, homogeneous, isotropic material properties, with Young modulus E and Poisson's ratio σ. Fourier series with spatial periods L and W are used to express the dependence of the variables in the horizontal directions.
The cell is moving from left to right. The level of deformation/stress is represented by a pseudo-color map according to the color bars at the right hand side of the figure. The green contour indicates the cell outline. The data are overlaid on the DIC image used to identify the cell body (see 
2.4). The black scale bars are 
long. (
a
), Tangential (horizontal) deformation in the direction parallel to cell speed, 
. (
b
), Tangential (horizontal) deformation in the direction perpendicular to cell speed, 
. (
c
), Normal (vertical) deformation, 
. (
d
), Tangential (horizontal) stress in the direction parallel to cell speed, 
. (e), Tangential (horizontal) stress in the direction perpendicular to cell speed, 
. (f), Normal (vertical) stress, 
. The arrows in panels , , and indicate the directions of positive (red) and negative (blue) deformation/stress. The 
and 
symbols in panels and indicate deformation/stress pointing respectively into the plane (blue, negative) and out of the plane (red, positive).
2.4). The black scale bars are long. (
a
), Tangential (horizontal) deformation in the direction parallel to cell speed, . (
b
), Tangential (horizontal) deformation in the direction perpendicular to cell speed, . (
c
), Normal (vertical) deformation, . (
d
), Tangential (horizontal) stress in the direction parallel to cell speed, . (e), Tangential (horizontal) stress in the direction perpendicular to cell speed, . (f), Normal (vertical) stress, . The arrows in panels , , and indicate the directions of positive (red) and negative (blue) deformation/stress. The and symbols in panels and indicate deformation/stress pointing respectively into the plane (blue, negative) and out of the plane (red, positive).
The Poisson's ratio is 
and the substratum thickness, 
, is equal to the length of the “synthetic cell”. The plots in the top row show the synthetic deformation field in the x direction (eq. 11, panel a), y direction (zero, panel b) and z direction (eq. 13, panel c). The second row shows the traction stresses calculated from the displacements in panels (a)–(c) by 3D Fourier TFM. (d), 
; (e), 
; (f), 
. The third row shows the traction stresses calculated from the displacements in panels (a)–(c) by 2D Fourier TFM under the assumption of zero normal displacements on the substratum's surface (
as in ref. [15]). (g), 
; (h), 
; (i), 
. The last row shows the traction stresses calculated from the displacements in panels (a)–(c) by 2D Fourier TFM under the assumption of zero normal stresses on the substratum's surface (
as in ref. [13]). (j), 
; (k), 
; (l), 
.
and the substratum thickness, , is equal to the length of the “synthetic cell”. The plots in the top row show the synthetic deformation field in the x direction (eq. 11, panel a), y direction (zero, panel b) and z direction (eq. 13, panel c). The second row shows the traction stresses calculated from the displacements in panels (a)–(c) by 3D Fourier TFM. (d), ; (e), ; (f), . The third row shows the traction stresses calculated from the displacements in panels (a)–(c) by 2D Fourier TFM under the assumption of zero normal displacements on the substratum's surface ( as in ref. [15]). (g), ; (h), ; (i), . The last row shows the traction stresses calculated from the displacements in panels (a)–(c) by 2D Fourier TFM under the assumption of zero normal stresses on the substratum's surface ( as in ref. [13]). (j), ; (k), ; (l), .
. Red lines and symbols, 
; blue lines and symbols, 
.–•–, 2D method with finite h and 
on the surface (ref. [13]);–▴–, 2D method with finite h and 
on the surface (ref. [15]);–▪–, Boussinesq solution with infinite h (refs. [10], [12]). (a), 
; (b), 
. The shaded patch represents the range of values of Poisson's ratio reported for gels customarily employed in TFM –.
Red lines and symbols, 
; blue lines and symbols, 
.–•–, 2D method with finite h and 
on the surface (ref. [13]);–▴–, 2D method with finite h and 
on the surface (ref. [15]);–▪–, Boussinesq solution with infinite h (refs. [10], [12]).
; blue lines and symbols, .–•–, 2D method with finite h and on the surface (ref. [13]);–▴–, 2D method with finite h and on the surface (ref. [15]);–▪–, Boussinesq solution with infinite h (refs. [10], [12]).
(a), 
assuming zero normal deformation on the surface of the substratum; (b), 
from Boussinesq's solution. The black×marks the combination of σ and 
used to calculate the traction stress maps in Figure 4. The horizontal lines mark the values of 
used to plot Figure 5.–––, 
;– – –, 
. The thick white contours correspond to 
. The vertical lines mark the values of σ used to plot Figure 6.– – –, 
;––––, 
.
assuming zero normal deformation on the surface of the substratum; (b), from Boussinesq's solution. The black×marks the combination of σ and used to calculate the traction stress maps in Figure 4. The horizontal lines mark the values of used to plot Figure 5.–––, ;– – –, . The thick white contours correspond to . The vertical lines mark the values of σ used to plot Figure 6.– – –, ;––––, .
(a), contour map of 
for 
, represented as a function of 
and 
. The black circle corresponds to the example cell in Figure 3. The thick white contour corresponds to 
. (b), contour map of 
for 
, represented as a function of σ and 
. The thick white (black) contours correspond to 
(0.2), and are well approximated by the magenta squares (circles) coming the eq. 19.
for , represented as a function of and . The black circle corresponds to the example cell in Figure 3. The thick white contour corresponds to . (b), contour map of for , represented as a function of σ and . The thick white (black) contours correspond to (0.2), and are well approximated by the magenta squares (circles) coming the eq. 19.
The four panels in the top row (a–d) show surface plots of 
as a function of the horizontal wavelengths of the strain/stress fields 
.(
a
), present 3D TFM method;(
b
), 2D TFM under the assumption of zero normal stresses on the substratum's surface (
as in ref. [13]);(
c
), 2D TFM under the assumption of zero normal displacements on the substratum's surface (
as in ref. [15]);(
d
), Boussinesq's traction cytometry assuming an infinitely-thick substratum (as in refs. [10], [12]). The symbol curves in these plots indicate the sections of 
represented in panel (e).(
e
), 
along the line 
from different traction TFM methods, represented as a function of 
.
as a function of the horizontal wavelengths of the strain/stress fields .(
a
), present 3D TFM method;(
b
), 2D TFM under the assumption of zero normal stresses on the substratum's surface ( as in ref. [13]);(
c
), 2D TFM under the assumption of zero normal displacements on the substratum's surface ( as in ref. [15]);(
d
), Boussinesq's traction cytometry assuming an infinitely-thick substratum (as in refs. [10], [12]). The symbol curves in these plots indicate the sections of represented in panel (e).(
e
), along the line from different traction TFM methods, represented as a function of .
Panels (a) and (b) display the apparent elastic modulus of the substratum (eq. 21) in the directions tangential and normal to the surface respectively. The data are plotted as a function of the Poisson's ratio, σ, and the substratum thickness h normalized with the wavenumber 
of the strain/stress fields. For simplicity, the 
case is represented but similar results are obtained for other combinations of the wavenumbers. The isolines plotted in panels (a) and (b) are respectively 
, and 
. Particularly, the The thick black contour in each panel represents the isoline 
corresponding to a two-fold increase in apparent stiffness. The vertical dashed lines indicate the values of the Poisson's ratio represented in Figure 11, 
.
of the strain/stress fields. For simplicity, the case is represented but similar results are obtained for other combinations of the wavenumbers. The isolines plotted in panels (a) and (b) are respectively , and . Particularly, the The thick black contour in each panel represents the isoline corresponding to a two-fold increase in apparent stiffness. The vertical dashed lines indicate the values of the Poisson's ratio represented in Figure 11, .
Panel (
a
) displays the ratio 
as a function of the substratum thickness h normalized with the inverse wavenumber 
of the strain/stress fields. For simplicity, the 
case is represented but similar results are obtained for other combinations of the wavenumbers.–––, normal direction and 
;––––, tangential direction and 
;– – –, normal direction and 
;– – –, tangential direction and 
. The black horizontal lines indicate the levels 
(no increase in apparent elastic modulus,– – –) and 
(two-fold increase,–– – ––). Panel (
b
) displays the sensing depth defined as the value of h that yields a two-fold increase in apparent elastic modulus compared to 
. The sensing depth is represented as a function of the Poisson's ratio for tangential and normal traction stresses.–––, normal direction;–––, tangential direction. The shaded patch represents the range of values of Poisson's ratio measured for polymer networks –.
as a function of the substratum thickness h normalized with the inverse wavenumber of the strain/stress fields. For simplicity, the case is represented but similar results are obtained for other combinations of the wavenumbers.–––, normal direction and ;––––, tangential direction and ;– – –, normal direction and ;– – –, tangential direction and . The black horizontal lines indicate the levels (no increase in apparent elastic modulus,– – –) and (two-fold increase,–– – ––). Panel (
b
) displays the sensing depth defined as the value of h that yields a two-fold increase in apparent elastic modulus compared to . The sensing depth is represented as a function of the Poisson's ratio for tangential and normal traction stresses.–––, normal direction;–––, tangential direction. The shaded patch represents the range of values of Poisson's ratio measured for polymer networks –.
(a) Vertical contour map of u obtained by applying a unit tangential synthetic deformation field at the free surface of the gel (eqs. 11–13 with 
and 
), and solving the elastostatic equation for different values of z. The deformation is plotted in the normal plane 
as a function of 
and 
. Thus, 
represents the bottom of the gel in contact with the coverslip and 
represents the free surface of the gel. (b) Same as (a) for the normal deformation w obtained by applying a unit normal deformation at the gel surface (eqs. 11–13 with 
and 
). (c) Same as (a) for 
. (d) Same as (b) for 
. In all panels, the data are normalized between −1 and 1, and the green contour represents the 10% iso-level.
and ), and solving the elastostatic equation for different values of z. The deformation is plotted in the normal plane as a function of and . Thus, represents the bottom of the gel in contact with the coverslip and represents the free surface of the gel. (b) Same as (a) for the normal deformation w obtained by applying a unit normal deformation at the gel surface (eqs. 11–13 with and ). (c) Same as (a) for . (d) Same as (b) for . In all panels, the data are normalized between −1 and 1, and the green contour represents the 10% iso-level.Similar articles
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References
-
- Li S, Guan J, Chien S (2005) Biochemistry and biomechanics of cell motility. ANNUAL REVIEW OF BIOMEDICAL ENGINEERING 7: 105–150. - PubMed
-
- Engler AJ, Sen S, Sweeney HL, Discher DE (2006) Matrix elasticity directs stem cell lineage specification. Cell 126: 677–689. - PubMed
-
- Discher D, Janmey P, Wang Y (2005) Tissue cells feel and respond to the stiffness of their substrate. Science 310: 1139. - PubMed
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