Analysis of mitochondrial mechanical dynamics using a confocal
fluorescence microscope with a bent optical fiber
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Yongbo Li*a, Satoshi Hondaa, Kentaro Iwamia, Yoshihiro Ohtab, and Norihiro Umeda*a
a
Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology,
Tokyo, Japan, 184-8588; bDepartment of Biotechnology and Life Science, Tokyo University of
Agriculture and Technology, Tokyo, Japan, 184-8588
*liyongbo2007@gmail.com or umeda@cc.tuat.ac.jp; phone & fax 81-42-388-7422;
S1. THEORY
S1.1 Contact model of BOF to a single mitochondrion
Figure S1 shows the model by which the BOF tip is contacted to a single mitochondrion. The
isolated mitochondrion is absorbed on a rigid glass substrate and indented by the flat-ended tip of the
BOF, which is wider than the mitochondrion.
This model has several advantages over conical or spherical tip models. First, the actual contact
area is more easily determined in the flat-ended tip model than in other models (see section 1).
Second, the relatively large diameter of the flat BOF tip reduces the local strain and slippage, the
presumed main sources of elastic moduli errors. Third, the proposed model is less likely to damage
the soft mitochondria than other models.
Figure S1 Model of contact between the flat-ended tip and a single mitochondrion. The isolated
mitochondrion adhered to the rigid glass is indented by the flat-ended BOF tip. The BOF tip
diameter is 2–4 μm, around or slightly exceeding the mitochondrion diameter of (1–3 μm).
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The experimental data were analyzed by traditional Hertzian contact theory. This theory, which is
universally applied to the data of cell mechanics (McKee, et al., 2011), relates the applied force to
the observed deformation as follows:
F=
3
4√𝑅𝐸
𝛿2 ,
3
(S1)
where F is the applied force, δ is the sample deformation and R is the radius obtained by
combining the radii of the mitochondrion (R1) and the BOF tip (R2) as shown in Eq. S2. For a flat
BOF tip (i.e., R2 = ∞), Eq. S2 reduces to the trivial case Eq. S3.
1
𝑅
=
1
𝑅1
1
+𝑅 ,
(S2)
2
R = 𝑅1 ,
(S3)
Similarly, the Young’s modulus E combines the Young's moduli of the mitochondrion (E1) and
BOF (E2) as shown in Eq. S4. In this equation, the Poisson's ratio of the mitochondrion (v1) is
assumed as the typical Poisson’s ratio of biomaterials; namely, 0.5 (Li, et al., 2008). Since the BOF
is made from glass, E2 and v2 are assumed as 50.9 GPa and 0.25, respectively. Therefore, (1-v22)/E2
can be approximated to zero and Eq. S4 reduces to Eq. S5.
1
E
=
1−𝑣12
𝐸1
E =
1−𝑣22
+
𝐸2
𝐸1
1−𝑣12
,
.
(S4)
(S5)
Substituting Eq. S3 and S5 into Eq. S1, the proposed model for the mitochondrial measurement
becomes
F=
4√𝑅1 𝐸1
3(1−𝑣12 )
3
𝛿2 .
(S6)
S1.2 Deformation of a single mitochondrion
The mitochondrial deformation can be computed from the indentation geometry, as revealed by
the AFM measurements (McKee, et al., 2011). As the BOF approaches a mitochondrion, the
translation (Z) of the PZT1 holding the BOF and the BOF deflection of BOF (d) are acquired.
2
The data of initial contact between the BOF and mitochondrion are designated Z0 and d0. The
initial contact positions are illustrated in Fig. S2(a). The useful data for measuring the elastic
modulus, when the BOF and mitochondrion are simultaneously deformed, are acquired beyond the
contact point. From the indentation geometry, the mitochondrial deformation (δ) is computed by
rearranging Eq. S7.
∆Z = ∆d + δ ,
(S7)
where ΔZ is the displacement of PZT-1 (ΔZ = Z – Z0) and Δd is the BOF deflection (Δd = d – d0).
Figure S2 Mitochondrial deformation measured by the indentation geometry. (a) Initial contact
position of BOF and mitochondrion (d0, Z0). (b) Geometry of the relationship between the BOF
deflection Δd and mitochondrial deformation δ. The piezo actuator displaces by Δz. The
positional shifts are related by Δz = Δd + δ.
S2. SPRING CONSTANT OF BOF
Here, we derive a theoretical expression for the spring constant of the BOF. To validate the theory,
the spring constant was also determined by a calibration experiment.
S2.1 Theoretical equation of spring constant
Figure S3(a) shows the structure of the BOF bent at an arbitrary angle (α). The lengths of the
unbent BOF and its bent part are denoted L1 and L2, respectively. The spring constant (k) of the
straight BOF is easily calculated by the traditional differential equation of the deflection curve (Gere,
2004), but calculating the k of the bent section is less straightforward.
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Figure S3 (a) Structure of the BOF. L1 and L2 denote the lengths of the straight and bent sections,
respectively. α represents the bending angle of the BOF. (b) The BOF deflects by δp under a load P.
To resolve these problems, we calculate k using Castigliano's theorem (Eq. S8), which states that
the partial derivative of the strain energy (Uj) of a structure with respect to any load (Pj) equals the
displacement (δj) under that load. Note that this theorem is applicable to structures of any shape, and
is thus suitable for deriving the k of the BOF used in the present study.
𝜕𝑈𝑗
𝜕𝑃𝑗
= δ𝑗 .
(S8)
Figure S3(b) shows the deflection of the BOF under a load P applied at point A. The strain energy
is given by Eq. S9.
𝐿
1
𝐿
U = 2𝐸𝐼 (∫0 1 𝑀1 2 dx1 + ∫0 2 𝑀2 2 dx2 ) ,
(S9)
where E is the Young's modulus, I is the second moment of the fiber's cross-sectional area. x1 is a
point between BO, the bending moment is 𝑀1 (𝑥1 ) = −𝑀𝑜 − 𝑃𝑥1 . MO is the bending moment at the
point O, 𝑀𝑂 = 𝑃𝑐𝑜𝑠𝛼 ∙ 𝐿2 . x2 is a point between OA, the bending moment is equal to 𝑀2 (𝑥2 ) =
−𝑃𝑐𝑜𝑠𝛼 ∙ 𝑥2 . Therefore, the deflection δP of point A is calculated as
δ𝑃 =
𝜕𝑈
𝜕𝑃
𝑃
= 3𝐸𝐼 [(𝐿1 + 𝐿2 𝑐𝑜𝑠𝛼)3 + 𝐿32 𝑐𝑜𝑠 2 𝛼(1 − 𝑐𝑜𝑠𝛼)].
(S10)
Finally, we evaluate k by Eq. S11:
𝑃
k = 𝛿 = (𝐿
𝑃
3𝐸𝐼
3
3
2
1 +𝐿2 𝑐𝑜𝑠𝛼) +𝐿2 𝑐𝑜𝑠 𝛼(1−𝑐𝑜𝑠𝛼)
,
(S11)
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where E is the Young's modulus of the BOF material. Because the glass fiber is a two-layer
structure composed of different materials, its E is difficult to determine. I is calculated as π𝑑 4 /64
and d is the fiber diameter (125 μm). Consequently, k is calculable by Eq. S11 when the BOF
parameters and E are known.
S2.2 Calibration method
As described in subsection 2.2 of main text, the BOF was coated with a metal film and its
deflection was measured with respect to an attached GND board. Obviously, this configuration is
ignored in the above theoretical derivation of k. For this reason, the spring constant of the BOF was
also accurately determined by calibrating the AFM cantilever's spring constant (Torii et al., 1996).
Figure S4 (a) illustrates the method for measuring the BOF's spring constant. The BOF is fixed
onto a pizeo stage (PZT). The BOF receives a small force through a straight fiber (quartz fiber,
Edmund Optics Inc.), which acts as a standard cantilever. Since both layers (cladding and core) of
the straight fiber consist of the same material, the Young’s modulus of this section is assumed to be
that of quartz (72 GPa). The diameter and length of the straight fiber is 600 μm and 141.34 mm,
respectively. Inserting these parameters into Eq. S12, the spring constant of the straight fiber (ks) is
calculated as 0.487 N/m.
𝑘𝑠 =
3𝐸𝜋𝑑4
64𝐿3
,
(S12)
The deflection of the straight fiber was detected by an optical level system. A beam of laser light
was reflected by the straight fiber's end and detected by a bi-photodetector. From these reflection
phenomena, the deflection of the straight fiber (δs) was measured.
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Figure S4 (a) Schematic of method for measuring the BOF spring constant. The BOF is fixed onto a
piezo stage (PZT). The straight fiber acts as a standard cantilever, and its deflection is measured
by an optical level system. Laser light reflected by the end of the straight fiber is collected by a
bi-photodetector (Bi-PD), which is sensitive to small changes induced by deflection. (b)
Geometric relationship between the BOF and the straight fiber when the two bodies are in contact.
The schematic shows that the PZT deflects by ΔZ = δs + δBOF, where δs and δBOF are the
deflections of the straight fiber and the BOF, respectively.
As the forces acting on the straight fiber and BOF are equal, we have ksδs = kδBOF, where δs and
δBOF denote the deflections of the straight fiber and BOF, respectively. Figure S4(b) shows the
geometric relationship when the straight fiber is deflected by a shift in the BOF. The PZT
displacement (ΔZ) is the sum of the deflections of the straight fiber and the BOF (δs and δBOF,
respectively; see Eq. S7). Hence, k is calculated as follows:
𝛿
𝑠
k = 𝑘𝑠 ∆𝑍−𝛿
,
𝑠
(S13)
In the calibration experiments we must vary the BOF length; hence, the BOF was not glued to the
fixing board (subsection 2.2 of main text). Figure S5 plots the spring constant versus the equivalent
BOF length (L), calculated as [(𝐿1 + 𝐿2 𝑐𝑜𝑠𝛼)3 + 𝐿32 𝑐𝑜𝑠 2 𝛼(1 − 𝑐𝑜𝑠𝛼)] . The BOF parameters,
measured under an optical microscope, are: L1 = 3.30 mm, 5.56 mm, 6.73 mm, 10.95 mm, L2 = 1.84
mm, α = 80.5°. The experimental results closely agreed with the fitting line, given by Eq. S14:
k = C𝐿−3 ,
(S14)
where C equals 3EI. From the fitting parameter C, E was determined as 50.9 ± 2.8 GPa.
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Figure S5 Spring constant as a function of equivalent length, equal to [(𝐿1 + 𝐿2 𝑐𝑜𝑠𝛼)3 +
𝐿32 𝑐𝑜𝑠 2 𝛼(1 − 𝑐𝑜𝑠𝛼)]. Experiments at each equivalent length were triplicated. The experimental
data are fitted to Eq. S14. From the fitting line, the Young's modulus of the BOF material was
determined as 50.9 ± 2.8 GPa, lower than that of quartz.
S2.3 Summary of BOF spring constant
In summary, we derived a theoretical expression (Eq. S11) for the spring constant of the BOF.
The derived equation is consistent with the results of the calibration experiments. The Young's
modulus of the BOF was 50.9 ± 2.8 GPa. As the spring constant was unaffected by the metal film
and the board glued to the BOF, it can be calculated by the derived equation.
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