Dynamics of traction stress field during cell division:Supplementary information

Hirokazu Tanimoto and Masaki Sano

1 Supplementary materials and methods

1.1 Cell culture

Axenically grown wild-typeDictyostelium discoideumcells (AX-2) were cultured as described previously

[1]: Cells were cultured in HL5 growing medium and kept in an exponential growth phase. Cells were

maintained on regular Petri dishes and subcultured at each time when cells became subconfluent, typically

every 2 to 3 days [2]. Cell culture and all experiments were carried out under 21 degree.

1.2 Substrate preparation and observation

Flexible poly-acrylamide substrates embedded with fluorescence beads (3% acrylamide, 0.25% bis, 0.28%

v/v of red fluorescence beads of a diameter 0.2µm) were prepared and calibrated with a standard way [3, 4].

The calibration of the substrate stiffness was performed for each time and the determined Young’s modulus

was approximately 800 Pa. We employed a confocal microscopy (TSC-SP5, Leica) equipped with a 63x

NA 1.4 Plan objective lens to visualize both the cell contour (transmission channel) and the fluorescence

beads (fluorescence channel, excitation wave length: 543 nm). The two-dimensional substrate deformation

was measured adopting Particle Image Velocimetry (PIV) method to the fluorescence image. The size of

the PIV grid was set to be 1.28µm (corresponding to 8 pixels) and shifted with an overlapping of 50 %.

1.3 Traction stress calculation in Fourier space

The traction stress was calculated from the substrate deformation in Fourier space. Since this computational

process is an inverse problem and the inverse of Green function amplifies high wave number measurement

noise, some filtering schemes to avoid the amplification are needed. Among several choices [5–8], we

adopted a simple low-pass filtering. The reasons are mainly two. (1) The measurement noise can be directly

estimated from the PIV of unstrained substrate and the strain signal had only low wave number component.

By comparing the strained data with the control, we could set the cut-off frequency of the low-pass filter so

that the filtered data does not contain high frequency components contaminated by the measurement noise.

(2) The low-pass filtering scheme ensures that the force dipole does not change during the calculation. Since

theNth moment is theNth derivative in Fourier space around the origin, the force dipole can be determined

only with the lowest Fourier components and thus is not altered during the stress-recovery process and

robust against the detail shape of the filter.

1

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

Figure S 1: Effects of shape of the low-pass filter on traction stress recovery. (a)-(e) Effects of cut-offfrequencyk0. The gaussian-type filters with differentk0 were adopted to calculate the stress field.k0 wasvaried from 15 (a) (corresponding wave-length: 5.5µm.) to 23 (j) (3.6µm.) with an interval of 2 andσ keptconstant (=4). k0 = 21 (d) is the cut-off frequency used in the main text. (f)-(j) Effects of functional form.Lorentzian-type filter with differentk0 were adopted to calculate the stress field.k0 was varied from 15 (f)to 23 (j) with an interval of 2. The scale parameter was determined so that the filter had the same half widthat half maximum of the gaussian-type filter withσ = 4.

We adopted the following gaussian-type low-pass filterF(|k|),

F(|k|) = 1 (|k| < k0)

exp(− (|k|−k0)2

σ2 ) (|k| ≥ k0)(1)

where|k| =√

k2x + k2

y, k0 is the cut-off frequency andσ2 is the variance. Although the force dipole, the

simplest quantity for the stress field asymmetry, is insensitive to the choice of cut-off frequency, this choice

can affect the fine structures of the traction stress. We plotted the traction stress (the same as Fig.2 -1

min in the main text) varying the cut-off frequency and the shape of low-pass filter (Fig. S1). To see the

effect of the cut-off frequency, we variedk0 with keepingσ the same value and plotted the filtered traction

stress patterns (Fig. S1 (a)-(e)). The characteristic distribution of the traction stress as four localized spots

qualitatively holds but their position more or less changes. In particular, the spot lying outside of the cell

(white arrow) moved toward cell inside with increasingk0. This result suggests that the outside stress spot

is not due to the wrong measurement nor an artifact of the inversion process but just due to the finite spatial

resolution of the data. We also tested Lorentzian-type filter, which decays much slower than the gaussian

(∼ k−2), and confirmed that the result is almost unchanged (Fig. S1 (f)-(j)).

2

Absolute force (nN)

Time (min)0T1 T2

F2F1

F3

Figure S 2: Characteristic parameters of traction stress dynamics. Time-evolution of the traction stresswas parameterized with six values.T1: The instant of first positive peak of the absolute forces (lies nearthe middle of phase 0).T2: The instant of first negative peak of the absolute force (corresponding to thebeginning of phase 1).F1: Difference of the absolute force betweenT1 andT2. F2: Difference of theabsolute force betweenT2 and the dividing instance.F3: The released absolute force on the occasion of thecell separation.F3 was decomposed into parallel (F3∥) and perpendicular (F3⊥) components with respect tothe division axis.

2 Supplementary text

2.1 Continuity of dipole axis

At the end of Phase 0 and the beginning of Phase 2, the two eigenvalues of dipole matrix are close and

thus the major axis of dipole frequently changes byπ/2 (Fig. 3, bottom column). For the information for

readers, we also identified the continuous dipole axis by choosing the closest one between two successive

frames and plotted its orientation as a dashed line in Fig.3 bottom column.

2.2 Statistics

To statistically characterize the traction stress dynamics, we parameterized the time-evolution of the abso-

lute forces with the following six values (Fig. S2).T1 is the time for the first positive peak of the absolute

forces and lies near the middle of phase 0.T2 is the time for the first negative peak of the absolute force and

corresponds to the beginning of phase 1.F1 is the difference of the absolute force between the two timings

T1 andT2. F2 is the difference of the absolute force between the two timingsT2 and the dividing instant.

F3 is the released absolute force on the occasion of the cell separation. This force was decomposed into

parallel and perpendicular components (F3∥ andF3⊥) with respect to the division axis. Note that the origin

of the time axis was set to be the instant when the cell division completed. The values of 5 cells and their

average and standard deviation are shown in Table S1.

3

T1 (min) T2 (min) F1 (nN) F2 (nN) F3∥ (nN) F3⊥ (nN)Cell 1 -8.4 -6.1 11.2 15.5 8.5 2.0

Cell 2 -6.8 -4.5 9.7 14.8 5.2 2.8

Cell 3* -5.8 -3.5 17.3 N.D. N.D. N.D.

Cell 4 -12.5 -7.7 18.6 19.9 3.1 0.4

Cell 5 -8.6 -2.8 18.9 17.4 7.4 2.4

mean± S.D. -8.4±2.5 -4.9±1.9 15.1±4.3 16.3±4.2 6.0±2.2 1.9±1.0

Table S 1: Statistics of traction stress dynamics. The characteristic parameters of the traction stress areshown for 5 cells. See text and Fig.S2 for the definition. The bottm column represents mean± S.D. (*: Asmall portion of dividing cells undergo repeated attaching and detaching during cytokinesis and we cannotevaluateF2 andF3 in this situation.)

3 Supplementary references

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