Supplementary material for “Emergence of bacterial glass”
Hisay Lama,1 Masahiro J. Yamamoto,2, 1 Yujiro Furuta,3, 4 Takuro Shimaya,1, 4 and Kazumasa A. Takeuchi1, 4, ∗2

1
Department of Physics, The University of Tokyo, Tokyo, Japan
National Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan3
Department of Physics, Tokyo Metropolitan University, Tokyo, Japan
4
Department of Physics, Tokyo Institute of Technology, Tokyo, Japan(Dated: July 4, 2024)

Mode-coupling theory exponent γ

Our bacterial glass transitions show a rapid increaseof relaxation times, τQ and τθ for translational and orientational degrees of freedom, respectively, as the area
fraction ϕ is increased (Fig. 2c symbols). The observed
dependence on ϕ is consistent with the power-law divergence that mode-coupling theories (MCT) predict [1–3](Fig. 2c dashed lines),−γQτQ ∼ (ϕQ,c − ϕ)τθ ∼ (ϕθc − ϕ)−γθ ,(S1)as well as with the Vogel-Fulcher-Tamman (VFT) law[1, 2, 4] (Fig. S6).
Here we focus on the exponent γ of the MCT powerlaws, Eq. (S1). Our fitting gives γQ = 1.6(3) andγθ = 1.5(13) (see main text). In contrast, for thermal systems, it is actually known that MCT generally
gives γ ≥ γ0 ≡ 1.76 . . . [1]. More precisely, for spherical particle systems near equilibrium, one may indeed
prove γ ≥ γ0 [1]. The situation is more involved forthe aspherical case, where the same inequality has notbeen proven from first principles, but by using a diagonalization approximation of the MCT memory kernel,
one can still show γ ≥ γ0 [5]. This inequality has also
been confirmed by simulations of different aspherical particle systems (e.g., [6, 7]), without any exception so far,to our knowledge. Therefore, it is reasonable to consider
that γ ≥ γ0 generally holds for systems near equilibrium.
Our estimate γQ = 1.6(3) seems to violate this bound,
thereby indicating the non-equilibrium nature of our bacterial system. This observation raises some interestingquestions that may deserve further investigation, such as
clarifying conditions to violate the inequality γ ≥ γ0 , how
generally this violation takes place in non-equilibrium or
active systems, etc.

∗kat@kaztake.org[1] W. Götze, Complex Dynamics of Glass-Forming Liquids:
A Mode-Coupling Theory, International series of monographs on physics (Oxford Univ. Press, New York, 2009).
[2] L. Berthier and G. Biroli, Rev. Mod. Phys. 83, 587 (2011).
[3] D. R. Reichman and P. Charbonneau, J. Stat. Mech. 2005,
P05013 (2005).

[4] G. L. Hunter and E. R. Weeks, Rep. Prog. Phys. 75,066501 (2012).
[5] R. Schilling and T. Scheidsteger, Phys. Rev. E 56, 2932
(1997).
[6] A. J. Moreno, S.-H. Chong, W. Kob, and F. Sciortino, J.
Chem. Phys. 123, 204505 (2005).
[7] P. Pfleiderer, K. Milinkovic, and T. Schilling, Europhys.
Lett. 84, 16003 (2008).

2SUPPLEMENTARY FIGURES

0.9
0.88
0.86
0.84
0.82
0.8
0.78
0.76

0100

200

300

400

500time (s)FIG. S1. Histogram of cell areas. The cell areas were evaluated for ϕ = 0.784(7), from the first three frames where thecell segmentation was carried out. The dashed line shows
the fitted log-normal distribution,
whoseprobability densityhi(log x−µ)21√
with µ = 1.37(2) andis given by 2πσx exp − 2σ2σ = 0.281(15) (the value of x is in the unit of µm2 ). Here theuncertainty corresponds to the 95% confidence interval fromthe fit. The mean cell area is 4.08 µm and the polydispersity
index is 1.07. Note that a few filamentous cells and cells at
the boundary of the region of interest were excluded from thishistogram.

FIG. S2. Growth of the area fraction ϕ. The area fraction ϕ was evaluated for each group of 500 images recorded
over ≈ 13 s. The time point at the center is used in thisplot. The dashed line shows a fit to the exponential growth
curve, ϕ(t) ∝ 2t/T , which estimated the doubling time atT = 45(5) min. Here the uncertainty corresponds to the 95%
confidence interval from the fit. Note that the origin of timeis set to be the moment at which we started the acquisition of
the first set of images, which was already roughly five hours
after the start of the experiment.

3A t = 6.7 s, ? = 0.784

B t = 248.2 s, ? = 0.817

C t = 329.3 s, ? = 0.845

10.9

D t = 396.8 s, ? = 0.859

E t = 437.2 s, ? = 0.873

F t = 491.5 s, ? = 0.887

0.8

0.7

0.6
FIG. S3. Spatial dependence of the area fraction ϕ. The area fraction ϕ was evaluated for each group of 500 images recorded
over ≈ 13 s, locally in a mesh composed of regions of 10 × 10 pixels (1.724 µm × 1.724 µm). Above each panel, the time at thecenter of each time interval and the mean value of ϕ (averaged over the time interval and the region of interest) are displayed.
Note that the origin of time is set to be the moment at which we started the acquisition of the first set of images, which wasalready roughly five hours after the start of the experiment. These indicate that uniform growth was indeed realized in ourdevice.

4= 0.784= 0.800
0.5

0.4
0.2
0-2
10

10

00-2
10

= 0.817
0.5

0-2
10

0.5

10

00-2
10

0.5

10

0

0.5

10-2
10

100

= 0.868

10= 0.846

00.2

0-2
100-2
10
10-2
10

100

100= 0.873

0-210= 0.837
0.6
0.4
0.20-210

10-21010= 0.833

1000-210= 0.837

0-210100.5

100= 0.841
0.5

100

= 0.8590.5

0.5

100.20.5

10= 0.8170.6
0.4= 0.845

0.5

0.5
0-2
1000.5= 0.8430-21010= 0.8090.4= 0.823= 0.837

0-2
10

100

0.6

0-2
10
1100

= 0.8640.5

1000-210100= 0.8770.5

1000-210

100FIG. S4. Results of fitting of the overlap function Q(∆t) by a stretched exponential function. Each data was fitted by aβQstretched exponential function plus an offset, Q(∆t) = fQ e−(∆t/τQ )+ aQ . See Fig. S5 for the obtained values of βQ .

51QQ,

0.8
0.6
0.4
0.2

0.8

0.85

0.9

FIG. S5. Stretched exponential parameter β. The estimatesof βQ and βθ from the stretched exponential fitting, Q(∆t) =βQβθfQ e−(∆t/τQ ) +aQ and Cθ (∆t) = fθ e−(∆t/τθ ) +aθ (Fig. S4and Fig. S7, respectively) are shown. The exponent values
are closer to 1 for lower area fractions ϕ, indicating that therelaxation becomes closer to that of simple liquids.

FIG. S6. Vogel-Fulcher-Tamman (VFT) fitting of the relaxation times. The data of τQ and τθ displayedin Fig. 2c are fitcϕ(dashed lines).
ϕVFT −ϕThe transition points are evaluated at ϕQ=0.908(12) forVFT
the translational relaxation and ϕθVFT = 0.870(35) for the ori-ted here with the VFT law, τ ∼ exp

entational relaxation. The two-step transition scenario is also
confirmed by the VFT fitting.

6= 0.784

1

1= 0.800

0.5

0.5

0.5
0-2
10

100= 0.817

10.5

0-2
10
1

100= 0.823

100= 0.837

10.5

0-2
10
1

100= 0.845

0.5

0-2
10

100

= 0.843

1
0.8

0-2
10
1

10

00-2
10

100

= 0.837

10-2
10

10

0

= 0.833

1
0.5

0-2
10

100

= 0.837

10-2
10

0.8

0.6

0.6

0.4

0.4-2

10

0

10

100

= 0.84110.8

10= 0.817

10.5

0.5

0.5

0-2
10

= 0.809

1-2100= 0.846

0.8

0.6

0.6

0.4
10-2

1000.4-210100FIG. S7. Results of fitting of the orientational correlation function Cθ (∆t) by a stretched exponential function. Each data wasβθ
fitted by a stretched exponential function plus an offset, Cθ (∆t) = fθ e−(∆t/τθ ) + aθ . See Fig. S5 for the obtained values ofβθ .

7ABCFIG. S8. A cage escape event in the orientation glass phase, ϕ = 0.873(4), different from the one shown in Fig. 3a-c. See also
Movie 7. In this event, the cell did not move across a border of microdomains, unlike the event shown in Fig. 3a-c and Movie 6.
A) Trajectory of the single cell for 0 ≤ t ≤ 7.16 s, drawn on the phase-contrast image taken at the last time frame. Note thatthe origin of time is different from that used in Fig. 3a-c. The positions at t = 0, 1.76, 3.53, 5.29, 7.06 s are shown by coloreddisks with labels 1, 2, · · · , 5, respectively. B,C) Time series of the displacement from the initial position, |∆⃗ri (t)| (B), and thatof the orientation θi (t) (C) of the cell tracked in panel A. These time series show a cage escape event during 4.5 s ≲ t ≲ 4.7 s.

8AB
10

3

1010

3

10

2

1

10

1

0

10

0

0.1
0.05
0.02
0.01

count

2

0.1
0.05
0.02
0.01

count

10

10

0

10

20

30

02

1020302

area ( m )area ( m )

CD
10

3

10

10

3

10

2

1

10

1

0

10

0

0.1
0.05
0.02
0.01

count

2

0.1
0.05
0.02
0.01

count

10

100102030
2

area ( m )

010

2030
2

area ( m )FIG. S9. Microdomain size distributions for different thresholds of |∇θ(⃗r, t)|2 (legend, in the unit of (rad/µm)2 ) and for different
area fractions ϕ. The dashed lines indicate the results of the fitting using all the displayed curves for each ϕ. The resulting
values of the characteristic area A0 are A0 = 7.5 ± 1.8 µm2 (A), A0 = 7.6 ± 2.6 µm2 (B), A0 = 7.5 ± 2.5 µm2 (C), andA0 = 8.6 ± 4.8 µm2 (D).

9MOVIE CAPTIONS

Movie 1: Uninterrupted Movie of bacteria undergoingglass transitions. The Movie starts from the active
fluid phase where bacteria were actively swarming,and lasts until they become completely jammed.
The Movie shows a central region of size 259 × 214
pixels (44.7 µm × 36.9 µm) and played at 20 times
the real speed. Scale bar: 10 µm. Note that thisMovie was taken from an experiment independent
from the other observations, using a different substrate (well diameter 76.3(4) µm, depth ≈ 1.4 µm).
Movie 2: Movie of bacteria at ϕ = 0.784(7) (active fluidphase). Scale bar 5 µm. Played at real speed.
Movie 3: Movie of bacteria at ϕ = 0.841(5) (active fluidphase, close to ϕθc ). Scale bar 5 µm. Played at realspeed.
Movie 4: Movie of bacteria at ϕ = 0.873(4) (orientationglass). Scale bar 5 µm. Played at real speed.
Movie 5: Movie of bacteria at ϕ = 0.887(4) (completeglass). Scale bar 5 µm. Played at real speed.
Movie 6: The cage escape event in the orientation glass
phase, ϕ = 0.873(4), shown in Fig. 3a-c. The left
panel shows the trajectory of the single cell drawn
on the phase-contrast image. The right panels show
the time series of the displacement from the initialposition, |∆⃗ri (t)| (top) and that of the orientation
θi (t) (bottom) of the cell shown in the left panel.
The cage escape event took place during 5 s ≲ t ≲7 s.
Movie 7: Another cage escape event in the orientation
glass phase, ϕ = 0.873(4), shown in Fig. S8. Theleft panel shows the trajectory of the single celldrawn on the phase-contrast image. The right panels show the time series of the displacement fromthe initial position, |∆⃗ri (t)| (top) and that of theorientation θi (t) (bottom) of the cell shown in theleft panel. The cage escape event took place during
4.5 s ≲ t ≲ 4.7 s.
Movie 8: Structure and evolution of microdomains at
ϕ = 0.784(7) (active fluid phase). The left andright panels show the orientation field θ(⃗r, t) and its
gradient squared |∇θ(⃗r, t)|2 , respectively, overlaid
on the phase-contrast image. Played at real speed.
See also Fig. 4.
Movie 9: Structure and evolution of microdomains at
ϕ = 0.841(5) (active fluid phase, close to ϕθc ).
The left and right panels show the orientation fieldθ(⃗r, t) and its gradient squared |∇θ(⃗r, t)|2 , respectively, overlaid on the phase-contrast image. Playedat real speed. See also Fig. 4.
Movie 10: Structure and evolution of microdomains at
ϕ = 0.873(4) (orientation glass). The left and rightpanels show the orientation field θ(⃗r, t) and its gradient squared |∇θ(⃗r, t)|2 , respectively, overlaid on
the phase-contrast image. Played at real speed. Seealso Fig. 4.
Movie 11: Structure and evolution of microdomains at
ϕ = 0.887(4) (complete glass). The left and rightpanels show the orientation field θ(⃗r, t) and its gradient squared |∇θ(⃗r, t)|2 , respectively, overlaid on
the phase-contrast image. Played at real speed. See
also Fig. 4.