Supporting information - Nature

Supporting information Nucleus factory on cavitation bubble for amyloid β fibril Kichitaro Nakajima,1 Hirotsugu Ogi,1, a) Kanta Adachi,1 Kentaro Noi,2 Masahiko Hirao,1 Hisashi Yagi,3 and Yuji Goto4 1)

Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan 2) Institute of Molecular Embryology and Genetics, Kumamoto University, 2-2-1 Honjo, Chuo-ku, Kumamoto 860-0811, Japan 3)

Center for Reserch on Green Sustainable Chemistry, Tottori University, 4-101 Koyama-cho minami, Tottori, Tottori 680-8550, Japan 4) Institute for Protein Research, Osaka University, Yamadaoka 3-2, Suita, Osaka 565-0871, Japan a)

Electronic mail: [email protected]

1. Bubble dynamics calculations by the Keller-Miksis equation To calculate bubble radius motion, we performed single-bubble simulation with the Keller-Miksis equation: 1−

𝑅 3 𝑅 𝑅 𝜌𝑅𝑅 + 1− 𝜌𝑅 ! = 1 + 𝑐 2 3𝑐 𝑐

𝑝! (𝑡) − 𝑃! − 𝑃 𝑡

2𝜎 𝑝! 𝑡 = 𝑃! + 𝑅!

𝑅 𝑅 2𝜎 𝑝! − 4𝜂 −  , 𝑐 𝑅 𝑅

(𝑆1)

!!

𝑅! 𝑅 𝑡

𝑃 𝑡 = 𝑃! sin 2𝜋𝑓 𝑡 +

+

   ,                    (𝑆2) 𝑅 𝑐

 .                    (𝑆3)

Here, 𝑅, 𝑅, 𝑅, 𝑅! , 𝑐, 𝜌, 𝜂  and  𝜎 denote radius, bubble-wall velocity and bubble-wall acceleration, sound velocity, equilibrium radius of bubble, density, viscosity and surface tension of water, respectively. pg(t), P(t), Pa and f denote the gas pressure inside bubble, acoustic pressure of ultrasonic wave, the pressure amplitude, and frequency of ultrasonic wave, respectively. Because bubble collapse occurs instantaneously (~10 ns), the process can be nearly regarded as adiabatic compression process. In adiabatic process, relationship between bubble radius and temperature of gas inside bubble can be written as following:

𝑇 𝑡 = 𝑇!

! !!!

𝑅! 𝑅 𝑡

.                      (𝑆4)

Here, γ denotes ratio of specific heat. Calculation results of the bubble radius motion and temperature change of gas inside bubble are shown in Fig. S8.

2. Calculation of solution-temperature field around the hot core For calculating temperature field generated by thermal diffusion from the hot spot, we consider a heat-affected region with radius RB (=1 mm), assuming that the temperature outside this region (r>RB) remains constant to be T∞.

When the bubble radius becomes minimum Rmin at

collapse, temperature of gas inside the bubble reaches the maximum. time t=0.

We set this moment to be

In the spherical coordinate system, we analytically derive temperature increase by

thermal diffusion from the heat source with radius Rmin and temperature 𝑇! +   ∆𝑇! . governing equation is given by

!" !"

−𝛼

! ! ! ! !"

𝑟!

!" !"

The

= 0 or

𝜕(𝑟𝑇) 𝜕 ! 𝑟𝑇 −𝛼 = 0  .              (𝑆5) 𝜕𝑡 𝜕𝑟 ! Here, α denotes thermal diffusivity of water.

Initial condition and boundary condition are written

by 𝑇 𝑟, 0 = 𝑇! +   ∆𝑇! (0 ≤ 𝑟 ≤ 𝑅!"# )  ,        (𝑆6)   𝑇 𝑟, 0 = 𝑇!   𝑅!"# < 𝑟 < 𝑅! ,                              (𝑆7)   𝑇 0, 𝑡 ≠ ∞,                                                                                                (𝑆8) 𝑇 𝑅, 𝑡 = 𝑇! .                                                                                                (S9) Here, ∆𝑇!  is the temperature increase of the hot water core, which is obtained by assuming that

temperature increase of gas inside bubble conducts to the water core with no dissipation: ∆𝑇! = !! !! !! !!

∆𝑇! , where ∆𝑇! denotes the temperature increase of the gas at collapse. Setting T’(r,t)≡T(r,t)-T∞ and introducing the variable transformation w=rT’, the thermal

diffusion equation and initial and boundary conditions (Eqs.(S6)-(S9)) can be rewritten as 𝜕𝑤 𝜕!𝑤 − 𝛼 ! = 0  ,                                                                                                                      (𝑆5)′ 𝜕𝑡 𝜕𝑟 𝑇 ! 𝑟, 0 =   ∆𝑇!    (0 ≤ 𝑟 ≤ 𝑅!"# ),                                                                (𝑆6)′ 𝑇′ 𝑟, 0 = 0   𝑅!"# < 𝑟 < 𝑅! ,                                                                        (𝑆7)′ 𝑇′ 0, 𝑡 ≠ ∞  ,                                                                                                                                    (𝑆8)′ 𝑇′ 𝑅! , 𝑡 = 0.                                                                                                                                      (𝑆9)′ General solution of Eq.(S5)’ is obtained by method of separation of variables: !

𝑤 𝑟, 𝑡 = 𝐴𝑠𝑖𝑛𝜆𝑟 + 𝐵𝑐𝑜𝑠𝜆𝑟 𝑒 !!! ! .                                          (𝑆10) Here, A, B and λ are constants.

Thus, T’ can be written as

𝑇 ! 𝑟, 𝑡 = 𝐴

𝑠𝑖𝑛𝜆𝑟 𝑐𝑜𝑠𝜆𝑟 !!"! ! +𝐵 𝑒 .                                    (𝑆11) 𝑟 𝑟

Because T’ takes a finite value at r=0, B=0. Thus, we have 𝑇 ! = 𝐴

!"#$% !

!

𝑒 !!" ! . Because a single

solution fails to satisfy all the condition equations, we adopt linear superposition of solutions and explored An to satisfy the conditions: ! !

𝑇 𝑟, 𝑡 =

𝐴! !!!

𝑠𝑖𝑛𝜆! 𝑟 !!! !! ! 𝑒 .                                                  (S12) 𝑟

From the condition (S9)’, 𝑠𝑖𝑛𝜆! 𝑅! = 0, which leads to 𝜆! =

!" !!

     (𝑛 = 1,2,3・・・). Thus, Eq.

(S12) can be rewritten as !

𝑇 ! 𝑟, 𝑡 =

𝐴!

sin 𝑛𝜋

𝑟 𝑅!

𝑟

!!!

𝑒

!!

!" ! ! !! .                                          (S13)

The initial condition, that is the step-like temperature distribution at t = 0, gives !

𝑇 ! 𝑟, 0 =

𝐴!

sin 𝑛𝜋

!!! !

⟺  

𝐴! sin 𝑛𝜋 !!!

𝑟

𝑟 𝑅!

=  

∆𝑇!  (0 ≤ 𝑟 ≤ 𝑅!"# ) .                          (S14)        0        (𝑅!"# ≤ 𝑟 ≤ 𝑅! )

𝑟 𝑟∆𝑇!  (0 ≤ 𝑟 ≤ 𝑅!"# ) =   .                                  (S14)′        0        (𝑅 𝑅! !"# ≤ 𝑟 ≤ 𝑅! )

We multiply Eq. (S14)’ by sin 𝑚𝜋 !! ! !

From

𝐴! sin 𝑛𝜋

!!! !! sin !

𝑛𝜋

! !!

and integrate it between r=0 and RB:

𝑟 𝑟  sin 𝑚𝜋  𝑑𝑟 =   ∆𝑇! × 𝑅! 𝑅!

 sin 𝑚𝜋

𝐴! ×

! !!

! !!

 𝑑𝑟 =

 0    (𝑚 ≠ 𝑛) !! !

𝑅! =   ∆𝑇! × 2

!!"#

 (𝑚 = 𝑛)

𝑟 sin 𝑛𝜋

!

!!"#

𝑟 sin 𝑚𝜋

!

𝑟  𝑑𝑟 .                              (S15) 𝑅!

 , Eq. (S15) can be rewritten as

𝑟  𝑑𝑟 .                                                  (S15)′ 𝑅!

By calculating integration of right hand of Eq. (S15)’, An is obtained as following: 𝐴! =  

2∆𝑇! 𝑅! 𝑅!"# 𝑅!"# sin 𝑛𝜋 − 𝑅!"# cos 𝑛𝜋   .                          (S16) 𝑛𝜋 𝑛𝜋 𝑅! 𝑅!

Then we obtain the temperature distribution around the bubble as following: 2∆𝑇! 𝑇 𝑟, 𝑡 = 𝑇!"# + 𝑟

!

!!!

𝑟 !! 𝐹!  sin 𝑛𝜋 𝑒 𝑅!

!" ! ! !!  ,                                          (S17)

where 𝐹! =

1 𝑅! 𝑅!"# 𝑅!"# sin 𝑛𝜋 − 𝑅!"# cos 𝑛𝜋 𝑛𝜋 𝑛𝜋 𝑅! 𝑅!

 .                                            (S18)

3. Estimation of overall solution temperature For roughly estimating temperature increase of bulk solution caused by bubble collapse, we calculated temporal and spatial average of temperature distribution around the bubble (Eq. (S17), (S18)): 𝑇!"# =    

!!"# !!"# 𝑇 ! !

𝑟, 𝑡  𝑑𝑡  𝑑𝑟

𝑅!"# ×𝑡!"#

                                 (S19)

Here, Rcav and tcav denote the half distance between bubbles and period of ultrasonic wave, respectively. The volume fraction of bubble is reported to be ~10-4 from the literature:

!!"# !!"!#$

= 10!! .

The total volume of solution is 500 µL (=5×10-7 m3) in our experiment. Thus, the total volume of cavitation bubbles is Vcav=5×10-11 m3. For example, at the acoustic pressure of 150 kPa and frequency of 29 kHz, the maximum radius of the bubble is Rmax = 53.29 µm, and the volume of one bubble, V1-cav., is 𝑉!!!"#. = is 𝑛!"#. =

!!"#. !!!!"#.

≈ 79.

! !!!!"#

!

= 6.34×10!!"  m! . Thus, the number of cavitation bubble, ncav.,

By assuming that bubbles are dispersed homogeneously, the volume of

sphere region affected by one bubble is 𝑉!"!#$ ÷ 79 = 6.3×10!!  m! , and the radius, Rcav., is

𝑅!"# = 1.1  mm. Using this value, we numerically computed the average temperature increase, Tave., based on Eq. (S19), and find that the overall temperature increase is less than 0.001 K; nearly unchanged.

This is

because the high temperature region is highly localized near the hot spot and exists within a very short time.

4. Supplementary Figures S1-S7

Fig.S1 Result of incubation experiments with various ambient temperatures (25, 37, 45 and 50 °C) and determination of activation energies for nucleation and growth. (a)Time course of ThT fluorescence intensity at each temperature. two-step model.

The solid lines denote fitted theory based on the

The Arrhenius plots for determining activation energies for (b) nucleation and (c)

growth of Aβ1-40 fibril, respectively.

(a) Schematic view of an experimental system.

(b) Top view of the reaction container. Fig.S2 (a) Schematic of the home-built experimental system for researching frequency and acoustic pressure dependences of aggregation reaction of Aβ1-40 peptide. We used two Langevin type ultrasonic transducers with fundamental frequencies of 28 and 40 kHz; the former was used for frequencies below 200 kHz, and the latter for those beyond 200 kHz.

One of them was strongly

fixed to the bottom of the container by a clump. The water was degassed by the degassing unit and its temperature was kept at 37 °C by the temperature controlling system. (b) Top view of the reaction container, showing five sample tubes in a circular pattern.

This measurement setup allows us to

perform ultrasonic irradiation experiments on five sample solutions simultaneously with nearly the same irradiation condition.

Rate constant for nucleation kn [1/h]

10-1

10-2

10-3 0

100

200

300

400

Acoustic pressure [kPa] Fig. S3 Relationship between the rate constant for nucleation kn and the acoustic pressure of the fundamental mode. No clear correlation is indicated between them.

Fig.S4 (a) Representative acoustic waveforms observed in sample tube under ultrasonic irradiation with fundamental frequencies of 19, 29 and 50 kHz, and (b) their corresponding FFT spectra.

Fig.S5 (a) Representative acoustic waveforms observed in sample tube under ultrasonic irradiation with fundamental frequencies of 71, 143, 208 and 239 kHz, and (b) their corresponding FFT spectra.

Fig.S6 (a) Representing waveforms and (b) their corresponding FFT spectra observed in experiments related to the acoustic-pressure dependence of Aβ aggregation reaction with fundamental frequency of 29 kHz. The averaged second-harmonics pressure values are shown in (a).

10

0 0

10

20

30

kn kg

10-2

10-3 10-1 10-4

Inc.

UI

SB

P-

Rate constant for growth kg [1/µM h]

Rate constant for nucleation kn [1/h]

Fluorescence intensity [arb. unit]

20

silica polystyrene (negative charged) polystyrene (positive charged)

P+

Time [h]

Fig.S7 (a) Evolution of ThT fluorescence intensity during incubation of Ab1-40 solutions involving silica beads, positively charged polystyrene beads, and negatively charged polystyrene beads.

(b)

Comparison of the reaction-rate constants among incubation (Inc.), optimized ultrasonic irradiation (UI), silica beads (SB), positively charged polystyrene beads (P+), and negatively charged polystyrene beads (P-).

Fig. S8 Changes in bubble radius and temperature of gas inside bubble calculated by the Keller-Miksis equation. (a) and (b) show the changes of radius and inside temperature, respectively, with three acoustic pressures of Pa =100 kPa (black line), 125 kPa (blue line), and 150 kPa (red line) at frequency of 29 kHz. (c) and (d) show the changes of radius and inside temperature, respectively, with three different frequencies of 20 kHz (red line), 50 kHz (blue line), and 200 kHz (black line) under acoustic pressure of Pa =150 kPa. Used parameters are shown as following: P0 = 101.3 kPa, R0= 2.0 µm, ρ = 993.9 kg / m3, σ = 7.275×10-2 N / m, µ = 0.685×10-3 Pa s, c = 1533 m / s, and T0 = 310.15 K.