Microscopic Dynamics of Liquid-Liquid Phase Separation and Domain Coarsening in a Protein Solution Revealed by X-Ray Photon Correlation Spectroscopy

doi:10.1103/PHYSREVLETT.126.138004

Microscopic Dynamics of Liquid-Liquid Phase Separation and Domain Coarsening

in a Protein Solution Revealed by X-Ray Photon Correl ation Spectroscopy

Anita Girelli ,

1

Hendrik Rahmann,

2

Nafisa Begam ,

1

Anastasia Ragulskaya,

1

Mario Reiser,

2,3

Sivasurender Chandran ,

1,4

Fabian Westermeier ,

5

Michael Sprung,

5

Fajun Zhang ,

1,*

Christian Gutt ,

2,†

and Frank Schreiber

1,‡

1

Institut für Angewandte Physik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

2

Department Physik, Universität Siegen, Walter-Flex-Strasse 3, 57072 Siegen, Germany

3

European X-Ray Free-Electron Laser XFEL, Holzkoppel 4,22869 Schenefeld, Germany

4

Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India

5

Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany

(Received 19 October 2020; accepted 23 February 2021; published 2 April 2021)

While the interplay between liquid-liquid phase separation (LLPS) and glass formation in biological

systems is highly relevant for their structure formation and thus function, the exact underlying mechanisms

are not well known. The kinetic arrest originates from the slowdown at the molecular level, but how this

propagates to the dynamics of microscopic phase domains is not clear. Since with diffusion, viscoelasticity,

and hydrodynamics, distinctly different mechanisms are at play, the dynamics needs to be monitored on the

relevant time and length scales and compared to theories of phase separation. Using x-ray photon

correlation spectroscopy, we determine the LLPS dynamics of a model protein solution upon low

temperature quenches and find distinctly different dynamical regimes. We observe that the early stage

LLPS is driven by the curvature of the free energy and speeds up upon increasing quench depth. In contrast,

the late stage dynamics slows down with increasing quench depth, fingerprinting a nearby glass transition.

The dynamics observed shows a ballistic type of motion, implying that visco elasticity plays an important

role during LLPS. We explore possible explanations based on the Cahn-Hilliard theory with nontrivial

mobility parameters and find that these can only partially explain our findings.

DOI: 10.1103/PhysRevLett.126.138004

Recent work suggests that structure formation in biology

can take place, inter alia, through liquid-liquid phase

separation (LLPS) [1–3]. Phase separation in crowded

environments thus represents a mechanism for intracellular

organization via the formation of biomolecular condensates

[4]. The biological functions of these condensates—includ-

ing steering biochemical reactions rates, sensing, or signal-

ing—are being intensely investigated [3]. LLPS is also

associated with a variety of diseases caused by a loss and/or

change of function of the condensates [5,6].

The state of the condensates depends on the dynamic

processes during their formation, often involving non-

equilibrium processes over a hierarchy of length and time

scales [2,7]. A case in point is the slowdown of the

dynamics on molecular length scales caused by concen-

tration and its influence on the dynamical and structural

properties of the condensate on mesoscopic length scales.

Ultimately, such a microscopic slowdown can lead to the

arrest of LLPS on larger length scales accompanied by the

formation of bicontinuous gel network structures [8–10].

The kinetics of arrested phase separations in protein

solutions have been studied successfully in the past,

demonstrating that the ensemble-averaged structure factor

ceases to develop further in q position and intensity upon

low temperature quenches [11–15]. However, the dynamics

of protein solutions en route to an arrested LLPS is largely

unknown, mainly because of the requirement to monitor an

exceptionally broad range of time and length scales

simultaneously. This, in turn, prevented the experimental

validation of models of the dynamics of critical phenomena

during LLPS, such as the Cahn-Hilliard equation and

related models, especially in the vicinity of glass-gel

transitions displaying large dynamical asymmetries

between the species involved [2].

LLPS is a general phenomenon, which is relevant not

only for protein systems but also in many other fields of

science [16]. LLPS domains have been studied with

different microscopy techniques [17,18]. Their macro-

scopic properties such as turbidity and viscosity [19–21]

were monitored, as well as their molecular properties

[20,22–27]. Other scattering techniques were used to

access the kinetics of the phase separation [14,15,19,21].

X-ray photon correlation spectroscopy (XPCS) employ-

ing coherent x-rays can resolve the collective dynamics on

the required length scales, ranging simultaneously from

nanometers to microns and timescales from microseconds

to hours [28–35]. Here, we demonstrate that a combination

of scanning techniques, large beams, and long sample

detector distances [36–38] allows us to reduce the required

x-ray doses to values below the critical dose of many

PHYSICAL REVIEW LETTERS 126, 138004 (2021)

protein systems [39]. More details on the experimental

parameters are provided in the Supplemental Material (SM)

[40]. With this approach, we are able to follow the

dynamics during an LLPS of γ globulin (Ig) in a concen-

trated aqueous polyethylene glycol (PEG) solution. The

experimental results are compared to simulations based on

the Cahn-Hilliard equation, taking the gel transition into

account (in the spirit of model C according to Ref. [2]; see

the SM) [2,45,46]. Our work paves the way for future

XPCS experiments exploring the full spatiotemporal win-

dow of LLPS, allowing one also to benchmark and guide

computer simulations of complex protein dynamics in

crowded environments.

XPCS experiments were conducted at the P10 Coherence

Applications Beamline at PETRA III, Deutsches Electronen-

Synchrotron, employing an x-ray beam of photon energy

8.54 keV, a size of 100 × 100 μm

2

, and a maximum photon

density of 10

7

photons=s=μm

2

. The key for performing low

dose XPCS experiments is to make use of large beams with a

sufficient degree of coherence. Time series of coherent

diffraction patterns were collected with an EIGER 4-mega-

pixel detector covering a q range from 0.003 to 0.05 nm

−1

.

Samples of Ig with PEG and NaCl have been prepared as

described in [15] and in the SM and quenched from 37 °C to

different quench temperatures T

q

below the binodal line.

Figure 1(a) displays the temporal evolution of the

scattering intensity as a function of scattering vector q

for a quench temperature of T

q

¼ 10 °C [for other quench

temperatures, see Figs. S1(a) and S1(b) in the SM],

capturing the LLPS process during the first 60 s. The

x-ray intensity increases rapidly during the early time of the

phase separation, and the position of the spinodal peak q

max

shifts to smaller values, indicating an increase in length

scales of the concentration fluctuations. Appropriately

normalizing the intensity and wave vectors by the respec-

tive peak intensities Iðq

max

Þ and positions q

max

, we obtain a

master plot as expected for spinodal decomposition [47]

[see inset, Fig. 1(a) and Figs. S1(c) and S1(d)]. Iðq

max

Þ

shows a rapid increase during the early stage, with a rate

determined by the quench depth, while at around 10 s the

growth starts to slow down considerably [Fig. 1(b)]. This

slowdown is more pronounced when quenched to lower

temperatures, while at higher temperatures, IðqÞ continues to

grow even beyond 40 s, albeit at a slower rate. Insights into

the dynamics during the LLPS are obtained by analyzing a

time series of the coherent x-ray speckle patterns. For this, we

calculate two-time correlation functions (TTCs) for specific

scattering vectors q and different quench depths via

Cðt

1

;t

2

;qÞ¼

h½Iðt

1

Þ −

¯

Iðt

1

Þ½Iðt

2

Þ −

¯

Iðt

2

Þi

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

½

I

2

ðt

1

Þ −

¯

I

2

ðt

1

Þ½I

2

ðt

2

Þ −

¯

I

2

ðt

2

Þ

q

; ð1Þ

with h·i being an average over detector pixels corresponding

to a specific range q  δq (calculated for q from 5 to 11 μm

−1

and δq ¼ 0.3 μm

−1

) and

¯

Iðt

1

Þ¼hIðt

1

Þi. From this quantity,

the time dependent g

2

ðt; t

w

;qÞ¼Cðt þ t

w

;t

w

;qÞ have

been extracted by horizontal cuts along t

1

, starting at the

diagonal of the respective TTC [48].

Figure 2 shows in the upper panel the TTCs and in the

lower panels the correlation function g

2

ðt; t

w

;qÞ for quench

temperatures of T

q

¼ 15, 8, and 4 °C (from left to right),

respectively. We identify three stages of the dynamics

during LLPS: the first stage appears directly after the

quench in the first 20 s, and its very fast dynamics is visible

only by a thin line in the TTC and by the final part of the

decay in the correlation function. After this early stage, the

TTC shows a pronounced slowing down of the dynamics

accompanied by a rising background level visible by the

appearance of a squarelike feature. This corresponds to a

second relaxation mode with a much slower relaxation time

appearing in the g

2

functions. The contribution of this

relaxation channel to the overall decay is increasing rapidly.

After ca. 40 s, the fast process in the LLPS is coming to an

end, as evidenced by the disappearance of the first decay.

(a)

0.005 0.01 0.015 0.02

q (nm

-1

)

0

100

200

300

400

I(q) (a.u.)

0

10

20

30

40

50

t

w

(s)

123

q/q

max

0

0.5

1

I(q)/I(q

max

)

(b)

FIG. 1. (a) Intensity as a function of scattering vector q for a

quench temperature T

q

¼ 10 °C; in the inset, the rescaled

intensity. The different colors correspond to different waiting

times t

w

as indicated in the color bar. The time t

w

¼ 0 is the time

at which the quench temperature T

q

was reached. (b) The

intensity at the peak position as a function of time for different

temperatures. The error bars are within the symbol size if they are

not visible.

FIG. 2. In the upper panels, the two-time correlation function

for T

q

¼ 15, 8, and 4 °C (left to right) at q ¼ 0 .005 nm

−1

are

displayed, and the lower panels show the corresponding g

2

functions at different waiti ng times t

w

.

PHYSICAL REVIEW LETTERS 126, 138004 (2021)

138004-2

In the third stage, later also called the late stage, the TTC

evidences a second slowing down process, with the

relaxation time depending on the final quench temperature

with slower dynamics visible for lower temperatures (from

T

q

¼ 15 °C to T

q

¼ 4 °C, on the top row). The g

2

ðt; t

w

;qÞ

functions have been fitted by a sum of Kohlrausch-

Williams-Watts (KWW) functions [49], resulting in the

following equation:

g

2

ðt; t

w

;qÞ¼A

1

exp



−



2

t

τ

1



γ

1



þ A

2

exp



−



2

t

τ

2



γ

2



;

ð2Þ

with decorrelation times τ

1

and τ

2

, relaxation amplitudes A

1

and A

2

, and KWW exponents γ

1

and γ

2

. We note that for

some g

2

functions a third exponential decay with small

amplitude and slow decay is needed to describe additional

tails in g

2

at very long timescales. However, due to the low

statistics of the third decay, we evaluate only the two

leading decays here.

Figure 3(a) displays the relaxation times as a function of

waiting time t

w

. We identify an exponential increase of the

relaxation time in the early stage of the LLPS and faster

dynamics with increasing quench depths [see Fig. 3(c) and

inset of Fig. 3(a)]. At this early phase of the spinodal

decomposition, the dynamics is driven by the curvature of

the free energy ∂

2

F=∂c

2

, which increases in magnitude

with deeper quenches (lower temperatures) and thus speeds

up the dynamics. This picture reverses when the slow

relaxation sets in [τ

2

in Fig. 3(c)]. Now the relaxation times

for the lower temperatures are considerably larger than the

high temperature quenches.

The transition from dynamics dominated by the fast

decay to dynamics dominated by the slow decay is

quantified by the nonergodicity parameter [50], here

defined as f ¼ A

2

=ðA

1

þ A

2

Þ. A rather sharp increase of

the nonergodicity parameter around t

w

¼ 30 s is observed

[Fig. 3(b)], with a later rise time and slower increase for

lower temperatures, suggesting that the transition to coars-

ening dynamics is already slowed down by the lower

mobility at low temperatures.

These three stages in the dynamics can be linked to the

evolution of the LLPS. The first corresponds to the early

stage of a spinodal decomposition, displaying an enhance-

ment of the concentration fluctuations and the formation of

an interface between two different phases. The third one is

the late stage in which the domains are growing. For our

system, this seems to be controlled by the mass transport

mechanism and by the mobility of the involved compo-

nents. In other cases in condensed matter physics, different

domain growth mechanisms have been identified [51]. The

second stage is a transition between the early and late stage.

Important for the assessment of the form of the dynamics

and relevant for computing quantities such as diffusion and

transport coefficients during the LLPS are the relaxation

rate ΓðqÞ¼1=τðqÞ and the KWW exponents, including

their q dependence [32]. We find in the early phase of the

LLPS a linear relationship Γ ∝ q [Fig. S2(a)] and γ

1

around

1.4 to 1.7 (Fig. S2) with no pronounced q dependence. In

contrast, the dynamics during the late coarsening stage also

displays a linear (Γ ∝ q) behavior but now with a pro-

nounced q dependence of the KWW exponents, which are

decreasing from values of γ

2

¼ 2 at small q values to

γ

2

¼ 1 at large values of q [Fig. S2(d)]. Our results clearly

show that the coarsening dynamics of the protein droplets

at this length scale are not governed by Brownian dynamics

but instead by a ballistic and partially cooperative motion of

the protein droplets driven by the spinodal decomposition.

Similar super diffusive ballistic types of dynamics have

been observed frequently in soft matter systems, e.g., in the

late stage of colloidal gelation processes [52–54], during

spinodal decomposition of colloidal systems [27], and in

the framework of MD simulations of metallic glasses [55]

with strongly interconnected clusters moving together.

Phenomenological models of single microcollapse events

(a)

10

1

10

2

t

w

(s)

10

0

10

1

10

2

(s)

5 10152025

2

4

6

8

(b)

0 1020304050

t

w

(s)

0

0.2

0.4

0.6

0.8

1

Non-ergodicity parameter

T

q

=15

o

C

T

q

=12

o

C

T

q

=10

o

C

T

q

=8

o

C

T

q

=6

o

C

T

q

=4

o

C

(c)

5

10

15

1

(s)

4 6 8 10121416

T

q

(

o

C)

100

200

300

2

(s)

t

w

=22s

t

w

=96s

FIG. 3. (a) Decorrelation time for q ¼ 0.005 nm

−1

as a function of waiting time t

w

. The open symbols correspond to the slow mode

(τ

2

) and the filled symbols to the fast one (τ

1

). Inset: linear representation. (b) Nonergodicity parameter f ¼ A

2

=ðA

1

þ A

2

Þ as a function

of t

w

. The legend also applies to (a). (c) Decorrelation time of the fast and slow modes as a function of quench temperature T

q

,

respectively.

PHYSICAL REVIEW LETTERS 126, 138004 (2021)

138004-3

have been put forward as possible explanations [56]

underlying such unusual dynamics and subsequently

extended to series of such intermittent events to account

for both q dependence of the relaxation rate and the KWW

exponents [57,58]. The connection of these phenomeno-

logical models to the dynamics during an LLPS is not

obvious, especially with regard to typical field theories

such as the Cahn-Hilliard equation (CHE) used to model

spinodal decomposition. A shared feature between colloi-

dal gels and the LLPS, investigated here, is the presence of

elastic deformations. These can arise in an LLPS from the

viscoelastic properties, which are caused by a dynamical

coupling of diffusion and stresses, and could explain

the similarities between these observations [57]. In fact,

the dynamical asymmetry of the two phases present in the

system can lead to viscoelastic phase separation [59].

To obtain more insight into the underlying dynamical

mechanisms, we compare the experimental data to numeri-

cal simulations of the temporal evolution of the CHE

[60,61] (here in 2D). The dynamics of the ordinary CHE

shows faster dynamics for quenches to lower temperatures.

As this is not what we observe in the experimental data, we

introduce a dependence of the mobility μ on protein density

ρðr; tÞ, representing the slowdown of the dynamics when the

density of the dense phase is increasing and its mobility

freezes out [45] in the spirit of model C according to Ref. [2].

The parameter c

g

sets the concentration at which the mobility

decreases (for details, see the SM). The simulations then

yield a time dependent real space configuration ρðr; tÞ of the

protein density [Fig. 4(c)], which is converted into an x-ray

speckle pattern by means of its Fourier components jρðq; tÞj

2

and analyzed with the same time correlation methods as

applied to the experimental data [45,46].

Figure 4(a) displays the temporal evolution of the

spatially averaged mobility ¯μðt

w

Þ in the dense phase and

Fig. S10 the spatially averaged density

¯

ρðt

w

Þ of the dense

and dilute phases for different values of c

g

. A point in the

2D image was considered part of the dense phase if its

concentration was higher than the initial concentration. The

mobility of the dense phase drops quickly during the LLPS

upon lowering the c

g

value. The densities in the dense and

dilute phases do not reach their equilibrium values any-

more, which is considered an arrest of the LLPS [45,46].

Typical real space configurations are shown in Fig. 4(b)

with the LLPS visible via the formation of domains of the

diluted phase (blue) in a host matrix of the dense phase

(red). The corresponding density profiles (see Fig. S9)

show the typical hallmarks of the spinodal decomposition,

with density fluctuations developing quickly and reaching

rather smooth profiles at the end of the LLPS for a mobility

that is independent of protein density and equal to 1. In

contrast, for c

g

¼ 0.6, we observe that the density fluctua-

tions become immobile when reaching the threshold value,

leading to smaller domains.

TTC and g

2

functions were computed following the

procedure of experimental data. Figure 4(c) displays the

TTC for the simulations and the corresponding g

2

func-

tions. We identify a fast dynamic process during the early

time of the LLPS, which represents the dynamics of

spinodal interface formation between the dense and diluted

phases. The dynamics quickly slows down with a sudden

appearance of a second much slower relaxation process

when the dense phase has formed. The relaxation times

during the first stage are determined by the quench depth

(smaller for deeper quenches) because the curvature of the

free energy is here the main driving force for the velocity of

interface formation. The corresponding KWW values γ

1

and γ

2

[Figs. S11(c) and S11(d)] are between 1.5 and 2.5,

which is in good agreement with our experimental values.

Comparing the real space data and the parameters

describing the dynamics, it is possible to see that the rise

(a)

10

2

10

3

t

w

0

0.2

0.4

0.6

0.8

1

(b)

(r,t

w

=10)

(r,t

w

=70)

(r,t

w

=90)

(r,t

w

=130)

0

0.5

1

(c)

FIG. 4. The simulated ¯μ of the dense phase (a) as a function of waiting time. In brown, the case with constant mobility and the quench

depth for this case was set to T

q

¼ 0.2 T

c

, with T

c

being the critical temperature [62,63]. In red, c

g

¼ 0.8 and T

q

¼ 0.12 T

c

, and in

yellow, c

g

¼ 0.6 and T

q

¼ 0.08 T

c

. (b) Examples of real space configurations for T

q

¼ 0.14 T

c

and mobility at different waiting times

with the densities indicated by the color bar. (c) Upper panel: two-time correlation function of the simulated data for q ¼ 1. From left to

right, the parameters T

q

and c

g

of the simulations are T

q

¼ 0.2 T

c

and c

g

¼ ∞, which correspond to a constant mobility set to 1,

T

q

¼ 0.12 T

c

and c

g

¼ 0.8, and T

q

¼ 0.08 T

c

and c

g

¼ 0.6. (c) Lower panel: corresponding g

2

functions at different waiting times t

w

.

PHYSICAL REVIEW LETTERS 126, 138004 (2021)

138004-4

of the nonergodicity parameter starts during the onset of

domain and coarsening dynamics. In this short transition

time (see Fig. 3), domain coarsening is progressing in

parallel with the final interface formation. After this

transition, the domain coarsening is taking place. The

dynamics seems to be composed of three processes: the

coalescence of different domains, domain growth, and

the spatial movement that is guided by the surface tension

and precedes the coalescence of two domains. Its decorre-

lation time is larger for deeper quenches because of the

lower transition concentration of the mobility [Fig. 4(c) and

Fig. S11], representing a situation in which the temper-

ature-dependent glass line of the dense host phase intersects

the spinodal phase region. For the lower value of c

g

, the

diluted phase is essentially trapped inside the frozen host

matrix [red in Fig. 4(b)].

Based on the simulation r esults, we can conclude that

thekineticsofthesystemsiscapturedbymodelCas

suggested before [2]. The d ynamics, however, is only

partially reproduced. While the slowdown of the late s tage

dynamics with increasing quench depth is correctly

described, as well as the presence of two relaxation

modes, model C does not reproduce the Γ ∝ q behavior

observed in the experiment or the q dependence of the

KWW values of the slow dynamics (Fig. S11). We

speculate that this is due to the neglect of the viscoelastic

properties of the system in the CHE, which would lead to

elastic deformation and motion inducing a linear

dispersion relation.

In conclusion, we demonstrated the technique of low

radiation dose XPCS and used it to study protein dynamics

during an LLPS. The method delivers simultaneously

information on the collective dynamics via XPCS and

the structural evolution via the ensemble averaged scatter-

ing IðqÞ. The two-time correlation maps provide a high

level of detail of the dynamics during a spinodal decom-

position of Ig in solution with PEG. We identify distinctly

different dynamical regimes of the LLPS with different

temperature behaviors. The early stage dynamics reflects

concentration fluctuation and interface formation and is

faster for lower temperatures, reflecting stronger quench

depths. In contrast, the later stage of coarsening is slower

for lower temperatures, which is caused by the reduced

mobility of the slowed down proteins comprising the host

matrix. With simulations, we were able to identify a

concentration and time dependence of the molecular-scale

mobility that connects the dynamics of the condensate to

molecular-scale quantities.

The authors acknowledge discussions with M. Oettel,

financial support from the DFG and the BMBF

(05K20VTA), and the allocation of beamtime by DESY.

N. B. acknowledges the Alexander von Humboldt-Stiftung

for a postdoctoral fellowship. A. R. acknowledges the

Studienstiftung des Deutschen Volkes for a Ph.D.

fellowship. C. G. acknowledges BMBF (Grants

No. 05K19PS1 and No. 05K20PSA) for financial support.

*

fajun.zhang@uni-tuebingen.de

†

christian.gutt@uni-siegen.de

‡

frank.schreiber@uni-tuebingen.de; https://publons.com/

researcher/2502617/frank-schreiber/

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Microscopic Dynamics of Liquid-Liquid Phase Separation and Domain Coarsening in a Protein Solution Revealed by X-Ray Photon Correlation Spectroscopy

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