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**on** **the** **phase** behavior

lysozyme: Theory

Christoph Gögelein, Dana Wagner, Frédéric Cardinaux, Gerhard Nägele,

Citati**on**: J. Chem. Phys. 136, 015102 (2012); doi: 10.1063/1.3673442

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THE JOURNAL OF CHEMICAL PHYSICS 136, 015102 (2012)

**on** **the** **phase** behavior

Christoph Gögelein, 1,a),b) Dana Wagner, 2,b) Frédéric Cardinaux, 2 Gerhard Nägele, 3

1 Max-Planck-Institut für Dynamik und Selbstorganisati**on**, Am Faßberg 17, 37077 Göttingen, Germany

2 C**on**densed Matter Physics Laboratory, Heinrich-Heine-University, Universitätsstraße 1,

40225 Düsseldorf, Germany

3 Institute

(Received 27 October 2011; accepted 6 December 2011; published **on**line 4 January 2012)

Salt, **the** properties **on**s.

We experimentally determined **the** effect **the**se additives **on** **the** **phase** behavior

soluti**on**s. Up**on** **the** additi**on** **the** fluid–solid transiti**on** **the** gas–liquid coexistence

curve (binodal) shift to lower temperatures **the** gap between **the**m increases. The experimentally

observed trends are c**on**sistent with our **the**oretical predicti**on**s based **on** **the** **the**rmodynamic

perturbati**on** **the**ory **the** Derjaguin-L**the** lysozyme-lysozyme

pair interacti**on**s. The values **the** parameters describing **the** interacti**on**s, namely **the** refractive indices,

dielectric c**on**stants, Hamaker c**on**stant

experimentally determined by independent experiments, including static light scattering, to determine

**the** sec**on**d virial coefficient. We observe that both, **the** potential

more repulsive, while sodium chloride reduces **the** repulsi**on**. © 2012 American Institute

[doi:10.1063/1.3673442]

I. INTRODUCTION

Proteins are complex macromolecules which play

a crucial role, e.g., as enzymes or structural units, not

**on**ly in various processes in living organisms, but also in

biotechnology. Many **the**se natural or industrial processes

require fur**the**r ingredients

solvents. In biotechnology, **the**se substances are

to modify **the** protein interacti**on**s **phase** behavior, for

example to favor crystallizati**on**. 1 There is a great variety

**the**y include i**on**s, such as sodium chloride

(NaCl) to regulate **the** electrostatic repulsi**on** or potential,

liquids, such as

against denaturati**on**, 2 protect proteins against freezing, 3, 4 or

inhibit protein aggregati**on**. 5 A thorough underst

**the** resulting changes in **the** interacti**on**s **phase** behavior

is beneficial for **the** design **on**trol **the**se processes.

For example, experience has shown that high quality crystals

are **on**s, 1 while

protein crystals with many defects or amorphous protein

precipitates (aggregates) are frequently obtained if **the** sample

is quenched deep into **the** crystal **phase**, in particular close

to **the** metastable gas– liquid coexistence curve where rapid

nucleati**on** occurs. 6, 7 Thus, underst**the** **phase** behavior

can guide **the** design

Despite **the** huge complexity

proteins such as lysozyme c**on**sidered here, we will

use a very simple spherically symmetric model. Although

this will not do full justice to **the** details **the** protein

a) Electr**on**ic mail: christoph.goegelein@ds.mpg.de.

b) Christoph Gögelein **on**tributed equally to this work.

structure **the** individual properties

it will allow us to apply c**on**cepts developed in s

physics to protein soluti**on**s. It will also shed light **on** general

properties

a coarse-grained model. On **the** o**the**r h**the** model will

fail to describe certain aspects

**the** specific properties **the**

protein–protein interacti**on**s, we use **the** Derjaguin-L

Verwey-Overbeek (DLVO) potential, a well-established

model potential in colloid physics. 8, 9 Based **on** this interacti**on**

model **the**rmodynamic perturbati**on** **the**ory (TPT), 10

we will calculate **the** **phase** behavior

soluti**on** with additives, namely added salt (NaCl),

**the**oretical

predicti**on**s to our experimental observati**on**s. For **the** additive

c**on**centrati**on**s studied here, lysozyme does not unfold or

undergo c**on**formati**on**al changes. 11–13 This allows us to test

**the** validity **the** simple model, to investigate **the**

effects **the**se additives **on** **the** interacti**on**s **the** **phase**

behavior, **the** trends

observed in our experiments.

In several previous studies, **the** properties

soluti**on**s, namely **the** protein–protein interacti**on**s **the**

**phase** behavior, were described using **the** DLVO potential.

Po**on** et al. 14 studied **the** fluid–solid transiti**on**, i.e., **the**

crystallizati**on** or solubility boundary, as a functi**on**

protein c**on**centrati**on** as well as **the** pH **the** soluti**on**

thus **the** protein charge. They found a universal crystallizati**on**

boundary if **the** salt c**on**centrati**on** is normalized by **the**

square **the** protein charge

**the** behavior **the** sec**on**d virial coefficient calculated using

**the** DLVO potential. A more detailed analysis was later

0021-9606/2012/136(1)/015102/12/$30.00 136, 015102-1

© 2012 American Institute

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015102-2 Gögelein et al. J. Chem. Phys. 136, 015102 (2012)

provided by Warren. 15 Similarly, computer simulati**on**s by

Pellicane et al. 16 based **on** **the** DLVO model showed that **the**

shape **the** gas–liquid coexistence curve can successfully be

predicted with **the** critical temperature str**on**gly depending **on**

**the** Hamaker c**on**stant. By c**on**trast, Broide et al. 17 report that

**the** DLVO model cannot fully explain **the**ir experimentally

observed **phase** behavior; **the** predicted height **the** Coulomb

barrier is too low to inhibit protein aggregati**on**. This suggests

that hydrati**on** forces are important to prevent lysozyme from

aggregating.

Despite **the** widespread use

**on** essays, **on**ly a few

studies were c**on**cerned with **the**ir effect **on** **the** **phase** behavior

**the** effect

**on** **the** protein interacti**on**s in aqueous soluti**on**s

bovine pancreatic trypsin inhibitor (BPTI). Up**on** increasing

**the** **on**centrati**on**, **the** repulsive interacti**on** increases.

A descripti**on** **the**se data based **on** **the** DLVO model requires

a Hamaker c**on**stant which is much smaller than expected

from **the** optical properties

**the**ir observati**on**s to a

small increase **the** effective protein size which is caused

by an enhanced hydrati**on** shell (

cut-**on**s

**the** hydrati**on** shell up**on** additi**on**

**the** same time, a decrease **the** core volume

using small-angle neutr**on** scattering, Sinibaldi et al. 20

observed, up**on** increasing **the** **on**tent, a significant

increase **the** hydrati**on** shell

0.6 nm, **the** core volume

**the** total effective protein volume increases

by at least about 13%. 20, 21 In **the** presence

Esposito et al. 22 fur**the**rmore observed an increase

**the** hydrodynamic radius with increasing temperature. Using

computer simulati**on**s, Vagenende et al. 12 found that

preferentially interacts with **the** hydrophobic patches **on** **the**

lysozyme surface. Glycerol molecules shield **the**se patches

with **the**ir hydrocarb**on** backb**on**e, allowing **the** lysozyme hydrati**on**

shell to grow. Moreover, Sedgwick et al. 23 studied **the**

effect **on** **the** **phase** behavior

lysozyme. They found that, based **on** **the** DLVO model, **the**

**phase** behavior can be described almost quantitatively if **the**

**the** dielectric c**on**stant

**on** are taken into account. The effect

DMSO **on** **the** fluid–solid boundary **on**s has

also been studied by Lu et al. 24 Both additives increase **the**

lysozyme solubility. Fur**the**rmore, c**on**sistent with **the** previously

menti**on**ed study, 23 Lu et al. 24 found that

DMSO decrease **the** protein critical supersaturati**on**, **the**reby

promoting nucleati**on**. In subsequent work, Gosavi et al. 25

studied **the** formati**on** **the**

crystal growth rate is enhanced up**on** additi**on**

Similar observati**on**s were made by Kulkarni

Fur**the**rmore, Arakawa et al. 3

a preferential hydrati**on**

indicating that DMSO does not bind to **the** lysozyme

surface. 4 This suggests that DMSO mainly affects **the** dielectric

properties **the** bulk soluti**on**. Never**the**less, DMSO

leads to denaturati**on** at volume fracti**on**s bey**on**d 70 vol.%. 13

It is hence still not clear whe**the**r **the** additi**on**

DMSO **on**ly changes **the** solvent dielectric c**on**stant

index or whe**the**r it also has a significant effect **on** **the**

protein, in particular **on** its hydrati**on**.

Combining **the**ory **the**

**phase** behavior **on**s with additives.

The samples c**on**tain 0.7 M or 0.9 M NaCl, up to 20 vol.%

**on**, we

experimentally determine both **the** fluid–solid transiti**on**

**the** metastable gas– liquid coexistence curves (binodals). Of

particular interest is **the**ir relative locati**on** in dependence **on**

**the** amount **on**

is expected to be enhanced in **the** vicinity **the**

gas–liquid critical point. 6 As we will show, **the** experimentally

determined **phase** diagram

c**on**sistent with our **the**oretical predicti**on**s. They are based **on**

**the** DLVO model

free parameters. The values **the** parameters describing **the**

DLVO model are ei**the**r taken from **the** existing literature or

determined by independent measurements, namely refractometry

to measure **the** refractive indices

(SLS) to determine **the** sec**on**d virial coefficient. These independent

measurements also provide detailed informati**on** **on**

**the** effect **the** additives **on** **the** protein interacti**on**s.

The organizati**on**

we describe **the** experimental procedures. Secti**on** III A introduces

**the** DLVO model for **the** protein–protein interacti**on**s

**the** values **on** this model

**on**al free parameters, in Sec. III B **the** **phase**

behavior is calculated using **the**rmodynamic perturbati**on** **the**ory.

Then **the** predicted **phase** behavior is compared to our experimental

observati**on**s. Finally, our findings are summarized

in Sec. IV.

II. MATERIALS AND EXPERIMENTAL METHODS

A. Materials **on**

We used three-times crystallized, dialyzed,

lyophilized hen egg-white lysozyme powder (Sigma

Aldrich L6876). An aqueous 50 mM sodium acetate buffer

was prepared

= 4.5. The protein powder was dissolved in this buffer to

result in a c**on**centrati**on** **on**

was several times passed through a filter with pore size

0.1 μm (Acrodisc syringe filter, low protein binding, Pall

4611) to remove impurities **on**

subsequently was c**on**centrated by a factor

an Amic**on** stirred ultra-filtrati**on** cell (Amic**on**, Millipore,

5121) with an Omega 10 k membrane disc, Pall OM010025.

The c**on**centrated soluti**on** was used as protein stock soluti**on**.

Its protein c**on**centrati**on** was determined in a quartz cell with

a path length **on** spectroscopy at a

wavelength **on** coefficient

Samples were prepared by mixing appropriate amounts

**on**, buffer, salt soluti**on** (3 M NaCl with

buffer), **on** (70 vol.%

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015102-3 Phase behavior

DMSO. Mixing was performed at a temperature above **the**

metastable gas– liquid coexistence to prevent **phase** separati**on**.

The **on**tained 0.9 M

**the** DMSO samples 0.7 M NaCl, **on**tained

buffer **on**centrati**on**s

In order to obtain protein c**on**centrati**on**s above that **the**

stock soluti**on**, we exploited **the** gas–liquid **phase** separati**on**.

A soluti**on** was quenched into **the** metastable gas– liquid coexistence

regi**on**, centrifuged for 10 min at a temperature below

**the** cloud point temperature, **the** protein-rich **phase** was

collected. The resulting protein c**on**centrati**on** **the** proteinrich

**phase** was calculated from **the** volume ratio **the** two

coexisting **phase**s, **the** initial protein c**on**centrati**on** **the**

protein c**on**centrati**on** **the** protein-poor **phase**, which was

determined via its refractive index.

B. Determinati**on** **the** **phase** boundaries

The gas–liquid coexistence curves (binodals) were determined

by cloud point measurements. The sample was filled

into a NMR tube with 5 mm diameter, sealed,

a water bath at a temperature above **the** demixing point. The

temperature subsequently was lowered stepwise **the** cloud

point temperature identified by **the** sample becoming turbid.

This measurement was repeated for samples with different

lysozyme volume fracti**on**s to determine **the** locati**on** **the**

binodals.

To determine **the** crystallizati**on** boundary, **the** sample

was filled into an x-ray capillary tube

curing glue (Norl**the**n mounted **on**to

a home-built temperature stage that allows for observati**on** by

optical microscopy (Nik**on** Eclipse 80i). For samples with a

low protein c**on**centrati**on**, it was necessary to lower **the** temperature

to +4 ◦ C to induce crystallizati**on**. Once crystals were

formed, **the** temperature was raised stepwise. The temperature

at which **the** crystals begin to melt was identified with

**the** transiti**on** temperature. Again, **the** experiments were repeated

with different lysozyme volume fracti**on**s to obtain **the**

fluid–solid coexistence curves.

C. Determinati**on** **the** refractive indices

We used a temperature-c**on**trolled Abbe refractometer

(Model 60L/R, Bellingham & Stanley) operated at a wavelength

**the** refractive indices **the**

samples, that is **the** protein soluti**on**s, n, **the** solvent,

n s . The refractive index increments, dn/dc p , were obtained

from linear fits to **the** dependence **on** **the** mass protein

c**on**centrati**on**, c p .

D. Determinati**on** **the** sec**on**d virial coefficient

SLS was performed with a 3D light scattering instrument

(LS-Instruments). Due to **the** low scattering intensity **the**

samples, **the** instrument was operated with a single beam with

wavelength λ = 633 nm. The samples were filled into cylindrical

glass cuvettes with a diameter

at least 10 min at typically 7 500 g prior to **the** measurements,

**the** temperature-c**on**trolled vat **the** instrument

which was filled with decalin. The samples used for **the**

SLS measurements were dilute with lysozyme c**on**centrati**on**s

between 2.8 mg/ml

We did not observe a dependence **the** scattered intensity,

I p , **on** **the** scattering angle, θ, c**on**sistent with **the** small

protein diameter σ = 3.4 nm 27, 28 compared to **the** inverse

**the** scattering vector q = (4πn/λ)sin (θ/2) so that q σ ≪ 1. 29

Therefore, **the** time-averaged intensity scattered by **the** protein

soluti**on**, 〈I p (c p )〉, was detected at a single scattering angle, θ

= 90 ◦ , corresp**on**ding to a magnitude **the** scattering vector

q ≈ 0.018 nm −1 . The excess scattering due to **the** protein is

expressed as **the** Rayleigh ratio,

R(c p ) = 〈I p(c p )〉−〈I s 〉 n 2

〈I t 〉 n 2 R t , (1)

t

which relates **the** difference in **the** average scattering intensities

**the** protein soluti**on** **the** solvent, 〈I p (c p )〉−〈I s 〉,

to **the** scattered intensity **the** toluene reference, 〈I t 〉, with

known Rayleight ratio, R t . The ratio **the** refractive indices

**the** protein soluti**on**, n, **the**

different sizes **the** scattering volumes in **the** two cases. The

temperature dependence

from **the** temperature dependence **the** intensity scattered by

a toluene sample. The absolute value R t = 1.40 × 10 −5 cm −1

at T = 35 ◦ C was taken from literature 30 **on**sistent with

o**the**r reports (see Ref. 31 **the**rein).

For dilute soluti**on**s, R(c p ) is related to **the** sec**on**d virial

coefficient, B 2 ,by

Kc p

R(c p ) = 1 (

1 + 2 N )

A B 2

c

M (0) M (1) p , (2)

where M (0) **the** molar mass M, but by

two different values (see below), N A is Avogadro’s c**on**stant

**on**stant given by

K = 4π 2 n 2 s

N A λ 4

( dn

dc p

) 2

. (3)

Note that **the** refractive indices **the** refractive index increments

in **the** above equati**on**s depend **on** temperature.

SLS measurements **on**centrati**on**s

hence allow to determine B 2 from **the** slope **the** c p dependence,

**the** molar mass M from **the** intercept at c p → 0.

The measurements were performed at different temperatures

T to obtain **the** temperature dependence

is expected to be temperature independent. The values

**the** molar mass, experimentally determined at different temperatures

from **the** intercept at c p → 0, lie in **the** range

14 800 g/mol M (0) 19 500 g/mol, while **the** expected

value is 14 320 g/mol. 32 The error in M (0) is mainly due to

**the** low scattered intensity, uncertainties in dn/dc p

light scattering experiments **on** **the** same dilute samples

did not indicate **the** presence

nei**the**r expected for aqueous soluti**on**s 7, 33 nor for

water 12 or DMSO–water mixtures. 13 To avoid that experimental

errors in **the** molar mass affect B 2 (T), we used **the** literature

value 14 320 g/mol in **the** term for **the** slope, 2N A B 2 /M (1) ,

i.e., M (1) = 14 320 g/mol, when calculating B 2 (T) based **on**

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015102-4 Gögelein et al. J. Chem. Phys. 136, 015102 (2012)

Eq. (2). 34 This renders B 2 (T) independent **the** absolute scattered

intensity, i.e., M (0) , which has a significant uncertainty,

III. RESULTS AND DISCUSSION

We first present **the** simple model used to quantify

**the** protein–protein interacti**on**s, namely **the** DLVO potential

model (Sec. III A 1), as well as **the** values

These are ei**the**r taken from **the** literature, or determined by

independent measurements (Sec. III A 2). Based **on** this interacti**on**

potential **the**se parameter values, we calculate **the**

**phase** diagram without any free parameter (Sec. III B 1). Our

experimentally determined **phase** diagrams (Sec. III B 2) are

**the**n compared to **the** **the**oretical predicti**on**s (Sec. III B 3).

A. Protein–protein interacti**on** potential

1. DLVO model

The DLVO pair potential, u(r), includes three parts: hardsphere

u 0 (r), repulsive electrostatic u el (r),

der Waals interacti**on**s u vdW (r),

u(r) = u 0 (r) + u el (r) + u vdW (r), (4)

where r is **the** center-to-center distance

The hard-sphere c**on**tributi**on** is

{

∞ , r0

= 3 ( )

4 k εp − ε 2

s

BT

+ 3hν ( )

e n

2

ε p + ε s 16 √ p − n 2 2

s

( )

2 n

2 p + n 2 3/2

, (9)

s

where A ν = 0 is **the** zero-frequency **the** frequencydependent

part, h **the** Planck c**on**stant, ε s **the** static dielectric

c**on**stants **the** solvent **the**ir

refractive indices, **the** characteristic

UV electr**on**ic adsorpti**on** frequency. 8

2. Values **the** model parameters

To quantitatively determine **the** potential u(r), we require

**the** values **the** refractive indices **on**stants,

**the** cut-**the** parameters describing

**the** protein. Lysozyme is an approximately spherical protein

with an effective diameter σ = 3.4 nm, 27, 28 molar mass M

= 14 320 g/mol, mass density ρ p = 1.351 g/cm 332

= 11.4 net positive elementary charges at **the** present

pH = 4.5. 37 The acidity c**on**stant

= 4.76 resulting in a dissociated buffer c**on**centrati**on**

= 27 mM at pH = 4.5.

a. Refractive indices **on**stants. The

protein index **on** is obtained by **the** linear extrapolati**on**,

n p = 6 M/(N A πσ 3 )dn/dc p + n s , where 6/(N A πσ 3 )

is **the** molar c**on**centrati**on**

significant dependence **on** compositi**on** or temperature

in **the** relevant parameter range, c**on**sistent with previous

findings. 38 Averaging dn/dc p over all tested temperatures

compositi**on**s yields n p = 1.72.

It is usually assumed that dry protein has a static

dielectric c**on**stant in **the** range

measurements with protein powders

for example 1 ≤ ε p ≤ 10 for wet lysozyme powder, 40 20

≤ ε p ≤ 37 for hydrated lysozyme crystals 41 **on** MD

simulati**on**s, ε p = 25.7 for lysozyme in aqueous soluti**on**. 42

The larger values are attributed to water or moisture leading

to hydrati**on** **on**. 39 For c**on**sistency

with previous work, 23, 39 we use a static dielectric c**on**stant

this way ε p **the** static c**on**tributi**on** to

**the** Hamaker c**on**stant A ν = 0 (Eq. (9)) **the** protein

attracti**on**. However, **the** effects are not significant, because,

as will be shown below, **the** main c**on**tributi**on** to **the** Hamaker

c**on**stant is due to **the** n**on**-zero frequency term A ν>0 .

The refractive index **the** solvent, n s , was measured

The refractive indices

larger than **the** **on**e

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015102-5 Phase behavior

FIG. 1. Temperature dependence **the** index **on** **the** solvent,

n s (T), for different added NaCl c**on**centrati**on**s, c s , as well as

DMSO c**on**tents (as indicated). The lines are linear fits to **the** experimental

data points.

n DMSO = 1.48, **on**

The dielectric c**on**stant **the** solvent, ε s , was taken

from literature 43–49 (Fig. 2). It decreases with increasing

temperature, which is due to **the** increasing **the**rmal moti**on**

**the** solvent molecules **the** alignment **the**

dipolar molecules by an external electric field. The dielectric

c**on**stants

smaller than **the** **on**e

ε DMSO = 48, **the** solvent

dielectric c**on**stant, ε s , decreases with increasing

DMSO c**on**tent, with **the** decrease being more pr**on**ounced

for

decrease **on**centrati**on**, c s ; adding

0.7 M NaCl lowers ε s more than adding 20 vol.%

(Fig. 2(c)). 47–49 This decrease is related to **the** decrease in **the**

electric polarizability **the** soluti**on** caused by **the** alignment

**on** **the** water molecules by **the** salt i**on**s.

Based **on** **the**se static dielectric c**on**stants, we calculate

**the** repulsive electrostatic interacti**on** part, u el (r),

(Eqs. (6) **the** different compositi**on**s, i.e., different

added salt, **on**tent. Up**on** increasing

**the** salt (NaCl) c**on**centrati**on** c s , **the** screening parameter, κ,

increases **the** c**on**tact value **the** electrostatic potential,

u el (σ ), decreases (Fig. 3). In additi**on**, u el (σ ) depends **on**

**the** soluti**on** compositi**on** since **the** Bjerrum length l B depends

**on** ε s . Adding

2(b)) **the** effect

being significantly str**on**ger.

We have determined **the** zero- **on**-zero frequency

parts **the** Hamaker c**on**stant (Eq. (9)) based **on** **the** above

values **the** refractive indices **on**stants.

The n**on**-zero frequency part, A ν>0 , is about ten times larger

than **the** zero-frequency part, A ν = 0 ,

(Fig. 4). Both parts decrease with increasing temperature as

well as increasing salt, **on**tent. Glycerol

**the** same decrease

a similar effect **on** n s (Eq. (9) **on**trast,

we find a less significant decrease **on** additi**on**

**the** weak effect

DMSO **on** ε s (Fig. 2(b)). Added salt (NaCl) causes a pr**on**ounced

decrease

(Fig. 2(c)). For our sample compositi**on**s, i.e., solvent mixtures

with added salt, **the** temperature dependence

available. However, since A ν = 0 is much smaller than A ν>0 ,

we neglect **the** effect **on**sider **on**ly **the** effect

**the**less, in our calculati**on**s

we take into account **the** effect **on** n s

A ν>0 (Fig. 4(b)), which represents **the** dominating c**on**tributi**on**

to **the** Hamaker c**on**stant. Glycerol **on**ly

similarly affect A ν>0 **the** Hamaker c**on**stant A, but

also **the** total DLVO pair potential, u(r)(seeFig.5).

FIG. 2. Temperature dependence **the** dielectric c**on**stant, ε s (T),

solvent mixtures: (a) water–

aqueous salt soluti**on**s. 47–49 The lines are linear fits. The fits that bel**on**g to

soluti**on**s c**on**taining 10 **on** five data points in

an extended temperature range 20 ◦ C ≤ T ≤ 80 ◦ C.

FIG. 3. Temperature dependence **the** c**on**tact value **the** reduced electrostatic

interacti**on** potential, βu el (σ ) = Z 2 l B /(σ [1 + κσ/2] 2 ), for different

added NaCl c**on**centrati**on**s, c s , as well as **on**s

(in vol.%, as indicated). Note **the** break in **the** ordinate axis.

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015102-6 Gögelein et al. J. Chem. Phys. 136, 015102 (2012)

duced sec**on**d virial coefficient,

b 2 (T ) = B 2(T )

B (0) . (11)

2

The latter is equal to **the** ratio **the** sec**on**d virial coefficient,

B 2 =− 1 ∫

dr (exp [−βu(r)] − 1) , (12)

2

FIG. 4. Temperature dependence **the** (a) zero-frequency, A ν = 0 ,

n**on**-zero frequency, A ν>0 , parts **the** Hamaker c**on**stant in Eq. (9) for

different compositi**on**s. (a) water–

dashed lines), water–DMSO mixtures without NaCl (green solid lines), aqueous

salt soluti**on**s with 0.7 M NaCl (black dotted line),

solid line), (b) water–**on**taining 0.7 M NaCl (black dotted

lines), water–**on**taining 0.9 M NaCl (blue solid lines),

**on**taining 0.7 M NaCl (green solid lines). The

numbers indicate **the** volume fracti**on**s (in vol.%)

b. Cut-**on**e potential parameter left

to be determined, namely **the** cut-

such that it reproduces **the** sec**on**d virial coefficient, B 2 (T),

which we have experimentally determined by static light scattering.

From a **the**oretical point **the** sec**on**d virial coefficient,

B 2 (T), is introduced by exp**the** reduced free energy

density, f(T, φ), in a power series **the** particle volume

fracti**on**, φ, according to

f (T,φ) = f0 id (φ) + 4b 2(T )φ 2 + O(φ 3 ) , (10)

which involves **the** ideal gas part **the** free energy density,

, **on**, **the** normalized or re-

f id

0

to **the** sec**on**d virial coefficient

2

= 2π(σ + δ) 3 /3. For **the** DLVO pair potential, u(r), (in Eq.

(4)) **the** normalized sec**on**d virial coefficient, b 2 , can be written

as

∫

3 ∞

b 2 = 1 −

dr r 2

(σ + δ) 3

σ +δ

× (exp [−β(u el (r) + u vdW (r))] − 1) . (13)

The sec**on**d virial coefficient characterizes **the** **the**rmodynamic

strength **the** pair interacti**on**s. Positive values

that repulsive interacti**on**s dominate, while negative values

corresp**on**d to dominating attracti**on**.

Using SLS, we determined **the** temperature dependence

**the** range 10 ◦ C < T < 40 ◦ C for all relevant soluti**on**

compositi**on**s, i.e., salt (NaCl) c**on**centrati**on**s, c s ,as

well as **on**s (Fig. 6). Outside

this temperature range, reliable SLS experiments cannot

be performed. With increasing temperature, as expected

b 2 (T) **the** repulsi**on** increases, which reflects **the** increasing

electrostatic (Fig. 3)

interacti**on**s (Fig. 4). This leads also to an increase

with increasing **on**centrati**on**s, in agreement

with previous observati**on**s, 23, 26, 50 with DMSO having a

significantly more pr**on**ounced effect. Fur**the**rmore, b 2 (T) increases

**on** lowering c s , due to **the** increased electrostatic repulsi**on**

(Figs. 6(a)

20 vol.% **the** investigated

temperature range 10 ◦ C < T < 40 ◦ C, indicating that

**the** lysozyme pair potential is predominantly repulsive at high

**on**centrati**on**s.

FIG. 5. DLVO pair potential, u(r), for NaCl c**on**centrati**on** c s = 0.7 M, protein

volume fracti**on** φ = 0.002,

soluti**on** (black line), a water–

water–DMSO mixture (5 vol.%, green line). The vertical lines indicate **the**

cut-**the** van der Waals attracti**on**.

FIG. 6. Temperature dependence **the** normalized sec**on**d virial coefficient,

b 2 (T), for different added NaCl c**on**centrati**on**s, c s , as well as

DMSO c**on**tents (as indicated), experimentally determined by static light

scattering. The solid lines represent fits assuming a linear temperature dependence

**the** cut-

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015102-7 Phase behavior

cut-**on**centrati**on**

DMSO interact differently with lysozyme. Compared to

little is known about **the** DMSO–lysozyme interacti**on**s,

apart from a preferential hydrati**on**

DMSO mixtures indicating a weak or no binding

to lysozyme. 3, 4 This suggests that DMSO mainly influences

**the** dielectric properties **the** soluti**on**, **on**sistent with

**the** observed weaker temperature dependence **the** cut-

length δ(T) in **the** case

FIG. 7. Temperature dependence **the** cut-

fits **on**s to **the** measured b 2 (T). (a) Different added

NaCl c**on**centrati**on**s, c s , **on**tents, **on**centrati**on**

c s = 0.7 M, but different **on**tents (as indicated).

The experimental b 2 (T) is fitted by **the** DLVO potential

predicti**on** (Eq. (13)), with **the** cut-**the** **on**ly

adjustable parameter. A c**on**stant δ does not reproduce **the** experimental

data satisfactorily. We thus assume a linear temperature

dependence **the** cut-

**the** coefficients δ 0

between experimental **the**oretical b 2 (T) in **the** temperature

range 10 ◦ C < T < 40 ◦ C (Fig. 6). The fitting is d**on**e by applying

**the** Levenberg–Marquardt method, 51 **the** improper

integrals appearing in **the** expressi**on** for **the** sec**on**d virial coefficient

are evaluated using **the** Chebyshev quadrature. For

all c**on**sidered compositi**on**s **the** temperature range **the**

light scattering experiments, δ(T) increases with increasing T

(Fig. 7). Its values are in **the** range 0.15 nm ≤ δ(T) ≤ 0.3 nm,

except for **the** soluti**on** with 0.7 M NaCl

for which unphysically large δ(T) are obtained. This range

values agrees with previously published values, which are in

**the** range

for δ(T) obtained for 0.7 M NaCl

are due to **the** positive-valued experimental b 2 (T), while **the**

DLVO model predicts a predominantly attractive potential

with T to reduce **the** attractive van der Waals c**on**tributi**on**. The

effect **the** van der Waals interacti**on** **on** b 2 (T) is c**on**trolled

by **the** Hamaker c**on**stant A **the** cut-

in **on**e **the** two hence can be compensated (or enhanced) by

a corresp**on**ding change in **the** o**the**r parameter.

The cut-**on** a

coarse-grained level, such as **the** solvati**on** shell, which might

c**on**tain both water **the**

c**on**densati**on** **on**s **on** **the** protein surface, i.e., in

**the** Stern layer. Its increase with increasing temperature

12, 20, 22

**on**tent is c**on**sistent with previous findings.

As menti**on**ed above,

i.e., **the** additi**on** **the** same effect

as 5 vol.% DMSO (Figs. 6(b) **the** optical

properties **the** two solvents suggest a similar effect

**on** A ν>0 **the** small effect

**on** **the** electrostatic interacti**on**s (Fig. 3),

also **on** **the** DLVO potential (Fig. 5). This results in a larger

B. Phase behavior

We have now determined all values **the** parameters determining

**the** DLVO model. We can thus predict **the** **phase**

behavior without any free parameters

experimental observati**on**s.

1. Phase behavior

**the**rmodynamic perturbati**on** **the**ory

Our predicti**on** **the** **phase** behavior

is based **on** expressi**on**s for **the** Helmholtz free energy

which are obtained by **the** sec**on**d-order TPT

Henders**on**. 10 Knowledge **the** free energy **the** fluid

solid **phase**s allows us to deduce all equilibrium **the**rmodynamic

properties, such as **the** fluid–solid

curves including **the** critical point. The Barker–

55, 56

Henders**on** **the**ory is known to yield quite accurate results.

It is **the**refore widely used to calculate **the** **phase** diagram

protein soluti**on**s. 57–61

Within TPT, **the** free energy, F, is exp

**the** perturbati**on** potential u p = u − u 0 , where u 0 is **the**

pair potential **the** total proteinprotein

pair potential (see Eq. (4)). The reference system is

an effective hard-sphere system with diameter σ + δ, which

takes into account **the** cut-

**the** free energy, F(N, V, T),

a volume V, we use **the** dimensi**on**less free energy density f

= βFv 0 /V, where v 0 = πσ 3 /6 is **the** particle volume. Exp

**the** free energy density, f, up to sec**on**d order in u p gives

f (φ,T) = f 0 (φ) + 12 ∫ ∞

(σ + δ) 3 φ′ 2

dr r 2 g 0 (r) βu p (r)

−

6

(σ + δ) 3 φ′ 2

( ∂φ

∂ 0

)T

σ +δ

∫ ∞

σ +δ

dr r 2 g 0 (r)

× [βu p (r)] 2 , (14)

where φ ′ = φ(1 + δ/σ ) 3 is **the** effective sphere volume fracti**on**,

**the** iso**the**rmal compressibility

**the** reference system with 0 its dimensi**on**less osmotic

pressure. Fur**the**rmore, g 0 (r) is **the** radial distributi**on**

functi**on** **the** fluid **phase**, **the** orientati**on**ally

averaged pair distributi**on** functi**on**

in **the** solid **phase**. For **the** radial distributi**on** functi**on** in

**the** fluid **phase**, we use **the** Verlet–Weis corrected 62 Percus–

Yevick soluti**on**, 63, 64 **the** orientati**on**ally averaged pair

distributi**on** functi**on** **the** expressi**on** by Kincaid 65 for a facecentered

cubic (fcc) crystal **phase**. The free energy density,

f 0 (φ), **the** effective hard-sphere reference system is sep-

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015102-8 Gögelein et al. J. Chem. Phys. 136, 015102 (2012)

arately defined in **the** fluid **the** fluid

**phase**, f 0 (φ) c**on**sists **the** ideal gas part,

αf0 id (φ) = φ′ [ln(φ ′ 3 /v 0 ) − 1] , (15)

where = h/ √ 2πm p k B T is **the** **the**rmal wavelength, m p **the**

protein mass **on**d, **the** interacti**on** part which

is approximated by **the** accurate Carnahan–Starling equati**on**

αf0 CS (φ) = 4φ′2 − 3φ ′ 3

. (16)

(1 − φ ′ ) 2

In **the** solid **phase**, f 0 (φ) is described by Wood’s equati**on**

state 67 **on** assuming a fcc crystalline lattice,

(

αf solid

0 (φ) = 2.1306 φ ′ + 3 φ ′ ln

φ ′ )

1 − φ ′ /φ cp

(

+ φ ′ 3

)

ln , (17)

v 0

where φ cp = π √ 2/6 is **the** fcc volume fracti**on** for closed

packing. To evaluate **the** free energy density, f(φ, T), in **the**

solid **phase**, **the** integrals in Eq. (14) are divided into intervals

centered around **the** crystal lattice sites, which is necessary

to sufficiently resolve **the** discreteness

with increasing φ.

The coexistence **phase**s requires

an identical temperature T, osmotic pressure

potential μ in both **phase**s. While **the** temperatures

**phase**s in c**on**tact are always identical under equilibrium c**on**diti**on**s,

**the** latter two c**on**diti**on**s determine **the** two volume

fracti**on**s, namely that **the** coexisting gas, φ g ,

**phase**, φ l ,

g (T,φ g ) = l (T,φ l ) with

( )

∂(f (T,φ) /φ)

(T,φ) = φ 2 , (18)

∂φ T

μ g (T,φ g ) = μ l (T,φ l ) with

( ) ∂f (T,φ)

βμ(T,φ) =

. (19)

∂φ T

Similarly, **the** coexistence **phase**s is

determined by **the** two c**on**diti**on**s,

f (T,φ f ) = s (T,φ s ), (20)

μ f (T,φ f ) = μ s (T,φ s ) , (21)

where φ f is **the** volume fracti**on** **the** fluid, **the** **on**e

**the** coexisting solid **phase**. To compute **the** **phase** coexistence

curves, **the** Newt**on**–Raphs**on** method with line search

is applied. 51

The critical point terminates **the** gas–liquid coexistence

curve. It is determined by **the** vanishing sec**on**d

derivatives **the** free energy density by, i.e.,

∂ 2 f (T c ,φ c )

∂ 2 φ

= 0

∂ 3 f (T c ,φ c )

∂ 3 φ

= 0 . (22)

The higher-order derivatives **the** free energy are calculated

using Ridder’s implementati**on**

FIG. 8. Phase diagrams with fluid–solid

**on**,

z −1 , as indicated. Coexistence curves with critical points calculated using

TPT are represented as solid lines with black filled circles, **the** data from

M**on**te Carlo simulati**on**s are represented as red open circles. 55, 68 The coexistence

curves are shown in **the** normalized density (ρσ 3 )-normalized potential

strength (βξ) plane. Note that **the** (βξ) axes run downwards. Fur**the**rmore,

no simulati**on** data for **the** fluid–solid

= 5.0

Before comparing our **the**oretical

we test **the** accuracy **on**.

No simulated **phase** diagrams are available for **the** DLVO

model with our parameter values. However, M**on**te Carlo

simulati**on**s 55, 68 have been used to determine **the** **phase** diagram

⎧

⎪⎨ ∞ ,

r0 is **the** (c**on**tact) strength, **the** range **the**

attracti**on**.

Using TPT, we calculate **the** **phase** diagram

The calculated fluid–solid coexistence curves agree with

**the** simulated **on**es for all investigated values

Fig. 8). Also **the** widths **the** gas–liquid coexistence curves

are well reproduced. The calculated gas–liquid binodals, however,

are located slightly above **the** simulati**on** data. We suppose

that **the** binodals are shifted because **the** mean-field type

TPT does not account for critical density fluctuati**on**s. Critical

fluctuati**on**s allow a system to probe density inhomogeneities

**the** homogeneous supercritical fluid

**phase** against **phase** separati**on**, **the**refore shifting **the** critical

point to lower temperatures. A renormalizati**on**-group correcti**on**

scheme has been derived 69 which, according to previous

calculati**on**s, 70 is expected to flatten **the** coexistence curves

around **the** critical point.

2. Experimentally determined **phase** behavior

We experimentally determined **the** fluid–solid

liquid coexistence curves for different c**on**tents

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015102-9 Phase behavior

FIG. 9. Experimental **phase** diagram **on**s with different

added NaCl c**on**centrati**on**s, c s , as well as different **on**tents

(as indicated), in dependence **on** **the** volume fracti**on**, φ,

T. The open circles, inverted triangles, pentag**on**s, stars,

triangles indicate **the** fluid–solid coexistence curves. The open squares, triangles,

diam**on**ds, hexag**on**s, **on**s indicate **the** gas–liquid coexistence

curves. The filled squares, triangles, **on**ds indicate **the** critical points.

(NaCl), **on**s

explored, **the** gas–liquid binodals lie within **the** fluid–solid

coexistence regi**on**s

for short-ranged attractive pair potentials. 17, 71–73 The binodal

widens rapidly up**on** cooling at volume fracti**on**s φ below **the**

critical volume fracti**on** φ c , while **the** data for φ>φ c suggest

almost flat binodals at volume fracti**on**s larger than **the** critical

point value. Similar shapes

73, 74

observed previously.

The effect **on** **the** **phase** behavior was studied

for 0.7

With increasing salt c**on**centrati**on**, c s , **the** fluid–solid

liquid coexistence curves move to higher temperatures,

**the** width **the** gas–liquid binodal increases. These findings

are c**on**sistent with **the** electrostatic screening by added

salt. Up**on** increasing c s , **the** screening length, κ −1 , decreases.

Thus **the** electrostatic repulsi**on** decreases **on** becomes

more important. This favors crystals compared to **the**

fluid **phase** **the** (metastable) gas–liquid **phase** separati**on**

compared to a homogeneous fluid. The fluid–solid as

well as **the** gas–liquid coexistence curves hence shift to higher

temperatures, **the** gas–liquid coexistence curve becomes

broader, as experimentally observed.

Up**on** additi**on**

curves shift to lower temperatures (see Fig. 9). The shift

**the** binodals is more pr**on**ounced, which widens **the** gap between

**the** two coexistence curves. This trend has been observed

before in o**the**r systems

range **on** 17, 55, 75 (Fig. 5). The critical point

is not **on**ly shifting to lower temperatures, but also to slightly

lower volume fracti**on**s. C**on**sidering **the** quantitative effect **on**

both curves, we find that **the** additi**on**

5 vol.% DMSO induce a similar shift. The higher efficiency

DMSO is c**on**sistent with its more pr**on**ounced effect **on** b 2 (T)

(Fig. 6).

FIG. 10. Experimental **phase** diagram **on**s with different

added NaCl c**on**centrati**on**s, c s , as well as different **on**tents

(as indicated), as a functi**on** **on**, φ, **on**d

virial coefficient, b 2 . The sec**on**d virial coefficient has been (a) experimentally

determined **on** **the** DLVO model. The filled

symbols indicate **the** fluid–solid (fs) coexistence curves, **the** open symbols

**the** gas–liquid (gl) coexistence curves.

We have investigated **the** relati**on** between **the** sec**on**d

virial coefficient, b 2 (T), **the** locati**on** **the** fluid–solid

**the** gas–liquid coexistence curves. For this purpose,

we parameterized **the** temperature dependencies **the**

experimentally determined sec**on**d virial coefficients (Fig. 6)

by **the** phenomenological relati**on** b 2 (T) = α 0 + α 1 T + α 2 T 2 ,

where α 1 **on** **the** compositi**on** **the** soluti**on**.

Using this relati**on**, **the** binodals are plotted in dependence

b 2 **on** leads to a collapse

**on**to a single curve, as it has been observed

previously. 14, 18, 74, 76 This indicates that in our system **the** binodals

are c**on**trolled by b 2 , i.e., by a single integral parameter

characterizing **the** potential, **the** details **the** potential,

such as its depth **on**trast, this scaling

b 2 is not observed for **the** fluid–solid transiti**on**. Never**the**less,

this transiti**on** occurs for b 2 values c**on**sistent with previous

reports. 77

If instead **the** b 2 values calculated based **on** **the** DLVO

model are used, **the** fluid–solid

curves do not scale (see Fig. 10(b)) **the** critical points

are different. This indicates that **the** scaling very sensitively

depends **on** **the** temperature dependence **on**

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015102-10 Gögelein et al. J. Chem. Phys. 136, 015102 (2012)

**the** model. Small errors might be introduced by **the** simplificati**on**s

implicit in **the** DLVO model. However, **the** main effect is

due to **the** extrapolati**on** **on**d **the** available temperature

range 10 ◦ C < T < 40 ◦ C(seeFig.6), which indicates

**the** need for a very reliable **on** **the**

temperature dependence **the** cut-

spurious effects, in **the** following we scale **the** temperature

c**on**centrati**on** axes by **the** critical temperatures **on**centrati**on**s.

This axis scaling also reduces artifacts due to **the** slight

shift in **the** binodals caused by **the** mean-field nature

(Fig. 8).

3. Comparis**on**

determined **phase** behavior

The experimental **phase** diagrams are now compared to

our TPT predicti**on**s based **on** **the** DLVO model. The axes

are scaled by **the** critical temperature, T c , **on**,

φ c , which matches all **the**oretical

points (see Fig. 11). (Note that in this representati**on** **the**

fluid–solid transiti**on** shifts to higher T/T c , in c**on**trast to **the**

representati**on** using **the** absolute temperature scale (Fig. 9).

This representati**on** reveals an increase **the** gap between **the**

fluid–solid **the** gas–liquid coexistence curves up**on** additi**on**

**the** experimental data

pr**on**ounced, in **the** range φ/φ c < 0.2 for **the** **the**oretical predicti**on**s.

Note that in **the** main part **the** **phase** diagram **the**

**the**oretically predicted gap actually decreases. In view **the**

successful test **the** TPT approach (Fig. 8), we attribute this

discrepancy to **the** inherent approximati**on**s **the** DLVO potential

**the** model for **the** cut-

The width **the** gas–liquid coexistence curve is experimentally

found to remain practically unchanged in this

representati**on** when **the**ory

predicts a slight increase **the** width. The **the**oretically

**the** largest **on**tent

our calculati**on**s do not reproduce **the** flat binodal regi**on**

around **the** critical point, which we observe experimentally.

As discussed above (see Fig. 8), this is attributed to **the** meanfield

character **the** **the**ory. This is fur**the**rmore c**on**sistent

with previous experiments 73 **on**s. 16 In particular,

similar gas–liquid coexistence curves have been **the**oretically

observed for colloidal dispersi**on**s with competing repulsive

**on**s,

fluctuati**on**s which become significant in an enlarged regi**on**

**the** **phase** diagram. 78

FIG. 11. Experimental **phase** diagram **on**s

with different added NaCl c**on**centrati**on**s, c s , as well as different

**on**tents (as indicated), as a functi**on** **on**,

φ/φ c , **the** critical point.

Solid lines represent **the** TPT calculati**on**s based **on** **the** DLVO model, open

symbols **the** experimental data for **the** stable fluid– solid (fs)

gas– liquid (gl) coexistence curves. Filled symbols mark **the** critical point.

IV. CONCLUSIONS

In a combined **the**oretical

**the** **phase** behavior **on**s

in **the** presence

in particular investigated **the** fluid–solid

curves. Up**on** **the** additi**on**

curves shift to lower temperatures **on**, **the** gap between

**the**se two curves increases, c**on**sistent with a decreasing

range **on** up**on** increasing **the** c**on**tent

Our **the**oretical calculati**on**s are based **on** a DLVO interacti**on**

potential. This models **the** proteins as hard spheres

with an isotropic electrostatic repulsi**on**, which is caused by,

e.g., uniformly distributed charges,

Waals attracti**on** characterized by **the** (effective) Hamaker

c**on**stant **on**stant is

calculated based **on** **the** experimentally determined, or in

**the** literature available, macroscopic optical

properties **the** solvent mixtures **the** protein, namely

**the**ir indices **on** **on**stants. The

cut-**the** sec**on**d virial

coefficient, which was experimentally determined by static

light scattering. We find cut-**the** range 0.15 nm

≤ δ(T) ≤ 0.3 nm, with **the** value increasing with increasing

temperature **on**tent **on**ounced,

with increasing DMSO c**on**tent, c**on**sistent with previous

27, 28, 52–54

studies. The cut-

effects, such as a hydrati**on** shell or c**on**densed counteri**on**s

**on** **the** protein surface. Our experiments thus indicate that

DMSO seems to affect **the** lysozyme interacti**on**s by mainly

lowering **the** solvent dielectric c**on**stant, while

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015102-11 Phase behavior

changes **the** hydrophobicity **the** lysozyme surface. This is

in accordance with recent computer simulati**on** results. 12

Having determined **the** values

**the** interacti**on** potential for different temperatures

compositi**on**s, added salt (NaCl) c**on**centrati**on**s as well as different

**on**tents. We found that

**the** repulsive interacti**on**, as quantified by

**the** sec**on**d virial coefficient b 2 . For a sufficiently high c**on**tent

**on**

is observed. The increase **the** repulsi**on** is significantly

str**on**ger for DMSO, although **the** optical

this difference to **the** specific molecular interacti**on**s

**the** additives with lysozyme, characterized by **the** cut-

length δ(T).

Based **on** **the** DLVO model

**the** **phase** behavior without any free parameters. Our calculati**on**s

are based **on** **the** TPT, which was successfully tested

**the**oretical predicti**on**s

were **the**n compared to our experimental observati**on**s, for **the**

fluid–solid **the** gas–fluid coexistence curves. Particular attenti**on**

was paid to **the** effect **on** **the**se curves,

**the** size **the** gap between **the**m. We find that **the** general

trends in **the** **phase** behavior, **the** effects **the**

additives, are also reproduced by **the** DLVO model, although

not **on** a quantitative level.

The strength **the** DLVO model lies in its simplicity,

for it allows to incorporate **the** effect

by taking into account **the**ir macroscopic optical

properties. More complex models are available, but

might need to be adapted to proteins

or additives. For example, **the**oretical calculati**on**s based **on**

anisotropic attractive interacti**on**s successfully describe both

**the** width **the** binodal **the** gap between **the** binodal,

**the** fluid–solid transiti**on** for aqueous protein soluti**on**s without

additives. 60, 79, 80 Anisotropic attractive interacti**on**s are indeed

induced by hydrophobic patches **on** **the** lysozyme surface,

model by Curtis et al. 81 Anisotropic attracti**on**s as well as

repulsi**on**s, e.g., as due to an anisotropic charge distributi**on**

or van der Waals interacti**on**, are not taken into account

in our DLVO model. Our model fur**the**rmore neglects n**on**electrostatic

c**on**tributi**on**s to **the** lysozyme pair potential that

originate from dispersi**on** forces between **the** micro-i**on**s

proteins. 58, 59 The range **on**-electrostatic

**on**s has not been determined experimentally,

e.g., through an effective potential **the**rmore,

hydrophobic interacti**on**s, as well as n**on**-rigid

n**on**-spherical shapes, specific surface properties, i**on**-specific

interacti**on**s, hydrati**on**, **the**r effects need to be explored

in future studies.

ACKNOWLEDGMENTS

We thank **the** Internati**on**al Helmholtz Research School

Biophysics

**the** Deutsche Forschungsgemeinschaft (DFG) for

financial support.

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