Gibbs, Volmer, and Crystal Nucleation -...

International School of Physics “Enrico Fermi” “Physics of Complex Colloids”

Varenna, Lake Como, July 3 - 13, 2012

Gibbs, Volmer, and Crystal Nucleation

Peter G. Vekilov

Department of Chemical and Biomolecular Engineering, Department of Chemistry, University of Houston

"FAST GROWTH Method Sets Crystal Size Record,"

LASER FOCUS WORLD, July 1999 Cover

Crystal from LLNL

KDP ~ 20 cm crystal grows in ~ 1 day

Ferritin ~ 700 mm crystal

grows in ~ 1 month

IPC, Sofia

Crystallization in Industry

Insulin as medication

In 2010, the worldwide insulin

market

$7.3 billion

projected to grow to $16 billion

by 2020

Other crystalline pharmaceuticals

acetaminophen,…, interferon, …

Products and intermediates in

chemical industry

adipic acid for nylon produced

in Texas

5,500,000 tons / year worldwide

shipped to New Jersey for nylon

production

Insulin Biosynthesis

G. Dodson, D. Steiner, Curr. Op.Struct. Biol. 8 (1998) 189

Crystals exclude proinsulin present in islet cells

• Single crystal

per vesicle

• Fast crystal growth

Crystals of Hemoglobin C in Red Blood Cells

Erythrocytes from HbC Transgenic Mice

• crystallization induced by 4 hour incubation in 3% NaCl, 37oC

• crystal dissolution induced by addition of 0.09 M NaCl solution

5 s original = 0.1 s

as played

J. E. Canterino, et al.,

Biophys. J. 95, 4025 (2008).

failure of nucleation;

nucleation of two crystals of apoferritin, which grow large;

nucleation of numerous crystals of insulin, with a broad size distribution;

amorphous precipitate in a lysozyme solution.

dense liquid droplets of hemoglobin A;

needle-like crystals of lysozyme.

Nucleation and the Outcome of Crystallization

Crystallization is a Miracle!

Crystallization

Nucleation

Crystallization and Nucleation

…

Josiah Willard Gibbs

(1839 - 1903)

… and this has been strictly shown for Gibbs only

Only Jesus and Gibbs never erred

E.D. Shchukin

Classical Nucleation Theory: Thermodynamics

n molecules from solution into a crystalline cluster

Solution—supersaturated: msoln > mcrystal, Dm = msoln – mcrystal > 0

Free energy gain = - nDm

Free energy loss = 6gn2/3

creation of new surface

DG(n) = - nDm + 6gn2/3

Gibbs JW On the equilibrium of heterogeneous substances

Trans. Connect. Acad. Sci. (1876) 3, 108; (1978) 16, 343

6gn2/3 DG

DG*

n

n*

DG(n)

- nDm ∆𝐺∗ =

32𝜸𝟑2

Dm2

𝑛∗ = 64𝜸𝟑2

Dm3

Max Volmer

(1885 - 1965)

Classical Nucleation Theory: Kinetics

Volmer M (1939) Kinetik der Phasenbildung (Steinkopff, Dresden)

Assumes that nuclei are perfect crystals

Predicts steep

J

Dm

𝐽 = 𝐽𝑜exp −∆𝐺∗

𝑘𝐵𝑇

Jo = n*Zn

n* – frequency of attachment, 300 s-1

Z – Zeldovich factor, 0.01

n – concentration of molecules, = 1018 cm-3

Jo = 1018 cm-3s-1

Mysteries of Protein Nucleation Kinetics

5 1 0 1 5 2 0 2 5 3 0 0

0 . 2

0 . 4

T L - L

T L - L

Clys= 50 mg/ml Clys = 80 mg/ml

T e m p e r a t u r e T [ ° C ]

Ho

mo

ge

ne

ou

s N

ucle

ati

on

R

ate

J [

cm-3 s

-1]

Non-monotonic behavior of J(T)

O. Galkin & P. G. Vekilov, PNAS 97, 6277 (2000)

Nucleation rate lower by 108 then prediction of CNT

J0(experiment) ~ 1010 cm-3s-1

J0(CNT) ~ 1018 cm-3s-1

)exp(*

0Tk

GJJ

B

D

Sharp leveling of J(Dm) Large data scatter

O. Galkin et al., JACS 122, 156 (2000)

Nucleation Rate of Dense Phase Droplets

Time [s]

Nu

mb

er

of

Dro

ple

ts

0 2 4 6 8 10 0

100

200

300

400

Nucleation Rate = 4.3 x 109 cm

-3s-1

Increase of droplet number at DT = 0.4 K

• J ~109 cm-3s-1

significantly higher than rates of crystal nucleation ~ 0.1 – 1 cm

-3s-1

• Jo = 1018 cm-3s-1

M. Shah, et al., J. Chem. Phys. 121 (2004) 7505

Determination of Nucleus Size

Nucleation theorem

D. Kashchiev, D. Oxtoby JCP 100 (1994) 7665

)1()/(

)(ln

B

* OTkd

Jdn

D

m

allows determination of nuclei sizes (10 4 1)

O. Galkin, P.G. Vekilov, JACS 122 (2000) 156

. .

0

0.2

0.4

2.5%

2.5 3.0 3.5

0.01

0.1

1

3%

4%

2.5% 3% 4%

L-L coexistence

L-L coexistence

Ho

mog

eneou

s N

ucle

ation R

ate

J [

cm

-3 s

-1]

Supersaturation s = In(C/C eq )

10.0oC

12.6oC

15.0oC

2.8 3.2 3.610

-2

10-1

100

Nu

cle

ati

on

Rate

J [cm

-3s

-1]

Supersaturation Dm / kBT

n* = 1

• Spinodal can be defined from n* 1

Solution-to-crystal Spinodal

L.F. Filobelo, et al., J. Chem. Phys. 123, 014904 (2005)

At and below spinodal DG* < kBT

0 100 200 300

-10

0

10

20

30

40

50

Tem

pera

ture

[oC

]

Concentration [mg/ml]

Solubility

Liquid-liquid (L-L) spinodal

L-L coexistence

Gelation line

J(T) reaches sharp max at solution-crystal spinodal

J(s) levels off at solution-crystal spinodal

Solution-to-crystal Spinodal

The Two-step Nucleation Mechanism

The Two-step Mechanism for Solution Crystallization

Protein Crystallization O. Galkin & P. G. Vekilov, PNAS 97, 6277 (2000)

P. G. Vekilov, Cryst. Growth & Design 4, 671 (2004)

L. Filobelo, et al.. J. Chem. Phys. 123, 014904 (2005)

Y.G. Kuznetsov, et al., J. Crystal Growth 232, 30 (2001)

Glycine, urea B. Garetz, et al., Phys. Rev. Lett. 89, 175501 (2002)

J.E. Aber, et al., Phys. Rev. Lett. 94, 145503 (2005) D.W. Oxtoby, Nature 420, 277 (2002)

Colloid crystals M. E. Leunissen, et al., Nature 437, 235 (2005)

J. R. Savage and A. D. Dinsmore, PRL 102, 198302 (2009) T. H. Zhang and X. Y. Liu, JPCB 111, 14001 (2007)

NaClO3

R.Y. Qian, G.D. Botsaris, Chem. Eng. Sci. 59, 2841 (2004)

Calcite nucleation in bulk solutions L. Gower, Chem. Rev. 108, 4551 (2008)

H. Coelfen, et al., Science 322, 1819 (2008) E. M. Pouget, et al., Science 323, 1455 (2009).

Clusters and HbS Polymer Nucleation

q(T) much stronger than R(T) contradicts 1-step nucleation and agrees with 2-step

Polymers are perpendicular to plane of polarization of polarized light

Dependencies of r, Vl and Nl of mesoscopic metastable clusters on C and T follow those of nucleated polymers

0

30

60

90

120

150

180

210

240

270

300

330

Time of Monitoring [min]

0 20 40 60

200

400

600

800

10oC 15

oC 20

oC

25oC 30

oC

Rad

ius

[n

m]

Clusters are precursors for polymer nuclei

O. Galkin, P.G. Vekilov, J. Mol. Biol. 336, 43 (2004) O. Galkin, et al., J. Mol. Biol. 365 425 (2007)

O. Galkin, et al., Biophys. J. 92, 902 (2007) P.G. Vekilov, Brit. J. Haematol. 139, 173 (2007)

0 5 10 15 20 250.0

0.2

0.4

0.6

q [s]

1/R

[s m

m-1

]

Aggregation Precedes Ordering in Biological Self-assembly

Hemoglobin assembly—from 2 a-chains, 2 b-chains and 4 heme-moieties after translocation a- and b-chains associate prior to folding K. Adachi, et al., J Biol Chem 277, 13415 (2002)

Hemes attach to a2b2 complex and then enter assigned slots G. Vasudevan, M. J. McDonald, Curr Protein Pept Sci 3, 461 (2002)

Nucleation of prion-protein fibers—via a disordered toxic fluid-like cluster

R. Krishnan, S. L. Lindquist, Nature 435, 765 (2005) A. Lomakin, et al., Proc. Natl. Acad. Sci. USA 93, 1125 (1996)

E. H.Koo, et al., Proc. Natl. Acad. Sci. USA 96, 9989 (1999)

Molten Oligomer Nucleus Fibril

The Two-step Mechanism

What are the precursors?

What are the consequences

for the nucleation kinetics?

Does it offer new “handles” for control?

Concentration

Str

uctu

re

…

The Two-step Nucleation Mechanism

s

olu

tion

d

ense

liq

uid

cr

yst

als

Fre

e E

ne

rgy G

Nucleation Reaction Coordinate Protein Concentration

Te

mp

era

ture

Solubility

Gelation

Binodal

Spinodal

*2GD

*1GD

0

LLG D

The Two-step Mechanism below the L-L Coexistence Line

Vivares, D.; et al., Acta Cryst. D 2005, 61, 819

Crystals do not nucleate inside dense liquid droplets!

.

10

0 µ

m

t = 4 h t = 6 h t = 8 h t = 10 h

t = 12 h t = 14 h t = 16 h

The Two-step Mechanism below the L-L Coexistence Line

The Two-step Nucleation Mechanism

s

olu

tion

d

ense

liq

uid

cr

yst

als

*2GD

*1GD

0

CGD

Fre

e E

ne

rgy G

Nucleation Reaction Coordinate Protein Concentration

Te

mp

era

ture

Solubility

Gelation

Binodal

Spinodal

?

*2GD

*1GD

0

LLG D

Atomic force microscopy, lumazine synthase

Evidence for Mesoscopic Clusters

O. Gliko, et al., JACS 127, 3433 (2005)

0

100

200

0 2.5 5 7.5 10 Surface Coordinate [mm]

He

igh

t [n

m]

0

100

200

0 2.5 5 7.5 10 Surface Coordinate [mm]

120 nm 75 nm

Evidence for Mesoscopic Clusters Dynamic light scattering, deoxy-HbS

10-4

10-2

100

102

104

0.0

0.5

1.0

Delay Time t [ms]

g2(t) – 1 G(t)

g2(t) - 1 = (G()exp(-t)d)2

W. Pan, et al., Biophys. J. 92, 267 (2007)

The Angular Dependence

q2 [mm-2] 0 200 400 600

0.0

0.5

1.0

1.5

2.0

2.5

Cluster peak

Deca

y r

ate

2 [m

s] q = 4pn/l sin(q/2)

q– scattering angle For freely diffusing clusters 2 = Dq2

Loose network is anisotropic environment 2 ≠ Dq2

Freely-diffusing clusters 2 = Dq2

Their size – from Einstein-Stokes law D = kT / 6pR2

Brownian Microscopy

10 1000

5

10

Exp. 1

Exp. 2

Exp. 3

Cluster Radius R2 [nm]

Co

nc

en

tra

tio

n n

2 [

10

7 c

m-3

]

Cluster lifetime is O(10 s) Reproduces results on R2 and n2 from dynamic

light scattering

50 mm

Y. Li, et al., Rev. Sci. Instr. 82, 053106 (2011)

Evidence for Mesoscopic Clusters in Protein Solutions

1E-4 0.01 1 1000.0

0.3

0.6

0.9

oxy HbS

C=169.6 mg/ ml

deoxy HbS

C=131.2 mg/ml

Delay Time t [ms]

g2(t

) – 1

HbS

molecules

Clusters

Dynamic light scattering determinations

Rad

ius

[n

m]

0 50 100 150

100

1000

deoxy-HbS 67.2 mg/ml

131.2

Time of Monitoring [min]

0 50 100 150

400

800

Lysozyme78 mg ml-1, 20 mM HEPES, pH = 7.8

LuSy 8.1 mg/mL 1.3 M phosphate pH = 8.7

0.15 M phosphate pH = 7.2

Cluster size n100 nm Cluster lifetime ~10 s

O. Gliko, et al., J. Amer. Chem. Soc. 127 (2005) 3433

Steady volume 10-8—10-3 V

0 50 100 150 200 250

10

100

Phenomenological Theory of Two-step Nucleation

0 1 2

t– mean first-passage time

J = t-1 , 2 – rate-limiting

)(

1

)()(

)(

)(

1

220

1

0 TuTuTu

Tu

Tut

D

D

Tk

G

U

UTC

Tk

GTCk

J

B

liquid

B

0

0

11

*

212

exp1),(

)exp(

TkECkC B/expexp1 110

D

2

2

21)(

**

2

spe

e

e TT

TT

TT

ETG

Single adjustable k2 reproduces 3 complex kinetic curves

W. Pan, et al., J. Chem. Phys. 122, 174905 (2005)

276 280 284 288 292 296 3000.0

0.1

0.2

0.3

0.4

0.5

Nu

cle

ati

on

Rate

J [

cm

-3s

-1]

Temperature T [K]

80 mg ml-1

50 mg ml-1

20 30 40 50 600.0

0.1

0.2

Nu

cle

ati

on

Rate

J [cm

-3s

-1]

Concentration C [mg ml-1]

T = 12.6 oC

Nucleation barrier on approach to spinodal

Viscosity inside dense liquid

D

D

Tk

G

U

UTC

Tk

GTCk

J

B

liquid

B

0

0

11

212

exp1),(

)exp(

The Pre-exponential Factor in the Nucleation Rate Law

)exp(*

0Tk

GJJ

B

D

From experiments: J0 ~ 1010 cm-3s-1

From classical theory

J0 ~ 1018 cm-3s-1

From phenomenological theory:

fractionvolumeclusterD

,1

)exp(

0

0

1

Tk

G

U

U

B

liquid

0 50 100 150 200 250

10-7

Vo

lum

e F

rac

tio

n

Time of Monitoring [min]

Low J0—due to nucleation within clusters

inside clusters 10 cP -1 108

Explains low Jo

Fe

O-

O-

O

O

N

N-

N-

N

CH3

CH3

CH3

CH3

CH2

CH2

H H

H H

(a)

(b)

So What: the Heme and Sickle Cell Polymerization

0 30 60 90 12010

-9

10-7

10-5

10-3

67 mg ml-1

96 121

121, 66 mM 131 178

Vo

lum

e F

rac

tio

n

2Time t [min]

Dependence of Cluster Volume Fraction on HbS Concentration

2 for hemoglobin S

Increases with C from 67 to 96 mg ml-1

Similar at 96, 121, 131, and 178 mg ml-1 as per theory

Heme addition:

2 higher by 80× The nucleation rate ??

60 80 100 120 140 160 180

10-6

10-5

10-4

10-3

Concentration C [mg ml-1]

Av

ara

ge

Vo

lum

e F

rac

tio

n

107

108

10

100

5 10 15 20 25 30 35

0.1

1

10

Nu

cle

ati

on

Rate

J [cm

-3s

-1]

Dela

y T

ime

q [s]

290 mg ml-1

, 0.16 mM

271, 0.26 267, 0.28

280 mg ml-1

;

265; 245

230 mg ml-1

; 220; 210; 201

Temperature T [oC]

Gro

wth

Rate

R [

mm

s-1]

The Nucleation Rate

In the presence of heme, J is ~ 100× faster

0

5

10

15

Mw

K/R

q

0 100 200 300 400-10

-8

-6

-4

-2

0

Protein Mass Concentration [mg ml-1]

Dg

/kBT

The Free-energy Excess in the Clusters

D𝐺𝑐𝑙𝑢𝑠𝑡𝑒𝑟𝑠 ~ 10 kBT

The clusters must consist of another chemical species

Protein oligomer!!

K /Rq ( /) /RT

)( VdVGH

L

DD

W. Pan, et al., J. Phys. Chem. B 114 (2010)

Microscopic Scenario of Cluster Formation

Clusters consist of mixture of monomers and oligomers

Cluster radius R is determined by the decay rate and diffusivity of the oligomers: R (D2/k2)

1/2

Cluster radius does not depend on concentration

The Oligomer Mechanism: Hydration

Separation [Å]

Fre

e E

ne

rgy [

kJ/

mol]

2 4 6 8 10 12 14

0

-40

40

80

10-5

10-3

10-1

101

103

10-7

10-5

10-3

10-1

2(t

)

t*sin2(q2)[ms]

150 mg ml-1

20 mM Hepes, pH 7.8

0.1M acetate, pH 4.5

20 mM phosphate, pH 7.8

Water structuring leads to secondary minima in interaction between nanoscopic solutes

Small ions, i.g., HPO42- , are known to disturb the hydration

shell

10-5

10-3

10-1

101

103

10-7

10-5

10-3

10-1

80 mg ml-1

No urea

0.2 M urea

0.5 M

1 M

2(t

)

t*sin2(q/2) [ms]

10-5

10-3

10-1

101

103

t*sin2(q/2) [ms]

150 mg ml-1

No urea

0.2 M urea

0.5 M

1 M

0 100 200 300 400 500

-10

-8

-6

-4

-2

0

5

10

0.5 M urea

no urea

DG

/Nk

BT

Lysozyme Concentration [mg ml-1]

0.5 M urea

Mw

K/R

q

Attraction between solvent exposed hydrophobic residues, “domain swapping”

Urea can be used to control degree of unfolding

The Oligomer Mechanism: Partial Unfolding

The Oligomer Mechanism: Partial Unfolding

peaks that exchange faster in concentrated than dilute solution.

normal H lysozyme dissolved in D2O 1H-15N HSQC spectrum (natural 15N abundance)

dilute (no clusters) and concentrated (with clusters) solutions

Robert Fox, Kevin McKenzie

UH-Biochemsitry

The Oligomer Mechanism: Partial Unfolding

Summary

A spinodal for the solution-to-crystal phase transition exists

The nucleation barrier in the vicinity of the spinodal is negligible

The nucleation rate reaches saturation or a maximum at the spinodal

Assembly of ordered arrays

crystals, oligomers, fibers, etc.

is preceded by association into disordered clusters

The precursor is a metastable mesoscopic liquid cluster

or a stable dense liquid droplet

The low volume fraction of the nucleation precursors delays nucleation by ~ 1010

Understanding and control of nucleation in solution requires insights into

the solution physical chemistry

So What?

Clusters are needed for nucleation of crystals. To enhance clusters:

moderate intermolecular attraction or repulsion

Partial protein conformational variability—necessary!

Crystal nucleation occurs in a spinodal regime

g is not important

Supersaturation increase--inconsequential