Kinetics and Thermodynamics of Multistep Nucleation and Self-assembly in Nanosize Materials

Brussels, March 25-26, 2010

The Two-step Mechanism of Nucleation of Crystals from Solution

Peter G. Vekilov, Oleg Galkin, Luis Filobelo, Weichun Pan, Anatoly Kolomeisky (Rice), Dimo Kashchiev (IPC), Vas Lubchenko Department of Chemical and Biomolecular Engineering,Department of Chemistry, University of Houston

NIH, NASA, NSF, Welch Foundation

The Goal: Crystals with “Just-right”…

Number PolymorphMorphologyHabitSizeSize distribution

Requires data on:

Solution PChem Phase

diagrams Metastable

statesNucleation

mechanismsGrowth

mechanismsAgglomeration…

Crystallization

Nucleation

Crystallization and Nucleation

Crystallization

Nucleation

Crystallization and Nucleation

Crystallization as Sequential Transition along Two Order Parameters

Concentration

Str

uct

ure

…

Classical viewpoint: direct nucleation along a “diagonal line” envisioned;

Protein Concentration

Tem

per

atu

re

Solubility

Gelation

Binodal

Spinodal

Two-step mechanism: suggested by t W & F, T & O for critical point for L-L phase separation for proteins

Everywhere else in phase diagram—classical crystal nucleation predicted P.R. ten Wolde, D. Frenkel,Science 277 (1997) 1975

V. Talanquer, D.W. Oxtoby, J. Chem. Phys. 109 (1998) 223

The Two-step MechanismConcentration

Str

uct

ure

…

It operates in all areas of the phase diagram

It may apply to all crystals (and other ordered solids) forming in solution

Protein Concentration

Tem

per

atu

re

Solubility

Gelation

Binodal

Spinodal

Galkin, O. & Vekilov, P. G. (2000) Proc. Natl. Acad. Sci. USA 97, 6277

Vekilov, P. G. (2004) Crystal Growth and Design 4, 671

5 10 15 20 25 300

0.2

0.4

TL-L

TL-L

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

Temperature T [°C]Hom

og

en

eou

s N

ucle

ati

on

Rate

J [

cm-3 s

-1]

The Nucleation Rate

Maximum in J(T) Exponential increase at intermediate DT’s; by weak decrease at

higher DT’s T of maximum shifts with concentration

Protein Concentration

Tem

per

atu

re

Solubility

Gelation

Binodal

Spinodal

Two steps:Which One is Rate Determining

Critical level of ordering

Critical size of ordered nucleus

Concentration

Str

uct

ure

…

*2G

*2G

*1G

*1G

s

olut

ion

de

nse

liqui

d

cr

ysta

ls

DGL-L

DGL-L

Fre

e E

ner

gy

G

Nucleation Reaction Coordinate

Rate of cluster formationJ1 ~ J01exp(– /kBT)

Rate of nucleation within clustersJ2 ~ J02exp(– / kBT)

Is > ?

Is J01 more important than ?

Is J02 more important than ?

*1G

*2G

*1G *

2G

*1G

*2G

Nucleation of Dense Liquid Droplets

3.53 s 10.58 s6.09 s

0.96 s 1.92 s 7.37 s

T – TL-L = 0.7 oC

T – TL-L = 1.3 oC

• Number of droplets increases with time• Faster nucleation at higher DT’s Characteristics of nucleation regime of droplet generation

Nucleation Rate of Dense Phase Droplets

Time [s]

Dro

ple

ts in

Vie

wfi

eld

0 2 4 6 8 100

100

200

300

400

Nucleation Rate = 4.3 x 109 cm-3s-1

• Number of droplets increases in time—nucleation regime

• Nucleation rate ~109 cm-

3s-1

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

3s-1

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

Structuring of dense liquid quasi-droplet is the rate determining stage Equilibrium between solution and clusters: msolution = mclusters Dm(solution,crystal) = Dm(clusters,crystal)

10.0oC12.6oC15.0oC

2.8 3.2 3.610-2

10-1

100

Nuc

leat

ion

Rat

e J

[cm

-3s-1

]Supersaturation / k

BT

n* = 1

• Spinodal can be defined from n* 1

• Spinodal – boundary between metastable and unstable two – phase areas

Other proteins: • Ferritin crystals grown at

s = 4.2, where n* 1 • Are protein crystals always

grown in spinodal regimes?

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

Why is the maximum in J(T) sharp?

20 mm

0 100 200 300

-10

0

10

20

30

40

50

Tem

per

atu

re [o C

]

Concentration [mg/ml]

Liquidus or solubility of crystals

L-L spinodal

L-L coexistenceSolution-crystalspinodal

Gelation line

Pre-exponential Factors and Barriers for Structuring

D.N. Petsev, et al., J. Phys. Chem. B 107 (2003) 3921

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

Why is the maximum in J(T) sharp?

0 100 200 300

-10

0

10

20

30

40

50

Tem

per

atu

re [o C

]

Concentration [mg/ml]

Liquidus

Liquid-liquid (L-L) spinodal

L-L coexistence

Gelation line

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

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

Tu

)exp(1

)exp(

0

1

22

TkG

UU

TkE

U

J

B

B

),( 1

122 TC

TCkU

TkECkC B/expexp1 110

2

2

21)(

*

2spe

e

e TT

TT

TT

ETE

276 280 284 288 292 296 3000.0

0.1

0.2

0.3

0.4

0.5

Nu

clea

tio

n R

ate

J [c

m-3s-1

]

Temperature T [K]

80 mg ml-1

50 mg ml-1

20 30 40 50 600.0

0.1

0.2

Nu

clea

tio

n R

ate

J [c

m-3s-1

]

Concentration C [mg ml-1]

Single adjustable k2 reproduces 3 complex kinetic curves

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

T = 12.6 oC

Nucleation barrier on approach to spinodal

Viscosity inside dense liquid

The Pre-exponential Factor in the Nucleation Rate Law

R.P. Sear, J. Phys. Chem. B 110 (2006) 21944

)exp(1

)exp(

0

1

22

TkG

UU

TkE

U

J

B

B

)exp(*

0 Tk

GJJ

B

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

From classical theoryJ0 ~ 1020

cm-3s-1

???

From phenomenological theory:

fractionvolumecluster,1

)exp(0

1

Tk

G

U

U

B

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

The Two-step Mechanism for Other Crystals

Glycine, ureaB. 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)

Charged colloid crystalsM. E. Leunissen, et al., Nature 437, 235

(2005)

NaClO3

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

NaCl nucleation from solution (MD simulation)

D. Zahn, Phys. Rev. Lett. 92, 040801 (2004)

Calcite nucleationL. Gower, Chem. Rev. 108, 4551 (2008)D. Gebauer, et al., Science 322, 1819 (2008)

Theoretical Justification of Generality of 2step MechanismJ. Lutsko, G. Nicolis, Phys. Rev. Lett., 96 (2006) 046102 For protein molecules

cba

d

ef

cba

d e f

For small molecules

Two-step barrier always lower than direct barrier

Reaction Coordinate

DG

/kBT

DG

/kBT

Clusters and HbS Polymer Nucleation

t=12 s n=1 t=15 s n=2 t=18 s n=3 t=21 s n=5

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 15oC 20oC

25oC 30oC

Rad

ius

[nm

] Clusters are precursors for polymer nucleiO. 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)

t=0 n=0

D. Kashchiev, et al.,J. Chem. Phys. 122 (2005) 244706

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 MechanismDensity

Str

uct

ure

… What are the precursors

above the L-L coexistence line?

What are the consequences for the nucleation kinetics?

Does it offer new “handles” for control?

Protein Concentration

Tem

per

atu

re

Solubility

Gelation

Binodal

Spinodal

What else may be precursor?

.

100

µm

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

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

Crystals Do Not Nucleate Within Liquid Droplets

Direct Observation of Clusters

0

100

200

0 2.5 5 7.5 10Surface Coordinate [m]

Hei

gh

t [n

m]

0

100

200

0 2.5 5 7.5 10Surface Coordinate [m]

lumazine synthase 3 mg/ml 1.3 M phosphate 24 C Clusters shrink in height:

Steps generated from clusters merge continuously with underlying lattice

O. Gliko, et al., J. Amer. Chem. Soc. 127 (2005) 3433 W. Pan, et al., Biophys. J. 92 (2007) 267

O. Gliko, et al., J. Phys. Chem. B 111 (2007) 3106

Clusters represent hidden dense liquid

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

Ra

diu

s[

nm

]

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 >15 ms to 10

s

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

Steady volume 10-8—10-3 V: not crystals

Fast decay rate indicates that clusters are hidden liquid

0 50 100 150 200 25010

100

The Angular Dependence

q2 [mm-2]0 200 400 600

0.0

0.5

1.0

1.5

2.0

2.5

Cluster peak

Dec

ay r

ate

G2

[ms]

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

q – scattering angle

For freely diffusing clustersG2 = Dq2

Loose network is anisotropic environmentG2 ≠ Dq2

Freely-diffusing clusters G2 = Dq2

Their size – from Einstein-Stokes law D = kT / 6R2

0

5

10

15

Mw

K/

R

0 100 200 300 400-10

-8

-6

-4

-2

0

Protein Mass Concentration [mg ml-1]

DG

/Nk B

T

The Free Energy Cost of Higher Concentration

DG ~ 10 kBT

Fraction of protein in clusters: exp(-10) = 4 × 10-5 an overestimate

Clusters must contain only a few molecules

Cluster lifetime must be O(tdiff) = 10 ms

K /R ( /) /RT

)( VdVGH

L

Clusters must contain a chemical species with slow decay rate

150 mg ml-1

10-7

10-5

10-3

10-1

80 mg ml-1

57o

77o

96o

116o

140o

10-5 10-3 10-1 101 10310-7

10-5

10-3

10-1

228 mg ml-1

10-5 10-3 10-1 101 103

334 mg ml-1

(a)

(c)

(b)

(d)

t sin2(q/2) [ms] t sin2(q/2) [ms]

2

2Open Question about the Clusters

Mesoscopic size

Macroscopic lifetimes

Cluster volume fraction less than suggested by equilibrium free energy

Cluster size independent of protein concentration

Clusters are fluid internally

Complicated dependence of cluster volume fraction on protein concentration

Cluster shoulder--non-exponential at high concentrations

Microscopic Scenario and the Cluster Radius

Clusters consist of off-equilibrium mixture of monomers and oligomers, kinetically stabilized

Cluster radius R is determined by the decay rate and diffusivity of the oligomers:

R (D2/k2)1/2

Since R ≃ 100 nm, oligomer lifetime

k2-1 10 ms

Cluster radius does not depend on concentration

The Cluster Lifetime

10-5 10-3 10-1 101 10310-7

10-5

10-3

10-1

228 mg ml-1

10-5 10-3 10-1 101 10310-7

10-5

10-3

10-1

*sin2(2) [ms]

334 mg ml-1

2

2

Cluster size fluctuates with characteristic time

The imprint of these fluctuations on the correlation functions scales with q2

t ≈ 10 msis the lower bond of cluster lifetime

2 R2

D1

The Oligomer Mechanism: Hydration

Separation [Å]

Fre

e E

nerg

y [

kJ/m

ol]

2 4 6 8 10 12 14

0

-40

40

80

10-5 10-3 10-1 101 10310-7

10-5

10-3

10-1

2()

*sin2(2)[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 10310-7

10-5

10-3

10-180 mg ml-1

No urea 0.2 M urea 0.5 M 1 M

2

()

*sin2(/2) [ms]10-5 10-3 10-1 101 103

*sin2(/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

G/N

kB

T

Lysozyme Concentration [mg ml-1]

0.5 M urea

Mw

K/

R

Attraction between solvent exposed hydrophobic residues, “domain swapping”

Urea can be used to control degree of unfolding

The Oligomer Mechanism: Domain Swapping

Summary and Conclusions

Assembly of ordered arrays crystals, oligomers, fibers, etc.

is preceded by association into disordered clusters

The precursor is a metastable mesoscopic liquid cluster

Rate of crystal nucleation is determined by structuring of dense quasi-droplet

Polymorph selection is determined by kinetics factors rather than by high barriers

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

Understanding and control of nucleation in solution requires insights into the solution physicochemical mechanism nano- and

mesostuctures

So What?

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

moderate intermolecular attraction or repulsionproper water structure around the protein molecules

Crystal nucleation occurs in a spinodal regime

g is not importantSimpler picture of nucleation and role of additives

Heterogeneous particles may affect polymorph selection via structural similarity