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