Gibbs, Volmer, and Crystal Nucleation

Transcription

1 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

2 KDP ~ 20 cm crystal grows in ~ 1 day "FAST GROWTH Method Sets Crystal Size Record," LASER FOCUS WORLD, July 1999 Cover Crystal from LLNL

3 Ferritin ~ 700 mm crystal grows in ~ 1 month IPC, Sofia

4 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

5 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

6 Crystals of Hemoglobin C in Red Blood Cells Erythrocytes from HbC Transgenic Mice crystallization induced by 4 hour incubation in 3% NaCl, 37 o C crystal dissolution induced by addition of 0.09 M NaCl solution J. E. Canterino, et al., Biophys. J. 95, 4025 (2008). 5 s original = 0.1 s as played

7 Nucleation and the Outcome of Crystallization 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.

8 Crystallization is a Miracle!

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10 Crystallization Crystallization and Nucleation Nucleation

11 Josiah Willard Gibbs ( ) Only Jesus and Gibbs never erred and this has been strictly shown for Gibbs only E.D. Shchukin

12 Classical Nucleation Theory: Thermodynamics Solution supersaturated: m soln > m crystal, Dm = m soln m crystal > 0 n molecules from solution into a crystalline cluster Gibbs JW On the equilibrium of heterogeneous substances Trans. Connect. Acad. Sci. (1876) 3, 108; (1978) 16, 343 Free energy gain = - ndm Free energy loss = 6gn 2/3 creation of new surface DG 6gn 2/3 DG(n) = - ndm + 6gn 2/3 n = 64γ3 2 Dm 3 DG * n * DG(n) n G = 32γ3 2 Dm 2 - ndm

13 Max Volmer ( )

14 Classical Nucleation Theory: Kinetics J = J o exp G k B T J Volmer M (1939) Kinetik der Phasenbildung (Steinkopff, Dresden) Assumes that nuclei are perfect crystals Predicts steep J o = n * Zn Dm n * frequency of attachment, 300 s -1 Z Zeldovich factor, 0.01 n concentration of molecules, = cm -3 J o = cm -3 s -1

15 Homogeneous Nucleation Rate J [cm -3 s -1 ] Mysteries of Protein Nucleation Kinetics Sharp leveling of J(Dm) O. Galkin et al., JACS 122, 156 (2000) Large data scatter Nucleation rate lower by 10 8 then prediction of CNT * DG J J0 exp( ) k T B J 0 (experiment) ~ cm -3 s -1 J 0 (CNT) ~ cm -3 s T L - L C lys = 50 mg/ml C lys = 80 mg/ml Non-monotonic behavior of J(T) 0. 2 O. Galkin & P. G. Vekilov, PNAS 97, 6277 (2000) T L - L T e m p e r a t u r e T [ C ]

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17 Number of Droplets Nucleation Rate of Dense Phase Droplets Increase of droplet number at DT = 0.4 K 400 M. Shah, et al., J. Chem. Phys. 121 (2004) Nucleation Rate = 4.3 x 10 9 cm -3 s -1 J ~10 9 cm -3 s -1 significantly higher than rates of crystal nucleation ~ cm -3 s -1 J o = cm -3 s Time [s]

18 Homogeneous Nucleation Rate J [cm -3 s -1 ] Determination of Nucleus Size % 3% 4% L-L coexistence Nucleation theorem n * D. Kashchiev, D. Oxtoby JCP 100 (1994) 7665 d(lnj) O(1) d( Dm / k T ) B allows determination of nuclei sizes (10 4 1) 2.5% 3% 4% O. Galkin, P.G. Vekilov, JACS 122 (2000) L-L coexistence Supersaturation s = In(C/C eq )

19 Solution-to-crystal Spinodal Nucleation Rate J [cm -3 s -1 ] o C 12.6 o C 15.0 o C n * = 1 At and below spinodal DG * < k B T L.F. Filobelo, et al., J. Chem. Phys. 123, (2005) Supersaturation Dm / k B T Spinodal can be defined from n * 1

20 Solution-to-crystal Spinodal Solubility Gelation line J(T) reaches sharp max at solution-crystal spinodal Temperature [ o C] L-L coexistence J(s) levels off at solution-crystal spinodal -10 Liquid-liquid (L-L) spinodal Concentration [mg/ml]

21 The Two-step Nucleation Mechanism

22 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, (2005) Y.G. Kuznetsov, et al., J. Crystal Growth 232, 30 (2001) Glycine, urea B. Garetz, et al., Phys. Rev. Lett. 89, (2002) J.E. Aber, et al., Phys. Rev. Lett. 94, (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, (2009) T. H. Zhang and X. Y. Liu, JPCB 111, (2007) NaClO 3 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).

23 1/R [s mm -1 ] Radius [nm] 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, V l and N l of mesoscopic metastable clusters on C and T follow those of nucleated polymers Clusters are precursors for polymer nuclei O. Galkin, P.G. Vekilov, J. Mol. Biol. 336, 43 (2004) O. Galkin, et al., J. Mol. Biol (2007) O. Galkin, et al., Biophys. J. 92, 902 (2007) P.G. Vekilov, Brit. J. Haematol. 139, 173 (2007) q [s] o C 15 o C 20 o C 25 o C 30 o C Time of Monitoring [min]

24 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, (2002) Hemes attach to a 2 b 2 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 Molten Oligomer Nucleus Fibril 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)

25 Structure The Two-step Mechanism Concentration What are the precursors? What are the consequences for the nucleation kinetics? Does it offer new handles for control?

26 Temperature Free Energy G solution dense liquid crystals The Two-step Nucleation Mechanism Solubility Gelation Binodal * DG 1 * DG 2 Spinodal Protein Concentration 0 DGL L Nucleation Reaction Coordinate

27 The Two-step Mechanism below the L-L Coexistence Line Vivares, D.; et al., Acta Cryst. D 2005, 61, 819

28 The Two-step Mechanism below the L-L Coexistence Line 100 µm t = 4 h t = 6 h t = 8 h t = 10 h Crystals do not nucleate inside dense liquid droplets! t = 12 h t = 14 h t = 16 h

29 Temperature Free Energy G solution dense liquid crystals The Two-step Nucleation Mechanism Solubility * DG 1? * DG 2 0 DG C Gelation Binodal * DG 1 * DG 2 Spinodal Protein Concentration 0 DGL L Nucleation Reaction Coordinate

30 Height [nm] Evidence for Mesoscopic Clusters Atomic force microscopy, lumazine synthase nm 75 nm 100 O. Gliko, et al., JACS 127, 3433 (2005) Surface Coordinate [mm] Surface Coordinate [mm]

31 Evidence for Mesoscopic Clusters Dynamic light scattering, deoxy-hbs g 2 (t) 1 G(t) g 2 (t) - 1 = ( G( )exp(- t)d ) Delay 10 0 Time t [ms] W. Pan, et al., Biophys. J. 92, 267 (2007)

32 Decay rate 2 [ms] The Angular Dependence Cluster peak q = 4pn/l sin(q/2) q scattering angle q 2 [mm -2 ] For freely diffusing clusters 2 = Dq 2 Loose network is anisotropic environment 2 Dq 2 Freely-diffusing clusters 2 = Dq 2 Their size from Einstein-Stokes law D = kt / 6p R 2

33 Concentration n 2 [10 7 cm -3 ] Brownian Microscopy 10 Exp. 1 Exp. 2 Exp mm Cluster Radius R 2 [nm] Cluster lifetime is O(10 s) Reproduces results on R 2 and n 2 from dynamic light scattering Y. Li, et al., Rev. Sci. Instr. 82, (2011)

34 Radius [nm] g 2 (t) 1 Evidence for Mesoscopic Clusters in Protein Solutions Dynamic light scattering determinations 0.9 oxy HbS C=169.6 mg/ ml deoxy HbS 0.6 C=131.2 mg/ml 0.3 HbS molecules Clusters 0.0 1E Delay Time t [ms] Cluster size n 100 nm Cluster lifetime ~10 s Steady volume V 100 Lysozyme78 mg ml -1, 20 mm HEPES, ph = LuSy 8.1 mg/ml 1.3 M phosphate ph = deoxy-hbs 67.2 mg/ml M phosphate ph = Time of Monitoring [min] O. Gliko, et al., J. Amer. Chem. Soc. 127 (2005) 3433

35 Phenomenological Theory of Two-step Nucleation t 1 u ( T) u t mean first-passage time J = t -1, 2 rate-limiting u1( T) ( T) u ( T) u 1 ( T * DG2 k2c1 T exp( ) kbt J 0 U DG 1 ( C T 1, ) 1 exp U0 kbt Viscosity inside dense liquid liquid C exp k C exp E / k T B ) Nucleation Rate J [cm -3 s -1 ] mg ml mg ml Temperature T [K] T = 12.6 o C Nucleation barrier on approach to spinodal DG * 2 ( T) e E * T T 2 T 1 T e e T T sp 2 2 Single adjustable k 2 reproduces 3 complex kinetic curves W. Pan, et al., J. Chem. Phys. 122, (2005) Nucleation Rate J [cm -3 s -1 ] Concentration C [mg ml -1 ]

36 The Pre-exponential Factor in the Nucleation Rate Law U U J J 1 0 J 0 * DG exp( ) k T From experiments: J 0 ~ cm -3 s -1 From classical theory J 0 ~ cm -3 s -1 k 2 B From phenomenological theory: C 1 ( C1, T ) 1 0 DG exp( k T liquid B DG2 T exp( ) k T U U 1 0 B 0 DG exp kbt liquid Volume Fraction ) cluster volume fraction, Time of Monitoring [min] inside clusters 10 cp Low J 0 due to nucleation within clusters Explains low J o

37 So What: the Heme and Sickle Cell Polymerization (a) O - O CH 3 O - O H H N N - Fe N - CH 3 H 3 C H N H CH 2 (b) CH 2 CH 3

38 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?? Volume Fraction Avarage Volume Fraction 67 mg ml , 66 mm Time t [min] Concentration C [mg ml -1 ]

39 The Nucleation Rate In the presence of heme, J is ~ 100 faster Nucleation Rate J [cm -3 s -1 ] Delay Time q [s] Growth Rate R [mm s -1 ] mg ml -1, 0.16 mm 271, , mg ml -1 ; 265; mg ml -1 ; 220; 210; Temperature T [ o C]

40 Dg/k B T The Free-energy Excess in the Clusters K /R q ( / )/RT DG H L dv D( V ) DG clusters ~ 10 k B T The clusters must consist of another chemical species M w K /R q Protein oligomer!! W. Pan, et al., J. Phys. Chem. B 114 (2010) Protein Mass Concentration [mg ml -1 ]

41 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 (D 2 /k 2 ) 1/2 Cluster radius does not depend on concentration

42 Free Energy [kj/mol] The Oligomer Mechanism: Hydration Separation [Å] 2 (t) mg ml mm Hepes, ph M acetate, ph mm phosphate, ph t*sin 2 (q 2)[ms] Water structuring leads to secondary minima in interaction between nanoscopic solutes Small ions, i.g., HPO 4 2-, are known to disturb the hydration shell

43 The Oligomer Mechanism: Partial Unfolding mg ml No urea 0.2 M urea 0.5 M 1 M 150 mg ml -1 No urea 0.2 M urea 0.5 M 1 M 2 (t) t*sin 2 (q/2) [ms] Attraction between solvent exposed hydrophobic residues, domain swapping Urea can be used to control degree of unfolding M w K /R q 10 5 t*sin2(q/2) [ms] 0.5 M urea 0.5 M urea no urea DG/Nk B T Lysozyme Concentration [mg ml -1 ]

44 The Oligomer Mechanism: Partial Unfolding normal H lysozyme dissolved in D 2 O 1 H- 15 N HSQC spectrum (natural 15 N abundance) dilute (no clusters) and concentrated (with clusters) solutions peaks that exchange faster in concentrated than dilute solution. Robert Fox, Kevin McKenzie UH-Biochemsitry

45 The Oligomer Mechanism: Partial Unfolding

46 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 ~ Understanding and control of nucleation in solution requires insights into the solution physical chemistry

47 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