Aggregation Effects - Spoilers or Benefactors of Protein Crystallization ? Adam Gadomski Institute of Mathematics and Physics University of Technology and Agriculture Bydgoszcz, Poland Berlin – September 2004

Plan of talk: 1. CAST OF CHARACTERS – a microscopic view: I. Crystal growth - a single-nucleus based scenario A. (Protein) Cluster-Cluster Aggregation – a short overview in terms of its microscopic picture B. Microscopic scenario associated with (diffusive) Double Layer formation, surrounding the protein crystal II. Crystal growth – a polynuclear path C. Smectic-pearl and entropy connector model (by Muthukumar) applied to protein spherulites 2. CAST OF CHARACTERS – a mesoscopic view: I. Crystal growth - a single-nucleus based scenario II. Crystal growth – a polynuclear path A. (Protein) Cluster-Cluster Aggregation – a cluster-mass dependent construction of the (cooperative) diffusion coefficient B. Fluctuational scenario associated with (diffusive) Double Layer formation – fluctuations within the protein (protein cluster) velocity field nearby crystal surface C. Protein spherulites’ formation – a competition-cooperation effect between biomolecular adsorption and “crystallographic registry” effects (towards Muthukumar’s view)

3. An attempt on answering the QUESTION: "Protein Aggregation - Spoiler or Benefactor in Protein Crystallization?" A. What do we mean by ‘Benefactor’: Towards constant speed of the crystal growth B. When ‘Spoiler’ comes? Always, if … it is not a ‘Benefactor’ 4. Conclusion and perspective Plan of talk (continued):

TO DRAW A (PROTEIN) CLUSTER-CLUSTER AGGREGATION* LIMITED VIEW OF PROTEIN CRYSTAL GROWTH __________ * Usually, an undesirable aggregation of (bio)molecules is proved experimentally to be a spoiling side effect for crystallization conditions OBJECTIVE:

N\** Relevant Variable* Dynamics N>1 Bo-Gi-On Protein crystallite’s individual volume – a stochastic variable v Thermodynamic potentials, and ‘forces’, a presence of entropic barriers N=1 Fr-Ste-Po Crystal radius RFluctuating protein velocity field – (algebraic) in-time- correlated fluctuations (Stokes- Langevin type) Sm-Ki-StCluster mass M (Flory-Huggins polymer-solution interaction parameter) Stochastic (e.g., Poisson) process N( t ), and its characteristics Legend to Table: Bo-Gi-On: Boltzmann-Gibbs-Onsager Sm-Ki-St : Smoluchowski-Kirkwood-Stokes Fr-Ste-Po: Frenkel-Stern-Poisson Routes of modeling – a summary

Effect of chain connectivity on nucleation [from: M. Muthukumar, Advances in Chemical Physics, vol. 128, 2004]

(A)aggregation on a single seed in a diluted solution, (B) agglomeration on many nuclei in a more condensed solution Matter aggregation models, leading to (poly)crystallization in complex entropic environments:

PIVOTAL ROLE OF THE DOUBLE LAYER (DL): Cl - ion DOUBLE LAYER surface of the growing crystal Na + ion water dipole Lysozyme protein random walk

Growth of smectic pearls by reeling in the connector (N = 2000). [from: M. Muthukumar, Advances in Chemical Physics, vol. 128, 2004]

GROWTH OF A SPHERE: mass conservation law (MCL)

EMPHASIS PUT ON A CLUSTER – CLUSTER MECHANISM: geometrical parameter (fractal dimension) interaction (solution) parameter of Flory-Huggins type - initial cluster mass - time- and size- dependent diffusion coefficient - characteristic time constant

MODEL OF GROWTH: emphasis put on DL effect Under assumptions [ A.G., J.Siódmiak, Cryst. Res. Technol. 37, 281 (2002) ]: (i) C=const (ii) The growing object is a sphere of radius: ; (iii) The feeding field is convective: ; (iv) The generalized Gibbs-Thomson relation: where: ; (curvatures !) and when (on a flat surface) : thermodynamic parameters i=1 capillary (Gibbs-Thomson) length i=2 Tolman length Growth Rule (GR) additional terms

DL-INFLUENCED MODEL OF GROWTH (continued, a.t. neglected): specification of and For A(R) from r.h.s. of GR reduces to For nonzero  -s: R~t is an asymptotic solution to GR – constant tempo ! velocity of the particles nearby the object Could v(R,t) express a truly mass-convective nature? What for?  - supersaturation dimensionless parameter;

DL-INFLUENCED MODEL OF GROWTH: stochastic part where Assumption about time correlations within the particle velocity field [see J.Łuczka et al., Phys. Rev. E 65, (2002)] K – a correlation function to be proposed; space correlations would be a challenge... Question: Which is a mathematical form of K that suits optimally to a growth with constant tempo?

DL-INFLUENCED MODEL OF GROWTH: stochastic part (continued) Langevin-type equation with multiplicative noise: Fokker-Planck representation: with and (Green-Kubo formula), with corresponding IBC-s

THE GROWTH MODEL COMES FROM MNET (Mesoscopic Nonequilibrium Thermodynamics, Vilar & Rubi, PNAS 98, (2001)) : a flux of matter specified in the space of cluster sizes where the energy (called: entropic potential) and the diffusion function The matter flux: Most interesting:(dispersive kinetics !) Especially, for readily small  it indicates a superdiffusive motion !

DL-INFLUENCED SCENARIO: when a.t. stands for an elastic contribution to the surface-driven crystal growth (  2 =0) Example:  =1 (1D case): c s (R)=c 0 (1 +  1 K 1 + y 1 ), where y 1 =  1 L eff ; here L eff =  y (1) =(L-L 0 )/L 0, L and L 0 are the circumferences of the nucleus at time t and t 0 respectively. In the case of (ideal) spherical symmetry we can write that y 1 =  1 (R-R 0 )/R 0. - positive or negative (toward auxetics) elastic term - specify different elastic-contribution influenced mechanisms linear (  =1), surfacional (  =2) or volumetric (  =3) - positive or negative dimensionless and system-dependent elastic parameter, involving e.g. Poisson ratio - elastic dimensionless displacement

POLYNUCLEAR PATH GRAIN (CLUSTER)-MERGING MECHANISM

TYPICAL 2D MICROSTRUCTURE: VORONOI-like MOSAIC FOR A TYPICAL POLYNUCLEAR PATH INITIAL STRUCTURE FINAL STRUCTURE

RESULTING FORMULA FOR VOLUME-PRESERVING d-DIMENSIONAL MATTER AGGREGATION – case A time derivative of the specific volume (inverse of the polycrystal density) hypersurface inverse term adjusting time- dependent kinetic prefactor responsible for spherulitic growth: it involves order- disorder effect

ADDITIONAL FORMULA EXPLAINING THE MECHANISM (to be inserted in continuity equation) - hypervolume of a single crystallite - independent parameters drift termdiffusion term surface - to - volume characteristic exponent scaling: holds !

AFTER SOLVING THE STATISTICAL PROBLEM is obtained USEFUL PHYSICAL QUANTITIES: TAKEN USUALLY FOR THE d-DEPENDENT MODELING where

AGAIN: THE GROWTH MODEL COMES FROM MNET - hypervolume of a single cluster (internal variable) Note: cluster surface is crucial! drift term diffusion term surface - to - volume characteristic exponent scaling: holds ! - independent parameters

GIBBS EQUATION OF ENTROPY VARIATION AND THE FORM OF DERIVED POTENTIALS (FREE ENERGIES) AS ‘STARTING FUNDAMENTALS’ OF CLUSTER-CLUSTER LATE- TIME AGGREGATION -internal variable and time dependent chemical potential -denotes variations of entropy S and (and f-unnormalized) (i) Potential for dense micro-aggregation (for spherulites): (ii) Potential for undense micro-aggregation (for non-spherulitic flocks):

 THERE ARE PARAMETER RANGES WHICH SUPPORT THE AGGREGATION AS A RATE-LIMITING STEP, MAKING THE PROCESS KINETICALLY SMOOTH, THUS ENABLING THE CONSTANT CRYSTALLIZATION SPEED TO BE EFFECTIVE (AGGREGATION AS A BENEFACTOR)  OUTSIDE THE RANGES MENTIONED ABOVE AGGREGATION SPOILS THE CRYSTALLIZATION OF INTEREST (see lecture by A.Gadomski)  ESPECIALLY, MNET MECHANISM SEEMS TO ENABLE TO MODEL A WIDE CLASS OF GROWING PROCESSES, TAKING PLACE IN ENTROPIC MILIEUS, IN WHICH MEMORY EFFECTS AS WELL AS NON-EXTENSIVE ‘LIMITS’ ARE THEIR MAIN LANDMARKS CONCLUSION & PERSPECTIVE

LITERATURE: -D.Reguera, J.M.Rubì; J. Chem.Phys. 115, 7100 (2001) - A.Gadomski, J.Łuczka; Journal of Molecular Liquids, vol. 86, no. 1-3, June 2000, pp J.Łuczka, M.Niemiec, R.Rudnicki; Physical Review E, vol. 65, no. 5, May 2002, pp /1-9 - J.Łuczka, P.Hanggi, A.Gadomski; Physical Review E, vol. 51, no. 6, pt. A, June 1995, pp A.Gadomski, J.Siódmiak; *Crystal Research & Technology, vol. 37, no. 2-3, 2002, pp ; *Croatica Chemica Acta, vol. 76 (2) 2003, pp.129–136 - A.Gadomski; *Chemical Physics Letters, vol. 258, no. 1-2, 9 Aug. 1996, pp.6-12; *Vacuum, vol 50. pp M. Muthukumar; Advances in Chemical Physics, vol. 128, 2004

ACKNOWLEDGEMENT !!! Thanks go to Lutz Schimansky-Geier for inviting me to present ideas rather than firm and well-established results...