1. ON (dis)ORDERED AGGREGATION OF PROTEINS Adam Gadomski & Jacek Siódmiak Institute of Mathematics and Physics University of Technology and Agriculture Bydgoszcz, Poland Workshop on Structure and Function of Biomolecules May 13 - 15, 2004, Będlewo near Poznań, Poland

2. OBJECTIVE TO PROPOSE A CONCEPTUAL AND THEORETICAL STRATEGY, BASED ON THE GROWTH RULE AND GROWTH MECHANISM, POSSIBLY OF USEFULNESS FOR QUALITY AND MANUFACTURE TESTS IN PROTEIN-BASED TECHNOLOGY AND PROTEIN-CLUSTER DESIGN

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

4. GENERAL RULE BASED ON THE GROWTH RATE - mechanism – dependent continuous function - systems main variables - control parameters - time (desirable behavior in time: )

5. ONE-NUCLEUS BASED SCENARIO GENERAL SCHEME FOR THE MASS CONSERVATION LAW - volume - surface - time - internal concentration (density) - external concentration - position vector

7. 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

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

9. deterministic: stochastic (an example): an (un)correlated noise Frenkel-like macroion velocity supersaturation parameter Growth rates for the DL-controlled on-one-nucleus-based aggregation model

10. MANY-NUCLEI BASED SCENARIO GRAIN (CLUSTER)-MERGING MECHANISM

11. RESULTING 2D-MICROSTRUCTURE: VORONOI-like MOSAIC FOR AGGREGATION INITIAL STRUCTUREFINAL STRUCTURE

12. RESULTING FORMULA FOR VOLUME-PRESERVING d-DIMENSIONAL MATTER AGGREGATION time derivative of the specific volume (inverse of the polycrystal density) hypersurface inverse term adjusting time- dependent kinetic prefactor responsible for spherulitic growth

13. 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 !

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

15. 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) CONCLUSION

16. LITERATURE: - A.Danch, A.Gadomski. a ; A.Gadomski, J.Łuczka b a Journal of Molecular Liquids, vol.86, no.1-3, June 2000, pp.249-257 b IBIDEM, pp. 237-247 - J.Łuczka, M.Niemiec, R.Rudnicki Physical Review E., vol.65, no.5, May 2002, pp.051401/1-9 - J.Łuczka, P.Hanggi, A.Gadomski Physical Review E., vol.51, no.6, pt.A, June 1995, pp.5762-5769 - A.Gadomski, J.Siódmiak *Crystal Research & Technology, vol.37, no.2-3, 2002, pp.281-291 *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; *Vacuume, vol50, pp.79-83 ACKNOWLEDGEMENT !!! This work was supported by KBN grant no. 2 P03B 032 25 (2003-2006).