Abstract
A superdiffusive random-walk action in the depletion zone around a growing protein crystal is considered. It stands for a dynamic boundary condition of the growth process and competes steadily with a quasistatic, curvature-involving (thermodynamic) free boundary condition, both of them contributing to interpret the (mainly late-stage) growth process in terms of a prototype ion-channeling effect. An overall diffusion function contains quantitative signatures of both boundary conditions mentioned and indicates whether the new phase grows as an orderly phase or a converse scenario occurs. This situation can be treated in a quite versatile way both numerically and analytically, within a generalized Smoluchowski framework. This study can help in (1) elucidating some dynamic puzzles of a complex crystal formation vs biomolecular aggregation, also those concerning ion-channel formation, and (2) seeing how ion-channel-type dynamics of non-Markovian nature may set properly the pace of model (dis)ordered protein aggregation.
Keywords: Protein solution, Protein crystal, Electrostatic double layer, Ion channel, Enhanced diffusion
References
- 1.Ducruix, A., Giege, R.: Crystallization of Nucleic Acids and Proteins, a Practical Approach. Oxford University Press, New York (1992)
- 2.McPherson, A.: Crystallization of Biological Macromolecules. Cold Spring Harbor Laboratory Press, New York (1999)
- 3.Miyazawa, S., Jernigan, R.L.: Estimation of effective interresidue contact energies from protein crystal structures: quasi-chemical approximation. Macromolecules 18, 534–552 (1985) [DOI]
- 4.Nicolis, G., Basios, V., Nicolis, C.: Pattern formation and fluctuation-induced transitions in protein crystallization. J. Chem. Phys. 120, 7708–7719 (2004) [DOI] [PubMed]
- 5.Vekilov, P.G., Lin, H., Rosenberger, F.: Unsteady crystal growth due to step-bunch cascading. Phys. Rev. E 55, 3202–3214 (1997) [DOI]
- 6.Hołyst, R.: Some features of soft matter systems. Soft Matter 1, 329–333 (2005) [DOI] [PubMed]
- 7.Gadomski, A., Siódmiak, J.: Kinetics of protein crystal growth in mass convection regime. Cryst. Res. Technol. 37, 281–291 (2002) [DOI]
- 8.Łuczka, J., Niemiec, M., Rudnicki, R.: Kinetics of growth process controlled by convective fluctuations. Phys. Rev. E 65, 051401.1–9 (2002) [DOI] [PubMed]
- 9.Siwy, Z., Fuliński, A.: Origin of 1/fα noise in membrane channel currents. Phys. Rev. Lett. 89, 158101.1–4 (2002) [DOI] [PubMed]
- 10.Bezrukov, S.M., Winterhalter, M.: Examining noise sources at the single-molecule level: 1/f noise of an open Maltoporin channel. Phys. Rev. Lett. 85, 202–205 (2000) [DOI] [PubMed]
- 11.Fuliński, A., Grzywna, Z., Mellor, I., Siwy, Z., Usherwood, P.N.R.: Non-Markovian character of ionic current fluctuations in membrane channels. Phys. Rev. E 58, 919–924 (1998) [DOI]
- 12.Siódmiak, J., Gadomski, A.: Computer model of biopolymer crystal growth and aggregation by addition of macromolecular units—a comparative study. Int. J. Mod. Phys. C 17, 1037–1054 (2006) [DOI]
- 13.Gadomski, A., Siódmiak, J.: On the elastic contribution to crystal growth in complex environments. Phys. Status Solidi B 242, 538–549 (2005) [DOI]
- 14.Chernov, A.A.: Crystal growth science between the centuries. J. Mater. Sci. Mater. Electron. 12, 437–449 (2001) [DOI]
- 15.Gadomski, A., Grzywna, Z., Łuczka, J.: An evolution equation applied to the perturbed sphere growth. Chem. Eng. Sci. 48, 3713–3721 (1993) [DOI]
- 16.Cabrera, N., Levine, M.M.: On the dislocation theory of evaporation of crystals. Philos. Mag. 1, 450–458 (1956) [DOI]
- 17.Gadomski, A., Łuczka, J.: Some remarks concerning spherulitic growth. Int. J. Quant. Chem. 52, 301–308 (1994) [DOI]
- 18.Tolman, R.C.: The effect of droplet size on surface tension. J. Chem. Phys. 17, 333–337 (1949) [DOI]
- 19.Gadomski, A., Łuczka, J.: Convection-driven growth in fluctuating velocity field. Acta Phys. Pol. B 24, 725–731 (1993)
- 20.Kurzynski, M.: The Thermodynamic Machinery of Life. Frontiers Collection. Springer, Berlin (2006)
- 21.Reguera, D., Rubí, J.M.: Nonequilibrium translational-rotational effects in nucleation. J. Chem. Phys. 115, 7100–7106 (2001) [DOI]
- 22.Gadomski, A., Rubí, J.M.: On the two principal curvatures as potential barriers in a model of complex matter agglomeration. Chem. Phys. 293, 169–177 (2003) [DOI]
- 23.Chernov, A.A.: Modern Crystallography III: Crystal Growth. Springer, Berlin (1984)
- 24.Chernov, A.A.: Crystals built of biological macromolecules. Phys. Rep. 288, 61–75 (1997) [DOI]
- 25.Pimpinelli, D.K., Villain, J.: Physics of Crystal Growth. Cambridge University Press, New York (1998)
- 26.Vainstein, M.H., Costa, I.V.L., Morgado, R., Oliveira, F.A.: Non-exponential relaxation for anomalous diffusion. Europhys. Lett. 73, 726–732 (2006) [DOI]
- 27.Gadomski, A., Łuczka, J.: Kinetics of temperature or pressure field induced phase transformation in lipid bilayers. Acta Phys. Pol. B 28, 1827–1842 (1997)
- 28.Málaga, C., Mandujano, F., Santamaría-Holek, I.: Mesoscopic constitutive relations for dilute polymer solutions. Physica A 369, 291–300 (2006) [DOI]
- 29.Kosińska, I.D., Fuliński, A.: Asymmetries and anomalous phenomena in ionic transport through nanochannels. Acta Phys. Pol. B 38, 1631–1645 (2007)
- 30.Liu, X.Y., Boek, E.S., Briels, W.J., Bennema, P.: Prediction of crystal growth morphology based on structural analysis of the solid-fluid interface. Nature 374, 342–345 (1995) [DOI]
- 31.Heinemann, S.H., Sigworth, F.J.: Open channel noise V. Fluctuating barriers to ion entry in gramicidin A channels. Biophys. J. 57, 499–514 (1990) [DOI] [PMC free article] [PubMed]
- 32.Jia, Y.W., Liu. X.Y.: Self-assembly of protein at aqueous solution surface in correlation to protein crystallization. Appl. Phys. Lett. 86, 023903–024100 (2005) [DOI]
- 33.Jia, Y.W., Liu. X.Y.: From surface self-assembly to crystallization: prediction of protein crystallization conditions. J. Phys. Chem. B 110, 6949–6955 (2006) [DOI] [PubMed]
- 34.Santamaría-Holek, I., Rubí, J.M., Gadomski, A.: Thermokinetic approach of single particles and clusters involving anomalous diffusion under viscoelastic response. J. Phys. Chem. B 111, 2293–2298 (2007) [DOI] [PubMed]
- 35.Sakmann, B., Neher, E.: Patch clamp techniques for studying ionic channels in excitable membranes. Annu. Rev. Physiol. 46, 455–472 (1984) [DOI] [PubMed]
- 36.Karle, J.: Some developments in anomalous dispersion for the structural investigation of macromolecular systems in biology. Int. J. Quant. Chem. 7, 357–367 (1980)
- 37.Li, H., Perozzo, M.A., Konnert, J.H., Nadarajah, A., Pusey, M.L.: Determining the molecular-packing arrangements on protein crystal faces by atomic force microscopy. Acta Crystallogr. D 55, 1023–1035 (1999) [DOI] [PubMed]
- 38.Kruszewska, N., Gadomski, A.: Towards a 2D model biopolymer polycrystalline aggregation based on Smoluchowski dynamics supplemented by computer experiment. Acta Phys. Pol. B 38, 1819–1835 (2007)
- 39.Stauffer, D., Aharony, A.: Introduction to Percolation Theory. Taylor and Francis, London (1994)
- 40.Siwy, Z., Apel, P., Baur, D., Dobrev, D.D., Korchev, Y.E., Neumann, R., Spohr, R., Trautmann, C., Voss, K.: Preparation of synthetic nanopores with transport properties analogous to biological channels. Surf. Sci. 532–535, 1061–1066 (2003) [DOI]
- 41.Pechkova, E., Nicolini, C.: Accelerated protein crystal growth by protein thin film template. J. Cryst. Growth 231, 599–602 (2001) [DOI]
- 42.Buff, F.P.: The spherical interface. I. Thermodynamics. J. Chem. Phys. 19, 1591–1594 (1951) [DOI]
- 43.Babin, V., Holyst, R.: Evaporation of a sub-micrometer droplet. J. Phys. Chem. B 109 11367–11372 (2005) [DOI] [PubMed]
- 44.Gadomski, A.: Diffusion-reaction approach applied to the ionic wave propagation through biomembrane channels. Cell. Mol. Biol. Lett. 1, 273–289 (1996)
- 45.Liebovitch, L.S., Fischbarg, J., Koniarek, J.P., Todorova, I., Wang, M.: Fractal model of ion-channel kinetics. Biochim. Biophys. Acta Biomembr. 896, 173–180 (1987) [DOI] [PubMed]