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Functionals linear in curvature and statistics of helical proteins. (English) Zbl 1119.82342

Summary: It is shown that proceeding from the spiral stationary form of the protein chains one can deduce, in a unique way, the explicit expression for the relevant free energy. Namely, the free energy density should be a linear function of the curvature \(k\) of the curve which describes the shape of the central line of the protein molecule. Minimization of this energy gives for the pitch-to-radius ratio of the helices the value 2\(\pi\). The model also enables one to estimate qualitatively the release of the free energy under the transition of the protein chain from the straight line form to the spiral form. The free energy we propose implies, in particular, that the effective bending energy of the protein chain is not proportional to \(k^{2}\), as it is usually adopted in the physics of semi-flexible polymers, but this energy is linear in the curvature \(k\). The relation of this model to the rigid relativistic particles and strings is briefly discussed. The consideration relies on proving the complete integrability of the variational equations for the functionals defined on smooth curves and dependent on the curvature of these curves.

MSC:

82D60 Statistical mechanics of polymers
81-XX Quantum theory
92C40 Biochemistry, molecular biology
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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[1] Chan, H. S.; Dill, K. A., Phys. Today, 46, 2, 24 (1993)
[2] Dill, K. A., Protein Sci., 8, 1166 (1999)
[3] Chothia, C.; Finkelstein, A. V., The classification and origins of protein folding patterns, Annu. Rev. Biochem., 59, 1007 (1990)
[4] Kratky, O.; Porod, G., Recl. Trav. Chim., 68, 237 (1949)
[5] Kholodenko, A., Ann. Phys. (N.Y.), 202, 186 (1990)
[6] Kholodenko, A.; Ballauff, M.; Aguero Granados, M., Physica A, 260, 267 (1998)
[7] Kamien, R. D., Rev. Mod. Phys., 74, 953 (2002)
[8] Hyde, S.; Anderson, S.; Larsson, K., The Language of Shape (1997), Elsevier: Elsevier Amsterdam
[9] Kleinert, H., Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics (1995), World Scientific: World Scientific Singapore · Zbl 0942.81038
[10] Maritan, A.; Micheletti, C.; Trovato, A.; Banavar, J. R., Nature, 406, 287 (2000)
[11] Banavar, J. R.; Maritan, A., Rev. Mod. Phys., 75, 23 (2003)
[12] Banavar, F. R.; Maritan, A.; Micheletti, C.; Seno, F., Geometrical aspects of protein folding
[13] Stasiak, A.; Maddocks, J. H., Nature, 406, 251 (2000)
[14] Pieranski, P., (Stasiak, A.; Katritch, V.; Kauffman, L. H., Ideal Knots (1998), World Scientific: World Scientific Singapore), 20-41
[15] Spivak, M., A Comprehensive Introduction to Differential Geometry (1979), Publish or Perish: Publish or Perish Houston · Zbl 0439.53005
[16] Barbi, M.; Lepri, S.; Peyrard, M.; Theodorakopoulos, N., Thermal denaturation of an helicoidal DNA model
[17] Harris, A. B.; Kamien, R. D.; Lubensky, T. C., Rev. Mod. Phys., 71, 1745 (1999)
[18] Nesterenko, V. V., J. Math. Phys., 32, 3315 (1991)
[19] Nesterenko, V. V., J. Math. Phys., 34, 5589 (1993)
[20] Isberg, J.; Lindström, U.; Nordström, H., Mod. Phys. Lett. A, 5, 2491 (1990)
[21] Iso, S.; Itoi, C.; Mukaida, H., Nucl. Phys. B, 346, 293 (1990)
[22] Nesterenko, V. V., J. Math. Phys., 34, 5589 (1993)
[23] Ambjörn, J.; Durhuus, B.; Jonsson, T., J. Phys. A, 21, 981 (1988)
[24] Pisarski, R. D., Phys. Rev. D, 34, 670 (1986)
[25] Plyushchay, M. S., Phys. Lett. B, 243, 383 (1990) · Zbl 1332.81110
[26] Batlle, C.; Gomis, J.; Pons, J. M.; Roman-Roy, N., J. Phys. A: Math. Gen., 21, 2693 (1989)
[27] Pauling, L.; Corey, R. B.; Brancon, H. R., Proc. Natl. Acad. Sci USA, 37, 2005 (1951)
[28] Kholodenko, A. L.; Nesterenko, V. V., J. Geom. Phys., 16, 15 (1995)
[29] Barbashov, B. M.; Nesterenko, V. V., Introduction to the Relativistic String Theory (1990), World Scientific: World Scientific Singapore · Zbl 0536.58035
[30] David, F.; Guitter, E., Europhys. Lett., 3, 1169 (1987)
[31] Chervyakov, A. M.; Nesterenko, V. V., Phys. Rev. D, 48, 5811 (1993)
[32] Nesterenko, V. V.; Feoli, A.; Scarpetta, G., J. Math. Phys., 36, 5552 (1995)
[33] Nesterenko, V. V.; Feoli, A.; Scarpetta, G., Class. Quantum Grav., 13, 1201 (1996)
[34] Duggal, K. L.; Bejancu, A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Mathematics and its Applications, vol. 364 (1996), Kluwer: Kluwer Dordrecht · Zbl 0848.53001
[35] Postnikov, M. M., Lectures on Geometry. Semester III: Smooth Manifolds (1987), Nauka: Nauka Moscow · Zbl 0639.53001
[36] Aminov, Yu. A., Differential Geometry and Topology of Curves (1987), Nauka: Nauka Moscow · Zbl 0628.53001
[37] Griffiths, P. A., Exterior Differential Systems and the Calculus of Variations (1983), Birkhäuser: Birkhäuser Boston · Zbl 0512.49003
[38] Nesterenko, V. V., J. Phys. A, 22, 1673 (1989)
[39] Schweber, S. S., An Introduction to Relativistic Quantum Field Theory (1961), Row Peterson: Row Peterson New York · Zbl 0111.43102
[40] Barbashov, B. M.; Nesterenko, V. V., Fortschr. Phys., 31, 535 (1983)
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