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Entropic Analysis of Protein Oscillations through Langevin Equations & Fokker-Planck Equations

Garrett Baughman, Preet Sharma

Abstract


Background: Protein oscillations have been one of the major highlights in the field of biophysics and bio-molecules. These oscillations can give us insights into complex bio-molecules and reveal their nature at a very fundamental level. They can also show us the dynamics involved in the functioning of bio-molecules through the nature of these oscillations. Method/Objective: In this article, we have described the basics of protein oscillations, giving a very fundamental approach to the physics of oscillations. We have also described some bio-systems in which protein oscillations play a vital role. In this article, we have used the Langevin equations and Fokker-Planck equations to describe the oscillation dynamics of proteins. Findings:Finally, we have shown the trend of an increase in the entropy of the oscillations by involving a perturbation term in the regular nature of oscillations. The entropy of protein oscillations is very important in understanding protein dynamics.

 

Doi: 10.28991/HEF-SP2022-01-05

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Keywords


Langevin Equations; Fokker-Plack Equations; Entropy; Proteins; Oscillation.

References


Sherratt, D. J. (2016). Oscillation helps to get division right. Proceedings of the National Academy of Sciences of the United States of America, 113(11), 2803–2805. doi:10.1073/pnas.1601792113.

Jülicher, F. (2001). Mechanical oscillations at the cellular scale. Comptes Rendus de l'Académie des Sciences-Series IV-Physics-Astrophysics, 2(6), 849-860. doi:10.1016/S1296-2147(01)01228-8.

Brauns, F., Pawlik, G., Halatek, J., Kerssemakers, J., Frey, E., & Dekker, C. (2020). Bulk-surface coupling reconciles Min-protein pattern formation in vitro and in vivo. bioRxiv. doi:10.1101/2020.03.01.971952.

Kim, J. K., & Tyson, J. J. (2020). Misuse of the Michaelis–Menten rate law for protein interaction networks and its remedy. PLoS Computational Biology, 16(10), 1008258. doi:10.1371/journal.pcbi.1008258.

Taniguchi, M., & Yoshida, H. (2011). 1.39 Unfolded Protein Response. Comprehensive Biotechnology (3rd Ed.), 508-520, Pergamon Press, Oxford, United Kingdom. doi:10.1016/B978-0-444-64046-8.00033-1.

Antonny, B. (2016). Composition, Physical Properties, and Curvature. Encyclopedia of Cell Biology, 201-207, Academic Press, Massachusetts, United States. doi:10.1016/B978-0-12-394447-4.10025-2.

Doolittle, W. F. (1998). A paradigm gets shifty. Nature, 392(6671), 15–16. doi:10.1038/32033.

Jo, H. H., Kim, Y. J., Kim, J. K., Foo, M., Somers, D. E., & Kim, P. J. (2018). Waveforms of molecular oscillations reveal circadian timekeeping mechanisms. Communications Biology, 1(1), 1. doi:10.1038/s42003-018-0217-1.

Nagel, D. H., & Kay, S. A. (2012). Complexity in the wiring and regulation of plant circadian networks. Current Biology, 22(16), 648. doi:10.1016/j.cub.2012.07.025.

Sehgal, A., Price, J. L., Man, B., & Young, M. W. (1994). Loss of circadian behavioral rhythms and per RNA oscillations in the Drosophila mutant timeless. Science, 263(5153), 1603–1606. doi:10.1126/science.8128246.

Bell-Pedersen, D., Cassone, V. M., Earnest, D. J., Golden, S. S., Hardin, P. E., Thomas, T. L., & Zoran, M. J. (2005). Circadian rhythms from multiple oscillators: Lessons from diverse organisms. Nature Reviews Genetics, 6(7), 544–556. doi:10.1038/nrg1633.

Gutierrez Lopez, D. E., Lashinger, L. M., Weinstock, G. M., & Bray, M. S. (2021). Circadian rhythms and the gut microbiome synchronize the host’s metabolic response to diet. Cell Metabolism, 33(5), 873–887. doi:10.1016/j.cmet.2021.03.015.

Brennen, C., & Winet, H. (1977). Fluid Mechanics of Propulsion by Cilia and Flagella. Annual Review of Fluid Mechanics, 9(1), 339–398. doi:10.1146/annurev.fl.09.010177.002011.

Goetz, J. G., Steed, E., Ferreira, R. R., Roth, S., Ramspacher, C., Boselli, F., Charvin, G., Liebling, M., Wyart, C., Schwab, Y., & Vermot, J. (2014). Endothelial cilia mediate low flow sensing during zebrafish vascular development. Cell Reports, 6(5), 799–808. doi:10.1016/j.celrep.2014.01.032.

Shah, A. S., Yehuda, B. S., Moninger, T. O., Kline, J. N., & Welsh, M. J. (2009). Motile cilia of human airway epithelia are chemosensory. Science, 325(5944), 1131–1134. doi:10.1126/science.1173869.

Litschel, T., Ramm, B., Maas, R., Heymann, M., & Schwille, P. (2018). Beating Vesicles: Encapsulated Protein Oscillations Cause Dynamic Membrane Deformations. Angewandte Chemie - International Edition, 57(50), 16286–16290. doi:10.1002/anie.201808750.

Man, Y., Ling, F., & Kanso, E. (2020). Cilia oscillations. Philosophical Transactions of the Royal Society B, 375(1792), 20190157. doi:10.1098/rstb.2019.0157.

Inaba, K. (2011). Sperm flagella: Comparative and phylogenetic perspectives of protein components. Molecular Human Reproduction, 17(8), 524–538. doi:10.1093/molehr/gar034.

Chew, J., Leypunskiy, E., Lin, J., Murugan, A., & Rust, M. J. (2018). High protein copy number is required to suppress stochasticity in the cyanobacterial circadian clock. Nature Communications, 9(1), 1. doi:10.1038/s41467-018-05109-4.

Mateos, J. L., de Leone, M. J., Torchio, J., Reichel, M., & Staiger, D. (2018). Beyond transcription: Fine-tuning of circadian timekeeping by post-transcriptional regulation. Genes, 9(12), 616. doi:10.3390/genes9120616.

Termonia, Y., & Ross, J. (1981). Oscillations and control features in glycolysis: numerical analysis of a comprehensive model. Proceedings of the National Academy of Sciences of the United States of America, 78(5), 2952–2956. doi:10.1073/pnas.78.5.2952.

Ibsen, K. H., & Schiller, K. W. (1967). Oscillations of nucleotides and glycolytic intermediates in aerobic suspensions of Ehrlich ascites tumor cells. Biochimica et Biophysica Acta (BBA)-Bioenergetics, 131(2), 405-407. doi:10.1016/0005-2728(67)90156-9.

Williamson, T., Adiamah, D., Schwartz, J. M., & Stateva, L. (2012). Exploring the genetic control of glycolytic oscillations in Saccharomyces Cerevisiae. BMC Systems Biology, 6, 108. doi:10.1186/1752-0509-6-108.

Schrödinger, E. (1944). What is life? The physical aspect of the living cell and mind. Cambridge University Press, Cambridge, United Kingdom.

Baughman, G., Baker, H. S., & Sharma, P. (2022). Quantum mechanical effects in protein oscillations. Biophysical Journal, 121(3), 187a. doi:10.1016/j.bpj.2021.11.1813.

Stöhr, M., & Tkatchenko, A. (2019). Quantum mechanics of proteins in explicit water: The role of plasmon-like solute-solvent interactions. Science Advances, 5(12), 24. doi:10.1126/sciadv.aax0024.

Ivanitskii, G. R. (2021). The Reason that Discussion of What Life Is from the Perspective of Physics Still Continues (A Response to the Article by A.M. Smolovich “A Hypothesis about the Physical Phenomenon of Life”). Biophysics (Russian Federation), 66(5), 872–878. doi:10.1134/S0006350921050080.

Khrennikov, A. (2022). Contextual Probability in Quantum Physics, Cognition, Psychology, Social Science, and Artificial Intelligence. From Electrons to Elephants and Elections, The Frontiers Collection, Springer, Cham, Switzerland. doi:10.1007/978-3-030-92192-7_28.

Leighton, R. B., & Sands, M. (1963). The Feynmann lectures on physics. Addison-Wesley, Boston, United States.

De Figueiredo, R. J. P. (1965). Forced Oscillations in Nonlinear Systems with Hysteresis. IEEE Transactions on Circuit Theory, 12(1), 52–58. doi:10.1109/TCT.1965.1082399.

Rossi, N. A., & Dunlop, M. J. (2018). Making Waves with Synthetic Oscillators. Cell Systems, 6(4), 406–407. doi:10.1016/j.cels.2018.04.001.

Danino, T., Mondragón-Palomino, O., Tsimring, L., & Hasty, J. (2010). A synchronized quorum of genetic clocks. Nature, 463(7279), 326–330. doi:10.1038/nature08753.

Stricker, J., Cookson, S., Bennett, M. R., Mather, W. H., Tsimring, L. S., & Hasty, J. (2008). A fast, robust and tunable synthetic gene oscillator. Nature, 456(7221), 516–519. doi:10.1038/nature07389.

Qian, H., Ge, H. (2013). Stochastic Processes, Fokker-Planck Equation. In: Dubitzky, W., Wolkenhauer, O., Cho, KH., Yokota, H. (eds) Encyclopedia of Systems Biology. Springer, New York, United States. doi:10.1007/978-1-4419-9863-7_279.

Peters, B. (2017). Chapter 16-Kramers theory. Reaction Rate Theory and Rare Events Simulations, 435-450, Elsevier, Amsterdam, netherlands. doi:10.1016/B978-0-44-456349-1.00016-7.

Chaudhari, P., & Soatto, S. (2018). Stochastic Gradient Descent Performs Variational Inference, Converges to Limit Cycles for Deep Networks. 2018 Information Theory and Applications Workshop (ITA). doi:10.1109/ita.2018.8503224.

Friedrich, R., & Peinke, J. (1997). Description of a Turbulent Cascade by a Fokker-Planck Equation. Physical Review Letters, 78(5), 863–866. doi:10.1103/PhysRevLett.78.863.

Tsang, K. Y., & Ngai, K. L. (1997). Dynamics of relaxing systems subjected to nonlinear interactions. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 56(1 SUPPL. A), 17. doi:10.1103/physreve.56.r17.

Bejan, A. (2022). Evolution, physics, and education. Biosystems, 215, 104663. doi:10.1016/j.biosystems.2022.104663.

Matsuno, K. (2022). Approaching biology through the retrocausal apex in physics. BioSystems, 213, 104634. doi:10.1016/j.biosystems.2022.104634.

Marinopoulou, E., Biga, V., Sabherwal, N., Miller, A., Desai, J., Adamson, A. D., & Papalopulu, N. (2021). HES1 protein oscillations are necessary for neural stem cells to exit from quiescence. IScience, 24(10), 103198. doi:10.1016/j.isci.2021.103198.

Bochter, M. S., Servello, D., Kakuda, S., D’Amico, R., Ebetino, M. F., Haltiwanger, R. S., & Cole, S. E. (2022). Lfng and Dll3 cooperate to modulate protein interactions in cis and coordinate oscillatory Notch pathway activation in the segmentation clock. Developmental Biology, 487, 42–56. doi:10.1016/j.ydbio.2022.04.004.

Bonny, M., Fischer-Friedrich, E., Loose, M., Schwille, P., & Kruse, K. (2013). Membrane Binding of MinE Allows for a Comprehensive Description of Min-Protein Pattern Formation. PLoS Computational Biology, 9(12), 1003347. doi:10.1371/journal.pcbi.1003347.

Wettmann, L., & Kruse, K. (2018). The min-protein oscillations in Escherichia coli: An example of self-organized cellular protein waves. Philosophical Transactions of the Royal Society B: Biological Sciences, 373(1747), 20170111. doi:10.1098/rstb.2017.0111.

Meinhardt, H., & De Boer, P. A. J. (2001). Pattern formation in Escherichia coil: A model for the pole-to-pole oscillations of Min proteins and the localization of the division site. Proceedings of the National Academy of Sciences of the United States of America, 98(25), 14202–14207. doi:10.1073/pnas.251216598.

Corbin, B. D., Yu, X. C., & Margolin, W. (2002). Exploring intracellular space: Function of the Min system in round-shaped Escherichia coli. EMBO Journal, 21(8), 1998–2008. doi:10.1093/emboj/21.8.1998.

Varma, A., Huang, K. C., & Young, K. D. (2008). The Min system as a general cell geometry detection mechanism: Branch lengths in Y-shaped Escherichia coli cells affect Min oscillation patterns and division dynamics. Journal of Bacteriology, 190(6), 2106–2117. doi:10.1128/JB.00720-07.

Goldstein, B., & Macara, I. G. (2007). The PAR Proteins: Fundamental Players in Animal Cell Polarization. Developmental Cell, 13(5), 609–622. doi:10.1016/j.devcel.2007.10.007.

Suzuki, A., & Ohno, S. (2006). The PAR-aPKC system: Lessons in polarity. Journal of Cell Science, 119(6), 979–987. doi:10.1242/jcs.02898.

Durney, C. H., Harris, T. J. C., & Feng, J. J. (2018). Dynamics of PAR Proteins Explain the Oscillation and Ratcheting Mechanisms in Dorsal Closure. Biophysical Journal, 115(11), 2230–2241. doi:10.1016/j.bpj.2018.10.014.

Tomé, T. (2006). Entropy production in nonequilibrium systems described by a Fokker-Planck equation. Brazilian journal of physics, 36, 1285-1289. doi:10.1590/S0103-97332006000700029.

Nicolas. G., & Prigogine I. (1977). Self-Organization in Nonequilibrium Systems. Wiley, New York, United States.

Capotosto, S., Smoot, B., Hallford, R., & Sharma, P. (2019). Entropy Production, Entropy Generation, and Fokker-Planck Equations for Cancer Cell Growth. Physics (Switzerland), 1(1), 147–153. doi:10.3390/physics1010014.

Capotosto, S., Smoot, B., Nemchen, Y., Hallford, R., & Sharma, P. (2020). Entropy Production in Protein Aggregation. Biophysical Journal, 118(3), 526a. doi:10.1016/j.bpj.2019.11.2888.

Dewar, R. (2003). Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states. Journal of Physics A: Mathematical and General, 36(3), 631–641. doi:10.1088/0305-4470/36/3/303.

Lucia, U. (2014). Entropy generation and the Fokker-Planck equation. Physica A: Statistical Mechanics and Its Applications, 393, 256–260. doi:10.1016/j.physa.2013.09.028.

Lucia, U. (2008). Statistical approach of the irreversible entropy variation. Physica A: Statistical Mechanics and Its Applications, 387(14), 3454–3460. doi:10.1016/j.physa.2008.02.002.

Lucia, U. (2009). Irreversibility, entropy and incomplete information. Physica A: Statistical Mechanics and Its Applications, 388(19), 4025–4033. doi:10.1016/j.physa.2009.06.027.

Lucia, U. (2010). Maximum entropy generation and κ-exponential model. Physica A: Statistical Mechanics and Its Applications, 389(21), 4558–4563. doi:10.1016/j.physa.2010.06.047.


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DOI: 10.28991/HEF-SP2022-01-05

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