Skip to main content
NCBI home page
Springer Nature - PMC COVID-19 Collection logoLink to Springer Nature - PMC COVID-19 Collection
. 2020 Jul 11;23(3):635–655. doi: 10.1515/fca-2020-0033

Fractional-Order Susceptible-Infected Model: Definition and Applications to the Study of COVID-19 Main Protease

Luciano Abadias 13,23,✉, Gissell Estrada-Rodriguez 33, Ernesto Estrada 23,43
PMCID: PMC8617368  PMID: 34849076

Abstract

We propose a model for the transmission of perturbations across the amino acids of a protein represented as an interaction network. The dynamics consists of a Susceptible-Infected (SI) model based on the Caputo fractional-order derivative. We find an upper bound to the analytical solution of this model which represents the worse-case scenario on the propagation of perturbations across a protein residue network. This upper bound is expressed in terms of Mittag-Leffler functions of the adjacency matrix of the network of inter-amino acids interactions. We then apply this model to the analysis of the propagation of perturbations produced by inhibitors of the main protease of SARS CoV-2. We find that the perturbations produced by strong inhibitors of the protease are propagated far away from the binding site, confirming the long-range nature of intra-protein communication. On the contrary, the weakest inhibitors only transmit their perturbations across a close environment around the binding site. These findings may help to the design of drug candidates against this new coronavirus.

Contributor Information

Luciano Abadias, Email: labadias@unizar.es.

Gissell Estrada-Rodriguez, Email: estradarodriguez@ljll.math.upmc.fr.

Ernesto Estrada, Email: estrada66@unizar.es.

References

  • [1].Protein Data Bank: The single global archive for 3D macromolecular structure data. Nucleic Acids Res. 2019;47:520–528. doi: 10.1093/nar/gky949. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • [2].Abadias L, Álvarez E. Uniform stability for fractional Cauchy problems and applications. Topol. Methods Nonlinear Anal. 2018;52(2):707–728. doi: 10.12775/TMNA.2018.038. [DOI] [Google Scholar]
  • [3].Abadias L, Lizama C, Miana PJ. Sharp extensions and algebraic properties for solutions families of vector-valued differential equations. Banach J. Math. Anal. 2016;10(1):169–208. doi: 10.1215/17358787-3345137. [DOI] [Google Scholar]
  • [4].Angstmann CN, Erickson AM, Henry BI, McGann AV, Murray JM, Nichols J A. Fractional order compartment models. SIAM J. Appl. Math. 2017;77(2):430–446. doi: 10.1137/16M1069249. [DOI] [Google Scholar]
  • [5].Bajlekova E. The abstract Cauchy problem for the fractional evolution equation. Fract. Calc. Appl. Anal. 1998;1(3):255–270. [Google Scholar]
  • [6].Berry H. Nonequilibrium phase transition in a self-activated biological network. Phys. Rev. E. 2003;67(3):1–9. doi: 10.1103/PhysRevE.67.031907. [DOI] [PubMed] [Google Scholar]
  • [7].Cooper A, Dryden D. Allostery without conformational change. Eur. Biophys J. 1984;11(2):103–109. doi: 10.1007/BF00276625. [DOI] [PubMed] [Google Scholar]
  • [8].Dai W, Zhang B, Su H, Li J, Zhao Y, et al. Science. 2020. Structure-based design of antiviral drug candidates targeting the SARS-CoV-2 main protease. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • [9].Diethelm K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. 2010. [Google Scholar]
  • [10].PLoS Computational Biology. 2011. [DOI] [PMC free article] [PubMed]
  • [11].Estrada E. The Structure of Complex Networks: Theory and Applications. 2012. [Google Scholar]
  • [12].Estrada E. Topological analysis of SARS CoV-2 Main Protease. Chaos. 2020;30:# 061102. doi: 10.1063/5.0013029. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • [13].Estrada E, Hatano N. Communicability in complex networks. Phys. Rev. E. 2008;77(3):1–12. doi: 10.1103/PhysRevE.77.036111. [DOI] [PubMed] [Google Scholar]
  • [14].Estrada E, Rodríguez-Velázquez JA. Subgraph centrality in complex networks. Phys. Rev. E. 2005;71(5):103–109. doi: 10.1103/PhysRevE.71.056103. [DOI] [PubMed] [Google Scholar]
  • [15].Fulger D, Scalas E, Germano G. Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys. Rev. E. 2008;77(2):1–7. doi: 10.1103/PhysRevE.77.021122. [DOI] [PubMed] [Google Scholar]
  • [16].Garrapa R. Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM J. Numer. Anal. 2015;53(3):1350–1369. doi: 10.1137/140971191. [DOI] [Google Scholar]
  • [17].Garrapa R, Popolizio M. Computing the matrix Mittag-Leffer function with applications to fractional calculus. J. Sci. Comput. 2018;77(1):129–153. doi: 10.1007/s10915-018-0699-5. [DOI] [Google Scholar]
  • [18].Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV. Mittag-Leffer Functions, Related Topics and Applications. Heidelberg: Springer; 2014. [Google Scholar]
  • [19].Gorenflo R, Mainardi F, Scalas E, Raberto M. Fractional Calculus and Continuous-Time Finance. III. The Diffusion Limit. Basel: Trends Math. Birkhäuser; 2001. pp. 171–180. [Google Scholar]
  • [20].Hu G, Zhou J, Yan W, Chen J, Shen B. The topology and dynamics of protein complexes: insights from intra-molecular network theory. Curr. Protein Pept. Sc. 2013;14(2):121–132. doi: 10.2174/1389203711314020004. [DOI] [PubMed] [Google Scholar]
  • [21].Latora V, Nicosia V, Russo G. Complex Networks: Principles, Methods and Applications. 2017. [Google Scholar]
  • [22].Lee AL, Whitley M. Frameworks for understanding long-range intra-protein communication. Curr. Protein Pept. Sc. 2009;10(2):116–127. doi: 10.2174/138920309787847563. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • [23].Lee C-H, Tenneti S, Eun DY. Proc. of the Twentieth ACM Internat. Symp. on Mobile Ad Hoc Networking and Computing. 2019. Transient dynamics of epidemic spreading and its mitigation on large networks; pp. 191–200. [Google Scholar]
  • [24].Lu S, Ji M, Ni D. and J. Zhang, Discovery of hidden allosteric sites as novel targets for allosteric drug design. Drug Discovery Today. 2019;23(2):359–365. doi: 10.1016/j.drudis.2017.10.001. [DOI] [PubMed] [Google Scholar]
  • [25].Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. 2010. [Google Scholar]
  • [26].Mainardi F, Gorenflo R. On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 2000;118(1–2):283–299. doi: 10.1016/S0377-0427(00)00294-6. [DOI] [Google Scholar]
  • [27].Matychyn I. arXiv Preprint. 2017. On computation of matrix Mittag-Leffler function. [Google Scholar]
  • [28].Mei W, Mohagheghi S, Zampieri S, Bullo F. On the dynamics of deterministic epidemic propagation over networks. Ann. Rev. Control. 2017;44:116–128. doi: 10.1016/j.arcontrol.2017.09.002. [DOI] [Google Scholar]
  • [29].Miotto M, Di Rienzo L, Corsi P, Raimondo D, Milanetti E. Simulated epidemics in 3D Protein structures to detect functional properties. J. Chem. Inf. Model. 2020;60:1884–1891. doi: 10.1021/acs.jcim.9b01027. [DOI] [PubMed] [Google Scholar]
  • [30].Mishra SK, Gupta M, Upadhyay DK. Fractional derivative of logarithmic function and its applications as multipurpose ASP circuit. Analog Integr. Circ. S. 2019;100(2):377–387. doi: 10.1007/s10470-018-1328-9. [DOI] [Google Scholar]
  • [31].Mugnolo D. Dynamical systems associated with adjacency matrices. Discrete Cont. Dyn-B. 2018;23(5):1945–1973. [Google Scholar]
  • [32].Ottemann KM, Xiao W, Shin Y K, Koshland DE. A piston model for transmembrane signaling of the aspartate receptor. Science. 1999;285:1751–1754. doi: 10.1126/science.285.5434.1751. [DOI] [PubMed] [Google Scholar]
  • [33].Paris RB. Exponential asymptotics of the Mittag–Leffler function. P. Roy. Soc. A-Math. Phy. 2002;458(2028):3041–3052. doi: 10.1098/rspa.2002.0977. [DOI] [Google Scholar]
  • [34].Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Boston: Academic Press; 1999. [Google Scholar]
  • [35].Sadeghi A, Cardoso JR. Some notes on properties of the matrix Mittag-Leffler function. Appl. Math. Comput. 2018;338:733–738. [Google Scholar]
  • [36].Gorbalenya A, Baker S. and R. Baric, Coronaviridae study group of the international committee on taxonomy of viruses, The species Severe acute respiratory syndrome-related coronavirus: classifying 2019-nCoV and naming it SARS-CoV-2. Nat. Microbiol. 2020;5:536–544. doi: 10.1038/s41564-020-0695-z. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • [37].Wu F, Zhao S, Yu B, Chen YM, Wang, et al. A new coronavirus associated with human respiratory disease in China. Nature. 2020;579:265–269. doi: 10.1038/s41586-020-2008-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • [38].Xu FY, Havenith M. Perspective: Watching low-frequency vibrations of water in biomolecular recognition by THz spectroscopy. J. Chem. Phys. 2015;143(17):# 170901. doi: 10.1063/1.4934504. [DOI] [PubMed] [Google Scholar]
  • [39].Zhang L, Lin D, Sun X, Curth U, Drosten C, et al. Crystal structure of SARS-CoV-2 main protease provides a basis for design of improved α-ketoamide inhibitors. Science. 2020;368(6489):409–412. doi: 10.1126/science.abb3405. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • [40].Zhou P, Yang X L, Wang X G, Hu B, Zhang L, et al. A pneumonia outbreak associated with a new coronavirus of probable bat origin. Nature. 2020;579:270–273. doi: 10.1038/s41586-020-2012-7. [DOI] [PMC free article] [PubMed] [Google Scholar]

Articles from Fractional Calculus & Applied Analysis are provided here courtesy of Nature Publishing Group

RESOURCES