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.
Article PDF
Similar content being viewed by others
References
Protein Data Bank: The single global archive for 3D macromolecular structure data. Nucleic Acids Res. 47 (2019), 520–528.
L. Abadias, E. Álvarez, Uniform stability for fractional Cauchy problems and applications. Topol. Methods Nonlinear Anal. 52, NoNo 2 (2018), 707–728.
L. Abadias, C. Lizama, P.J. Miana, Sharp extensions and algebraic properties for solutions families of vector-valued differential equations. Banach J. Math. Anal. 10, No 1 (2016), 169–208.
C.N. Angstmann, A.M. Erickson, B.I. Henry, A.V. McGann, J.M. Murray, J. A. Nichols, Fractional order compartment models. SIAM J. Appl. Math. 77, No 2 (2017), 430–446.
E. Bajlekova, The abstract Cauchy problem for the fractional evolution equation. Fract. Calc. Appl. Anal. 1, No 3 (1998), 255–270.
H. Berry, Nonequilibrium phase transition in a self-activated biological network. Phys. Rev. E. 67, No 3 (2003), 1–9.
A. Cooper, D. Dryden, Allostery without conformational change. Eur. Biophys J. 11, No 2 (1984), 103–109.
W. Dai, B. Zhang, H. Su, J. Li, Y. Zhao, et al. Structure-based design of antiviral drug candidates targeting the SARS-CoV-2 main protease. Science (2020), 10.1126/science.abb4489.
K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type Springer Science & Business Media (2010).
K.H. DuBay, J.P. Bothma, P.L. Geissler, Long-range intraprotein communication can be transmitted by correlated side-chain fluctuations alone. PLoS Computational Biology 7, NoNo 9 (2011).
E. Estrada, The Structure of Complex Networks: Theory and Applications Oxford University Press (2012).
E. Estrada, Topological analysis of SARS CoV-2 Main Protease. Chaos 30, (2020), # 061102, In Press.
E. Estrada, N. Hatano, Communicability in complex networks. Phys. Rev. E. 77, No 3 (2008), 1–12.
E. Estrada, J.A. Rodríguez-Velázquez, Subgraph centrality in complex networks. Phys. Rev. E. 71, No 5 (2005), 103–109.
D. Fulger, E. Scalas, G. Germano, Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys. Rev. E. 77, No 2 (2008), 1–7.
R. Garrapa, Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM J. Numer. Anal. 53, No 3 (2015), 1350–1369.
R. Garrapa, M. Popolizio, Computing the matrix Mittag-Leffer function with applications to fractional calculus. J. Sci. Comput. 77, No 1 (2018), 129–153.
R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffer Functions, Related Topics and Applications Springer Heidelberg, (2014).
R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional Calculus and Continuous-Time Finance. III. The Diffusion Limit. Mathematical Finance (Konstanz, 2000) 171–180 Trends Math. Birkhäuser Basel, (2001).
G. Hu, J. Zhou, W. Yan, J. Chen, B. Shen, The topology and dynamics of protein complexes: insights from intra-molecular network theory. Curr. Protein Pept. Sc. 14, No 2 (2013), 121–132.
V. Latora, V. Nicosia, G. Russo, Complex Networks: Principles, Methods and Applications Cambridge University Press (2017).
A.L. Lee, M. Whitley, Frameworks for understanding long-range intra-protein communication. Curr. Protein Pept. Sc. 10, No 2 (2009), 116–127.
C.-H. Lee, S. Tenneti, D.Y. Eun, Transient dynamics of epidemic spreading and its mitigation on large networks. Proc. of the Twentieth ACM Internat. Symp. on Mobile Ad Hoc Networking and Computing (2019), 191–200.
S. Lu, M. Ji, D. Ni, and J. Zhang, Discovery of hidden allosteric sites as novel targets for allosteric drug design. Drug Discovery Today 23, No 2 (2019), 359–365.
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models World Scientific (2010).
F. Mainardi, R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, No 1–2 (2000), 283–299.
I. Matychyn, On computation of matrix Mittag-Leffler function. arXiv Preprint arXiv:1706.01538 (2017).
W. Mei, S. Mohagheghi, S. Zampieri, F. Bullo, On the dynamics of deterministic epidemic propagation over networks. Ann. Rev. Control. 44, (2017), 116–128.
M. Miotto, L. Di Rienzo, P. Corsi, D. Raimondo, E. Milanetti, Simulated epidemics in 3D Protein structures to detect functional properties. J. Chem. Inf. Model. 60, (2020), 1884–1891.
S.K. Mishra, M. Gupta, D.K. Upadhyay, Fractional derivative of logarithmic function and its applications as multipurpose ASP circuit. Analog Integr. Circ. S. 100, No 2 (2019), 377–387.
D. Mugnolo, Dynamical systems associated with adjacency matrices. Discrete Cont. Dyn-B 23, No 5 (2018), 1945–1973.
K.M. Ottemann, W. Xiao, Y. K. Shin, D.E. Koshland, A piston model for transmembrane signaling of the aspartate receptor. Science 285, (1999), 1751–1754.
R.B. Paris, Exponential asymptotics of the Mittag–Leffler function. P. Roy. Soc. A-Math. Phy. 458, No 2028 (2002), 3041–3052.
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications Academic Press Boston, (1999).
A. Sadeghi, J.R. Cardoso, Some notes on properties of the matrix Mittag-Leffler function. Appl. Math. Comput. 338, (2018), 733–738.
A. Gorbalenya, S. Baker, 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. 5, (2020), 536–544.
F. Wu, S. Zhao, B. Yu, Y.M. Chen, Wang, et al. A new coronavirus associated with human respiratory disease in China. Nature 579, (2020), 265–269.
F.Y. Xu, M. Havenith, Perspective: Watching low-frequency vibrations of water in biomolecular recognition by THz spectroscopy. J. Chem. Phys. 143, No 17 (2015), # 170901.
L. Zhang, D. Lin, X. Sun, U. Curth, C. Drosten, et al., Crystal structure of SARS-CoV-2 main protease provides a basis for design of improved α-ketoamide inhibitors. Science 368, No 6489 (2020), 409–412.
P. Zhou, X. L. Yang, X. G. Wang, B. Hu, L. Zhang, et al., A pneumonia outbreak associated with a new coronavirus of probable bat origin. Nature 579, (2020), 270–273.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Abadias, L., Estrada-Rodriguez, G. & Estrada, E. Fractional-Order Susceptible-Infected Model: Definition and Applications to the Study of COVID-19 Main Protease. Fract Calc Appl Anal 23, 635–655 (2020). https://doi.org/10.1515/fca-2020-0033
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2020-0033