
This study analyzed the role of electric charge in human viral infections. Examples of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), dengue, Ebola, influenza A, and respiratory syncytial virus (RSV) are presented. Charge distribution in SARS-CoV-2 and electrostatic interactions of SARS-CoV-2 with its receptor, angiotensin-converting enzyme 2 (ACE2), were evaluated, and the mean time required for respired SARS-CoV-2 virus attachment was evaluated. The virus–cell attachment modality of all of the above viruses was calculated. The impact of electric charge on other viral-related processes, such as replication of virion material, release, and immune response, was also discussed. Special charge conditions in virus treatments were also indicated.
Citation: Piotr H. Pawłowski, Piotr Zielenkiewicz. The role of electric charge in SARS-CoV-2 and other viral infections[J]. AIMS Biophysics, 2024, 11(2): 166-188. doi: 10.3934/biophy.2024011
[1] | Shaimaa A. M. Abdelmohsen, D. Sh. Mohamed, Haifa A. Alyousef, M. R. Gorji, Amr M. S. Mahdy . Mathematical modeling for solving fractional model cancer bosom malignant growth. AIMS Biophysics, 2023, 10(3): 263-280. doi: 10.3934/biophy.2023018 |
[2] | Mati ur Rahman, Mehmet Yavuz, Muhammad Arfan, Adnan Sami . Theoretical and numerical investigation of a modified ABC fractional operator for the spread of polio under the effect of vaccination. AIMS Biophysics, 2024, 11(1): 97-120. doi: 10.3934/biophy.2024007 |
[3] | Yasir Nadeem Anjam, Mehmet Yavuz, Mati ur Rahman, Amna Batool . Analysis of a fractional pollution model in a system of three interconnecting lakes. AIMS Biophysics, 2023, 10(2): 220-240. doi: 10.3934/biophy.2023014 |
[4] | Marco Menale, Bruno Carbonaro . The mathematical analysis towards the dependence on the initial data for a discrete thermostatted kinetic framework for biological systems composed of interacting entities. AIMS Biophysics, 2020, 7(3): 204-218. doi: 10.3934/biophy.2020016 |
[5] | Mehmet Yavuz, Kübra Akyüz, Naime Büşra Bayraktar, Feyza Nur Özdemir . Hepatitis-B disease modelling of fractional order and parameter calibration using real data from the USA. AIMS Biophysics, 2024, 11(3): 378-402. doi: 10.3934/biophy.2024021 |
[6] | Larisa A. Krasnobaeva, Ludmila V. Yakushevich . On the dimensionless model of the transcription bubble dynamics. AIMS Biophysics, 2023, 10(2): 205-219. doi: 10.3934/biophy.2023013 |
[7] | Carlo Bianca . Differential equations frameworks and models for the physics of biological systems. AIMS Biophysics, 2024, 11(2): 234-238. doi: 10.3934/biophy.2024013 |
[8] | Mohammed Alabedalhadi, Mohammed Shqair, Ibrahim Saleh . Analysis and analytical simulation for a biophysical fractional diffusive cancer model with virotherapy using the Caputo operator. AIMS Biophysics, 2023, 10(4): 503-522. doi: 10.3934/biophy.2023028 |
[9] | Bertrand R. Caré, Pierre-Emmanuel Emeriau, Ruggero Cortini, Jean-Marc Victor . Chromatin epigenomic domain folding: size matters. AIMS Biophysics, 2015, 2(4): 517-530. doi: 10.3934/biophy.2015.4.517 |
[10] | David Gosselin, Maxime Huet, Myriam Cubizolles, David Rabaud, Naceur Belgacem, Didier Chaussy, Jean Berthier . Viscoelastic capillary flow: the case of whole blood. AIMS Biophysics, 2016, 3(3): 340-357. doi: 10.3934/biophy.2016.3.340 |
This study analyzed the role of electric charge in human viral infections. Examples of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), dengue, Ebola, influenza A, and respiratory syncytial virus (RSV) are presented. Charge distribution in SARS-CoV-2 and electrostatic interactions of SARS-CoV-2 with its receptor, angiotensin-converting enzyme 2 (ACE2), were evaluated, and the mean time required for respired SARS-CoV-2 virus attachment was evaluated. The virus–cell attachment modality of all of the above viruses was calculated. The impact of electric charge on other viral-related processes, such as replication of virion material, release, and immune response, was also discussed. Special charge conditions in virus treatments were also indicated.
Numerous mathematical models have been established to predict and study the biological system. In the past four decades, there have been far-reaching research on improving cell mass production in chemical reactors [1]. The chemostat model is used to understand the mechanism of cell mass growth in a chemostat. A chemostat is an apparatus for continuous culture that contains bacterial populations. It can be used to investigate the cell mass production under controlled conditions. This reactor provides a dynamic system for population studies and is suitable to be used in a laboratory. A substrate is continuously added into the reactor containing the cell mass, which grows by consuming the substrate that enters through the inflow chamber. Meanwhile, the mixture of cell mass and substrate is continuously harvested from the reactor through the outflow chamber. The dynamics in the chemostat can be investigated by using the chemostat model [2].
Ordinary differential equations (ODEs) are commonly used for modelling biological systems. However, most biological systems behaviour has memory effects, and ODEs usually neglect such effects. The fractional-order differential equations (FDEs) are taken into account when describing the behaviour of the systems' equations. A FDEs is a generalisation of the ODEs to random nonlinear order [3]. This equation is more effective because of its good memory, among other advantages [4]–[8]. The errors occuring from the disregarded parameters when modelling of phenomena in real-life also can be reduced. FDEs are also used to efficiently replicate the real nature of various systems in the field of engineering and sciences [9]. In the past few decades, FDEs have been used in biological systems for various studies [5], [6], [10]–[17].
Since great strides in the study of FDEs have been developed, the dynamics in the chemostat can be investigated using the mathematical model of the chemostat in the form of FDEs. Moreover, there have been few studies on the expansion of the chemostat model with fractional-order theory. Thus, we deepen and complete the analysis on the integer-order chemostat model with fractional-order theory and discuss the stability of the equilibrium points of the fractional-order chemostat model. Next, the bifurcation analysis for the fractional-order chemostat model is conducted to identify the bifurcation point that can change the stability of the system. The analysis identifies the values of the fractional-order and the system parameters to ensure the operation of the chemostat is well-controlled.
Recently, there are many approaches to define fractional operators such as by Caputo, Riemann-Liouville, Hadamard, and Grunwald-Letnikov [3]. However, Caputo is often used due to its convenience in various applications [18]. Caputo is also useful to encounter an obstacle where the initial condition is done in the differential of integer-order [19]. In this paper, we applied Caputo derivative to define the system of fractional-order. The Caputo derivative for the left-hand side is defined as
where Г denotes the function of gamma, n is an integer, where
The Adams-type predictor-corrector method is one of the technique that have been proposed for fractional-order differential equations [19]–[21]. The Adams-type predictor-corrector method is a analysis of numerical algorithm that involves two basics steps: predictor and corrector. The predictor formula can be described as
meanwhile the corrector formula can be determined by
The predictor-corrector method is also called as the PECE (Predict, Evaluate, Correct, Evaluate) method [22]. The procedure of the predictor-corrector method can be explained as follows
(i) Calculate the predictor step,
(ii) Evaluate
(iii) Calculate the corrector step,
(iv) Evaluate
The procedure repeatedly predicts and corrects the value until the corrected value becomes a converged number [21]. This method is able to maintain the stability of the properties and has good accuracy. Moreover, this method also has lower computational cost than other methods [23]. The algorithm of the Adams-type predictor-corrector method proposed by [22] is shown in Appendix.
The conditions of stability for integer-order differential equations and fractional-order differential equations are different. Both systems could have the same steady-state points but different stability conditions [24], [25]. The stability condition for fractional-order differential equations can be stated by Theorem 1 and the Routh-Hurwitz stability condition as described by Proposition 1.
Theorem 1
Proposition 1
or
The stability theorem on the fractional-order systems and fractional Routh-Hurwitz stability conditions are introduced to analyze the stability of the model. The fractional Routh-Hurwitz stability conditions is specifically introduced for the eigenvalues of the Jacobian matrix that obtained in quadratic form. The proof of this proposition is shown in Appendix.
Bifurcation can be defined as any sudden change that occurs while a parameter value is varied in the differential equation system and it has a significant influence on the solution [27]. An unstable steady-state may becomes stable and vice versa. A slight changes in the parameter value may change the system's stability. Despite the steady-state point and the eigenvalues of the system of fractional-order are similar as the system of integer-order, the discriminant method used for the stability of the steady-state point is different. Accordingly, the Hopf bifurcation condition in the fractional-order system is slightly different as compared with the integer-order system.
Fractional order α can be selected as the bifurcation parameter in a fractional-order system, but this is not allowed in an integer-order system. The existence of Hopf bifurcation can be stated as in Theorem 2.
Theorem 2
(i) 1. The characteristic equation of chemostat system has a pair of complex conjugate roots,
(ii) Critical value
(iii)
Proof. Condition (i) is not easy to obtain due to the selected parameter's value. However, this condition can be managed under some confined conditions. In fact, the washout steady-state solution of the chemostat model has two negative real roots. The remaining two roots depend on the characteristic of the polynomial from the no-washout steady-state solution.
Condition (ii) can be satisfied with the existence of critical value α* and when
The integer system required p = 0 for the bifurcation's operating condition. For the fractional-order system, the operating condition of the system will change into
In studying the dynamic process of chemostat, the parameters such as
Theorem 3
(i) The characteristic equation of chemostat system has a pair of complex conjugate roots,
(ii) Critical value
(iii)
Proof. This theorem can be proved in the same way as Theorem 2. Therefore, condition (i) can be guaranteed. Condition (ii) can be satisfied with the existence of critical value
For condition (iii), the condition of
Firstly, determine the steady-states, Jacobian matrix and eigenvalues of the fractional-order chemostat model. The stability properties of the fractional-order chemostat model were estimated by using the stability and bifurcation analyses with FDEs by referring to Theorem 1 and Proposition 1. Then, determine the bifurcation point of fractional-order by referring to Theorem 2 and determine the bifurcation point of parameter values by referring to Theorem 3. Next, plot the phase portrait of fractional-order chemostat model by using Adam-types predictor-corrector method to study the dynamic behaviour of the system. Figure 1 depicts the flowchart of this research. This flowchart can be applied to all problems with suitable parameter values.
An integer-order chemostat model that considered a variable yield coefficient and the Monod growth model from [1] is studied in this section. The chemostat system can be written as
with the initial value of
Let Eq. (3.2) equal to zero in order to find the steady-state solutions
By solving Eq. (3.4), the following solutions are obtained
From
Hence, the solutions of steady-state for the chemostat model are
(i) Washout:
(ii) No Washout:
where
The steady-state solutions are physically meaningful if their components are positive. Therefore,
The solution of steady-state in Equation (3.8) represents the washout situation, where the cell mass is wholly removed from the reactor and where the substrate concentration is at the same stock as in the beginning. This state must always be unstable in order to ensure that the cell mass is able to grow in the chemostat. This is because the cell mass will be continuously removed from the chemostat if the washout steady-state is stable. The Jacobian matrix for the washout steady-state solution can be written as
The eigenvalues of this matrix are
The eigenvalues in
The eigenvalues of the Jacobian matrix in terms of the characteristic polynomial are
where
and
The eigenvalues of the no-washout steady-state solution were evaluated with Routh-Hurwitz condition in Proposition 1. Based on the eigenvalues in Eq. (3.15) and by referring to the study by [5], the eigenvalues' condition can be simplified as the following two cases
(i) If b>0 or equivalent to
(ii) If b<0 or equivalent to
Then, if
The parameter values of the fractional-order chemostat model are provided in Table 1. The initial substrate concentration, S0 and ρ were assumed as non-negative values to ensure that the steady-state solutions were physically meaningful. The stability diagram of the steady-state solutions is plotted in Figure 2.
Parameters | Description | Values | Units |
k | Saturation constant | 1.75 | gl−1 |
Q | Dilution rate | 0.02 | l2gr−1 |
µ | Maximum growth rate | 0.3 | h−1 |
γ | Constant in yield coefficient | 0.01 | – |
β | Constant in yield coefficient | 5.25 | lg−1 |
S0 | Input concentration of substrate | 1 | gl−1 |
The washout steady-state solution is stable if Q > 0 and
The steady-state solutions of the fractional-order chemostat model for the parameter values given in Table 1 are
(i) Washout:
(ii) No Washout:
The eigenvalues obtained from the washout steady-state solution are
and the eigenvalues from the no-washout steady-state solution are
Based on Eq. (3.22) to Eq. (3.25), these satisfied the first condition of Hopf bifurcation in Theorem 2. There exists a pair of complex conjugate roots and the other eigenvalues are negative real roots. The transversality condition as the third condition is also satisfied. The eigenvalues of the washout steady-state solution based on the chemostat system is not imaginary, and so there is no existence of Hopf bifurcation in the washout steady-state solution. According to Theorem 2, the critical value of the fractional-order as stated in the second condition can be obtained as
where
Value of p and q are obtained from
The running state of the fractional-order chemostat system when fractional order α at the Hopf bifurcation point is shown. The fractional-order chemostat system changed its stability once Hopf bifurcation occurred. Therefore, we conjecture that the system of fractional-order chemostat may be lost or gain its stability when the fractional order α is less than the Hopf bifurcation point, or
The initial concentration of the substrate, S0, was chosen as the control parameter, while fractional order α was fixed. The solutions of steady-state of the fractional-order chemostat model with S0 as the control parameter are
(i) Washout:
(ii) No Washout:
The eigenvalues obtained from the washout steady-state solution are
and the eigenvalues from the no-washout steady-state solution are
These satisfied the first condition of Hopf bifurcation in Theorem 3. There exist a pair of complex conjugate roots in terms of S0, and the other eigenvalues were negative real roots in terms of S0. The transversality condition as the third condition is also satisfied. According to Theorem 3, the critical value of the fractional order as stated in the second condition can be obtained as follows
By referring to the study by [18], Eq. (3.37) can also be calculated as
From the calculations, the critical value of the initial concentration of the substrate is
The change in the running state when the value of the initial substrate concentration passes through the Hopf bifurcation point is shown. The stability of the fractional-order chemostat system changed once Hopf bifurcation occurred. In
The Hopf bifurcation points of system of fractional-order chemostat and system of integer-order chemostat are different.
The stability analysis of the fractional-order chemostat model was conducted based on the stability theory of FDEs. The integer-order chemostat model was extended to the FDEs. There are two steady-state solutions obtained, which are washout and no-washout steady-state solutions. The Hopf bifurcation of the order of α occured at the solutions of steady-state when the Hopf bifurcation conditions is fulfilled. The results show that the increasing or decreasing the value of α may stabilise the unstable state of the chemostat system. Therefore, the running state of the fractional-order chemostat system is affected by the value of α. The Hopf bifurcation of the initial concentration of the substrate, S0, also occurred when the Hopf bifurcation condition is fulfilled. As the evidence from the phase portrait plots, increase the value of the initial substrate concentration may destabilise the stable state of the chemostat system. The value of the initial substrate should remain at
[1] | Viral Infections. Physiopedia (2022) . Available from: http:///index.php?title=Viral_Infections&oldid=290753 |
[2] |
Betts MJ, Russell RB (2003) Amino acid properties and consequences of subsitutions. Bioinformatics for Geneticists.Wiley 289-316. https://doi.org/10.1002/0470867302.ch14 ![]() |
[3] |
Lipfert J, Doniach S, Das R, et al. (2014) Understanding nucleic acid-ion interactions. Annu Rev Biochem 83: 813-841. https://doi.org/10.1146/annurev-biochem-060409-092720 ![]() |
[4] |
Ma Y, Poole K, Goyette J, et al. (2017) Introducing membrane charge and membrane potential to T cell signaling. Front Immunol 8: 1513. https://doi.org/10.3389/fimmu.2017.01513 ![]() |
[5] |
Zhao X, Ma X, Dupius JH, et al. (2022) Negatively charged phospholipids accelerate the membrane fusion activity of the plant-specific insert domain of an aspartic protease. J Biol Chem 298: 101430. https://doi.org/10.1016/j.jbc.2021.101430 ![]() |
[6] |
Cruz-Chu ER, Malafeev A, Pajarskas T, et al. (2014) Structure and response to flow of the glycocalyx layer. Biophys J 106: 232-242. https://doi.org/10.1016/j.bpj.2013.09.060 ![]() |
[7] | Debye P, Hückel E (1923) The theory of electrolytes. I. Lowering of freezing point and related phenomena (PDF). Physikalische Zeitschrift 24: 185-206. |
[8] |
Michen B, Graule T (2010) Isoelectric points of viruses. J Appl Microbiol 109: 388-397. https://doi.org/10.1111/j.1365-2672.2010.04663.x ![]() |
[9] |
Heffron J, Mayer BK (2021) Virus isoelectric point estimation: theories and methods. J Appl Environ Microb 87: e02319-e02320. https://doi.org/10.1128/AEM.02319-20 ![]() |
[10] |
Luisetto M, Tarro G, Edbey K, et al. (2021) Coronavirus COVID-19 surface properties: electrical charges status. Int J Clin Microbiol Biochem Technol 4: 016-027. https://doi.org/10.12688/f1000research.108667.2 ![]() |
[11] |
Cavezzi A, Menicagli R, Troiani E, et al. (2022) COVID-19, cation dysmetabolism, sialic acid, CD147, ACE2, viroporins, hepcidin and ferroptosis: a possible unifying hypothesis. F1000Res 11: 102. https://doi.org/10.12688/f1000research.108667.2 ![]() |
[12] | Northwestern University.“Research exposes new vulnerability for SARS-CoV-2: Electrostatic interactions enhance the spike protein's bond to host cells.”. ScienceDaily (2020) . Available from: https://www.sciencedaily.com/releases/2020/08/200811120227.htm |
[13] |
Leung WWF, Sun Q (2020) Electrostatic charged nanofiber filter for filtering airborne novel coronavirus (COVID-19) and nano-aerosols. Sep Purif Technol 250: 116886. https://doi.org/10.1016/j.seppur.2020.116886 ![]() |
[14] |
Corrêa Giron C, Laaksonen A, Barroso da Silva FL (2020) On the interactions of the receptor-binding domain of SARS-CoV-1 and SARS-CoV-2 spike proteins with monoclonal antibodies and the receptor ACE2. Virus Res 285: 198021. https://doi.org/10.1016/j.virusres.2020.198021 ![]() |
[15] |
Chavda VP, Bezbaruah R, Deka K, et al. (2022) The delta and omicron variants of SARS-CoV-2: What we know so far. Vaccines 10: 1926. https://doi.org/10.3390/vaccines10111926 ![]() |
[16] | Carabelli AM, Peacock TP, Thorne LG, et al. (2023) SARS-CoV-2 variant biology: immune escape, transmission and fitness. Nat Rev Microbiol 21: 162-177. https://doi.org/10.1038/s41579-022-00841-7 |
[17] |
Chavda VP, Patel AB, Vaghasiya DD (2022) SARS-CoV-2 variants and vulnerability at the global level. J Med Virol 94: 2986-3005. https://doi.org/10.1002/jmv.27717 ![]() |
[18] |
Chavda VP, Ghali ENHK, Yallapu MM, et al. (2022) Therapeutics to tackle Omicron outbreak. Immunotherapy 14: 833-838. https://doi.org/10.2217/imt-2022-0064 ![]() |
[19] |
Chavda VP, Vuppu S, Mishra T, et al. (2022) Recent review of COVID-19 management: diagnosis, treatment and vaccination. Pharmacol Rep 74: 1120-1148. https://doi.org/10.1007/s43440-022-00425-5 ![]() |
[20] |
Polatoğlu I, Oncu-Oner T, Dalman I, et al. (2023) COVID-19 in early 2023: Structure, replication mechanism, variants of SARS-CoV-2, diagnostic tests, and vaccine & drug development studies. MedComm 4: e228. https://doi.org/10.1002/mco2.228 ![]() |
[21] |
Basu D, Chavda VP, Mehta AA (2022) Therapeutics for COVID-19 and post COVID-19 complications: an update. Curr Res Pharmacol Drug Discov 3: 100086. https://doi.org/10.1016/j.crphar.2022.100086 ![]() |
[22] |
Lancet T (2023) The COVID-19 pandemic in 2023: far from over. Lancet 401: 79. https://doi.org/10.1016/s0140-6736(23)00050-8 ![]() |
[23] |
Arbeitman CR, Rojas P, Ojeda-May P, et al. (2021) The SARS-CoV-2 spike protein is vulnerable to moderate electric fields. Nat Commun 12: 5407. https://doi.org/10.1038/s41467-021-25478-7 ![]() |
[24] |
Božič A, Podgornik R (2024) Changes in total charge on spike protein of SARS-CoV-2 in emerging lineages. Bioinformatics Adv 4: vbae053. https://doi.org/10.1093/bioadv/vbae053 ![]() |
[25] |
Javidpour L, Božič A, Naji A, et al. (2020) Electrostatic interaction between SARS-CoV-2 virus and charged electret fibre. Soft Matter 17: 4296-4303. https://doi.org/10.1039/D1SM00232E ![]() |
[26] |
Zhang Z, Zhang J, Wang J (2022) Surface charge changes in spike RBD mutations of SARS-CoV-2 and its variant strains alter the virus evasiveness via HSPGs: a review and mechanistic hypothesis. Front Public Health 10: 952916. https://doi.org/10.3389/fpubh.2022.952916 ![]() |
[27] |
Bar-On YM, Flamholz A, Phillips R, et al. (2020) SARS-CoV-2 (COVID-19) by the numbers. elife 9: e57309. https://doi.org/10.7554/eLife.57309 ![]() |
[28] |
Berman HM, Westbrook J, Feng Z, et al. (2000) The protein data bank. Nucleic Acids Res 28: 235-242. https://doi.org/10.1093/nar/28.1.235 ![]() |
[29] |
Sayers EW, Bolton EE, Brister JR, et al. (2022) Database resources of the national center for biotechnology information. Nucleic Acids Res 50: D20-D26. https://doi.org/10.1093/nar/gkab1112 ![]() |
[30] |
Jiao LG, Zan LR, Zhu L, et al. (2019) Accurate computation of screened Coulomb potential integrals in numerical Hartree–Fock programs. Comput Phys Commun 244: 217-227. https://doi.org/10.1016/j.cpc.2019.06.001 ![]() |
[31] | SARS-CoV-2 variants of concern and variants under investigation in England Technical briefing 15. Public Health England (2021) . Available from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/993879/Variants_of_Concern_VOC_Technical_Briefing_15.pdf |
[32] | SARS-CoV-2 variants of concern and variants under investigation in England Technical briefing 23. Public Health England, PHE2 (2021) . Available from: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1018547/Technical_Briefing_23_21_09_16.pdf |
[33] |
Adamczyk Z, Batys P, Barbasz J (2021) SARS-CoV-2 virion physicochemical characteristics pertinent to abiotic substrate attachment. Curr Opin Colloid Interface Sci 55: 101466. https://doi.org/10.1016/j.cocis.2021.101466 ![]() |
[34] |
Berger O, Edholm O, Jähnig F (1997) Molecular dynamics simulations of a fluid bilayer of dipalmitoylphosphatidylcholine at full hydration, constant pressure, and constant temperature. Biophys J 72: 2002-2013. https://doi.org/10.1016/S0006-3495(97)78845-3 ![]() |
[35] |
Krebs F, Scheller C, Grove-Heike K (2021) Isoelectric point determination by imaged CIEF of commercially available SARS-CoV-2 proteins and the hACE2 receptor. Electrophoresis 42: 687-692. https://doi.org/10.1002/elps.202100015 ![]() |
[36] | Pawłowski PH (2021) SARS-CoV-2 variant Omicron (B.1.1.529) is in a rising trend of mutations increasing the positive electric charge in crucial regions of the spike protein S. Acta Biochim Pol 69: 263-264. https://doi.org/10.18388/abp.2020_6072 |
[37] |
Cotten M, Phan M (2022) Evolution to increased positive charge on the viral spike protein may be part of the adaptation of SARS-CoV-2 to human transmission. iScience 26: 106230. https://doi.org/10.1016/j.isci.2023.106230 ![]() |
[38] |
Lu Y, Zhao T, Lu M, et al. (2021) The analyses of high infectivity mechanism of SARS-CoV-2 and its variants. COVID 1: 666-673. https://doi.org/10.3390/covid1040054 ![]() |
[39] |
Pawłowski PH (2021) Additional positive electric residues in the crucial spike glycoprotein S regions of the new SARS-CoV-2 variants. Infect Drug Resist 14: 5099-5105. https://doi.org/10.2147/IDR.S342068 ![]() |
[40] | Bromage E The risks-know them-avoid them (2020). Available from: https://www.erinbromage.com/post/the-risks-know-them-avoid-them |
[41] | Islam MA, Ford Versypt AN (2022) Mathematical modeling of impacts of patient differences on COVID-19 lung fibrosis outcomes. bioRxiv [Preprint] . https://doi.org/10.1101/2022.11.06.515367 |
[42] | Coulomb CA (1785) Premier mémoire sur l'électricité et le magnétisme" [First dissertation on electricity and magnetism]. Histoire de l'Académie Royale des Sciences [History of the Royal Academy of Sciences] (in French) : 569-577. https://www.academie-sciences.fr/pdf/dossiers/Coulomb/Coulomb_pdf/Mem1785_p569.pdf |
[43] |
Kobayashi Y, Suzuki Y (2012) Compensatory evolution of net-charge in influenza A virus hemagglutinin. PLoS One 7: e40422. https://doi.org/10.1371/journal.pone.0040422 ![]() |
[44] |
Xia S, Liu M, Wang C, et al. (2020) Inhibition of SARS-CoV-2 (previously 2019-nCoV) infection by a highly potent pan-coronavirus fusion inhibitor targeting its spike protein that harbors a high capacity to mediate membrane fusion. Cell Res 30: 343-355. https://doi.org/10.1038/s41422-020-0305-x ![]() |
[45] |
Brunton B, Rogers K, Phillips EK, et al. (2019) TIM-1 serves as a receptor for Ebola virus in vivo, enhancing viremia and pathogenesis. PLoS Negl Trop Dis 13: e0006983. https://doi.org/10.1371/journal.pntd.0006983 ![]() |
[46] | Fahimi H, Allahyari H, Hassan ZM, et al. (2014) Dengue virus type-3 envelope protein domain III; expression and immunogenicity. Iran J Basic Med Sci 11: 836-843. |
[47] |
Saad-Roy CM, Arinaminpathy N, Wingreen NS, et al. (2020) Implications of localized charge for human influenza A H1N1 hemagglutinin evolution: insights from deep mutational scans. PLoS Comput Biol 16: e1007892. https://doi.org/10.1371/journal.pcbi.1007892 ![]() |
[48] |
Feng Z, Xu L, Xie Z (2022) Receptors for respiratory syncytial virus infection and host factors regulating the life cycle of respiratory syncytial virus. Front Cell Infect Microbiol 12: 858629. https://doi.org/10.3389/fcimb.2022.858629 ![]() |
[49] |
Petrache HI, Tristram-Nagle S, Gawrisch K, et al. (2004) Structure and fluctuations of charged phosphatidylserine bilayers in the absence of salt. Biophys J 86: 1574-1586. https://doi.org/10.1016/S0006-3495(04)74225-3 ![]() |
[50] | Varki A, Schauer R (2009) Sialic Acids. Essentials of Glycobiology. Cold Spring Harbor (NY): Cold Spring Harbor Laboratory Press. |
[51] |
Lupala CS, Lil X, Lei J, et al. (2021) Computational simulations reveal the binding dynamics between human ACE2 and the receptor binding domain of SARS-CoV-2 spike protein. Quant Biol 9: 61-72. https://doi.org/10.15302/J-QB-020-0231 ![]() |
[52] |
Pawłowski PH (2021) Charged amino acids may promote coronavirus SARS-CoV-2 fusion with the host cell. AIMS Biophys 8: 111-121. https://doi.org/10.3934/biophy.2021008 ![]() |
[53] |
Romano M, Ruggiero A, Squeglia F, et al. (2020) A structural view of SARS-CoV-2 RNA replication machinery: RNA synthesis, proofreading and final capping. Cells 9: 1267. https://doi.org/10.3390/cells9051267 ![]() |
[54] |
Silva JRA, Urban J, Araújo E, et al. (2022) Exploring the catalytic mechanism of the RNA cap modification by nsp16-nsp10 complex of SARS-CoV-2 through a QM/MM approach. Int J Mol Sci 23: 300. https://doi.org/10.3390/ijms23010300 ![]() |
[55] |
Kirchdoerfer RN, Ward AB (2019) Structure of the SARS-CoV nsp12 polymerase bound to nsp7 and nsp8 co-factors. Nat Commun 10: 2342. https://doi.org/10.1038/s41467-019-10280-3 ![]() |
[56] |
Subissi L, Posthuma CC, Collet A, et al. (2014) One severe acute respiratory syndrome coronavirus protein complex integrates processive RNA polymerase and exonuclease activities. Proc Natl Acad Sc USA 111: E3900-E3909. https://doi.org/10.1073/pnas.1323705111 ![]() |
[57] |
Bianchi M, Borsetti A, Ciccozzi M, et al. (2021) SARS-Cov-2 ORF3a: mutability and function. Int J Biol Macromol 15: 820-826. https://doi.org/10.1016/j.ijbiomac.2020.12.142 ![]() |
[58] |
Scheller C, Krebs F, Minkner R, et al. (2020) Physicochemical properties of SARS-CoV-2 for drug targeting, virus inactivation and attenuation, vaccine formulation and quality control. Electrophoresis 41: 1137-1151. https://doi.org/10.1002/elps.202000121 ![]() |
[59] |
Bohan D, Ert HV, Ruggio N, et al. (2021) Phosphatidylserine receptors enhance SARS-CoV-2 infection. PLoS Pathog 17: e1009743. https://doi.org/10.1371/journal.ppat.1009743 ![]() |
[60] | Alberts B, Johnson A, Lewis J, et al. (2002) Molecular Biology of the Cell. New York: Garland Science. |
[61] |
Jalloh S, Olejnik J, Berrigan J, et al. (2022) CD169-mediated restrictive SARS-CoV-2 infection of macrophages induces pro-inflammatory responses. PLoS pathogens 18: e1010479. https://doi.org/10.1371/journal.ppat.1010479 ![]() |
[62] |
Watanabe Y, Allen JD, Wrapp D, et al. (2020) Site-specific glycan analysis of the SARS-CoV-2 spike. Science 369: 330-333. https://doi.org/10.1126/science.abb9983 ![]() |
[63] |
von Glasow R, Sander R (2001) Variation of sea salt aerosol pH with relative humidity. Geophys Res Lett 28: 247-250. https://doi.org/10.1029/2000GL012387 ![]() |
[64] |
Field RD, Moelis N, Salzmann J, et al. (2021) Inhaled water and salt suppress respiratory droplet generation and COVID-19 incidence and death on US coastlines. Mol Front J 5.01n02: 17-29. https://doi.org/10.1142/S2529732521400058 ![]() |
[65] |
Duran-Meza AL, Villagrana-Escareño MV, Ruiz-García J, et al. (2021) Controlling the surface charge of simple viruses. PLoS One 16: e0255820. https://doi.org/10.1371/journal.pone.0255820 ![]() |
[66] |
Vega-Acosta JR, Cadena-Nava RD, Gelbart WM, et al. (2014) Electrophoretic mobilities of a viral capsid, its capsid protein, and their relation to viral assembly. J Phys Chem B 118: 1984-1989. https://doi.org/10.1021/jp407379t ![]() |
[67] |
Bockstahler LE, Kaesberg P (1962) The molecular weight and other biophysical properties of bromegrass mosaic virus. Biophys J 2: 1962. https://doi.org/10.1016/s0006-3495(62)86836-2 ![]() |
[68] |
Johnson MW, Wagner GW, Bancroft JB (1973) A titrimetric and electrophoretic study of cowpea chlorotic mottle virus and its protein. J Gen Virol 19: 263-273. https://doi.org/10.1099/0022-1317-19-2-263 ![]() |
[69] |
van der Schoot P, Bruinsma R (2005) Electrostatics and the assembly of an RNA virus. Phys Rev E 71: 061928. https://doi.org/10.1103/PhysRevE.71.061928 ![]() |
[70] |
Belyi VA, Muthukumar M (2006) Electrostatic origin of the genome packing in viruses. Proc Natl Acad Sci 103: 17174-17178. https://doi.org/10.1073/pnas.0608311103 ![]() |
[71] |
Hagan MF (2009) A theory for viral capsid assembly around electrostatic cores. J Chem Phys 130: 114902. https://doi.org/10.1063/1.3086041 ![]() |
[72] |
Lorenzo-Leal AC, Vimalanathan S, Bach H (2022) Adherence of SARS-CoV-2 delta variant to a surgical mask and N95 respirators. Future Sci OA 8: FSO808. https://doi.org/10.2144/fsoa-2022-0025 ![]() |
[73] |
Javidpour L, Božič A, Najili A, et al. (2021) Electrostatic interaction between SARS-CoV-2 virus and charged electret fibre. Soft Matter 17: 4296-4203. https://doi.org/10.1039/D1SM00232E ![]() |
[74] |
Ren C, Haghighat F, Feng Z, et al. (2023) Impact of ionizers on prevention of airborne infection in classroom. Build Simul 16: 749-764. https://doi.org/10.1007/s12273-022-0959-z ![]() |
[75] |
Fantini J, Azzaz F, Chahinian H, et al. (2023) Electrostatic surface potential as a key parameter in virus transmission and evolution: How to manage future virus pandemics in the post-COVID-19 era. Viruses 15: 284. https://doi.org/10.3390/v15020284 ![]() |
[76] |
Wood JP, Magnuson M, Touati A, et al. (2021) Hook Evaluation of electrostatic sprayers and foggers for the application of disinfectants in the era of SARS-CoV-2. PLoS One 16: e0257434. https://doi.org/10.1371/journal.pone.0257434 ![]() |
[77] |
Kalra RS, Kandimalla R (2021) Engaging the spikes: heparan sulfate facilitates SARS-CoV-2 spike protein binding to ACE2 and potentiates viral infection. Sig Transduct Target Ther 6: 39. https://doi.org/10.1038/s41392-021-00470-1 ![]() |
[78] |
Pawłowski P, Szutowicz I, Marszałek P, et al. (1993) Bioelectrorheological model of the cell. 5. Electrodestruction of cellular membrane in alternating electric field. Biophys J 65: 541-549. https://doi.org/10.1016/S0006-3495(93)81056-7 ![]() |
[79] |
Igakura T, Stinchcombe JC, Goon PK, et al. (2003) Spread of HTLV-I between lymphocytes by virus-induced polarization of the cytoskeleton. Science 299: 1713-1716. https://doi.org/10.1126/science.1080115 ![]() |
1. | Xiaomeng Ma, Zhanbing Bai, Sujing Sun, Stability and bifurcation control for a fractional-order chemostat model with time delays and incommensurate orders, 2022, 20, 1551-0018, 437, 10.3934/mbe.2023020 |
Parameters | Description | Values | Units |
k | Saturation constant | 1.75 | gl−1 |
Q | Dilution rate | 0.02 | l2gr−1 |
µ | Maximum growth rate | 0.3 | h−1 |
γ | Constant in yield coefficient | 0.01 | – |
β | Constant in yield coefficient | 5.25 | lg−1 |
S0 | Input concentration of substrate | 1 | gl−1 |