Peplomer bulb shape and coronavirus rotational diffusivity

Recently, we calculated the rotational diffusivity of the coronavirus particle in suspension as a function of peplomer population, from first principles, using general rigid bead-rod theory (Fig. 12 of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875). We did so by beading the capsid and then also by replacing each of its bulbous spikes with a single bead (Fig. 1). One of the challenges of ab initio calculations from general rigid bead-rod theory on coronaviruses is that the peplomer arrangement is not known. However, we do know that the spikes are charge-rich.^2,32. Q. Yao, P. S. Masters, and R. Ye, “ Negatively charged residues in the endodomain are critical for specific assembly of spike protein into murine coronavirus,” Virology 442(1), 74–81 (2013). https://doi.org/10.1016/j.virol.2013.04.0013. R. Ye, C. Montalto-Morrison, and P. S. Masters, “ Genetic analysis of determinants for spike glycoprotein assembly into murine coronavirus virions: Distinct roles for charge-rich and cysteine-rich regions of the endodomain,” J. Virol. 78(18), 9904–9917 (2004). https://doi.org/10.1128/JVI.78.18.9904-9917.2004 It also seems reasonable to assume that they are charged identically. Furthermore, we know that the coronavirus spikes are not anchored into its hard capsid, but rather just into its elastic viral membrane (Sec. 1 of Ref. 44. M. A. Tortorici and D. Veesler, “ Structural insights into coronavirus entry,” Adv. Virus Res. 105, 93–116 (2019). https://doi.org/10.1016/bs.aivir.2019.08.002). The coronavirus spikes are, thus, free to rearrange under their own electrostatic repulsions. This is why coronavirus spikes normally present microscopically as uniformly distributed over the capsid. In our previous work, we followed the well-known polyhedral solutions to the Thomson problem for singly charged particles repelling one another over a spherical surface.^5–75. J. J. Thomson, “ XXIV. On the structure of the atom: An investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure,” London, Edinburgh, Dublin Philos. Mag. J. Sci., Ser. 6 7(39), 237–265 (1904). https://doi.org/10.1080/147864404094631076. D. J. Wales and S. Ulker, “ Structure and dynamics of spherical crystals characterized for the Thomson problem,” Phys. Rev. B 74(21), 212101 (2006). https://doi.org/10.1103/PhysRevB.74.2121017. D. J. Wales, H. McKay, and E. L. Altschuler, “ Defect motifs for spherical topologies,” Phys. Rev. B 79(22), 224115 (2009). https://doi.org/10.1103/PhysRevB.79.224115 By Thomson problem, we mean determination of how identically charged particles repel and then spread over a sphere by minimizing system potential energy. This minimum system electrostatic potential energy, when divided by the sphere area, is not to be confused with surface energy.

Since each coronavirus spike is a glycoprotein trimer, each spike bulb is triangular (Fig. 14 of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875). By replacing each coronavirus spike bulb with a single bead (see circle in Fig. 14 of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875), our prior work neglects this triangularity. In this present work, we replace each bulbous coronavirus spike (Fig. 2) with a bead triplet (Fig. 3), with each bead identical and charged identically. We must, thus, replace the well-known polyhedral solutions to the single-bead Thomson problem with our new solutions to the triple-bead Thomson problem. In this work, we thus use minimum potential energy peplomer arrangements for our coronavirus model particles.

Since coronavirus bulbs are trimers, they not only translate into a set of centroidal positions relative to one another but also twist into a set of orientations relative to one another. Our potential energy minimization for our triply beaded peplomers thus yields both triplet positions and triplet orientations (Fig. 4). This new potential energy minimization yields a set of bead positions for the triply beaded peplomers whose centroid positions differ, of course, from the bead positions for the singly beaded counterpart of the same $N_p.$ In other words, the polyhedra of centroids differ from the well-known Thomson solutions used in Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875.

The challenge in determining the rotational diffusivity of a virus particle, from first principles, begins with modeling its intricate geometry with beads, locating the position of each bead. Once overcome, the next challenge is to use this geometry to arrive at the transport properties for the SARS-CoV-2 particle. From these, we deepen our understanding of how these remarkable particles align their peplomers both for long enough, and often enough, to infect.¹1. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875

Whereas our prior work relied on the Thomson solution for point charges (Fig. 1), here, we work with triads of point charges each spaced rigidly and equilaterally (Fig. 4). We, thus, complicate the energy minimization with the length of this equilateral triangle, $r_{\Delta }$. From Table X of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875, mindful of Fig. 8 of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875, we get

$$\frac{1}{10}\le \frac{r_{\Delta }}{r_p}\le \frac{7}{25},$$

(1)

and in this work, we choose $r_{\Delta }/r_p=0.19$ for our energy minimization. To compare with our previous work, we match the dimensionless virus radius of Fig. 12 of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875, $r_v/r_c=5/4$. Using the energy minimization to arrange and orient the coronavirus spikes relative to one another, we next arrive, from first principles, at the rotational diffusivity of the coronavirus particles with triple beaded peplomers in suspension.

For this work, we chose general rigid bead-rod theory for its flexibility and accuracy (Sec. I of Refs. 88. M. A. Kanso, A. J. Giacomin, C. Saengow, and J. H. Piette, “ Macromolecular architecture and complex viscosity,” Phys. Fluids 31(8), 087107 (2019); Editor's pick. Errata: Eq. (21) should be “ $a^2{\nu }^2+\frac{2}{3}(6b-9)a\nu +\frac{1}{9}(36b^2-123b+81)=0$;” in Table XIV, $n_0-n_s$ should be ${\eta }_0-{\eta }_s$. In Table XV, ${\psi }_{1,0}$ should be ${\Psi }_{1,0}$, and $\mathit{nKT}$ should be $\mathit{nkT}$. In Table IV, Macromolecule 21 entry should be $\left[-\frac{1}{2}L,-\frac{\sqrt{3}}{2}L,0\right]$; $\left[L,0,0\right]$; $\left[0,0,0\right]$; $\left[-\frac{1}{2}L,\frac{\sqrt{3}}{2}L,0\right]$ and Macromolecule 17 entry should be multiplied by $L$. In Eq. (44), $\eta ″$ should be $2\eta ″$. https://doi.org/10.1063/1.5111763 and 99. M. A. Kanso, “ Polymeric liquid behavior in oscillatory shear flow,” Masters thesis (Polymers Research Group, Chemical Engineering Department, Queen's University, Kingston, Canada, 2019).). Using general rigid bead-rod theory, we follow the method of Sec. II of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875 to construct our virus particles from sets of beads whose positions are fixed relative to one another. For example, the SARS-CoV-2 particle geometry is a spherical capsid surrounded by a constellation of protruding peplomers. We take our bead-rod models of virus particles to be suspended in a Newtonian solvent. To any such collection of bead masses, we can associate a moment of inertia ellipsoid (MIE) whose center is the center of mass and whose principal moments of inertia match those of the virus particle. The MIE, thus, determines the orientability of the virus particle, and thus, the virus rotational diffusivity. We use Eqs. (3)–(13) in Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875 for the method of computing the rotational diffusivity (see Footnote 2 of p. 62 of Ref. 1010. R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, 2nd ed. ( John Wiley & Sons, Inc., New York, 1987), Vol. 2; Errata: On p. 409 of the first printing, the (n + m)! in the denominator should be (n + m)!; in Table 16.4–1, under L entry “length of rod” should be “bead center to center length of a rigid dumbbell;” in the Fig. 14.1–2 caption, “Multibead rods of length $L$” should be “Multibead rods of length L + d.”)

$$D_r\equiv \frac{1}{6\lambda },$$

(2)

or [Eq. (23) of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875]

$${\lambda }_0{D}_r=\frac{\nu }{72},$$

(3)

which we will use for our results below. Symbols, dimensional, or nondimensional are defined in Table I or Table II, following the companion paper for singly beaded peplomer for SARS-CoV-2 particle.¹1. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875

In this paper, we focus on small-amplitude oscillatory shear flow (SAOS). For this flow field, for the molecular definition of small amplitude, general rigid bead-rod theory yields [Eq. (32) of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875]

$${\lambda \dot{\gamma }}^0\ll \frac{1}{\nu \sqrt{2}},$$

(4)

whose left side is the macromolecular Weissenberg number.

The polymer contributions to the complex viscosity,^11,1211. R. B. Bird and A. J. Giacomin, “ Who conceived the complex viscosity?,” Rheol. Acta 51(6), 481–486 (2012). https://doi.org/10.1007/s00397-012-0621-212. A. J. Giacomin and R. B. Bird, “ Erratum: Official Nomenclature of The Society of Rheology-η” J. Rheol. 55(4), 921–923 (2011). https://doi.org/10.1122/1.3586815

$${\eta }^{*}\equiv {\eta }^{\prime }-{i\eta }^{″},$$

(5)

are [Eqs. (40) and (41) of Ref. 88. M. A. Kanso, A. J. Giacomin, C. Saengow, and J. H. Piette, “ Macromolecular architecture and complex viscosity,” Phys. Fluids 31(8), 087107 (2019); Editor's pick. Errata: Eq. (21) should be “ $a^2{\nu }^2+\frac{2}{3}(6b-9)a\nu +\frac{1}{9}(36b^2-123b+81)=0$;” in Table XIV, $n_0-n_s$ should be ${\eta }_0-{\eta }_s$. In Table XV, ${\psi }_{1,0}$ should be ${\Psi }_{1,0}$, and $\mathit{nKT}$ should be $\mathit{nkT}$. In Table IV, Macromolecule 21 entry should be $\left[-\frac{1}{2}L,-\frac{\sqrt{3}}{2}L,0\right]$; $\left[L,0,0\right]$; $\left[0,0,0\right]$; $\left[-\frac{1}{2}L,\frac{\sqrt{3}}{2}L,0\right]$ and Macromolecule 17 entry should be multiplied by $L$. In Eq. (44), $\eta ″$ should be $2\eta ″$. https://doi.org/10.1063/1.5111763]

$$\frac{{\eta }^{\prime }-{\eta }_s}{{\eta }_0-{\eta }_s}={\left(\frac{1}{2b/a\nu }+1\right)}^{-1}\left(\frac{1}{2b/a\nu }+\frac{1}{1+{\left(\lambda \omega \right)}^2}\right),$$

(6)

and

$$\frac{{\eta }^{″}}{{\eta }_0-{\eta }_s}={\left(\frac{1}{2b/a\nu }+1\right)}^{-1}\frac{\lambda \omega }{1+{\left(\lambda \omega \right)}^2},$$

(7)

where $\lambda \omega$ is the Deborah number. In this paper, we plot the real and minus the imaginary parts of the shear stress responses to small-amplitude oscillatory shear flow as functions of frequency, following Ferry (Secs. 2.A.4.-2.A.6. of Ref. 1313. J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed. ( Wiley, New York, 1980).) or Bird et al. (Sec. 4.4 of Ref. 1414. Bird, R. B. R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, 1st ed. ( Wiley, New York, 1977), Vol. 1.).

As $\omega \to 0$, for the polymer contribution to the zero-shear viscosity, we get

$$\frac{{\eta }_0-{\eta }_s}{\mathit{nkT}\lambda }=\frac{a\nu }{2}+b=b\left[1+\frac{2b}{a\nu }\right]{\left(\frac{2b}{a\nu }\right)}^{-1},$$

(8)

which we use in the table of Sec. V below.

As shown by Kirchdoerfer,¹⁵15. R. N. Kirchdoerfer, N. Wang, J. Pallesen, D. Wrapp, H. L. Turner, C. A. Cottrell, K. S. Corbett, B. S. Graham, J. S. McLellan, and A. B. Ward, “ Stabilized coronavirus spikes are resistant to conformational changes induced by receptor recognition or proteolysis,” Sci. Rep. 8(1), 15701 (2018). https://doi.org/10.1038/s41598-018-34171-7 each trimeric peplomer head, consisting of three glycoproteins, is well-approximated by an equilateral triangle when viewed along the spike axis. In the general rigid bead-rod model,¹1. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875 this trimer is replaced with a sphere. Here, we approximate the trimer by considering an identical point charge at each vertex of the equilateral triangle.

From Fig. 5, we learn that the detailed triangular structure of the peplomer head and its singly beaded counterpart share the same qualitative behavior. For both, the rotational diffusivity, ${\lambda }_0{D}_r$, descends monotonically with $N_p$. However, the detailed triangular structure of the peplomer head reduces significantly ${\lambda }_0{D}_r$ of the coronavirus particle. Specifically, at the measured peplomer population of $N_p=74$, we see a reduction in ${\lambda }_0{D}_r$ of $39\%$. On close inspection, Fig. 5 also reveals

$$\frac{{\lambda }_0{D}_r\left(3N_p\right)}{{\lambda }_0{D}_r\left(N_p\right)}\approx 1,$$

(14)

that is, the dimensionless rotational diffusivity of a coronavirus with $N_p$ singly beaded peplomers has about the same dimensionless rotational diffusivity of a coronavirus with ${\frac{1}{3}N}_p$ triply beaded peplomers.

From Fig. 6, we learn that the elasticity, ${\eta }^{″}/\left({\eta }_0-{\eta }_s\right)$, of the coronavirus particle suspension is slight and that the detailed triangular structure of the peplomer head slightly reduces this elasticity. From Table III, we see that the corresponding $b$ is nearly zero so that the polymer contribution to the real part of the complex viscosity is constant, $\left(\eta \prime -{\eta }_s\right)/\mathit{nkT}\lambda =\left({\eta }_0-{\eta }_s\right)/\mathit{nkT}\lambda =3/2$. From Table III, we learn that the detailed triangular structure of the peplomer head increases the relaxation time, $\lambda$, and thus, decreases the zero-shear viscosity, ${\eta }_0$. From the rightmost column of Table III, we learn that the detailed triangular structure of the peplomer head decreases the zero-shear value of the first normal stress coefficient, ${\Psi }_{1,0}$.

Whereas much prior work on fluid physics related to the virus has attacked transmission,^17–4117. T. Dbouk and D. Drikakis, “ On coughing and airborne droplet transmission to humans,” Phys. Fluids 32(5), 053310 (2020). https://doi.org/10.1063/5.001196018. R. Bhardwaj and A. Agrawal, “ Likelihood of survival of coronavirus in a respiratory droplet deposited on a solid surface,” Phys. Fluids 32(6), 061704 (2020). https://doi.org/10.1063/5.001200919. S. Verma, M. Dhanak, and J. Frankenfield, “ Visualizing the effectiveness of face masks in obstructing respiratory jets,” Phys. Fluids 32(6), 061708 (2020). https://doi.org/10.1063/5.001601820. T. Dbouk and D. Drikakis, “ On respiratory droplets and face masks,” Phys. Fluids 32(6), 063303 (2020). https://doi.org/10.1063/5.001504421. S. Chaudhuri, S. Basu, P. Kabi, V. R. Unni, and A. Saha, “ Modeling the role of respiratory droplets in Covid-19 type pandemics,” Phys. Fluids 32(6), 063309 (2020). https://doi.org/10.1063/5.001598422. Y. Y. 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Kota, “ Can face masks offer protection from airborne sneeze and cough droplets in close-up, face-to-face human interactions?—A quantitative study,” Phys. Fluids 32(12), 127112 (2020). https://doi.org/10.1063/5.003507241. H. Wang, Z. Li, X. Zhang, L. Zhu, Y. Liu, and S. Wang, “ The motion of respiratory droplets produced by coughing,” Phys. Fluids 32(12), 125102 (2020). https://doi.org/10.1063/5.0033849 this paper focuses on the ab initio calculation of coronavirus transport properties. Specifically, we have determined the rotational diffusivity, the property governing the particle alignment for cell attachment (see Sec. I of Ref. 11. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875). Although our work is mainly curiosity driven, it may deepen our understanding of drug, vaccine, and cellular infection mechanisms.

Chaurasia et al.⁴²42. V. Chaurasia, Y.-C. Chen, and E. Fried, “ Interacting charged elastic loops on a sphere,” J. Mech. Phys. Solids 134, 103771 (2020). https://doi.org/10.1016/j.jmps.2019.103771 (see also Chaurasia⁴³43. V. Chaurasia, “ Variational formulation of charged curves confined to a sphere,” Ph.D. thesis (Department of Mechanical Engineering, University of Houston, Houston, Texas, USA, 2018).) developed a framework to find equilibrium solutions of a system consisting of flexible structures, specifically charged elastic loops constrained to a sphere. Their framework could be used to model flexible peplomers with uniformly charged heads. We leave this daunting task for a future study.

Since the coronavirus capsid can be ellipsoidal (Fig. 3. of Ref. 4444. N. Zhu, D. Zhang, W. Wang, X. Li, B. Yang, J. Song, X. Zhao, B. Huang, W. Shi, R. Lu, P. Niu, F. Zhan et al., “ A novel coronavirus from patients with pneumonia in China, 2019,” N. Engl. J. Med. 382(8), 727–733 (2020). https://doi.org/10.1056/NEJMoa2001017), called pleomorphism, we must eventually consider this too. Whereas this work considered the detailed triangular structure of the peplomer head as triads of three point-charges, we could also consider uniformly charged triangular rigid peplomers constrained to a sphere. By uniformly charged triangular, we mean that the charge would be uniformly distributed over the edges of the triangle rather than point charges at its vertices. We leave this task for a later date.

One cognate transport problem is the transient translation and twist of coronavirus spikes rearranging freely under their own electrostatic repulsions, for instance, the transient following the extraction of a single spike. This paper is, of course, silent on this interesting problem, which we leave for another day.

As in our previous work,¹1. M. A. Kanso, J. H. Piette, J. A. Hanna, and A. J. Giacomin, “ Coronavirus rotational diffusivity,” Phys. Fluids 32(11), 113101 (2020). https://doi.org/10.1063/5.0031875 we have used repulsions of charged particles over the surfaces of spheres for both the capsid and the peplomer heads of the coronavirus to arrive at its transport properties. It has not escaped our attention that our solutions to the Thomson problem can also be used to calculate the Young's modulus of the coronavirus particle [Eq. (3a) of Ref. 4545. Bowick, M. A. Cacciuto, D. R. Nelson, and A. Travesset, “ Crystalline order on a sphere and the generalized Thomson problem,” Phys. Rev. Lett. 89(18), 185502 (2002). https://doi.org/10.1103/PhysRevLett.89.185502] and that by extension this Young's modulus will depend upon peplomer population. We leave this calculation for another day. When using the references cited herein, it is best to be mindful of corresponding ganged errata in Ref. 4646. Kanso, M. A. V. Chaurasia, E. Fried, and A. J. Giacomin, “ Peplomer bulb shape and coronavirus rotational diffusivity,” PRG Report No. 076, QU-CHEE-PRGTR-2021-76 (Polymers Research Group, Chemical Engineering Dept., Queen's University, Kingston, Canada, 2021)..