Coronavirus rotational diffusivity

Abstract

Just 11 weeks after the confirmation of first infection, one team had already discovered and published [D. Wrapp et al., "Cryo-EM structure of the 2019-nCoV spike in the prefusion conformation," Science 367(6483), 1260-1263 (2020)] in exquisite detail about the new coronavirus, along with how it differs from previous viruses. We call the virus particle causing the COVID-19 disease SARS-CoV-2, a spherical capsid covered with spikes termed peplomers. Since the virus is not motile, it relies on its own random thermal motion, specifically the rotational component of this thermal motion, to align its peplomers with targets. The governing transport property for the virus to attack successfully is thus the rotational diffusivity. Too little rotational diffusivity and too few alignments are produced to properly infect. Too much, and the alignment intervals will be too short to properly infect, and the peplomer is wasted. In this paper, we calculate the rotational diffusivity along with the complex viscosity of four classes of virus particles of ascending geometric complexity: tobacco mosaic, gemini, adeno, and corona. The gemini and adeno viruses share icosahedral bead arrangements, and for the corona virus, we use polyhedral solutions to the Thomson problem to arrange its peplomers. We employ general rigid bead-rod theory to calculate complex viscosities and rotational diffusivities, from first principles, of the virus suspensions. We find that our ab initio calculations agree with the observed complex viscosity of the tobacco mosaic virus suspension. From our analysis of the gemini virus suspension, we learn that the fine detail of the virus structure governs its rotational diffusivity. We find the characteristic time for the adenovirus from general rigid bead-rod theory. Finally, from our analysis of the coronavirus suspension, we learn that its rotational diffusivity descends monotonically with its number of peplomers.

FIG. 1.

General rigid bead–rod model of tobacco mosaic virus, N = 12.

FIG. 2.

General rigid bead–rod model of gemini virus, N = 22. See Fig. 2 of Ref. .

FIG. 3.

General rigid bead–rod model of adenovirus, N_c = 252, N_p = 12, and r_$v$/r_c = 5/4.

FIG. 4.

Connections between adenovirus particle dimensions and its general rigid bead–rod model.

FIG. 5.

General rigid bead–rod model of coronavirus, N_c = 256, N_p = 74, and r_$v$/r_c = 5/4.

FIG. 6.

The dimensionless complex viscosity of tobacco mosaic virus suspension predicted by the general rigid bead–rod theory vs experimental data (Fig. 14.5-1 of Ref. 22). The data are for solutions of tobacco mosaic virus of M = 3.9 × 10⁷ g/mol. The red points represent data taken at 310 K, and the blue ones are taken at 298.2 K. The solvent viscosity at the two temperatures is η_s = 3.43 × 10⁻³ Pa s and η_s = 5.16 × 10⁻³ Pa s, respectively. The general rigid bead–rod theory predictions are computed using a value [η]₀ = 18 cm³/g as the best fit with the data. The solid curve describes (η′ − η_s)/(η₀ − η_s), and the dashed one describes η″/(η₀ − η_s).

FIG. 7.

Gemini virus complex viscosity comparison: two osculating beads (green) and twin icosahedra (blue). The solid curve describes (η′ − η_s)/(η₀ − η_s), and the dashed one describes η″/(η₀ − η_s).

FIG. 8.

Connections between coronavirus particle dimensions and its general rigid bead–rod model. For the peplomer bulb, we have bead radius $r_b\equiv \frac{1}{2}d$ so that for the peplomer height, we have $r_v-r_c=r_p+r_b=r_p+\frac{1}{2}d$ (see Table X). The peplomer head radial position is thus the center to center distance between the peplomer head and the capsid (r_$v$ − r_b).

FIG. 9.

Tobacco mosaic (black), gemini (green), adeno (blue), and corona (red) complex viscosity comparison. The solid curve describes (η′ − η_s)/(η₀ − η_s), and the dashed one describes η″/(η₀ − η_s).

FIG. 10.

Corona (red) elastic complex viscosity, η″/(η₀ − η_s), curve.

FIG. 11.

Effect of capsid osculated beadings on complex viscosity (N_c = 16, 32, 64, 128, 256, 510, N_p = 74, r_$v$/r_c = 5/4, and L = d). N_c = 16 (black), N_c = 32 (blue), N_c = 64 (red), N_c = 128 (green), N_c = 256 (yellow), and N_c = 510 (magenta). The solid curve describes (η′ − η_s)/(η₀ − η_s), and the dashed one describes η″/(η₀ − η_s).

FIG. 12.

Dimensionless rotational diffusivity λ₀D_r from Eq. (23) vs peplomer population N_p (N_c = 256).

FIG. 13.

Dimensionless structure-dependent characteristic time, λ_c/λ₀, vs peplomer population, N_p (N_c = 256), from Eq. (40).

FIG. 14.

In this work, we replace the peplomer bulk with sphere (a) inscribed in the trimer, thus neglecting its triangularity. Future work shall improve upon this by inscribing the trimer (b) into three osculating beads.