Recent advancements in viral hydrodynamics afford the calculation of the transport properties of particle suspensions from first principles, namely, from the detailed particle shapes. For coronavirus suspensions, for example, the shape can be approximated by beading (i) the spherical capsid and (ii) the radially protruding peplomers. The general rigid bead-rod theory allows us to assign Stokesian hydrodynamics to each bead. Thus, viral hydrodynamics yields the suspension rotational diffusivity, but not without first arriving at a configuration for the cationic peplomers. Prior work considered identical peplomers charged identically. However, a recent pioneering experiment uncovers remarkable peplomer size and charge heterogeneities. In this work, we use energy minimization to arrange the spikes, charged heterogeneously to obtain the coronavirus spike configuration required for its viral hydrodynamics. For this, we use the measured charge heterogeneity. We consider 20 000 randomly generated possibilities for cationic peplomers with formal charges ranging from 30 to 55. We find the configurations from energy minimization of all of these possibilities to be nearly spherically symmetric, all slightly oblate, and we report the corresponding breadth of the dimensionless rotational diffusivity, the transport property around which coronavirus cell attachment revolves.

## I. INTRODUCTION

^{1–5}For coronavirus suspensions, for example, the shape can be approximated by beading (i) the spherical capsid (Sec. VII of Ref. 1) and then (ii) the radially protruding peplomers (Sec. VII of Refs. 1 and 2). The general rigid bead-rod theory then allows us to proceed by assigning Stokesian hydrodynamics to each bead (Refs. 6 and 7; EXAMPLE 16.7–1 of Ref. 8 or EXAMPLE 13.6–1 of Ref. 9). Table I classifies chronologically prior literature on coronavirus hydrodynamics and defines the novelty of this work. Although our work is motivated mainly by curiosity, its many applications have not escaped our attention. Viral hydrodynamics thus then yields the suspension rotational diffusivity, but not without first arriving at a configuration for the cationic peplomers [see Fig. 1(b) of Ref. 10] Prior work considered identical peplomers, charged identically.

^{1–3}However, recent pioneering experiment uncovers remarkable heterogeneity of both peplomer size [see Fig. 1(a) of Ref. 10; Introduction of Ref. 11] and charge [see Fig. 1(b) of Ref. 10] With permission, we reproduce these seminal results in Fig. 1. By size heterogeneity, we mean that the molecular weights of the peplomer protein trimers vary, and specifically, from Fig. 1, we learn that

Method . | Peplomer population . | Hydrodynamic interactions . | Capsid shape . | Peplomer shape (beads) . | Peplomer charge distribution . | Reference . |
---|---|---|---|---|---|---|

Analytical | $ 10 \u2264 N p \u2264 100 $ | Spherical | 1 | Uniform | 1 | |

Analytical | N = 74 _{p} | Spherical | 3 | Uniform | 2 | |

Analytical | N = 74 _{p} | Ellipsoidal | 1 | Uniform | 3 | |

Molecular dynamics | N = 26 _{p} | $ A \u226a 0.1 $ | Spherical | 1 | Uniform | 5 |

Analytical | N = 74 _{p} | $ 0.09 \u2264 A \u2264 0.11 $ | Spherical | 1 | Uniform | 17 |

Analytical | N = 74 _{p} | Spherical | 1 | Nonuniform | This work |

Method . | Peplomer population . | Hydrodynamic interactions . | Capsid shape . | Peplomer shape (beads) . | Peplomer charge distribution . | Reference . |
---|---|---|---|---|---|---|

Analytical | $ 10 \u2264 N p \u2264 100 $ | Spherical | 1 | Uniform | 1 | |

Analytical | N = 74 _{p} | Spherical | 3 | Uniform | 2 | |

Analytical | N = 74 _{p} | Ellipsoidal | 1 | Uniform | 3 | |

Molecular dynamics | N = 26 _{p} | $ A \u226a 0.1 $ | Spherical | 1 | Uniform | 5 |

Analytical | N = 74 _{p} | $ 0.09 \u2264 A \u2264 0.11 $ | Spherical | 1 | Uniform | 17 |

Analytical | N = 74 _{p} | Spherical | 1 | Nonuniform | This work |

*Q*is the integer-valued with mean value $ Q \xaf = 42.29$ electron units. Tables II and III, respectively, define our dimensional and dimensionless symbols. In this paper, exploiting the general rigid bead-rod theory, we carry these pioneering experiments to the implied distribution of rotational diffusivities. The peplomer count on a coronavirus particle,

*N*, matters. For the special cases of identical peplomers, charged identically, Fig. 12 of Ref. 1 and Fig. 5 of Ref. 2 give the dimensionless rotational diffusivities for $ 10 \u2264 N p \u2264 100$. In this work, we use energy minimization

_{p}^{12,13}for spreading the spikes, charged heterogeneously to obtain the coronavirus spike configuration required for its viral hydrodynamics. For this, we use the measured charge heterogeneity [Fig. 1(b) of Ref. 10]. We consider 20 000 randomly generated possibilities for cationic peplomers with formal charges ranging from 30 to 55. By random generation, we mean that we randomize the formal charge of each of the

*N*= 74 peplomers over the measured range. We rely on our previous literature review (see Table X of Ref. 1) to choose

_{p}*N*= 74. Calculations of the rotational diffusivity of coronavirus suspensions initially excluded interferences of Stokes flow velocity fields between nearby peplomers.

_{p}^{1–3}More recently, the method for incorporating hydrodynamic interactions was advanced (Sec. III of Ref. 15) and used (Sec. V of Refs. 15 and 16) and even applied to coronavirus suspensions.

^{17}However, in this work, we neglect hydrodynamic interactions for formal charge distribution of coronavirus peplomer protein trimers, leaving this detail for another day. Our work here is about the heterogeneity of formal charge from one peplomer to another, which is separate from the local charge distribution over the surface of an individual peplomeric cation, which has been simulated (see Fig. 2 of Ref. 18) but not measured. These beautiful distributions are averaged over many peplomers and are thus silent on peplomer-to-peplomer heterogeneity. The heterogeneity of this local charge distribution is yet to be discovered. Our formulation of the general rigid bead-rod theory is limited to axisymmetric viruses. By axisymmetry, we mean that both the virus particle and its moment of inertia ellipsoid have at least one axis of symmetry (Refs. 19 and 20, Chap. 13 of Ref. 9, and Chap. 16 of Ref. 8). Furthermore, if the virus particle structure is axisymmetric, at least two of its principal moments of inertia equate at any angle from the molecular axis. Since the coronavirus particle structure is axisymmetric, so will its moment of inertia ellipsoid. Our usage of axisymmetric is not to be confused with the common geometric meaning of continuous rotational symmetry about an axis. In the tradition of the transport sciences, we define the rotatory diffusivity as (see Footnote 2 of p. 62 of Ref. 8)

Name . | Unit . | Symbol . |
---|---|---|

Angular frequency | $ t \u2212 1 $ | ω |

Augmented energy functional | $ ML 2 / t 2 $ | $ E \u0302 $ |

Bead friction coefficient | M/t | ζ |

Complex viscosity | M/Lt | $ \eta * $ |

Dielectric permittivity | $ T 4 I 2 / ML 3 $ | ϵ |

Element for Kronecker delta | $ t \u2212 1 $ | $ \delta ( s ) $ |

Kinetic molecular energy per molecule | $ ML 2 / t 2 $ | kT |

Length of the spike of each peplomer | L | $\u2113$ |

Mass | M | m |

Mean value of charge distribution | e | $ Q \xaf $ |

Minus the imaginary part of the complex viscosity | M/Lt | $ \eta \u2033 $ |

Molecular weights of the peplomer protein trimers | M/mol | M _{p} |

Nearest bead center-to-center distance | L | L |

Number of dumbbells per unit volume | $ 1 / L $ | n |

Point charge | e | Q |

Real part of the complex viscosity | M/Lt | $ \eta \u2032 $ |

Relaxation modulus | M/L | G(s) |

Relaxation time of rigid dumbbell | t | λ _{0} |

Relaxation time of solution | t | λ |

Rotational diffusivity | $ t \u2212 1 $ | D _{r} |

Rotatory diffusivity | $ L 2 / t $ | $ D rot $ |

Shear rate amplitude | $ t \u2212 1 $ | $ \gamma \u0307 0 $ |

Solvent viscosity | M/Lt | η _{s} |

Time difference | t | $ s \u2261 t \u2212 t \u2032 $ |

Total electrostatic energy | $ ML 2 / t 2 $ | E |

Translational diffusivity | $ L 2 / t $ | D _{tr} |

Viscosity, zero-shear | M/Lt | η _{0} |

Zero-shear first normal stress difference | M/L | $ \psi 1 , 0 $ |

Name . | Unit . | Symbol . |
---|---|---|

Angular frequency | $ t \u2212 1 $ | ω |

Augmented energy functional | $ ML 2 / t 2 $ | $ E \u0302 $ |

Bead friction coefficient | M/t | ζ |

Complex viscosity | M/Lt | $ \eta * $ |

Dielectric permittivity | $ T 4 I 2 / ML 3 $ | ϵ |

Element for Kronecker delta | $ t \u2212 1 $ | $ \delta ( s ) $ |

Kinetic molecular energy per molecule | $ ML 2 / t 2 $ | kT |

Length of the spike of each peplomer | L | $\u2113$ |

Mass | M | m |

Mean value of charge distribution | e | $ Q \xaf $ |

Minus the imaginary part of the complex viscosity | M/Lt | $ \eta \u2033 $ |

Molecular weights of the peplomer protein trimers | M/mol | M _{p} |

Nearest bead center-to-center distance | L | L |

Number of dumbbells per unit volume | $ 1 / L $ | n |

Point charge | e | Q |

Real part of the complex viscosity | M/Lt | $ \eta \u2032 $ |

Relaxation modulus | M/L | G(s) |

Relaxation time of rigid dumbbell | t | λ _{0} |

Relaxation time of solution | t | λ |

Rotational diffusivity | $ t \u2212 1 $ | D _{r} |

Rotatory diffusivity | $ L 2 / t $ | $ D rot $ |

Shear rate amplitude | $ t \u2212 1 $ | $ \gamma \u0307 0 $ |

Solvent viscosity | M/Lt | η _{s} |

Time difference | t | $ s \u2261 t \u2212 t \u2032 $ |

Total electrostatic energy | $ ML 2 / t 2 $ | E |

Translational diffusivity | $ L 2 / t $ | D _{tr} |

Viscosity, zero-shear | M/Lt | η _{0} |

Zero-shear first normal stress difference | M/L | $ \psi 1 , 0 $ |

Name . | Symbol . |
---|---|

Amplitude, mean value, and variance | $ A , \mu , \sigma 2 $ |

of normal distribution | |

Aspect ratio | c/a |

Capsid-sphere | $c$ |

Coefficient in Eqs. (19) and (20) | a |

Coefficient in Eqs. (19) and (20) | b |

Coefficient in Eqs. (19) and (20) | ν |

Augmented Lagrangian | $ E \u0302 $ |

Charge | q _{i} |

Electrostatic energy | $ E \u0303 $ |

Wales's electrostatic energy | E _{w} |

Electrostatic energy relative to Wales's energy | F |

Deborah number, oscillatory shear | $ D e \u2261 \lambda \omega $ |

Principal moment of inertia | $ I 1 , I 2 , I 3 $ |

Normal distribution | f |

Probability distribution function | pdf |

Sphere on which peplomers lie | $s$ |

Total number of beads | N |

Total number of peplomers | N _{p} |

Weissenberg number | $ W i \u2261 \lambda \gamma \u0307 0 $ |

Name . | Symbol . |
---|---|

Amplitude, mean value, and variance | $ A , \mu , \sigma 2 $ |

of normal distribution | |

Aspect ratio | c/a |

Capsid-sphere | $c$ |

Coefficient in Eqs. (19) and (20) | a |

Coefficient in Eqs. (19) and (20) | b |

Coefficient in Eqs. (19) and (20) | ν |

Augmented Lagrangian | $ E \u0302 $ |

Charge | q _{i} |

Electrostatic energy | $ E \u0303 $ |

Wales's electrostatic energy | E _{w} |

Electrostatic energy relative to Wales's energy | F |

Deborah number, oscillatory shear | $ D e \u2261 \lambda \omega $ |

Principal moment of inertia | $ I 1 , I 2 , I 3 $ |

Normal distribution | f |

Probability distribution function | pdf |

Sphere on which peplomers lie | $s$ |

Total number of beads | N |

Total number of peplomers | N _{p} |

Weissenberg number | $ W i \u2261 \lambda \gamma \u0307 0 $ |

Quantity x
. | Equation . | Figure . | Parameters of the normal distribution $ f = A e \u2212 ( x \u2212 \mu ) 2 / 2 \sigma 2$ . | ||
---|---|---|---|---|---|

. | . | . | A
. | μ
. | σ
. |

Q | Eq. (2) | Fig. 3 | 0.1153 | 41.586 | 3.5251 |

$ log \u2009 ( F \u2212 min F ) $ | Eq. (31) | Fig. 5(b) | 0.6082 | −6.3755 | 0.6423 |

I _{1} | Fig. 7 | 3.0800 | 158.22 | 0.1310 | |

I _{2} | 3.9447 | 158.56 | 0.1007 | ||

I _{3} | 3.0926 | 158.91 | 0.1304 | ||

$ \lambda / \lambda 0 $ | Eq. (11) | Fig. 11 | 1.5296 | 3.1644 | 0.2669 |

$ \lambda 0 D r $ | Eq. (16) | Fig. 12 | $ 9.1324 \xd7 10 5 $ | $ 5.2670 \xd7 10 \u2212 4 $ | $ 4.4882 \xd7 10 \u2212 7 $ |

Quantity x
. | Equation . | Figure . | Parameters of the normal distribution $ f = A e \u2212 ( x \u2212 \mu ) 2 / 2 \sigma 2$ . | ||
---|---|---|---|---|---|

. | . | . | A
. | μ
. | σ
. |

Q | Eq. (2) | Fig. 3 | 0.1153 | 41.586 | 3.5251 |

$ log \u2009 ( F \u2212 min F ) $ | Eq. (31) | Fig. 5(b) | 0.6082 | −6.3755 | 0.6423 |

I _{1} | Fig. 7 | 3.0800 | 158.22 | 0.1310 | |

I _{2} | 3.9447 | 158.56 | 0.1007 | ||

I _{3} | 3.0926 | 158.91 | 0.1304 | ||

$ \lambda / \lambda 0 $ | Eq. (11) | Fig. 11 | 1.5296 | 3.1644 | 0.2669 |

$ \lambda 0 D r $ | Eq. (16) | Fig. 12 | $ 9.1324 \xd7 10 5 $ | $ 5.2670 \xd7 10 \u2212 4 $ | $ 4.4882 \xd7 10 \u2212 7 $ |

In this paper, we depart from the said transport tradition of using the rotatory
diffusivity, $ D rot$,
and frame our results in terms of the rotational diffusivities, *D _{r}*, of coronavirus particles.

## II. METHOD

^{21,22}or nearest macromolecules. With the general rigid bead-rod theory, we thus locate beads to sculpt an approximation of the coronavirus particle shapes. In this way, using the general rigid bead-rod theory, we can model any virus macromolecular architecture (see Fig. 9 of Ref. 7). For the general rigid bead-rod theory, Hassager arrived at general equation of the shear relaxation function [Eq. (48) of Ref. 19]:

*I*

_{1},

*I*

_{2}, and

*I*

_{3}are the dimensionless principal moment of intertia (see Tables III and IV of Ref. 23). For axisymmetric macromolecules,

*I*

_{1}=

*I*

_{2}. And the respective definitions for oblate and prolate for such molecules are

## III. OSCILLATORY SHEAR FLOW

^{25,26}

*et al.*(Sec. 4.4 of Ref. 28) As $ \omega \u2192 0$, for the polymer contribution to the zero-shear viscosity, we get

*x*by

*A*,

*μ*, and $ \sigma 2$ are, respectively, the amplitude, mean value, and variance of

*x*.

## IV. PEPLOMER CONFIGURATIONAL HETEROGENEITY

### A. Kinematics

*N*be the number of peplomers attached to the capsid-sphere $c$ of radius

_{p}*r*. Let $\u2113$ be the length of the spike of each peplomer attached normal to $c$ at the point of contact on $c$. Therefore, each peplomer must lie on a sphere $s$ of radius $ r s = r c + \u2113$, as shown in Fig. 2. Denoting the position vector of

_{c}*i*th bead by $ r s r i$, the dimensionless position vector $ r i$ must obey

*N*scalar equations

_{p}### B. Energetics

*i*th bead, where $ Q \xaf = 42.29$ electron unit is the mean value of the experimentally measured heterogeneous distribution of the charges on the peplomers by Miler

*et al.*,

^{14}as shown in Figs. 1(b) and 3. Interestingly, the charge distribution follows a normal distribution, parameters of which are provided in Table IV. Here,

*q*denotes the dimensionless charge on

_{i}*i*th peplomer. The total electrostatic energy of such

*N*peplomers, constrained to a sphere $s$ of radius

_{p}*r*, is given by

_{s}*ϵ*is the dielectric permittivity. We use the quantity

*r*distance apart, as the energy scale, and define the dimensionless total electrostatic energy of

_{s}*N*peplomers by

_{p}### C. Equilibrium equations

*q*, $ i = 1 , \u2026 , N p$ and number

_{i}*N*of peplomers. The following are the details of the constrained minimization approach. We define the augmented energy functional by

_{p}*ρ*, $ i = 1 , \u2026 , N p$, are the dimensionless Lagrange multipliers required to enforce the kinematic constraints in Eq. (24). We obtain the Euler–Lagrange equations by differentiating Eq. (28) with the dimensionless position vectors $ r i , \u2009 i = 1 , \u2026 , N p$, and equating the resulting expressions to $0$,

_{i}In the Euler–Lagrange equation [Eq. (29)],
the first term on the left-hand side is the total dimensionless electrostatic force
exerted on the *i*th bead by all the remaining beads. For an
equilibrium, this electrostatic force is balanced by the second term in Eq. (29), $ \u2212 \rho i r i$, which is the
dimensionless reactive force between the capsid and *i*th bead
required to satisfy Eq. (24), that is, to
maintain contact of that bead with the capsid. Therefore, the term $ \u2212 \rho r i , \u2009 i = 1 , \u2026 , N p$, in Eq. (29) is a measure of the interaction between
the capsid and *i* the bead. Notably, if all the peplomer
beads are positively charged, they will repel each other. Therefore, for equilibrium, the
electrostatic interaction between the peplomer beads and the capsid must be attractive to
counter the repulsive forces between the beads. In this respect, we follow Refs. 12 and 29,
wherein a Lagrange multiplier is used to represent the adhesive reaction between a
semiflexible polymer and a rigid spherical substrate.^{30} as well used Lagrange multipliers to describe the adhesive
reaction between a generalized surface and confined semiflexible polymers. Another example
of this approach appears in the work of Ref. 31,
who introduced a Lagrange multiplier to predict adhesive reactions.

For each value of *i* in the vector equation [Eq. (29)], there are three scalar equations. Thus,
Eq. (29) yields $ 3 N p$ scalar
equations. We solve a total of $ 4 N p$ equations, $ 3 N p$ equilibrium
equations in Eq. (29), and *N _{p}* constraints in Eq. (24), simultaneously to determine $ 4 N p$ unknowns, $ 3 N p$ scalar
components of

*N*position vectors in $ r i$, and

_{p}*N*in scalar Lagrange multipliers

_{p}*ρ*, $ i = 1 , 2 , \u2026 , N p$. We use the

_{i}*Levenberg–Marquardt*Algorithm from the

*fsolve*package of MATLAB to solve the system of equations with $ 10 \u2212 16$ error tolerance to determine the energy minimizing configuration for given values of $ Q \xaf q i , \u2009 i = 1 , \u2026 , N p$ charges.

^{32}wherein one seeks to find a state that distributes

*N*electrons over a unit sphere as evenly as possible, with minimum electrostatic energy. Wales

_{p}*et al.*

^{33,34}solved this problem for identically charged particles by minimizing the dimensionless energy functional

*N*, where $ r i , \u2009 i = 1 , \u2026 , N p$, are the dimensionless position vectors of unit point charges on a unit sphere. By contrast, our energy minimization solves the problem for particles not charged identically (

_{p}*i*th peplomer has

*q*dimensionless charge). Our energy expression [Eq. (27)] recovers the energy expression [Eq. (30)] of Wales

_{i}*et al.*

^{33,34}for

*q*= 1, $ i = 1 , \u2026 , N p$, and thus, we recover the Thompson solution (energy minimization over a spherical surface about which much has been written

_{i}^{10}) as it should (see Sec. VII of Ref. 7) As far as we know, we are the first to perform an energy minimization of heterogeneously charged particles over the surface of a sphere.

*et al.*,

^{33,34}we define a dimensionless electrostatic energy

*E*are defined in Eqs. (27) and (30), respectively. In this work, we restrict our attention to

_{w}*N*= 74, for which Wales

_{p}*et al.*

^{33,34}calculated $ E w = 2387.07$.

## V. RESULTS

Since *Q* is integer-valued, we next abstract from Fig. 1 the formal charge distribution line spectrum of
coronavirus peplomer protein trimers. We give this in Fig. 3, from which we glean Eq. (2), as we
must. Whereas Fig. 3 and its companion [Eq. (2)] tell us the charge heterogeneity, these are
silent on how these peplomer charges will arrange themselves over the coronavirus surface.
In this work, we examine 20 000 possible arrangements of number *N _{p}* = 74 of peplomers, which we call states. For a given state, we
randomly draw $ Q \xaf q i , \u2009 i = 1 , \u2026 , N p$, charges from
the discrete, integer-valued bin of charges in the interval $ 30 \u2264 Q \u2264 55$ with probability as per
experimental measurement,

^{14}as shown in Fig. 3. Next, we follow the energy minimization method of Sec. IV C to determine the minimum energy configuration of that state. We repeat the process of randomly drawing $ Q \xaf q i , \u2009 i = 1 , \u2026 , N p$, charges from the discrete, integer-valued bin 20 000 times and determine the energy minimizing configuration of each state. Figure 4 illustrates one such energy-minimized state, with color coding for the formal charges. Although considering a larger number of states than 20 000 will give us a more comprehensive understanding of the effect of charge heterogeneity, we restrict our attention to 20 000 states due to computational cost limitations. We expect that the practically chosen number of states is large enough to derive meaningful conclusions.

*F*, defined in Eq. (31), of the 20 000 reasonable states is provided in Fig. 5(a). The red horizontal line in the figure corresponds to

*F*=

*0 at which, using Eq. (31), the dimensionless electrostatic energy $ E \u0303$, defined in Eq. (27), of our heterogeneously charged system is equal to the dimensionless electrostatic energy*

*E*, defined in (30) of a identically charged system with unit charge for

_{w}*N*= 74. We find 687 states with $ E < E w$. Moreover, we find that the energy of states shown in Fig. 5(a) follows a log –normal distribution, as shown in the histogram in Fig. 5(b). Sorting on these minimum energies, E, we construct Fig. 6, which inflect at about the 10 000 state. The lowest value of

_{p}*F*(vertical intercept of Fig. 6), defined in Eq. (31), is $ F = \u2212 4.2625 \xd7 10 \u2212 4$. We next calculate the dimensionless moments of inertia tensor of each state, from which we calculate the dimensionless principal moment of inertia

*I*

_{1},

*I*

_{2}, and

*I*

_{3}. We arrange those quantities into the three histograms of Fig. 7, from which we learn that

Equations (35) and (36) are the main results of this work. Recall
that the rotational diffusivity is the transport property around which coronavirus cell
attachment revolves. From Eq. (36), we learn
that through its heterogeneity of peplomer formal charges [Eq. (2)], even if all capsids are spherical, and all of these with *N _{p}* = 74, the coronavirus attacks with a distribution of
rotational diffusivities. We find Fig. 12 to be
intrinsically beautiful.

## VI. CONCLUSION

We began this paper by distinguishing two forms of higher energy coronavirus peplomer
arrangement arising for fixed peplomer population: (i) those caused by the heterogeneity of
charge from one peplomer to the next and (ii) those caused by local energy minima at a
uniform charge (less probable than the lowest energy configuration^{33,34}). In this paper, we focused on the former romanette
because this charge heterogeneity has been recently observed and quantified experimentally
(Fig. 1. of Ref. 14 reprinted as Fig. 1). For the romanette (i), we have examined the lowest energy (and
thus the most probable) configuration for each of 20 000 random charge arrangements (see one
of these in Fig. 4). By contrast, romanette (ii) would
involve examining lower probability arrangements (higher energy and local energy minima)
arising for any fixed charge arrangement (be it with or without^{33,34} charge heterogeneity). Romanettes (i) and (ii) yield
multiplicities of relaxation time, thus multiplicities of coronavirus transport properties.
We leave the intriguing problem of the relaxation time distribution implied by romanette
(ii) for another day. In this paper, we calculate the transport properties of coronavirus
suspensions from the first principles, namely, from the detailed particle shapes. For this,
we approximated the shape by beading (i) the spherical capsid and then (ii) the radially
protruding peplomers (see gray and colored beads of Fig. 4, respectively). Following the general rigid bead-rod theory, we assigned
Stokesian hydrodynamics to each bead position. Our work, called viral hydrodynamics, thus
then yields the suspension rotational diffusivity equation [Eq. (36)], but not without first arriving at a
configuration for the cationic peplomers. From the general rigid bead-rod theory, we learn
that the transport properties depend upon the macromolecular moments of inertia. By
transport properties, we mean the rotational diffusivity, the relaxation time [Eq. (11)] upon which the rotational diffusivity
depends, and the complex viscosity equations [Eqs. (18)–(20)]. In this work, we considered 20 000 randomly generated
possibilities for cationic peplomer arrangements with formal charges ranging from 30 to 55
[Eq. (2)]. We find the configurations from
energy minimization of all of these possibilities to be nearly spherically symmetric (Figs. 7 and 8), all
slightly oblate, and we report the corresponding breadth of the dimensionless relaxation
time distribution in Fig. 11 and Eq. (35) and rotational diffusivity in Fig. 12 and Eq. (36), the transport property around which coronavirus cell attachment revolves. In
this paper, exploiting the general rigid bead-rod theory, we carried the pioneering
experiments on peplomer formal charge heterogeneity through to what this implies about the
rotational diffusivity, and specifically about its heterogeneity. From Fig. 12 and Eq. (36), we
learn that the coronavirus attacks with a distribution of rotational diffusivities. In this
paper, we have neglected the interferences of velocity profiles of nearby peplomers.^{17} We call these interferences hydrodynamic
interactions. We leave the exploration of how formal charge distribution of coronavirus
peplomer protein trimers is affected by hydrodynamic interactions for future work. We would
account for such hydrodynamic interactions following the method of Sec. II of Ref. 10 with $ A \u2261 L / 2 d \u2248 1 / 10$ [Eq. (8) of Ref. 10]. Our work is silent about the charge distribution
over the surfaces of individual peplomers, recently quantified by molecular dynamics
simulation (see Fig. 2. of Ref. 18). We leave this
intriguing task for the future. Docking coronavirus with a receptor site requires not just
one but two adjacent peplomers to align and attach. For this probability, see Eq. (1) of
Ref. 1. This has yet to be calculated for peplomers
charged identically, and our work is silent on how this probability would be affected by the
recently measured peplomer formal charge heterogeneity. We leave this daunting task for
another day.

## ACKNOWLEDGMENTS

This research was undertaken, in part, thanks to support from the Canada Research Chairs
program of the Government of Canada for the Natural Sciences and Engineering Research
Council of Canada (NSERC) Tier 1 Canada Research Chairs in Rheology, and in *Physics of Fluids*. This research was also undertaken, in part,
thanks to support from the Discovery Grant program of the Natural Sciences and Engineering
Research Council of Canada (NSERC) (A. J. Giacomin), Vanier Canada Graduate Scholarship (M.
A. Kanso), and the Mitacs Research Training Award (A. J. Giacomin and M. A. Kanso). V.
Chaurasia and E. Fried gratefully acknowledge support from the Okinawa Institute of Science
and Technology Graduate University with subsidy funding from the Cabinet Office, Government
of Japan. We are grateful to the anonymous referees for helping us improve the
manuscript.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Vikash Chaurasia:** Conceptualization (equal); Data curation (lead); Formal
analysis (lead); Resources (equal); Software (lead); Validation (lead); Visualization
(lead); Writing – original draft (equal); Writing – review & editing (equal). **Mona Kanso:** Project administration (lead); Writing – original draft
(equal). **Eliot Fried:** Conceptualization (equal); Funding acquisition (equal);
Supervision (equal). **Alan Jeffrey Giacomin:** Conceptualization (lead); Funding
acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal);
Supervision (lead); Validation (equal); Writing – original draft (lead); Writing – review
& editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*k*, and of columns 7 and 11, by

*k*

^{−1}, where

*k*= 52

^{2}. In FIG. 11, the ordinate should be divided by

*k*

^{−1}. In Fig. 7, the ordinate and abscissa should be divided by

*k*.

*Recent Advances in Rheology: Theory, Biorheology, Suspension and Interfacial Rheology*

^{3}.” Above Eq. (2.80), “(77)” should be “(2.77).”

*Dynamics of Polymeric Liquids*

_{ν1}” should be “R

_{ν2}.”

*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*