Hydrodynamics of spike proteins dictate a transport-affinity competition for SARS-CoV-2 and other enveloped viruses

Many viruses, such as SARS-CoV-2 or Influenza, possess envelopes decorated with surface proteins (a.k.a. spikes). Depending on the virus type, a large variability is present in the surface-proteins number, morphology and reactivity, which remains generally unexplained. Since viruses’ transmissibility depends on features beyond their genetic sequence, new tools are required to discern the effects of spikes functionality, interaction, and morphology. Here, we postulate the relevance of hydrodynamic interactions in the viral infectivity of enveloped viruses and propose micro-rheological characterization as a platform for virus differentiation. To understand how the spikes affect virion mobility and infectivity, we investigate the diffusivity of spike-decorated structures using mesoscopic-hydrodynamic simulations. Furthermore, we explored the interplay between affinity and passive viral transport. Our results revealed that the diffusional mechanism of SARS-CoV-2 is strongly influenced by the size and distribution of its spikes. We propose and validate a universal mechanism to explain the link between optimal virion structure and maximal infectivity for many virus families.


List of Supplementary figures and tables
1. Rigid multiblob methodology.

Rotational and traslational diffusivity of SARS-CoV-2
Supplementary Table 6: Comparative translational and rotational diffusivities computed with D t virion formula and real values of viscosity, temperature and energy for SARS-CoV-2. Supplementary Table 7: Actual diffusion of SARS-CoV-2 virion per maximum and minimum spikes number 26 ± 15.

Viruses binding affinity information.
Supplementary

Rigid multiblob methodology
In one of his celebrated papers, Einstein used the linear response theory to demonstrate that the translational diffusion coefficient of a colloid Ω is related to its translational mobility [1]. The same arguments can be easily extended to include the rotational diffusion [2]. In both cases the diffusion coefficients, D t and D r , are porportional to the the mobilities, where k B T is the thermal energy and Tr denotes the trace operator. The mobility components yield the linear and angular velocities of a colloid (u and ω) in response to applied forces and torques (f and τ ), For colloids larger than a few nanometers the mobility components can be calculated using the Stokes equations to a good approximation [3,4]. In this limit, the fluid velocity and pressure, v and p, obey the Stokes equations with viscosity η while for boundary conditions one can assume that the fluid velocity obeys the no-slip condition at the colloid's surface and that it decays to zero at infinity. We have assumed through this work that the virion, including its spikes, behaves like rigid body, thus the no-slip condition for a virion located at q is quite simple, These partial differential equations are closed by the balance of force and torque. The integral of the fluid traction, −λ, over the surface of the virion balance the external forces and torques apply to the virion [5] To solve the Stokes problem and compute the mobilities we use the rigid multiblob method [6]. We discretize the surface of the virion with N markers or blobs of radius a and with position r i . The blobs are subject to constraint forces, λ i , that ensure the rigid motion of the whole virion. Evaluating the no-slip condition at the blobs, as in collocation methods, leads to a linear system of equations for the unknowns u, ω and λ i , In the no-slip equation, Eq. (8), the blob mobility matrix (M B ) ij couples the force acting on the blob j to the flow generated at the blob i. We use the regularized Rotne-Prager mobility where G(r, r ) is Green's function of the Stokes equation, i.e. the Oseen kernel. The Rotne-Prager mobility has a closed analytical expression [7,8]. Given the forces and torques the linear system (8)- (10) can be solved to compute the particle velocities or, equivalently, the mobility matrix components from Eq. (2).

Dimensionless diffusivityD t andD r
The theoretical translational diffusion of a sphere is and the rotational is Furthermore, the translational mobility is and the rotational mobility is The ratio between computed mobility M t and theoretical mobility M o t is given by, for translational, andD for rotational.
3 Dimensionless diffusivityD t andD r for virions with spherical or ellipsoidal envelope with spikes 3.1 Sphere with spikes D t andD r of virions with spikes are calculated as the ratio between the mobility of the virions with spikes M t for translational and M r for rotational, divided by the single sphere M t | sphere and M r | sphere respectively,D The Eq.18 and Eq.19 are the computed dimensionless diffusivities for virions with spherical envelope.

Ellipsoid with spikes
Similar to the single ellipsoid and the sphere with spikes, in this case,D t andD r are the dimensionless ratio between computed mobility of the particle with spikes M t , M r divided by the calculated mobility of the single ellipsoid M t | ellip and M r | ellip The Eq.20 and Eq.21 are the computed dimensionless diffusivities for virions with ellipsoid envelope.

Optimal resolution determination
We calculate an optimal resolution to simulate the virions computing the rotational and translational mobilities of a sphere, ellipsoid, and different spikes morphologies. The optimal resolution is given when the difference between simulations converges. The resolution is given for the radius of the sphere divided by the distance between particles (ro) We test six different resolutions of 0.9 (12 particles), 1.8 (42 particles), 3.6(162 particles), 7.2(642 particles), 14.5(2562 particles) and 29 (10242 particles).
The errors ofD t andD r are in Supplementary Table 1. we registered the error for all the resolutions, and choose the favorable resolution as the best trade off between accuracy and computational cost. The resolution of 29 provided the better approximation, with a rather high computational cost. Therefore, we selected as a reasonable resolution the corresponding 14.5, with errors on the order of 1% inD t , and 3% forD r , for an spherical envelope. We remark that acceptable results can be already obtained with resolutions of 7.2, having errors of 2% forD t , and 6% forD r .

Ellipsoid -Equivalent radius R e
For ellipsoids, we calculated an equivalent radius R e of a sphere with equivalent volume. The volume of an ellipsoid with principal axis length a, b, and c, is The equivalent radius is then given by

Optimal resolution test
To determine an optimal resolution, are tested six different resolutions for sphere and ellipsoid envelopes. To calculate the mobility is important to consider the blob radius for the spheres and ellipsoids (section 4.2.1).
Supplementary  In the computational methodology RMB, we consider particles are located at a distance r o , and set a blob radius of r b = ro/2 for spheres and r b = 0, 75ro for ellipsoids to ensure the accuracy of the method.
Supplementary Figure 1: blob radius ro have overloaps, to minimize the overloap is defined a blob radius of ro/2 for sphere, and 0.75ro for ellipsoids.

Resolution envelope and spikes for SARS-CoV-2
We test four different resolution using spikes and envelope with the SARS-CoV-2 configuration. The main is identify the resolution differences using the spikes and envelope together. In Supplementary Table 3, is illustrated the outcomes for the dimensionless diffusivity and the computed radius. Considering the resolution of 29 as the best approximation, an optimal resolution of 14.5 have a maximum error of 0.7%. This is in contrast with coarser resolutions of 3.6 with errors on the order of 20% forD r .
Supplementary Table 3: Resolution test of normalized ellipsoid envelope with N s =26 tetra-rod spikes. The spikes length is l s /R=0.5 and tetrahedron width is w s /R=0.31. Where R is the largest axis of the ellipsoidal envelope.
ResolutionD tDr R t R r Figure   29 0

Tetrahedral spike resolution
For tetrahedral spikes we also conduct a separate calibration of the tetrahedron alone. We consider a tetrahedron of with a = w s , inter-blob distance r o , height is H tetra = √ 6 3 a, and circumscribed in a sphere of radius R tetra = w s (3/8) 1/2 See Supplementary Figure 2. Then, we characterize them using its equivalent radius (R e ), defined by the radius of a sphere with equivalent volume of the tetrahedron, V tetra = a 3 6 √ 2 , leading to The diffusivities for the single spikes are reduced as for the spheres and ellipsoids, with respect to a reference mobility Since we do not count with an analytical expression for the mobility of a tetrahedron, we define the reference mobility, as the one of a sphere of radius R e | solid , where R e | solid is the equivalent radius of a solid tetrahedron of width a. In Supplementary Table 4, we present the computed mobilities for different resolutions R tetra /r o , and the convergence criteria for each case. As a convergence criteria, we use the relative difference with respect to the highest resolution achieved, before the calculation become computationally prohibitive. Using a spike resolution of 2.5 we obtain an error of 3% for M t and 8% for M r , further improvement is achieved with resolutions of 4.9 with M t and M r errors of 1% and 4% respectively. The better resolutions reduce the error, but, requires a higher computational cost.
Supplementary Table 4: Different resolution of a discretized tetrahedron. The mobilities for the different resolutions are reduced using the mobility of a sphere of radius R e , for a solid tetrahedron of edge a. As a convergence estimator, we measure variation on the reduced mobility with respect to the largest resolution simulated. Resolution

Rotational and traslational diffusivity of SARS-CoV-2
For practical applications, virion diffusivities, D t | virion , can be easily retrieved using the known diffusivities of their respective plain envelope (sphere or ellipsoid), D i | envelope , and the reduced diffusivityD t , such that For SARS-CoV-2, we compute the diffusivities for G-form and D-form, the results are in Supplementary Table 6. Supplementary

Random distribution
The random distribution of spikes brings different mobility results. We compute ten replicas using the SARS-CoV-2 morphology with different number of spikes (12,15,18,21,24,26,30,33,36,39,42,50 and 100) and a resolution of 14.5, to estimate the error bars with the standard deviation. We calculate the fitting curve using the function in the Eq 28 of the mean of random replicas and just one of the replicas. The difference between the fitting functions illustrates an error of 2% in the number of spikes.
Supplementary  However, it is evident that at large N s , both random and uniform distribution converge due to the packing of the spikes in the envelope.
In Supplementary Figure 3 we present the dimensionless diffusivities of a SARS-CoV-2, using random and homogeneous distributions with different number of spikes. For the homogeneous distributions we required equidistant location of the spikes, thus we tested N s equal to 12, 42, and 162. Models with 100 spikes cannot be constructed directly without further optimization on spikes location, therefore for simplicity we calculated it as the interpolation between N s = 42 and N s = 162. For the random distribution are illustrated the mean of the diffusivities and the error bars. We can observe the error bars reduced as the number of spikes increases. Furthermore, the homogeneous distribution leads to diffusivities that are approximately 9.6% smaller than the random ones. However, as the number of spikes increases, the diffusivities of both homogeneous and random converged (difference of around 0.74%) due to the crowding of the spikes, leading to a close packing.

Size of the spikes
If the length and width of the spikes increase,D t andD r diminishes in a consistent fashion. Moreover, the rotational diffusivity exhibits a greater reduction than the translational one because of its dependency on the radius. Both, the length and the width are dimensionless, divided by the envelope radius. In Supplementary Figure 4 and 5, we present the variation on bothD t andD r for virion models with rod and tetra spikes, respectively. For rod type the size l s is varied, whereas for tetra type the radius R tetra (see Supplementary Figure 2) is varied. We can observe that the dimensionality of the spikes affects the scaling of diffusion with the spike size. Comparing Supplementary Figure 4 and 5, rod shapes (that are dominantly one dimensional) showed a weaker variation onD t andD r as the size increases. In contrast, tetra-shape spikes displayed a strong reduction on diffusivity. Overall, we observed that the virions with bulkier and larger spikes (compared to the envelope size) had an intrinsic diffusional penalty. Therefore, the regulation on the number of spikes suggests a possible alternative to compensate for this reduction in mobility.
Supplementary Figure 4: Dimensionless rotational and translational diffusivity for different spikes length l s /R. The simulation is with ellipsoid envelope, N s =12, uniform distribution and rod spikes.
Supplementary Figure 5: Dimensionless rotational and translational diffusivity for different tetraspikes radius R tetra . The simulation is with ellipsoid envelope, N s =12, uniform distribution and tetrahedron spikes.

Virions with spikes
Other viruses with similar morphologies as SARS-CoV-2 are tested and compared. The tested virions are SARS-CoV-2, HIV, MHV, SARS-CoV, Dengue, Lassa, Influenza and Herpes Simplex.

Fitting of the excess diffusivity functions
The results were plotted according the normalized diffusionD t andD r . For example: where N ∞ s is the number of spikes at which the virion mobility saturates, b is a fitting parameter that depend on the characteristic size of the spike and,D To determine the accuracy of the proposed fitting equation we computed the coefficient of determination R 2 d . The meanȳ isȳ and the coefficient is According the Eq.30 the calculated data f i is the fitting function (Eq. 28), the observed data y i are the dimensionless values of diffusion (D i ), and the mean of the observed dataȳ (Eq. 29) is for the dimensionless diffusion. The results summarized in Supplementary Table 13 show values of R d = 0.99 between dimensionless diffusion and the fitting function introduced.
Supplementary  10 Viruses binding affinity information.

Spikes concentration
We approximate the molar concentration of spikes as [S] =moles S/l, to compute the saturation function [S]/([S] + K D ) using the number of spikes N s , the length of the spike l s and the radius of the envelope R. The reaction volume is given by 4/3π((R + l S ) 3 − R 3 ), and the number of moles N s /NA. In Supplementary Table 14, we summarize the magnitude of the binding affinities between spikes proteins and cell receptors for the different viruses modelled. Since the smaller the value of K D , the smaller the amount of spikes needed to achieve an effective binding. Viruses with lower K D will reach saturation earlier.
Supplementary 11 Mesh construction 11.1 Sphere The construction of a sphere of radius R and distance between blobs r b starts with the coarser representation of an icosphere with 12 vertex located at R and distance r o between them, as shown in Supplementary Figure 9. One step refinement of this structure is conducted by taking the middle points of the segments connecting two adjacent vertex, and projecting those points radially, such that their new position satisfies R 2 = x 2 + y 2 + z 2 . This refining procedure can be repeated until the distance between vertex is smaller or equal to the target inter-particle distance (r o ≤ r b ).
Supplementary Figure 9: Blob representation of spheres using 12, 42, and 162 blobs in a uniform distribution

Tetrahedron
The tetrahedron construction starts with the primary coarse surface with only four vertex. Following the same methodology that for the spheres, the structure is refined by splitting in half the edges between two vertex and adding a new point in that position. This addition must be applied to all the edges of the primary surface. This procedure is repeated until the distance between adjacent points is smaller or equal that the target resolution. Tetrahedron with different three different resolutions are presented in SI.10 Supplementary Figure 10: Blob representation of tetrahedrons using 10, 34 and 130 blobs in a uniform distribution.

Rods
The rods construction consist in linking beads one by one, equidistant and all the centroids in the same angle. In Supplementary Figure 14 we have the ratio between the transportations time τ t and τ r , the results show a scale of 1000 seconds for the majority of the virions, but for the herpes simplex the scale is in 100 seconds and for dengue is for 10000 seconds. These differences are given by the radius of the virion.
Supplementary Figure 14: Time of transportation ratio τ t /τ r 13 Type of spikes Five types of spike morphologies are constructed using the RMB methodology: rod, tetra, sphere, rod-tetra, and rod-sphere. Rod, tetra, and sphere are characterize by a single lenght, whereas rodtetra and rod-sphere require two. parameters Supplementary Figure 15: Dimensionless rotational diffusivity for different type of spikes. Diffusivities are calculated using the ratio between the type of spike and the rod spike. Except for the sphere spike in a sphere envelope, all diffusivities are lower than the rod spikes.
Supplementary Figure 16: Dimensionless translational diffusivity for different type of spikes. Diffusivities are calculated using the ratio between the type of spike and the rod spike. Except for the sphere spike in a sphere envelope, all diffusivities are lower than the rod spikes.