key: cord-0292159-17yxpka5
authors: Moreno, Nicolas; Moreno-Chaparro, Daniela; Usabiaga, Florencio Balboa; Ellero, Marco
title: Hydrodynamics of spike proteins dictate a transport-affinity competition for SARS-CoV-2 and other enveloped viruses
date: 2022-01-03
journal: bioRxiv
DOI: 10.1101/2022.01.03.474721
sha: d76101d95406d72dd5e6538184474133f11e6b8c
doc_id: 292159
cord_uid: 17yxpka5

Many viruses, such as SARS-CoV-2 or Influenza, possess spike-decorated envelopes. Depending on the virus type, a large variability is present in spikes number, morphology and reactivity, which remains generally unexplained. Since viruses’ transmissibility depend 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 viruses differentiation. To understand how the spikes affect virion mobility and infectivity, we investigate the diffusivity of spike-decorate 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.

Using the RMB we discretize the virion as a set of rigidly connected blobs (see Figure 1 .c), and consider that it moves as 85 a rigid object. We construct virion models with spherical and ellipsoidal E of size R, and investigate the effects of spikes 86 shape on mobility using five different spikes inspired by the morphologies reported in the literature for various viruses 87 (i.e. HIV, MVH, Denge, SARS-CoV, Lassa, Herpes, Influenza). We adopt the following labelling for the investigated 88 shapes: rod, sphere, tetra, rod-sphere, and rod-tetra. Rod, tetra, and sphere shapes posess only one characteristic size: 89 length (l s /R) or width (w s /R), whereas for rod-tetra and rod-sphere both l s /R and w s /R are defined (see SI Section 7, 90 Figures 4 and 5). To account for S distribution on the surface of E, we construct virions with both homogeneous and 91 randomly localized S. For SARS-CoV-2 [11] a realistic representation is constructed using an ellipsoidal E, with rod-tetra 92 (prefusion) and rod (postfusion) S, localized in random configurations. 93 Based on the currently known structures [11] of SARS-CoV-2 in their original (D-form) and mutated (G-form) strains, 94 we give, in Table 1 , a breakdown of the numerically-estimated diffusion coefficients for three different media viscosities: 95 water, blood, and mucus (see SI. Table 12 a Table 4 and 5, we compiled the estimated diffusivities of SARS-CoV-2 on the range of 102 the variance on number of spikes reported. In the remaining, to streamline the discussion, we introduce a reduced translationalD t = D t | virion /D t | envelope and 104 rotationalD r = D r | virion /D r | envelope diffusion coefficients. Where D| envelope is the calculated diffusivity for an envelope 105 without spikes. Reduced diffusivities allow us to rationalize the results in terms of spikes count and morphology. The 106 results described herein correspond to resolutions with discretization errors below 3%. while convergence results are 107 presented in the Section 4 of the supporting information (SI Table 1 in E, whereas keeping virion transport unaffected. In contrast, the distribution of S showed to have a more important 115 role on virion transport. For all the spike shapes investigated we found a relative increment on bothD t andD r , when 116 spikes are localized at random positions around the surface relative to the homogeneous case (see SI Section 6 S. Figure   117 3 and S. Table 8 ). In particular, for rod-tetra type (used for SARS-CoV-2), the increment onD r ranged from 10% to 118 3%. We speculate that the sparsity and randomness of S favour the ability of the virion to explore its surroundings to 119 reach binding receptors. This is consistent with all-atoms simulations [25] of S, which suggested that the conformational 120 freedom of the spikes in E may increase the infectivity of the virus by providing mechanical robustness, facilitating 121 motions to avoid antibodies access, and increasing the avidity when binding the cell [25].

Spikes shape and size 123

In Figure 2 .a, we compare the reduced translation and rotational diffusivity for the five types of spikes studied for virions 124 with 12 homogeneously distributed spikes, and fixed spike size, for both spherical and ellipsoidal envelopes. The presence 125 of spikes induces a reduction in the translational diffusion of the virions between 20 to 30 per cent (compared to the naked 126 envelope), whereas the impact in rotational diffusion is more significant, ranging from 50 to 70 per cent. Significantly, 127 the spike shape determines the extent of the reduction on bothD t andD r . Overall, larger spikes affect the transport 128 properties of the virion strongly. The small but noticeable differences in the diffusion between globular and tetrahedral 129 spikes indicate a characteristic transport signature that can be further exploited for virus identification. In SI Section 13 130 (S. Figure 15 and 16), we show the relative differences in mobility for the S shapes evaluated.

131 Figure 2 : Effect of S morphology on the mobility of virions. a. Comparison of the reduced translational and rotational diffusion for the different S shapes for virions with N s = 12 spikes, and l s /R = 0.4, homogeneously distributed. Ellipsoidal envelopes exhibit slightly larger deviations on the rotational diffusion than spherical envelopes due to their small asymmetry. The differences inD t between S shapes do not reveal a significant difference. In contrast, forD r , small but observable differences indicate the potential use of the rotational diffusion of virions as a rheological biomarker. b. Variation onD t andD r with N s for fully postfusion (rod), fully prefusion (rod-tetra) and mixed postfusion/prefusion S. The diffusion coefficient values for each N s are obtained from ten independent realizations with S randomly distributed on the envelope. The fraction of postfusion S in the mixed case corresponds to the original D-forms of SARS-CoV-2, N s | post /N s = 0.7. Fully prefusion case is consistent with mutated G-forms characterized by N s | pre /N s ∼ 1. Regardless of the differences in S morphology, the reduction in bothD t andD r exhibits the same functional dependence as N s increases. However, the magnitude of the mobility reduction is larger for the full-prefusion case.

Regarding the effect of the size of the spike (l s /R and/or w s /R) we found, as expected, that larger spikes considerably 132 reduce the diffusion of the virions in all cases. Nevertheless, the dimensionality of the spikes affects the scaling of diffusion 133 with the spike size. For instance, comparing rod-and tetra-type shapes, rod shapes (that are dominantly one dimensional) 134 showed a weaker variation onD t andD r as the size increases (see SI Figure 4 ). In contrast, tetra-shape S displayed a 135 strong reduction on diffusivity (see SI Figure 5 ). Overall, we observed that the virions with bulkier and larger spikes 136 (compared to the envelope size) had an intrinsic diffusional penalty. Therefore, the regulation on the number of spikes 137 suggests a possible alternative to compensate for this reduction in mobility. where increasing N s no longer significantly affect the diffusion rate of the virions.

To explore further the decay in mobility due to N s , we modelled various enveloped viruses (Figure 3 .a) using the 150 characteristic sizes reported in the literature (in SI Table 10 we compile N s , size, and S shape for de different virions, 151 and in Tables 11-12 the estimated reduced difusivities). Remarkably, we identified that the form of the decay inD t and 152D r (see SI Figure 6 and 7) is consistent in other viruses, as shown forD r in Figure 3 .a. However, the magnitude of the 153 drop in mobility depends on the S shape and size. For example, Herpes virions with l s /R = 0.22 can exhibit upto 23% 154 drop inD r , whereas SARS-CoV-2 with l s /R = 0.52 showed reduction up to 70%.

Considering that the diffusivity of the virions varies between one corresponding to an envelope without spikes (D i = 1) 156 and one with closely packed spikes (

for i = t, r. This expression varies from 1 when N s = 0 to 0 when N s = N ∞ s . The term N ∞ s is the number of S at which 158 the virion mobility saturates,D ∞ i , and the effect of N s is negligible. Based on the computed translational and rotational 159 diffusivities we postulate the following expression to describe the excess diffusion dependency with N s

where N ∞ s and b are fitting parameters that depend on the characteristic size of the spike. The first term on the right-161 hand side of (1) accounts for a linear dependency on the number of spikes before reachingD ∞ i . Similar linear dependence 162 on N s has been identified for ligand-receptor interactions using functionalized colloids [26] . The second term on the right-163 hand side of (1) describes an exponential decay in the diffusivity that is controlled by the shape-dependent parameter, Table 2 and SI ( Figure 8 166 and Table 13 Section 9). In Figure 3 .b, we depict the variation in ∆D r for different viruses along with the experimentally 167 reported N s . Except for HIV and MHV (with ∆D r ∼ 0.5), independently of the type, we identify a characteristic trend 168 in the mobility, with a mean ∆D t ∼ 0.12 and ∆D r ∼ 0.08 among all the viruses. This suggests the existence of a general 169 transport mechanism across enveloped viruses. We speculate that this mobility regime coincides with condition where 170 reactivity and mobility balance out. For each type of virion, the shape and size of S determines hydrodynamically how 171 much the mobility can change, while the reactivity of S sets the extent of such reduction inD t andD r . Thus, this 172 interplay is conserved across virus families. In the following section we address this hypothesis.

173 Table 2 : Summary of characteristic N s , l s , w s , and spike volume V s /R 3 for different virions, along with the fitted parameters N ∞ s (±2) and b (±2 for ∆D t and ±3 for ∆D r ) from (1). All the parameters obtained from fitting of (1) lead to a determination coefficient As the viral spikes of SARS-CoV-2 mutate into new variants with potentially different affinities but equivalent spikes 175 size, it is necessary to elucidate the relative balance between affinity and mobility. To consider only affinity effects, it is Table 2 and SI Section 9.

Filled markers indicate the experimentally reported N s for the different viruses. Viruses share an equivalent diffusional state independently of N s . This ground state is likely induced as a balance between their geometrical features and spikes reactivity. The higher ∆D r in HIV and MHV can be explained due to an enhanced mobility of S on the E surface. Such effect is not currently accounted in our model. c. Saturation function change with N s for various virus based on their binding constant K D to cellular receptors. The magnitudes of K D are summarized in SI Table 14 

In SI Section 12 (Figures 12-14) , we have summarized translational and rotational time scales for the different virion 204 types, for the first entry stage.

If we consider that the energy of binding between RBDs and a cell receptor is bind , the strength of all the interactions 206 combined (upto a first order approximation) can be expressed as Γ = N s bind . In general, increasing the number of spikes 207 should favor virion avidity, whereas reducing the diffusion rate of the virion as shown in Figure 3 .b. For convenience, we 208 introduce a geometrically constraint avidityΓ, given by Now, we define an infectivity parameter I = ∆D rΓ that considers the interplay between the transport and reactivity. results provide an initial step for further microrheological characterization of viral solutions that can serve as a tool to 235 identify potential biomarkers and overall gaining an understanding of viral functionality. 236 We show how the interplay between spikes distribution, shape, size affect the mobility of the virions. We postulate 237 that transport properties of the virions roots from geometrical constraints that can explain the differences in spikes 238 density across a variety of virus. Thus, these geometrical constraints, along with the affinity of the RBD of the spikes, 239 may indicate how different virus families exist on optimal evolutionary conditions. The saturation values on S affinity 240 and along with I, justify the lower N s reported for SARS-CoV-2 and its variants. From an evolutionary standpoint, 241 SARS-CoV-2 may have reached an optimal avidity/mobility balance that ensures a large mobility due to a moderate 242 value of N s and a large affinity to the receptors groups thanks to a high saturation value. dimensions of E and S respectively. In Section 11 of SI we describe the construction of discrete morphologies for E (SI 372 Figure 9 ) and S (SI Figure 10 and 11). We use resolutions fine enough to compute the mobilities with errors below 3%, 373 see SI Section 4 for convergence results.

The translational [32] and rotational [33] diffusivity of can be then computed using the numerical approximation of M , as of an equivalent sphere with the same volume and equivalent radius R e (see SI Section 4.1 for detailed description of R e ).

For the SARS-CoV-2 ellipsoidal envelopes investigated, with ratios between the minor and major radius of the principal 394 axes of 0.9 and 0.7, we obtain D ellip /D o ∼ 0.98. This is consistent with semianalytical derivations of drag coefficients [34] 395 that lead to ratios on the order of 0.99.

Spikes shape

To investigate the effect of shape, we construct a model of spikes consistent with the morphologies presented in Figure 1 .c.

In Figure 5 , we present the reported morphology and size for the different virions investigated. SI Table 10 to discretization, we test various ratios between spike size l s , and inter-blob distance r o . In SI Section 4.3 (Table 3) and Table 3 : Resolution test for SARS-CoV-2 models with N s =26 tetra-rod spikes. The spikes rod length is l s /R=0.5 and tetrahedron width is w s /R=0.31. The resolution R/r o and computed reduced coefficients are presented. The resolution adopted in our simulations is highlighted. ResolutionD tDr R t R r Figure   29 0 Spikes distribution 417 We compare the mobility of virions having homogeneously and randomly distributed spikes to elucidate the effects of 

Isotropically distributed spikes preserve the symmetry of the envelopes in a way that the hydrodynamic forces and torques 428 originated due to the presence of spikes is kept balanced across the axes. As the distribution of S becomes anisotropic, 429 this balance is broken, favouring virion mobility along some axes. At low spikes densities random configurations exhibit 430 overall lower mobility reduction than homogeneous distributions. As the number of spikes increases, the packing of the 431 spikes become more restricted, and the symmetry-breaking effects of random distributions vanish. Owning the symmetry 432 of spherical envelopes, the effect of spikes distribution appears more significant compared to the ellipsoidal ones. 

Architecture of the SARS coronavirus 261 prefusion spike