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M. Martín-Bravo*, J. M. Gomez Llorente & J. Hernández-Rojas Departamento de Física e IUdEA, Universidad de La Laguna, 38205, Tenerife, Spain * [email protected] — (+34) 922 31 82 53 Unveiling latent instabilities in the empty Satellite Tobacco Necrosis Virus capsid by performing a low-frequency normal modes analysis using a novel coarse-grained model INTRODUCTION A minimal interaction model between axially symmetric bodies [1] is generalized to account for the interaction between viral capsomers with a given rotation symmetry axis (Fig. 1) [2]. In this model, the capsid is represented as an interacting many-body system whose composing elements are capsid subunits (capsomers), which are treated as three- dimensional rigid bodies with rotational symmetry around an axis of order n > 2. THE MODEL The total interaction potential energy V is written as a sum of pairwise capsomer–capsomer interactions V ij . It can be proved that if binary interactions are of very short range, all contributions vanish at the capsid equilibrium configuration except for p 0 F 0 (r ij ), which accounts for the depth of the potential well. The cross terms of V ij (2) break the I h symmetry (from V ij (0) + V ij (1) ) into I for the full capsid and completes the missing non-diagonal elements of the Hessian matrix. These term are: RESULTS CONCLUSIONS Fig. 1 Equilibrium configuration of a pair of pentamers in a regular dodecahedron around a twofold symmetry axis, showing in pink the vectors used in our model. ACKNOWLEDGMENTS Ministerio de Economía y Competitividad, Gobierno de España, for financing the project FIS2016-79596-P, Ministerio de Ciencia e Innovación, Gobierno de España, for financing the project PID2019-105225GB-I00 and Ministerio de Economía, Industria y Competitividad, Gobierno de España, for financing the grant BES-2017-081104. REFERENCES [1] J. M. Gomez Llorente, J. Hernández-Rojas and J. Bretón, Soft Matter, 10, 3560 (2014). [2] M. Martín-Bravo, J. M. Gomez Llorente and J. Hernández-Rojas. Soft Matter, 16, 3443 (2020). [3] E. C. Dykeman and O. F. Sankey, Phys. Rev. Lett. 100, 028101 (2008). [4] Nikesh Patel, Eric C. Dykeman, Robert H. A. Coutts, George P. Lomonossoff, David J. Rowlands, Simon E. V. Phillips, Neil Ranson, Reidun Twarock, Roman Tuma and Peter G. Stockley. PNAS, 112(7), 2227 (2015). A minimal coarse-grained model for the interaction energy of viral capsids as a sum of pairwise interactions between properly chosen equivalent subunits (trimers and pentamers for T = 1 capsids) has been designed as the most complete form within the linear response regime. The binary interaction is anisotropic and leads to a full Hessian matrix of independent force constants. This model is able to reflect instabilities and can be regarded as a promising tool to theoretically interpret many experimental data such as those provided by the Atomic Force Microscopy or even to better understand processes far from equilibrium such as the capsid self-assembly. Moreover, it is straightforward to generalize the model beyond the linear regime or to include interactions with other capsomers like hexamers. Fig. 2 Capsid frequencies separated by symmetry species. Pentons in the left and trimers in the right. Dotted lines are used for the target frequencies from [3] and solid lines give the model results. Black colour is used for the frequencies included in the fit and red for the frequencies excluded from the target set. Turquoise is used for the additional frequencies predicted by the coarse-grained model. The capsid configuration is found to be a global minimum of energy only for trimers (regular icosahedron). For pentamers (regular dodecahedron), it is just a local minimum. Pentamers join each other but twisted, which impedes the STNV capsid from assembling. This is consistent with the theoretically and experimentally known properties of the STNV capsid since the assembly is possible only in presence of genetic material forming trimers at room temperature [4], which would justify the unstable mode found for pentamers and the inability of the trimer capsid to detect the instability. Tab. 1 Optimal values of the normal-mode frequencies (cm −1 ) for the binary interaction for trimers and pentamers. The model performance is evaluated by fitting the spectrum to the full-atom results for the Satellite Tobacco Necrosis Virus (STNV) [3]. Both pentons and trimers provide fair fits (~5% error) for the capsid configuration (Fig. 2). However, the parameters obtained from the fit lead to an imaginary frequency in the normal mode spectrum of the binary system alone (Fig. 1) to show an imaginary frequency mode (Tab. 1), which makes the capsomer-pair equilibrium configuration unstable along this mode coordinate (a transition state).
