Electrostatics-Driven Inflation of Elastic Icosahedral Shells as a Model for Swelling of Viruses.

Article Electrostatics-Driven Inflation of Elastic Icosahedral Shells as a Model for Swelling of Viruses 2
An
ze Lo
sdorfer Bo
zi
c1,* and Antonio Siber 1 Department of Theoretical Physics, Jo
zef Stefan Institute, Ljubljana, Slovenia and 2Institute of Physics, Zagreb, Croatia ABSTRACT We develop a clear theoretical description of radial swelling in virus-like particles that delineates the importance of electrostatic contributions to swelling in the absence of any conformational changes. The model couples the elastic parameters of the capsid—represented as a continuous elastic shell—to the electrostatic pressure acting on it. We show that different modifications of the electrostatic interactions brought about by, for instance, changes in pH or solution ionic strength are often sufficient to achieve the experimentally observed swelling (
10% of the capsid radius). Additionally, we derive analytical expressions for the electrostatics-driven radial swelling of virus-like particles that enable one to quickly estimate the magnitudes of physical quantities involved. INTRODUCTION Molecular interactions in viruses regulate their stability with respect to chemical and physical influences such as changes in pH, ionic strength, and temperature. The integrity of viruses in a changing environment is important for their successful propagation from cell to cell and for their survival in the inactive, compact state between two hosts. At the same time, the molecular interactions involved in viral stability and integrity need to be sufficiently ‘‘soft’’ and weak to enable the dynamics of the viral life cycle, which in most cases includes the disassembly of viruses, i.e., the disintegration of their protein shell (capsid) and the release of their genome (either a DNA or RNA molecule in single- or double-stranded form). These interactions are encoded primarily in the physicochemical properties of the capsid proteins, in the nature of the packaged genome, and in some cases in the properties of proteins that serve to condense the genome and pack it more efficiently (1–5). There is not much more to an assembled virus other than the interactions that keep it together—once outside the host cell, the virus can be thought of as a macromolecular complex held together by electrostatic- and entropy-derived forces. And yet, the roles of these different interactions have yet to be fully elucidated. There have been attempts to study viruses on a molecular level (6–11), which is a Submitted May 24, 2018, and accepted for publication July 30, 2018. *Correspondence: Editor: Nathan Baker. https://doi.org/10.1016/j.bpj.2018.07.032 Ó 2018 Biophysical Society. 822 Biophysical Journal 115, 822–829, September 4, 2018 daunting task, as viruses contain a huge number of atoms. For instance, even very small viruses, such as satellite tobacco mosaic virus (STMV) or southern bean mosaic virus, contain more than 106 atoms (9,10). Furthermore, capsid integrity depends on the molecules that surround it, such as water and dissolved ions, which need to be taken into account in molecular simulations (7,11). All this requires the knowledge of a lot of parameters that determine the different atomic interactions involved. Other attempts at understanding the contributions of interactions involved in capsid assembly and stability use either various coarse-graining methods (4,12–15) or simplified continuum models (16–20) and are based on some generalized form of interactions. This is possible because, even though the nature of capsid proteins influences the mechanics of the capsids, there exist more generic aspects of the physics of the capsid shells as well. These overarch both the chemical specificity of their protein constituents and the myriad of molecular interactions involved (21). The generic aspects of capsid mechanics should be prominent at the spatial scales typical for viruses (
10–500 nm), and it is important to clearly separate them from protein- and molecule-specific interactions. In this way, we can elucidate the background physical principles that do not depend on the molecular details but only on a small number of parameters characterizing the elastic response of a virus shell to changes in the environment. This should be of help in identifying the wider space of physical possibilities and potentialities available in the course of viral evolution. Virus Swelling as Shell Inflation What is more, coarse-grained approaches may at present be the most reliable way to extract the relevant energy scales and forces involved in viral life cycles. Such approaches have led to important results regarding generic aspects of both electrostatic interactions in viruses (16,22,23) as well as their elasticity (13,17,24–28) and have enabled classifications of viruses according to their electrostatic (29,30) and elastic (31–33) nature. This study is a step further in this direction and couples different aspects of physics of viruses to explain the radial expansion— swelling—of viruses within a simple generic framework, connecting together the relevant elastic and electrostatic energy scales. Swelling is a quite common phenomenon in viruses (34–36), often observed also as a side effect of the procedures applied to study the stability and (dis)assembly of viruses (37). It has been studied in detail, particularly in the case of cowpea chlorotic mottle virus (CCMV) (38–42), as well as other plant viruses such as brome mosaic virus (43), STMV (44), and southern bean mosaic virus (45–47). Swelling is often triggered by the changes in the environment that modify the electrostatic interactions in the system, known to be of key importance for the fixation of the capsid shape and structure (48). Swelling can thus arise due to changes in the pH or the ionic strength of the solution, release of bound ions (such as Ca2þ), or different modifications of charge on the capsids. The main aim of our work will be to elucidate the more universal physical principles that can drive capsid swelling when the latter is sufficiently small and no conformational changes occur. Depending on the stiffness of the capsid and on the magnitude of perturbation from the equilibrium state, such changes may be observable or not, so the effect could be more prominent in some virus species than in others (33). In this way, we will provide a complementary view of radial swelling in capsids, and we shall explain and quantify the effects within a simple theoretical framework. METHODS Swelling as expansion of an elastic icosahedral shell under pressure As a capsid can be viewed, at least approximately, as an elastic shell (24), it should elastically deform in response to (small) forces acting on it. A sufficiently small swelling could thus be viewed as an elastic response of the capsid to the extra forces acting on it, a description that we will use in our work. This excludes situations in which swelling involves a significant conformational change of the (...truncated)