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A Statistical-Thermodynamic Model of Viral Budding

Shelly Tzlil ^*, Markus Deserno , William M. Gelbart  and Avinoam Ben-Shaul ^* 

^* Department of Physical Chemistry and The Fritz Haber Research Center, The Hebrew University, Jerusalem, Israel; Max Planck Institute for Polymer Research, Mainz, Germany; Department of Chemistry and Biochemistry, University of California, Los Angeles, California; and Department of Biochemistry and Molecular Biophysics, Columbia University, New York, New York

Correspondence: Address reprint requests to A. Ben-Shaul, Dept. of Physical Chemistry, Hebrew University, Jerusalem 91904, Israel. Tel.: 972-2-658-5271; Fax: 972-2-651-3742; E-mail: abs{at}fh.huji.ac.il.

We present a simple statistical thermodynamic model for budding of viral nucleocapsids at the cell membrane. The membrane is modeled as a flexible lipid bilayer embedding linker (spike) proteins, which serve to anchor and thus wrap the membrane around the viral capsids. The free energy of a single bud is expressed as a sum of the bending energy of its membrane coat, the spike-mediated capsid-membrane adhesion energy, and the line energy associated with the bud's rim, all depending on the extent of wrapping (i.e., bud size), and density of spikes in the curved membrane. This self-energy is incorporated into a simple free energy functional for the many-bud system, allowing for different spike densities, and hence entropy, in the curved (budding) and planar membrane regions, as well as for the configurational entropy of the polydisperse bud population. The equilibrium spike densities in the coexisting, curved and planar, membrane regions are calculated as a function of the membrane bending energy and the spike-mediated adhesion energy, for different spike and nucleocapsid concentrations in the membrane plane, as well as for several values of the bud's rim energy. We show that complete budding (full wrapping of nucleocapsids) can only take place if the adhesion energy exceeds a certain, critical, bending free energy. Whenever budding takes place, the spike density in the mature virions is saturated, i.e., all spike adhesion sites are occupied. The rim energy plays an important role in determining the size distribution of buds. The fraction of fully wrapped buds increases as this energy increases, resulting eventually in an all-or-nothing mechanism, whereby nucleocapsids at the plasma membrane are either fully enveloped or completely naked (just touching the membrane). We also find that at low concentrations all capsids arriving at the membrane get tightly and fully enveloped. Beyond a certain concentration, corresponding approximately to a stoichiometric spike/capsid ratio, newly arriving capsids cannot be fully wrapped; i.e., the budding yield decreases.

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