Mathematical determination of the HIV-1 matrix shell structure and its impact on the biology of HIV-1

Since its discovery in the early 1980s, there has been significant progress in understanding the biology of type 1 human immunodeficiency virus (HIV-1). Structural biologists have made tremendous contributions to this challenge, guiding the development of current therapeutic strategies. Despite our efforts, there are unresolved structural features of the virus and consequently, significant knowledge gaps in our understanding. The superstructure of the HIV-1 matrix (MA) shell has not been elucidated. Evidence by various high-resolution microscopy techniques support a model composed of MA trimers arranged in a hexameric configuration consisting of 6 MA trimers forming a hexagon. In this manuscript we review the mathematical limitations of this model and propose a new model consisting of a 6-lune hosohedra structure, which aligns with available structural evidence. We used geometric and rotational matrix computation methods to construct our model and predict a new mechanism for viral entry that explains the increase in particle size observed during CD4 receptor engagement and the most common HIV-1 ellipsoidal shapes observed in cryo-EM tomograms. A better understanding of the HIV-1 MA shell structure is a key step towards better models for viral assembly, maturation and entry. Our new model will facilitate efforts to improve understanding of the biology of HIV-1.