This proposed research project aims to examine the human immunodeficiency virus type 1 (HIV-1). Mathematical models will be developed specifically for the first stage of the virus formation, called the nucleation stage, of HIV-1 capsid. The capsid acts as a protective shell for the genetic material (DNA or RNA) inside the virus and is in its weakest stage during maturation. After the capsid matures, the virus is able to attack new cells and replicate its DNA or RNA, leading to virus spread throughout the body. Therefore it is of great interest to characterize the favorable, restrictive, or even prohibitive conditions for capsid development so that these deterministic features can be targeted with antiviral therapies. This research will be conducted in collaboration with the computational virology expertise of Dr. Xiufen Zou at Wuhan University in China and will be validated with biological experiment data using the resources from The (China) State Key Laboratory of Virology.
Previous work has modeled the process of viral capsid assembly using one large-size dynamical system, combining both the nucleation and elongation phase. This approach has overlooked the effects and experimental evidence of these two stages. Investigating the nucleation stage first gives this model a unique advantage for characterizing the conditions required to start capsid formation and produce the building blocks for the mature capsid. Numerical simulations of the capsid protein assembly will be conducted using a 6-species dynamical systems model and will be implemented as a simulation code package. Deterministic and stochastic factors in the biological processes of viral capsid assembly will be examined as well as the stability of equilibria and sensitivity to model parameters. While in vitro experiments can be conducted to investigate these conditions, certain biological parameters in this process are immensely difficult to measure, so the mathematical model will enable us to analyze sensitivity of these parameters to outside factors, such as drug treatments. This NSF EAPSI award is funded in collaboration with the Chinese Ministry of Science and Technology.
NSF EAPSI 2014 Fellowship Grant IIA-1415117 Project Report Farrah Sadre-Marandi, Colorado State University There are two main outcomes from this project: a mathematical model for HIV-1 gag protein intracellular trafficking and assembly as published in paper  and a math model for HIV-1 viral capsid nucleation as shown in preprint .  Yuanbin Wang, Jingyin Tan, Farrah Sadre-Marandi, Jiangguo Liu, Xiufen Zou, Mathematical modeling for intracellular transport and binding of HIV-1 gag proteins, Math. Biosci., 262(2015), pp.198--205.  Farrah Sadre-Marandi, Yuewu Liu, Jiannguo Liu, Simon Tavener, Xiufen Zou, A kinetic model for HIV-1 viral capsid nucleation, In preparation (To be submitted in March 2015) With the support from this grant, I, as a Ph.D. student of Drs. James Liu and Simon Tavener in the Department of Mathematics at Colorado State University had an 8-week research visit to Dr. Xiufen Zou's research group in the Department of Mathematics at Wuhan University, China. During this visit, I had also interaction with researchers at the "Mathematics in Action: Modeling and analysis in molecular biology and electrophysiology" conference at Soochow University in Suzhou, China. This visit has strengthened our research collaboration in mathematical biology, specifically on development and validation of mathematical models for HIV-1 viral protein trafficking and assembly. Among the two main outcomes of this research project, a mathematical model for the intracellular trafficking and trimerization of HIV-1 gag proteins is formulated, as discussed in paper . The results in  show that active transport of the gag proteins inside an infected host cell, along with their trimerization near cell membrane, play important roles in the assembly of immature HIV-1 virions. However, it is known that the HIV-1 virions need to go through a maturation process for them to become infectious. During the maturation, HIV-1 capsid (CA) proteins assemble into a conic shell (called capsid) that protects the viral genome RNA. After the capsid forms, a virus is able to attack new cells and replicate its DNA or RNA, causing the cell to become infected. Therefore, it is of great interest to understand the formation of the capsid, with the goal of developing innovative antiviral therapies that can break or control capsid formation, as well as synthetic materials. It is known that there are two stages in the viral capsid assembly: nucleation and elongation. Some existing work model the whole process of viral capsid assembly using one large-size dynamical system, combining both the nucleation and elongation phases. This approach has overlooked the effects and experimental evidence of these two stages. This research project investigates a novel efficient approach for representing nuclei growth, independent of capsid elongation. This research project examines the association and dissociation pathways of lower order CA multimers, e.g., monomers, dimmers, trimers, in the nucleation stage. A 6-species dynamical system is used to model this process. The parameters in this model are estimated using published in vitro experimental data. Sensitivity analysis further examines how the system behavior relies on these parameters. Our results reveal that CA dimers (two joined proteins) indeed play an important role in the nucleation stage, which agrees with the findings in biological experiments. This research project advances knowledge within the fields of virology, medicine, and mathematics. It is well known that viruses are virulent to their host; one main goal of this project is to model the growth of viral capsids to provide insights for development of novel antiviral therapies. Alternatively, since capsids serve as a platform for synthetic manipulation, viruses can also be used to benefit the society. Icosahedral- and tube-shaped virus-like capsids have already been used in creating batteries, gene therapy, serving as a drug delivery vehicle for the anticancer drug doxorubicin, and the development of self-assembling nanostructure materials. Studying the factors that make retroviral assembly so robust could encourage the advancement of these technologies as well as giving scientists information about a new shape of capsid with different parameters and sensitivities to allow a new subset of synthetic manipulation.