Model of human immunodeficiency virus budding and self-assembly: Role of the cell membrane

Budding from the plasma membrane of the host cell is an indispensable step in the life cycle of the human immunodeficiency virus (HIV), which belongs to a large family of enveloped RNA viruses, retroviruses. Unlike regular enveloped viruses, retrovirus budding happens concurrently with the self-assembly of the main retrovirus protein subunits (called Gag protein after the name of the genetic material that codes for this protein: Group-specific AntiGen) into spherical virus capsids on the cell membrane. Led by this unique budding and assembly mechanism, we study the free energy profile of retrovirus budding, taking into account the Gag-Gag attraction energy and the membrane elastic energy. We find that if the Gag-Gag attraction is strong, budding always proceeds to completion. During early stage of budding, the zenith angle of partial budded capsids, $\alpha$, increases with time as $\alpha \propto t^{1∕2}$. However, if the Gag-Gag attraction is weak, a metastable state of partial budding appears. The zenith angle of these partially spherical capsids is given by ${\alpha }_0\simeq {({\tau }^2∕\kappa \sigma )}^{1∕4}$ in a linear approximation, where $\kappa$ and $\sigma$ are the bending modulus and the surface tension of the membrane, and $\tau$ is a line tension of the capsid proportional to the strength of Gag-Gag attraction. Numerically, we find ${\alpha }_0<0.3\pi$ without any approximations. Using experimental parameters, we show that HIV budding and assembly always proceed to completion in normal biological conditions. On the other hand, by changing Gag-Gag interaction strength or membrane rigidity, it is relatively easy to tune it back and forth between complete budding and partial budding. Our model agrees reasonably well with experiments observing partial budding of retroviruses including HIV.

Figure 1

Electron microscopic images of partial budding of HIV-1 viruses. (a) Reprinted from Ref. 2, with permission by Annual Reviews. Bar: $100\mathrm{nm}$. (b) Reprinted from Ref. 4, with permission by Journal of Virology. Bar: $500\mathrm{nm}$.

Figure 2

(Color online) Schematic illustrations of two different types of virus budding. (a) Budding of retroviruses. Capsid proteins (Gags) are first attracted to the membrane, then self-assemble and bud on the membrane at the same time. This is the system studied in this paper. (b) Budding of regular enveloped viruses. Capsid proteins first self-assemble into complete capsids inside the cell, then bud on the membrane [5]. (a) Also shows the cylindrical coordinate system $(r,h,\varphi )$ used in the model (the azimuthal angle $\varphi$ is not shown).

Figure 3

(Color online) The schematic illustration of the total free energy density as a function of the capsid size $\alpha$. The left- and right-hand profiles correspond to strong or weak Gag-Gag attraction, respectively. Here $\tau$ is the line tension of the rim of a partially budded capsid. $\tau$ is proportional to the strength of Gag-Gag attraction. ${\tau }_c$ is the threshold line tension at which the local minimum at ${\alpha }_0$ appears.

Figure 4

(Color online) Numerical result of the optimal membrane shape (dark gray, blue online) around a capsid (light gray, red online). The upper panels are plotted in the “soft membrane” regime. $\tilde{\sigma }=0.1$ or $r_s=10R$. Here the membrane takes the catenoid shape. The lower panels are plotted in the “stiff membrane” regime. $\tilde{\sigma }=10$ or $r_s=0.1R$. The left-hand panels are plotted at $\alpha =0.25\pi$. The right-hand panels are plotted at $\alpha =0.75\pi$. $r_c$ are shown for comparison with corresponding $r_s$.

Figure 5

(Color online) Numerical result of the dimensionless membrane elastic energy ${\tilde{\epsilon }}_m={\epsilon }_m∕\kappa$ as a function of $\alpha$. The 11 sets of data points are at $\tilde{\sigma }={10}^{-5},{10}^{-4},{10}^{-3},\dots ,{10}^5$. They are labeled correspondingly as $-5,-4,-3,\dots ,5$. The right axis ${\tilde{\epsilon }}_m∕\tilde{\sigma }$ is for all $\tilde{\sigma }⩾1$ data points (light gray, red online), while the left axis ${\tilde{\epsilon }}_m∕{\tilde{\sigma }}^2\ln (1∕\tilde{\sigma })$ is for all $\tilde{\sigma }<1$ data points (dark gray, blue online), as indicated by the two arrows. The curve represents the analytical asymptotic solution (30) with an additional factor 1.3 (dark gray, blue online), fitting the data points for $\tilde{\sigma }⪡1$.

Figure 6

(Color online) An effective “phase diagram” in the plane of two dimensionless parameters, $\tilde{\sigma }=R∕r_s$ and $\tilde{\tau }=R\tau ∕\kappa$. In the upper-left-hand part, the free energy density decreases monotonically with $\alpha$, while in the lower-right-hand part, it has a local minimum, as shown in the insets. The numerical data points mark the “phase boundary” at which the local minimum appears. The dotted-dashed line and the solid lines fit the data points using $\tilde{\tau }=0.11\tilde{\sigma }$ and $\tilde{\tau }=0.065{\tilde{\sigma }}^2\ln (1∕\tilde{\sigma })$, respectively.

Figure 7

(Color online) The capsid size ${\alpha }_0$ as a free energy local minimum is shown as a function of $\tilde{\tau }∕\tilde{\sigma }$ and $\tilde{\tau }∕{\tilde{\sigma }}^2\ln (1∕\tilde{\sigma })$ at two limits of $\tilde{\sigma }$. The dots are numerical results taken at $\tilde{\sigma }={10}^2,{10}^3,{10}^4,{10}^5$ for the upper panel and $\tilde{\sigma }={10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5}$ for the lower panel. The curves are analytical results of Eq. (40) at $\tilde{\sigma }⪢1$ and Eq. (37) at $\tilde{\sigma }⪡1$ with additional numerical factors of 2 and 1.5, respectively. The range of $\tilde{\tau }$ plotted corresponds to the lower-right-hand “phase” in the “phase diagram” of Fig. 6.

Figure 8

(Color online). The local minimum ${\tilde{f}}_0$, maximum ${\tilde{f}}_m$, and barrier ${\tilde{f}}_m-{\tilde{f}}_0$ of the free energy density. The values of $\tilde{\sigma }$ and $\tilde{\tau }$ plotted are the same as in Fig. 7. The circles (blue) are the numerical result of ${\tilde{f}}_0$, fitted by the solid lines (blue) using Eq. (41) at $\sigma ⪢1$ and Eq. (38) at $\sigma ⪡1$ with additional numerical factor 1.7 and 3.2, respectively. The squares (green) are the numerical result of ${\tilde{f}}_m$, marked by the dashed lines (green) at their zero $\tilde{\tau }$ values. The triangles (red) are numerical result of the barriers, ${\tilde{f}}_m-{\tilde{f}}_0$, fitted by the dotted-dashed lines (red) using Eqs. (46, 47).