Global bifurcation in a virus, defective genomes, satellite RNAs tripartite system: breakdown of a coexistence quasi-neutral curve

Abstract

The dynamics of wild-type (wt) RNA viruses and their defective viral genomes (DVGs) have been extensively studied both experimentally and theoretically. This research has paid special attention to the interference effects of DVGs on wt accumulation, transmission, disease severity, and induction of immunological responses. This subject is currently a highly active. However, viral infections involving wt, DVGs and other subviral genetic elements, like viral RNA satellites (satRNAs) have received scarce attention. Satellites are molecular parasites genetically different from the wt virus, which exploit the products of the latter for their own replication in as much as DVGs do, and thus they need to coinfect host cells along with the wt virus to complete their replication cycle. Here, we analyze a mathematical model describing the initial replication phase of a wt virus producing DVGs and coinfecting with a satRNA. The model has three different dynamical regimes depending upon the wt replication rate ($α$), the fraction of DVGs produced during replication ($ω$), and the replication rate of the satRNA ($β$): ($i$) full extinction when $β> α(1 - ω)$; ($ii$) a bistable regime with full coexistence governed by a quasi-neutral curve of equilibria and full extinction when $β= α(1 - ω)$; and ($iii$) a scenario of bistability separating full extinction from wt-DVGs coexistence with no satRNA when $β< α(1 - ω)$. The transition from scenarios ($i$) to ($iii$) occurs through the creation and destruction of a quasi-neutral curve of equilibria in a global bifurcation that we name as \textit{quasi-neutral nullcline confluence} (QNC) bifurcation: at the bifurcation value, two nullcline hypersurfaces coincide, giving rise to the curve of equilibria.

Figures (10)

• Figure 1: (a) Schematic diagram of the main interactions modeled for the tripartite system wild-type helper virus (HV), its interfering defective particles (DIPs), and a satellite RNA (satRNA) coinfecting the same cell. The HV RNA genome is translated into the viral RNA-dependent RNA polymerase (RdRp). The RdRp replicates the HV genomes producing DIPs at a rate $\omega$. The RdRp also supports the replication of the DIPs and the satRNA. It is assumed that all the viral agents compete for cellular resources such as nucleotides. Panels (b-d) display phase portraits for three qualitatively different scenarios which depend on the effective replication rate of the HV $\alpha (1-\omega)$ and the replication rate of the satRNA $\beta$. (c) Quasi-neutral curve $\Gamma$ (red curve with locally stable (solid line) and unstable (dashed line) branches). Here, black dots denote local asymptotically stable equilibrium points. White dots are equilibrium points of saddle type with three-dimensional stable manifolds and an unstable one. Blue orbits belong to the basin of attraction of the origin $Q_0$; grey orbits to the basin of attraction of the satRNA-extinction equilibrium point $Q_1$, and the red ones to the basin of attraction of coexistence equilibria $Q_2$.
• Figure 2: Projection of solutions of Eqs. \ref{['eq1']}-\ref{['eq2']} onto the plane $(V,S)$ for $\alpha(1-\omega) = \beta$. The arrows indicate the direction of the orbits in positive time.
• Figure 3: (a) Projection of the quasi-neutral curve $\Gamma$ (red) onto the space $(V,S,D)$. $\Gamma$ is contained in the invariant plane $\Pi_{DV}$ (blue surface) and embedded into the projection of $\mathcal{U}$ onto $(V,S,D)$ (grey tetrahedron). (b) Schematic projection in the space $(V,S,D)$ of the heteroclinic connections among $Q_2$-points in $\Gamma$. Notice that, in particular, these connections belong to the homoclinic invariant manifold of the curve $\Gamma$ itself. The heteroclinic connections (straight lines) go in infinite time from the unstable equilibrium points in $\Gamma_r$ (dashed red curve) to the locally attracting equilibrium points of the piece of curve $\Gamma_a$ (in solid red color). The dark-green point in $\Gamma$ separates both branches, being tangent to the plane $S/V=c^*$ and having two zero eigenvalues in its jacobian matrix. The equilibrium points $Q_1^1$ and $Q_1^2$ (in light blue color) represent the intersection of $\Gamma$ with the invariant plane $\{ S=0\}$.
• Figure 4: Coordinates of equilibrium points $Q_2$ filling $\Gamma$ (upper panels) and the corresponding eigenvalues of their differential matrix (lower panels). The parameter values used satisfy conditions in \ref{['cond:eq:coex']} and are given by $\alpha=0.875$, $\beta = 0.7$, $\omega = 0.2$, $\gamma = 0.6$, $\kappa = 0.3$, $\varepsilon = 0.1$ and $\varepsilon_p = 0.01$. Panels (a) and (b) are related, respectively, to the two possible expressions for $V_2$ given by \ref{['fS_numerical']}. The point $(S_2^*,V_2^*)$, marked with a cross, is given by expression \ref{['double:0:eigval:coord']} and corresponds to the $(V,S)$-coordinates of $Q_2^*$.
• Figure 5: Time series for values of $\alpha$ such that $|\mu| > 0.1$ (far from the global bifurcation value), with $\alpha < \alpha^*$ ($\mu<0$) (a); and $\alpha > \alpha^*$ ($\mu>0$) (b). Panels in (b) display the bistability scenario where both the origin and the satRNA-extinction equilibrium point $Q_1$ are locally asymptotically stable. Compare the transient times far from the bifurcation shown in this figure with those transients found close to the bifurcation (Figs. \ref{['alpha<alpha*']}, \ref{['pOm_TS']}, \ref{['mu-5_alpha>alpha*']}).

Theorems & Definitions (19)

• Lemma 1
• Proposition 1: Non-existence of periodic orbits
• Lemma 2
• Lemma 3
• Lemma 4
• Proposition 2: SatRNA extinction equilibrium points
• Remark 1
• Proposition 3: Coexistence equilibrium points
• Proposition 4
• Proposition 5