Impact of Latently Infected Cells on Strain Archiving Within HIV Hosts

Abstract

Latently infected cells are a barrier to HIV eradication on therapy due to long half-lives of between 6 and 44 months. The mechanism behind this long term maintenance is unclear although bystander proliferation and asymmetric division have both been put forward for consideration in mathematical models. The latently infected cell reservoir seems to act as an archive for strains of HIV no longer dominant in the blood, such as wild-type virus when the individual is on therapy. This is particularly significant when patients wish to come off medication and wild-type virus re-emerges.

We use a two target cell model capable of producing low-level viral load on therapy and include latent cells and two strains of virus, wild-type and drug resistant, to investigate the impact of two possible mechanisms of latent cell reservoir maintenance on strain archiving. We find that although short term (less than a year) archiving of viral strains is possible in a model with no mechanism for reservoir maintenance, both bystander proliferation and asymmetric division of latent cells allow archiving to occur over much longer timescales (2 or more years). We suggest that regardless of the mechanism involved, latent cell reservoir maintenance allows strain archiving to occur. We interpret our results for clinical consideration.

Appendices

Appendix A: Parameter Derivations

1.1 A.1 Secondary Target Cell Parameters

Values for the cell production rate λ ₂ and the infection rate β ₂ for the secondary target cells were calculated using the following constraints (Callaway and Perelson 2002). Constraint 1: given e=1 and h=0 (drug 100 % effective in main target cell and completely ineffective in second target cell) the viral load should be 100 virions per ml. Constraint 2: given e=1 and h=0.5 (drug half as effective in second target cell) the viral load should be 0. Assuming e=1 gives

$$ v_w=\frac{\lambda_2 k}{ac}-\frac{d}{\beta_2(1-h)}. $$

Using the two constraints and parameter values from Table 1, we find λ ₂=1.533 and β ₂=10⁻⁴.

1.2 A.2 Mutation Rate

The HIV mutation rate per base per replication has been found to be around 3×10⁻⁵ (Perelson and Ribeiro 2008). As there are approximately 50 different reverse transcriptase inhibitor resistance mutations (Bennett et al. 2009), we have used a mutation rate of 5×10⁻⁴.

1.3 A.3 Asymmetric Division Terms from Model 3

Upon activation by an antigen, the latent cells divide to produce two daughter cells. There are three possible outcomes of this: two latently infected cells are created, one latently infected cell and one productively infected cell are created, or two productively infected cells are created. We assume that the parent cell is lost when the two new cells are created. If we take the probability of a daughter cell being latently infected to be p _l then the expected number of new latent cells upon activation is $(2\times p_{l}^{2}+1\times 2p_{l}(1-p_{l})-1 )L_{i}=(2p_{l}-1)L_{i}$ and the expected number of active cells is (2×(1−p _l )²+1×2p _l (1−p _l ))L _i =2(1−p _l )L _i .

Appendix B: Steady State Calculations

From the differential equations for latently infected cells (1), it is clear that if f>0 and P is negative there are no steady states. This is because the left-hand side of the differential equation would be positive meaning that the concentration of latent cells will always grow. The analysis below assumes that if f>0 then P>0.

It is not possible to obtain explicit expressions for steady states when ϵ>0. However, we can find series solutions of the form $T=\sum_{i=0}^{\infty} \epsilon^{i} T_{i}$, where T=S,I,V is a given state variable. Below we give the O(1) solutions which are equivalent to finding the steady states of the corresponding competition model (ϵ=0). Setting ϵ=0 and the left-hand sides of the system of ordinary differential equations (1) to zero, we obtain the following four steady states: disease free, dominant wild-type, dominant resistant, and ‘co-existence’. The steady state and R _0i expressions for each model can be obtained by substituting in the relevant forms for P and Q.

• 1. The disease free steady state is S ₁=λ ₁/d,S ₂=λ ₂/d with all other state variables equal to zero.

• 2. The dominant wild-type steady state depends on the roots of the quadratic:

  

  (2)

  where the basic reproductive ratios are $R_{01w}=\frac{\lambda_{1} (1-e)\beta_{1} k}{acd} (1-f +\frac{fQ}{P} )$, $R_{02w}=\frac{\lambda_{2}(1-eh)\beta_{2}k}{acd}$ and R _0w =R _01w +R _02w . The other variables are

  

  (3)

  All variables also have O(ϵ) terms and are therefore non-zero.

• 3. The dominant mutant steady state depends on the roots of the quadratic:

  

  (4)

  where $R_{01m}=\frac{\lambda_{1} r\beta_{1} k}{acd} (1-f +\frac{fQ}{P} )$, $R_{02m}=\frac{\lambda_{2}r\beta_{2}k}{acd}$ and R _0m =R _01m +R _02m . The other variables are

  

  (5)

  Again all variables have O(ϵ) terms and are non-zero.

• 4. The co-existence steady state is given by:

  

  (6)

  All variables have additional terms of O(ϵ).

The existence and stability of the steady states are discussed below.

• 1. The disease free steady state is stable if the basic reproductive ratio max(R _0w ,R _0m )<1, otherwise it is unstable.

• 2. When the respective basic reproductive ratio R _0i >1 for i=w,m, there is a single real positive root of the quadratic in v _i leading to a biologically realistic dominant strain steady state.

• 3. The co-existence steady state only exists if h≠1. As the parameter values used give both P and Q positive the steady state expressions for S ₁ and S ₂ are both positive if (1−e)<r<(1−eh). In addition providing v _w and v _m are positive, the remaining variables are also positive. Therefore, in order to ascertain whether the co-existence steady state is biologically realistic the constraints for v _w >0 and v _m >0 need to be investigated. Numerical solutions were obtained for a range of parameter values and the region of parameter space for which the co-existence steady state was stable was vanishingly small (data not shown).