Virion model

The deterministic model of HIV intracellular replication describes the concentration of each species as a continuous variable, and so in fact describes the average of the population. The assumption that reactions occur at a constant rate is value when considering large numbers of molecules of each interacting species, as is being simulated in the large combined model described above. However, when simulating very small populations of molecules however, the reacting molecules will not come in to contact at a constant rate and a deterministic model that uses constant reaction rates would not describe this system very well as one would expect to see a large variation from the average behaviour of a larger system. The discrete stochastic formulation considers the exact number of molecules present in the system at that time and the probability of each possible reaction occurring within a certain time interval. This assumes that the probability of the reaction A+B⇀kC firing in time interval [t, t + dt) is exponentially distributed with mean A(t)B(t)dt.

For simulating cell surface presentation of virion-derived peptides, we used a stochastic model equivalent to the equations of Table ^S3, which uses elementary chemical reactions to describes the copy numbers of each molecular species as discrete variables. The complete set of reactions is given in Table 3. The peptides are produced as described in equation ¹, and connected up to the MHC peptide filtering model using the supply term g_i[P_i]_cyt as before.

Chemical reaction network model of peptide filtering. The reactions are extended from³³ to include the degradation of protein j, and proteasomal cleavage and ER translocation from the cytosol of peptide i. The superscripts denote the compartment containing the molecules (cyt - cytoplasm; cs - cell surface), with no superscript denoting ER. The superscript is omitted from Prot_j, which is always in the cytoplasm.

To perform the stochastic simulations, we used the Visual GEC software http://research.microsoft.com/gec, which uses Gillespie’s stochastic simulation algorithm⁶⁴ (SSA) to simulate chemical reaction networks. As the algorithm is stochastic, each trajectory produced can only be considered as one sample from a distribution of samples. Therefore, we always used 300 independent trajectories, and computed statistics based on these such as the mean copy number, or the probability of there being at least one copy.