Figure 1

Graphical description of simulation model. We consider an expanding network of infected hosts. The first host in the network is infected with $N_v$ copies of $\overrightarrow{s^0}=\vec{0}$, corresponding to the wild-type (WT) strain. Each new host added to the network is infected with $N_v$ copies of a single viral strain derived from the quasispecies within an existing host chosen at random (based on evidence that most infections are initiated by a single virus strain). $\overrightarrow{b^{\kappa }}$ is a “field” that acts on the proteomic sites within host $\kappa$ and represents the genetically determined immune response within that host. The “consensus” viral strain is extracted from every productively infected host, $i\in 1,2,...,M$, and added to the in silico population ensemble, mimicking the way these sequences were sampled from a real population. For example, in this figure, infection in host $\kappa$ is seeded with $N_v$ copies of strain $\overrightarrow{m^3}$, which is randomly chosen from the quasispecies within host 3. At a randomly chosen generation of viral quasispecies evolution within host $\kappa$, the consensus strain $\overrightarrow{s^{\kappa }}$ is derived from the quasispecies and added to the population ensemble.Reuse & Permissions

Figure 2

Immune pressure facilitates exploration of the virus in sequence space. Viral exploration of sequence space in our simulations is depicted using a lower dimensional representation of the intrinsic fitness landscape of the protein p17, computed by applying principal component analysis (PCA) to sequences resulting from an equilibrium sampling of $H_{\mathrm{int}}\left[\vec{s}\right]$ (cf. Supplemental Notes 1 and 2). We only focus on the primary basin relevant to our simulations (cf. Supplemental Fig. S2). Different colors represent contours of the free energy computed as $A(x,y)=-\log P(x,y)$, where $P(x,y)$ is the normalized density of sequences at point $(x,y)$ on the $[P{C}_1,P{C}_2]$ plane. Low values correspond to regions of high fitness. (a) In the absence of immune pressure ($\left\{b_i^{\kappa }\right\}=0$), the population ensemble extracted from our simulations consists of only WT sequences, which are represented by a single bullet ($•$), located at (0,0). (b) In the presence of immune pressure, the population ensemble consists of sequences that explore different parts of the landscape. Each bullet ($•$) represents the most frequent strain in a particular host.Reuse & Permissions

Figure 3

Comparison of one- and two-body mutational probabilities at low and intermediate mutation rates.(a), (c) $\mu =5\times {10}^{-5}$/site/generation, $N_v=15$ 000. The one- and two-body mutational probabilities (average per sequence) computed from the population ensemble resulting from our simulations, ${\langle s_i\rangle }_{\mathrm{dyn}}$ and ${\langle s_i{s}_j\rangle }_{\mathrm{dyn}}$, are compared with their counterparts, ${\langle s_i\rangle }_{\mathrm{int}}$ and ${\langle s_i{s}_j\rangle }_{\mathrm{int}}$, both computed from an equilibrium sampling of the intrinsic fitness Hamiltonian $H_{\mathrm{int}}\left[\vec{s}\right]$, which agree with values computed from sequences used to infer the maximum entropy model. At small mutation rates, escape mutations are rarely sampled. The viral quasispecies within each host stays frozen near the ground state [cf. Fig. 2a] and mutations are not selected at the population level, resulting in ${\langle s_i\rangle }_{\mathrm{dyn}},{\langle s_i{s}_j\rangle }_{\mathrm{dyn}}\approx 0$. (b), (d) For intermediate mutation rates $\mu \in ({10}^{-4},{10}^{-2})$ (here, $\mu =5\times {10}^{-3}$/site/generation and $N_v=15$ 000), we find that immune selection leads to the accumulation of mutations across the viral proteome, and at the population level the one- and two-body mutational probabilities ${\langle s_i\rangle }_{\mathrm{dyn}}$ and ${\langle s_i{s}_j\rangle }_{\mathrm{dyn}}$ correlate monotonically with their counterparts ${\langle s_i\rangle }_{\mathrm{int}}$ and ${\langle s_i{s}_j\rangle }_{\mathrm{int}}$. At higher mutation rates ($\mu$ $>{10}^{-2}$/site/generation), the quasispecies within most hosts become unable to survive selection as deleterious mutations are rapidly accumulated.Reuse & Permissions

Figure 4

Variational analysis using the Feynman bound predicts that the prevalence landscape is correlated with the fitness landscape. A numerical comparison between the variational estimate of the prevalence Hamiltonian $H_T\left[\vec{s}\right]$ [Eq. (9)] to the intrinsic Hamiltonian $H_{\mathrm{int}}\left[\vec{s}\right]$ for the case for 2500 subtype B p17 sequences. Both $H_T\left[\vec{s}\right]$ and $H_{\mathrm{int}}\left[\vec{s}\right]$ have units of dimensionless energy. $H_T\left[\vec{s}\right]$ is a variational estimate that is related to the logarithm of the prevalence of the strain $\vec{s}$, whose relation to $H_{\mathrm{int}}\left[\vec{s}\right]$ is corrupted by phylogeny and immune pressure. The parameters $\mu =5\times {10}^{-3}$/site/generation, $n_{\max }=6$, $N_v=15000$ were used in the simulations. Each point on the plot corresponds to one p17 sequence. The sequences were downloaded from the Los Alamos sequence database [12] and converted to the binary code. ${\rho }_{\mathrm{sp}}$ is the standard rank correlation computed from Spearman's test [31] and $p$ is the corresponding significance value. The line is computed from fitting an ordinary least squares regression model.Reuse & Permissions