Parametric inference of REB1 fitness landscape.

<p>(A) Histogram of energies of intergenic sites calculated using the REB1 energy matrix (dashed line) and the neutral distribution of sequence energies expected from the mono- and dinucleotide background model (solid line; see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#s4" target="_blank">Methods</a> for details). The color bar on the bottom indicates the percent deviation between the two distributions (red is excess, green is depletion relative to the background model). The vertical bars show the distribution of functional sites <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683-Chen1" target="_blank">[31]</a>, which correspond to the low-energy excess in the distribution of intergenic sites. (B) From top to bottom: REB1 energy matrix, the sequence logo obtained from the energy matrix by assuming a Boltzmann distribution at room temperature at each position in the binding site (), and the sequence logo based on the alignment of observed REB1 genomic sites. (C) Histogram of binding site energies and its prediction based on the three fits (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683.e042" target="_blank">Eq. 5</a>). (D) Fitness function inference. Dots represent data points (as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi-1003683-g006" target="_blank">Fig. 6</a>); also shown are the unconstrained fit to the Fermi-Dirac function of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683.e060" target="_blank">Eq. 7</a> (“UFD”; solid red line), constrained fit to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683.e060" target="_blank">Eq. 7</a> with (“CFD”; dashed black line), and fit to an exponential fitness function (“EXP”; dashed green line). Error bars in (D) are calculated as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi-1003683-g006" target="_blank">Fig. 6</a>.</p

Comparison of fitness function models.

<p>For each TF, we show the AIC differences between the unconstrained Fermi-Dirac fit (“UFD”), the constrained Fermi-Dirac fit with (“CFD”), and the exponential fit (“EXP”). Also shown are Akaike weights , which indicate the relative likelihood of each model.</p

The monomorphic limit and steady state of a Wright-Fisher model of population genetics.

<p>In (A)–(C) we show results from simulations at various mutation rates, using a fitness function with , , and . Each mutation rate data point is an average over independent runs, as described in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#s4" target="_blank">Methods</a>. Colors from green to orange correspond to increasing mutation rates. (A) Observed steady-state distributions for various mutation rates. The steady state predicted using <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683.e030" target="_blank">Eq. 3</a> is shown in gray. (B) Fitness functions predicted using observed distributions in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683.e076" target="_blank">Eq. 8</a>. The exact fitness function is shown in gray. Inferred fitness functions are matched to the exact one by using the known population size , and setting the maximum fitness to 1.0 for each curve. (C) For each mutation rate, the total variation distance (TVD) between and , and the average number of unique sequences in the population (the degree of polymorphism) are shown. The predicted bound on mutation rate required for monomorphism is shown as a dashed line. In (D)–(F) we show simulations in the monomorphic regime which have not reached steady state, with the same parameters as in (A)–(C) and . Colors from blue to red correspond to the increasing number of generations.</p

Tests of site-specific selection.

<p>We divide binding sites for each TF into two groups: those regulating essential genes and those regulating nonessential genes. (A) Comparison of mean binding energies of sites regulating essential () and nonessential genes () for each TF in the data set. (B) Comparison of variance in binding energies for sites regulating essential () and nonessential () genes. (C) Mean Hamming distance between corresponding sites in <i>S. cerevisiae</i> and <i>S. paradoxus</i> for sites regulating essential () versus nonessential genes (). (D) Mean squared difference in binding energy between corresponding sites in <i>S. cerevisiae</i> and <i>S. paradoxus</i> for sites regulating essential () versus nonessential genes (). In (A)–(D), 25 TFs were used; black diagonal lines have slope one. In (A),(C),(D), vertical and horizontal error bars show the standard error of the mean in each group. Points lacking error bars have only one sequence in that group. (E) Normalized histogram of TF binding site sequence entropies, divided into 16 essential and 109 nonessential TFs, for 125 TFs in Ref. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683-Chen1" target="_blank">[31]</a>.</p

Fitness and selection strength as functions of energy and inverse temperature.

<p>Energy is measured with respect to the chemical potential . Top row uses ; bottom row uses . (A,D) Logarithm of Fermi-Dirac fitness versus energy for several values of ; note that the high-energy tail looks distinctly different when is nonzero. (B,E) Per-unit-energy selection strength versus energy for several values of ; note that the relative ordering of selection strength curves depends on the value of . (C,F) Sign of derivative of selection strength with respect to , as a function of and . Black boundary in (C) is the curve , where is the Lambert W-function; the boundaries in (F) are the curves and where , are the solutions to (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683.e148" target="_blank">Eq. 14</a>) with .</p

Qualitative behavior of fitness landscapes.

<p>Shown are plots of for 12 TFs, which, according to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683.e076" target="_blank">Eq. 8</a>, equals the logarithm of fitness up to an overall scale and shift. For each TF, sequences are grouped into 15 equal-size energy bins between the minimum and maximum energies allowed by the energy matrix. Shown also are the total number of binding sites for each TF. Error bars are calculated as , where is the fraction of sites falling into a given bin out of total sites, as would be expected if the sequences were randomly distributed according to the observed distribution.</p

Summary of unconstrained Fermi-Dirac landscape fits to TF binding site data.

<p>Columns show maximum-likelihood value of , , and . The last column shows whether most binding site energies are lower than the inferred chemical potential , near it, or above it (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003683#pcbi.1003683.s002" target="_blank">Table S2</a> for details).</p

Contingency and Entrenchment of Drug-Resistance Mutations in HIV Viral Proteins

The ability of HIV-1 to rapidly mutate leads to antiretroviral therapy (ART) failure among infected patients. Drug-resistance mutations (DRMs), which cause a fitness penalty to intrinsic viral fitness, are compensated by accessory mutations with favorable epistatic interactions which cause an evolutionary trapping effect, but the kinetics of this overall process has not been well characterized. Here, using a Potts Hamiltonian model describing epistasis combined with kinetic Monte Carlo simulations of evolutionary trajectories, we explore how epistasis modulates the evolutionary dynamics of HIV DRMs. We show how the occurrence of a drug-resistance mutation is contingent on favorable epistatic interactions with many other residues of the sequence background and that subsequent mutations entrench DRMs. We measure the time-autocorrelation of fluctuations in the likelihood of DRMs due to epistatic coupling with the sequence background, which reveals the presence of two evolutionary processes controlling DRM kinetics with two distinct time scales. Further analysis of waiting times for the evolutionary trapping effect to reverse reveals that the sequences which entrench (trap) a DRM are responsible for the slower time scale. We also quantify the overall strength of epistatic effects on the evolutionary kinetics for different mutations and show these are much larger for DRM positions than polymorphic positions, and we also show that trapping of a DRM is often caused by the collective effect of many accessory mutations, rather than a few strongly coupled ones, suggesting the importance of multiresidue sequence variations in HIV evolution. The analysis presented here provides a framework to explore the kinetic pathways through which viral proteins like HIV evolve under drug-selection pressure

Contingency and Entrenchment of Drug-Resistance Mutations in HIV Viral Proteins

The ability of HIV-1 to rapidly mutate leads to antiretroviral therapy (ART) failure among infected patients. Drug-resistance mutations (DRMs), which cause a fitness penalty to intrinsic viral fitness, are compensated by accessory mutations with favorable epistatic interactions which cause an evolutionary trapping effect, but the kinetics of this overall process has not been well characterized. Here, using a Potts Hamiltonian model describing epistasis combined with kinetic Monte Carlo simulations of evolutionary trajectories, we explore how epistasis modulates the evolutionary dynamics of HIV DRMs. We show how the occurrence of a drug-resistance mutation is contingent on favorable epistatic interactions with many other residues of the sequence background and that subsequent mutations entrench DRMs. We measure the time-autocorrelation of fluctuations in the likelihood of DRMs due to epistatic coupling with the sequence background, which reveals the presence of two evolutionary processes controlling DRM kinetics with two distinct time scales. Further analysis of waiting times for the evolutionary trapping effect to reverse reveals that the sequences which entrench (trap) a DRM are responsible for the slower time scale. We also quantify the overall strength of epistatic effects on the evolutionary kinetics for different mutations and show these are much larger for DRM positions than polymorphic positions, and we also show that trapping of a DRM is often caused by the collective effect of many accessory mutations, rather than a few strongly coupled ones, suggesting the importance of multiresidue sequence variations in HIV evolution. The analysis presented here provides a framework to explore the kinetic pathways through which viral proteins like HIV evolve under drug-selection pressure