Exploring the Complexity of the HIV1 Fitness Landscape
Although fitness landscapes are central to evolutionary theory, so far no biologically realistic examples for largescale fitness landscapes have been described. Most currently available biological examples are restricted to very few loci or alleles and therefore do not capture the high dimensionality characteristic of real fitness landscapes. Here we analyze largescale fitness landscapes that are based on predictive models for in vitro replicative fitness of HIV1. We find that these landscapes are characterized by large correlation lengths, considerable neutrality, and high ruggedness and that these properties depend only weakly on whether fitness is measured in the absence or presence of different antiretrovirals. Accordingly, adaptive processes on these landscapes depend sensitively on the initial conditions. While the relative extent to which mutations affect fitness on their own (main effects) or in combination with other mutations (epistasis) is a strong determinant of these properties, the fitness landscape of HIV1 is considerably less rugged, less neutral, and more correlated than expected from the distribution of main effects and epistatic interactions alone. Overall this study confirms theoretical conjectures about the complexity of biological fitness landscapes and the importance of the high dimensionality of the genetic space in which adaptation takes place.
Published in the journal:
. PLoS Genet 8(3): e32767. doi:10.1371/journal.pgen.1002551
Category:
Research Article
doi: 10.1371/journal.pgen.1002551
Summary
Although fitness landscapes are central to evolutionary theory, so far no biologically realistic examples for largescale fitness landscapes have been described. Most currently available biological examples are restricted to very few loci or alleles and therefore do not capture the high dimensionality characteristic of real fitness landscapes. Here we analyze largescale fitness landscapes that are based on predictive models for in vitro replicative fitness of HIV1. We find that these landscapes are characterized by large correlation lengths, considerable neutrality, and high ruggedness and that these properties depend only weakly on whether fitness is measured in the absence or presence of different antiretrovirals. Accordingly, adaptive processes on these landscapes depend sensitively on the initial conditions. While the relative extent to which mutations affect fitness on their own (main effects) or in combination with other mutations (epistasis) is a strong determinant of these properties, the fitness landscape of HIV1 is considerably less rugged, less neutral, and more correlated than expected from the distribution of main effects and epistatic interactions alone. Overall this study confirms theoretical conjectures about the complexity of biological fitness landscapes and the importance of the high dimensionality of the genetic space in which adaptation takes place.
Introduction
The fitness landscape is one of the central concepts in evolutionary biology. Ever since Sewall Wright [1], it has been used to study and conceptualize the process of longterm evolutionary adaptation. Fundamentally, knowledge of fitness landscapes is required to translate microevolutionary adaptation (i.e. changes in gene frequencies) into macroevolutionary change (i.e. speciation events and largescale phenotypic modifications). One of the major limitations of the concept of fitness landscapes, however, is the near complete lack of knowledge of any largescale and biologically realistic fitness landscapes. Most of the landscapes currently available are restricted to very few loci or alleles [2], [3], [4], [5], [6], [7], [8]. Due to their limitation in size, these landscapes do not allow the study of properties that might arise from the high dimensionality that is characteristic for real fitness landscapes. Current examples for largescale landscapes are based on RNA secondary structure [9] or enzymatic activity of RNA [10]. However, the relation of RNA structure to fitness is unclear and the relation between enzymatic activity and fitness is often highly nonlinear [11].
The centrality of the concept of fitness landscapes for evolutionary biology, combined with the absence of good biological examples has necessitated the study of theoretically conceived and idealized fitness landscapes, often tailored to the particular question studied. The socalled NK landscapes are an example for a broad class of theoretical fitness landscapes [12], which have tunable ruggedness ranging from smooth, singlepeaked MountFujiyamalike landscapes to maximally rugged uncorrelated landscapes, in which the fitness of each sequence is independent of the fitness of its neighbors. These NK landscapes have been used, among other things, to study properties of landscapes arising from high dimensionality [13]. Landscapes based on neutral networks [14], [15] reconcile Kimura's neutral theory [16] with natural selection, and have been used to explain phenomena such as punctuated equilibria observed for example in the evolution of the antigenic profile of influenza [17]. The related holey landscapes, which consist of a network of highfitness genotypes with embedded fitnessholes, have been very influential as models of speciation [18]. All these examples have been tremendously valuable in studying processes of evolutionary adaptation, but are purely conceptual and it is unclear to what extent they reflect properties of real fitness landscapes.
Recent progress in high throughput data generation now allows measuring both fitness and genotype for a large number of mutants [19], [20], [21]. Combining such data sets with appropriate computational methods enables for the first time the reconstruction of largescale and biologically realistic fitness landscapes. Here we analyze fitness landscapes that are based on predictive models for fitness of HIV in an in vitro replication assay [21]. These models predict fitness based on estimated effects of individual mutations (main effects) and of pairwise combinations of mutations (epistasis) and can thus be considered as a quadratic approximation to the real HIV fitness landscape.
Results/Discussion
The fitness landscapes analyzed here are based on statistical models that are based on extensive measurements of in vitro replicative fitness. These models allow to predict the fitness of HIV from amino acid sequences (see Materials and Methods and ref. [21]). The entire landscape consists of approximately 2^{1800}≅10^{600} fitness values. Clearly, it is impossible to generate all these values despite the fact that the predictive model would allow in principle to compute the fitness for any sequence. Therefore, we describe the properties of the fitness landscapes by using summary statistics based on different types of random or directed walks on these landscapes. Specifically we use such walks to compute three measures that characterize different properties of the landscapes: ruggedness, correlation length and neutrality (see Materials and Methods).
Ruggedness refers to the number of local fitness optima; i.e. genotypes whose fitness exceeds that of every one of its neighbors. We determine ruggedness as the number of different local optima reached by adaptive walks that climb the fitness landscape by means of steepest ascent from random positions on the landscape. These adaptive walks always move to that neighboring sequence, which has the highest fitness of all the neighboring sequences. Local optima act as attractors for such steepestascent walks: if a walk is started within the “attraction domain” of the optimum, the walk will converge to this optimum. Depending on the structure of the landscape, such walks need not end up in the same optima, even if they are started from similar initial conditions. Conversely, walks that end up in the same optima need not originate from similar areas of the fitness landscape. We use such simple hillclimbing walks here as tools to analyze structural properties of the underlying fitness landscape such as ruggedness or the attraction domain of the local optima. To characterize the process of adaptation of populations evolving on these fitness landscapes such hillclimbing walks have limited validity and may overly simplify more complex aspects of evolution. The correlation length quantifies to what extent proximity in sequence space translates into similarity in fitness. To measure correlation length, we perform random walks, which start at a random genotype in the landscape and then randomly move in each step to neighboring genotypes. Recording the fitness values along such a random walk we then determine correlation length as the characteristic distance over which the autocorrelation of fitness decays. Neutrality measures to what extent populations can move on the landscape without changing their fitness. To measure neutrality, we perform quasineutral walks, where random steps to neighboring genotypes are only accepted if they do not change fitness by more than a defined small threshold value. We determine neutrality as the maximal distance from the starting genotype that is attained by such a neutral walk.
We first explore these measures for a reference landscape (RL), which is based on the model that best predicts replicative capacity in the drugfree environment (see Materials and Methods and ref. [21]). We then examine different variations of the RL (see Materials and Methods) in order to explore how these measures depend on the features of the underlying landscape. These variations include landscapes based on fitness measured in the presence of different antiretroviral drugs; landscapes, in which the strength of epistasis is reduced; and landscapes in which the coefficients determining main effects and pairwise epistasis in the RL are randomized (see below).
The RL is characterized by a large number of optima, a large correlation length and considerable neutrality (Figure 1). For an increasing number of starting points we find an increasing number of optima, with only a weak saturation of the increase up to 10^{5} different starting points. For the 10^{5} starting points tested in Figure 1, on average every fourth one leads to a different optimum. By contrast, in a completely smooth landscape such walks would always converge to the same optimum. The large number of optima indicate (Figure 1A) a high degree of ruggedness and hence a multitude of basins of attraction (i.e. the set of starting points from which adaptive walks end up in a given optimum). Moreover, the starting genotype of an adaptive walk has a strong influence on the longterm evolutionary trajectory and neighboring starting points can lead to completely different trajectories (Figure 1B). The basins of attraction of the different optima differ greatly in structure. Some optima have an attraction domain that is confined to a small region of sequence space and is sparsely distributed within this region (type α in Figure 1B). Other optima have an attraction domain, which also covers a small region of sequence space, but is densely distributed in that region (type β). Finally, optima of the last type have an attraction domain that spans a large part of sequence space but is only sparsely distributed (type γ). The long correlation length of random walks on the RL (Figure 1C) indicate that almost all loci characterizing a genotype (here, genotypes consist of 404 loci, see Materials and Methods) have to be mutated until the memory of the initial fitness value is lost.
In the landscapes considered here, mutations may have extremely small effects, but they are never completely neutral. To define a sensible concept of neutrality we therefore need to define a threshold for the maximal fitness effect that a mutation is allowed to have to be considered neutral. The exploration range of the resulting quasineutral walks strongly depends on the magnitude of this threshold (see Figure 1D). If this threshold is 10^{−4} or lower, the exploration range is very small with a maximal distance of 5–10 mutations. For thresholds of 10^{−3} or higher, on the other hand, neutral walks can reach considerable distances of 100 mutations or more. Thus, although there are no fully neutral mutations in the RL, the landscape is characterized by large networks over which fitness changes only minimally.
Comparing the RL to the corresponding bestfit landscapes for 15 different environments each characterized by the presence of a different antiretroviral drug (see Materials and Methods), we find that ruggedness, correlation length, and neutrality are of similar magnitude in all these environments (Figure 2). However, the nodrug environment exhibits less neutrality and longer correlation length than all environments with antiretrovirals present. Interestingly, there are also consistent differences between drugclasses. Figure 2 shows results for the 3 main classes of antiretroviral drugs: protease inhibitors (green), nucleoside reverse transcriptase inhibitors (blue), and nonnucleoside reverse transcriptase inhibitors (cyan). For instance, the landscape is particularly rugged in the presence of protease inhibitors. Interestingly, protease inhibitors are also known to have the most complex resistance profiles [22], [23], [24], in the sense that resistance against them is mediated by a large number of interacting mutations.
To assess the impact of the strength of epitasis relative to that of main effects, we consider alternative landscapes in which fitness interactions between mutations are weaker. We chose the RL as a reference because it has the highest predictive power (see [21] and Figure S1). However, this landscape might overestimate the role of epistasis for statistical reasons (see Materials and Methods). A landscape based on a more conservative estimate of the role of epistasis can be obtained by fitting a hierarchical model that first estimates the effects of individual mutations (“main effects”) and then uses pairwise interactions between mutations (“epistasis”) to explain the remaining variance in the biological data (see Materials and Methods). This hierarchical landscape (HL) has a predictive power almost equal to that of the RL (see Materials and Methods). As both the RL and the HL represent equally valid approximations of the true biological fitness landscape, the true magnitude of epistasis will presumably lie between these two extremes. To further explore the role of epistasis we can reduce its magnitude beyond the realistic range by reducing all epistatic interactions in the HL by a factor ε (0≤ε≤1). The predictive power of the corresponding landscapes (HL_{ε}) decreases for small ε, but even for ε = 0 the predictive power is only reduced by 13% (see Materials and Methods). This continuum of the HL_{ε} landscapes allows us to study the effect of the relative strength of epistasis on ruggedness, correlation length and neutrality.
We find that ruggedness and neutrality consistently increase with the magnitude of epistatic effects (Figure 3). With increasing epistasis, the HL_{ε} gradually shift from a singlepeaked smooth landscape without neutral networks (ε = 0) to a very rugged landscape with large quasineutral networks (ε = 1). In contrast to the other two measures, correlation length in the HL_{ε} depends only weakly on epistasis. All three measures continue to exhibit the same type of dependence on epistasis when switching from the HL to the more epistatic RL (Figure 3). Taken together, these results indicate that the relative strength of pairwise epistasis is a major determinant of the structure of fitness landscapes.
An intuition for the impact of the strength of epistasis on ruggedness can be obtained as follows: If main effects dominate, a given mutation is always either beneficial or deleterious, independent of its background. However, if epistatic interactions dominate, a change in the genetic background can turn a beneficial mutation into deleterious one and vice versa. Thus the landscape only has one peak if main effects dominate, but may have multiple peaks if epistatic effects dominate. Note that epistasis need not necessarily increase ruggedness. For example, this would not be the case if most epistatic interactions were of the same sign (as has often been assumed [25], [26]). Thus the increase in ruggedness with epistasis is a particular feature of the landscapes studied here. The fact that neutrality increases with epistasis might seem contradictory at first, given that epistasis contributes to the selective effects of mutations. One should note, however, that nontrivial neutrality (i.e. a mutation being neutral in some genetic backgrounds but not in others) requires epistasis by definition. This type of nontrivial neutrality is responsible for the observed increase in neutrality with increasing epistasis. In contrast to the trivial type of neutrality that would be due to synonymous mutations, the neutrality observed here is exclusively due to the cancelling out of selective effects. Finally, the correlation length of a random walk decreases with the strength of epistatic effects for the following reason: In the case of independent loci, correlation is lost if on average every locus has been mutated. If loci interact epistatically, then a mutation at one locus affects the fitnesscontribution of the mutations at other loci as well and hence the number of changes required to loose correlation decreases. Given these interpretations, our results (that ruggedness and correlation length are high) suggest that epistasis is strong enough to increase ruggedness, but too weak to strongly affect correlation length. Interestingly Fontana et al. [9], [27] found that landscapes in which fitness is predicted by RNA secondary structure combine neutrality and ruggedness similarly to the landscapes described here. However, these RNAderived landscapes exhibit short correlation lengths, in contrast to our HIV landscapes. As correlation length decreases with the strength of epistatic interaction (see Figure 3), one reason for this difference might be that the epistatic or pairwise interactions are much stronger in RNA landscapes than in the HIV landscapes analyzed here.
The strong impact of the relative strength of main effects and epistasis raises the question whether the properties of fitness landscapes also depend on the detailed correlation structure between different epistatic effects and main effects or whether they are only determined by the distributions of these effects. In order to address this question, we use three different schemes to randomize the main and epistatic effects underlying the RL (Figure 4) and then measure ruggedness, neutrality and correlation length for the different types of randomized landscapes. We find that, despite its large number of peaks, the RL is still considerably smoother (smaller number of peaks) than the randomized landscapes (see Figure 4A). Furthermore, the RL is also less neutral and more correlated (see Figure 4C and 4B). This implies that the fitness landscape of HIV is considerably less rugged, less neutral, and more correlated than expected from the distribution of main effects and epistatic interactions alone. Moreover, it suggests that, although the overall strength of epistasis is an important factor, knowledge of the distributions of main effects and epistatic interactions does not fully characterize fitness landscapes in general, because the correlational structure between epistatic effects plays an essential role in determining the properties of the landscapes.
It should be noted that the structure of the fitness landscapes discussed here might be affected by selection biases in the data used for the development the fitnessprediction model. The viral isolates have been obtained from HIVinfected individuals and therefore the mutations found in these isolates do not represent a random sample of all possible mutations. On the one hand, because all isolates harbour replication competent viruses, the sample is biased against lethal or highly deleterious mutations. On the other hand, most viral isolates carry drug resistance mutations. These resistance mutations are beneficial in the presence, but typically detrimental in absence of drugs. Hence, in the drug free environment (or in an environment containing drugs to which a give mutation does not confer resistance) the isolates may be enriched in deleterious mutations. In any event, the mutations found in the isolates represent the standing variation of mutations that are present on the level of the host population. Clearly, however, it is likely that the complete fitness landscape of HIV does contain much more fitnessholes/troughs than the landscapes described here, because of the observation bias against lethal mutants.
Comparing the fitness landscape of HIV with various theoretical landscapes that have been used to study evolutionary processes [12], [14], [15], [18] shows that these classical landscapes can capture certain aspects of a real landscape while failing to describe others. For example, the fitness landscape of HIV resembles uncorrelated landscapes with regard to the high ruggedness. However, unlike uncorrelated landscapes, it is characterized by considerable neutrality and large correlation length. In these respects, the HIV fitness landscape is closer to neutral landscapes (such as holey landscapes) or singlepeaked MountFujiyamalike landscapes. Finally, the structure of the attraction domains—in particular the existence of attraction domains which are at the same time very sparsely and widely distributed—strongly contrasts the situation in low dimensional spaces. Overall, these results highlight the complexity and the high dimensionality that need to be taken into account to describe adaptive processes in real biological systems.
Materials and Methods
Fitness Landscapes
The fitnesslandscapes analyzed here are based on models that predict the fitness of HIV from amino acid sequences. Fitness is measured as the reproductive capacity (RC) of HIVderived amplicons (representing all of Protease (PR) and most of Reverse Transcriptase (RT)) inserted into a constant backbone of a resistance test vector. The models are then trained to predict this fitness from the aminoacid sequence of the amplicons. Although the fitness, which is predicted by these models, is an invitro RC, we could show in [28] that this predicted RC is significantly correlated to HIV virus load in vivo. Details on the experimental measurement of the RC values and on inferring the predictor have been published in [29] and [21]. Here, we briefly reiterate the principles of the models fitted to the data.
In essence, the predictor is based on fitting the data consisting of amino acid sequences (s), coded here as a binary string, and the corresponding RC values (w) with the following model(M1)For the purpose of this paper, s_{ij} denotes the presence (s_{ij} = 1) or absence (s_{ij} = 0) of allele j at position i. (Although the present work is restricted to this simple binary case, a more general definition is used in the data fitting procedure [21]: If an ambiguity in the population sequencing is consistent with several amino acids at a given position, then s_{ij} denotes the probability of allele j at position i.) Thus s is a valid sequence only if for all positions i, . In total, there are 1859 alleles at 404 positions. The vast majority (1848/1859) of these alleles are amino acids (thus not all possible amino acids at the 404 positions are allowed); only 11 alleles correspond to either insertions or deletions. The model parameters I, s_{ij} and ε_{ij;kl} can be interpreted as intercept, main effects (effects of individual alleles on their own), and epistatic effects (effects of an allele at one locus in combination with an allele at another locus). As the number of parameters exceeds the number of datapoints, the model M1 has been fitted to the data on the basis of a machine learning approach (generalized kernel ridge regression). With this approach overfitting is no concern because the subdataset on which the predictor is evaluated is independent from the subdataset from which the predictor is inferred (see [21]).
Note that equation (M1) can also be written as a second order cluster expansion [30] of the logfitness(M2)where S_{j} denotes the allele at position j of the aminoacid sequence, the impact on logfitness of allele S_{i} at position i, and denotes the combined impact of allele S_{i} at position i and S_{j} at position j. The firstorder effects in equation (M2) correspond to the maineffects in equation (M1) and the second order effects to the epistatic effects in equation (M1). For instance if i is a biallelic locus with alleles S_{i}/S_{i}' and k/k' denote the position corresponding to those alleles in the binary representation used above, then = m_{k} and = m_{k'}.
The different landscapes are all based on model M1, but differ with respect to the relative weight that is given to epistasis and main effects:

The reference landscape RL is obtained by fitting the full model M1 to the data. Thus main effects and epistatic effects are fitted simultaneously. As main and epistatic effects are given the same weight in model fitting, while epistatic effects greatly outnumber main effects, this approach will explain the variance in RCs using mainly epistasis. Therefore this approach tends to overestimate the role of epistasis relative to that of main effects.

The hierarchic landscape (HL) avoids this overestimation of epistatic effects by first fitting model M1 only with the main effects (i.e. the ε_{ij;kl} in M1 are set to 0) and then fitting the residuals of the first fit only with the epistatic effects (i.e. m_{ij} in M1 are set to 0). This fit may however underestimate the magnitude of epistatic effects, because they are only used to explain that part of the variance, which cannot be explained by main effects.

The HL_{ε} are derived from the HL by scaling the epistatic effects by a factor ε (i.e. by replacing the ε_{ij;kl} with ε_{ij;kl} ε).
Figure S1 shows the predictive power of the different models.
Environments
If not stated otherwise, the RC values underlying the fitnesslandscapes RL, HL and HL_{ε} are measured in the absence of drugs. In addition we consider 15 alternative versions of the RL based on RC values measured in the presence of 15 different single drugs. The drugs used here are the protease inhibitors amprenavir (AMP), indinavir (IDV), lopinavir (LPV), nelfinavir (NFV), ritonavir (RTV), and saquinavir (SQV), the 6 nucleoside reverse transcriptase inhibitors abacavir (ABC), didanosine (ddI), lamivudine (3TC), stavudine (d4T), zidovudine (ZDV), and tenofovir (TFV) and the nonnucleoside reverse transcriptase inhibitors delavirdine (DLV), efavirenz (EFV), and nevirapine (NVP). For each drug, the replicative capacity of a virus on drugs was given by the interpolated value measured at the drug concentration at which the NL43 based control virus has 10% of its replicative capacity in the absence of drug (i.e. the IC90 for NL43 is used as the reference drug concentration for every subsequent measurement) [21].
Characteristics of Fitness Landscapes
The landscapes are characterized by adaptive, neutral and random walks. Each walk consists of a series/succession of genotypes s^{0}→s^{1}→s^{2}→s^{3}…. . The different types of walks differ with respect to the updating rule (i.e. on how genotype s^{k+1} is determined from s^{k}). Unless stated otherwise the start genotype of each walk is chosen randomly, i.e. at each position one of the possible alleles is chosen randomly and independently from alleles at the other positions.
The ruggedness of the fitness landscapes is measured as the number of different endpoints reached from a prespecified number of steepestascent hill climbing walks (SAHCW) starting from different, random start genotypes. In each step s^{k}→s^{k+1} of a SAHCW, the fitness of all single mutants of s^{k} is determined. If the single mutant with the maximal fitness (s^{max}) is less fit than s^{k}, the walk is terminated, as s^{k} represents a local maximum. Otherwise, the fittest single mutant s^{max} is chosen as the next genotype in the walk (s^{k+1} = s^{max}).
The neutrality of fitness landscapes is measured as the range explored by quasineutral walks (QNW) of a prespecified length L (typically 1000 steps). In each step s^{k}→s^{k+1} of a QNW, a single random aminoacid substitution is performed on the genotype s^{k}, yielding the genotype s^{k}'. If the logfitness of s^{k}' differs by less than ε from the logfitness of s^{k}, then s^{k}' is chosen as the next step in the QNW (s^{k+1} = s^{k}'). If the difference is larger than or equal to ε, the step is rejected and an alternative single mutant of s^{k} is probed in the same way for its neutrality etc. If after 10^{4} trials, no quasineutral mutation has been found, the QNW stays at s^{k} (s^{k+1} = s^{k}). The range of such a neutral walk is determined as the maximal Hamming distance of one the genotypes s^{1}….s^{L} from the startgenotype s_{0}.
The correlation length of a fitness landscape is measured as the inverse decay rate of the autocorrelation of the logfitness along random walks (RW). Specifically, a prespecified number (typically 10^{5}) of random walks are initiated each from a different random start genotype. In each step of a given RW a single randomly chose amino acid substitution is performed. The autocorrelation after k steps is then determined aswhere the brackets refer to averaging over all random walks performed. An exponential decay is then fitted to these autocorrelation coefficients by performing a linear leastsquare fit according toThe correlation length is then given by 1/β.
Supporting Information
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Genetika Reprodukční medicínaČlánek vyšel v časopise
PLOS Genetics
2012 Číslo 3
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