The Complexity of Fitness Landscapes

Additive fitness landscapes have a single peak. In contrast, random (uncorrelated or rugged) fitness landscapes have no correlation between the fitness of adjacent alleles. Random fitness and additivity can be considered as two extremes with regard to the amount of structure in the fitness landscapes. The degree of additivity, roughly how close the landscape is to an additive landscape, is of interest for comparing landscapes in different settings. Since we work with qualitative information we use the qualitative measure of additivity, a value ranging from 0 to 1 for fitness landscapes, where the value is 1 for additive landscapes and close to 0 for a random fitness landscape [53]. The mean values were 0.33 for TEM-50 and 0.57 for TEM-85, using the landscapes where the measure applies. The complete statistics show that the TEM-85 landscapes have considerably less additivity than for additive landscapes and considerably more additivity than expected for random fitness landscapes. The results for TEM-50 point in the same direction (see materials and methods).

In an additive landscape where TEM-85 confers the greatest fitness advantage, fitness should always increase with the addition of more mutations. However, in the cefotaxime and ceftazidime landscapes where TEM-85 does confer the greatest fitness advantage, there are seven instances in each where fitness increases via reversion of a previously existing mutation. Overall, there are several occurrences of sign epistasis in 14 out of 15 landscapes. The cefoxitin landscape (Figure S26) has nearly no sign epistasis because it is almost flat, in the sense that there are no significant fitness differences for most alleles. For TEM-50 sign epistasis also occurred in 14 out of 15 landscapes and the ampicillin landscape was flat (Figure S1).

Adaptive Trajectories in Single-antibiotic and Fluctuating Environments

TEM-85 landscapes.

In contrast, two of the 15 adaptive landscapes that include bla_TEM-85 related alleles contain pathways of consecutively increasing resistance between bla_TEM-1 and bla_TEM-85 (Figures S16, S17, S18, S19, S20, S21, S22, S23, S24, S25, S26, S27, S28, S29, S30). Cefotaxime (Figure 1a) and ceftazidime (Figure 1b) can individually select for the evolution of bla_TEM-85 with either two or three pathways, respectively. TEM-85 is the allele of greatest fitness for both cefotaxime and ceftazidime. The cefotaxime landscape has 2 peaks and the ceftazidime landscape has 4 peaks. We computed the probabilities for a population going to fixation at TEM-85, rather than at suboptimal peaks, using a basic model which assumes that available beneficial mutations are equally likely to occur and go to fixation. For cefotaxime the probability for fixation at TEM 85 was 75%, and for ceftazidime 12.5%. When all landscapes were simultaneously considered, which is appropriate under circumstances of fluctuating selection, we found 15,716 pathways between bla_TEM-1 and bla_TEM-85.

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Figure 1. Adaptive landscapes of TEM-85.

These diagrams show the pathways through which the bla_TEM-85 can evolve in a single antibiotic. 1a. (Left) The TEM-85 adaptive landscape in cefotaxime with pathways to TEM-85 indicated. 1b. (Right) The TEM-85 adaptive landscape in ceftazidime with pathways to TEM-85 indicated.

https://doi.org/10.1371/journal.pone.0056040.g001

These results are consistent with Weinreich et al. [52] in that when a single environment is considered, there are few pathways through which evolution can proceed. These results are also consistent with the study by Bergstrom et al. [3] in which they found that random fluctuations of antibiotics can accelerate the evolution of resistance.

Antibiotic Cycles

The complexity of the adaptive landscapes makes it possible to identify cycles of antibiotics that are likely to effectively manage resistance. The somewhat frequent increases in resistance that result from reversions of mutations indicate that the evolution of resistance is sometimes reversible. Based on this observation we investigated approaches for cycling antibiotics as a method for managing resistance. We determined whether it was possible to cyclically restore susceptibility to a sequence of antibiotics by alternating exposure to those antibiotics as follows: We first identified the alleles that were local optima in the presence of at least one antibiotic, and that were also adaptive valleys in the presence of at least one other antibiotic. Next we identified the antibiotic environments where those local optima and valleys exist. We then identified pathways that formed closed loops within those adaptive landscapes; the pathways returned to the allele where they had begun. For bla_TEM-50 landscapes the antibiotics we identified were cefepime, cefprozil, and ceftazidime (Figure 2). We found that 41,961 different cycles exist in those landscapes. For the bla_TEM-85 landscapes, we identified the antibiotics cefprozil, ceftazidime, cefotaxime, and ampicillin as the most appropriate choices for antibiotic cycling and with those, we identified 1770 cycles (Figure S31). These results indicate that there are numerous routes for resistance to be reversed when those three antibiotics are cycled, which is an indication that this approach is robust. If the order of antibiotics is perturbed, the effects of cycling those antibiotics should be consistent. One caveat is that in the case of TEM-85, there are very few reversions that increase fitness when the allele bla_TEM-85 has been reached (Figure S31), allowing the potential for “escape” from the cycling regimen. Generating adaptive landscapes for more antibiotics may ameliorate this situation.

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Figure 2. Example of one possible outcome from antibiotic cycling.

These diagrams show that by alternating the antibiotics cefepime, ceftazidime, and cefprozil susceptibility to those antibiotics can be restored in bacterial populations expressing variant alleles present in TEM-50 adaptive landscapes. 2a. (Top left) The TEM-50 adaptive landscape in cefepime. Yellow peaks indicate the adjacent alleles that are important during cefepime selection. 2b. (Top right) The TEM-50 adaptive landscape in ceftazidime. Orange peaks indicate the adjacent alleles that are important during ceftazidime selection. 2c. (Bottom left) The TEM-50 adaptive landscape in cefprozil. Red peaks indicate the adjacent alleles that are important during cefprozil selection. 2d. (Bottom right) Composite cycle: The yellow arrow indicates the direction of selection in the presence of cefepime. The red arrows indicate the direction of selection in the presence of cefprozil. The orange arrow indicates the direction of selection in the presence of ceftazidime. Rotation of these antibiotics results in cyclical renewal of antibiotic susceptibility.

https://doi.org/10.1371/journal.pone.0056040.g002

Recommendations for Further Development of Cycling

It is likely that effective cycling programs will be specific to local environments and the specific resistance alleles in that environment. The identification of an effective antibiotic cycling program will require identifying the resistance alleles currently circulating in the local environment, determining the fitness of each of those alleles with respect to the set of antibiotics being considered, then identifying those drugs that will result in a repeating cyclic path through the local adaptive landscape. Additionally, optimizing the order in which antibiotics are applied [11], [12] and the duration for which the antibiotics are administered are key [3], [10].

The required experiments are neither difficult nor time consuming. The analyses will be facilitated by user-friendly programs that are under development. Although the introduction of unexpected novel alleles, either by mutation or from sources external to the local environment, may disrupt an effective cycling program, identification of those alleles and their addition to the analysis is likely permit effective modification of the cycling program. Applying the suggested antibiotic cycling scheme will be made even more practical as it builds upon thorough analyses of many adaptive landscapes. Moreover, in pathogens such as malaria and HIV where fewer and more predictable resistance mechanisms exist, this method may be more easily implemented.

Mutants and Susceptibility Testing

We used QuikChange® site-directed mutagenesis (Stratagene) to generate all mutant constructs from the bla_TEM-1 gene in plasmid pBR322 (Table 1). Those mutant alleles were expressed in Escherichia coli strain DH5-αE. We performed Kirby Bauer Disk Diffusion Susceptibility testing [56] for 10 replicates of each of the 32 strains under 15 commonly prescribed β-Lactam antibiotics (Table 2).

Identifying Paths and Cycles

In order to compute all possible combinations of pathways, we have represented the fitness graph of a drug as a possibly cyclic directed graph . The nodes of the graph represent the alleles, and the directed edges of the graph are determined by the statistical analysis of the resistance differences among alleles. Nodes do not have costs associated to them, however costs are associated with each edge and are determined by the resistance difference between the two nodes that are connected by the edge.

Since we consider a biallelic system, the genotypes can be represented by a string of 0′s and 1′s, where the zero-string 0000 represents the wild-type. A fitness graph compares the fitness ranks of mutational neighbors. Roughly, consider the zero-string as the starting point and each non-zero position of a string as an event, i.e., that a mutation has occurred. Under these assumptions the fitness graph coincides with the Hasse-diagram of the power set of events, except that each edge in the Hasse-diagram is replaced with an arrow toward the string with greater fitness.

For a formal definition, a fitness graph for a biallelic L-loci population is a directed graph where each node corresponds to a string (which represents a genotype). The fitness graphs has levels. Each string such that corresponds to a node on level in the fitness graph. In particular, the node representing the zero-string is at the bottom, the nodes representing strings with exactly one non-zero position are one level above, the nodes representing strings with exactly two non-zero positions are on the next level, and the 1-string is at the top. Moreover, the nodes are ordered from left to right according to the lexicographic order where 1>0 of the corresponding strings. A directed edge connects each pair of nodes such that the corresponding strings differ in exactly one position. The edge is directed toward the node representing the more fit of the two genotypes.

Fitness graphs reflect coarse properties of fitness landscapes, including sign epistasis. For a complete analysis of fitness landscapes other methods are necessary. The most fine-scaled approach to epistasis is the geometric theory of gene interaction [44].

Once graph is built and the edge costs are computed and associated with their corresponding edges, path queries in the graph can then be computed. To compute all the pathways, from the initial starting node (0000 in our trials), a search expansion is performed by adding each connected node as a child to the current node in a search tree representation. Since backward edges are possible, a mechanism to detect cycles is included by making sure that expanded nodes do not appear twice in a pathway. When a cycle is detected, the last node of the cycle is not expanded. The overall expansion continues until all the search tree branches reach the target node 1111. The final search tree will represent all possible pathways.

In addition to the search tree, we use a priority queue Q that maintains ranked all the current leaves of the search tree to be expanded. We use a Dijkstra-like cost-to-come sorting function in Q, which represents the accumulated costs of the pathways since the source node. The priority queue ranks all available leaf nodes to be expanded and the node with lowest cost is always expanded first. This guarantees that each node is reached in the order of appearance in the shortest path from the source node to it. This guarantees that the shortest cycles are always found first. Since cycle determination is important in our research, all identified cycles are stored and saved for later analysis.

Although our experiments involved searches with different graphs (single or multiple drugs), searches with either forward or backward edges and searches with different starting nodes (1111 for backwards pathways), the described search method was the same and handled well all situations. The used notation in our figures shows backward edges in red and forward edges in green.

Degree of Additivity

The degree of additivity, roughly how close a landscape is to being a completely additive landscape, can be measured in different ways. We used the qualitative measure of additivity which ranges from 0 to 1 for fitness landscapes. For a formal definition: The set consist of all double mutants such that both corresponding single mutations are beneficial.

The set consists of all double mutants in B_p which are more fit than at least one of the corresponding single mutants.

The qualitative measure of additivity is the ratio .

for an additive landscapes. For random fitness landscapes, the measure is expected to be close to zero in this setting. Indeed, using standard arguments in the Orr-Gillespie approach, the wild-type has very high fitness also in the new environment, in comparison with a randomly generated genotype. By definition, fitness is uncorrelated for a random fitness landscape, so that double mutants combining beneficial mutations are expected to be no more fit than randomly generated genotypes. It follows that the qualitative measure is close to zero, and a more precise estimate is that it should be less than 3% for random landscapes in this context. The derivation of this result will be published elsewhere. The conclusion depends on an analysis of TEM data from the record of clinically found mutants.

For the 15 TEM-85 landscapes, the qualitative measure applies for 9 out of the 15 landscapes. The result is 0, 0, 0, 1/3, 5/6, 1, 1, 1, 1. The mean value is 0.57. This result deviates considerably from expectations for additive and random landscapes. Indeed, if all landscapes were additive, the result should be 1 in each case modulo measurement errors. For random landscapes, non-zero values are expected to be rare. For TEM-50 the qualitative measure applies for 3 landscapes out of 15 and the corresponding data is 0,0,1. The mean value is 0.33. From the qualitative measure alone, we have an indication that the landscapes are neither all additive, nor all random, also for TEM-50 (even if the data set is small).

The qualitative measure of additivity is useful for comparing a fitness landscape with other empirical landscapes, as well as with additive and random (or uncorrelated) landscapes. The measure is robust in the sense that small differences in the environment, such as (moderate) changes of the concentration of antibiotics, have no impact. Quantitative measures may be more sensitive. However, one should not over interpret the qualitative measure. This is a coarse measure, since it depends on fitness ranks only.

Probabilities

By the SSWM assumption we were able to consider fixation of beneficial mutations as independent events. Therefore, we computed the probability for a trajectory as the product of the probabilities of its steps. The probabilities for single substitutions can be determined by the following well-established estimate:

The probability that a beneficial mutation j will be substituted at the next step in adaptation is:where is the fitness contribution of mutation and where there are beneficial mutations in total. However, we used the simplified assumption that fitness is equal for available beneficial mutations, so that this probability equals .