Fitness landscapes are central in the theory of adaptation. Recent work compares global and local properties of fitness landscapes. It has been shown that multi-peaked fitness landscapes have a local property called reciprocal sign epistasis interactions. The converse is not true. We show that no condition phrased in terms of reciprocal sign epistasis interactions only, implies multiple peaks. We give a sufficient condition for multiple peaks phrased in terms of two-way interactions. This result is surprising since it has been claimed that no sufficient local condition for multiple peaks exist. We show that our result cannot be generalized to sufficient conditions for three or more peaks. Our proof depends on fitness graphs, where nodes represent genotypes and where arrows point toward more fit genotypes. We also use fitness graphs in order to give a new brief proof of the equivalent characterizations of fitness landscapes lacking genetic constraints on accessible mutational trajectories. We compare a recent geometric classification of fitness landscape based on triangulations of polytopes with qualitative aspects of gene interactions. One observation is that fitness graphs provide information not contained in the geometric classification. We argue that a qualitative perspective may help relating theory of fitness landscapes and empirical observations.

The development of reliable methods for restoring susceptibility after antibiotic resistance arises has proven elusive. A greater understanding of the relationship between antibiotic administration and the evolution of resistance is key to overcoming this challenge. Here we present a data-driven mathematical approach for developing antibiotic treatment plans that can reverse the evolution of antibiotic resistance determinants. We have generated adaptive landscapes for 16 genotypes of the TEM β-lactamase that vary from the wild type genotype “TEM-1” through all combinations of four amino acid substitutions. We determined the growth rate of each genotype when treated with each of 15 β-lactam antibiotics. By using growth rates as a measure of fitness, we computed the probability of each amino acid substitution in each β-lactam treatment using two different models named the Correlated Probability Model (CPM) and the Equal Probability Model (EPM). We then performed an exhaustive search through the 15 treatments for substitution paths leading from each of the 16 genotypes back to the wild type TEM-1. We identified optimized treatment paths that returned the highest probabilities of selecting for reversions of amino acid substitutions and returning TEM to the wild type state. For the CPM model, the optimized probabilities ranged between 0.6 and 1.0. For the EPM model, the optimized probabilities ranged between 0.38 and 1.0. For cyclical CPM treatment plans in which the starting and ending genotype was the wild type, the probabilities were between 0.62 and 0.7. Overall this study shows that there is promise for reversing the evolution of resistance through antibiotic treatment plans.

Darwinian fitness is a central concept in evolutionary biology. In practice, however, it is hardly possible to measure fitness for all genotypes in a natural population. Here, we present quantitative tools to make inferences about epistatic gene interactions when the fitness landscape is only incompletely determined due to imprecise measurements or missing observations. We demonstrate that genetic interactions can often be inferred from fitness rank orders, where all genotypes are ordered according to fitness, and even from partial fitness orders. We provide a complete characterization of rank orders that imply higher order epistasis. Our theory applies to all common types of gene interactions and facilitates comprehensive investigations of diverse genetic interactions. We analyzed various genetic systems comprising HIV-1, the malaria-causing parasite Plasmodium vivax, the fungus Aspergillus niger, and the TEM-family of β-lactamase associated with antibiotic resistance. For all systems, our approach revealed higher order interactions among mutations.

The evolution of antibiotic resistance among bacteria threatens our continued ability to treat infectious diseases. The need for sustainable strategies to cure bacterial infections has never been greater. So far, all attempts to restore susceptibility after resistance has arisen have been unsuccessful, including restrictions on prescribing [1] and antibiotic cycling [2], [3]. Part of the problem may be that those efforts have implemented different classes of unrelated antibiotics, and relied on removal of resistance by random loss of resistance genes from bacterial populations (drift). Here, we show that alternating structurally similar antibiotics can restore susceptibility to antibiotics after resistance has evolved. We found that the resistance phenotypes conferred by variant alleles of the resistance gene encoding the TEM β-lactamase (bla TEM) varied greatly among 15 different β-lactam antibiotics. We captured those differences by characterizing complete adaptive landscapes for the resistance alleles bla TEM-50 and bla TEM-85, each of which differs from its ancestor bla TEM-1 by four mutations. We identified pathways through those landscapes where selection for increased resistance moved in a repeating cycle among a limited set of alleles as antibiotics were alternated. Our results showed that susceptibility to antibiotics can be sustainably renewed by cycling structurally similar antibiotics. We anticipate that these results may provide a conceptual framework for managing antibiotic resistance. This approach may also guide sustainable cycling of the drugs used to treat malaria and HIV.

2 ðDÞ. They show that if k is a complete NP kernel and if D is a separable Hilbert space, then for any scalar multiplier invariant subspace M of Hðk; DÞ there exists an auxiliary Hilbert space E and a multiplication operator F : Hðk; EÞ ! Hðk; DÞ such that F is a partial isometry and M ¼ FHðk; EÞ. Such multiplication operators are called inner multiplication operators and they satisfy FF * ¼ the orthogonal projection onto M. In this paper, we shall show that for many interesting complete NP kernels the analogy with the Beurling-Lax-Halmos theorem can be strengthened. We show that for almost every z 2 @B d the nontangential limit fðzÞ of the multiplier f:B d ! BðE; DÞ associated with an inner multiplication operator F is a partial isometry and that rank fðzÞ is equal to a constant almost everywhere. The result applies to certain weighted Dirichlet spaces and to the space H 2 d , which is determined by the kernel k l ðzÞ ¼ 1 1Àhz;li d ; l; z 2 B d . In particular, our result implies that the curvature invariant of Arveson (Proc. Natl. Acad. Sci. USA 96 (1999), 11,096-11,099) of a pure contractive Hilbert module of finite rank is an integer. This answers a question