Rank orders of genotypes with respect to fitness are central in evolutionary biology. Key concepts, such as peaks and mutational trajectories, refer to an underlying rank order. The impact of suboptimal peaks has been discussed ever since fitness landscapes were introduced (Wright, 1932). A major advantage with rank orders, as compared to other statistics for fitness landscape, is that order data is robust for variation in experimental methods and measurements. In fact, one does not necessarily need to measure fitness at all. If a rank order can be established by a competition experiment or by any type of rank order preserving fitness proxy, then one can apply theory on rank orders. Results from different empirical studies can therefore be compared; fitness or log-fitness, radius or area, change per minute or generation, makes no difference if rank orders are considered.

The concept of sign epistasis introduced by Weinreich et al. (2005), has been important for recent work on rank orders in evolutionary biology (Kaznatcheev, 2019; Crona et al., 2017; Wu et al., 2016; de Visser and Krug, 2014; Crona et al., 2013; Poelwijk et al., 2011). A system has sign epistasis if the sign of the effect of a mutation, whether positive or negative, depends on genetic background. The graphs in Figure 1 illustrate the absence of sign epistasis (A), sign epistasis (B), and reciprocal sign epistasis (C) for two-locus systems. Each arrow points toward the genotype of higher fitness. The importance of rank order data for two-locus subsystems is that they carry information about global aspects of fitness landscapes, and therefore about long-term evolution. For instance, all multi-peaked fitness landscapes have reciprocal sign epistasis (Poelwijk et al., 2011). Conversely, if subsystems of type C exist, but no systems of type B, then the landscape is multi-peaked (Crona et al., 2013).

Figure 1

Download asset Open asset

The squares representing a two-locus subsystem (see the orange square in Figure 2) are clearly informative. However larger rectangles can improve the precision of the analysis. Each arrow in Figure 2 points toward the genotype of higher fitness. The undirected edges connect mutational neighbors, that is genotypes that differ at one locus only. The blue rectangles in Figure 2 concern replacements of the type $00\mapsto 11$ for the second and third loci. If the sign of the effect of such a replacement depends on background, then the long arrows have different directions (Figure 2B and D). Such a perturbation is similar to sign epistasis, except that two loci are replaced.

Figure 2

Download asset Open asset

We propose a broader perspective on perturbations, that takes advantage of all rectangles in the cube (or hypercube). Differently expressed, in addition to single mutations it is sometimes useful to consider the effects of double mutations and any higher order mutations, as well as replacements $0\mapsto 1$ and $1\mapsto 0$ for any selected subset of loci. If the sign of the effect of such a replacement depends on background, then the system has a rectangular perturbation. As proof of principle, we demonstrate that the large rectangles give new insights for two empirical studies (as compared to an analysis of sign epistasis). The first concerns antimalarial drug resistance (Ogbunugafor and Hartl, 2016), whereas the second example involves bacteria adapting to a methanol environment (Chou et al., 2011).

As in Crona et al. (2017), fitness is additive if the fitness effects of individual mutations sum, otherwise the system has epistasis (or gene interactions). By a rank order induced, or signed, interaction, we mean that the rank order reveals epistasis. Questions about how different approaches to signed interactions relate and how it is possible to determine whether a rank order is compatible with additive fitness were briefly discussed by the authors. As eLife reviewers pointed out, the signed interactions in Crona et al. (2017) are not generalizations of sign epistasis but there is clearly some connection. The remaining open problems were the starting point for this work. The main idea for resolving the problems was to consider rectangles such as those in Figure 2 (no knowledge of Walsh-coefficient, polytopes, or similarly is assumed here).

We consider biallelic $n$-locus systems. The fitness of a genotype $g$ is denoted $w_g$. For simplicity, we assume that there exists a total order of the genotypes with respect to fitness, here referred to as a rank order (no two genotypes have equal fitness).

For two-locus systems, the expression $u=w_{00}+w_{11}-w_{10}-w_{01}$ measures epistasis. If it easy to verify that $u=0$ if fitness is additive. By definition, the system has an interaction if $u\ne 0$, and a signed interaction if the rank order implies that $u>0$ or $u<0$. For instance the order $w_{11}>w_{00}>w_{10}>w_{01}$ implies that $u>0$.

For a systematic treatment of order perturbations, it is necessary to write the information in compact form. Notice that a rank order where ${\displaystyle w_{00}>w_{10}}$ and ${\displaystyle w_{01}<w_{11}}$ implies sign epistasis, as the effect of a mutation at the first locus can increase or decrease fitness. Reciprocal sign epistasis for a two-locus system can be defined as a system with two peaks (Figure 1C). In particular, parallel arrows point in different directions.

It is straightforward to a check that a two-locus system has sign epistasis exactly if at least one of the following expressions is negative, and reciprocal sign epistasis if both of them are negative (Figure 1).

A negative sign corresponds to that a pair of parallel arrows that point in different directions. Large systems can be analyzed similarly.

A system has a rectangular perturbation if the sign of the effect of replacing a subset of loci, according to the rule $0\mapsto 1$ and $1\mapsto 0$, depends on background. The size of the perturbation refers to the number of loci replaced.

In particular, a rectangular perturbation of size 1 is a case of sign epistasis. The expression $u$ was used to analyze gene interactions in the two-locus case, and a similar approach works in general. We define $𝒞$ as the set of all expressions of the type

for genotypes $g,g^{\prime },g^{\prime \prime },g^{\prime \prime \prime }$ such that $r=0$ if fitness is additive. For instance, for n = 3, the expressions

associated with two of the rectangles in Figure 2 (i.e., the leftmost side and the blue rectangle) are included in $𝒞$. Note that each expression in $𝒞$ can be obtained by going around a rectangle in the cube and assigning coefficients with alternating signs along the way (each $w$-coefficient is +1 or –1).

We call the elements in $𝒞$ circuits, with reference to theory presented in Beerenwinkel et al. (2007b) (see also 'Materials and methods'; no knowledge of circuits is assumed). By a rank-order-induced circuit, or signed circuit, we mean that the rank order implies that the circuit is positive or negative (exactly as in the two-locus case).

For $n=3$, consider the circuit $w_{111}+w_{000}-w_{100}-w_{011}$ and the related expressions (the four genotypes from the circuit can be combined in two different ways):

If the rank order implies that expression (3) is negative, then the system has sign epistasis, and if expression (4) is negative, the system has an order perturbation of size 2. The short blue arrows in Figure 2 point in different directions in the first case, and the long blue arrows do so in the latter. Similarly, any rectangular perturbation corresponds to two parallel arrows (along opposite sides of a rectangle) that point in different directions.

Remark 1. The relation between signed circuits and sign epistasis can be summarized as follows for $n=2$.

1. A rank order of the genotypes 00, 10, 01 and 11 with respect to fitness implies that $u>0$ or $u<0$ exactly if the system has sign epistasis.

2. From the information that the rank order implies $u>0$ (or $u<0$) alone, one cannot determine whether there are one or two order perturbations, that is, whether or not the system has reciprocal sign epistasis.

3. The system has reciprocal sign epistasis if both expressions (1) and (2) are negative, and sign epistasis if at least one of them is negative.

Remark 2. A similar observation holds for signed circuits and any $n$.

1. Each case of sign epistasis is associated with a signed interaction for a circuit in $𝒞$, that is the rank order of genotypes with respect to fitness implies that the form is positive or negative.

2. More generally, each rectangular perturbation is associated with a signed circuit interaction for a circuit in $𝒞$.

3. Each circuit in $𝒞$ corresponds exactly to two potential rectangular perturbations.

In particular, in order to check a three-locus system for perturbations it is sufficient to check the signs of 24 expressions, including expression (3) and (4), associated with 12 circuits (see 'Materials and methods').

Applications

The first application concerns a study of the malaria-causing parasite Plasmodium vivax (Ogbunugafor and Hartl, 2016). The original study investigates a four-locus system exposed to different concentrations of the anti-malarial drug pyrimethamine (PYR). The quadruple mutant denoted 1111 has the highest degree of drug resistance, whereas the genotype 0000 has the highest fitness among all genotypes in the drug-free environment. Several concentrations of the drug were tested in the original study. We compared the highest concentration of the drug and the drug-free environment. Sign epistasis was equally frequent in both fitness landscapes. However, the drug-free environment had about twice as many perturbations of size 2 and 3, as the drug-exposed environment. The summary statistic indicates that (Table 1) adaptation is more difficult in the drug-free environment, because few replacements of pieces of code (regardless of size) are universally beneficial.

Table 1

Perturbation size	1	2	3	Size (1-3)
Drug-exposed	55	21	5	81
Drug-free	54	39	9	102
Expressions checked	112	72	16	200

This finding agrees well with the authors’ observation that resistance development is a relatively straightforward process, whereas reversed evolution from the mutant 1111 back to the ancestral type is difficult. Interestingly, the complete statistics of order perturbations was better able to distinguish the landscapes than sign epistasis alone.

The second application concerns a study of the bacterium Methylobacterium extorquens, which adapts to a methanol environment (Chou et al., 2011). For the sake of the argument, we assume that the published measurements are exact (a discussion about the impact of measurement errors can be found in 'Materials and methods'). The fitness landscape does not have sign epistasis. However, the system has size 2 perturbations (Table 2). In particular $w_{1000}>w_{0100}$ and $w_{1011}<w_{0111}$, so that the sign of the effect of the change $10\mapsto 01$ at the first pair of loci depends on background. The perturbations show that the rank order is incompatible with additive fitness.

Table 2

Perturbation of size	1	2
Number of perturbations	0	3
Expressions checked	112	72

A complete analysis of perturbations for an $n$-locus system requires an investigation of $2\left(\frac{6^n}{8}-4^{n-1}+2^{n-3}\right)$ expressions for $n\ge 3$ (Theorem 1, 'Materials and methods'). Statistics on order perturbations should always be considered estimates because of measurement errors. However, that is true for all rank-order-based concepts, and a peak count is sensitive to noise. One could naively believe that a broader perspective makes the problem worse. By contrast, rectangular perturbations can sometimes be helpful in separating signal and noise.

For instance, assume that fitness is additive and that all mutations decrease fitness (Figure 3). The graphs show the proportion of unexpected observations, that is increased fitness, for replacements of the type $0\mapsto 1$, $00\mapsto 11$ and $000\mapsto 111$ for two error distributions (errors follow a normal distribution in the upper graph and the Student’s t-distribution with df = 3 in the lower, see 'Materials and methods' for more details). In both cases, the proportion of unexpected results decreases by the number of loci replaced. The reason is that the noise level is similar in all cases, whereas the effect size is not.

Figure 3

Download asset Open asset

This observation can be useful. For instance, it is difficult to exclude additive fitness for a collection of detrimental mutations by observing sign epistasis alone (apparent sign epistasis can result from noise). However, an investigation of the three types of replacements described could be conclusive. In particular, similar proportions of unexpected results would constitute a strong argument against additive fitness.

Rank orders and perturbations are important for problems beyond natural selection. Constraints on the orders in which mutation occur are known for development of cancer drug resistance (Hosseini et al., 2019; Beerenwinkel et al., 2007a). In brief, a typical case would be that some mutation $B$ is not selected for unless a mutation $A$ has occurred, even though $AB$ has high fitness. If the constraint holds universally, it can be described as: $w_{01\ast }<w_{00\ast }<w_{10\ast }<w_{11\ast }$, where _* is an arbitrary sequence. For an $n$-locus system, such a constraint implies $2^{n-2}$ order perturbations if fitness is additive for the remaining loci.

A rank order of genotypes, from highest to lowest fitness, is informative about gene interactions (Crona et al., 2017). We introduced rectangular perturbations to resolve problems on how different approaches to rank order induced (or signed) interactions relate. Sign epistasis concerns the effects of single mutations, and the new perturbations also concern the effects of multiple mutations.

We have established that sign epistasis implies that a system has signed circuit interactions, but the converse is not true. Strictly speaking, signed circuits are not generalizations of sign epistasis, neither are signed circuits refinements of sign epistasis. However, sign epistasis can be seen as rectangular perturbations of minimal size.

A rank order is compatible with additive fitness unless there is sign epistasis for a two-locus system. We have provided a counterexample for the analogous claim for three loci. For four-locus systems, a rank order is compatible with additive fitness unless there are rectangular perturbations. However, counterexamples exist for five loci (Maclagan, 1998; Kraft et al., 1959). In general, only a very small proportion of all rank orders are compatible with additive fitness, and the theoretical understanding for the property is limited.

As a proof of principle, we applied rectangular perturbations to empirical studies on antimalarial drug resistance (Ogbunugafor and Hartl, 2016) and on bacteria adapting to a methanol environment (Chou et al., 2011). Rectangular perturbations have the capacity to detect epistasis when conventional rank-order-based methods cannot. The perturbations capture evolutionary important properties beyond local aspects. A complete analysis of rectangular perturbations requires an investigation of a large number of expressions (of the order $6^n$). Often, it is meaningful to consider selected perturbations, depending on the context.

In general, rank orders are quite informative about gene interactions, and also regarding evolutionary potential (in a qualitative sense) provided one assumes the Strong Selection Weak Mutation regime (Gillespie, 1984). If the available information from an empirical study is a rank order, for instance from a competition experiment, then it is obviously useful to have methods for interpreting rank orders (further motivations are discussed in Crona et al., 2017). However, rank-orders methods have obvious limitations. Rank orders are insufficient for determining the effect of genetic recombination (de Visser et al., 2009; Otto and Lenormand, 2002). Evolutionary predictability is sensitive for population parameters, and the importance of accessible mutational trajectories is not universal (de Visser and Krug, 2014; Krug, 2019).

It would be interesting to determine the extent to which rectangular perturbations are helpful for relating local and global properties of fitness landscapes, similar to results on sign epistasis (Weinreich et al., 2005; Poelwijk et al., 2011; Crona et al., 2013), and also in analyzing statistical aspects further. Algorithms and exact formulas for potential perturbations are provided in the 'Materials and methods'.

All data generated or analyzed during this study are included in the manuscript and supporting files.

1. 1. Albers PCH
   2. de Vries H

   (2001) Elo-rating as a tool in the sequential estimation of dominance strengths

   Animal Behaviour 61:489–495.

   https://doi.org/10.1006/anbe.2000.1571

   • Google Scholar
2. 1. Beerenwinkel N
   2. Eriksson N
   3. Sturmfels B

   (2007a) Conjunctive bayesian networks

   Bernoulli 13:893–909.

   https://doi.org/10.3150/07-BEJ6133

   • Google Scholar
3. 1. Beerenwinkel N
   2. Pachter L
   3. Sturmfels B

   (2007b)

   Epistasis and shapes of fitness landscapes

   Statistica Sinica 17:1317–1342.

   • Google Scholar
4. 1. Boyd R
   2. Silk JB

   (1983) A method for assigning cardinal dominance ranks

   Animal Behaviour 31:45–58.

   https://doi.org/10.1016/S0003-3472(83)80172-9

   • Google Scholar
5. 1. Bradley RA
   2. Terry ME

   (1952) Rank analysis of incomplete block designs: I. the method of paired comparisons

   Biometrika 4:324–345.

   https://doi.org/10.2307/2334029

   • Google Scholar
6. 1. Chou HH
   2. Chiu HC
   3. Delaney NF
   4. Segrè D
   5. Marx CJ

   (2011) Diminishing returns epistasis among beneficial mutations decelerates adaptation

   Science 332:1190–1192.

   https://doi.org/10.1126/science.1203799

   • PubMed
   • Google Scholar
7. 1. Crona K
   2. Greene D
   3. Barlow M

   (2013) The peaks and geometry of fitness landscapes

   Journal of Theoretical Biology 317:1–10.

   https://doi.org/10.1016/j.jtbi.2012.09.028

   • PubMed
   • Google Scholar
8. 1. Crona K
   2. Gavryushkin A
   3. Greene D
   4. Beerenwinkel N

   (2017) Inferring genetic interactions from comparative fitness data

   eLife 6:e28629.

   https://doi.org/10.7554/eLife.28629

   • PubMed
   • Google Scholar
9. 1. de Visser JA
   2. Park SC
   3. Krug J

   (2009) Exploring the effect of sex on empirical fitness landscapes

   The American Naturalist 174:S15–S30.

   https://doi.org/10.1086/599081

   • PubMed
   • Google Scholar
10. 1. de Visser JA
    2. Krug J

    (2014) Empirical fitness landscapes and the predictability of evolution

    Nature Reviews Genetics 15:480–490.

    https://doi.org/10.1038/nrg3744

    • PubMed
    • Google Scholar
11. 1. Gillespie JH

    (1984) Molecular evolution over the mutational landscape

    Evolution 38:1116–1129.

    https://doi.org/10.2307/2408444

    • Google Scholar
12. Book

    1. Grimaldi RP

    (2006)

    Discrete and Combinatorial Mathematics (5th Edition)

    Pearson Education India.

    • Google Scholar
13. 1. Hosseini SR
    2. Diaz-Uriarte R
    3. Markowetz F
    4. Beerenwinkel N

    (2019) Estimating the predictability of Cancer evolution

    Bioinformatics 35:i389–i397.

    https://doi.org/10.1093/bioinformatics/btz332

    • PubMed
    • Google Scholar
14. 1. Kaznatcheev A

    (2019) Computational complexity as an ultimate constraint on evolution

    Genetics 212:245–265.

    https://doi.org/10.1534/genetics.119.302000

    • PubMed
    • Google Scholar
15. 1. Kraft CH
    2. Pratt JW
    3. Seidenberg A

    (1959) Intuitive probability on finite sets

    The Annals of Mathematical Statistics 30:408–419.

    https://doi.org/10.1214/aoms/1177706260

    • Google Scholar
16. Preprint

    1. Krug J

    (2019) Accessibility percolation in random fitness landscapes

    arXiv.

    https://arxiv.org/abs/1903.11913

    • Google Scholar
17. 1. Maclagan D

    (1998) Boolean term orders and the root system bn

    Order 15:279–295.

    https://doi.org/10.1023/A:1006207716298

    • Google Scholar
18. 1. Ogbunugafor CB
    2. Hartl D

    (2016) A pivot mutation impedes reverse evolution across an adaptive landscape for drug resistance in plasmodium vivax

    Malaria Journal 15:40.

    https://doi.org/10.1186/s12936-016-1090-3

    • PubMed
    • Google Scholar
19. 1. Otto SP
    2. Lenormand T

    (2002) Resolving the paradox of sex and recombination

    Nature Reviews Genetics 3:252–261.

    https://doi.org/10.1038/nrg761

    • PubMed
    • Google Scholar
20. 1. Poelwijk FJ
    2. Tănase-Nicola S
    3. Kiviet DJ
    4. Tans SJ

    (2011) Reciprocal sign epistasis is a necessary condition for multi-peaked fitness landscapes

    Journal of Theoretical Biology 272:141–144.

    https://doi.org/10.1016/j.jtbi.2010.12.015

    • Google Scholar
21. 1. Searles D
    2. Slinko A

    (2015) Noncoherent initial ideals in exterior algebras

    Beiträge Zur Algebra Und Geometrie / Contributions to Algebra and Geometry 56:759–762.

    https://doi.org/10.1007/s13366-015-0239-5

    • Google Scholar
22. 1. Thurstone LL

    (1927) The method of paired comparisons for social values

    The Journal of Abnormal and Social Psychology 21:384–400.

    https://doi.org/10.1037/h0065439

    • Google Scholar
23. Conference

    1. Wauthier F
    2. Jordan M
    3. Jojic N

    (2013)

    Efficient ranking from pairwise comparisons

    International Conference on Machine Learning. pp. 109–117.

    • Google Scholar
24. 1. Weinreich DM
    2. Watson RA
    3. Chao L

    (2005) Perspective: sign epistasis and genetic costraint on evolutionary trajectories

    Evolution 59:1165–1174.

    https://doi.org/10.1111/j.0014-3820.2005.tb01768.x

    • Google Scholar
25. Conference

    1. Wright S

    (1932)

    The roles of mutation, inbreeding, crossbreeding, and selection in evolution

    Proceedings of the Sixth International Congress of Genetics. pp. 356–366.

    • Google Scholar
26. 1. Wu NC
    2. Dai L
    3. Olson CA
    4. Lloyd-Smith JO
    5. Sun R

    (2016) Adaptation in protein fitness landscapes is facilitated by indirect paths

    eLife 5:e16965.

    https://doi.org/10.7554/eLife.16965

    • PubMed
    • Google Scholar

Author details

1. Kristina Crona

   Mathematics and Statistics, American University, Washington DC, United States

   Contribution

   Conceptualization, Formal analysis, Methodology

   For correspondence

   kcrona@american.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-1819-474X

• 1,404

  views

• 147

  downloads

• 16

  citations

Views, downloads and citations are aggregated across all versions of this paper published by eLife.