key: cord-0303953-r7g734ud
authors: Ogbunugafor, C. Brandon
title: The mutation effect reaction norm (MERN) highlights environmentally dependent mutation effects and genetic interactions
date: 2021-09-27
journal: bioRxiv
DOI: 10.1101/2021.09.23.461533
sha: c984b0cd5a95addeff0a7769359e5ef2657ab43b
doc_id: 303953
cord_uid: r7g734ud

Measuring the fitness effects of mutations and epistasis remain central yet provocative concepts in evolutionary and population genetics. In addition to the baseline complexity that arises from the notion that genetic information can interact in a nonlinear way, recent studies have revealed that interactions can change as a function of environmental context. Here I propose the fusion of measurements of the effect of mutations and physiological epistasis with the reaction norm, a central abstraction used to depict genotype by environment interactions. In doing so, I formalize the notion of a “mutation effect reaction norm” (MERN) as an instrument through which one can analyze or depict the phenotypic consequences of interactions between mutations across environmental contexts. I demonstrate its utility through a discussion of the signature of mutations that undermine reverse evolution of antimicrobial resistance. In closing, I argue that the mutation effect reaction norm may help us resolve the dynamism of evolution across fitness seascapes through specific insight into how mutation effects and interactions are modulated by environmental context.

While breakthroughs in the various subfields of genomics 23 continue to improve our understanding of the architecture of 24 complex traits and phylogenetic relationships between or-25 ganisms, recent findings have refined our lenses and cre- 26 ated new questions. For example, modern perspectives from 27 evolutionary genetics are increasingly driven by notions that 28 complex traits are the product of interactions between many 29 individual SNPs, which inform applications of the infinitesi- 30 mal model and other iterations of how we study the relation- 31 ship between genotype and phenotype [1, 2] . 32 For decades, interactions between mutations have played 33 a meaningful role in related discussions surrounding con- 34 ceptual challenges in population and evolutionary genetics. 35 The term "physiological epistasis" has been used to describe 36 "any situation in which the genotype at one locus modifies 37 the phenotypic expression of the genotype at another locus 38 [3]." Within this definition, an expansive literature exists that 39 has examined epistasis in adaptive landscapes [4, 5, 6] . with 40 respect to protein biophysics [7, 8, 9] , in terms of genomic 41 architecture [10, 11] , and many other arenas. 42 The broader notion that mutations may interact with other 43 parcels of genetic information in a spurious or nonlinear 44 fashion casts a shadow over much of modern genetics [12, 45 13, 14] . In some sense, nonlinear interactions between muta-46 tions might challenge simple models of genotype-phenotype 47 mapping, plausibly contributing to phenomenon like phan-48 tom heritability [15, 16, 17] . Relatively underexplored in 49 conversations about how epistasis may craft complex phe- 50 notypes are theoretical treatments of how environmental gra-51 dients may influence the interactions between mutations or 52 SNPs of interest that craft complex traits. 53 Conveniently, an abstraction exists in the evolutionary biol-54 ogy and ecology canons-the reaction norm (also known as 55 the "norm of reaction")-to describe how the environment 56 shapes the performance (phenotype) of genotypes [18] . The 57 reaction norm is widely applied in evolutionary biology in 58 quantitative genetics [19, 20] , in discussions of phenotypic 59 plasticity [21, 22] , and other subtopics. 60 While several studies have examined how environments 61 can tune nonlinear interactions between mutations [23, 24, 62 25, 26, 27] , there have been no formal attempts to integrate 63 how the environment tunes nonlinear interactions between 64 mutations. In this study, I introduce the "mutation effect re-65 action norm," an abstraction that combines the reaction norm 66 with calculations of mutation effect and physiological epista-67 sis. It demonstrates how mutation effects and epistatic coeffi-68 cients can change appreciably across environmental contexts 69 of various kinds. To demonstrate its utility, I explore data 70 sets corresponding to a collection of alleles carrying muta-71 tions associated with antimicrobial drug resistance. I analyze 72 these data using the mutation effect reaction norm framework 73 and diagnose the signature of the effect of a compensatory 74 ratchet mutation whose effect is specific to environment. phenotype of interest. It was pioneered for use in the study of 126 epistasis in a 2013 study that both provided a primer for the 127 calculation and analyzed several combinatorially complete 128 data sets [4] . It has since been further applied to study of 129 higher-order epistasis across a larger sampling of empirical 130 data sets [36] .

The Walsh-Hadamard transform implements phenotypic 132 measurements into a vector, then a Hadamard matrix, which 133 is scaled by a diagonal matrix. The result is a set of co-134 efficients that measure the degree to which the genotype-135 phenotype map, perhaps encoded as an adaptive landscape, is 136 linear, or second order, third, and so forth. This vector of phenotypes corresponding to each of those 162 variants can be multiplied by a (16 × 16) square matrix, which 163 is the product of a diagonal matrix V and a Hadamard matrix 164 H. These are defined recursively:

n is the number of loci (n = 4 in this Plasmodium falci-166 parum DHFR setting corresponding to resistance mutations). 167 This multiplication gives the following:

and 2 above and is the Walsh coefficient, the measure of 170 the interaction between mutations. Using this formulation, 171 we compute values for every possible interaction between 172 bits in each string.

In addition to the aforementioned references where this approach were described [4, 37] , this method is also detailed 175 in the supporting information, which also contains a spread-176 sheet that outlines the calculation and provides a means for 177 even inexperienced users to calculate interaction coefficients.

Having outlined the method used to calculate the Walsh- We can calculate higher-order epistasis using several mi- Relatedly: 0011 0101, 0110, 1001, 1100 are second-order 229 or pairwise interactions; 0111, 1010, 1011, 1101, 1110 are 230 third-order interactions; and 1111 is a fourth-order interac-231 tion, the interaction between the four mutations that consti-232 tute the quadruple mutant, IRNL. For even more clarity, one 233 can replace the 0s with asterisks (*) to emphasize that those 234 sites represent mutation interaction effects across all possible 235 combinations of those sites. For example, the pairwise effect 236 coefficient corresponding to "0110" truly means the average 237 effect of the C59R and S018N mutations, across all other 238 backgrounds. We can depict this effect as "*11*".

One can average the interaction coefficients within an order 240 to facilitate comparisons between orders (e.g., are third order 241 effects stronger than pairwise effects across environments?). 242 Though the data in this study are not normalized, it is of-243 ten prudent to compute a normalized version of the epistatic 244 coefficients, and then take the absolute value. The normaliza-245 tion standardizes the value relative to those in the set, which 246 is useful to contextualize effects relative to the entire set. 247 For a given epistatic coefficient , we define the normalized 248 epistatic coefficient E, as in prior studies of in silico adaptive 249 landscapes [40]:

In this study, we only use the absolute values of the 251 epistatic coefficients for all analyses, without normalization. 252 While the absolute mean masks negative values, this cal-253 culation offers a window into the magnitude of nonlinear-254 ity in a genotype-phenotype map, across environments. One 255 can take the average value of the epistatic coefficients across 256 orders (average all pairwise, three-way interactions, etc.), 257 which facilitates comparisons between orders (in the case of 258 the mutation effect reaction norm, across environments). 259 We label these order-averaged effects with the term "abso-260 lute mean." This provides mean values for each order, which 261 calculates the overall contribution of, for example, 1st order 262 effects and higher-order (3 order, 4 ℎ order, etc.) effects.

And we can examine how the order of effects changes 264 across environmental gradients, representing a kind of mu-265 tation effect reaction norm for higher-order epistasis. To offer a clearer understanding of how environments 321 shape higher-order interactions, I provide a mutation ef-322 fect reaction norm corresponding to the normalized abso-323 lute mean values of mutation effects, organized by order 324 (Fig. 2B and 2D ). These represent magnitude differences be-325 tween orders of effects and communicate the overall presence 326 of higher-order interactions across environmental gradients 327 (drug type and concentration in this case). Of particular interest are the effects of S108N in P. falci-331 parum dihydrofolate reductase ( Fig. 2A, 2C ). The effects of 332 an orthologous mutation was described in a study of reverse 333 evolution of antifolate resistance in Plasmodium vivax [42] . 334 Note that in both pyrimethamine and cycloguanil, the mu-335 tation effect has a similar pattern: a negative effect at low 336 drug concentrations, with a sign change (from negative to 337 positive) as drug concentrations increase towards 100 g/ml, 338 after which it then decreases as the drug concentration in-339 creases. This reflects a single mutation whose interaction 340 effect changes from negative at low drug concentrations to 341 positive effects at higher drug concentrations.

This "flip" of the sign of a mutation effect is reflective of 343 a mutation that could be described as compensatory. That is, 344 the mutation corresponding to S108N is conditionally bene-345 ficial, offering positive epistatic interaction in drug environ-346 ments with higher concentrations (Fig. 2A) , where many al-347 leles are growing poorly. These mutations restore growth in 348 genetic backgrounds that are growing poorly. These muta-349 tions also serve as ratchets that undermine the reversal of evo-350 lution (from the IRNL quadruple mutant towards the NCSI 351 wildtype in this setting).

ing the S018N mutation can influence evolution. In 3A, 354 we observe that alleles that contain S018N have a signifi-355 cantly higher growth rate, across all drug concentrations of 356 both pyrimethamine (Kruskal-Wallis: 5A, pyrimethamine, p 357 = 0.0002;). Notably, the S108N mutation is an orthologue of 358 a mutation, S117N, that has been described as a "pivot" mu-359 tation, that both dictates the direction of adaptive evolution, 360 and precludes reversal [42]. 361 Fig. 3B is a hypergraph summary of the predicted evo-362 lutionary trajectories in pyrimethamine. Predictions were 363 made from the rank orders of alleles outlined in Figure 364 1. That is, starting from NCSI, evolution follows a path 365 of increasing growth rate, a proxy for reproductive fit-366 ness in this setting. Figure 3B depicts "forward" evo-367 lution starting from the wild type (NCSI) allele evolv-368 ing at 10 6

Pyrimethamine, as summarized in previ-369 ous studies [29] . In addition, Figure 3B shows the pre- To demonstrate the utility of the abstraction, I use 389 empirical fitness landscapes as test data sets. I apply previ-390 ously developed mathematical methods introduced to mea-391 sure higher-order epistasis on binary-encoded combinatori-392 ally complete data sets, across categorical and both continu-393 ous environments. The purpose of this examination is to con-394 solidate these perspectives into a single abstraction that sim-395 plifies the notion that epistatic effects can be depicted across 396 environments.

With this mutation effect reaction norm, several sub-399 questions can be examined, such as how multi-dimensional 400 environments tune the phenotypic effects of mutations. We 401 observe this through comparing the topography of the muta-402 tion effect reaction norms for a suite of SNPs associated with 403 resistance to pyrimethamine and cycloguanil (one dimen-404 sion: antifolate compounds used to treat malaria that are sim-405 ilar in structure), across a range of drug concentrations (sec-406 ond dimension). In this case, we observe how and why the 407 fitness landscapes (composed of different haplotypes) differ 408 for the two landscapes. Though the two drugs have a similar 409 The study's examination of S108N's interactions serves 429 as an example of how tracking the mutation effects allows 430 one to identify the (i) quantitative range of environments in 431 which a mutation has effects, (ii) the genotypic context in 432 which a given mutation is compensatory and (iii) the degree 433 to which the effect is compensatory or not. I further argue 434 that the story that we've revealed about S108N likely applies 435 to many compensatory ratchet mutations: their effects are of-436 ten not binary-as in, they are beneficial in an environment 437 or not-but rather, have stories that are more nuanced. This 438 is consistent with studies of bacterial translation machinery 439 have reflected the dubiousness of compensatory mutations in 440 evolution [44].

The term "compensatory" implies that a mutation appears 442 on genetic backgrounds where its effects are restorative. That 443 is, one might believe that compensatory mutations must oc-444 Figure 3 : The signature of a compensatory ratchet mutation. Here we depict how the S108N mutation's effect across environments plays a critical role in "forward" evolution and undermines "reverse" evolution. (A) In pyrimethamine, alleles containing the S018N mutation grow significantly better than any set of alleles containing any other single mutation (Kruskal-Wallis: 5A, pyrimethamine, p = 0.0002). (B) The hypercube represents the combinatorial set of 16 alleles as described in the Methods. Based on the rank-order in Figure 1 , the predicted pathways of stepwise evolution from the wild-type genotype (NCSI) through the fitness landscape at a high drug concentration (black arrows) at 106 uM, and reversal in a drugless environment (dashed arrow). In this scenario, the compensatory nature of the S108N mutation provides a compensatory ratchet, that helps evolution evolve up a fitness peak in the high drug concentration (black arrows) but prevents it from reversing towards wildtype (NCSI) in a drugless environment (dashed arrow).

cur downstream in an adaptive landscape. As summarized 445 in the adaptive landscape in Figure 3B , the S108N muta-446 tion can occur early in an adaptive trajectory (e.g., first step).

Consequently, the language of epistatic "ratchet" applies as 448 well, even though studies that have outlined them often de-449 scribe ratchets in terms more biophysical than are explored 450 in this study [45, 46] . For that reason, we can call mutations 451 like S108N-whose environment specific effects (discussed 

This has practical relevance and informs public health and 469 biomedical approaches to addressing the antimicrobial resis-470 tance problem. Studies have revealed why resistance man-471 agement approaches that attempt to drive populations of re-472 sistant microbes "back" to more susceptible forms can be 473 challenging [52, 53, 54] . And other studies have proposed 474 methods to reverse the effects of resistance [55] . Table 1 describes the differences between the reaction 476 norm and the mutation effect reaction norm, including how 477 one can interpret the information contained within them. 478 This mutation effect reaction norm abstraction offers a dif-479 ferent view of gene by environment interactions, by decon-480 structing genotype-phenotype maps in terms of the often pe-481 culiar interactions between genes and mutations across en-482 vironments. This can offer a "mutation-centric" view of 483 genotype-phenotype maps where the object of interest is not 484 entire haplotypes, but rather, the individual interactions be-485 tween the mutations that compose those haplotypes. . I argue that the key to 543 properly characterizing these mutations resides in an under-544 standing of how environmental context shapes their effect. 545 That is, host structure, demographics (e.g., age), and other 546 factors may influence how a given SARS-CoV-2 mutation 547 interacts with others, creating a variant of concern. 548 Notably absent from my introduction of the mutation ef-549 fect reaction norm are mathematical formalisms that more 550 directly link the shape of underlying reaction norms to that 551 of the mutation effect reaction norms. While this study pro-552 poses the utility for an abstraction that depicts how environ-553 ments tune interactions between mutations, it doesn't firmly 554 ground them in quantitative genetic terms, like the reaction 555 norm has been in the past [70, 19, 71] . The existence of that 556 studies have utilized rank orders of genotypes to infer ge-557 netic interactions [41, 33] , those that have examined the fate 558 of mutations in fluctuating environments [72] and others that 559 mathematically resolve the shape of fitness landscapes [73] 560 suggest that such formalisms may exist. An analytical rec-561 onciliation between these existing treatments constitutes an 562 obvious future direction of inquiry.

These gaps notwithstanding, the mutation effect reaction 564 norm may allow us to tell more meaningful stories of muta-565 tion effects, that describe many aspects of how a mutation or 566 SNP of interest may have arisen, and/or how we would ex-567 pect it to influence phenotypes of interest. While true predic-568 tion in evolutionary genetics remains mostly a fantasy, future 569 efforts to properly predict evolution must include a consider-570 ation for how environments tune interactions between muta-571 tions.

Reaction norm or norm of reaction The performance, phenotype or trait value for different alleles, strains, mutants, variants, or forms, or populations, across an environmental gradient (continuous or other)

Used to identify and measure the effects of genes, environments, gene x environment interactions, phenotypic plasticity, and other related properties in quantitative genetics, evolutionary genetics, and ecology.

The phenotypic effect of individual or collections of mutations across a set of environments (continuous or other)

Used to measure how environments influence the effect of individual mutations, the interaction between suites of mutations, and epistatic interactions. It can help to explain how the environmental sculpts the topography of adaptive landscapes via environment-dependent mutation effects. 

The in-592 finitesimal model: Definition, derivation, and impli-593 cations

An expanded view of complex traits: From polygenic 597 to omnigenic

Genotypic 599 context and epistasis in individuals and populations

Should evolutionary geneticists 603 worry about higher-order epistasis? Current Opinion 604 in Genetics & Development

Selection bi-606 ases the prevalence and type of epistasis along adaptive 607 trajectories

Mapping the fitness landscape 610 of gene expression uncovers the cause of antagonism 611 and sign epistasis between adaptive mutations

ness-epistasis link shapes the fitness landscape of a ran-616 domly drifting protein

Epistasis in 619 protein evolution

Stability-mediated epistasis re-624 stricts accessible mutational pathways in the functional 625 evolution of avian hemoglobin

Epistasis be-628 tween new mutations and genetic background and a test 629 of genetic canalization

Epistasis correlates to 632 genomic complexity

Epistasis and its con-635 tribution to genetic variance components

Epistasis: what it means, what it doesn't 638 mean, and statistical methods to detect it in humans

Dependence of epistasis 685 on environment and mutation severity as revealed byin 686 SilicoMutagenesis of phage t7

Multi-environment fitness 689 landscapes of a tRNA gene

692 Molecular basis of differential resistance to cycloguanil 693 and pyrimethamine in plasmodium falciparum malaria

Adap-698 tive landscape by environment interactions dictate evo-699 lutionary dynamics in models of drug resistance

702 Competition along trajectories governs adaptation rates 703 towards antimicrobial resistance

Detecting high-706 order epistasis in nonlinear genotype-phenotype maps

Molecular 709 ensembles make evolution unpredictable

Rank orders and signed interactions in 713 evolutionary biology. eLife, 9:e51004

Inferring fitness land-715 scapes by regression produces biased estimates of epis-716 tasis

Uninter-719 pretable interactions: epistasis as uncertainty. bioRxiv

The influence of higher-order 723 epistasis on biological fitness landscape topography

Vinod Krishna, and Rama Ran-726 ganathan. The context-dependence of mutations: A 727 linkage of formalisms

Proteostasis environment shapes higher-order epis-738 tasis operating on antibiotic resistance

Lexical landscapes as large 742 in silico data for examining advanced properties of fit-743 ness landscapes

The rank ordering of genotypic 745 fitness values predicts genetic constraint on natural se-746 lection on landscapes lacking sign epistasis

A pivot 749 mutation impedes reverse evolution across an adap-750 tive landscape for drug resistance in plasmodium vivax

From fitness land-753 scapes to seascapes: non-equilibrium dynamics of se-754 lection and adaptation

Evolu-758 tionary stalling and a limit on the power of natural se-759 lection to improve a cellular module

An epistatic ratchet constrains the direc-764 tion of glucocorticoid receptor evolution

Artem Dubovet-767 skyi, Kesava-Phaneendra Cherukuri

Enzyme 770 evolution: An epistatic ratchet versus a smooth re-771 versible transition

Dollo on dollo's law: Irreversibil-774 ity and the status of evolutionary laws

Reversing opin-777 ions on dollo's law

Hidden randomness between fitness land-781 scapes limits reverse evolution

Compensatory mutations and epistasis for 784 protein function. Current Opinion in Structural Biol-785 ogy

Reversion is most likely under high muta-788 tion supply, when compensatory mutations don't fully 789 restore fitness costs. bioRxiv

Mar-791 tin J. Blaser, and Lars Engstrand. Long-term persis-792 tence of resistant enterococcus species after antibiotics 793 to eradicate helicobacter pylori

Antibiotic re-796 sistance and its cost: is it possible to reverse resistance? 797

Little evidence for re-801 versibility of trimethoprim resistance after a drastic re-802 duction in trimethoprim use

Multidrug evo-805 lutionary strategies to reverse antibiotic resistance. Sci-806 ence

Detecting epistasis with the marginal epista-809 sis test in genetic mapping studies of quantitative traits

Caging and uncaging 812 genetics

CRISPR perturba-815 tion of gene expression alters bacterial fitness under 816 stress and reveals underlying epistatic constraints

Understanding, 819 predicting and manipulating the genotypic evolution 820 of antibiotic resistance

Predic-826 tive biology: modelling, understanding and harnessing 827 microbial complexity

Molecular foundations of precambrian unifor-832 mitarianism. bioRxiv, preprint

Contin-835 gency and chance erase necessity in the experimental 836 evolution of ancestral proteins. eLife, 10:e67336

Delayed commitment to evolutionary fate in 840 antibiotic resistance fitness landscapes

843 Enzyme efficiency but not thermostability drives ce-844 fotaxime resistance evolution in TEM-1 β-lactamase

Price equation captures 848 the role of drug interactions and collateral effects in the 849 evolution of multidrug resistance. eLife

The pop-852 ulation genetics of collateral resistance and sensitivity

Initiating a 857 watch list for ebola virus antibody escape mutations

Caroline Col-860 ijn

The origins and poten-863 tial future of SARS-CoV-2 variants of concern in 864 the evolving COVID-19 pandemic

Quantitative genetics of reaction norms

Gene reg-870 ulation, quantitative genetics and the evolution of reac-871 tion norms

Fate of a mutation in a 874 fluctuating environment

Finding the boundary between evolutionary 878 basins of attraction, and implications for wright's fit-879 ness landscape analogy

I would like to thank R. Oomen, M. Miller-Dickson, R. 573 Guerrero, D. Weinreich, and P. Pennings for helpful discus-574 sions on this topic. I would like to thank M. Rebolleda-575