Getting higher on rugged landscapes Inversion mutations open access to tter adaptive peaks in NK tness landscapes

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Getting higher on rugged landscapes: Inversion

mutations open access to ﬁtter adaptive peaks

in NK ﬁtness landscapes

Leonardo Trujillo1,*, Paul Banse1, Guillaume Beslon1

Universit´e de Lyon, INSA-Lyon, INRIA, CNRS, Universit´e Claude Bernard Lyon 1, ECL, Universit´e

Lumi`ere Lyon 2, LIRIS UMR5205, Lyon, France

*leonardo.trujillo@inria.fr

Abstract

Molecular evolution is often conceptualised as adaptive walks on rugged ﬁtness landscapes,

driven by mutations and constrained by incremental ﬁtness selection. It is well known that

epistasis shapes the ruggedness of the landscape’s surface, outlining their topography (with

high-ﬁtness peaks separated by valleys of lower ﬁtness genotypes). However, within the

strong selection weak mutation (SSWM) limit, once an adaptive walk reaches a local peak,

natural selection restricts passage through downstream paths and hampers any possibility of

reaching higher ﬁtness values. Here, in addition to the widely used point mutations, we

introduce a minimal model of sequence inversions to simulate adaptive walks. We use the

well known NK model to instantiate rugged landscapes. We show that adaptive walks can

reach higher ﬁtness values through inversion mutations, which, compared to point

mutations, allows the evolutionary process to escape local ﬁtness peaks. To elucidate the

eﬀects of this chromosomal rearrangement, we use a graph-theoretical representation of

accessible mutants and show how new evolutionary paths are uncovered. The present model

suggests a simple mechanistic rationale to analyse escapes from local ﬁtness peaks in

molecular evolution driven by (intragenic) structural inversions and reveals some

consequences of the limits of point mutations for simulations of molecular evolution.

Author summary

Ninety years ago, Wright translated Darwin’s core idea of survival of the ﬁttest into rugged

landscapes—a highly inﬂuential metaphor—with peaks representing high values of ﬁtness

separated by valleys of lower ﬁtness. In this picture, once a population has reached a local

peak, the adaptive dynamics may stall as further adaptation requires crossing a valley. At

the DNA level, adaptation is often modelled as a space of genotypes that is explored

through point mutations. Therefore, once a local peak is reached, any genotype ﬁtter than

that of the peak will be away from the neighbourhood of genotypes accessible through point

mutations. Here we present a simple computational model for inversion mutations, one of

the most frequent structural variations, and show that adaptive processes in rugged

landscapes can escape from local peaks through intragenic inversion mutations. This new

escape mechanism reveals the innovative role of inversions at the DNA level and provides a

step towards more realistic models of adaptive dynamics, beyond the dominance of point

mutations in theories of molecular evolution.

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arXiv:2210.06804v1 [q-bio.PE] 13 Oct 2022

Introduction

The fitness landscape is a very influential metaphor introduced by Wright [1, Fig 2] to

describe evolution as explorations through a “field of possible genes combinations”,

where high values of fitness are represented as peaks separated by valleys of lower

fitness. The topography of the fitness landscape has important evolutionary

consequences, e.g. speciation via reproductive isolation [2]. Within this framework, the

evolution of any population can be conceptualised as adaptive walks driven by

successive mutations constrained by incremental or neutral fitness steps. Thus, in the

absence of additional evolutionary forces such as drift or environmental variations, once

a population reaches a local peak, natural selection hampers any further mutational

paths that decrease fitness. However, there are empirical evidences showing that

populations do not stop indefinitely at a local peak and can explore alternative

trajectories on the landscape [3–7], ergo, the following question arises: How does

evolution escape from a local peak to a fitter one?

Since it has been formulated, considerable progress have been made on this

question [8

–

22]. However, conventional theoretical approaches still state that a genotype

mutates into another through point mutations (e.g. single-nucleotide variations). If one

takes a look at the molecular scale of DNA and the different mutation types, this may

seem contradictory as it is well known that many other kinds of variation operators

(including insertions, deletions, duplications, translocations and inversions) act on the

genome. Hence, a fundamental aspect of this challenge is to understand—at the scale of

molecular evolution—the roles played by these different mutation types. Indeed, there is

a gap between the theoretical models, that account for a very limited set of mutations

types—typically only point mutations—and the reality of molecular evolution, where

multiple variation operators act on the sequence.

As a contribution to bridge this gap, we present a minimal DNA-inspired

mechanistic model for inversion mutations, and explore their relationship with the

escape dynamics from local fitness peaks. Inversion mutations are one of the most

frequent chromosomal rearrangements [23, Ch. 17.2] with lengths covering a wide range

of sizes. For example, in Long Term Evolution Experiments with E. coli, chromosomal

rearrangements have been characterised by optical mapping (hence limiting the

resolution to rearrangements larger than 5000 bp) [24]. In this study, 75% of evolved

populations showed inversion events—ranging in size from ∼164 Kb to ∼1.8 Mb [24]

(for other examples of large inversions in different clades, see [25, Table 1 ]). With the

development of novel sequencing technologies [26–28], it has been possible to identify

intragenic (submicroscopic) DNA inversions, for example, an inversion of seven

nucleotides in mitochondrial DNA, resulting in the alteration of three amino acids and

associated to an unusual mitochondrial disorder [29, Fig 1]. Intragenic inversions have

also been suggested to be an important mechanism implied in the evolution of

eukaryotic cells [30]. Although chromosomal inversions are ubiquitous in many

evolutionary processes [23–27,29–38], very little is known about their theoretical

description and computational simulation at the sequence level, as models generally

focus on very large inversions (typically larger than a single gene), hence on their effect

on synteny (or the deleterious effects at breakpoints), but neglect the possibility that

small inversions occur inside coding sequences [39].

Here, we simulate a representation of molecular evolution of digital organisms

(replicators), each of which contains a single piece of DNA. We engineer a

computational method to cartoon the double-stranded structure of DNA, and simulate

inversion-like mutations consisting of a permutation of a segment of the complementary

strand, which is then exchanged with the main strand segment (see Methods, schema 6).

For the sake of simplicity, we consider digital genotypes made up of binary nucleotides

(i.e. a binary alphabet {0,1}instead of the four-nucleotides alphabet {A,T,C,G}). The

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sequences are arranged in circular strings with constant number of base-pairs. In an

abstract sense, the model mimics the molecular evolution of some viruses [40] and

(animal) mitochondrial DNA [41] with compact genomes and closed double-stranded

DNA circles [40,42,43]. It is very important to emphasise that our computational model

simulates intragenic-like mutations [29,30,44]. We are modelling asexual replication,

therefore recombination is not considered. To build rugged fitness landscapes, we adopt

the well known Kauffman NK model, where N denotes the length of the genome and

parameterizes the “epistatic” coupling between nucleotides [45–48]. We do not include

environmental changes, so the landscape remains constant through the simulations.

Finally, it is worth mentioning that all our simulations were conducted in the

evolutionary regime of strong selection weak mutations (SSWM) [49, Ch. 5].

Results

Who is next to whom: the mutational network

We first study how inversion mutations can increase the number of accessible mutants.

For this we translate the canonical notion of neighbour genotypes (see Methods, Eq. 1)

into a graph theory approach, and analyse the simplest and most familiar geometric

object in molecular evolutionary theory: the discrete space of binary sequences (unless

explicitly stated we will henceforth consider only binary alphabets). Thinking

topologically, all the sequences xwith Nbinary-nucleotides xi∈ {0,1},∀i= 1, . . . , N

and N∈N≥2, define the set X ∈ {0,1}Nof 2Npossible genotype combinations. A

canonical measure to characterise the topology of the set Xis the Hamming distance

dH(x,x0) :=

i=1

(xi−x0

i)2.

A convenient way to organise such a set is by graphs connecting two sequences

and

that differ by one point mutation (i.e. dH(x,x0) = 1). This is the so-called Hamming

graph H(N, 2)—a special case of the hypercube graph QN(the well-known graph

representation of the genotype space). On the other hand, the Hamming distance for

inversion mutations forms a set of integers satisfying

0≤dH(x,x0)≤N,

meaning that, contrary to point mutations, the Hamming distance of inversion

mutations range from zero to N(see Methods). Note that if the inversion spans the

entire chromosome, then dH(x,x0) = N, all loci have changed, but it also implies that

5’— 3’ becomes 3’—5’ and vice versa and nothing has changed biologically.

We propose that, for a sequence

x∈ X

, the mutation operation (i.e. the mechanistic

representation of point or inversion mutations) build the set Nν(x) of accessible

mutants x0∈ Nν(x),∀x06=x,=⇒dH(x,x0)6= 0 (the subindex νdenotes the type of

mutation:

for point mutations and

for inversion mutations). Therefore, the number

of neighbouring mutants can be reformulated as

Dν(x) = |Nν(x)|.

Inversion’s combinatorics is not trivial since it involves the permutation of a

subsequence and its flips between each strand (see Methods, schemata 2and 6).

Nevertheless, from the algorithmic point of view, for a given genotype

the mutational

operations can be used to enumerate all the accessible mutants (see Methods,

Algorithm 1:Mutate). Also, the combinatorics can be represented as a directed

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Fig 1. Atlas of accessible-mutants. Example of the total enumeration of inversion mutations, represented as graphs of

accessible mutants, for each one of the 24genotypes (central red nodes) with size N= 4. Edges colour quantifies the

Hamming distance

(

x,x0

), between the central nodes

(wild-types) and their mutants

. Each wild type is labeled in red

and its number of accessible mutants

is also displayed. Let us remark that this enumeration also depends on the fact that

in our model the sequences are circular (i.e. periodic boundary conditions: xN+i=xi,∀i∈ {1, . . . , N}).

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Table 1. Enumeration of accessible-mutants. Number of neighboring mutants accessible via inversion mutations DI

and via point mutations

for all genomes of size

N∈

[[2

10]] (subscripts numbers denote the number of occurrence of each

value of DIand DP).

Genome size Number of accessible-mutants Number of accessible-mutants

Nvia inversions DIvia point mutations DP

2 22,3224

3 56,7238

4 74,810,132(See Fig 1) 416

5 1320,1710,212532

6 1630,1718,182,2212,312664

7 2570,2942,3714,4327128

8 2816,2952,30112,322,348,3648,4616,5728256

9 396,4018,41234,45162,5236,5336,6418,7329512

10 45100,46150,47420,502,5240,53200,6240,6350,7720,912101024

multigraph of mutations m(

)

,∀x∈ X

(see the mathematical definition in [50, p.8]), i.e.

the ordered triple

m=(V(m), E(m), Im),

where V(m) = x∪ Nν(x) is the set of vertices (formed by a given genotype xand its

mutated genotypes {x’}∈X), E(m) is the set of directed edges (from genotype xto a

mutated genotype

) and

X −→ X

is an incidence relation that associates to each

element of

(m) an ordered pair of

(m). In fact, the incidence relation

corresponds

to a mutation operation (see Methods, Eqs. (3), (4), and (5)). As an example, in Fig 1

we display the atlas of accessible-mutants for N= 4, constructed by calculating all

inversion mutations for each one of the 24(wild-type) sequences (central red dots). In

this example (see also Table 1), it is verified that the number of accessible-mutants for

inversion mutations

(

)

,∀x∈ {

is in the set

{

}

(unlike the case for point

mutations, which must be a singleton, i.e. the single-value set:

DP(x)∈ {4},∀x∈ {0,1}4). We can also verify that min(DI(x)) ≥max(DP(x)). In

Table 1we show the enumeration of accessible-mutants for inversions and point

mutations for genome sizes ranging from

= 2 to 10. From Fig 1and Table 1, we can

see that the combinatorics of the inversion mutations is not trivial. We can verify that

the maximum number of accessible mutants is equal to

N2−N

+ 1, which corresponds

to the trivial cases of genotypes xwith xi= 0,∀i∈ {0,...N}and

= 1

,∀i∈ {

,...N}

. Note that for a circular sequence of size

, the total number of

inversion mutations is N2, while for point mutations this number is equal to N.

However, the number of mutants accessible by inversions is lower than the total number

of inversions mutations (DI< N2). This is due to “degenerate” inversion mutations:

several inversions—occurring between different loci and/or for different interval

sizes—may mutate the initial sequence to the same accessible mutant (see the multiple

edges in Fig 1). In Fig 1, we can also verify that there are loops (an “edge” joining a

vertex to itself), that is, “invariant inversions” that preserves the nucleotide sequence

after the inversion operation (i.e. dH(x,x0) = 0). It can easily be shown that the

fraction of invariant inversions converges to 1/N (see S1 Text, section 1). A very

important consequence of inversions is that mutated sequences can differ with the wild

type by more than one nucleotide, i.e.

(

x,x0

)

1 (edges colours in Fig 1denote the

values of the Hamming distance). This result allows us to gain a first insight of how

inversions can promote the escape from local fitness peaks: they can “connect”, in a

single mutational event, genotypes that are at two or more point-mutational steps away.

It is pertinent to remark that the combinatorics of inversions for alphabets with size

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Gettinghigheronruggedlandscapes:InversionmutationsopenaccesstotteradaptivepeaksinNKtnesslandscapesLeonardoTrujillo1,*,PaulBanse1,GuillaumeBeslon11UniversitedeLyon,INSA-Lyon,INRIA,CNRS,UniversiteClaudeBernardLyon1,ECL,UniversiteLumiereLyon2,LIRISUMR5205,Lyon,France*leonardo.trujillo@inria.frAbs...

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Getting higher on rugged landscapes Inversion mutations open access to tter adaptive peaks in NK tness landscapes

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