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Theories of Lethal Mutagenesis: From
Error Catastrophe to Lethal Defection
Héctor Tejero, Francisco Montero and Juan Carlos Nuño
Abstract RNA viruses get extinct in a process called lethal mutagenesis when
subjected to an increase in their mutation rate, for instance, by the action of muta-
genic drugs. Several approaches have been proposed to understand this phenome-
non. The extinction of RNA viruses by increased mutational pressure was inspired
by the concept of the error threshold. The now classic quasispecies model predicts
the existence of a limit to the mutation rate beyond which the genetic information of
the wild type could not be efficiently transmitted to the next generation. This limit
was called the error threshold, and for mutation rates larger than this threshold, the
quasispecies was said to enter into error catastrophe. This transition has been
assumed to foster the extinction of the whole population. Alternative explanations of
lethal mutagenesis have been proposed recently. In the first place, a distinction is
made between the error threshold and the extinction threshold, the mutation rate
beyond which a population gets extinct. Extinction is explained from the effect the
mutation rate has, throughout the mutational load, on the reproductive ability of the
whole population. Secondly, lethal defection takes also into account the effect of
interactions within mutant spectra, which have been shown to be determinant for the
understanding the extinction of RNA virus due to an augmented mutational pressure.
Nonetheless, some relevant issues concerning lethal mutagenesis are not completely
understood yet, as so survival of the flattest, i.e. the development of resistance to
lethal mutagenesis by evolving towards mutationally more robust regions of
sequence space, or sublethal mutagenesis, i.e., the increase of the mutation rate
below the extinction threshold which may boost the adaptability of RNA virus,
H. Tejero (&)F. Montero
Department of Biochemistry and Molecular Biology I, Universidad Complutense
de Madrid, 28040 Madrid, Spain
e-mail: hector.tejero.81@gmail.com
J.C. Nuño
Department of Applied Mathematics, Universidad Politécnica de Madrid,
28040 Madrid, Spain
H. Tejero
Translational Bioinformatics Unit, Centro Nacional de Investigaciones Oncológicas,
28029 Madrid, Spain
©Springer International Publishing Switzerland 2015
Current Topics in Microbiology and Immunology
DOI 10.1007/82_2015_463
increasing their ability to develop resistance to drugs (including mutagens). A better
design of antiviral therapies will still require an improvement of our knowledge
about lethal mutagenesis.
Contents
1 Introduction..............................................................................................................................
2 The Error Threshold and the Error Catastrophe .....................................................................
2.1 The Limits of the Error Catastrophe ..............................................................................
3 Mutation–Selection-Based Models ..........................................................................................
3.1 Lethality, Extinction and Error Catastrophe...................................................................
4 Lethal Defection ......................................................................................................................
4.1 Stochastic Extinction Model...........................................................................................
4.2 Interference, Complementation and Lethal Defection....................................................
5 Problems of Lethal Mutagenesis .............................................................................................
5.1 Resistance and Survival of Flattest ................................................................................
5.2 Sublethal Mutagenesis ....................................................................................................
6 Concluding Remarks ...............................................................................................................
References ......................................................................................................................................
1 Introduction
Lethal mutagenesis is taken to be the extinction of a micro-organism by accumu-
lation of mutations due to the treatment with mutagenic drugs. This phenomenon
has been verified empirically in a large number of RNA viruses (see Chap. 14), and
consequently, its viability and potential application as an antiviral therapy is beyond
doubt. However, in recent years, there has been a growing debate over how this
phenomenon is produced, and which theoretical model can best explain it. The aim
of this article is to review and compare the different models which have been put
forward to explain this phenomenon.
2 The Error Threshold and the Error Catastrophe
In the classic quasispecies model, the error threshold is the mutation rate beyond
which the master phenotype, which is the phenotype with the highest replicative
capacity, is extinguished (Chap. 1). In this way, the population becomes exclusively
composed of mutant phenotypes, i.e. all the phenotypes that differ from the master
(Eigen 1971). Simultaneously to the disappearance of the master phenotype, the
population delocalises over the sequence space which, in finite populations, can be
interpreted as the quasispecies beginning to drift over the whole sequence space.
H. Tejero et al.
Given that this phenomenon takes place as a phase transition, with respect to the
mutation rate, it is said that beyond the error threshold, the population enters into
error catastrophe.
The definition and phenomenology of the error threshold and the entry into error
catastrophe have not been free from controversy (Tejero et al. 2011). The problem
largely arises from the nature of the models that were originally studied, mainly in a
single-peak landscape in the absence of back mutations from the mutant phenotype
to the master phenotype, factors on which it is extremely dependent (Hermisson
et al. 2002; Schuster 2010). The differences that are observed for the phenome-
nology and definition of error catastrophe, and error threshold, can also be observed
in the interpretation of both concepts (Tejero et al. 2011).
Both the loss of the master phenotype and the delocalisation of the quasispecies
over the sequence space have been related to an information “crisis”or “loss”, since
the population is unable to maintain the information contained in the master phe-
notype (Biebricher and Eigen 2005; Eigen 2002; Eigen and Schuster 1979). Since
the mutation rate from the master to the mutant phenotype is determined by the size
of the sequences that compose them, the entry into error catastrophe establishes a
maximum limit to the amount of information that a self-replicative system can
maintain at a given mutation rate (Wilke 2005; Takeuchi et al. 2005; Obermayer
and Frey 2010). After the RNA viruses have been conceptualised as quasispecies
(Domingo et al. 1978), the possibility of pushing RNA viruses into error catas-
trophe by means of mutagenic drugs (Eigen 1993,2002) was the origin of the first
lethal mutagenesis experiments, as well as the first explanation for the extinction of
the virus in these conditions (Cameron and Castro 2001; Holland et al. 1990; Loeb
et al. 1999).
2.1 The Limits of the Error Catastrophe
Although the concept of error catastrophe was the first explanation for lethal
mutagenesis, several objections were later raised to this explanation. In the classic
quasispecies model, all the mutants have the capacity to reproduce themselves, that
is to say there are no lethal mutants. Some papers have shown that if all the
mutations were lethal, and therefore all the mutants were unable to reproduce
themselves, error catastrophe could not happen (Summers and Litwin 2006; Bull
et al. 2005). Earlier Wagner and Krall (1993) had shown, using a different model,
that error catastrophe could not take place in the presence of lethal genotypes, a
result which was confirmed by Wilke (2005). Subsequently, his results were jus-
tifiably criticised by Takeuchi and Hogeweg (2007), as they showed that the model
used by Wagner and Krall contained a series of very limiting restrictions and that it
was these restrictions which were responsible for the results obtained. To consider
that all mutations are lethal, or to consider that they are all absent, are extreme,
highly unlikely situations. When intermediate situations are studied, it can be seen
that the presence of lethality does not prevent error catastrophe from taking place,
Theories of Lethal Mutagenesis: From Error …
but it does cause it to happen at much higher mutation rates, that is to say it
increases the error threshold (Sardanyes et al. 2014; Bonnaz and Koch 1998;
Takeuchi and Hogeweg 2007; Tejero et al. 2010). This result appears to be inde-
pendent of the distribution of lethality over the sequence space. Of course, from the
point of view that entry into error catastrophe is the mechanism that underlies lethal
mutagenesis, this result may appear to be counter-intuitive, since we would expect
an increase in the proportion of lethal mutations to facilitate extinction caused by
accumulation of mutations, and not the reverse. What was required was a clarifi-
cation of the relation between the error threshold and the extinction threshold in
quasispecies models since, strangely enough, this relation had not yet been defined,
something which can be explained by the fact that the quasispecies models had
been formulated under a constant population restriction.
Wilke was the first to correctly point out that most of the models for quasispecies
and entry into error catastrophe considered constant population conditions which,
consequently, by definition, ruled out the possibility of extinction (Wilke 2005). At
around the same time, Bull et al. (2005) explicitly showed the need to differentiate
between extinction, which involves the disappearance of the whole population, and
entry into error catastrophe, in which it is the master phenotype that disappears. In
other words, it is important to distinguish between the error threshold, namely the
mutation rate at which the population enters into error catastrophe, and the
extinction threshold, namely the mutation rate at which the population becomes
extinct. Thus, both papers not only ruled out the possibility that lethal mutagenesis
is caused by the entry of the quasispecies into error catastrophe, but they also
suggested that the error catastrophe could in fact hinder or delay the extinction of a
virus by lethal mutagenesis. The reason for this is that error catastrophe, ultimately,
is caused by what is known as the “survival of the flattest”or, in other words, the
dominance, at high mutation rates, of phenotypes with a lower replicative capacity
but a higher tolerance to mutations (Cowperthwaite et al. 2008; Tejero et al. 2011;
Bull et al. 2005,2007). Specifically, Tejero et al. showed that the error threshold
can be reformulated in terms of a selection coefficient that depends on the mutation
rate and which is as follows:
sQ
m
ðÞ¼
Ak
AmQm
1ð1Þ
where Akand Amare the replicative capacities of the mutant and master phenotypes,
respectively, and Qmis the quality factor, namely the probability that the master
sequence is copied correctly. This reformulation shows that the error threshold is
the value of the quality factor, Qm, for which the selection coefficient, s, becomes
zero. In other words, the selection coefficient is positive for mutation rates higher
than the error threshold and, therefore, the mutant phenotype has a selective
advantage over the master phenotype. To put it another way, before the error
threshold natural selection favours the master phenotype, but beyond it natural
selection favours the mutant phenotype.
H. Tejero et al.
The consideration that entry into error catastrophe is caused by the survival of
the flattest phenotypes provides an explanation as to why lethality displaces the
error threshold towards lower mutation rates. In the aforementioned papers, the
introduction of lethality affected only the mutant phenotype, decreasing its muta-
tional robustness. Because of this, the mutant phenotype becomes less competitive
and, therefore, higher mutation rates are required in order to allow the mutant
phenotype to dominate the master phenotype due to the survival of the flattest.
Furthermore, the supposed absence of error catastrophe in the presence of lethality,
together with the fact that it was considered to be a phenomenon different from
extinction, led to the positing of new theoretical models which would be able to
explain lethal mutagenesis.
3 Mutation–Selection-Based Models
The first model which tried to explain lethal mutagenesis in RNA viruses without
explicitly taking error catastrophe into account was formulated by Bull et al. (2007).
It explored in-depth the idea that it is necessary to differentiate between demo-
graphic processes, such as extinction, and genetic–evolutionary processes, such as
error catastrophe. The mutation–selection equilibrium, which is the result of the
interaction between the relative biological fitness of the alleles in the population and
their mutation rates, determines composition and, more importantly in this case,
average biological fitness at a stationary state. An increase in the mutation rate
displaces the mutation–selection equilibrium towards a greater diversity in the
population and, assuming that all the mutations are deleterious, towards a lower
average biological fitness. Under a series of conditions, basically the absence of
density or frequency-dependent selection, the population´s reproductive capacity
can be taken to be the product of the absolute biological fitness of the master
species expressed in Wrightian terms, R, and the average biological fitness,
wðUÞ.
When this reproductive capacity falls below 1, the populations start to be extin-
guished. Consequently, the condition required in order for a population to become
extinct can be described as:
R
wðUÞ\1ð2Þ
The extinction threshold would be the mutation rate at which this condition is
complied with. In order to determine this threshold, it is necessary to know the
dependence of the average biological fitness,
wðUÞ, with respect to the genomic
mutation rate U, and to do this, it is necessary to define both the landscape of
biological fitness in which the population evolves, and a series of assumptions
about phenotypic mutation frequencies.
In Bull et al. (2007,2008), a multiplicative fitness landscape was considered. As
neither back mutations nor compensatory mutations were assumed, under these
conditions, it was confirmed that the average biological fitness at a stationary state
Theories of Lethal Mutagenesis: From Error …
is solely dependent on the genomic mutation rate U, according to
w¼eU(Kimura
and Maruyama 1966). In this way, it is possible to determine an extinction
threshold as follows:
Uex ¼log Rð3Þ
where U
ex
is the extinction threshold, and Ris the absolute Wrightian fitness.
It should be noted that, according to this model, the extinction threshold does not
depend on the deleterious effects of the mutations. That is, ultimately, and without
taking finite size effects into account, a virus in which all the mutations are lethal
(s= 1) will have the same extinction threshold as a virus in which all the mutations
are only slightly deleterious (s≈0). Despite this, although the extinction threshold
does not depend on the deleterious effect of the mutations in this model, the time
that the virus takes to reach the mutation–selection equilibrium does depend on it
(Bull et al. 2013). Thus, the greater the deleterious effect of the mutations, the faster
the average biological fitness of the population decreases, and the quicker it starts to
become extinct. Naturally, as will be commented below, the fact that the extinction
threshold isindependent of the deleterious effect of mutations comes from not
con-sidering compensatory or back mutations. Shortly afterwards, Chen and
Shaknovich (2009) studied a model in which the fitness landscape was based on an
experimental distribution of the effect that mutations have on the thermal stability of
proteins. It was subsequently verified that this fitness landscape is compatible with
the distributions of the mutational effects obtained experimentally in RNA viruses
(Wylie and Shakhnovich 2011). In this way, and as oppose to the multiplicative
landscape in the absence of back mutations, this fitness landscape permits beneficial
and compensatory mutations. The stability of a protein determines the percentage of
time that the protein is folded. Ultimately, the biological fitness of a virus is the
product of this value for a series of genes. Due to the complexity of the model used,
it is not possible to obtain an explicit expression of the extinction threshold.
However, under conditions of conservative replication, similar to those of RNA
viruses, the error threshold obtained is approximately lineal (and not exponential)
with respect to absolute fitness. Furthermore, in this paper, the differences between
conservative and semiconservative replication are compared, showing that the
critical mutation rate is less in the latter. The same phenomenon can be observed
when considering semiconservative replication in a multiplicative model where
there are only deleterious mutations (Bull and Wilke 2008). In this case, the
extinction threshold is defined as:
Ud[lnð2Þln 1 þ1
R
ð4Þ
This, as it can be seen, establishes a maximum limit on the critical rate of delete-
rious mutations U
d
= ln(2) ≈0.69. This dependence should also appear when
studying other possible replicative modes, since it is known that these affect the
mutational load (Sardanyés et al. 2009).
H. Tejero et al.
The same model was used to show that the lower the population size, the lower
the mutational rate needed to extinguish the population (Wylie and Shakhnovich
2012), as was to be expected.
Lastly, Martin and Gandon (2010) used a fitness landscape based on a multi-
variate Gaussian function. Although this landscape represents a single optimum
phenotype, genetically it is very rugged, and one in which epistasis and compen-
satory mutations occur. Finally, the idea that a given percentage of the mutations
are lethal was considered. In this paper, the infective dynamic of the virus in the
population using a SIR model was also examined (Nowak and May 2000). By
doing this, the reproductive capacity depends on the demographic conditions: the
lower the amount of virus, the higher the amount of susceptible cells and, therefore,
the greater its reproductive capacity. However, this effect is probably offset by an
increase in mutational meltdown phenomena due to finite populations. Furthermore,
Wylie and Shaknovich have shown that, in their fitness landscape, the percentage of
lethal mutations clearly increases when the population size decreases, which could
help to offset the demographic effect. The chief finding of this paper is the deri-
vation of a complex expression of the extinction threshold, which depends on the
mutational effect and the growth rate of the optimum phenotype.
When the deleterious effect of the mutations is due, above all, to lethal muta-
tions, the critical mutation rate depends lineally on the ratio between the Malthusian
fitness (r0) and the fraction of lethal mutants (pL). This can be expressed as:
Uex r0
pL
ð5Þ
Taking into account that the Malthusian fitness is the logarithm of the Wrightian
fitness (Wu et al. 2013), the expression can be reduced to an equivalence of (3)
UexpLlog RðÞ ð6Þ
in which the product UexpLis the rate of lethal mutations by genome. However,
when the deleterious mutations are chiefly non-lethal, the extinction threshold
depends on the quotient of the square of the absolute biological fitness and a factor
that is proportional to the deleterious effect of the mutations. These results confirm
what has been discussed above: if the compensatory mutations are only of small
importance (in this case due to the greater importance of the lethal mutations), the
extinction threshold’s dependence on the effect of the mutations is extremely low.
Otherwise, the higher the deleterious effect of the mutations, the lower the mutation
rate needed to extinguish the virus.
These three models are used to study different biological fitness landscapes,
replication modes and infection dynamics. Figure 1shows the effect of the different
biological fitness landscapes on the extinction threshold. As has been commented,
the extinction threshold obtained by Bull et al. (2007) is a lower limit due to the
absence of compensatory mutations. The lower is the cost of mutations in Martin
and Gandon (2010), the higher is the extinction threshold. Finally, the biophysical
Theories of Lethal Mutagenesis: From Error …
fitness landscape proposed by Chen and Shakhnovich (2009) determines an
extinction threshold similar to that in which the average effect of mutations is not
very high. Thus, despite these quantitative differences, the fundamental idea is the
same in all the models: extinction by lethal mutagenesis is produced because an
increase in the mutation rate displaces the mutation–selection equilibrium until the
reproductive capacity of the population falls below a threshold beyond which it
cannot maintain itself over time. Within this general framework, the differences
between the three models, especially with respect to the biological fitness land-
scapes, give rise to three different predictions for the extinction threshold.
1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
R
Uex
Bull et al, 2007, s =0.25
Chen & Shaknovich, 2009
M & G, 2010. pL=0.2,s=0.1
M & G, 2010. pL=0.2,s=0.8
M & G, 2010. pL=0.7,s=0.1
Fig. 1 Extinction threshold, U
ex
, as a function of the Wrightian fitness, R. Results of simulating a
simple stochastic model using a Gillespie algorithm (1977) with the three different fitness
landscapes presented in the main text. Symbols in the upper box describe the three models used in
the simulations; references are given in the text. Each point is obtained from 50 independent
repetitions. The extinction threshold was taken as the mutation rate beyond which the populations
got extinct in all simulations before a given final time. Therefore, this estimation must be
considered an upper bound. The maximum population allowed in all the simulations was 1000
individuals. No cell infection dynamics was considered in Martin and Gandon (2010) model (M
&G). In this model, p
L
is the fraction of lethal mutations and
sis the average selection coefficient
of the mutations. In the (Bull et al. 2007) model, sis the selection coefficient. Simulations were
carried out using Matlab®. Code is available at request
H. Tejero et al.
3.1 Lethality, Extinction and Error Catastrophe
According to some authors, an increase in the mutation rate can extinguish a
population either by an excess of non-viable mutants or by an accumulation of
errors “without limits”(Schuster 2011), in other words, by entry into error catas-
trophe. A similar idea had previously been proposed in Biebricher and Eigen
(2006). Consequently, this leads to consider the relationship between entry into
error catastrophe and extinction in the presence of lethality. When a quasispecies
model is considered in a single-peak fitness landscape in the absence of lethality,
extinction and entry into error catastrophe are mutually exclusive events (Bull et al.
2007,2008). When the mutation rate increases, the population either becomes
extinct or enters into error catastrophe, depending on whether or not the mutant
phenotype is stable demographically, that is to say, whether it can self-maintain.
Not only does this mean that entry into error catastrophe cannot explain population
extinction due to an accumulation of mutations, but also that quite the reverse is
true, in that only entry into error catastrophe can prevent the population from
becoming extinct, as was mentioned earlier (Bull et al. 2007). The introduction of a
fraction of lethal mutants 1 −p, uniformly distributed over the sequence space, does
not change this situation. The population either becomes extinct or enters into error
catastrophe, depending on whether or not the mutant phenotype is stable demo-
graphically (Tejero et al. 2010) which, in this case, depends on the value of the
fraction of lethal mutants.
This situation changes if we study a lethality distribution in which the
self-replicative species has n“lethal positions”, i.e. positions whose mutation
generates lethal mutants. In this case, entry into error catastrophe and extinction are
no longer mutually exclusive, since the lethality is not distributed uniformly. If the
number of lethal positions in the sequence is low, the mutant phenotype can
dominate the master phenotype when the mutation rate exceeds the error threshold.
If the mutation rate continues to increase, the population—which would be in “error
catastrophe”—will accumulate lethal mutants until its average productivity is less
than zero and it becomes extinct. If, on the other hand, there are a high number of
lethal positions, the population becomes extinct before entering error catastrophe
(see Fig. 2). In this way, it is possible to define a “critical lethality”, beyond which
the error threshold disappears (Tejero et al. 2010). However, the mutation rate
needed for the population to become extinct beyond the error threshold is always
higher than the rate which occurs at high lethality rates. This means that the entry
into “error catastrophe”hinders population extinction by lethal mutagenesis, as has
been mentioned earlier (Bull et al. 2007) which, in turn, is a consequence of the fact
that the dominance of the mutant phenotype over the master one is due to its greater
mutational robustness. Lastly, it is important to highlight that, both before and after
crossing the error threshold, extinction occurs because the average replicative
capacity is lower than the degradation rate of the population. In this sense, we think
that it is not possible to talk of two types of extinction, one associated with the entry
into error catastrophe and the other independent of error catastrophe and, in fact,
Theories of Lethal Mutagenesis: From Error …
Fig. 2shows a large region where the population is in a situation of error catas-
trophe but does not become extinct.
4 Lethal Defection
In conjunction with the models presented in the previous section, which are based
solely on the effect of the mutation–selection equilibrium on the demographic
behaviour of a quasispecies, there is also an alternative model in which the par-
ticipation of a subpopulation of mutant viruses, known as defectors, is considered
an essential factor in the extinction of a population. This section provides a brief
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
1
2
3
4
5
6
fn, fraction of lethal positions
U
Uc
Uex
Fig. 2 Extinction and error thresholds as a function of the fraction of lethal mutations. Ucand Uex
are the error and extinction thresholds expressed in terms of the genomic mutation rate,
U¼m1qðÞ, where mis the genomic length and qthe per digit quality factor. The Wrightian
fitness of the master phenotype was Rm¼10 and that of the mutant phenotype was Rk¼3. The
figure can be interpreted as a phase diagram. Below the error and the extinction thresholds, the
population stays in a mutation–selection equilibrium. Above the extinction threshold, the
population gets extinct. Above the error threshold, the population survives but enters into error
catastrophe. In this case, critical lethality occurs at f
n
= 0.5, so no error threshold appears for
f
n
> 0.5. See main text for further details
H. Tejero et al.
introduction to the experimental basis of this hypothesis and the theoretical model
which explains it.
One of the characteristics of viral quasispecies is the suppression of variants with
high biological fitness by the spectrum of mutants which accompany them (de la
Torre and Holland 1990; Domingo et al. 2006,2012). This phenomenon partly
motivated the study of the interfering capacity of pre-extinction genomes in FMDV
(Gonzalez-Lopez et al. 2004). Subsequent work with LCMV indicated that viral
infectivity was lost much earlier than replicative capacity during the extinction by
mutagenesis (Grande-Pérez et al. 2005) (see also Chap. 10). On the basis of these
results, the hypothesis was made that the mutagenesis of a RNA virus would result
in a subgroup of interfering mutant viruses, known as defectors, which would play a
crucial role in the extinction of the virus. This hypothesis was called lethal defection
(Grande-Pérez et al. 2005). Later experiments have confirmed the important role
that defective and defector mutants play in the extinction of an RNA virus by
increased mutation rate (see Chap. 14).
4.1 Stochastic Extinction Model
The origin of the lethal defection hypothesis is closely connected to the develop-
ment of computational models and in silico studies (Grande-Pérez et al. 2005). The
first model which was proposed to explain extinction by lethal defection was the
stochastic extinction model (Iranzo and Manrubia 2009). Basically, the lethal
defection model of Manrubia and Iranzo considered that viruses code two pheno-
typic traits: replicative capacity and infectivity, the latter being associated with the
capacity to code the proteins needed for replication. The first trait acts solely in cis,
while the second acts in trans. In virology, the terms interaction in cis or in trans are
used to describe whether the action of a genetically coded element (a genomic
sequence, secondary or tertiary RNA structure, protein, etc.) takes place with the
genome that codes it, in which case it is called interaction in cis, or whether it takes
place with other genomes, in which case it is an interaction in trans (see also
Chap. 14). During replication, the viruses accumulate mutations that may affect
both their replicative capacity and their infectivity. These mutations may be ben-
eficial or deleterious. However, the effect of the mutations on both traits is coupled,
since the model considers that the loss of infectivity by mutations means that the
mutations simultaneously decrease their replicative capacity and vice versa, and the
restoration of the infective capacity means that their replicative capacity increases.
The viruses that code functional proteins are called viable, while those that have
undergone mutations are called defective. In this way, the population’s reproductive
capacity is proportional to the number of viable viruses, and when these disappear,
the population becomes extinct.
This model predicts that when a mutation rate is high enough, the population
becomes extinct regardless of the presence of defectives in the population and at the
same mutation rate value. This value is equivalent to the deterministic extinction
Theories of Lethal Mutagenesis: From Error …
threshold. However, when the population size is small enough and the mutation rate
is relatively low, the behaviour is qualitatively different: if there are no defectives,
the population continues to exist over time, but if they are present, the population
eventually becomes extinct, since the defective mutants displace the viable viruses
(Grande-Pérez et al. 2005). This extinction is known as stochastic extinction (Iranzo
and Manrubia 2009). Ultimately, what occurs is the fixation of the defective
mutants as a consequence of the genetic drift effect and, therefore, the smaller the
population size, the higher the likelihood of its occurrence.
Although the model of lethal defection by stochastic extinction can explain the
extinction of a population by accumulation of defective viruses, it has several limi-
tations. Firstly, the model is very dependent on the coupling of the effect of the
mutations on the two phenotypic traits and the defective viruses not having a higher
replicative capacity that the viable ones. If either of these assumptions are not com-
plied with, the defectives will get fixed, extinguishing the population, at any mutation
rate. Secondly, as the authors themselves acknowledge (Manrubia et al. 2010), this
model does not explain the action of lethal defection in lytic infections, even though a
large part of the experimental phenomenology on which the lethal defection
hypothesis is based comes from lytic viruses (Gonzalez-Lopez et al. 2004; Perales
et al. 2007). Furthermore, the model only explains extinction when the population
sizes are relatively small (Iranzo and Manrubia 2009). In this regard, although the
population sizes of RNA viruses can be extremely large in lytic and persistent
infections, both inside and outside the cells, it is also true that not all viral genomes
inside a cell are replicating, and this could significantly decrease the effective size of
the population, thus facilitating stochastic extinction. (See also Chaps. 10 and 14 for
discussion of the lethal defection model in connection with experimental results.)
4.2 Interference, Complementation and Lethal Defection
Interference and complementation interactions in RNA virus quasispecies are a
phenomenon whose importance is increasingly recognised (Shirogane et al. 2013;
Manrubia 2012; Perales et al. 2012) (Chap. 10). In this regard, the effect of
defectors or defective mutants on lethal mutagenesis can also be explained by the
complementation–interference interactions they establish with the rest of the virus.
It must be recalled that, in most of the cases in which a high-fitness virus interacts in
trans with a mutant virus with lower fitness, the fitness of the former decreases,
whereas that of the mutant increases. The fitness loss of the higher fitness variant is
called interference. Complementation is, on the other hand, the gain of fitness by the
mutant virus. Thus, interference and complementation can be seen as two sides of
the same interaction. In virology, complementation can also refer to the process by
which two low-fitness viruses complement each other to increase their fitness. This
case will not be considered here.
Several authors have studied, theoretically and computationally, the effect of com-
plementation on the mutation–selection equilibrium of RNA viruses (Froissart et al.
H. Tejero et al.
2004;WilkeandNovella2003;GaoandFeldman2009; Novella et al. 2004). The main
result of all these papers is that complementation displaces the mutation–selection
equilibrium to higher frequencies of the mutant allele. Essentially, what happens is that
complementation, by permitting the mutant allele to partially take advantage of the
biological fitness of the wild-type allele, decreases the selective disadvantage of this
allele with respect to the wild-type one, which in turn decreases the effect of natural
selection. This also explains why, in a quasispecies, complementation makes error
threshold occur at lower mutation rates (SardanyésandElena2010).
In Moreno et al. (2012), a computational model based on interference–com-
plementation interactions was developed to find out to what extent lethal defection
can explain the effect of the initial MOI on lethal mutagenesis in certain viruses.
Besides considering various phenotypic traits, the model was characterised by its
consideration of a partial trans interaction and for coupling, in a nested way, the
intra- and extracellular dynamics of RNA virus infections. This study also showed
how the presence of trans interactions increases the percentage of inhibition of the
virus titre, caused by an increase in the mutation rate. A simplified version of this
model shows that the interfering action of the defective viruses causes the popu-
lation to become extinct at lower mutation rates, and that this effect depends on the
degree of trans interaction in the quasispecies (Tejero 2013).
The main criticism of a lethal defection model based on interference–comple-
mentation interactions is based on the fact that when population size decreases
during the extinction process, interactions in the quasispecies become increasingly
infrequent. Ultimately, this means that although the viral load may be lower in the
presence of interference at intermediate mutation rates, the population’s extinction
threshold does not depend on lethal defection (Steinmeyer and Wilke 2009).
However, this criticism does not take into account the fact that when the extra-
cellular dynamic is considered, even when a single virus infects a cell, interference
may be produced during the intracellular replication process among the progeny of
the virus. The importance of this phenomenon will vary, depending on the repli-
cation mode of the virus (Sardanyés et al. 2009; Sardanyés and Elena 2011).
Despite its differences, and although the importance of lethal defection can
depend on the biological characteristics of the virus and of the infection, the two
models of lethal defection predict that it will always decrease both the extinction
and the error threshold.
5 Problems of Lethal Mutagenesis
5.1 Resistance and Survival of Flattest
Like any other antiviral therapy, lethal mutagenesis will face the development of
resistances. In this case, the resistances will be selected for their capacity to mitigate
the effect of the mutations on the fitness of the virus. In addition to the classic
Theories of Lethal Mutagenesis: From Error …
drug‐resistant mutants, which can recognise and prevent the incorporation of a
specific mutagen, and the fidelity mutants, i.e. viruses which have a lower muta-
tion rate both in the presence and absence of mutagen drugs (see the Chap. 13
by Marco Vignuzzi in this book), RNA viruses may develop a resistance to lethal
mutagenesis by evolving towards mutationally more robust regions of sequence
space. As has been discussed in Sect. 2.1, when discussing the possibility that entry
into error catastrophe hinders extinction by lethal mutagenesis, this mechanism is
known as the survival of the flattest. Although this phenomenon has been observed
experimentally (Codoñer et al. 2006; Sanjuánetal.2007; Graci et al. 2011), it has not
been proved that treatment with mutagens increases the mutational robustness of
LCMV (Martínetal.2008), nor has it been proved that a virus has escaped mutagenic
treatments by increasing its robustness. Moreover, the only theoretical study that has
addressed this point considers robustness unlikely to affect lethal mutagenesis, chiefly
because of the difference in the timescales of acquisition of mutational robustness and
increased mutagenesis (O’Dea et al. 2010).
5.2 Sublethal Mutagenesis
The second possible problem raised by mutagen therapies is “sublethal mutagen-
esis”or, to put it another way, an increase in the mutation rate which is not enough
to extinguish the virus. Taking into account the role of mutations in evolution, some
authors have suggested that sublethal doses of a mutagenic drug may increase the
adaptability of a virus, which could be counterproductive clinically (Pillai et al.
2008), as it facilitates escape from the immune system or the appearance of mutants
that are resistant to drugs, irrespective of whether they are mutagenic or
non-mutagenic. This possibility has been demonstrated experimentally in the case
of mutations that make the FMDV less sensitive to ribavirin. Thus, while it is
possible to obtain a ribavirin-resistant mutant when it is subjected to serial passages
in the presence of increasing concentrations of ribavirin (Sierra et al. 2007), when
faced with high concentrations of ribavirin from the start, the virus does not
dominate and the population becomes extinct (Perales et al. 2009). Another
example of this effect is the change in the result with respect to interactions with
other drugs, such as replication inhibitors (Iranzo et al. 2011). In this case, it can be
seen that the combined action of certain dosages of mutagens and inhibitors is
antagonistic—the mutagen diminishes the effect of the inhibitor—instead of syn-
ergistic due, mainly, to the increased probability that mutants resistant to the
inhibitor will appear. (This point is treated in Chap. 14.)
Finally, sublethal mutagenesis can also occur due to the appearance of resistant
mutants (either fidelity or drug-resistant ones) which convert lethal doses of
mutagen into sublethal doses, with the risks mentioned above. Drug resistances are
usually partial resistances which often allow the virus to gain biological fitness in
the presence of the drug thanks to compensatory mutations. The consequence of a
resistant or fidelity mutant appearing is that it will incorporate fewer mutations per
H. Tejero et al.
genome, which will displace the quasispecies to a region of sublethal mutagenesis,
with the additional problem, at least in theory, that in this region an increase in
adaptability will make the acquisition of compensatory mutations more likely.
6 Concluding Remarks
In this chapter, we have reviewed the main contributions of mathematical modelling
to the theory of lethal mutagenesis produced in the last decades. Since the pioneer
works of Manfred Eigen and Peter Schuster in the seventies of the twentieth cen-
tury, the modelling of molecular evolution has achieved a period of great success
that has allowed to disentangle reasonably the complex behaviour of populations of
error-prone replicators (e.g. RNA viruses) in terms of the mutation rate and the
fitness landscape. We find here one of the most appealing examples of how
mathematical modelling has fostered new questions and concepts, and conse-
quently, it has suggested new experiments. Besides, the mathematical formulation
of these biological problems has attracted the attention of scientists from other
fields, namely mathematics, physics and chemistry that have accelerated the pro-
gress of this theory more than ever before. This interdisciplinary interest has
allowed the application of similar models to study the evolution of some kind of
tumour lineages (Soléand Deisboeck 2004; Soléet al. 2014).
One of the central concepts of the quasispecies theory is the error threshold
which, as it has been extensively described in the previous sections, quantifies the
mutation rate below which the information codified in the master phenotype is
surely maintained in the next generations (see also Chaps. 1 and 5). On the contrary,
if the mutation rate is above this threshold, the presence of the wild phenotype in
the population is no longer assured. This mathematical result was immediately
borrowed by virologists as a possible therapy to drop the infectiveness of some
RNA viruses by using mutagenic drugs. The strategy seems to be evident: to
increase the mutation rate of the virus in order to cause the disappearance of the
wild-type (more infective) copy. Implicitly, it was assumed that the extinction of the
master was equivalent to the extinction of the whole population. Unfortunately,
reality is more complex than mathematical models portray. As some experiments
with mutagens presented the feasibility of lethal mutagenesis, the debate about the
real meaning of the error catastrophe and its role as a therapy against virus infec-
tions was taken up again. To shed some light in the discussion, new mathematical
models considered other factors that are essential for the extinction of the quasi-
species, concretely the extinction threshold and the phenotype interactions within
the quasispecies distribution. As this article has pointed out, a clear distinction
between the error and extinction thresholds is required if we want to design an
optimal therapy based on lethal mutagenesis. Furthermore, the existence of inter-
actions among phenotypes can modify both thresholds and the relation between
them and consequently the response of the whole population to a hypothetical
Theories of Lethal Mutagenesis: From Error …
increase of the mutation rate. These, among others, are open questions that have to
be answered necessarily by a close collaboration between theoretical and experi-
mental groups.
Acknowledgments This paper has been supported in part by Grants no. BFU2012-39816-C02-02
from MINECO (Spain). Héctor Tejero was supported by AP2006-01044, from MEC (Spain) and
Marie‐Curie Career Integration Grant (CIG). CIG334361. Guillaume Martin and Sylvain Gandon
kindly gave us the code necessary to simulate its model.
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