Departamento de Bioquímica y Biología Molecular I, Universidad Complutense de Madrid, Avd. Complutense s/n, 28040 Madrid, Spain

Abstract

Background

The quasispecies model is a general model of evolution that is generally applicable to replication up to high mutation rates. It predicts that at a sufficiently high mutation rate, quasispecies with higher mutational robustness can displace quasispecies with higher replicative capacity, a phenomenon called "survival of the flattest". In some fitness landscapes it also predicts the existence of a maximum mutation rate, called the error threshold, beyond which the quasispecies enters into error catastrophe, losing its genetic information. The aim of this paper is to study the relationship between survival of the flattest and the transition to error catastrophe, as well as the connection between these concepts and natural selection.

Results

By means of a very simplified model, we show that the transition to an error catastrophe corresponds to a value of zero for the selective coefficient of the mutant phenotype with respect to the master phenotype, indicating that transition to the error catastrophe is in this case similar to the selection of a more robust species. This correspondence has been confirmed by considering a single-peak landscape in which sequences are grouped with respect to their Hamming distant from the master sequence. When the robustness of a classe is changed by modification of its quality factor, the distribution of the population changes in accordance with the new value of the robustness, although an error catastrophe can be detected at the same values as in the general case. When two quasispecies of different robustness competes with one another, the entry of one of them into error catastrophe causes displacement of the other, because of the greater robustness of the former. Previous works are explicitly reinterpreted in the light of the results obtained in this paper.

Conclusions

The main conclusion of this paper is that the entry into error catastrophe is a specific case of survival of the flattest acting on phenotypes that differ in the trade-off between replicative ability and mutational robustness. In fact, entry into error catastrophe occurs when the mutant phenotype acquires a selective advantage over the master phenotype. As both entry into error catastrophe and survival of the flattest are caused by natural selection when mutation rate is increased, we propose differentiating between them by the level of selection at which natural selection acts. So we propose to consider the transition to error catastrophe as a phenomenon of intra-quasispecies selection, and survival of the flattest as a phenomenon of inter-quasispecies selection.

1. Background

The quasispecies model was developed by Eigen in 1971

Furthermore, both the loss of the master sequence and the delocalization of the quasispecies over the sequence space have been related to an information "crisis" or "meltdown"

Another implication of the quasispecies theory that we would like to emphasize is the "survival of the flattest" concept

Both transition to error catastrophe and survival of the flattest are related to the behaviour of quasispecies at high mutation rates

This brief review of recent theoretical papers on this subject shows that the interpretation and meaning of error catastrophe, and its consequences, is a contentious subject. In this paper we will show that what has been called an entry into error catastrophe should actually be considered to be as a specific case of survival of the flattest. Accordingly, the transition into error catastrophe is caused by natural selection when the selection pressure is an increment in mutation rate. For this reason, an error threshold can only take place in fitness landscapes with flat enough regions. Thus, entry into error catastrophe is caused neither by the dominance of mutation over selection nor because natural selection ceases to operate.

In section 2.1 we show throughout a very simplified model that the entry into error catastrophe is formally equivalent to the survival of the flattest. In section 2.2 we study how changes the population distribution beyond the error threshold when the mutational robustness of some Hamming classes is changed by modifying its quality factor. In section 2.3 the possible relationship between the entry into error catastrophe and the survival of the flattest is considered. Finally, these and other previous results are interpreted in the discussion by considering the transition to error catastrophe as a specific case of intra-quasispecies survival of the flattest.

2. Results

2.1. A simple model to show that entry into error catastrophe is caused by survival of the flattest

In order to show that the transition to error catastrophe is caused by the selection of a flatter phenotype at high mutation rates, the simplest quasispecies model that displays an error threshold can be used, applying some purely formal modifications. This model has a master phenotype, composed of a single sequence with an amplification factor A_{m}, and a mutant phenotype, which combines all the other sequences and has an amplification factor A_{k}. The value of the degradation factor, D, is assumed to be the same for all the sequences and, without loss of generality, is set to D = 0. Throughout this section, the classic notation in which q is the quality factor ^{ν}, ν being the sequence length. As the mutant phenotype is composed of all the mutant sequences except the master sequence, back mutation can be neglected as a first approximation. This means that the probability of a mutant sequence producing another mutant sequence is unity. Assuming that the dynamics takes place in a flux reactor

where ϕ is the output flux of information carriers. Assuming the constraint of a constant population, N, this results in:

where

The definition of the error threshold is not a trivial point, as has been discussed under the Introduction. Throughout this paper the error threshold will be considered to be the value of q at which a "fitness threshold" takes place. This kind of threshold can induce the loss of the master type, i.e. a "wild-type threshold", or the delocalization over the whole sequence space, i.e. a "degradation threshold" _{m }

The system of differential equations (1) can be expressed in terms of relative fitness and selection coefficients referred to the master phenotype, as usually done in population genetics. A coefficient of selection,

To find an explicit relationship between the selection coefficient and other parameters in this case, the system can be reformulated by making a variable change:

which is equivalent to dividing the equations of the system (1) by the effective fitness of the master phenotype, i.e. the product of its replicative ability and its mutational robustness: _{m }q^{ν }

As _{m }q^{ν }

The selection coefficient can be obtained from this system of differential equations,, and in this case it is a function of the quality factor and the sequence length:

Substituting s in equation (5) results in:

Thus, the selection coefficient is the quotient between the effective fitness of the mutant phenotype over that of the master phenotypes, minus one. Actually, the effective fitness of the mutant phenotype should be considered as

Keeping the constraint of constant population, N, the flux term is now:

Taking this into account, and inserting equation (3) into equation (6), it follows that the error threshold is the value of q for which s = 0. That is to say, for q < q_{c }the mutant phenotype has a selective advantage over the master phenotype, so s > 0 (Figure

Phenotype Fraction vs the selection coefficient (s)

**Phenotype Fraction vs the selection coefficient (s)**. The amplification factors of the master and mutant phenotype are A_{m }= 10 and A_{k }= 2, respectively. In both cases the degradation factor is D = 1, and the sequence length is ν = 20. The selection coefficient is evaluated as in equation 6.

2.2. Beyond the error threshold the population evolves to more mutational robust regions of the sequence space

In the previous section, entry into error catastrophe was shown to be the result of natural selection acting on the different effective fitness of the master and the mutant phenotypes. This can also be observed too by using the classical extended model

that can be linearized following

The matrix W is given by _{ij }determine the probability of obtaining any class i sequence from a class j sequence. The largest eigenvalue, λ, is the mean fitness of the population at the steady state, and its associated right eigenvector v is the population distribution in this state. As this problem is not analytically tractable, even for small sequence lengths, it was solved numerically using MATLAB^{®}.

Figure

Hamming class fraction vs the selection coefficient (s)

**Hamming class fraction vs the selection coefficient (s)**. The amplification factors of the master sequence, i.e. the zero Hamming class, is A_{m }= 10, and that of every other Hamming class H_{i }is A_{i }= 2. The degradation factor is the same for all the Hamming classes and is D = 1. The sequence length is ν = 20. The selection coefficient is evaluated following equation 6, that is to say considering the mutant phenotype as a whole, not taking into account that it is composed by Hamming classes.

Beyond the error threshold, the population is delocalized over the whole sequence space. This delocalization is due to the fact that all the sequences of the mutant phenotype have the same amplification factor and the same quality factor, so natural selection ceases to operate. However, this is an extreme restriction, and no general conclusion should be derived from it. When either the amplification factor or the robustness of some mutant Hamming classes is changed, the population distribution beyond the error threshold departs from the uniform distribution. However, the error threshold is not necessarily modified by these changes.

We will now study how the population distribution beyond the error threshold changes when the robustness of some Hamming classes is modified. As the number of sequences is an intrinsic property of the classification, and depends on the Hamming distance with respect to the master sequence, in order to change the robustness of each Hamming class, we assume that the quality factor _{i}, depends on its sequence, as previously done in

In a first approach, the quality factor of each sequence depends on its Hamming distance with respect to the master sequence, so the robustness of each Hamming class can be tuned. By using a truncated mutation landscape the quality factor depends on the Hamming distance in accordance with:

where q_{k }is the quality factor of the sequences of the Hamming class k, and K_{q }a constant we use to change the quality factor of the Hamming classes beyond an arbitrary threshold k_{c}. Figure _{c }= 10. For K_{q }= 1, then q2 = q1, so the quality factor is the same for all the sequences, i.e. the classical model. However, for K_{q} < 1, then q_{2 }> q_{1}, and the average Hamming distance of the population at error catastrophe increases with respect to the homogeneous case (K_{q }= 1), whereas for K_{q }>, 1 then q_{2 }< q_{1}, and the average Hamming distance at error catastrophe decreases with respect to the homogeneous case. However, for any K_{q}, the error threshold does not change as, at least for k_{c }= 10, the relative effective fitness of the mutant phenotype as a whole with respect to the master phenotype is not affected by the variation in the quality factor. Figure _{q }= 0.1 and K_{q }= 3 compared with K_{q }= 1. As we said before, the error threshold does not change. On the other hand, the population distribution at error catastrophe is displaced to greater or lesser Hamming distances, so the average Hamming distance beyond the error threshold increases with respect to the case q1 = q2 (Figure

Average Hamming distance at the error catastrophe when robustness distribution changes

**Average Hamming distance at the error catastrophe when robustness distribution changes**. The quality factor of each sequence depends on the Hamming distance with the master sequence k, according to _{k }_{c }_{k }_{q}_{c }_{c }_{q }is a parameter that modifies the quality factor and the robustness of the Hamming classes. (see text for details). The amplification and degradation factors of the master, and mutant sequences as well as the sequence length are the same as in figure 2.

Changes in population distribution beyond the error threshold for different values of Kq

**Changes in population distribution beyond the error threshold for different values of Kq**. Part A. Hamming class distribution beyond the error threshold for different values of Kq. All the distributions have been obtained for q = 0.8. The population for Kq = 1 is the uniform distribution obtained in classical error catastrophe (see text for details). For a greater or lesser value of Kq, the population is displaced to nearest or farther Hamming classes, respectively. Part B. Average Hamming distance as a function of 1-q, for different values of Kq. The changes in Kq does not modify the error threshold, but the average Hamming distance at the error catastrophe is modified as a consequence of the displacement of the population showed in part A of the figure.

2.3. Two competing quasispecies with different robustness

In previous sections we showed that the entry intro error catastrophe implies the selection of a flatter phenotype when the population confronts a higher mutational pressure. In this section we will consider the possible relationship between terror catastrophe within a 12 quasispecies and the survival of the flattest between quasispecies. To study this relationship a simplified model of two quasispecies, A and B, similar to those presented in _{Am}, A_{Ak}, A_{Bm }and A_{Bk}, in each of which the first subscript specifies the quasispecies and the second the phenotype. The degradation factor, D, is the same for all of them and equal to zero.

The _{i }is an arbitrary parameter inversely related to robustness _{Am }and a_{Ak}, respectively and, in a similar way, those of quasispecies B have a robustness inversely related to a_{Bm }and a_{Bk}.

Taking this into account, it is possible to obtain the following system of linear differential equations:

Assuming a constant population size N, the flux term φ_{0 }in these is given by:

Figure _{Ak }= 0; a_{Bk }= 0). In this figure three regimes, clearly differentiated by two transitions, can be distinguished. Quasispecies A is dominant in the first and the third regime, whereas quasispecies B is dominant in the second. Between these three regimes, two "survival of the flattest"-like transitions take place. The first one at q = 0.9719 and the second one at q = 0.9589 (Figure _{cA }= 0.9659 and q_{cB }= 0.9539 are obtained. Therefore, quasispecies A enters into error catastrophe at a value of q which lies between the values of q of the two "survival of the flattest"-like transitions. In fact, if quasispecies A can displace quasispecies B at q = 0.9589 (Figure _{cB}, but as the robustness of the mutant phenotypes of both quasispecies is the same (a_{Ak }= a_{Bk }= 0), the mutant phenotype of quasispecies A outcompetes that of quasispecies B as the amplification factor of the former is higher.

Population fraction for two competing quasispecies as a function of genotypic mutation rate

**Population fraction for two competing quasispecies as a function of genotypic mutation rate**. The figure shows the population fraction obtained from equations 12 at the steady state as a function of genotypic mutation rate. Colored lines show the frontiers between the three phases in which alternate quasispecies dominate. In phase I and III, quasispecies A is selected, whereas in phase II, quasispecies B is selected. The amplification factors of quasispecies A are A_{Am }= 10 and A_{Ak }= 6. Those of quasispecies B are A_{Bm }= 8 and A_{Bk }= 5. The value of a_{i}, which determines the effect of the genotypic mutation rate on the phenotypic mutation rate are a_{Am }= 15, a_{Ak }= 0, a_{Bm }= 7, and a_{Bk }= 0.

3. Discussion

3.1. The entry into error catastrophe is an specific case of survival of the flattest

The aim of this paper has been to show that the transition into error catastrophe is the result of natural selection acting on the differences in the trade-off between replicative ability and mutational robustness, also called effective fitness. Thus, entry into error catastrophe is a specific case of survival of the flattest. To show this more clearly, a minimal quasispecies model that displays an error threshold was reformulated in section 2.1 in terms of relative effective fitness and selection coefficients. Using this new formulation, we showed that the error threshold is the value of q for which the selection coefficient, s, equals zero (Figure

Equation 6 shows that the selection coefficient is the quotient between the effective fitness of the mutant and the master phenotype. As back mutation has been neglected, the probability of going from any mutant sequence to any other sequence of the mutant phenotype is unity, so it does not appear in the numerator.

In section 2.2 we showed with the classical extended model in which sequences are grouped in Hamming classes that the transitions characteristic of an error threshold can be observed when the selection coefficient obtained in the previous section equals zero (Figure

Beyond the error threshold, a quasispecies is delocalized over the sequence space in such a way as to produce a uniform population distribution. This is hardly surprising, as the mutant phenotype comprises all but one of the sequences of the sequence space, so it is effectively flat both with respect to the amplification factor and the quality factor. However, the mutational robustness of some mutant sequences can be modified by considering a quality factor that depends on Hamming distance. In this case, a uniform distribution is not obtained beyond the error threshold, and the population evolves in such a way that the regions of the mutant phenotype with a higher quality factor are more populated. That is to say, a population evolves to regions with a greater mutational robustness, as has been shown previously for other similar situations

3.2. Revisiting previous results

Error catastrophe and error thresholds have been extensively studied since they were first postulated in 1971

3.2.1. Revisiting the effect of lethality

We have previously studied the effect of lethality on the error threshold, and shown that an increase in lethality decreases the error threshold, i.e. the quasispecies enters into error catastrophe at greater mutation rates

The same paper studied the changes in the error catastrophe distribution caused by the introduction of lethality. Briefly, the introduction of lethality decreases the average Hamming distance at the error threshold, which subsequently increases linearly with the mutation rate. As a consequence, the uniform distribution is only obtained at q = 0.5. This result is analogous to the result obtained in section × of this paper, in which the effect of the quality factor that depends on the Hamming distance was considered. The introduction of the lethality scheme used in

3.2.2. Revisiting the effect of neutrality and canalization

Since error threshold is a selective transition to more robust phenotypes when the mutation rate is increased, we will study in this section how error threshold is affected by considering neutrality and/or canalization in the master and mutant phenotypes. For the sake of clarity, canalization is considered to be the existence of different genotypes that produce the same phenotype

As far as we aware, the first account of neutrality in the quasispecies theory was given by Eigen and co-workers

The introduction of canalization in the master phenotype through neutral networks led the introduction of the distinction between genotypic and phenotypic error thresholds

Finally, it is possible to define a selection coefficient analogous to the one obtained in section × when neutrality is introduced in the master phenotype.

When, following

which is equal to zero at the error threshold

Equation 16 shows that the effect of increasing the neutral network of the master phenotype is to increase its mutational robustness, and therefore the effective fitness of the master phenotype, at higher mutation rates. As a consequence the mutant phenotype outcompetes the master phenotype at a higher mutation rate than in the case with no neutrality, so error threshold decreases

3.2.3. Revisiting multiple error thresholds

Several papers have described the existence of multiple error thresholds, also known as error cascades, in complex fitness landscapes

From the point of view of the survival of the flattest and natural selection, the appearance of multiple error thresholds is just a consequence of the existence of multiple phenotypes with different trade-offs between replicative ability and mutational robustness

Finally, the existence of multiple error thresholds shows that delocalization over a given region of the sequence space is essentially dependent on the phenotype's degeneration. Therefore, delocalization can take place over the whole sequence space, or just over some limited regions corresponding to the neutral network of a given phenotype, as in the case of the so-called genotypic error catastrophe commented above.

3.3. Error catastrophe, survival of the flattest, and levels of selection

The main purpose of this paper has been to show that the so-called entry into error catastrophe is a specific form of survival of the flattest, that is to say: it is the consequence of natural selection acting in systems of self-replicating species at high mutation rates. As they are essentially equivalent phenomena, and thus have the same cause and features, we propose differentiating between them by considering that they refer to the two different levels of selection that appears in quasispecies models. The first selection level is that of individual self-replicating species, grouped in phenotypes, which, through selection and mutation, determine the population distribution of the quasispecies. The second is a higher level in which some quasispecies can compete with others, through their emerging biological fitness derived from the interaction of their components

Therefore, we propose to use the term "survival of the flattest" to refer to situations in which two quasispecies compete, and there is no mutational coupling between them. This is either because the number of mutations between them implies that the possibility of obtaining one from the other in a reasonable time is negligible

The study of the system of differential equations presented in section 2.4 shows that the changes produced by natural selection in the internal population structure of a quasispecies, i.e. entry into error catastrophe, can modify the result of natural selection on the immediately higher level of selection, that of competition between quasispecies. In fact, it can be said that entry into error catastrophe induces a survival of the flattest.

3.4. ¿Survival of the flattest vs. survival of the fittest?

At a metaphorical level, survival of the flattest and survival of the fittest are often used to denote two completely different concepts _{i}. Accordingly, mutational robustness may play a key role in determining the effect of natural selection at high mutation rate conditions _{i}, the product of the replicative ability and mutational robustness (the effective fitness) are compared, and the transition to error catastrophe arises naturally as a result of natural selection acting on the difference of fitness.

3.5. Information crisis? What information crisis?

As briefly commented in the introduction, error catastrophe has been related either to an information crisis

The so-called "information crisis" beyond the error threshold sometimes is associated to the loss of the master sequence

In the first place, as Eigen has shown, natural selection implies displacements in the 'information space', which can take the form of phase transitions

Finally, the loss of the master sequence obviously implies the loss of the information contained in it. However, any natural selection process implies changes in the informational characteristics of the population. Obviously, if the result of natural selection is the disappearance of a given phenotype of the population, that implies an information loss, but, at the same time, it implies an increment in biological fitness. Again, no critical consequence can be expected.

A similar reasoning can be applied to the idea of a "breakdown of evolutionary adaptation"

In the specific case of the single-peak fitness landscape, entry into error catastrophe would be the most extreme case of selection for mutational robustness within a quasispecies. On the one hand, there is a master phenotype which, as it is made up of a single genotype, the master sequence, has null mutational robustness. On the other hand, there is a mutant phenotype which, as it is made up of the rest of the 2^{ν }-1 possible sequences, has a virtually infinite mutational robustness (and thus, a_{iK }= 0 in section 2.3). When confronting a mutational pressure, an infinite mutational robustness is probably one of the best adaptations that a system can reach. Although the mean fitness is insensitive to further increases in mutation rate beyond the error threshold, the condition of which Hermisson et al.

3.6. Error threshold and RNA viruses

The error threshold concept acquired great importance, among other reasons, when the quasispecies concept was applied to RNA viruses

Conclusions

The main conclusion of this paper is that the entry into error catastrophe is a specific case of survival of the flattest acting on phenotypes which differ in the trade-off between replicative ability and mutational robustness. In fact, the entry into error catastrophe takes place when the mutant phenotype acquires a selective advantage over the master phenotype. Moreover, beyond the error thresholds, changing the quality factor of some sequences modifies the population distribution at the error catastrophe, displacing it towards the flatter regions of the mutant phenotype. However, the value of the error threshold is not altered by these changes in the mutant phenotype as it depends on its effective fitness as a whole. Both neutrality and some lethality schemes increase the effective fitness of the master phenotype with respect to the mutant phenotype, so error threshold decreases. Taking this into account the notion of crisis information beyond error threshold does not make sense.

As both entry into error catastrophe and survival of the flattest are caused by natural selection when mutation rate is increased, we propose differentiating between them by the level of selection at which natural selection act. Thus, we propose to use the term "survival of the flattest" to refer to situations in which two quasispecies compete, and there is no mutational coupling between them; and the term "entry into error catastrophe" as the displacement of a phenotype with high replicative ability but less robustness by another flatter phenotype, when they are mutationally coupled and within the same quasispecies.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

HT conceived the study, participated in its design and drafted the manuscript. HT and AM carried out the numerical and computational studies. FM participated in its design of the study, coordinated it and helped to draft the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This paper has been supported in part by Grants no. BFU2009-12895-C02-02 from MEC (Spain). Hector Tejero is supported by AP2006-01044, from MEC (Spain); and Arturo Marín is supported by Consejería de Educación de la Comunidad de Madrid (Spain) and Fondo Social Europeo (FSE). We would like to thank Esteban Domingo for useful discussion and critical reading of this manuscript, and Athel Cornish-Bowden for the final improving of the written English.