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Quasispecies Theory and Emerging Viruses: Challenges and Applications

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The dawn of quasispecies theory revolutionized our understanding of viral evolution and pathogenesis. This theory conceptualises viruses as dynamic populations of closely related but genetically diverse variants that constantly mutate and adapt to environmental pressures. Quasispecies dynamics govern key aspects of virus-host interactions, such as adaptive evolution, immune evasion, drug resistance, and host tropism. In this article, we discuss the fundamental role of quasispecies theory in elucidating viral fitness landscapes, shaping antiviral strategies, and predicting viral emergence and evolution. We provide a concise overview of the original quasispecies model and its latest advancements, which enable the study of the connection between viral dynamics and the significant genetic diversity exhibited by viruses. We then point out some key features of virus dynamics that need to be incorporated into quasispecies theory. We continue with examples of convergence between theory and real viruses by discussing theoretical results supported by RNA virus data and vice versa. Next, we discuss the need to extend the concept of sequence space beyond the classical hypercube towards more complex, multidimensional connected sequence spaces that we have called ultracubes. Finally, we highlight the necessity of developing multi-scale models to understand how viral evolutionary dynamics within a host can affect epidemiological patterns. We also examine the limitations of quasispecies theory in predicting virus evolution and emergence.
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Quasispecies Theory and Emerging
Viruses: Challenges and Applications
Josep Sardanyes * , Celia Perales , Esteban Domingo , Santiago Elena
Posted Date: 8 July 2024
doi: 10.20944/preprints202407.0559.v1
Keywords: Quasispecies; Virology; Evolutuion; Fitness landscapes; Genetic heterogeneity; Modelling
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Review
Quasispecies Theory and Emerging Viruses:
Challenges and Applications
Josep Sardanyés 1,2,*, Celia Perales 3,4, Esteban Domingo 5 and Santiago F. Elena 6,7
1 Centre de Recerca Matemàtica, Edifici C, Campus de Bellaterra, Cerdanyola del Vallès, 08193 Barcelona,
Spain
2 Dynamical Systems and Computational Virology, CSIC Associated Unit I2SysBio-CRM
3 Department of Molecular and Cell Biology, Centro Nacional de Biotecnología, CSIC, Cantoblanco, 28049
Madrid, Spain
4 Department of Clinical Microbiology, Instituto de Investigación Sanitaria, Fundación Jiménez Díaz
University Hospital-Universidad Autónoma de Madrid, 28040 Madrid, Spain
5 Microbes in Health and Welfare Program, Centro de Biología Molecular “Severo Ochoa”, CSIC-
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
6 Instituto de Biología Integrativa de Sistemas (I2SysBio), CSIC-Universitat de València, Paterna, 46980
València, Spain
7 The Santa Fe Institute, Santa Fe, New Mexico 87501, USA
* Correspondence: author: J. Sardanyés (jsardanyes@crm.cat)
Abstract. The dawn of quasispecies theory revolutionized our understanding of viral evolution and
pathogenesis. This theory conceptualises viruses as dynamic populations of closely related but genetically
diverse variants that constantly mutate and adapt to environmental pressures. Quasispecies dynamics govern
key aspects of virus-host interactions, such as adaptive evolution, immune evasion, drug resistance, and host
tropism. In this article, we discuss the fundamental role of quasispecies theory in elucidating viral fitness
landscapes, shaping antiviral strategies, and predicting viral emergence and evolution. We provide a concise
overview of the original quasispecies model and its latest advancements, which enable the study of the
connection between viral dynamics and the significant genetic diversity exhibited by viruses. We then point
out some key features of virus dynamics that need to be incorporated into quasispecies theory. We continue
with examples of convergence between theory and real viruses by discussing theoretical results supported by
RNA virus data and vice versa. Next, we discuss the need to extend the concept of sequence space beyond the
classical hypercube towards more complex, multidimensional connected sequence spaces that we have called
ultracubes. Finally, we highlight the necessity of developing multi-scale models to understand how viral
evolutionary dynamics within a host can affect epidemiological patterns. We also examine the limitations of
quasispecies theory in predicting virus evolution and emergence.
Keywords:
The quasispecies theory, conceived by Manfred Eigen and Peter Schuster more than fifty years
ago (Eigen, 1971; Eigen & Schuster, 1979; Schuster & Swetina, 1988; Eigen et al., 1989) was developed
to investigate the dynamics of biological information in replicators subjected to exceptionally high
mutation rates. This theory is a cornerstone in understanding prebiotic evolution and the genetic
diversity of viruses (Domingo et al., 1999; Domingo, 2001; Mas et al., 2004; Biebricher & Eigen, 2005;
Elena et al., 2010; Domingo et al., 2021). Moreover, it also extends to the dynamics of cancer cells
(Solé, 2003; Solé & Deisboek, 2004; Brumera et al., 2006) and has been linked to the conformational
diversity of prions (Li et al., 2010). At its core, the theory posits that viral populations exist not as
static entities with a single genome, but rather as dynamic distributions of closely related mutant
genomes known as mutant swarms or quasispecies. Viral populations are characterized by high
mutation rates due to the limited template-copying fidelity of RNA-dependent RNA polymerases
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(RdRp) and RNA-dependent DNA polymerases (RdDp), leading to the continuous generation of
genetic variants. The collective behavior of these variants, along with multiple selection pressures
such as heterogeneity in susceptible target cells, host immune systems, and potential antiviral
therapiesand bottleneck events at various scales, together shape the evolutionary trajectories of the
virus. Thus, understanding the quasispecies dynamics is essential for elucidating viral pathogenesis,
transmission dynamics, and the emergence of drug resistance. The molecular quasispecies theory has
driven the development of a comprehensive theory of virus evolution, supported and enriched by
numerous experimental and clinical studies. Figure 1 shows a brief outline of some key contributions
to viral quasispecies.
Figure 1. Milestones in quasispecies theory in Virology. Time arrow showing some key
achievements in the development of quasispecies theory for viruses since its birth in 1971. The colors
indicate whether the results have been achieved from theoretical/computational research (blue), from
experimental data (violet), and/or from data from infected patients (green). CChMVd:
Chrysanthemum chlorotic mottle viroid; CSVd: Chrysanthemum stunt viroid; FMDV: foot-and-
mouth disease virus; HCV: hepatitis C virus; HIV-1: human immunodeficiency virus type-1; LCMV:
lymphocytic choriomeningitis virus; VSV: vesicular stomatitis virus; PV: poliovirus.
Research on bacteriophage Qβ by Charles Weissmann and colleagues in the 1970s (Domingo et
al., 1978) coincided with the development of quasispecies theory by Eigen and Schuster. During the
1980s and 1990s, scientists compared genomic sequences of clones from natural viral isolates or
experimental viral populations. The introduction in the 2000s of ultra-deep sequencing significantly
advanced the understanding of the genetic complexity of the mutant swarm. Recent studies on SARS-
CoV-2, for example, have identified numerous low-frequency mutations in virus isolates, some with
functional significance (Martínez-González et al., 2022; Delgado et al., 2024). The quasispecies concept
has profound implications for clinical and public health interventions. By viewing viral populations
as dynamic ensembles of genetic variants, rather than static entities, the quasispecies theory
underscores the challenges to eradicating viral infections through conventional interventions.
Antiviral therapies that target a single viral genotype may exert a selective pressure, favoring the
emergence of drug-resistant mutants within the cloud of mutants forming the quasispecies. This
evolutionary resilience necessitates the development of novel therapeutic strategies that take into
account the complex evolutionary dynamics of viral populations. Additionally, insights from the
quasispecies theory have led to advancements in experimental and computational techniques for
characterizing viral diversity. These improvements allow for more precise monitoring of viral
evolution and help design programs to reduce the spread of viral pathogens.
Two related key concepts in quasispecies theory are the so-called sequence spaces and fitness
landscapes. The sequence space is a multidimensional discrete space, also called a hypercube, where
each node corresponds to a given genotype, which is connected to the neighbouring genotypes by
single-point mutations (Figure 2a) (Schuster & Stadler, 2023).
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Figure 2. Simulating the evolutionary dynamics of digital quasispecies in silico. (a) Error threshold
(orange lines) in the single-peak fitness landscape (illustrated with a 3-bits hypercube) for geometric
and stamping machine replication modes. The size of the balls is proportional to the genotype’s
fitness. (b) Fitness landscape with antagonistic epistasis where the effects of mutations are less severe
in combination than individually. Here we also display the error threshold for this landscape for
geometric and stamping machine replication. All the diagrams show the stationary distributions of
the master sequence (000, thick lines) and the pool of mutants (thin lines) averaged over 200 replicas.
See Sardanyés et al. (2009) for further details.
Generically, the dimension of a hypercube is nL, n being the number of letters in the alphabet (4
for nucleotides) and L the length of the genome. That is, the dimension of the sequence space for an
RNA virus with a short genome, such as bacteriophage MS2, with 3569 nucleotides, results in a
hypercube of dimension 43569. This is a huge sequence space that can be explored by the virus through
the processes of point mutations or recombination events that do not alter the genome length (Figure
3a). Each of the nodes of this hypercube thus corresponds to a different genotype with a different
fitness. It is known that point mutations impact the fitness of the viral genotypes, typically being
deleterious or lethal (Elena and Moya, 1999; de la Peña et al., 2000; Sanjuan et al., 2004; Carrasco et
al., 2007; Fernández et al., 2007; Sanjuan, 2010). The quasispecies is expected to be located in a given
region or regions of this multidimensional space after selection takes place. As we discuss below,
these hypercubes may not accurately describe the complexity of quasispecies when considering other
genetic processes, such as deletions or insertions, which result in viral genomes of varying lengths.
Hence, the quasispecies should be expected to live in a much more complex sequence space that we
define as the ultracube and will explain below.
The distribution of fitnesses along the hypercube nodes defines the so-called fitness landscape.
A fitness landscape is a conceptual model used in evolutionary biology that metaphorically
represents the multidimensional space where each point corresponds to a specific genotype and is
associated with a quantitative value that represents its fitness. For example, in the most visually
appealing two-dimensional representation, the coordinates in the plane represent genotypes, and the
elevation over the surface indicates the fitness of that genotype. In simpler terms, a fitness landscape
summarizes how well an organism with a particular genotype is adapted to its environment and e.g.,
performs in replication. Fitness landscapes can vary in complexity, ranging from simple, smooth
landscapes with a single peak representing the optimal genotype to rugged landscapes with multiple
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peaks and valleys, indicating the presence of different adaptive solutions or evolutionary pathways
(Kauffman & Levin, 1987; Schuster, 2016; Schuster & Stadler, 2023). The shape of the fitness landscape
is influenced by several factors, including the genetic architecture of traits, the nature of
environmental conditions, and the interactions among different genotypes. Having a precise fitness
landscape of viral quasispecies is an extremely challenging problem. For real viruses, fitness
landscapes are increasingly viewed as very rugged and dynamic (Domingo et al., 2023).
Theoretical Quasispecies: What Has Been Explored and Existing gaps
The population dynamics of quasispecies has been extensively studied since the original
contributions by Eigen and Schuster. Several mathematical and computational models have been
used to investigate various aspects of viral quasispecies. Here, we discuss the main contributions of
quasispecies theory to the field of RNA viruses, highlighting several processes that have not yet been
investigated in detail from a theoretical or computational point of view and may be worth exploring.
The original Eigen-Schuster quasispecies model is given by the set of differential equations

 =
=1
 () .
This mathematical model describes the time change of the fraction of the population of the ith
mutant sequence (= 1, . . . , ) , and being very large. Here
is the replication rate of the
th mutant,  is the probability of having a mutation , and () =
=1
denotes the
average fitness of the population. One key aspect of quasispecies models is that the sequences inhabit
a fitness landscape; in this case, each sequence has a given fitness value, which determines its rate of
replication. An infecting wild-type (wt) virion can produce an enormous progeny inside the host,
giving rise to a large amount of different sequences embedded in a more or less complex fitness
landscape. Despite such complexity, the quasispecies model can be explored in a simpler setting,
allowing to mathematically characterize phenomena of interest that may also occur in more complex
scenarios.
One of such useful simplifications is to consider a quasispecies formed by only two populations
of genomes: the wt and the pool of mutants. This simple model easily illustrates a fundamental
consequence of quasispecies theory: the error catastrophe or error threshold. This two-populations
model assumes that the quasispecies is embedded in a single-peak fitness landscape (Swetina &
Schuster, 1982; Solé, 2003; Bull et al., 2005; Solé et al., 2006) and the wt sequence, 0 with high fitness
0, produces deleterious mutants that are grouped into an average mutant sequence, 1, all with
equal fitness
1<
0. This model also assumes that backward mutations are negligible. This is a first
good assumption due to the enormous size of the sequence space. This model can be written taking
as variables these two sequence types, with:
0
 =
0 0(1 ) (0,1) 0 ,
1
 =
0 0+
11 (0,1) 1 .
Here is the mutation rate, and (0,1) is an outflow term given by the average fitness and
which keeps a constant population (another unrealistic assumption for real viral quasispecies). This
oversimplified model allows the calculation of the error threshold occurring when mutation
overcomes the critical value = 1
1/
0. When < the quasispecies is composed by both the
wt and mutant sequences, while for > the entire quasispecies is only composed by mutants. In
other words, this critical mutation marks a transition between a phase wherein the genetic
information carried by the wt sequence is preserved and a phase in which this information is no
longer maintained (Eigen, 1971; Biebricher & Eigen, 2005; Bull et al., 2005; Solé et al., 2006; Sardanyés
& Elena, 2010; Solé et al., 2021). Interestingly, more complex versions of the model that incorporate
epistatic fitness landscapes and different modes of genomic replication (e.g., geometric amplification
versus stamping machine) still preserve the existence of an error threshold, although its actual value
strongly depends on these assumptions and can be even bigger under realistic combinations of
parameters (Figure 2), as we discuss below. Moreover, oversimplifications allowing for mathematical
modelling are sometimes meaningful. For instance, research on quasispecies in HCV infected patients
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revealed an inverse correlation between viral load and quasispecies complexity (Mas et al., 2004).
Such a feature was reproduced with the two-populations model described above using in silico
quasispecies evolution (Solé et al., 2006).
The original Eigen-Schuster quasispecies model was built upon several assumptions that might
not hold for real viruses. It assumes continuous, well-mixed populations of replicators, constant
population, geometric replication, and determinism. During the last decades, considerable efforts
have been made to extend the initial quasispecies theory to more realistic scenarios for RNA viruses.
Examples comprise the study of other key features of viral quasispecies that are summarised next
and which have been investigated separately or in combination. These include finite populations
(Nowak & Schuster, 1989; Solé & Deisboek, 2004; Sardanyés, 2008), stochastic effects (Solé et al., 2006;
Ben-Ari & Schinazi, 2016), spatially-extended quasispecies (Altemeyer & McCaskill, 2001; Aguirre &
Manrubia, 2008; Sardanyés, 2008; Sardanyés & Elena, 2011), viral complementation (Sardanyés &
Elena, 2010), and recombination (Jacobi, 2006; Saakian et al., 2019). The investigation of asymmetric
modes of replication, identified in RNA viruses either indirectly from mutant distributions (Denhardt
& Silver, 1966; Chao et al., 2002; García-Villada & Drake, 2012) or from direct RNA quantification of
positive and negative strands during the progress of infection (Martínez et al., 2011; Schulte et al.,
2015), have been also studied within the quasispecies framework (Sardanyés et al., 2009; Sardanyés
& Elena, 2011).
Another relevant theoretical result of quasispecies theory is the so-called survival-of-the-flattest
effect. This effect is mainly produced because fast replicating genomes that produce low-fitness
offspring can be outcompeted by slow replicating genomes with moderate fitness, provided the latter
inhabit a region of sequence space characterized by high neutrality and connectivity (Schuster &
Swetina, 1988; van Nimwegen et al., 1999; Wilke et al., 2001a). This theoretical prediction was first
demonstrated in artificial life experiments (Wilke et al., 2001b) and later described in experiments
with competing viroids (Codoñer et al., 2006) and withVSV (Sanjuán et al., 2007) under mutagenic
conditions. This effect was later explored mathematically, and the outcompetition of the fit
quasispecies by the flat one at an increasing mutation rate was shown to be given by an abrupt
transition (Sardanyés et al., 2008). Flat-like quasispecies may underlie failures in resolving chronic
infections by antiviral agents, despite the absence of specific inhibitor-resistance substitutions
(Gregori et al., 2024). Concerning the fitness landscapes, the Swetina-Schuster single-peak one is, of
course, an oversimplification of how mutations impact fitness in viral quasispecies. During the last
decades, more complex fitness landscapes have been studied for quasispecies (Schuster, 2016). These
include different fitness functions (Saakian & Hu, 2006), dynamic fitness landscapes (Wilke &
Ronnewinkel., 2001; Wilke et al., 2001) and non-linear interactions among mutations, i.e., epistasis
(Sardanyés et al., 2009; Elena et al., 2010; Sardanyés & Elena, 2011).
As we have discussed, the original quasispecies theory has been significantly refined over the
past decades to better align with real virology. Relevant experimental and clinical results involving
RNA viruses have been interpreted through the lens of these theoretical models. Despite this fact,
there are still missing ingredients in quasispecies theory that could play a crucial role in the
evolutionary dynamics of viruses. For example, some processes, such as RNA viral synthesis or viral
protein production and maturation, involve time lags. In this sense, several studies have revealed
that the SARS-CoV-2 replication complex has an elongation rate of 150 to 200 nucleotides per second,
being more than twice as fast as the poliovirus polymerase complex (Campagnola et al., 2022).
Nucleotide incorporation time, determined by differences at and around viral RNA polymerase
catalytic sites, may influence the fidelity of template copying (Yin et al., 2023). In this sense, it is
known that time lags can profoundly affect the dynamics of nonlinear systems, causing self-sustained
oscillations and chaos (Mackey & Glass, 1977). Also, during their intracellular phase, viruses
synthesize distinct molecules, constructing viral factories for genome replication and morphogenesis.
The central element of these factories is the replication organelle (RO), where viral replication
complexes produce multiple copies of the viral genome. Viral factories often consist of remodeled
cell membranes with functional compartments for replication, assembly, and egress, and they
frequently recruit other cellular elements like mitochondria and the cytoskeleton, which interact with
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the RO. Unlike DNA viruses, almost all RNA viruses form factories exclusively in the cytoplasm
(Kopek et al., 2007; Harak & Lohmann, 2015). For instance, HCV induces remodeling of reticulum
endoplasmatic membranes forming double-membraned vesicles, and later on, it induces the
formation of multi-membrane vesicles that are composed of several concentric membrane bilayers.
These membranous rearrangements are produced by the action of viral nonstructural proteins.
Hence, the process of virus genomes amplification can be highly compartmentalized inside the cell
and possibly also induce time lags in viral replication and virions assembly.
For RNA viruses, especially those infecting plants, replication processes may exhibit periodic
fluctuations at the within-tissue or within-host levels across different time scales, primarily due to
temperature changes (Honjo et al., 2020). Additionally, the mammalian brain possesses an
endogenous central circadian clock that regulates both central and peripheral cellular activities. At
the molecular level, this day-night cycle triggers the expression of upstream and downstream
transcription factors that affect the immune system and modulate the severity of viral infections over
time. The role of circadian systems in regulating viral infections and the host response to viruses is
thus of great clinical importance (Zandi et al., 2023), and thus fluctuating parameters may be included
in theoretical quasispecies investigations.
Another unexplored process in quasispecies theory is how the evolutionary dynamics of
quasispecies at the within-cell/within-host levels scale up to the population-epidemiological levels.
This question, which we discuss below, can be addressed by developing multi-scale models including
the rapid quasispecies evolution (fast dynamics) and the impact of the continuous synthesis of
heterogeneous mutants at a population scale (slow dynamics). The mathematical results on slow-fast
systems may be thus relevant to tackle this problem. In this sense, the connection of scales in virology
is extremely important to understand virus pathogenesis and potential zoonoses and epidemic
spread among different hosts (Solé et al., 2021). This subject can be further explored combining
dynamical systems theory with complex networks theory.
The limits of predictability of quasispecies are also an extremely challenging and open problem
in virology. That is, how the dynamical population structure of the quasispecies may impact disease
outbreaks and the emergence of new variants with epidemic and pandemic potential (see Section on
Impact of quasispecies populations in emerging viral diseases below).
Quasispecies and RNA Viruses: Genomic Heterogeneity, Adaptation and Clinical Implications
of Genetic Information Thresholds
Initial experimental evidence of viral quasispecies dynamics involved studies mainly with
bacteriophage Qβ, VSV, FMDV, or LCMV [reviewed in Domingo et al. (2021)]. These experiments
have robustly confirmed high mutation rates during genome replication, the heterogeneity of viral
populations, fitness variations among biological clones, rapid changes in sequence space occupation,
limited tolerance to increased error rates, to cite a few. The early studies with bacteriophage Qβ by
Weissmann and colleagues were carried out simultaneously with the development of quasispecies
theory by Eigen and Schuster in the 1970s. In the 1980s and 1990s, the primary approach was to
compare genomic sequences of molecular or biological clones from natural viral isolates or viral
populations subjected to various experimental evolution designs. The advent of ultra-deep
sequencing techniques greatly broadened the capacity to probe the complexity and dynamics of viral
quasispecies. This is exemplified by results on HCV (Mas et al., 2004; Perales, 2020), and, more
recently, in SARS-CoV-2 displaying multitudes of low frequency mutations in isolates of the virus,
some of them endowed with functional relevance (Martínez-González et al., 2022; Delgado et al.,
2024; Martínez-González et al., 2024).
During recent years, the development of new high-throughput sequencing techniques (e.g., Cir-
Seq) along with novel bioinformatic algorithms have shown the relevance of an abundant component
of viral quasispecies that was previously largely ignored: the defective viral genomes (DVGs)
(Vignuzzi & López, 2019; González-Aparicio & López, 2024; (Olmo-Ulceda et al., 2022; Hillung et al.,
2024). Broadly speaking, DVGs are nonstandard genomes generated during error-prone replication
that contain deletions, insertions, duplications, inversions, and potential hypermutated viral
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genomes contributed by cellular editing activities. These major-effect mutations render genomes
unable to self-replicate and coinfection with a wt (also dubbed helper) virus is needed for the
defective genomes to persist in the population. Several recent studies have described the temporal
dynamics of DVGs during the course of infection for several viruses, including poliovirus (PV) and
dengue virus (DENV) (Rangel et al., 2023), SARS-CoV-2 (Zhou et al., 2023) and other
betacoronaviruses (Hillung et al., 2024). Consistently, these studies show that some DVGs are
pervasively maintained in the viral population in cell cultures but also in patients, suggesting a
possible selective role. For instance, deletions affecting the receptor binding domain and the S1/S2
cleavage site in SARS-CoV-2 Omicron variant, may have increased host cell ACE2 receptor
recognition, thus enhancing the infection, and allowing this variant to become dominant (Campos et
al., 2022). Indeed, it has been suggested that DVGs might confer some advantage to viruses, such as
serving as reservoirs of genetic variability, decoys for immune responses, regulators of translational
shut-down, or mediators of persistent infections (González-Aparicio & López, 2024).
A particularly notable type of DVG are the copy-back and snap-back (cb/sb) variants, which
consist of small RNA molecules with a hairpin-loop structure created by template-switching from
positive to negative templates during replication.
As we have mentioned, an increase in population mutational load should concomitantly result
in a decrease in the average population fitness as most mutations with phenotypic effects are
deleterious or lethal (Elena & Moya, 1999; Biebricher & Eigen, 2005; Domingo-Calap et al., 2009;
Sanjuán, 2010). Given this, and together with the aforementioned prediction of the existence of a
critical mutation rate beyond which the population enters the error catastrophe regime (Eigen, 1971;
Eigen et al., 1989), innovative antiviral strategies have been proposed aiming to push down
replication fidelity and forcing the viral population to cross the error threshold.
Quasispecies Complexity: From the Hypercube to the Ultracube
In the study of viral evolution and coevolution, hypercubes provide a useful conceptual
framework for comprehending the sequence space of viral genomes (Eigen et al., 1989; Sardanyés &
Solé, 2013; Schuster, 2016). This sequence space is a mathematical multidimensional construct where
each dimension corresponds to a nucleotide position within the viral genome (Figs. 2 and 3a). A
hypercube, in particular, is a discrete geometric representation of this space, with vertices
representing every possible sequence configuration in which a viral genome of length L can exist.
Each vertex is connected to adjacent vertices by edges that signify single nucleotide changes,
illustrating the potential mutational pathways a virus can traverse while retaining the same length.
This model effectively visualizes the extensive diversity and evolutionary potential of viral
populations. The structure of the hypercube allows researchers to analyze the distribution and
dynamics of viral quasispecies within this space, providing valuable insights into how viruses adapt
to environmental pressures and develop resistance to antiviral treatments. The foundational works
of Eigen and Schuster (Eigen & Schuster, 1979) were instrumental in introducing the concepts of
sequence space and hypercubes within the framework of quasispecies theory, offering a
mathematical basis for understanding the evolutionary landscape of viruses (Swetina & Schuster,
1982). However, viral quasispecies may be embedded within more complex and entangled sequence
spaces, including the full spectrum of mutant sequences together with, e.g., subgenomic sequences
and DVGs. These length variants may form other subpopulations of clouds of mutants spanning
lower- or higher-dimensional hypercubes arising from the full-genome sequence space, thus being
connected through deletions or insertions. Hence, a more realistic geometric space for viruses may
include connected hypercubes of different dimensions: we call these sequence spaces, in which a
given node can represent a hypercube itself, as ultracubes (Figure 3b). This view enlarges the size of
the sequence space beyond the single-point mutation hypercube dimension. As for the hypercube, a
quasispecies may not span all this sequence space but be located on some specific regions of this
ultracube in a mutation-selection balance or in a transient towards a mutation-selection balance.
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Figure 3. Virus genotypic complexity: from the hypercube to the ultracube. (a) Classical view of
quasispecies evolution in a hypercube schematized with 3-bits sequences. Each node of the network
connects two genotypes via a single-point mutation. Sequences evolve to first neighbours by single
bit (nucleotide) substitutions during replication. Homologous recombination may allow genotypes to
jump to further neighbors (blue arrow). (b) Example of a sequence space for binary genomes of length
five considering deletions (blue dashed arrows) and insertions (green solid arrows) during
replication. These processes produce mutants and connect hypercubes of different dimensions, giving
rise to a more complex sequence space that we label ultracube and which can be conceived as a
multiplex network. For clarity, we do not display all the nodes but exemplify some processes of
deletion and insertion, which give rise to a set of connected hypercubes of dimension 5 (gray), 4
(black), 3 (blue), 2 (red), and 1 (orange). (c) Schematic diagram of connected hypercubes of different
dimensions illustrated schematically as multilayered networks.
Multilayer Models for Multi-Scale Virus Dynamics: Integrating Quasispecies into Virus
Epidemiology and Ecology
A new area of complex networks theory that has been quickly developing over the last decade
is the study of so-called multilayer networks. Each layer contains nodes connected by intralayer edges
that describe rules of interactions between the nodes for this particular layer. In addition,
dependencies across layers are represented by interlayer edges. It is straightforward to place and
visualize viruses into a multilayer; for illustrative purposes, let’s discuss here a simple 3-layer case.
The bottom layer would represent the quasispecies generated within a single infected individual. At
this level, the ultracube (Figure 3b) would be an appropriate representation. The equations describing
the dynamics at this bottom level could be those described in the previous sections incorporating
mutants of different nature. A middle layer would represent local host populations. At this level,
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each node corresponds to a particular host, and the edges represent the contact network among the
hosts that determines transmission dynamics. The equations governing the virus’ dynamics at this
intermediate level could be the well-known SIR epidemiological model, and the network can show
any topology such as scale-free (Pastor-Satorras & Vespignani, 2001). The interlayer edges connecting
the bottom and the middle layers represent the probability that a given viral genotype is actually
present in each infected host, which indeed depends on the replicative advantage at the bottom level
and the mutation rate. Coinfections are allowed (two interlayer edges pointing towards the same
individual). The upper layer represents the epidemiological level. In this layer, nodes represent, e.g.,
communities, whereas edges represent the connectivity between these communities (e.g., airport
traffic connections, vector flights, polen and seed dispersal, etc.). The dynamics at this level can be
modeled using phylogeography tools. The interlayer edges connecting the middle and the upper
layers would represent the probability that an infected individual will move from one community to
another.
This multilayer representation allows us to study not only the dynamics at each layer but also
the entire multilayer system and infer properties such as multilayer modularity, robustness to
perturbation or phase transitions. For instance, viral genotypes closely connected in the quasispecies
bottom layer may be found in individuals in the middle layer that also form a transmission cluster
(or module). For instance, ecological multilayer networks for plant-aphid and plant-aphid-
parasitoids (Pocock et al., 2012) show nontrivial stability properties that result in quantitative
predictions about the persistence or extinction probabilities that would not be shown up by other
modeling approaches (Pilosof et al., 2017). Information between nodes in a particular layer can be
transmitted via two different paths: those involving only intralayer edges and those involving both
intra- and interlayer edges. This means that catastrophic failures in a particular layer can easily be
transmitted to the rest of layers (Cellai et al., 2013; Boccaletti et al., 2014). The epidemic spread of
infectious diseases in multilayer networks has received some attention (Dickison et al., 2012;
Mendiola et al., 2012; Zhao et al., 2014; Stella et al., 2017). In particular, a topic that has been studied
is the interaction between two genetic variants of the same virus (Funk & Jansen, 2010; Sahneh &
Scoglio, 2014; Sanz et al., 2014; Zhao et al., 2014).
When multiple variants are spreading in a host scale-free network and compete for hosts, they
obviously influence each other’s dynamics. In such a situation, for example, the derived epidemic
threshold for each variant is substantially different than predicted using monolayer simple
coinfection models. Indeed, two new thresholds arise from these models (Sahneh & Scoglio, 2014):
(1) the survival threshold determines a continuous phase transition from extinction to existence
during competition between both variants. (2) The absolute-dominance threshold denotes the critical
point where one of the variants fully outcompetes the other. Between these two thresholds,
coexistence emerges as a property of the interconnected structure of the multilayer.
Impact of Quasispecies Populations in Emerging Viral Diseases
The emergence of viral disease is an unpredictable event since it is the result of several
interconnected factors: the adaptive potential of viruses circulating in different host species,
environmental modifications, or political and sociological circumstances, among others. These factors
can affect virus traffic and facilitate encounters with potential new hosts (Morse, 1993; Morens &
Fauci, 2000; Smolinski et al., 2003). Viral emergence can be regarded as a facet of complex biological
behavior since its occurrence cannot be anticipated by the sum contribution of the multiple
underlying factors (Solé, 2000; Solé et al., 2018; Solé et al., 2021). In this scenario, quasispecies provide
prime material for adaptation following chance encounters between viruses and potential new host
species to play a triggering role in an emergence. Indeed, in many documented examples, too
numerous to describe here, mutant spectra hide minority populations with altered cell tropism or
host range (Domingo, 2020). All the genotypic complexity portrayed by the ultracube connections
(Figure 3) add to the lottery of which variants (and when) they may come into contact with a
susceptible individual of a different host species, to initiate an infection and to produce sufficient
transmissible progeny to achieve the status of emergent disease.
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John Holland and colleagues were the first to point out a number of medical implications of a
highly dynamic RNA world coexisting with a relatively more static DNA-based cellular biosphere
(Holland et al., 1982). They underlined high mutation rates as a source of atypical viral forms capable
of invading new tissues and organs, to establish persistent or inapparent infections, or to be selected
in response to medical interventions. These facets of viral dynamics have been amply documented in
the 1940s that have followed their prescient publication. Joshua Lederberg warned of the human
vulnerability in the face of the adaptive potential of RNA viruses: “Abundant sources of genetic
variation exist for viruses to learn new tricks, not necessarily confined to what happens routinely or
even frequently“ (Lederberg, 1993). In May 2003, the World Health Organization (WHO) released a
report titled SARS: Status of the outbreak and lessons for the immediate future(WHO, 2003). In
this report, the WHO cautioned that SARS could continue to pose a threat for several reasons,
including e.g., limited understanding of SARS epidemiology and insufficient diagnostic tests, and the
high mutation rates of coronaviruses, which elevate the risk of future outbreaks. In 2004, Snell (2004)
discussed the risk that SARS and influenza have on human health worldwide. Two years later, SARS
was addressed as an emerging infectious disease with a potential to cause future outbreaks (Tapper,
2006). By 2009, Woo et al. (2009) wrote about coronavirus diversity, phylogeny and interspecies
jumps, stating that before the SARS outbreak only ten complete coronavirus genomes were available.
By 2008, this number increased to twenty-six genomes including human and bat coronaviruses,
among others. The study highlighted the rapid ability of coronaviruses to jump between species and
noted the presence of closely related coronaviruses in more distantly related organisms. It stressed
that such interspecies transmission from zoonotic outbreaks could potentially infect humans and lead
to devastating outbreaks (Woo et al., 2009).
Future Challenges in Experimental and Clinical Quasispecies
As often happens in science, new techniques pose new questions. The availability of deep
sequencing methodology to probe into the composition of virus populations has opened new
questions, in addition to solving old ones such as the definitive confirmation that mutant spectra are
the biological reality of viral populations, so far without documented exceptions. Among the new
challenges are: (1) capturing the depth of minority genomes that until now have escaped detection;
(2) how they feed the dominant ones, be them viable or defective; (3) how the latter modulate
behavior of the ensemble; (4) which is the time frame in which minority subpopulations can be
replaced by others, relative to the time required for selective forces (metabolic modifications, signal
effectors, antibodies, drugs, etc.) to reach the extracellular and intracellular environments or even
individual viral factories; (5) what is the extent of heterogeneity within individual cells; and (6)
capacity of mutant spectrum composition to inform of long-term evolutionary events. These are but
some of many highly relevant questions. What is at stake is to better understand viruses, the diseases
they produce, and the means to combat them. In short, a challenge is to shift from a consensus
sequence-centered understanding of evolution into a mutant spectrum-centered understanding of
evolution, and harmonizing the conclusions drawn from both.
Therapies based on the error catastrophe concept have been brought forward. Maintenance of
inheritable genetic information conditioned to a limitation in error introduction has also been
documented with catalytic ribozymes (Papastavrou et al., 2024). Lethal mutagenesis approaches
exploit the intrinsic balance between mutation rates and viral viability, offering a promising avenue
for combating viral infections. As a difference from the error threshold, which causes a shift in the
sequence space, i.e., the master sequence is replaced by the mutant swarm, lethal mutagenesis
involves virus extinctions (Bull et al., 2005). Virus extinction by lethal mutagenesis has been
documented with several RNA viruses and mutagenic base and nucleoside analogues, which are
licensed for clinical use [reviewed in Perales (2019)]. These include, the pioneering works with HIV-
1 (Loeb et al., 1999; Dapp et al., 2013), PV (Crotty et al., 2001), FMDV (Perales et al., 2011; de Ávila et
al., 2017), LCMV (Grande-Pérez et al., 2002; Ruiz-Jarabo et al., 2003), and HCV (Ortega-Prieto et al.,
2013; de Ávila et al., 2016) in cell cultures. Virological, biochemical and structural studies suggest that
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lethal mutagenesis is, at least in part, the mechanism of some antiviral agents currently used to treat
COVID-19 (Gordon et al., 2021; Kabinger et al., 2021; Somovilla et al., 2023; García-Crespo et al., 2024).
Synergistic antiviral combinations that include one or two lethal mutagens (a strategy termed
synergistic lethal mutagenesis) constitute a promising avenue for the treatment of COVID-19,
extensible to other emergent RNA viral infections (García-Crespo et al., 2024). Additionally, lethal
defection, characterized by the extinction of viral populations due to the emergence of DVGs, has
been observed for LCMV in cell cultures (Grande-Pérez et al., 2005). For this virus, two extinction
pathways have been identified: one at high mutagen concentrations, resulting in the complete loss of
infectivity and replication ability of the quasispecies, and another at lower mutagen concentrations,
where replication persists while the infective class becomes extinct due to the presence of defectors.
For a long time, the aforementioned cb/sb mutants have been known for their negative impact
on virus accumulation, as they retain the replication signals of both polarities and have a replication
advantage owed to their very short genomes. In recent years, the antiviral efficiency of synthetic
cb/sb, known as therapeutic interfering particles (TIPs), has been proven against SARS-CoV-2 in
animal models. These antiviral effects have involved a decrease in virulence and milder edema
symptomatologies (Chaturvedi et al., 2021) as well as a decreased viral transmission (Chaturvedi et
al., 2022) in hamsters. Thus, the continuous expansion of theoretical quasispecies concepts, and their
scrutiny with viral experimental systems, now reinforced with the new tools of ultra-deep
sequencing, is paving the way towards innovations for the control of highly heterogeneous and
dynamic cellular pathogens.
Contributions: JS, CP, ED, and SFE: conceptualization, manuscript writing, funding acquisition.
Funding: JS has been supported by grant PID2021-127896OB-I00 from MCIN/AEI/10.13039/501100011033
”ERDF A way of making Europe” and from the María de Maeztu Program for Units of Excellence in R&D grant
CEX2020-001084-M. JS also thanks the Generalitat de Catalunya CERCA Program for institutional support. ED
and CP have been supported by grant PID2020-113888RB-I00 from MCIN/AEI/10.13039/501100011033 and
202220I116 and by the European Commission-Next Generation EU (regulation EU 2020/2024) through the CSIC
Global Health Platform (PTI Salud Global); by grants PI21/00139 from Instituto de Salud Carlos III, and project
525/C/2021 from Fundació “La Marató de TV3”, grants 202136-30 and 202136-31. ED and CP also acknowledge
the project S2018/BAA-4370 (PLATESA2 from Comunidad de Madrid/FEDER) and Institutional grants from the
Fundación Ramón Areces and Banco Santander to the CBMSO are also acknowledged. The team at CBMSO
belongs to the Global Virus Network (GVN). SFE has been supported by grant PID2022-136912NB-I00 funded
by MCIN/AEI/10.13039/501100011033 and by “ERDF a way of making Europe”, and by grant CIPROM/2022/59
funded by Generalitat Valenciana.
Acknowledgements: We want to thank the members of the research groups Genetic Variability of RNA viruses
(CBMSO), CNB-CSIC, EvolSysVir (I2SysBio), and the Knowledge Transference Unit at the CRM for useful
discussions. We also want to thank J. Tomás Lázaro for useful comments on the article. We also acknowledge
the hospitality of the Institut de Mathemátiques de Jussieu (Sorbonne Université, Université Paris Cité) where
part of this manuscript was developed.
Competing Interests: The authors declare no competing interests.
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