subsection{Introduction x_{i}f_{i}$. The probability that genotype $j$ results

subsection{Introduction to Article II}Contrary to what presented in Article I, works from this point on, started as an attempt to investigate the origin and evolution of phenotypic robustness in living systems. This is a different, under-investigated aspect of phenotypic robustness. In fact, many previous mentioned studies aimed to mark the long-term effects of phenotypic robustness on evolvability and innovability. Indeed, except for some few studies, the mechanisms by which robustness might be established during evolution are far from clear and overall little explored. Phenotypic robustness seems to be an individual quality that should oppose the short-term adaptation process of populations per se even excluding possible fitness costs. Is robustness an adaptation in historical sense, i.

e. has it been shaped by natural selection? Or simply a by-product of evolution? A complete treatment of adaptation within evolutionary theory should try to “endogenize” citep{okasha2006evolution}phenotypic robustness, rather than to treat it as a given. According to this, we adopted a theoretical approach; We tried to fill the gap of a rigorous mathematical theory that should precede any experimental plan or hypothesis. We tried to avoid simple linear reasoning and trade-off based hypothesis. Instead, we focused on what we could directly derive from standard evolutionary models.

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In particular, we elaborated on two famous models representing two very different approaches to model construction: The Quasi-species model citep{eigen1989molecular}, originally conceived as an allele-based modellization, like the majority of evolutionary models in the history of evolutionary biology, and the Price’s equation citep{price1970selection}, is a phenotype-bases theorem that actually subsumes gene-based theories, since alleles and genotypes can be thought of as phenotypic characters themselves.The Quasi-species is a multi-allele mutation-selection model that has the form of a dynamical system describing the frequency change of a particular sequence type ($i$) during time. The term “quasi-species” refers to an ensemble of similar genomic sequences generated by a mutation-selection process.

Imagine a sufficiently large population of sequences of length $L$ and of different type $i$. Denote by $x_{i}$ the relative abundance of the $i$th sequence type, thus we have $sum_{i} x_{i}=1$. The population structure is given by the vector $vec x= ( x_{0}, x_{1},…

x_{n})$. Denote by $f_{i}$ the fitness (growth rate) of  the $i$th sequence type. The fitness landscape is given by the vector $vec f= (f_{0},f{_1},..

.,f_{n})$. The average population fitness is $psi=sum_{i} x_{i}f_{i}$. The probability that genotype $j$ results in genotype $i$ by mutation is given by $q_{ij}$.

$Q=q_{ji}$, where Q is the mutation matrix. Each element of Q represents the probability of a sequence $j$ to mutate in a sequence $i$ per replication. The quasi species equation is given by:egin{equation}dot x_{i}=sum_{j=0}^n x_{j}f_{j}q_{ji}- x_{i}psiend{equation}This means that sequence $i$ is obtained by replicating sequence $j$ at rate $f_{j}$ times the probability that replication of sequence $j$ generates sequence $i$.This model is no more than a deterministic mutation-selection multi-allele model describing the frequency change of the $i$th sequence ($dot x_{i}$).Differently, the Price’s equation is an “a posteriori” description of the change over time (generations) of the population mean trait value (note that the trait described could be the fitness itself or a variance instead of a mean, or the frequency of an allele) not focused on the dynamic but on the different states of the system at different arbitrary times. The Price equation is based on a different approach, tracking the change in the mean value of the population phenotype (note that the character described could be the fitness itself or a variance instead of a mean), not a frequency of a single pheno(geno)type. If a positive covariance between fitness and the phenotypic value of a certain trait exist, we can define adaptation as the increase of the mean of that particular character value over a definite time interval. This approach is very useful if we have to deal with quantitative traits and continuous character values differently from what happens with the quasi-species model where we have discrete values.

In addition, the Price’s equation contains all the evolutionary relevant factors and is a more complete and comprehensive description of the evolutionary process. Here we adopt the Price’s equation as a model to describe the effects of phenotypic robustness on adaptation considering the following form citep{okasha2006evolution}: egin{equation}Delta overline{z}=cov(w_i,z’_i)+EDelta z_iend{equation}Where$Delta overline{z}$ is the change in the population mean character value, $z’_i$ is the mean of the offspring’s character values, $w_i$ the relative fitness, and  $EDelta z_i$ is the transmission bias, namely the change in mean character value due to other factors rather than selection, i.e genetic or environmental mutations, drift etc…Beyond the transmission bias, what really matters in evolution is the covariance between the parent’s fitness $w_i$ (offspring number) and the offspring phenotype, $z’_i$.  Price equation is a theorem rather than a theory describing what is actually going on rather than make a simplified model of basic properties of the system.

Indeed, almost every evolutionary phenomenon can be find in such equation which can be properly decomposed to describe it. Irrespective of the model, either Quasi-species or Price, interpretation of our results is based on three principal key premises, that should correspond to situations found in the majority of the biological cases. First, we consider an infinitesimal model citep{barton2016infinitesimal} perspective, where each phenotypic trait is the result of a very high number of genetic determinants citep{turelli2017fisher}.

In other words, we consider that a huge number of loci can affects each trait of an organism, although to a different degree. The infinitesimal model finds its origin in a seminal paper R.A.

Fisher, where he showed that, if many genes affect a trait, then the random sampling of alleles at each gene produces a continuous, normally distributed phenotype in the population citep{aymler1918correlation}. As the number of genes grows very large, the contribution of each gene becomes correspondingly smaller, leading in the limit to Fisher’s famous “infinitesimal model” citep{barton2016infinitesimal}. Despite the revival of interest (both theoretical and empirical) in “evolutionary quantitative genetics” in recent decades, the infinitesimal model itself has received little attention. However, recent advances in genome wide association studies (GWAS) highlight the possibility that virtually all the genome can affect every trait, that the genetic determinants are widespread through the genome and are highly interconnected such as that even apparently non-related peripheral factors can have a tiny effect on a given trait.

This phenomenon has recently been marked as the extit{omnigenetic} model (Boyle et al., 2017). One important consequence of this model is that the high number of genetic determinants dramatically increase the phenotypic mutation rate of a given trait. In the following work, we will show how this can affect our interpretation of the role of phenotypic robustness on adaptive dynamics.Second, we adopted the universal pleiotropy view, which is a natural consequence of the infinitesimal model, where each locus can affects virtually all traits (Boyle et al., 2017). It is reasonable to think that an organism is the result of the interaction of its interdependent parts rather than simply the resulting sum of independent factors.

There are arguably millions of traits one can describe in a complex organism, but the number of genes is generally much lower. Inevitably, exactly for the same principle that a phenotype corresponds to multiple genotypes, there are genes that must affect multiple traits. This phenomenon of one gene (or one mutation) affecting multiple traits is known as pleiotropy citep{allen2000adaptation, orr2005genetic}. Pleiotropy is a central topic in genetics and has broad implications for evolution citep{aymler1918correlation, allen2000adaptation, wagner2011pleiotropic, barton2016infinitesimal}. Pleiotropy is the cause of trade-offs among the adaptations in different traits, because a mutation that is advantageous to one trait may be disadvantageous for another trait. The quantitative modelling of this idea led to the so called “cost of complexity” hypothesis, which posits that complex organisms are inherently less evolvable or adaptable to changing environments with respect to simple organisms, because mutations have more pleiotropic effects (Fisher 1930). Recent advances in GWAS highlighted the fact that universal pleiotropy might seem the most likely scenario.

Both the infinitesimal model and the universal pleiotropy are more and more supported by empirical evidence as suggested by recent works (Boyle et al., 2017).Third, and most important point, in this work we elaborated on a particular aspect of the G-P map, namely the many to one relationship between genotypes and phenotypes, which is usually not taken into account in the standard interpretation of these evolutionary models. Accounting for phenotypic robustness in the Quasi-species and Price models lead us to the results presented and discussed in the following article.ibliographystyle{apalike}ibliography{BibIntro}subsection{Article II}{f Article in preparation,to be submitted to an evolutionary biology journal (e.

g., extit {J. Teor.