subsection{Genotype between the size of the genotype

subsection{Genotype networks}A phenotype may be realized by a number of different genotypes, which are said to form a extit {neutral network}  citep{fontana1993rna, kauffman1993origins, schuster1994sequences, gruner1996analysis, fontana2002modelling}, also called a genotype network citep{wagner2011origins, payne2014robustness}. Genotype networks resides in the genotype space. A genotype space comprises all sequences of a given length $L$. This is astronomically large, comprising $20^L$ protein genotypes (for amino acid sequences) or $4^L$ RNA or DNA genotypes (for nucleotide sequences). Evolutionary change takes place in populations of organisms, and each member of a population is considered having a single genotype.

It is thus useful to think of a population as a collection of genotypes in the genotype space. The members of this population “explore” this space through mutations. An especially important class of mutations is point mutations, which transform a genotype into one of its neighbours, the set of genotypes that differ only for one amino acid or one nucleotide from the original genotype. Typical genotype networks i) are vast (count very many genotypes), ii) extend widely across the genotype space (genotypes of the neutral network can differ significantly in their sequences), but iii) occupy a vanishing small volume of the genotype space (the ratio between the size of the genotype network and the size on the genotype space).

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Thus, a network can be traversed through many small mutational steps with little or no phenotypic change. Existence of these vast genotype networks was first suggested by computational models of phenotypic formation citep{lipman1991modelling, schuster1994sequences}, but they had been also observed in real macromolecules citep{babajide1997neutral}. It is important to remark that the definition of neutral genotype network adopted here is derived from a broader definition of neutrality citep{wagner2012role}.

In most cases mutations arising in a genotype are not strictly neutral, i.e. they can actually slightly affect the phenotype and thus fitness. For example, weakly deleterious mutations are more abundant than neutral mutations in most macromolecules, but they are often accompanied by compensatory genotypic changes that allow a preservation of the phenotype.

The simultaneous occurrence of multiple mutations can help a population “tunnel” through a region of low fitness in genotype space, and thus help preserve a phenotype citep{sawyer2007prevalence, eyre2002quantifying}. In other words, a combination of non-strictly neutral mutations can lead to an effective neutral phenotype due to their epistatic interaction.Another central feature of the genotype space is the genotype’ s neighbourhood, that is the collection of those genotypes that can be reached from a given genotype through one or few mutations. Neighbourhoods are important from both a qualitative and a quantitative point of view. This is because the set of different phenotypes in a neighbourhood is easily accessible by mutation. The size of this set is thus a simple measure of how phenotypically variable a genotype is in response to mutations citep{wagner2008robustness}.Genotype networks allow individuals in a population to preserve their phenotypes while changing their genotypes in many small mutational steps. This could lead to an increase in the population genetic variation even in the presence of a high selective pressure.

This neutral genetic variation is cryptic since is not manifested at the phenotypic level, however, as neighbourhoods of different genotypes typically contain different novel phenotypes citep{ wagner2005distributed}, a population of different genotypes on a genotype network can access a more vast sets of different novel phenotypes citep{espinosa2011phenotypic}.subsection{Phenotypic robustness and evolvability}Phenotypic robustness has the appearances of an attribute of the genotype-phenotype map that should oppose the adaptation process of populations. This is because there is a tension between the need of biological complex systems to evolve in a changing environment and the need to preserved their complex phenotypes from mutations citep{draghi2010mutational}.Consider a particular genotype of a genotype network.

We can define the mutational robustness of its associated phenotype as the proportion of neutral genotype neighbours at one or more mutational step of distance. This proportion of neutral genotype neighbours can be different for different genotypes in the neutral network. In fact, computational studies citep{wilke2001evolution, wagner2005circuit} show that in general neutral networks in genotype space can be arranged as “the galaxies in our universe”. This means that there are zones of high interconnection (high neutrality) and zones with lower connectivity. However, if we consider a population of individuals distributed in a neutral network, we can define a mean phenotype robustness, which is the mean of proportion of neutral neighbours of each genotype of a given population.

This scenario highlights some properties of robustness that are very important for adaptation dynamics. Adaptation dynamics (and ultimately evolution) are population phenomena but can be influenced by an individual property, namely phenotypic robustness, in several ways. Firstly, a saw before, highly robust phenotypes allow populations to access greater phenotypic variability, namely the total of neighbours with new different phenotypes. Secondly, phenotype robustness allows the accumulation of cryptic genetic variation that could be exapted citep{gould2002structure} to new mutational or environmental perturbations citep{wagner2008robustness, hayden2012environmental}. Thirdly, phenotypic robustness allows a faster cryptic exploration of the genotype space, increasing the probability to find a new superior phenotype even in the absence of substantial phenotypic variation. All these properties, deriving from the structures of neutral network, have been demonstrated computationally or experimentally for molecules such as RNAs citep{lipman1991modelling, gruner1996analysis}, rybozymes citep{stelling2004robustness, tanner1996determinants} and proteins citep{lipman1991modelling, rost1997protein}, while for whole organisms some evidence are provided in this work (see Chapter 1).subsection{Origin and evolution of phenotypic robustness}The present work is structured as a paper collection articulated in chapter, one for each article. Three out of four articles are in advance stage of preparation and have not been submitted yet.

Each chapter contains an introduction to the corresponding article, to allow the reader to familiarize with the most important concepts and theoretical or technical tools specific of the work, and to logically connect one article to the others.As illustrated above, most of the preceding studies aimed at exploring the long-term effects of phenotypic robustness on adaptation and evolvability. However, the lack of in vivo experimental evolution evidence, lead us to design an experiment to test the above-mentioned effects on evolvability in a real, organism level living system (Article I).

 In addition, except for some few studies, the mechanisms by which robustness might be established during evolution are far from clear and overall little explored citep{masel2009robustness, rigato2016enhancing}. Given that phenotypic robustness seems a quality that would oppose the adaptation process, how such a feature of living systems can be maintained throughout generations without, apparently, any short-term benefits and in a continuously changing environment? Is robustness an adaptation in historical sense, i.e.

a feature that has been shaped by natural selection? Or is it simply a by-product of evolution? Long term beneficial effects on evolvability cannot explain why high levels of phenotypic robustness are preserved in complex living system. A more complete treatment of adaptation within evolutionary theory should try to include phenotypic robustness as an evolvable parameter, rather than to treat it as a given. Accordingly, we tried to explore the possible causes of the origin and widespread persistence of phenotypic robustness in evolving complex living systems. We started adopting a theoretical approach with the aim of exploring the relation between phenotypic robustness, mutation, selection, adaptation and complexity either through deterministic models (Article II) and simulations accounting for stochasticity (Article III). Finally, we tried to test predictions of the theoretical model and to find evidences from empirical data supporting theoretical results (Article IV).ibliographystyle{apalike}ibliography{BibIntro}

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