Phenotypes can be robust and evolvable if mutations have non-local effects on sequence constraints

Non-Poissonian Bursts in the Arrival of Phenotypic Variation Can Strongly Affect the Dynamics of Adaptation

Modeling the rate at which adaptive phenotypes appear in a population is a key to predicting evolutionary processes. Given random mutations, should this rate be modeled by a simple Poisson process, or is a more complex dynamics needed? Here we use analytic calculations and simulations of evolving populations on explicit genotype-phenotype maps to show that the introduction of novel phenotypes can be "bursty" or overdispersed. In other words, a novel phenotype either appears multiple times in quick succession or not at all for many generations. These bursts are fundamentally caused by statistical fluctuations and other structure in the map from genotypes to phenotypes. Their strength depends on population parameters, being highest for "monomorphic" populations with low mutation rates. They can also be enhanced by additional inhomogeneities in the mapping from genotypes to phenotypes. We mainly investigate the effect of bursts using the well-studied genotype-phenotype map for RNA secondary structure, but find similar behavior in a lattice protein model and in Richard Dawkins's biomorphs model of morphological development. Bursts can profoundly affect adaptive dynamics. Most notably, they imply that fitness differences play a smaller role in determining which phenotype fixes than would be the case for a Poisson process without bursts.

The non-deterministic genotype-phenotype map of RNA secondary structure

Selection and variation are both key aspects in the evolutionary process. Previous research on the mapping between molecular sequence (genotype) and molecular fold (phenotype) has shown the presence of several structural properties in different biological contexts, implying that these might be universal in evolutionary spaces. The deterministic genotype-phenotype (GP) map that links short RNA sequences to minimum free energy secondary structures has been studied extensively because of its computational tractability and biologically realistic nature. However, this mapping ignores the phenotypic plasticity of RNA. We define a GP map that incorporates non-deterministic (ND) phenotypes, and take RNA as a case study; we use the Boltzmann probability distribution of folded structures and examine the structural properties of ND GP maps for RNA sequences of length 12 and coarse-grained RNA structures of length 30 (RNAshapes30). A framework is presented to study robustness, evolvability and neutral spaces in the ND map. This framework is validated by demonstrating close correspondence between the ND quantities and sample averages of their deterministic counterparts. When using the ND framework we observe the same structural properties as in the deterministic GP map, such as bias, negative correlation between genotypic robustness and evolvability, and positive correlation between phenotypic robustness and evolvability.

Maximum mutational robustness in genotype-phenotype maps follows a self-similar blancmange-like curve

Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organized as bricklayer's graphs, so-called because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere sums-of-digits function from number theory. Interestingly, genotype-phenotype maps for RNA secondary structure and the hydrophobic-polar (HP) model for protein folding can exhibit phenotype robustness that exactly attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components from the bricklayer's graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula and associated bounds for the transition probabilities between such phenotypes.

Probabilistic Genotype-Phenotype Maps Reveal Mutational Robustness of RNA Folding, Spin Glasses, and Quantum Circuits

Recent studies of genotype-phenotype (GP) maps have reported universally enhanced phenotypic robustness to genotype mutations, a feature essential to evolution. Virtually all of these studies make a simplifying assumption that each genotype maps deterministically to a single phenotype. Here, we introduce probabilistic genotype-phenotype (PrGP) maps, where each genotype maps to a vector of phenotype probabilities, as a more realistic framework for investigating robustness. We study three model systems to show that our generalized framework can handle uncertainty emerging from various physical sources: (1) thermal fluctuation in RNA folding, (2) external field disorder in spin glass ground state finding, and (3) superposition and entanglement in quantum circuits, which are realized experimentally on a 7-qubit IBM quantum computer. In all three cases, we observe a novel biphasic robustness scaling which is enhanced relative to random expectation for more frequent phenotypes and approaches random expectation for less frequent phenotypes.

Robustness and stability of spin-glass ground states to perturbed interactions

Across many problems in science and engineering, it is important to consider how much the output of a given system changes due to perturbations of the input. Here, we investigate the glassy phase of ±J spin glasses at zero temperature by calculating the robustness of the ground states to flips in the sign of single interactions. For random graphs and the Sherrington-Kirkpatrick model, we find relatively large sets of bond configurations that generate the same ground state. These sets can themselves be analyzed as subgraphs of the interaction domain, and we compute many of their topological properties. In particular, we find that the robustness, equivalent to the average degree, of these subgraphs is much higher than one would expect from a random model. Most notably, it scales in the same logarithmic way with the size of the subgraph as has been found in genotype-phenotype maps for RNA secondary structure folding, protein quaternary structure, gene regulatory networks, as well as for models for genetic programming. The similarity between these disparate systems suggests that this scaling may have a more universal origin.

Predicting phenotype transition probabilities via conditional algorithmic probability approximations

Unravelling the structure of genotype-phenotype (GP) maps is an important problem in biology. Recently, arguments inspired by algorithmic information theory (AIT) and Kolmogorov complexity have been invoked to uncover simplicity bias in GP maps, an exponentially decaying upper bound in phenotype probability with the increasing phenotype descriptional complexity. This means that phenotypes with many genotypes assigned via the GP map must be simple, while complex phenotypes must have few genotypes assigned. Here, we use similar arguments to bound the probability P(x → y) that phenotype x, upon random genetic mutation, transitions to phenotype y. The bound is [Formula: see text], where [Formula: see text] is the estimated conditional complexity of y given x, quantifying how much extra information is required to make y given access to x. This upper bound is related to the conditional form of algorithmic probability from AIT. We demonstrate the practical applicability of our derived bound by predicting phenotype transition probabilities (and other related quantities) in simulations of RNA and protein secondary structures. Our work contributes to a general mathematical understanding of GP maps and may facilitate the prediction of transition probabilities directly from examining phenotype themselves, without utilizing detailed knowledge of the GP map.

The structure of genotype-phenotype maps makes fitness landscapes navigable

Fitness landscapes are often described in terms of 'peaks' and 'valleys', indicating an intuitive low-dimensional landscape of the kind encountered in everyday experience. The space of genotypes, however, is extremely high dimensional, which results in counter-intuitive structural properties of genotype-phenotype maps. Here we show that these properties, such as the presence of pervasive neutral networks, make fitness landscapes navigable. For three biologically realistic genotype-phenotype map models-RNA secondary structure, protein tertiary structure and protein complexes-we find that, even under random fitness assignment, fitness maxima can be reached from almost any other phenotype without passing through fitness valleys. This in turn indicates that true fitness valleys are very rare. By considering evolutionary simulations between pairs of real examples of functional RNA sequences, we show that accessible paths are also likely to be used under evolutionary dynamics. Our findings have broad implications for the prediction of natural evolutionary outcomes and for directed evolution.

Phenotype Bias Determines How Natural RNA Structures Occupy the Morphospace of All Possible Shapes

Morphospaces-representations of phenotypic characteristics-are often populated unevenly, leaving large parts unoccupied. Such patterns are typically ascribed to contingency, or else to natural selection disfavoring certain parts of the morphospace. The extent to which developmental bias, the tendency of certain phenotypes to preferentially appear as potential variation, also explains these patterns is hotly debated. Here we demonstrate quantitatively that developmental bias is the primary explanation for the occupation of the morphospace of RNA secondary structure (SS) shapes. Upon random mutations, some RNA SS shapes (the frequent ones) are much more likely to appear than others. By using the RNAshapes method to define coarse-grained SS classes, we can directly compare the frequencies that noncoding RNA SS shapes appear in the RNAcentral database to frequencies obtained upon a random sampling of sequences. We show that: 1) only the most frequent structures appear in nature; the vast majority of possible structures in the morphospace have not yet been explored; 2) remarkably small numbers of random sequences are needed to produce all the RNA SS shapes found in nature so far; and 3) perhaps most surprisingly, the natural frequencies are accurately predicted, over several orders of magnitude in variation, by the likelihood that structures appear upon a uniform random sampling of sequences. The ultimate cause of these patterns is not natural selection, but rather a strong phenotype bias in the RNA genotype-phenotype map, a type of developmental bias or "findability constraint," which limits evolutionary dynamics to a hugely reduced subset of structures that are easy to "find."

Gene duplication and subsequent diversification strongly affect phenotypic evolvability and robustness

We study the effects of non-determinism and gene duplication on the structure of genotype-phenotype (GP) maps by introducing a non-deterministic version of the Polyomino self-assembly model. This model has previously been used in a variety of contexts to model the assembly and evolution of protein quaternary structure. Firstly, we show the limit of the current deterministic paradigm which leads to built-in anti-correlation between evolvability and robustness at the genotypic level. We develop a set of metrics to measure structural properties of GP maps in a non-deterministic setting and use them to evaluate the effects of gene duplication and subsequent diversification. Our generalized versions of evolvability and robustness exhibit positive correlation for a subset of genotypes. This positive correlation is only possible because non-deterministic phenotypes can contribute to both robustness and evolvability. Secondly, we show that duplication increases robustness and reduces evolvability initially, but that the subsequent diversification that duplication enables has a stronger, inverse effect, greatly increasing evolvability and reducing robustness relative to their original values.

From genotypes to organisms: State-of-the-art and perspectives of a cornerstone in evolutionary dynamics

Understanding how genotypes map onto phenotypes, fitness, and eventually organisms is arguably the next major missing piece in a fully predictive theory of evolution. We refer to this generally as the problem of the genotype-phenotype map. Though we are still far from achieving a complete picture of these relationships, our current understanding of simpler questions, such as the structure induced in the space of genotypes by sequences mapped to molecular structures, has revealed important facts that deeply affect the dynamical description of evolutionary processes. Empirical evidence supporting the fundamental relevance of features such as phenotypic bias is mounting as well, while the synthesis of conceptual and experimental progress leads to questioning current assumptions on the nature of evolutionary dynamics-cancer progression models or synthetic biology approaches being notable examples. This work delves with a critical and constructive attitude into our current knowledge of how genotypes map onto molecular phenotypes and organismal functions, and discusses theoretical and empirical avenues to broaden and improve this comprehension. As a final goal, this community should aim at deriving an updated picture of evolutionary processes soundly relying on the structural properties of genotype spaces, as revealed by modern techniques of molecular and functional analysis.

On the evolution and development of morphological complexity: A view from gene regulatory networks

How does morphological complexity evolve? This study suggests that the likelihood of mutations increasing phenotypic complexity becomes smaller when the phenotype itself is complex. In addition, the complexity of the genotype-phenotype map (GPM) also increases with the phenotypic complexity. We show that complex GPMs and the above mutational asymmetry are inevitable consequences of how genes need to be wired in order to build complex and robust phenotypes during development. We randomly wired genes and cell behaviors into networks in EmbryoMaker. EmbryoMaker is a mathematical model of development that can simulate any gene network, all animal cell behaviors (division, adhesion, apoptosis, etc.), cell signaling, cell and tissues biophysics, and the regulation of those behaviors by gene products. Through EmbryoMaker we simulated how each random network regulates development and the resulting morphology (i.e. a specific distribution of cells and gene expression in 3D). This way we obtained a zoo of possible 3D morphologies. Real gene networks are not random, but a random search allows a relatively unbiased exploration of what is needed to develop complex robust morphologies. Compared to the networks leading to simple morphologies, the networks leading to complex morphologies have the following in common: 1) They are rarer; 2) They need to be finely tuned; 3) Mutations in them tend to decrease morphological complexity; 4) They are less robust to noise; and 5) They have more complex GPMs. These results imply that, when complexity evolves, it does so at a progressively decreasing rate over generations. This is because as morphological complexity increases, the likelihood of mutations increasing complexity decreases, morphologies become less robust to noise, and the GPM becomes more complex. We find some properties in common, but also some important differences, with non-developmental GPM models (e.g. RNA, protein and gene networks in single cells).

Using small samples to estimate neutral component size and robustness in the genotype-phenotype map of RNA secondary structure

In genotype-phenotype (GP) maps, the genotypes that map to the same phenotype are usually not randomly distributed across the space of genotypes, but instead are predominantly connected through one-point mutations, forming network components that are commonly referred to as neutral components (NCs). Because of their impact on evolutionary processes, the characteristics of these NCs, like their size or robustness, have been studied extensively. Here, we introduce a framework that allows the estimation of NC size and robustness in the GP map of RNA secondary structure. The advantage of this framework is that it only requires small samples of genotypes and their local environment, which also allows experimental realizations. We verify our framework by applying it to the exhaustively analysable GP map of RNA sequence length L = 15, and benchmark it against an existing method by applying it to longer, naturally occurring functional non-coding RNA sequences. Although it is specific to the RNA secondary structure GP map in the first place, our framework can probably be transferred and adapted to other sequence-to-structure GP maps.

Model genotype-phenotype mappings and the algorithmic structure of evolution

Cancers are complex dynamic systems that undergo evolution and selection. Personalized medicine approaches in the clinic increasingly rely on predictions of tumour response to one or more therapies; these predictions are complicated by the inevitable evolution of the tumour. Despite enormous amounts of data on the mutational status of cancers and numerous therapies developed in recent decades to target these mutations, many of these treatments fail after a time due to the development of resistance in the tumour. The emergence of these resistant phenotypes is not easily predicted from genomic data, since the relationship between genotypes and phenotypes, termed the genotype-phenotype (GP) mapping, is neither injective nor functional. We present a review of models of this mapping within a generalized evolutionary framework that takes into account the relation between genotype, phenotype, environment and fitness. Different modelling approaches are described and compared, and many evolutionary results are shown to be conserved across studies despite using different underlying model systems. In addition, several areas for future work that remain understudied are identified, including plasticity and bet-hedging. The GP-mapping provides a pathway for understanding the potential routes of evolution taken by cancers, which will be necessary knowledge for improving personalized therapies.