The organization of biological sequences into constrained and unconstrained parts determines fundamental properties of genotype-phenotype maps

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.

Genotype-phenotype mapping with polyominos made from DNA origami tiles

The correlation between genetic information and characteristics of a living cell-its genotype and its phenotype-constitutes the basis of genetics. Here, we experimentally realize a primitive form of genotype-phenotype mapping with DNA origami. The DNA origami can polymerize into two-dimensional lattices (phenotype) via blunt-end stacking facilitated by edge staples at the seam of the planar DNA origami. There are 80 binding positions for edge staples, which allow us to translate an 80-bit long binary code (genotype) onto the DNA origami. The presence of an edge staple thus corresponds to a "1" and its absence to a "0." The interactions of our DNA-based system can be reproduced by a polyomino model. Polyomino growth simulations qualitatively reproduce our experimental results. We show that not only the absolute number of base stacks but also their sequence position determine the cluster size and correlation length of the orientation of single DNA origami within the cluster. Importantly, the mutation of a few bits can result in major morphology changes of the DNA origami cluster, while more often, major sequence changes have no impact. Our experimental realization of a correlation between binary information ("genotype") and cluster morphology ("phenotype") thus reproduces key properties of genotype-phenotype maps known from living systems.

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.

Fast free-energy-based neutral set size estimates for the RNA genotype-phenotype map

The genotype–phenotype (GP) map of RNA secondary structure links each RNA sequence to its corresponding secondary structure. Previous research has shown that the large-scale structural properties of GP maps, such as the size of neutral sets in genotype space, can influence evolutionary outcomes. In order to use neutral set sizes, efficient and accurate computational methods are needed to compute them. Here, we propose a new method, which is based on free energy estimates and is much faster than existing sample-based methods. Moreover, this approach can give insight into the reasons behind neutral set size variations, for example, why structures with fewer stacks tend to have larger neutral set sizes. In addition, we generalize neutral set size calculations from the previously studied many-to-one framework, where each sequence folds into a single energetically preferred structure, to a fuller many-to-many framework, where several low-energy structures are included. We find that structures with high neutral sets in one framework also tend to have large neutral sets in the other framework for a range of parameters and thus the choice of GP map does not fundamentally affect which structures have the largest neutral set sizes.

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.

On the emergence of structural complexity in RNA replicators

The RNA world hypothesis relies on the ability of ribonucleic acids to spontaneously acquire complex structures capable of supporting essential biological functions. Multiple sophisticated evolutionary models have been proposed for their emergence, but they often assume specific conditions. In this work, we explore a simple and parsimonious scenario describing the emergence of complex molecular structures at the early stages of life. We show that at specific GC content regimes, an undirected replication model is sufficient to explain the apparition of multibranched RNA secondary structures-a structural signature of many essential ribozymes. We ran a large-scale computational study to map energetically stable structures on complete mutational networks of 50-nt-long RNA sequences. Our results reveal that the sequence landscape with stable structures is enriched with multibranched structures at a length scale coinciding with the appearance of complex structures in RNA databases. A random replication mechanism preserving a 50% GC content may suffice to explain a natural enrichment of stable complex structures in populations of functional RNAs. In contrast, an evolutionary mechanism eliciting the most stable folds at each generation appears to help reaching multibranched structures at highest GC content.

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.

On the nature and origin of biological information: The curious case of RNA

Biological information is most commonly thought of in terms of biology's Central Dogma where DNA is viewed as a linearized code used to synthesize proteins. Using DNA's chemical cousin, RNA, as a case study we consider how biological information operates outside the linear arrangement of its polymeric subunits. Much like individual pieces of a jigsaw puzzle, particular structures enable biomolecules to undergo precise molecular interactions with one another based on their respective shapes. By exploring the relationship between sequence and structure in RNA we argue that biological information finds its ultimate functional fulfillment in the three-dimensional structural arrangement of its atoms. We show how recurrent structural RNA motifs-operating at the tertiary level of a molecule-provide robust building blocks for the formation of new structural configurations and thereby convey the information required for emergent biological functions. We posit that these same RNA structures, guided by their respective thermodynamic stabilities, experience selective pressure to maintain particular three-dimensional architectures over and above pressures to maintain a particular sequence of nucleotides. Ultimately, this framework for understanding the nature of biological information provides a useful paradigm for understanding its origins and how biological information can result from chaotic prebiotic conditions.

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

The mapping between biological genotypes and phenotypes plays an important role in evolution, and understanding the properties of this mapping is crucial to determine the outcome of evolutionary processes. One of the most striking properties observed in several genotype-phenotype (GP) maps is the positive correlation between the robustness and evolvability of phenotypes. This implies that a phenotype can be strongly robust against mutations and at the same time evolvable to a diverse range of alternative phenotypes. Here, we examine the causes for this positive correlation by introducing two analytically tractable GP map models that follow the principles of real biological GP maps. The first model is based on gene-like GP maps, reflecting the way in which genetic sequences are organized into protein-coding genes, and the second one is based on the GP map of RNA secondary structure. For both models, we find that a positive correlation between phenotype robustness and evolvability only emerges if mutations at one sequence position can have non-local effects on the sequence constraints at another position. This highlights that non-local effects of mutations are closely related to the coexistence of robustness and evolvability in phenotypes, and are likely to be an important feature of many biological GP maps.

Adding levels of complexity enhances robustness and evolvability in a multilevel genotype-phenotype map

Robustness and evolvability are the main properties that account for the stability and accessibility of phenotypes. They have been studied in a number of computational genotype-phenotype maps. In this paper, we study a metabolic genotype-phenotype map defined in toyLIFE, a multilevel computational model that represents a simplified cellular biology. toyLIFE includes several levels of phenotypic expression, from proteins to regulatory networks to metabolism. Our results show that toyLIFE shares many similarities with other seemingly unrelated computational genotype-phenotype maps. Thus, toyLIFE shows a high degeneracy in the mapping from genotypes to phenotypes, as well as a highly skewed distribution of phenotypic abundances. The neutral networks associated with abundant phenotypes are highly navigable, and common phenotypes are close to each other in genotype space. All of these properties are remarkable, as toyLIFE is built on a version of the HP protein-folding model that is neither robust nor evolvable: phenotypes cannot be mutually accessed through point mutations. In addition, both robustness and evolvability increase with the number of genes in a genotype. Therefore, our results suggest that adding levels of complexity to the mapping of genotypes to phenotypes and increasing genome size enhances both these properties.

The emergence of the two cell fates and their associated switching for a negative auto-regulating gene

Background

Decisions in the cell that lead to its ultimate fate are important for fundamental cellular functions such as proliferation, growth, differentiation, development, and death. These cell fate decisions can be influenced by both the gene regulatory network and also environmental factors and can be modeled using simple gene feedback circuits. Negative auto-regulation is a common feedback motif in the gene circuits. It can act to reduce gene expression noise or induce oscillatory expression and is thought to lead to only one cell fate. Here, we present experimental and modeling data to suggest that a self-repressor circuit can lead to two cell fates under specific conditions.

Results

We show that the introduction of inducers capable of binding and unbinding to a self-repressing gene product (protein), thus regulating the associated gene, can lead to the emergence of two cell states. We suggest that the inducers can alter the effective regulatory binding and unbinding speed of the self-repressor regulatory protein to its destination DNA without changing the gene itself. The corresponding simulation results are consistent with the experimental findings. We propose physical and quantitative explanations for the origin of the two phenotypic cell fates.

Conclusions

Our results suggest a mechanism for the emergence of multiple cell fates. This may explain the heterogeneity often observed among cell states, while illustrating that altering gene regulation strength can influence cell fates and their decision-making processes without genetic changes.

Electronic supplementary material

The online version of this article (10.1186/s12915-019-0666-0) contains supplementary material, which is available to authorized users.

On the networked architecture of genotype spaces and its critical effects on molecular evolution

Evolutionary dynamics is often viewed as a subtle process of change accumulation that causes a divergence among organisms and their genomes. However, this interpretation is an inheritance of a gradualistic view that has been challenged at the macroevolutionary, ecological and molecular level. Actually, when the complex architecture of genotype spaces is taken into account, the evolutionary dynamics of molecular populations becomes intrinsically non-uniform, sharing deep qualitative and quantitative similarities with slowly driven physical systems: nonlinear responses analogous to critical transitions, sudden state changes or hysteresis, among others. Furthermore, the phenotypic plasticity inherent to genotypes transforms classical fitness landscapes into multiscapes where adaptation in response to an environmental change may be very fast. The quantitative nature of adaptive molecular processes is deeply dependent on a network-of-networks multilayered structure of the map from genotype to function that we begin to unveil.

Structural properties of genotype-phenotype maps

The map between genotype and phenotype is fundamental to biology. Biological information is stored and passed on in the form of genotypes, and expressed in the form of phenotypes. A growing body of literature has examined a wide range of genotype-phenotype (GP) maps and has established a number of properties that appear to be shared by many GP maps. These properties are 'structural' in the sense that they are properties of the distribution of phenotypes across the point-mutation network of genotypes. They include: a redundancy of genotypes, meaning that many genotypes map to the same phenotypes, a highly non-uniform distribution of the number of genotypes per phenotype, a high robustness of phenotypes and the ability to reach a large number of new phenotypes within a small number of mutational steps. A further important property is that the robustness and evolvability of phenotypes are positively correlated. In this review, I give an overview of the study of GP maps with particular emphasis on these structural properties, and discuss a model that attempts to explain why these properties arise, as well as some of the fundamental ways in which the structure of GP maps can affect evolutionary outcomes.

Distribution of genotype network sizes in sequence-to-structure genotype-phenotype maps

An essential quantity to ensure evolvability of populations is the navigability of the genotype space. Navigability, understood as the ease with which alternative phenotypes are reached, relies on the existence of sufficiently large and mutually attainable genotype networks. The size of genotype networks (e.g. the number of RNA sequences folding into a particular secondary structure or the number of DNA sequences coding for the same protein structure) is astronomically large in all functional molecules investigated: an exhaustive experimental or computational study of all RNA folds or all protein structures becomes impossible even for moderately long sequences. Here, we analytically derive the distribution of genotype network sizes for a hierarchy of models which successively incorporate features of increasingly realistic sequence-to-structure genotype-phenotype maps. The main feature of these models relies on the characterization of each phenotype through a prototypical sequence whose sites admit a variable fraction of letters of the alphabet. Our models interpolate between two limit distributions: a power-law distribution, when the ordering of sites in the prototypical sequence is strongly constrained, and a lognormal distribution, as suggested for RNA, when different orderings of the same set of sites yield different phenotypes. Our main result is the qualitative and quantitative identification of those features of sequence-to-structure maps that lead to different distributions of genotype network sizes.

Adaptive multiscapes: an up-to-date metaphor to visualize molecular adaptation

Background

Wright’s metaphor of the fitness landscape has shaped and conditioned our view of the adaptation of populations for almost a century. Since its inception, and including criticism raised by Wright himself, the concept has been surrounded by controversy. Among others, the debate stems from the intrinsic difficulty to capture important features of the space of genotypes, such as its high dimensionality or the existence of abundant ridges, in a visually appealing two-dimensional picture. Two additional currently widespread observations come to further constrain the applicability of the original metaphor: the very skewed distribution of phenotype sizes (which may actively prevent, due to entropic effects, the achievement of fitness maxima), and functional promiscuity (i.e. the existence of secondary functions which entail partial adaptation to environments never encountered before by the population).

Results

Here we revise some of the shortcomings of the fitness landscape metaphor and propose a new “scape” formed by interconnected layers, each layer containing the phenotypes viable in a given environment. Different phenotypes within a layer are accessible through mutations with selective value, while neutral mutations cause displacements of populations within a phenotype. A different environment is represented as a separated layer, where phenotypes may have new fitness values, other phenotypes may be viable, and the same genotype may yield a different phenotype, representing genotypic promiscuity. This scenario explicitly includes the many-to-many structure of the genotype-to-phenotype map. A number of empirical observations regarding the adaptation of populations in the light of adaptive multiscapes are reviewed.

Conclusions

Several shortcomings of Wright’s visualization of fitness landscapes can be overcome through adaptive multiscapes. Relevant aspects of population adaptation, such as neutral drift, functional promiscuity or environment-dependent fitness, as well as entropic trapping and the concomitant impossibility to reach fitness peaks are visualized at once. Adaptive multiscapes should aid in the qualitative understanding of the multiple pathways involved in evolutionary dynamics.

Reviewers

This article was reviewed by Eugene Koonin and Ricard Solé.

The genotype-phenotype map of an evolving digital organism

To understand how evolving systems bring forth novel and useful phenotypes, it is essential to understand the relationship between genotypic and phenotypic change. Artificial evolving systems can help us understand whether the genotype-phenotype maps of natural evolving systems are highly unusual, and it may help create evolvable artificial systems. Here we characterize the genotype-phenotype map of digital organisms in Avida, a platform for digital evolution. We consider digital organisms from a vast space of 10¹⁴¹ genotypes (instruction sequences), which can form 512 different phenotypes. These phenotypes are distinguished by different Boolean logic functions they can compute, as well as by the complexity of these functions. We observe several properties with parallels in natural systems, such as connected genotype networks and asymmetric phenotypic transitions. The likely common cause is robustness to genotypic change. We describe an intriguing tension between phenotypic complexity and evolvability that may have implications for biological evolution. On the one hand, genotypic change is more likely to yield novel phenotypes in more complex organisms. On the other hand, the total number of novel phenotypes reachable through genotypic change is highest for organisms with simple phenotypes. Artificial evolving systems can help us study aspects of biological evolvability that are not accessible in vastly more complex natural systems. They can also help identify properties, such as robustness, that are required for both human-designed artificial systems and synthetic biological systems to be evolvable.

The phenotype of an organism comprises the set of morphological and functional traits encoded by its genome. In natural evolving systems, phenotypes are organized into mutationally connected networks of genotypes, which increase the likelihood for an evolving population to encounter novel adaptive phenotypes (i.e., its evolvability). We do not know whether artificial systems, such as self-replicating and evolving computer programs—digital organisms—are more or less evolvable than natural systems. By studying how genotypes map onto phenotypes in digital organisms, we characterize many commonalities between natural and artificial evolving systems. In addition, we show that phenotypic complexity can both facilitate and constrain evolution, which harbors lessons not only for designing evolvable artificial systems, but also for synthetic biology.