The genes for Vma3 and Vma11 must therefore share a common ancestry. Thornton and his colleagues concluded that early in the evolution of fungi, an ancestral gene for ring proteins was accidentally duplicated. Those two copies then evolved into Vma3 and Vma By comparing the differences in the genes for Vma3 and Vma11, Thornton and his colleagues reconstructed the ancestral gene from which they both evolved.
They then used that DNA sequence to create a corresponding protein—in effect, resurrecting an million-year-old protein.
The scientists called this protein Anc. They wondered how the protein ring functioned with this ancestral protein. To find out, they inserted the gene for Anc. They also shut down its descendant genes, Vma3 and Vma Normally, shutting down the genes for the Vma3 and Vma11 proteins would be fatal because the yeast could no longer make their rings. But Thornton and his co-workers found that the yeast could survive with Anc. It combined Anc. Experiments such as this one allowed the scientists to formulate a hypothesis for how the fungal ring became more complex.
Fungi started out with rings made from only two proteins—the same ones found in animals like us. The proteins were versatile, able to bind to themselves or to their partners, joining up to proteins either on their right or on their left. Later the gene for Anc. These new proteins kept doing what the old ones had done: they assembled into rings for pumps.
But over millions of generations of fungi, they began to mutate. Some of those mutations took away some of their versatility. Vma11, for example, lost the ability to bind to Vma3 on its clockwise side. Vma3 lost the ability to bind to Vma16 on its clockwise side. These mutations did not kill the yeast, because the proteins could still link together into a ring. They were neutral mutations, in other words. But now the ring had to be more complex because it could form successfully only if all three proteins were present and only if they arranged themselves in one pattern.
Thornton and his colleagues have uncovered precisely the kind of evolutionary episode predicted by the zero-force evolutionary law. Over time, life produced more parts—that is, more ring proteins.
And then those extra parts began to diverge from one another. The fungi ended up with a more complex structure than their ancestors had. But it did not happen the way Darwin had imagined, with natural selection favoring a series of intermediate forms.
Instead the fungal ring degenerated its way into complexity. Gray has found another example of constructive neutral evolution in the way many species edit their genes. When cells need to make a given protein, they transcribe the DNA of its gene into RNA, the single-stranded counterpart of DNA, and then use special enzymes to replace certain RNA building blocks called nucleotides with other ones.
RNA editing is essential to many species, including us—the unedited RNA molecules produce proteins that do not work.
But there is also something decidedly odd about it. Why don't we just have genes with the correct original sequence, making RNA editing unnecessary? The scenario that Gray proposes for the evolution of RNA editing goes like this: an enzyme mutates so that it can latch onto RNA and change certain nucleotides.
This enzyme does not harm the cell, nor does it help it—at least not at first. Doing no harm, it persists. Later a harmful mutation occurs in a gene. It shields the cell from the harm of the mutation, allowing the mutation to get passed down to the next generation and spread throughout the population. The evolution of this RNA-editing enzyme and the mutation it fixed was not driven by natural selection, Gray argues. David Speijer, a biochemist at the University of Amsterdam, thinks that Gray and his colleagues have done biology a service with the idea of constructive neutral evolution, especially by challenging the notion that all complexity must be adaptive.
But Speijer worries they may be pushing their argument too hard in some cases. On one hand, he thinks that the fungus pumps are a good example of constructive neutral evolution. In other cases, such as RNA editing, scientists should not, in his view, dismiss the possibility that natural selection was at work, even if the complexity seems useless.
Gray, McShea and Brandon acknowledge the important role of natural selection in the rise of the complexity that surrounds us, from the biochemistry that builds a feather to the photosynthetic factories inside the leaves of trees. Yet they hope their research will coax other biologists to think beyond natural selection and to see the possibility that random mutation can fuel the evolution of complexity on its own. This article was produced in collaboration with Quanta Magazine , an editorially independent division of SimonsFoundation.
Daniel W. McShea and Robert N. University of Chicago Press, This article and more information about Quanta Magazine are available at www. Already a subscriber? Sign in. Thanks for reading Scientific American. Create your free account or Sign in to continue. See Subscription Options. Discover World-Changing Science. Conventional wisdom holds that complex structures evolve from simpler ones, step-by-step, through a gradual evolutionary process, with Darwinian selection favoring intermediate forms along the way.
But recently some scholars have proposed that complexity can arise by other means—as a side effect, for instance—even without natural selection to promote it. Even past natural mass extinctions were only temporary setbacks that may have created even more opportunities for diversity in the long run. As organisms evolve more complicated systems of development, they may, however, become less able to modify certain aspects of their anatomy. For example, nearly all mammals — from giraffes to humans — are stuck with just seven neck bones.
Whenever different numbers develop or evolve, they bring other anatomical problems. Birds are entirely different, and seem to evolve new numbers of neck vertebrae with remarkable ease: Swans alone have between 22 and But in general, while evolution produces new species, the flexibility of the body plans of those species may decrease with rising complexity.
Quite often, closely related species end up being selected along similar paths. Take mammals. They come from a common ancestor, and have taken strikingly similar forms even though they have evolved on different continents. Clearly, there is an apparent contradiction at the heart of evolutionary biology.
On one hand, the mechanisms of evolution have no predisposition for change in any particular direction. On the other hand, let those mechanisms get going, and beyond some threshold, the interwoven ecological and developmental systems they generate tend to yield more and more species with greater maximum complexity. So can we expect more diversity and complexity going forward? We are now at the beginning of a sixth mass extinction , caused by humans and showing no signs of stopping — wiping out the results of millions of years of evolution.
Despite this, humans themselves are too numerous, widespread and adaptable to be at serious risk of extinction any time soon. It is far more likely that we will extend our distribution yet further by engineering habitable biospheres on other planets. On other planets, we may one day find alien life. Instead, we want to understand how complexity evolves when it does, i. Complex morphologies evolved from simpler morphologies. Therefore, to understand the evolution of complexity, we will ask how often mutations increase morphological complexity when the morphology is simple and how often they do it when morphology is already complex.
In this line of thinking, recent work on a specific organ teeth suggests there is a mutational asymmetry. It is easier to experimentally manipulate development to produce a decrease in morphological complexity than to produce an increase [ 25 , 26 ].
Similar views have been expressed for natural variation in populations [ 27 ]. In this article we use a general mathematical model of embryonic development to explore whether this mutational asymmetry is a consequence, or a side-effect, of how genes and cells need to be wired into networks to produce complex robust morphologies.
We will also ask if another side-effect of complex robust morphologies is to have a complex relationship between genetic and phenotypic variation, or genotype-phenotype map GPM.
The GPM is just an association, or map. The GPM does not tell which morphologies are possible or how they form in development. The GPM only shows which of the morphologies that are possible through development are associated with which specific genetic variants.
One of the major tenets of developmental evolutionary biology or evo-devo is that the GPM is very complex [ 28 — 35 ]. Morphology is constructed during development through complex networks of interactions between gene products, cells, and biophysical processes. In other words, genes do not have intrinsic effects on morphology.
Genes affect morphology because they affect the dynamics of the networks of gene, cell and biophysics where they are embedded. In other words, the GPM does not depend only on genes or gene interactions, it also depends on cell interactions and biophysics.
In this article, as elsewhere [ 30 , 36 ], a GPM is considered to be complex when small genetic changes can lead to relatively large morphological changes. Complex GPMs have been suggested to hamper evolution [ 20 , 37 , 38 ]: With a complex GPM, small genetic changes can lead to relatively large morphological changes and, on average, the similarity between parents and offspring decreases.
In that case adaptive phenotypic variations in the parents are less likely to pass to their offspring, thus hampering evolution [ 20 , 37 , 38 ]. Studying how the complexity of the GPM differs between simple and complex morphologies is thus relevant to understand how complex morphologies can arise and evolve. There are many theoretical studies on the general properties of the GPM. Most of these studies do not consider morphology and development, rather the other phenotypic levels and the processes other than development by which those levels are constructed.
The mapping between the primary genotype and secondary phenotype structures of RNA can be considered a GPM that has been extensively studied by mathematical modeling [ 39 — 43 ]. Similar, but coarser model studies exist for protein structure [ 44 — 46 ] and gene networks within single cells [ 47 — 50 ].
Only few GPM studies consider genetic and mechanical interactions between cells in a spatial context and how these lead to complex multicellular phenotypes i.
The studies that do, only consider specific organs [ 51 — 56 ] or only consider gene networks and cell signaling [ 57 — 62 ] without mechanical interactions or cell behaviors. Herein, we complement these latter approaches by using a general modeling framework for development, EmbryoMaker [ 63 ], which considers gene networks and cell signaling, but also mechanical interactions and cell behaviors.
This modeling approach is not specific to any organ. Instead, it considers animal development in general. The study of this model should inform us about properties of the GPM that are shared between phenotypic levels RNA, protein, single cell gene networks, and morphology and highlight the properties not shared between these levels. Despite the striking complexity of animals, some maintain that their development can be accomplished by a finite number of cell behaviors and interactions [ 64 — 67 ].
These cell behaviors are cell division, cell adhesion, cell death, cell growth, cell contraction, extracellular matrix secretion, signaling and extracellular signal reception and cell differentiation [ 64 — 67 ]. Cell migration and cell shape changes that result from certain patterns of cell contraction and adhesion could also be considered as cell behaviors.
In addition to cell behaviors, development involves gene product and cell interactions. Gene products interact in networks to regulate each other, cell behaviors, and cell interactions [ 68 ]. Cell interactions during development occur mainly through molecular signals extracellularly diffusing or membrane-bound and forces. Signaling often involve extracellularly diffusive molecules [ 68 ], while forces are generated by cell behaviors e. Both signaling and forces can lead to changes in the gene expression [ 68 ].
Altering gene expression can result in regulatory changes in the behaviors and the biophysical properties of cells. The gene expression in certain cells is also regulated by cell behaviors, since cell behaviors can induce cell movements, which alter the location of cells in space.
This cell movements will influence the spatial allocation of extracellular signals that elicit gene expression changes in cells [ 64 ].
As in [ 64 ], we use the concept of developmental mechanism. A developmental mechanism is defined as a gene network, the cell behaviors, and cell bio-physical properties it regulates. Developmental mechanisms can be seen as necessary for transforming morphologies over developmental time, i.
The building of morphological complexity can be seen as a sequence of such morphological transformations over developmental time Fig 2.
In turn, morphological evolution can be seen as changes in these transformations between generations. A Initial condition used in every simulation. Cylinders represent epithelial cells, the apical side in blue and the basal side in purple. The pink spheres represent mesenchymal cells. B The network diagrams in the center represent two idealized developmental mechanisms. Together with the initial condition depicted in their left they lead to the transformation of the initial morphology into other morphologies, on the right.
In the initial morphology, color shows the starting level of expression of gene 1. In contrast, in the resulting morphology color shows the position along the z-axis. C One of the developmental mechanism found in this research left. We show four different pattern transformations time points for two different gene products.
The morphological and gene expression changes result from the dynamics of the developmental mechanism left. If as suggested above, morphological transformations in development involve gene networks and a finite number of cell behaviors and cell interactions, then any computational model that includes them could potentially reproduce the range of morphological transformations possible in animal development.
In this work, we used one such model, EmbryoMaker [ 63 ] to perform an ensemble study. We randomly wired gene products, cell behaviors, and cell mechanical properties into a huge number of developmental mechanisms i.
We explored whether mutational asymmetry and complex GPMs are a general property of most or all, of the developmental mechanisms that are able to produce complex morphologies in such an ensemble. We are fully aware that developmental mechanisms found in nature are not completely random, but the outcome of evolution.
Nevertheless, a random sample of developmental mechanisms provides a general view on possible developmental mechanisms that is not restricted to those that have actually occurred in nature or to those that happen to be known to current developmental biology.
In other words, by modeling a large number of different in silico morphologies and their statistical analysis we aim at understanding general principles of development and the GPM. EmbryoMaker is a general computational model of animal development that can simulate any gene network, most animal cell behaviors division, adhesion, apoptosis, etc.
Each cell has a number of variables that include its position in 3D space, its size, many mechanical properties, and the level of expression of genes. These variables take continuous values and change according to a set of differential equations one equation per variable in each cell, see S1 Text , and Fig 3. Some of these equations calculate how the expression of a gene in a cell changes due to the other genes expressed in a cell and incoming extracellular signals see Fig 3C and 3D.
Secretion and diffusion of extracellular signals are calculated by other differential equations. Additional differential equations calculate how cell positions change due to pushing and pulling by neighboring cells at a given moment, and how movement is affected by the mechanical properties of each cell see Fig 4.
Similarly, a set of differential equations are used to calculate how the mechanical properties of cells change due gene regulation and incoming forces. In addition to variables that can change over developmental time , each equation takes a set of parameters that do not change over developmental time and are supposed to be determined genetically. These parameters determine, for example, how strongly a gene product regulates another, the diffusivity of an extracellular signal, or how strongly a gene product regulates a specific cell behavior according to its concentration.
A EmbryoMaker models three types of elements, epithelial and mesenchymal cells, and extracellular nodes. There are two types of epithelial nodes, apical in violet and basal blue , which together form a cylinder or epithelial cell. Mesenchymal cells and extracellular matrix components are made of single spherical nodes.
B Cell contraction. Nodes of a cell can change their size by decreasing or increasing their equilibrium radius p EQD.
This can result in epithelial cells with a conical shape. C Extracellular signaling. D Gene product transcription regulates the expression of a gene. E-J Cell behaviors implemented in EmbryoMaker.
E Cell adhesion. Two cells whose radius of adhesion p ADD , blue sphere overlap, are considered to be in contact. If they are in contact and if they adhere to each other, they will come closer together until they reach their radius of equilibrium p EQD , purple sphere.
F Extracellular matrix secretion. ECM can be secreted by any cell that expresses a gene product regulating this cell behavior. G Cell division. H Epithelial-mesenchymal transition EMT. I Apoptosis. Cell undergoing apoptosis will gradually decrease their size until they reach a minimal size and are completely eliminated. A When two nodes are at a distance d smaller than d ADD they experience an attractive force, when they are closer than d EQD they experience a repulsive force.
The values of these radii can change over time as a result of gene expression or external pressures. The direction of the interaction is from the center of one node to the center of the other red arrows.
The interaction of a spherical node either mesenchymal or from the ECM with the apical or basal side of an epithelial node is always parallel to the apical-basal axis of the cylinder and perpendicular to it when the interaction is lateral.
B-D Depict mechanical forces specific for epithelia. B The two nodes composing a cylinder are connected by an unbreakable spring black line. Elastic forces will always follow the direction of that spring. The spring has an equilibrium distance d EQS , if the distance between the centers of the nodes d in a epithelial cylinder are closer than d EQS left figure in B , elastic forces will repulse the nodes red arrows.
If the distance between the centers of the nodes is greater than d EQS , elastic forces will attract them this distance is again the sum of the mechanical property, p EQS , of the two nodes that is itself a variable of the model. D The bending rotational force applies in the direction connecting the two epithelial nodes from the same side see supplementary.
Cells are represented by nodes in space and these can take a spherical shape, for mesenchymal cells, or a cylindrical shape, for epithelial cells. In addition, the extracellular matrix ECM is also represented by spherical nodes. Cell behaviors are represented by discrete rules on these elements see Fig 4. For example, cell division is implemented as duplication of a cell element. EmbryoMaker is a modeling framework.
By providing a specific gene network and the cell behaviors and mechanical properties it regulates i. To simulate a morphology, EmbryoMaker requires the specification of a developmental mechanism i. All the simulations in this article start from the same very simple initial condition, a small flat epithelial sheet with one gene expressed in a gradient across the sheet.
Different developmental mechanisms can have a different number of genes and, thus, a different number of parameters. For example, the two developmental mechanisms depicted in Fig 2 , have one parameter for each gene interaction 7 and 10 respectively and one parameter for the regulation of cell division by gene 6 in one developmental mechanism and gene 7 in the other developmental mechanism.
In the ensemble of random developmental mechanisms, we will also study whether there is a general relationship between morphological complexity and robustness. We will restrict our analysis to robustness understood in the narrow sense of developmental instability. Developmental instability reflects the difference between the morphologies of individuals which share the same genotype and environment [ 71 — 73 ].
Developmental instability can also decrease the efficiency of natural selection because it diminishes the likelihood that adaptive variation in the parents passes to their offspring. We found that some of the developmental mechanisms in the ensemble, i.
The frequency of these morphologies in the ensemble, however, decreases with their complexity Fig 5. In other words, developmental mechanisms producing complex morphologies are much rarer than the developmental mechanisms producing simple morphologies. In addition, we found that a large proportion of the interactions in each developmental mechanism were not necessary for producing the observed morphology, i. This is perhaps not surprising, given the fact that the developmental mechanisms were randomly built.
The superfluous interactions of each developmental mechanism were pruned. The rest of the analysis in this article considers only these pruned developmental mechanisms. Unsurprisingly, the number of non-superfluous interactions necessary for the development of a morphology increases with the complexity of such a morphology S1B Text.
The histograms show the distribution of complexity for AV and OPC in the morphologies found in the ensemble. The right panels show example morphologies and their complexities. As in a previous study [ 10 ], we found that the developmental mechanisms that can lead to complex morphologies tend to have higher developmental instability than the developmental mechanisms that can only lead to simple morphologies see S1C Text. To measure developmental instability, each developmental mechanism was simulated ten times.
All simulations in EmbryoMaker include noise, i. We call each simulation of a developmental mechanism a twin. We took the morphological distance between twins as a measure of developmental instability of the underlying developmental mechanism.
The morphological distance between twins was calculated via three methods: 1 as the procrustes distance between homologous cells between morphologies Homologous morphological distance or HMD ; 2 as the average differences in the local convexity between the homologous cells of different morphologies Convexity morphological distance or CMD ; or 3 as the average minimal distances between each cell in a morphology and each cell in the other morphology Euclidian Morphological distance or EMD; see S1D Text and examples in S1E Text.
See Methods 3 for a more detailed description of these measures of morphological distance. Each developmental mechanism in the ensemble was simulated with a specific random combination of parameter values. To better understand these developmental mechanisms, we performed an iso-morphological random walk in some of the developmental mechanisms in the ensemble.
We call these developmental mechanisms the parental set. The developmental mechanisms in the parental set were chosen to produce morphologies evenly spaced along the possible range of morphological complexities in the ensemble. In each step of the walk an IS-mutations was introduced to the developmental mechanism. The step was accepted if it did not change the resulting morphology beyond a small threshold morphological distance see Fig 6 and Methods 2.
IS-mutations are defined as mutations that change the value of a parameter, but do not change which genes interact, or which cell behaviors are regulated by gene products i. The iso-morphological random walks estimate the region of the parameter space of each developmental for which a specific morphology will form.
We found that the simpler the morphology, the larger this region Fig 6 and S1F Text. Conversely, the more complex a morphology is, the smaller this region. In other words, even when a given developmental mechanism can produce a complex morphology, this is only possible for a small range of values within its developmental parameters.
Thus, the most complex morphologies of a developmental mechanism are only possible for small regions of the parameter space of such mechanism. Note that each developmental mechanism may include a different network of interactions and that the strength of each interaction is specified by a different parameter.
Based on this the number of developmental parameters can differ between developmental mechanisms S1G Text. The figure shows the number of accepted steps in each iso-morphological random walk of each developmental mechanism Y axis versus the complexity of the morphology produced by each developmental mechanism. The Y axis, thus, is a measure of the region of the parameter space where a morphology forms.
We performed an iso-morphological random walk for the developmental mechanisms in the parental set that are very stable developmentally i. EMD distance between parental twins less than 0. In each walk we mutated, one at a time and chosen randomly, gene-gene interactions or gene-cell behavior interactions in each developmental mechanism.
If a mutation did not change in a significant way the phenotype when compared to the original parental morphology in the walk the mutation was kept, and a new mutation was applied. If the mutation did change the phenotype, this mutation was reversed. This process of mutation was iterated times per developmental mechanism see Methods 2. This way we calculated the proportion of mutations that changed the phenotype: the more mutations changed the parental phenotype, the smaller the region of the parameter space where a developmental mechanism can produce its parental phenotype.
We performed 10 random walks per developmental mechanisms. To minimize the effect of random developmental noise in our results each mutant was simulated 5 times see S1 Text for details.
The morphological distance between the final morphology of each of these 5 simulations and the parental was measured using CMD.
The average of these distances was used to evaluate whether a mutation was accepted or not. In order to be considered different to the parent, the CMD had to be 0. See—Methods 2. Our study found that there is a global degeneracy in the space of developmental mechanisms, i. This is seen by calculating the morphological distances between morphologies in the ensemble. Degeneracy, however, occurs primarily for simple morphologies and it is very rare for complex morphologies see S1H and S1I Text.
In addition, these figures also show that complex morphologies produced by different developmental mechanisms are more different from each other than the simple morphologies produced by different developmental mechanisms. Moreover, complex morphologies tend to be very different from simple ones. To ensure that these results were not due to the higher developmental instability of complex morphologies, the morphological distances in these figures S1H and S1I Text were calculated between the mean morphologies of each pair of developmental mechanisms in the ensemble.
This mean morphology was calculated by averaging the morphologies of all the twins of a developmental mechanism in the ensemble to calculate this average we used the position of homologous cells between the morphologies; see -Methods 3. Notice, that simple morphologies can be quite different from each other see S1J Text , so our results do not stem from how we define complexity. To study mutational asymmetry, we performed a mutational screening of a random sample of developmental mechanisms in the ensemble Methods 2.
Each developmental mechanism in the ensemble, what we call a parent, was IS-mutated to produce offspring developmental mechanisms. Each mutant differed from its parent in only one parameter value. The figures show a clear mutational asymmetry. The higher the complexity of the parent, the larger the proportion of offspring that are simpler than their parents. Most of the developmental mechanisms producing complex morphologies in the ensemble had some offspring with the minimal possible complexity a flat epithelium.
Thus, many complex morphologies are one mutation away from the simplest morphologies. However, the reverse is not true as the vast majority of simple parents were not one mutation away from producing complex morphologies see Fig 7 and S1K—S1N Text. This mutational asymmetry was even more evident when the mutational analysis was done with Topological or T-mutations see S1L Text. T-mutations are defined, as in [ 36 ], as mutations that change which genes interact with which other genes, cell behaviors, or cell mechanical properties, i.
The plot shows the distribution of the difference in complexity between each parent and its mutant offspring in the Y-axis versus the complexity of the parent X axis. Parents with similar complexity are clumped together in X bins and offspring with similar complexities are clumped together in Y bins. The gray scale in each box indicates which proportion of the offspring of parents of a given complexity in X exhibit a given complexity.
The plot shows that most offspring have a complexity similar to that of their parents. It also shows that, for complex morphologies, there are more offspring that are simpler than their parents than offspring that are more complex than their parents i.
Notice that even the most complex parents can have very simple offspring but that simple parents rarely have very complex offspring see also S1L—S1N Text. Offspring was generated by mutating each parameter of the parent. Each parameter was IS-mutated the same number of times and each mutant offspring had only one mutation. Offspring is arranged along the y-axis according to their complexity minus their parental complexity and along the x-axis according to the complexity of their parents.
The x-y plane is divided in square bins. For A the size of each bin is 0. The darkness of each bin represents the natural logarithm of the relative abundance of offspring of a given parental complexity x-axis.
To calculate the relative abundance, for each column x-axis , we divide the number of offspring falling in each bin by the total number of offspring in that column. Thus, the relative abundance of each column in a plot sum 1.
On the right we show some examples of offspring morphologies. The red asterisks mark the complexity of the examples. See S1 Text for details. To study the GPM we use the mutational screening of the previous section to calculate the regression between genetic and morphological distances among the IS-mutant offspring of each parent developmental mechanism as in Fig 7.
We calculated one GPM regression per developmental mechanism and parameter i. The distance in parameter values between two mutants of the same parent was taken as a proxy for their genetic distance details in Fig 8 and S1 Text. In that case, the GPM regression coefficient is relatively large. Conversely, the GPM regression coefficient should be small if morphological distance increases slowly with genetic distance a simpler GPM , most mutations have small gradual morphological effects.
In addition, such a regression should also be small if developmental instability is so large that twins are as different from each other as they are different to their non-twin brothers or parents. In this latter case, as seen in Fig 8C , the low regression arises from the noisiness of the plot. In addition, such a regression should also be small if developmental instability is so large that twins are as different from each other as to the other non-twin mutant offspring or parents.
Each point in the Y axis shows the average morphological distance between all the offspring with a parameter value and all the offspring with another parameter value at a specific genetic distance. The morphologies in C have a higher developmental instability than the morphologies on B , but since the morphological distances between offspring do not change with genetic distance, they both have very small regression coefficients.
D-F Plots of the GPM regressions coefficients of each parent and parameter against the parent complexity. In other words, the slopes of the GPM plots as the ones shown in A, B and C are plotted against the complexity of the parent. In this plot there would be one point per parent and parameter but those are binned into boxes of 0. The line in the box shows the median and the gray diamond the average for that interval.
The whiskers extent 1. Spearman correlations. See section 6 in S1 Text. However, non-linear regression measures are unlikely to be robust to noise and the small number of points per parameter in our study. It should also be noted, the form of these non-linearities could be different for each developmental mechanism and parameters. Thus, even if enough points would be available, it would be difficult to compare the different parameters and developmental mechanisms.
This is not the case with a linear regression. Note also that the y-axis in the GPM plots is the morphological distance, not the value of any morphological trait, and that two morphologies can be quite different from each other, yet at the same morphological distance from a third e.
Each offspring was simulated 10 times to control for noise. Fig 8 shows that the complexity of the GPM increases with the complexity of the morphology. The higher the complexity, the larger the morphological changes produced by small genetic changes. The result in Fig 8 is not due to the higher developmental instability of complex morphologies. If that were the case, the morphological distance between twins would be roughly as large as the morphological distance between brothers i. For the developmental mechanisms producing complex morphologies, the GPM regression coefficients would necessarily be small, as described above.
That is the contrary of what we observe and, thus, the developmental mechanisms producing complex morphologies have inherently more complex GPMs. We found that many developmental mechanisms had parameters that could be changed without leading to any morphological change see Fig 8B , even after totally superfluous interactions were eliminated see first section of the results. We also found that some of the interactions associated with those parameters could not be deleted without the morphology changing dramatically.
In other words, these interactions are not superfluous, but rather required for the development of a morphology. However, they do not contribute much to the morphological variation. Based on this, we re-analyzed the GPM plots for the parameters of each developmental mechanism that have a large contribution to morphological variation, e.
The same occurs if we only focus on the GPM regression for the proliferation rate parameter i. The result that the GPM plots of complex morphologies have larger regression coefficients implies that more diverse morphologies are accessible by mutation from complex morphologies. In other words, if the regression is high it means that mutants are more morphological different from each other, thus overall there is a higher diversity of morphologies.
This means that a larger disparity of morphologies is possible from genetic variation in the developmental mechanisms that can lead to complex morphologies than in the developmental mechanisms that cannot. Our results can be summarized as follows: 1 Most developmental mechanisms do not produce complex morphologies; 2 Those that do, can only produce them for a relatively narrow range of their parameter values; 3 complex morphologies are developmentally unstable; 4 Simple morphologies can be produced by different developmental mechanisms while complex morphologies often cannot; 5 Complex morphologies are more different from each other than simple morphologies; 6 Mutational asymmetry is common and increases with morphological complexity, that is mutations are more likely to decrease than increase complexity; 7 Developmental mechanisms that lead to complex morphologies tend to have more complex GPMs than developmental mechanisms that lead to simple morphologies; and 8 The developmental mechanisms of complex morphologies, when mutation occurs, can lead to more diverse morphological variation than the developmental mechanisms that can only produce simple morphologies.
Some of our results were also found in GPM models at other phenotypic levels. This is an interesting result since different phenotypic levels are quite diverse and form via very different mechanisms e.
Moreover, our model differs from other models, like RNA models, because it does not include a proper genotype, but a set of parameters, as utilized in gene network and other development models [ 51 — 62 ].
In this discussion we will call the specific combination of parameters in an individual a genotype i. An additional difference between our model and most other GPM models is that the dimensionality of the genotype and the phenotype is not fixed. In other words, the ensemble has developmental mechanisms with different number of genes and interactions and morphologies with different number of cells each of them can vary along the x, y, and z coordinates and, thus, there are 3N c dimension per morphology, where N c is the number of cells.
One property found in most, if not all, GPM models is that different phenotypes have very different frequencies [ 41 — 50 , 74 — 76 ]. That is, some phenotypes are associated with many different genotypes while others are associated with only a few genotypes. This is also the case in our study; however, in our study we also found that the common phenotypes happen to be simple and the complex phenotypes happen to be rare. Three other studies have found a similar relationship between frequency and phenotypic complexity [ 74 , 76 , 77 ].
These studies also found that the individuals with complex phenotypes tend to mutate into a larger diversity of other phenotypes than the individuals with simple morphologies. The earliest of these studies uses a computational model of the development of a specific organ teeth [ 74 ]. The second study is a model of small mutable computer programs [ 76 ]. The first study, thus, applies only to a specific organ and the related developmental mechanisms, while the second model has no direct biological analogue.
The third study [ 77 ] provides some analytical arguments for the lower frequency of complex morphologies and shows that this applies to four relatively simple GPM models; the RNA model, a finances model, a model of circadian clocks, and a very simple model of branching in plants. Another property found in many GPM models [ 41 — 45 , 47 — 50 , 75 , 76 ] is that the neutral networks of the most common phenotypes are intertwined and percolate the genotypic space.
Neutral networks are sets of genotypes that lead to the same phenotype and that can be transformed into each other through simple mutations that do not change that phenotype [ 40 — 43 , 47 — 49 , 75 , 76 , 78 ]. The intertwining and percolation of the common phenotypes means that genotypes of rare phenotypes can be transformed into the genotypes of common phenotypes by just one, or few, mutations [ 39 — 42 , 44 , 49 , 79 ]. In our development model, there is also intertwining and percolation of the neutral networks of the most common phenotypes, that is the simplest ones.
In fact, the simpler the morphology the larger its neutral network as shown in the iso-morphological random walks, Fig 6. In our case, any genotype is only one or few mutations away from a genotype of a simple morphology. In other words, most morphologies can be transformed into very simple morphologies by a single mutation in the underlying developmental mechanism.
However, there is a strong mutational asymmetry. Most simple morphologies cannot be transformed into complex morphologies by a single mutation in the underlying developmental mechanism. In fact, in most cases, many mutations would be required for that to occur, i.
This is the mutational asymmetry shown in Fig 7. In addition, we found that complex morphologies tend to form clusters in the parameter space i. This is because complex morphologies can be transformed into very simple ones by a single mutation, but they can also be transformed into slightly less complex morphologies.
The latter, however, are not as likely to be transformed into the former by single mutations due to the mutational asymmetry. These slightly less complex morphologies are found in large areas of the parameter space and are made possible by many more developmental mechanisms.
Thus, the parameter space can be seen as having a structure in respect to morphological complexity see Fig 9. Complex morphologies form clusters of neutral networks, with the less complex morphologies bordering the more complex, while all of them are in contact with the neutral networks of the simplest morphologies. Idealization of the developmental parameter space of developmental mechanisms.
Each colored region represents a parameter region i. As in our results, the simpler morphologies occupy larger regions of the space and most such regions are in contact with the region producing the simplest morphology in white. The regions with complex morphologies tend to neighbor regions that also produce complex morphologies.
To our knowledge no previous study has explored the relationship between morphological complexity and developmental instability, or between morphological complexity and the complexity of the GPM. One evolutionary implication of our results is that in those lineages where complexity increases in evolution, it does so at a progressively slower rate as complexity increases. Thus, on one hand, as complexity increases it becomes more difficult to change development to produce even more complex morphologies due to the mutational asymmetry , and on the other hand, this complexity is less likely to be passed between generations and selected due to a complex GPM and higher developmental instability.
It follows that the evolution of morphological complexity would gradually slow until it effectively stops. Our results also imply that the evolution of complex and simple morphologies is qualitatively different. Complex morphologies evolve under a complex GPM and higher developmental instability. This reduces the efficiency of natural selection. From a classic neo-Darwinian paradigm this would imply that complex morphologies should evolve less or more slowly, at least [ 20 , 38 ].
Complex morphologies produce a higher morphological diversity, e. In other words, the offspring of complex individuals spread across large regions of the morphospace. This higher morphological disparity allows, in principle, for adaptation to a larger diversity of selective pressures on morphology as suggested previously in studies based on tooth development [80 ]. Thus, the differences between complex and simple morphologies cannot be reduced to differences in evolution rates.
The differences are in the tempo and mode of evolution. Simpler lineages can evolve faster, but only within a smaller region of the morphospace, while complex lineages may evolve in wider regions of the morphospace.
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