INSERM, UMR-S 722, Paris, F-75018, France

INSERM, UMR-S 738, Paris, F-75018, France

Univ Paris Diderot, Sorbonne Paris Cité, UMR-S 722 INSERM, F-75018, Paris, France

Univ Paris Diderot, Sorbonne Paris Cité, UMR-S 738 INSERM, F-75018, Paris, France

Institut Claude Bernard, IFR2, F-75018, Paris, France

Division of Biology, University of California San Diego, La Jolla, California, USA

Abstract

Background

The emergence of organismal complexity has been a difficult subject for researchers because it is not readily amenable to investigation by experimental approaches. Complexity has a myriad of untested definitions and our understanding of its evolution comes primarily from static snapshots gleaned from organisms ranked on an intuitive scale. Fisher's geometric model of adaptation, which defines complexity as the number of phenotypes an organism exposes to natural selection, provides a theoretical framework to study complexity. Yet investigations of this model reveal phenotypic complexity as costly and therefore unlikely to emerge.

Results

We have developed a computational approach to study the emergence of complexity by subjecting neural networks to adaptive evolution in environments exacting different levels of demands. We monitored complexity by a variety of metrics. Top down metrics derived from Fisher's geometric model correlated better with the environmental demands than bottom up ones such as network size. Phenotypic complexity was found to increase towards an environment-dependent level through the emergence of restricted pleiotropy. Such pleiotropy, which confined the action of mutations to only a subset of traits, better tuned phenotypes in challenging environments. However, restricted pleiotropy also came at a cost in the form of a higher genetic load, as it required the maintenance by natural selection of more independent traits. Consequently, networks of different sizes converged in complexity when facing similar environment.

Conclusions

Phenotypic complexity evolved as a function of the demands of the selective pressures, rather than the physical properties of the network architecture, such as functional size. Our results show that complexity may be more predictable, and understandable, if analyzed from the perspective of the integrated task the organism performs, rather than the physical architecture used to accomplish such tasks. Thus, top down metrics emphasizing selection may be better for describing biological complexity than bottom up ones representing size and other physical attributes.

Background

The evolution of the complexity of organisms has been a challenge for Darwinian theories of evolution

Any attempt to understand the evolution of complexity must rely on a meaningful definition of complexity coupled to some quantitative methods of estimation. Initial estimates of complexity have been based on the number of nucleotides, genes or cell types in a genome, but such bottom up estimates often fail to have useful properties

The most integrated vision of complexity comes from Fisher's geometric model of adaptation

Using Fisher geometric model of adaptation, several theoretical studies have also analyzed the consequences of phenotypic complexity on evolution. All of them found higher complexity to be costly. The cost results from the difficulty of having to optimize many phenotypes simultaneously and it is manifested by the decreasing fraction of beneficial mutations as dimensionality increases

Although previous studies have used Fisher's geometrical model to examine the effect of complexity on evolution, none have allowed dimensionality to change as a result of evolution and adaptation. To characterize both the selective forces acting on the emergence of complexity and the underlying mechanisms, we have designed an evolutionary system in which complexity was free to emerge depending on its costs and benefits. Although experimental studies of complexity with real biological organisms are possible

Methods

Models

Neural networks were chosen over other models

We used as the reference function Legendre polynomials. These functions were chosen because they could be readily ranked in term of complexity by the Order of the Legendre Polynomial (OLP). Higher OLP's both require more parameters and have a higher Kolmogorov complexity

Model of neural networks and environmental challenge

**Model of neural networks and environmental challenge**. All networks were evolved under an asexual mutation-selection-drift process. Fitness of an individual network was obtained by providing an input gradient and retrieving from the network a response that was then matched to a response function or environmental challenge. Networks varied in their number of nodes and connections and the weight of each connection and nodes could be mutated to generate heritable variation. Response functions were described by Legendre Polynomials. A higher Order of the Legendre Polynomial reflected a more intense environmental challenge. Populations of networks with identical size were adapted independently to each of the environmental challenges.

The evolution of complexity in our networks was monitored and compared to standard bottom up metrics such as network size with three additional metrics: Information Complexity (IC), Principal Component Phenotypic Complexity (PCPC), and Effective Phenotypic Complexity (EPC) (see detailed methods). IC measures the information content of the environment that is stored in a network by selection

A geometrical model of adaptation and the derived estimates of complexity

**A geometrical model of adaptation and the derived estimates of complexity**. A) A geometric model of adaptation is used as a reference to characterize complexity. In this model, an organism is defined by a number of idealized independent phenotypes (here 3). The number of phenotypes is what we will call phenotypic complexity. The model assumes the existence of an optimal combination of phenotype having maximal fitness. The more organisms are distant from that optimal combination, the lower is their fitness. B) To estimate phenotypic complexity, one can analyze a set of fitness-linked-phenotypes in a collection of mutants and perform a principal component analysis (PCA). The distribution of variance explained by the different axes of the PCA is directly linked to phenotypic complexity. For instance, if there is indeed a single phenotype (case 1), a single axis will explain all variance, while if complexity is indeed 2 or 3, 2 or 3 axis will be necessary to explain the phenotypic variance of mutants. C) Mathematical derivation from the geometric model have proved the existence of some fitness equilibrium and that the fitness at these equilibrium is a direct function of the effective population size and the phenotypic complexity. Hence if we record the average fitness of populations of different population size at equilibrium, we can estimate phenotypic complexity.

Detailed methods

Networks

Neural networks consisted of single input and output cells connected by a series of neurons, or nodes (Figure _{j }

(different activation function provided similar results but with much lower efficiency of adaptation (data not shown)), where

_{i }_{ij }_{j }_{0 }_{j }

For each input value, the network provides an output value that can be compared to a reference. Rather than using the response to a single input value to define the fitness of the network, we used the response to 100 different input values. Fitness was hence determined by testing the response of each network to a gradient of values _{0 }= g(i)

where

Function

For our purposes, we normalized them to be bounded by 0.1 and 0.9 in interval [-1,1].

Legendre polynomials of high order require many parameters to be defined and as such have a higher complexity than simple ones (in terms of Kolmogorov complexity

Network evolution

Populations of networks were initially monomorphic, starting with a network having random weights (sampled in a uniform distribution between -0.5 and 0.5). The population of 500 networks was then submitted to a model of asexual evolution with discrete generations. Each generation, a network had a 1% probability of mutating one of its weights. The quantitative value of the mutated weight was then shifted by a random normal deviate of mean 0 and standard error 0.1. Using classical Wright-Fisher population genetics formalism, networks contributed to the next generation of networks according to their respective fitness.

The above evolutionary process represents our primary process of adaptation, however two other slightly modified evolutionary processes were used. We call the first modified process the "intense selection" evolutionary process. For that process, a smaller population size was used (50), but every 100 000 generations, the intensity of selection was increased, by increasing constant K, to set fitness back to 3%. This promoted an intense selective pressure that allowed the emergence of very high fitness clones that would have otherwise required very high population size (and massive amount of computer time) to emerge, as mutation of effects smaller than the inverse of population size behave as neutral mutations. Using this protocol fitness as high as 0.99999 were sometimes reached, while a population size of about 10^{5 }would be required for this level of fitness to be reached. The final evolutionary process we used can be called "adaptive dynamics" _{i }_{0}, the mutant either immediately invaded the population and became the new resident or disappeared. The probability of invasion P(_{0}→_{i}

This protocol provided an exact solution for populations having a small mutation rate by population size product. This evolutionary process allowed a faster computing than the process simulating a whole population; it limited the effects of high mutation rate by population size product that may lead to confusing effects and provided a direct access to the whole line of descent of the final clones. It allowed to follow of the coupled changes in fitness and complexity through the adaptive walk.

Similar levels of complexity were reached under all of these evolutionary algorithms. Our results are therefore robust and are not resulting from some specific selection favoring genetic robustness due to a high mutation supply.

In the first dataset (Figure

Fitness reached by networks as a function of genome size and environmental challenge

**Fitness reached by networks as a function of genome size and environmental challenge**. Boxplot of fitness reached at the end of adaptation. Network size, 19, 14, 9, 6, 4, is presented with a gray level, lighter gray representing larger network sizes. Larger Network size facilitates adaptation to higher fitness. To uncover the difference between highly adapted networks, the scale used is -log(1-Fitness).

Estimates of networks complexity as a function of both network size and environmental challenge

**Estimates of networks complexity as a function of both network size and environmental challenge**. A) Average Effective Phenotypic Complexity (EPC) estimated for different network sizes and environmental challenges (OLP). Network sizes of 4, 6, 9, 14 and 19 nodes are represented as decreasing shades of gray. Error bars are 95% confidence intervals. B) and C) same as A but for the metrics Principal Component Phenotypic Complexity (PCPC) and Informational Complexity (IC). D) Networks adapted to an OLP of 2 and having similar fitness and PCPC are presented with their respective value of PCPC and IC. On the networks graphed, the width of a connection reflects the impact of the underlying weight on fitness. A large width reflects a weight that impaired fitness largely when mutated. Large differences in the internal structuring of networks affected their IC but not their phenotypic complexity that remained more linked to the function performed.

Restricted pleiotropy and complexity

**Restricted pleiotropy and complexity**. A) Matrix illustrating pleiotropic effect of mutating weights of connections and nodes on network phenotype. The pleiotropic effect of a mutation was measured through the mean and variance of its effects on networks phenotypes. An average pleiotropy was computed for each connection and node weights in the network (generating matrix A) and averaged over all weights to compute the network pleiotropy. B) Correlation between the estimates of PCPC of phenotypic complexity and network pleiotropy for all networks. In red, a power law fit to the data.

Optimization of phenotypic complexity (PCPC)

**Optimization of phenotypic complexity (PCPC)**. A) Evolution of fitness against PCPC for 50 19-nodes networks evolved on OLP 4 during the adaptive process. Gray line represents the fitness versus PCPC trajectory of the population and black dots the position of the most common genotype in the population each 10^{4 }generations. High PCPC was required to reach high fitness. B) Same as in A except that the average pleiotropy of networks was plotted instead of their PCPC. C) Final fitness reached under an "intense selection" adaptation process after 10^{6 }generations in two environments (OLP 3, light gray and OLP 8, dark gray) for networks of varying sizes as a function of evolved complexity PCPC. For each OLP a different range of optimal PCPC evolved and maximized fitness. D) Distribution of PCPC estimated on 100 random networks and 100 networks adapted to an OLP of 8 (top panel). Distribution of the PCPC estimated on networks, derived from the previous ones, after adaptation to OLP 4. PCPC converged, either up or down, towards an intermediate and optimal value.

Distribution of changes in complexity (PCPC) fixed during the adaptive process A) 12 populations of size 100 were adapted to an OLP of 4 in three independent replica from 4 networks previously adapted to an OLP of 8 for 10 million generations

**Distribution of changes in complexity (PCPC) fixed during the adaptive process A) 12 populations of size 100 were adapted to an OLP of 4 in three independent replica from 4 networks previously adapted to an OLP of 8 for 10 million generations**. The populations were evolved under an "adaptive dynamics" process of adaptation (low mutation rate such that populations were always monomorphic unless a single mutant occurred and got either lost of fixed). The changes in PCPC of all the fixed mutations were then recorded. We focused on changes occurring while fitness of the network population changed from 10% to 60%, to avoid any effect due to the stabilization of the changes in PCPC in early and late phases of adaptation. The red dotted line represents the mean change in PCPC. B) Same as in A with a population size of 10 000. C) Same as in A on 5 networks (in 5 replicates each) adapting from an OLP of 4 to an OLP of 8 with a population of 100. Changes in PCPC were recorded while fitness changed from 5% to 30%. D) Same as in C with a population size of 10 000.

Network complexity

Principal component phenotypic complexity

For a given network, the different outputs of 1,000 mutations having more than 1% effect on fitness were recorded (using all mutations provided similar estimates, but due to the existence of fully neutral mutations resulted in a higher estimation variance and in the failure to estimate PCPC for some networks). For each mutation and for each of the P = 100 fitness linked phenotype (i = 1, 2,..., 100), the deviation between the mutant and wild type output N(g(i)) was recorded. A principal component analysis (PCA) was then performed in R, with a correlation approach. The eigen values, _{i}_{i}_{i}

_{i}_{i}

PCPC can be used to estimate the number of effective dimensions in Fisher's geometric model even when individuals sharing the same fitness are not equidistant from the optimal phenotypic combination (they define circular fitness isoclines in two dimensions)

Effective Phenotypic Complexity

In Fisher geometric model of adaptation, population evolves towards equilibrium fitness values defined by population size and the dimension of the phenotypic space or EPC. The best network at the end of adaptation was then used to initiate new populations of reduced size (6, 10, 30, 60 and 100), which were evolved for 10 millions of generation to be sure that equilibrium would be reached. Fitness was recorded over the last 5 million generations and the observed decay of fitness with population size was fitted by the theoretical prediction to estimate EPC

Indeed EPC is a composite index which corresponds to the ratio of dimensionality and an epistasis parameter named Q in reference

Informational Complexity

The quantity of information stored in a genome and transmitted to the next generation has been used as a measure of organismal complexity

For a given weight, j, the fraction of populations evolving with size N having the i^{th }value for that weight and the associated fitness f_{j,i }is just:

the entropy for that weight is then

and the information content

The information content of a network was hence defined as the sum over all the w weights

The larger the N, the higher the information content, as any slightly deleterious mutation is easily filtered by natural selection. To be able to uncover the differences among networks and to be able to compute the solutions we chose an N of 100.

Network Pleiotropy

To estimate pleiotropy, we used a method similar to the one used to estimate PCPC. We sampled one hundred mutations per mutable entity, w_{i}, of the network and averaged their effect, to obtain _{i }by studying the coefficient of variation of these

Similarly to PCPC, a pleiotropy of 3 meant that the mutation had a distribution of effects on traits equivalent to a mutation that would affect 3 traits identically and none of the other traits (Figure

Network Modularity

To estimate modularity, we used the bipartite leading eigen vector approach

Results

Networks were evolved under an asexual mutation-selection-drift process and selected to match OLP's of the order of 2, 3, 4, 5, 6, 7 and 8. We examined network sizes of 4, 6, 9, 14 and 19 nodes. For each combination of network size and OLP, thirty to eighty populations were started from a random network and evolved until they either reached fitness of 0.99 or evolved for at least 10 million generations at a population size of 500. Populations of networks with an absolute fitness of less than a threshold of 15% were not retained, as they do not match properly the imposed challenge (other threshold values were used and provided similar results). With low OLP's, all networks evolved to high fitness values and matched accurately their reference function. With increasing OLP's, fewer networks reached a fitness greater than 15%, and those that did, especially the smaller ones, attained a lower fitness. For instance, at an OLP of 8, only 29% of the networks with 4 nodes exceeded 15% while 82% of the networks with 19 nodes did; of these, average fitness was 74% in the 4-node networks and 96% in the 19-node ones. Hence, increased network size facilitates adaptation to increased environmental challenge (Figure

PCPC, EPC and IC were all found to correlate positively with the size of the network and the environmental demand (OLP) (Figures

Despite the fact that they are measured with radically different methods, the two estimates of phenotypic complexity PCPC and EPC correlated strongly (r = 0.69). IC correlated to the other two metrics, but to a lesser degree (r = 0.50 with PCPC and r = 0.34 with EPC). Network size was able to explain 85%, 20% and 8% of the variance of IC, PCPC and EPC, while OLP explained 2%, 40%, and 57% of the variance, respectively (Figure

Although network size did correlate with all three metrics of complexity, its inability to explain the variance in PCPC and EC revealed its limitations in influencing the evolution of complexity. This effect was further illustrated when networks were challenged with the simplest demand of OLP equal 2 (Figures

To elucidate how connections within a network evolved, we examined the effects of changing randomly the connection and node weights on each of the 100 traits used to estimate the network fitness (see Methods). A change in a weight is equivalent to a mutation and its effects were quantified as a pleiotropic coefficient, based on the coefficient of variation of the magnitude change at each trait (Figure

The negative correlation between pleiotropy and our three metrics of complexity indicated that complexity evolved by incorporating mutations with restricted effects on the response of the networks. To determine whether the restriction also created structuring in which particular group of weights interacted preferentially with different subsets of phenotypes, we searched for modularity by applying the bipartite leading eigen vector approach to the matrix of connections from weight to phenotypes

Previous theoretical studies, analyzing the consequences of phenotypic complexity on evolution, have indicated that high complexity generates some costs such as an increased drift load

However, an ever-increasing complexity, which could in theory lead to an ever-increasing fitness, was not observed (Figure

Discussion

By evolving adaptive networks with different physical properties under different environmental conditions, we have been able to identify the determinants controlling the evolution of complexity. The three different measures of complexity we used correlated positively with one another, yet our analysis reveals that they captured different facets of complexity. We found that while IC captured the physical architecture used to accomplish a given task, PCPC and EPC were most useful in describing the integrated task the organism performs.

The measure of informational complexity or IC, being still connected to the physical constituents of the networks, was found to be mostly driven by network size and not to correlate well with the environmental challenge. For large networks, IC responded more to the environmental challenge (Figure

The estimates of phenotypic complexity, PCPC and EPC strongly correlated with each other, which is remarkable because it indicates that mathematical derivations from an idealized model of adaptation such as Fisher's quantifies complexity in a manner similar to a statistical model of principal components. Although this may seem intuitive as both estimates are supposed to measure phenotypic complexity as defined in Fisher's geometric model, nothing suggested initially that such a model should apply to our networks. The high correlation we observed relies on the robustness of the estimators of complexity derived from Fisher's geometric model that we used. Other measures of dimensionality based on the distribution of mutation effects in Fisher's geometric model have been proposed

Both PCPC and EPC responded consistently to the demands of the environment. Because the environment is perceived by the networks through the fitness consequences of natural selection, our results shift the evolutionary focus of complexity from bottom up or physical measures of complexity (genome size, number of cell types, network size) to top down or more ecological ones in which complexity is linked to the ecological niche. Physical measures of complexity alone are not adequate to capture complexity of the task performed because their mapping to phenotype was modulated as we found by pleiotropy.

Pleiotropy has been defined a century ago

The link between restricted pleiotropy and the emergence of phenotypic complexity provides both a mechanistic interpretation of complexity and a selective hypothesis underlying its evolution. Previous analyses of phenotypic complexity have mostly focused on the consequences of complexity on evolution, rather than on the selective forces acting on it. As such it appeared costly to have a high complexity due to a limited number of beneficial mutations

Phenotypic complexity and restricted pleiotropy appeared to be under stabilizing selection due to a balance between their benefits and costs. Increasing complexity allows the organisms to wonder in a larger phenotypic space and closer to the optimal combination of phenotypes, but it also leads to a higher drift load

Finally, restrictive pleiotropy's link to complexity is consistent with Ohno's hypothesis for the evolution of complexity via gene duplication

Conclusions

Using a model of adaptive neural networks, we have shown that phenotypic complexity evolved as a function of the demands of the selective pressures, rather than the physical properties of the network architecture, such as functional size. The phenotypic complexity we observed resulted from a selective balance between the costs associated with the optimization of many independent traits and the benefit provided by the exploration of a larger phenotypic space. Our model suggests hence both a selective process for the emergence of phenotypic complexity and a mechanistic model allowing its evolution: the emergence of restricted pleiotropy. Our results therefore show that complexity may be more predictable, and understandable, if analyzed from the perspective of the integrated task the organism performs, rather than the physical architecture used to accomplish such tasks. Thus, top down metrics emphasizing selection may be better for describing biological complexity than bottom up ones representing size and other physical attributes.

Authors' contributions

HLN and OT conceived and design the experiments. HLN performed the experiments. HLN, LC and OT wrote the paper.

Acknowledgements

We thank the Centre de Biomodélisation of the Institut Claude Bernard (IFR2) for supplying the computation, E. Comets for statistical advices, E. Denamur, F. Mentré for hosting this research, PA. Gros, O. Martin, G. Martin and JP. Uzan for discussions, J. Zhang for kindly providing a software to compute bipartite modularity and two anonymous reviewers for their constructive comments. This work was supported by Agence Nationale de la Recherche, ANR-08-GENM-023 ("EvoGeno") and NSF grant DEB- 0748903