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Conceived and designed the experiments: LC OT OS. Performed the experiments: OT OS. Analyzed the data: LC OT JU. Wrote the paper: LC OT OS JU.

The authors have declared that no competing interests exist.

Various definitions of biological complexity have been proposed: the number of genes, cell types, or metabolic processes within an organism. As knowledge of biological systems has increased, it has become apparent that these metrics are often incongruent.

Here we propose an alternative complexity metric based on the number of genetically uncorrelated phenotypic traits contributing to an organism's fitness. This metric, phenotypic complexity, is more objective than previous suggestions, as complexity is measured from a fundamental biological perspective, that of natural selection. We utilize a model linking the equilibrium fitness (drift load) of a population to phenotypic complexity. We then use results from viral evolution experiments to compare the phenotypic complexities of two viruses, the bacteriophage X174 and vesicular stomatitis virus, and to illustrate the consistency of our approach and its applicability.

Because Darwinian evolution through natural selection is the fundamental element unifying all biological organisms, we propose that our metric of complexity is potentially a more relevant metric than others, based on the count of artificially defined set of objects.

A persistent question in biology is how organismal complexity changes through the course of evolution

Phenotypic complexity quantifies the number of genetically uncorrelated phenotypic traits contributing to an organism's fitness. A phenotypic trait contributes to an organism's fitness only to the extent that natural selection acts upon that trait. Thus an organismal phenotype that is no longer under selection (for example during an evolutionary transition from a generalist to specialist lifestyle), although expressed by the organism, contributes nothing to organismal complexity. Secondly, if two phenotypes contribute to complexity, they must be genetically separable: some mutations must exist that affect one phenotype but not the other. If no such mutations exist, then although we may perceive two phenotypes under selection, these phenotypes contribute only a single trait toward determining phenotypic complexity. As an example consider the affinity of an enzyme for a substrate, and the rate at which that substrate is converted to product. If there are no mutations that affect one of these traits but not the other, then these two phenotypes are considered one, until the organism gains the genetic complexity to generate variation in one phenotype without affecting the second, for example by evolving functionally separate domains in the enzyme. Phenotypic complexity is thus a combined description of how natural selection perceives organismal phenotypes and how phenotypic variation is generated by the organism. This concept was first articulated by Orr, and followed later by others

An important aspect of measuring complexity in this manner is that both the organism and the environment affect the metric. An organism with many phenotypes, but living in simple environment could thus be just as complex as a simpler organism in the same environment. For example, if one organism is capable of metabolizing both lactose and glucose, while second can metabolize only glucose, the first organism will only be designated as more complex when there is a possibility that lactose will be present in the environment.

Recent population genetic theory

The phenotypic model used to link drift load to phenotypic complexity was first formalized by R. A. Fisher

Fitness varies along two phenotypic axes, with the maximum fitness located, for convenience, at the origin of these axes. Any individual in a population (black point) can thus be described by its phenotypic values, which determine the fitness of that organism. At any specific fitness, there are a number of other phenotypic combinations that have equivalent fitness; the values of these phenotypic combinations establish the fitness isoclines (black circle). From the optimum, fitness declines monotonically according to the structure of the landscape (see text). Each mutation (arrow) is drawn from a distribution centered on the phenotypic position of each individual, resulting in offspring with new phenotypic combinations and fitness values (white point).

A population of individuals can be represented as a collection of points in FGM and the phenotypic values of each point allow ascertainment of the fitness of each individual. Individual fitness then determines the probability of each individual surviving and reproducing the next generation. Evolution is thus described in FGM by following the collection of points over many generations. To generate novel genetic variation, mutations are drawn from an assumed distribution that is centered on the phenotypic position of each individual (

FGM makes a set of theoretical predictions about how adaptation tends to occur, and many of these have been corroborated by experimental results. The greater frequency of small-sized beneficial mutations

The utility of FGM lies in the fact that it does not require any particular assumptions about the map between phenotype and genotype, and that the specific predictions about how fitness changes during evolution appear to be robust. In the present paper, we further refine previous predictions derived from FGM

The link between drift load and phenotypic complexity under FGM was first investigated by Hartl and Taubes

In which ν = 2⋅N_{e}−1 in the diploid case and 2⋅N_{e}−2 in the haploid, and ρ(f), the density function,

If fitness is assumed to be a linearly decreasing function of the phenotypic distance to the optimum, then we find that the average fitness is given by (_{eq} is the equilibrium fitness (drift load) expressed as a fraction of the maximum attainable fitness of the organism, n_{e} is the _{e} is the effective population size. This confirms the results obtained by Poon and Otto who approximated F_{eq}(n_{e}, N_{e}) as 2N_{e}/(2N_{e}+n_{e})

Although earlier studies on FGM model have used such linear fitness functions (for the sake of mathematical simplicity), recent experimental studies do not seem to support the use of such a function

In an effort to explore fitness functions more compatible with experimental data, we studied the following family of functions. f(d) = exp(−(d^{Q})), in which fitness is an exponentially decaying function of the distance to the optimum to the power of Q. Q is a parameter that modifies the concavity of the fitness decline. As organisms move away from the optimum the effect of the mutation tend to have bigger effect if Q>1 and smaller effect if Q<1. In such a case the fitness equilibriums are (

Thus in the haploid case on which we will focus later:

The validity of these results was confirmed by an individual based model of simulation analogous to one used previously

Results are shown for populations of size 100 (black), ten (grey), and three (white). An exponential fitness decline in which Q = 1 was used (yielding a fitness function of f(d) = exp(−d)). Circles indicate the average fitness reached in the simulation model; curves indicate the analytical results.

The implementation of FGM requires several assumptions in regards to the biology of the organism. The distributions of the mutations and the shape of the fitness function are required, and the geometry of the fitness isoclines needs to be symmetrical. However, we show below that the equilibrium drift load is fairly insensitive to these strict assumptions.

First, as equation (4) suggests, the fitness equilibrium is independent of the mutational properties. As long as mutation is assumed to be isotropic, only the convergence time to equilibrium, and not the equilibrium fitness value, is affected by the distribution of mutational effects (data not shown). Second, although the results depend on the shape of the fitness function (linear or exponential-type), they are independent of the slope: equilibrium values will be the same if f(d) = exp(−αd^{Q}) (_{i}x_{i}
^{Q}), where X = (x_{0}, x_{1}, …, x_{n}) is the coordinate of an individual in FGM, and α_{i} are positive parameters (

The equilibrium drift load seems to be a robust property of FGM that is determined by the number of dimensions of phenotypic space, the population size and the fitness function (especially its curvature). An accurate estimate of phenotypic complexity can thus be obtained if it is possible to estimate equilibrium fitness values (drift load) for several population sizes, as well as the amount of curvature in the fitness function.

We used two sets of evolution experiments in which both fitness equilibrium values and fitness curvature have been investigated (

Each point indicates the mean fitness of a population. The VSV populations are shown in dark grey and the ΦX174 are shown in white. Some points have been displaced on the x-axis for clarity. The VSV populations were transferred at effective population sizes of four, ten, and 60; the ΦX174 populations were transferred at effective population sizes of 15, 50, 150, 500, and 1250. The dotted lines specify the maximum likelihood estimate of the f_{ref} value (the maximum attainable fitness); the dark dotted line indicates the value for VSV and the lighter dashed line indicates the value for ΦX174.

To estimate the curvature of the fitness function we performed a mutation accumulation analysis for high and low fitness clones and showed that the distribution of deleterious mutations was similar at both ends of the fitness range spanning a 300-fold difference. This suggests that there is very little curvature of the fitness function.

We used a second set of data from the literature, in which populations were evolved for 20 transfers at different effective sizes. Novella et al.

Additionally, an impressive set of data using site directed mutagenesis in VSV suggests that the concavity of fitness function is slightly upward

We wish to use the previous mathematical results to estimate phenotypic complexity from experimental data. However, there are two unknown parameters in the experimental system that affect the equilibrium drift load in a population: phenotypic complexity and the maximum attainable fitness that can be reached by the viruses in the laboratory environment (this parameter has been scaled to one in the previous derivations). Using methods from statistical physics, we can find the distribution of population fitness at equilibrium (shown above), and thus derive a likelihood model that gives the probability of the observed data for each couplet (n_{e}/Q, f_{ref}), in which n_{e} is the phenotypic complexity, Q a parameter of the curvature of the fitness surface, and f_{ref} the maximum attainable fitness. We also take into account the noise in our experimental assessment of fitness values; especially for high fitness populations, noise in the estimates of fitness can alter the estimation of f_{ref}, as this parameter is by definition higher than all fitness measures. Thus rather than using the probability of the point estimate of fitness, we integrated the probability between plus (f_{+}) and minus (f_{−}) one standard deviation of the point estimate. In Appendix C we show that

We applied the maximum likelihood estimator to the experimental estimates of population fitness for ΦX174 and VSV, and using a likelihood ratio test we defined 95% confidence intervals (CI), which we list here in parentheses. For ΦX174 we found n_{e}/Q = 45 (42−49), and f_{ref} = 1.245 (1.23−1.26), whereas for VSV we found n_{e}/Q = 10 (8−12) and f_{ref} = 1.98 (1.94−2.05) (_{ref} is calculated per generation relative to the ancestral virus for ΦX174 and relative to a reference strain for VSV. As no strong signature of curvature in the fitness surface has been found for either virus, we assume that Q is approximately one.

To understand how biological complexity changes during the course of evolution, a metric is needed. Previously, measures such as the number of genes, cell types or metabolic processes have been proposed, but they often lead to incongruent results. Organisms with more cell types do not necessarily have more genes. Here we suggest that a metric unifying biological systems has not been appropriately identified. To circumvent this problem, we have developed a metric of biological complexity termed phenotypic complexity (n_{e}). We have quantified this metric in the viruses ΦX174 and VSV by utilizing a population genetic model that describes how phenotypic complexity affects the drift load that a population experiences.

Phenotypic complexity (n_{e}) is a measure of the number of genetically uncorrelated phenotypes that are acted upon by natural selection. Because Darwinian evolution through natural selection is the fundamental element unifying all biological organisms, we propose that n_{e} is potentially a more relevant metric than those previously suggested.

Using recent theoretical results we have analytically quantified the dependency of the drift load (equilibrium fitness) on the effective population size and phenotypic complexity. The linear fitness function that has been employed previously to simplify the mathematical analyses is no longer necessary. Such a function makes strong assumptions about the form of the fitness landscape; specifically, mutational effects become very large as fitness is reduced. Thus at low fitness most mutations are either lethal or of very large effect, a scenario which is incompatible with what we have previously observed

We have therefore studied a more general family of fitness functions of the form f(d) = exp(−(d^{Q})), and found that F_{eq}(N_{e}, n_{e}) = (1−(2⋅N_{e}−1)^{−1})^{(ne/Q)}. It appears that this equation remains valid over a much wider range of conditions than those used in the canonical FGM, in which mutations are required to be isotropic and fitness isoclines are symmetric about the origin. An interesting feature of this formula is that it does not require a model in which mutations can affect all phenotypic traits simultaneously. In the initial formulation of FGM, all phenotypic axes intersect at the origin of each axis. This original FGM can be modified slightly such that some phenotypes are grouped into separate phenotypic modules, and within a module, all phenotypes again intersect at each other's origin. Any mutation that occurs within a module can affect only other phenotypes within that module, and none that lie outside of it (_{e} dimensions has the same drift load function as a landscape composed of m independent modules of size n_{e,i} with ∑_{i} n_{e,i} = n_{e} because we have

Hence the drift load formula that we have obtained seems to be robust to many of the assumptions underlying FGM.

Recently, another theoretical study developed a framework to estimate phenotypic complexity

One of the central FGM hypotheses that we have so far not addressed is the single-peaked nature of the landscape. Although FGM contains few assumptions about the nature of the genotypic landscape, the model explicitly requires a phenotypic landscape containing a single peak; without this, then the fitness function, f(d), cannot be described by a decreasing function. However, recent experimental evidence over large evolutionary time scales strongly suggests that while the genotypic landscape may contain multiple peaks, the phenotypic landscape is generally much less complex. Several experimental studies using microbes have shown that a considerable amount of phenotypic convergence occurs during evolution

As discussed previously, the quantity denoted by n_{e} is the number of genetically uncorrelated phenotypes that are influenced by the action of natural selection. The dimensions enumerated by n_{e} are thus genetically orthogonal to each other, and analogous to the axes needed to describe the variation among multiple phenotypes measured on a collection of individuals and mutants in a principal component analysis. However, the number of axes enumerated by n_{e} is filtered by natural selection, while in a PCA analysis the number of axes is limited only by the number of independent phenotypes that are measured. Because each phenotype is optimized at a value determined by each organism's ecological environment, there is a dependence of phenotypic complexity on the complexity of the ecological niche experienced by each organism; if natural selection does not act on a phenotype, then that phenotype does not contribute to the complexity metric. Finally, although the estimates of n_{e} arise from an idealized model of phenotypic evolution; as Orr suggested previously, estimates of phenotypic complexity using FGM can be viewed as “effective” estimates of phenotypic complexity _{e}, in which two populations with different numbers of individuals and different sex ratios might have the same effective population size and therefore respond similarly to the different population genetic forces. Thus two organisms, although they may differ in both the underlying genetic mechanisms and in the complexity of the environment in which they live, may have similar phenotypic complexities. The utility of the concept lies not in the implications it makes about specific phenotypes or genetic details, but in that it enables a general quantification of how an organism is affected by natural selection (the

Unsurprisingly, our estimates of phenotypic complexity are orders of magnitude smaller than either the number of nucleotides or even the number of amino acids encoded by the genomes of these organisms (5386 bp and 11,161bp in ΦX174 and VSV, respectively). This agrees with the concept of phenotypic complexity that we have defined. Although mutations that occur at one nucleotide or amino acid do not affect those at another (

Although we have only two estimates, we can briefly consider them from a comparative standpoint: although the genome size of ΦX174 is half of VSV, our estimate of phenotypic complexity quantifies ΦX174 as being approximately four-fold more complex. It is notable, then, that ΦX174 contains approximately twice the number of genes as VSV. Additionally, the lifestyle of ΦX174 is arguably much more elaborate than that of VSV. ΦX174 interacts with several host factors in order to perform transcription and replication; 13 host factors are required for replication alone

These estimates suggest that, for very simple organisms such as viruses, phenotypic complexity correlates well with the number of genes in an organism, and more specifically, with the number of interactions characteristic of that organism

Although the theory presented here appears to be quite robust, it is too early to conclude that it is an accurate reflection of the underlying biology. To be studied in an FGM framework, organisms need to present at least one phenotypic property to selection. Additionally, populations, even those of very small size, should evolve towards a fitness equilibrium that is explicitly dependent on population size. We found data in the literature consistent with this expectation for one organism, VSV. We now provide further support for population size-dependent fitness equilibria by evolving populations of the bacteriophage ΦX174. Together, these two data sets suggest that evolutionary analyses using an FGM framework are a valid approach. Moreover, the use of very simple organisms such as viruses is useful for gaining insight into metrics of complexity, as for such simple organisms, gene number is likely to be a very good correlate of organismal complexity, and this should be reflected by the metric. Although our observations are currently limited to two viral species, it is clear that from both a qualitative level (

Here we have presented a top-down approach to quantifying biological complexity. This can be contrasted with previously proposed metrics of complexity, which have relied on physically measurable quantities of the organism (bottom-up approaches). Two important conceptual differences separate these two approaches. Most importantly, phenotypic complexity is dependent on both the organism and the environmental context. An organism is not complex because it has many measurable phenotypes; it is complex because it has many phenotypes on which natural selection acts. Secondly, phenotypic complexity does not rely on artificially constructed concepts such as genes

However, phenotypic complexity remains an inherently abstract metric. It cannot aid in identifying the specific characteristics contributing to the complexity of an organism. Instead, it addresses the complexity with which natural selection views an organism, and the complexity with which an organism is capable of generating novel phenotypic variation. For this reason, testing how phenotypic complexity compares to more traditional metrics of complexity (for example, the numbers of genes, protein interactions, or cellular pathways) may provide significant insight into biological systems. Finally, phenotypic complexity (and the resulting equilibrium drift load) affords a unique opportunity to contrast the action of natural selection between different organisms or different environments in a very general and unconstrained manner.

The details of experimental evolution of ΦX174 have been described previously

Sella and Hirsh _{e}−1 in the diploid case and 2N_{e}−2 in the haploid, and ρ(f) is the density function of fitness value f.

In an n-dimensional space, the density ρ(^{(n−1)}, where Ω(^{(n/2)}/Γ(^{(n−1)} d

If fitness is defined as f(^{(n−1)} d^{n}
^{−1} d

If fitness is defined as: f(^{Q}) we have

This expression is independent of mutational properties.

Note that if f(^{Q})

Let us assume that f = exp(−^{Q}) where _{1},_{2},…,_{n}) is the position in the n-dimensional space and α_{i} are positive numbers. We then have ellipsoidal fitness isoclines of semi-axes R/α_{i}. As the volume of such an ellipsoid is

we therefore find the same value of 〈f〉, as the constant cancels out in the ratio of integrals. More generally, if fitness is defined as

We can show through recursions that this defines volumes

Sella and Hirsh showed that the probability of being at fitness

Using the previous derivations with f(^{Q}), we find the probability that

Because we do not know maximum fitness f_{ref}, we must estimate it and therefore fitness b and a be used relative to f_{ref}.

We would like to thank Art Poon, Dan Weinreich, and Thomas Berngruber for valuable discussions.