Power Laws in Genomic Quantities

A few years ago it was noticed that the distributions of gene-family sizes in fully-sequenced genomes follow power-law distributions.^1,2 Since then different authors have shown that there is in fact a large array of genomic features that show power law distributions. Almost all of these concern the distributions of genomic features within a single genome. For instance, it is shown in reference 3 that the number of genomic occurrences of DNA words, protein folds, superfamilies, and families all follow power-law distributions. Power-law distributions are also found in the structure of the ‘protein universe’;⁴ the number of protein families per fold is power-law distributed, and so is the number of different assigned biological functions per fold.³ Power-laws also appear in the structure of cellular interaction and regulatory networks. For example, the number of genes that a given gene interacts with is power-law distributed. This holds both when one defines ‘interaction’ between genes on the level of the proteins that they encode^5,7 or if one defines it at the level of coregulation of the expression of the genes.⁸ The experimental data on transcription regulatory networks is rather incomplete but they also suggest that the number of genes regulated per transcription factor might have power-law tails.^9,10 Finally, power-laws also appear in cellular metabolic networks; the number of substrates that any given substrate interacts with is power-law distributed.^11,12

Comparing Genomic Features across Genomes

Note that almost all the power-law distributions just mentioned refer to statistics that are taken over a single genome or cellular network. The statistics of genomic features across genomes has been much less (if at all) investigated. To a large extent this may be because until recently there simply weren't enough fully-sequenced genomes to obtain meaningful statistics across genomes. However, this situation is changing rapidly.

There are currently about 150 fully-sequenced microbial genomes in GenBank and this number appears to grow exponentially as I have shown in reference 13.

Figure 1 shows an updated plot of the number of fully sequenced microbial genomes as a function of time (see Methods section). The current number of available microbial genomes is only large enough to allow for meaningful cross-genome comparisons of the most basic statistics of gene-content and organization and this is what I will focus on in this chapter. However, the exponential fit in Figure 1 predicts that the number of sequenced genomes doubles roughly every 17 months. This implies that by 2010 we may have as many as 3000 fully-sequenced microbial genomes available. It is therefore clear that much more detailed comparative genomic analyses than the ones presented in this chapter will become possible over the next decade.

Figure 1

The number of fully-sequenced microbial genomes submitted to the genbank database as a function of time (in years). The vertical axis is shown on a logarithmic scale. The black line is the least-square fit to an exponential. n = 2^{(t-1993.4)/1.38}

In reference 13 I compared the number of genes in high-level functional categories across all sequenced genomes and showed that they follow power-laws as a function of the total number of genes in the genome. In this chapter I will recapitulate these results and augment them in several ways. In particular, I have extended the analysis to all functional categories that are represented by at least 1 gene in each bacterial genome and have recalculated the observed exponents based on the latest genomic data. Second, I will go into more detail regarding the implications of the observed scaling laws for the general organization of gene-content across genomes and discuss an evolutionary model that relates the observed scaling behavior in gene content to fundamental constants of the evolutionary process. Finally, I will discuss theoretical explanations for the category of transcription regulatory genes which shows approximately quadratic scaling with the total number of genes in the genome.

Scaling in Functional Gene-Content Statistics

To count and compare the number of genes in different functional categories for all sequenced genomes one needs to first define a set of functional categories and then annotate all genomes in terms of these functional categories. I used the biological process hierarchy of the Gene Ontology¹⁴ to define functional categories, and Interpro annotations of fully-sequenced genomes to associate genes with GO categories. The details of the annotation procedure are described in the Methods section. The result is a count of the number of genes associated with each of the GO categories in the biological process hierarchy for each of the sequenced genomes.

The set of genomes in this study consists of 116 bacteria, 15 archaea, and 10 eukaryota. In this chapter I will focus solely on the bacterial data since this is the only kingdom for which there is sufficient data to obtain meaningful statistics. The reader is referred to reference 13 for a discussion of the observed scaling laws in archaea and eukaryota.

There are 154 GO categories in the biological process hierarchy that have at least 1 associated gene in each of the 116 bacterial genomes. I will refer to these categories as the ‘ubiquitous’ categories. The results for a selection of 6 of these ubiquitous GO categories are shown in Figure 2. The figure shows the dependence of the number of genes in each category on the total number of genes with annotation in the genome.

Figure 2

The number of genes involved in (uper panel) signal transduction (red), carbohydrate metabolism (blue), and DNA repair (green), and (lower panel) any biological process (blue), transcription regulation (red), and protein biosynthesis (green) as a function (more...)

The upper panel shows the categories “signal transduction” (red), “carbohydrate metabolism” (blue), and “DNA repair” (green), while the lower panel shows the categories “transcription regulation” (red), “biological process” (blue), and “protein biosynthesis” (green). Each dot represents the counts in a single bacterial genome and both axes in Figure 2 are shown on a logarithmic scale. Note that in reference 13 a similar plot was shown but with the horizontal axis representing the total number of genes rather than the total number of annotated genes. Since the total number of annotated genes is to a very good approximation proportional the total number of genes, the results are virtually identical whether one uses the total number of genes or the total number of annotated genes. I decided to use the total number of annotated genes on the horizontal axis in Figure 2 partly to illustrate this fact. In addition, the genome size in bacteria is also to a very good approximation proportional to the total number of genes in the genome. Thus, if we had used the genome size instead of the number of annotated genes on the horizontal axis Figure 2 would again have looked virtually identical.

The dots of each color in Figure 2 fall approximately on a straight line. Thus, the logarithms of the number of genes n_c in a category c and the total number of genes g (or the number of annotated genes or the genome size) are approximately linearly related:

In other words, the number of genes n_c in a category increases as a power-law in the total number of genes g:

For the 6 functional categories shown, the exponents of the best power-law fits are indicated in the figure caption. The fits were obtained using the procedure described in the Methods section. The exponents range from α = 0.16 for protein biosynthesis to α = 1.95 for signal transduction.

To further show the range and variation of the observed exponents Figure 3 shows the inferred exponents and their 99% posterior probability intervals for a selection of 20 functional categories.

Figure 3

The 99% posterior probability intervals for the scaling exponents of a selection of functional categories. The functional categories are indicated on the left of each bar and the dark section of the bar indicates the 99% posterior probability interval (more...)

The exponents range from close to zero to roughly 2. Note that, for a category c with exponent α_c the relative proportion p_c of genes in the genome scales as p_c = λ_cg^α-1. That is, when α_c <1the proportion of genes in the category will decrease with genome size, while for αc>1 the proportion of genes in the category will increase with genome size. Thus, for a category_c with exponent close to 2, the proportion p_c will increase almost linearly with genome size. The behavior of different categories thus ranges from categories where the number of genes is almost constant with genome size ( α_c ≈ 0), to categories where the proportion of genes in the genome increases linearly with genome size ( αc ≈ 2).

The general picture that emerges from Figure 3 is that the proportion of genes in essential low-level functional categories such as protein biosynthesis and DNA replication decreases with genome size, whereas the proportion of genes that play regulatory roles such as genes involved in signal transduction and transcription regulation increases approximately linearly with genome size. In between these extremes is a number of categories, including different metabolic functions, for which the exponent is roughly 1, indicating that the genomic percentage of genes in these categories is roughly independent of genome size.

Consequences for the Topology of the Transcription Regulatory Network

The approximately quadratic scaling of the number of transcription regulatory genes also has some interesting consequences for the structure of the transcription regulatory network as a function of genome size. The class of transcription regulatory genes consists for the most part of DNA-binding transcription factors that regulate transcription through the binding to specific regulatory motifs in intergenic regions. We can imagine the transcription regulatory network by a set of arrows pointing from each of the regulators to each of the genes that they regulate. The total number of arrows in this network for a genome of size g is the product of the number of genes g times the average number of incoming arrows per gene <r(g)>, i.e. <r(g)> represents the average number of different regulators regulating each gene in a genome of size g. Note that we can also write the total number of arrows as the number of transcription factors n_tr (g)in a genome of size g times the average number of genes <n(g)> that each regulator regulates. We thus have

where the equation on the right follows from the quadratic scaling of the number of regulators:n_tr(g)∝g². To elucidate what the equation on the right implies, let us consider what follows if we assume that either n(g)> = constant or <r(g)> = constant. In the first case the average number of genes regulated per transcription factor, i.e., the regulon size, is independent of genome size. In that case the average number of different transcription factors <r(g)> regulating each gene must be increasing linearly with genome size, i.e.,<r(g)>.∝g. If on the other hand <r(g)> were constant with genome size, then the average regulon size <n(g)> should be decreasing with genome size, i.e, <n(g)>∝1/g. Between these two extremes there is a continuum of solutions where <r(g)> increases more slowly, and <n(g)> decreases more slowly, but such that still <r(g)>/<n(g)>∝g.

There is currently very little data to decide if real regulatory networks are closer to the limit where <r(g)> increases linearly, or closer to the limit where <n(g)>∝1/g. One piece of indirect evidence is the dependence of the number of operons and the amount of intergenic region on genome size. If the average number of transcription factors <r(g)> regulating each gene were to increase with genome size, then one might expect that, as genome size increases, the average operon size should decrease and that the amount of intergenic region per gene should increase. It is of course a nontrivial task to identify the number of operons from genome sequence alone. However, as a proxy we may consider runs of consecutive genes that are located on the same strand of the DNA (see ref. 16 for a method of estimating operon number using this statistic). Since all genes in an operon necessarily have to be transcribed in the same direction, a decrease in operon number would likely be reflected by a decrease in the average length of runs of consecutive genes on the same strand. Figure 4 (upper panel) shows this average length of iso-strand genes as a function of genome size for all bacterial genomes that are currently in the NCBI database of fully-sequenced genomes.

Figure 4

The average length of runs of genes that are transcribed from the same strand (upper panel) and the average number of intergenic bases per gene (lower panel) as a function of the total number of genes in the genome. Each dot corresponds to a bacterial (more...)

The figure suggests a slight decrease of the length of these runs, consistent with what was reported in reference 16, but there is a large amount of variation and the trend is far from convincing, i.e,r²=0.23 under simple regression. Note also that the drop in operon size is at most a factor of two between the largest and smallest genomes, while total gene number increases almost a factor of 20. The lower panel shows the average number of intergenic bases per gene as a function of the total number of genes in the genome. In this case a correlation between genome size and the amount of intergenic region is completely absent, i.e, r² = 0.0005.Thus these two statistics provide little evidence that <r(g)> increases substantially with genome size. However, one cannot exclude the possibility that in small genomes a large proportion of genes is not transcriptionally regulated at all, and that as genome size increases this proportion drops dramatically. This would still lead to a substantial increase of <r(g)> with genome size. It does seem plausible, however, that larger genomes may have a larger number of ‘specialized’ regulons that typically regulate a smaller number of genes compared to the more general regulators that one expects to find in all organisms, and that <n(g)> thus decreases with genome size.

Quality of the Fits

The power-laws observed in Figure 2 are observed for the large majority of the 154 ubiquitous functional categories. I assessed the quality of the power-law fits by a measure F that measures the fraction of the variance in the data that is explained by the power-law fit (see Methods). Figure 5 shows the cumulative distribution of F for all 154 ubiquitous functional categories.

Figure 5

Cumulative distribution of the quality of the power-law fits for the 154 ubiquitous functional categories. The horizontal axis shows the fraction F of the variance in the data explained by the fit, and the vertical axis shows the percentage of categories (more...)

As can be seen from Figure 5 about two thirds of the categories have more than 90% of the variance explained by the fit. More than 95% of the categories have more than 80% of the variance explained by the fit. We thus see that most ubiquitous functional categories follow scaling laws like the ones shown in Figure 2.

Principle Component Analysis

Instead of considering the scaling behavior of one functional category at a time, we can consider all functional categories at once. We can consider a 154-dimensional ‘function space’ in which each axis represents an ubiquitous functional category, and represent the ‘functional gene content’ of a genome by a point in this 154-dimensional space. That is, if we number the categories 1 through 154, and n_c is the number of genes in category c, then ñ= (n₁,n₂,...,n₁₅₄) represents the functional gene content of the genome as a point in the function space. The set of all sequenced genomes thus forms a scatter in this function space.

We can now ask what the shape is of this cloud in function space. To do this, it is convenient to again consider all axes on logarithmic scales. That is, we consider the scatter of points →x with x_c = log(n_c).The results in the previous section showed that, to a good approximation, almost all functional categories obey the linear equations

where g is the total number of genes in the genome. If this equation were to hold exactly for all categories, then the numbers x_c and x_c', for any two categories c and c', would also be linearly related:

Thus, the statement that equation (4) holds for all categories is equivalent to the statement that the scatter of points in function space falls on a straight line. Of course, since equation (4) holds only approximately for each category, the scatter of points falls only approximately on a line. To illustrate this Figure 6 shows three projections of the scatter of points onto three-dimensional subspaces each representing three functional categories.

Figure 6

Three ‘slices’ through the scatter of genomes in function space. Each point corresponds to a genome, and its components along the three axes correspond to the logarithms of the gene numbers in each of the three categories that the axes (more...)

That is, each of these three figures shows the scatter of points with respect to only three of the 154 axes. The functional categories that were used for the projections are indicated in each of the plots and in the caption. As can be seen, the scatter of points indeed approximately falls on a straight line in each of these projections.

The extent to which the scatter of points falls on a line can be quantified most directly using principle component analysis.¹⁷ In principle component analysis one aims to represent a scatter of points in a high-dimensional space by a scatter in a much lower dimensional space. To this end it finds an ordered set of orthogonal coordinate axes in the n-dimensional space that have the property that, for any m ≤ n, the sum of the squared distances of the data points to their projections on the first m axes of the coordinate system is minimized (i.e., no set of m axes has a lower sum of squared distances). That is, the first principle axis is chosen such that the average squared-distance of the data points to this axis is minimized. The second principle axis is chosen such that the average squared-distance of the data points to the surface spanned by the first and second principle axis is minimized. And so on for the further principle components.

The ability of this coordinate system to represent the scatter of points is again measured by the fraction of the variance in the data that is captured by consecutive axes. For the scatter of 116 bacterial genomes in the function space of 154 ubiquitous categories, the first principle axis captures 22% of all the variance in the data. The second principle axis captures 6% of the variance, the third 5% of the variance, etcetera. Thus, the amount of variance explained drops sharply between the first and second principle axis. After that, the amount of variance explained by the data decreases smoothly, dropping between 5 and 10 percent between consecutive axes. As many as 55 axes are needed to cover 95% of the variance in the data. These statistics suggest that only the first principle axis captures an essential characteristic of gene-content in bacterial genomes. Once this largest component of the variance is taken into account, one needs almost as many dimensions as there are genomes to explain the remaining variance in gene-content.

Table 1 summarizes the statistics of the first three principle axes. Each of the axes is a vector ã whose direction in function space is reflected by the relative sizes of the components a_c. In the leftmost column of Table 1 I have listed, for each axis, the 4 categories with the highest components a_c and the 4 categories with the lowest components a_c (redundant categories were omitted). The values of these components ac are shown in the second column. The projection of all genomes onto one of the principle axes gives another vector →v where v_g is the component of genome g in the direction of the principle axis. The third column in Table 1 shows, for each principle axis, the 4 genomes with the highest components and the 4 genomes with the lowest components v_g. The components themselves are shown in column 4. These first three principle axes are also indicated as the red, blue, and green lines in Figure 6.

Table 1

Summary of the first three principle axes.

As can be easily derived from equation (4), the components a_c of the first principle axis correspond precisely to the inferred exponents of Figure 3. Thus, the entire collection of scaling laws is summarized by this single vector. In summary, a large fraction of the variation in functional gene-content among all bacterial genomes can be summarized by a single vector ã which encodes how the numbers of genes in different functional categories increase and decrease as the total size of the genome varies. That is, as the genome size increases or decreases the numbers of genes in each functional category c increase or decrease as g .The vector ã thus reflects a basic functional architecture of gene-content that holds across all bacterial genomes.

The meaning of the second and third principle axis is less clear, and given that they only capture a relatively small amount of the variance it is not clear that they are very meaningful at all. For both these axes the genomes at the extremes tend to be small parasitic organisms. This suggests that these axes may reflect different types of parasitic lifestyles.

Evolutionary Interpretation

What is the origin of the scaling laws discussed in the previous sections? In this section I will show that the observed scaling laws in fact suggest that there are fundamental constants in the evolutionary dynamics of genomes.

Consider a particular genome with numbers of genes n_c in different functional categories c. Consider next the evolutionary history of this genome. With this I mean that the current genome can be followed back in time through the life of the cell the genome was taken from, through the cell division that produced it, through the life of its ancestral cell, and its ancestors, and so on. In this way the history of the genome can be traced back all the way to an ancestral prokaryotic cell from which all currently existing bacteria stem. During this evolutionary history the numbers n_c have of coursed increased and decreased in ways that are unknown to us. That is, there are (unknown) functions of time n_c(t) that describe the evolution of the numbers of genes in each category c in the genome under study. Similarly, there will be a function g(t) that describes the evolutionary history of the total number of genes in the genome.

One can now ask what constraints the functions g(t) and n_c(t) should obey such that the observed scaling laws hold. To this end it is convenient to write the dynamics of n_c(t) and g(t) in terms of effective duplication and deletion rates. That is, we write

and

In these equations, β_c(t) is the (time-dependent) average duplication rate of genes in category c, β(t) is the average duplication rate of all genes, δ_c(t) is the average deletion rate of genes in category c, and δ(t) is the average deletion rate of all genes. I have also defined the differences of duplication rates and deletion rates as ρ_c(t) and ρ(t). Notice that, since in the above equations ρ_c(t) and ρ(t) can be arbitrary functions of time, any time-dependent function can still be obtained as the solution of the above equations. Formally solving these equations we obtain

and

where I have written the integrals as the time averages of ρ_c and ρ times the total time t. Note that these averages are a function of the evolutionary history of the genome under study. If we now express n_c(t) in terms of g(t) we obtain

Note that, although this equation may appear to already imply the general scaling relations that were found, in fact it is completely general and holds for any set of evolutionary histories n_c(t) and g(t). The equation is nothing more than a way of rewriting the relation between nc(t) and g(t) in terms of the averages <ρ_c> and <ρ. of this genome's history.

The observed scaling relations only follow from (10) if the same equation holds for all genomes. That is, if the variables n_c(0), g(0) and <ρ_c>/<ρ> are the same for all evolutionary histories. This requirement is illustrated in Figure 7.

Figure 7

The requirement that < ρc>/< ρ> be equal for all evolutionary histories thus demands that if one takes the average of the duplication minus deletion rate of genes in category c over the entire evolutionary (more...)

The figure shows the evolutionary histories of a set of genomes in a phylogenetic tree. Each leaf of the tree represents one of the bacterial genomes, and at the root is the common ancestor genome of all the genomes at the leafs. We thus have a separate equation (10) for each of the genomes at the leafs. The initial numbers nc(0)and g(0) in each of these equations are just the numbers of genes in category c and in the whole genome of the common ancestor, and it is therefore clear that the variables n_c(0) and g(0) are indeed the same for all the equations.^*

Much more interesting is the requirement that the ratios < ρ_c>/< ρ> are the same for all evolutionary histories. These different evolutionary histories are indicated as colored lines in Figure 7. The requirement that <ρ_c>/<ρ> be equal for all evolutionary histories thus demands that if one takes the average of the duplication minus deletion rate of genes in category c over the entire evolutionary history of a genome, and divides this by the average duplication minus deletion rate of all genes in the genome, then this ratio always comes out the same, independent of what evolutionary history is averaged over. That is, the ratios <ρ_c>/<ρ> are universal evolutionary constants and correspond precisely to the exponents α_c = <ρ_c>/<ρ> of the observed scaling laws.

So what determines <ρ_c> and <ρ> ? It seems reasonable to assume that the rate at which genes are duplicated and deleted mostly reflects the effects of selection. That is, as a first approximation we may assume that the rate at which duplications and deletions are introduced during evolution are approximately the same for all genes but that different genes have different probabilities of being selectively beneficial when duplicated or deleted. The rate ρ_c(t) is then given by some overall rate γ at which duplications and deletions are introduced^* times the difference between the fraction of genes ƒ⁺_c(t) in the category that would benefit the organism when duplicated and the fraction ƒ^-_c(t) of genes in the category that would benefit the organism when deleted:

We then have

The fractions ƒ⁺_c(t) and ƒ^-_c(t) of course depend on the selective pressures of the environment at time t. Thus, as one follows the genome through its evolutionary history, the demand for genes of a certain function fluctuates up and down, and this will be reflected in the fluctuations of ƒ⁺_c(t) - ƒ^-_c(t). It seems intuitively clear that the size of the fluctuations is going to depend strongly on the functional class. That is, one would expect that the demand for genes that provide an essential and basic function fluctuates relatively little. For instance, one would expect that even as the selective environment changes it is rare that existing protein biosynthesis genes become dispensable or that duplicates of these genes become desirable. On the other hand, the desirability of transcription regulatory genes and signal transducers is going to depend crucially on the selective environment in which the organism finds itself. It is thus not implausible that, if one averages over sufficiently long evolutionary times that the ratio <ƒ⁺_c - ƒ^-_c>/<ƒ⁺ - ƒ^-> always reaches the same limit, and that this limit is large for highly environment-dependent categories such as transcription regulation, and small for environment-insensitive categories such as protein biosynthesis and replication. The main open question that remains is the origin of the precise numerical values of the ratios <ƒ⁺_c - ƒ^-_c>/<ƒ⁺ - ƒ^-> for different functional categories.

Methods

Power-Law Fitting

In order to fit the data to power-law distributions I used a Bayesian procedure that is described in reference 13. The main advantage of this fitting procedure with respect to simple regression is that the results are explicitly rotationally invariant. That is, the best line that the fitting procedure produces doesn't depend on the orientation of the coordinate axes with respect to the scatter of data points.

The result is that the posterior distribution P(α | D)dα for the slope α of the line, given the data D, is given by

where n is the number of genomes in the data, s_xx is the variance in x-values (logarithms of the total gene numbers), s_yy the variance in y-values (logarithms of the number of genes in the category), s_yx is the covariance, and C is a normalizing constant. For each of the 154 ubiquitous categories we then calculated the 99% posterior probability interval for the slope from equation.¹³

Quality of the Power-Law Fits

To calculate the quality of the power-law fits, I first log-transform the data points (g_i,n_i) to (x_i,y_i) = (log(g_i),log(n,_i)). The best power-law fit is a straight line y = αx+β in the (x,y) plane. For each data point (x_i,y_i) I then find the distance d_i(l) to this line, and the distance d_i(c) to the center of the scatter of points. That is, with <x> the average of the x-values, and <y> the average of the y-values, the distance d_i(c) is given by

and the distance to the line is given by

I then define the quality of the fit F as the fraction of the variance that is explained by the fit.

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Footnotes

*

Note that in principle nothing keeps a genome from increasing n_c by more than 1 gene when g is increased by 1 gene. That is, some genes in other categories than c may be removed and replaced by genes in category c.

*

One caveat is that we implicitly assume that n_c(t) > 0 for all t. A slightly more complex treatment is needed for categories that were not present in the ancestor, or that disappeared and reappeared during the evolutionary history of some genomes.

*

In reality the rate at which duplications are introduced is unlikely to equal the rate at which delections are introduced. We ignore this complication for notational simplicity. The theoretical development with this complication would be analogous.