Quantitative measure of randomness and order for complete genomes

We propose an order index, $\varphi$, which gives a quantitative measure of randomness and order of complete genomic sequences. It maps genomes to a number from 0 (random and of infinite length) to 1 (fully ordered) and applies regardless of sequence length. The 786 complete genomic sequences in GenBank were found to have $\varphi$ values in a very narrow range, ${\varphi }_g={0.031}_{-0.015}^{+0.028}$. We show this implies that genomes are halfway toward being completely random, or, at the “edge of chaos.” We further show that artificial “genomes” converted from literary classics have $\varphi$’s that almost exactly coincide with ${\varphi }_g$, but sequences of low information content do not. We infer that ${\varphi }_g$ represents a high information-capacity “fixed point” in sequence space, and that genomes are driven to it by the dynamics of a robust growth and evolution process. We show that a growth process characterized by random segmental duplication can robustly drive genomes to the fixed point.

Figure 1

(Color online) (a) Log-log plot of order index, $\varphi$, vs length of random sequence for $p=0.5$ and $k=2–6$. (b) Same as (a); for $k=4$ and $p=0.20–0.50$. (c) Semi-log plot of $\varphi$ vs $N_{\mu }$, number of random point mutations, for an initially ordered 20 Mb, $p=0.5$ sequence. The intersection of the lines is the critical point where sequence becomes random. (d) Same as (c); initial sequence is genome of E. coli.

Figure 2

(a) Interval distributions of three 4-mers with frequencies $f_u=\overline{f}=L/256$ (○), $2\overline{f}$ (◻), and $\overline{f}/2$ (△), respectively, in E. coli ($L=4.6\text{Mb}$, $p=0.5$; solid symbols) and in corresponding random sequence (hollow symbols); solid lines express Eq. (7). (b) Average interval $x_u$ of $k$-mer $u$ versus $d_u$, where $-d_u$ is the slope of the linear regression of the logarithm of the interval distribution of $u$; $x_u=d_u$ if the distribution is exponential. Data for all $k$-mers, $k=2$ to 6, in E. coli are shown; for each $k$ the number of data and the mean value of $x_u$ are both equal to $4^k$.

Figure 3

(a) Order index, $\varphi$, vs sequence length, $L$, for 384 prokaryotic genomes (gray ◼’s), 402 eukaryotic chromosomes (black ●’s), and random sequences (line composed of solid △’s). In (b)–(f): box (gray for prokaryotes; black for eukaryotes) height is given by 25% to 50% values and the range represents 10% to 90% values; numbers above boxes are numbers of sequences in group; all $\varphi$’s are averaged over $k=2$ to 6. (b) $\varphi$ vs $\text{log}L$. (c) $\varphi$ vs fractional A/T content, $p$. (d) Ratio of ${\varphi }_{cd}$ (for genic parts) to ${\varphi }_{ncd}$ (nongenic) vs $\text{log}L$. (e) Ratio of ${\varphi }_{mRNA}$ (mRNA segments) to ${\varphi }_{nmRNA}$ (non-mRNA), averaged over classes of eukaryotes. (f) Ratio of equivalent mutation rate, ${\mu }_{eq}$, to critical mutation rate, ${\mu }_c$, vs $\text{log}L$.

Figure 4

$P$ values of (a) data boxes from Fig. 3b and (b) from Fig. 3c being part a normal distribution defined by the right-hand side of Eq. (8). The horizontal dashed line indicates $P=0.05$.

Figure 5

(Color online) Left panel, average segmental index, ${\overline{\varphi }}_l$ vs segment length $l$ for the human chromosome I (solid square) and the R. baltica chromosome (circle). Right panel, category (number of chromosomes given above box) statistics of pairwise Hamming distances between $k=5$ $\vec{s}$ vectors; legends for boxes same as in Fig. 3.

Figure 6

Box plots of $\varphi$ for 50 kb segments obtained from partition of a chromosome and grouped in 0.05 $p$ intervals; numeral indicates number of segments represented by each box. (a) E. coli; (b) S. pombe chr. III; (c) A. thaliana chr. I; (d) D. melanogaster chr. X; (e) C. elegans chr. I; (f) H. sapiens chr. I.

Figure 7

(Color online) The function $I\left(z\right)$ [Eq. (10)] plotted as a function of $z={\varphi }^{0.21}$; $\varphi$; ${\text{log}}_{10}{L}_{eq}\left(\varphi \right)$; ${\mu }_{eq}\left(\varphi \right)$ (in units of ${\text{b}}^{-1}$). Coordinates for 384 prokaryotic and 402 eukaryotic chromosomes are shown in gray and black, respectively, and those for the six classics (stars), two long repeats of short sentences (diamonds), five bichromosome concatenates (hexagons), and two $\pi$ sequences (triangles).

Figure 8

Order index versus $p$ of model sequences generated from a minimum growth model [46]. Lengths of the sequences are approximately 2 Mb and each $p$-interval bin gives the result of 20 sequences.