# Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons

^{*}

## Abstract

**:**

_{B}= ~1.7) was close to that expected of a diffusion limited aggregation process, confirming prior predictions as to a possible mechanism for development of this structure.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Predicted Promoters

#### 2.2. GPNs and Thresholding

_{ij}) were evaluated as the number of bp shared. These weighted edge values were used to form the adjacency matrix,

**A**. A network or graph G was generated based on the matrix

**A**(see Supplementary Figure 1). Networks were visualized using Pajek [51,52], a program used to work with large networks. Projections were rendered with the Kamada and Kawai [53] algorithm.

#### 2.3. Assessment of Fractal Structure

_{B}), or shortest path length between nodes, each node is colored in a fashion such that neighbors of like color are no further away than the current box length. Then the network is renormalized by collapsing adjacent nodes into a single node if they share the same color (see Supplementary Figure 3). This enforces the graph coloring rule that no two adjacent nodes can share the same color. The value N

_{B}then gives the minimum number of boxes of length l

_{B}required to cover the graph of N

_{B}nodes, and is equal to the graph size (node count) following renormalization. Considering a range of box lengths, a plot of l

_{B}versus N

_{B}on a log-log scale will be linear for networks with a fractal topology (see Supplementary Figure 3). On a normalized series of graphs with minimum size N, the fractal dimension d

_{B}is obtained from linear regression of the log-log transformation of the general scaling relation:

^{2}

_{d}) was used to assess the fit of the renormalization data to this fractal model.

#### 2.4. Information Entropy

_{b,l}and is calculated across the set of promoter sequences within a GPN or regulon. R

_{Sequence}represents the amount of information present at position l, and is calculated as the observed uncertainty minus the maximum possible uncertainty (2) under a 2-bit system:

_{RMean}denotes the arithmetic mean of R

_{Sequence}(l) across the l positions in the promoter footprint. We used the entropy calculator from the HCV database website [60] maintained by the Los Alamos National Laboratory to obtain summary estimates of R

_{Sequence}for each regulon, and WebLogo [61] to obtain figures of the sequence logos [59].

#### 2.5. Footprint Size, Information Entropy, and Fractal Dimension

_{RMean}), and fractal dimension (d

_{B}) using regression. MATLAB (The MathWorks Inc., Natick, MA, USA) was used to fit the linear model Y = βX + A + ε in which x and y were the independent and dependent variables, respectively. All three pair-wise relationships were explored, separately, given the three parameters F, I

_{RMean}, and d

_{B}. We examined relationships in which neither variable was transformed and in which both variables were log

_{10}-transformed; a linear fit in the latter circumstance indicates a power-law relationship, and this type of association has been found among numerous genome properties [4]. The fit was judged by the coefficient of determination (R

^{2}), and the corresponding correlation coefficient (r) is also reported.

## 3. Results

#### 3.1. Visual Pattern

#### 3.2. Fractal Dimensions

_{B}, Table 1). The average fractal dimension was d

_{B}= 2.118. The lowest observed was d

_{B}= 1.534 (Figure 1M) for a long linear symmetry; the highest observed was d

_{B}= 3.415 (Figure 1J) for a highly dense network. The fit of the fractal relationship was generally high for these (R

^{2}

_{d}= 0.906–0.978).

#### 3.3. Footprint Symmetry and Information Entropy

_{6})TCAC-3’ and complementary strand 3’-CACT(n

_{6})AGTG-5’.

_{RMean}, ranged between 0.927 (Fur) and 1.557 (−35 box of AlgU).

#### 3.4. Scaling of Footprint Size, Information (I_{RMean}), and Fractal Dimension

_{B}) scaled negatively with footprint size (F) (Figure 2 and Figure 3; Table 2). The power-law relationship (log-log transformation) was strongest (R

^{2}= 0.569) and significant (P = 0.002). The relation can be seen in Figure 2 in that the sequences near the base are generally longer than near the top. The smallest footprints ranged in size from 9–13 bp and displayed the highest fractal dimensions (group mean d

_{B}= 2.937), and these were topologically densest, containing numerous edges between very similar promoters (Figure 1). This contrasted with the larger footprint GPNs which generally had lower fractal dimensions (group mean d

_{B}= 1.790) and a greater likelihood of visually evident bilateral or radial symmetry.

_{RMean}) also scaled negatively with footprint size (Figure 2 and Figure 3; Table 2). Again, the power-law relationship gave the best and significant fit (R

^{2}= 0.795; P = 0.001). The relationship can be seen in Figure 2 whereby the smaller promoter motifs tend to have greater sequence conservation overall.

## 4. Discussion

_{B}) implies the rate at which the network changes in size (log

_{10}of the number of nodes) with each change of box length (log

_{10}of the relevant scale in number of edges). Thus a GPN with d

_{B}= 3.0 (i.e., line with slope = −3.0) drops in size much more quickly as box length is increased compared to one with d

_{B}= 2.0. Those with d

_{B}= 3.0 (e.g., Figure 1G and J) contain dense groups of highly related promoters, and on renormalization the size of the groups changes rapidly whereas change on renormalization is more gradual for the GPNs with d

_{B}= 2.0 (e.g., Figure 1A).

## 5. Conclusion

## Acknowledgments

## References

- Watson, J.D.; Crick, F.H.C. A Structure for Deoxyribose Nucleic Acid. Nature
**1953**, 171, 737–738. [Google Scholar] [CrossRef] [PubMed] - Li, W.; Marr, T.G.; Kaneko, K. Understanding long-range correlations in DNA sequences. Phys. D: Nonlinear Phenom.
**1994**, 75, 392–416. [Google Scholar] [CrossRef] - Mantegna, R.N.; Buldyrev, S.V.; Goldberger, A.L.; Havlin, S.; Peng, C.; Simons, M.; Stanley, H.E. Linguistic features of noncoding DNA sequences. Phys. Rev. Lett.
**1994**, 73, 3169–3172. [Google Scholar] [CrossRef] - Luscombe, N.M.; Qian, J.; Zhang, Z.; Johnson, T.; Gerstein, M. The dominance of the population by a selected few: Power-law behavior applies to a wide variety of genomic properties. Genome Biol.
**2002**, 3, research0040.1–research0040.7. [Google Scholar] [CrossRef] [PubMed] - Li, W. Features, patterns, correlations in DNA and protein texts. Available online: http://www.nslij-genetics.org/dnacorr/ (accessed on 11 November 2011).
- Molina, N.; Van Nimwegen, E. Scaling laws in functional genome content across prokaryotic clades and lifestyles. Trends Genet.
**2009**, 25, 243–247. [Google Scholar] [CrossRef] - Van Nimwegen, E. Scaling laws in the functional content of genomes. Trends Genet.
**2003**, 19, 479–484. [Google Scholar] [CrossRef] - Cattani, C. Fractals and hidden symmetries in DNA. Math. Probl. Eng.
**2010**, 2010, 507056. [Google Scholar] [CrossRef] - Lebedeva, D.V.; Filatova, M.V.; Kuklinb, A.I.; Islamovb, A.K.; Kentzingerc, E.; Pantinaa, R.; Toperverga, B.P.; Isaev-Ivanova, V.V. Fractal nature of chromatin organization in interphase chicken erythrocyte nuclei: DNA structure exhibits biphasic fractal properties. FEBS Lett.
**2005**, 579, 1465–1468. [Google Scholar] [CrossRef] [PubMed] - Lebedev, D.V.; Filatov, M.V.; Kuklin, A.I.; Islamov, A.K.; Stellbrink, J.; Pantina, R.A.; Denisov, Y.Y.; Toperverg, B.P.; Isaev-Ivanov, V.V. Structural hierarchy of chromatin in chicken erythrocyte nuclei based on small-angle neutron scattering: Fractal nature of the large-scale chromatin organization. Crystallogr. Rep.
**2008**, 53, 110–115. [Google Scholar] [CrossRef] - Lieberman-Aiden, E.; van Berkum, N.L.; Williams, L.; Imakaev, M.; Ragoczy, T.; Telling, A.; Amit, I.; Lajoie, B.R.; Sabo, P.J.; Dorschner, M.O.; et al. Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science
**2009**, 326, 289–293. [Google Scholar] [CrossRef] [PubMed] - Adam, R.L.; Silvab, R.C.; Pereira, F.G.; Leite, N.J.; Lorand-Metze, I.; Metze, K. The fractal dimension of nuclear chromatin as a prognostic factor in acute precursor B lymphoblastic leukemia. Cell. Oncol.
**2006**, 28, 55–59. [Google Scholar] [PubMed] - Bedin, V.; Adam, R.L.; de Sá, B.C.S.; Landman, G.; Metze, K. Fractal dimension of chromatin is an independent prognostic factor for survival in melanoma. BMC Cancer
**2010**, 10. [Google Scholar] [CrossRef] [PubMed] - Ferro, D.P.; Falconi, M.A.; Adam, R.L.; Ortega, M.M.; Lima, C.P.; de Souza, C.A.; Lorand-Metze, I.; Metze, K. Fractal characteristics of May-Grunwald-Giemsa stained chromatin are independent prognostic factors for survival in multiple myeloma. PLoS One
**2011**, 6. [Google Scholar] [CrossRef] [PubMed] - Kauffman, S. The Origins of Order: Self-Organization and Selection in Evolution; Oxford University Press: New York, NY, USA, 1993. [Google Scholar]
- Davidson, E.; Levin, M. Gene regulatory networks. Proc. Natl. Acad. Sci. USA
**2005**, 102, 4935. [Google Scholar] [CrossRef] [PubMed] - Karlebach, G.; Shamir, R. Modelling and analysis of gene regulatory networks. Nat. Rev. Mol. Cell Biol.
**2008**, 9, 770–780. [Google Scholar] [CrossRef] [PubMed] - Barabasi, A.-L.; Oltvai, Z. Network biology: Understanding the cell’s functional organization. Nat. Rev. Genet.
**2004**, 5, 101–113. [Google Scholar] [CrossRef] [PubMed] - Babu, M.M.; Luscombe, N.M.; Aravind, L.; Gerstein, M.; Teichmann, S.A. Structure and evolution of transcriptional regulatory networks. Curr. Opin. Struct. Biol.
**2004**, 14, 283–291. [Google Scholar] [CrossRef] [PubMed] - Barabasi, A.-L.; Albert, R. Emergence of scaling in random networks. Science
**1999**, 286, 509–512. [Google Scholar] - Jeong, H.; Tombor, B.; Albert, R.; Oltvai, Z.N.; Barabási, A.-L. The large-scale organization of metabolic networks. Nature
**2000**, 407, 651–654. [Google Scholar] [CrossRef] - Giot, L.; Bader, J.S.; Brouwer, C.; Chaudhuri, A.; Kuang, B.; Li, Y.; Hao, Y.L.; Ooi, C.E.; Godwin, B.; Vitols, E.; et al. A protein interaction map of Drosophila melanogaster. Science
**2003**, 302, 1727–1736. [Google Scholar] [CrossRef] - Yook, S.H.; Oltvai, Z.N.; Barabasi, A.-L. Functional and topological characterization of protein interaction networks. Proteomics
**2004**, 4, 928–942. [Google Scholar] [CrossRef] [PubMed] - Rocha, L.B.; Adam, R.L.; Leite, N.J.; Metze, K.; Rossi, M.A. Shannon’s entropy and fractal dimension provide an objective account of bone tissue organization during calvarial bone regeneration. Microsc. Res. Tech.
**2008**, 71, 619–625. [Google Scholar] [CrossRef] [PubMed] - Goldberger, A.L.; Amaral, L.A.N.; Hausdorff, J.M.; Ivanov, P.Ch.; Peng, C.-K.; Stanley, H.E. Fractal dynamics in physiology: Alterations with disease and aging. Proc. Natl. Acad. Sci. USA
**2002**, 99, 2466–2472. [Google Scholar] [CrossRef] [PubMed] - Galvão, V.; Mirandab, J.G.V.; Andradeb, R.F.S.; Andrade, J.S., Jr.; Gallose, L.K.; Maksee, H.A. Modularity map of the network of human cell differentiation. Proc. Natl. Acad. Sci. USA
**2010**, 107, 5750–5755. [Google Scholar] [CrossRef] [PubMed] - West, G.B.; Woodruff, W.H.; Brown, J.H. Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals. Proc. Natl. Acad. Sci. USA
**2002**, 99, 2473–2478. [Google Scholar] [CrossRef] [PubMed] - West, G.B.; Brown, J.H. The origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization. J. Exp. Biol.
**2005**, 208, 1575–1592. [Google Scholar] [CrossRef] [PubMed] - Savage, V.M.; Allen, A.P.; Brown, J.H.; Gillooly, J.F.; Herman, A.B.; Woodruff, W.H.; West, G.B. Scaling of number, size, and metabolic rate of cells with body size in mammals. Proc. Natl. Acad. Sci. USA
**2007**, 104, 4718–4723. [Google Scholar] [CrossRef] - Weaver, R. Molecular Biology, 4th ed.; McGraw-Hill: Boston, MA, USA, 2007. [Google Scholar]
- Collado-Vides, J. Grammatical model of the regulation of gene expression. Proc. Natl. Acad. Sci. USA
**1992**, 89, 9405–9409. [Google Scholar] [CrossRef] - Hawley, D.; McClure, W. Compilation and analysis of Escherichia coli promoter DNA sequences. Nucleic Acids Res.
**1983**, 11, 2237–2255. [Google Scholar] [CrossRef] - Huerta, A.; Collado-Vides, J. Sigma70 promoters in Escherichia coli: Specific transcription in dense regions of overlapping promoter-like signals. J. Mol. Biol.
**2003**, 333, 261–278. [Google Scholar] [CrossRef] - Lu, L.; Jia, H.; Dröge, P.; Li, J. The human genome-wide distribution of DNA palindromes. Funct. Integr. Genomics
**2007**, 7, 221–227. [Google Scholar] [CrossRef] [PubMed] - Janga, S.C.; Collado-Vides, J. Structure and evolution of gene regulatory networks in microbial genomes. Res. Microbiol
**2007**, 158, 787–794. [Google Scholar] [CrossRef] [PubMed] - Aldrich, P.R.; Horsley, R.K.; Ahmed, Y.A.; Williamson, J.J.; Turcic, S.M. Fractal topology of gene promoter networks at phase transitions. Gene Reg. Syst. Biol.
**2010**, 4, 75–82. [Google Scholar] [CrossRef] - Vallabhajosyula, R.R.; Chakravarti, D.; Lutfeali, S.; Ray, A.; Raval, A. Identifying hubs in protein interaction networks. PLoS One
**2009**, 4. [Google Scholar] [CrossRef] [PubMed] - Supekar, K.; Menon, V.; Rubin, D.; Musen, M.; Greicius, M.D. Network analysis of intrinsic functional brain connectivity in Alzheimer’s disease. PLoS Comput. Biol.
**2008**, 4. [Google Scholar] [CrossRef] [PubMed] - de Nooy, W.; Mrvar, A.; Batagelj, V. Exploratory Social Network Analysis with Pajek; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Mandelbrot, B.B. The Fractal Geometry of Nature; W.H. Freeman: San Francisco, CA, USA, 1983. [Google Scholar]
- Aldrich, P.R. Diffusion limited aggregation and the fractal evolution of gene promoter networks. Netw. Biol.
**2011**, 1, 99–111. [Google Scholar] - Song, C.; Havlin, S.; Makse, H.A. Origins of fractality in the growth of complex networks. Nat. Phys.
**2006**, 2, 275–281. [Google Scholar] [CrossRef] - Wittgenstein, L. Philosophical Investigations, Translated by G.E.M. Anscombe, 3rd ed.; MacMillan Publishing Co.: New York, NY, USA, 1958. [Google Scholar]
- Ferrer i Cancho, R.; Solé, R.V. Least effort and the origins of scaling in human language. Proc. Natl. Acad. Sci. USA
**2003**, 100, 788–791. [Google Scholar] [CrossRef] - Ferrer i Cancho, R. Zipf’s law from a communicative phase transition. Eur. Phys. J. B
**2005**, 47, 449–457. [Google Scholar] [CrossRef] - Shannon, C.E.A. Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423, 623–656. [Google Scholar] [CrossRef] - Steyvers, M.; Tenenbaum, J. The large-scale structure of semantic networks: Statistical analyses and a model of semantic growth. Cogn. Sci.
**2005**, 29, 41–78. [Google Scholar] [CrossRef] [PubMed] - Münch, R.; Hiller, K.; Grote, A.; Scheer, M.; Klein, J.; Schobert, M.; Jahn, D. Virtual Footprint and PRODORIC: An integrative framework for regulon prediction in prokaryotes. Bioinformatics
**2005**, 21, 4187–4189. [Google Scholar] [CrossRef] [PubMed] - Münch, R.; Hiller, K.; Barg, H.; Heldt, D.; Linz, S.; Wingender, E.; Jahn, D. Prodoric Database. 2003. Available online: http://prodoric.tu-bs.de/ (accessed on 11 November 2011).
- Gama-Castro, S.; Jiménez-Jacinto, V.; Peralta-Gil, M.; Santos-Zavaleta, A.; Peñaloza-Spinola, M.I.; Contreras-Moreira, B.; Segura-Salazar, J.; Muñiz-Rascado, L.; Martínez-Flores, I.; Salgado, H.; et al. RegulonDB (version 6.0): Gene regulation model of Escherichia coli K-12 beyond transcription, active (experimental) annotated promoters and Textpresso navigation. Nucleic Acids Res.
**2008**, 36, D120–D124. [Google Scholar] [CrossRef] [PubMed] - Batagelj, V.; Mrvar, A. Pajek – program for large network analysis. Connections
**1998**, 21, 47–57. [Google Scholar] - Batagelj, V.; Mrvar, A. Networks/Pajek. Available online: http://vlado.fmf.uni-lj.si/pub/networks/pajek/. (accessed on 11 November 2011).
- Kamada, T.; Kawai, S. An algorithm for drawing general undirected graphs. Inf. Process. Lett.
**1989**, 31, 7–15. [Google Scholar] [CrossRef] - Song, C.; Havlin, S.; Makse, H.A. Self-similarity of complex networks. Nature
**2005**, 433, 392–395. [Google Scholar] [CrossRef] [PubMed] - Song, C.; Gallos, L.K.; Havlin, S.; Makse, H.A. How to calculate the fractal dimension of a complex network: The box covering algorithm. J. Stat. Mech.
**2007**, P03006. [Google Scholar] [CrossRef] - Hagberg, A.; Schult, D.; Swart, P. Exploring Network Structure, Dynamics, and Function Using NetworkX. In Proceedings of the 7th Python in Science Conference (SciPy2008), Pasadena, CA USA, 19–24 August 2008; pp. 11–15. [Google Scholar]
- Hagberg, A.; Schult, D.; Swart, P. NetworkX. Available online: http://networkx.lanl.gov/ (accessed on 11 November 2011).
- Schneider, T.D.; Stormo, G.D.; Gold, L.; Ehrenfeucht, A. Information content of binding sites on nucleotide sequences. J. Mol. Biol.
**1986**, 188, 415–431. [Google Scholar] [CrossRef] - Schneider, T.D.; Stephens, R.M. Sequence logos: A new way to display consensus sequences. Nucleic Acids Res.
**1990**, 18, 6097–6100. [Google Scholar] [CrossRef] - Los Alamos National Laboratory. HCV Sequence Database: Entropy. Available online: http://hcv.lanl.gov/content/sequence/ENTROPY/entropy_one.html (accessed on 11 November 2011).
- Crooks, G.E.; Hon, G.; Chandonia, J.M.; Brenner, S.E. WebLogo: A sequence logo generator. Genome Res.
**2004**, 14, 1188–1190. [Google Scholar] [CrossRef] - National Center for Biotechnology Information (NCBI). Taxonomy. Available online: http://www.ncbi.nlm.nih.gov/guide/taxonomy/ (accessed on 11 November 2011).
- Hook-Barnard, I.G.; Hinton, D.M. Transcription initiation by mix and match elements: Flexibility for polymerase binding to bacterial promoters. Gene Reg. Syst. Biol.
**2007**, 1, 275–293. [Google Scholar] [CrossRef] - Papp, P.P.; Chattoraj, D.K.; Schneider, T.D. Information analysis of sequences that bind the replication initiator RepA. J. Mol. Biol.
**1993**, 233, 219–230. [Google Scholar] [CrossRef] [PubMed] - Schneider, T.D. Reading of DNA sequence logos: Prediction of major groove binding by information theory. Methods Enzymol.
**1996**, 274, 445–455. [Google Scholar] [PubMed] - Liebovitch, L.S. Fractals and Chaos; Oxford University Press: New York, NY, USA, 1998. [Google Scholar]
- Zipf, G.K. Human Behaviour and the Principle of Least Effort: An Introduction to Human Ecology; Addison–Wesley: Cambridge, MA, USA, 1949. [Google Scholar]
- Ferrer-I-Cancho, R.; Forns, N. The self-organization of genomes. Complexity
**2010**, 15, 34–36. [Google Scholar] [CrossRef] - Solé, R.V. Genome size, self-organization and DNA’s dark matter. Complexity
**2010**, 16, 20–23. [Google Scholar] [CrossRef] - Misteli, T. Self-organization in the genome. Proc. Natl. Acad. Sci. USA
**2009**, 106, 6885–6886. [Google Scholar] [CrossRef] [PubMed] - Rajapakse, I.; Scalzo, D.; Tapscott, S.J.; Kosak, S.T.; Groudine, M. Networking the nucleus. Mol. Syst. Biol.
**2010**, 6. [Google Scholar] [CrossRef]

**Figure 1.**The nuclei of fourteen gene promoter networks (GPNs) representing in each case the promoter footprint of a transcription factor binding in the genome of one of three bacterial species: Bacillus subtilis (BS), Escherichia coli (EC), and Pseudomonas aeruginosa (PA). Each network is an x-section taken from the upper phase transition of a serial extraction. Promoter predictions were obtained from the Virtual Footprint database [48]. The transcription factors defining each regulon are as follows: (

**A**) DegU, BS; (

**B**) Anr-Dnr(37), PA; (

**C**) ArgR, EC; (

**D**) Hpr, BS; (

**E**) ResD, BS; (

**F**) SigB(n14), BS; (

**G**) AlgU, PA; (

**H**) FleQ, PA; (

**I**) Fur, PA; (

**J**) PvdS, PA; (

**K**) DeoR, EC; (

**L**) CpxR, EC; (

**M**) Crp, EC; (

**N**) MarA, EC. See Table 1 for more details.

**Figure 2.**Sequence logos [59] for the fourteen regulons examined in this study. Each row of letters was rendered using WebLogos [61] and represents the sequence conservation across the set of promoters within a GPN. The height of each letter denotes the position-specific information entropy (R

_{Sequence}). Regulon id’s are as follows: (

**A**) PvdS; (

**B**) AlgU(-35); (

**C**) FleQ; (

**D**) ResD; (

**E**) Hpr; (

**F**) DeoR; (

**G**) Anr-Dnr(37); (

**H**) DegU; (

**I**) SigB(n14); (

**J**) MarA; (

**K**) Fur; (

**L**) ArgR; (

**M**) CpxR; and (

**N**) Crp. The order from bottom to top represents GPNs ranked according to increasing fractal dimension (d

_{B}) of their phase transition LCC x-section.

**Figure 3.**Relationships between footprint size, information entropy, and fractal dimension across fourteen nuclei of GPNs from three bacterial species.

**Table 1.**Promoter prediction settings for Virtual Footprint downloads along with footprint size, information entropy, and the outcome of fractal analyses.

Image | Regulon | Species | Library | S | X | F | I_{RMean} | d_{B} | R^{2}_{d} |
---|---|---|---|---|---|---|---|---|---|

A | DegU | BS | Prod | 0.8 | 17 | 21 | 1.032 | 1.837 | 0.921 |

B | Anr-Dnr(37) | PA | Prod | 1.0 | 13 | 14 | 1.295 | 1.953 | 0.921 |

C | ArgR | EC | Prod | 0.7 | 13 | 14 | 1.175 | 1.640 | 0.978 |

D | Hpr | BS | Prod | 0.8 | 16 | 19 | 1.081 | 2.120 | 0.914 |

E | ResD | BS | Prod | 0.2 | 12 | 13 | 1.551 | 2.599 | 0.960 |

F | SigB(n14) | BS | Prod | 0.8 | 20 | 32 | 0.967 | 1.831 | 0.945 |

G | AlgU(-35) | PA | Prod | 0.5 | 9 | 10 | 1.557 | 3.064 | 0.959 |

H | FleQ | PA | Prod | 0.5 | 10 | 11 | 1.410 | 2.669 | 0.965 |

I | Fur | PA | Prod | 0.9 | 15 | 19 | 0.927 | 1.665 | 0.972 |

J | PvdS | PA | Prod | 0.3 | 8 | 9 | 1.541 | 3.415 | 0.938 |

K | DeoR | EC | Reg | 1.0 | 14 | 16 | 1.109 | 2.040 | 0.906 |

L | CpxR | EC | Prod | 1.0 | 14 | 16 | 1.192 | 1.561 | 0.966 |

M | Crp | EC | Prod | 0.6 | 17 | 22 | 0.964 | 1.534 | 0.977 |

N | MarA | EC | Reg | 1.0 | 16 | 21 | 1.084 | 1.719 | 0.938 |

_{RMean}, arithmetic mean across base positions of the position-specific information (R

_{Sequence}) for a regulon of promoters; d

_{B}, fractal dimension of GPN for upper phase transition x-section as calculated by method of Song et al. [55]; R

^{2}

_{d}, coefficient of determination for regression of log-log transformed plot of l

_{B}versus N

_{B}.

**Table 2.**Relationships between footprint size, information (I

_{RMean}), and fractal dimension across nuclei of fourteen GPNs from three bacterial species based on linear regression of original data and log-log transformations of data.

x | y | β | A | R^{2} | r | F_{stat} | P |
---|---|---|---|---|---|---|---|

F | d_{B} | –0.064 | 3.205 | 0.431 | –0.656 | 9.079 | 0.010 |

log)
_{10}(F | log)
_{10}(d_{B} | –0.560 | 0.986 | 0.569 | –0.754 | 15.821 | 0.002 |

F | I_{RMean} | –0.031 | 1.726 | 0.669 | –0.818 | 24.245 | <0.001 |

log)
_{10}(F | log)
_{10}(I_{RMean} | –0.472 | 0.643 | 0.795 | –0.892 | 46.551 | <0.001 |

I_{RMean} | d_{B} | 2.236 | –0.579 | 0.734 | 0.857 | 33.088 | <0.001 |

log_{10}(I_{RMean}) | log(d_{B}) | 1.168 | 0.225 | 0.696 | 0.834 | 27.463 | <0.001 |

_{B}, fractal dimension of GPN for upper phase transition x-section; I

_{RMean}, average Shannon’s Information index based on measures of sequence entropy; β, slope of regression line (power-law coefficient for log-log scaling); A, intercept of regression line; R

^{2}, coefficient of determination for regression; r, correlation coefficient; F

_{stat}, F statistic for regression; P, P-value for regression.

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Aldrich, P.R.; Horsley, R.K.; Turcic, S.M.
Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons. *Symmetry* **2011**, *3*, 750-766.
https://doi.org/10.3390/sym3040750

**AMA Style**

Aldrich PR, Horsley RK, Turcic SM.
Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons. *Symmetry*. 2011; 3(4):750-766.
https://doi.org/10.3390/sym3040750

**Chicago/Turabian Style**

Aldrich, Preston R., Robert K. Horsley, and Stefan M. Turcic.
2011. "Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons" *Symmetry* 3, no. 4: 750-766.
https://doi.org/10.3390/sym3040750