Abstract
Networks exhibiting “accelerating” growth have total link numbers growing faster than linearly with network size and either reach a limit or exhibit graduated transitions from nonstationary-to-stationary statistics and from random to scale-free to regular statistics as the network size grows. However, if for any reason the network cannot tolerate such gross structural changes then accelerating networks are constrained to have sizes below some critical value. This is of interest as the regulatory gene networks of single-celled prokaryotes are characterized by an accelerating quadratic growth and are size constrained to be less than about 10,000 genes encoded in DNA sequence of less than about 10 megabases. This paper presents a probabilistic accelerating network model for prokaryotic gene regulation which closely matches observed statistics by employing two classes of network nodes (regulatory and non-regulatory) and directed links whose inbound heads are exponentially distributed over all nodes and whose outbound tails are preferentially attached to regulatory nodes and described by a scale-free distribution. This model explains the observed quadratic growth in regulator number with gene number and predicts an upper prokaryote size limit closely approximating the observed value.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albert, R., Barabási, A.L., 2002. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97.
Barabási, A.L., Albert, R., 1999. Emergence of scaling in random networks. Science 286, 509–512.
Barabási, A.L., Albert, R., Jeong, H., 1999. Mean-field theory for scale-free random networks. Physica A 272 (1–2), 173–187.
Barabási, A.L., Jeong, H., Néda, Z., Ravasz, E., Schubert, A., Vicsek, T., 2001. Evolution of the social network of scientific collaborations. Eprint cond-mat/0104162.
Barabási, A.L., Jeong, H., Néda, Z., Ravasz, E., Schubert, A., Vicsek, T., 2002. Evolution of the social network of scientific collaborations. Physica A 311, 590–614.
Bentley, S.D., et al. 2002. Complete genome sequence of the model actinomycete Streptomyces coelicolor A3(2). Nature 417 (6885), 141–147.
Bhan, A., Galas, D.J., Dewey, T.G., 2002. A duplication growth model of gene expression networks. Bioinformatics 18 (11), 1486–1493.
Cases, I., de Lorenzo, V., Ouzounis, C.A., 2003. Transcription regulation and environmental adaptation in bacteria. Trends Microbiol. 11 (6), 248–253.
Casjens, S., 1998. The diverse and dynamic structure of bacterial genomes. Annu. Rev. Genet. 32, 339–377.
Cheng, X., Wang, H., Ouyang, Q., 2002. Scale-free network model of node and connection diversity. Phys. Rev. E 65, 066115.
Cherry, J.L., 2003. Genome size and operon content. J. Theoret. Biol. 221, 401–410.
Chung, F., Lu, L., Dewey, T.G., Galas, D.J., 2003. Duplication models for biological networks. J. Comput. Biol. 10 (5), 677–687.
Consortium, I.H.G.S., 2001. Initial sequencing and analysis of the human genome. Nature 409 (6822), 860–921.
Croft, L.J., Lercher, M.J., Gagen, M.J., Mattick, J.S., 2003. Is prokaryotic complexity limited by accelerated growth in regulatory overhead? arXive:q-bio.MN/0311021. See http://arxiv.org/abs/ q-bio.MN/0311021.
de Jong, H., 2002. Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9 (1), 67–103.
Dorogovtsev, S.N., Mendes, J.F.F., 2000. Scaling behaviour of developing and decaying networks. Europhys. Lett. 52 (1), 33–39.
Dorogovtsev, S.N., Mendes, J.F.F., 2001a. Effect of the accelerating growth of communications networks on their structure. Phys. Rev. E 63, 025101 (R)
Dorogovtsev, S.N., Mendes, J.F.F. 2001b. Language as an evolving word web. Proc. R. Soc. London B 268, 2603–2606.
Dorogovtsev, S.N., Mendes, J.F.F., 2001c. Scaling properties of scale-free evolving networks: continuous approach. Phys. Rev. E 63, 056125.
Dorogovtsev, S.N., Mendes, J.F.F., 2002. Evolution of networks. Adv. Phys. 51 (4), 1079–1187.
Erdös, P., Renyi, A., 1960. On the evolution of random graphs. Publ. Math. Inst. Hungarian Acad. Sci. 5, 17–61.
Faloutsos, M., Faloutsos, P., Faloutsos, C., 1999. Power-law relationships of the Internet topology. In: Chapin, L., Sterbenz, J.P.G., Parulkar, G., Turner, J.S. (Eds.), Proceedings of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication. ACM Press, New York, pp. 251–262.
Gagen, M.J., Mattick, J.S., 2003. Failed “nonaccelerating” models of prokaryote gene regulatory networks. arXive: q-bio.MN/0312022. See http://arxiv.org/abs/q-bio.MN/0312022.
Goh, K.-I., Kahng, B., Kim, D., 2002. Fluctuation-driven dynamics of the Internet topology. Phys. Rev. Lett. 88 (10), 108701.
Hartwell, L.H., Hopfield, J.J., Leibler, S., Murray, A.W., 1999. From molecular to modular cell biology. Nature 402 (Suppl), C47-C52.
Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabási, A.-L., 2000. The large-scale organization of metabolic networks. Nature 407, 651–654.
Kunin, V., Ouzounis, C.A., 2003. The balance of driving forces during genome evolution in Prokaryotes. Genome Res. 13, 1589–1594.
Liu, Z., Lai, Y.-C., Ye, N., 2002. Statistical properties and attack tolerance of growing networks with algebraic preferential attachment. Phys. Rev. E 66, 036112.
Lukashin, A.V., Lukashev, M.E., Fuchs, R., 2003. Topology of gene expression networks as revealed by data mining and modeling. Bioinformatics 19 (15), 1909–1916.
Madan Babu, M., Teichmann, S.A., 2003. Evolution of transcription factors and the gene regulatory network in Escherichia coli. Nucleic Acids Res. 31 (4), 1234–1244.
Mathews, C.K., van Holde, K.E., Ahern., K.G., 2000. Biochemistry. Addison-Wesley, San Francisco.
Mattick, J.S., 2001. Non-coding RNAs: The architects of eukaryotic complexity. EMBO Rep. 2 (11), 986–991.
Mattick, J.S., 2003. Challenging the dogma: the hidden layer of non-protein-coding RNAs in complex organisms. BioEssays 25, 930–939.
Mattick, J.S., Gagen, M.J., 2001. The evolution of controlled multitasked gene networks: the role of introns and other noncoding RNAs in the development of complex organisms. Mol. Biol. Evol. 18 (9), 1611–1630.
Mattick, J.S., Gagen, M.J., 2005. Accelerating networks in biology, engineering, and society. Science 307, 856–858.
Rosenfeld, N., Elowitz, M.B., Alon, U., 2002. Negative autoregulation speeds the response times of transcription networks. J. Mol. Biol. 323, 785–793.
Sen, P., 2004. Accelerated growth in outgoing links in evolving networks: deterministic vs stochastic picture. Phys. Rev. E 69, 046107. arXive:cond-mat/0310513. See http://arxiv.org/abs/cond-mat/ 0310513.
Shen-Orr, S.S., Milo, R., Mangan, S., Alon, U., 2002. Network motifs in the transcriptional regulation network of Escherichia coli. Nat. Genet. 31, 64–68.
Stover, C.K., et al., 2000. Complete genome sequence of Pseudomonas aeruginosa PAO1, an opportunistic pathogen. Nature 406 (6799), 959–964.
Thieffry, D. 1998. From specific gene regulation to genomic networks: a global analysis of transcriptional regulation in Escherichia coli. BioEssays 20 (5), 433–440.
van Nimwegen, E., 2003. Scaling laws in the functional content of genomes. Trends Genet. 19 (9) 479–484.
Vázquez, A., 2000. Knowing a network by walking on it: emergence of scaling. Eprint cond-mat/0006132.
Vázquez, A., 2003. Growing network with local rules: preferential attachment, clustering hierarchy, and degree correlations. Phys. Rev. E 67, 056104.
Vázquez, A., Pastor-Satorras, R., Vespignani, A., 2002. Large-scale topological and dynamical properties of the Internet. Phys. Rev. E 65, 066130.
Venter, J.C., et al., 2001. The sequence of the human genome. Science 291 (5507), 1304–1351.
Vogel, J., Bartels, V., Tang, T.H., Churakov, G., Slagter-Jäger, J.G., Hüttenhofer, A., Wagner, E.G.H., 2003. RNomics in Escherichia coli detects new sRNA species and indicates parallel transcriptional output in bacteria. Nucleic Acids Res. 31 (22), 6435–6443.
Wagner, A., 2001. How to reconstruct a large genetic network from n gene perturbations in fewer than n 2 easy steps. Bioinformatics 17 (12), 1183–1197.
Warren, P.B., ten Wolde, P.R., 2003. Statistical analysis of the spatial distribution of operons in the transcriptional regulation network of Escherichia coli. q-bio.MN/0310029 (http://arxiv.org/abs/ q-bio.MN/0310029).
Watts, D.J., 1999. Networks, dynamics, and the Small-World phenomenon. Am. J. Sociol. 105 (2), 493–527.
Yanai, I., Camacho, C.J., DeLisi, C., 2000. Predictions of gene family distributions in microbial genomes: evolution by gene duplication and modification. Phys. Rev. Lett. 85 (12), 2641–2644.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gagen, M.J., Mattick, J.S. Inherent size constraints on prokaryote gene networks due to “accelerating” growth. Theory Biosci. 123, 381–411 (2005). https://doi.org/10.1016/j.thbio.2005.02.002
Received:
Accepted:
Issue date:
DOI: https://doi.org/10.1016/j.thbio.2005.02.002