Abstract
Mathematical models are useful for providing a framework for integrating data and gaining insights into the static and dynamic behavior of complex biological systems such as networks of interacting genes. We review the dynamic behaviors expected from model gene networks incorporating common biochemical motifs, and we compare current methods for modeling genetic networks. A common modeling technique, based on simply modeling genes as ON—OFF switches, is readily implemented and allows rapid numerical simulations. However, this method may predict dynamic solutions that do not correspond to those seen when systems are modeled with a more detailed method using ordinary differential equations. Until now, the majority of gene network modeling studies have focused on determining the types of dynamics that can be generated by common biochemical motifs such as feedback loops or protein oligomerization. For example, these elements can generate multiple stable states for gene product concentrations, state-dependent responses to stimuli, circadian rhythms and other oscillations, and optimal stimulus frequencies for maximal transcription. In the future, as new experimental techniques increase the ease of characterization of genetic networks, qualitative modeling will need to be supplanted by quantitative models for specific systems.
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Abel, T., K. C. Martin, D. Bartsch and E. R. Kandel (1998). Memory suppressor genes: inhibitory constraints on the storage of long-term memory. Science 279, 338–341.
Adams, C. C. and J. L. Workman (1995). Binding of disparate transcriptional activators to nucleosomal DNA is inherently cooperative. Mol. Cell. Biol. 15, 1405–1421.
Angel, P., K. Hattori, T. Smeal and M. Karin (1988). The jun proto-oncogene is positively autoregulated by its product, Jun—AP-1. Cell 55, 875–885.
Arkin, A., J. Ross and H. McAdams (1998). Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected E. coli cells. Genetics 149, 1633–1648.
Arkin, A., P. Shen and J. Ross (1997). A test case of correlation metric construction of a reaction pathway from measurements. Science 277, 1275–1279.
Bagley, R. J. and L. Glass (1996). Counting and classifying attractors in high dimensional dynamical systems. J. Theor. Biol. 183, 269–284.
Banks, H. and J. M. Mahaffy (1978). Stability of cyclic gene models for systems involving expression. J. Theor. Biol. 74, 323–334.
Bartsch, D., A. Casadio, K. A. Karl, P. Serodio and E. R. Kandel (1998). CREB1 encodes a nuclear activator, a repressor, and a cytoplasmic modulator that form a regulatory unit critical for long-term facilitation. Cell 95, 211–223.
Bartsch, D., M. Ghirardi, P. Skehel, K. A. Karl, S. Herder, M. Chen, C. Bailey and E. R. Kandel (1995). Aplysia CREB2 represses long-term facilitation: relief of repression converts a transient facilitation into long-term functional and structural change. Cell 83, 979–992.
Bazhan, S. I., V. A. Likhosvai and O. E. Belova (1995). Theoretical analysis of the regulation of interferon expression during priming and blocking. J. Theor. Biol. 175, 149–160.
Blattner, F. R. et al. (1997). The complete genome sequence of Escherichia coli K-12. Science 277, 1453–1462.
Bliss, R. D., P R. Painter and A. G. Marr (1982). Role of feedback inhibition in stabilizing the classical operon. J. Theor. Biol. 97, 177–93.
Blumenfeld, H., L. Zablow and B. Sabatini (1992). Evaluation of cellular mechanisms for modulation of Ca2+ transients using a mathematical model of fura-2 Ca2+ imaging in Aplysia sensory neurons. Biophys. J. 63, 1146–1164.
Boden, J. (1997). Programming the Drosophila embryo. J. Theor. Biol. 188, 391–445.
Busenberg, S. and J. M. Mahaffy (1985). Interaction of spatial diffusion and delays in models of genetic control by repression. J. Math. Biol. 22, 313–333.
Byrne, J. H. et al. (1991). Neural and molecular bases of nonassociative and associative learning in Aplysia. Ann. New York Acad. Sci. 627, 124–149.
The C. elegans Sequencing Consortium (1998). Genome sequence of the nematode C. elegans: a platform for investigating biology. Science 282, 2012–2018.
Carrier, T. A. and J. D. Keasling (1997). Mechanistic modeling of prokaryotic mRNA decay. J. Theor. Biol. 189, 195–209.
Carrion, A. M., W. A. Link, F. Ledo, B. Mellstrom and J. R. Naranjo (1999). DREAM is a Ca2+-regulated transcriptional repressor. Nature 398, 80–84.
Castano, J., R. Kineman and L. S. Frawley (1996). Dynamic monitoring and quantification of gene expression in single, living cells: a molecular basis for secretory cell heterogeneity. Mol. Endocrinol. 10, 599–605.
Cole, S. T. et al. (1998). Deciphering the biology of Mycobacterium tuberculosis from the complete genome sequence. Nature 393, 537–544.
Collado-Vides, J., B. Magasanik and J. D. Gralla (1991). Control site location and transcriptional regulation in Escherichia coli. Microbiol. Rev. 55, 371–394.
Collins, F. S., A. Patrinos, E. Jordan, A. Chakravati, R. Gesteland and L. Walters (1998). New goals for the U.S. human genome project: 1998–2003. Science 282, 682–689.
Dash, P. K. and A. N. Moore (1996). Characterization and phosphorylation of CREB-like proteins in Aplysia central nervous system. Brain Res. Mol. Brain Res. 39, 43–51.
Doedel, E. (1981). AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congr. Num. 30, 265–284.
Donachie, W. D. (1993). The cell cycle of Escherichia coli. Ann. Rev. Microbiol. 47, 199–230.
Dorman, C. J. (1995). DNA topology and the global control of bacterial gene expression: implications for the regulation of virulence gene expression. Microbiology (Reading) 141, 1271–1280.
Edwards, D. R. (1994). Cell signaling and the control of gene transcription. Trends Pharmacol. Sci. 15, 239–244.
Eisen, M. B., P. T. Spellman, P. O. Brown and D. Botstein (1998). Cluster analysis and display of genome-wide expression patterns. Proc. Natl. Acad. Sci. USA 95, 14863–14868.
Femino, A. M., F. S. Fay, K. Fogarty and R. H. Singer (1998). Visualization of single RNA transcripts in situ. Science 280, 585–590.
Gartner, K. (1990). A third component causing variability beside environment and genotype. A reason for the limited success of a 30 year long effort to standardize laboratory animals? Lab. Anim. 24, 71–77.
Gerhold, D., T. Rushmore and C. T. Caskey (1999). DNA chips: promising toys have become powerful tools. Trends Biochem. Sci. 24, 168–173.
Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 61, 2340–2361.
Glass, L. and S. A. Kauffman (1973). The logical analysis of continuous, non-linear biochemical control networks. J. Theor. Biol. 39, 103–129.
Glazewski, S., A. L. Barth, H. Wallace, M. McKenna, A. Silva and K. Fox (1999). Impaired experience-dependent plasticity in barrel cortex of mice lacking the α and δ isoforms of CREB. Cereb. Cortex 9, 249–256.
Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period (PER) protein. Proc. R. Soc. Lond. Ser. B 261, 319–324.
Goodwin, B. C. (1965). Oscillatory behavior of enzymatic control processes. Adv. Enzyme Reg. 3, 425–439.
Griffith, J. S. (1968a). Mathematics of cellular control processes. I. Negative feedback to one gene. J. Theor. Biol. 20, 202–208.
Griffith, J. S. (1968b). Mathematics of cellular control processes. II. Positive feedback to one gene. J. Theor. Biol. 20, 209–216.
Guckenheimer, J. and P. Holmes (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York: Springer-Verlag.
Guzowski, J. and J. L. McGaugh (1997). Antisense oligodeoxynucleotide-mediated disruption of hippocampal cAMP response element binding protein levels impairs consolidation of memory for water maze training. Proc. Natl. Acad. Sci. USA 94, 2693–2698.
Hammond, B. J. (1993). Quantitative study of the control of HIV-1 gene expression. J. Theor. Biol. 163, 199–221.
Hauri, D. C. and J. Ross (1995). A model of excitation and adaptation in bacterial chemotaxis. Biophys. J. 68, 708–722.
Hevroni, D. et al. (1998). Hippocampal plasticity involves extensive gene induction and multiple cellular mechanisms. J. Mol. Neurosci. 10, 75–98.
Hicks, K. A. and A. D. Grossman (1996). Altering the level and regulation of the major sigma subunit of RNA polymerase affects gene expression and development in Bacillus subtilus. Mol. Microbiol. 20, 201–212.
Hilsenbeck, S. G., W. E. Friedrichs, R. Schiff, P. O’Connell, R. K. Hansen, C. K. Osborne and S. A. Fuqua (1999). Statistical analysis of array expression data as applied to the problem of tamoxifen resistance. J. Natl. Cancer Inst. 91, 453–459.
Howard, P. and R. Maurer (1995). A composite Ets/Pit-1 binding site in the prolactin gene can mediate transcriptional responses to multiple signal transduction pathways. J. Biol. Chem. 270, 20930–20936.
Hunter, J., A. Kassam, C. Winrow, R. Rachubinski and J. Capone (1996). Crosstalk between the thyroid hormone and peroxisome proliferator-activated receptors in regulating peroxisome proliferator-responsive genes. J. Mol. Cell. Endocr. 116, 213–221.
Ishiura, M., S. Kutsuna, S. Aoki, H. Iwasaki, C. Andersson, A. Tanabe, S. Golden, C. Johnson and T. Kondo (1998). Expression of a gene cluster kaiABC as a circadian feedback process in cyanobacteria. Science 281, 1519–1523.
Itoh, K., B. Stevens, M. Schachner and R. D. Fields (1995). Regulated expression of the neural cell adhesion molecule L1 by specific patterns of neural impulses. Science 270, 1369–1372.
Iyer, V. et al. (1999). The transcriptional program in the response of human fibroblasts to serum. Science 283, 83–87.
Jacobs, C. and L. Shapiro (1998). Microbial asymmetric cell division: localization of cell fate determinants. Curr. Opin. Gen. Dev. 8, 386–391.
Jacob, F. and J. Monod (1961). On the regulation of gene activity. Cold Spring Harb. Symp. Quant. Biol. 26, 193–211, 389–401.
Karin, M. (1994). Signal transduction from the cell surface to the nucleus through the phosphorylation of transcription factors. Curr. Opin. Cell Biol. 6, 415–424.
Keizer, J. (1987). Statistical Thermodynamics of Equilibrium Processes, New York: Springer-Verlag.
Keller, A. (1994). Specifying epigenetic states with autoregulatory transcription factors. J. Theor. Biol. 170, 175–181.
Keller, A. (1995). Model genetic circuits encoding autoregulatory transcription factors. J. Theor. Biol. 172, 169–185.
Kerszberg, M. (1996). Accurate reading of morphogen concentrations by nuclear receptors: a formal model of complex transduction pathways. J. Theor. Biol. 183, 95–104.
Kerszberg, M. and J. Changeux (1994). A model for reading morphogenetic gradients: autocatalysis and competition at the gene level. Proc. Natl. Acad. Sci. USA 91, 5823–5827.
Kientzle, M. J. (1946). Properties of learning curves under varied distributions of practice. J. Exp. Psychol. 36, 187–211.
King, M. L. (1996). Molecular basis for cytoplasmic localization. Dev. Genet. 19, 183–189.
Ko, M. (1991). A stochastic model for gene induction. J. Theor. Biol. 153, 181–194.
Koh, B. T., R. B. Tan and M. G. Yap (1998). Genetically structured mathematical modeling of trp attenuator mechanism. Biotech. Bioeng. 58, 502–509.
Kouzarides, T. and E. Ziff (1988). The role of the leucine zipper in the fos—jun interaction. Nature 336, 646–651.
Lamprecht, R., S. Hazvi and Y. Dudai (1997). cAMP response element-binding protein in the amygdala is required for long-but not short-term conditioned taste aversion memory. J. Neurosci. 17, 8443–8450.
Lee, C., K. Bae and I. Edery (1998). The Drosophila CLOCK protein undergoes daily rhythms in abundance, phosphorylation, and interactions with the PER-TIM complex. Neuron 21, 857–867.
Leloup, J. C. and A. Goldbeter (1998). A model for circadian rhythms in Drosophila incorporating the formation of a complex between the PER and TIM proteins. J. Biol. Rhythms 13, 70–87.
Liu, B. Z., J. H. Peng, Y. C. Sun and Y. W. Liu (1997). A comprehensive dynamical model of pulsatile secretion of the hypothalamo-pituitary-gonadal axis in man. Comput. Biol. Med. 27, 507–513.
Luo, C., J. Loros and J. C. Dunlap (1998). Nuclear localization is required for function of the essential clock protein FRQ. EMBO J. 17, 1228–1235.
MacDonald, N. (1989). Biological Delay Systems: Linear Stability Theory, Cambridge: Cambridge University Press.
MacLeod, M. (1996). A possible role in chemical carcinogenesis for epigenetic, heritable changes in gene expression. Mol. Carcinog. 15, 241–250.
Mahaffy, J. M. (1984). Cellular control models with linked positive and negative feedback and delays. I: The models. J. Theor. Biol. 106, 89–102.
Mahaffy, J. M., D. A. Jorgensen and R. L. van der Heyden (1992). Oscillations in a model of repression with external control. J. Math. Biol. 30, 669–691.
Mahaffy, J. M. and C. V. Pao (1984). Models of genetic control by repression with time delays and spatial effects. J. Math. Biol. 20, 39–57.
Martin, K., D. Michael, J. Rose, M. Barad, A. Casadio, H. Zhu and E. R. Kandel (1997). MAP kinase translocates into the nucleus of the presynaptic cell and is required for long-term facilitation in Aplysia. Neuron 18, 899–912.
McAdams, H. and A. Arkin (1997). Stochastic mechanisms in gene expression. Proc. Natl. Acad. Sci. USA 94, 814–819.
McAdams, H. and A. Arkin (1998). Simulation of prokaryotic genetic circuits. Ann. Rev. Biophys. Biomed. Struct. 27, 199–224.
McAdams, H. and L. Shapiro (1995). Circuit simulation of genetic networks. Science 269, 650–656.
Merrow, M. W., N. Garceau and J. C. Dunlap (1997). Dissection of a circadian oscillation into discrete domains. Proc. Natl. Acad. Sci. USA 94, 3877–3882.
Mestl, T., C. Lemay and L. Glass (1996). Chaos in high-dimensional neural and gene networks. Physica D 98, 33–52.
Mestl, T., E. Plahte and S. W. Omholt (1995a). A mathematical framework for describing and analyzing gene regulatory networks. J. Theor. Biol. 176, 291–300.
Mestl, T., E. Plahte and S. W. Omholt (1995b). Periodic solutions in systems of piecewise linear differential equations. Dyn. Stab. Systems 10, 179–183.
Meyer, T., G. Waeber, J. Lin, W. Beckmann and J. Habener (1993). The promoter of the gene encoding cAMP response element binding protein contains cAMP response elements: evidence for positive autoregulation of gene transcription. Endocrinology 132, 770–780.
Meyers, S. and P. Friedland (1984). Knowledge-based simulation of genetic regulation in bacteriophage λ. Nucleic Acids Res. 12, 1–9.
Molina, C., N. Foulkes, E. Lalli and P. Sassone-Corsi (1993). Inducibility and negative autoregulation of CREM: an alternative promoter directs the expression of ICER, and early response repressor. Cell 75, 875–886.
Novak, B., A. Csikasz-Nagy, B. Gyorffy, K. Chen and J. J. Tyson (1998a). Mathematical model of the fission yeast cell cycle with checkpoint controls at the G1/S, G2/M, and metaphase/anaphase transitions. Biophys. Chem. 72, 185–200.
Novak, B., A. Csikasz-Nagy, B. Gyorffy, K. Chen and J. J. Tyson (1998b). Model scenarios for evolution of the eukaryotic cell cycle. Philos. Trans. R. Soc. Lond. Ser. B 353, 2063–2076.
Okayama, H., A. Nagata, S. Jinno and H. Murakami (1996). Cell cycle control in fission yeast and mammals: identification of new regulatory mechanisms. Adv. Cancer Res. 69, 17–62.
O’Leary, F., J. Byrne and L. Cleary (1995). Long-term structural remodeling in Aplysia sensory neurons requires de novo protein synthesis during a critical time period. J. Neurosci. 15, 3519–3525.
Omholt, S. W., X. Kefang, O. Andersen and E. Plahte (1998). Description and analysis of switchlike regulatory networks exemplified by a model of cellular iron homeostasis. J. Theor. Biol. 195, 339–350.
Plahte, E., T. Mestl and S. W. Omholt (1994). Global analysis of steady points for systems of differential equations with sigmoid interactions. Dyn. Stab. Syst. 9, 275–291.
Plahte, E., T. Mestl and S. W. Omholt (1998). A methodological basis for description and analysis of systems with complex switch-like interactions. J. Math. Biol. 36, 321–348.
Polach, K. J. and J. Widom (1996). A model for the cooperative binding of eukaryotic regulatory proteins to nucleosomal target sites. J. Mol. Biol. 258, 800–812.
Ptashne, M. (1992). A Genetic Switch: Phage λ and Higher Organisms, Cambridge: Cell Press/Blackwell.
Reppert, S. M. (1998). A clockwork explosion! Neuron 21, 1–4.
Richards, J., H. Bachinger, R. Goodman and R. Brennan (1996). Analysis of the structural properties of cAMP-responsive element binding protein (CREB) and phosphorylated CREB. J. Biol. Chem. 271, 13716–13723.
Roberts, R. C., C. D. Mohr and L. Shapiro (1996). Developmental programs in bacteria. Curr. Top. Dev. Biol. 34, 207–257.
Robertson, B. D. (1992). Genetic variation in pathogenic bacteria. Trends Genet. 8, 422–427.
Rosen, R. (1968). Recent developments in the theory of control and regulation of cellular processes, in International Review of Cytology, G. H. Bourne (Ed.), New York: Academic Press.
Rossant, J. and N. Hopkins (1992). Of fin and fur: mutational analysis of vertebrate embryonic development. Genes Dev. 6, 1–13.
Sabry, J., T. O’Connor and M. W. Kirschner (1995). Axonal transport of tubulin in Ti1 Pioneer neurons in situ. Neuron 14, 1247–1256.
Sassone-Corsi, P. (1995). Transcription factors responsive to cAMP. Ann. Rev. Cell Dev. Biol. 11, 355–377.
Sassone-Corsi, P., J. C. Sisson and I. M. Verma (1988). Transcriptional autoregulation of the proto-oncogene fos. Nature 334, 314–319.
Scheper, T., D. Klinkenberg, C. Pennartz and J. van Pelt (1999). A mathematical model for the intracellular circadian rhythm generator. J. Neurosci. 19, 40–47.
Schnitzer, M. J. and S. M. Block (1997). Kinesin hydrolyses one ATP per 8-nm step. Nature 388, 386–389.
Seo, H., C. Yang, H. Kim and K. Kim (1996). Multiple protein factors interact with the cis-regulatory elements of the proximal promoter in a cell-specific manner and regulate transcription of the dopamine β-hydroxylase gene. J. Neurosci. 16, 4102–4112.
Serfling, E. (1989). Autoregulation—a common property of eukaryotic transcription factors? Trends Gen. 5, 131–133.
Shapiro, L. and R. Losick (1997). Protein localization and cell fate in bacteria. Science 276, 712–718.
Sheng, H., R. D. Fields and P. Nelson (1993). Specific regulation of immediate-early genes by patterned neuronal activity. J. Neurosci. Res. 35, 459–467.
Smith, H. (1987a). Oscillations and multiple steady states in a cyclic gene model with repression. J. Math. Biol. 25, 169–190.
Smith, H. (1987b). Monotone semiflows generated by functional differential equations. J. Diff. Eqs. 66, 420–442.
Smolen, P., D. A. Baxter and J. H. Byrne (1998). Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory systems. Am. J. Physiol. 274, C531–C542.
Smolen, P., D. A. Baxter and J. H. Byrne (1999a). Effects of macromolecular transport and stochastic fluctuations on the dynamics of genetic regulatory systems. Am. J. Physiol., 277, C777–C790.
Smolen, P., D. A. Baxter and J. H. Byrne (1999b). Modeling clarifies the role of delays and feedback in circadian oscillators. Soc. Neurosci. Abstr., 25, 867.
Snoussi, E. H. and R. Thomas (1993). Logical identification of all steady states: The concept of feedback loop characteristic states. Bull. Math. Biol. 55, 973–991.
Somogyi, R. and C. Sniegoski (1996). Modeling the complexity of genetic networks: understanding multigenic and pleiotropic regulation. Complexity 1, 45–63.
Somogyi, R. and C. Sniegoski (1997). The gene expression matrix: towards the extraction of genetic network architectures, Proceedings of the Second World Congress of Nonlinear Analysis, Elsevier Science.
Spiro, P. A., J. Parkinson and H. Othmer (1997). A model of excitation and inhibition in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA 94, 7263–7268.
Stehle, J. H., N. Foulkes, C. Molina, V. Simonneaux, P. Pevet and P. Sassone-Corsi (1993). Adrenergic signals direct rhythmic expression of transcriptional repressor CREM in the pineal gland. Nature 365, 314–320.
Thayer, M. J., S. J. Tapscott, R. L. Davis, W. E. Wright, A. B. Lassar and H. Weintraub (1989). Positive autoregulation of the myogenic determination gene MyoD1. Cell 58, 241–248.
Thomas, R. (1994). The role of feedback circuits: positive feedback circuits are a necessary condition for positive real eigenvalues of the Jacobian matrix. Ber. Bunsenges. Phys. Chem. 98, 1148–1151.
Thomas, R. and R. d’Ari (1990). Biological Feedback, Boca Raton, FL: CRC Press.
Thomas, R., D. Thieffry and M. Kauffman (1995). Dynamical behaviour of biological regulatory networks—I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull. Math. Biol. 57, 247–276.
Tomita, M. et al. (1999). E-CELL: a software environment for whole-cell simulation. Bioinformatics 15, 72–84.
Tully, T., T. Preat, S. Boynton and M. Del Vecchio (1994). Genetic dissection of consolidated memory in Drosophila melanogaster. Cell 79, 35–47.
Tyson, J. and H. G. Othmer (1978). The dynamics of feedback control circuits in biochemical pathways. Prog. Theor. Biol. 5, 2–62.
Vandien, S. J. and J. D. Keasling (1998). A dynamic model of the E. coli phosphatestarvation response. J. Theor. Biol. 190, 37–49.
Von Heijne, G., C. Blomberg and H. Liljenstrom (1987). Theoretical modeling of protein synthesis. J. Theor. Biol. 125, 1–4.
Walker, W., L. Fucci and J. Habener (1995). Expression of the gene encoding transcription factor CREB: regulation by follicle-stimulating hormone-induced cAMP signaling in primary rat sertoli cells. Endocrinology 136, 3534–3545.
Wen, X. L., S. Fuhrman, G. Michaels, D. Carr, S. Smith, J. Barker and R. Somogyi (1998). Large-scale temporal gene expression mapping of central nervous system development. Proc. Natl. Acad. Sci. USA 95, 334–339.
Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos, Heidelberg: Springer-Verlag.
Winzeler, E. A. et al. (1998). Direct allelic variation scanning of the entire yeast genome. Science 281, 1194–1197.
Wolf, D. M. and F. H. Eeckman (1998). On the relationship between genomic regulatory element organization and gene regulatory dynamics. J. Theor. Biol. 195, 167–186.
Wong, P., S. Gladney and J. D. Keasling (1997). A mathematical model of the lac operon-inducer exclusion, catabolite repression, and diauxic growth on glucose and lactose. Biotech. Prog. 13, 132–143.
Yagil, G. and E. Yagil (1971). On the relation between effector concentration and the rate of induced enzyme synthesis. Biophys. J. 11, 11–27.
Yin, J., M. Del Vecchio, H. Zhou and T. Tully (1995). CREB as a memory modulator: induced expression of a dCREB2 activator isoform enhances long-term memory in Drosophila. Cell 81, 107–115.
Yin, J. and T. Tully (1996). CREB and the formation of long-term memory. Curr. Opin. Neurobiol. 6, 264–267.
Yisraeli, J. K., S. Sokol and D. A. Melton (1990). A two-step model for the localization of maternal mRNA in Xenopus oocytes: involvement of microtubules and microfilaments in the translocation and anchoring of Vg1 mRNA. Development 108, 289–298.
Yuh, C. H., H. Bolouri and E. H. Davidson (1998). Genomic cis-regulatory logic, experimental and computational analysis of a sea urchin gene. Science 279, 1896–1902.
Zimmerman, S. and A. P. Minton (1993). Macromolecular crowding: biochemical, biophysical, and physiological consequences. Ann. Rev. Biophys. Biomol. Struct. 22, 27–65.
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Smolen, P., Baxter, D.A. & Byrne, J.H. Modeling transcriptional control in gene networks—methods, recent results, and future directions. Bull. Math. Biol. 62, 247–292 (2000). https://doi.org/10.1006/bulm.1999.0155
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DOI: https://doi.org/10.1006/bulm.1999.0155