Computational biology seeks to integrate experimental data with predictive mathematical models—testing hypotheses which result from the former through simulations of the latter. Such models should ideally be approachable and accessible to the widest possible community, motivating independent studies. One of the most commonly modeled biological systems involves a gene family critical to segmentation in Drosophila embryogenesis—the segment polarity network (SPN). In this paper, we reduce a celebrated mathematical model of the SPN to improve its accessibility; unlike its predecessor our reduction can be tested swiftly on a widely used platform. By reducing the original model we identify components which are unnecessary; that is, we begin to detect the core of the SPN—those mechanisms that are essentially responsible for its characteristic behavior. Hence characteristic behavior can scale up; we find that any solution of our model (defined as a set of conditions for which characteristic behavior is seen) can be converted into a solution of the original model. The original model is thus made more accessible for independent study through a more approachable reduction which maintains the robustness of its predecessor. 1. Introduction It is widely acknowledged that mathematical models can provide valuable biological predictions [1, 2]. As levels of biological data increase, the scale of representative mathematical models must increase accordingly. This increase in scale will add computational complexity, posing a significant challenge to in silico testing. Hence, there is an ever increasing need for approaches which can facilitate the efficient study of large-scale models. A suitable approach is to reduce the dimension of large-scale models [3]. Predictions of a reduced model will ideally translate to the original model [4]. In this paper, we demonstrate such an approach for a model system in developmental biology. Over the course of Drosophila embryogenesis (when the formation of embryonic cells and organs occurs [5]), a cascade of maternal and zygotic segmentation genes establishes the bodily layout [6]. Normal development of segmental structures in the Drosophila embryo relies on a selective expression of the segment polarity (SP) genes [7–10] (which are initiated by the previously expressed pair-rule genes [11] at the beginning of gastrulation [12]) along the anterior-posterior axis. Products of SP genes (e.g., mRNAs and proteins) form a complex network of interaction which gives cells their distinctive identities [13] and functions [14, 15], refines
References
[1]
H. de Jong, “Modeling and simulation of genetic regulatory systems: a literature review,” Journal of Computational Biology, vol. 9, no. 1, pp. 67–103, 2002.
[2]
J. Jaeger, S. Surkova, M. Blagov et al., “Dynamic control of positional information in the early Drosophila embryo,” Nature, vol. 430, no. 6997, pp. 368–371, 2004.
[3]
Y.-C. Liu, C.-L. Lin, and C.-H. Chuang, “An approach for model reduction of biochemical networks,” Computational Biology Journal, vol. 2013, Article ID 263973, 14 pages, 2013.
[4]
M. Sunn？ker, G. Cedersund, and M. Jirstrand, “A method for zooming of nonlinear models of biochemical systems,” BMC Systems Biology, vol. 5, article 140, 2011.
[5]
J. W. Bodnar and M. K. Bradley, “Programming the Drosophila embryo 2: from genotype to phenotype,” Cell Biochemistry and Biophysics, vol. 34, no. 2, pp. 153–190, 2001.
[6]
S. Surkova, D. Kosman, K. Kozlov et al., “Characterization of the Drosophila segment determination morphome,” Developmental Biology, vol. 313, no. 2, pp. 844–862, 2008.
[7]
A. M. Arias, N. E. Baker, and P. W. Ingham, “Role of segment polarity genes in the definition and maintenance of cell states in the Drosophila embryo,” Development, vol. 103, no. 1, pp. 157–170, 1988.
[8]
A. Hidalgo and P. Ingham, “Cell patterning in the Drosophila segment: spatial regulation of the segment polarity gene patched,” Development, vol. 110, no. 1, pp. 291–301, 1990.
[9]
A. Wagner, “Robustness, evolvability, and neutrality,” FEBS Letters, vol. 579, no. 8, pp. 1772–1778, 2005.
[10]
G. Stoll, M. Bischofberger, J. Rougemont, and F. Naef, “Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico,” BioSystems, vol. 102, no. 1, pp. 3–10, 2010.
[11]
M. Chaves, R. Albert, and E. D. Sontag, “Robustness and fragility of Boolean models for genetic regulatory networks,” Journal of Theoretical Biology, vol. 235, no. 3, pp. 431–449, 2005.
[12]
D. Kosman, J. Reinitz, and D. H. Sharp, “Automated assay of gene expression at cellular resolution,” in Proceedings of the 1998 Pacific Symposium on Biocomputing, pp. 6–17, 1998.
[13]
A. Bejsovec and E. Wieschaus, “Segment polarity gene interactions modulate epidermal patterning in Drosophila embryos,” Development, vol. 119, no. 2, pp. 501–517, 1993.
[14]
B. O. Palsson, Systems Biology: Properties of Reconstructed Networks, Cambridge University Press, 2006.
[15]
A.-L. Barabasi and Z. N. Oltvai, “Network biology: understanding the cell’s functinal organization,” Nature Reviews Genetics, vol. 5, pp. 101–114, 2004.
[16]
M. D. Schroeder, M. Pearce, J. Fak et al., “Transcriptional control in the segmentation gene network of Drosophila,” PLoS Biology, vol. 2, no. 9, article e271, 2004.
[17]
N. H. Patel, B. Schafer, C. S. Goodman, and R. Holmgren, “The role of segment polarity genes during Drosophila neurogenesis,” Genes & Development, vol. 3, no. 6, pp. 890–904, 1989.
[18]
T. C. Lacalli, D. A. Wilkinson, and L. G. Harrison, “Theoretical aspects of stripe formation in relation to Drosophila segmentation,” Development, vol. 104, no. 1, pp. 105–113, 1988.
[19]
B. Sanson, C. Alexandre, N. Fascetti, and J.-P. Vincent, “Engrailed and hedgehog make the range of wingless asymmetric in Drosophila embryos,” Cell, vol. 98, no. 2, pp. 207–216, 1999.
[20]
S. DiNardo, J. Heemskerk, S. Dougan, and P. H. O'Farrell, “The making of a maggot: patterning the Drosophila embryonic epidermis,” Current Opinion in Genetics and Development, vol. 4, no. 4, pp. 529–534, 1994.
[21]
A. Martinez-Arias and P. A. Lawrence, “Parasegments and compartments in the Drosophila embryo,” Nature, vol. 313, no. 6004, pp. 639–642, 1985.
[22]
B. Prud'homme, R. de Rosa, D. Arendt et al., “Arthropod-like expression patterns of engrailed and wingless in the annelid Platynereis dumerilii suggest a role in segment formation,” Current Biology, vol. 13, no. 21, pp. 1876–1881, 2003.
[23]
G. von Dassow, E. Meir, E. M. Munro, and G. M. Odell, “The segment polarity network is a robust developmental module,” Nature, vol. 406, no. 6792, pp. 188–192, 2000.
[24]
G. von Dassow and G. M. Odell, “Design and constraints of the Drosophila segment polarity module: robust spatial patterning emerges from intertwined cell state switches,” Journal of Experimental Zoology, vol. 294, no. 3, pp. 179–215, 2002.
[25]
N. T. Ingolia, “Topology and robustness in the Drosophila segment polarity network,” PLoS Biology, vol. 2, no. 6, article e123, 2004.
[26]
R. Albert and H. G. Othmer, “The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster,” Journal of Theoretical Biology, vol. 223, no. 1, pp. 1–18, 2003.
[27]
W. Ma, L. Lai, Q. Ouyang, and C. Tang, “Robustness and modular design of the Drosophila segment polarity network,” Molecular Systems Biology, vol. 2, article 70, 2006.
[28]
A. Dayarian, M. Chaves, E. D. Sontag, and A. M. Sengupta, “Shape, size, and robustness: feasible regions in the parameter space of biochemical networks,” PLoS Computational Biology, vol. 5, no. 1, Article ID e1000256, 2009.
[29]
H. Meinhardt, “Hierarchical inductions of cell states: a model for segmentation in Drosophila,” Journal of Cell Science, vol. 4, supplement, pp. 357–381, 1986.
[30]
J. W. Bodnar, “Programming the Drosophila embryo,” Journal of Theoretical Biology, vol. 188, no. 4, pp. 391–445, 1997.
[31]
T. Schlitt and A. Brazma, “Current approaches to gene regulatory network modelling,” BMC Bioinformatics, vol. 8, no. 6, article S9, 2007.
[32]
U. Gritzan, V. Hatini, and S. DiNardo, “Mutual antagonism between signals secreted by adjacent Wingless and Engrailed cells leads to specification of complementary regions of the Drosophila parasegment,” Development, vol. 126, no. 18, pp. 4107–4115, 1999.
[33]
E. Meir, E. M. Munro, G. M. Odell, and G. von Dassow, “Ingeneue: a versatile tool for reconstituting genetic networks, with examples from the segment polarity network,” Journal of Experimental Zoology, vol. 294, no. 3, pp. 216–251, 2002.
[34]
C. J. Tomlin and J. D. Axelrod, “Biology by numbers: mathematical modelling in developmental biology,” Nature Reviews Genetics, vol. 8, no. 5, pp. 331–340, 2007.
[35]
A. Crudu, A. Debussche, and O. Radulescu, “Hybrid stochastic simplifications for multiscale gene networks,” BMC Systems Biology, vol. 3, article 89, 2009.
[36]
N. Barkai and S. Leibler, “Robustness in simple biochemical networks,” Nature, vol. 387, no. 6636, pp. 913–917, 1997.
[37]
K. Subramanian and C. Gadgil, “Robustness of the Drosophila segment polarity network to transient perturbations,” IET Systems Biology, vol. 4, no. 2, pp. 169–176, 2010.
[38]
S. Dan？, M. F. Madsen, H. Schmidt, and G. Cedersund, “Reduction of a biochemical model with preservation of its basic dynamic properties,” FEBS Journal, vol. 273, no. 21, pp. 4862–4877, 2006.
[39]
A. Saadatpour, I. Albert, and R. Albert, “Attractor analysis of asynchronous Boolean models of signal transduction networks,” Journal of Theoretical Biology, vol. 266, no. 4, pp. 641–656, 2010.
[40]
A. Naldi, P. T. Monteiro, and C. Chaouiya, “Efficient handling of large signalling-regulatory networks by focusing on their core control,” in Proceedings of the 10th International Conference on Computational Methods in Systems Biology, pp. 288–306, 2012.
[41]
A. Naldi, E. Remy, D. Thieffry, and C. Chaouiya, “Dynamically consistent reduction of logical regulatory graphs,” Theoretical Computer Science, vol. 412, no. 21, pp. 2207–2218, 2011.
[42]
A. Naldi, E. Remy, D. Thieffry, and C. Chaouiya, “A reduction of logical regulatory graphs preserving essential dynamical properties,” in Proceedings of the 7th International Conference on Computational Methods in Systems Biology, pp. 266–280, 2009.
[43]
M. Salathé and O. S. Soyer, “Parasites lead to evolution of robustness against gene loss in host signaling networks,” Molecular Systems Biology, vol. 4, article 202, 2008.
[44]
W. Liebermeister, U. Baur, and E. Klipp, “Biochemical network models simplified by balanced truncation,” FEBS Journal, vol. 272, no. 16, pp. 4034–4043, 2005.
[45]
H. Kitano, “Computational systems biology,” Nature, vol. 420, no. 6912, pp. 206–210, 2002.
[46]
O. Radulescu, A. N. Gorban, A. Zinovyev, and A. Lilienbaum, “Robust simplifications of multiscale biochemical networks,” BMC Systems Biology, vol. 2, article 86, 2008.
[47]
H. Kitano, “Biological robustness,” Nature Reviews Genetics, vol. 5, no. 11, pp. 826–837, 2004.
[48]
M. Apri, M. de Gee, and J. Molenaar, “Complexity reduction preserving dynamical behavior of biochemical networks,” Journal of Theoretical Biology, vol. 304, pp. 16–26, 2012.
[49]
D. J. Irons and N. A. M. Monk, “Identifying dynamical modules from genetic regulatory systems: applications to the segment polarity network,” BMC Bioinformatics, vol. 8, article 413, 2007.
