Frame-based mathematical models – a tool for the study of molecular genetic systems

This paper reviews existing approaches for reconstructing frame-based mathematical models of molecular genetic systems from the level of genetic synthesis to models of metabolic networks. A frame-based mathematical model is a model in which the following terms are specified: formal structure, type of mathematical model for a particular biochemical process, reactants and their roles. Typically, such models are generated automatically on the basis of description of biological processes in terms of domain-specific languages. For molecular genetic systems, these languages use constructions familiar to a wide range of biologists in the form of a list of biochemical reactions. They rely on the concepts of elementary subsystems, where complex models are assembled from small block units (“frames”). In this paper, we have shown an example with the generation of a classical repressilator model consisting of three genes that mutually inhibit each other’s synthesis. We have given it in three different versions of the graphic standard, its characteristic mathematical interpretation and variants of its numerical calculation. We have shown that even at the level of frame models it is possible to identify qualitatively new behaviour of the model through the introduction of just one gene into the model structure. This change provides a way to control the modes of behaviour of the model through changing the concentrations of reactants. The frame-based approach opens the way to generate models of cells, tissues, organs, organisms and communities through frame-based model generation tools that specify structure, roles of modelled reactants using domain-specific languages and graphical methods of model specification.

1. Alon U. Network motifs: theory and experimental approaches. Nat Rev Genet. 2007;8(6):450-461. doi 10.1038/nrg2102

2. Beal J., Lu T., Weiss R. Automatic compilation from high-level bio logically-oriented programming language to genetic regulatory networks. PLoS One. 2011;6(8):e22490. doi 10.1371/journal.pone.0022490

3. Büchel F., Rodriguez N., Swainston N., Wrzodek C., Czauderna T., Keller R., Mittag F., … Saez-Rodriguez J., Schreiber F., Laibe C., Dräger A., Le Novère N. Path2Models: large-scale generation of computational models from biochemical pathway maps. BMC Syst Biol. 2013;7:116. doi 10.1186/1752-0509-7-116

4. Cornish-Bowden A. An automatic method for deriving steady-state rate equations. Biochem J. 1977;165(1):55-59. doi 10.1042/bj1650055

5. Courtot M., Juty N., Knüpfer C., Waltemath D., Zhukova A., Dräger A., Dumontier M., … Wilkinson D.J., Wimalaratne S., Laibe C., Hucka M., Le Novère N. Controlled vocabularies and semantics in systems biology. Mol Syst Biol. 2011;7(1):543. doi 10.1038/msb.2011.77

6. Cox R.S., Madsen C., McLaughlin J., Nguyen T., Roehner N., Bart ley B., Bhatia S., … Voigt C.A., Zundel Z., Myers C., Beal J., Wipat A. Synthetic Biology Open Language Visual (SBOL Visual). Version 2.0. J Integr Bioinform. 2018;15(1):20170074. doi 10.1515/jib-2017-0074

7. Der B.S., Glassey E., Bartley B.A., Enghuus C., Goodman D.B., Gor don D.B., Voigt C.A., Gorochowski T.E. DNAplotlib: programmable visualization of genetic designs and associated data. ACS Synth Biol. 2017;6(7):1115-1119. doi 10.1021/acssynbio.6b00252

8. Dräger A., Zielinski D.C., Keller R., Rall M., Eichner J., Palsson B.O., Zell A. SBMLsqueezer 2: context-sensitive creation of kinetic equations in biochemical networks. BMC Syst Biol. 2015;9(1):68. doi 10.1186/s12918-015-0212-9

9. Ebrahim A., Lerman J.A., Palsson B.O., Hyduke D.R. COBRApy: COnstraints-Based Reconstruction and Analysis for Python. BMC Syst Biol. 2013;7(1):74. doi 10.1186/1752-0509-7-74

10. Elowitz M.B., Leibler S. A synthetic oscillatory network of transcriptional regulators. Nature. 2000;403(6767):335-338. doi 10.1038/35002125

11. Fadeev S.I., Likhoshvai V.A. On hypothetical gene networks. Sib Zh Ind Mat. 2003;6(3):134-153 Funahashi A., Matsuoka Y., Jouraku A., Morohashi M., Kikuchi N., Ki tano H. CellDesigner 3.5: a versatile modeling tool for biochemical networks. Proc IEEE. 2008;96(8):1254-1265. doi 10.1109/JPROC.2008.925458

12. Galdzicki M., Clancy K.P., Oberortner E., Pocock M., Quinn J.Y., Ro driguez C.A., Roehner N., … Villalobos A., Wipat A., Gennari J.H., Myers C.J., Sauro H.M. The Synthetic Biology Open Language (SBOL) provides a community standard for communicating de signs in synthetic biology. Nat Biotechnol. 2014;32(6):545-550. doi 10.1038/nbt.2891

13. Harris L.A., Hogg J.S., Tapia J.-J., Sekar J.A.P., Gupta S., Korsun sky I., Arora A., Barua D., Sheehan R.P., Faeder J.R. BioNetGen 2.2: advances in rule-based modeling. Bioinformatics. 2016;32(21):3366 3368. doi 10.1093/bioinformatics/btw469

14. Herold S., Krämer D., Violet N., King R. Rapid process synthesis supported by a unified modular software framework. Eng Life Sci. 2017;17(11):1202-1214. doi 10.1002/elsc.201600020

15. Hoops S., Sahle S., Gauges R., Lee C., Pahle J., Simus N., Singhal M., Xu L., Mendes P., Kummer U. COPASI – a COmplex PAthway SImulator. Bioinformatics. 2006;22(24):3067-3074. doi 10.1093/bioinformatics/btl485

16. Hucka M., Bergmann F.T., Hoops S., Keating S.M., Sahle S., Schaff J.C., Smith L.P., Wilkinson D.J. The Systems Biology Markup Language (SBML): language specification for level 3 version 1 core. J Integr Bioinform. 2015;12(2):382-549. doi 10.2390/biecoll-jib-2015-266

17. Kanehisa M. KEGG: Kyoto Encyclopedia of Genes and Genomes. Nucleic Acids Res. 2000;28(1):27-30. doi 10.1093/nar/28.1.27

18. Karr J.R., Phillips N.C., Covert M.W. WholeCellSimDB: a hybrid re lational/HDF database for whole-cell model predictions. Database (Oxford ). 2014;2014:bau095. doi 10.1093/database/bau095

19. Kazantsev F.V., Akberdin I.R., Bezmaternykh K.D., Likhoshvai V.A. The tool for automatic generation of gene networks mathematical models. Vestnik VOGiS. 2009;13(1):163-169

20. Kazantsev F.V., Akberdin I.R., Lashin S.A., Ree N.A., Timonov V.S., Ratushnyi A.V., Khlebodarova T.M., Likhoshvai V.A. MAMMOTh: a new database for curated mathematical models of biomolecular systems. J Bioinform Comput Biol. 2018;16(01):1740010. doi 10.1142/S0219720017400108

21. King E.L., Altman C. A schematic method of deriving the rate laws for enzyme-catalyzed reactions. J Phys Chem. 1956;60(10):1375-1378. doi 10.1021/j150544a010

22. Kolmykov S., Yevshin I., Kulyashov M., Sharipov R., Kondrakhin Y., Makeev V.J., Kulakovskiy I.V., Kel A., Kolpakov F. GTRD: an integratedview of transcription regulation. Nucleic Acids Res. 2021; 49(D1):D104-D111. doi 10.1093/nar/gkaa1057

23. Kolodkin A., Simeonidis E., Balling R., Westerhoff H.V. Understanding complexity in neurodegenerative diseases: in silico reconstruction of emergence. Front Physiol. 2012;3:291. doi 10.3389/fphys.2012.00291

24. Kolodkin A., Simeonidis E., Westerhoff H.V. Computing life: add logos to biology and bios to physics. Prog Biophys Mol Biol. 2013; 111(2-3):69-74. doi 10.1016/j.pbiomolbio.2012.10.003

25. Kolpakov F., Akberdin I., Kiselev I., Kolmykov S., Kondrakhin Y., Kulyashov M., Kutumova E., Pintus S., Ryabova A., Sharipov R., Yevshin I., Zhatchenko S., Kel A. BioUML – towards a universal research platform. Nucleic Acids Res. 2022;50(W1):W124-W131. doi 10.1093/nar/gkac286

26. Lakhova T.N., Kazantsev F.V., Mukhin A.M., Kolchanov N.A., Matushkin Y.G., Lashin S.A. Algorithm for the reconstruction of mathematical frame models of bacterial transcription regulation. Mathematics. 2022;10(23):4480. doi 10.3390/math10234480

27. Likhoshvai V.A., Matushkin Y.G., Ratushnyi A.V., Anako E.A., Igna tieva E.V., Podkolodnaya O.A. Generalized chemokinetic method for gene network simulation. Mol Biol. 2001;35(6):919-925. doi 10.1023/A:1013254822486

28. Likhoshvai V.A., Matushkin Y.G., Fadeev S.I. Problems in the theory of the functioning of genetic networks. Sib Zh Ind Mat. 2003;6(2): 64-80 Malik-Sheriff R.S., Glont M., Nguyen T.V.N., Tiwari K., Ro berts M.G., Xavier A., Vu M.T., … Park Y.M., Buso N., Rodriguez N., Hucka M., Hermjakob H. BioModels – 15 years of sharing computational mo dels in life science. Nucleic Acids Res. 2019;48(D1):D407-D415. doi 10.1093/nar/gkz1055

29. Miller A.K., Marsh J., Reeve A., Garny A., Britten R., Halstead M., Cooper J., Nickerson D.P., Nielsen P.F. An overview of the CellML API and its implementation. BMC Bioinformatics. 2010;11:178. doi 10.1186/1471-2105-11-178

30. Moodie S., Le Novère N., Demir E., Mi H., Villéger A. Systems Biology Graphical Notation: Process Description language level 1 version 1.3. J Integr Bioinform. 2015;12(2):263. doi 10.1515/jib-2015-263

31. Norsigian C.J., Pusarla N., McConn J.L., Yurkovich J.T., Dräger A., Palsson B.O., King Z. BiGG Models 2020: multi-strain genomescale models and expansion across the phylogenetic tree. Nucleic Acids Res. 2019;48(D1):D402-D406. doi 10.1093/nar/gkz1054

32. Orth J.D., Thiele I., Palsson B.Ø. What is flux balance analysis? Nat Biotechnol. 2010;28(3):245-248. doi 10.1038/nbt.1614

33. Rigden D.J., Fernández X.M. The 2025 Nucleic Acids Research da tabase issue and the online molecular biology database collection. Nucleic Acids Res. 2025;53(D1):D1-D9. doi 10.1093/nar/gkae1220

34. Shapiro B.E., Levchenko A., Meyerowitz E.M., Wold B.J., Mjols ness E.D. Cellerator: extending a computer algebra system to include biochemical arrows for signal transduction simulations. Bioinformatics. 2003;19(5):677-678. doi 10.1093/bioinformatics/btg042

35. Shapiro B.E., Meyerowitz E.M., Mjolsness E. Using cellzilla for plant growth simulations at the cellular level. Front Plant Sci. 2013;4:408. doi 10.3389/fpls.2013.00408

36. Tiwari K., Kananathan S., Roberts M.G., Meyer J.P., Sharif Sho han M.U., Xavier A., Maire M., … Ng S., Nguyen T.V.N., Glont M., Hermjakob H., Malik‐Sheriff R.S. Reproducibility in systems biology modelling. Mol Syst Biol. 2021;17(2):e9982. doi 10.15252/msb.20209982

37. Vorontsov I.E., Eliseeva I.A., Zinkevich A., Nikonov M., Abramov S., Boytsov A., Kamenets V., … Medvedeva Y.A., Jolma A., Kolpa kov F., Makeev V.J., Kulakovskiy I.V. HOCOMOCO in 2024: are build of the curated collection of binding models for human and mouse transcription factors. Nucleic Acids Res. 2024;52(D1):D154 D163. doi 10.1093/nar/gkad1077

38. Wittig U., Rey M., Weidemann A., Kania R., Müller W. SABIO-RK: an updated resource for manually curated biochemical reaction kinetics. Nucleic Acids Res. 2018;46(D1):D656-D660. doi 10.1093/nar/gkx1065

39. Zhabotinsky A.M., Zaikin A.N. Autowave processes in a distributed chemical system. J Theor Biol. 1973;40(1):45-61. doi 10.1016/00225193(73)90164-1