Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A formal methods approach to predicting new features of the eukaryotic vesicle traffic system


Vesicle traffic systems (VTSs) transport cargo among the intracellular compartments of eukaryotic cells. The compartments are viewed as nodes that are labeled by their chemical identity and the transport vesicles are similarly viewed as labeled edges between the nodes. Several interesting questions about VTSs translate to combinatorial search and synthesis problems. We present novel encodings for the problems based on Boolean satisfiability (SAT), satisfiability modulo theories and quantified Boolean formula of the properties over vesicle traffic systems. We have implemented the presented encodings in a tool that searches for the networks that satisfy properties related to transport consistency conditions using these solvers. In our numerical experiments, we show that our tool can search for networks of sizes that are relevant to real cellular systems. Our work illustrates the potential of novel biological applications of SAT solving technology.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    There are total 68,715 simple (no parallel edges) 3-connected graphs with 8 nodes.

  2. 2.


  1. 1.

    Alberts, B., Johnson, A., Lewis, J., Walter, P., Raff, M., Roberts, K.: Molecular biology of the cell 4th edition: international student edition. Routledge (2002).

  2. 2.

    Alur, R., Bodik, R., Juniwal, G., Martin, M.M., Raghothaman, M., Seshia, S.A., Singh, R., Solar-Lezama, A., Torlak, E., Udupa, A.: Syntax-guided synthesis. In: 2013 Formal Methods in Computer-Aided Design, IEEE, pp. 1–8 (2013)

  3. 3.

    Baker, R.W., Hughson, F.M.: Chaperoning snare assembly and disassembly. Nat. Rev. Mol. Cell Biol. 17(8), 465 (2016)

  4. 4.

    Barlow, L., Dacks, J.: Seeing the endomembrane system for the trees: evolutionary analysis highlights the importance of plants as models for eukaryotic membrane-trafficking. In: Seminars in cell & developmental biology. Elsevier (2017)

  5. 5.

    Barrett, C., Tinelli, C.: Satisfiability modulo theories. In: Handbook of Model Checking, pp. 305–343. Springer (2018)

  6. 6.

    Benedetti, M., Mangassarian, H.: Qbf-based formal verification: experience and perspectives. J. Satisf. Boolean Model. Comput. 5, 133–191 (2008)

  7. 7.

    Bexiga, M.G., Simpson, J.C.: Human diseases associated with form and function of the golgi complex. Int. J. Mol. Sci. 14(9), 18670–18681 (2013)

  8. 8.

    Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic model checking without BDDs. In: International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pp. 193–207. Springer (1999)

  9. 9.

    Biere, A., Cimatti, A., Clarke, E.M., Fujita, M., Zhu, Y.: Symbolic model checking using sat procedures instead of bdds. In: Proceedings of the 36th Annual ACM/IEEE Design Automation Conference, pp. 317–320. ACM (1999)

  10. 10.

    Biere, A., Cimatti, A., Clarke, E.M., Strichman, O., Zhu, Y.: Bounded model checking. Adv. Comput. 58, 117–148 (2003)

  11. 11.

    Biere, A., Heule, M., van Maaren, H.: Handbook of Satisfiability, vol. 185. IOS press, Amsterdam (2009)

  12. 12.

    Biere, A., Heule, M., van Maaren, H., Walsh, T.: Conflict-driven clause learning sat solvers. Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, pp. 131–153 (2009)

  13. 13.

    Bjesse, P., Leonard, T., Mokkedem, A.: Finding bugs in an alpha microprocessor using satisfiability solvers. In: International Conference on Computer Aided Verification, pp. 454–464. Springer (2001)

  14. 14.

    Braell, W.A., Balch, W.E., Dobbertin, D.C., Rothman, J.E.: The glycoprotein that is transported between successive compartments of the golgi in a cell-free system resides in stacks of cisternae. Cell 39(3), 511–524 (1984)

  15. 15.

    Büning, H.K., Bubeck, U.: Theory of quantified boolean formulas. Handb. Satisf. 185, 735–760 (2009)

  16. 16.

    Burri, L., Lithgow, T.: A complete set of snares in yeast. Traffic 5(1), 45–52 (2004)

  17. 17.

    Cardelli, L., Češka, M., Fränzle, M., Kwiatkowska, M., Laurenti, L., Paoletti, N., Whitby, M.: Syntax-guided optimal synthesis for chemical reaction networks. In: International Conference on Computer Aided Verification, pp. 375–395. Springer (2017)

  18. 18.

    Chin, G., Chavarria, D.G., Nakamura, G.C., Sofia, H.J.: Biographe: high-performance bionetwork analysis using the biological graph environment. BMC Bioinf. 9(6), S6 (2008)

  19. 19.

    Clarke, E., Kroening, D., Lerda, F.: A tool for checking ANSI-C programs. In: TACAS, pp. 168–176. Springer (2004)

  20. 20.

    Cocucci, E., Gaudin, R., Kirchhausen, T.: Dynamin recruitment and membrane scission at the neck of a clathrin-coated pit. Mol. Biol. Cell 25(22), 3595–3609 (2014)

  21. 21.

    Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing, pp. 151–158. ACM (1971)

  22. 22.

    de Moura, L., Bjorner, N.: Z3: An efficient smt solver. In: TACAS, LNCS, vol. 4963, pp. 337–340. Springer Berlin Heidelberg (2008).

  23. 23.

    de Moura, L.M., Bjørner, N.: Efficient e-matching for SMT solvers. In: Automated Deduction—CADE-21, 21st International Conference on Automated Deduction, Bremen, Germany, July 17–20, 2007, Proceedings, vol. 4603, pp. 183–198. Springer (2007)

  24. 24.

    Dacks, J.B., Field, M.C.: Evolution of the eukaryotic membrane-trafficking system: origin, tempo and mode. J. Cell Sci. 120(17), 2977–2985 (2007)

  25. 25.

    Davletov, B., Connell, E., Darios, F.: Regulation of snare fusion machinery by fatty acids. Cell. Mol. Life Sci. 64(13), 1597–1608 (2007)

  26. 26.

    De Moura, L., Bjørner, N.: Z3: An efficient smt solver. In: International conference on Tools and Algorithms for the Construction and Analysis of Systems, pp. 337–340. Springer (2008)

  27. 27.

    Di Giovanni, J., Iborra, C., Maulet, Y., Lévêque, C., El Far, O., Seagar, M.: Calcium-dependent regulation of snare-mediated membrane fusion by calmodulin. J. Biol. Chem. 285, Jbc-M109 (2010)

  28. 28.

    Dunn, S.J., Martello, G., Yordanov, B., Emmott, S., Smith, A.: Defining an essential transcription factor program for naive pluripotency. Science 344(6188), 1156–1160 (2014)

  29. 29.

    Faini, M., Beck, R., Wieland, F.T., Briggs, J.A.: Vesicle coats: structure, function, and general principles of assembly. Trends Cell Biol. 23(6), 279–288 (2013)

  30. 30.

    Fisher, J., Köksal, A.S., Piterman, N., Woodhouse, S.: Synthesising executable gene regulatory networks from single-cell gene expression data. In: International Conference on Computer Aided Verification, pp. 544–560. Springer (2015)

  31. 31.

    Fries, E., Rothman, J.E.: Transient activity of golgi-like membranes as donors of vesicular stomatitis viral glycoprotein in vitro. J. Cell Biol. 90(3), 697–704 (1981)

  32. 32.

    Furukawa, N., Mima, J.: Multiple and distinct strategies of yeast snares to confer the specificity of membrane fusion. Sci. Rep. 4, 4277 (2014)

  33. 33.

    Giacobbe, M., Guet, C.C., Gupta, A., Henzinger, T.A., Paixao, T., Petrov, T.: Model checking gene regulatory networks. In: TACAS (2015)

  34. 34.

    Gissen, P., Maher, E.R.: Cargos and genes: insights into vesicular transport from inherited human disease. J. Med. Genet. 44(9), 545–555 (2007)

  35. 35.

    Goldberg, E.I., Prasad, M.R., Brayton, R.K.: Using sat for combinational equivalence checking. In: Design, Automation and Test in Europe, 2001. Conference and Exhibition 2001. Proceedings, pp. 114–121. IEEE (2001)

  36. 36.

    Gomes, C.P., Selman, B., McAloon, K., Tretkoff, C.: Randomization in backtrack search: exploiting heavy-tailed profiles for solving hard scheduling problems. In: AIPS, pp. 208–213 (1998)

  37. 37.

    Gubbels, M.J., Duraisingh, M.T.: Evolution of apicomplexan secretory organelles. Int. J. Parasitol. 42(12), 1071–1081 (2012)

  38. 38.

    Guerra, J., Lynce, I.: Reasoning over biological networks using maximum satisfiability. In: PPCP, pp. 941–956. Springer (2012)

  39. 39.

    Gulwani, S.: Automating string processing in spreadsheets using input-output examples. In: Proceedings of the 38th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2011, Austin, TX, USA, January 26–28, 2011, pp. 317–330. ACM (2011)

  40. 40.

    Gupta, A., Mani, S., Shukla, A.: Synthesis for vesicle traffic systems. In: Češka, M., Šafránek, D. (eds.) Computational Methods in Systems Biology, pp. 93–110. Springer, Cham (2018)

  41. 41.

    Gupta, A., Shukla, A., Srivas, M., Thattai, M.: Smt solving for vesicle traffic systems in cells. In: SASB (2017)

  42. 42.

    He, Y., Linder, M.E.: Differential palmitoylation of the endosomal snares syntaxin 7 and syntaxin 8. J. Lipid Res. 50(3), 398–404 (2009)

  43. 43.

    Heule, M., Verwer, S.: Exact dfa identification using sat solvers. Grammatical inference: theoretical results and applications pp. 66–79 (2010)

  44. 44.

    Hirst, J., Miller, S.E., Taylor, M.J., von Mollard, G.F., Robinson, M.S.: Epsinr is an adaptor for the snare protein vti1b. Mol. Biol. Cell 15(12), 5593–5602 (2004)

  45. 45.

    Jena, B.P.: Intracellular organelle dynamics at nm resolution. Methods Cell Biol. 90, 19–37 (2008)

  46. 46.

    Jussila, T., Biere, A.: Compressing bmc encodings with qbf. Electron. Notes Theor. Comput. Sci. 174(3), 45–56 (2007)

  47. 47.

    Kahn, R.A.: Toward a model for arf gtpases as regulators of traffic at the golgi. FEBS Lett. 583(23), 3872–3879 (2009)

  48. 48.

    Kautz, H., Selman, B.: Pushing the envelope: Planning, propositional logic, and stochastic search. In: Proceedings of the National Conference on Artificial Intelligence, pp. 1194–1201 (1996)

  49. 49.

    Koksal, A.S., Pu, Y., Srivastava, S., Bodik, R., Fisher, J., Piterman, N.: Synthesis of biological models from mutation experiments. In: Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ’13, pp. 469–482. ACM, New York, NY, USA (2013).

  50. 50.

    Lodish, H., Darnell, J.E., Berk, A., Kaiser, C.A., Krieger, M., Scott, M.P., Bretscher, A., Ploegh, H., Matsudaira, P., et al.: Molecular Cell Biology. Macmillan, Basingtoke (2008)

  51. 51.

    Lonsing, F., Biere, A.: DepQBF: a dependency-aware QBF solver. J. Satisf. Boolean Model. Comput. 7, 71–76 (2010)

  52. 52.

    Mangla, K., Dill, D.L., Horowitz, M.A.: Timing robustness in the budding and fission yeast cell cycles. PLoS One 5(2), e8906 (2010)

  53. 53.

    Mani, S., Thattai, M.: Stacking the odds for golgi cisternal maturation. Elife 5, e16,231 (2016)

  54. 54.

    McNew, J.A., Parlati, F., Fukuda, R., Johnston, R.J., Paz, K., Paumet, F., Söllner, T.H., Rothman, J.E.: Compartmental specificity of cellular membrane fusion encoded in snare proteins. Nature 407(6801), 153 (2000)

  55. 55.

    Mima, J., Hickey, C.M., Xu, H., Jun, Y., Wickner, W.: Reconstituted membrane fusion requires regulatory lipids, snares and synergistic snare chaperones. EMBO J. 27(15), 2031–2042 (2008)

  56. 56.

    Mishev, K., Dejonghe, W., Russinova, E.: Small molecules for dissecting endomembrane trafficking: a cross-systems view. Chem. Biol. 20(4), 475–486 (2013)

  57. 57.

    Müller, M.P., Goody, R.S.: Molecular control of rab activity by gefs, gaps and gdi. Small GTPases 9(1–2), 5–21 (2018)

  58. 58.

    Munro, S.: Organelle identity and the organization of membrane traffic. Nat. Cell Biol. 6(6), 469–472 (2004)

  59. 59.

    Nakatsukasa, K., Kanada, A., Matsuzaki, M., Byrne, S.D., Okumura, F., Kamura, T.: The nutrient stress-induced small gtpase rab5 contributes to the activation of vesicle trafficking and vacuolar activity. J. Biol. Chem. 289, jbc-M114 (2014)

  60. 60.

    Nickel, W., Rabouille, C.: Unconventional protein secretion: Diversity and consensus. In: Seminars in cell & developmental biology. Elsevier (2018)

  61. 61.

    Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving sat and sat modulo theories: from an abstract davis-putnam-logemann-loveland procedure to dpll (t). J. ACM (JACM) 53(6), 937–977 (2006)

  62. 62.

    Paczkowski, J.E., Richardson, B.C., Fromme, J.C.: Cargo adaptors: structures illuminate mechanisms regulating vesicle biogenesis. Trends Cell Biol. 25(7), 408–416 (2015)

  63. 63.

    Paoletti, N., Yordanov, B., Hamadi, Y., Wintersteiger, C.M., Kugler, H.: Analyzing and synthesizing genomic logic functions. In: International Conference on Computer Aided Verification, pp. 343–357. Springer (2014)

  64. 64.

    Polishchuk, R., Mironov, A.: Structural aspects of golgi function. Cell. Mol. Life Sci. CMLS 61(2), 146–158 (2004)

  65. 65.

    Progida, C., Bakke, O.: Bidirectional traffic between the golgi and the endosomes-machineries and regulation. J. Cell Sci. 129(21), 3971–3982 (2016)

  66. 66. QDIMACS standard ver. 1.1. (2018).

  67. 67.

    Richardson, E., Zerr, K., Tsaousis, A., Dorrell, R.G., Dacks, J.B.: Evolutionary cell biology: functional insight from “endless forms most beautiful”. Mol. Biol. Cell 26(25), 4532–4538 (2015)

  68. 68.

    Rink, J., Ghigo, E., Kalaidzidis, Y., Zerial, M.: Rab conversion as a mechanism of progression from early to late endosomes. Cell 122(5), 735–749 (2005)

  69. 69.

    Robbins, H.E.: A theorem on graphs, with an application to a problem of traffic control. Am. Math. Mon. 46(5), 281–283 (1939)

  70. 70.

    Rosenblueth, D.A., Muñoz, S., Carrillo, M., Azpeitia, E.: Inference of boolean networks from gene interaction graphs using a sat solver. In: ICACB, pp. 235–246. Springer (2014)

  71. 71.

    Rothman, J.E.: The machinery and principles of vesicle transport in the cell. Nat. Med. 8(10), 1059–1063 (2002)

  72. 72.

    Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. J. Comput. Syst. Sci. 4(2), 177–192 (1970)

  73. 73.

    Shavit, Y., Yordanov, B., Dunn, S.J., Wintersteiger, C.M., Otani, T., Hamadi, Y., Livesey, F.J., Kugler, H.: Automated synthesis and analysis of switching gene regulatory networks. Biosystems 146, 26–34 (2016)

  74. 74.

    Shimizu, H., Woodcock, S.A., Wilkin, M.B., Trubenová, B., Monk, N.A., Baron, M.: Compensatory flux changes within an endocytic trafficking network maintain thermal robustness of notch signaling. Cell 157(5), 1160–1174 (2014)

  75. 75.

    Shukla, A., Bhattacharyya, A., Kuppusamy, L., Srivas, M., Thattai, M.: Discovering vesicle traffic network constraints by model checking. PloS one 12(7), e0180,692 (2017)

  76. 76.

    Solar-Lezama, A.: The sketching approach to program synthesis. In: Programming Languages and Systems, 7th Asian Symposium, APLAS 2009, Seoul, Korea, December 14–16, 2009. Proceedings, pp. 4–13. Springer (2009)

  77. 77.

    Soos, M.: The cryptominisat 5 set of solvers at sat competition 2016. SAT Compet. 2016, 28 (2016)

  78. 78.

    Sorensson, N., Een, N.: Minisat v1. 13-a sat solver with conflict-clause minimization. SAT 2005(53), 1–2 (2005)

  79. 79.

    Stenmark, H.: Rab gtpases as coordinators of vesicle traffic. Nat. Rev. Mol. Cell Biol. 10(8), 513–525 (2009)

  80. 80.

    Stephan, P., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: Combinational test generation using satisfiability. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 15(9), 1167–1176 (1996)

  81. 81.

    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time (preliminary report). In: Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, pp. 1–9. ACM (1973)

  82. 82.

    Stoops, E.H., Caplan, M.J.: Trafficking to the apical and basolateral membranes in polarized epithelial cells. J. Am. Soc. Nephrol. 25(7), 1375–1386 (2014)

  83. 83.

    Tang, D., Wang, Y.: Cell cycle regulation of golgi membrane dynamics. Trends Cell Biol. 23(6), 296–304 (2013)

  84. 84.

    Velev, M.N., Bryant, R.E.: Effective use of boolean satisfiability procedures in the formal verification of superscalar and vliw microprocessors. J. Symb. Comput. 35(2), 73–106 (2003)

  85. 85.

    Weber, T., Zemelman, B.V., McNew, J.A., Westermann, B., Gmachl, M., Parlati, F., Söllner, T.H., Rothman, J.E.: Snarepins: minimal machinery for membrane fusion. Cell 92(6), 759–772 (1998)

  86. 86.

    Weill, U., Arakel, E.C., Goldmann, O., Golan, M., Chuartzman, S., Munro, S., Schwappach, B., Schuldiner, M.: Toolbox: creating a systematic database of secretory pathway proteins uncovers new cargo for copi. Traffic 19(5), 370–379 (2018)

  87. 87.

    Wells, W.A.: The discovery of synaptic vesicles (2005)

  88. 88.

    Yoon, T.Y., Munson, M.: Snare complex assembly and disassembly. Curr. Biol. 28(8), R397–R401 (2018)

  89. 89.

    Yordanov, B., Dunn, S.J., Kugler, H., Smith, A., Martello, G., Emmott, S.: A method to identify and analyze biological programs through automated reasoning. NPJ Syst. Biol. Appl. 2, 16,010 (2016)

  90. 90.

    Yordanov, B., Wintersteiger, C.M., Hamadi, Y., Kugler, H.: Smt-based analysis of biological computation. In: NASA Formal Methods Symposium, pp. 78–92. Springer (2013)

  91. 91.

    Zerial, M., McBride, H.: Rab proteins as membrane organizers. Nat. Rev. Mol. Cell Biol. 2(2), 107 (2001)

  92. 92.

    Zhou, K., Sumigray, K.D., Lechler, T.: The arp2/3 complex has essential roles in vesicle trafficking and transcytosis in the mammalian small intestine. Mol. Biol. Cell 26(11), 1995–2004 (2015)

Download references

Author information

Correspondence to Ankit Shukla.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

AG was partly supported in part by MPG Partner group grant. AS was partly supported by Austrian Science Fund (FWF) Project W1255-N23.

The natural VTSs

The natural VTSs

Fig. 5

A found-in-nature VTS. Nodes and edges are labelled with sets of molecules. \(\hat{}\) indicates that the molecule is active

Here we will present the two VTS collected from the literature.

Mammalian VTS

The Fig. 5 represent mammalian SNARE map created by studying the wide array of literature. To construct the map, we have assumed that vesicles only contain a single active v-SNARE, and we have attributed t-SNAREs and inactive v-SNAREs that travel between compartments to one of the known vesicles that go between the same source and target compartments. In order to identify the active SNARE complex involved in any particular vesicle fusion, we used two criteria. The SNARE complex is formed in vivo. In most papers, this is determined by immunoprecipitation of the SNARE complex from the relevant cell fraction. Blocking SNARE complex formation (for example, using antibodies against these SNAREs, or using cytosolic forms of these SNAREs) blocks the specific transport step. Note that these vesicles have been collected from multiple cell types, and any given cell type is likely to contain only a subset of the vesicles in the map.

In this figure, the rectangles represent compartments, the identities of compartments are written within ER = endoplasmic reticulum, ERGIC = ER-Golgi intermediate compartment, RE = recycling endosome, EE = early endosome, LE=late endosome, LYS = lysosome, PM = plasma membrane. The arrows represent vesicle edges.

Fig. 6

Yeast VTS

Yeast VTS

In Fig. 6, we present the yeast VTS. We have borrowed the VTS from [16]. It has been adapted from the paper by separating the v and the t SNAREs. It is clear that it is an incomplete description of the VTS. For example, the inactive molecules were not reported in the reference. We are currently searching for more literature that can help us complete all known information about the VTS.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bhattacharyya, A., Gupta, A., Kuppusamy, L. et al. A formal methods approach to predicting new features of the eukaryotic vesicle traffic system. Acta Informatica (2019).

Download citation