No Access Submitted: 24 July 2015 Accepted: 27 November 2015 Published Online: 22 December 2015
J. Chem. Phys. 143, 244103 (2015); https://doi.org/10.1063/1.4937937
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Many stochastic models of biochemical reaction networks contain some chemical species for which the number of molecules that are present in the system can only be finite (for instance due to conservation laws), but also other species that can be present in arbitrarily large amounts. The prime example of such networks are models of gene expression, which typically contain a small and finite number of possible states for the promoter but an infinite number of possible states for the amount of mRNA and protein. One of the main approaches to analyze such models is through the use of equations for the time evolution of moments of the chemical species. Recently, a new approach based on conditional moments of the species with infinite state space given all the different possible states of the finite species has been proposed. It was argued that this approach allows one to capture more details about the full underlying probability distribution with a smaller number of equations. Here, I show that the result that less moments provide more information can only stem from an unnecessarily complicated description of the system in the classical formulation. The foundation of this argument will be the derivation of moment equations that describe the complete probability distribution over the finite state space but only low-order moments over the infinite state space. I will show that the number of equations that is needed is always less than what was previously claimed and always less than the number of conditional moment equations up to the same order. To support these arguments, a symbolic algorithm is provided that can be used to derive minimal systems of unconditional moment equations for models with partially finite state space.
This work has received funding from the German Research Foundation (DFG) as part of the Transregional Collaborative Research Center “Automatic Verification and Analysis of Complex Systems” (SFB/TR 14 AVACS, http://www.avacs.org/), by the European Research Council (ERC) under Grant No. 267989 (QUAREM), by the Austrian Science Fund (FWF) under Grant Nos. S11402-N23 (RiSE) and Z211-N23 (Wittgenstein Award) and by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement No. 291734.
  1. 1. J. Paulsson, Phys. Life Rev. 2, 157 (2005). https://doi.org/10.1016/j.plrev.2005.03.003, Google ScholarCrossref
  2. 2. T. Jahnke, Multiscale Model. Simul. 9, 1646 (2011). https://doi.org/10.1137/110821500, Google ScholarCrossref
  3. 3. B. Munsky and M. Khammash, J. Chem. Phys. 124, 044104 (2006). https://doi.org/10.1063/1.2145882, Google ScholarScitation, ISI
  4. 4. T. Jahnke and W. Huisinga, J. Math. Biol. 54, 1 (2007). https://doi.org/10.1007/s00285-006-0034-x, Google ScholarCrossref
  5. 5. P. Bokes, J. King, A. Wood, and M. Loose, J. Math. Biol. 64, 829 (2012). https://doi.org/10.1007/s00285-011-0433-5, Google ScholarCrossref
  6. 6. R. Grima, D. Schmidt, and T. Newman, J. Chem. Phys. 137, 035104 (2012). https://doi.org/10.1063/1.4736721, Google ScholarScitation
  7. 7. S. Engblom, Appl. Math. Comput. 180, 498 (2006). https://doi.org/10.1016/j.amc.2005.12.032, Google ScholarCrossref
  8. 8. J. Goutsias and G. Jenkinson, Phys. Rep. 529, 199 (2013). https://doi.org/10.1016/j.physrep.2013.03.004, Google ScholarCrossref
  9. 9. J. Hespanha, IEE Proc.: Control Theory Appl. 153, 520 (2006). https://doi.org/10.1049/ip-cta:20050088, Google ScholarCrossref
  10. 10. A. Singh and J. Hespanha, IEEE Trans. Autom. Control 56, 414 (2011). https://doi.org/10.1109/TAC.2010.2088631, Google ScholarCrossref
  11. 11. J. Hasenauer, V. Wolf, A. Kazeroonian, and F. Theis, J. Math. Biol. 69, 687 (2014). https://doi.org/10.1007/s00285-013-0711-5, Google ScholarCrossref
  12. 12. A. Andreychenko, L. Mikeev, and V. Wolf, J. Coupled Sys. Multiscale Dyn. 3(2), 156 (2015). https://doi.org/10.1166/jcsmd.2015.1073, Google ScholarCrossref
  13. 13. D. Gillespie, Physica A 188, 404 (1992). https://doi.org/10.1016/0378-4371(92)90283-V, Google ScholarCrossref
  14. 14. A. Milias-Argeitis, S. Engblom, P. Bauer, and M. Khammash, J. R. Soc. Interface 12(113), 20150831 (2015). https://doi.org/10.1098/rsif.2015.0831, Google ScholarCrossref
  15. 15. H. El Samad, M. Khammash, L. Petzold, and D. Gillespie, Int. J. Robust Nonlinear Control 15, 691 (2005). https://doi.org/10.1002/rnc.1018, Google ScholarCrossref
  16. 16. B. Munsky, B. Trinh, and M. Khammash, Mol. Syst. Biol. 5, 318 (2009). https://doi.org/10.1038/msb.2009.75, Google ScholarCrossref
  17. 17. G. Neuert, B. Munsky, R. Tan, L. Teytelman, M. Khammash, and A. van Oudenaarden, Science 339, 584 (2013). https://doi.org/10.1126/science.1231456, Google ScholarCrossref
  18. 18. G. Rieckh and G. Tkacik, Biophys. J. 106, 1194 (2014). https://doi.org/10.1016/j.bpj.2014.01.014, Google ScholarCrossref
  19. 19. J. Ruess, F. Parise, A. Milias-Argeitis, M. Khammash, and J. Lygeros, Proc. Natl. Acad. Sci. U. S. A. 112, 8148 (2015). https://doi.org/10.1073/pnas.1423947112, Google ScholarCrossref
  20. 20. C. Zechner, M. Unger, S. Pelet, M. Peter, and H. Koeppl, Nat. Methods 11, 197 (2014). https://doi.org/10.1038/nmeth.2794, Google ScholarCrossref
  21. 21. C. Zechner, J. Ruess, P. Krenn, S. Pelet, M. Peter, J. Lygeros, and H. Koeppl, Proc. Natl. Acad. Sci. U. S. A. 109, 8340 (2012). https://doi.org/10.1073/pnas.1200161109, Google ScholarCrossref
  22. 22. J. Vilar and S. Leibler, J. Mol. Biol. 331, 981 (2003). https://doi.org/10.1016/S0022-2836(03)00764-2, Google ScholarCrossref
  23. 23. J. Hespanha, available at http://www.ece.ucsb.edu/~hespanha/software (2007). Google Scholar
  24. 24. D. Schnoerr, G. Sanguinetti, and R. Grima, J. Chem. Phys. 143, 185101 (2015). https://doi.org/10.1063/1.4934990, Google ScholarScitation, ISI
  25. 25. See supplementary material at http://dx.doi.org/10.1063/1.4937937 for the symbolic algorithm for the derivation of minimal systems of moment equations. Google Scholar
  26. 26. A. Hindmarsh, P. Brown, K. Grant, S. Lee, R. Serban, D. Shumaker, and C. Woodward, ACM Trans. Math. Software (TOMS) 31, 363 (2005). https://doi.org/10.1145/1089014.1089020, Google ScholarCrossref
  27. 27. J. Ruess, A. Milias-Argeitis, and J. Lygeros, J. R. Soc., Interface 10, 20130588 (2013). https://doi.org/10.1098/rsif.2013.0588, Google ScholarCrossref
  28. 28. A. Singh, B. Razooky, R. Dar, and L. Weinberger, Mol. Syst. Biol. 8, 607 (2012). https://doi.org/10.1038/msb.2012.38, Google ScholarCrossref
  29. 29. F. Bertaux, S. Stoma, D. Drasdo, and G. Batt, PLoS Comput. Biol. 10, e1003893 (2014). https://doi.org/10.1371/journal.pcbi.1003893, Google ScholarCrossref
  30. 30. R. Grima, J. Chem. Phys. 136, 154105 (2012). https://doi.org/10.1063/1.3702848, Google ScholarScitation
  31. 31. J. Ruess and J. Lygeros, ACM Trans. Model. Comput. Simul. (TOMACS) 25, 8 (2015). https://doi.org/10.1145/2688906, Google ScholarCrossref
  32. 32. B. Munsky, Z. Fox, and G. Neuert, Methods 85, 12 (2015). https://doi.org/10.1016/j.ymeth.2015.06.009, Google ScholarCrossref
  33. 33. H. Qian and E. Elson, Biophys. Chem. 101, 565 (2002). https://doi.org/10.1016/S0301-4622(02)00145-X, Google ScholarCrossref
  34. 34. S. Kou, B. Cherayil, W. Min, B. English, and X. Sunney Xie, J. Phys. Chem. B 109, 19068 (2005). https://doi.org/10.1021/jp051490q, Google ScholarCrossref
  35. 35. A. Andreychenko, L. Bortolussi, R. Grima, P. Thomas, and V. Wolf, e-print arXiv:1509.09104 [q-bio.QM] (2015). Google Scholar
  36. 36. M. Soltani, C. Vargas-Garcia, and A. Singh, IEEE Trans. Biomed. Circuits Syst. 9, 518 (2015). https://doi.org/10.1109/TBCAS.2015.2453158, Google ScholarCrossref
  37. 37. P. Thomas and R. Grima, Phys. Rev. E 92, 012120 (2015). https://doi.org/10.1103/PhysRevE.92.012120, Google ScholarCrossref
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