Rational Design of Robust Biomolecular Circuits: from Specification to Parameters

  • Marc Hafner
  • Tatjana Petrov
  • James Lu
  • Heinz KoepplEmail author


Despite the early success stories synthetic biology, the development of larger, more complex synthetic systems necessitates the use of appropriate design methodologies. In particular, the integration of smaller circuits in order to perform complex tasks remains one of the most important challenges faced in synthetic biology. We propose here a methodology to determine the region in the parameter space where a given dynamical model works as desired. It is based on the inverse problem of finding parameter sets that exhibit the specified behavior for a defined topology. The main issue we face is that such inverse mapping is highly expansive and suffers from instability: small changes in the specified dynamic property could lead to large deviations in the parameters for the identified models. To solve this issue, we discuss regularized maps complemented by local analysis. With a stabilized inversion map, small neighborhoods in the property space are mapped to small neighborhoods in the parameter space, thereby finding parameter vectors that are robust to the problem specification. To specify dynamic circuit properties we discuss Linear Temporal Logic (LTL). We apply these concepts to two models of the cyanobacterial circadian oscillation.


Robustness Inverse problems Robust control Optimal control Circuit design Formal verification 



Marc Hafner, Tatjana Petrov and James Lu acknowledge the funding of (the Swiss initiative for systems biology) for their IPhD and BIP (JL) projects. Tatjana Petrov gratefully acknowledges Barbara Jobstmann for her helpful comments and suggestions to the LTL part. Heinz Koeppl acknowledges the support from the Swiss National Science Foundation, grant no. 200020-117975/1 and grant no. PP00P2_128503/1.


  1. 1.
    Bagheri N, Stelling J, Doyle FJ (2007) Quantitative performance metrics for robustness in circadian rhythms. Bioinformatics 23(3):358–364CrossRefGoogle Scholar
  2. 2.
    Balzer R (1985) A 15 year perspective on automatic programming. IEEE Trans Software Eng SE-11(11):1257–1268Google Scholar
  3. 3.
    Calzone L, Fages F, Soliman S (2006) BIOCHAM: an environment for modeling biological systems and formalizing experimental knowledge. Bioinformatics 22(14):1805–1807CrossRefGoogle Scholar
  4. 4.
    Carlson JM, Doyle J (2002) Complexity and robustness. Proc Natl Acad Sci U S A 99(Suppl 1): 2538–2545CrossRefGoogle Scholar
  5. 5.
    Chen M-T, Weiss R (2005) Artificial cell-cell communication in yeast saccharomyces cerevisiae using signaling elements from arabidopsis thaliana. Nat Biotechnol 23(12):1551–1555CrossRefGoogle Scholar
  6. 6.
    Chomsky N (1956) Three models for the description of language. IRE Trans Inform Theor 2:113–124CrossRefGoogle Scholar
  7. 7.
    Clarke EM, Emerson EA, Sistla AP (1986) Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Trans Progr Lang Sys 8:244–263zbMATHCrossRefGoogle Scholar
  8. 8.
    Clodong S, Duhring U, Kronk L, Wilde A, Axmann I, Herzel H, Kollmann M (2007) Functioning and robustness of a bacterial circadian clock. Mol Syst Biol 3:90CrossRefGoogle Scholar
  9. 9.
    Cornish-Bowden A (2004) Fundamentals of enzyme kinetics, 3rd edn. Portland Press, LondonGoogle Scholar
  10. 10.
    Daniels BC, Chen Y-JJ, Sethna JP, Gutenkunst RN, Myers CR (2008) Sloppiness, robustness, and evolvability in systems biology. Curr Opin Biotech 19(4):389–395CrossRefGoogle Scholar
  11. 11.
    Daubechies I, Defrise M, De Mol C (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm Pur Appl Math 57(11):1413–1457zbMATHCrossRefGoogle Scholar
  12. 12.
    Dayarian A, Chaves M, Sontag ED, Sengupta AM (2009) Shape, size, and robustness: feasible regions in the parameter space of biochemical networks. PLoS Comput Biol 5(1):e1000256 + Google Scholar
  13. 13.
    Desharnais J, Edalat A, Panangaden P (2002) Bisimulation for labelled Markov processes. Inf Comput 179(2):163–193MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ditty JL, Williams SB, Golden SS (2003) A cyanobacterial circadian timing mechanism. Annu Rev Genet 37(1):513–543CrossRefGoogle Scholar
  15. 15.
    Doyle F, Gunawan R, Bagheri N, Mirsky H, To T (2006) Circadian rhythm: a natural, robust, multi-scale control system. Comput Chem Eng 30(10–12):1700–1711CrossRefGoogle Scholar
  16. 16.
    Doyle J, Csete M (2005) Motifs, control, and stability. PLoS Biol 3(11):e392 + Google Scholar
  17. 17.
    D’Silva V, Kroening D, Weissenbacher G (2008) A survey of automated techniques for formal software verification. IEEE Trans Comput Aid D 27(7):1165–1178CrossRefGoogle Scholar
  18. 18.
    Eissing T, Allgöwer F, Bullinger E (2005) Robustness properties of apoptosis models with respect to parameter variations and intrinsic noise. IEEE Proc Syst Biol 152(4):221–228CrossRefGoogle Scholar
  19. 19.
    El-Samad H, Khammash M (2006) Regulated degradation is a mechanism for suppressing stochastic fluctuations in gene regulatory networks. Biophys J 90(10):3749–3761CrossRefGoogle Scholar
  20. 20.
    El-Samad H, Kurata H, Doyle JC, Gross CA, Khammash M (2005) Surviving heat shock: control strategies for robustness and performance. Proc Natl Acad Sci USA 102(8):2736–2741CrossRefGoogle Scholar
  21. 21.
    Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403(6767):335–338CrossRefGoogle Scholar
  22. 22.
    Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Mathematics and its applications, vol 375. Kluwer Academic Publishers Group, DordrechtGoogle Scholar
  23. 23.
    Fages F, Rizk A (2008) On temporal logic constraint solving for analyzing numerical data time series. Theor Comput Sci 408(1):55–65MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Fainekos GE, Pappas GJ (2007) Robust sampling for MITL specifications. In: FORMATS 2007 Proceedings of the 5th international conference on Formal modeling and analysis of timed systems, Springer-Verlag Berlin, HeidelbergGoogle Scholar
  25. 25.
    Fukunaga K (1990) Introduction to statistical pattern recognition. Computer science and scientific computing, 2nd edn. Academic Press, San Diego CAGoogle Scholar
  26. 26.
    Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403(6767):339–342CrossRefGoogle Scholar
  27. 27.
    Gonze D, Goldbeter A (2006) Circadian rhythms and molecular noise. Chaos 16(2):026110 + Google Scholar
  28. 28.
    Gonze D, Halloy J, Goldbeter A (2002) Robustness of circadian rhythms with respect to molecular noise. Proc Natl Acad Sci U S A 99(2):673–678CrossRefGoogle Scholar
  29. 29.
    Grumberg O, Veith H (eds) (2008) 25 Years of model checking - history, achievements, perspectives. Lecture notes in computer science, vol 5000. Springer, Berlin HeidelbergGoogle Scholar
  30. 30.
    Hafner M, Koeppl H, Hasler M, Wagner A (2009) ‘Glocal’ robustness analysis and model discrimination for circadian oscillators. PLoS Comput Biol 5(10):e1000534 + Google Scholar
  31. 31.
    Hamza J, Jobstmann B, Kuncak V (2010) Synthesis for regular specifications over unbounded domains. In: Conference on formal methods in computer aided design (FMCAD), pp 101–110Google Scholar
  32. 32.
    Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Am Stat Assoc 58(301):13–30MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Johnson CH (2004) Global orchestration of gene expression by the biological clock of cyanobacteria. Genome Biol 5(4):217CrossRefGoogle Scholar
  34. 34.
    Kitano H (2007) Towards a theory of biological robustness. Mol Syst Biol 3:137CrossRefGoogle Scholar
  35. 35.
    Kramer BP, Viretta AU, Baba MD, Aubel D, Weber W, Fussenegger M (2004) An engineered epigenetic transgene switch in mammalian cells. Nat Biotechnol 22(7):867–870CrossRefGoogle Scholar
  36. 36.
    Kwiatkowska MZ, Norman G, Sproston J (2003) PCTL model checking of symbolic probabilistic systems. Technical Report CSR-03-2. University of Birmingham, School of Computer ScienceGoogle Scholar
  37. 37.
    Laidler KJ (1987) Chemical kinetics, 3rd edn. Prentice Hall, New York, NYGoogle Scholar
  38. 38.
    McAdams HH Arkin A (1997) Stochastic mechanisms in gene-expression. Proc Natl Acad Sci U S A 94(3):814–819CrossRefGoogle Scholar
  39. 39.
    Mehra A, Hong CI, Shi M, Loros JJ, Dunlap JC, Ruoff P (2006) Circadian rhythmicity by autocatalysis. PLoS Comput Biol 2(7):e96CrossRefGoogle Scholar
  40. 40.
    Mihalcescu I, Hsing W, Leibler S (2004) Resilient circadian oscillator revealed in individual cyanobacteria. Nature 430(6995):81–85CrossRefGoogle Scholar
  41. 41.
    Moles CG, Mendes P, Banga JR (2003) Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Res 13(11):2467–2474CrossRefGoogle Scholar
  42. 42.
    Mori T, Williams DR, Byrne MO, Qin X, Egli M, Mchaourab HS, Stewart PL, Johnson CH (2007) Elucidating the ticking of an in vitro circadian clockwork. PLoS Biol 5(4):e93CrossRefGoogle Scholar
  43. 43.
    Morohashi M, Winn AE, Borisuk MT, Bolouri H, Doyle J, Kitano H (2002) Robustness as a measure of plausibility in models of biochemical networks. J Theor Biol 216(1):19–30MathSciNetCrossRefGoogle Scholar
  44. 44.
    Nakajima M, Imai K, Ito H, Nishiwaki T, Murayama Y, Iwasaki H, Oyama T, Kondo T (2005) Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro. Science 308(5720):414–415CrossRefGoogle Scholar
  45. 45.
    Newman MEJ, Girvan M, Farmer JD (2002) Optimal design, robustness, and risk aversion. Phys Rev Lett 89(2):028301 + Google Scholar
  46. 46.
    Nishiwaki T, Satomi Y, Kitayama Y, Terauchi K, Kiyohara R, Takao T, Kondo T (2007) A sequential program of dual phosphorylation of KaiC as a basis for circadian rhythm in cyanobacteria. EMBO J 26(17):4029–4037CrossRefGoogle Scholar
  47. 47.
    Nowak MA, Boerlijst MC, Cooke J, Smith JM (1997) Evolution of genetic redundancy. Nature 388(6638):167–171CrossRefGoogle Scholar
  48. 48.
    Ouyang Y, Andersson CR, Kondo T, Golden SS, Johnson CH (1998) Resonating circadian clocks enhance fitness in cyanobacteria. Proc Natl Acad Sci U S A 95(15):8660–8664CrossRefGoogle Scholar
  49. 49.
    Pattanayek R, Williams DR, Pattanayek S, Mori T, Johnson CH, Stewart PL, Egli M (2008) Structural model of the circadian clock KaiB-KaiC complex and mechanism for modulation of KaiC phosphorylation. EMBO J 27(12):1767–1778CrossRefGoogle Scholar
  50. 50.
    Pnueli A (1977) The temporal logic of programs. In Ann IEEE Simp Found 46–57, IEEE, New York, NYGoogle Scholar
  51. 51.
    Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1992) Numerical recipes in C: the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge UKGoogle Scholar
  52. 52.
    Purnick PEM, Weiss R (2009) The second wave of synthetic biology: from modules to systems. Nat Rev Mol Cell Biol 10(6):410–422CrossRefGoogle Scholar
  53. 53.
    Rand D, Shulgin B, Salazar J, Millar A (2006) Uncovering the design principles of circadian clocks: mathematical analysis of flexibility and evolutionary goals. J Theor Biol 238(3):616–635MathSciNetCrossRefGoogle Scholar
  54. 54.
    Ruoff P (1992) Introducing temperature-compensation in any reaction kinetic oscillator model. J Interdiscipl Cycle Res 23(2):92–99Google Scholar
  55. 55.
    Rust MJ, Markson JS, Lane WS, Fisher DS, O’Shea EK (2007) Ordered phosphorylation governs oscillation of a three-protein circadian clock. Science 318(5851):809–812CrossRefGoogle Scholar
  56. 56.
    Stelling J, Gilles ED, Doyle FJ (2004a) Robustness properties of circadian clock architectures. Proc Natl Acad Sci U S A 101(36):13210–13215CrossRefGoogle Scholar
  57. 57.
    Stelling J, Sauer U, Szallasi Z, Doyle FJ, Doyle J (2004b) Robustness of cellular functions. Cell 118(6):675–685CrossRefGoogle Scholar
  58. 58.
    Stengel RF (1986) Stochastic optimal control. Wiley, New York. Theory and applicationGoogle Scholar
  59. 59.
    Stricker J, Cookson S, Bennett MR, Mather WH, Tsimring LS, Hasty J (2008) A fast, robust and tunable synthetic gene oscillator. Nature 456(7221):516–519CrossRefGoogle Scholar
  60. 60.
    Swain PS, Elowitz MB, Siggia ED (2002) Intrinsic and extrinsic contributions to stochasticity in gene expression. Proc Natl Acad Sci U S A 99(20):12795–12800CrossRefGoogle Scholar
  61. 61.
    Swinburne IA, Miguez DG, Landgraf D, Silver PA (2008) Intron length increases oscillatory periods of gene expression in animal cells. Gene Dev 22(17):2342–2346CrossRefGoogle Scholar
  62. 62.
    Thattai M, van Oudenaarden A (2001) Intrinsic noise in gene regulatory networks. Proc Natl Acad Sci U S A 98(15):8614–8619CrossRefGoogle Scholar
  63. 63.
    Tigges M, Marquez-Lago TT, Stelling J, Fussenegger M (2009) A tunable synthetic mammalian oscillator. Nature 457(7227):309–312CrossRefGoogle Scholar
  64. 64.
    Wagner A (2000) Robustness against mutations in genetic networks of yeast. Nat Genet 24(4):355–361CrossRefGoogle Scholar
  65. 65.
    Wagner A (2007) Robustness and evolvability in living systems: (princeton studies in complexity), 1st edn. Princeton University Press, Princeton NJGoogle Scholar
  66. 66.
    Weber W, Schuetz M, Dénervaud N, Fussenegger M (2009) A synthetic metabolite-based mammalian inter-cell signaling system. Mol Biosyst 5(7):757–763CrossRefGoogle Scholar
  67. 67.
    Weiss R, Knight T (2001) Engineered communications for microbial robotics. In: Condon A, Rozenberg G, (eds) DNA Computing. Lecture Notes in Computer Science, vol 2054. chapter 1. Springer, Berlin/Heidelberg, pp 1–16Google Scholar
  68. 68.
    Wolf J, Becker-Weimann S, Heinrich R (2005) Analysing the robustness of cellular rhythms. Syst Biol 2(1):35–41CrossRefGoogle Scholar
  69. 69.
    Zamora Sillero E, Hafner M, Ibig A, Stelling J, Wagner A (2010) Efficient characterization of high-dimensional parameter spaces for systems biology. Submitted to BMC Systems BiologyGoogle Scholar
  70. 70.
    Zarzer C (2009) On Tikhonov regularization with non-convex sparsity constraints. Inverse Probl 25(2):025006MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Marc Hafner
  • Tatjana Petrov
  • James Lu
  • Heinz Koeppl
    • 1
    Email author
  1. 1.Swiss Federal Institute of Technology Zurich (ETHZ)ZurichSwitzerland

Personalised recommendations