Problem solving by inverse methods in systems biology

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2 Problem solving by inverse methods in systems biology Peter Schuster Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA High-erformance comutational and systems biology HiBi 2010 Twente,

3 Web-Page for further information: htt://

4 Heinz W. Engl, Christoh Flamm, Phili Kügler, James Lu, Stefan Müller, Peter Schuster Inverse roblems in systems biology. Inverse Problems 25: (51) Free download until December 31, 2010.

5 What is (comutational) systems biology? Systems biology is an attemt to understand integral systems like cells and whole organisms and their roerties by means of the knowledge from molecular biology. The goal of the comutational aroach is rediction of changes in henotyes from known changes in molecular structures and environmental conditions. The current methods aly a combination of bottom-u techniques like using data from in vitro measurements on isolated molecules and to-down data on time series of gene acitivities and metabolite concentrations from array studies.

6 1. From biochemical kinetics to quantitative biology 2. Forward and inverse roblems in reaction kinetics 3. Modeling biochemical reaction kinetics 4. Examles of inverse bifurcation analysis and design 5. Problems and ersectives of systems biology

7 1. From biochemical kinetics to quantitative biology 2. Forward and inverse roblems in reaction kinetics 3. Modeling biochemical reaction kinetics 4. Examles of inverse bifurcation analysis and design 5. Problems and ersectives of systems biology

8 Biochemical kinetics Conventional enzyme kinetics Theory of bioolymers, macroscoic roerties 1958 Gene regulation through reressor binding Allosteric effects, cooerative transitions Theory of cooerative binding to nucleic acids Revival of biochemical kinetics in systems biology

9 Biochemical kinetics Conventional enzyme kinetics Theory of bioolymers, macroscoic roerties 1958 Gene regulation through reressor binding Allosteric effects, cooerative transitions Theory of cooerative binding to nucleic acids Revival of biochemical kinetics in systems biology

10 Biochemical kinetics Conventional enzyme kinetics Theory of bioolymers, macroscoic roerties 1958 Gene regulation through reressor binding Allosteric effects, cooerative transitions Theory of cooerative binding to nucleic acids Revival of biochemical kinetics in systems biology

11 Biochemical kinetics Conventional enzyme kinetics Theory of bioolymers, macroscoic roerties 1958 Gene regulation through reressor binding Allosteric effects, cooerative transitions Theory of cooerative binding to nucleic acids Revival of biochemical kinetics in systems biology

12 Biochemical kinetics Conventional enzyme kinetics Theory of bioolymers, macroscoic roerties 1958 Gene regulation through reressor binding Allosteric effects, cooerative transitions Theory of cooerative binding to nucleic acids Revival of biochemical kinetics in systems biology

13 Biochemical kinetics Conventional enzyme kinetics Theory of bioolymers, macroscoic roerties 1958 Gene regulation through reressor binding Allosteric effects, cooerative transitions Theory of cooerative binding to nucleic acids Revival of biochemical kinetics in systems biology

14 A model genome with 12 genes Regulatory gene Structural gene Regulatory rotein or RNA Enzyme Metabolite Sketch of a genetic and metabolic network

15 A B C D E F G H I J K L 1 Biochemical Pathways The reaction network of cellular metabolism ublished by Boehringer-Mannheim.

16 The citric acid or Krebs cycle (enlarged from revious slide).

17 The bacterial cell as an examle for the simlest form of autonomous life The human body: cells = eukaryotic cells bacterial (rokaryotic) cells, and 200 eukaryotic cell tyes The satial structure of the bacterium Escherichia coli

18 1. From biochemical kinetics to quantitative biology 2. Forward and inverse roblems in reaction kinetics 3. Modeling biochemical reaction kinetics 4. Examles of inverse bifurcation analysis and design 5. Problems and ersectives of systems biology

19 dx dt = Kinetic differential equations f x; k); x= ( x, K, x ); k = ( k, K, k ( 1 n 1 m ) Reaction diffusion equations x 2 = D x + f ( x; k) t () Solution curves: x t x i (t) Parameter set k ( T,, H, I, K ) ; j= 1, 2, K,m j General conditions: T,, H, I,... Initial conditions : x(0) Concentration Time t Boundary conditions: boundary... S, normal unit vector... u Dirichlet : Neumann : x S = g( r, t) x = uˆ x S g( r, t) u = The forward roblem of chemical reaction kinetics (Level I)

20 dx dt Kinetic differential equations = f ( x; k); x= ( x1, K, xn ); k= ( k1, K, km ) Genome: Sequence I G Reaction diffusion equations x t = D General conditions : T,, H, I,... Initial conditions : 2 Parameter set x + f ( x; k) k j ( I G ; T,, H, I, K ) ; j = 1, 2, K, m x(0) Concentration Solution curves: x() t x i (t) Time t Boundary conditions : boundary... S, normal unit vector... u Dirichlet : Neumann : x S = g( r, t) x = uˆ x S g( r, t) u = The forward roblem of biochemical reaction kinetics (Level I)

21 Genome: Sequence I G Inverse roblem: Parameter determination Parameter set k j ( I G ; T,, H, I, K ) ; j = 1, 2, K, m d x dt = Kinetic differential equations f ( x; k); x = ( x1, K, xn ); k = ( k1, K, km Reaction diffusion equations x t Data from measurements x i (t ) j = D 2 x + f ( x; k) General conditions : T,, H, I,... Initial conditions : Boundary conditions : boundary Dirichlet : Neumann : x(0)... S, normal unit vector... u x S = g ( r, t) x = uˆ x S g( r, t) u = x (t j); j= 1, 2,..., N ) The inverse roblem of biochemical reaction kinetics (Level I) Concentration Time t

22 Genome: Sequence I G Inverse roblem: Parameter determination Parameter set k j ( I G ; T,, H, I, K ) ; j = 1, 2, K, m d x dt = Kinetic differential equations f ( x; k); x = ( x1, K, xn ); k = ( k1, K, km Reaction diffusion equations x t Data from measurements x i (t ) j = D 2 x + f ( x; k) General conditions : T,, H, I,... Initial conditions : Boundary conditions : boundary Dirichlet : Neumann : x(0)... S, normal unit vector... u x S = g ( r, t) x = uˆ x S g( r, t) u = x (t j); j= 1, 2,..., N ) The inverse roblem of biochemical reaction kinetics (Level I) Concentration Time t

23 The arameter identification roblem

24 The arameter identification roblem

25 The forward roblem of bifurcation analysis in cellular dynamics (Level II)

26 Inverse roblem: Design of bifurcation behavior The inverse roblem of bifurcation analysis in cellular dynamics (Level II)

27 Inverse roblem: Design of bifurcation behavior The inverse roblem of bifurcation analysis in cellular dynamics (Level II)

28 Why should one be interested in inverse bifurcation analysis? 1. Identification of arameters that are involved in changes of qualitative behavior. 2. Reverse engineering of dynamical systems, e.g. arresting a cyclic rocess in a certain hase.

29 A dynamical system with an oscillatory regime between a saddle node - invariant cycle (SNIC) bifuraction and a Hof bifurcation. Oscillatory regime

30 1. From biochemical kinetics to quantitative biology 2. Forward and inverse roblems in reaction kinetics 3. Modeling biochemical reaction kinetics 4. Examles of inverse bifurcation analysis and design 5. Problems and ersectives of systems biology

31 Ten reaction stes An enzyme catalyzed addition reaction, A + B C in the flow reactor

32 Combination of influx and outflux into one reversible reaction Eight reaction stes A model reaction network with 12 comlexes: C = { ø, A, B, C, EA, EB, EAB, E+A, E+B, EA+B, EB+A, E+C }

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36 Kinetic differential equation of the reaction network with mass action kinetics

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38 A + B X 2 X Y Y + X D Stoichiometric equations SBML systems biology marku language Sequences Vienna RNA Package da dt dx dt db = = k1 ab dt = k ab k 1 2 x 2 k 3 x y Kinetic differential equations Structures and kinetic arameters d y dt = k 2 x 2 k 3 x y ODE Integration by means of CVODE dd dt = k 3 x y x i (t) Solution curves Concentration Time t The elements of the simulation tool MiniCellSim SBML: Bioinformatics 19: , 2003; CVODE: Comuters in Physics 10: , 1996

39 Peter Schuster Prediction of RNA secondary structures: From theory to models and real molecules. Re.Prog.Phys. 69:

40 1. From biochemical kinetics to quantitative biology 2. Forward and inverse roblems in reaction kinetics 3. Modeling biochemical reaction kinetics 4. Examles of inverse bifurcation analysis and design 5. Problems and ersectives of systems biology

41 ( ) ( ) ( ) ( ) ( ) { } s s s i s i s i m m n x x x x f x Σ Σ = = Σ = = = ; P P P ; P P, bifurcation manifold P ;,, ;,, ; ; 1 1 K K K & R Inverse bifurcation analysis J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Alication to simle gene systems. AMB Algorithms for Molecular Biology 1, no.11, 2006.

42 The bifurcation manifold

43 ( ) ( ) ( ) ( ) ( ) { } s s s i s i s i m m n x x x x f x Σ Σ = = Σ = = = ; P P P ; P P, bifurcation manifold P ;,, ;,, ; ; 1 1 K K K & R ( ) ( ) ( ) ( ) oerator forward, ) (, ) ( K s i s i F F F s Σ = π J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Alication to simle gene systems. AMB Algorithms for Molecular Biology 1, no.11, 2006.

44 Defininition of the forward oerator F()

45 ( ) ( ) ( ) ( ) ( ) { } s s s i s i s i m m n x x x x f x Σ Σ = = Σ = = = ; P P P ; P P, bifurcation manifold P ;,, ;,, ; ; 1 1 K K K & R ( ) ( ) ( ) ( ) oerator forward, ) (, ) ( K s i s i F F F s Σ = π ( ) i i i F c F J s s ) ( 0 and subject to ) ( min ) ( min u low =... formulation of the inverse roblem J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Alication to simle gene systems. AMB Algorithms for Molecular Biology 1, no.11, 2006.

46 Iterative solution for min J()

47 Examles of inverse bifurcation analysis and design Oscillatory behavior in Escherichia coli The reressilator The mitotic cell cycle Time scales and oscillatory bursts Circadian rhythms.

48 k =1, 2,3 Switch or oscillatory behavior in Escherichia coli T.S. Gardner, C.R. Cantor, J.J. Collins. Construction of a genetic toggle switch in Escherichia coli. Nature 403: , M.R. Atkinson, M.A. Savageau, T.J. Myers, A.J. Ninfa. Develoment of genetic circuitry exhibiting toggle switch or oscillatory behavior in Escherichia coli. Cell 113: , 2003.

49 Inverse bifurcation analysis of switch or oscillatory behavior in Escherichia coli J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Alication to simle gene systems. AMB Algorithms for Molecular Biology 1:11, 2006.

50 Examles of inverse bifurcation analysis and design Oscillatory behavior in Escherichia coli The reressilator The mitotic cell cycle Time scales and oscillatory bursts Circadian rhythms.

51 An examle analyzed and simulated by MiniCellSim The reressilator: M.B. Ellowitz, S. Leibler. A synthetic oscillatory network of transcritional regulators. Nature 403: , 2002

52 Stable stationary state Hof bifurcation Increasing inhibitor strength Limit cycle oscillations Bifurcation to May-Leonhard system Fading oscillations caused by a stable heteroclinic orbit

53 Proteins mrnas e+07 2e+07 3e+07 4e+07 5e e+07 2e+07 3e+07 4e+07 5e e+07 2e+07 3e+07 4e+07 5e e+07 2e+07 3e+07 4e+07 5e+07 The reressilator limit cycle

54 Proteins mrnas e+06 1e e+06 1e e+06 1e e+06 1e+08 The reressilator heteroclinic orbit (logarithmic time scale)

55 P 1 start start P 2 P 3 The reressilator limit cycle

56 2 <0 2 Bifurcation from limit cycle to stable heteroclinic orbit at 2 =0 P 1 P 2 1 Stable heteroclinic orbit 2 >0 2 P 2 1 Unstable heteroclinic orbit P 3 P 2 The reressilator heteroclinic orbit

57 All ossible scenarios of reressilator dynamics

58 α α, β = β, h = h, δ = δ i = i i i Inverse bifurcation analysis of the reressilator model S. Müller, J. Hofbauer, L. Endler, C. Flamm, S. Widder, P. Schuster. A generalized model of the reressilator. J. Math. Biol. 53: , 2006.

59 Inverse bifurcation analysis of the reressilator model J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Alication to simle gene systems. AMB Algorithms for Molecular Biology 1:11, 2006.

60 Examles of inverse bifurcation analysis and design Oscillatory behavior in Escherichia coli The reressilator The mitotic cell cycle Time scales and oscillatory bursts Circadian rhythms.

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62 d dt [ RB] = k [ E2F1] 11 1 RB K 1 + [E2F1] J11 + [ RB] φ m J [ RB] d dt [ E2F1] = k P + k [ E2F1] 2 2 a + J12 1 φ 2 2 E2F1 Km2 + [E2F1] J12 + [ RB] [ E2F1] d dt [ AP1] = F + k [ E2F1] m J15 + [RB] φap1 J15 J11 + [ RB' ] J [ AP1] A simle dynamical cell cycle model J.J. Tyson, A. Csikasz-Nagy, B. Novak. The dynamics of cell cycle regulation. Bioessays 24: , 2002

63 A simle dynamical cell cycle model J.J. Tyson, A. Csikasz-Nagy, B. Novak. The dynamics of cell cycle regulation. Bioessays 24: , 2002

64 Inverse bifurcation analysis of a dynamical cell cycle model J. Lu, H.W. Engl, P. Schuster. Inverse bifurcation analysis: Alication to simle gene systems. AMB Algorithms for Molecular Biology 1:11, 2006.

65 Sarsity romoting functional

66 Examles of inverse bifurcation analysis and design Oscillatory behavior in Escherichia coli The reressilator The mitotic cell cycle Time scales and oscillatory bursts Circadian rhythms.

67 Neurons (nerve cells) and endocrine signalling

68 Bifurcation analysis of neuronal burst dynamics

69 Examles of inverse bifurcation analysis and design Oscillatory behavior in Escherichia coli The reressilator The mitotic cell cycle Time scales and oscillatory bursts Circadian rhythms.

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71 Bifurcation analysis of circadian rhymths

72 Bifurcation analysis of circadian rhymths

73 1. From biochemical kinetics to quantitative biology 2. Forward and inverse roblems in reaction kinetics 3. Modeling biochemical reaction kinetics 4. Examles of inverse bifurcation analysis and design 5. Problems and ersectives of systems biology

74 Challenges of quantitative biology 1. Validation of data from different sources 2. Low article numbers and stochasticity 3. Conformational heterogeneity of biomolecules 4. Satial heterogeneity of cells and cell organelles 5. High dimensionality of molecular dynamical systems

75 Challenges of quantitative biology 1. Validation of data from different sources 2. Low article numbers and stochasticity 3. Conformational heterogeneity of biomolecules 4. Satial heterogeneity of cells and cell organelles 5. High dimensionality of molecular dynamical systems

76 Challenges of quantitative biology 1. Validation of data from different sources 2. Low article numbers and stochasticity 3. Conformational heterogeneity of biomolecules 4. Satial heterogeneity of cells and cell organelles 5. High dimensionality of molecular dynamical systems

77 Challenges of quantitative biology 1. Validation of data from different sources 2. Low article numbers and stochasticity 3. Conformational heterogeneity of biomolecules 4. Satial heterogeneity of cells and cell organelles 5. High dimensionality of molecular dynamical systems

78 Challenges of quantitative biology 1. Validation of data from different sources 2. Low article numbers and stochasticity 3. Conformational heterogeneity of biomolecules 4. Satial heterogeneity of cells and cell organelles 5. High dimensionality of molecular dynamical systems

79 The bacterial cell as an examle for the simlest form of autonomous life The human body: cells = eukaryotic cells bacterial (rokaryotic) cells, and 200 eukaryotic cell tyes The satial structure of the bacterium Escherichia coli

80 Challenges of quantitative biology 1. Validation of data from different sources 2. Low article numbers and stochasticity 3. Conformational heterogeneity of biomolecules 4. Satial heterogeneity of cells and cell organelles 5. High dimensionality of molecular dynamical systems

81 Suitable systems for uscaling 1. Linear systems via large eigenvalue roblems 2. Cascades 3. Cyclic systems 4. Sufficiently simle networks and flux analysis

82 Coworkers Heinz Engl, Phili Kügler, James Lu, Stefan Müller, RICAM Linz, AT Universität Wien Walter Fontana, Harvard Medical School, MA Martin Nowak, Harvard University, MA Christian Reidys, Nankai University, Tien Tsin, China Christian Forst, Los Alamos National Laboratory, NM Ivo L.Hofacker, Christoh Flamm, Universität Wien, AT Stefanie Widder, Lukas Endler, Rainer Machne, Universität Wien, AT

83 Acknowledgement of suort Universität Wien Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) Project No. Mat05 Universität Wien and the Santa Fe Institute

84 Thank you for your attention!

85 Web-Page for further information: htt://

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