Introduction. Dagmar Iber Jörg Stelling. CSB Deterministic, SS 2015, 1.

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1 Introduction Dagmar Iber Jörg Stelling CSB Deterministic, SS 2015, 1

2 Origins of Systems Biology On this assumption of the passage of blood, made as a basis for argument, and from the estimation of the pulse rate, it is apparent that the entire quantity of blood passes from the veins to the arteries through the heart, and likewise through the lungs. Systems approach by William Harvey,

3 Computational Approaches to Biology Bioinformatics / Computational Biology CSB CCGACCATGCTGGCTTC CGGGCGCCCCCTGGTGC E A CAGGCATAAATTCAGTG AGAGCCGGAACTAGTCC B D GTGAGGCGCGCGTTGCC H. Lodish et al., Molecular Cell Biology, 5 th ed., C Linearly Structure and Interacting encoded function of components: genetic components networks and information (proteins,...) their behavior 3

4 Cellular Networks: Challenges High numbers of Protein interactions in Drosophila: Giot et al. (2003) Science 302: 1727 components Dynamic, non-linear interactions Self-modifying system Low molecular copy numbers: 'noise' Spatial organization 4

5 Cellular Networks: Challenges Complexity: Protein interactions in Drosophila: J. Giot et al. (2003) Science 302: 1727 Many components Dynamic interactions Self-modifying system Spatial organization Uncertainty: Incomplete inventory Few quantitative data Conflicting hypotheses Molecular 'noise' 5

6 Systems Biology & Complexity 6

7 Systems Biology & Uncertainty Unknown function Figure from: H. Lodish et al., Molecular Cell Biology, 5 th ed., ~ 50% of gene functions unknown in all (model) organisms. 7

8 Systems Biology & Uncertainty Figure from: C. von Mering et al. (2002) Nature 417: 399. Example: Protein-protein interactions, different studies. General phenomenon: Trade-offs coverage - accuracy. 8

9 Systems Biology & Uncertainty Experiment Model Emphasis on massive data generation as a prerequisite. Representation of biology in a 'true' mathematical model. Efficient way for making statements about biology? 9

10 Systems Biology & Uncertainty Main limitations for the development of mechanistic mathematical models for biological systems: Unknown parameter values and model structures for (nearly all) biological systems of interest. Stochastic effects as a consequence of low molecular copy numbers and environmental fluctuations. Effects of heterogeneous spatial distribution of cellular components / complicated cellular geometries. 10

11 Example: 'Probabilistic Dynamics' Experimental Simulation M. Kaltenbach, S. Dimopoulos & J. Stelling. FEBS Letters 583: (2009). 11

12 Example: Stochastic Noise M.J. Dunlop, R.S. Cox, J.H. Levine, R.M. Murray & M.B. Elowitz. Nat. Genetics 40: 1493 (2008). 12

13 Example: Spatial Effects B: w/o localization B: with localization B. di Ventura, C. Lemerle, K. Michalodimitrakis & L. Serrano. Nature 443: 527 (2006). Positive feedback circuit with high initial B, low A: Effect of kinase / phosphatase localization to opposite cell poles. 13

14 Science 284: 87 (1999). Every good model starts from a question. The modeler should always choose the correct level of detail to answer the question. Use the right level of description to catch the phenomena of interest. Don t model bulldozers with quarks. 14

15 Reverse Engineering / Systems Identification Reverse engineering / systems identification: Given a set of experimental data / prior knowledge, reconstruct and understand the network structure and function. Inference from system behavior in contrast to measurements of all individual interactions and subsequent assembly of facts. D.W. Selinger et al. (2003). On the complete determination of biological systems. Trends Biotechnol. 21,

16 Possible Aim: Whole-Cell Models

17 Computational Systems Biology Part of computational science: Development of methods and tools that help solving problems in (natural) science. Most problems in systems biology are mathematically and computationally hard (and cannot be solved in general). Domain knowledge (biological reality) as well as methods knowledge are needed to understand the problems and to develop real-world solutions. 17

18 Cohen JE, PLoS Biol 2(12): e439 (2004). 18

19 Aims of the Course Provide advanced mathematical and computational methods for the analysis of biological systems in real world settings State of the art, open problems. Focus on the interplay of theoretical and experimental approaches to typical questions in systems biology. Focused on examples from cell to organismal biology. 19

20 Course Structure Introduction / review ODEs (1) Structural network analysis (2) Network structures and data (2) Identification and experimental design (4) for dynamic (ODE) models Uncertainty quantification and network (2) inference Spatial effects and PDEs (3) 20

21 Modus Operandi Credits for this course: 6 ECTS. Session examination: Oral exam of 20 mins. Exercises: Will help a lot in learning the material, but not mandatory (no testate). Combination of analytic and computer (Matlab) exercises. Start next week. Materials via: 21

22 ODE-based Models: A Brief Review 22

23 ODE-based Model Development: Approach Pictogram Reaction list Approximations Figure from: Aldridge et al. (2006) Nature Cell Biology 8: Pathway diagram Differential equations 23

24 Reaction Kinetics: Dynamic Systems Reaction network System of elementary reactions: 1, j X 1 n, j X n k j 1, j X 1 n, j X n Law of mass action System of differential equations: dc i t dt q = j=1 k j i, j i, j l c l t l, j Equivalence to: d c t dt = N r t 24

25 Reaction Kinetics: Dynamic Models d c t dt = N r c t,u t, p Reactand concentrations c(t) To be determined. Stoichiometric matrix N Systems invariant. Reaction rates r Time- and state-dependent: Kinetic rate law r( ) From reaction structure. Parameters (kinetic constants) p Identification. Inputs u(t) Additional (time-varying) influences. 25

26 ODE Models: General Form d x t dt = f x t,u t, p System of ordinary, first-order, linear or nonlinear differential equations (ODEs) characterized by: Right hand sides f(x(t),u(t),p) = function in. System states x(t) = n x x 1 state vector. Parameters p = n p x 1 parameter set. Inputs u(t) = n u x 1 input vector. R n x 26

27 ODE Models: Solution d x t dt = f x t, p, x t 0 =x 0 Existence and uniqueness of solution to the initial value problem (IVP) of finding x(t) with given x 0 guaranteed. Three possible ''solution'' methods: Analytical Only applicable for simple systems. Numerical Always possible for well-posed IVPs. Graphical Qualitative analysis methods. 27

28 ODE Models: Analytical Solution d x t dt = f x t, p, x t 0 =x 0 Determine a closed-form formula g(x 0,p) that can be evaluated at any (time) point t to determine x(t). Example: Linear homogeneous system of ODEs: d x dt = A x, x 0 = i=1 n x n x i v i x t = i v i e t i i=1 Solutions for all linear systems in terms of exponential functions exp(λ i t) and harmonic functions sin(ω i t+φ i ). 28

29 ODE Models: Numerical Solution d x t dt = f x t, p, x t 0 =x 0 Replace differential equation with algebraic equation whose solution approximates that of the ODE. Example: Explicit finite-difference approximation x i t t x i t f i x t, p t Many different numerical algorithms for integration (e.g. Euler, Adams-Gear, Runge-Kutta,...). 29

30 ODE Models: Graphical ''Solution'' x 2 x(t) d x t dt = f x t, p x 0 x x t 0 =x 0 x 1 Derivatives dx(t)/dt define vector field in state space Solution x(t) obtained by following the field from x 0. Qualitative analysis in two-dimensional systems Determine characteristic features of the vector field. 30

31 Simple Dynamic Systems: Kinetics (Bio)chemical reaction networks ODE models Simplifications / assumptions (separation of time- and concentration-scales) Derivation of rate laws. Example: Gene G bound by transcription factor T: Without repression: Competitive repressor R: Cooperative binding: [G T ] = [G ]T [T ] [T ] K [G T ] = [G T ] [G ] T [T ] [T ] K 1 [ R]/ K I = [G]T [T ] n [T ] n K n 31

32 Feedback Systems Signal +/- Subsystem Output Circular patterns of interactions can establish feedback loops with positive or negative net effect. Intertwined feedback loops Complex dynamics. 32

33 Feedback Systems Negative Feedback u Positive Feedback u X X Y P E Y Y E Y P Single steady state. Multiple steady states. Homeostasis, rejection of perturbations. Bistability, hysteresis Switches, decisions. 33

34 Negative Feedback: Example System u X Y P E Y Protein X: Phosphatase that dephosphorylates Y P. Protein Y: Dephosphorylated form activates degradation of X Negative feedback. Input signal u: Control of production rate for X. 34

35 Negative Feedback: Example System u R 1 X R 2 Y P R 3 E R 4 Y Two-state (ODE) model: Michaelis-Menten kinetics d [ X ] = k dt 1 u k 2 [Y ] [ X ] d [Y ] dt = k 3 [ X ] [Y ] T [Y ] K M3 [Y ] T [Y ] k 4 [ E ][Y ] K M4 [Y ] Re-scaling: Sigmoidal Goldbeter-Koshland function. 35

36 Negative Feedback: Phase Plane Analysis Example trajectory Y-Nullcline Steady state X-Nullcline 36

37 Positive Feedback: Example System u X Y E Y P Protein X: Phosphatase that dephosphorylates Y P. Protein Y: Dephosphorylated form activates production of X Positive feedback. Input signal u: Control of production rate for X. 37

38 Positive Feedback: Example System u R 1 X R 3 R 2 Y R 4 E R 5 Y P Two-state (ODE) model: Michaelis-Menten kinetics d [ X ] dt d [Y ] dt = k 1 u k 2 [Y ] k 3 [ X ] = k 4 [ X ] [Y ] T [Y ] K M4 [Y ] T [Y ] k 5[ E ] [Y ] K M5 [Y ] 38

39 Positive Feedback: Phase Plane Analysis Example trajectory Y-Nullcline Stable steady state Stable steady state X-Nullcline Unstable steady state 39

40 Relations Between Spaces Parameter space Flux space State space 40

41 Positive Feedback: Stability Classification of steady states (nodes) according to directions of the vector field: unstable node stable node saddle point (unstable) Stability: Global vs. local (w.r.t. 'small' perturbations). 41

42 Positive Feedback: Bifurcation Diagram [X] ss u crit1 u u crit2 History dependence of the system's state (here with respect to changes in the input): Hysteresis. Used, for example, to establish (computer) memory. 42

43 Combined Feedback: Activator-Inhibitor Protein X: Phosphatase u that dephosphorylates Y P. Protein Y: Y form activates X production of X Positive feedback (on X). Y E Z Y P Protein Z: Production mediated by Y, activates degradation of X Negative feedback (on X). Input signal u: Control of X. 43

44 Activator-Inhibitor: Simplification Three state variables X, Y u Y R 4 X E R 5 Z Y P and Z No simple geometric interpretation of the system behavior. Simplification: Quasi steady-state assumption for species of protein Y. Module already contained in simple feedback system. 44

45 Activator-Inhibitor: Phase Plane Analysis Example trajectory Unstable steady state Stable limit cycle X-Nullcline Z-Nullcline 45

46 Activator-Inhibitor: Bifurcation Diagram stable unstable [X] ss stable u crit1 u u crit2 Bifurcations between single stable steady state and stable oscillations (dotted lines amplitude of oscillations). 46

47 Stability Analysis: Systematic Approach Convert nonlinear ODE system to linear ODE system Linearization around operating point (steady state). d x t dt = f x t, p, t d x dt A x Characterize solution for linear homogeneous system Analysis of local stability properties. Systematic evaluation of system behavior Existence & stability of steady states as a function of parameters Bifurcation analysis. 47

48 Stability Analysis: Linearization d x t dt = f x t, p,t, x t 0 =x 0 Determine a closed-form formula g(x,p,t) that can be evaluated at any (time) point t to determine x(t). Solution for linear homogeneous system of ODEs: d x dt = A x, x 0 = i=1 n x n x i v i x t = i v i e t i i=1 Solutions for all linear systems in terms of exponential functions exp(λ i t) and harmonic functions sin(ω i t+φ i ). 48

49 Stability Analysis: Linear ODE Systems Matrix notation (two dimensions) with system matrix A: d x ' dt = A x ' y, x ' = x 1', y= y 1 2, x 2 ' y A= a 11 a 12 a 21 a 22 Local stability of steady states x ss of interest: d x ss dt = A x ss y = 0 y = A x ss With deviation from steady state solution x = x' - x ss Canonical form of linear ODEs for stability analysis. d x dt = A x 49

50 Stability Analysis: Linear ODE Systems Canonical solution for homogeneous linear ODEs: d x dt = A x n x x t = i=1 i v i e i t Eigenvectors v i Direction of system movement. Eigenvalues λ i Exponential part (time-dependent). Constants α i Adjusted to meet initial conditions. Dynamics from eigenvalue decomposition of A: A v = v A = v v 1 50

51 Stability Analysis: Linear ODE Systems Canonical solution for homogeneous linear ODEs: d x dt = A x n x x t = i=1 i v i e i t Local system stability means x(t) 0 for t : Real parts of all eigenvalues have to be negative (otherwise some solutions do not converge locally). Complex eigenvalues associated with oscillations: exp i =exp exp i =exp cos i sin 51

52 Systems Identification Most important aspects of the system? Complete knowledge on components / interactions? Exact mechanisms of interactions? Level of detail for the mathematical descriptions? Modeling approach (qualitative / mechanistic /...)? Experimental data for identification & validation? 52

53 Identification for ODE Models d x t dt = f x t,u t, p, t System of ordinary, first-order, linear or nonlinear differential equations (ODEs) characterized by: f(x(t),u(t),p) Mechanistic uncertainties. System states x(t) (Experimentally observable). Parameters p Parameter estimation. Inputs u(t) Control for experimental design. 53

54 Parameter Identification: Estimation N p = i=1 [e i T Q i e i ] min e i = e i x, p, t i = x p,t i x M t i General formulation as an optimization problem: Minimization of deviation between model and experiment: Φ( p ) : Identification functional for parameter set p. e i (x, p, t i ) : Error vector for time point t i (i = 1... N). x( p, t i ) : Model prediction (state vector) at time t i. x M ( t i ) : Experimental data (measurement) at time t i. 54

55 Optimization: Local vs. Global Optima Φ( p ) N p = i=1 Gobal optimization: p* that minimizes Φ( p ) over all p [e i T Q i e i ] min possible values of p No efficient algorithms known. Local optimization: Φ( p* ) Φ( p ) for all p close to p*. 55

56 Introduction: Summary Uncertainty in biology as a major factor that limits systems biology modeling (and poses interesting mathematical challenges). Systems identification beyond traditional parameter estimation Structural model uncertainties, experimental design, spatial effects... Challenges: Modeling approaches as well as advanced identification / experimental design methods. Review of ODE systems: Fundamental kinetics, feedback systems, linear systems and stability, parameter estimation approaches Foundation for everything that follows... 56

57 "All models are wrong but some models are useful." George Box (1979) Robustness in the strategy of scientific model building. 57