Model I dentification: Model I dentification: A Key Challenge in A - PowerPoint PPT Presentation

Model I dentification: Model I dentification: A Key Challenge in A Key Challenge in Com putational Com putational System s Biology System s Biology Eberhard O.Voit Eberhard O.Voit Departm ent of Biom edical Engineering Departm ent of Biom edical Engineering Georgia I nstitute of Technology and Em ory University Georgia I nstitute of Technology and Em ory University Atlanta, Georgia Atlanta, Georgia The 2 nd nd I nternational Sym posium I nternational Sym posium The 2 on Optim ization and System s Biology ( OSB'0 8 ) on Optim ization and System s Biology ( OSB'0 8 ) Lijiang, China, 3 1 October Lijiang , China, 3 1 October – – 3 Novem ber 2 0 0 8 3 Novem ber 2 0 0 8 1

Overview Overview Systems Biology and Optimization Choice of a Suitable Model Bottom-up and Top-down Model Estimation Technical Issues Dynamic Flux Estimation Open Problems 2

3 Model match System s Biology System s Biology Data explain measure Biological System

4 optimize Optim ization Optim ization Model manipulate match extrapolate Data explain measure Biological System

5 optimize Optim ization Optim ization Model manipulate match extrapolate System s Biology System s Biology Data explain measure Biological System

6 optimize Optim ization Optim ization Model manipulate match extrapolate System s Biology System s Biology Data explain measure Focus today Focus today Biological System

Application: Pathw ay Modeling Application: Pathw ay Modeling “Local” Data Literature, Brenda, de novo Experiments Understanding (Enzyme Kinetics) Extrapolation Local Processes Model Manipulation Optimization “Global” Data inverse “inverse Internet, ” problem ” problem Model de novo Experiments “ Structure (Microarrays, Proteomics, Mass Spec, NMR, Time Series Literature, KEGG, de novo Experiments 7

8 Overview of Modeling Process Overview of Modeling Process

Form ulation of a Form ulation of a Dynam ical System s Model Dynam ical System s Model + – V i V i + − dX − & = = i X V V X i i i i dt + = + com plicated V V ( X , X ,..., X , X ,..., X ) + + i i 1 2 n n 1 n m inside outside Big Problem: Where do we get functions from? 9

Sources of Functions for Sources of Functions for Com plex System s Models Com plex System s Models Physics: Functions come from theory Biology: No theory available Solution 1: Educated guesses: growth functions Solution 2: “Partial” theory: Enzyme kinetics Solution 3: Generic approximation 10

W hy not Use “ “True True” ” Functions? Functions? W hy not Use A+B P+Q ⎛ ⎞ ⎛ ⎞ num.1 num.1 num.2 × ⎜ ⎟ ⎜ ⎟ (A) (B) - (P)(Q) ⎝ ⎠ ⎝ ⎠ coef. AB coef. AB num.1 = v ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ constant coef. A coef. A coef. B × + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (A) (B) ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ coef. A coef. AB coef. AB coef. AB ⎛ ⎞ ⎛ ⎞ coef. AB coef.P coef. AP coef. A E + + × × ⎜ ⎟ ⎜ ⎟ (A)(B) (P) ⎝ ⎠ ⎝ ⎠ (1) coef. AB coef. AP coef. A coef. AB k 12 k 41 ⎛ ⎞ k 21 k 14 coef.Q constant coef. A + × × ⎜ ⎟ (Q) A+B EA EQ P+Q ⎝ ⎠ constant coef. A coef. AB k 43 (2) (4) k 32 ⎛ ⎞ ⎛ ⎞ coef.AP coef.A coef.BQ coef.B EPQ + × + × ⎜ ⎟ ⎜ ⎟ (A)(P) (B)(Q) k 23 ⎝ ⎠ ⎝ ⎠ coef. A coef. AB coef. B coef. AB k 34 EAB ⎛ ⎞ coef.PQ coef.Q constant coef. A (3) + ⎜ × × × ⎟ (P)(Q) ⎜ ⎟ ⎝ ⎠ coef.Q constant coef. A coef. AB ⎛ ⎞ coef.ABP + ⎜ ⎟ (A)(B)(P) ⎝ ⎠ coef. AB ⎛ ⎞ coef.BPQ coef.BQ coef.B + ⎜ × × ⎟ ⎜ ⎟ (B)(P)(Q) ⎝ ⎠ coef. BQ coef. B coef. AB from Schultz (1994) 11

W hy not Use Linear Functions? W hy not Use Linear Functions? Example: Heartbeat modeled as stable limit cycle sin(t) vdP(t) 4 4 2 2 0 0 0 30 60 0 30 60 time time System of non-linear System of linear differential equations differential equations 12

Form ulation of a Nonlinear Model Form ulation of a Nonlinear Model for Com plex System s for Com plex System s Challenge: Linear approximation unsuited Infinitely many nonlinear functions + − dX − & = = i V V X Solution w ith Potential: i i i dt + and V i – in a Savageau (1969): Approximate V i logarithmic coordinate system, using Taylor theory. Result: Canonical Modeling; Biochemical Systems Theory . 13

Exam ple Exam ple Adenine Excretion as a Function of Plasma Adenine Concentration Adenine Metabolites 0.5 2 Log of Excretion of Adenine Metabolites 0 Excretion of 1.5 -0.5 1 -1 0.5 -1.5 -2 0 0 20 40 60 80 -2 -1 0 1 2 Concentration and Log of Concentration of Plasma Adenine 14

Result: S- - system system Result: S & = α − β g h g g + h h + i , n m i , n m X X i 1 X i 2 ... X X i 1 X i 2 ... X + + i i 1 2 n m i 1 2 n m Each term is represented as a product of power-functions. Each term contains and only those variables that have a direct effect; others have exponents of 0 and drop out. α ’s and β ’s are rate constants , g ’s and h ’s kinetic orders . I m portant: Each term contains exactly those variables that have a direct effect; others have exponents of 0 and drop out. 15

Mapping Mapping Structure Param eters Structure Param eters X 1 X 2 X 3 X 4 α g X 21 2 1 g 41 = 0 X 1 X 2 X 3 X 4 g X α g X 21 41 g 41 < 0 2 1 4 16

Alternative Form ulations Alternative Form ulations W ithin BST W ithin BST S-system Form : & = α − β g h g g + h h + i , n m i , n m X X X ... X X X ... X i 1 i 2 i 1 i 2 + + i i 1 2 n m i 1 2 n m + – V i 1 V i 1 + ∑ dX ∑ & − = = − i X V V X i i ij ij dt – + V i ,q V i ,p 17

Alternative Form ulations Alternative Form ulations S-system Form : & = α − β g h g g + h h + i , n m i , n m X X i 1 X i 2 ... X X i 1 X i 2 ... X + + i i 1 2 n m i 1 2 n m + – V i 1 V i 1 + ∑ dX ∑ & = = − − i X V V X i i ij ij dt – + V i ,q V i ,p Generalized Mass Action Form : ∑ ∏ & = ± γ f X X ijk i ik j 18

Exam ple of Canonical Model Design Exam ple of Canonical Model Design X 2 X 2 X 2 X 3 X 3 X 3 X 0 X 0 X 0 X 1 X 1 X 1 X 4 X 4 X 4 & = − 0 . 75 0 . 3 GMA / S: X 8 X 5 X X 2 ( t 0 ) = 1 2 1 2 & = − 0 . 3 0 . 5 0 . 2 GMA / S: X 5 X 5 X X X 3 ( t 0 ) = 0.5 3 2 3 4 & − = − 0 . 5 1 0 . 8 GMA / S: X 4 ( t 0 ) = 6 12 4 X X X X 4 1 4 4 GMA / S: X 0 = 1.1 (constant) & = − − − − 0 . 9 0 . 75 0 . 5 1 GMA: X 20 X X 8 X 12 X X X 1 ( t 0 ) = 0.8 1 0 3 1 1 4 − − & = − 0 . 9 0 . 64 0 . 45 S-system: 20 19 X 1 ( t 0 ) = 0.8 X X X X X 1 0 3 1 4 19

Exam ple of Canonical Model Design Exam ple of Canonical Model Design X 2 X 2 X 2 X 3 X 3 X 3 X 0 X 0 X 0 X 1 X 1 X 1 X 4 X 4 X 4 & = − − − − 0 . 9 0 . 75 0 . 5 1 GMA: X 20 X X 8 X 12 X X X 1 ( t 0 ) = 0.8 1 0 3 1 1 4 − − & = − 0 . 9 0 . 64 0 . 45 S-system: 20 19 X 1 ( t 0 ) = 0.8 X X X X X 1 0 3 1 4 10 10 GMA system S-system X 2 X 2 5 5 X 3 X 3 X 4 X 4 X 1 X 1 0 0 0 4 8 0 4 8 20 time time

Doable Size Doable Size Sphingolipid pathw ay ( purely m etabolic) 1. Many metabolites 2. Many reactions 3. Many stimuli and agents regulate several enzymes of lipid metabolism 4. Some in vivo experiments Alvarez, Sims, Hannun, Voit 21 JTB , 2004; Nature, 2005

Applications Applications Pathways: purines, glycolysis, citric acid, TCA, red blood cell, trehalose, sphingolipids, ... Genes: circuitry, regulation,… Genome: explain expression patterns upon stimulus Growth, immunology, pharmaceutical science, forestry, ... Metabolic engineering: optimize yield in microbial pathways Dynamic labeling analyses possible Math: recasting, function classification, bifurcation analysis,... Statistics: S-system representation, S-distribution, trends; applied to seafood safety, marine mammals, health economics 22

Advantages of Canonical Models Advantages of Canonical Models Prescribed model design: Rules for translating diagrams into equations; rules can be automated Direct interpretability of parameters and other features One-to-one relationship between parameters and model structure simplifies parameter estimation and model identification Simplified steady-state computations (for S-systems), including steady-state equations, stability, sensitivities, gains Simplified optimization under steady-state conditions Efficient numerical solutions and time-dependent sensitivities In some sense minimal bias of model choice and minimal model size; easy scalability 23

Flow Chart of Flow Chart of System s I dentification Strategy System s I dentification Strategy dX j V i = R i ( S i , M i ) = f k ( X j , V i ) dt p 1 , p 2 , p 3 , … 24 Voit, Drug Discovery Today , 2004