Modeling metabolic networks: Advantages and limitations of approximated mathematical formalisms

1. Modeling metabolic networks: Advantages and limitations of approximatedmathematical formalisms Albert Sorribas Grup de Bioestadística i Biomatemàtica Departament de Ciències Mèdiques Bàsiques Institut de Recerca Biomèdica de Lleida (IRBLLEIDA) Universitat de Lleida

2. Summary • Goals and strategies in analyzing complex (metabolic) networks • The need for a systemic perspective • The role of mathematical models • The need of an appropriate mathematical formalism • Alternative representations using approximate representations • General ideas • The power-law formalism as a modeling tool for complex networks • Definition and properties • System analysis using power-law models • Model indentification from systemic data • Conclusions and challenges 2008 Oeiras

3. The need for a systemic analysis of metabolic networks Systems biology is an emerging field that enables us to achieve in-depth understanding at the system level. For this, we need to establish methodologies and techniques that enable us to understand biological systems as systems, which means to understand:(1) the structure of the system, such as gene/metabolic/signal transduction networks and physical structures,(2) the dynamics of such systems,(3) methods to control systems, and (4) methods to design and modify systems to generate desired properties. However, the meaning of ‘‘system-level understanding ’’is still ambiguous. … Systems biology is both an old and new field in biology … … Concepts such as robustness and feedback control were already discussed at that time and extensively investigated. Kitano, H. (2002) Curr.Gen.41:1-10 2008 Oeiras

4. System-level understanding of metabolic networks • Understand complex metabolic networks • Gene regulatory networks, Signal transduction, Apoptosis, Cell cycle, Oxidative stress, etc. • Adaptative response to changes in environmental conditions • Characterize pathological situations • Common questions (a mathematical formalism must be able of answering those questions) • Identify key features in complex systems (robustness, structural properties, …) • How are they regulated? Which are the key components? • Relate different levels of complexity: genomics, biochemistry, physiology. • Which are the optimal expression profiles in a given situation? • Why a given expression profile leads to a pathological situation? • Make predictions (what happens if…) on future observations 2008 Oeiras

5. Quantitative approaches:A Systems Biology perspective • Use abstraction to simplify the problem. • Concentrate in class of systems. • Build-up mathematical models of complex systems. • Use an appropriate formalism • Use data to constraint the model. • Use biological knowledge to challenge the model. • Concentrate in deriving general rules and in understanding design and operational principles. • Design principles: Which are the evolutionary advantages of a given regulatory network? • Operational principles: Given a regulatory network, which are the optimal expression profiles to adapt to environmental changes? 2008 Oeiras

6. Mathematical models of complex systems • Mathematical models are a (simplified) representation of the actual system. • The challenge for system reconstruction should not aim to have an exact picture of reality. • If we concentrate in an exact picture of reality we would came out with an object (model) as complex as reality. • Mathematical models are (incomplete) abstractions (and result from conceptual incomplete models of reality) • We need a certain level of abstraction to be able of understanding a complex network. • It is possible to understand reality without knowing every single piece of evidence in every imaginable situation. 2008 Oeiras

7. The need for appropriate mathematical formalisms • Structural complex systems analysis • Graph theory, Boolean networks, Statistical and similar analysis (Identify connectivity properties, free-scale networks, etc.) • Quantitative approaches • Stocihiometric analysis techniques • Kinetic modeling • Detailed kinetic descriptions (lack of information, complicated descriptions, too many parameters, are they real in vivo? …) • Alternative strategies based on approximated representations (abstraction and simplification help in analyzing complex systems). 2008 Oeiras

8. Some uses of mathematical modelsThe user perspective • Fit experimental data to derive parameter values that characterize the processes of interest • Reconstruct and identify the topology of reactions and regulation in biological pathways and circuits • Analyze design principles • Optimize specific properties of the system • Integrate different levels of the cellular response and create a network that accounts for the dynamic behavior of genes, proteins and metabolites 2008 Oeiras

9. Basic mathematical models

10. Basic mathematical modelsNode equations • Node equations • Aggregated node equations Network structure: mij Regulatorystructure: vr Dynamicmodel: select a mathematicalrepresentationforvr 2008 Oeiras

11. ExampleNode equations 4 3 (-) 1 2 5 X3 6 X5 X1 X2 X4 (+) 2008 Oeiras

12. ExampleAggregatednode equations 4 3 (-) 1 2 5 X3 6 X5 X1 X2 X4 (+) 2008 Oeiras

13. Steady-state equations 4 3 (-) 1 2 5 X3 6 X5 X1 X2 X4 (+) 2008 Oeiras

14. Methods based on the stoichiometric matrix • Flux balance analysis • Find the optimal flux distribution constrained to some goal (maximum growth, maximum flux,..) • Predict the effect of knocking-out a given gene • As the steady-state equation must be fulfilled, fluxes must be changed to match the effect of knocking-out a gene • Advantages • Genome-wide models • Ready to go from a simple conceptual description • Independent of detailed kinetic information • Problems • Does not include regulatory information • Does not include metabolite levels • Does not include dynamic changes 2008 Oeiras

15. Flux balance analysis 4 4 3 3 (-) 1 1 2 2 5 5 X3 X3 6 6 X5 X5 X1 X1 X2 X2 X4 X4 Different systems Same stoichiometry (+) 2008 Oeiras

16. Basic mathematical modelsNode equations • Node equations • Include regulatory structure • Which are the metabolites that affect each reaction? • Which is the influence of a change in a metabolite on the properties of each reaction? Network structure: mij Regulatorystructure: vr 2008 Oeiras

17. Regulatory effects 4 3 • Use kinetic equations? • Lack of information, complicated representations • Use approximated representations (-) 1 2 5 X3 6 X5 X1 X2 X4 (+) 2008 Oeiras

18. Regulatory effects 4 3 • Use kinetic equations • Which is the available information? • Is the in vitro information relevant? • How many parameters are required? • Can they be indentified? (-) 1 2 5 X3 6 X5 X1 X2 X4 (+) 2008 Oeiras

19. Mathematical formalisms based on approximate representations • Consider a process that depends on different metabolites and parameters • Homogeneous function of the enzyme 2008 Oeiras

20. The power-law formalism 2008 Oeiras

21. The power-law formalism 2008 Oeiras

22. Example X3 X4 v2 X2 X1 (-) X12 The power-law formalism X5 is the enzyme in reaction v2 2008 Oeiras

23. 9 X8 (+) (-) X7 X1 5 4 (+) (+) (-) 1 (-) (-) X5 (+) (-) Building-up models: Generalized Mass Action (GMA model) 2 (-) X2 3 X6 X6 (+) (+) 6 8 (-) (-) Thyroid hormone metabolism Sorribas & Gonzalez (1999) J.Theor.Med. 2:19-38 X3 X4 7 Automatic model generation from the scheme

24. Matrix representation of a GMA model From a given scheme, we can automatically generate these matrices. The GMA analysis is straightforward using these matrices 2008 Oeiras

25. Steady-state characterization in GMA models • GMA analysis is based on sensitivity theory. • It provides a complete steady-state characterization through parameter sensitivity (robustness) and log-gains (response to environmental changes). • Dynamic responses are analyzed through numerical simulations. Logarithmic gains 2008 Oeiras

26. A simple example on design principles (-) X3 X5 X1 X2 X4 (+) • Which are the requirements (design principles) for havingan increase of X3 as a response to an increase of X5? • Can we design a system in which an increase in X5 willproduce a decrease of X3? 2008 Oeiras

27. GMA model 4 3 5 1 2 6 2008 Oeiras

28. Logarithmic gains GMA model 4 3 5 1 2 6 2008 Oeiras

29. Design principle for a positive log-gain 4 3 5 1 2 6 Systemic response Designprinciple 2008 Oeiras

30. Design of specific systemsResponse to a 10% increase of X5 over the reference state 2008 Oeiras

31. Steady-state characterization in GMA systems Metabolite sensitivities Flux sensitivities 2008 Oeiras

32. Use of sensitivity analysis • Check model consistency • High sensitivity may indicate a ill-defined model. • Compare design performance • If design (a) has a lower sensitivity than design (b), then design (a) can be a better choice for the considered function • Relate local and global properties • Sensitivity is a global property that depends on the underlying processes on the network 2008 Oeiras

33. Purine metabolism in man Kinetic-ordersensitivities 2008 Oeiras Curto et al. (1998) Math.Biosc. 151:1-49

34. Analytical methods provided by the power-law formalismBiochemical Systems Theory (BST) • Automatic model generation from a conceptual scheme • Algebraic methods for analyzing model characteristics and design • Mathematical controlled comparisons (design principles) • Quantitative modeling and analysis • Sensitivity analysis (assess model robustness) • Parameter scanning (operational principles) • Simulation • Optimization • Canonical modeling strategies (recasting non-linear models into power-law models) http://www.udl.es/Biomath/PowerLaw/ 2008 Oeiras

35. Alternative kinetic formalisms based on approximated representations

36. Mathematical formalisms based on approximate representations Kinetic-order Linear (1) Collect constant terms 2008 Oeiras

37. Mathematical formalisms based on approximate representations Linear (1) Taylor series Reversing the result (log)linear 2008 Oeiras

38. Mathematical formalisms based on approximate representations 2008 Oeiras

39. The Saturable and Cooperative formalism • Consider a transformation: • Aproximate by Taylor series and return to cartesian coordinates Sorribas A, Hernández-Bermejo B, Vilaprinyo E, Alves R. Cooperativity and saturation in biochemical networks: a saturable formalism using Taylor series approximations. Biotechnol Bioeng. (2007) 97(5):1259-1277. 2008 Oeiras

40. Linear (2) Linear (1) (log)linear Lin-log Power-law A family picture of the different formalisms based on Taylor series Saturable and cooperative

41. Mixed AC inhibiton with fractal kinetics k2 k1 E + S ES E + P k-1 + + I I k3 k-3 a k3 a k-3 a k1 EI ESI a k-1 Using the SC formalism(unknown rate-laws) I v1 S 2008 Oeiras

42. Mathematical formalisms based on approximate representations • Common characteristics of all these representations • Exact representations at a given operational point • Same operational point values for fluxes and metabolites • Local sensitivity at the operational point (kinetic-order, elasticity) 2008 Oeiras

43. Are those important limitations for analyzing systemic design? • Characterization of systemic properties at a given steady-state can be made with approximated representations as they are exact at that point. • If we can prove that a given design is better (based on criteria of functional effectiveness) independently of the parameter values, then this conclusion holds for any steady state. Issues of accuracy are not relevant in this case. • Accuracy issues become relevant when “goodness” of design depends on parameter values because calculations of systemic behavior away from the operational point become less acurate. 2008 Oeiras

44. Characterization from systemic measurements • Collect information of the integrated behavior of the target system • Steady-state values at different conditions • Dynamic response to perturbations • Fitting a model to dynamic data • Non-linear regression, Neural networks, Alternative regression, etc. • In any of those cases, the resulting representation is no longer local. It smoothes the available data within a given range (for instance by using a least-squares criteria) • Hernández-Bermejo B, Fairén V, Sorribas A. Power-law modeling based on least-squares criteria: consequences for system analysis and simulation. Math Biosci. 2000,167(2):87-107. • Hernández-Bermejo B, Fairén V, Sorribas A. Power-law modeling based on least-squares minimization criteria. Math Biosci. 1999, 161(1-2):83-94. 2008 Oeiras

45. Parameter estimation from dynamic data • Which are the limitations of approximated formalisms? • Which information is contained in dynamic data? • Can we identify a model from a single experiment? • Experimental design for model identification from dynamic data • Which are the minimum requirements in collecting dynamic data? • Simulated examples 2008 Oeiras

46. Which kind of information is contained in dynamic data? X2 (-) • Define a reference model using classical kinetics • At t=0 increase X3 and run the simulation • From the simulations compute Xi(t) and vi(t) X1 X3 X1 X2 v2 v3 X1 X1 v2 X2 X2 2008 Oeiras X1

47. A single S-system can fit the experiment (-) X3 X1 X2 v2 v3 2008 Oeiras

48. Several models can fit a single experiment • The information contained in a single experiment allows fitting approximated models as the variation of metabolites and fluxes is limited • In that case, approximated models are a good representation • Different formalisms can provide a good model • However, since the information of a single experiment is limited, in most cases various models would be able of fitting the same data • In general, more information would be required to uniquely identify the best model for the actual system 2008 Oeiras

49. Which kind of information is contained in dynamic data? X2 (-) • Define a reference model using classical kinetics • At t=0 decrease X3 and run the simulation • From the simulations compute Xi(t) and vi(t) X3 X1 X2 X1 v2 v3 X1 v2 X2 X1 X2 2008 Oeiras X1

50. A different S-system can fit each experiment (-) X3 X1 X2 v2 v3 X1 X2 X2 X1 2008 Oeiras