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What? Why? How? An Introduction to Modeling Biological Systems Eberhard O.Voit Department of Biomedical Engineering Georgia Institute of Technology and Emory University Atlanta, Georgia 11 th International Conference on Molecular Systems Biology June 21-25, 2009 Shanghai, China

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Points to Ponder (1) What is a model? What is modeling? Conceptual Physical Maps and blueprints Mathematical models Does modeling change over time? Euclid and computers

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Points to Ponder (2.1) Why modeling? Prediction Manipulation, optimization Explanation (counterintuitive behavior; chains of causes) Bookkeeping Organize thoughts Organize data Identify outliers

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Points to Ponder (2.2) Why modeling? (a)(b) X 2 X 1 X 2 X 1 2.0 212 2.0 1 15.0 2 4.0 111 XXX XXXX 1 = 0.9 or 1 = 1.02 X 1 X 2 03060 0 0.75 1.5 (c) time 0120240 0 1.5 3 X 1 X 2 (d) time

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Points to Ponder (3.1) What is a good/bad model?

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Points to Ponder (3.2) What is a good/bad model? “The best” model? Example: Heart Purpose Correctness Simplicity vs. complexity Degree of detail Range of applicability Qualitative vs. quantitative results

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Points to Ponder (4) Limitations of models Assumptions Simplifications Extrapolation Complexity masking problems, errors

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Points to Ponder (5) Theory of biology Specific predictions; population vs. individual General predictions; qualitative vs. quantitative Design and operating principles

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Points to Ponder (6) Type of model Components Methods Use

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Steps of a Typical Analysis Model conception and formulation Parameter estimation Concept of a steady state Stability Sensitivities, gains, robustness Dynamics Bolus experiments Persistent changes in system components

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Reality Simplification and Abstraction Ignore details Omit components, factors Hypothesize Approximation Represent complex processes with simple(r) functions Linear, nonlinear, piecewise Model Conception Reality Abstraction Graph Equations Analysis Reality

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Criteria of a Good Model Capture the essence of the system under realistic conditions Be qualitatively and quantitatively consistent with key observations In principle, allow analyses of arbitrarily large systems Be generally applicable Be characterized by measurable quantities Allow simple translation of results back to subject area Have a mathematical form that is amenable to analysis

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Formulation of a Model for Complex Systems Tenets of systems analysis : Each component of the system may potentially depend on all other components and outside factors. To “understand” the system, we need to know how every component changes over time. Dynamic changes in a system component are driven by inputs and outputs.

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Needs Language / Notation: “Convenient” math Theory Methods of analysis Must be mathematics

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Convenient Math Two emerging dogmata: 1. Not all mathematical approaches are equally useful. 2. All laws in nature are approximations. Two pieces of conventional wisdom: 1. In math it’s either right or wrong. 2. Laws in nature are true and absolute.

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Components of a Systems Model Variables Dependent Independent Time Change Processes Flow of material Signals Parameters and constants

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Variables Dependent: Variable is affected by the action of the system typically changes over time may or may not affect other variables Names: X, Y 4, Z i Independent: Variable is not affected by the action of the system typically constant over time sometimes external and under experim. control may or may not affect other variables examples: inputs, enzymes may change from one “experiment” to the next

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Change Mathematics: time is independent variable Systems modeling: time is often implicit What about time and change? Typical equation: Change in X 3 over time =

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Change (cont’d) Fluxes are functions of variables, thus: new notation Don’t see t anymore, but variables do change over time. Example from enzyme kinetics:

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Example: Radioactive Decay “The change in X is directly proportional to the present amount of X, the proportionality is quantified by k, and the change is in the negative direction (decrease).” Why does this describe radioactive decay over time? “Solution” to the differential equation is X(t) =X 0 exp(–kt), because Equation: Change (cont’d)

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Processes Very important to distinguish Flow of material (mass is moving): Solid, heavy arrows and Flow of information (signals, modulation) : Dashed, thin arrows Essentially any interaction between variables or between system and environment Confusion may lead to wrong model structure; often difficult to diagnose.

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Formulation of a Model for Complex Systems Translation into a diagram : X1X1 V1+V1+ V1–V1– XiXi Vi+Vi+ Vi–Vi–

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Formulation of a Model for Complex Systems Translation into math : X1X1 V1+V1+ V1–V1– XiXi Vi+Vi+ Vi–Vi– insideoutside very complex 1 i

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Formulation of a Model for Complex Systems Savageau: Approximate it per Taylor but in log-space Result: What can we do with this “very complex” function ? i

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S-systems The change in each system component is described as a difference between two terms, one describing all contributions to growth or increase in the variable, the other one describing all contributions to loss or decrease in the variable. 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. ’s and ’s are rate constants, g’s and h’s kinetic orders.

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Alternative Power-Law Formulations S-system Form: XiXi Vi1+Vi1+ Vi1–Vi1– V i,p + V i,q – Generalized Mass Action Form:

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Meaning of Parameters Kinetic orders g i j, h i j : Effect of variable X j on production or degradation of variable X i. Rate constants i and i : Magnitude of production and degradation fluxes of X i.

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Meaning of Parameters Kinetic orders g i j, h i j : Effect of variable X j on production or degradation of variable X i. Rate constants i and i : Magnitude of production and degradation fluxes of X i.

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Parameter Values Experience and educated guesses. Data needs, advantages and limitations of the various approaches. Estimation of parameters from traditional rate laws. Estimation of kinetic orders from steady-state data. Estimation of parameters from dynamic data. Talk during Conference.

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Comment on Parameter Estimation Parameter estimation is arguably the hardest part of modeling Very different options: flux-versus-concentration data rate laws dynamic data Dynamic data contain the most information, but are the most difficult to evaluate

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Steps of a “Typical Analysis” Model conception and formulation Parameter estimation Computations at a steady state Stability Sensitivities, gains, robustness Dynamics Bolus experiments Persistent changes in system components

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Can we compute the steady state(s) of the system? Does the system have a steady state, where no variable changes in value? How is the steady state of the system affected by inputs? (“Gains”) Steady-State Analysis Justitia of Biel Can the system tolerate a slightly changed structure? (“Sensitivities”)

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Can the system tolerate a large perturbation? (Change in environment) Can the system tolerate a small perturbation? (Normal fluctuations in milieu) Can the system tolerate a slightly changed structure? (Mutation, Disease) Stability Castellers of Nens del Vendrell Method: Eigenvalue analysis.

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How does the system respond to changed input? Where is the system going from here? How does the system respond to a slightly changed structure? Dynamics How can we optimize the performance of the system ? How can we intervene in the function of the system? Georgio de Chirico (1914)

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Dynamical Analyses Types: Bolus Persistent change in input Exogenous supply of (dependent) metabolite Changes in structure Methods: Algebraic analyses Numerical analyses Simulations Almost all done per computer! This is the fun part!

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Modeling Process Ideas Reality CheckDraft (?) Model Refinement DataAnalyze Data

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Example X 1 X 2 X 5 X 3 X 4 Question: What affects production of X 1 ? Answer: X 5 and X 3 Thus: Degradation analogously:

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Example (numerical) X 1 X 2 X 5 X 3 X 4

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X 1 X 2 X 5 X 3 X 4 Example: Focus on Inhibition

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X 1 X 2 X 5 X 3 X 4 g 13 is the parameter that characterizes the strength of the inhibition. g 13 is negative or zero. Ifg 13 =0, then there is no inhibition. Example: Focus on Inhibition

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Case Study: Purine Metabolism Start: Lots of data (kinetic, physiological, clinical, …) Which data are (most) relevant? Decent idea about pathway structure Questions: Do pieces fit together? Can we make reliable predictions? How do diseases relate to metabolism? What would be good drug targets?

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First Model PRPP IMP Xa UA GMP GDP GTP S-AMP AMP ADP ATP HX XMP 5- -P v 16 v 15 v 14 v 13 v 12 v 11 v 10 v8v8 v7v7 v9v9 v3v3 v2v2 v1v1 v6v6 v5v5 v4v4 Equations Parameters Analysis Stoichiometry faulty Verdict: Revise!

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Second Model Refinements More Data Analysis Verdict: Revise!

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“Final” Model Numerous iterations of refinements and comparisons Verdict: Cautious optimism Result: Model consistent with literature information New classification of purine-related mental diseases

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What can we do with such a model? o Analyze normal metabolic state: study responses o Bolus experiments: study response to inputs o Changes in enzyme activities: study metabolic diseases o Change fluxes: Screen for drug treatments Intended effects Side effects o Redistribute fluxes diseases metabolic engineering

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Optimization (Citric acid again) Task: Reroute flux in an optimal fashion; e.g., maximize citric acid output

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Pathway Optimization with S-systems (Voit, 1992) Optimization under steady-state (batch) conditions becomes Linear Program even though (nonlinear) kinetics is taken into account: maximize log(flux) [or log(variable)] subject to: Steady-state conditions in log(variables) Constraints on log(variables) Constraints on log(fluxes)

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Pathway Optimization (cont’d) Hatzimanikatis, Bailey, Floudas, 1996: Use these features for optimization of pathway structure Great Advantage: Methods of Operations Research applicable very well understood applicable for over 1,000 simultaneous variables robust and efficient incomparably faster than nonlinear methods Torres, Voit, …: Applications (e.g., citric acid, ethanol, glycerol, L-carnitine)

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Pathway Optimization (cont’d) Recent extensions: Optimize dynamics over time horizons (with Ernandi-Radhakrishnan) Optimize Generalized Mass Action systems (alternative power-law systems), using dynamic programming and branch-and-bound methods (with E. Gatzke, USC)

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Method of Controlled Mathematical Comparisons Crucial consequence for many purposes: Structure determined by parameter values Identification of structure becomes parameter estimation Comparison of two alternative systems allows characterization of the role of some mechanism In contrast to models such as polynomials, the relationship between S-system parameters and structural features of a pathway is essentially one-to-one.

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Recall: Search for Design Principles X 4 X 7 X 1 X 5 X 2 X 6 X 3 X 4 X 7 X 1 X 5 X 2 X 6 X 3 Exploration of Design Principles: What is the effect of feedback inhibition, everything else being equal?

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Controlled Mathematical Comparisons 1.Construct a system model and an alternative that differs in one feature of interest (e.g., feedback inhibition). 2.Select all parameter values the same but adjust parameter values associated with this feature such that both systems have same steady state and as many other features as possible. 3.Study sensitivities, dynamics etc. (e.g., response time) 4.Differences are caused by the parameter (mechanism) of interest. 5.New variations on this theme in Schwacke and Voit (2004). 6.Applications mainly in gene circuitry (Savageau et al.) and some generic metabolic pathways. 7.Results are almost independent of specific parameters and elucidate general design and operating principles.

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Summary Given fully parameterized model equations, study: Steady-state Sensitivities, gains Stability Dynamics: bolus, mutation, scenarios, simulations Some analyses could be done by hand, but computer analysis is much more convenient

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