Mathematical Modeling of Metabolism, Signaling Pathways, and Gene Expression Edda Klipp Max Planck Institute for Molecular Genetics Ihnestr. 73, 14195 Berlin, Germany http://www.molgen.mpg.de/~ag_klipp

Content Techniques Introduction to Modeling - Kinetics of individual reactions Construction of networks - Systems equations, system simplifications, conservation relations - Analysis of networks and cellular reaction systems - Time constants, time hierarchy - Metabolic control analysis - Computer based modeling tools  Datenbases Biological objects - Enzyme reactions, Ligand interactions Metabolism Signal transduction pathways Gene expression - Cell cycle

Systems Biology Biological research in the 1900s followed a reductionistic approach: detect unusual phenotype  isolate/purify 1 protein/gene,  determine its function However, it is increasingly clear that discrete biological function can only rarely be attributed to an individual molecule.  The new task is to understanding the structure and dynamics of the complex intercellular web of interactions that contribute to the structure and function of a living cell.

Systems Biology Development of high-throughput data-collection techniques, e.g. microarrays, protein chips, yeast two-hybrid screens allow to simultaneously interrogate all cell components at any given time.  there exists various types of interaction webs/networks - protein-protein interaction network - metabolic network - signaling network - transcription/regulatory network ... These networks are not independent but form „network of networks“. Barabasi & Oltvai, Nature Rev Gen (2004)

What is a model? Abstraction Yeast, mouse – as models for human Verbal explanation A sequence of letters ATTCGAGGTATA for DNA sequence Wiring scheme (e.g. Metabolic network) Mathematical description: Boolean Network Differential Equations Stochastic Equations Abstraction Simplified representation allowing for understanding

Why modeling? Experimental observations: many simple and complex processes isolated enzymatic reaction : temporal prozesses in metabolic networks pattern of gene expression and regulation Even the behavior of simple systems can usually not be predicted intuitively and from experience. The behavior of complex dynamical processes can not predicted with sufficient precision just from experience. For prediction and explanation of processes one needs a model.

Why modeling? Advantages Time scales may be stretched or compressed. Solution algorithms / computer programs can often be used independently of the actually modeled system. Costs of modeling are lower than for experiments. Representation of quantities that are experimentally hidden. No risk for real systems, no interactions investigation/system.

Why modeling? Burning questions (there are more…) How is cellular response to environmental changes and stress regulated? How should a cell be treated to yield a high output of a desired product (Biotechnology) Where should a drug operate to cure a disease (Health care)? Is our knowledge about a network/pathway complete?

Example 1: African Sleeping Disease Epidemias in East and Central Africa (3-4 Mill. sick) Parasit (Flagellat) in blood Carrier: Tsetse fly NO THERAPY Glykosom Parasit Red Blood cell Mitochondrium Blood Of Host

Example 2: Metabolic Oscillations Experimental Observation: Addition of KCN to yeast cells causes metabolic oscillations (measured as NADH fluorescence) What happens after mixing of populations that oscillate out of phase? NADH Fluorescence + KCN + KCN Time Time MIX Time NADH Fluoresczence

Example 2: Metabolic Oscillations Explanatory Model Oscillations: What is the Oscillator? How occurs synchronization? Ethanol Acetaldehyde Glucose Ethanol Acetaldehyde Glucose Acetaldehyd Acetaldehyd

Example 3: Gene Expression Regulation Whole-Genome MicroArrays De Risi et al., 1997 Science 278 During diauxic shift at glucose starvation Yeast cells switch their metabolism from fermentation to respiration. In this time, they also change the expression pattern of about 1700 of the 6000 genes in an ordered manner. - red: induced expression - green: reduced expression (measured as fluorescently marked cDNA)

Example 3: Gene Expression Regulation Bioinformatic: - Clustering of genes according to expression profiles - Reconstruction of networks Modeling: Establishing a dynamic model for changes of concentrations or activities - Prediction of changes of metabolism - Optimization approach: Prediction of optimal gene expression under certain conditions

Example 4: Biosynthesis Optimization Glucose Amino acid Synthesis Glucose-6-P Fructose-6-P 6-Phosphogluconat Histidin PRPP Erythrose-4-P Glycerinaldehyd-3-P + Dihydroxyacetonphosphat Glycin Serin Chorismat Phenylalanin Cystin Cystein Phosphoenolpyruvat Tyrosin Tryptophan Isoleucin Pyruvat Alanin Valin How to increase the synthesis of specific amino acids? Threonin Lysin Leucin Homoserin Aspartat Oxalacetat Methionin Asparagin a-Ketoglutarat Metabolic models Flux balance analysis Metabolic control analysis Lysin Hydroxyprolin Prolin Glutamat Arginin Glutamin

Model and Modeling The analysis of model systems generates a language, to describe complex processes. Investigation starts with: - Ordering, classification - Choice of prototypes - Systematics e.g. Enzyme reactions They may have 1, 2, 3,... substrates; they catalyze transfer of different groups,... Phosphofructokinase is an example for kinases Enzyme nomenclature (EC 1.1.2.7)

Model and Modeling Distinction of complex processes: reversible – irreversible periodic – non-periodic deterministic – stochastic If the system is known at one time then it is known forever Probability distribution for system states Other system-relevant categories: stable – unstable robust – fragile active - inactive

Adequateness of Models Theoretical models are frequently mathematical models. but mathematical formalism is not first aim of a model, instead start with consideration of problem. Translation of the biological problem into a model. Models are not “right” or “wrong”. Their usefulness is a compromise between Adequatness (reflection of reality) and handling/Simplicity. „If we don‘t make the models, we will not know, why they are wrong.“

Purpose of a Model A model reflects only specific aspects of the reality. Aim of a model: give answers to specific questions!!! Model development is subjective and selective. Prediction of system behavior: precise results for input/output but system itself might maybe be treated as black box. Function of an object: model be very realistic, parts/structure/relations of the object/process are important (glass box). Generality: model application to many different objects/processes Simplicity: e.g. for mathematical treatment – contradiction to realistic

Model Development in 5 Steps Defining the scope – - Formulating the PROBLEM Distribution of molecules on both sides of a membrane Ai Ao dAi/dt = f(Ai, Ao, C, p) Establishing a simple model as „Cartoon“ in mathematical terms - Model verification: can it in principle explain the observation? - Solving the related (mathematical) problems - Model validation: determine parameters from experimental data - Sensitivity analysis: depends the result on parameter choice? Prediction / Comparison of results with real systems (EXPERIMENT) - Difference ?!? - Iterative improvement of model (model structure, parameters, …)

Structure of the system fast Sext S1 S2 S3 S4 S5 Smito S6 Boundary of the system slow slow Variables, parameters, constants State variables - set of variables describing the system completely Dimension of the systems = number of independent state variables How many variables are used in my model? Units of variables and parameters etc. fit together?

Variables, Parameters, Constants V, d – Variable, p – Constant – Parameter, v, S – Variables Variables – variable quantities, for which the model establishes a relation Constant – quantity with a fixed value Parameter – quantity to which a value can be assigned This value is changeable and depends on measurements. If a quantity is a variable, a parameter or a constant depends on the model.

Analysis of Dimensions State variables are a set of variables that describe a system completely. They are independent of each other; each of them is necessary to describe the system states completey. Dimension of a systems = number of independent state variables How many variables has my model? too few – system is underdetermined too many – system is underdetermined and probably contratictory Question/Task Can we reduce the number of state variables, e.g. by changing system limits? Which units do the variables have?

What determines system behavior? Two fundamental causes for changes of the system behavior: Influences of the environment (input) Processes within the system – the structure of the system determines how exogenous or endogenous signals will be processed General rule: assign system limits such that the output does NOT feedback To the input of the system. General rule: different system structures can produce similar system behavior Structure determines behavior, not the other way around.

Direction of discovery known to be predicted Structure Function Protein interactions Biochemical action Metabolic pathways Concentration changes Enzyme sets Influence of perturbations Possible behavior, bifurcations : : Function Structure Transmission of a signal Sequence of signaling compounds Time course of concentrations Possible protein interactions

Concept of state The state of a system is a snapshot of the system at a given time that contains enough information to predict the behaviour of the system for all future times. The state of the system is described by the set of variables that must kept track of in a model. Different models of gene regulation have different representations of the state: Boolean model: a state is a list containing for each gene involved, of whether it is expressed („1“) or not expressed („0“) Differential equation model: a list of concentrations of each chemical entity Probabilistic model: a current probability distribution and/or a list of actual numbers of molecules of a type Each model defines what it means by the state of a system. Given the current state the model predicts what state/s can occur next.

Kinetics – change of state A B Deterministic, continuous time and state: e.g. ODE model concentration of A decreases and concentration of B increases. Concentration change in per time interval dt is given by Probabilistic, discrete time and state : transformation of a molecule of type A into a molecule of type Sorte B. The probability of this event in a time interval dt is given by a – number of molecules of type A Deterministic, discrete time and state : e.g. Boolean network model Presence (or activity) of B at time t+1 depends on presence (or activity) of A at time t

Biological processes are complex phenomena Central dogma of molecular biology: Gene mRNA Proteines Cellular processes

Cellular Reaction Systems are Networks Behavior is dependent on the individual kinetics of single reactions the topology of the network Metabolism Graph theoretical: Nodes + Edges  Routes, Measures for Connectivity Stoichiometric: + number of molecules Moiety conservation, Flux distribution Kinetic: + Kinetics + Concentrations Time-dependent behavior, Steady States, Control Analysis, Bifurcations Signaling

Basic Elements of Biochemical Networks Transport Reaction Reaction Glucose1-P v1 Glucose6-P v2 Fructose6-P Metabolite Metabolite Glucose- Phosphat- isomerase Phospho- glucomutase Metabolite Design of structured dynamic models 1. Determination of system limits G1P G6P F6P v1 v2 System extern Concentration change = Production – Degradation + Transport 2. Balancing Rate as function of concentrations and parameters 3. Assignment of Kinetics

Basic Elements of Biochemical Networks v1 v2 v3 v4 v5 Systems equations r – number of reactions Si – metabolite concentrations vj – reaction rates nij – stoichiometric coefficients Network properties Individual reaction properties Matrix representation

Data Bases GO (Gene Ontology) http://www.geneontology.org, functional description of gene products KEGG (Kyoto Enzyclopedia of Genes and Genomes) http://www.genome.ad.jp/kegg/, reference knowledge base offering information about genes and proteins, biochemical compounds and reactions, and pathways BRENDA (Comprehensive Enzyme Information System) http://www.brenda.uni-koeln.de, curated database containing functional data for individual enzymes NCBI (National Center for Biotechnology) http://www.ncbi.nlm.nih.gov/ ,provides several databases: - molecular databases, with information about nucleotide sequences, proteins, genes, molecular structures, and gene expression - taxonomy database: names and lineages of more than 130,000 organisms SPAD (Signaling PAthway Database) http://www.grt.kyushu-u.ac.jp/spad/index.html, information about signaling pathways (schemes, links) JWS Online, Model database http://jjj.biochem.sun.ac.za/database/index.html, published models,implemented in Mathematica® Models can be simulated Biomodels, Model database http://www.biomodels.net/, published models,implemented in SBML

Modeling Tools http://sbml.org BALSA BASIS BIOCHAM BioCharon biocyc2SBML BioGrid BioModels BioNetGen BioPathway Explorer Bio Sketch Pad BioSens BioSPICE Dashboard BioSpreadsheet BioTapestry BioUML BSTLab CADLIVE CellDesigner Cellerator CellML2SBML Cellware CL-SBML COPASI Cytoscape DBsolve Dizzy E-CELL ecellJ ESS FluxAnalyzer Fluxor Gepasi INSILICO discovery JACOBIAN Jarnac JDesigner JigCell JWS Online Karyote* KEGG2SBML Kinsolver* libSBML MathSBML MesoRD MetaboLogica MetaFluxNet MMT2 Modesto Moleculizer Monod Narrator NetBuilder Oscill8 PANTHER Pathway PathArt PathScout PathwayLab Pathway Tools PathwayBuilder PaVESy PNK Reactome ProcessDB PROTON pysbml PySCeS runSBML SBML ODE Solver SBMLeditor SBMLmerge SBMLR SBMLSim SBMLToolbox SBToolbox SBW SCIpath Sigmoid* SigPath SigTran SIMBA SimBiology Simpathica SimWiz SmartCell SRS Pathway Editor StochSim STOCKS TERANODE Suite Trelis Virtual Cell WinSCAMP XPPAUT http://sbml.org

Modeling of biochemical reaction networks Content (metabolism): Components Main properties of catalyzed reactions Structure of networks Stoichiometric analysis, flux balance analysis System simplifications Control theory Definitions, Theorems Typical control structures in metabolic networks

Literature Mathematical Physiology, Springer, 1998 James Keener, James Sneyd Computational Cell Biology, Springer, 2002 Editors: Christopher Fall, Eric Marland, John Wagner, John Tyson Systems Biology in Practice. Concepts, Implementation and Application, VCH Wiley, 2005 Edda Klipp, Ralf Herwig, Axel Kowald, Christoph Wierling, Hans Lehrach Systems Biology. Properties of Reconstructed Networks, Cambridge Univ Press, 2006 Bernhard Palsson System Modeling in Cellular Biology. From concepts to nuts and bolts, MIT Press, 2006 Editors: Zoltan Szallai, Jörg Stelling, Vipul Periwal An Introduction to Systems Biology. CRC Press, 2006 Uri Alon Heinrich, R. & Schuster, S., 1996, The Regulation of Cellular Systems, Chapman & Hall Fell, D. 1997, Understanding the Control of Metabolism, Portland Press