Mathemati cal Modeling with Version 2.0 Introducti on Mathemati cal Modeling with © TIKANIS GmbH Users Manual Version 2.0 f(x) Documentation written by Thomas Maiwald1,2, Oliver Eberhardt2, Sherwin K. Sy3, and Julie Blumberg4 1 Department of Systems Biology, Harvard Medical School, Boston, MA, USA 2 Freiburg Center for Systems Biology, University of Freiburg, Germany 3 Department of Pharmaceutics, University of Florida, Gainesville, FL, USA 4 University Hospital Freiburg, Germany Contact: [email protected] Document version: 2010-12-16 Copyright 2010, all rights reserved, www.potterswheel.de 2 Table of Contents 1. Introduction 4 1.1 Reactions, variables, and parameters 5 1.2 Key modeling questions 8 2. Overview and notation 9 3. Model creation 10 4. Data import 12 4.1 Meta information 13 4.2 Data matrix 15 4.3 Measurement error 16 4.4 Creating a model-data couple 16 5. Fitting 16 6. Goodness-of-fit 17 7. Model refinement 20 8. Analysis and model prediction 21 9. Design of new experiments 23 10. Model exploration in the visual model designer 24 11. Rule based modeling and naming convention 24 12. Implicit and explicit multi-compartment modeling 25 13. Configuration 26 14. Macros and custom Matlab programs 26 15. Reporting 27 16. Model stiffness 27 17. Bibliography 28 3 Introduction The program PottersWheel has been developed to provide an intuitive and yet powerful framework for data-based modeling of dynamical systems like biochemical reaction networks. Its key functional- ity is multi-experiment fitting, where several experimental data sets from different laboratory condi- tions are fitted simultaneously in order to improve the estimation of unknown model parameters, to check the validity of a given model, and to discriminate competing model hypotheses. New experi- ments can be designed interactively. Models are either created text based or using a visual model designer. Dynamically generated and compiled C files provide fast simulation and fitting procedures. Each function can either be accessed using a graphical user interface or via command line, allowing for batch processing within custom Matlab scripts. PottersWheel is designed as a Matlab toolbox, comprises 250.000 lines of Matlab and C code. The PottersWheel software requires Matlab 2006a or higher on any Linux, Mac, or Windows PC. The Matlab optimization toolbox and a few external programs are recommended, depending on the intended use of PottersWheel. A detailed installation description and introductory videos are given atwww.potterswheel.de where the software can freely be downloaded for academic usage. 1. Introduction Modeling aims to formulate a set of mathematical equations which are able to predict computa- tionally the dynamic behavior of a real system, e.g. of a biochemical network. Entities of the system should preferably have a representation within the model in contrast to black-box modeling. This approach deepens the understanding about the network during the process of hypothesis genera- tion, mathematical formulation, and comparison with experimental data potentially leading to new hypotheses. At the same time, laboratory measurements too complex to be understood by visual inspection are brought into relation. Beyond that, a formalized model is helpful to unambiguously communicate a hypothesis to colleagues. Finally, a suitable model renders possible new approaches to identify therapeutic strategies and to assess the feasibility of a research agenda minimizing the cost in terms of animals, time, and money. Modeling approaches differ in the choice of possible model structures. Regression models use alge- braic equations (1), Boolean models comprise logical gates (2), Bayesian networks (3) and stochastic models (4,5) are based on probability distributions and mechanistic models are expressed using either ordinary differential equations (ODEs) if spatial effects are neglected (6) or partial differential equations (PDEs) in the more general case (7). Each approach implies assumptions about the sys- tem at hand, requires a certain type of measurements, and permits conclusions at different levels of detail and accuracy. Moreover, the burden of computational implementation and analysis varies strongly between the modeling concepts. Here, we focus on mechanistic modeling of biochemical networks using ODEs and the PottersWheel modeling framework (8). This approach assumes a sufficiently large number of molecules not be- low 1000 and no spatial effects, requires time-resolved measurements, allows predicting transient 4 Introduction system responses to changed environmental conditions, and often simplifies interpretation due to a one-to-one relationship to physiological entities. The numerical effort lies between regression/ Boolean models and stochastic/PDE models and can usually be handled by a single PC or a small computer cluster, given that suitable algorithms are applied. In the following we introduce key concepts and the PottersWheel terminology based on the JAK/ STAT signal transduction pathway as an example for a biochemical reaction network (9). 1.1 Reactions, variables, and parameters In the simple irreversible reaction „A to B“ assuming mass action kinetics, the reaction rate is pro- portional to the time-dependent concentration of species A, denoted as [A](t), often abbreviated as A(t), neglecting the square brackets, or just A. The proportionality is expressed using a rate coeffi- or just . ௗሾ஺ሿ ோ ܣ ሶ ൌ െ݇ ȉܣ ሿܣሶሿ ൌ െ݇ ȉ ሾܣݒ ൌ ௗ௧ ൌ ሾ cient , leading to the following ordinary differential equation (ODE) for the flux of the reaction: The time-dependent concentration for A is thus simply an exponential decay starting from an initial concentration at t=0: ି௞௧ ଴ -solutionȉ݁ is not feasible. Instead, numerical integra ܣ ሺݐሻ ൌܣIn a more complex situation, the above analytical tion is used to determine the concentration-time profiles. Several kinetic laws apart from mass ac- tion can be used, e.g. Michaelis-Menten and Hill kinetics (10). 5 Introduction k 1 k 2 k 4 k k 3 8 k 5 k 6 k 7 Figure 1. Mathematical formalization of a biological hypothesis. The left panel shows a cartoon of the JAK/ STAT signal transduction pathway. The right panel displays the related formalized and simplified mathematical model created and visualized using the PottersWheel model designer. In contrast to the cartoon, the receptor appears in the model three times: once for each modification state: the free receptor (R), the phosphorylated receptor (pR), and the internalized receptor iR. Every reaction is shown as a small rectangle with incoming flux from the reactants in light gray, outgoing flux to the products in dark gray, and in dashed red the effects of enzymes which are not consumed or produced within the reaction, also called modifiers. S denotes STAT. The prefixes p and np define phosphorylation in the cytoplasm and the nucleus, and the underscore „_“ rep- resents the bond of two species. Figure 1 visualizes the formalization of a biological hypothesis into a reaction network. PottersWheel automatically creates a corresponding set of differential equations, in this case ܴሶ ൌ െ݇ଵ ȉܴȉܧܱܲ൅݇ଶ ȉ ݌ܴ ݌ܴሶ ൌ െ݇ଶ ȉ݌ܴ൅݇ଵ ȉ ܴ ȉ ܧܱܲെ݇ଷ ȉ ݌ܴ ଓܴሶ ൌ ൅݇ଷ ȉ ݌ܴ ܵሶ ൌ െ݇ସ ȉܵȉ݌ܴ൅଼݇ ȉ ݊ܵ ݌ܵሶ ൌ Ȃ ʹ ȉ ݇ହ ȉ݌ܵȉ݌ܵ൅݇ସ ȉܵȉ݌ܴ ݌̴ܵ݌ܵሶ ൌ െ݇଺ ȉ ݌̴ܵ݌ܵ൅݇ହ ȉ ݌ܵ ȉ ݌ܵ ݊݌̴ܵ݊݌ܵሶ ൌ െ݇଻ ȉ ݊݌̴ܵ݊݌ܵ ൅ ݇଺ ȉ ݌̴ܵ݌ܵ ݊ܵሶ ൌ െ଼݇ ȉ ݊ܵ ൅݇଻ ȉ ʹ ȉ ݊݌̴ܵ݊݌ܵ The network is equivalent to the differential equations and describes the structure of the mathemat- ical model, also called its topology. Together with specific values for the dynamic parameters k1, …, 6 Introduction k8, initial values for each species R0 = R(t=0), …, nS0 = nS(t=0), and information about the external Epo stimulation for each time point, EPO(t), the model can be integrated, i.e. the concentration-time profiles can be determined computationally as shown in Figure 2. Table 1 provides an overview of important model elements. Figure 2: Model trajectory with measured data and main user interface. Left: The experimental data (blue) of the phosphorylated receptor is shown over time with error bars corresponding to one standard-deviation. The parameters of the JAK/STAT model have been calibrated in order to minimize the distance of the model trajectory (red) to the data. The model cannot be rejected as a wrong model as long as the χ2 (Chi-Square) value does not exceed a threshold value depending on the desired significance level (see section 3.3). As a rule of thumb, the Chi-Square value should not exceed the number of fitted data points N. Whether the model is „true“ cannot be concluded, as every mathematical formulation of process in nature is only an ide- alization. Right: Shown is the main user interface with two windows comprising the observed variables (left) and internal variables (right) of a fitted JAK/STAT model. JAK/STAT Element Name Description Example model All model species which Dynamic vari- Cytoplasmic and R, pR, iR, S, X occur as a reactant or able nuclear proteins pS, pS_pS product Externally controlled Driving input U species only occurring as Ligands, drugs EPO variable modifiers Observation Sum of dynamic vari- Proteins detected pS_obs = pS + Y variable (ob- ables that can be mea- e.g. by Western 2 * pS_pS servable) sured experimentally blotting Dynamic variables or Derived vari- total_nS = nS Z functions thereof of spe- able + 2 * npS_npS cial interest 7 Introduction JAK/STAT Element Name Description Example model Cell compartments Cytoplasm, C Compartment A distinct reaction space like cytoplasm or Nucleus organs like liver Dynamic pa- A parameter occurring in Rate and dissocia- k k , k , k , … rameter a reaction tion constants 1 2 3 Initial protein con- Start value pa- Value of dynamic vari- X centration before R(t=0), S(t=0) 0 rameter ables at time zero stimulation Scaling param- S eter Table 1. Overview of important model elements.