1. Andrzej Mizera Institute of Fundamental Technological Research Polish Academy of Sciences amizera@ippt.gov.pl November 22, 2011 University of Luxembourg 1 Methods for Construction and Analysis of Computational Models in Systems Biology

2. Outline November 22, 2011 University of Luxembourg 2 Introduction: Systems Biology Part I: Model construction Parameter Estimation Model validation Model identifiability Part II: Higher-level model analysis methodologies Quantitative submodel comparison Techniques for model modifications (extensions & simplifications) model refinement of ODE-based self-assembly models Summary

3. Introduction November 22, 2011 University of Luxembourg 3 Biological processes have long been seen as static systems comprising a vast number of loosely linked, highly detailed, molecular devices. http://micro.magnet.fsu.edu/cells/animals/animalmodel.html Systems biology essentially advocates a departure from the reductionist viewpoint, while putting emphasis on the holistic approach towards the analysis of a biological system. ATP synthase — a marvellous rotary engine of the cell Nature Reviews Molecular Cell Biology 2, 669-677 (September 2001) Molecular Cell Biology. 4th edition. Lodish H, Berk A, Zipursky SL, et al. New York: W. H. Freeman; 2000. Action of muscle Ca2+ ATPase

4. Introduction November 22, 2011 University of Luxembourg 4 Systems biology: emerging research field, aims to study biochemical and biological systems from a holistic perspective, with the goal to provide a comprehensive, system-level understanding of cellular behaviour. “... the objective of systems biology is defined as the understanding of network behaviour, and in particular their dynamic aspects, which requires the utilization of mathematical modelling tightly linked to experiment.” - WTEC Panel Report on International Research and Development in Systems Biology, 2005 -

5. Introduction November 22, 2011 University of Luxembourg 5 Systems biology, unlike “traditional” biology, focuses on high-level concepts. The very terminology of systems biology is “foreign” to “traditional” biology and it marks its drastic shift in research paradigm. network component robustness Systems biology efficiency control regulation hierarchical design signalling synchronisation parallelism competition

6. Introduction November 22, 2011 University of Luxembourg 6 The key concepts in systems biology have already been studied for a long time in computer science (albeit from different perspectives). A key contribution that computer science brings to systems biology is the ability to manipulate, analyse, and reason about such system-level concepts and structures. For example, mathematical modelling, formal system specifications, control design, and others are by now mainstream techniques in systems biology.

7. Introduction November 22, 2011 University of Luxembourg 7 My research concerns a number of challenges of computational modelling in Systems Biology. It is focused on the development and utilization of different methodologies having their origins in the fields of computer science and mathematics. The presented methodologies are applied in the modelling of two biological processes chosen as modelling case studies: 1) the heat shock response mechanism in eukaryotic cells (HSR) 2) the in vitro self-assembly of intermediate filaments from tetrameric vimentin. The developed methodologies are general in nature: can be applied for the modelling of other biological processes as well.

8. November 22, 2011 University of Luxembourg 8 PART I: Model construction

9. The heat shock response November 22, 2011 University of Luxembourg 9 Cell’s response to elevated temperatures Intense research on HSR in the last years HSR is a very well-conserved regulatory network across all eukaryotes (bacteria have a similar mechanism) Good candidate for deciphering the engineering principles of regulatory networks Heat shock proteins are very potent chaperones (sometimes called the “master proteins” of the cell) Involved in a large number of regulatory processes Also in anti-inflammatory processes Found in extra-cellular environment, which may suggest they are used for signaling Major role in the resilience of cancer cells; attractive as targets for cancer treatment Tempting for a biomodelling, SysBio project because it involves relatively few main actors (at least in a first, simplified presentation)

10. Heat shock response: main actors November 22, 2011 University of Luxembourg 10 Heat shock proteins (HSP) Very potent chaperones Main task: assist the refolding of misfolded proteins Several types of them, we treat them all uniformly in our model with hsp70 as a base denominator Heat shock elements (HSE) Several copies found upstream of the HSP-encoding gene, used for the transactivation of the HSP-encoding genes Treat uniformly all HSEs of all HSP-encoding genes Heat shock factors (HSF) Proteins acting as transcription factors for the HSP-encoding gene Trimerize, then bind to HSE to promote gene transcription Generic proteins Consider them in two states: correctly folded and misfolded Under elevated temperatures, proteins tend to misfold, exhibit their hydrophobic cores, form aggregates, lead eventually to cell death (see Alzheimer, vCJ, and other diseases) Various bonds between these metabolites

11. The molecular model for HSR November 22, 2011 University of Luxembourg 11 Heat shock geneHSEHeat shock geneHSE HSP HSF RNA pol HSP:HSF MFP 37  C 42  C

12. The new molecular model November 22, 2011 University of Luxembourg 12 Transcription 1. HSF+HSF  HSF 2 2. HSF+HSF 2  HSF 3 3. HSF 3 +HSE  HSF 3 :HSE 4. HSF 3 :HSE  HSF 3 :HSE+HSP Backregulation 5. HSP+HSF  HSP:HSF 6. HSP+HSF 2  HSP:HSF+HSF 7. HSP+HSF 3  HSP:HSF+2HSF 8. HSP+HSF 3 :HSE  HSP:HSF+2HSF+HSE Response to stress 9. PROT  MFP 10. HSP+MFP  HSP:MFP 11. HSP:MFP  HSP+PROT Protein degradation 12. HSP  

13. The new molecular model November 22, 2011 University of Luxembourg 13 Several criteria were followed when introducing this molecular model: 1) as few reactions and reactants as possible; 2) include the temperature-induced protein misfolding; 3) include HSF in all its three forms: monomers, dimers, and trimers; 4) include the HSP-backregulation of the transactivation of the HSP-encoding gene; 5) include the chaperon activity of HSP; 6) include only well-documented, textbook-like reactions and reactants.

14. The Law of Mass Action November 22, 2011 University of Luxembourg 14 Introduced by Guldberg and Waage in 1864: Waage, P.; Guldberg, C. M. Forhandlinger: Videnskabs-Selskabet i Christiana 1864, 35 The rate of any given reaction is proportional to the concentration of reactants. For example: S A + S B  P In modern chemistry derived using statistical mechanics

15. The mathematical model November 22, 2011 University of Luxembourg 15

16. The mathematical model November 22, 2011 University of Luxembourg 16 Modelling of the heat-induced misfolding Adapted from Pepper et al (1997), based on studies of Lepock (1989, 1992) on differential calorimetry  (T)=(1-0.4/e T-37 )  14.5  10 -7  1.4 T-37 s -1 Formula valid for temperatures between 37 °C and 45 °C, gives a generic protein misfolding rate per second.

17. The mathematical model November 22, 2011 University of Luxembourg 17 Our model contains 3 mass-conservation relations: The total number of HSF molecules is constant, The total number of heat shock elements (HSE) is constant, and the total number of Protein molecules (PROT) is conserved.

18. Parameter estimation November 22, 2011 University of Luxembourg 18 Data readily available for the goal: Kline, Morimoto (1997) – heat shock of HeLa cells at 42 °C for up to 4 hours, data on DNA binding (HSF 3 :HSE) Requirements for the model: 17 kinetic constants, 10 initial values to estimate, but: 3 conservation relations available; the initial value of the variables of the model is a steady state for temperature set to 37 °C, which gives 7 more independent algebraic equations (each of them quadratic). Altogether: 17 independent values Other conditions: total HSF somewhat low, refolding a fast reaction, HSPs long-lived proteins

19. Parameter estimation November 22, 2011 University of Luxembourg 19 Parameter estimation performed with (www.copasi.org) Hoops, S., Sahle, S., Gauges, R., Lee, C., Pahle, J., Simus, N., Singhal, M., Xu, L., Mendes, P., and Kummer, U. (2006). COPASI — a COmplex PAthway SImulator. Bioinformatics 22, 3067-74. User-friendly Stochastic and deterministic time course simulation Steady state analysis Sensitivity analysis Metabolic control analysis Mass conservation analysis Optimization of arbitrary objective functions SBML-based Excellent for parameter estimation FREE for non-commercial use

20. Strategy for parameter estimation November 22, 2011 University of Luxembourg 20 Ideal approach: Solve analytically the steady state equations at 37 °C. Use the solution to decrease the number of parameters and initial values to the independent ones. Do parameter estimation on the remaining independent variables to fit the model based on the data at 42 °C. Problem: the steady state (37 °C) equations cannot be solved because of they high overall degree. Solution: use the Kline-Morimoto experimental data to fit parameters and ask also that the fluctuations at 37°C are (close to) 0. Duplicate the model (the same parameter values!) and run both at the same time (37 & 42 °C).

21. Parameter estimation results November 22, 2011 University of Luxembourg 21

22. Predictions and validation November 22, 2011 University of Luxembourg 22 Higher the temperature, higher the response Prolonged transcription at 43 °C confirmed Unlike previous models Heat shock removed at the peak of the response confirms a more rapid attenuation phase All data is in relative terms with respect to the highest value in the graph so that it can be easily compared

23. Predictions and validation November 22, 2011 University of Luxembourg 23 Experiment: two waves of heat shock, the second applied after the level of HSP has peaked Observation: the second heat shock response much milder than the first The reason is that the cell is better prepared to deal with the second heat shock Therapeutic consequences have been suggested: “train” the cell for heat shock by an initial milder heat shock The model prediction is in line with the experimental observation Dotted line: heat shock at 42 °C for two hours, behavior followed up to 20 hours Continuous line: heat shock at 42 °C for two hours, followed by a second wave of heat shock after the level of HSP has peaked

24. Model identifiability November 22, 2011 University of Luxembourg 24 The problem of model identifiability: the question of the uniqueness of the set of parameters that fulfil the imposed conditions. By repeating from scratch the whole parameter estimation procedure, we obtained several different sets of parameter values that result both in a good fit of the model to the experimental data, as well as in initial values that are steady-states of the model at 37 °C. However, all these parameter sets failed the model validation tests with respect to the qualitative observations concerning the behaviour of cells under stress. A new thorough method of searching for alternative numerical model fits based on systematic parameter scan in the space determined by the considered ranges of parameter values Latin Hypercube Sampling - provides samples which are uniformly distributed over each parameter while the number of samples is independent of the number of parameters

25. Model identifiability November 22, 2011 University of Luxembourg 25 Strategy: look for alternative models fits that are in agreement with the experimental data of Kline and Morimoto (1997), and satisfy the steady-state condition for the initial values. Sampled N = 100.000 sets of parameter values For each set, we estimated numerically the steady state of the model for a temperature value of 37 °C. We then set the initial state of the model as the calculated steady state. Simulated the model for 14400s at a temperature of 42 °C. Classified as non-responsive those parameter samples that led to low DNA binding level at the peak of the response, and excluded them from further analysis. Only 31.506 out of the 100.000 samples were responsive, already a result pointing to difficulties in finding satisfactory alternative numerical fits. For each model, we made a scatter plot for each variable and each parameter where we plotted the steady state values of each variable at 37 °C, against the values of the parameter.

26. Model identifiability November 22, 2011 University of Luxembourg 26 37 °C : most of the alternative fits predicted high levels of gene transcription in the absence of the heat shock none of the sampled models reached such a low level of HSF as the reference model the reference model is one of the very few models in which most of the HSF molecules are sequestered by HSP, i.e. at 37°C the response mechanism is turned off the basic model reaches the lowest values for HSF dimers – in agreement with the observation that HSF dimers are unnoticeable in biological experiments

27. Model identifiability November 22, 2011 University of Luxembourg 27 Our reference model obtained the lowest score of around 12, while the 13 best fits of the sampled models were in the range between 300 and 1000. All the other models had much worse scores, of more than 1000. Conclusions While it is likely that a model of this size is not uniquely identifiable, our parameter scan showed that finding parameter values satisfying our model constraints is far from being easy. Conclusions While it is likely that a model of this size is not uniquely identifiable, our parameter scan showed that finding parameter values satisfying our model constraints is far from being easy. 42 °C :

28. November 22, 2011 University of Luxembourg 28 PART II: Higher-level model analysis methodologies

29. Model decomposition and quantitative submodel comparison November 22, 2011 University of Luxembourg 29 Various experimental investigations of a given biochemical system often lead to generation of a large variety of alternative molecular designs and models. How to compare their functionality, efficiency, and robustness? Comparing alternative models for a given biochemical system is, in general, a very difficult problem: the models may focus on different aspects of the same system, may consist of very different species and reactions, numerical setups of the associated computational models play a crucial role in the quantitative comparison. It involves a deep analysis of the underlying network of reactions, the biological assumptions as well as the numerical setup. Unbiased methods to objectively compare the alternative designs are needed.

30. Model decomposition and quantitative submodel comparison November 22, 2011 University of Luxembourg 30 The problem becomes somewhat simpler when the alternative designs are actually submodels of a larger model: the underlying networks are similar, although not identical, and the biological constrains are given by the larger model. E.g., submodels are considered when striving to obtain a global picture of a large system’s architecture, i.e. understanding the interactions between various components. Similar problems have been encountered in engineering sciences, e.g., in control theory. One strategy is to adapt specific methods originating from these disciplines.

31. Model decomposition and quantitative submodel comparison November 22, 2011 University of Luxembourg 31 Three methods were developed: 1. Local submodels comparison El. Czeizler, Eu. Czeizler, R.-J. Back, and I. Petre. Control strategies for the regulation of the eukaryotic heat shock response. In P. Degano and R. Gorrieri (Eds.), Computational Methods in Systems Biology, LNCS, vol. 5688, pp. 111-125, Heidelberg, 2009. Springer-Verlag. 2. Discrete approach for comparing continuous submodels El. Czeizler, A. Mizera, and I. Petre. A Boolean approach for disentangling the numerical contribution of modules to the system-level behavior of a biomodel. Submitted, 2011. 3. A statistical method for quantitative submodel comparison A. Mizera, El. Czeizler, and I. Petre. Methods for biochemical model decomposition and quantitative submodel comparison. Israel Journal of Chemistry, 51(1):151-164, 2011.

32. Model decomposition and quantitative submodel comparison November 22, 2011 University of Luxembourg 32 These methods are: using the control-based decomposition of the reference model, comparing knockdown mutants of the reference model, i.e., submodels missing one or several of the modules, illustrated on the eukaryotic heat shock response case study.

33. Model decomposition November 22, 2011 University of Luxembourg 33

34. Model decomposition November 22, 2011 University of Luxembourg 34

35. Local submodels comparison November 22, 2011 University of Luxembourg 35 Aim: biologically unbiased analysis, i.e., elimination of accidental differences between models. Question: initial conditions in the alternative models? Taking them from the reference model  biased comparison Local submodel comparison: initial setup of each submodel constitutes a steady state in the absence of a trigger  unbiased comparison The method: Two initial biological constraints are imposed on the models: identical chemistry (reaction rates) and mass constants (conservation relations) initial values form a steady state under physiological conditions

36. Local submodels comparison November 22, 2011 University of Luxembourg 36 Comparison of the mutants Based on the numerical observations one can summarize the contribution of the modules to some performance indicators of the network. E.g., in the case of the HSR model these are: economical use of the cellular resources, speed of response to a heat shock, effectiveness of the response, scalability.

37. Discrete approach for comparing continuous submodels November 22, 2011 University of Luxembourg 37 Three conditions are imposed on the each knockdown mutant model: 1. kinetic rate constants: the numerical prediction of the mutant model fits in with the experimental data 2. initial conditions: form a steady state under physiological conditions 3. mass constants: are chosen to be identical to those of the reference model All three constraints come as natural consequences of the fact that we consider all knockdown mutants as viable alternatives.

38. Discrete approach for comparing continuous submodels November 22, 2011 University of Luxembourg 38 For each knockdown mutant a Boolean formula is associated that describes the mutant’s control architecture (negation and conjunction of variables associated to regulating mechanisms). Boolean formulas describing all mutants that show given behavioural properties are written. Example: which knockdown mutants can be both effective and economic? The numerical comparison of the mutants is then performed by analysing the Boolean formulas associated with various behavioural properties.

39. Statistical method for quantitative submodel comparison November 22, 2011 University of Luxembourg 39 Statistical sampling of the reference model and mutant behaviour is performed: the parameter value space of the reference model is scanned (e.g. Latin Hypercube Sampling) each coordinate of the sampled parameter vectors is associated with one of the parameters in the reference model For each sampled parameter vector: parameters of the reference model and the submodel are set in accordance with the considered vector, initial conditions of the reference model and the submodel are determined independently of each other by a systemic property (e.g. being in a steady state in a given setup).

40. Statistical method for quantitative submodel comparison November 22, 2011 University of Luxembourg 40 For each sampled parameter vector: one numerical instance for the mutant and one for the reference model numerical simulations are run for both of them in order to evaluate their functional effectiveness The obtained results for each variant are summarized by use of some statistical measures (e.g. the moving median technique).

41. November 22, 2011 University of Luxembourg 41 Refinement of ODE-based self-assembly models

42. November 22, 2011 University of Luxembourg 42 Challenge: construction of models capable of capturing the evolution of filaments of certain length up to some arbitrarily chosen value. R. Kirmse et al. A Quantitative Kinetic Model for the in Vitro Assembly of Intermediate Filaments from Tetrameric Vimentin. Journal of Biological Chemistry, 282(25):18563-18572, 2007. monomer coiled-coil dimer tetramer ULF simple model extended model

43. Refinement of ODE-based self-assembly models November 22, 2011 University of Luxembourg 43 The proposed methodology of self-assembly model refinement is a particular instance of formal model refinement – a topic extensively studied in Computer Science, especially in connection to formal software specifications. Model refinement: starting from an abstract model of a system, a more detailed model is constructed the refinement mechanism preserves already proven systemic quantitative properties of the original model (e.g. model fit, stochastic semantics)

44. Refinement of ODE-based self-assembly models November 22, 2011 University of Luxembourg 44 Our method: an instance of data refinement, where one replaces a variable with a set of other variables in a way that introduces more details into the model, while keeping the model constraints unchanged. A. Mizera, Eu. Czeizler, and I. Petre. Self-assembly models of variable resolution. To appear in Transactions on Computational Systems Biology, 2012. a generic formal model for the process of self-assembly is presented the notion of model resolution is introduced – a self-assembly mathematical model is of resolution n if it allows for capturing the dynamics of the number (or concentration) of components that are exactly of size i, where 0 ≤ i ≤ n. model refinement procedures preserving the fit to experimental data for a family of self- assembly ODE models are developed to the best of our knowledge, this is the first time formal refinement is considered in the context of ODE-based mathematical models

45. Refinement of ODE-based self-assembly models November 22, 2011 University of Luxembourg 45 increasing the model resolution: an exact, constructive method based on analytical investigations of the ODEs is developed decreasing the model resolution is more challenging: method based on symbolical computations but requires numerical investigations and simulations of the models case study: the in vitro self-assembly of intermediate filaments

46. November 22, 2011 University of Luxembourg 46 Summary

47. November 22, 2011 University of Luxembourg 47 Issues related to model construction methodologies: parameter estimation, model validation, model identifiability, choice of deterministic vs stochastic modelling framework Discuss and develop new techniques for the problem of model comparison. methodologies for model decomposition special case: comparison between submodels of a certain model The problem of model modifications: various techniques useful for applying simplifications or extensions to an already fitted and validated mathematical model in such a way that the desired properties of the model are retained (fitting experimental data is computationally expensive!). computational heuristics for simplifying a biological model model refinement

48. Summary November 22, 2011 University of Luxembourg 48 prof. Ion Petre Computational Biomodelling Laboratory Åbo Akademi University, Department of Information Technologies Turku Centre for Computer Science prof. Barbara Gambin Institute of Fundamental Technological Research Polish Academy of Sciences