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Optimal Design of Dynamic Experiments Julio R. Banga IIM-CSIC, Vigo, Spain “The Systems Biology Modelling Cycle (supported by BioPreDyn)” EMBL-EBI (Cambridge, UK), May 2014

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Optimal Experimental Design (OED) Introduction Introduction OED: Why, what and how? OED: Why, what and how? Model building cycle Model building cycle OED to improve model calibration OED to improve model calibration Formulation and examples Formulation and examples OED software and references OED software and references

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Modelling – considerations Models starts with questions (purpose) and level of detail Models starts with questions (purpose) and level of detail We need a priori data & knowledge to build a 1st model We need a priori data & knowledge to build a 1st model We then plan and perform new experiments, obtain new data and refine the model We then plan and perform new experiments, obtain new data and refine the model We repeat until stopping criterion satisfied We repeat until stopping criterion satisfied

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Modelling – considerations We plan and perform new experiments, obtain new data and refine the model We plan and perform new experiments, obtain new data and refine the model But, how do we plan these experiments? But, how do we plan these experiments? Optimal experimental design (OED) Optimal experimental design (OED) Model-based OED Model-based OED

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OED – Why? We want to build a model We want to build a model We want to use the model for specific purposes (“models starts with questions”) We want to use the model for specific purposes (“models starts with questions”) OED allow us to plan experiments that will produce data with rich information content OED allow us to plan experiments that will produce data with rich information content

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OED - What ? We need to take into account: We need to take into account: (i) the purpose of the model, (i) the purpose of the model, (ii) the experimental degrees of freedom and constraints, (ii) the experimental degrees of freedom and constraints, (iii) the objective of the OED: (iii) the objective of the OED: Parameter estimation Parameter estimation Model discrimination Model discrimination Model reduction Model reduction …

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OED – simple example We want to build a 3D model of an object from 2D pictures We want to build a 3D model of an object from 2D pictures

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OED – simple example We want to build a 3D model of an object from 2D pictures We want to build a 3D model of an object from 2D pictures

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OED – simple example We want to build a 3D model of an object from 2D pictures We want to build a 3D model of an object from 2D pictures (i) purpose of the model: a rough 3D representation of the object, (i) purpose of the model: a rough 3D representation of the object, (ii) experimental constraints: we can only take 2D snapshots (ii) experimental constraints: we can only take 2D snapshots (iii) degrees of freedom: pictures from any angle (iii) degrees of freedom: pictures from any angle “Experiment” with minimum number of pictures? “Experiment” with minimum number of pictures?

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OED – simple example

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OED – basic considerations There is a minimum amount of information in the data needed to build a model There is a minimum amount of information in the data needed to build a model This depends on: This depends on: how detailed we want the model to be how detailed we want the model to be how complex the original object (system) is how complex the original object (system) is safe assumptions we can make (e.g. symmetry in 3D object -> less pictures needed) safe assumptions we can make (e.g. symmetry in 3D object -> less pictures needed)

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Static model Dynamic model

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OED for dynamic models We need time-series data with enough information to build a model We need time-series data with enough information to build a model Data from different contexts, with enough time resolution Data from different contexts, with enough time resolution

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Reverse engineering (dynamic) Mario

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Main characteristics: Non-linear, dynamic models ( i.e. batch or semi-batch processes) Nonlinear constraints ( safety and/or quality demands) Distributed systems (T, c, etc.) Coupled transport phenomena Thus, mathematical models consist of sets of ODEs, DAEs, PDAEs, or even IPDAEs, with possible logic conditions (transitions, i.e. hybrid systems) PDAEs models are usually transformed into DAEs (I.e. discretization methods, like FEM, NMOL, etc.) Dynamic process models

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Experiment Data Model Solver Fitted Model Model building

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Experiment Data Model Solver Fitted Model Identifiability Analysis Parameter Estimation Optimal Experimental Design Model building

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Model building cycle OED New experiments New data Model selection and discrimination Parameter estimation Prior information

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Experimental degrees of freedom and constraints Initial conditions Dynamic stimuli: type and number of perturbations Measurements What? When? (sampling times, experiment duration) How many replicates? How many experiments? Etc. Experimental design

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Examples Bacterial growth in batch culture Bacterial growth in batch culture 3-step pathway 3-step pathway Oregonator Oregonator

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Concentration of microorganisms Concentration of growth limiting substrate Example: Example: Bacterial growth in batch culture

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Concentration of microorganisms Concentration of growth limiting substrate Yield coefficient Decay rate coefficient Maximum growth rate Michaelis-Menten constant Example: Example: Bacterial growth in batch culture

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Experimental design: Initial conditions? What to measure? (concentration of microorganisms and substrate?) When to measure? (sampling times, experiment duration) How many experiments? How many replicates? Etc.

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Example: Example: Bacterial growth in batch culture Case A: 1 experiment 11 equidistant sampling times Duration: 10 hours Measurements of S and B

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Example: Example: Bacterial growth in batch culture Parameter estimation using GO method (eSS)

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Example: Example: Bacterial growth in batch culture Case B: 1 experiment 11 equidistant sampling times Duration: 10 hours Measurements of S only

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Example: Example: Bacterial growth in batch culture Good fit for substrate! But bad predictions for bacteria…

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Example: Example: Bacterial growth in batch culture Measuring both B and S…

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Example: Example: Bacterial growth in batch culture Measuring only S…

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Example: Example: Bacterial growth in batch culture So, for this case of 1 experiment, we should measure both B and S But confidence intervals are rather large… What happens if we consider a second experiment? (same experiment, but with different initial condition for S) max : e-001 e-002 (23%); Ks : e+000 e+000 (50%); Kd : e-002 e-002 (177%); Y : e-001 e-001 (34%);

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Example: Example: Bacterial growth in batch culture 1 st experiment2 nd experiment max : e-001 e-002 (9%) Ks : e+000 e-001 (17%) Kd : e-002 e-002 (62%) Y : e-001 e-002 (13%); Great improvement with a second experiment ! BUT, can we do even better?

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Example: Example: Bacterial growth in batch culture 1 st experiment2 nd experiment max : e-001 e-002 (9%) Ks : e+000 e-001 (17%) Kd : e-002 e-002 (62%) Y : e-001 e-002 (13%); Great improvement with a second experiment ! BUT, can we do even better? OPTIMAL EXPERIMENTAL DESIGN

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Example: simple biochemical pathway C.G. Moles, P. Mendes y J.R. Banga, Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Research., 13:

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Kinetics described by set of 8 ODEs with 36 parameters

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Parameter estimation: 36 parameters measurements: concentrations of 8 species 16 experiments (different values of S y P) Example: simple biochemical pathway

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Initial conditions for all the experiments Experiments (S, P values) 21 measurements per experiment, t f = 120 s

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Example: simple biochemical pathway Multi-start local methods fail… Multi-start SQP

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Example: simple biochemical pathway Parameter estimation: again (some) global methods can fail too…

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Example: simple biochemical pathway Parameter estimation: best fit looks pretty good but…

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Contours [p1, p6] Contours [p1, p4] Example: simple biochemical pathway Identifiability problems…

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Practical identifiability problems are often due to data with poor information content Need: more informative experiments (data sets) Solution: optimal design of (dynamic) experiments

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What about identifiability? I.e. can the parameters be estimated in a unique way? Identifiability: A.Global a priori (theoretical, structural) B.Local a priori (local) C.Local a posteriori (practical) (A) is hard to evaluate for realistic nonlinear models (B) and (C) can be estimated via the FIM and other indexes... (C) takes into account noise etc.

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Parametric sensitivities Fisher information matrix (FIM)

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Checking identifiability and other indexes… Compute sensitivities (direct decoupled method) : Build FIM, covariance and correlation matrices Analyse possible correlations among parameters

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Sensitivities w.r.t. p1 and p6 are highly correlated (i.e. The system exhibits rather similar responses to changes in p1 and p6 for the given experimental design)

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p1 & p6 p1 & p4

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Checking identifiability and other indexes… Compute sensitivities (direct decoupled method) : Build FIM, covariance and correlation matrices Analyse possible correlations among parameters Compute confidence intervals Check FIM-based criterions (practical identifiability) Singular FIM: unidentifiable parameters, non-informative experiments Large condition number of FIM means lower practical identifiability Rank parameters v Finally, use OED to improve experimental design

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Model structure & parameters Parametric sensitivities Ranking of parameters Practical Identifiability analysis Optimal experimental design (New) Experiments Model calibration

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Balsa-Canto, E., Alonso, A. A., & Banga, J. R. (2010). An iterative identification procedure for dynamic modeling of biochemical networks. BMC Systems Biology 4:11

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Optimal (dynamic) experimental design Design the most informative experiments, facilitating parameter estimation and improving identifiability How? Define information criterion Optimize it modulating experimental conditions “If you want to truly understand something, try to change it.” Kurt Lewin, circa 1951

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Optimal (dynamic) experimental design Computational approaches that are applicable to support the optimal design of experiments in terms of how to manipulate the degrees of freedom (controls) of experiments, what variables to measure, why to measure them, when to take measurements. Experiment Data

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Optimal (dynamic) experimental design Information content measured with the FIM We will use scalar functions of the FIM (“alphabetical” criteria) Find experiments which maximize information content

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Some FIM-based Criterions... D-criterion (determinant of F ), which measures the global accuracy of the estimated parameters E-criterion (smallest eigenvalue of F ), which measures largest error Modified E-criterion (condition number of F ), which measures the parameter decorrelation A-criterion (trace of inverse of F ), which measures the arithmetic mean of estimation error

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A criterion = D criterion = E criterion = Modified-E criterion =

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E criterion = E-optimality: max the min eigenvalue of FIM (minimizes the largest error)

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Modified-E criterion = Maximize decorrelation between parameters (make contours as circular as possible)

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Calculate the dynamic scheme of measurements so as to generate the maximum amount and quality of information for model calibration purposes. OED as a dynamic optimization problem When to measure? (Optimal sampling times) Which type of dynamic stimuli?

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Calculate time-varying control profiles (u(t)), sampling times, experiment duration and initial conditions (v) to optimize a performance index (scalar measure of the FIM): System dynamics (ODEs, PDEs): Experimental constraints: OED as a dynamic optimization problem

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Back to example: Back to example: Bacterial growth in batch culture Experimental design: Initial conditions? What to measure? (concentration of microorganisms and substrate?) When to measure? (sampling times, experiment duration) How many experiments? How many replicates? Etc.

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Example: Example: Bacterial growth in batch culture 1 st experiment2 nd experiment max : e-001 e-002 (9%) Ks : e+000 e-001 (17%) Kd : e-002 e-002 (62%) Y : e-001 e-002 (13%); Great improvement with a second experiment ! BUT, can we do even better? OPTIMAL EXPERIMENTAL DESIGN

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Example: Example: Bacterial growth in batch culture Let us design the second experiment in an optimal way: Criteria: E-optimality (minimize the largest error) Degress of freedom we can ‘manipulate’ in the second experiment: Initial concentrations of S and B Duration of experiment

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Example: Example: Bacterial growth in batch culture OED of second experiment 1 st experiment2 nd experiment after OED max : e-001 e-002 (4.3%) Ks : e+000 e-001 (6%) Kd : e-002 e-003 (6%) Y : e-001 e-002 (2.8%); free initial conditions cb0:[1,5] cs: [5 40], free experiment duration [6,15] h, ns=11

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Example: Example: Bacterial growth in batch culture Two arbitrary experiments After OED of second experiment max : e-001 e-002 (4.3%) Ks : e+000 e-001 (6%) Kd : e-002 e-003 (6%) Y : e-001 e-002 (2.8%); free initial conditions cb0:[1,5] cs: [5 40], free experiment duration [6,15] h, ns=11 max : e-001 e-002 (9%) Ks : e+000 e-001 (17%) Kd : e-002 e-002 (62%) Y : e-001 e-002 (13%);

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Example: Example: Bacterial growth in batch culture Correlation matrix after OED of second experiment free initial conditions cb0:[1,5] cs: [5 40], free experiment duration [6,15] h, ns=11

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Example: Example: Bacterial growth in batch culture After OED of second experiment… free initial conditions cb0:[1,5] cs: [5 40], free experiment duration [6,15] h, ns=11 > Correlation between parameters has substantially improved (in general) > Kd and Y are still highly correlated but the size of the confidence ellipse is much smaller

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Example: Example: Bacterial growth in batch culture 5% 7% 6% 3% max : e-001 e-002 (4.3%) Ks : e+000 e-001 (6%) Kd : e-002 e-003 (6%) Y : e-001 e-002 (2.8%); Robust confidence intervals are similar to those obtained by the FIM

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Example: OED for the simple biochemical pathway Moles, C. G., Pedro Mendes and Julio R. Banga (2003) Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Research 13(11):

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Original experimental design: 16 experiments (different S and P values) Result: large E-criterion and modified E-criterion

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Some FIM-based criterions for the original design... Very large modified E-criterion indicates large correlation among (some) parameters, making the identification of the system hard Can we improve this by an alternative (optimal) design of experiments?

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Improve experimental design by solving OED Find the values of S and P for a set of N exp experiments which e.g. maximize E-criterion s.t. constraints (dynamics plus bounds)

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Improved experimental design: 16 experiments with optimal S and P E-criterion improved (> one order of magnitude) Other criteria also improved

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Improved design by solving OED problem:

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Original Design (E-crit= 60, logD=161)Improved Design (E-crit= 320, logD=181)

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Example: OED for Oregonator reaction The Oregonator is the simplest realistic model of the chemical dynamics of the oscillatory Belousov-Zhabotinsky (BZ) reaction (Zhabotinsky, 1991; Gray and Scott, 1991; Epstein and Pojman, 1998)

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Oregonator reaction: highly nonlinear, oscillatory kinetics

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Oregonator reaction: identifiability problems Villaverde, A., J. Ross, F. Morán, E. Balsa-Canto, J.R. Banga (2011) Use of a Generalized Fisher Equation for Global Optimization in Chemical Kinetics.Journal of Physical Chemistry A115(30):

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Oregonator reaction: OED E-optimality criterion

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Example: OED for Oregonator reaction E-optimality criterion: improved 3 orders of magnitude Villaverde, A., J. Ross, F. Morán, E. Balsa-Canto, J.R. Banga (2011) Use of a Generalized Fisher Equation for Global Optimization in Chemical Kinetics.Journal of Physical Chemistry A115(30):

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Example: OED for Oregonator reaction

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OED conclusions Currently, most experiments are designed based on intuition of experimentalists and modellers Currently, most experiments are designed based on intuition of experimentalists and modellers Model-based OED can be used to: Model-based OED can be used to: Improve model calibration Improve model calibration Discriminate between rival models Discriminate between rival models OED is a systematic and optimal approach OED is a systematic and optimal approach OED can take into account practical limitations and constraints by incorporating them into the formulation OED can take into account practical limitations and constraints by incorporating them into the formulation

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Check identifiability Use proper optimization methods for parameter estimation Use optimal experimental design Main tips for dynamic model building Take-home messages

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“All models are wrong, but some are useful” --- Statistician George E. P. Box Main tips for dynamic model building

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“All models are wrong, but some are useful” --- Statistician George E. P. Box The practical question is: How wrong do they have to be to not be useful? Main tips for dynamic model building

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Software for dynamic model building and OED

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A few selected references… Ashyraliyev M, Fomekong-Nanfack Y, Kaandorp JA & Blom JG (2009a). Systems biology: parameter estimation for biochemical models. FEBS J 276: 886–902. Ashyraliyev M, Fomekong-Nanfack Y, Kaandorp JA & Blom JG (2009a). Systems biology: parameter estimation for biochemical models. FEBS J 276: 886–902. Balsa-Canto, E. and Julio R. Banga (2011) AMIGO, a toolbox for Advanced Model Identification in systems biology using Global Optimization. Bioinformatics 27(16): Balsa-Canto, E. and Julio R. Banga (2011) AMIGO, a toolbox for Advanced Model Identification in systems biology using Global Optimization. Bioinformatics 27(16): Balsa-Canto, E., Alonso, A. A., & Banga, J. R. (2010). An iterative identification procedure for dynamic modeling of biochemical networks. BMC Systems Biology 4:11. Balsa-Canto, E., Alonso, A. A., & Banga, J. R. (2010). An iterative identification procedure for dynamic modeling of biochemical networks. BMC Systems Biology 4:11. Bandara, S., Schlöder, J. P., Eils, R., Bock, H. G., & Meyer, T. (2009). Optimal experimental design for parameter estimation of a cell signaling model. PLoS computational biology, 5(11), e Bandara, S., Schlöder, J. P., Eils, R., Bock, H. G., & Meyer, T. (2009). Optimal experimental design for parameter estimation of a cell signaling model. PLoS computational biology, 5(11), e Banga, J.R. and E. Balsa-Canto (2008) Parameter estimation and optimal experimental design. Essays in Biochemistry 45:195–210. Banga, J.R. and E. Balsa-Canto (2008) Parameter estimation and optimal experimental design. Essays in Biochemistry 45:195–210. Balsa-Canto, E., A.A. Alonso and J.R. Banga (2008) Computational Procedures for Optimal Experimental Design in Biological Systems. IET Systems Biology 2(4): Balsa-Canto, E., A.A. Alonso and J.R. Banga (2008) Computational Procedures for Optimal Experimental Design in Biological Systems. IET Systems Biology 2(4): Chen BH, Asprey SP (2003) On the Design of Optimally Informative Dynamic Experiments for Model Discrimination in Multiresponse Nonlinear Situations. Ind Eng Chem Res 2003, 42: Chen BH, Asprey SP (2003) On the Design of Optimally Informative Dynamic Experiments for Model Discrimination in Multiresponse Nonlinear Situations. Ind Eng Chem Res 2003, 42: Jaqaman K., Danuser G. Linking data to models: data regression. Nat. Rev. Mol. Cell Bio.7: Jaqaman K., Danuser G. Linking data to models: data regression. Nat. Rev. Mol. Cell Bio.7: Kremling A, Saez-Rodriguez J: Systems Biology - An engineering perspective. J Biotechnol 2007, 129: Kremling A, Saez-Rodriguez J: Systems Biology - An engineering perspective. J Biotechnol 2007, 129: Mélykúti, B., E. August, A. Papachristodoulou and H. El-Samad (2010) Discriminating between rival biochemical network models: three approaches to optimal experiment design. BMC Systems Biology 4:38. Mélykúti, B., E. August, A. Papachristodoulou and H. El-Samad (2010) Discriminating between rival biochemical network models: three approaches to optimal experiment design. BMC Systems Biology 4:38. van Riel N (2006) Dynamic modelling and analysis of biochemical networks: Mechanism-based models and model-based experiments. Brief Bioinform 7(4): van Riel N (2006) Dynamic modelling and analysis of biochemical networks: Mechanism-based models and model-based experiments. Brief Bioinform 7(4): Villaverde, A.F. and J.R. Banga (2014) Reverse engineering and identification in systems biology: strategies, perspectives and challenges. J. Royal Soc. Interface 11(91): Villaverde, A.F. and J.R. Banga (2014) Reverse engineering and identification in systems biology: strategies, perspectives and challenges. J. Royal Soc. Interface 11(91): (review papers in yellow) (review papers in yellow)

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