Optimal Design of Dynamic Experiments Julio R. Banga IIM-CSIC, Vigo, Spain julio@iim.csic.es “The...

Optimal Design of Dynamic Experiments

Julio R. Banga

IIM-CSIC, Vigo, Spain

julio@iim.csic.es

“The Systems Biology Modelling Cycle (supported by BioPreDyn)” EMBL-EBI (Cambridge, UK), 12-15 May 2014

Optimal Experimental Design (OED)

Introduction OED: Why, what and how? Model building cycle OED to improve model calibration

Formulation and examples OED software and references

Modelling – considerations

Models starts with questions (purpose) and level of detail

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 repeat until stopping criterion satisfied

Modelling – considerations

We plan and perform new experiments, obtain new data and refine the model

But, how do we plan these experiments?

Optimal experimental design (OED) Model-based OED

OED – Why?

We want to build a model 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 - What ?

We need to take into account: (i) the purpose of the model, (ii) the experimental degrees of freedom

and constraints, (iii) the objective of the OED:

Parameter estimation Model discrimination Model reduction …

OED – simple example

We want to build a 3D model of an object from 2D pictures

OED – simple example

We want to build a 3D model of an object from 2D pictures

OED – simple example

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, (ii) experimental constraints: we can only take

2D snapshots (iii) degrees of freedom: pictures from any

angle “Experiment” with minimum number of

pictures?

OED – simple example

OED – basic considerations

There is a minimum amount of information in the data needed to build a model

This depends on: how detailed we want the model to be how complex the original object

(system) is safe assumptions we can make (e.g.

symmetry in 3D object -> less pictures needed)

Static model

Dynamic model

OED for dynamic models

We need time-series data with enough information to build a model

Data from different contexts, with enough time resolution

Reverse engineering (dynamic) Mario

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

ExperimentData

Model

Solver

Fitted Model

Model building

ExperimentData

Model

Solver

Fitted ModelIdentifiability Analysis

Identifiability Analysis

Parameter Estimation

Optimal Experimental Design

Model building

Model building cycle

OED

New experimen

ts

New data

Model selection

and discriminatio

n

Parameter estimation

Prior informatio

n

Experimental degrees of freedom and constraints

Initial conditionsDynamic stimuli: type and number of perturbations

Measurements

What? When? (sampling times, experiment duration)How many replicates?

How many experiments?

Etc.

Experimental design

Examples

Bacterial growth in batch culture

3-step pathway

Oregonator

Concentration of microorganisms

Concentration of growth limiting substrate

Example: Bacterial growth in batch culture

Concentration of microorganisms

Concentration of growth limiting substrate

Yield coefficient

Decay rate coefficient

Maximum growth rate

Michaelis-Menten constant

Example: Bacterial growth in batch culture

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.

Example: Bacterial growth in batch culture

Case A: 1 experiment11 equidistant sampling timesDuration: 10 hoursMeasurements of S and B

0 5 102

4

6

8

10

12

14

Time

obsB

0 5 10

0

5

10

15

20

25

30

Time

obsS

Example: Bacterial growth in batch culture

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time

obsB

fit cb

data3

data cs

Parameter estimation using GO method (eSS)

Example: Bacterial growth in batch culture

Case B: 1 experiment11 equidistant sampling timesDuration: 10 hoursMeasurements of S only

0 1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

30

Time

obsS

Example: Bacterial growth in batch culture

0 1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

30

Time

obsS

0 1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

Time

cb

Predicted

Real

Good fit for substrate! But bad predictions for bacteria…

Example: Bacterial growth in batch culture

kd vsks

ks

kd

3 4 5 6 7 8 9 10

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

kd vsmumax

mumax

kd

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1ks vsmumax

mumax

ks

0.2 0.3 0.4 0.5 0.6 0.7 0.8

3

4

5

6

7

8

9

10

yield vskd

kd

yiel

d

0.04 0.06 0.08 0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1yield vsks

ks

yie

ld

3 4 5 6 7 8 9 10

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

yield vsmumax

mumax

yie

ld

0.2 0.4 0.6 0.8

0.4

0.6

0.8

1

Measuring both B and S…

Example: Bacterial growth in batch culture

kd vsks

ks

kd

3 4 5 6 7 8 9 10

0.04

0.06

0.08

0.1kd vsmumax

mumax

kd

0.2 0.4 0.6 0.8

0.04

0.06

0.08

0.1ks vsmumax

mumax

ks

0.2 0.4 0.6 0.8

4

6

8

10

yield vskd

kd

yiel

d

0.04 0.06 0.08 0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

yield vsks

ks

yie

ld

3 4 5 6 7 8 9 10

0.3

0.4

0.5

0.6

0.7

0.8

0.9

yield vsmumax

mumax

yie

ld

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Measuring only S…

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)

mmax : 4.0940e-001 9.4155e-002 (23%);Ks : 6.6525e+000 3.3475e+000 (50%);Kd : 3.9513e-002 7.0150e-002 (177%);Y : 4.8276e-001 1.6667e-001 (34%);

Example: Bacterial growth in batch culture

0 2 4 6 8 10

2

4

6

8

10

12

14

Time

obsB

0 2 4 6 8 10

5

10

15

20

25

30

Time

obsS

0 2 4 6 8 102

3

4

5

6

7

8

Time

obsB

0 2 4 6 8 100

5

10

15

Time

obsS

1st experiment 2nd experiment

mmax : 3.9542e-001 3.4730e-002 (9%)Ks : 5.3551e+000 9.1440e-001 (17%)Kd : 4.1657e-002 2.5753e-002 (62%)Y : 4.8529e-001 6.1227e-002 (13%);

Great improvement with a second experiment !BUT, can we do even better?

Example: Bacterial growth in batch culture

0 2 4 6 8 10

2

4

6

8

10

12

14

Time

obsB

0 2 4 6 8 10

5

10

15

20

25

30

Time

obsS

0 2 4 6 8 102

3

4

5

6

7

8

Time

obsB

0 2 4 6 8 100

5

10

15

Time

obsS

1st experiment 2nd experiment

mmax : 3.9542e-001 3.4730e-002 (9%)Ks : 5.3551e+000 9.1440e-001 (17%)Kd : 4.1657e-002 2.5753e-002 (62%)Y : 4.8529e-001 6.1227e-002 (13%);

Great improvement with a second experiment !BUT, can we do even better?OPTIMAL EXPERIMENTAL DESIGN

Example: simple biochemical pathway

C.G. Moles, P. Mendes y J.R. Banga, 2003. Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Research., 13:2467-2474.

Kinetics described by set of 8 ODEs with 36 parameters

Parameter estimation:

36 parameters

measurements: concentrations of 8 species

16 experiments (different values of S y P)

Example: simple biochemical pathway

Example: simple biochemical pathway

Initial conditions

for all the experiments

Experiments (S, P values)

21 measurements per experiment , tf = 120 s

Example: simple biochemical pathway

Multi-start local methods fail…

Multi-start SQP

Example: simple biochemical pathway

Parameter estimation: again (some) global methods can fail too…

tiempo

Conce

ntr

aci

ón E

1

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

tiempo

Conce

ntr

aci

ón M

2

Example: simple biochemical pathway

Parameter estimation: best fit looks pretty good but…

Contours [p1, p6]

Contours [p1, p4]

Example: simple biochemical pathway

Identifiability problems…

Practical identifiability problems are often due to data with poor information content

Need: more informative experiments (data sets)

Solution: optimal design of (dynamic) experiments

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.

Parametric sensitivities

Fisher information matrix (FIM)

N

iiIii

TI tStWtSFIM

1

1

00 ,, ppptxxj

iij p

xS

Checking identifiability and other indexes…

Compute sensitivities (direct decoupled method) :

Build FIM , covariance and correlation matrices

Analyse possible correlations among parameters

N

iiIii

TI tStWtSFIM

1

1

00 ,, ppptxxj

iij p

xS

1C FIM , ; 1,ijij ij

ii jj

CR i j R i j

C C

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)

p1 & p6 p1 & p4

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

N

iiIii

TI tStWtSFIM

1

1

00 ,, ppptxxj

iij p

xS

1C FIM , ; 1,ijij ij

ii jj

CR i j R i j

C C

Model structure & parameters

Parametric sensitivities

Ranking of parameters

Practical Identifiability analysis

Optimal experimental design

(New) Experiments Model

calibration

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

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

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.

ExperimentData

N

iiIii

TI tStWtSFIM

1

1

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

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

FJ max

)(/)( min minmax FFF J

)1 Ftrace( J

FIMJ min max

A criterion =

D criterion =

E criterion =

Modified-E criterion =

FIMFIM

min

maxmin

FIMminmax

FIMdetmax

1min FIMtrace

2

1

A-optimality

E-optimality

D-optimality

E criterion =

E-optimality: max the min eigenvalue of FIM

(minimizes the largest error)

FIMminmax

2

1

A-optimality

E-optimality

D-optimality

Modified-E criterion =

Maximize decorrelation between parameters

(make contours as circular as possible)

FIMFIM

min

maxmin

2

1

A-optimality

E-optimality

D-optimality

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?

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

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.

Example: Bacterial growth in batch culture

0 2 4 6 8 10

2

4

6

8

10

12

14

Time

obsB

0 2 4 6 8 10

5

10

15

20

25

30

Time

obsS

0 2 4 6 8 102

3

4

5

6

7

8

Time

obsB

0 2 4 6 8 100

5

10

15

Time

obsS

1st experiment 2nd experiment

mmax : 3.9542e-001 3.4730e-002 (9%)Ks : 5.3551e+000 9.1440e-001 (17%)Kd : 4.1657e-002 2.5753e-002 (62%)Y : 4.8529e-001 6.1227e-002 (13%);

Great improvement with a second experiment !BUT, can we do even better?OPTIMAL EXPERIMENTAL DESIGN

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

Example: Bacterial growth in batch culture

OED of second experiment1st experiment 2nd experiment after

OED

mmax : 3.9950e-001 1.7133e-002 (4.3%)Ks : 4.9530e+000 2.9647e-001 (6%)Kd : 5.0859e-002 2.9936e-003 (6%)Y : 5.0544e-001 1.4074e-002 (2.8%);

0 2 4 6 8 10

2

4

6

8

10

12

14

Time

obsB

0 2 4 6 8 10

5

10

15

20

25

30

Time

obsS

0 5 10 15

1.2

1.4

1.6

1.8

2

2.2

2.4

Time

obsB

0 5 10 15

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time

obsS

free initial conditions cb0:[1,5] cs: [5 40], free experiment duration [6,15] h, ns=11

Example: Bacterial growth in batch culture

Two arbitrary experiments

After OED of second experiment

mmax : 3.9950e-001 1.7133e-002 (4.3%)Ks : 4.9530e+000 2.9647e-001 (6%)Kd : 5.0859e-002 2.9936e-003 (6%)Y : 5.0544e-001 1.4074e-002 (2.8%);

free initial conditions cb0:[1,5] cs: [5 40], free experiment duration [6,15] h, ns=11

mmax : 3.9542e-001 3.4730e-002 (9%)Ks : 5.3551e+000 9.1440e-001 (17%)Kd : 4.1657e-002 2.5753e-002 (62%)Y : 4.8529e-001 6.1227e-002 (13%);

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

mumax ks kd yield

mumax

ks

kd

yield

Crammer Rao based correlation matrix for global unknowns

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

mumax ks kd yield

mumax

ks

kd

yield

Crammer Rao based correlation matrix for global unknowns

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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

-1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

2.5

3mumax vs kd

mumax

kd

-1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

2.5

3ks vs kd

ks

kd

-1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

2.5

3kd vs yield

kd

yiel

d

0.95 1 1.05

0.94

0.96

0.98

1

1.02

1.04

1.06

mumax vs kd

mumax

kd

0.95 1 1.05

0.94

0.96

0.98

1

1.02

1.04

1.06

ks vs kd

ks

kd

0.95 1 1.05

0.95

1

1.05

kd vs yield

kd yi

eld

> 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

Example: Bacterial growth in batch culture

0.37 0.38 0.39 0.4 0.41 0.42 0.430

10

20

30

40

50

60Monte-Carlo based confidence interval

mumax

4.4 4.6 4.8 5 5.2 5.40

10

20

30

40

50

60Monte-Carlo based confidence interval

ks

0.048 0.049 0.05 0.051 0.052 0.053 0.0540

10

20

30

40

50Monte-Carlo based confidence interval

kd

0.48 0.49 0.5 0.51 0.52 0.53 0.540

10

20

30

40

50Monte-Carlo based confidence interval

yield

5%

7%

6%

3%

mmax : 3.9950e-001 1.7133e-002 (4.3%)Ks : 4.9530e+000 2.9647e-001 (6%)Kd : 5.0859e-002 2.9936e-003 (6%)Y : 5.0544e-001 1.4074e-002 (2.8%);

Robust confidence intervals are similar to those obtained by the FIM

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):2467-2474

Original experimental design:

16 experiments (different S and P values)

Result: large E-criterion and modified E-criterion

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?

Improve experimental design by solving OED

Find the values of S and P for a set of Nexp experiments

which e.g. maximize E-criterion s.t. constraints

(dynamics plus bounds)

Improved experimental design:

16 experiments with optimal S and P

E-criterion improved (> one order of magnitude)

Other criteria also improved

Improved design by solving OED problem:

Improved design by solving OED problem:

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

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)

Oregonator reaction: highly nonlinear, oscillatory kinetics

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):8426-8436.

Oregonator reaction: OED

E-optimality criterion

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):8426-8436.

Example: OED for Oregonator reaction

OED conclusions

Currently, most experiments are designed based on intuition of experimentalists and modellers

Model-based OED can be used to: Improve model calibration Discriminate between rival models

OED is a systematic and optimal approach

OED can take into account practical limitations and constraints by incorporating them into the formulation

Check identifiability

Use proper optimization methods for parameter estimation

Use optimal experimental design

Main tips for dynamic model building

Take-home messages

“All models are wrong, but some are useful”--- Statistician George E. P. Box

Main tips for dynamic model building

“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

Software for dynamic model building and OED

http://www.iim.csic.es/~amigo/

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.

Balsa-Canto, E. and Julio R. Banga (2011) AMIGO, a toolbox for Advanced Model Identification in systems biology using Global Optimization. Bioinformatics 27(16):2311-2313.

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), e1000558.

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):163-172.

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, 42:1379-1390.

Jaqaman K., Danuser G. Linking data to models: data regression. Nat. Rev. Mol. Cell Bio.7:813-819.

Kremling A, Saez-Rodriguez J: Systems Biology - An engineering perspective. J Biotechnol 2007, 129:329-351

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):364-374.

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):20130505

(review papers in yellow)

http://www.iim.csic.es/~gingproc/software.html

Thank you!

More info?

julio@iim.csic.es