Supplementary Information

Rational selection of experimental readout and intervention

sites for reducing uncertainties in computational model

predictions

Robert J Flassig∗1, Iryna Migal1, Esther van der Zalm1, LiisaRihko-Struckmann1 and Kai Sundmacher1,2

1Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg,Germany

2Process Systems Engineering Group, Otto von Guericke University, Magdeburg,Germany

January 7, 2015

Contents

1 In silico example: model equations 21.1 Profile likelihood and criterion space for different readout setups . . . . . 2

2 D. salina chlorophyll fluorescence induction model: Criterion space forreducing uncertainties in the model parameters 32.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Classical vs. Profile likelihood based sensitivity indices and entropies . . . 4

∗flassig@mpi-magdeburg.mpg.de

1

1 In silico example: model equations

The model equations of the in silico example are given as

A(t) = u(t) − (k11 + k21)A(t) (1)

B(t) = k11A(t) − k12B(t) (2)

C(t) = k12A(t) − (k22 + k23)C(t) (3)

D(t) = k12B(t) + k22C(t) − dD(t) (4)

with stimulus u(t) = 1, states A, B, C, D (which may reflect protein concentrations) andparameters k11, k12, k21, k22, k23 and d. We further assumed that A(t = 0) ≡ B(t = 0) ≡C(t = 0) ≡ D(t = 0) ≡ 0.

1.1 Profile likelihood and criterion space for different readout setups

Here we show the criterion space derived from profile likelihood samples (Figs. S1,S2) .

D AD BD CD

0.4 0.6 0.8 1 1.2 1.40

50

100

150

A

B

CD

entropy

sensitivity

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

20

40

60

80

100

120

140

A

B

CD

entropy

sensitivity

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

5

10

15

20

25

30

35

40

45

50 A

B

C

D

entropy

sensitivity

1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.240

10

20

30

40

50

60

70

80

90

100

A

B

C D

entropy

sensitivity

0 5 10 15 200

1

2

State A

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State B

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State C

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State D

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State A

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State B

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State C

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State D

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State A

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State B

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State C

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State D

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State A

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State B

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State C

time [1]

conc

entra

tion

[a.u

.]

0 5 10 15 200

1

2

State D

time [1]

conc

entra

tion

[a.u

.]

-2 lo

g(L)

-2

log(

L)

Supplementary Figure S1: Criterion space and profile likelihoods for different readoutsetups (one and two readouts).

2

ABD ACD BCD ABCD

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

−2 0 20

1

2

3

-2 lo

g(L)

-2

log(

L)

-2 lo

g(L)

-2

log(

L)

-2 lo

g(L)

-2

log(

L)

-2 lo

g(L)

-2

log(

L)

Supplementary Figure S2: Profile likelihoods for different readout setups (three and fourreadouts).

2 D. salina chlorophyll fluorescence induction model: Cri-terion space for reducing uncertainties in the model pa-rameters

Here we compare profile likelihood based information indices and classical sensitivityindices, which are derived from classical sensitivity analysis (s. Sec. 2.1).

2.1 Sensitivity analysis

In general, sensitivity analysis quantifies how a change in a model parameter influencesthe model prediction. In the following we restrict the discussion to state trajectorypredictions only. As with PLS indices, sensitivity analysis can be used to determine,which predictions or state variables are most affected by parameter uncertainties. Forthe classical local sensitivity analysis, a system of ordinary differential equations is solvedto calculate the sensitivities

dS

dt= JS + Fθ, (5)

where S =dxjdθi

denotes the sensitivity matrix, J =dfjdxj

is the Jacobian of the given system

and Fθ =dfjdθi

. xj indicates the state variable. To be able to compare all sensitivity valuessij it is useful to normalize them. To ensure the positivity of the results the sensitivitiesare also squared

sijk =

(sij(tk)

θixi(tk)

)2

. (6)

These coefficients should qualitatively correspond to the PLS indices, although nonlin-earity features of the model and parameter interdependencies are neglected. Further, wemay as well calculate classical PLS entropy, by using sijk instead of the profile likelihoodbased sijk indices. Note that in the main text we introduce the PLS index of parameter

3

θi at time point tk for one specific prediction as sik. Here the prediction refers to the setof unmeasured model states j (= a set of predictions), which is the reason for introducingthe third subscript-index j.

2.2 Classical vs. Profile likelihood based sensitivity indices and en-tropies

The relation between classical sijk and profile likelihood based sensitivity indices sijk isdepicted in Fig. S3 for the model of D. salina. We see that classical based indices showqualitative differences compared to profile likelihood based ones (Fig. S4).In Fig. S5, the criterion space is depicted for classical and profile likelihood basedsensitivity indices. When only looking at the sensitivity indices, the classical basedapproach favorsPQ,Q−

B, Q2−B and A?, in contrast to profile likelihood based predictions,

which only favor Q−B and Q2−

B . Based on the profile likelihood criterion space, one wouldfavor PQ for an additional readout with regard to equal effects on uncertainty reductionand distribution along the parameters. Classical based criteria would favor A∗ and Q2−

B .In Fig. 4 and also in Tab. 4 of the main document, the performance on PQ and Q−

B isillustrated in terms of the profile likelihood.

4

0 0.5 100.51

A∗

k1

0 0.5 100.51

Q−A

k1

0 0.5 100.51

Q−B

k1

0 0.5 100.51

Q2−B

k1

0 0.5 100.51

PQ

k1

0 0.5 100.51k

2

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51k

3

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51k

4

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51k

5

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51k

6

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51k

7

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51k

8

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51k

9

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51k

10

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

PQ

0

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51

0 0.5 100.51r

2

time, [s ]0 0.5 100.51

time, [s ]0 0.5 100.51

time, [s ]0 0.5 100.51

time, [s ]0 0.5 100.51

time, [s ]

profile likelihood classic

Supplementary Figure S3: Comparison of sensitivity indices for all model states andparameters. Sensitivity indices based on classical, local sensitivity equtions (blue) andprofile likelihood samples (black) were calculated and scaled for qualitative comparison.

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

t ime, [s ]

A∗

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t ime, [s ]

Q− A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

t ime, [s ]

Q− B

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

t ime, [s ]

Q2−

B

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

t ime, [s ]

PQ

Supplementary Figure S4: Trajectories of internal model states along the profile likeli-hood of each parameter. Calculated time profiles for all parameter sets along the profilelikelihood of each parameter are presented in black. Red curves indicate the lower andupper bounds of the trajectories. Green curves represent the model simulation with theoptimal parameter set.

6

10−4

10−3

10−2

10−1

100

10−3

10−2

10−1

100

PLS entropy Ji , to t

PLSindex

si,t

ot/max(s

{i,t

ot})

A*

Q-

A

Q-

B

Q2-

BPQ

A∗

Q−A

Q−B Q2−

B

PQ

profile likelihood classic

Supplementary Figure S5: Parameter contribution to the prediction sensitivity index.Normalized sensitivity indices are plotted versus the sensitivity entropy. The analysiswas performed based on classical, local sensitivity equations (blue) as well as on profilelikelihood (black). Index i refers to the states.

7