Supplementary MaterialStatistical model checking based calibration and

analysis of bio-pathway models

Sucheendra K. Palaniappan1, Benjamin M. Gyori2, Bing Liu3, David Hsu1,2, andP.S. Thiagarajan1,2

1 School of Computing, National University of Singapore, 117417, Singapore2 NUS Graduate School for Integrative Sciences and Engineering, National University of

Singapore, 117417, Singapore3 Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA

1 Pathways

1.1 The reaction diagrams and ODE models

Figure 1 show the reaction diagrams of the segmentation clock network. Due to the largesize of thrombin-dependent MLC phosphorylation pathway, we only show its majorsignaling transduction in Figure 2. The corresponding ODE models of the three pathwaycan be found in the BioModels database4,5.

Fig. 1. Segmentation clock pathway

1.2 The repressilator pathway

In addition to the case studies reported in the main text, we also studied the repressilatorpathway.

4 Segmentation clock network: http://www.ebi.ac.uk/biomodels-main/BIOMD00000002015 Thrombin-dependent MLC phosphorylation pathway: http://www.ebi.ac.uk/biomodels-

main/BIOMD0000000088

2 Palaniappan et al.

thrombin

Thrombin receptor Membrane G12

p115RhoGEF

Rho RhoGAP

Rho-kinase

MLC MLC-p

MYPT1-p MYPT1 CPI-17

RGS Gq

PLCβ

IP3 DAG

PKC

Ca2+

CaM

MLCK

Fig. 2. Thrombin-dependent MLC phosphorylation pathway

The repressilator is a synthetic gene network originally introduced by [1]. The net-work consists of three genes linked in an inhibitory cycle. The ODE model of thepathway, consists of 3 mRNA transcripts and 3 associated protein products. With anappropriate combination of parameter values, the proteins show sustained oscillations.By specifying the properties of the oscillations (see Table 1), we attempt to recover 9unknown parameters. It is interesting to note that specification of the dynamics can bemade without access to experimental data, based only on qualitative prior knowledge.All the properties/quantitaive experimental data were required to hold with a high prob-ability. We fixed the range of the unknown parameters as shown in Table 2. The initialstates of all the species were assumed to be uniformly distributed in a range 5% aroundthe nominal initial concentration.

m1,m2,m3 represent mRNA transcripts of 3 genes and p1, p2, p3 are the proteinproducts for each mRNA respectively.

dmidt

= −γmi +α

1 + kpnjdpidt

= β(mi − pi)

Depending on the values of the parameters α, β, γ, k, n, the protein products show sus-tained oscillations. We assumed 9 parameters corresponding to the parameters α, γ andk for each of the mRNA transcripts to be unknown.

Statistical model checking for parameter estimation 3

2 Statistical model checking

This section presents additional details about our case studies in the statistical modelchecking chapter.

Objective function We add the term∑(Jqlty+∑i∈O Jiexp)

k=1

n+k

nk

Jqlty+∑

i∈O Jiexp

to our original objective

value. Here n+k , nk denote the number of samples evaluating to true and the total num-ber of samples needed to verify the kth PBLTL formula. This is done for faster conver-gence.

Species name Property

p1 [p1 ≤ 0.1]∧ ∼ F≤4([1.3 ≤ p1 ≤ 1.5]∧F≤10[1.3 ≤ p1 ≤ 1.5]∧F ([1.3 ≤ p1 ≤ 1.5]∧F ([0.3 ≤p1 ≤ 0.4] ∧ F ([1.05 ≤ p1 ≤ 1.15] ∧ F ([0.35 ≤ p1 ≤ 0.45] ∧ F ([1 ≤ p1 ≤ 1.1] ∧ F ([0.35 ≤p1 ≤ 0.45]))))))

p2 [1.9 ≤ p2 ≤ 2.1] ∧ F≤10[0.2 ≤ p2 ≤ 0.3] ∧ F ([0.2 ≤ p2 ≤ 0.3] ∧ F ([1.15 ≤ p2 ≤ 1.25] ∧F ([0.3 ≤ p2 ≤ 0.4]∧F ([1.0 ≤ p2 ≤ 1.1]∧F ([0.35 ≤ p2 ≤ 0.45]∧F ([0.95 ≤ p2 ≤ 1.05]))))))

p3 [p3 ≤ 0.2]∧F≤10[1.55 ≤ p3 ≤ 1.7]∧F ([1.55 ≤ p3 ≤ 1.7]∧F ([0.275 ≤ p3 ≤ 0.375]∧F ([1 ≤p3 ≤ 1.2] ∧ F ([0.35 ≤ p3 ≤ 0.45] ∧ F ([1 ≤ p3 ≤ 1.2] ∧ F ([0.35 ≤ p3 ≤ 0.45]))))))

Table 1. Repressilator pathway: Properties

Table 2. Repressilator pathway: Unknown parameters with range : SRES

Parameter range SRES, r =0.8

SRES, r =0.9

α1 [0, 100] 81.21886 86.15479α2 [0, 100] 51.95532 68.90892α3 [0, 100] 75.57755 73.12696γ1 [0, 200] 189.7099 178.5928γ2 [0, 200] 88.04731 130.0404γ3 [0, 200] 163.9563 156.9079k1 [0, 16] 10.86995 11.73143k2 [0, 16] 8.125588 12.15338k3 [0, 16] 11.99097 14.44549

4 Palaniappan et al.

Table 3. Repressilator pathway: Unknown parameters with range : GA

Parameter range GA, r = 0.8 GA, r = 0.9

α1 [0, 100] 33.65326 20.56796α2 [0, 100] 59.20567 89.14866α3 [0, 100] 62.52711 81.10954γ1 [0, 200] 78.78442 49.09057γ2 [0, 200] 100.5856 158.5545γ3 [0, 200] 128.8992 167.4489k1 [0, 16] 11.65575 12.81053k2 [0, 16] 6.617118 7.022426k3 [0, 16] 10.86312 15.0332

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m1

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m2

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m3

0 5 10 150

0.5

1

1.5

2

time(seconds)

p1

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

p2

0 5 10 150

0.5

1

1.5

2

time(seconds)

p3

(a)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(b)Fig. 3. (a)Time profile of all the species in the repressilator pathway based on the best parametersreturned by SRES based parameter estimation,(b) objective value vs number of generations, r=0.8

0 5 10 150

0.5

1

1.5

2

time(seconds)

m1

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m2

0 5 10 150

1

2

3

time(seconds)

m3

0 5 10 150

0.5

1

1.5

2

time(seconds)

p1

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

p2

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

p3

(a)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(b)Fig. 4. (a)Time profile of all the species in the repressilator pathway based on the best parametersreturned by SRES based parameter estimation,(b) objective value vs number of generations, r=0.9

Statistical model checking for parameter estimation 5

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m1

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m2

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m3

0 5 10 150

0.5

1

1.5

2

time(seconds)

p1

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

p2

0 5 10 150

0.5

1

1.5

2

time(seconds)

p3

(a)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(b)Fig. 5. (a)Time profile of all the species in the repressilator pathway based on the best parametersreturned by GA based parameter estimation,(b) objective value vs number of generations, r=0.8

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m1

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m2

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

m3

0 5 10 150

0.5

1

1.5

time(seconds)

p1

0 5 10 150

0.5

1

1.5

2

2.5

time(seconds)

p2

0 5 10 150

0.5

1

1.5

2

time(seconds)

p3

(a)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(b)Fig. 6. (a)Time profile of all the species in the repressilator pathway based on the best parametersreturned by GA based parameter estimation,(b) objective value vs number of generations, r=0.9

6 Palaniappan et al.

0 50 100 150 2000

0.2

0.4

0.6

0.8Notch protein

0 50 100 150 2000

0.05

0.1

0.15

0.2nuclear NicD

0 50 100 150 2000

1

2

3

4Lunatic fringe mRNA

0 50 100 150 2000

0.5

1

1.5

2active ERK

0 50 100 150 2000

2

4

6

8Dusp6 mRNA

0 50 100 150 2000

5

10

15

20

25

Axin2 mRNA

(a)

0 50 100 150 2000

0.5

1

1.5cytosolic NicD

0 50 100 150 2000

5

10

15

20Dusp6 protein

0 1 2 3 4 5 6

x 104

0

1

2

3

4

5

6

7

(b)Fig. 7. Segmentation clock (a)Parameter estimation results - training and test data - SRES algo-rithm (b) objective value vs number of generations, r=0.8

Statistical model checking for parameter estimation 7

Table 4. Segmentation Clock pathway: Unknown parameters with range : SRES

ID Parameter range SRES, r =0.9

SRES, r =0.8

k1 KdN [0, 2.8] 2.270008 1.85477k2 vsN [0, 0.46] 0.1777086 0.225461k3 vdN [0, 5.64] 3.681801 3.01694k4 kt1 [0, 0.2] 0.06868474 0.106655k5 kt2 [0, 0.2] 0.1398081 0.122804k6 KdNan [0, 0.002] 0.001274139 0.00162818k7 V dNan [0, 0.2] 0.06987804 0.106778k8 KdMF [0, 1.536] 1.299962 1.39512k9 KIG1 [0, 5] 1.901268 1.96934k10 vsF [0, 6] 3.204958 2.35898k11 vmF [0, 3.84] 3.108714 3.09863k12 KdF [0, 0.74] 0.4161383 0.250136k13 vdF [0, 0.78] 0.6714567 0.626846k14 ksF [0, 0.6] 0.4291293 0.287691k15 kd2 [0, 14.124] 13.31986 4.662k16 vMB [0, 3.28] 1.755259 1.43224k17 KaB [0, 1.4] 0.9412496 1.18731k18 vMXa [0, 1] 0.9813186 0.995318k19 ksAx [0, 0.04] 0.03362519 0.0365767k20 vdAx [0, 1.2] 0.295083 0.0586986k21 KdAx [0, 1.26] 1.098116 0.504046k22 kt3 [0, 1.4] 0.1988066 0.0875287k23 kt4 [0, 3] 1.390139 2.46001k24 ksDusp [0, 1] 0.9668858 0.660403k25 vdDusp [0, 4] 3.27664 2.23029k26 KdDusp [0, 1] 0.6531061 0.0311686k27 kcDusp [0, 2.7] 0.8591396 2.35226k28 KaFgf [0, 1] 0.5732082 0.0352701k29 KaRas [0, 0.206] 0.1417966 0.114468k30 KdRas [0, 0.2] 0.1218366 0.108022k31 KaMDusp [0, 1] 0.925967 0.679978k32 KdMDusp [0, 1] 0.645723 0.959026k33 VMsMDusp [0, 1.8] 1.744184 1.34448k34 VMdMDusp [0, 1] 0.8397297 0.777251k35 VMaRas [0, 9.936] 6.891038 8.06544k36 VMdRas [0, 0.82] 0.5663031 0.354376k37 VMaErk [0, 6.6] 6.327112 6.37508k38 VMaX [0, 3.2] 1.090576 3.09787k39 VMdX [0, 1] 0.6747395 0.537238

8 Palaniappan et al.

Species name PropertyNotch protein (([0.45 ≤ Notch protein ≤ 0.55] ∧ F3([ Notch protein ≤ 0.05])) ∧ (F ([ Notch protein

≤ 0.05]∧ F ([0.10 ≤Notch protein≤ 0.15]∧ F ([Notch protein≤ 0.05]∧ F ([0.10 ≤Notchprotein≤ 0.15]))))))

nuclear NicD (([ nuclear NicD ≤ 0.012]) ∧ (F ([0.07 ≤ nuclear NicD ≤ 0.08] ∧ F ([ nuclear NicD ≤0.012] ∧ F ([0.07 ≤ nuclear NicD ≤ 0.08] ∧ F ([ nuclear NicD ≤ 0.012]))))))

Lunatic fringe mRNA (([ Lunatic fringe mRNA ≤ 0.4]) ∧ (F ([ Lunatic fringe mRNA ≥ 2.2] ∧ F ([ Lunaticfringe mRNA ≤ 0.4] ∧ F ([ Lunatic fringe mRNA ≥ 2.2] ∧ F ([ Lunatic fringe mRNA≤ 0.4]))))))

active ERK ([ active ERK ≤ 0.27]∧F3([1.9 le active ERK ≤ 2.2]))∧(F ([1.9 le active ERK ≤ 2.2]∧F ([active ERK ≤ 0.27] ∧ F ([1.9 le active ERK ≤ 2.2] ∧ F ([ active ERK ≤ 0.27])))))

Dusp6mRNA ([Dusp6mRNA≤ 1]) ∧ (F ([Dusp6mRNA≥ 5.5] ∧ F ([Dusp6mRNA≤ 1] ∧ F ([Dusp6mRNA≥ 5.5] ∧ F ([ Dusp6mRNA≤ 1])))))

Table 5. Segmentation pathway: Properties, additional constraints were added to limit the numberof crests and troughs

Species name PropertyDusp6 protein (([ Dusp6 protein ≤ 0.5]) ∧ (F ([ Dusp6 protein ≥ 9] ∧ F ([ Dusp6 protein ≤ 0.5] ∧ F ([

Dusp6 protein≥ 9] ∧ F ([ Dusp6 protein≤ 0.5]))))))cytosolic nicD (([ cytosolic nicD≤ 0.5])∧(F ([ cytosolic nicD≥ 1]∧F ( cytosolic nicD≤ 1]∧F ([ cytosolic

nicD ≥ 1] ∧ F ([ cytosolic nicD ≤ 1]))))))Table 6. Segmentation pathway:Test

0 50 100 150 2000

0.2

0.4

0.6

0.8Notch protein

0 50 100 150 2000

0.02

0.04

0.06

0.08

0.1nuclear NicD

0 50 100 150 2000

1

2

3

4Lunatic fringe mRNA

0 50 100 150 2000

0.5

1

1.5

2active ERK

0 50 100 150 2000

2

4

6

8Dusp6 mRNA

0 50 100 150 2000

5

10

15

20

25

Axin2 mRNA

(a)

0 50 100 150 2000

0.5

1

1.5cytosolic NicD

0 50 100 150 2000

10

20

30Dusp6 protein

0 1 2 3 4 5 6

x 104

0

1

2

3

4

5

6

7

(b)Fig. 8. Segmentation clock (a)Parameter estimation results - training and test data - SRES algo-rithm (b) objective value vs number of generations, r=0.9

Statistical model checking for parameter estimation 9

Species name Property

thrombinRactive F≤5([0.004 ≤ thrombinRactive ≤ 0.0046]∧F ([thrombinRactive ≤ 0.0006]))∧F ([0.004 ≤thrombinRactive ≤ 0.0046] ∧ F ([thrombinRactive ≤ 0.0006] ∧XG([thrombinRactive ≤0.0006])))

3IP3.IP3R F≤5([0.0007 ≤ 3IP3.IP3R ≤ 0.0009] ∧ F ([3IP3.IP3R ≤ 0.00009])) ∧ F (([0.0007 ≤3IP3.IP3R ≤ 0.0009]) ∧ F ([3IP3.IP3R ≤ 0.00009] ∧XG([3IP3.IP3R ≤ 0.00009])))

Table 7. MLC dependent Thrombin pathway: Properties

0 500 10000

0.005

0.01

0.015

0.02

0.025

RGS2

0 500 10000

0.01

0.02

0.03

0.04

0.05

Rho.GTP

0 500 10000

0.05

0.1

0.15

0.2

PKC.DAG

0 500 10000

1

2

3

4

MLC

0 500 10000

0.5

1

1.5

x 10−3 CPI−172

0 500 10000

0.2

0.4

0.6

0.8

1

Ca_super_2_plus__endsuper_

0 500 10000

0.02

0.04

0.06

0.08

0.1

p115RhoGEF.GTP_alpha_

0 500 10000

0.2

0.4

0.6

0.8

1

MYPT1PPase

2

0 500 10000

1

2

3

4

5x 10

−3thrombinR active

0 500 10000

0.2

0.4

0.6

0.8

1x 10

−33IP3.IP3R

(a)

0 200 400 600 800 10000

0.002

0.004

0.006

0.008

0.01Rho−kinase.MLC

0 200 400 600 800 10000

0.005

0.01

0.015

MYPT1.Rho−kinase2

0 2 4 6 8 10

x 104

0

1

2

3

4

5

6

7

8

9

(b)Fig. 9. Thrombin dependant MLC pathway (a)Parameter estimation results - training and test data- SRES algorithm (b) objective value vs number of generations, r=0.8

2.1 Sensitivity analysis

Our sensitivity analysis is based on the a global sensitivity analysis technique calledmulti-parametric sensitivity analysis (MPSA) [2]. Figure 11 illustrates our approach.We assume a prior (uniform) distribution for the range of values that each parametercan take. We discretize the value space of each parameter intoNSens intervals. We thendraw NSens samples from the resulting set of hypercubes using the Latin hypercubesampling technique. For each sampled parameter hypercube, run the statistical modelchecking procedure by sampling an initial state and a parameter value combinationwithin the hypercube, and compute an objective value according to Equation (2) in themain text. We define a threshold value θ as the average of these objective values. Wethen classify the sampled parameter hypercubes as “good” or “bad” by comparing their

10 Palaniappan et al.

Table 8. Thrombin dependent MLC pathway: Unknown parameters with range

Parameter range SRES, r = 0.8 (SRES), r =0.9

k1 [2.5, 10] 6.826E + 00 7.701E + 00k2 [7.5, 30] 9.549E + 00 2.638E + 01k3 [2, 8] 6.339E + 00 2.500E + 00k4 [0.005, 0.02] 8.600E − 03 1.300E − 02k5 [0.0011, 0.0044] 2.340E − 03 2.305E − 03k6 [0.5, 2] 8.986E − 01 7.410E − 01k7 [0.003, 0.012] 1.076E − 02 3.686E − 03k11 [0.025, 0.1] 5.299E − 02 4.557E − 02k12 [0.0051, 0.0204] 1.952E − 02 1.288E − 02k13 [11.755, 47.02] 2.139E + 01 3.366E + 01k15 [10, 40] 3.546E + 01 1.139E + 01k16 [0.05, 0.2] 1.619E − 01 1.034E − 01k17 [0.0075, 0.03] 2.032E − 02 1.124E − 02k18 [0.026, 0.104] 9.003E − 02 7.846E − 02k22 [1.415, 5.66] 2.073E + 00 2.037E + 00k23 [0.4965, 1.986] 1.408E + 00 1.275E + 00k24 [6.623, 26.492] 1.585E + 01 1.769E + 01k25 [0.02, 0.08] 6.527E − 02 5.646E − 02k27 [0.0051, 0.0204] 7.999E − 03 1.080E − 02k28 [0.00585, 0.0234] 1.178E − 02 1.182E − 02k29 [0.0002, 0.0008] 4.223E − 04 5.492E − 04k30 [5, 20] 1.653E + 01 1.511E + 01k31 [0.25, 1] 9.286E − 01 5.040E − 01k32 [7.5, 30] 2.617E + 01 2.429E + 01k33 [0.5, 2] 1.075E + 00 1.213E + 00k34 [0.003, 0.012] 6.274E − 03 6.497E − 03k35 [0.00665, 0.0266] 2.217E − 02 2.442E − 02k36 [0.0011, 0.0044] 4.043E − 03 2.155E − 03k37 [12.6005, 50.402] 3.964E + 01 4.202E + 01k38 [0.5, 2] 1.391E + 00 9.488E − 01k39 [1.26005, 5.0402] 3.279E + 00 3.352E + 00k40 [0.5, 2] 1.388E + 00 7.517E − 01k41 [15, 60] 3.372E + 01 5.546E + 01k42 [0.5, 2] 1.053E + 00 1.539E + 00k43 [1.5, 6] 4.247E + 00 2.175E + 00k44 [0.5, 2] 1.472E + 00 1.790E + 00k45 [0.00665, 0.0266] 1.157E − 02 2.287E − 02k46 [9.9205, 39.682] 1.584E + 01 1.469E + 01k47 [5, 20] 5.942E + 00 1.795E + 01

Statistical model checking for parameter estimation 11

Table 9. Thrombin dependent MLC pathway: Unknown parameters with range

Parameter range SRES, r = 0.8 SRES, r =0.9

k51 [2, 8] 3.483E + 00 5.743E + 00k52 [5, 20] 8.005E + 00 1.294E + 01k53 [24, 96] 7.028E + 01 7.180E + 01k54 [0.075, 0.3] 2.386E − 01 1.734E − 01k55 [1.25, 5] 3.504E + 00 2.951E + 00k56 [0.05, 0.2] 1.273E − 01 1.554E − 01k57 [5, 20] 8.269E + 00 1.861E + 01k58 [75, 300] 1.848E + 02 1.020E + 02k60 [5, 20] 1.057E + 01 8.574E + 00k61 [25, 100] 6.458E + 01 6.760E + 01k62 [5, 20] 6.884E + 00 1.093E + 01k63 [22.5, 90] 2.687E + 01 4.650E + 01k64 [5, 20] 1.272E + 01 1.518E + 01k65 [250, 1000] 6.840E + 02 5.570E + 02k66 [5, 20] 9.962E + 00 6.764E + 00k67 [85, 340] 1.218E + 02 1.719E + 02k78 [0.00015, 0.0006] 4.736E − 04 5.103E − 04k79 [0.05, 0.2] 1.757E − 01 1.744E − 01k82 [0.15, 0.6] 2.447E − 01 3.346E − 01k83 [0.25, 1] 7.778E − 01 6.465E − 01k85 [15, 60] 2.193E + 01 3.161E + 01k86 [1.5, 6] 4.458E + 00 1.929E + 00k94 [0.02, 0.08] 6.215E − 02 5.822E − 02k95 [2.45, 9.8] 5.096E + 00 3.950E + 00k96 [2, 8] 7.389E + 00 6.733E + 00k106 [1.97, 7.88] 7.644E + 00 3.708E + 00k107 [1.97, 7.88] 6.653E + 00 6.957E + 00k108 [1.97, 7.88] 6.852E + 00 4.176E + 00k109 [0.25, 1] 3.115E − 01 4.359E − 01

12 Palaniappan et al.

Table 10. Thrombin dependent MLC pathway: Unknown parameters with range

Parameter range SRES, r = 0.8 SRES, r =0.9

k110 [50, 200] 8.727E + 01 1.220E + 02k111 [0.31, 1.24] 4.793E − 01 8.308E − 01k115 [2.25495, 9.0198] 6.914E + 00 2.332E + 00k116 [0.64, 2.56] 6.515E − 01 1.637E + 00k117 [8.3085, 33.234] 2.494E + 01 9.268E + 00k121 [0.64, 2.56] 2.362E + 00 2.333E + 00k122 [0.64, 2.56] 1.952E + 00 1.962E + 00k130 [4.33, 17.32] 1.299E + 01 1.423E + 01k134 [0.73, 2.92] 2.281E + 00 2.296E + 00k135 [0.05, 0.2] 9.592E − 02 1.627E − 01k136 [8.7525, 35.01] 3.363E + 01 2.962E + 01k137 [0.01415, 0.0566] 4.928E − 02 3.357E − 02k138 [8.7525, 35.01] 2.966E + 01 2.660E + 01k139 [29.0495, 116.198] 1.069E + 02 3.791E + 01k140 [0.975, 3.9] 2.587E + 00 2.647E + 00k141 [14.3975, 57.59] 2.168E + 01 3.605E + 01k142 [8, 32] 1.135E + 01 1.387E + 01k143 [4.6585, 18.634] 1.303E + 01 9.563E + 00k144 [3.79325, 15.173] 1.159E + 01 9.394E + 00k145 [29.0495, 116.198] 4.372E + 01 5.949E + 01k146 [0.975, 3.9] 2.438E + 00 1.570E + 00k147 [14.3975, 57.59] 4.269E + 01 4.167E + 01k151 [0.975, 3.9] 2.692E + 00 1.415E + 00k153 [0.975, 3.9] 1.845E + 00 2.073E + 00k155 [0.1, 0.4] 3.673E − 01 2.815E − 01k163 [1.835, 7.34] 4.735E + 00 6.936E + 00k188 [1.835, 7.34] 4.718E + 00 3.770E + 00k190 [1.835, 7.34] 4.841E + 00 4.866E + 00k192 [1.835, 7.34] 4.662E + 00 4.859E + 00k194 [1.835, 7.34] 4.500E + 00 3.740E + 00k196 [0.005, 0.02] 8.390E − 03 9.265E − 03k197 [0.005, 0.02] 8.950E − 03 7.214E − 03

Statistical model checking for parameter estimation 13

0 500 10000

0.005

0.01

0.015

0.02

0.025

RGS2

0 500 10000

0.01

0.02

0.03

0.04

0.05

Rho.GTP

0 500 10000

0.05

0.1

0.15

0.2

PKC.DAG

0 500 10000

1

2

3

4

MLC

0 500 10000

0.5

1

1.5x 10

−3 CPI−172

0 500 10000

0.2

0.4

0.6

0.8

1

1.2

Ca_super_2_plus__endsuper_

0 500 10000

0.02

0.04

0.06

0.08

0.1p115RhoGEF.GTP_alpha_

0 500 10000

0.2

0.4

0.6

0.8

1

MYPT1PPase

2

0 500 10000

1

2

3

4

5

6x 10

−3thrombinR active

0 500 10000

0.2

0.4

0.6

0.8

1x 10

−33IP3.IP3R

(a)

0 200 400 600 800 10000

0.005

0.01

0.015

0.02Rho−kinase.MLC

0 200 400 600 800 10000

0.002

0.004

0.006

0.008

0.01

0.012

MYPT1.Rho−kinase2

0 2 4 6 8 10

x 104

0

1

2

3

4

5

6

7

8

9

(b)Fig. 10. Thrombin dependant MLC pathway (a)Parameter estimation results - training and testdata - SRES algorithm (b) objective value vs number of generations, r=0.9

14 Palaniappan et al.

objective value to θ. If an objective value is greater than θ, we say its correspondingparameter hypercube is “good”. Otherwise, it is “bad”. We then plot two cumulativefrequency curves for each parameter. The X-axis is the value range of a parameter.For the “good” curve of this parameter, the Y-axis is the cumulative number of “good”samples. For the “bad” curve, the Y-axis is the cumulative number of “bad” samples.We next compute the Kolmogorov-Smirnov statistic of the two curves, which is themaximum difference in Y-axis between the two curves. We define our global sensitivityas the returned value. The sensitivity values are between 0 and 1. The larger the value,the higher is the sensitivity of the corresponding parameter.

Sensitivityanalysis

specification

Sampleparameterinterval

combination

Sample✓ and x0

Simulate onetime-step

Modelchecker

PBLTLspecification

True / False

Sequentialhypothesis

test

Objectivevalue

Sensitivities

Undetermined?

More samples?

Nsens times

MPSA

Statisticalmodel checking

Fig. 11. Statistical model checking based sensitivity analysis

We now look at sensitivity analysis results obtained using our proposed method.In each experiment, we run MPSA with 5 · 104 random samples from the parameterspace. In addition to the sensitivity analysis on the segmentation clock pathway that wasreported in the main text, we performed sensitivity analysis on the simple repressilatormodel and the thrombin dependent MLC pathway.

We show results on a simple repressilator model. The repressilator is a syntheticgene network originally introduced by [1]. The network consists of three genes linkedin an inhibitory cycle. The ODE model of the pathway, consists of 3 mRNA transcriptsand 3 associated protein products (more details on the model are given in the Supple-

Statistical model checking for parameter estimation 15

mentary material). With an appropriate combination of parameter values, the proteinsshow sustained oscillations. We start by specifying the properties of the oscillationsfor all the proteins. We wish to understand which parameters affect the properties ofthese oscillations significantly. The results obtained using our method (Figure 12(a)-(1))shows that the oscillations are most sensitive to the parameters γ1, γ2 and γ3. The anal-ysis took ∼3 minutes. Next, we explored the most sensitive parameter with respect tothe oscillation pattern observed for one of the proteins, p1. The results in Figure 12(a)-(2) show that in addition to γ1, γ2 and γ3, the parameter k3 has a significant effect onthis trend. Next, we show that sensitivities can be quantified with respect to qualitativeproperties alone. Consider a case where one does not know the dynamic ranges of theconcentration of protein p1, but only the notion that there are oscillations observed, withthe levels of p1 oscillating between low (defined as p1 having concentration less than0.5) and high (defined as p1 having concentration higher than 1.0). We also added theadditional constraint that there must be a minimum of 3 crests. The corresponding for-mula is [p1 ≤ 0.5]∧F ([p1 ≥ 1]∧F ([p1 ≤ 0.5]∧F ([p1 ≥ 1]∧F ([p1 ≤ 0.5]∧F ([p1 ≥1]∧F ([p1 ≤ 0.5])))))), Figure 12(a)-(3) shows that γ1, γ2 and γ3 are the most sensitiveparameters with respect to this qualitative oscillation property.

Finally we performed sensitivity analysis on the thrombin dependent MLC path-way. We evaluated the sensitivities with respect to the same set of properties as usedfor parameter estimation. The analysis took ∼10 hours. The results can be found in Fig-ure 13. The method identified the set of parameters (k28, k30,k31, k136, k155 and k196) towhich the specified properties show the most sensitivity. These parameters are the onesinvolved in activation of p115Rho, Rho-kinase, and further activation of componentssuch as MYPT1. Please refer to the supplementary for details of these parameters. Itis well known that the sustained contraction of cells depends on sustained Rho-kinaseactivation which can be induced by the transient phosphorylation of MLC and MYPT1.The sensitivity analysis result is consistent with these insights.

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

Parameters

Sen

sitiv

ity

(a)-(1)

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

Parameters

Sen

sitiv

ity

(a)-(2)

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

Parameters

Sen

sitiv

ity

(a)-(3)Fig. 12. Sensitivity analysis results of the represillator pathway

References1. Elowitz, M., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature

403(6767) (2000) 335–3382. Cho, K.H., Shin, S.Y., Kolch, W., Wolkenhauer, O.: Experimental design in systems biology,

based on parameter sensitivity analysis using a Monte Carlo method: A case study for theTNFα-mediated NF-κB signal transduction pathway. Simulation 79(12) (2003) 726–739

16 Palaniappan et al.

0 20 40 60 80 1000

0.02

0.04

0.06

0.08

Parameters

Sen

sitiv

ity

Fig. 13. Sensitivity analysis results of the thrombin-dependent MLC phosphorylation pathwaywith respect to all properties

Supplementary Material Statistical model checking based calibration and analysis of bio-pathway models