Systems biology and beyond From data to mathematical models

Systems biologyand beyond

–From data to mathematical

modelsZuse Institute Berlin

Susanna RoblitzComputational Systems Biology Group

ZIB

DFG Research Center

March 1st, 2013 Matheon

February 28, 2013

Overview

Matheon

1. Who we are and what we are doing

2. A typical research example: Mathematical modelling of thehuman menstrual cycle

3. Further research topics

4. Software

MSB Seminar 2 Susanna Roblitz

Who we are...

Matheon

Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin http://www.zib.de

� Founded by P. Deuflhard in 1986 as a research institute forComputational Mathematics

� Basic funding by the State of Berlin; strong support by third partyfunding

� About 200 employees (50% research, 50% service)

� R&D branches: Numerical Mathematics, Discrete Mathematics,Computer Science

� High-level services for high-performance computing


http://www.zib.de

Group Members

MatheonHead

Susanna RoblitzStaff Students

Rainald Ehrig Julia Plontzke

Claudia Stotzel Gabriel Muller

Thomas Dierkes Dany Pascal Moualeu

Mascha Berg


Activities

Matheon

AlgorithmsParameter identification

Deterministic/stochastic modelling

Discrete/continuous modelling

Endocrinological networks

Metabolic networks

Biosensors

Applications

SoftwareBioPARKIN (C++/Python)

POEM (Matlab)


Mathematical modelling of thehuman menstrual cycle


The human menstrual cycle

Matheon

(http://www.websters-online-dictionary.org/definitions/Menstrual Cycle)


Aims

Matheon

First step

� a model for the idealized cycle of an idealized woman

� calculation of hormone profiles and follicle development overtime

What can we use such a model for?

� find parametrizations for individual women, study of infertilityor genetic differences

� simulation of external effects (administration of drugs, designof hormone therapies)


Overview

Matheon

Models available before

� smaller compartment models: fewer (though essential)compartments coupled by coarse interactions

[Schlosser/Selgrade 2000, Harris 2001]

� small biochemical reaction models (such as receptor binding):often not fully integrated into total model

[Clement 2001]

ZIB Model GynCycle

� fully integrated compartment model: containing most relevantcouplings, both physiological and biochemical ones

� modifications for special purposes:� model reduction (to focus on specific aspects)� model expansion (to catch further effects)


Systems biology approach

Matheon

� the cycle as a whole system rather than only focusing onindividual parts

� investigate how the components function together

� find reliable abstraction levels that display the most importantmechanisms

Mathematical task::

Simulation

Validation

Modelling

� Model development and parametrization such that simulationresults match given measurement data


Data


Data for normal cycle

Matheon

−10 0 10 20 30 400

20

40

60

80

100

IU/L

LH

−10 0 10 20 30 400

5

10

15

20

IU/L

FSH

−10 0 10 20 30 400

100

200

300

400

500

pg/m

l

E2

−10 0 10 20 30 400

5

10

15

20

25

30

ng/m

l

P4

measurement values for 12 women over one cycle (Pfizer R&D)


Data for single dose nafarelin

Matheon

4 5 6 7 8 90

0.5

1

1.5

2

2.5

4 5 6 7 8 90

50

100

150

4 5 6 7 8 95

10

15

20

25

30

35

40

nafarelin LH FSH

4 5 6 7 8 90

50

100

150

200

250

300

4 5 6 7 8 90.4

0.5

0.6

0.7

0.8

0.9

1

GnRH-Agonist: activa-

tion of GnRH receptor

resulting in (initially) in-

creased secretion of FSH

and LH, followed by a

drop in gonadotropin se-

cretion caused by recep-

tor downregulationE2 P4


Data for single and multiple dose cetrorelix

Matheon

0 20 40 600

5

10

15

20

25

0 20 40 60 800

2

4

6

8

cetrorelix LH

0 20 40 602.5

3

3.5

4

4.5

5

5.5

6

6.5

0 20 40 60 800

50

100

150

GnRH-Antagonist: com-

petitive and reversible

binding to GnRH recep-

tors, immediate drop in

gonadotropin secretion

FSH E2


Typical problems

Matheon

� different physical units, sometimes not even convertible

� missing measurement errors

� missing information about the cycle day

� averaged data for women with different cycle length or indifferent stages of the cycle

ideal measurement: data for a single woman together with thelength of her cycle, the stage of the cycle (days since last menses),and measurement errors


Model Development


Components and compartments considered

Matheon

Compartments: blood, ovaries,uterus, pituitary, hypothalamusComponents:

� Estradiol

� Progesterone

� Inhibin A and B

� LH + receptor binding

� FSH + receptor binding

� GnRH + receptor binding

� 6 follicular stages

� 6 luteal stages (corpusluteum)

HYPOTHALAMUS

PITUITARY

CORPUS LUTEUM

OVARIES

inhibin

activin

follistatin

FSH

LH

GnRH

estradiol

progesterone

estradiol

progesterone

TEUM

ovulation


Differential equations

MatheonAbstract mathematical formulation:

y 1ptq � f pt, yptq, pq, ypt � 0q � y0

Production, clearance, synthesis and release usually depend onother components. To model stimulation or inhibition we use

positive Hill functions negative Hill functions

H�pSptq,T , nq :� Sptqn

Sptqn�T n H�pSptq,T , nq :� T n

Sptqn�T n

TSHtL

m

H+HSHtL,T,nL

n=10

n=5

n=2

TSHtL

m

H-HSHtL,T,nL

n=10

n=5

n=2


Current model

Matheon

��

��

Lut1 Lut2Sc2OvFPrF

GnRH antagonist

CENTRAL COMPARTMENT

GnRH antagonist

DOSING COMPARTMENT

PERIPHERAL COMPARTMENT

GnRH antagonist

GnRH agonist

DOSING COMPARTMENT

GnRH Ant−RecComplex

inactive GnRH−Rec

complex

complex

active GnRH−Rec

active Ago−Rec

complex

GnRH agonist

CENTRAL COMPARTMENT

AF1 AF2 AF3 AF4 Sc1 Lut3 Lut4

inactive

GnRH Receptors

GnRH Receptors

active

inactive Ago−Rec

complex

GnRH (G)

Progesterone (P4)

Estradiol (E2)

Inhibin B (IhB)

Inhibin A (IhA)

effective IhA (IhA )e

free LH receptors

LH(R )

LH receptor complex

(LH−R)

desensitized rec.

LH,des

pit

pituitary LH

(LH )blood

serum LH

pit(FSH )pituitary FSH

blood(FSH )

serum FSH free FSH receptors

(R )FSH

FSH receptor complex

(FSH−R)

(R )

FSH,des(R )desensitized rec.

(LH )

(freq)

( s )

foll. LH sensitivity

(mass)GnRH mass

GnRH frequency

2007: 49 DDEs, 208 parameters, 9 identifiable [Reinecke,Deuflhard (2007)]

2012: 33(+8) ODEs, 114 parameters, 63 identifiable [Roblitz et al. (2013)]


Parameter Identification


A nonlinear least squares problem

Matheon

Model ypt, pq � py1pt, pq, . . . , ynpt, pqqParameters p � pp1, ..., pqqData x � px1, ..., xmq,m ¥ qmostly m " q, data compression

0 5 10 15−2

0

2

4

6

8

10

12

t

y

measurement values

model function

residuesLeast squares formulation:

}F ppq}22 �

m

i�1

�xi � ypti , pq

δxi

2

Ñ min

δxi : measurement accuracy for the xi (never forget!)relative measurement accuracy:

δxi � εxxi

εx mostly 10�1 to 10�3 in experiments


Gauss-Newton algorithm

Matheon

� solution of the nonlinear least squares problem byerror-oriented global Gauss-Newton method

}F 1pppkqq∆ppkq � F pppkqq}2 Ñ min

ppk�1q � ppkq � λk∆ppkq, k � 0, 1, 2, . . .

[Deuflhard: Newton Methods for Nonlinear Problems, 2004]

� sequence of linear least squares problems with pm � qqJacobian matrix F 1ppq

� good initial guess required (model decomposition)� F 1ppq gives us some hints whether the current combination of

model and data will permit an actual identification of theparameters


Identifiability

Matheon

� the rows of F 1ppq contain the sensitivities of the measuredcomponents w. r. t. the parameters p � pp1, . . . , pqq

F 1ijppq �

B

Bpjyki pti , pq, i � 1, . . . ,m, ki P t1, . . . , nu

� solution of linear LSQ problems by QR factorization withcolumn pivoting

F 1ppqΠ � QR, r11 ¥ r22 ¥ . . . ¥ rqq

� detection of linear dependencies by monitoring thesubcondition numbers

scj � r11{rjj

identifiable parameters: εxscj 1 [Deuflhard/Sautter 1980]

� estimation of incompatibility factor κ 1 as asymptoticconvergence rate


Simulation Results


Results

Matheon

� normal cycle simulation

0 10 20 300

50

100

150

mIU

/mL

LH

0 10 20 300

5

10

15

20

mIU

/mL

FSH

0 10 20 300

5

10

15

20

25

30

ng/m

L

P4

0 10 20 300

100

200

300

400

500

pg/m

L

E2

� simulating the effect ofbirth control pills

0 50 100 150 200 250 3000

20

40

60

80

100

120

day

LH P4 estrogens


Applications

MatheonAnalyzing the role of dose and timing of certain drugs

� single dose agonist (nafarelin)

−20 0 20 40 60 80 1000

50

100

150

200

day−20 0 20 40 60 80 1000

50

100

150

200

day−20 0 20 40 60 80 1000

50

100

150

200

day

� multiple dose agonist (nafarelin)

−20 0 20 40 60 80 100 120 1400

5

10

15

20

days

ng/m

L

datasimulated P4

� single dose antagonist(cetrorelix)

−30 −20 −10 0 10 200

100

200

300

days

pg/m

L

dataE2


Perspective

Matheon

PAEON: Model-Driven Computation of Treatments for InfertilityRelated Endocrinological Diseases

� 3 years EU project

� development ofpatient-specific models

� cooperation partners:Universita di Roma ”LaSapienza”,Hochschule Luzern,Universitatsspital Zurich,

Med. Hochschule Hannover


Further topics


ToxoMod

Matheon

Model-based spectrometer calibration for toxin determination in food

A: antibodyX: analyte (mycotoxins)F: fluorescence-marked analyte

A+F+Xk1

GGGGGGBFGGGGGG

k2

AFk3

GGGGGGBFGGGGGG

k4

AF2

k5 ÓÒ k6 k9 ÓÒ k10

AXk11

GGGGGGGBFGGGGGGG

k12

AFX

k7 ÓÒ k8

AX2

Polarisation of fluorescent light:

P �pF rF sQF � pAF rAF sQAF � 2pAF2

rAF2sQAF2

rF sQF � rAF sQAF � 2rAF2sQAF2

aokin spectrometer FP470


BovCycle

MatheonA mathematical model of the bovine estrous cycle15 ODEs, 60 parameters

Inhibin

Estradiol

FSH Pituitary

LH Blood

FSH Blood

+

−

−

+−

+

+

−

+

−

T

T

T

T

T

T +

T

TT

T

+

TT

T

T

T

T

+

+

T +

++

+

+

+

T

+

T

T

T

T+

+T

+

α

Oxytocin Enzymes

PGF2

Follicles

Corpus Luteum

IOF

Progesterone

GnRH Hypothalamus GnRH Pituitary

LH Pituitary

Cooperations:FU Veterinary MedicineU Wageningen

0 10 20 30 400

0.2

0.4

0.6

0.8

1

days

rela

tive level

FSH

LH

P4

E2

0 10 20 30 400

0.2

0.4

0.6

0.8

1

days

rela

tive level

FSH

LH

P4

E2

analysis of follicular wave patterns

−25 −20 −15 −10 −5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 950

0.2

0.4

0.6

0.8

1

1.2

days

Fo

ll

1st PGFadmin

2nd PGF admin cow1

cow2

cow3

cow4

cow5

cow6

numerical validation of synchronization

protocols

[Boer, Stotzel, Roblitz, Deuflhard, Veerkamp, Woelders, J. Theoret. Biol. 2011][Stotzel, Plontzke, Heuwieser, Roblitz, Theriogenology 2012]


Discrete/continuous modeling

Matheon

Logic-based models:

� discrete variables and parameters� efficient analysis methods

� modularization – state spaceanalysis

� model checking –parameter/state space analysis

1010110001

00001

01001

01101

01201

11201

10201

00000010000110001200

11200

11210

11110

11010

01010

01020

01110 01210

01120

00020

00120

00220

01220

01221

01211

PGF

CL

LH

Foll

FSH

Integrated approach:

� parameteridentification

� comprehensive statespace analysis

with A. Bockmayr, H. Siebert

(FU Berlin)


Deterministic/stochastic modeling

MatheonRare events in chemical reaction systems

A B

mutually repressing gene pair (e.g. bacteriophage-λ) with two competing proteins

[Gardner et al., Nature 403 (2000)]

ordinary differential equationsA1 � c1{pc2 � Bβq � c3AB 1 � c4{pc5 � Aγq � c6B

continuous-time discrete-spaceMarkov chain (SSA [Gillespie, 1977])

0 100 200 3000

50

100

150

200

250

300

0 100 200 3000

50

100

150

200

250

300


Deterministic/stochastic modeling

Matheononly 5 transitions between tpA,Bq : A ¡ Bu and tpA,Bq : A Buwithin 5 � 104 steps ñ poor statistics

29971 : 20016 (theoretically 1:1)

0 100 200 3000

50

100

150

200

250

300

350

A

B

0 1 2 3 4 5x 10

6

0

50

100

150

200

250

300

time

A

0 1 2 3 4 5x 10

6

0

50

100

150

200

250

300

time

B

reduced description of the dynamical system in terms of nearlyinvariant (metastable) sets/functions

Qc � 10�5

��0.3288 0.32880.3094 �0.3094


Deterministic/stochastic modeling: CME

MatheonChemical Master Equation (CME)

Btppx , tq �R

r�1x�νr¥0

αr px � νr qppx � νr , tq �R

r�1x�νr¥0

αr pxqppx , tq

Solution by meshfree discrete Galerkin methods

0 100 200 3000

50

100

150

200

250

300

350

0 100 200 3000

50

100

150

200

250

300

350

0 100 200 3000

50

100

150

200

250

300

350

0 100 200 3000

50

100

150

200

250

300

350

application to a model for differentiation ofprogenitor cells into bone or cartilage cells(with S. Waldherr, FU)


Software


BioPARKIN

Matheon

integrated software environment for simulation and parameteridentification www.bioparkin.zib.de


www.bioparkin.zib.de

BioPARKIN

Matheon� C++ stand alone library for numerical routines� Graphical user interface for intuitive model handling� cross-platform development for Windows, Mac OS X, Linux� open source� SBML import and export� efficient integrator for differential-algebraic equations (LIMEX

Linearly Implicit Euler method with EXtrapolation, highlycited in Google-Scholar)http://www.zib.de/en/numerik/software/codelib/ivpode.html

� sensitivity analysis based on variational equations� parameter estimation via Gauss-Newton methods (tunable, in

particular scaling of species and parameters possible)� output of information on identifiability of parameters� time-shifting of data and concatination of IVPs for

multi-experiment simulations


http://www.zib.de/en/numerik/software/codelib/ivpode.html

Thank you for your attention!

Contact:Zuse Institute Berlin

Computational Systems Biology Grouphttp://www.zib.de/en/numerik/csb.html


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Systems biology and beyond From data to mathematical models