Systems biology and beyond From data to mathematical models
Transcript of 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
MSB Seminar 3 Susanna Roblitz
Group Members
MatheonHead
Susanna RoblitzStaff Students
Rainald Ehrig Julia Plontzke
Claudia Stotzel Gabriel Muller
Thomas Dierkes Dany Pascal Moualeu
Mascha Berg
MSB Seminar 4 Susanna Roblitz
Activities
Matheon
AlgorithmsParameter identification
Deterministic/stochastic modelling
Discrete/continuous modelling
Endocrinological networks
Metabolic networks
Biosensors
Applications
SoftwareBioPARKIN (C++/Python)
POEM (Matlab)
MSB Seminar 5 Susanna Roblitz
Mathematical modelling of thehuman menstrual cycle
MSB Seminar 6 Susanna Roblitz
The human menstrual cycle
Matheon
(http://www.websters-online-dictionary.org/definitions/Menstrual Cycle)
MSB Seminar 7 Susanna Roblitz
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)
MSB Seminar 8 Susanna Roblitz
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)
MSB Seminar 9 Susanna Roblitz
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
MSB Seminar 10 Susanna Roblitz
Data
MSB Seminar 11 Susanna Roblitz
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)
MSB Seminar 12 Susanna Roblitz
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
MSB Seminar 13 Susanna Roblitz
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
MSB Seminar 14 Susanna Roblitz
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
MSB Seminar 15 Susanna Roblitz
Model Development
MSB Seminar 16 Susanna Roblitz
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
MSB Seminar 17 Susanna Roblitz
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
MSB Seminar 18 Susanna Roblitz
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)]
MSB Seminar 19 Susanna Roblitz
Parameter Identification
MSB Seminar 20 Susanna Roblitz
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
MSB Seminar 21 Susanna Roblitz
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
MSB Seminar 22 Susanna Roblitz
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
MSB Seminar 23 Susanna Roblitz
Simulation Results
MSB Seminar 24 Susanna Roblitz
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
MSB Seminar 25 Susanna Roblitz
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
MSB Seminar 26 Susanna Roblitz
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
MSB Seminar 27 Susanna Roblitz
Further topics
MSB Seminar 28 Susanna Roblitz
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
MSB Seminar 29 Susanna Roblitz
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]
MSB Seminar 30 Susanna Roblitz
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)
MSB Seminar 31 Susanna Roblitz
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
MSB Seminar 32 Susanna Roblitz
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
MSB Seminar 33 Susanna Roblitz
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
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350
application to a model for differentiation ofprogenitor cells into bone or cartilage cells(with S. Waldherr, FU)
MSB Seminar 34 Susanna Roblitz
Software
MSB Seminar 35 Susanna Roblitz
BioPARKIN
Matheon
integrated software environment for simulation and parameteridentification www.bioparkin.zib.de
MSB Seminar 36 Susanna Roblitz
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
MSB Seminar 37 Susanna Roblitz
Thank you for your attention!
Contact:Zuse Institute Berlin
Computational Systems Biology Grouphttp://www.zib.de/en/numerik/csb.html
MSB Seminar 38 Susanna Roblitz