ADAPT: Analysis of Dynamic Adaptations in Parameter Trajectories

Data Integration in the Life Sciences
Feb. 5, 2015, Lorentz Center, Leiden
Natal van Riel
Systems Biology and Metabolic Diseases
n.a.w.v.riel@tue.nl, GEM-Z 3.109, tel. 040 247 5506

Objectives
• Follow-up on parameter estimation
• Propagation of Uncertainty
• ADAPT
/ biomedical engineering PAGE 22/5/2015
SlideShare
http://www.slideshare.net/natalvanriel
measuring
modelling

Today’s team
• Karen van Eunen (UMCG)
• Yared Paalvast (UMCG)
• Bert Groen (UMCG)
• Yvonne Rozendaal (TU/e)
• Natal van Riel (TU/e)
/ biomedical engineering PAGE 35-2-2015

Longitudinal - Treatment in time

Preclinical study of pharmaceutical
intervention
• data: control, treated for 1, 2, 4, 7, 14, and 21 days
0 10 20
0
100
200
Hepatic TG
Time [days]
[umol/g]
0 10 20
0
1
2
3
Hepatic CE
Time [days]
[umol/g]
0 10 20
0
2
4
6
Hepatic FC
Time [days]
[umol/g]
0 10 20
0
50
100
Hepatic TG
Time [days]
[umol]
0 10 20
0
0.5
1
1.5
Hepatic CE
Time [days]
[umol]
0 10 20
0
2
4
Hepatic FC
Time [days]
[umol]
0 10 20
0
1000
2000
3000
Plasma CE
Time [days]
[umol/L]
0 10 20
0
1000
2000
3000
HDL-CE
Time [days]
[umol/L]
0 10 20
0
500
1000
1500
Plasma TG
Time [days]
[umol/L]
0 10 20
6
8
10
12
VLDL clearance
Time [days]
[-]
0 10 20
100
200
300
400
ratio TG/CE
Time [days]
[-]
0 10 20
0
5
10
15
VLDL diameter
Time [days]
[nm]
0 10 20
0
1
2
3
VLDL-TG production
Time [days]
[umol/h]
0 10 20
1
2
3
Hepatic mass
Time [days]
[gram]
0 10 20
0
0.2
0.4
DNL
Time [days]
[-]
Grefhorst et al. Atherosclerosis, 2012, 222: 382– 389

Modelling

Understanding (modeling) progressive diseases and effect of
treatment-in-time
Challenges:
• Many factors involved
• Different biological levels, many details unknown
• Dynamic interactions of molecular species, cells,
tissues/organs
• Multiple time scales (orders of magnitude different) - molecular
mechanisms governing cell behaviour versus gradual
(patho)physiological changes induced by a progressive disease
or therapeutic intervention
• In vivo values of parameters unknown

ADAPT
Analysis of Dynamic Adaptations in Parameter Trajectories
? ? ?

Data integration via dynamic
network models

System identification
M.C. Escher

Mechanism-based models for data integration
• Physical / biological interpretation of model variables and
parameters
• Structure based on known
physics and biology
• Parameter values estimated
from experimental data
(parameter identification)
biology physics
model model
scheme equations

‘Fitting’ of model to data
• Known from linear regression
• Which ‘estimator’?
• Which algorithm?
• What are the underlying principles?
• What is the effect of the uncertainty (‘noise’) in the data
• Can we get more out of this than a line through some
datapoints?
• Can we generalize this? (nonlinear, dynamic)
uu y
y u

Parameter Estimation
• Minimize the sum of squared model errors by varying model
parameters
• The parameter value for which criterion is minimal is the best
(most likely) estimate for the parameters
parameters
+
-
MODEL ERROR
input
MODEL OUTPUT
MODEL
( ) ( | ) ( )d k y k k  

Dynamic systems and models
• Dynamic system (state-space representation)
• outputs:
• initial conditions:
• Stoichiometry matrix N
u2
u1 1 S1
S3S2
S4
3
4 5
2
1 2 3 4 5v v v v v
1
2
3
4
1 0 1 1 0
1 1 0 0 0
1 1 0 0 0
0 0 0 1 1
S
S
S
S
  
  
 
 
 
N

Dynamic systems and models
• Network structure and stoichiometry are fixed
• Variables: concentrations S (in x)
reaction rates v (in f)
• Parameters Vmax, Km, …
• In general, output y(t) cannot be calculated analytically, but
results from numerical simulation
• Matlab ODE suite, e.g. ode45, ode15s
• Mathematical model: continuous time
• Computational model: discrete time
( , , )x f x u t
y(t)u(tk)u(t) u(k)~
interpolate
y(tk)
1 2
1
2
( ) ( )
( )
( )
max
m
u t S t
v t V
K S t


A ‘driving’ / ‘forcing’ function
measured data is interpolated and used as input
Cubic spline
interpolation

Data interpolation
Matlab
• Linear interpolation
interp1
• Cubic Spline interpolation
csaps
0 30 60 90 120 150 180
5
5.5
6
6.5
7
7.5
8
8.5
time [min]
G[mmol/L]
raw data
spline interpolation
0 30 60 90 120 150 180
5
5.5
6
6.5
7
7.5
8
8.5
time [min]
G[mmol/L]
raw data
linear interpolation

Parameter estimation for Dynamic models
• Error model
• Maximum Likelihood Estimation
2
2
1 1
( ) ( | )
( )
n N
i i
i k ik
d k y k 
 
 
 
  
 

( ) ( | )i id k y k  
( | ) ( )i iy k k  
2
ˆ 0
ˆ arg min ( )

  



Unknowns to be estimated
• Initial conditions of dynamic models x0 often not known for
biological / biomedical systems
• If measured → uncertainty / error
• So typically
• But potentially not all parameters/initial conditions need to be
estimated
0[ , ]p x 
0[ ', ']p x  0 0' 'p p x x 

Parameter estimation for Dynamic models
• Parameter estimation: nesting of 2 numerical schemes

Examples

A theoretical example
• A metabolic system with
metabolite controlled,
negative transcriptional
feedback
• A progressive
perturbation acting on
the gene/protein circuit
encoding the repressor
• Time scales relevant to this phenotype:
• Metabolic network – seconds
• Gene regulatory circuit – minutes/hours
• Progressive adaptation to the perturbation – days…
R1
u2
u1 1 S1
S3S2
S4
3
4 5
2
7
6
Van Riel et al. (2013) Interface Focus, 3(2): 20120084

A theoretical example
Experimental data:
• metabolic profile (S1, S2, S3, S4)
• 5 stages / 5 ‘snapshots’
(time 1, 2, 3, 4, 5)
R1
u2
u1 1 S1
S3S2
S4
3
4 5
2
7
6

R1
u2
u1 1 S1
S3S2
S4
3
4 5
2
7
6
Case 1: one model for each stage
• Transcription:
• Simulate steady-state xss
• Infer values for from the data for stage 2, 3, 4, 5
• Stoichiometry matrix
• ODE model
1 2
1 max
1i
u S
v V
K R


 
( )
( ), , ( )
d t
f t t
dt

x
N x p u
6 6 4
6 0.01
v k S
k
 

6
ˆk
1 0 1 1 0
1 1 0 0 0
1 1 0 0 0
0 0 0 1 1
  
  
 
 
 
N

Estimate transcription rate k6 for the time
points after the perturbation
R1
u2
u1 1 S1
S3S2
S4
3
4 5
2
7
6
• Statistically acceptable fits and
accurate parameter estimates
0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
days
S1
S2
S3
S4
1 2 3 4 5

Results case 1
• Case 1:
• Metabolic level: topology and interaction kinetics known
• Gene / protein level: topology known, kinetic parameters
unknown (changing)
• Kinetic parameters of the gene/protein circuit estimated from
experimental observations at the metabolic level (metabolic
profiling) during the different stages of progression
• Resulting in 5 separate simulation models (one for each stage)
stage 1 stage 5

Case 2: Lacking information at gene/protein
level
• Next, a more challenging but common scenario is explored:
• Metabolic level: topology known, uncertainty in interaction
kinetics (kinetic parameters)
• Gene / protein level: from functional genomics studies we
know that the intervention affects a gene/protein controlling
reaction 1 (but molecular details are lacking)
• Same experimental observations, reflecting progressive
metabolic adaptations after an intervention at time 0 (stage 1)
u2
u1 1 S1
S3S2
S4
3
4 5
2

Analyze the data as individual ‘snapshots’
• Metabolic network without feedback
• The unknown adaptation at gene/protein level is translated into
an unknown, but inferable value for the metabolic rate constant
• However, like in the approach with case 1, this ignores the fact
that the snapshots are linked
1 1 1 2
ˆv k u Smax
1 1 2
4( )m
V
v u S
K f S


 
( )
( ), , ( )
d t
f t t
dt

x
N x p u
u2
u1 1 S1
S3S2
S4
3
4 5
2
phenomenological parameter
k1 (‘undermodeling’)

Identifiability and Uncertainty

The Elephant in the Room,
Banksy exhibition, 2006

Bootstrapping
• Sampling based method
Vanlier et al. Math Biosci. 2013 Mar 25

Example cont’d – case 2
• Monte Carlo (drawing samples from the data distribution)
• MLE (weighting with the data variance)
u2
u1 1 S1
S3S2
S4
3
4 5
2
Simulation of
the five
models,
with the
mean value
of the
ensemble of
parameter k1
for the
different
stages.
k1

ADAPT

Time-continuous description of the data
• ADAPT accounts for uncertainty in the data
• ADAPT accounts for potential differences in dynamic behavior
Gaussian distribution
Sampling replicates from error model
( , )d d N

Modelling phenotype transition (1)
34
treatment
disease progression
 longitudinal discrete data: different phenotypes

Introducing time-dependent parameters
35
 steady state model

Parameter trajectory estimation
36
 iteratively calibrate model to data: estimate parameters over time
minimize difference between data and model simulation

37

38

Modelling phenotype transition
 longitudinal discrete data: different phenotypes
 estimate continuous data: ensemble of cubic smooth spline
 incorporate uncertainty in data: multiple describing functions

Estimated parameter trajectories

Results with ADAPT
• Using the model of the metabolic network to integrate and
connect metabolomic data obtained at different stages of
progressive adaptations after an intervention
u2
u1 1 S1
S3S2
S4
3
4 5
2
Van Riel et al. (2013) Interface Focus, 3(2): 20120084

ADAPT of lipoprotein and lipid metabolism
• Connecting the longitudinal data
• Taking into account uncertainties
• Calculating unobserved quantities
Tiemann et al. (2013) PLoS
Comput Biol. 9: e1003166

Literature
• Hijmans BS, Tiemann CA, Grefhorst A, Boesjes M, van Dijk TH, Tietge UJ, Kuipers F,
van Riel NA, Groen AK, Oosterveer MH. A systems biology approach reveals the
physiological origin of hepatic steatosis induced by liver X receptor activation. FASEB
Journal, 2014 Dec 4. [Epub ahead of print]
• Tiemann CA, Vanlier J, Hilbers PA, and van Riel NA. Parameter adaptations during
phenotype transitions in progressive diseases. BMC Syst Biol. 5:174, 2011.
• Tiemann CA, Vanlier J, Oosterveer MH, Groen AK, Hilbers PAJ, and van Riel NAW.
Parameter trajectory analysis to identify treatment effects of pharmacological
interventions. PLoS computational biology 9: e1003166, 2013.
• van Riel NA, Tiemann CA, Vanlier J, and Hilbers PA. Applications of analysis of
dynamic adaptations in parameter trajectories. Interface Focus 3(2): 20120084, 2013.
Systems Biology of Disease Progression
http://www.youtube.com/watch?v=x54ysJDS7i8

ADAPT: Analysis of Dynamic Adaptations in Parameter Trajectories

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