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ADAPT: Analysis of Dynamic Adaptations in Parameter Trajectories

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Part of the Training Course: Data Integration in the Life Sciences.
from 2 Feb 2015 through 6 Feb 2015, Lorentz Center, Leiden

Organized by ERA-Net program for Systems Biology Applications (ERASysApp, https://www.erasysapp.eu/) and the Dutch systems biology and bioinformatics community (BioSB, http://biosb.nl).

http://www.lorentzcenter.nl/lc/web/2015/684/description.php3?wsid=684&venue=Snellius

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ADAPT: Analysis of Dynamic Adaptations in Parameter Trajectories

  1. 1. 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
  2. 2. Objectives • Follow-up on parameter estimation • Propagation of Uncertainty • ADAPT / biomedical engineering PAGE 22/5/2015 SlideShare http://www.slideshare.net/natalvanriel measuring modelling
  3. 3. 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
  4. 4. Longitudinal - Treatment in time / biomedical engineering PAGE 45-2-2015
  5. 5. Preclinical study of pharmaceutical intervention • data: control, treated for 1, 2, 4, 7, 14, and 21 days / biomedical engineering PAGE 55-2-2015 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
  6. 6. Modelling / biomedical engineering PAGE 65-2-2015
  7. 7. 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 / biomedical engineering PAGE 75-2-2015
  8. 8. ADAPT Analysis of Dynamic Adaptations in Parameter Trajectories / biomedical engineering PAGE 85-2-2015 ? ? ?
  9. 9. / biomedical engineering PAGE 92/5/2015 Data integration via dynamic network models
  10. 10. System identification / biomedical engineering PAGE 105-2-2015 M.C. Escher
  11. 11. 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) / biomedical engineering PAGE 115-2-2015 biology physics model model scheme equations
  12. 12. ‘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) / biomedical engineering PAGE 122/5/2015 uu y y u
  13. 13. 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 / biomedical engineering PAGE 135-2-2015 parameters + - MODEL ERROR input MODEL OUTPUT MODEL ( ) ( | ) ( )d k y k k  
  14. 14. Dynamic systems and models • Dynamic system (state-space representation) • outputs: • initial conditions: • Stoichiometry matrix N / biomedical engineering PAGE 145-2-2015 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
  15. 15. / biomedical engineering PAGE 155-2-2015 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
  16. 16. Data interpolation Matlab • Linear interpolation interp1 • Cubic Spline interpolation csaps / biomedical engineering PAGE 165-2-2015 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
  17. 17. Parameter estimation for Dynamic models • Error model • Maximum Likelihood Estimation / biomedical engineering PAGE 175-2-2015 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 ( )      
  18. 18. / biomedical engineering PAGE 185-2-2015 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 
  19. 19. / biomedical engineering PAGE 195-2-2015 Parameter estimation for Dynamic models • Parameter estimation: nesting of 2 numerical schemes
  20. 20. / biomedical engineering PAGE 205-2-2015 Examples
  21. 21. 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… / biomedical engineering PAGE 212/5/2015 R1 u2 u1 1 S1 S3S2 S4 3 4 5 2 7 6 Van Riel et al. (2013) Interface Focus, 3(2): 20120084
  22. 22. A theoretical example Experimental data: • metabolic profile (S1, S2, S3, S4) • 5 stages / 5 ‘snapshots’ (time 1, 2, 3, 4, 5) / biomedical engineering PAGE 225-2-2015 R1 u2 u1 1 S1 S3S2 S4 3 4 5 2 7 6
  23. 23. 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 / biomedical engineering PAGE 235-2-2015 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
  24. 24. Estimate transcription rate k6 for the time points after the perturbation / biomedical engineering PAGE 245-2-2015 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
  25. 25. 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) / biomedical engineering PAGE 255-2-2015 stage 1 stage 5
  26. 26. 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) / biomedical engineering PAGE 265-2-2015 u2 u1 1 S1 S3S2 S4 3 4 5 2
  27. 27. 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 / biomedical engineering PAGE 275-2-2015 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’)
  28. 28. / biomedical engineering PAGE 285-2-2015 Identifiability and Uncertainty
  29. 29. / biomedical engineering PAGE 295-2-2015 The Elephant in the Room, Banksy exhibition, 2006
  30. 30. Bootstrapping • Sampling based method / biomedical engineering PAGE 302/5/2015 Vanlier et al. Math Biosci. 2013 Mar 25
  31. 31. Example cont’d – case 2 • Monte Carlo (drawing samples from the data distribution) • MLE (weighting with the data variance) / biomedical engineering PAGE 315-2-2015 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
  32. 32. / biomedical engineering PAGE 322/5/2015 ADAPT
  33. 33. Time-continuous description of the data • ADAPT accounts for uncertainty in the data • ADAPT accounts for potential differences in dynamic behavior / biomedical engineering PAGE 335-2-2015 Gaussian distribution Sampling replicates from error model ( , )d d N
  34. 34. Modelling phenotype transition (1) 34 treatment disease progression  longitudinal discrete data: different phenotypes
  35. 35. Introducing time-dependent parameters 35  steady state model
  36. 36. Parameter trajectory estimation 36  steady state model  iteratively calibrate model to data: estimate parameters over time minimize difference between data and model simulation
  37. 37. Parameter trajectory estimation 37  steady state model  iteratively calibrate model to data: estimate parameters over time
  38. 38. Parameter trajectory estimation 38  steady state model  iteratively calibrate model to data: estimate parameters over time
  39. 39. Modelling phenotype transition  longitudinal discrete data: different phenotypes  estimate continuous data: ensemble of cubic smooth spline  incorporate uncertainty in data: multiple describing functions / biomedical engineering PAGE 395-2-2015
  40. 40. Estimated parameter trajectories / biomedical engineering PAGE 402/5/2015
  41. 41. 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 / biomedical engineering PAGE 415-2-2015 u2 u1 1 S1 S3S2 S4 3 4 5 2 Van Riel et al. (2013) Interface Focus, 3(2): 20120084
  42. 42. ADAPT of lipoprotein and lipid metabolism • Connecting the longitudinal data • Taking into account uncertainties / biomedical engineering PAGE 425-2-2015 • Calculating unobserved quantities Tiemann et al. (2013) PLoS Comput Biol. 9: e1003166
  43. 43. 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. / biomedical engineering PAGE 432/5/2015 Systems Biology of Disease Progression http://www.youtube.com/watch?v=x54ysJDS7i8

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