How data assimilation helps to illuminate complex biology

Transcription

1 How data assimilation helps to illuminate complex biology The dynamic elastic-net Maik Kschischo Department of Mathematics and Technology University of Applied Sciences Koblenz RheinAhrCampus Remagen Joseph-Rovan-Allee Remagen kschischo@rheinahrcampus.de RISDA 2017 Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

2 Overview 1 Some challenges for modelling in molecular systems biology 2 The dynamic elastic net: An algorithm for state and model error estimation in imperfect models 3 Test examples 4 Outlook Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

3 How to model a complex dynamical reaction system? Pathways in Cancer (KEGG hsa05200, 10/23/25, Kaneshisa Lab) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

4 Modelling subnetworks Pathways in Cancer (KEGG hsa05200, 10/23/25, Kaneshisa Lab) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

5 Crosstalk and external feedback loops Pathways in Cancer (KEGG hsa05200, 10/23/25, Kaneshisa Lab) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

6 Challenges for Modelling Kahm et al., PLoS Comput Biol 8, 2012; Tsigkinopoulou et al., Trends in Biotechnology, 2016 We have only incomplete or uncertain information about the Reaction network Reaction rate functions (dynamic laws of the reactions) Parameter values Data: Molecular concentration measurements over time (e.g. protein) Not all the molecular substances can directly be measured Measurements are often very noisy Some proteins can not be identified Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

7 Challenges for Modelling Kahm et al., PLoS Comput Biol 8, 2012; Tsigkinopoulou et al., Trends in Biotechnology, 2016 We have only incomplete or uncertain information about the Reaction network Reaction rate functions (dynamic laws of the reactions) Parameter values Data: Molecular concentration measurements over time (e.g. protein) Not all the molecular substances can directly be measured Measurements are often very noisy Some proteins can not be identified Methods for modelling under uncertain and incomplete information are needed! Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

8 Outline 1 Some challenges for modelling in molecular systems biology 2 The dynamic elastic net: An algorithm for state and model error estimation in imperfect models 3 Test examples 4 Outlook Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

9 Systems models and influence graphs State vector: x = (x 1,..., x n ) T Output (measured): y = (y 1,..., y m ) T u Ordinary differential equation (ODE) model Influence graph ẋ = f (x, u) y = h(x) Each state variable x i is a vertex Draw a directed edge x i x j iff f j x i 0 Legend: known input u state variable: x i model system systems border interaction Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

10 The nominal model and the true system a u b u c u Legend: nominal model system true system nominal model with unknown inputs known input u nominal systems border hidden state variable unknown input nominal systems state variable: interaction in nominal system hidden interaction Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

11 Defining the model error Ordinary differential equation (ODE) models Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

12 Defining the model error Ordinary differential equation (ODE) models The nominal model (the draft) x = f ( x, u) ỹ = h( x) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

13 Defining the model error Ordinary differential equation (ODE) models The nominal model (the draft) x = f ( x, u) ỹ = h( x) True model (with unkown error) ẋ = f (x, u) + w y = h (x) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

14 Defining the model error Ordinary differential equation (ODE) models The nominal model (the draft) x = f ( x, u) ỹ = h( x) True model (with unkown error) ẋ = f (x, u) + w y = h (x) Model error = hidden input to nominal model w(t) = ẋ(t) f (x(t), u(t)) Note: x(t) is the trajectory of the true system. Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

15 Data assimilation problem with structural model error What we have: The nominal model (a model draft) Data: y(t k ) for some time points t k, k = 1,..., N Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

16 Data assimilation problem with structural model error What we have: The nominal model (a model draft) Data: y(t k ) for some time points t k, k = 1,..., N What we want to know: 1 Where is the nominal model wrong? Set of target nodes W = {i {1,..., n} w i 0} 2 The dynamics of the model error w(t) 3 The unknown state trajectory x(t) of the true system Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

17 Data assimilation problem with structural model error What we have: The nominal model (a model draft) Data: y(t k ) for some time points t k, k = 1,..., N This is an ill posed problem! What we want to know: 1 Where is the nominal model wrong? Set of target nodes W = {i {1,..., n} w i 0} 2 The dynamics of the model error w(t) 3 The unknown state trajectory x(t) of the true system Different models/model errors could explain the data equally well = (Non-)observability of the model error Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

18 Data assimilation with structural model error Additional assumptions The model error signal w(t) 1 is a smooth function 2 a sparse vector The set of target nodes W = {i {1,..., n} wi 0} is small Most parsimonious explanation for observed discrepancy between the nominal model output and the data Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

19 Estimating the model error using the Dynamic Elastic-Net Which simple error signal could fit the data? Data: Measure y(t k ) for some time points t k, k = 1,..., N Idea: Build an observer system (copy of the true system): ˆx = f (ˆx(t), u(t)) + ŵ(t) ŷ(t) = h(ˆx(t)) Infer the simplest ŵ(t) that explains the data: { N } min y(t k ) ŷ(t k ) 2 Q(t ŵ k ) + R[ŵ] (t) k=1 L1 and L 2 regularisation (dynamic elastic-net) induces sparsity and smoothness R[ŵ] = α 1 ŵ(t) 1 + α 2 2 ŵ(t) 2 2 Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

20 Outline 1 Some challenges for modelling in molecular systems biology 2 The dynamic elastic net: An algorithm for state and model error estimation in imperfect models 3 Test examples 4 Outlook Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

21 Testing the algorithm Example 1: The Jak-Stat-model from Swameye et al., PNAS, Nominal Model of Swameye et al.: Epo binds the EpoR (receptor) Cytoplasmatic STAT5 ist activated Phosphorylated STAT5 dimerises Export to the nucleous Swameye et al., PNAS, 100, 2001, Figure 1 on page Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

22 Testing the algorithm Example 1: The Jak-Stat-model from Swameye et al., PNAS, Nominal Model of Swameye et al.: Epo binds the EpoR (receptor) Cytoplasmatic STAT5 ist activated Phosphorylated STAT5 dimerises Export to the nucleous Missing reaction: Recycling of nuclear STAT (Raue et al., Bioinformatics, 2009.) Swameye et al., PNAS, 100, 2001, Figure 1 on page Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

23 Testing the algorithm Example 1: The Jak-Stat-model from Swameye et al., PNAS, Swameye et al., PNAS, 100, 2001, Figure 1 on page Nominal Model of Swameye et al.: Epo binds the EpoR (receptor) Cytoplasmatic STAT5 ist activated Phosphorylated STAT5 dimerises Export to the nucleous Missing reaction: Recycling of nuclear STAT (Raue et al., Bioinformatics, 2009.) Could this have been learned from the data? Apply the dynamic elastic-net Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

24 The Jak-Stat-System Detecting nucleocytoplasmatic cycling (data points from Swameye et al., 2001.) Legend: known input, data, nominal model, dynamic elastic-net a 1 pepor (a.u.) c b 1 total STAT5p (a.u.) c 1 total STAT5 (a.u.) ^y u(t) 0.5 ^y d x 1 x 2 x 3 x t (min) u(t) x^1 x^2 x^3 x^4 ^w 1 ^w 2 ^w 3 ^w 4 e AUC (a.u.) t (min) 0 ^w 1 ~ y 1 model error (a.u.) ^w 2 ^w 4 ^w t (min) f ~ y t (min) STAT5, cyt. (a.u.) x^ t (min) t (min) x~ ~ 1 x t (min) STAT5p, cyt. (a.u.) x^2 STAT5p-dimer, cyt. (a.u.) STAT5p-dimer, nucl.(a.u.) x~ 3 x^3 x~ 4 x^ t (min) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

25 Testing the algorithm Example 2: The UVB-system A model for photomorphogenic UV-B signalling in plants (Ouyang, X. et al. Proc. Natl. Acad. Sci. USA, 2014) 11 protein concentrations x(t) participating in 10 reactions The system was perturbed by a known model error Can we recover this error w(t) and the true state x(t)? Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

26 The UVB-system Detecting the error and the true state Legend: data, nominal model, dynamic elastic-net a x 5 x 7 b 40 c x x 9 10 x 3 x 6 (a.u.) x 2 x 1 10 x 11 x 4 x 10 x t (min) 2x 5 +2x 4 +x 8 (a.u.) t (min) d 20 e 25 f x 7 (a.u.) (w-ŵ)/a (a.u.) t (min) g AUC (a.u.) h 0.1 ^w 9 ^w 6 i ^w w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 8 w 9 w 10 w^ 11 t (min) 2x5+2x3+x1 (a.u.) x 8 (a.u.) state error (a.u.) t (min) t (min) t (min) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

27 Testing the limitations All single and pairwise combinations in the UVB-system a x1 3 x3 3, Na 3 x4 5, Na 5, 3 6 x5 5, 3 5, 3 5, 6 5 x6 6, 5 6, 3 6, Na 6, 5 6 x7 7, 3 7, 3 7, 8 7, 5 8, 6 7 x8 8, 7 5, 3 8, 6 8, 5 8, 6 8, 7 8 x9 9, 3 9, 3 9, 6 9, 5 9, 6 9, 7 9, 8 9 x1 x3 x4 x5 x6 x7 x8 x9 b c AUC (a.u.) AUC (a.u.) true target nodes: (9,1) no constraints w^ 3 =0 w ^ 1 w ^ 2 w ^ 3 w ^ 4 w ^ 5 w ^ 6 w ^ 7 w ^ 8 w ^ 9 w ^ 10 w^ 11 Some model errors are non-observable Simple algorithm to explore suboptimal solutions (next slide) The true target nodes are found amongst the highest ranking suboptimal solutions Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

28 Exploring suboptimal solutions A heuristic algorithm 1 Init B = I n (identity matrix) 2 Solve the DEN using the observer system ˆx = f (ˆx(t), u(t)) + Bŵ(t) and estimate ŵ(t) and Ŵ = {k ŵ k (t) 0} 3 If the squared error N k=1 y(t k) ŷ(t k ) 2 Q(t k ) is too big, then STOP. 4 Find the index k Ŵ of the smallest nonzero signal ŵ k (t) (as measured by L 1 norm) 5 Delete the k-th column in B and the corresponding row of ŵ and GOTO 2. Result: A set of suboptimal solutions of the DEN providing candidate model errors and state estimates. Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

29 Summary and Outlook Dynamic elastic-net Estimates the nodes in a network, where the model could be wrong and Estimates the true system state trajectory, even for incomplete or incorrect models Non-observable model error No unique solution possible Greedy approach to explore suboptimal solutions Further research Network properties and model error observability Optimal experimental design to improve observability of model error Assign uncertainty/confidence (Engelhardt et al., submitted) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

30 Bayesian Dynamic Elastic-Net (BDEN) JAK-STAT signalling pathway (Swameye et al., 2003) Uncertainty quantification for state and model error estimates (Engelhardt et al., submitted) Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23

31 Acknowledgement Thanks for the great collaboration Holger Fröhlich (1,2) Benjamin Engelhardt (1) (1) Rheinische Friedrich-Wilhelms-Universität Bonn, Institute for Computer Science, Algorithmic Bioinformatics, c/o Bonn-Aachen International Center for IT (2) UCB Biosciences GmbH For some details: Engelhardt, B., Fröhlich, H. and Kschischo, M Learning (from) the errors of a systems biology model. Scientific Reports 6: Maik Kschischo (RheinAhrCampus) Dynamic elastic-net RISDA / 23