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SYSMETAB Non Stationary Metabolic flux analysis in isotope labeling experiments using the adjoint approach S. Mottelet 1 , G. Gaullier 2 Universit e de Technologie de Compi` egne 1 Laboratoire TIMR EA 4297 2 Laboratoire LMAC EA 2222 S.

  1. SYSMETAB Non Stationary Metabolic flux analysis in isotope labeling experiments using the adjoint approach S. Mottelet 1 , G. Gaullier 2 Universit´ e de Technologie de Compi` egne 1 Laboratoire TIMR EA 4297 2 Laboratoire LMAC EA 2222 S. Mottelet, G. Gaullier (UTC) SYSMETAB 1 / 16

  2. The project 2013-2016 Post-doctoral fellows G. Sadaka (2013-2014), G. Gaullier (2015-2016) This work was performed, in partnership with the SAS PIVERT, within the frame of the French Institute for the Energy Transition (Institut pour la Transition Energ´ etique - ITE) P .I.V.E.R.T. (www.institut-pivert.com) selected as an Investment for the Future (”Investissements d’Avenir”). This work was supported, as part of the Investments for the Future, by the French Government under the reference ANR-001-01” Work Package 4 ”Metabolism of lipids from the plant to microorganisms” (sub-task 4.1.22 ”Metabolic Modelling” joint work with INRA-BIMLip, UMR 6022 GEC - UTC). S. Mottelet, G. Gaullier (UTC) SYSMETAB 2 / 16

  3. Outline A model problem and some important notions Non stationary MFA : motivation, software Sysmetab, the adjoint approach and other innovations Trends and conclusion S. Mottelet, G. Gaullier (UTC) SYSMETAB 3 / 16

  4. A model problem and some important notions Monod/Michaelis kinetics Continuous enzymatic reaction Q v Q � ! S � ! P � ! v ( S ) = V max S Q < V max K + S , S 0 = � v ( S ) + Q , t > 0 Metabolic stationary state P 0 = � S 0 , t !1 S ( t ) = S ⇤ lim S ( 0 ) = P ( 0 ) = 0 . How to identify K et V max when the metabolic stationary state is reached and maintained ? � ! feed the reactor with a mix of two isotopomers of S = S 1 + S 2 : Q 1 v 1 Q 1 Q 2 v 2 Q 2 � ! S 1 � ! P 1 � ! � ! S 2 � ! P 2 � ! Q 1 = α Q , Q 2 = ( 1 � α ) Q , S = S 1 + S 2 , P = P 1 + P 2 . S. Mottelet, G. Gaullier (UTC) SYSMETAB 4 / 16

  5. A model problem and some important notions Monod/Michaelis kinetics Continuous enzymatic reaction Q v Q � ! S � ! P � ! v ( S ) = V max S Q < V max K + S , S 0 = � v ( S ) + Q , t > 0 Metabolic stationary state P 0 = � S 0 , t !1 S ( t ) = S ⇤ lim S ( 0 ) = P ( 0 ) = 0 . How to identify K et V max when the metabolic stationary state is reached and maintained ? � ! feed the reactor with a mix of two isotopomers of S = S 1 + S 2 : Q 1 v 1 Q 1 Q 2 v 2 Q 2 � ! S 1 � ! P 1 � ! � ! S 2 � ! P 2 � ! Q 1 = α Q , Q 2 = ( 1 � α ) Q , S = S 1 + S 2 , P = P 1 + P 2 . S. Mottelet, G. Gaullier (UTC) SYSMETAB 4 / 16

  6. A model problem and some important notions Monod/Michaelis kinetics Continuous enzymatic reaction Q v Q � ! S � ! P � ! v ( S ) = V max S Q < V max K + S , S 0 = � v ( S ) + Q , t > 0 Metabolic stationary state P 0 = � S 0 , t !1 S ( t ) = S ⇤ lim S ( 0 ) = P ( 0 ) = 0 . How to identify K et V max when the metabolic stationary state is reached and maintained ? � ! feed the reactor with a mix of two isotopomers of S = S 1 + S 2 : Q 1 v 1 Q 1 Q 2 v 2 Q 2 � ! S 1 � ! P 1 � ! � ! S 2 � ! P 2 � ! Q 1 = α Q , Q 2 = ( 1 � α ) Q , S = S 1 + S 2 , P = P 1 + P 2 . S. Mottelet, G. Gaullier (UTC) SYSMETAB 4 / 16

  7. A model problem and some important notions Monod/Michaelis kinetics Continuous enzymatic reaction Q v Q � ! S � ! P � ! v ( S ) = V max S Q < V max K + S , S 0 = � v ( S ) + Q , t > 0 Metabolic stationary state P 0 = � S 0 , t !1 S ( t ) = S ⇤ lim S ( 0 ) = P ( 0 ) = 0 . How to identify K et V max when the metabolic stationary state is reached and maintained ? � ! feed the reactor with a mix of two isotopomers of S = S 1 + S 2 : Q 1 v 1 Q 1 Q 2 v 2 Q 2 � ! S 1 � ! P 1 � ! � ! S 2 � ! P 2 � ! Q 1 = α Q , Q 2 = ( 1 � α ) Q , S = S 1 + S 2 , P = P 1 + P 2 . S. Mottelet, G. Gaullier (UTC) SYSMETAB 4 / 16

  8. A model problem and some important notions Isotopic labeling S 1 = 13 C 6 H 12 O 6 , S 2 = 12 C 6 H 12 O 6 Under the hypothesis justifying the Monod/Michaelis kinetics S S 0 = � V max K + S + Q , S 1 S 0 1 = � V max K + S + α Q , S 2 S 0 2 = � V max K + S + ( 1 � α ) Q . Isotopic labeling experiment I t < T m , α = 0 until the metabolic stationary state is reached I t ≥ T m , α = 1 until the isotopic stationary state is reached S. Mottelet, G. Gaullier (UTC) SYSMETAB 5 / 16

  9. A model problem and some important notions Isotopic labeling S 1 = 13 C 6 H 12 O 6 , S 2 = 12 C 6 H 12 O 6 Under the hypothesis justifying the Monod/Michaelis kinetics S S 0 = � V max K + S + Q , S 1 S 0 1 = � V max K + S + α Q , S 2 S 0 2 = � V max K + S + ( 1 � α ) Q . Isotopic labeling experiment I t < T m , α = 0 until the metabolic stationary state is reached I t ≥ T m , α = 1 until the isotopic stationary state is reached S. Mottelet, G. Gaullier (UTC) SYSMETAB 5 / 16

  10. A model problem and some important notions Isotopic labeling S 1 = 13 C 6 H 12 O 6 , S 2 = 12 C 6 H 12 O 6 Under the hypothesis justifying the Monod/Michaelis kinetics S S 0 = � V max K + S + Q , S 1 S 0 1 = � V max K + S + α Q , S 2 S 0 2 = � V max K + S + ( 1 � α ) Q . Isotopic labeling experiment I t < T m , α = 0 until the metabolic stationary state is reached I t ≥ T m , α = 1 until the isotopic stationary state is reached S. Mottelet, G. Gaullier (UTC) SYSMETAB 5 / 16

  11. A model problem and some important notions For t � 50 a first order kinetics is observed for S 1 and S 2 S 0 1 = � µ S 1 + α Q , S 0 2 = � µ S 2 + ( 1 � α ) Q , V max KQ S ⇤ = µ = K + S ⇤ , V max � Q µ can be estimated from S 1 and/or S 2 for t 2 [ 50 , 60 ] S. Mottelet, G. Gaullier (UTC) SYSMETAB 6 / 16

  12. A model problem and some important notions For t � 50 a first order kinetics is observed for S 1 and S 2 S 0 1 = � µ S 1 + α Q , S 0 2 = � µ S 2 + ( 1 � α ) Q , V max KQ S ⇤ = µ = K + S ⇤ , V max � Q µ can be estimated from S 1 and/or S 2 for t 2 [ 50 , 60 ] S. Mottelet, G. Gaullier (UTC) SYSMETAB 6 / 16

  13. A model problem Actual state variables in MFA Isotopomer fractions : s i = S i S Complete kinetics (metabolic/isotopic) S 0 = � v ( S ) + Q , ( Ss 1 ) 0 = � v ( S ) s 1 + α Q , ( Ss 2 ) 0 = � v ( S ) s 2 + ( 1 � α ) Q . the reaction rate is shared proportionally to isotopomer fractions Sole isotopic kinetics S ⇤ s 0 1 = � v ( S ⇤ ) s 1 + α Q , S ⇤ s 0 2 = � v ( S ⇤ ) s 2 + ( 1 � α ) Q . S. Mottelet, G. Gaullier (UTC) SYSMETAB 7 / 16

  14. A model problem Actual state variables in MFA Isotopomer fractions : s i = S i S Complete kinetics (metabolic/isotopic) S 0 = � v ( S ) + Q , ( Ss 1 ) 0 = � v ( S ) s 1 + α Q , ( Ss 2 ) 0 = � v ( S ) s 2 + ( 1 � α ) Q . the reaction rate is shared proportionally to isotopomer fractions Sole isotopic kinetics S ⇤ s 0 1 = � v ( S ⇤ ) s 1 + α Q , S ⇤ s 0 2 = � v ( S ⇤ ) s 2 + ( 1 � α ) Q . S. Mottelet, G. Gaullier (UTC) SYSMETAB 7 / 16

  15. A model problem Actual state variables in MFA Isotopomer fractions : s i = S i S Complete kinetics (metabolic/isotopic) S 0 = � v ( S ) + Q , ( Ss 1 ) 0 = � v ( S ) s 1 + α Q , ( Ss 2 ) 0 = � v ( S ) s 2 + ( 1 � α ) Q . the reaction rate is shared proportionally to isotopomer fractions Sole isotopic kinetics S ⇤ s 0 1 = � v ( S ⇤ ) s 1 + α Q , S ⇤ s 0 2 = � v ( S ⇤ ) s 2 + ( 1 � α ) Q . S. Mottelet, G. Gaullier (UTC) SYSMETAB 7 / 16

  16. Non stationary MFA Motivation Why make some experiments once the metabolic state is reached ? I The particular form of the kinetics (and associated parameters) disappears and is replaced by asymptotic reaction rates, called fluxes I Remaining unknowns : fluxes and asymptotic concentration of metabolites called pool sizes I If the reaction network is exhaustive, the model is reliable Why make some experiments before the isotopic stationary state is reached ? I Cost of the labeled substrate I Time constants can be very large for some organism I Qualitative and quantitative gain of information (narrower confidence intervals for fluxes) S. Mottelet, G. Gaullier (UTC) SYSMETAB 8 / 16

  17. Non stationary MFA Software INCA/MFA suite (Young et al., Vanderbilt Univ., Nashville, 10/2013) I BFGS inspired, Matlab implementation, only MS and H1-RMN measurements, closed source Sysmetab (Mottelet et al., UTC, 1/2016) I BFGS inspired / feasible directions (FSQP), Scilab implementation, adjoint approach, open-source influx si (Sokol et al., LISBP Toulouse, 4/2016) I Gauss-Newton inspired / feasible directions (NLSIC), R/C++ implementation, open-source S. Mottelet et al., Metabolic Flux Analysis in Isotope Labeling Experiments using the Adjoint Approach, IEEE/ACM Transactions on Computational Biology and Bioinformatic , 2016 (to appear, DOI : 10.1109/TCBB.2016.2244299) S. Mottelet, G. Gaullier (UTC) SYSMETAB 9 / 16

  18. Non stationary MFA The identification problem in continuous time State equation for t 2 [ 0 , T ] D ( s ⇤ ) x 0 ( t ) = f ( x ( t ) , v ) , x ( t ) 2 R N is the vector of cumomers x 1 ( t ) , . . . , x n ( t ) Observation at times τ i , i = 1 . . . m n X y ( τ i ) = C 0 + Cx ( t ) = C 0 + C k x k ( τ i ) , i = 1 . . . m k = 1 Cost function (where z i is the measurement at time τ i ) m X k y ( τ i ) � z i k 2 J ( v , s ⇤ ) = i = 1 Identification problem : find v , s ⇤ ( p parameters) minimizing J ( v , s ⇤ ) S. Mottelet, G. Gaullier (UTC) SYSMETAB 10 / 16

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