Controllability Metrics in Markov Decision Linear Models of Gene - PowerPoint PPT Presentation

The model Controllabilities Minimal Intervention Controllability Metrics in Markov Decision Linear Models of Gene Networks Dan Goreac 1 Journées ALEA, CIRM, March 21 st 2017 1 UPEM. Based on papers joint with T. Diallo, M. Martinez (UPEM), E. -P. Rotenstein (UAIC, Iasi, Romania), C. A. Grosu (UAIC, Iasi, Romania) Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Outline The model 1 Biochemical Reactions Mathematical Model The Questions Controllabilities 2 Metric By Observability Riccati Formulation of the Metric Main Theoretical Results Minimal Intervention 3 Optimization Problems Back to Lambda Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Lambda Phage Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Reference Model Biochemical reactions K 1 K 2 K 3 � R 2 , D (+ R 2 ) � DR 2 , D (+ R 2 ) � DR ∗ 2 R 1 2 , K 4 � DR 2 R 2 , DR 2 + P K t K d DR 2 (+ R 2 ) → DR 2 + P + rR 1 , R 1 → . Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Reference Model Biochemical reactions K 1 K 2 K 3 � R 2 , D (+ R 2 ) � DR 2 , D (+ R 2 ) � DR ∗ 2 R 1 2 , K 4 � DR 2 R 2 , DR 2 + P K t K d DR 2 (+ R 2 ) → DR 2 + P + rR 1 , R 1 → . Groups of reaction 2 , DR 2 R 2 ) T trend : DNA mechanism of the host E-Coli ( D , DR 2 , DR ∗ K 1 K d � R 2 , R 1 → update : 2 R 1 K t rare : DR 2 + P → DR 2 + P + rR 1 Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Reference Model Biochemical reactions K 1 K 2 K 3 � R 2 , D (+ R 2 ) � DR 2 , D (+ R 2 ) � DR ∗ 2 R 1 2 , K 4 � DR 2 R 2 , DR 2 + P K t K d DR 2 (+ R 2 ) → DR 2 + P + rR 1 , R 1 → . Groups of reaction 2 , DR 2 R 2 ) T trend : DNA mechanism of the host E-Coli ( D , DR 2 , DR ∗ K 1 K d � R 2 , R 1 → update : 2 R 1 K t rare : DR 2 + P → DR 2 + P + rR 1 Some mathematical models ( d 1 , d 2 , d 3 , d 4 , x 1 , x 2 ) : pure jump, 2-scale PDMP, Marked-point, discrete model Reaction speeds can be "chosen". Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Trend At time n , trend (DNA occupation) is L n E.g. If L n = D : D k 2 → DR 2 , D k 3 → DR ∗ 2 Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Trend At time n , trend (DNA occupation) is L n E.g. If L n = D : D k 2 → DR 2 , D k 3 → DR ∗ 2 � DR 2 If L n = D then L n + 1 = with proportional probability DR ∗ � 2 k 2 k 2 + k 3 k 3 k 2 + k 3 Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Trend At time n , trend (DNA occupation) is L n E.g. If L n = D : D k 2 → DR 2 , D k 3 → DR ∗ 2 � DR 2 If L n = D then L n + 1 = with proportional probability DR ∗ � 2 k 2 k 2 + k 3 k 3 k 2 + k 3 In general, since only one type of occupation, one gets basis vectors e 1 ( D ) , e 2 ( DR 2 ) ... e p Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Trend At time n , trend (DNA occupation) is L n E.g. If L n = D : D k 2 → DR 2 , D k 3 → DR ∗ 2 � DR 2 If L n = D then L n + 1 = with proportional probability DR ∗ � 2 k 2 k 2 + k 3 k 3 k 2 + k 3 In general, since only one type of occupation, one gets basis vectors e 1 ( D ) , e 2 ( DR 2 ) ... e p Take ∆ M n + 1 = L n + 1 − ” average ” (in fact E [ L n + 1 / F n ]) To make it simple , assume L n + 1 is completely independent of L n and has 0 − mean ∆ M n + 1 = L n + 1 Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Update K 1 K d � R 2 , R 1 2 R 1 → Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Update K 1 K d � R 2 , R 1 2 R 1 → continuous with choice of speed (u) : x � 1 = − k 1 ( u ) x 2 1 − k d ( u ) x 1 + k − 1 ( u ) x 2 x � 2 = k 1 ( u ) x 2 1 − k − 1 ( u ) x 2 Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Update K 1 K d � R 2 , R 1 2 R 1 → continuous with choice of speed (u) : x � 1 = − k 1 ( u ) x 2 1 − k d ( u ) x 1 + k − 1 ( u ) x 2 x � 2 = k 1 ( u ) x 2 1 − k − 1 ( u ) x 2 linearized − 1 x 2 + b 1 · u 1 = − 2 k eq 1 x eq 1 x 1 − k eq d x 1 + k eq x � 2 = 2 k eq 1 x eq 1 x 1 − k eq − 1 x 2 + b 2 · u x � Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Update K 1 K d � R 2 , R 1 2 R 1 → continuous with choice of speed (u) : x � 1 = − k 1 ( u ) x 2 1 − k d ( u ) x 1 + k − 1 ( u ) x 2 x � 2 = k 1 ( u ) x 2 1 − k − 1 ( u ) x 2 linearized − 1 x 2 + b 1 · u 1 = − 2 k eq 1 x eq 1 x 1 − k eq d x 1 + k eq x � 2 = 2 k eq 1 x eq 1 x 1 − k eq − 1 x 2 + b 2 · u x � or, again dX x , u = [ A ( γ s ) X x , u + B s u s ] ds s s Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Update K 1 K d � R 2 , R 1 2 R 1 → continuous with choice of speed (u) : x � 1 = − k 1 ( u ) x 2 1 − k d ( u ) x 1 + k − 1 ( u ) x 2 x � 2 = k 1 ( u ) x 2 1 − k − 1 ( u ) x 2 linearized − 1 x 2 + b 1 · u 1 = − 2 k eq 1 x eq 1 x 1 − k eq d x 1 + k eq x � 2 = 2 k eq 1 x eq 1 x 1 − k eq − 1 x 2 + b 2 · u x � or, again dX x , u = [ A ( γ s ) X x , u + B s u s ] ds s s discrete X x , u n + 1 = A n ( ω ) X x , u + Bu n + 1 n Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Rare (and Synthesis) � � � � K 2 K t � DR 2 , + R 2 D DR 2 + P → DR 2 + P + rR 1 . Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Rare (and Synthesis) � � � � K 2 K t � DR 2 , + R 2 D DR 2 + P → DR 2 + P + rR 1 . discrete � � � � � � � � x 1 , n + 1 x 1 , n 0 r x 1 , n = + x 2 , n + 1 x 2 , n 0 0 x 2 , n Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Rare (and Synthesis) � � � � K 2 K t � DR 2 , + R 2 D DR 2 + P → DR 2 + P + rR 1 . discrete � � � � � � � � x 1 , n + 1 x 1 , n 0 r x 1 , n = + x 2 , n + 1 x 2 , n 0 0 x 2 , n Continuous f ( config. DNA γ , reaction speeds u , fast variable X ) + B s u s ] ds + � dX x , u = [ A ( γ s ) X x , u E C ( γ s − , θ ) X x , u s − � q ( d θ ds ) s s Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks

The model Biochemical Reactions Controllabilities Mathematical Model Minimal Intervention The Questions Rare (and Synthesis) � � � � K 2 K t � DR 2 , + R 2 D DR 2 + P → DR 2 + P + rR 1 . discrete � � � � � � � � x 1 , n + 1 x 1 , n 0 r x 1 , n = + x 2 , n + 1 x 2 , n 0 0 x 2 , n Continuous f ( config. DNA γ , reaction speeds u , fast variable X ) + B s u s ] ds + � dX x , u = [ A ( γ s ) X x , u E C ( γ s − , θ ) X x , u s − � q ( d θ ds ) s s Discrete + Bu n + 1 + ∑ p X x , u n + 1 = A n ( ω ) X x , u i = 1 � ∆ M n + 1 , e i � C i , n ( ω ) X x , u n n Dan Goreac Controllability Metrics in Markov Decision Linear Models of Gene Networks