Asymptotic convergence study of a Partial Integro-Differential - PowerPoint PPT Presentation

Asymptotic convergence study of a Partial Integro-Differential Equation (PIDE) used to model gene regulatory networks. Manuel P´ ajaro Di´ eguez Process Engineering Group, IIM-CSIC. Spanish Council for Scientific Research. Eduardo Cabello 6, 36208 Vigo, Spain Granada, May, 9th 2017 Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 1 / 35

Index 1 Introduction 2 System description 3 From CME to PIDE 4 Convergence to equilibrium Exponential convergence (1D) Exponential convergence evidence (nD) 5 Conclusions 6 References Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 2 / 35

Introduction Index 1 Introduction 2 System description 3 From CME to PIDE 4 Convergence to equilibrium Exponential convergence (1D) Exponential convergence evidence (nD) 5 Conclusions 6 References Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 3 / 35

Introduction Study of self regulated gene expression networks usually involve low copy numbers. Stochastic processes. Chemical Master Equation (CME), its solution is not available in the most cases. Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 4 / 35

Introduction Study of self regulated gene expression networks usually involve low copy numbers. Stochastic processes. Chemical Master Equation (CME), its solution is not available in the most cases. We derive the partial integral differential (PIDE) model, proposed by Friedman et al., as the continuous counterpart of one master equation with jump processes. Friedman, N., Cai, L., and Xie, X. S. (2006). Linking stochastic dynamics to population distribution: An analytical framework of gene expression. Phys. Rev. Lett. , 97(16), 168302. Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 4 / 35

Introduction Study of self regulated gene expression networks usually involve low copy numbers. Stochastic processes. Chemical Master Equation (CME), its solution is not available in the most cases. We derive the partial integral differential (PIDE) model, proposed by Friedman et al., as the continuous counterpart of one master equation with jump processes. Using entropy methods we study the convergence to equilibrium, we prove (1D PIDE) or find numerical evidences (nD PIDE) of exponential stability. Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 4 / 35

System description Index 1 Introduction 2 System description 3 From CME to PIDE 4 Convergence to equilibrium Exponential convergence (1D) Exponential convergence evidence (nD) 5 Conclusions 6 References Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 5 / 35

� ✤ � ✤ � � ✤ � System description Gene Regulatory Network k ǫ i k i k i k i m x on � mRNA DNAi off DNAi on X i ⇋ k i off γ i γ i x ( x ) m ∅ ∅ X J Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 6 / 35

✤ � ✤ � ✤ � � � System description Gene Regulatory Network k ǫ i k i k i k i m x on � mRNA DNAi off DNAi on X i ⇋ k i off γ i γ i x ( x ) m ∅ ∅ X J Reaction Steps k i γ i m c i ( x ) m 1. ∅ − − − − → mRNA i 3. mRNA i − → ∅ k i γ i x ( x ) x 2. mRNA i − → mRNA i + X i 4. X i − − − → ∅ Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 6 / 35

System description Self regulation mechanism: c ( x ) = [1 − ρ ( x )] + ρ ( x ) ε , with ε = k ε k m ∈ (0 , 1) the transcrip- x H x H + K H the Hill type function, where K = k off tional leakage constant and ρ ( x ) = k on is an equilibrium constant and H the Hill coefficient. Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 7 / 35

System description Self regulation mechanism: c ( x ) = [1 − ρ ( x )] + ρ ( x ) ε , with ε = k ε k m ∈ (0 , 1) the transcrip- x H x H + K H the Hill type function, where K = k off tional leakage constant and ρ ( x ) = k on is an equilibrium constant and H the Hill coefficient. � − ( x − y ) � Protein production in bursts: ω ( x − y ) = 1 b exp the conditional probability for b protein level to jump from a state y to x . Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 7 / 35

From CME to PIDE Index 1 Introduction 2 System description 3 From CME to PIDE 4 Convergence to equilibrium Exponential convergence (1D) Exponential convergence evidence (nD) 5 Conclusions 6 References Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 8 / 35

From CME to PIDE Chemical Master Equation (CME) c ( n + 1) k m c ( n + 1) k m ∂ p mn m − 1 m + 1 m = k m c ( n )( E − 1 − 1) p mn n + 1 n + 1 n + 1 m ∂ t m γ m ( m + 1) γ m + γ m ( E 1 m − 1) mp mn ( m − 1) k x ( n + 1) γ x ( n + 1) γ x ( n + 1) γ x ( m + 1) k x mk x + k x m ( E − 1 − 1) p mn n + γ x ( E 1 n − 1) np mn c ( n ) k m c ( n ) k m m − 1 m m + 1 with E m and E n being step operators n n n m γ m such that: ( m + 1) γ m ( m − 1) k x ( m + 1) k x n γ x mk x n γ x n γ x E − 1 m p mn = p m − 1 n E 1 m mp mn = ( m + 1) p m +1 n c ( n − 1) k m c ( n − 1) k m m − 1 m + 1 E − 1 m n p mn = p mn − 1 n − 1 n − 1 n − 1 E 1 m γ m n np mn = ( n + 1) p mn +1 ( m + 1) γ m Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 9 / 35

From CME to PIDE Master equation deduction n g i g i n g n +1 1 g − n n n Transition probabilities g n i := · · · · · · n − 1 n n + 1 k m c ( i ) ω ( n − i ) r n := γ x n r n r n +1 P ( t + ∆ t , n ) = (1) Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 10 / 35

From CME to PIDE Master equation deduction n g i n g i 1 g − g n +1 n n n Transition probabilities g n i := · · · · · · n − 1 n n + 1 k m c ( i ) ω ( n − i ) r n := γ x n r n r n + 1 n − 1 � g n P ( t + ∆ t , n ) = i P ( t , i )∆ t + r n +1 P ( t , n + 1)∆ t (1) i =0 Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 10 / 35

From CME to PIDE Master equation deduction n g i g i n 1 g n + 1 g − n n n Transition probabilities g n i := · · · · · · n − 1 n n + 1 k m c ( i ) ω ( n − i ) r n := γ x n r n r n + 1   n − 1 ∞ � � g n g i P ( t + ∆ t , n ) = i P ( t , i )∆ t + r n +1 P ( t , n + 1)∆ t + P ( t , n )  1 − r n ∆ t − n ∆ t (1)  i =0 i = n +1 Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 10 / 35

From CME to PIDE Master equation deduction n g i g i n 1 g n + 1 g − n n n Transition probabilities g n i := · · · · · · n − 1 n n + 1 k m c ( i ) ω ( n − i ) r n := γ x n r n r n + 1   n − 1 ∞ � � g n g i P ( t + ∆ t , n ) = i P ( t , i )∆ t + r n +1 P ( t , n + 1)∆ t + P ( t , n )  1 − r n ∆ t − n ∆ t (1)  i =0 i = n +1 Master equation with jump processes n − 1 ∞ ∂ P ( t , n ) � � g n g i = i P ( t , i ) − n P ( t , n ) + r n +1 P ( t , n + 1) − r n P ( t , n ) (2) ∂ t i =0 i = n +1 Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 10 / 35

From CME to PIDE Continuous formulation (PIDE) Master equation with jump processes n ∞ ∂ P ( t , n ) � � g n g i = i P ( t , i ) − n P ( t , n ) + r n +1 P ( t , n + 1) − r n P ( t , n ) (3) ∂ t i =0 i = n Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 11 / 35

From CME to PIDE Continuous formulation (PIDE) Master equation with jump processes n ∞ ∂ P ( t , n ) � � g n g i − n P ( t , n ) + r n +1 P ( t , n + 1) − r n P ( t , n ) = i P ( t , i ) ∂ t i =0 i = n � �� � ≈ � x k m 0 ω ( x − y ) c ( y ) P ( t , y ) d y (3) The integer indexes n and i are substituted by real x and y respectively: n � x � x � g n 0 g x i P ( t , i ) ≈ y p ( t , y ) d y = k m 0 ω ( x − y ) c ( y ) P ( t , y ) d y i =0 Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 11 / 35

From CME to PIDE Continuous formulation (PIDE) Master equation with jump processes n ∞ ∂ P ( t , n ) � � g n g i i P ( t , i ) − + r n +1 P ( t , n + 1) − r n P ( t , n ) = n P ( t , n ) (3) ∂ t i =0 i = n � �� � ≈ k m c ( x ) p ( t , x ) The integer indexes n and i are substituted by real x and y respectively: n i P ( t , i ) ≈ � x � x � g n 0 g x y p ( t , y ) d y = k m 0 ω ( x − y ) c ( y ) P ( t , y ) d y i =0 ∞ n P ( t , n ) ≈ � ∞ x p ( t , x ) d y = k m c ( x ) p ( t , x ) � ∞ � g y g i ω ( y − x ) d y = k m c ( x ) p ( t , x ) x x i = n Manuel P´ ajaro (IIM-CSIC) PIDE model for gene circuits (Stability) Granada, May, 9th 2017 11 / 35