Modeling biochemical signal transduction in heterogeneous cell - PowerPoint PPT Presentation

Modeling biochemical signal transduction in heterogeneous cell populations Steffen Waldherr, Jan Hasenauer, and Frank Allgöwer Institute for Systems Theory and Automatic Control Universität Stuttgart August 28, 2011

The big picture Intracellular signal transduction Heterogeneous Common response Stimulus Ψ 0 . 5 µ M Response Φ Cellular heterogeneity p 1 / 21

Main points Heterogeneous cellular properties lead to a heterogeneous response from a common stimulus Heterogeneity formulated in terms of probability densities May not be a simple density, nonparametric description needed Response of non-interacting heterogeneous populations is linear ! Linear is easy Makes analysis algorithms computationally efficient 2 / 21

Outline Examples of heterogeneity in cellular signaling 1 Construction of heterogeneous signaling models 2 Simulation and analysis of heterogeneous signaling models 3 3 / 21

Outline Examples of heterogeneity in cellular signaling 1 Construction of heterogeneous signaling models 2 Simulation and analysis of heterogeneous signaling models 3 4 / 21

Apoptosis death time distributions Observations Cells in a clonal population die at different times Some cells survive completely Heterogeneity in protein amounts of caspases and death receptors observed Institute for Cell Biology and Immunology, Universität Stuttgart 5 / 21

Heterogeneous apoptotic signaling Cell death signalling Apoptotic Death time stimulus distribution 10 ng/ml TNF initial concentration XIAP 1 Heterogeneous caspase & Θ 0 ( XIAP ) 0.5 receptor amounts 0 10 5 10 6 10 7 10 8 6 / 21

Outline Examples of heterogeneity in cellular signaling 1 Construction of heterogeneous signaling models 2 Simulation and analysis of heterogeneous signaling models 3 7 / 21

The intracellular signaling model Single cell mathematical model as starting point: ˙ x = Sv ( x , p ) , x ( 0 ) = x 0 Assumption: Pathway components x and structure S , v ( · , · ) are the same for all cells, but parameters p and initial state x 0 may be different. Ensemble model for N cells indexed i , i = 1 , . . . , N : x ( i ) = Sv ( x ( i ) , p ( i ) ) , x ( i ) ( 0 ) = x ( i ) ˙ 0 Key assumptions / simplifications Heterogeneity only in parameters and initial conditions No interactions among cells 8 / 21

Formulating the cellular heterogeneity Probability density function Φ for parameter values Φ p For any given cell i : p ( i ) ∼ Φ Φ( p ) is also the number density of cells in the population with parameter p Ensemble model for cell population x ( i ) = Sv ( x ( i ) , p ( i ) ) , x ( i ) ( 0 ) = x ( i ) ˙ 0 � Prob ( p ( i ) ∈ P , x ( i ) ∈ X ) = Φ( x , p ) dpdx 0 P×X 9 / 21

The response distribution Response of individual cell y ( i ) is a function of the state trajectory: y ( i ) = h x ( i ) ( t , p ( i ) , x ( i ) 0 ) , p ( i ) � � Examples: One concentration at time t k : y = x j ( t k ) Time point at which a threshold is crossed: y = inf { t : x j ( t ) ≥ 0 . 5 x k ( 0 ) } Response heterogeneity can be described by a probability density function Ψ( y ) : � Prob ( y ( i ) ∈ Y ) = Ψ( y ) dy Y Ψ( y ) is also the number density of cells with response y . 10 / 21

Model for heterogeneous populations Heterogeneous Common response Stimulus Model for cell i x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ y ( i ) ∼ Ψ Ψ x ( i ) ( 0 ) = x ( i ) 0 y ( i ) = h � x ( i ) ( t ) , p ( i ) � 0 . 5 µ M y 0 , p ( i ) ∼ Φ x ( i ) Φ Cellular heterogeneity p 11 / 21

Linearity of heterogeneous populations Linearity F ( a + b ) = F ( a ) + F ( b ) 12 / 21

Linearity of heterogeneous populations Linearity F ( a + b ) = F ( a ) + F ( b ) x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ Ψ 1 x ( i ) ( 0 ) = x ( i ) 0 y ( i ) = h x ( i ) ( t ) , p ( i ) � � 0 . 5 µ M Φ 1 12 / 21

Linearity of heterogeneous populations Linearity F ( a + b ) = F ( a ) + F ( b ) x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ Ψ 2 x ( i ) ( 0 ) = x ( i ) 0 y ( i ) = h x ( i ) ( t ) , p ( i ) � � 0 . 5 µ M Φ 2 12 / 21

Linearity of heterogeneous populations Linearity F ( a + b ) = F ( a ) + F ( b ) x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ Ψ 1 + Ψ 2 x ( i ) ( 0 ) = x ( i ) 0 y ( i ) = h x ( i ) ( t ) , p ( i ) � � 0 . 5 µ M Φ 1 + Φ 2 12 / 21

On linearity Reminder: No interactions among cells! F 13 / 21

On linearity Reminder: No interactions among cells! F 13 / 21

On linearity Reminder: No interactions among cells! F F ( 3 × + 2 × ) = 3 × + 2 × 13 / 21

Formulation as partial differential equation – state density function Modeling approach Probability density function Θ for extended state (= concentrations + parameters) � Prob ( x ( t ) ∈ X , p ∈ P ) = Θ( t , x , p ) dxdp X×P Resulting equation Fokker-Planck equation with a drift term only ∂ Θ( t , x , p ) = − div ( x , p ) ( Sv ( x , p )Θ( t , x , p )) ∂ t Initial condition Θ( 0 , x , p ) = Φ( x , p ) 14 / 21

Outline Examples of heterogeneity in cellular signaling 1 Construction of heterogeneous signaling models 2 Simulation and analysis of heterogeneous signaling models 3 15 / 21

Simulating heterogeneous cell populations x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ ? x ( i ) ( 0 ) = x ( i ) 0 y ( i ) = h � x ( i ) ( t ) , p ( i ) � 0 . 5 µ M Φ p 16 / 21

Simulating heterogeneous cell populations Numerical simulation x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ x ( i ) ( 0 ) = x ( i ) y ( i ) Ψ 0 y ( i ) = h � x ( i ) ( t ) , p ( i ) � 0 . 5 µ M Density estimation x ( i ) , p ( i ) Parameter sampling Φ 16 / 21

Density estimation of the response distribution Histogram 1 N ( y k + 1 − y k )# { i : y k ≤ y ( i ) ≤ y k + 1 } Ψ( y ) = Naive estimator (“Sliding histogram”): Ψ( y ) = 1 Nh # { i : y − h 2 ≤ y ( i ) ≤ y + h 2 } Kernel density estimator: 0.6 Υ( y | t, Θ) 0.4 N Ψ( y ) = 1 0.2 � K y ( i ) ( y ) N 0 0 1 2 3 4 5 6 7 8 9 10 i = 1 y 17 / 21

Parameter estimation from population snapshot data x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ Ψ x ( i ) ( 0 ) = x ( i ) 0 y ( i ) = h � x ( i ) ( t ) , p ( i ) � 0 . 5 µ M Response ? 18 / 21

Optimizing over the cellular heterogeneity x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ Ψ sim x ( i ) ( 0 ) = x ( i ) Ψ meas Φ j 0 y ( i ) = h � x ( i ) ( t ) , p ( i ) � Ansatz densities Φ j Optimization problem: min c j � Ψ meas − � k j = 1 c j Ψ sim Φ j � BMC Bioinformatics 12:125 (2011) 19 / 21

Optimizing over the cellular heterogeneity x ( i ) = Sv ( x ( i ) , p ( i ) ) ˙ � k j = 1 c j Ψ sim x ( i ) ( 0 ) = x ( i ) Ψ meas 0 Φ j y ( i ) = h � x ( i ) ( t ) , p ( i ) � Estimated density � k j = 1 c j Φ j Optimization problem: min c j � Ψ meas − � k j = 1 c j Ψ sim Φ j � BMC Bioinformatics 12:125 (2011) 19 / 21

Wrap-up Heterogeneous cellular properties lead to a heterogeneous response from a common stimulus Heterogeneity formulated in terms of probability distributions Simulation by parameter sampling and density estimation Response of non-interacting heterogeneous populations is linear ! Makes repeated simulations computationally cheap Optimizing for measured response distribution, ...? 20 / 21

The big picture (again) Intracellular signal transduction Heterogeneous Common response Stimulus Ψ 0 . 5 µ M Response Φ Cellular heterogeneity p 21 / 21