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

Response Ψ p Φ 0.5 µM

Intracellular signal transduction Heterogeneous response Common Stimulus Cellular heterogeneity

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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

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Outline

1

Examples of heterogeneity in cellular signaling

2

Construction of heterogeneous signaling models

3

Simulation and analysis of heterogeneous signaling models

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Outline

1

Examples of heterogeneity in cellular signaling

2

Construction of heterogeneous signaling models

3

Simulation and analysis of heterogeneous signaling models

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Apoptosis death time distributions

Institute for Cell Biology and Immunology, Universität Stuttgart

Observations

Cells in a clonal population die at different times Some cells survive completely Heterogeneity in protein amounts of caspases and death receptors observed

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Heterogeneous apoptotic signaling

0.5 1

Θ0(XIAP) 105 106 107 108

10 ng/ml TNF

Cell death signalling Death time distribution Apoptotic stimulus Heterogeneous caspase & receptor amounts

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Outline

1

Examples of heterogeneity in cellular signaling

2

Construction of heterogeneous signaling models

3

Simulation and analysis of heterogeneous signaling models

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The intracellular signaling model

Single cell mathematical model as starting point: ˙ x = Sv(x, p), x(0) = x0 Assumption: Pathway components x and structure S, v(·, ·) are the same for all cells, but parameters p and initial state x0 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)

Key assumptions / simplifications

Heterogeneity only in parameters and initial conditions No interactions among cells

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Formulating the cellular heterogeneity

Probability density function Φ for parameter values

p Φ

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) Prob(p(i) ∈ P, x(i) ∈ X) =

• P×X

Φ(x, p)dpdx

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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 tk: y = xj(tk) Time point at which a threshold is crossed: y = inf{t : xj(t) ≥ 0.5xk(0)}

Response heterogeneity can be described by a probability density function Ψ(y): Prob(y(i) ∈ Y) =

• Y

Ψ(y)dy Ψ(y) is also the number density of cells with response y.

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Model for heterogeneous populations

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

y Ψ p Φ 0.5 µM

Model for cell i Heterogeneous response Common Stimulus Cellular heterogeneity y(i) ∼ Ψ x(i)

0 , p(i) ∼ Φ

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Linearity of heterogeneous populations

Linearity

F(a + b) = F(a) + F(b)

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Linearity of heterogeneous populations

Linearity

F(a + b) = F(a) + F(b)

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

Ψ1 Φ1

0.5 µM

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Linearity of heterogeneous populations

Linearity

F(a + b) = F(a) + F(b)

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

Ψ2 Φ2

0.5 µM

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Linearity of heterogeneous populations

Linearity

F(a + b) = F(a) + F(b)

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

Ψ1 + Ψ2 Φ1 + Φ2

0.5 µM

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On linearity

Reminder: No interactions among cells! F

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On linearity

Reminder: No interactions among cells! F

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On linearity

Reminder: No interactions among cells! F

F(3 × + 2 × ) = 3 × + 2 ×

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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) =

• X×P

Θ(t, x, p)dxdp

Resulting equation

Fokker-Planck equation with a drift term only ∂Θ(t, x, p) ∂t = −div(x,p)(Sv(x, p)Θ(t, x, p)) Initial condition Θ(0, x, p) = Φ(x, p)

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Outline

1

Examples of heterogeneity in cellular signaling

2

Construction of heterogeneous signaling models

3

Simulation and analysis of heterogeneous signaling models

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Simulating heterogeneous cell populations

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

?

p Φ 0.5 µM

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Simulating heterogeneous cell populations

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

0.5 µM

Φ

x(i), p(i) y(i)

Ψ

Parameter sampling Numerical simulation Density estimation

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Density estimation of the response distribution

Histogram Ψ(y) = 1 N(yk+1 − yk)#{i : yk ≤ y(i) ≤ yk+1} Naive estimator (“Sliding histogram”): Ψ(y) = 1 Nh#{i : y − h 2 ≤ y(i) ≤ y + h 2} Kernel density estimator: Ψ(y) = 1 N

N

• i=1

Ky(i)(y)

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6

y Υ(y|t, Θ)

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Parameter estimation from population snapshot data

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

Response Ψ

?

0.5 µM

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Optimizing over the cellular heterogeneity

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

Ansatz densities Φj

Ψsim

Φj

Optimization problem: mincj Ψmeas − k

j=1 cjΨsim Φj

Ψmeas

BMC Bioinformatics 12:125 (2011)

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Optimizing over the cellular heterogeneity

˙ x(i) = Sv(x(i), p(i)) x(i)(0) = x(i) y(i) = h

• x(i)(t), p(i)

Estimated density k

j=1 cjΦj

k

j=1 cjΨsim Φj

Optimization problem: mincj Ψmeas − k

j=1 cjΨsim Φj

Ψmeas

BMC Bioinformatics 12:125 (2011)

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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, ...?

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The big picture (again)

Response Ψ p Φ 0.5 µM

Intracellular signal transduction Heterogeneous response Common Stimulus Cellular heterogeneity

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