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Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi Imperial College London Theoretical Systems Biology Group 20/02/2013 The ERK MAP pathway In eukaryotes Mitogen-activated protein (MAP) kinases

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Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase

Sarah Filippi

Imperial College London Theoretical Systems Biology Group

20/02/2013

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The ERK MAP pathway

  • In eukaryotes Mitogen-activated protein (MAP)

kinases take a central role in regulating biological processes ranging from gene expression, cell cycle, differentiation and proliferation of celles all the way to apoptosis.

  • MEK and MAPK (also called ERK) need to be

phosphorylated at two phosphorylation sites (its serine and threonine residues) in order to be active.

  • We focus on the study of the process of

phosphorylation and dephosphorylation of ERK by respectively MEK and MKP .

Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 1 of 28

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Two phosphorylation mechanisms

Processive phosphorylation Distributive phosphorylation

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Two phosphorylation mechanisms

  • It has been demonstrated that in vitro phosphorylation and

dephosphorylation of ERK occur through a distributive mechanism.

  • We study the MEK-ERK cascade in vivo in PC12 cell lines

using a combination of quantitative image cytometry.

  • The data consist of quantitative measurements of fluorescent

intensity of activated ERK and doubly phosphorylated MEK for hundreds of individual cells at 24 time points separated by 2 minutes intervals.

  • Toni et al, 2012 have used the model selection tool based on

ABC-SMC fitting the average of the data and concluded that the model with a distributive mechanism for both phosphorylation and dephosphorylation of ERK is the most likely as soon as the prior range is large.

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Cell-to-cell variability

Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 4 of 28

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Cell-to-cell variability

Aim of this work

Elucidate the cell-to-cell variability in phosphorylation of the ERK MAP kinase Summary of the presentation:

  • 1. Model of the phosphorylation dynamics of the ERK MAP kinase
  • 2. Model selection based on the averaged data
  • 3. Different modelisation of the cell-to-cell variability: intrinsic and

extrinsic noise

Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 5 of 28

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Model

The model contains 9 species:

  • double phosphorylated MEK which acts as a kinase denoted as

M

  • un-, single and double phosphorylated ERK denoted

respectively by E, pE and ppE

  • the species Pt represents the (unknown) phosphotase activity.
  • the complexes E · M and pE · M
  • and the complexes pE · Pt and ppE · Pt

Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 6 of 28

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Model

Phosphorylation can happen in a processive or distributive way: Processive: E + M

k4

− ⇀ ↽ −

k3

E · M

k5

− → pE · M

k6

− → ppE + M Distributive: E + M

k4

− ⇀ ↽ −

k3

E · M

k7

− → pE + M

k9

− ⇀ ↽ −

k8

pE · M

k6

− → ppE + M as well as dephosphorylation: Processive: ppE + Pt

k

′ 4

− ⇀ ↽ −

k′

3

ppE · Pt

k

′ 5

− → pE · Pt

k

′ 6

− → E + Pt Distributive: ppE + Pt

k

′ 4

− ⇀ ↽ −

k′

3

ppE · Pt

k

′ 7

− → pE + Pt

k

′ 9

− ⇀ ↽ −

k′

8

pE · Pt

k

′ 6

− → E + Pt Number of reactions: Model DD DP PD PP Reactions 14 12 12 10

Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 7 of 28

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Model

Initial conditions: We assume that at the beginning of the experiment there is only M, E, ppE, and Pt in the cells. Dynamics of double phosphorylated MEK (M): Ø

g(k1,t)

− − − − → M M

k2

− → Ø where g is a function of time, modelled as a sigmoid function. Other constraint: The amount of Pt as well as the amount of E is constant over the duration of the experiment (see Ozaki et al., 2010).

Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 8 of 28

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Measurements

  • We have at our disposal the concentration of double

phosphorylated MEK (M) on single cells at different time points as well as the concentration of double phosphorylated ERK (ppE): Xt = [M]t + [E.M]t + [pE.M]t and Yt = [ppE]t + [ppE.Pt]t

  • For each time point t, we measure independently in Nt distinct

cells both Xt and Yt.

  • Notation: x∗

t,i and y∗ t,i denote the measured quantities, where

the index 1 ≤ i ≤ Nt corresponds to different cells.

Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 9 of 28

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Parameter inference and model evidence using SMC sampler

  • Fit the averages: ¯

x∗

t = 1 Nt

Nt

i=1 x∗ t,i and ¯

y∗

t = 1 Nt

Nt

i=1 y∗ t,i.

  • Assuming a Gaussian measurement error independent for each

time point with constant variance σ2, the likelihood function is f({¯ x∗

t , ¯

y∗

t }t|θ) =

  • t

Φ(¯ x∗

t ; xt(θ), σ2)Φ(¯

y∗

t ; yt(θ), σ2),

  • The SMC sampler enables us to sample from the posterior

distribution p(θ|{¯ x∗

t , ¯

y∗

t }t) = 1

Z π(θ)f({¯ x∗

t , ¯

y∗

t }t|θ)

in a sequential way and to obtain the evidence Z i.e the probability of the model given the data.

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Model selection on the average data

  • DD

PD DP PP −350 −300 −250 −200 log evidence

⇒ phosphorylation and dephosphorylation of ERK occur through a distributive mechanism

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Modelling cell-to-cell variability

  • Intrinsic noise: the differences in measurement between the

cells is due to the stochasticity of the process

  • Extrinsic noise: each cell is associated with a slighly different

parameter

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Extrinsinc noise

At each time point t, each cell i is associated with a parameter θt,i. The parameters {θt,i}t,i are distributed according to a log normal distribution with mean µ and variance Σ.

μ Σ θ x y

t,i t,i t,i Hyper parameters Parameter Species i = 1, ..., Nt

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Posterior distribution

The parameters to infer are the so called hyper parameters µ and Σ and the posterior distribution is then p(µ, Σ|{x∗

t,i, y∗ t,i}t,i) ∝ π(µ)π(Σ)

  • t

Nt

  • i=1

h({x∗

t,i, y∗ t,i}t,i|µ, Σ)

where h({x∗

t,i, y∗ t,i}t,i|µ, Σ) =

  • f({x∗

t,i, y∗ t,i}t,i|θ)p(θ|µ, Σ)dθ

which can not be computed exactly.

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Approximation of the likelihood function

  • To approximate the likelihood function we use the Unscented

Transform which tells us how the moments of θ are transformed by the non-linear function f.

  • Assuming that θ is distributed according to a log-normal

distribution of mean µ and variance Σ, the UT method estimated the mean ˜ µ and the variance ˜ Σ of the distribution of the species.

  • We assume that the species are distributed according to a

log-normal distribution of mean ˜ µ and variance ˜ Σ

  • Computational cost: to estimate ˜

µ and ˜ Σ, the ODE system need to be solved 2*nb of parameters +1 times. The posterior distribution is then p(µ, Σ|{x∗

t,i, y∗ t,i}t,i) ∝ π(µ)π(Σ)

  • t

Nt

  • i=1

Ψ({x∗

t,i, y∗ t,i}t,i|˜

µ, ˜ Σ)

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Parameter inference for the DD model

We used SMC sampler to infer the hyper parameters assuming that both phosphorylation and dephosphorylation of ERK occur through a distributive mechanism. Some computational details:

  • There are 16 parameters and 4 unknown initial conditions

⇒ 40 hyper parameters to infer

  • For each parameter, the ode system need to be solved 81

times.

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Simulated data for parameters in the posterior distribution

500 1000 1500 2000 2500 200 600 1000

average ERK

times m_traj 500 1000 1500 2000 2500 100000 250000

variance ERK

times m_traj 500 1000 1500 2000 2500 600 1000 1400

average MEK

times m_traj 500 1000 1500 2000 2500 2e+05 6e+05

variance MEK

times m_traj

Red: Median trajectory; Black: 5 and 95 quantiles

Remark: the extrinsinc noise can explain cell-to-cell variability

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Intrinsic noise

  • the differences in measurement between the cells is due to the

stochasticity of the process

  • use Linear Noise Approximation (LNA): continuous

approximation to Markov jump processes defined by the chemical master equation

  • the LNA estimates the variances of the species and the

covariances between them

  • for every parameter value θ, a set of ODEs describes the

evolution of average behaviour xt(θ) and yt(θ) as well as the evolution of the variance between species Σt(θ) The likelihood has the following form: f({x∗

t,i, y∗ t,i}t,i|θ) =

  • t

Nt

  • i=1

Φ({x∗

t,i, y∗ t,i}t,i; (xt(θ), yt(θ)), Σt(θ))

Computationally: ODE system with 54 species and 20 parameters

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Illustration of the approximation by Linear Noise Approximation

Hundreds independent trajectories simulated using the Gillespie algorithm

100 200 300 400 200 400 600 800 1000 1200 1400 time

ERK

100 200 300 400 400 600 800 1000 1200 1400 1600 time

MEK

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Comparison of LNA and Gillespie simulation

100 200 300 400 200 400 600 800 1000 1200 1400 time

m_ERK

100 200 300 400 500 1000 1500 2000 2500 time

v_ERK

100 200 300 400 400 600 800 1000 1200 1400 1600 time

m_MEK

100 200 300 400 500 1000 1500 time

v_MEK

Black: Average and variance of 100 simulated trajectories using Gillepie; Red: evolution of average and variance given by the LNA Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 20 of 28

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Not possible to explain the variability with intrinsic noise alone

  • use SMC sampler to fit the data with this model

500 1000 1500 2000 2500 500 1000

average ERK

times m_traj 500 1000 1500 2000 2500 100000 250000

variance ERK

times m_traj 500 1000 1500 2000 2500 600 1000 1400

average MEK

times m_traj 500 1000 1500 2000 2500 0e+00 2e+05 4e+05

variance MEK

times m_traj

⇒ fit of the average behaviour but variances far too small

  • Latin hypercube sampling: no stable solution show a variance

as high as the observed ones.

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Can the heterogeneity be explained by the intrinsic noise and a variability in the initial conditions ?

Fix all the parameters according to the posterior distribution and sample the initial condition according to a uniform distribution with large range.

500 1000 1500 2000 2500 5000 15000

average ERK

times m_traj 500 1000 1500 2000 2500 100000 250000

variance ERK

times m_traj 500 1000 1500 2000 2500 500 1500

average MEK

times m_traj 500 1000 1500 2000 2500 400 800

variance MEK

times m_traj

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Can the heterogeneity be explained by the intrinsic noise and a variability in the initial conditions ?

Average among the previous trajectories and total variance

500 1000 1500 2000 2500 2000 5000

average average ERK

times m_traj 500 1000 1500 2000 2500 0.0e+00 2.0e+07

variance tot ERK

times v_traj 500 1000 1500 2000 2500 600 1000 1400

average average MEK

times m_traj 500 1000 1500 2000 2500 1e+05 3e+05

variance tot MEK

times v_traj

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Can the heterogeneity be explained by the intrinsic noise and a variability in the initial conditions ?

Fix all the parameters according to the posterior distribution and sample the initial condition according to a log normal distribution with parameters given by the posterior of extrinsinc noise.

500 1000 1500 2000 2500 500 1500

average ERK

times m_traj 500 1000 1500 2000 2500 1000 2500

variance ERK

times m_traj 500 1000 1500 2000 2500 500 1500

average MEK

times m_traj 500 1000 1500 2000 2500 500 1000 1500

variance MEK

times m_traj

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Can the heterogeneity be explained by the intrinsic noise and a variability in the initial conditions ?

Average among the previous trajectories and total variance

500 1000 1500 2000 2500 200 600 1200

average average ERK

times m_traj 500 1000 1500 2000 2500 100000 250000

variance tot ERK

times v_traj 500 1000 1500 2000 2500 600 1000 1400

average average MEK

times m_traj 500 1000 1500 2000 2500 0e+00 2e+05 4e+05

variance tot MEK

times v_traj

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No change in variance when intrinsic noise is added to extrinsinc noise.

Sample each parameter independently according to log normal distribution and simulate the intrinsinc noise model.

500 1000 1500 2000 2500 200 600 1000 1400

average average ERK

times m_traj 500 1000 1500 2000 2500 100000 250000

variance tot ERK

times v_traj 500 1000 1500 2000 2500 600 1000 1400

average average MEK

times m_traj 500 1000 1500 2000 2500 0e+00 2e+05 4e+05

variance tot MEK

times v_traj

Black: Extrinsinc noise alone; Red: Extrinsinc and Intrinsinc noise Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 26 of 28

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Conclusion

Extrinsinc noise explains cell-to-cell variability in phosphorylation

  • f the ERK MAP kinase

Future work

  • Study the impact of the extrinsinc/intrinsic noise in the four

models.

  • Analyse the impact of different stimulus intensity on the

cell-to-cell variability

  • Determine a stimulus which would enable to control cell-to-cell

variability

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Thanks for listening!

Acknowledgements:

  • Chris Barnes

(University College London)

  • Paul Kirk
  • Michael Stumpf

s.filippi@imperial.ac.uk Theoretical Systems Biology group at Imperial College London www.theosysbio.bio.ic.ac.uk

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