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 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
Two phosphorylation mechanisms Processive phosphorylation Distributive phosphorylation Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 2 of 28
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. Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 3 of 28
Cell-to-cell variability Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 4 of 28
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
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
Model Phosphorylation can happen in a processive or distributive way: k 4 k 5 k 6 − ⇀ Processive: E + M − E · M − → pE · M − → ppE + M ↽ k 3 k 4 k 9 k 6 k 7 − − ⇀ ⇀ Distributive: E + M − E · M − → pE + M − pE · M − → ppE + M ↽ ↽ k 3 k 8 as well as dephosphorylation: ′ ′ ′ k k k 4 − 5 6 ⇀ Processive: ppE + Pt − ppE · Pt − → pE · Pt − → E + Pt ↽ k ′ 3 ′ ′ ′ ′ k k k k 4 9 − ⇀ 7 − ⇀ 6 Distributive: ppE + Pt → pE + Pt → E + Pt ↽ − ppE · Pt − ↽ − pE · Pt − k ′ k ′ 3 8 Model DD DP PD PP Number of reactions: Reactions 14 12 12 10 Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 7 of 28
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 ): g ( k 1 , t ) Ø − − − − → M k 2 → Ø M − 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
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): X t = [ M ] t + [ E . M ] t + [ pE . M ] t and Y t = [ ppE ] t + [ ppE . Pt ] t • For each time point t , we measure independently in N t distinct cells both X t and Y t . • Notation: x ∗ t , i and y ∗ t , i denote the measured quantities, where the index 1 ≤ i ≤ N t corresponds to different cells. Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 9 of 28
Parameter inference and model evidence using SMC sampler t = 1 � N t t = 1 � N t • Fit the averages: ¯ t , i and ¯ x ∗ i = 1 x ∗ y ∗ i = 1 y ∗ t , i . N t N t • Assuming a Gaussian measurement error independent for each time point with constant variance σ 2 , the likelihood function is � t ; x t ( θ ) , σ 2 )Φ(¯ t ; y t ( θ ) , σ 2 ) , f ( { ¯ t , ¯ Φ(¯ x ∗ y ∗ t } t | θ ) = x ∗ y ∗ t • The SMC sampler enables us to sample from the posterior distribution t } t ) = 1 p ( θ |{ ¯ t , ¯ Z π ( θ ) f ( { ¯ t , ¯ x ∗ y ∗ x ∗ y ∗ t } t | θ ) in a sequential way and to obtain the evidence Z i.e the probability of the model given the data. Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 10 of 28
Model selection on the average data ● ● −200 −250 log evidence −300 −350 DD PD DP PP ⇒ phosphorylation and dephosphorylation of ERK occur through a distributive mechanism Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 11 of 28
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 Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 12 of 28
Extrinsinc noise At each time point t , each cell i is associated with a parameter θ t , i . The parameters { θ t , i } t , i are distributed according to a log normal distribution with mean µ and variance Σ . μ Σ Hyper parameters Parameter θ t,i y x Species t,i t,i i = 1, ..., Nt Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 13 of 28
Posterior distribution The parameters to infer are the so called hyper parameters µ and Σ and the posterior distribution is then N t � � p ( µ, Σ |{ x ∗ t , i , y ∗ t , i } t , i ) ∝ π ( µ ) π (Σ) h ( { x ∗ t , i , y ∗ t , i } t , i | µ, Σ) t i = 1 where � h ( { x ∗ t , i } t , i | µ, Σ) = f ( { x ∗ t , i } t , i | θ ) p ( θ | µ, Σ) d θ t , i , y ∗ t , i , y ∗ which can not be computed exactly. Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 14 of 28
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 µ and the variance ˜ estimated the mean ˜ Σ of the distribution of the species. • We assume that the species are distributed according to a µ and variance ˜ log-normal distribution of mean ˜ Σ µ and ˜ • Computational cost: to estimate ˜ Σ , the ODE system need to be solved 2*nb of parameters +1 times. The posterior distribution is then N t µ, ˜ � � p ( µ, Σ |{ x ∗ t , i , y ∗ t , i } t , i ) ∝ π ( µ ) π (Σ) Ψ( { x ∗ t , i , y ∗ t , i } t , i | ˜ Σ) t i = 1 Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 15 of 28
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. Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 16 of 28
Simulated data for parameters in the posterior distribution average ERK average MEK 1400 1000 m_traj m_traj 1000 600 200 600 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 times times variance ERK variance MEK 250000 6e+05 m_traj m_traj 100000 2e+05 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 times times Red: Median trajectory; Black: 5 and 95 quantiles Remark: the extrinsinc noise can explain cell-to-cell variability Elucidating cell-to-cell variability in phosphorylation of the ERK MAP kinase Sarah Filippi 17 of 28
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