Modelling nucleocytoplasmic transport with application to the - PowerPoint PPT Presentation

Modelling nucleocytoplasmic transport with application to the intracellular dynamics of the tumor suppressor protein p53 L. Dimitrio Supervisors: R. Natalini (IAC-CNR) and J. Clairambault (INRIA) 5 Septembre 2012

Summary 1. A model for p53 intracellular dynamics ◮ biology of p53 - basics ◮ a new model to reproduce its dynamics 2. A model for protein transport within the cell ◮ biology of intracellular transport - basics ◮ locating a single microtubule

What is p53? In 1979 a protein of molecular mass of 53 kDa was isolated. It was named p53 .

p53 roles: the Guardian of the Genome After a stress p53 acts as a transcription factor : ◮ blocks the cell cycle progress. ◮ repairs the DNA . ◮ launches apoptosis (programmed cell death). It has a huge network of interactions- hard to model!

Healthy or Stressed cell In healthy cells p53 is dangerous , Mdm2 keeps a balanced cellular level of p53. ◮ Mdm2 induces degradation of p53 and blocks its nuclear import. ◮ p53 transcribes the mRNA of Mdm2. In stressed cells p53 concentration rises to prevent the transmission of harmful mutations .

Two different “states” Healthy cells or Stressed cells

How to switch from a state to the other? Healthy cells : blocked import + increased degradation

How to switch from a state to the other? Stressed cells : modifications block p53-Mdm2 interactions. Principal factor in case of DNA damage: ATM

p53 dynamics the p53-Mdm2 network has an oscillatory behavior Figure: in vitro experiments A time-lapse movie of one cell nucleus after exposure to a 5 Gy gamma dose of a MCF7 breast cancer cell line Oscillations and variability in the p53 system Geva-Zatorsky et al. , Molecular Systems Biology 2006 doi : 10 . 1038 / msb 4100068

Mathematical models of p53 Why study p53? ◮ explain oscillations (which mechanism): HOW? ◮ understanding its behaviour: WHY? Literature ODE models ❀ mean concentrations - depend on time ◮ Use delay : du dt ( t ) = f ( t − τ ) ◮ Use negative and positive feedback . Lev-Bar-Or et al. 2001, Monk et al. 2003, Ma et al. 2005, Ciliberto et al. 2005, Chickarmane et al. 2007, Ouattara et al. 2010

Mathematical models of p53 Introducing space : ◮ “Operations” in Nucleus and Cytoplasm are not homogeneous ( transcription-translation-degradation depends on compartment ). ◮ Temporal dynamics: different space scales (p53’s “radius” is 2,4 nm - diameter of a cell can be 30 µ m ) Sturrock et al. - JTB 2011, Sturrock et al. - Bull Math Biol. 2012

Model: biological hypotheses Cytoplasm ubiquitination p53 Mdm2 phosphorylation dephosphorylation translation by ATM p53_p Mdm2 mRNA Nucleus ubiquitination p53 Mdm2 phosphorylation dephosphorylation ATM p53_p Mdm2 mRNA transcription

Mathematical Model Model variables (nuclear and cytoplasmic concentrations) ◮ [ p 53] ( n ) and [ p 53] ( c ) ◮ active p53: [ p 53 p ] ( n ) and [ p 53 p ] ( c ) ◮ [ Mdm 2] ( n ) and [ Mdm 2] ( c ) ◮ [ mdm 2 R NA ] ( n ) and [ mdm 2 R NA ] ( c ) All variables diffuse within each compartment

The Model: Nucleus  dephosphorylation ubiquitination   diffusion � �� � � �� �   ∂ [ p 53] [ p 53 p ] � �� � [ p 53]   = k dph K dph + [ p 53 p ] + d p ∆[ p 53] − k 1 [ Mdm 2]   ∂ t K 1 + [ p 53]     [ p 53]   − k 3 ATM    K ATM + [ p 53]    ∂ [ Mdm 2]   = d m ∆[ Mdm 2] − δ m [ Mdm 2]    ∂ t p53 − dependent synthesis � �� �   ([ p 53 p ]) 4  ∂ [ mdm 2 RNA ]   = k Sm + k Sp + d mRNA ∆[ mdm 2 RNA ]  ([ p 53 p ]) 4 + K Sp  ∂ t     − δ mRNA [ mdm 2 RNA ]      phosphorylation by ATM   � �� �    ∂ [ p 53 p ] [ p 53] [ p 53 p ]   = k 3 ATM K ATM + [ p 53] + d p ′ ∆[ p 53 p ] − k dph   ∂ t K dph + [ p 53 p ]

The Model: Cytoplasm  ∂ [ p 53] [ p 53 p ] [ p 53]  = k S + k dph K dph + [ p 53 p ] + d p ∆[ p 53] − k 1 [ Mdm 2]   ∂ t K 1 + [ p 53]     [ p 53]   − k 3 ATM K ATM + [ p 53] − δ p 53 [ p 53]            translation    � �� �  ∂ [ Mdm 2]  = d m ∆[ mdm 2] + k tr [ mdm 2 RNA ] − δ m [ mdm 2] ∂ t    ∂ [ mdm 2 RNA ]    = d mRNA ∆[ mdm 2 RNA ] − k tr [ mdm 2 RNA ]   ∂ t     − δ mRNA [ mdm 2 RNA ]          ∂ [ p 53 p ] [ p 53] [ p 53 p ]   = k 3 ATM K ATM + [ p 53] + d p ′ ∆[ p 53 p ] − k dph   ∂ t K dph + [ p 53 p ]

Kedem-Katchalsky boundary conditions  ∂ [ p 53] ( n ) ∂ [ p 53] ( c ) = p p 53 ([ p 53] ( c ) − [ p 53] ( n ) ) = − d p  d p    ∂ n ∂ n   d p ′ ∂ [ p 53 p ] ( n ) = − d p ′ ∂ [ p 53 p ] ( c )   = p pp [ p 53] ( c )   p  ∂ n ∂ n  ∂ [ Mdm 2] ( n ) ∂ [ Mdm 2] ( c ) = p mdm 2 ([ Mdm 2] ( c ) − [ Mdm 2] ( n ) ) = − d m d m  ∂ n ∂ n    ∂ [ mdm 2 RNA ] ( n ) ∂ [ mdm 2 RNA ] ( c )  = − p mRNA [ mdm 2 RNA ] ( n ) = − d mRNA   d mRNA   ∂ n ∂ n    on the common boundary Γ. A. Cangiani and R. Natalini. A spatial model of cellular molecular trafficking including active transport along microtubules. Journal of Theoretical Biology, 2010.

The Spatial Environment(s!) The spatial environment is the cell ◮ compartmental model (ODE system) NUCLEUS ← → CYTOPLASM ◮ spatial model (PDE system): 1D and 2D domains Where Ω 1 =Nucleus, Ω 2 =Cytoplasm and Γ the common boundary, Γ = Ω 1 ∩ Ω 2 .

ODE system: exchange between compartments Let S be one of the species S = p 53 , Mdm 2 , mdm 2 RNA , or p 53 p , S ( n ) its nuclear concentration, S ( c ) its cytoplasmic concentration. d S ( n ) = Nuclear Reactions − ρ S V r ( S ( n ) − S ( c ) ) dt d S ( c ) = Cytoplasmic Reactions + ρ S ( S ( n ) − S ( c ) ) dt where V r = cytoplasmic volume nuclear volume

ODE system: positivity of solutions and sustainend oscillations Proposition The positive quadrant is invariant for the flow of the system if ATM > 0 . Numerics Sustained oscillations appear for ATM min < ATM < ATM max . 0.4 3.5 p53 p 0.35 3 Mdm2 0.3 2.5 nuclear Mdm2 concentration 0.25 2 0.2 1.5 0.15 1 0.1 0.5 0.05 0 0 0 100 200 300 400 500 0 0.5 1 1.5 2 2.5 3 3.5 nuclear p53 p TIME

Supercritical Hopf bifurcation and oscillations ◮ ATM and oscillations: existence of a supercritical Hopf Bifurcation 36 2.5 34 2 32 concentration period (min) 1.5 30 28 1 26 0.5 24 0 22 0 20 40 60 80 100 10 20 30 40 50 60 70 80 90 ATM ATM red dotted curve: unstable equilibrium point + marked curve: amplitude of oscillations blue curve: period of the oscillations (minutes) 0.3 ◮ Hypothesis of the Hopf 0.2 bifurcation theorem satisfied 0.1 l 0 by our model -numerical −0.1 −0.2 proof −0.3 −0.4 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02

Simulations in a 1-dimensional PDE system Ω Ω 1 2 a b c Evolution of cytoplasmic concentrations 3 p53 p mdm2 2.5 1.6 2 1.4 1.5 1.2 concentration 1 1 0.8 0.6 0.5 0.4 0.2 0 0 50 100 150 200 250 300 350 400 450 500 0 0 10 20 30 40 50 TIME ATM Figure: Simulations of the 1-dimensional PDE system; Left : temporal evolution of p53 nuclear concentrations. Right : ‘Bifurcation diagram’ over ATM

The 1-dimensional environment does not permit a ‘spatial’ analysis 5 8 7 DIFF=1200 4 DIFF=50 6 concentration concentration 5 3 4 2 3 2 1 1 0 0 0 200 400 600 800 1000 1200 0 100 200 300 400 500 DIFFUSION COEFF ( µ m 2 /min) TIME Figure: Simulations of the 1-dimensional PDE system; Left :‘Bifurcation diagram’ over the diffusion coefficients . Right : temporal evolution of p53 nuclear concentrations for different diffusion values

Simulations in a 2-dimensional PDE system p53 oscillations D S = 10 µ m 2 / s D mRNA = 0 . 1 µ m 2 / s p S = 0 . 16 µ m / s Mdm2 oscillations Volume ratio ( C : N ) = 10 : 1

Oscillations appear for realistic diffusion and permeability values Parameter Description Ref. values values for oscillations 300 µ m 2 Vol > 0( µ m 2 ) Vol Total area of the simulations domain 10 2 ≤ V r ≤ 100 V r Volume ratio Cytoplasm:Nucleus p i 10 µ m / min 5 ≤ p S ≤ 5000( µ m / min) Protein permeabilities 600 µ m 2 / min 10 ≤ D S ≤ 1000( µ m 2 / min) D i Protein diffusion coefficients Table: Parameter ranges of spatial values for which oscillations occurs. the ratio “protein diffusion:mRNA diffusion” has been fixed to 100:1. 400 0.4 0.35 V r =3 350 V r =10 0.3 V r =18 concentration period (min) 0.25 300 0.2 250 0.15 0.1 200 0.05 0 150 0 10 20 30 40 50 60 70 0 100 200 300 400 500 TIME (min) Volume Ratio D. Fusco et al., Curr. Biol 2003, Shav Tal et al. Science 2004, Hong et al. J Biomater Nanobiotechnol 2010

The geometry of the domain does not influence the dynamics of the system

Conclusion - Part I ◮ Spatial physiological model that reproduces the oscillations ◮ ATM as a ‘natural’ bifurcation value ◮ Oscillations appear for realistic diffusion and permeability values ◮ The geometry of the domain does not influence the dynamics of the system