Pattern formation at cellular membranes by phosphorylation and dephosphorylation of proteins

Transcription

1 Pattern formation at cellular membranes by phosphorylation and dephosphorylation of proteins Sergio Alonso Department of Mathematical modeling and data analysis Physikalisck-Technische Bundesanstalt & Department of Applied Physics Universitat Politècnica de Catalunya BCAM Workshop on Nonlinear Dynamics in Biological Systems Bilbao, June 2014

2 Phosphorylation of proteins Phosphorylation: An enzyme (Kinase) adds a phosphate group to a protein Kinase M M P Dephosphorylation: An enzyme (phosphatase) removes the phosphate group from a protein M M Phosphatase P

3 Enzyme kinetics Substrate reacts into a product S E P E S d[p]/dt = ke [S] Which implies: k+1 k -1 Michaelis-Menten: S+E k+1 k-1 C C P+E k+2 Product is rapidly removed or used k-2= 0 k+2 Conservation of total enzyme: [C]+[E] =[E]0 E P

4 Enzyme kinetics The law of mass action: d[s]/dt = k-1[c] - K+1 [S][E] E S k+1 k -1 d[e]/dt = (k-1 + k+2)[c] - K+1 [S][E] C d[c]/dt = - (k-1 + k+2)[c] + K+1 [S][E] d[p]/dt = k+2 [C] Conservation of total enzyme: d[c]/dt+d[e]/dt =0 [C]+[E] =[E]0 k+2 E P

5 Enzyme kinetics The Quasi-steady approximation d[c]/dt = 0 With the condition: [C]+[E] =[E]0 The concentration of complex is [C] = [S][E]0 / (Km + [S]) where Km = (k-1+k+2) / k+1 And the velocity of reaction is [P] d[p]/dt= K2 [C] d[p]/dt= k2[e]0 [S]/ (Km + [S]) k2[eo]

6 Protein activation Goldbeter-Koshland mechanism of Phosphorylation and dephosphorylation by enzimatic reactions E2 C2 W C1 E1 Incorporate the enzymatic dynamics: Two MichaelisMenten dynamics W + E1 W* + E2 k+1 k-1 k+3 C1 C2 E2 W P E1 E2: Phosphatase E1: Kinase k+2 W* + E1 k+4 W + E2 k-3

7 Protein activation Assuming Michaelis-Menten dynamics for both enzymes R1 = V 1 E 1 / ( K 1 + m ) R-1 = V2 E2 / ( K2 + p ) The two processes are R-1 m P Equations governing concentrations m and p m m p = g 1 + g2 t k1+ m k 2+ p m R1 p m p =+ g 1 g 2 t k1+ m k 2+ p With g1 = V1 E1, g2 = V2 E2 and T=m+p

8 Protein activation m m p = g 1 +g t k1+ m 2 k 2 + p One solution but two activation states p m p =+ g 1 g 2 t k1+ m k 2+ p Tunning of the control parameter g1 large change in the response Important for activation of proteins T=m+p m g1

9 Induction of complexity The Goldbeter-Koshland mechanism produce monotonous dependence on parameters Interactions between enzymes and products may induce complex dynamics Bistable switch Oscillations Kholodenko Nat Rev Mol Cell Biol (2006)

10 Spatially distributed enzymes Polarized cells may produce internal gradients Tropini et al. BMC Biophysics (2012) Distribution of enzymes may be different Kholodenko Nat Rev Mol Cell Biol (2006)

11 Saturation at the membrane Binding of proteins depends on the available space There is a saturation concentration (S) at the membrane (here S=1) One stable solution from three possible m Monotonous behaviour m m p = g 1 + g (1 m) t k1+ m 2 k 2+ p p m p =+ g 1 g 2 (1 m) t k1+ m k 2+ p T=m+p g1

12 Saturation at the membrane Binding of proteins depends on the available space m m p = g 1 (1 m) + g 2 (1 m) t k 1+ m k 2+ p p m p =+ g 1 (1 m) g 2 (1 m) t k1 + m k 2+ p Unbinding depends on the available space (binding of m the kinase) Two solutions T=m+p Monotonous behaviour g1

13 Saturation at the membrane Intermediate cytosolic state: protein is in the cytosol but not phosphoralyzed Unbinding and binding depend on the available space m m = g 1 (1 m) + g 3 c (1 m) t k 1+ m p m p =+ g 1 (1 m) g 2 t k1 + m k 2+ p c p = g 3 c (1 m)+ g 2 t k 2+ p Three solutions Non-monotonous behaviour T=m+p+c

14 Simple model with Spatial diffusion Cycling dynamics of membrane-bound (m) cytosolic (c) and phosphorylated (p) proteins. m m 2 = g 1 (1 m) + g 3 c (1 m)+ D m m t k 1+ m p m p 2 =+ g 1 (1 m) g 2 + Dc p t k1 + m k 2+ p c p 2 = g 3 c (1 m)+ g 2 + Dc c t k 2+ p The model does not consider dynamics of the kinases or phosphatases Diffusion at the membrane (Dm) is smaller than in the cytoplasm (D)

15 Model in 1D First we consider a one dimensional approach for the cell Living Cell Mass-conserved model (total number of proteins conserved) Typically for model of cell polarity: Otsuji et al PLOS Comp. Biol (2007), Mori et al. Biophys. J (2008), Goryachev et al FEBS Lett (2008)

16 Stability analysis The linear stability analysis shows a region of long-wave instability where spontaneous domain formation is possible and a region of bistability m m 2 = γ 1 (1 m) + γ 3 c (1 m)+ D m m t k 1+ m p m 2 =+ γ 1 (1 m) γ 2 p+ Dc p t k1+ m c 2 = γ 3 c (1 m)+ γ 2 p+ D c c t Bistability Long wave instability T T

17 Stability analysis Long wave instability Bistability x x T T S. Alonso, Bär Phys. Biol. (2010)

18 Model in 2D m (1 m ) + Dm 2 m k +m t c = γ 3c(1 m ) + γ 2 p + D 2 c Same type of model t m = γ 3 c (1 m ) γ 1 Two dimensional domain m (1 m ) t p = γ 1 γ 2 p + D 2 p k+m Coarsening High concentration of protein Characteristic length D tν

19 Two and three dimensional models Some studies neglected vertical gradients or considered diffusion in two dimensions D D Here we consider the effects of the bulk in the process of pattern formation

20 Model of binding-unbinding Linear binding-unbinding t m=γ 3 c γ 1 m t c= γ 3 c+γ 1 m m o= c o= γ1+ γ3 γ1 T γ 1 +γ 3 Linear binding-unbinding and diffusion t m=γ 3 c γ 1 m m o= 2 t c= γ 3 c+γ 1 m+d c γ 3T Effective reaction rate γ 3 = γ3 L γ3ℓ ℓL γ 3 L 1 T γ1+ γ3 L 1 c o= γ1 T γ1+ γ3 L 1

21 Effective reaction rates Linear Two effective reaction rates are defined γ 2 2 γ2 L γ 3 3 γ3 L With two characteristic lengths D 2 = γ2 3 = x

22 Simple model of 3D structure The phase diagram is recovered with effective parameter values Regions of spontaneous domain formation 3D numerical simulations Phase diagram with effective parameters S. Alonso, Bär Phys. Biol. (2010)

23 Examples of membrane proteins Rho family regulation of lymphocytes Tybulewicz, Henderson Nat rev. (2009) Self-organization of Cdc42 in yeasts Goryachev, Pokhilko FEBS Lett. (2008) Min-proteins in Escherichia coli Meacci, Kruse Phys. Biol (2005) PAR proteins in C. elegans embryo Goehring et al, Science (2011)

24 MARCKS proteins Myristoylated alanine-rich C kinase substrate Characteristics High concentration in living cells: 10 µm and higher Charged and unfolded protein Hydrophilic membrane protein Binding to the membrane Effector domain (electrostatic interactions, with PIP2 or PS) Myristate inserts hydrophobically Unbinding from the membrane Phosphorilation by protein kinase C (PKC), activated by: Ca2+, phospholipid (PS), DAG Calmodulin: activated by Ca 2+ A Gambhir Biophys. J. (2004)

25 MARCKS are common in cells MARCKS is common in cells: In-vivo experiments: In White blood cells, Embryonic chick neurons, Bovine luteal cells (ovaries), Fibroblasts, Rat hippocampal neuron cells, Bovine muscle cells, Spermatozoa MARCKS extracted for in-vitro experiments:mutated e-coli to express human MARCKS, Bovine brain, Porcino brain MARCKS functions: Phagositosis (Allen and Anderem J. Exp Med 1995) Human acrosomal Exocytosis (Rodriguez Peña et al. PLOS one 2013) Actin filament assembly (Li and Chen Mol. Biol Cell 2008) Cell movement (Kalwa and Michel J. Biol. Chem 2011)

26 Three types of experiments In vitro experiments: MARCKS binding to vesicles Change fraction of phopholipids at the vesicles Measure the binding rate to the vesicles In vitro experiments: MARCKS binding to Langmuir monolayers Change fraction of phospholipids at the monolayer Change on the lateral pressure of the monolayer In vivo experiments Induccion of calcium signals Measure of binding and unbinding of proteins

27 Langmuir Monolayers Consider a phospholipid monolayer There is MARCKS proteins at the water subphase MARCKS binds to the monolayer Monolayer is a simple model of biomembrane where we can vary the composition

28 Monolayers: Experiments Deposition of the monolayer. MARCKS in the bulk Diffusion and translocation of MARCKS to the monolayer Introduction of PKC Phosphorylation of MARCKS by PKC

29 Experimental results The membrane-bound MARCKS increase the lateral pressure of the monolayer The lateral pressure of the monolayer oscillates after the introduction of PKC However there are too few ingredients to generate oscillations

30 Monolayers changes with MARCKS We know, however, that domains of the monolayer change with membrane-bound MARCKS Low MARCKS concentration High MARCKS concentration U. Dietrich, P. Krüger, T. Gutberlet, J. A. Käs, Bio. et Biophys. Acta, (2009).

31 Mechanism of oscillation We consider the translocation of MARCKS and PKC between monolayer and bulk and the diffusion in the bulk (+ MARCKS) (- MARCKS) The monolayer state controls the translocation dynamics

32 Numerical simulations Accumulation is observed without PKC Transitory oscillations are obtained for initial conditions similar to the experimental studies S. Alonso et al, Biophys. J (2011)

33 Dynamics of MARCKS in living cells Temporal dynamics Experiments with living cells: rat insulin-secreting cell line (ISN-1) and chinese hamster ovary-k1 cells. Spatial distribution Experiments with hippocampal neuron cones. Presence of domains of PIP2 and MARCKS. Binding of PKC activated by calcium at the membrane and unbinding of MARCKS. 4 µm S. Ohmori et al J. Biol Chem (2000) H. Mogami et al. J Biol Chem.(2003) T Laux et al. J Cell Bio (2000)

34 Domain formation in Living cells MARCKS Domains of MARCKS and PKC in insuling-secreting cell Mogami J Biol Chem 2003 CaM Co-distribution of MARCKS with CAM in smooth muscle cells Gallant J Cell Science 2005 Co-distribution of MARCKS with PIP2. Hippocampal neuron cultures Laux J Cell Bio 2000 Domains of MARCKS and PKC In spermatozoa Rodriguez Peña PLOS one 2013

35 Experimental realization MARCKS response to a calcium spike in insulinsecreting cells MARCKS-GFP proteins are located mainly at the membrane The induction of calcium signals produces the translocation of the MARCKS-GFP proteins to the cytoplasm H. Mogami et al. J Biol Chem.(2003)

36 Experimental realization MARCKS and PKC response to a calcium spike The induction of calcium signals produces the binding of PKC-RFP at the membrane and the translocation of the MARCKS-GFP proteins to the cytoplasm MARCKS PKC H. Mogami et al. J Biol Chem.(2003)

37 Experimental results PKC after calcium spikes MARCKS and PKC after calcium spikes H. Mogami et al. J Biol Chem.(2003)

38 Living cells: Processes We consider different process involving the membrane and enzymes To numerically integrate the resulting reactiondiffusion equations we need the division of the terms depending where are active

39 Reaction-diffusion model Dynamics at the membrane and its immediate vicinity, where all the translocation processes occur Dynamics in the bulk of the cytoplasm

40 Reproducing experiments Response to a calcium spike MARCKS PKC

41 Reproducing experiments Response to a calcium spike Points: experiments Solid lines: numerical simulations Response to a set of calcium spikes

42 Spontaneous domain formation Homogeneous initial condition Small random perturbations grow with time Domain formation at the membranes Coarsening Gradients in the cytosol Complementary domain formation of PKC

43 Bistability-induced domains Step initial condition Domain formation at the membranes Gradients in the cytosol Domain formation of PKC Small random perturbation do not produce the formation of domains

44 Phase diagram Phase diagram of the model for different concentrations of MARCKS and PKC Examples of the dynamics

45 Robustness Changes in the simulations of spatial parameters: Size of the cell Diffusions Dc= 5 µm2s-1 Cytosol Membrane Dm= 0.1 µm2s-1 c

46 Conclusions Spatial aspects activation of proteins Monolayers with proteins in the subphase Domain formation is obtained when spatial localization of kinases and phosphatases is considered in a phosphorylation/dephosphorylation process Changes on the monolayer due to the attachment of MARCKS produce oscillations in the lateral pressure of the monolayer and therefore on the concentration of bound proteins Modeling living cells A model of MARCKS binding to membranes fitted to experimental results predicts the formation of domain at the membranes of living cells

47 Bibliography Spatial aspects activation of proteins S. Alonso, H.-Y. Chen, M. Bär and A. S. Mikhailov, Self-Organization processes at active interfaces, Eur. Phys. J. Special Topics 191, (2010). Monolayers with proteins in the subphase S. Alonso and M. Bär, Phase separation and bistability in a threedimensional model for protein domain formation at biomembranes, Phys. Biol (2010). S. Alonso, U. Dietrich, C. Händel, J. A. Käs and M. Bär. Oscillations in the lateral pressure of lipid monolayers induced by nonlinear chemical dynamics of the second messengers MARCKS and Protein kinase C, Biophys. J. 100, (2011). Modeling living cells S. Alonso and M. Bär, Modeling domain formation of MARCKS and protein kinase C at cellular membranes, EPJ Nonlinear Biomedical Physics 2, 1 (2014).

48 Acknowledgments Collaborations Markus Bär (PTB, Berlin) Karin John (Université Joseph Fourier, Grenoble) Josef A. Käs, Undine Dietrich, Chris Händel (Universität Leipzig, Leipzig) Funds Collaborative Research Center (SFB 910): Control of self-organizing nonlinear systems: Theoretical methods and concepts of application. Collaborative Research Center (SFB 555): Complex non-lineal processes