Membran-Kontinuumsmodelle Biegeelastizität, Vesikel, Formumwandlungen, Fluktuationen. Biophysik der Zelle

Transkript

1 Membran-Kontinuumsmodelle Biegeelastizität, Vesikel, Formumwandlungen, Fluktuationen Biophysik der Zelle

2 Due to the amphipathic nature of phospholipids, these molecules spontaneously assemble to form closed bilayers

3 The Tanford Modell of opposing forces

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5 Lipid membranes are soft (a) Fluctuating liposome (50µm) (b) Aspiratrion of a giant vesicle (32µm) into a micropipette from Mouritsen: Springer 2005

6 Elasticity of Soft Interfaces The three modes of deformation Bending area dilation (condensation) shear: surface extension at constant area " 1 M = K " = K # $A B + 1 % ' $% $ ' " = K # # R 1 R 2 & A % & % * ), ( $% + τ:tension [N/m] M: torque [N/m]

7 Elastic moduli of red blood cells

8 Die Biegeelastizität von Membranen Wenn eine zusätzliche Fläche Δa erzeugt wird, muß ein energetischer Preis bezahlt werden.!g = c 0 + c 1 "!a + c 2 "(!a) Konstante bei Δa=0 soll Minimum sein = c 2!("a) 2 := k 2!("a)2 Wird die Membran um einen Zylinder mit Radius R gewickelt, so erzeugt man eine zusätzliche Oberfläche. Die Oberfläche pro Kopfgruppe wächst relativ um:!a a = 2" # dr 2" # R = d R!a = d " a R!G = k 2! d 2 a 2 R 2 = 1 2! R 2

9 Wird die Membran in eine Kugel mit Radius R geformt, so gilt folgendes:!a a 8"R # dr = = 2d 4"R 2 R!a = 2d " a R!G = k 2 4d 2 a 2 R 2 = 2! R 2 Die elastische Energie, ein Vesikel zu formen hängt demnach nicht vom Radius ab: Gel = " da#g = 8$% Wie groß ist κ? Grobe Abschätzung: Energie, die benötigt wird, ein Vesikel zu formen, das den Radius einer Lipidkettenlänge l besitzt? G 1 = 8"# Die Energie müsste aber auch ungefähr äquivalent zur Oberflächenenergie eines Öltröpfchens desselben Radius ein G 2 = 4"l 2 # Öl$Wasser! = l2 2 " Öl #Wasser l=1.3 nm und σ=0,05 J/m 2! = 0,4 "10 #19 J tatsächlicher Wert! = 0,6"10 #19 J =15 k B T

10 The bending constant is of order κ J The bending constant of a membrane matches the needs of a cell: A shape change can be induced by few ATP energy equivalents (ca. 20 kt / ATP) (kt= J) "F = 8#$ % 400k B T R A cylindrical corrugation with radius R of a cell surface with area A=1000 µm 2 requires an energy "F = # 2R 2 $ A i.e. for R smaller than 10 µm is the energy larger than kt and the shape is stable.

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12 The curvature model The membrane as a two-dimensional surface R(s1; s2) is parametrized by two internal coordinates (s1; s2) embedded in the three-dimensional space. Any such surface can locally be characterized by the two radii of curvature R1 and R2 from which one forms the mean curvature : H = 1 " % $ ' 2 # R 1 R 2 & and the Gaussian curvature: K G = 1 R 1 R 2

13 The curvature model The basic assumption of the curvature model is that bending such a membrane costs locally the energy f = " ( 2 2H ) 2 + " G K (1) where κ and κ G are the ordinary and the Gaussian bending rigidity. Then the total free energy of a closed membrane configurations, i.e., vesicles, is F k = # f " da = $ 2 A # A ( 2H) 2 " da Due to the mathematical Gauss-Bonnet theorem the integral of the second term of (1) depends only on the topology (i.e. number of handles g) of a vesicle but not on its shape, i.e. # K " da = 4$ 1% g A ( ) (2)

14 Minimal surfaces P-surface cosx+ cosy+ cosz = 0 plummers nightmare

15 Shapes of minimal free energy We are now looking for the shapes which minimize the curvature energy (2) for given area A and volume V. These constraints are added with Lagrange multipliers Σ and P to the energy, which leads to The stationary shapes depend only on one parameter which is the reduced volume The shapes do not depend on κ and R => scale invariance

16 The spontaneous curvature model if we assume a preferred curvature, C 0, the free energy turns into F k = " 2 + A $ 1 & + 1 # 2 % R 1 R 2 R 0 ' ) ( 2 * da

17 The area difference model The curvature is correlated with an area difference between the interior and exterior leaflet of the bilayer. For bilayer small thicknesses d >>R it can be approximated : The correct form of the bilayer feature is to add to F an area difference elasticity, Numerical minimization of F k at fixed A in, A and V, no exchange of lipids yields phase diagram of equilibrium shape

18 Phase diagram for closed vesicle membranes: shape is characterized by dimension less parameters v and a ( sphere : ) K. Berndl, J.Käs, R.Lipowsky, E.Sackmann & U. Seifert Europhys. Lett. (1990), Vol 13 p 659-

19 Comparison of the spontaneous curvature model and the area difference model

20 Vesicles with holes and handles genus=2 genus=1

21 Starfish Vesicles

22 Thermal fluctuations Helfrich interaction potential

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26 The induced shape changes according to the curvature model

27 Charge induced phase transition Domains formed in mixed vesicles of DMPA and DMPC induced by adsorption of polylysine

28 Shape transformations can split off sub-compartments budding transition The vesicle forms a daughter vesicle that is still connected by a thin neck fission requires the break-up of the membrane

29 Mechanism of lipid shell formation shell formation around GPIanchored proteins (purple) Specific transmembrane proteins (blue) have the ability to dynamically associate with two or more condensed complexes On the inner membrane surface, the predominant condensed complex is composed of cholesterol and PS. R. Anderson & K. Jacobson Science 296 (2002)

30 Phasediagram of Cholesterol/DOPC/Sphingomyelin Mixtures

31 Lipid Cholesterol Complexes with Specific Stoichiometry Schematic view of a lipid membrane containing cholesterol and phospholipids. Clusters composed of gray and black circles represent condensed complexes. The stoichiometry of the complex (C c P p ) n is c=1, p=2, n=2

32 Domain induced budding Theorie: R. Lipowsky

33 Motivation für die Untersuchung von Lipid/Wasser Phasen Verständnis der lateralen Komposition biologischer Membranen Rafts und der Steuerung ihrer räumlichen Organisation Exo-Endozytose

34 Budding through line tension between the phases

35 Pore formation Any open edge of a membrane patch exposed to water costs a large energy. Rim energy energy of pore formation

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37 The role of non-lamellar structures in membrane morphology and function chain packing model - lateral pressure profile

38 Membrane Adhesion: Vesicle aspiration technique For details: E. A. Evans, "Detailed mechanics of membrane-membrane adhesion and separation I. Continuum of molecular cross-bridges" Biophys. J. (1985),Vol 48, p

39 Surface force apparatus J. N. Israelachvili, Intermolecular and surface forces Academic Press, New York, 1992

40 Transfersome - Liposome in der Medizin Die Haut als Transportbarriere Transfersome Eine geeignete Lipidmischung ermöglicht die Bildung von gekrümmten cups und damit die leichte Anpassung und Haut-Permeabilität der Liposome. Quelle: IDEA-AG

41 Receptor-mediated endocytosis generally occurs via clathrin-coated pits and vesicles

42 Zellen importieren Cholesterin über LDL Rezeptor-vermittelte Endozytose

43 Components that participate in budding of coated vesicles

44 A clathrin-coated pit on the cytosolic face of the plasma membrane

45 Structure of a clathrin-coated vesicle

46 Model for formation of a clathrin-coated pit and selective incorporation of integral membrane proteins

47 Specific fusion of intracellular vesicles involves a conserved set of fusion proteins

48 Phagocytosis: Macrophage eats red blood cells Physics: Receptor induced membrane wetting

49 Phagocytosis: Neutrophil eats bacterium

50 Transport vom trans-golgi- zur Zelloberfläche: Regulierte Exozytose sekretorischer Vesikel

51 Nervenzellen übertragen Signale durch Exocytose von Synaptischen Vesikel Neurotransmitter: Acetylcholin Glutamat Aminobuttersäure Effizientes Recycling der Vesikel erlaubt bis zu 1000 Signale pro Sekunde

52 Hormone-induced release of Ca 2+ from the ER is mediated by IP 3

53 Phosphatidyl inositol signaling pathway

54 Modification of a common phospholipid precursor generates several second messengers: synthesis of DAG and IP 3

55 The effects of many hormones are mediated by second messengers

56 Common signaling pathways are initiated by different receptors in a class

57 CREB links camp signals to transcription

58 Cell-surface receptors belong to four major classes

59 Cell-surface receptors belong to four major classes