MAE 545: Lecture 17 (11/19) Mechanics of cell membranes

Transcription

1 MAE 545: Lecture 17 (11/19) Mechanics cell membranes

2 Membrane deformations stretch bend shear change. Phillips et al., Physical Biology Cell

3 a model sometimes known as Helfrich Canham Evans free. involves a ne We Elastic Deformations Membranes Produced byκproteins have introduced notations κ1 and to represent princi units κ pal curvatures w surface at point interest and y may be 0 in Surrounding Membrane Proteins Induce Elastic Deformations curvature as thought as outcome diagonalizing matrix κij that appears length, an in Equation The is defined As argued at beginning this mean chapter,curvature liveliness mem- as κ = (κ1 + κ )/. we see that K branes What owes much activity ir proteins.us Thetohyposis this to equation really instructs do is to visit every point 10 0 kb Tand. W we explore nowsurface, is that find energetics surrounding membrane point, on its curvature, compute κ at that equilibriumand contributes to functioning a wide class membrane proteins. Section n to add up over all points on membrane. Our examples choice are mechanosensitive proteins because Note that bending introduced availability quantitative data free such as those shown in Fig- in Equation 11.7 There IsSince a Fre ure This class ion channelsparameter, is gated by involves a new material Kb, presence bending rigidity. tension in surrounding generally, Bilayer units κ aremembrane. 1/lengthHowever, (as canmore be seen fromany definition w membrane protein that undergoes some conformational change that curvature as second derivative, length/length ) and da has units alters shape that it presents to surrounding membrane will, and since overall unit expression Yet anor length is an,ty deform that membrane. The interesting idea explored in this section thickn back typical in range is that we this see membrane deformation feeds to with protein and canvalues that K b has units deformed alter its10 0 conformational an equil kb T. Wepreference. will describe how bending rigidity is sider measured in equilibrium To see how membrane deformation might couple to protein func(for mama Section : Energy penalty for tion, we begin by considering afigure membrane protein like that shown membrane), t changes. Springs are schematically in Figure 11.44(A). For1mamatical convenience, we used to illustrate idea rethickness is overall t 0 that consider one-dimensional geometry shown in Figure 11.44(B), Energy There Is1a FreePenalty for Changing a free Lipiden change and hcost that characterize where we define functions han + (x) (x) to t Bilayer MscL. Different curves correspond to different length tails for lipids in surrounding membrane. The particular cases are tails with 16, 18, and 0 carbon atoms in ir backbone. (Adapted from E. Perozo et al., Nat. Struct. Biol. 9:696, 00.) Membrane deformation undeformed w0w 0 w deformed w Z K w(x, x ) w p E = gdx dx w. The basis a lipid its equilibrium height upper and lower leaflets 0 lipidfrom for this figure is idea that, value, like ir partners, membrane. w0lipid Yet anor type membrane springiness results from changing as indicated schematically in Figure If we con deformed hydrophobic regionw and is changed to w Kt 60kB T /nmsider an equilibrium hydrophobic Chapter0 11 BIOLOGICAL MEMBANES region446 protein (for mamaticalprotein convenience, we define w as half-width Figure 11.1: Energy penalty for membrane), n contribution such variations to h+(x ) changes. Springs are used to illustrate idea that re is overall free budget is given by an cost to change chap11.tex page 446[#0] "!! a lipid from its equilibrium Kt w(x, y) w0 G [w(x, y)] = da, (11.8) value, w0. w0 Membrane proteins can locally deform lipid 446 (A) Chapter 11 BIOLOGICAL MEMBANES h (x ) x (B). Phillips et al., Physical Biology Cell Figure 11.44: Protein-induced membrane deformation. (A) Protein in a membrane. (B) Schematic showing nature deformations in vicinity a membrane protein. The heights upper and lower leaflets are defined by two fields, h+ (x) and h (x). The lipids near protein are deformed as 3 a result a hydrophobic matching to hydrophobic chap11.tex page 446[#0] patches protein. The lipid schematic ignores fact that lipids are fluctuating, resulting in many different 5/10/01

4 > Osmotic pressure p = p in p out = k B T ( ) < Water flows in cell until mechanical equilibrium is reached. > Water flows out cell until concentrations become equal. = 4

5 Osmotic pressure > p = p in p out = k B T ( ) A =4 V = The radius swollen cell can be estimated by minimizing free. E = A B A p V A E =8 B 4 p Water flows in cell until mechanical equilibrium is reached. > = p 4B + A =8 V =4 = B Membrane tension A A = B = p (Young-Laplace equation) 5 p = (1/ 1 +1/ )

6 Osmotic pressure < p = p in p out = k B T ( ) Total concentration molecules inside a cell (vesicle) Water flows out cell until concentrations become equal. = = N V Preferred cell (vesicle) volume V 0 = N Energy cost for modifying volume E v = E v = Z V V 0 k B T p(v )dv apple V N ln V 0 (V V 0 ) E v = 1 k BT V 0 V V 0 V0 6

7 Area difference between lipid layers Length difference for D example on left out in ' w 0 ` = `out `in =( + w 0 /)' ( w 0 /)' ` = w 0 ' = w 0` Area difference between lipid layers in 3D Z 1 A = A out A in = w 0 da Lipids can move within a given layer, but flipping between layers is unlikely. This sets a preferred area difference A 0. Non-local bending E = k r Aw 0 ( A A 0 ) 7 k r 3apple 60k B T