MAE 545: Lecture 15 (11/12)

Mechanics of cell membranes

2

“chap11.tex” — page 441[#15] 5/10/2012 16:41

stretch

thicknesschange

shear

bend

Figure 11.13: The geometry ofmembrane deformation. From top tobottom we illustrate stretching of amembrane, bending of a membrane,thickness deformation of a membrane,and shearing of a membrane.

which proteins can influence the thickness of the surrounding bilayer.Finally, to understand the various shapes of red blood cells, we willhave to consider shear deformations of the cell membrane and itsassociated spectrin network. To get a sense geometrically for howsuch deformations work, we will repeatedly appeal to the square patchof membrane shown in Figure 11.13. There are many subtleties lay-ered on top of the treatment here, but a full treatment of this richtopic would take us too far afield, and we content ourselves with thepictorial representations shown here.

Membrane Stretching Geometry Can Be Described by a Simple AreaFunction

The top image in Figure 11.13 illustrates the first class of defor-mations we will consider, namely, when the area of the patch ofmembrane is increased by an amount!a. Just as the parameter !L wasintroduced in Section 5.4.1 (p. 216) to characterize the homogeneousstretching of a beam, the parameter !a will provide a simple wayto characterize the change in the area of a membrane. To be explicitabout the fact that the amount of stretch could in principle vary atdifferent points on the membrane, we introduce a function !a(x, y)

that tells us how the area of the patch of membrane at position (x, y)

is changed upon deformation.

Membrane Bending Geometry Can Be Described by a Simple HeightFunction, h(x,y)

To consider bending deformations, we treat surfaces as shown inFigure 11.14. We lay down an x–y grid on the reference plane and we

ON THE SPRINGINESS OF MEMBRANES 441

Membrane deformations

R. Phillips et al., Physical Biology of the Cell

3

Energy cost for stretching and shearing

isotropicdeformation

undeformedsquare patch L

A = L2

patch area

L+�L

L+�L

E

A=

B

2

✓�A

A

◆2

⇡ B

2

✓2�L

L

◆2

bulk modulusB ⇠ 0.2N/m

L

(lipid bilayer)

sheardeformation

L

✓L

E

A=

µ✓2

2

shear modulusµ ⇠ 10�5N/m

(spectrin network)

anisotropicstretching

L(1 + �1)

L(1 + �2)

E

A⇡ B

2(�1 + �2)

2 +µ

2(�1 � �2)

2

�1,�2 ⌧ 1

(shearing can be interpreted as anisotropic stretching)

4

Metric for measuring distances along curves

~r(x1)0

parameter describing position along the curve x

1

function describing shape of the curve ~r(x1)

~

t(x1) =d~r(x1)

dx

1

local tangentto the curve

metric for measuring lengths

d`

2 = d~r

2 = ~

t

2�dx

1�2

= g

�dx

1�2

g = ~t 2

d` =pgdx

1

~

t(x1)

Example

~r(x1) ~

t(x1)

~r(x

1) = R

�cos(!x

1), sin(!x

1)

�

g(x1) = R

2!

2

d` = R!dx

1

~

t(x

1) = R!

�� sin(!x

1), cos(!x

1)

�

5

Strain for deformation of beams

~r(x1)0 0

~r

0(x1)

undeformed beam deformed beam

g =�d~r/dx

1�2

g

0 =�d~r

0/dx

1�2

d` =pgdx

1d`

0 =p

g

0dx

1 = d`(1 + ✏)

straind`02 � d`2 = (2✏+ ✏2)d`2 ⇡ 2✏ d`2

✏ =d`02 � d`2

2d`2=

1

2g�1 (g0 � g)

Energy cost for stretching/compressing

E =

Z �pgdx

1� 1

2k✏

2

6

Metric tensor for measuring distances on surfacesparameters describing

position along the surface

function describing shape of the surface

local tangent vectors to the surface

x

1, x

2

~r(x1, x

2)

0

0~r(x1

, x

2)

~t1

~t2 ~n

~

ti =@~r

@x

i

~n =~t1 ⇥ ~t2|~t1 ⇥ ~t2|

unit normal vector of the surface

metric tensor for measuring lengths

d`

2 = d~r

2 =X

i,j

~

ti · ~tjdxidx

j =X

i,j

gijdxidx

j

gij = ~ti · ~tj =✓

~t1 · ~t1, ~t1 · ~t2~t2 · ~t1 ~t2 · ~t2

◆

g = det(gij) = |~t1 ⇥ ~t2|2

area element

↵

~

t1dx1

~

t2dx2

dA = |~t1||~t2| sin↵dx1dx

2

dA =pg dx

1dx

2

7

~tx

~ty

~n

dA = dxdy

gij = ~ti · ~tj =✓

1, 00, 1

◆

Examples~r(x, y) = (x, y, 0)

~

t

x

=@~r

@x

= (1, 0, 0)

~ty =@~r

@y= (0, 1, 0)

~n =~tx

⇥ ~ty

|~tx

⇥ ~ty

|= (0, 0, 1)

~n

~t�

~tz gij = ~ti · ~tj =✓

R2, 00, 1

◆~t� =

@~r

@�= R(� sin�, cos�, 0)

~tz =@~r

@z= (0, 0, 1)

~n =

~t� ⇥ ~tz|~t� ⇥ ~tz|

= (cos�, sin�, 0)

dA = Rd�dz

~r(�, z) = (R cos�, R sin�, z)

~n

~t�

~t✓

~r(✓,�) = R(sin ✓ cos�, sin ✓ sin�, cos ✓)

~t✓ =

@~r

@✓= R(cos ✓ cos�, cos ✓ sin�,� sin ✓)

~t� =

@~r

@�= R sin ✓(� sin�, cos�, 0)

~n =

~t✓ ⇥ ~t�|~t✓ ⇥ ~t�|

= (sin ✓ cos�, sin ✓ sin�, cos ✓)

gij = ~ti · ~tj =✓

R2, 00, R2 sin2 ✓

◆

dA = R2 sin ✓ d✓d�

8

Strain tensor for deformation of membranes

0~r(x1

, x

2)

undeformed membrane deformed membrane

0 ~r

0(x1, x

2)

gij =@~r

@x

i· @~r

@x

jg

0ij =

@~r

0

@x

i· @~r

0

@x

j

strain tensor

uij =1

2

X

k

(g�1)ik�g0kj � gkj

�

X

k

(g�1)ikgkj =X

k

gik(g�1)kj = �ij

inverse metric tensor

Energy cost for stretching/compressing

E =

Zpgdx

1dx

2 1

2

2

4(B � µ)(X

i

uii)2 + 2µ

X

i,j

u

2ij

3

5

g = det(gij)

d`

2 =X

i,j

gijdxidx

jd`

02 =X

i,j

g

0ijdx

idx

j

9

“chap11.tex” — page 441[#15] 5/10/2012 16:41

stretch

thicknesschange

shear

bend

Figure 11.13: The geometry ofmembrane deformation. From top tobottom we illustrate stretching of amembrane, bending of a membrane,thickness deformation of a membrane,and shearing of a membrane.

which proteins can influence the thickness of the surrounding bilayer.Finally, to understand the various shapes of red blood cells, we willhave to consider shear deformations of the cell membrane and itsassociated spectrin network. To get a sense geometrically for howsuch deformations work, we will repeatedly appeal to the square patchof membrane shown in Figure 11.13. There are many subtleties lay-ered on top of the treatment here, but a full treatment of this richtopic would take us too far afield, and we content ourselves with thepictorial representations shown here.

Membrane Stretching Geometry Can Be Described by a Simple AreaFunction

The top image in Figure 11.13 illustrates the first class of defor-mations we will consider, namely, when the area of the patch ofmembrane is increased by an amount!a. Just as the parameter !L wasintroduced in Section 5.4.1 (p. 216) to characterize the homogeneousstretching of a beam, the parameter !a will provide a simple wayto characterize the change in the area of a membrane. To be explicitabout the fact that the amount of stretch could in principle vary atdifferent points on the membrane, we introduce a function !a(x, y)

that tells us how the area of the patch of membrane at position (x, y)

is changed upon deformation.

Membrane Bending Geometry Can Be Described by a Simple HeightFunction, h(x,y)

To consider bending deformations, we treat surfaces as shown inFigure 11.14. We lay down an x–y grid on the reference plane and we

ON THE SPRINGINESS OF MEMBRANES 441

Membrane deformations

R. Phillips et al., Physical Biology of the Cell

10

“chap11.tex” — page 442[#16] 5/10/2012 16:41

use the variable h to characterize the height of the membrane abovethat plane at the point of interest. The geometry of the membrane iscaptured by its height h(x, y) at every point in the plane. Note that incases where the deformations of the membrane are sufficiently severe(that is, there are folds and overlaps), this simple description willnot suffice and we would have to work using an intrinsic treatmentof the geometry without reference to the planar reference coordinatesdescribed here.

x

yh

Figure 11.14: The height functionh(x, y). The surface of the membrane ischaracterized by a height at each point(x, y). This height function tells us howthe membrane is disturbed locally fromits preferred flat reference state.

Once we have the height function in hand, we can then computethe curvature, which we will see is the key way that we will cap-ture the extent of bending deformations. As with our treatment ofbeams, we are going to see that the energetics of bending a lipidbilayer membrane will depend upon the curvature of the membrane.To explore the idea of membrane curvature, we take the approachshown in Figure 11.15. We can cut through our surface with a plane,and in so doing, the intersection of the surface with that planeresults in a curve. We compute the curvature of that space curvein exactly the same way we did in Chapter 10 (see Figure 10.4 onp. 386) by finding the circle that best fits the curve at the pointof interest. However, there is a problem with this story. The valuewe get for the curvature depends upon the orientation of the planewe use to cut the surface. Each such plane will result in a differ-ent curve and a correspondingly different curvature. The way aroundthis impasse is a beautiful theorem that states that there is one par-ticular choice of two orthogonal planes for which the curvature willtake two extreme values, one high and one low. These are the so-called principal curvatures. This theorem guarantees that it takes twonumbers to capture the curvature of a surface, as opposed to the

h

h

hx

x

y

y

x

R(x)

R1(x,y)

R2(x,y)

(A) (B)

Figure 11.15: The curvature of space curves and surfaces. (A) The curvature of a curve is found by making the best fit of a circleto the point at which we are computing the curvature. (B) The curvature of a surface is obtained by finding the best circle alongtwo orthogonal directions on the surface. This figure shows the intersection between a surface and a plane parallel to the y-axisand a second intersection between the surface and a plane parallel to the x-axis.

442 Chapter 11 BIOLOGICAL MEMBRANES

Curvature of surfaces

R. Phillips et al., Physical Biology of the Cell

curvature forspace curves

curvature for surfaces depends on the orientation

maximal and minimal curvatures are called principal curvatures and

they appear in orthogonal directions

11

Surfaces of various principal curvatures

r

r1

R1=

1

R2=

1

r

1

R1=

1

r

1

R2= 0

1

R1> 0

1

R2< 0

1

R1

1

R2

R 1=R 2

R1 =

�R2

12

Curvature of curves

~r(x1)0

parameter describing position along the curve x

1

function describing shape of the curve ~r(x1)

~

t(x1) =d~r(x1)

dx

1

local tangentto the curve

metric for measuring lengths

~

t(x1)

~n(x1)

local unit normal vector to the curve

R(x1)

g = ~t 2

curvature of curve

~n(x1)

1

R

= K =1

g

✓~n · d

2~r

d(x1)2

◆

Example

~r(x1)

~

t(x1)

~r(x

1) = R

�cos(!x

1), sin(!x

1)

�

~n(x1)

~n(x

1) =

�cos(!x

1), sin(!x

1)

�

g(x1) = R

2!

2

K = � 1

R

13

Curvature tensor for surfacesparameters describing

position along the surface

function describing shape of the surface

local tangent vectors to the surface

x

1, x

2

~r(x1, x

2)

0

0~r(x1

, x

2)

~t1

~t2 ~n

~

ti =@~r

@x

i

~n =~t1 ⇥ ~t2|~t1 ⇥ ~t2|

unit normal vector of the surface

metric tensor for measuring lengthsgij = ~ti · ~tj

curvature tensor for surfaces

Kij =X

k

�g

�1�ik

✓~n · @

2~r

@x

k@x

j

◆

principal curvatures correspond to the eigenvalues of curvature tensor

mean curvature

1

R1,1

R2

1

2

✓1

R1+

1

R2

◆=

1

2

X

i

Kii =1

2tr(Kij)

Gaussian curvature1

R1R2= det(Kij)

14

~tx

~ty

~n

gij = ~ti · ~tj =✓

1, 00, 1

◆

Examples Kij =X

k

�g

�1�ik

✓~n · @

2~r

@x

k@x

j

◆

~r(x, y) = (x, y, 0)

~

t

x

=@~r

@x

= (1, 0, 0)

~ty =@~r

@y= (0, 1, 0)

~n =~tx

⇥ ~ty

|~tx

⇥ ~ty

|= (0, 0, 1)

~n

~t�

~tz gij = ~ti · ~tj =✓

R2, 00, 1

◆~t� =

@~r

@�= R(� sin�, cos�, 0)

~tz =@~r

@z= (0, 0, 1)

~n =

~t� ⇥ ~tz|~t� ⇥ ~tz|

= (cos�, sin�, 0)

~r(�, z) = (R cos�, R sin�, z)

Kij =

✓� 1

R , 00, 0

◆

Kij =

✓0, 00, 0

◆

~n

~t�

~t✓

~r(✓,�) = R(sin ✓ cos�, sin ✓ sin�, cos ✓)

~t✓ =

@~r

@✓= R(cos ✓ cos�, cos ✓ sin�,� sin ✓)

~t� =

@~r

@�= R sin ✓(� sin�, cos�, 0)

~n =

~t✓ ⇥ ~t�|~t✓ ⇥ ~t�|

= (sin ✓ cos�, sin ✓ sin�, cos ✓)

gij = ~ti · ~tj =✓

R2, 00, R2 sin2 ✓

◆

Kij =

✓� 1

R , 00, � 1

R

◆

15

Examples for Gaussian curvature

1

R1R2< 0

1

R1R2= 0

1

R1R2> 0

16

Bending energy for deformation of membranes

0~r(x1

, x

2)

undeformed membrane deformed membrane

0 ~r

0(x1, x

2)

bending strain tensor Energy cost of bending

Kij =X

k

�g

�1�ik

✓~n · @

2~r

@x

k@x

j

◆K

0ij =

X

k

�g

0�1�ik

✓~n

0 · @

2~r

0

@x

k@x

j

◆

bij = K 0ij �Kij

(local measure of deviation from preferred curvature)

E =

Zpgdx

1dx

2

1

2 tr(bij)

2 + G det(bij)

�

17

Bending energy

E =

ZdA

"

2

✓1

R1+

1

R2� C0

◆2

+G

R1R2

#

r

1

R1=

1

R2=

1

r

C0 = 0

Example: bending energy for a sphere

bending rigidity ⇠ 20kBT

G ⇠ �0.8Gaussian

bending rigidity

mean curvature H =1

2

✓1

R1+

1

R2

◆

Gaussian curvature G =

1

R1R2

spontaneous curvature C0

E = 4⇡ (2+ G) ⇠ 300kBT

bending energy is independent of the sphere radius!

Helfrich free energy

18

Schwarz minimal surface

E =

ZdA

"

2

✓1

R1+

1

R2� C0

◆2

+G

R1R2

#Bending energy

1

R1+

1

R2= 0

1

R1R2< 0

Gaussian bending rigidity has to be negative for stability of membranes

G

Such surfaces would be preferred for

positive Gaussian bending rigidity,

when C0=0.

19

Gauss-Bonet theorem

ZdA

R1R2= 4⇡ (1� g)

For closed surfaces the integral over Gaussian curvature only

depends on the surface topology!

g = 0 g = 1 g = 2 g = 3

CHAPTER 1. PHENOMENOLOGY OF MEMBRANES

Figure 1.7: Representative shapes from the stomatocyte–discocyte–echinocyte sequence of red bloodcells obtained from experiments (left images) and theory (right plots). See also Sec. 3.3. (AfterRef. [6].)

Figure 1.8: Schematic illustration of the great diversity in cell shape found in nature, with E. coli,which roughly has the shape of a cylinder of length 2 µm and thickness/height 1 µm, as a “measurementstick”. For details, see Fig. 2.8 of Ref. [3]. (After Ref. [3].)

10

It is hard to experimentally measure the Gaussian bending rigidity for cells, because cell deformations

don’t change the topology!