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Leading Edge

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Differential Geometry Meets the Cell

Wallace F. Marshall1,*

1Department of Biochemistry and Biophysics, University of California, San Francisco, 600 16th Street, San Francisco, CA 94158, USA

*Correspondence: wallace.marshall@ucsf.edu

http://dx.doi.org/10.1016/j.cell.2013.06.032

A new study by Terasaki et al. highlights the role of physical forces in biological form by showing

that connections between stacked endoplasmic reticulum cisternae have a shape well known in

classical differential geometry, the helicoid, and that this shape is a predictable consequence of

membrane physics.

Cells are beautiful structures whose form

is tailored to function, but what specifies

theform?Acentury

Wentworth Thompson

physical principles

tension could dictate biological form. But

then the genome happened, and with it

came the desire to explain away ques-

tions of cellular structure by telling

ourselves that geometry is encoded in

the genome. Although the genome is not

a blueprint that explicitly encodes shape,

the genome does encode proteins that

sculpt cellular structures, for example by

dictating membrane curvature (Mim and

Unger, 2012). The existence of such pro-

teins goes against the concepts of

D’Arcy Thompson and appeared to be a

final nail in the coffin of his Pythagorean

approach to cell biology. But a paper by

Terasaki et al. (2013) in this issue of Cell

breathes new life into the old dream of

mathematical biology

that the connections between endo-

plasmic reticulum (ER) sheets mimic a

well-known class of mathematical sur-

facesandthatthisshapeisinfactpredict-

able from simple physical rules governing

membrane energetics.

In their paper, Terasaki et al. (2013)

explored the structure of the ER using a

new imaging method and obtained a

beautiful new structure. In professional

secretory cells, the ER forms stacks of

membrane cisternae, apparently because

this is an efficient way to pack a lot of ER

membrane into a small volume inside the

cell. Although these stacks have been

seen for decades, it was never quite clear

how the edges of the cisternae were con-

nected together into one continuous

lumen. Using a new method of serial sec-

tion scanning electron microscopy, the

ago,

proposed

such

D’Arcy

that

surfaceas

bydiscovering

authors discovered a novel arrangement

of membranes forming the connectors,

which turned out to resemble a mathe-

matical object known as a Helicoid,

discovered by Jean Baptise Meusnier in

1776. To visualize a helicoid, start with a

fixed axis, draw a line segment perpen-

dicular to the axis, and then rotate the

line segment around the axis while

moving along the axis to sweep out a

surface (Figure 1).

Why would ER stack connectors take

this shape? Terasaki et al. (2013) con-

structed a simple physical model giving

the total energy of the ER shape as the

sum of two fundamental energetic terms

and then solved for the shape that mini-

mizes both energetic contributions simul-

taneously. The first energy is the bending

energy of the sheet surface, which, for

symmetrical surfaces like the paired

membranes of an ER cisterna, is lowest

energy when its average curvature is

zero (Helfrich, 1973). For surfaces, one

can measure curvature in two orthogonal

directions. The ‘‘average curvature’’ re-

fers to the average of the curvatures in

these two directions. There are two ways

to get an average curvature of zero: either

the surface is flat, like a tabletop, or the

surface curves up in one direction and

down in the orthogonal direction, like a

saddle. So the closer the ER membrane

could get to one of these zero average

curvature shapes, the more energetically

favorable it would be.

The second energy term that must be

considered is the shape of the edge of

the sheet. The path of the edge in 3D

space is called the edge line, and the

authorsusedifferential

methods to derive an energy for bending

the edge of a membrane sheet composed

geometry

of two parallel membranes. Their result is

that the edge line will be at its lowest

energy when it has a constant, nonzero

curvature.

Thus, the lowest energy state for the

system would be if the membrane surface

has zero average curvature and the edge

line has a constant nonzero curvature.

Any deviations from these conditions will

impose an energetic cost whose value

depends on the detailed physical proper-

ties of the membrane system. But regard-

less of the physical details, the lowest

energy state can be described quite sim-

ply without having to know any numerical

parameterswhatsoever:

energy state is a shape whose edge has

constant curvature and whose surface

has zero curvature. In fact, the shape

with these properties is well known in

theclassical literature

geometry: the helicoid! The helical shape

of the edge has constant curvature,

whereas the surface itself has zero

average curvature because it has a

saddle-like shape at every point.

The fact that the helicoid shape is pre-

dicted without having to know the value

of any physical constants is one of the

most beautiful and surprising results of

this paper. It is not very shocking that

physical properties of cellular compo-

nents contribute to their shape—even

the most committed geneticist would

have to grant this point. But usually

when one talks about modeling the rela-

tion between physical forces and bio-

logical structures, one ends up having to

know, or at least estimate, the value of

various physical constants like elastic

moduli, rate constants, and so on. But in

this case, the fundamental form of the

helicoidal shape does not rely on knowing

thelowest

ofdifferential

Cell 154, July 18, 2013 ª2013 Elsevier Inc. 265

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such constants, and so in this

sense, it may be said that the

shape of the ER connectors

comesfrom

rather than physics.

How unusual are structures

likethis?Helicoid-shaped

defects have been predicted

to occur in liquid crystals

(Kamien and

1996), but helicoids are just

one example of a larger class

of surfaces, known as mini-

mal surfaces, characterized

by zero average curvature.

Whereas helicoids are not

seen that often, another mini-

malsurface,

arises in a huge number of

contexts. The gyroid contains

helicaltwistssimilartothehe-

licoid but, whereas the heli-

coid is periodic along one

axis, the gyroid is periodic in

three perpendicular

Gyroids are seen in many

self-assembled

such as in diblock copoly-

mers, for example (Bates

and Fredrickson, 1999). Bio-

logicallyoccurring

have been reported in butter-

fly scales (Michielsen and Stavenga,

2008). Although the scales themselves

are formed of chitin, the chitin deposition

occurs in invaginations of the plasma

membrane separated

smooth ER (Ghiradella 1989), suggesting

that, again, the elastic properties of bio-

logical membranes may drive the forma-

tion of complex-looking minimal surfaces.

Under conditions of stress or viral infec-

tion, ER can form periodic structures

mathematics

Lubensky,

the gyroid,

axes.

surfaces,

gyroids

by tubulesof

(Snapp et al., 2003; Goldsmith et al.,

2004), some of which represent triply

periodic minimalsurfaces

et al., 2006). An important difference is

that these prior descriptions were in path-

ological system, whereas Terasaki et al.

(2013) have found a minimal surface to

describe normal ER stacks in healthy

cells.

MaybeERstack connectorsare special

cases in that simple math and physics

(Almsherqi

won’t

most cellular structure. The

jury is still out. The fact

that Terasaki et al. (2013)

needed advanced

scopy methods to visualize

their structure suggests that

wemightnotseerecognizable

mathematical forms because

we don’t yet know how to

look for them.

explainall or even

micro-

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Almsherqi, Z.A., Kohlwein, S.D., and

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Figure 1. Visualizing a Helicoid

(A–D)Toconstructahelicoid,startwithanaxis(A)andthendrawahelixaround

it, like the snake in a caduceus. Draw a line running from the axis through the

helix and perpendicular to the axis (B). Draw similar lines through all points on

the helix,so that thelines trace outasurfaceas they rotate up thehelix (C). The

result is a helicoid (D), the structure discovered in the connections between ER

cisternae by Terasaki et al. (2013). In the ER, successive stacked cisternae

would be fused to the helicoid periodically as indicated by arrows.

266 Cell 154, July 18, 2013 ª2013 Elsevier Inc.