Membranes by the Numbers

Rob Phillips∗
Department of Applied Physics and Division of Biology and Biological
Engineering, California Institute of Technology, Pasadena, California, U.S.A

$\ast$ E-mail<EMAIL_ADDRESS>
## The Quantitative Membrane Landscape

The pace at which biology is advancing is staggering. Just as there was a
short 50 year gap between the invention of manned flight by the Wright
Brothers and the beginning of the space age, in the little more than a half
century since the discovery of the structure of DNA and its interpretation
through the genetic code, the life sciences have entered their own age,
sometimes dubbed “the genome age”. But there is more to living matter than
genomes. While the genome age has unfolded, a second biological revolution has
taken place more quietly. This other success story in the emergence of modern
biology is the unprecedented and detailed microscopic view of cellular
structures that has been garnered as a result of the emergence of new ways to
visualize cells. Both electron and optical microscopy have afforded an
incredible view of the cellular interior. In addition, the use of techniques
for profiling the molecular contents of cells has provided a detailed,
quantitative view of the proteomes and lipidomes of both cells and the viruses
that infect them meaning that, in broad brush strokes, we know both what
molecular components the cell is made of and how the cellular interior looks.
A particularly fertile example that serves as the backdrop for the present
chapter is given by our ever improving understanding of the membrane
organization associated with the organelles and plasma membranes of cells of
many kinds.[1]

The goal of this chapter is to develop a feeling for membranes in the form of
biological numeracy. That is, for the many different ways we can think about
membranes whether structurally, mechanically or electrically, we will try to
formulate those insights in quantitative terms. The strategy used here is to
move back and forth between a data-based presentation in which key
quantitative facts about membranes are examined, and a rule-of-thumb and
simple-estimate mentality, in which we attempt to reason out why those numbers
take the values they do. For those cases in which we introduce hard data, our
device will be to use the so-called BioNumbers ID (BNID).[2] Some readers will
already be familiar with the PMID (Pubmed ID) that links the vast biological
literature and databases. Similarly, the BioNumbers database provides a
curated source of key numbers from across biology. By simply typing the
relevant BNID into your favorite search engine, you will be directed to the
BioNumbers website where both the value of the parameter in question will be
reported as well as a detailed description from the primary literature of how
that value was obtained. Unfortunately, my presentation is representative
rather than encyclopedic. There is much more that could have (and should have)
been said about the fascinating question of membrane numeracy. Nevertheless,
the hope is that this gentle introduction will inspire readers to undertake a
more scholarly investigation of those topics they find especially interesting,
while still providing enough quantitative insights to develop intuition about
membranes.

There are many conceivable organizational principles for providing biological
numeracy for membranes. The strategy to be adopted here is to organize the
numbers that characterize membranes along several key axes, starting with
their sizes and shapes, turning then to their chemical makeup, followed in
turn by some key themes such as the mechanics of membrane deformations, the
transport properties of various molecular species across and within membranes
and the electrical properties of membranes. In particular, depending upon the
context, there are many different ways of thinking about membranes (see Figure
1) and each of these different pictures of a membrane has its own set of
characteristic parameters. Once these parameters are in hand, we then attempt
to make sense of all of these numbers in a section on membrane Fermi problems
with the ambition of this section being to give an order-of-magnitude feeling
for the numbers that characterize membranes.[3, 4] The notion of a Fermi
problem refers to the penchant of Enrico Fermi to find his way to simple
numerical estimates for complex phenomena of all kinds in short order. The
chapter closes with a look to the future that lays out my views of some of the
key challenges that await the next generation of scientists trying to further
the cause of membrane numeracy.

Figure 1: The many quantitative faces of a membrane. Depending upon the
experiments being done or the questions being asked, the way we characterize
membranes is different. When thinking about mechanical deformations of a
membrane, we will characterize it in terms of elastic constants. Mass
transport across and within membranes is described by permeability and
diffusion coefficients, respectively. When describing changes in the membrane
potential, we characterize the membrane in terms of its conductivity and
capacitance. Statistical mechanics teaches us to think about membranes from
the standpoint of their fluctuations which interestingly contribute to the
membrane tension. Each section of the chapter explores one of these ways of
characterizing membranes from the point of view of biological numeracy.

## The Geometrical Membrane: Size and Shape

An inspiring episode from the history of modern science that relates deeply to
biological numeracy was the unfolding of our understanding of lipids and the
kinds of extended structures they make both in the laboratory and in living
cells. In his book “Ben Franklin Stilled the Waves”, Charles Tanford gives a
charming and insightful tour of this development starting with the efforts of
Franklin who was intrigued by the capacity of lipids when spread on water to
“still the waves”. Indeed, this fascination led Franklin to a famous
experiment in which a spoonful of oil was seen to cover nearly half an acre of
Clapham Common near London, giving a first indication of the molecular
dimensions of lipids.

Franklin’s insights into the structural significance of thin films of lipids
led a century later to the emergence of more formal laboratory methods for
studying lipid monolayers. In a short 1890 paper on the subject, Lord Rayleigh
notes “In view, however, of the great interest which attaches to the
determination of molecular magnitudes, the matter seemed well worthy of
investigation”. To that end, he performed a table-top version of the Franklin
experiment concluding that for a film of olive oil he could actually compute
the thickness of a monolayer, reporting a lipid length of 1.63 nm.[5] Agnes
Pockels in a letter to Lord Rayleigh published in Nature only a year later
described her efforts with a trough and force measuring balance to explore
surface tension of films on water surfaces.[6] But above all, the study of the
“determination of molecular magnitudes” entered a new stage as a result of a
tour de force investigation by Irving Langmuir that really gave a first
detailed molecular view of lipid molecules and the kinds of collective
structures they can form.

Langmuir walks us through his experiments and deep musings about the shape of
lipids in his paper entitled “The Constitution and Fundamental Properties of
Solids and Liquids. II. Liquids”. Here I reproduce a lengthy but interesting
series of quotes from that paper, where Langmuir says: “In order to determine
the cross-sections and lengths of molecules in oil films, experiments similar
to those of Marcelin were undertaken. The oil, or solid fat, was dissolved in
freshly distilled benzene, and, by means of a calibrated dropping pipet, one
or two drops of the solutions were placed upon a clean water surface in
photographic tray. The maximum area covered by the film was measured. Dividing
this area by the number of molecules of oil on the surface, the area of water
covered by each molecule is readily obtained. The results are given in the
first column of Table I.” Langmuir’s Table I is reproduced here as our own
Figure 2 and shows the impressive outcome of his work, providing not only key
numbers but also a much-needed object lesson in the power of indirect
experimental methods. He then goes on to tell the reader how he found the
lengths of these same molecules noting, “The volume of each molecule is found
by dividing the “molecular volume” of the oil (M/$\rho$) by the Avogadro
constant N. By dividing this volume by the cross-section of each molecule, the
length of the molecule in a direction perpendicular to the surface can be
obtained.”

Figure 2: Lipid sizes as obtained by Irving Langmuir.[7] This table shows that
already a century ago, indirect methods had yielded a quite modern picture of
lipid geometry.

Langmuir then goes on to say: “It is interesting to compare these lengths with
the cross-sections. As a rough approximation we may assume that the dimensions
of the molecule in directions parallel to the surface can be found by taking
the square root of cross-section. This is equivalent to assuming that each
molecule in the surface film occupies a volume represented by a square prism
with its axis vertical. The length of the square side, which we shall refer to
as the average diameter, is given in the second column of Table I, while the
height of the prism (or the length of the molecule) is given in the third
column.” Again, the reader is encouraged to refer to Figure 2 to see
Langmuir’s results. He then proceeds telling us “It is seen at once that the
molecules are very much elongated. Thus the length of the palmitic acid
molecule is about 5.2 times the average diameter. The results prove that the
molecules arrange themselves on the surface with their long dimension vertical
as is required by the theory.”[7] Langmuir went much farther commenting on the
significances of the different lengths and areas emboldening him even to think
about the role of unsaturated bonds in determining molecular shape. Indeed,
one of my favorite aspects of these experiments from Langmuir is that they led
him to understand both the number of tails and their degree of saturation
truly providing a detailed molecular picture of these molecules. This work
went even farther in the hands of Gorter and Grendel who used similar trough
experiments to hypothesize that biological membranes are lipid bilayers, a
subject we will take up again in the section on “The Electrical Membrane”,
though I note that there are subtleties about the Gorter and Grendel approach
that continue to escape me since in their analysis, they did not account in
any way for the fraction of the membrane that is taken up by membrane
proteins.[8]

What we see from this short historical interlude is that already at the
beginning of the twentieth century, long before tools such as x-ray
diffraction and nuclear magnetic resonance had made their way onto the scene
of modern biological science, scientists had already gleaned a detailed view
of the makeup of lipids and started to synthesize a view of how they assemble
in cell membranes. The same story already told by experiments using Langmuir
troughs has been retold much more accurately on the basis of x-ray and
electron microscopy experiments.[9, 10] Indeed, an assessment of the current
state of the art for the same kinds of questions originally broached by
Langmuir can be seen in Table 1.

Lipid | Area/lipid (nm2) | thickness (nm)
---|---|---
DLPE | 0.51 $\pm$ 0.005 | 2.58
DOPS | 0.65 $\pm$ 0.005 | 3.04
DMPC | 0.61 $\pm$ 0.005 | 2.54
DLPC | 0.63 $\pm$ 0.005 | 2.09
POPC | 0.68 $\pm$ 0.015 | 2.71
diC22:1PC | 0.69 $\pm$ 0.0005 | 3.44
DOPC | 0.72 $\pm$ 0.005 | 2.68
Table 1: Summary of modern version of measured lipid geometric parameters to
be compared to those from Langmuir shown in Figure 2. All values taken from
[9]

The rules of thumb that emerge from a century of study of these molecules is
that we should think of lipid masses as being in the range of many hundreds of
Daltons up to thousands of Daltons for the largest lipids. The lengths of
these lipids vary with tail lengths of $\approx$ 2-2.5 nm. The tail-length
rule of thumb can be articulated more precisely in terms of the number of
carbons in the tail ($n_{c}$) as $l_{c}=n_{c}l_{cc}$, where the length of a
carbon-carbon bond is approximately $l_{cc}\approx 0.13$ nm.[11] The cross
sectional areas of lipids can be captured by a rule of thumb that the area per
lipid is $\approx$ 0.25 - 0.75 nm2. Note that the use of a single cross-
sectional area is overly facile because lipids can have much richer shapes
than the “square prism with its axis vertical” described by Langmuir. Indeed,
because lipids can have shapes more like wedges, this can lead to spontaneous
curvature, a topic that we will not delve into more deeply here, but that is
critical to understanding the relation between membrane shape and lipid
geometry. These rules of thumb are based upon a host of different
measurements, with the thickness and area per molecule found here (BNID
101276, 104911, 105298, 105810, 105812). We have traveled a very long way
since the days of Langmuir, since we can now order designer lipids with
specific chemical properties and even with special groups attached making
these lipids fluorescently labeled.

A higher-level view of the structure of cell membranes has been developing on
the basis of electron cryo-microscopy which offers an unprecedented view of
the very same structural features already explored a century ago using the
kinds of indirect methods described above. Figure 3 provides a collage of
electron cryo-microscopy images of bacterial cell membranes. We see that in
most of these cases, the inner and outer membranes are easily resolved and
that they have a thickness of roughly 5 nm (BNID 104911). To be more precise,
we should bear in mind that in quoting numbers such as a membrane thickness of
5 nm, of course, we are talking about a characteristic dimension since the
interaction of the lipids with the surrounding proteins can induce thickness
variations due to the effect of hydrophobic matching of the proteins and
lipids.[12, 13, 14] Since the bacteria themselves are several microns in
length and a bit less than a micron in diameter, we can make a simple estimate
of the overall membrane area of the inner and outer membranes by thinking of
the bacterium as a spherocylinder with a characteristic volume of $1~{}\mu
m^{3}\approx 1fL$ and a corresponding surface area of 5-10 $\mu m^{2}$.

Figure 3: Electron cryo-microscopy images of bacterial cell walls. The
Caulobacter crescentus cell gives an impression of overall cell dimensions
while the higher-resolution images of other bacteria zoom in on their
membranes. Note that these are gram-negative bacteria meaning that their
external membrane architecture consists of an inner membrane, a cell wall and
an outer membrane. (Images courtesy of Grant Jensen and his laboratory
members)

The membranes of eukaryotic organisms are typically more heterogeneous and
complex than those shown in Figure 3. Figure 4 gives several examples coming
from electron microscopy to make that point. First, such cells, like their
prokaryotic counterparts, have an external plasma membrane that separates them
from the rest of the world. But as seen in Figure 4(A), even the cell surface
can adopt extremely complex geometries as exemplified by the microvilli. One
of my favorite examples in all of biology is shown in Figure 4(B) where we see
the outer segment of a photoreceptor with its dense and regular array of
membrane stacks. However, it is perhaps the spectacular organellar membranes
(see Figure 4(C)) that give a sense of the great challenges that remain in
understanding membrane shape in cells.[15] Structural complexity similar to
that found in the mitochondria abounds in other organelles such as the
endoplasmic reticulum.[16, 17]

Figure 4: Eukaryotic membrane structures. (A) Apical surface of intestinal
epithelial cells showing the dense membrane folds around the microvilli. The
sugar chains extending outwards from the surface of the membrane can also be
seen as a fuzzy layer above the microvilli. (B) Stacks of membranes packed
with photoreceptors in the outer segment of a rod cell. (C) Thin section of a
mitochondrion surrounded by rough endoplasmic reticulum from the pancreas of a
bat. (All figures adapted from “Physical Biology of the Cell”, Garland Press,
2012)

Our brief foray into the size and shape of membranes and the molecules that
make them up would of course be woefully incomplete without also commenting
briefly on the role proteins play in our modern view of biological membrane
structures. Though early ideas about cell membranes painted a picture of a sea
of lipids dotted with membrane proteins, the modern view has turned out to be
altogether different. “A picture is emerging in which the membrane resembles a
cobblestone pavement, with the proteins organized in patches that are
surrounded by lipidic rims, rather than icebergs floating in a sea of
lipids”.[18] As a rule of thumb, we can think of the protein densities in
bacterial membranes as being $\sigma\approx 10^{5}$ proteins/$\mu$m2. This can
be used in turn to estimate the typical center-to-center protein spacing in
the cell membrane as $d\approx\sigma^{-{1\over 2}}\approx 3~{}nm$, a result
that is uncomfortably tight given that typical protein sizes are themselves
$3-5$ nm as seen in Figure 5. The question of mean membrane-protein spacing is
also of great interest in the context of organellar membranes, with a hint at
what can be expected in these cases given by a classic study on synaptic
vesicles.[18]

Figure 5: Sizes and shapes of membrane proteins.[19] Top and side views of
several notable membrane proteins. Note the 10 nm scale bar, though the
membrane thickness can also be used as a scale marker as indicated in Figure
3. (Images courtesy of David Goodsell and adopted from “Cell Biology by the
Numbers”, Garland Press, 2015)

## The Chemical Membrane

With each passing generation, our understanding of the structures of the cell
is becoming more and more refined. As shown in the previous section, we have
learned a huge amount about the structures of membranes and the molecules that
make them up. But what about the specific chemistry of these membranes? One of
the ways that our picture of the membranes of living cells has been
transformed is through the ability to count up the molecules of different
kinds, both the lipids making up the plasma membrane and organelles and of the
many proteins that decorate these membranes. In many ways, the development of
a census of lipid composition of membranes is an astonishing achievement and
has revealed not only that these membranes are heterogeneous, but that the
cell “cares” about its lipid composition.[18, 20, 21, 22, 23, 24, 25, 26, 27,
28] Though there is still much left to be understood about precisely how cells
keep track of their membrane composition and why they “care”, in this section
of the chapter we focus on what has been learned thus far about these chemical
effects from a quantitative perspective. For a pedagogical review, see chapter
4 of Buehler’s interesting book. [1]

The same membrane strategy used to separate the interior of cells from the
extracellular medium is also used for separating the cellular interior into a
collection of membrane-bound organelles such as the nucleus, the endoplasmic
reticulum, the Golgi apparatus and mitochondria. Each of these membrane
systems is host to lipids that come in different shapes, sizes and
concentrations. There are hundreds of distinct types of lipid molecules found
in these membranes and, interestingly, their composition varies from one
organelle to the next. This is highly intriguing since these distinct membrane
systems interact directly through intracellular trafficking by vesicles. This
same heterogeneity applies to the asymmetric plasma membrane, with different
classes of lipids occupying the outer and cytosolic leaflets of the membrane
(i.e. the two faces of the lipid bilayer).

Experimentally, the study of lipid diversity is a thorny problem. Sequencing a
set of single or double bonds along a carbon backbone requires very different
analytic tools than sequencing nucleotides in DNA or amino acids along
proteins. Still, the omics revolution has hit the study of lipids too. The use
of careful purification methods coupled with mass spectrometry have made
inroads into the lipid composition of viral membranes, synaptic vesicles, and
organellar and plasma membranes from a number of different cell types. Indeed,
the numbers in this section owe their existence in no small measure to the
maturing field of lipidomics, based in turn upon impressive advances in mass
spectrometry. As noted above, we remain largely in the fact-collection stage
of this endeavor since a conceptual framework that allows us to understand in
detail the whys and wherefores of lipid compositions and how they change with
growth conditions is quite immature.

Figure 6: Lipids in yeast. The top panel shows the relative proportions of
different types of lipid as a function of the physiological state of the cells
as revealed by the inset in the upper right. That inset shows the result of
cellular growth as measured by spectrophotometry and leading to the optical
density (OD) as a function of time. The lower panel shows the diversity of
different phospholipids. These lipids exhibit both different tail lengths and
degree of saturation as shown by the schematics of the lipids in the lower
panel. The abbreviations used in the figure are: CL: cardiolipin; Erg:
Ergosterol; IPC: inositolphosphorylceramide; MIPC: mannosyl-inositol
phosphorylceramide; M(IP)2C: mannosyl-di-(inositolphosphoryl) ceramide; PA:
phosphatidic acid; PC: phosphatidylcholine; PE: phosphatidyl-ethanolamine; PI:
phosphatidylinositol; PS: phosphatidylserine; TAG: Triacylglycerols; DAG:
diacylglycerol; LPC: Lysophosphatidylcholine Adapted from “Cell Biology by the
Numbers”, Garland Press, 2015. Data in top panel adapted from [27] and data in
bottom panel adapted from [22].

Perhaps the simplest question we can pose about lipids at the outset is how
many there are in a typical cell membrane. A naive estimate for a bacterial
cell can be obtained by noting that the area of the bacterial cell membrane is
roughly 5 $\mu$m2, and recalling further that many bacteria have both an inner
and outer membrane. To effect the estimate, we take

$N_{lipids}={4\times\mbox{membrane area}\over\mbox{area per
lipid}}\approx{20\times 10^{6}~{}\mbox{nm}^{2}\over
1/4~{}\mbox{nm}^{2}}\approx 8\times 10^{7},$ (1)

where the factor of 4 accounts for the fact that we have two lipid bilayers
because of the presence of both an inner and outer membrane. This estimate is
flawed, however, because we failed to account for the fraction of the membrane
area that is taken up by proteins rather than lipids. As was seen in the
previous section on size and shape, a useful rule of thumb is that 1/4 of the
membrane area is taken up by proteins [20], so our revised estimate of the
number of lipids in a cell membrane would be reduced by 25%. Further, note
that we used an area per lipid on the low side and if we amended that estimate
to a value of $\approx$ 0.5 nm2 per lipid, this would also bring our estimate
down by a factor of two. Literature values reported for the bacterium E. coli
claim roughly $2\times 10^{7}$ lipids per E. coli cell, squaring
embarrassingly well with our simple estimate, and leaving us with a useful
rule of thumb for the lipid density of

$\sigma\approx{2\times 10^{7}~{}\mbox{lipids}\over 5~{}\mu\mbox{m}^{2}\times
4~{}\mbox{leaflets}}\approx
10^{6}{\mbox{lipids}\over\mu\mbox{m}^{2}~{}\mbox{leaflet}}.$ (2)

[29] Given our estimate of $2\times 10^{7}$ lipids per bacterial cell, we can
make a corresponding estimate of the fraction of the cell’s dry mass that is
lipids. As a basis for comparison, we recall that the number of proteins per
bacterial cell is $\approx 3\times 10^{6}$. [29, 4, 19] If these proteins have
an average mass of 30,000 Da, this means the total protein mass is roughly
$10^{11}$ Da or $0.15$ pg, corresponding to roughly 1/2 of the dry mass of a
bacterial cell. For our $2\times 10^{7}$ lipids, each with a mass of roughly
$1000$ Da, this means that the lipids contribute an approximate mass of
$2\times 10^{10}$ Da, corresponding to 20% of the protein mass, or 1/10 of the
dry mass of the cell.

What about the composition of membranes? In broad brush strokes, what has been
learned in lipidomic studies is that in most mammalian cells, phospholipids
account for approximately 60% of total lipids by number and sphingolipids make
up another 10%. Non-polar sterol lipids range from 0.1% to 40% depending on
cell type and which subcellular compartment is under consideration. The
primary tool for such measurements is the mass spectrometer. In the mass
spectrometer each molecule is charged and then broken down, such that the
masses of its components can be found and from that its overall structure
reassembled. Such experiments make it possible to infer both the identities
and the number of the different lipid molecules. Absolute quantification is
based upon spiking the cellular sample with known amounts of different kinds
of lipid standards. One difficulty following these kinds of experiments, is
the challenge of finding a way to present the data such that it is actually
revealing. In particular, in each class of lipids there is wide variety of
tail lengths and bond saturations. Figure 6 makes this point by showing the
result of a recent detailed study of the phospholipids found in budding yeast.
In Figure 6(A), we see the coarse-grained distribution of lipids over the
entire class of species of lipids found while Figure 6(B) gives a more
detailed picture of the diversity even within one class of lipids.[22] Studies
like the one presented above for yeast have also been done in other eukaryotes
as shown in Figure 7. [21, 30] Data like this shows that the subject is even
more interesting than one might first expect because we see that lipid
composition is different for different organelles. As noted earlier, this is
especially intriguing given the fact that these different organelles are in
dynamical contact as a result of intracellular trafficking, calling for a
mechanistic and quantitative description of how these composition
heterogeneities are maintained. All of these measurements leave us with much
left to understand since as noted at the beginning of this section, the
question of how cells regulate and control their lipid composition and why
they care remains unanswered.

Figure 7: Organellar lipids in mammalian cells. (A) Lipid production is spread
across several organelles. The text associated with each organelle shows the
site of synthesis for the major lipids. The main organelle for lipid
biosynthesis is the endoplasmic reticulum (ER), which produces the bulk of the
structural phospholipids and cholesterol. (B) The lipid composition of
different membranes also varies throughout the cell. The graphs show the
composition out of the total phospholipid for each membrane type in a
mammalian cell. As a measure of sterol content, the molar ratio of cholesterol
to phospholipid is indicated. SM: sphingomyelin; R: remaining lipids. For more
detailed notation see caption of Figure 6. (Adapted from [21])

## The Mechanical Membrane

Electron microscopy images make it abundantly clear that whether we think of
the stacked membrane discs making up the outer segment of a photoreceptor or
the tortuous folds of the endoplasmic reticulum of a pancreatic cell,
biological membranes are often severely deformed. But as we all know from
everyday experience, changing the shape of materials usually costs energy. As
a result of membrane deformations, energetic costs resulting from both
membrane stretching and bending are incurred. The aim of this part of the
chapter is to give a quantitative view of the energetic cost of these
deformations.[31] These two different membrane deformation mechanisms are
indicated schematically in Figure 8.

Figure 8: The mechanics of membrane deformations. One of the deformation modes
is changing the membrane area by stretching. The second mode of membrane
deformation considered here is membrane bending.

A natural mechanical question we might imagine starting with is the energetic
cost associated with bending the membrane. The free energy cost to deform a
tiny patch of membrane is codified in the form of the so-called Helfrich-
Canham-Evans free energy.[11] For a tiny patch of membrane with area $\Delta
A_{patch}$, the free energy cost to bend it is given by

$\mbox{energy to bend a membrane patch}={\kappa_{B}\over 2}({1\over
R_{1}}+{1\over R_{2}})^{2}\Delta A_{patch},$ (3)

where $\kappa_{B}$ is the membrane bending rigidity and $R_{1}$ and $R_{2}$
are the principal radii of curvature of the patch of membrane. Note that the
membrane bending rigidity has units of energy since the unit of the factor in
parentheses is $1/\mbox{area}$ which is cancelled by $\Delta A_{patch}$ which
has units of area. The values of $R_{1}$ and $R_{2}$ characterize the
curvature of the surface at the point of interest. Specifically, if we visit a
particular point on the surface, we can capture the curvature by using two
orthogonal circles whose radii are chosen so that those two circles most
closely follow the shape of the surface at that point. Given the free energy
in eqn. 3, we can find the total free energy of a given deformed membrane
configuration by adding up the contribution from each little patch as

$E_{bend}={\kappa_{B}\over 2}\int dA({1\over R_{1}(x,y)}+{1\over
R_{2}(x,y)})^{2},$ (4)

where now we acknowledge that the curvature (as measured by $R_{1}$ and
$R_{2}$) is potentially different at each point on the surface. Of course, the
scale of this energy is dictated by the bending rigidity $\kappa_{B}$. Our
discussion has neglected a second topological contribution to the membrane
deformation energy related to the Gaussian curvature, though clearly such
terms will be of interest in the context of the topologically rich membrane
structures found in cell organelles.[11]

Figure 9: Elastic moduli characterizing membrane bending and stretching. (A)
Values for the membrane bending rigidity. Each value corresponds to a
different lipid with the values showing a range of tail lengths and tail
saturation. (B) Values for the area stretch modulus. All values obtained using
pipette aspiration experiments.[32]

A wide range of experiments on a variety of different lipids suggest the rule
of thumb that the bending modulus ($\kappa_{B}$) for lipid bilayers is in the
range 10-25 kBT.[32, 33] Characteristic values of the membrane bending
rigidity for phospholipid bilayers are shown in Figure 9(A). We will freely
use kBT for our energy units and note the conversion factors $k_{B}T\approx
4.1~{}pN~{}nm\approx 4.1\times 10^{-21}~{}J$. The presence of sterols in lipid
bilayers can increase those numbers to $\approx 100$ kBT.[34] Interestingly,
even measurements on biological membranes derived from the ER and Golgi
apparatus report a membrane rigidity of $\kappa_{B}\approx 3\times 10^{-19}$ J
$\approx 75$ kBT (BNID 110851), only a factor of three larger than the values
for phospholipid bilayers reported in Figure 9(A).[35, 36]

Another important question we can ask about membrane deformations is the
energy cost for changing the area of the membrane as seen in Figure 8. When we
stretch a membrane away from its equilibrium area, a consequence is the
development of a tension in that membrane. One way to understand the magnitude
of membrane tensions is by appealing to a so-called constitutive equation
which loosely speaking relates force and membrane geometry. In particular, the
mechanics of membrane stretching is often described by the constitutive
equation

$\tau=K_{A}{\Delta A\over A_{0}},$ (5)

where $K_{A}$ is the area stretch modulus and $\Delta A$ is the area change.
To figure out the tension, we compute the change in area, normalize by the
total unstressed area $A_{0}$ and then multiply by the modulus $K_{A}$. In
general, when we change the area of a patch of membrane by some amount $\Delta
A$, the corresponding free energy cost can be written as

$\mbox{stretching energy}={K_{A}\over 2}({\Delta A\over A_{0}})^{2}A_{patch},$
(6)

where note that the units of the area stretch modulus $K_{A}$ are energy/area.
Several examples of the values adopted by the area stretch modulus are shown
in Figure 9(B), which gives the interesting insight that for a range of tail
lengths and degrees of saturation, the area stretch modulus is nearly
constant.

The actual magnitudes of the tensions in the membranes of both vesicles and
cells can vary over a wide range and even the underlying mechanistic origins
of these tensions are different depending upon what regime of tension we are
considering. Interestingly, the energetics of area change is a subtle one in
the same way as the energetics of stretching a polymer like DNA is.
Specifically, let’s remind ourselves of the subtleties associated with DNA
stretching as a prelude to thinking about membrane stretching.[11, 4] In the
“force free” state, DNA will be folded up and compact since such states have
lower free energy in part because the entropy of the compact conformation is
higher. To stretch DNA, the free energy cost can be thought of as being almost
entirely entropic, meaning that with increasing stretch, there are fewer and
fewer configurations available to the DNA and hence the entropy decreases,
resulting in a net increase in free energy. It is only when the DNA is
stretched to its full contour length that we enter a different regime that
actually involves molecular bond stretching. Because the mechanisms in these
regimes are different, it should not surprise us that they are actually
characterized by different mechanical stiffnesses. Similar intuition emerges
for the membrane case.

By analogy with polymer stretching, we can think of the energetic cost
associated with membrane deformations in much the same way. That is, for a
floppy (low tension) membrane, stretching the membrane has an associated free
energy cost that results from “pulling out the wrinkles”, and is effectively
entropic.[11] At higher tensions, the actual bond stretching effect
intervenes. Though very few systematic insights have been obtained for
thinking about the membranes within cells, a series of rigorous, systematic
studies in lipid bilayers have set the standard in the field.[32] At even
higher tensions, lipid bilayer membranes will actually rupture with the
rupture tensions occurring between 5 mN/m and 10 mN/m depending upon the type
of lipids in question.[37]

Though there are fewer systematic measurements for cellular membranes, some
clever experiments have shed light on this topic as well. The tension measured
in ER membrane networks has a value of $1.3\times 10^{-2}$ mN/m while that
measured in the Golgi membrane is given by $0.5\times 10^{-2}$ mN/m.[38] These
numbers are quite small as can be seen by comparing them to the membrane
rupture tension which is a thousand times larger with a range of $5-10$ mN/m
as noted above.[37] Note also that the subject of membrane tension is a tricky
one in the cellular setting because measured tensions have many contributions
including from the underlying cytoskeleton and the battery of molecular motors
associated with it.[39] There is an excellent review featuring both a clear
discussion of the different methods as well as the range of measured
tensions.[40] Table 1 of that review includes an exhaustive listing of
measured membrane tensions as well as the caveats associated with each such
measurement.

## The Dynamic Membrane

Perhaps the defining feature of biological membranes is that they serve as
barriers between some compartment of interest (the cytoplasm, the Golgi
apparatus, the nucleus, the endoplasmic reticulum, etc) and the rest of the
world. The very word ”barrier” points toward underlying molecular rules that
determine the rate at which molecules cross through or move within membranes,
and thereby regulate how a cell distinguishes itself from the environment. In
this section, we begin by exploring the permeability of biological membranes
to various molecular species. After that, we then turn to the diffusive
properties of molecules within the membrane.

One of the key ways we characterize membrane permeability is to ask the
question of how many molecules cross a given area of membrane each second, a
quantity defined as the flux, $j$. In particular if we have a difference in
concentration of some species across the membrane given by $\Delta c$, then in
the simplest model the flux is given by

$j=-p\Delta c,$ (7)

where the parameter $p$ is the permeability of interest here. Note that a more
rigorous treatment of the flux invokes the chemical potential difference
across the membrane, though for our purposes this simple linearization
suffices.[41, 42] The units of the permeability can be deduced by noting first
that the units of $j$ are

$\mbox{units of}~{}j={\mbox{number of molecules}\over L^{2}T}.$ (8)

Here we adopt the standard strategy when examining units of physical
quantities of using the symbol $L$ to signify units of length and $T$ to
signify the units of time.[43] Given these conventions, the units of
concentration are

$\mbox{units of}~{}c={\mbox{number of molecules}\over L^{3}}.$ (9)

The requirement that the units on the two sides of the equation balance
implies that the units of the permeability itself are

$\mbox{units of}~{}p={{\mbox{number of molecules}\over
L^{2}T}\over{\mbox{number of molecules}\over L^{3}}}={L\over T}.$ (10)

In the remainder of the paper, we will report units of permeability in nm/s,
though often one finds values reported in cm/s as well.

The first and probably most important thing we should say about the numerical
values adopted by membrane permeability is that there is no such thing as the
membrane permeability. That is, the rate at which molecules pass across
membranes is an extremely sensitive function of which molecules we are
discussing as well as the type of molecules making up the membrane itself.[44,
37, 10] Figure 10 makes this point clear by reporting the range of values for
permeability for a number of different molecular species revealing a more than
10-order-of-magnitude range of permeabilities, with the membrane being
effectively impermeable to ions such as Na+ and K+, while for water molecules,
the permeability is ten orders of magnitude larger. Though this doesn’t rival
the 30 order of magnitude range that is found for electrical conductivities of
different materials, these numbers still imply a huge difference in the
transport properties of different molecules across membranes.

Figure 10: Range of membrane permeabilities. Permeability coefficients for a
number of different lipid species showing the huge dynamic range in
permeability.

How are such permeabilities measured? One approach to measuring these membrane
permeabilities is the use of radioactive tracer molecules. By setting up a
membrane separating two aqueous regions with different compositions, one can
measure the accumulation of the tracer in one region as a result of flux from
the other region over time.[44] A second important set of measurements for
water permeability were performed using giant unilamellar vesicles using the
so-called micropipette aspiration method where an osmotic pressure is applied
across the membrane and the resulting flux of water across the membrane is
measured. Here the idea is that a pipette with a characteristic diameter of
several microns is used to grab onto a vesicle with a diameter of roughly 10
$\mu$m or larger. By applying a suction pressure, the tension of the vesicle
can be monitored. Further, by using video microscopy, the volume of the
vesicle can be carefully monitored, giving a sense of the rate at which the
vesicle is inflated as a result of mass transport of water across the
membrane. The results of such measurements for a set of different lipid types
are shown in Figure 11, with values entirely consistent with those shown
schematically in Figure 10.

Figure 11: Range of membrane permeabilities for water. Measurements made at 21
∘ C using the micropipette aspiration technique in conjunction with video
microscopy to monitor vesicle size.[37]

The classic work of Hodgkin and Huxley offered many important insights. To my
mind, one of the most interesting arguments that they made is a testament to
the role of clear theoretical (and quantitative) thinking in biology. In
particular, they argued that the membrane permeability to ions such as Na+ and
K+ must change transiently and substantially to permit key ions across the
otherwise impermeable membrane (see Figure 10 to get a sense of the extremely
low permeability of charged ions). Specifically, they introduced a highly
nonlinear permeability response that suggested that there must be molecules in
the membrane of the cell that could selectively change the permeability in
response to changes in driving forces such as the membrane potential,
effectively hypothesizing the existence of ion channels before they were
known.

We now know that biological membranes are littered with batteries of different
channels and pumps whose job it is to transiently alter the permeability of
the membrane or to actively transport molecular species across it. These
membrane proteins are responsible for many physiologically important functions
including the transport of ions and sugars such as glucose and lactose that
are critical to the cellular economy. Ions typically pass across ion channels
at rates between $10^{7}$ and $10^{9}$ ions per second, though of course this
rate depends upon the concentration difference across the membrane itself
(BNID 103163,103164). Glucose transporters have a much lower characteristic
rate of several hundred sugars per second (BNID 102931, 103160) while
bacterial lactose transporters have a characteristic rate of 20-50 sugars per
second (BNID 103159). Though here we report on the rates associated with
several well-known membrane proteins, more generally, the rates at which the
various membrane proteins that are responsible for transport operate are not
that well known, with a dearth of modern data spanning the range of different
membrane transporters (BNID 103160).[45]

A second kind of membrane dynamics different from the transport across the
membrane described above is diffusion of molecules laterally within the
membrane. As already noted throughout the chapter, the membrane is a highly
heterogeneous composite of lipids and proteins and when thinking about the
diffusive dynamics within the membrane, we need to do so on a molecule-by-
molecule basis. Since we are thinking about membranes, the first class of
molecules we might be interested in characterizing are the lipids
themselves.[46, 47, 48, 49, 50] For example, in eukaryotic cell membranes, by
using the clever method of fluorescence-recovery-after-photobleaching (FRAP),
a lipid diffusion constant of $0.9$ $\mu$m2/s was measured.[46] This diffusion
constant is roughly ten-fold lower than the values that would be found in a
model lipid bilayer membrane.[51] More recent measurements confirm these
classic numbers (see Figure 4 of ref. [50], for example).

It is of great interest to characterize the in-plane diffusion not only of the
lipids themselves, but also of the proteins that populate those membranes.
Figure 12 gives examples of membrane diffusion constants for several different
membrane proteins. Further, we need to acknowledge the large differences in
lateral diffusion coefficients between model membranes such as are found in
giant unilamellar vesicles where the values of diffusion coefficients for
membrane proteins are 1-10 $\mu m^{2}/s$ [52] and those in native membranes
where membrane proteins are characterized by diffusion coefficients that are
several of orders of magnitude lower with values of 0.01-0.1 $\mu
m^{2}/s$.[53, 54, 55, 50] However, these measurements are more nuanced than
first meets the eye and the results for several membrane proteins have been
shown to depend upon the time scales over which the diffusion is
characterized.[54] In particular, using the FCS method which probes diffusion
on short length and time scales, both the TAR receptor and TetA (a
tetracycline antiporter) were found to have diffusion constants of 4.2
$\mu$m2/s and 9.1 $\mu$m2/s, respectively, to be contrasted with the values of
0.017 $\mu$m2/s and 0.086 $\mu$m2/s, respectively found when using the FRAP
measurement. Indeed, as we will note in the final section of the chapter, the
question of how best to move from biological numeracy in model membranes to
biological membranes with their full complexity is one of the key challenges
of the coming years of membrane research.

Figure 12: Range of diffusion coefficients. (A) Diffusion coefficients for
different membrane proteins measured using fluorescence correlation
spectroscopy in giant unilamellar vesicles showing dependence on protein size.
The red line is a fit using the Saffman-Delbrück model which characterizes
membrane diffusion as a function of the size of the diffusing molecule.[56,
57] (B) Diffusion coefficients for different membrane proteins measured using
fluorescence recovery after photobleaching (FRAP) in the E. coli cell
membrane. The red line is an empirical fit as a function of the number of
transmembrane helices in the protein. The names refer to particular membrane
proteins used in the experiments. (A) adapted from [52] and (B) adapted from
[53].

## The Electrical Membrane

A membrane has many different properties as shown in Figure 1. So far, our
picture of membranes has focused on their mechanical and transport properties.
However, our discussion of action potentials and the pathbreaking work of
Hodgkin and Huxley already hinted at the view that membranes can also be
thought of as circuit elements. Specifically, part of this chapter’s very
business is to illustrate some of the different abstract ways of describing
membranes and what effective parameters to attribute to them. We now jettison
the view of a membrane as a mechanical object, instead focusing on it as a
collection of resistors and capacitors as shown in one of the panels of Figure
1.

The picture already developed under the heading of the “Electrical Membrane”
in Figure 1 tells us that in the presence of an electric potential, a lipid
bilayer behaves as an array of resistors and capacitors in parallel. One way
to measure the electrical conductance across a membrane patch is to form a
lipid bilayer membrane across a hole separating two solutions. Then, different
voltages are applied across the membrane and the current-voltage
characteristics are measured, with the membrane conductance then determined by
using the slope of these current-voltage curves. In our discussion of the
electrical membrane we characterize electrical properties on a per unit area
of membrane basis. For the conductance, a series of measurements like those
described above for a number of different charged species result in a range of
values for the bare membrane conductance of roughly 1-5 nS/cm2.[58, 59] To get
a sense of how small the membrane conductance is, note that if we consider a
characteristic conductance of 1 nS for an ion channel such as the
mechanosensitive channels found in bacteria [60], if we normalize by the area
this means that the channel conductance is more than ten orders of magnitude
larger than that of the membrane itself.

But membranes have more electrical properties than their conductance alone.[4]
Capacitance is a measure of the ability of a circuit element to store charge.
A local disruption of charge neutrality is permitted near surfaces. In
particular, in this setting, the capacitance is defined as the ratio of the
excess charge on either side of the membrane and the membrane potential,
$C=q/V_{mem}$. The capacitance of a patch of the cell membrane can be
approximated by thinking of it as a parallel plate capacitor. The charge on
the capacitor plates is ${\pm}\sigma A_{patch}$, where $\sigma$ is the excess
charge per unit area of membrane, and $A_{patch}$ is the area. The electric
field inside a parallel plate capacitor is uniform and equal to
$\sigma/(\varepsilon_{0}D)$, where $D$ is the dielectric constant of the
material between the plates. Therefore the potential drop across the membrane
is $V_{mem}=\sigma d/\varepsilon_{0}D$, where $d$ is the thickness of the
membrane, or the distance between the plates of the parallel plate capacitor.
Dividing the charge by the membrane voltage leads to the formula,
$C=\varepsilon_{0}DA_{patch}/d$, for the capacitance of a patch of membrane.
Since the cell membrane has a thickness of $d\approx 5$ nm and a dielectric
constant $D_{mem}=2$, its capacitance is predicted to be
$C_{area}=C/A_{patch}\approx 0.4\,\mu$F/cm2. The typical measured value for
the capacitance per unit area in cell membranes is $C_{area}=1\,\mu$F/cm2.[61,
62, 63]

We have already discussed the century long quest to understand the size of
lipid molecules and the membranes they make up. We learned that one branch of
these investigations passed through the enormously impressive work of Pockels,
Rayleigh and Langmuir. Amazingly, a completely independent line of enquiry in
the hands of Fricke related to the electrical properties of membranes led to
nearly the same result.[61] Using these ideas, we can recast the measured
value of the membrane capacitance as a result for the membrane thickness as

$d={\epsilon\over(C/A)}={2\epsilon_{0}\over(C/A)}\approx{2\times 8.8\times
10^{-12}{F\over m}\over 0.4\times 10^{-2}{F\over m^{2}}}\approx 4~{}nm,$ (11)

a beautiful result astonishingly close to the value obtained using the
equation of state of monolayers by Pockels, Rayleigh and Langmuir. Note that
to obtain this result, we rewrote the conventional membrane capacitance of
$0.4~{}\mu F/cm^{2}$ in the more appropriate SI units as $0.4\times
10^{-2}~{}F/m^{2}$. Further, whereas Frick used a relative dielectric constant
of 3, the estimate used here is based upon the value of 2. In light of the
measurement of the membrane capacitance, scientists such as Fricke realized
that this would provide yet another sanity check on the membrane
thickness.[61] In this era where many scientists seem almost to have scorn for
the idea of figuring things out without seeing them directly, the
determination of the thickness of lipid bilayers long before the advent of
direct techniques such as electron microscopy should give readers pause before
casually dismissing results that come from indirect measurements.

## The Fermi Membrane: Thinking up membranes

So far, this chapter has been an ode to biological numeracy in the context of
membranes, showing us the many different ways in which we can quantitatively
describe our hard-earned understanding of these fascinating structures. These
numbers are summarized in Table 2. But in the abstract, such numbers are often
boring and sometimes useless, or worse yet, misleading. To my mind, numbers
that characterize the world around us are only really interesting when put in
the context of some argument or reflection. For example, we know that if we
drop an object near the surface of the Earth, in the first second, it will
fall roughly 5 meters. So what? In the powerful hands of Newton, this
innocuous number became part of his inference of the law of universal
gravitation. There is a direct intellectual line from a knowledge of the
radius of the Earth and the distance to the moon to Newton’s estimate leading
him to further trust the idea that the force of gravity falls off as the
square of the distance. In that case, he realized that the distance to the
moon is roughly 60 times larger than the radius of the Earth, meaning that the
acceleration of the moon as it “falls” towards the center of the Earth should
be $(60)^{2}$ = 3600 times smaller than that associated with that apocryphal
apple falling from Newton’s tree. To finish off his estimate, he asked the
question of how far the moon falls compared to how far the apple falls when
watched for the same time and found them to “answer pretty nearly”, with the
moon falling roughly 1/3600 as much in one second as the 5 m a falling body at
the surface of the Earth falls in that same time interval. But what does this
have to do with our quantitative musings on membranes? To my mind, it
illustrates how powerful simple numerical arguments can sometimes be to help
us see whether our way of thinking is consonant with the known facts about a
system.

membrane parameter | range of parameter values | BNID
---|---|---
lipid length |  $\approx$ 2.5 - 3.5 nm | see Table 1
lipid area |  $\approx$ 1/4 - 3/4 nm2 | see Table 1
number of lipids per cell (bacterium) | $\approx 2\times 10^{7}$ | 100071
bending rigidity |  $10-25$ kBT | 105297
area stretch modulus | 200-250 mN/m (or $\approx$ 50 kBT/nm2) | 112590, 112659
membrane tension | $10^{-4}-1~{}k_{B}T/nm^{2}$ | 110849,112509, 112519
rupture tension | 1-2 $k_{B}T/nm^{2}$ | 112489, 110911
membrane permeability (water) | 10-50 $\mu$m/s | 112488
membrane capacitance | $\approx 1\mu F/cm^{2}$ | 110759, 109244, 110802
membrane resistance |  $0.1-1.5\times 10^{9}$ $\Omega$ cm2 | 110802
membrane potential | 100 mV | 109775, 107759
diffusion constant (lipid) |  $\approx$ 1 $\mu$m2/s | 112471, 112472
diffusion constant (membrane protein) |  $\approx$ 0.02 - 0.2 $\mu$m2/s | 107986
Table 2: Membranes by the numbers. A summary of the key numbers about
membranes discussed throughout the chapter/ for easy reference. Numbers
reported are “typical” values and should be used as a rule of thumb. For a
more detailed description of parameter values, the reader should use the
Bionumbers database through the relevant BNID. Also see Box 1 of [14].

Inspired by the long tradition of simple estimates when faced with numerical
magnitudes to describe the world around us, we now examine the ways in which
the numbers presented throughout the chapter can help us to better understand
membranes and the biological processes that take place at them. Indeed, we are
inspired by the notion of the so-called Fermi problems introduced at the
beginning of the chapter where the goal is to try to develop simple numerical
estimates for various quantities of interest by pure thought. Not only does
the Fermi approach allow us to estimate key magnitudes, but even more
importantly, it is one of the most powerful ways I know to make sure that the
stories we tell about our data actually make sense. In this section, we ask
ourselves whether we can understand some of the numerical values reported
throughout this chapter as well as what key scaling results we should bear in
mind when thinking about membranes. We pass through each of the sections of
the chapter in turn, each time taking the opportunity to reflect on the
numbers we have seen.

Size and shape redux. In the first part of the chapter, we considered
different ways of characterizing the size and shapes of membranes and the
molecules that make them up. This led us to the fascinating experiments of
Langmuir that used the relationship between tension and area as a way of
determining the size and shape of lipids. Here, our aim is to use order-of-
magnitude thinking to try and put those numbers in perspective. As an example
from everyday life where a simple numerical estimate of the Fermi type can
help us build intuition by giving us a sense of the relative sizes of
membranes and the cells they envelop, we consider the fuselage of an airplane.
One of the most popular tourist destinations in Seattle is the factory of
Boeing where one can see giant airplanes such as the 747, 777 and 787 in the
process of assembly. As part of that tour one is treated to the view of a
cross-section of a 747 fuselage which gives a sense of just how thin the skin
of an airplane really is. For the perceptive flyer, this same observation can
be made upon entering the plane by looking at the fuselage near the door. What
one notices is that the the exterior shell of the plane is less than a
centimeter thick while the overall diameter of that very same fuselage is
roughly 5 m, resulting in an aspect ratio of 1:500. Interestingly, the aspect
ratio of cell membrane width to cell size is quite comparable to those of an
airplane fuselage. For a 2 micron cell size, typical of a bacterium, the 4 nm
thickness of its cell membrane implies a similar aspect ratio of 1:500.

Concentrations. The section on concentrations reminded us that cell membranes
are made up of molecules and that even in tiny bacterial cells, there are tens
of millions of such molecules of hundreds of different types. A very simple
order of magnitude result that emerges from these numbers is a naive estimate
of the rates of lipid synthesis. Specifically, if the membrane area has to
double during the cell cycle, this tells us that the number of lipids in the
cell membrane has to double. For a bacterium such as E. coli, this means that
if a typical bacterium has $2\times 10^{7}$ lipids and the cell cycle is
roughly 2000 seconds, then the rate of lipid synthesis is roughly

$\mbox{bacterial lipid synthesis rate}={\mbox{number of lipids}\over\mbox{cell
cycle time}}\approx{2\times 10^{7}~{}\mbox{lipids}\over 2\times
10^{3}~{}\mbox{s}}\approx 10^{4}~{}\mbox{lipids/s}.$ (12)

It is deeply interesting to think of how the many different types of lipids
are each synthesized with the correct rates to maintain the overall
concentration distribution.

Another critical concern in our discussion of the chemistry of membranes was
how to think about the relative abundance of lipids and proteins. One of the
interesting ways to broach this question is through reference to the fraction
of genomes that is devoted to membrane proteins. We can examine this question
both from a genomic point of view and from a proteomic point of view.
Scientists have become increasingly adept at reading genomes and as a result,
by recognizing features such as transmembrane alpha helices, it is possible to
estimate the fraction of proteins that are membrane proteins with a rule of
thumb being that roughly 1/4 of the protein coding genes correspond to
membrane proteins.[64] From a proteomic point of view, this question can be
addressed by asking what are the copy numbers of these different membrane
proteins. Given that a bacterium such as E. coli has several million proteins
in total, what fraction of those proteins are in the membrane? To give a
feeling for the answer to that question, we ask about the copy numbers of some
key membrane proteins. Specifically, we consider membrane transporters,
components of the ATP synthesis machinery and the receptors of chemotaxis to
give an idea of the molecular census for some of the most important classes of
membrane proteins. Transport of sugars across the cell membrane is one of the
most critical activities of growing bacteria. Recent ribosome profiling
measurements and mass spectrometry measurements tell us that the number of
copies of sugar transporters for glucose (ptsI proteins, a component of the
phosphoenolpyruvate-dependent sugar phosphotransferase system) have a copy
number of between roughly 3000 copies per cell and 15,000 copies per cell
depending upon the growth conditions.[65, 66] We examine the relevance of
these numbers in the context of membrane dynamics below. ATP synthase is one
of the most important of membrane protein components in almost all cell types.
In E. coli the ATP synthase complex is built up of many different subunits.
For those subunits that come with a stoichiometry of one molecule per complex,
their copy number ranges between 3000 and 10,000 copies per cell.[65, 66]
Knowing these numbers provides a powerful sanity check on the rate of ATP
production per cell since with roughly 3000 such synthases, each rotating at
about 300 turns per second (BNID 104890), this means that over a cell cycle of
2000 s, on the order of $10^{9}$ ATPs will be generated, comparable to the
number needed to run the cellular economy. [4, 19] Finally, for the chemotaxis
receptors such as Tar and Tsr, the copy numbers can be as low as several
thousand and as high as nearly 40,000 per cell (BNID 100182).[67, 68] These
numbers give us a sense that if roughly 1000 of the 4000 or so E. coli
proteins are membrane proteins and each comes with a copy number of roughly
1000, then a first simple estimate is that there are a total of $10^{6}$
membrane proteins distributed across the inner and outer membranes of these
cells.

Membrane mechanics. Our section on membrane mechanics gave us a basis for
thinking about many key processes that take place in cell biology. One such
example that begins to shed light on the free energy demands associated with
sculpting membranes into different shapes is that of membrane vesicles. From
the standpoint of the energetic description given in eqn. 4, we can make a
simple estimate of the free energy cost required to create spherical vesicles
such as those found at synapses. Since for a sphere the two radii of curvature
are equal and have a value $R$ and the total area of each such sphere is $4\pi
R^{2}$, eqn. 3 instructs us to sum up

$E_{vesicle}={\kappa_{B}\over 2}({1\over R}+{1\over
R})^{2}A_{sphere}={\kappa_{B}\over 2}{4\over R^{2}}4\pi R^{2}.$ (13)

This implies the fascinating and for many people, counterintuitive result,
that the energetic cost for vesicle formation due to membrane bending is
$E_{vesicle}=8\pi\kappa_{B}\approx 250-500~{}k_{B}T$, completely independent
of the size of the vesicle.

A second example from membrane mechanics is to try to estimate the strain
suffered by a membrane at the time of rupture. To estimate this magnitude, we
can use

$\tau_{rupture}=K_{A}{\Delta A_{rupture}\over A},$ (14)

where the subscript rupture indicates the value of the parameter at rupture.
If we use the values provided in table 2, we can estimate the rupture strain
as

${\Delta A_{rupture}\over A}={\tau_{rupture}\over
K_{A}}\approx{5~{}\mbox{mN/m}\over 200~{}\mbox{mN/m}}\approx 2.5\%.$ (15)

Often people are surprised by how small the rupture strains really are since
we have an impression that lipid bilayers are floppy, squishy and highly
deformable materials.

Membrane dynamics. In the section on the Dynamic Membrane, we considered the
permeability of membranes to various molecular species. One simple estimate
that we can do to get a sense of the meaning of the permeability is to ask how
many molecules cross the cell membrane each second given some concentration
difference. Given the concept provided in eqn. 7, we can estimate

${dN\over dt}=j\times A,$ (16)

Given a typical membrane permeability for water of order $p\approx
100~{}\mu$m/s and considering a typical concentration difference of salt
across the cell membrane when cells are subjected to an osmotic shock of order
100 mM $\approx 10^{8}~{}\mbox{molecules}/\mu$m3, [69] for example, we can
make the simple estimate that

$j\times A=p\Delta cA\approx 100~{}\mu m/s\times 10^{8}/\mu m^{3}\times 5\mu
m^{2}\approx 5\times 10^{10}s^{-1}.$ (17)

These numbers are interesting to contrast with the rate of transport of
molecules across ion channels. Specifically, given the conductivity of a
channel such as the mechanosensitive channel of large conductance (MscL), we
find that the opening of a single channel yields a flow rate of several
molecules per nanosecond, quite comparable to the flow rate rate of water
across the membrane itself.[70, 60]

One of the most interesting estimates concerning membrane dynamics that we can
consider focuses on the mass and energy economy of a cell. To this day, I
still marvel at the fact that one can take 5 mL of liquid containing some
salts and sugars, inoculate that solution with a single bacterium, and 12
hours later one will find as many as $10^{9}$ cells per mL of solution.
Effectively, what has happened is that the molecules in the medium have been
taken up by that bacterium, used to construct building materials and energetic
molecules such as ATP and then used them to construct a new cell. This process
repeats over and over again every 20 or so minutes. These observations raise
an obvious Fermi question: is the rate of membrane transport of sugar
molecules, for example, fast enough to keep up with the needs of the cell to
reproduce.[19] To approach that question, we consider the flux of sugar across
the membrane using the numbers presented above, namely, that there are
3000-15,000 sugar transporters per cell, each of which is able to take up
sugars at a rate of several hundred sugars/sec (BNID 102931, 103160, 100736).
We can get a feeling for the number of sugars taken up per cell cycle as

$\mbox{flux of
sugar}=(10^{4}~{}\mbox{transporters})\times(300~{}\mbox{sugars/transporter
sec})\times(2\times 10^{3}~{}\mbox{seconds})\approx 6\times
10^{9}~{}\mbox{sugars}.$ (18)

This number is of the right order, though probably on the low side of what is
needed to power the cellular economy and raises interesting questions about
possible rate-limiting steps in cellular growth.[4, 19]

Just as we did in the section of the chapter on the Dynamic Membrane, it is of
interest to focus not only on the dynamics across the membrane, but also on
the dynamics of molecules within the membrane. Specifically, one question of
interest is how long does it take molecules to travel across the cell membrane
given the measured diffusion constants? To answer this question, we appeal to
the simple estimate that the time scale for diffusing a distance $L$ is given
by

$t_{diffusion}\approx{L^{2}\over D}.$ (19)

For a bacterial cell with dimensions of several microns, this means that the
diffusion time to explore the membrane is

$t_{diffusion}\approx{1~{}\mu m^{2}\over 1~{}\mu m^{2}/s}\approx 1~{}s,$ (20)

where we have taken a diffusion coefficient for a lipid of 1 $\mu m^{2}/s$.
This characteristic time scale is confirmed in fluorescence-recovery-after-
photobleaching (FRAP) experiments (see [53], for example).

The Electrical Membrane. The electric fields across biological membranes are
surprisingly high as can be estimated by using

$E\approx{V\over d}\approx{100~{}\mbox{mV}\over
4~{}\mbox{nm}}\approx{100\times 10^{-3}~{}\mbox{V}\over 4\times
10^{-9}~{}\mbox{m}}\approx 2.5\times 10^{7}~{}{\mbox{V}\over\mbox{m}}.$ (21)

Note that this field is an order of magnitude higher than the electric fields
associated with dielectric breakdown in the atmosphere. And yet, fields five
times as high have been measured in membranes with no evidence for any
anomalous behavior.[71]

This section had as its ambition to give a sense of how the numbers summarized
in Table 2 can be used to develop intuition. [3, 4] In fact, more than
anything, this brief section is an invitation to others to look for meaning in
the hard won outcome of the recent work to extend membrane numeracy.

## The Missing Membrane Numbers

As a final send off of this brief ode to biological numeracy for membranes, we
reflect on the state of our art and how it can be improved. Despite a long
list of truly amazing successes, there are still many things not to like about
the current status of biological numeracy, not only in terms of how well we
actually know the numbers, but also in terms of what those numbers might mean
for a deeper understanding of biological systems. The goal in this final
section is to make an attempt at critiquing both this article and the current
state of the art with the aim of suggesting future directions. Though the “by
the numbers” approach has become something of a cliche, my opinion remains
that there is much to be gained by pushing hard with this approach on each of
the many diverse and wonderful facets of biology.[72, 73, 74, 75, 76, 19, 77]

One of the first weaknesses of biological numeracy in the membrane setting
(and beyond) is the need to establish measurements of sufficient precision
that we can confidently report on measured values. For example, there is
already much evidence that biological membranes “care” about their lipid
composition. It would be a powerful addition to our ability to ferret out
molecular mechanisms to be able to examine these membrane compositions for all
organelles as a function of time and for a variety of different environmental
conditions. First steps in this direction have been made in thinking about
proteomes with one of my favorites reporting on the proteome of E. coli in
more than twenty distinct conditions.[66] Absent accurate and reproducible
measurements in the membrane setting, we are handcuffed in our efforts to
construct a fruitful dialogue between theory and experiment.[78, 79]

A second important challenge for the future of membrane numeracy is the vast
differences between model membranes and the real world of plasma and
organellar membranes. Effectively each and every section of this chapter -
size and shape, composition, mechanics, transport, electrical properties - is
bereft of any deep understanding of how all of the heterogeneities of real
membranes might alter the numbers, and what the significances of such
alterations might be. The advent of mass spectrometry in conjunction with ever
more sophisticated microscopies as a window onto membrane composition have
left in their wake a host of mysteries and challenges. As highlighted in
Figures 6 and 7, and indicated widely in other literature,[18, 20, 21, 22, 23,
24, 25, 26, 27, 28] cells care about their lipid composition. What is lacking
is a conceptual framework that tells us what these numbers really mean in
terms of biological function, what they imply about the regulation of lipid
biochemistry and perhaps most importantly, what they imply about the evolution
of life.

Another example that strikes me as an exciting challenge to our current
thinking broadly concerns the question of cellular shape, and the shapes of
organelles, more specifically. The images shown in Figure 4 make clear the
great diversity of membrane shapes. The study of mitochondria as a concrete
example presents challenges at every turn.[15] My personal favorite remains
the intriguing membrane structures found in the outer segments of
photoreceptors (see Figure 4B). In the context of the ideas presented in this
chapter, one of the ways that people have attacked questions of shape
traditionally has been through the approach of free energy minimization.[80,
11] But there are interesting, novel alternatives that are now in play. One
approach focuses on the role of dynamics where there is an interplay between
differential growth and the cost of elastic deformations as characterized by
the kinds of mechanical parameters reported here.[81, 82]

Thus far our discussion has largely focused on the physical properties of
membranes. But there is another interesting angle on membranes that is more
related to their evolutionary significance. Interestingly, one of the simplest
acts of biological numeracy, namely, counting, can provide evolutionary
insights. Specifically, the number of membranes surrounding an organelle is
perhaps the best indicator of its evolutionary origins, with the argument
being made that more than one such membrane means that organelle has an
endosymbiotic origin and more than two such membranes might imply nested
symbioses.[83, 84, 85]

We are in the midst of a biological revolution. The pace of discovery in the
study of living matter is dizzying in all corners of biology. The central
thesis of this article is enlightenment through biological numeracy. That is,
as part of our attempt to make sense of the living world, we can sharpen our
questions and be more rigorous in our demands about what it means to really
understand something.[78, 79] One of the ways of placing those demands is to
ask for an interplay between our experimental data and our theoretical
understanding of biological processes. The study of biological membranes is
one of the most important areas for future work and in many ways has not kept
pace with insights into genomes and the proteins they code for because of a
want of appropriate tools. It is hoped that the chapters in this book will
serve as an inspiration for the development of the tools that will make
membrane numeracy as sophisticated as is our understanding of nucleic acids
and proteins.

Acknowledgments

One of the best parts of being a member of the scientific enterprise is all
the smart and interesting people we get to interact with. In preparing this
chapter I sent out a survey to many experts in membrane biology and biophysics
and was overwhelmed with the thoughtful responses that I received from many
colleagues. I am grateful to Olaf Andersen, Patricia Bassereau, Joel Dacks,
Markus Deserno, Evan Evans, Ben Freund, Jay Groves, Christoph Haselwandter,
Liz Haswell, KC Huang, Ron Kaback, Heun Jin Lee, Mike Lynch, Bill Klug, Jane
Kondev, Ron Milo, Uri Moran, John Nagle, Phil Nelson, Bert Poolman, Tom
Powers, Doug Rees, James Saenz, Pierre Sens, Victor Sourjik, Stephanie
Tristram-Nagle, and Tristan Ursell for useful discussions. I am especially
grateful to Olaf Andersen, Markus Deserno, Christoph Haselwandter, James
Saenz, Pierre Sens and Tristan Ursell who have been patient and persistent in
advancing my membrane education, though obviously all shortcomings in this
chapter are due to my failure to absorb that education and are no fault of
their own. I am privileged to be entrusted by the National Science Foundation,
the National Institutes of Health, The California Institute of Technology and
La Fondation Pierre Gilles de Gennes with the funds that make the kind of work
described here possible. Specifically I am grateful to the NIH for support
through award numbers DP1 OD000217 (Director s Pioneer Award), R01 GM085286
and R01 GM084211. I am also grateful to the Kavli Institute for Theoretical
Physics where much of this chapter was written. More generally, this article
is part of an adventure that I have undertaken with Ron Milo and Nigel Orme
(our illustrator) and generously funded by the Donna and Benjamin Rosen
Bioengineering Center at Caltech. Finally and sadly, since the completion of
this chapter, my friend and collaborator Bill Klug was brutally murdered in
his office by a former graduate student. I had asked Bill to join me in the
writing of this chapter, but he was too busy during this summer and instead of
having the happy presence of his name as a co-author I instead have the solemn
and unhappy duty to dedicate this short piece to him, kind and intellectually
deep, above all a family man, he will be deeply missed.

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