In this section we explore how protein crowding and phase separation [130] affect the way that proteins laterally organize. Our previous discussions in this chapter focused on specific interactions between proteins that are mediated by bilayer mechanics. In a mechanism similar to how crowding in the cytosolic environment affects chemical reactions and equilibria [131, 132], we hypothesize that there may be important effects from protein and lipid crowding in a membrane environment,
a) b) c)
Figure 2.9: Depictions of crowding and depletion zones in two dimensions. a) A crowded arrange- ment of discs (gray) in a finite area (blue). b) The same discs as in (a), now with small red zones around each disc showing the excluded zone for the particle shown just under the box. The relative sizes here reflect the different between the area of lipids and nominal membrane proteins. c) The same discs as in (a), now with red zones around each disc showing the excluded zone for the par- ticle shown just under the box. The relative sizes here reflect crowding of membrane proteins by membrane proteins. The red zones indicate regions where another crowding particle cannot enter, and in regions where red zones overlap, other crowding particles are completely excluded.
as shown schematically in Fig.2.9, where the density of proteins is such that neighboring proteins are generally within a few nanometers of each other [91]. In fact, during the composition of this thesis, new and interesting research on the effects of membrane protein crowding was published [133], and advocated for a more systematic look at the effects of crowding in membranes. We will discuss in detail the effects of ‘ideal’ crowding, that is, crowding from a dilute ‘gas’ of proteins, as well as some approximate schemes for handling weakly non-ideal crowding. As we show in the following few subsections, the statistical mechanical ensemble one chooses significantly affects how crowding tension in the membrane exerts its effect.
First, consider what general forces are at work that give a lipid bilayer its coarse structure.
Hydration energy of the lipid tails, though arguably due to the reduction in entropy of water when in contact with those tails [134], is essentially an enthalpic effect that causes lipids to aggregate.
The smectic nature of the liquid crystal is due to its amphiphilic qualities, but again is generally an enthalpic effect [57]. Once the lipids have aggregated into a smectic liquid crystal, the tails themselves entropically explore space similar to a worm like chain polymer [57], and hence push against the surrounding tails to gain volume in which to wander around. The result is a competition between this tail entropy, which prefers more area per lipid, and the hydration forces, that assemble a membrane into a structure with minimum surface free energy, and a well defined area per lipid molecule. From this energetic competition the membrane is able to support lateral tension. We discuss the course structure of a bilayer to build intuition for why lipids do not simply form a molecular gas when in an aqueous solution, and by way of contrast to the two dimensional gas of
membranes we are about to discuss, that exist in the bilayer itself.
2.10.1 Ideal Crowding Tension
Consider a nearly spherical vesicle3 with total areaAand a dilute (ideal) solution of N membrane proteins embedded within it. The proteins’ sampling of configuration space includes the entire metric area of the membrane, which we denote as A, as opposed to some flat plane projection of that metric area. The partition function is then
Z =AN, (2.48)
where we have neglected the factor of 1/N!, because it does not affect our calculations. Then the free energy due to entropy is
Gideal=−N kBTln(A). (2.49)
The spectrum of thermal undulations on a membrane is heavily weighted towards wavelengths much longer than the size of a typical protein, and on the length scale of a protein, the bilayer is exceptionally flat due to the bending stiffness (see Section 4.5 for details). Considering that the proteins can explore the metric area of the membrane, flattening of these long wavelength thermal fluctuations does not make more area available to the proteins; only a change in the area per lipid adds configurational area. Thus an appropriate description of the bilayer is a linear elastic material, where the metric area of the lipid bilayer at zero lateral tension isAo and with an areal strain the area becomes A=Ao(1 +φ), where φ is the areal strain. In the ideal limit, the area taken up by proteins is negligible compared to the total membrane area, and hence this equation forAis valid.
When stretched, the membrane elastically stores energy Gstretch=Ao
Z φ
0
τ(φ0)dφ0 =AoKA
2 φ2, (2.50)
resulting in the total free energy for the vesicle with dilutely embedded proteins G=Gideal+Gstretch =−N kBTln(Ao(1 +φ)) +AoKA
2 φ2. (2.51)
The equilibrium value of areal strain is found by solving
∂G
∂φ = 0 (2.52)
forφ, yielding
φ= 1 2
r 1 +4
δ −1
! ' 1
δ, (2.53)
3We assume that the vesicle has some amount of free area, but that the shape is roughly that of a sphere.
whereδ =KAAo/N kBT is a dimensionless measure of stretch energy compared to thermal entropic energy. In the dilute limit, where δ → ∞, the crowding tension on the bilayer is independent of bilayer properties, and is given by
τc =kBT N
Ao =kBT c. (2.54)
Note that this tension is always positive, meaning it pulls on the bilayer in an attempt to make more configurational area available to the diffusing proteins. In the dilute solution for typical values of the stretch modulus in phospholipid bilayers (e.g. KA'60kBT /nm2),δ 1.
2.10.2 Slightly Non-Ideal Crowding Tension
Keeping the same physical picture and ensemble, where the number of lipids and proteins is fixed, we now consider a more crowded vesicle where the area available to each protein to wander around is actually reduced by the presence of the other proteins. The area of the lipids in such a vesicle is
A`=Ao(1 +φ), (2.55)
whereAois the unstressed area of thelipids. The total area of the proteins is then simplyAp =N α, whereα is the area per protein, such that the total area of the vesicle isAtot=A`+Ap. Then the available area for any one protein to wander around is
Aent =Atot−N α
pmax =A`−N α p−max1 −1
=A`
1−N α
A` p−max1 −1
, (2.56)
the constant pmax=π√
3/6'0.907 is the maximum 2D packing fraction for discs. Effectively, this is subtracting from the total area of the vesicle N hexagonal unit cells of area α/pmax, to give a more accurate measure of the area in which any one protein can wander. From this equation, we can see that the ideal gas assumption is indeed the dilute limitN α/A` 1. Then the partition function of this system is
Z = A`−N α p−max1 −1N
. (2.57)
This partition function is the popular approximation [135] that leads to the equation of state for a 2D hard disc gas
τc=kBT c
1−cmaxc , (2.58)
when the derivative is taken with respect to Atot, but we are asserting that proteins are incom- pressible, and it is onlyA`that can change. The parametercmax =pmax/αis the maximum packing concentration of discs in 2D. Then the entropic component of the free energy is
Gnon-ideal=−N kBTln A`−N α p−max1 −1
. (2.59)
Like before, the elastic energy stored in the bilayer is Gstretch=AoKA
2 φ2, (2.60)
so that the total free energy is
G=Gideal+Gstretch =−N kBTln Ao(1 +φ)−N α p−max1 −1
+AoKA
2 φ2. (2.61) Again we solve∂G/∂φ= 0 to find the entropic areal strain
φ= 1 2
"r
(−1)2+4
δ +−1
#
, (2.62)
where the new constant,
= N α
Ao p−max1 −1
, (2.63)
is the ratio of the total close-packed interstitial area to the total area of unstressed lipids, and δ is defined immediately after eqn.2.53. The value= 1 signifies that all available lipid is packed into the interstitial sits of the hexagonal protein crystal. The value of is generally between 0 and 1, however, values greater than 1 are possible, simply corresponding to the case where all protein is in a close-packed disc crystal and there is stressed lipid in the interstitial area. For a situation where the proteins are nearly close-packed, and hence '1, φ'δ−1/2. Generally, this equation of state does a good job of accounting for the first few terms of the real Virial expansion of the non-ideal hard disc gas, and contains a singularity in the right place (i.e.when the packing fraction reaches pmax), however, the order of the singularity is not correct [135].
Examiningδwe find it is a constant,KAao/kBT ∼35, connected to the stretch stiffness per lipid (ao is the unstressed area per lipid), multiplied by the molar ratio of lipid to protein,N`/N. If the area per protein is 10 nm2 and each protein is occupying twice as much area as the minimum unit cell (α/pmax), then the lipid associated with each protein uses∼12 nm2, or∼20 lipids. Using these values, the molar ratio of lipid to protein is twenty, and then the δ ' 700, ' 0.09, φ = 0.0016, and τc = 0.09kBT /nm2.
Let us estimate what this means for the crowding tension felt by lipids in the bilayer. If the proteins are fairly densely packed (only 25% more area than the minimum unit cell) then for a 10 nm2 protein, the area of lipid is∼3.8 nm2 or about 6 lipids per protein. Thenδ '220,'0.27, φ'0.0062, and τc '0.36kBT /nm2. Already this suggests some interesting possibilities for lipids in the bilayer, if there are organizational principles in membranes that are dependent on the lateral tension felt by lipids, this protein induced crowding tension may be important. Additionally, this means that in a protein dense membrane, there will be a chemical potential benefit for lipids to join the bilayer, distinct from the aforementioned hydrophobic effects. This crowding tension is felt by the lipids, but the overall effect on embedded proteins remains unclear, although we will explore
0 0.5
1 200
600400 1000800
0.04 0.08 0.12 0.16
0
Figure 2.10: Plot showing the effects of material constants and protein concentration on the osmotic areal strain in the bilayer. This plots show how the areal strain (φ) in the slightly non-ideal crowding tension model depends on the dimensionless material constantδ, and the dimensionless protein area occupancy . Multiplyφby ∼60kBT /nm2 to get the corresponding osmotic lateral tension.
two physical scenarios below.
2.10.3 Proteins that Change Area
Given the analysis of Section2.10.1, we question what overall osmotic tension is felt by an embedded membrane protein, and more specifically, we ask what happens if a protein changes area? As an initial inquiry, and to build intuition, we will simply assume that there areN−1 crowding proteins with areaα and one area-changing protein with areaαc. Recall that in both of the previous cases, the area available for proteins to explore was the bilayerarea. Given the rather small areal strains that we would expect from the previous analysis, let us assume the area per lipid is constant, then the canonical partition function is
Z =ANo , (2.64)
whereAo is the area of the bilayer. So long as the number of proteins and lipids remains constant in the system, a change in protein area, represented by a change inαc, does not change the metric area of the bilayer. Thus, to lowest order, the net crowding tension felt by embedded proteins is zero. Alternatively said, as far as an area-changing protein is concerned, the positive tension on the membrane due to expansion of the protein gas is exactly balanced by the negative osmotic tension of those same crowding proteins, as schematically shown in Fig. 2.11b.
In this problem, the ensemble we choose plays a critical role. One example of an ensemble where the area changing proteindoes feel a crowding tension, is when the total area of the vesicle is held constant, and the protein areaαc is allowed to change, necessitating that more lipid be added. In
this case, the partition function would be
Z = (Ao+α(o)c −αc)N, (2.65)
where α(o)c −αc is the area of lipid added as the initial protein area,α(o)c , changes. Then the free energy is
G=−N kBTln(Ao+α(o)c −αc), (2.66) and the crowding tension would be
τc =−kBTN
Ao · 1 1−α(o)cA−oαc
| {z }
'1
=−kBT c, (2.67)
where the approximation indicated by the bracket is valid for any sufficiently large system. Thus we see that this ensemble generates a net compressive tension, or alternatively, this could be viewed as the tension felt by an external membrane reservoir. In other words, the current ensemble becomes relevant if there is a reservoir of lipid from which lipids can be added to the vesicle to hold the area constant during the area change of the protein. The current physical scenario is similar to a conserved volume container with gas, where if one of the gas particles changes size, that change in volumeisavailable to the configurational volume of the other molecules. However, in a bilayer, the medium in which the particles are diffusing is essentially incompressible, thus a change in protein size translates to a change in ‘container’ size. From the perspective of crowding tensions that affect protein conformations, this is a dubious physical picture (as discussed in the following section) because it holds the total vesicle area constant, when in reality, on the time scales for protein conformational area changes (on the order of microseconds [43]), we want to hold the total number of lipids and proteins constant.
2.10.4 Choosing an Ensemble
While there are myriad scenarios where the choice of ensemble has little effect on the overall thermodynamics of a system, this does not seem to be one them. Due to the fact that we are calculating the configurational entropy of proteins within a certain available area, and that area depends on the ensemble we choose, the ensemble plays a key role in the physical interpretation of crowding tension. Additionally, the time scale of interest affects what ensemble is reasonable. For instance, consider a large vesicle that encloses machinery capable of producing new lipids over time - part of what one might call a simple model of a cell. Simultaneously, this vesicle has some population of membrane proteins that can change their areal footprint. If the time required to produce and incorporate a group of lipids, whose area is comparable to the area change of the protein, is much longer than the time scale of the protein conformational change, it seems reasonable that the right ensemble to choose is one of fixed protein and lipid number, as we did above.
In fact, we can estimate and compare these time scales, at least in the microbial setting. For instance, consider the nominal dimensions of an E. coli bacterium (a pill of length ∼ 2µm and radius ∼ 0.5µm), whose area corresponds to approximately 107 lipids per leaflet4, with the area per lipid∼0.6 nm2 [136]. Under the best, and hence fastest, growth conditions, the bacterium will divide approximately every ∼ 1200s (20 minutes), meaning that, at their fastest, lipids must be incorporated at a rate of approximately one lipid per 100µs. On the other hand, the well studied channel protein MscL has a gating time scale of∼5µs[43], and for this particular channel, a rather large change in area upon gating ofO(10 nm2) =O(10 lipids) [21, 50,19]. Thus on the time scale of protein conformational change, the area of the lipids added, as compared to the area change of the protein, is small. Hence, it seems that the appropriate ensemble for studying the effects of crowding on channel gating, even for proteins whose area change is a tenth of MscL, is the fixed particle number ensemble, in which the net crowding tension on the channel is zero (in the ideal limit), as shown in Fig.2.11. It should be noted that this equilibrium world view may be inaccurate in the fast-paced setting of a dividing cell.
If on the other hand, we are concerned with the behavior of this system on much longer time scales, the fixed particle number ensemble is likely not appropriate. While it is difficult to know the
‘right’ ensemble in the various contexts relevant to a cell, we can make a general and interesting comment, rooted in a basic understanding of configurational entropy. A simple calculation of configurational entropy shows that the maximum degree of entropy is reached when half of the available sites for a protein to occupy are filled. This suggests that, at least from an entropic perspective, there is an energetic driving force that prefers the area taken up by lipids and the area taken up by proteins in the membrane, to be equal. With no claims of causality, it is interesting to note that cellular measurements of the relative areas of lipid and protein are approximately one to one in various biological membranes [91], in line with this notion of maximizing the configurational entropy.
An interesting result of the fixed particle number ensemble in the ideal limit is that regardless of dimensionality, particles immersed in an incompressible medium (like water or a bilayer) experience zero net osmotic forces per unit length (2D) or area (3D). This is not to say that there are no effects from crowding; indeed an excluded area force exists, as we will show in the next section.
2.10.5 Excluded Area Effects
This section does not present any new work, and indeed, excluded volume forces have been a topic of intense research interest for decades [137]. In the context of lipid bilayers, there has been some work exploring the effects of excluded area forces, due to the depletion of lipids between proteins [93], however our discussion here is more centered on excluded area forces arising from a population of crowding proteins. This calculation is included for the sake of completeness in the context of
4This assumes the leaflet is made up entirely of lipids, but considering the mass ratio of protein and lipid in many membranes is roughly one to one [91], this is correct within a factor of two.
a) b)
Figure 2.11: Schematic showing how protein expansion tension couples to the bilayer equation of state. a) The upper layer represents the protein configurational space, while the lower layer is a bilayer. Proteins, shown as blue circles, thermally wander around in the upper layer and exert an expansive tension on the boundary as shown by the arrows. Demanding that the area of the upper and lower layers be equal, is equivalent to saying that the proteins remain in the bilayer; then the expansive tension couples directly to the elastic equation of state of the bilayer below. b) Upon insertion of an area changing element, it is clear that the crowding proteins in the upper layer exert a compressive tension on this element, while those same proteins couple to the equation of state in the lower bilayer, causing an equal and opposite expansive tension. Thus there is zero net tension on the area changing element.
discussing crowding effects. While it might seem that the previous calculations of crowding tension affect how we think of excluded area forces, the following calculations demonstrate that these forces remain intact.
Consider a box of areaA with N diffusing particles of radius R and two blocks a distance D apart, as shown in Fig.2.12. Choosing either the fixed particle or fixed area ensemble does not affect this situation because no particles are changing size and the particle numbers are being conserved.
The distance between the blocks does not affect how much area is available to the blocks, hence we need not consider the conformational entropy of the blocks. However, the area available to each diffusing particle is
Aent =A−N αp−max1 −Θ(2R−D)D` (2.68) where ` is the length of the blocks, α = πR2 is the area of the diffusing particles, and Θ is the Heaviside function. The partition function is then
Z = A−N αp−max1 −Θ(2R−D)D`N
, (2.69)
and the free energy is
G=−N kBTln A−N αp−max1 −Θ(2R−D)D`
. (2.70)