As a transition to the next chapter, which will discuss phase separation and formation of lipid domains, we take a look at how proteins with a hydrophobic mismatch that is better suited to a lipid domain, energetically segregate themselves into a domain. This has relevance to both the specific interactions we discussed, in that the hydrophobic mismatch considered here is the same mechanical effect we discussed earlier. Likewise, the osmotic tension of proteins in a domain will be an important effect, and hence this section serves to demonstrate one particular union between specific mechanical and non-specific entropic effects.
Consider a domain whose lipids occupy an area AD and proteins with fixed area α. Due to the membrane thickness difference between the domain and the surrounding membrane, there is an energy cost per unit boundary lengthγ around the domain [130,138]. We assume that due to the difference in membrane deformation around the protein outside and inside the domain, there is a fixed energy benefit −µ for the protein to enter the domain — this can be calculated from membrane mechanics and is generally related to the difference of the squares of the hydrophobic mismatch in the two lipid regions and the protein in question, although strictly speaking there are contributions from the size of the protein, relative to the size of the lipid domain. Given these assumptions, we see that the free energy forN proteins to enter a domain atT = 0 is
G= 2√ πγhp
AD+N α−p AD
i
−N µ, (2.73)
where the first terms accounts for the change in unfavorable lipid domain boundary length upon entry of a protein, and the second term is the elastic benefit of a protein entering a domain. We define the dimensionless constant5
χ= γα µ√
AD
, (2.74)
which can be interpreted as the ratio of the compressive tension on the domain from the unfavor- able boundary, to effective tension on the protein when entering the domain. We also define the dimensionless protein concentration as
η= N α AD
. (2.75)
For a domain that contains proteins, this parameter measures the area taken up by proteins relative to the area taken up by lipids within the lipid phase boundary. Finally, a natural energy scale can be defined, such that the energy is normalized by
Go=βµAD
α , (2.76)
whereβ = 1/kBT, yielding the dimensionless energy G=Goh
2√ πχp
1 +η−1
−ηi
. (2.77)
Recall that the maximum packing density ispmax, hence the maximum dimensionless particle num- ber isηmax= p−max1 −1−1
'9.74. This energy function has a number of interesting features. Upon examination, one finds that the energy decreases with increasing protein concentration as long as χ < π−12, up toηmax. For values of χ greater than this critical value, there is an energy barrier for proteins to join the domain. For a given domain size and line tension around the phase boundary, this means that proteins that are too large (α) and/or proteins whose chemical potential from hydrophobic mismatch (µ) is too small will be energetically inhibited from joining the domain, up
5We apologize for the redundancy in parameter naming, but there are only so many Greek and Latin letters.
to a point. Above this critical value, the maximum of the energy barrier is located atχ=π−12, and hence if the dimensionless protein concentration in the domain is greater than this value, the free energy change for other proteins to join the domain again becomes negative. Hence, this has the appearance of nucleation a problem, with a twist. If the domain size is large, and the protein size is small, and/or the chemical potential is very favorable, any addition of protein is energetically beneficial. Although, above the critical value ofχ, there is a critical protein concentration, below which addition of the proteins to the domain is inhibited. Generically, this means that larger do- mains are more promiscuous than smaller domains, about which proteins are energetically favored to enter. Thus, the values of the parameter χ determine a domain’s ability to segregate proteins by hydrophobic mismatch, with higher values of chemical potential µ corresponding to stronger enrichment of the corresponding protein in the domain.
To include the effects of entropy in the ideal limit6, we define the fixed concentration of protein outside the domain asco and the concentration of protein inside the domain is
c= N
AD+N α = 1 α · 1
1
η+ 1 ' η α = N
AD
, (2.78)
where to get analytical answers, we have taken the limit of low protein concentration,η→0, such that the entropic component of the chemical potential for a protein to enter the domain is
µc(n) =kBTln c
co
=kBTln n
coAD
, (2.79)
where n is the integration variable for the number of proteins, as shown below. Making use of this result from the statistical mechanics of an ideal gas, allows us to skip the laborious spatial integrals that pertain to the position of each protein in the system. The free energy contribution from entropy is then
Gc=β XN n=1
µc(n) = ln
"
N! (coAD)N
#
. (2.80)
Then the total free energy is Gtot=Goh
2√ πχp
1 +η−1
−ηi + ln
"
N! (coAD)N
#
, (2.81)
and the mode concentration of proteins in the domain, in the limit of low protein concentration and large domain area, is found by solving∂Gtot/∂N = 0 , to give
N
AD 'coeβµ(1−χ√
π) =coeβ
„
µ−γαq π
AD
«
, (2.82)
6It is a relatively straightforward extension to include non-ideal terms in the entropy.
0.5 0.6 0.7 0.8 0.9 1
1 2 3 4 5 6
Figure 2.13: Normalized concentration enrichment as a function of the domain area forα= 10 nm2 andγ = 0.4kBT /nm. Note that as the domain area grows, the concentration enrichment increases, but the rate of that increase drops past a certain domain area, as indicated by the dashed line.
and further
c co =eβ
„
µ−γαq π
AD
«
, (2.83)
with c'N/AD. As domain area grows larger, and the marginal cost in line tension for adding a new protein shrinks, and we approach the maximum enrichment factor
c
co =eβµ. (2.84)
Figure 2.13 plots the concentration enrichment, normalized by this maximum enrichment, as a function of the initial lipid domain size. More work remains to better understand how the mechanics of a phase separated bilayer, and the hydrophobic mismatch of an embedded protein work in concert to segregate specific proteins into domains, but this section serves as good launching point.