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.
proteins that arise from crowding in the membrane. We calculated an approximation for the depletion force due to crowding, finding it has a relevant scale of around one pico-Newton. We found that likely on the time scales of protein conformational changes, the appropriate choice of ensemble for crowding is fixed particle number, and thus likely there is a corresponding zero net crowding tension. On the other hand, crowding likely serves as an energetic bias that prefers the addition of lipids, and might even be at work setting the coarse lipid-protein areal ratio in membranes. Lastly, we combined a subset of this information to begin to understand how lipid domains might be able to both specifically select and enrich the concentration of certain proteins within their borders. All of these projects are ongoing, and we hope that the varied stories each of these physical effects tell will come to some convergence in our future work.
Chapter 3
Volumetric Flow in Protein Channels
“A scientist prizes what he does not understand at least as much as what he does.” – Unknown
3.1 Not All Are for Ions
As a cell grows and eventually divides, a host of processes are in play to manage cell size and partition internal components. Broadly speaking, the genome must be replicated and partitioned, new membrane and cell wall (where applicable) must be synthesized, and throughout the process of continual growth, cell and organelle volume must be managed in accordance with the rate of production of lipids, cell wall material, and internal components [15, 14]. Permeation of water through the membrane is a passive process [140], which when driven by osmotic gradients can serve as a mechanism for volume management. Though beyond that, cells have a number of membrane channels whose main purpose is to move water and osmolytes [141], for instance the well know protein family of aquaporins [142, 143] and aquaglyceroporins [144, 145] is found in nearly all organisms. In some cases, the channel activity is coupled to membrane tension [71,73], indicating that transmembrane pressure may be the basal physiological regulator of their conformational state. In particular, bacteria [69, 11, 18, 24] and, by homology, certain plants species [17] have developed a set of membrane tension sensitive channels that, by virtue of their ability to rescue microbes from osmotic shock or regulate organelle morphology in plants, respectively, are ostensibly involved in volume management. In both the microbe and plant cases (and aquaporins as well), absence or mutations of these channels has serious deleterious effects on the organism under certain physiological conditions. Yet, despite their clearly important role in cellular physiology, there are two aspects of these channels that are relatively unexplored. First, as we have discussed at length in Chapters 1 and 2 of this thesis, details of the mechanical interaction between lipids and these channels remain largely at a hypothetical stage, where in Chapter1we discussed the possible effects of lipid mechanics on channel function. The second set of questions, which we will explore in more detail in this chapter, revolve around the volumetric flow properties of these channels in a physical sense. More specifically, the manner in which water osmolytes move through the channel pore present us with fertile ground for experimentation. A few well-formed questions have motivated our thinking on this topic, in particular: What is the relationship between pressure gradient, pore size and flow through the channel? Does the flow follow ‘typical’ low Reynolds number properties
of a small pipe? Does the flow exhibit any effects from the fact that the pore is only an order of magnitude larger (or less) than the size of the particles passing through it? Do thermal fluctuations of the water and solute molecules affect the flow properties of the channel?
A huge amount of experimental effort has been expended to understand the gating properties [21,22,50,49] and structure [11,18,146,19] of the microbial channels, as well as propose possible functions in vivo [147, 148, 149]. In these explorations, electrophysiology has been an indispens- able tool in measuring the conformational states of channels (i.e. conducting or non-conducting states) as well as probing what physiological factors cause these changes in conformation [21, 50]
(e.g. transmembrane voltage, tension, or ligand concentration). In accordance with their proposed function as regulators of volume and/or pressure (depending the physical viewpoint), and the fact that the membrane is essentially impermeable on time scales of channel gating [43], these channels must be sensitive to the pressure acting on the membrane. Indeed, in a system where the enclosed volume and membrane surface area are conserved on time scales relevant to volume flux through these channels, membrane tension is a reporter of pressure and enclosed volume. In particular, the Laplace-Young relation [65] relates the mean curvature of a surface to the pressure drop across it and surface tension on it (in the absence of bending). Using this relationship, many important material properties of membranes have been gleaned through the use of micropipette aspiration [140,26, 150].
In this chapter we perform a series of calculations aimed at estimating channel flow proper- ties and constraining the range of parameters that might be seen in experiments that attempt to measure those properties. On the experimental side, we will discuss at length how to measure the ensemble flow properties of the bacterial mechanosensitive channel of large conductance (MscL), using both wild-type and mutant forms, to gain a better understanding of both the lipid-related gating behavior and the nanoscopic flow across a membrane. Additionally, we will present prelim- inary experimental results on pressure-driven water flux across a bilayer, as a proof of concept of the proposed volumetric channel flow measurement.