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.
highly permeable membrane selec!vely permeable membrane s!ff cell wall
hypo-osmo!c shock equilibra!on
closed channel open channel
wt E. coli
osmolytes
cell death
∆ msc
osmolytes
Figure 3.1: Hypo-osmotic shock in E. coli. Schematic showing how bacteria cope with rapid reduction in the external, absolute osmotic pressure. First row: cells initially in equilibrium with the surrounding osmotic pressure are subject to hypo-osmotic shock. Water rapidly permeates across the membrane increasing cell volume and raising membrane tension. At some critical tension, mechanosensitive channels open, increasing the influx of water, but also allowing the cell to jettison some portion of its internal osmolytes, thereby returning to equilibrium with the external osmotic pressure. Second row: upon hypo-osmotic shock, cells that lack mechanosensitive channels increase in volume, membrane tension increases, and without alleviation of the internal osmotic pressure and corresponding membrane tension, the membrane ruptures resulting in cell death.
movement on the scale of a single cell.
The low copy-number [112] (∼ 10 per cell) bacterial channel MscL is an excellent target for studying how water and osmolytes move across a membrane when a pressure gradient, and corre- sponding membrane tension, are present. It has been conclusively shown that membrane tension is the key physical parameter that regulates the gating of this channel [21,50]. The channel does not exhibit any strong ion selectivity, again suggesting that its purpose is not to move ions, per se, but to allow flow of water and/or osmolytes. In the microbial setting, the pressure gradient across the inner membrane couples to the radius of curvature of the membrane to produce a lateral tension on the membrane surface. At relatively low membrane tension the channel adopts a closed, non-conducting conformation, while at tensions near the rupture tension of a typical bilayer, the channel transitions to an open state, and spends relatively little time in sub-conducting states [21, 50, 49]. For this reason, we can reasonably approximate the channel, mechanically coupled to the membrane, as a two-state system whose equilibrium probability for being in either state is influenced by the bilayer tension. As we will show in the following subsections, experimental techniques are available to quantitatively control the tension on giant unilamellar vesicles in such
a) c)
b)
Figure 3.2: Schematics showing the hypothetical structural transitions of MscL. a) Side and top views of the transmembraneα-helices (TM1 and TM2) as they transition from the closed to pro- posed open state, in an iris-like mechanism. b) View of the pore cross-section during the same three stages of the closed to open transition, showing that the pore gets wider and shorter going from closed to open. c) Detailed view, with dimensions, of the orientation and resultant pore size of the transmembraneα-helices in the open state as measured from electron paramagnetic resonance.
Figure adapted from [19].
a way that we can control which channel conformation dominates.
In the open state, the channel pore is relatively large [146, 19, 75], approximately 3 nm in diameter (as compared to the pore sizes of channels that conduct single chains of water [145] or ions [78,152]) and∼2.5 nm in length [49,146,19], as schematically shown in Fig.3.2. With a pore of this size conducting ∼4nS1, the electrical conductance is∼10−100 times larger than typical ion channels [153], and in fact is so large that even small proteins can pass through [75, 154]. If our goal is to construct an experiment capable of measuring the water and osmolyte flux through a channel, the large size of this channel’s pore makes it a good target for studying how a variety of osmolytes, with different sizes and hydration shells (e.g. glycerol, sugars, urea, amino acids and small peptides) affect channel flow [148, 149]. Similarly, the large pore means that the flux per channel will be relatively high, in comparison to the aforementioned water transport channels, and thus we expect the contribution to the volume change of a vesicle or cell from each channel to be relatively high.
Additionally, this particular channel has the advantage that it does not exhibit significant
1By comparison a copper wire with these dimensions would conduct a whopping 170mS, or 40 million times more current!
transitions to, nor latency in, non-conducting states [49], like those seen in other mechanosensitive channels [24]. This feature will prove crucial to our experimental construction, since it means that for a given tension, the equilibrium probability that the channel is in the open state remains constant in time, or put another way, the channel is a well behaved two-state system (closed and open). This gives us confidence that the behavior of the channel is described well by the relatively simple dose-response curve shown in Fig. 3.3. The free energy difference between the open and closed states has three main contributions, as were discussed earlier in Chapter 1. The increase in channel area couples to membrane tension to give a free energy contribution of the form −τ∆A, while the change in the deformation of the surrounding membrane and internal structure of the protein give contributions ∆Gmemand ∆Gprot, respectively. Measurements of the area change upon gating span a range from 6.5−7 nm2 [21, 50] up to ∼ 20 nm2 [23]. The free energy difference coming from membrane deformation has already been discussed in detail in Chapter1. In contrast, relatively little is known about the free energy component coming from rearrangements of the protein’s internal structure or hydration of the pore [34, 98], although this contribution will be discussed in later sections of this chapter. Measurements of the free energy of channel gating, in different lipid bilayer compositions, range from∼15−20kBT [21,50] up to∼50kBT [23], and we will use 20kBT as the typical value. Together, these contributions yield the open state probability as a function of tension given by
Po(τ) = 1
1 +e(∆Gmem+∆Gprot−τ∆A)/kBT. (3.1) Though we do not know the precise molecular origins of the protein free energy ∆Gprot, we do know that its contribution can be made to favor the open state via single point mutation of residues lining the channel pore; the use of such mutants will be discussed later in this chapter. With these facts in mind, we are now in a position to (conceptually) build an experiment, using MscL, that measures the ensemble volumetric flow through a protein pore, though first we will build intuition for how bilayers with and without volumetric channels respond to osmotic shock.