Molecular motion across membranes

A crucial aspect of biochemical change is the rate at which species are transported across a membrane, so we need to understand the kinetic factors that facilitate or impede transport. 

Consider the passive transport of an uncharged species A across a lipid bilayer of thickness /. To simplify the problem, we assume that the concentration of A is always maintained at [A] = [A]o on one surface of the membrane and at [A] = 0 on the other surface, perhaps by a perfect balance between the rate of the process that produces A on one side and the rate of another process that consumes A completely on the other side. Then 3[A]/3t = 0 because the two boundary conditions ensure that the interior of the membrane is maintained at a constant but not necessarily uniform concentration, and eqn 8.12 simpUfies to 

Before using this simple result we need to take into account the fact that the concentration of A on the surface of a membrane is not always equal to its concentration measured in the bulk solution, which we assume to be aqueous. This difference arises from the significant difference in the solubility of A in an aqueous environment and in the solution-membrane interface. One way to deal with this problem is to define a partition coefficient k (kappa) as 

We see, as intuition would surest, that the flux is high when the concentration of A in the bulk solution is high and the membrane is thin. 

In spite of the assumptions that led to its final form, eqn 8.19 describes adequately the passive transport of many nonelectrolytes through membranes of blood cells. In many cases, however, eqn 8.19 imderestimates the flux, which suggests that the membrane is more permeable than expected. However, because the permeability increases only for certain species, we can infer that in these cases, transport is facilitated by carrier molecules. One example is the transporter protein that carries glucose into cells. But we issue a word of caution there is little justification for supposing that D in the membrane is equal to its value in aqueous solution or that k has any particular value, and the conclusion that facilitated transport is involved needs additional evidence before it can be accepted. 

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Practically speaking, this concept explains the basis for the establishment of partial pressure equilibrium of anesthetic gas between the lung alveoli and the arterial blood. Gas molecules will move across the alveolar membrane until those in the blood, through random molecular motion, exert pressure equal to their counterparts in the lung. Similar gas tension equilibria also will be established between the blood and other tissues. For example, gas molecules in the blood will diffuse down a tension gradient into the brain until equal random molecular motion (equal pressure) occurs in both tissues. 

Note, however that the concepts about the lipid membrane as the isotropic, structureless medium are oversimplified. It is well known [19, 190] that the rates and character of the molecular motion in the lateral direction and across the membrane are quite different. This is true for both the molecules inserted in the lipid bilayer and the lipid molecules themselves. Thus, for example, while it still seems possible to characterize the lateral movement of the egg lecithin molecule by the diffusion coefficient D its movement across the membrane seems to be better described by the so-called flip-flop mechanism when two lipid molecules from the inner and outer membrane monolayers of the vesicle synchronously change locations with each other [19]. The value of D, = 1.8 x 10 8 cm2 s 1 [191] corresponds to the time of the lateral diffusion jump of lecithin molecule, Le. about 10 7s. The characteristic time of flip-flop under the same conditions is much longer (about 6.5 hours) [19]. The molecules without long hydrocarbon chains migrate much more rapidly. For example for pyrene D, = 1.4x 10 7 cm2 s1 [192]. 

In looking at the various physical properties of these cations (cf. Table 1) one realizes especially the differences in size, which manifest themselves clearly in energy parameters. However, the effective sizes of the solvated ions — i. e. the Stokes radii — apart from showing a reversed order do not differ greatly. Any simple model based on Brownian motion of totally solvated metal ions under the influence of an electric field gradient could not explain pronounced specificities of electric transport across membranes. In realizing this fact physiologists proposed the idea of a carrier, which via its peculiar molecular architecture could specifically interact with an ion of a... 

Studies of the effect of permeant s size on the translational diffusion in membranes suggest that a free-volume model is appropriate for the description of diffusion processes in the bilayers [93]. The dynamic motion of the chains of the membrane lipids and proteins may result in the formation of transient pockets of free volume or cavities into which a permeant molecule can enter. Diffusion occurs when a permeant jumps from a donor to an acceptor cavity. Results from recent molecular dynamics simulations suggest that the free volume transport mechanism is more likely to be operative in the core of the bilayer [84]. In the more ordered region of the bilayer, a kink shift diffusion mechanism is more likely to occur [84,94]. Kinks may be pictured as dynamic structural defects representing small, mobile free volumes in the hydrocarbon phase of the membrane, i.e., conformational kink g tg ) isomers of the hydrocarbon chains resulting from thermal motion [52] (Fig. 8). Small molecules can enter the small free volumes of the kinks and migrate across the membrane together with the kinks. 

Another important class of proteins that contain water channels are the aquaporins, which regulate the flow of water in and out of cells. They will let water through but not salts or other dissolved substances, and as such, they act as molecular water filters. Water transport occurs via a chain of nine hydrogen-bonded molecules (Fig. 6.13). But if this chain were to permit rapid transmembrane proton motion, that would disturb the delicate charge balance across the membrane. So aquaporin must somehow disrupt the potential proton wire that threads through it. The mechanism has been much debated, but it now seems that the inhibition of proton transport is dominated by electrostatic repulsion by positively charged groups in a narrow constriction in the middle of the pore [72]. 

Fluctuations of interfaces are directly relevant to a number of interfacial phenomena. One example, ion transfer across a liquid-liquid interface, will be discussed in Section 6.1. Another example is the behavior of monolayers of surfactants on water surfaces. Surface fluctuations are also fundamental to several processes in water-membrane systems, such as unassisted ion transport across lipid bilayers and the hydration forces acting between two membranes. Here, however, the problem is more complicated because not only capillary waves but also bending motions of the whole bilayer have to be taken into account. Furthermore, the concept of the surface tension is less clear in this case. This topic is discussed in Molecular Dynamics Studies of Lipid Bilayers. 

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