Conductivities and ion–ion interactions

The remaining problem is to account for the c^1/2 dependence of the Kohlrausch law (eqn 21.29). In Section 5.9 we saw something similar: the activity coefficients of ions at low concentrations also depend on c^1/2 and depend on their charge type rather than their specific identities. That c^1/2 dependence was explained in terms of the properties of the ionic atmosphere around each ion, and we can suspect that the same explanation applies here too. To accommodate the effect of motion, we need to modify the picture of an ionic atmosphere as a spherical haze of charge. Because the ions forming the atmosphere do not adjust to the moving ion immediately, the atmosphere is incompletely formed in front of the moving ion and incompletely decayed behind the ion (Fig. 21.19). The overall effect is the displacement of the centre of charge of the atmosphere a short distance behind the moving ion. Because the two charges are opposite, the result is a retardation of the moving ion. This reduction of the ions’ mobility is called the relaxation effect. A confirmation of the picture is obtained by observing the conductivities of ions at high frequencies, which are greater than at low frequencies: the atmosphere does not have time to follow the rapidly changing direction of motion of the ion, and the effect of the field averages to zero. The ionic atmosphere has another effect on the motion of the ions. We have seen that the moving ion experiences a viscous drag. When the ionic atmosphere is present this drag is enhanced because the ionic atmosphere moves in an opposite direction to the central ion. The enhanced viscous drag, which is called the electrophoretic effect, reduces the mobility of the ions, and hence also reduces their conductivities. The quantitative formulation of these effects is far from simple, but the Debye Hückel–Onsager theory is an attempt to obtain quantitative expressions at about the same level of sophistication as the Debye–Hückel theory itself. The theory leads to a Kohlrausch-like expression in which

K =A+BΛ°_m

With

Where ε is the electric permittivity of the solvent (Section 18.3) and q = 0.586 for a 1,1-electrolyte (Table 21.7). The slopes of the conductivity curves are predicted to depend

on the charge type of the electrolyte, in accord with the Kohlrausch law, and some comparisons between theory and experiment are shown in Fig. 21.20. The agreement is quite good at very low ionic strengths, corresponding to very low molar concentrations (less than about 10⁻³ M, depending on the charge type).

Controlled transport of molecules and ions across biological membranes is at the heart of a number of key cellular processes, such as the transmission of nerve im pulses, the transfer of glucose into red blood cells, and the synthesis of ATP by oxidative phosphorylation (Impact I7.2). Here we examine in some detail the various ways in which ions cross the alien environment of the lipid bilayer. Suppose that a membrane provides a barrier that slows down the transfer of molecules or ions into or out of the cell. We saw in Impact I7.2 that the thermo dynamic tendency to transport an ion through the membrane is partially determined by a concentration gradient (more precisely, an activity gradient) across the membrane, which results in a difference in molar Gibbs energy between the inside and the outside of the cell, and a transmembrane potential gradient, which is due to the different potential energy of the ions on each side of the bilayer. There is a tendency, called passive transport, for a species to move down concentration and membrane potential gradients. It is also possible to move a species against these gradients, but now the flow must be driven by an exergonic process, such as the hydrolysis of ATP. This process is called active transport.

The transport of ions into or out of a cell needs to be mediated (that is, facilitated by other species) because the hydrophobic environment of the membrane is in hospitable to ions. There are two mechanisms for ion transport: mediation by a carrier molecule and transport through achannel former, a protein that creates a hydrophilic pore through which the ion can pass. An example of a channel former is the poly peptide gramicidin A, which increases the membrane permeability to cations such as H⁺, K⁺, and Na⁺.

Ion channels are proteins that effect the movement of specific ions down a mem brane potential gradient. They are highly selective, so there is a channel protein for Ca²⁺, another for Cl⁻, and so on. The opening of the gate may be triggered by potential differences between the two sides of the membrane or by the binding of an effector molecule to a specific receptor site on the channel. Ions such as H+, Na+, K+, and Ca2+ are often transported actively across membranes by integral proteins called ion pumps. Ion pumps are molecular machines that work by adopting conformations that are permeable to one ion but not others depending on the state of phosphorylation of the protein. Because protein phosphorylation requires dephosphorylation of ATP, the conformational change that opens or closes the pump is endergonic and requires the use of energy stored during metabolism.

Let’s consider some of the experimental approaches used in the study of ion chan nels. The structures of a number of channel proteins have been obtained by the now traditional X-ray diffraction techniques described in Chapter 20. Information about the flow of ions across channels and pumps is supplied by the patch clamp technique. One of many possible experimental arrangements is shown in Fig. 21.21. With mild suction, a ‘patch’ of membrane from a whole cell or a small section of a broken cell can be attached tightly to the tip of a micropipet filled with an electrolyte solution and containing an electronic conductor, the so-called patch electrode. A potential differ ence (the ‘clamp’) is applied between the patch electrode and an intracellular electronic conductor in contact with the cytosol of the cell. If the membrane is permeable to ions at the applied potential difference, a current flows through the completed circuit. Using narrow micropipette tips with diameters of less than 1 µm, ion currents of a few picoamperes (1 pA = 10−12 A) have been measured across sections of membranes containing only one ion channel protein. A detailed picture of the mechanism of action of ion channels has emerged from analysis of patch clamp data and structural data. Here we focus on the K+ ion channel protein, which, like all other mediators of ion transport, spans the membrane bilayer (Fig. 21.22). The pore through which ions move has a length of 3.4 nm and is divided into two regions: a wide region with a length of 2.2 nm and diameter of 1.0 nm and a narrow region with a length of 1.2 nm and diameter of 0.3 nm. The narrow region is called the selectivity filter of the K⁺ ion channel because it allows only K+ ions to pass. Filtering is a subtle process that depends on ionic size and the thermodynamic tendency of an ion to lose its hydrating water molecules. Upon entering the selectivity filter, the K+ ion is stripped of its hydrating shell and is then gripped by carbonyl groups of the protein. Dehydration of the K+ ion is endergonic (∆dehydG7 =+203 kJ mol−1), but is driven by the energy of interaction between the ion and the protein. The Na⁺ ion, though smaller than the K+ ion, does not pass through the selectivity filter of the K+ ion channel because interactions with the protein are not sufficient to compensate for the high Gibbs energy of dehydration of Na+ (∆dehydG7 =+301 kJ mol−1). More specifically, a dehydrated Na+ ion is too small and cannot be held tightly by the protein carbonyl groups, which are positioned for ideal interactions with the larger K+ ion. In its hydrated form, the Na+ ion is too large (larger than a dehydrated K+ ion), does not fit in the selectivity filter, and does not cross the membrane. Though very selective, a K⁺ ion channel can still let other ions pass through. For example, K⁺ and Tl⁺ ions have similar radii and Gibbs energies of dehydration, so Tl⁺ can cross the membrane. As a result, Tl⁺ is a neurotoxin because it replaces K+ in many neuronal functions.

The efficiency of transfer of K+ ions through the channel can also be explained by structural features of the protein. For efficient transport to occur, a K+ ion must enter the protein, but then must not be allowed to remain inside for very long so that, as one K+ion enters the channel from one side, another K+ ion leaves from the opposite side. An ion is lured into the channel by water molecules about halfway through the length of the membrane. Consequently, the thermodynamic cost of moving an ion from an aqueous environment to the less hydrophilic interior of the protein is minimized. The ion is ‘encouraged’ to leave the protein by electrostatic interactions in the selectivity filter, which can bind two K+ ions simultaneously, usually with a bridging water molecule. Electrostatic repulsion prevents the ions from binding too tightly, minimizing the residence time of an ion in the selectivity filter, and maximizing the trans port rate. Now we turn our attention to a very important ion pump, the H+-ATPase responsible for coupling of proton flow to synthesis of ATP from ADP and Pi (Impact I7.2). Structural studies show that the channel through which the protons flow is linked in tandem to a unit composed of six protein molecules arranged in pairs of α and β sub units to form three interlocked αβ segments (Fig. 21.23). The conformations of the three pairs may be loose (L), tight (T), or open (O), and one of each type is present at each stage. A protein at the centre of the interlocked structure, the γ subunit shown as a white arrow, rotates and induces structural changes that cycle each of the three segments between L, T, and O conformations. At the start of a cycle, a T unit holds an ATP molecule. Then ADP and a Pi group migrate into the L site and, as it closes into T, the earlier T site opens into O and releases its ATP. The ADP and Pi in the T site meanwhile condense into ATP, and the new L site is ready for the cycle to begin again. The proton flux drives the rotation of the γ subunit, and hence the conformational changes of the α/β segments, as well as providing the energy for the condensation reaction itself. Several key aspects of this mechanism have been confirmed experimentally. For example, the rotation of the γ subunit has been observed directly by using single-molecule spectroscopy (Section 14.6).

Fig. 21.19 (a) In the absence of an applied f ield, the ionic atmosphere is spherically symmetric, but (b) when a field is present it is distorted and the centres of negative and positive charge no longer coincide. The attraction between the opposite charges retards the motion of the central ion.

Fig. 21.20 The dependence of molar conductivities on the square root of the ionic strength, and comparison (straight lines) with the dependence predicted by the Debye–Hückel–Onsager theory.

Fig. 21.21 A representation of the patch clamp technique for the measurement of ionic currents through membranes in intact cells. A section of membrane containing an ion channel (shown as a green rectangle) is in tight contact with the tip of a micropipette containing an electrolyte solution and the patch electrode. An intracellular electronic conductor is inserted into the cytosol of the cell and the two conductors are connected to a power supply and current measuring device.

Fig. 21.22 A schematic representation of the cross-section of a membrane-spanning K⁺ ion channel protein. The bulk of the protein is shown in beige. The pore through which ions move is divided into two regions: a wide region with a length of 2.2 nm and diameter of 1.0 nm, and a narrow region, the selectivity filter, with a length of 1.2 nm and diameter of 0.3 nm. The selectivity filter has a number of carbonyl groups (shown in dark green) that grip K⁺ ions. As explained in the text, electrostatic repulsions between two bound K⁺ ions ‘encourage’ ionic movement through the selectivity filter and across the membrane.

Fig. 21.23 The mechanism of action of H+ ATPase, a molecular motor that transports protons across the mitochondrial membrane and catalyses either the formation or hydrolysis of ATP.

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مواضيع ذات صلة

Monitoring the progress of a reaction

The measurement of diffusion coefficients

Solutions of the diffusion equation

Diffusion with convection

The drift speed

Strong electrolytes

Conductance and conductivity

Superconductors

Induced magnetic moments

Magnetic susceptibility

Nonlinear optical phenomena

Light emission by solid-state lasers and light-emitting diodes