In the Classroom
Illustrating Chemical Kinetic Principles with Ion Motions in Thin Membranes Michael E. Starzak Department of Chemistry, State University of New York at Binghamton, Binghamton, NY 13902-6014
Bilayer membranes, with two layers of molecules, are easily formed from many molecules that have polar and nonpolar regions. These amphipathic molecules spontaneously order themselves so that their polar heads face the aqueous solutions that bound the membrane. The long hydrocarbon chains of the molecules for the two layers lie parallel to each other to make a liquid crystalline phase that forms an insulating barrier to limit diffusion of water and ions between the two aqueous solutions. Some materials can dissolve in the hydrophobic membrane to facilitate the transport of both water and ions. Membrane channel molecules have a lipophilic outer surface, so that the molecule can “dissolve” in the membrane, and a polar interior that spans the membrane and accepts ions and water. For example, gramicidin is a small peptide that dissolves in the membrane as a single helix. Two of these conformers dimerize in a head-to-head fashion to span the bilayer membrane (Fig. 1). The hollow interior of this helix is lined with the polar carboxyls of the peptide backbone to create a waterlike channel for water and ions. When an electrical potential difference is applied across the membrane, approximately 107–108 metal cations pass through a single channel each second. Since each ionic charge is 1.6 × 10{19 C, these flows are equivalent to currents of 10{12 to 10{11 A. With sensitive current detectors, the formation of individual dimeric gramicidin channels, a molecular event, is observed as a current pulse when ions pass through the formed channel (1, 2). Channel kinetics have two disparate time scales. The dimerization of two gramicidins to form an operational channel and channel dissociation are slow (>ms), whereas the kinetics of ion permeation through the channels is rapid (ca. 100 ns). Metal ions diffuse through the channel but their net charge excludes them from the hydrophobic membrane phase. Some organic ions may dissolve in the membrane because, despite their net charge, they are extremely hydrophobic. The tetraphenylborate anion (TPB{) (3) has four phenyl groups around boron to lower its energy barrier for entry into the membrane phase. The charged molecule can diffuse between the interfacial states or move in response to an applied electrical potential. Certain organic ionophores bind metal cations to create a membrane soluble ion complex that can also be moved by an applied electric field. Since these charged species are most stable in the membrane interfacial region, applied electrical potentials can move them between the two interfaces (Fig. 2). Because the ions reside in two interfacial states, the system becomes a simple two-state kinetic system. The relative state populations are controlled by an applied transmembrane potential. If this potential is changed, ions will move between the interfacial states to establish the new equilibrium ion populations. The ion motions will generate a detectable current in the external circuit. Ions moving toward an interface will attract oppositely charged solution ions toward the interface to maintain current continuity in all phases. The ion does not have to leave the membrane to generate a current at the electrodes.
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Figure 1. The dimer of single helix conformers of gramicidin in a bilayer membrane. The molecules are joined via hydrogen bonds through formyl groups at the ends of the chain. The interior of the single helix can accommodate both water and metal cations.
An exponential current decay is observed in the external circuit following a change in potential. This is a firstorder kinetic process that implies a mechanism where each complex translates independently. The rate constant for the first-order kinetics describes a spatial motion of the ion rather than a change from reactant to product so that the rate constant can be modeled at the molecular level using particle dynamics. Rate constants or time constants for homogeneous reaction systems are experimental parameters determined from the change of concentration with time. Models are then used to calculate values for the rate constant that can be compared with the experimental rate constant. An accurate match then justifies the selected molecular mechanism. Ion translation also has a decay time that is determined from the decay of current. This time is not the transit time (the
Journal of Chemical Education • Vol. 74 No. 4 April 1997
Figure 2. Orientation of a lasalocid molecule in a bilayer membrane. The molecular size can be compared with the full bilayer length (left) and a single membrane molecule (right).
In the Classroom time it actually takes the ion to move from one interface to the other). It is a summation of transit time and the average time the ion rests in its initial state before it begins to move. This random time at rest is responsible for the observed exponential decay. Transit time can be measured if the distance the ion travels and its velocity are known. Experimental determination of both decay and transit times gives more information about the mechanism than a single observation of decay time for a homogeneous reaction system. The ion transit velocity can be measured experimentally with laser Doppler velocimetry (4), which measures the Doppler frequency difference produced when light is scattered by a moving particle. The frequency difference is directly proportional to the particle velocity. A Doppler system is block diagrammed in Figure 3. A single laser beam is split into two equal parts that are displaced, reflected and then refocused on the optical axis to form an interference pattern of light and dark regions. The Doppler difference frequency for a single particle equals the modulation frequency of scattered light intensity as the particle crosses the light and dark regions. Ions in the membrane move a distance much shorter than the distance between the interference fringes. The Doppler difference frequency is the net frequency difference produced by Doppler scattering from the two exciting beams that intersect the moving particle at different angles. These Doppler-shifted scattered light beams are mixed by the photomultiplier to produce the difference frequency.
Figure 3. Laser Doppler velocimetry. A single laser beam is split evenly into two equal beams that are displaced and then refocused onto the optical axis to create an interference pattern. The different components of Doppler scatter from the two intersecting beams combine to give the net Doppler frequency difference.
Boltzmann Statistics The Boltzmann fraction of ions in each of the two interfacial states is determined by the energies of the two states; that is, their electrical potential times the charge (zeψ). If N ions are absorbed in the membrane with a transmembrane potential ψt, the ions at the interfaces 1 and 2 will have potentials ψ1 and ψ2, respectively. The fractional populations for each state depend on the energy difference, so relative potentials of 0 and ψ for states 2 and 1, respectively, determine the fractional populations or probabilities for each state:
f1 =
e– 1+
f2 =
zeψ kT zeψ e– kT
=
e–
zeψ kT
eV / kT
q ;
1 1 zeψ = q 1 + e– kT
(1)
where q is the two state partition function. As the potential difference between the two states increases with increasing transmembrane potential, the fractional population for the lower energy state will increase rapidly relative to the higher energy state population (Fig. 4). A potential difference of only 125 mV is sufficient to populate only one of the interfaces. The transmembrane potential can be varied to generate any distribution when the total number of ions in the membrane remains constant. The situation when ions are allowed to flow through the membrane is more complicated and is not considered here. First Order Kinetics Because very few ions dissolve in the membrane, interactions between ions are minimal. This is a model for a unimolecular reaction where the rate is proportional to the
Figure 4. The change in populations of the lower energy state (upper curve) and higher energy state as the transmembrane potential expressed as eV/ kT is increased. The lower potential state is essentially 100% populated when the potential is only 125 mV.
number of reactant molecules at any time. The irreversible rate of decrease, c (t), of ions at one interface is
dc = {kc dt
(2)
This loss of ion is related to the flux of ions from one side of the membrane to the other. However, flux has units of moles of ions per second per unit area, whereas the rate has units of moles per unit volume per second. The two “rates” can be reconciled by noting that flux is the product of the decay velocity (a time-averaged velocity) and the concentration: J = vd c
(3)
The decay velocity includes both the transit velocity and the random waiting time that precedes it.
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In the Classroom This mole flux can be converted into a charge flux, a current per unit membrane area, by multiplying the expression by the charge per mole, zF: I = zFJ = zFvd c
(4)
where z is the net charge on the ion and F is Faraday’s constant. The flux equation is converted into the rate equation by dividing both sides of the equation by the distance, λ, traveled by the ion:
dc = J = { vd c = {kc λ dt λ
(5)
The decay velocity divided by the distance traveled is the rate constant for the transition. Detailed Balance at Equilibrium When the electric potential on the membrane is changed, the state populations evolve until they reach the equilibrium Boltzmann populations. Although transitions still occur, the forward and reverse rates are equal to maintain the equilibrium populations. This is the condition of detailed balance at equilibrium. Since the Boltzmann populations differ for different transmembrane potentials, the forward and reverse rate constants must also depend on potential to maintain detailed balance. With a field that favors motion from side 1 to side 2, the reverse rate constant is reduced by an exponential factor,
k r = k f exp
{ze∆ψ kT
Is a Rate Constant Constant? The ion kinetics have assumed a constant transit velocity and a constant decay velocity or rate constant for the intramembrane transition. An ion undergoes some Brownian motion within its interfacial state until it acquires sufficient energy. The ion will then accelerate toward the opposite interface to give a range of transit velocities rather than a constant transit velocity. The rate constant will vary during the initial stages of the reaction. This problem is not significant because the ion reaches its transit velocity in a short time relative to the time scale for the transition. The relaxation to the constant, or terminal, velocity is calculated using a one-dimensional Langevin equation, a restatement of Newton’s force equation in terms of the velocity, v = dx/dt, 2
Fapplied + A(t) – fv = ma = m d x2 dt ∆ψ + A(t) – fv = m dv d dt
vt = zeE f
(7)
f = 6πηrF(r)
(9)
The time to reach this stationary velocity is determined by solving the time-dependent equation using the integratft ing factor technique. The integrating factor, e m , combines the force and friction terms in a single differential, ft ft d ve m = zeE m e m dt
(10)
Both sides are integrated to give the velocity as a function of time: v,t
ft d v(t)e m = eE m
0,0
t
ft
e m dt 0
ft m ft v(t)e m – 0 = eE m f em – 1
(11)
ft v(t) = eE 1 – e{ m f
The velocity reaches its mass-independent terminal value after an exponential decay with a time constant, τ = m/f directly proportional to the ion mass. The actual times to reach a terminal velocity (or constant rate constant) can be estimated from Stokes law. For example, an ionophore complex of molecular weight 6,000 g/mol (10{23 kg/molecule) and a frictional coefficient of 10{11 kg/s has a relaxation time of
τ =m/ f =
10{23 kg = 10{12 s 10{11 kg s{1
(12)
compared to transit times of 10{3 s. The experimental transit velocity and the rate constant are effectively constant. A similar phenomenon occurs in homogeneous chemical kinetics. A diatomic molecule in a heat bath of inert atoms or molecules dissociates with first order kinetics. However, a microscopic view of the reaction requires collisions with inert gas molecules that change the reactive molecule’s vibrational energy level. Most of the collisions take energy away from the molecule. However, a favorable series of collisions may excite the molecule to a vibrational level with sufficient energy to break the bond—that is, to react. The microscopic kinetics involves each vibrational level, and the kinetics for a system with N vibrational states, including the state that dissociates, is described by N – 1 exponential decays with decay constants, λi,
where f is the frictional coefficient and the applied force is the product of the net charge of the complex multiplied by
412
(8)
The velocity is independent of the mass of the moving complex. Stokes law relates the frictional coefficient to the viscosity of the medium η, the radius r, and a form factor F(r) that corrects for the shape of the complex:
(6)
Interestingly, in order to have constant rate constants and reach the proper equilibrium, these rate constants that reflect equilibrium Boltzmann statistics are used even when the system is evolving to equilibrium. Direct observation of the forward and reverse rate constants is required to test this interesting consequence of detailed balance. The ion/ membrane system is a particularly good candidate for such tests because the initial conditions (i.e., the distance from equilibrium) are controlled by the applied potential.
ze
the homogeneous electric field (E = ∆ψ/d) in a membrane of thickness d. The random force, A(t), produced by thermal interactions, is small relative to the driving force and will be ignored to simplify the analysis. The terminal velocity is attained when dv/dt = 0,
Journal of Chemical Education • Vol. 74 No. 4 April 1997
eq
fi(t) = f i +
N–1
Σ
j=1
C j e{λ it
(13)
In the Classroom This microscopic solution, with its many decay constants, differs from the single exponential decay observed experimentally for the dissociation. Experiment and theory are reconciled because one of the decay constants is significantly smaller than the others and dominates the kinetics (5). The remaining exponentials decay rapidly relative to the time scale of the experiment, and the microscopic system is also characterized by a single decay. An experimenter, observing the system at very short times while fitting it to a single exponential, could conclude that his rate constant was not constant at these times. Fortunately, these rapidly decaying exponentials have usually disappeared on the time scale of most experiments. This is the discrete state analog of the membrane-ion evolution to its terminal velocity. Conclusions The translational kinetics of membrane soluble ions in thin membranes illustrate some unusual properties of simple kinetic systems. The system is unusually flexible because the initial and equilibrium states are controlled externally by the magnitude and polarity of an applied electrical potential. The current decays have a millisecond time scale and the instrumentation to detect these currents can easily be constructed as a joint electronics–kinetics laboratory experiment.
The system encourages students to make a more critical examination of some fundamental ideas in chemical kinetics. The distinction between the actual time for a molecule to react and the time constant for the reaction is clarified. The fact that the rate at any time is proportional to the time interval rather than the total elapsed time can be illustrated. Students normally have difficulty with this statistical concept because it is outside their experience. In fact, it is worthwhile to ask the students to predict the current if all ions move synchronously on application of the transmembrane potential, so that they can see the difference. Finally, it is easier to visualize kinetic transitions as spatial changes. This context, in turn, makes it easier to understand abstract chemical concepts like detailed balance and the connection between microscopic transitions and the observable rate constant for the system. Literature Cited 1. 2. 3. 4.
Starzak, M. Prog. Surface Sci. 1994, 46, 61–104. Andersen, O. S. Physiol. Rev. 1992, 72, S89–S158. Benz, R. Biophys. J. 1988, 54, 25–33. Macias, F.; Starzak, M. In Membrane Electrochemistry; Vodyanoy, I.; Blank, M., Eds.; ACS Advances in Chemistry Series 235; American Chemical Society: Washington, DC, 1993. 5. Starzak, M. Mathematical Methods in Chemistry and Physics; Plenum: New York, 1989.
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