One example of a periodic or oscillating driving force in a living cell is the transmembrane electric field. Although the transmembrane potential of a cell has been considered constant in the past, recent analyses suggest that locally it may exhibit large amplitude oscillations of fluctuations when time (is) resolved to ms or us, (microsecond), levels. Many membrane integral proteins have been shown to be electrically active

For examples, the VDAC (voltage-dependent anion channel) from the outer membrane of the mitochondria [18], the Na-channel/batrachotoxin complex [19], and the acetylcholine receptor [20] open in specific ranges of transmembrane electric potential and close in other ranges. These field dependent conformational changes may be coupled to ligand binding processes for energy and signal transductions [12-15]. Let us consider a simple two state conformational transition,

(formulae) (1)

The equilibrium constant of the reaction is K = k(1)/k(-1). The properties which make a protein responsive to an electric field are its electric moment (u) and polarizability (a). The molar electric moment of P(1) is M(1) = u(1) + a(1)E and of P(2) , M(2) = u(2) + a(2)E. The change in molar electric moment for the conformational transition is (delta)M = M(2) – M(1). An electric field of strength E will shift the equilibrium according to the generalized van’t Hoff equation,


The energy involved in this transaction is (delta)M*E. This energy can be utilized to {missing}

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{ Fig. 2. Computer analysis of the cyclic kinetic scheme of eqn. 3). (A) Numerical integration of an ac field stimulated pumping of a neural substrate is shown. The MLAB computer program of National Institutes of Health was used. In the computation an interaction energy (delta)M*E of 12.6kJ/mol was used. Changes in the concentration of S(in) are plotted against stimulation time. The frequency and the strength of the ac field were adjusted so that there is an apparent resonance of the enzyme conformational transitions and the oscillating field. A fine feature of oscillation can be seen on top of the steady rise in the concentration of S(in). This rise would level off when the concentration gradient balanced the interaction energy. For the set of rate constants and other parameters used here, the resonance occurred around 300 Hz. (B) When an ac field of 1500 Hz was used, the resonance was broken and there was a fluctuation of the concentration of S(in) but no net accumulation of S(in) in the cytoplasmic side was detected. See refs. 12-15 for details.} drive an endergonic reaction. Equation (3) shows one such design for a membrane transport protein or enzyme.

(formulae) (3)

Here the protein is assumed to be a membrane integral protein and the activity of the protein is to transport the substrate S from the extracellular medium into the cytoplasm of the cell. For convenience only, two conditions will be introduced in our discussion. First, (delta)M of the P(1) to P(2) transition has a positive value. In other words, a positive electric field will favor the P(2) state and a negative electric field the P(1) state. Second, the affinity of S(out) for P(2) is much greater than that of S(in) for P(1). When these two conditions are imposed, a periodic perturbation by an oscillating electric field (ac field) will drive a clockwise flux of the enzyme and hence the transport of S(out) from the extracellular space into S(in) of the cytoplasm. Figure 2 gives some results of the computer analysis for eqn. (3). These computations and p. 324

{Fig. 3. Summary of the behavior of the cyclic kinetic scheme of eqn. (3) in response to an oscillating electric field. (A) In the scheme the affinity of S(out) for P(2) is assumed to be much greater than the affinity of S(in) for P(1). The substrate binding and dissociation steps are fast compared to the conformational transition steps. With these conditions, [P(1)] > [P(e)2] and [P(2)S] > [P(1)S] under zero field. (B) When the ac field is in its positive phase, it induces a large flux of P(1) – P(2) and a small flux of P(1)S – P(2)S. The catalytic wheel turns clockwise. (C) When the ac field is in its negative phase, it induces a large flux of P(2)S – P(1)S and a small flux of P(2) – P(1). The catalytic wheel again turns clockwise. These results indicate that the wheel turns only in one direction regardless of the polarity of the stimulating ac field. This means that the system can capture energy from the oscillating field for driving an endergonic reaction such as energy transduction or signal transduction. The direction of the revolving wheel is determined by the affinity of S to P(1) and P(2). If the affinity is greater for P(1) than for P(2), the wheel will turn counterclockwise.} analyses also reveal many interesting properties of eqn. (3) which are summarized in Fig. 3. Figure 3 indicates that, regardless of the polarity of the induced membrane potential during the experiment, the fluxes are directed towards the cytoplasm. This means that eqn. (3) is a molecular pump which is driven by an oscillating electric field by direct coulombic coupling to the enzyme conformational equilibrium. No interaction between the substrate and the electric field need be presumed. Several remarkable points are listed below.

(1) The scheme would transduce energy only if the four state scheme has inherent asymmetries. We have already mentioned the difference in the affinity of S for P(1) and P(2). This equivalent to the interaction energy discussed by Jencks [21]. Other asymmetry requirements have been discussed elsewhere [13,16,22].

(2) The frequency of the applied field must match the kinetic attributes of the system. When the ligand binding and dissociation steps are fast, the optimum frequency of the field for driving the reaction is determined by the conformational transition steps.

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(3) There is also an optimum field strength for the energy coupling. A field of strength exceeding this value tends to lock the enzyme into certain states, thus preventing its effective turnover [16].

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