INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL THE RATE CONSTANT THE CAPTURE RADIUS DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

Proton translocation across biomembranes is a key step in biological energy conversion in mitochondria and chloroplasts. The translocation is carried out by proton pumps, membrane enzymes that utilize redox energy (or light energy in photosynthetic systems) to move protons from one side of the membrane to the other, thereby creating electrochemical gradients. Later, the free energy stored in proton gradients is utilized in synthesis of ATP or other energy-requiring processes (Skulachev, 1988; Cramer and Knaff, 1990)

Proton pumps, such as cytochrome oxidase (CcO) (Wikström, 1998), are very efficientthey are capable of pumping more than 10³ H⁺ per second (Babcock and Wikström, 1992). The corresponding time scale for overall processing of a single proton is as short as 1 ms. Moreover, some proton-transfer reactions during the catalytic cycle of CcO occur on even a shorter time scale of 0.1 ms, and the reprotonation of some protolytic sites of the enzyme from the bulk is believed to occur even faster (Karpefors et al., 1999, 2000; Kotelnikov et al., 2001). Similar time scales of proton transfer reactions have been reported for bacteriorhodopsin (Heberle, 2000).

The high rate of proton pumping raises the question about the maximum possible rate of supply of protons by the organelle's medium to the pump, given the fact that the concentration of available protons is limited and the diffusion occurs with only a finite speed (e.g., in the bulk free protons diffuse with coefficient D_b ~ 10⁴ cm²s¹, see review in Gutman and Nachliel, 1997). Although the diffusion control has not been directly implicated in proton pumps, such a diffusion-limited kinetics does seem to occur in a number of proton-conducting channels (DeCoursey and Cherny, 1999). The absence of the diffusion bottleneck in CcO is ascribed to a proton-collecting antenna surrounding the entrance of the proton-conducting channel, which enhances the proton influx from the solution to the channel entrance (Gutman and Nachliel, 1997; Brandsburg-Zabary et al., 2000). Hence, the rate of proton supply is modified by the properties of the surface of the protein, or that of the surrounding membrane. The lateral membrane diffusion has been also suggested to occur in other proton-conducting systems (DeCoursey and Cherny, 1999). The relevant question then is how to correctly estimate such a diffusion-limited and surface-enhanced rate of proton supply from the medium to the entrance of the proton-conducting channel.

The above question is complicated by the fact that the nature of the protons participating in the process is not always known with certainty. For instance, under physiological conditions, the concentration of free protons in the organelles is of an order of 10⁷ to 10⁸M. Hence, in a typical organelle with dimension of 1 µm, there will be only an order of one free proton or less(!) on average. The splitting of water is energetically unfavorable, hence the involved protons must belong to buffer molecules that contain acidic or basic groups. These buffer molecules can be both mobile and immobile, i.e., fixed on the surface of membrane proteins and on the membrane itself (Gutman and Nachliel, 1997; Brandsburg-Zabary et al., 2000). The presence of fixed and mobile buffers changes the effective concentration and mobility of the protons (Junge and McLaughlin, 1987).

The question of how the membrane surface can modify the proton transport is also of interest in a more general context of chemiosmotic coupling (Nicholls and Ferguson, 1992; Ferguson, 1995). Previously, it has been suggested that the diffusion of protons along the membrane may play a major role in translocation of protons between the generators, such as CcO or bacteriorhodopsin, and consumers, such as ATP synthase (Scherrer, 1995; Teissié, 1996; Gabriel and Teissié, 1996; Nachliel et al., 1996; Antonenko and Pohl, 1998; Krasinskaya et al., 1998). Long-distance migration of protons along membranes has been observed in purple membranes and reconstituted bacteriorhodopsin (Heberle and Dencher, 1992; Alexiev et al., 1994, 1995; Heberle et al., 1994; Scherrer et al., 1994; Nachliel et al., 1996; Riesle et al., 1996). Long-range proton diffusion has also been observed along lipid (Gabriel et al., 1994) and stearic acid monolayers (Slevin and Unwin, 2000; Slevin and Unwin proposed a similar model to what is discussed in this paper, and analyzed it numerically. They found that, for stearic acid monolayers, the surface diffusion coefficient is 1.2 × 10⁵ cm²s¹). The effect of the surface on the proton transport depends on its propertiesdiffusion coefficient, nature of the lipid head groups, their pK_a, concentration, etc. (Scherrer, 1995; Teissié, 1996). Because these properties are not easily defined, since the proposal of the chemiosmotic theory, there has been a continuing debate about whether the surface participates in the proton transport.

To formulate an exact criterion of when and how the surface can affect the proton transport is also a nontrivial theoretical problem. Various aspects of the effect of reduced dimensionality have been discussed by many authors in a general context of diffusion-controlled ligand-receptor binding (Adam and Delbrück, 1968; Berg and Purcell, 1977; Berg and Blomberg, 1976; Hardt, 1979, 1981; Schranner and Richter, 1978; Berg, 1985). It has been recognized that, when diffusion is limited to one or two dimensions, for example, DNA molecule or a membrane, the rate of finding the target can be significantly increased.

In our previous paper (Georgievskii et al., 2002), a phenomenological model was developed that describes diffusion of protons near the entrance of a proton-conducting channel. The transport of protons occurs both through the bulk and along the membrane surface, which can exchange protons with the bulk. There are two regimes of the proton transport. In the first regime, the exchange between the bulk solution and the membrane is so fast that a local equilibrium is always established between the surface and bulk concentrations at neighboring points. This case is most likely to occur in a typical biological environment. In the second regime, the exchange is slow, so that the local equilibrium is not established. A rigorous solution of the model was obtained with no restrictions on the exchange kinetics.

In this paper, we present a detailed analysis and application of the fast exchange regime to the relevant experimental studies of proton migration along biological membranes. The plan of the paper is as follows. In the next section the model is presented and the fast exchange approximation is introduced. In the following section, the solution of the model is described and simplified derivation for the fast exchange regime is given in Appendix A. The concept of the capture radius is introduced in the next to last section. In the last section, we discuss the results and parameters of our model, and give numerical estimates for pure unbuffered and buffered aqueous solutions and the rate expression derived in this paper. The concept of proton lifetime on the surface, for which an explicit formula is derived in Appendix B, the extension of the theory to buffered solutions, and the capture radius are also discussed in the last section.

	MODEL

TOP ABSTRACT INTRODUCTION MODEL THE RATE CONSTANT THE CAPTURE RADIUS DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

We consider the following idealized model. A half-space filled with a solution of protons (free or attached to buffer molecules) is limited by a membrane. The membrane is populated with protolytic groups and can exchange protons with the bulk. The protons can diffuse both in the bulk and on the surface of the membrane. The diffusion coefficients on the surface and in the bulk are different. There is a specific finite lifetime of the protons on the surface, during which protons can migrate along the surface. The protons are dynamically exchanged between the surface and the bulk. On the surface of the membrane, there is a sink (or a source) of the protons of molecular size through which protons are removed (supplied) to the system. (For simplicity we will talk about the removal of the protons from the system. The results for the reversed process are identical in the approximation considered in this paper.) We model this sink of the protons as an absorbing spot on the surface, such that, after a proton gets to this spot, it is immediately removed from the system. Thus, the absorbing conditions are ideal, and there is no reverse reaction with the sink. The absorbing spot models the entrance of the proton channel, through which protons are pumped out of the system. The questions in which we are interested are: 1) what is the maximum (diffusion-limited) rate at which protons can be collected (pumped out) from such a system, and 2) at what distance from the absorbing spot the equilibrium concentration both in the bulk and on the surface will be established. The former question is related to the efficiency of proton pumps and the latter to the discussion of the nature of chemiosmotic coupling between the source and the sink of the protons on the membrane. The effects of the finite size of the organelle on the results can be easily incorporated and will be discussed in the paper.

In a system of a finite size, after some transient period, a quasi-stationary flow and a distribution of the concentration of protons are established. In our idealized infinite model, these conditions can be described as completely stationary by assuming that there is a compensating source of the protons at infinity such that the equilibrium is maintained at an infinitely large distance from the sink.

The stationary proton concentration in the bulk, n(x, y, z), obeys the three-dimensional diffusion equation

(1)

(The symbols used in the equations and their meanings are listed in Table 1.) The boundary condition on the surface depends on the properties of the surface. We assume that the coupling between the bulk and surface protons is described by the Langmuir kinetics of adsorption/desorption. The surface groups can be protonated from the bulk with a bimolecular rate constant k_on (in units of M¹s¹) and deprotonated with a monomolecular rate of desorption k_off (in units of s¹). The rate of protonation is proportional to ₀  , where (x, y) is the stationary surface density of protons captured at a given point (x, y) at the surface, and ₀ is the concentration of the protolytic groups on the surface. We will consider the case when the surface protolytic groups are far from saturation,   ₀, a regime that is practically the most important one. Then the proton flux from the bulk to the surface is given by the equation

(2)

where D_b is the bulk diffusion coefficient. This equation is the boundary condition to Eq. 1. It relates the bulk concentration of protons and its normal derivative at z = 0 to the surface concentration.

The second assumption of the model is that there exists a long-range connectivity between the protonatable groups on the surface (as described, e.g., in the percolation model [Rupley and Careri, 1991; Gutman and Nachliel, 1995, 1997]), resulting in a Brownian proton migration along the surface. The stationary concentration of the protons on the surface satisfies the two-dimensional diffusion equation,

(3)

where D_s is the surface diffusion coefficient. The bulk diffusion flux D_bn/z at z = 0 is a source of protons on the surface.

The proton channel entrance, which serves as a sink for both the bulk and surface protons, is modeled by a circle of radius r₀ on the membrane surface. Eqs. 1-3 supplemented by the absorbing boundary condition inside the channel entrance and the equilibrium condition at infinity were solved in our previous paper (Georgievskii et al., 2002).

Qualitatively, there are two regimes of proton transport, depending on the rate with which protons are exchanged between the surface and the bulk. In the fast-exchange regime, each of the two Langmuir terms in the right-hand side of Eq. 2 is much larger than the total diffusional flux on the surface, the term on the left-hand side. Then the surface density and the local bulk density of the protons in every point on the surface are essentially in equilibrium and are related as

(4)

The equilibrium association constant L₀ in Eq. 4 is expressed in terms of the parameters of the Langmuir kinetics,

(5)

where pK refers to the protolytic surface groups. Note that, in this definition, 10^pK has a dimension of inverse concentration, M¹, where M = mole/liter = N_A × 10³ cm³ and N_A is the Avogadro number. Because ₀ is the number of groups per unit surface, L₀ has a dimension of length. The physical meaning of L₀ is the height of a column in the bulk equilibrium solution that contains the same number of protons as a spot on the surface with the same area as the column's cross-section. Eq. 5 is written for a pure (unbuffered) solution. In the presence of a buffer, it is modified as discussed in the Discussion, Buffers.

Eq. 4 is a modified boundary condition to Eq. 1 (replacing Eq. 2) and it provides a direct coupling between the surface and bulk diffusion. The analysis shows (Georgievskii et al., 2002) that the formal condition for the local equilibrium described by Eq. 4 is expressed in terms of two length parameters characterizing the system. The first parameter,

(6)

is of the order of an effective size of the collecting antenna, or a capture radius (see the Capture Radius). Another parameter,

(7)

is the average distance that a proton migrates along the surface during its surface dwell time,

(8)

The formal condition for the fast exchange reads L_sb L_s, whereas the opposite case, L_sb L_s, corresponds to the slow exchange regime. In this paper, we will consider in more detail the case of the fast exchange because it is most relevant to the proton transport in biological systems.

Using the cylindrical symmetry of the problem and the coupling condition 4, we rewrite Eq. 3 as

(9)

where r is the distance on the surface, measured from the center of the proton channel, n  n(r, z), n'_z = n/z at z = 0, and L_sb is given by Eq. 6. A simplified method to solve the coupled Eq. 1 and 9 with appropriate boundary conditions is presented in Appendix A. The solution to these equations is the function n(r, z) given by Eqs. A1, A6, A8, and A14. The function n(r)  n(r, z = 0) necessary to calculate the surface proton flux in the next section is given by Eq. A15.

	THE RATE CONSTANT

TOP ABSTRACT INTRODUCTION MODEL THE RATE CONSTANT THE CAPTURE RADIUS DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

The total rate of proton transport to the channel consists of two proton fluxes, k = k_b + k_s. The bulk proton flux k_b through the channel is given by

(10)

The surface proton flux k_s through the border of the channel entrance is

(11)

The expressions for the stationary distributions of both the surface and bulk protons are given in Appendix A and Eq. 13.

The mechanism of proton transport to the proton channel is different for different values of L_sb. If L_sb  r₀, the proton transport occurs mainly via absorption of protons from the bulk solution by the channel entrance, i.e., k_b k_s. In this situation, one can neglect the surface diffusion and replace Eq. 3 with n'_z = 0, r > r₀. This is the standard model for proton transfer. The rate constant of such a process is given by (Crank, 1990)

(12)

More interesting is the opposite case, L_sb r₀. The distribution of the surface density is given by the equation

(13)

which is obtained by substituting Eq. A15 into Eq. 4. Here, C₀  1.1229 (see Eq. A13), and _eq is the equilibrium concentration of protons on the surface,

(14)

Inserting Eq. 13 into Eq. 11, we obtain the rate constant due to the surface diffusion mechanism,

(15)

The rate constant k_s depends on the size of the proton channel entrance in a much weaker, logarithmic fashion than the rate constant k_b for the bulk diffusion mechanism, Eq. 12. As shown by Georgievskii et al. (2002), in the slow-exchange regime, when L_s L_sb r₀, the expression for the rate has the same form as Eq. 15, but, in the logarithmic factor, the length L_sb is replaced with L_s. In a different context, Berg (1985) obtained an expression for surface-enhanced diffusion flux similar to our Eq. 15.

For the purpose of comparison, the above equation is rewritten in the form equivalent to the bulk rate, Eq. 12,

(16)

where R₀ is the effective radius of the channel, a formal parameter that tells what the radius of the channel should be, in the absence of the surface diffusion, to provide the same absorption rate as in the presence of the surface diffusion. The expression for R₀ thus obtained is

(17)

where R_c is the capture radius (see Eq. 20). This expression is valid for R_c r₀. The effect of the surface diffusion can now be measured directly by comparing the effective radius of the channel R₀ with the actual radius r₀.

	THE CAPTURE RADIUS

TOP ABSTRACT INTRODUCTION MODEL THE RATE CONSTANT THE CAPTURE RADIUS DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

The above-introduced effective radius of the channel R₀ is a more or less formal parameter, which is useful for comparison with the usual bulk diffusion. It should not be confused with the actual size of the area on the membrane surface from which protons are collected to the channel in the process of the coupled surface-bulk diffusion. The size of such an area is described in terms of the capture radius, R_c. The latter is defined as the distance from which a proton once on the surface will be absorbed by the channel.

The capture radius R_c can be also understood as a distance between a generator and a consumer of protons on the surface of the membrane at which the exchange between the two occurs mainly via the surface, before equilibrating with the bulk. The coupling between the source and the sink in this case is said to be local. The opposite case of delocalized coupling is realized when the source and the sink are each in equilibrium with the bulk. The rate of absorption by the consumer in this case will neither depend on the rate of proton generation (assuming infinite volume), nor on the distance between the generator and the consumer (Ferguson, 1995).

The probability of capture by the channel via the surface diffusion is described by the function,

(18)

The deviation of (r) from the equilibrium value _eq is due to absorption by the channel, and is a measure of the probability that the proton will be absorbed by the channel from a specified position. At the boundary of the absorbing channel,  = 0 and the probability is exactly one. At an infinitely large distance from the channel,  = _eq and the probability of absorption is zero.

From Eq. A15, we find that

(19)

The capture radius defined formally as the distance at which the probability of capture decreases to zero is

(20)

One should notice that, because the dependence of the probability of capture on the distance is logarithmic, the capture radius should be understood only as defining an order of magnitude of the relevant characteristic length (say, increasing R_c by a factor of e will change the probability only by 1/ln(L_sb/r₀) 1, because we assume L_sb r₀). Thus, the constant C₀ ~ 1, Eq. A13, is essentially irrelevant in the above expression for R_c. It is recalled that, here, we treated the case L_sb L_s. In a general case, as shown by Georgievskii et al. (2002), the capture radius is

(21)

and the capture probability is given by

(22)

This expression is valid for R_c r₀ and r₀ < r  R_c.

	DISCUSSION AND CONCLUSIONS

TOP ABSTRACT INTRODUCTION MODEL THE RATE CONSTANT THE CAPTURE RADIUS DISCUSSION AND CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

Summary

The surface can enhance the flow of protons to a target, such as the entrance of a proton channel, via its low-pK protonatable groups capable of catching protons from the bulk and then shuttling them along the surface. If there are only a few such groups, then, quantitatively, the enhancement can be expressed in terms of so-called virtual bimolecular rate constants introduced by Gutman and coworkers (Yam et al., 1988). When there are many such groups, and the connectivity between them is established, the transport to the target on the surface is a combined surface-bulk diffusion. The model discussed in this paper refers to the latter type of proton transfer.

The theory developed allows one to evaluate the maximum (diffusion-limited) rate at which a proton pump can translocate protons across a membrane. The model is phenomenological. It assumes that the protons are collected by a proton channel in the membrane, and that they can diffuse both along the surface and from the bulk toward the entrance of the channel. There is a dynamic proton exchange between the surface and the bulk, which is described by the Langmuir adsorption/desorption kinetics. Such a coupling results in a nontrivial coupled surface and bulk proton diffusion in the system. The surface was assumed to be infinite, homogeneous, and characterized by a surface proton diffusion coefficient, D_s, pK_a of the surface protolytic groups, pK, their density ₀ (the number of groups per unit surface), and the parameters of the Langmuir kineticsthe first-order rate constant of desorption, k_off, and the second-order rate constant of adsorption, k_on. The radius of the proton sink on the surface is r₀. The bulk protons are characterized by the bulk diffusion coefficient D_b and the equilibrium bulk density n₀.

In this phenomenological description, we do not specify the precise nature of the protonatable groups on the surface, but rather characterize them by their pK_a, their density ₀, and the proton diffusion coefficient D_s. The protonatable groups on the surface of the protein (Gutman and Nachliel, 1997), and the protonatable lipid head groups of the membrane both can play the role of the proton-transmitting elements on the surface (Scherrer, 1995; Teissié, 1996). In this section, quantitative estimates will be made with the use of parameters from Table 2, where the values of pK_a and ₀ are characteristic of carboxylates on the surface of proteins such as cytochrome c oxidase and bacteriorhodopsin, and are in the range of values for protonatable lipid head groups studied by Scherrer (1995).

Modeling the channel entrance as a plain sink of a given radius r₀ with free proton diffusion in its vicinity may look, at a first glance, as an oversimplification. Indeed it has been suggested that the entrance of protein proton channels may be surrounded by negative charges that provide a guiding electrostatic potential, which extends to some distances, leading to both directed and enhanced bulk diffusion toward the channel entrance. The range of action of the attracting potential, however, may be expected to be only of the order of the Coulomb cage radius, say, 7-14 Å. Because the phenomena described by the present model typically occur on a much larger scale, as we demonstrate below, this external "short-range" potential can be described in our phenomenological model by simply increasing the effective radius of the channel from a few angstroms to 10-15. The rate of surface transport, as we saw, only weakly, logarithmically, depends on the channel radius. Thus, the increase of the effective size of the channel may be insignificant. However, the relative contribution of the bulk and the surface transport may be indeed affected, because the bulk contribution is proportional to r₀. This uncertainty in r₀ should be kept in mind when estimates are made using this theory.

Within the model developed, the theory also allowed us to formulate a criterion of when the coupling between a source and a sink on the membrane surface is localized, i.e., the transport between the two occurs before the protons generated by the source are fully equilibrated with the bulk.

Both the issue of diffusion-limited transport and the range of delocalization of the surface protons are of interest in a general context of chemiosmotic theory of energy transduction.

The rate

In this paper, we treated the case of the fast exchange between the surface and the bulk, a regime in which the surface and bulk protons are in local equilibrium. Formally, the condition for the fast exchange is L_sb L_s. The study of the opposite case of the slow exchange (Georgievskii et al., 2002), L_sb  L_s, yields the expression for the rate similar to the one obtained here.

In the regime where the surface diffusion dominates the transport, in both cases of the fast and slow exchange between the surface and the bulk, the rate can be written in the form

(23)

where R_c is essentially the maximum of two lengths, L_s and L_sb (the factor C₀ in the exact definition is close to unity and can be neglected). The logarithmic factor is expected to be roughly in the range of one to ten. Due to the logarithmic dependence, the radius of the channel r₀ has a minor effect on the rate.

The effect of the surface on the maximum rate of absorption by the channel is described in terms of the effective radius of the channel, R₀, as defined in Eq. 17. When R₀ is much larger than the actual radius of the channel, r₀, the surface-modified diffusion enhances the rate by a factor of R₀/r₀. Analytical results could be obtained in the regime where the surface dominates proton transport, R₀/r₀ 1, and the protonation of the surface is far from saturation, _{eq 0}. The latter condition makes the coupling between the bulk and the surface linear.

Because the surface diffusion coefficient typically is not expected to be much different from that of the bulk (the reported values for D_s/D_b are in the range of 10³ to 1 [Heberle et al., 1994; Scherrer et al., 1994; Slevin and Unwin, 2000]), but the surface density can be much higher than the equivalent bulk density (the measure is given by the equilibrium constant L₀, see Eqs. 4 and 5), the surface can provide an extremely high output of protons. For example, using D_s = 10⁵ cm²s¹ and other parameters from Table 2, we obtain _dw = 4 µs by Eq. 8, L₀ = 170 µm by Eq. 5, L_sb = 17 µm by Eq. 6, L_s = 660 Å by Eq. 7, R_c = 17 µm by Eq. 20, and R₀ = 2 µm by Eq. 17. Thus, the enhancement effect would be R₀/r₀ ~ 10⁴, provided that the surface available for the proton diffusion was infinite.

If the proton diffusion on the surface around the channel entrance is restricted by a finite area, such as a cluster of protonatable groups of dimension L_max and L_max < R_c, then the enhancement factor is less than R₀/r₀. In this case, a proton, once adsorbed by the cluster with a high probability, will be dragged into the channel before returning to the bulk. The cluster, therefore, will act as an absolutely absorbing disk for which the rate of absorption is given by Eq. 12 where r₀ should be replaced with L_max. The effective radius of the channel, R^eff, will be equal to L_max, and the enhancement effect will be L_max/r₀.

In the opposite limit of L_max > R_c the surface proton flow is independent of L_max, and R^eff is gradually approaching R₀ somewhere between R₀ and R_c. This behavior is schematically depicted in Fig. 1. One can see, therefore, that the length parameter R_c is a critical parameter that determines the rate, although formally the rate depends only logarithmically on it.

View larger version (9K): [in this window] [in a new window]   FIGURE 1   Schematic dependence of the effective channel radius Reff on Lmax, the maximum dimension of the surface area available for the proton diffusion. The effective radius is equal to Lmax when Lmax is smaller than the capture radius Rc, and reaches its limiting value R0 at Lmax > Rc.

Above, we gave estimates of the effective radius of the channel and the surface enhancement effect for a pure unbuffered solvent. Buffers can change these estimates dramatically. However, the surface is still predicted to dominate the transport up to a high buffer concentration (see Buffers subsection).

In Fig. 2, the proton flux into the channel due to the surface-diffusion mechanism is shown as a function of the surface-diffusion constant. Within the assumed range of D_s, the rate of proton supply to a channel may vary from 10⁵ up to 4 × 10⁷ s¹. In cytochrome c oxidase, the rates of internal proton transfer in different redox states of the enzyme were estimated in the previous work from this group (Kotelnikov et al., 2001) to be (10²-10⁴) s¹, so that, even at the lowest diffusional mobility on the surface, the calculated rate of proton delivery is still sufficiently high to prevent a bottleneck in CcO turnover. Further, in the work cited, the proton channel was represented by a single protonatable group that was in the fast equilibrium with the bulk at the apparent protonation rate of 5 × 10¹¹ M¹s¹, which is an order of magnitude higher than that expected for a diffusion-controlled protonation from the bulk. From Fig. 2, it is seen that, at the lowest diffusion constant, the flux is 10⁵ s¹, which corresponds to the apparent bimolecular protonation rate from the bulk of 10¹² M¹s¹ at pH = 7. Thus, the results shown in Fig. 2 are consistent with our previous findings.

View larger version (13K): [in this window] [in a new window]   FIGURE 2   The dependence of the rate of the surface-mediated proton supply to the channel entrance, ks, on the surface diffusion coefficient, Ds, calculated by Eq. 15 with parameters from Table 2 for pure solution.

The high surface rate is due to a relatively high concentration of protons on the surface, their high mobility, and, most importantly, to the reduced dimensionality of diffusion. The latter results in practical independence of the rate of absorption on the actual radius of the channel. In contrast, the bulk diffusion to the channel is limited by the small size of the channel and a relatively low bulk density of protons.

An increased effective radius of the channel, R₀, is equivalent to the effect of a proton-collecting antenna (Gutman and Nachliel, 1997). In the antenna model, special properties of the groups surrounding the entrance to the channel are required (high mobility, overlap of their Coulomb wells, etc.). We find that, in fact, any surface area surrounding the channel that conducts protons can already significantly increase the output of protons through the channel. That is, if the enhancement of the infinite surface is significant, R₀/r₀ 1, a patch of a similar surface with dimensions of L_max < R_c can also serve as an enhancement factor of the output of the channel. In this case, the proton conducting patch of the surface is exactly equivalent to the antenna model of Gutman and Nachliel (1997; see also the Capture Radius subsection). In contrast, when L_max > R_c, only a part of the proton-collecting collecting cluster will work as a collecting antenna, see Fig. 1. The main point here is that no special properties, such as an extremely fast exchange between the groups of the antenna, are in fact required. The two-dimensional nature of the antenna can significantly relax the requirements on the transport and the collecting properties of the groups making up the antenna.

It is well recognized that the effect of the surface can be significant only if the dwell time on the surface is relatively large. In this case, the proton can migrate a long distance on the surface during its dwell time, L_s r₀, and L_s is often tacitly assumed to be the natural length that determines the size of the region on the surface from which protons can migrate to a target. We find, however, that such a region, in fact, can be much larger than L_s, and is equal to L_sb instead when L_sb L_s. Moreover, we find that this is a typical situation. When the exchange between the surface and the bulk is fast (in the sense discussed in Model section and below in the Parameters and Assumptions subsection), the proton can many times desorb and be readsorbed by the surface, staying, on average, close to the surface, and hence make use of the reduced dimensionality of the space in which its migration occurs.

The surface depletion rate and the dwell time

The above result can be qualitatively understood if it is recognized that the actual average time that the proton spends at or near the surface, _l, is not the same as the inverse of the desorption rate, k. Indeed, the proton that desorbs from the surface can be recaptured again by the surface groups. Therefore, the actual time during which a proton will be localized in the vicinity of the surface will be defined by how quickly the proton migrates into the bulk from the surface (we assume that the desorption rate k_off is high, see Eq. 29). A detailed description of such a surface-depletion kinetics is given in Appendix B. The probability of finding a proton on the surface decays with time as t^1/2, and the depletion time _l, apart from the numerical factor of the order of one, is

(24)

where L₀ is given by Eq. 5. This time is directly related to the coupled surface-bulk diffusion of the proton, and is typically much longer than k. For pure unbuffered solutions and the parameters from Table 2, we obtain _l ~ 1 s.

The capture radius

The capture radius R_c describes the maximum distance at which a source and a sink on the surface can exchange protons without full equilibration with the bulk. Thus, if the distance between the donor and acceptor is less than R_c, the coupling is localized.

Eq. 19 and a more general Eq. 22 provide expressions for the probability of capture of a proton that was initially localized on the surface at a distance r from the channel. We find that the capture radius R_c is the larger of two lengths, L_sb and L_s.

Recently, the question of the nature of communication between a source and a sink on the membrane surface has been directly addressed experimentally by studying two connected ion channels (Antonenko and Pohl, 1998). The reported distance of "direct" coupling between the donor and acceptor is of the order of 100 µm, which is in a remarkably good agreement with our estimate of R_c = 17 µm in the Rate subsection above. Similar distances in the micrometer range are measured by other authors (Heberle and Dencher, 1992; Heberle et al., 1994; Alexiev et al., 1994, 1995; Scherrer et al., 1994). Lateral proton movement on a macroscopic scale has been also observed both in pure aqueous solutions and buffered solutions (Gabriel et al., 1994, and references therein). We note that, according to the present theory, R_c is a function not only of the properties of the surface but also those of the buffers (see the Buffers subsection below).

It should be noticed that a distance as long as L_max = 100 µm between the source and the sink requires a very long lifetime even if proton diffusion on the surface is as fast as in the bulk. For D_s = 10⁴ cm²s¹, one obtains L_max²/4D_s = 0.25 s. This long time clearly can not be a simple desorption time _dw (= 4 µs by the estimate given in the Rate subsection). However, it is very close to our estimate of the surface depletion time, _l ~ 1 s, given in the previous subsection. Thus, the long distance of direct and local coupling can be explained by the coupled surface-bulk diffusion considered in this paper.

In contrast to our calculations and the above-cited experimental data, Gutman and coworkers derived from their experimental results that the proton-collecting antenna consisted of a few (typically 2-3) surface groups in a close vicinity of the fluorescein probe (Flu) and its dimension did not exceed 30 Å (Gutman and Nachliel, 1997). This disagreement might be a consequence of the way the experimental data were described. A notable example is the work on cytochrome c oxidase (Marantz et al., 1998). The Flu protonation on the enzyme surface was found to be mediated by one carboxylate (COO) and one histidine (His_near) located within 10 Å from the probe. In addition, two moieties of 15 distant carboxylates and 10 distant histidines were found to affect the dynamics of Flu protonation indirectly, via proton exchange with COO and His_near. Obviously, in this case, the antenna extends far beyond the 10 Å distance and involves many surface groups. Thus, what we describe in terms of the surface diffusion, Gutman and Nachliel describe in terms of shuttling protons between distant and nearby groups. In the work on bacteriorhodopsin (Checover et al., 1997), it is also obtained that the size of the antenna is comparable to the dimension of the molecule. In other experiments by Gutman and Nachliel (1997), where the antenna dimension was found to be small, the contributions of distant groups might not be properly resolved because the kinetic constants for all numerous proton-transfer reactions were extracted from a single kinetic curve, the protonation dynamics of the probe.

Parameters and assumptions

The phenomenological parameters of this theory listed in Table 2 are actually not all independent. For instance, the ratio of _on and k_off for the unbuffered solution is expressed in terms of pK, Eq. 5, which runs typically from 4 to 6.5 (see, e.g., data for cytochrome c [Marantz and Nachliel, 1999], bacteriorhodopsin [Nachliel and Gutman, 1996], and cytochrome c oxidase [Marantz et al., 1998]). The absolute value of k_off is expected to strongly depend on pK (Gutman and Nachliel, 1995), whereas that of _on on the bulk diffusion coefficient D_b, see Eq. 25.

It is recognized that the surface diffusion coefficient can depend on both pK and k_off. The potential field around a binding site on the surface is characterized by two energy barriers. The first one is for the proton displacement in the direction perpendicular to the surface (Teissié, 1996), and the second is for the lateral movement. The first barrier is always associated only with pK, but the second will also depend on the distance between the surface sites and on how much their Coulomb cages overlap (Matthew and Richards, 1982; Gutman and Nachliel, 1990, 1997; Peitzsch et al., 1995). If the sites are close enough, the lateral barrier will be small (due to overlaps of the neighboring wells), and, for some range of values of pK, the surface diffusion coefficient should be independent of pK. At some high pK, however, D_s should be expected to decrease exponentially with the increase of pK, because the barrier for the lateral translation will increase. The understanding of all intricate relations between pK, D_s, k_off, and _on requires a detailed microscopic model of the surface and solving the diffusion equation in the Coulomb potential, similar to the approach developed by Agmon (1988) for bulk reactions, which is beyond the scope of the present paper.

The parameters L_sb and L_s are also related. The rate constant of the diffusion-controlled protonation of a single surface group from the bulk can be written as (Crank, 1990)

(25)

where d is the reaction radius. We disregard other factors due to electrostatic interactions (Gutman and Nachliel, 1995) and adopt the rate expression for the reaction on a half-sphere. Inserting Eqs. 25, 5, and 7 into 6, we obtain

(26)

(27)

Thus, L_sb depends quadratically on L_s, and the ratio L_sb/L_s increases linearly with L_s.

It can be seen from the above relations that, to have an enhancement effect in the transport rate k_s, the dwell time on the surface should be sufficiently large, _dw r₀²/D_s, so that L_s r₀ (see Eqs. 7 and 8). Indeed, the first factor in Eq. 26 is 1 because both the reaction radius and the target size are of the same order or smaller than the separation between the surface groups. Therefore, if L_s  r₀, we obtain L_sb  r₀ as well, and, according to the results of the Rate Constant section, no enhancement effect would be obtained in this case. However, typically, L_s is expected to be large and the condition L_s r₀ to be satisfied.

In Eq. 27, the first factor is ~10³ for the unbuffered solution. For the reported values of D_s/D_b = 10³  10⁰ (Heberle et al., 1994; Scherrer et al., 1994) one has L_sb L_s, i.e., only the fast exchange regime is realized (see the Introduction). Thus, for the unbuffered solutions, we obtain the estimate for L_s using d = 6 Å,

(28)

Under the same conditions L_sb = 0.17  170 µm, and therefore typically we have L_sb L_s.

Our final remark is on the conditions under which the surface and bulk protons are in local equilibrium, resulting in Eq. 4. It is qualitatively clear that it should be the case when the exchange rate between the surface and the bulk is sufficiently high. It is not immediately obvious, however, what this rate should be compared with (to determine if it is indeed sufficiently high). The relevant estimate can be obtained as follows. The formal condition could be directly derived from Eqs. 2 and 3, where the terms describing the surface-bulk exchange kinetics must dominate over the surface diffusion term. The analysis shows that k should be short compared with the characteristic diffusion time,

(29)

Thus, the condition for the fast exchange can be stated as _{dw c}. Using the definitions in Eqs. 7, 8, and 21, we obtain L_sb L_s.

It is interesting to note that, because L_sb is increasing quadratically with L_s, see Eq. 26, the greater the length L_s is, the better condition L_sb L_s is satisfied. The latter is a condition for the local equilibrium between the surface and the bulk protons, which requires a fast exchange with the bulk, i.e., a "short" dwell time. However, with increasing L_s the dwell time of the proton on the surface is increasing instead(!). The seeming contradiction is resolved when it is recognized that, although with increasing L_s the dwell time _dw does increase, the corresponding time scale of the diffusion, _c, with which _dw should be compared, increases even more rapidly due to its quartic dependence on L_s.

Buffers

The present theory is easily generalized to the case when an unsaturated buffer is present in the solution,

(30)

and when the collisional proton exchange between HB and the surface groups is the dominant pathway of the proton movement.

Reaction 30 introduces proton generation/consumption in the bulk (Nunogaki and Kasai, 1988) with a characteristic time,

(31)

where k'_b and k'_d are the proton-binding and dissociation rate constants, respectively. In deriving Eq. 31, we assumed that [H⁺]  const. Here and below, the primed parameters refer to the buffered solutions. Using the parameters from Table 2, we calculate the characteristic diffusion length
We will see in this section that this length is much longer than the capture radius in our problem, hence the exchange reaction 30 is very slow on a relevant time scale and can be neglected. The above estimate for ', Eq. 31, was obtained under the assumption that the proton diffusion is sufficiently fast in comparison with that of the buffer molecules and, therefore, [H⁺]  const. In the oposite limit of slow proton diffusion, the supply of protons to the reaction region will be a limiting factor, and reaction 30 can be neglected as well.

In this situation, the concentration of the protonated buffer molecules, [HB], plays the role of the concentration of free protons in the solution. The proton flux F from the bulk to the surface can be written in a way identical to Eq. 2,

(32)

where n now stands for [HB] and the primed constants related to the collisional proton exchange between the buffer and the membrane groups, which we assume to be the main mechanism of proton supply to the surface. Then the rate constants obey the relation

(33)

With the above substitutions, one can then use all the derived formulas in this paper.

The diffusion coefficient D'_b now describes the diffusion of the buffer molecules in the solution. The parameter L'₀ for the buffered solution, which comes into the expression L'_sb = L'₀(D_s/D'_b) (cf. Eq. 6) can be written as

(34)

where _eq is independent of the buffer. Substituting Eq. 14 into Eq. 34, one finds

(35)

where L₀, as before, relates to the unbuffered solution, Eq. 5.

This situation is different as compared to the pure unbuffered solution, because the ratio [H⁺]_eq/[HB]_eq can be several orders of magnitude smaller than one. Hence, L'_sb, the capture radius R'_c (Eq. 21 with primed parameters), and the effective radius of the channel,

(36)

will be significantly decreased as compared to the pure unbuffered solution. Thus, the buffer can be a major factor that determines whether the proton transfer occurs via the buffer in the bulk solution or through the surface-diffusion mechanism.

To discuss this issue quantitatively, we will write the expressions for L'_sb and L'_s in the form

(37)

(38)

where [B] = [BH] + [B]  [B] is the total concentration of the buffer, and

(39)

(40)

The parameters [B]₀ and Q are useful measures of enhancement of the proton transport. According to the present theory, the enhancement effect in the buffered solutions occurs when L'_sb is large, L'_sb r₀. However, L'_sb is rapidly decreasing with