Soluble fatty acid binding proteins (FABPs) are thought to facilitate exchange of fatty acids between intracellular membranes. Although many FABP variants have been described, they fall into two general classes. “Membrane-active” FABPs exchange fatty acids with membranes during transient collisions with the membrane surface, whereas “membrane-inactive” FABPs do not. We used modeling of fatty acid transport between two planar membranes to examine the hypothesis that these two classes catalyze different steps in intracellular fatty acid transport. In the absence of FABP, the steady-state flux of fatty acid from the donor to the acceptor membrane depends on membrane separation distance (d) approaching a maximum value (J max) as d approaches zero. J max is one-half the rate of dissociation of fatty acid from the donor membrane, indicating that newly dissociated fatty acid has a 50% chance of successfully reaching the acceptor membrane before rebinding to the donor membrane. For larger membrane separations, successful transfer becomes less likely as diffusional concentration gradients develop. The mean diffusional excursion of the fatty acid into the water phase (dm) defines this transition. For d≪dm, dissociation from the membrane is rate limiting, whereas for d≫dm, aqueous diffusion is rate limiting. All forms of FABP increase dmby reducing the rate of rebinding to the donor membrane, thus maintaining J max over larger membrane separations. Membrane-active FABPs further increaseJ max by catalyzing the rate of dissociation of fatty acids from the donor membrane, although frequent membrane interactions would be expected to reduce their diffusional mobility through a membrane-rich cytoplasm. Individual FABPs may have evolved to match the membrane separations and densities found in specific cell lines.
- lipid binding proteins
- rate constants
- kinetic models
most unesterified fatty acid in the cytoplasm is bound to cellular membranes (20, 34). Fatty acids enter and exit cells through the plasma membrane and must bind to membranes before they can be metabolized. Thus intracellular fatty acid transport may be viewed as transfer of the fatty acid from one membrane to another (e.g., from a “donor” membrane such as the plasma membrane to an “acceptor membrane” such as the outer mitochondrial membrane). This process may occur in a single step or may involve binding to intermediate membranes along the way.
Long-chain fatty acids are minimally soluble in water (36) and have a strong affinity for membranes (20). These properties restrict their movement through the cytoplasm, which is comprised predominantly of water and is typically rich in membranes. For example, liver cells contain >10 m2 of membranes per cubic centimeter of volume (4), sufficient to bind the vast majority of unesterified fatty acid molecules in the cytoplasm (20). Yet, hepatocytes efficiently transfer fatty acids from the plasma membrane to intracellular sites of metabolism (16, 20).
Soluble fatty acid binding proteins (FABPs) are present within most nucleated cells, where they are thought to mediate intracellular fatty acid transport (9, 30, 32, 37, 42). To date, 14 members of the FABP gene family have been identified (30). They may be divided into two general classes based on their ability to interact with membranes. Many forms of FABP (e.g., from heart, adipocyte, and intestine) transfer fatty acids to and from membranes during brief collisional encounters (7, 10, 12, 13, 31). This process involves an interaction between a positively charged region of the binding protein and negative charges on the membrane surface, resulting in a conformational change that permits direct transfer of fatty acid between the protein binding site and the membrane (5, 7, 10, 11,30). We will refer to this FABP class as being membrane active. The second class of FABP molecules lacks the ability to interact with membranes and is therefore able to bind only fatty acid monomers that are already in aqueous solution. The best studied example of this class is liver FABP (12, 31). We refer to this class as being membrane inactive.
The precise function(s) of FABPs remain poorly defined. More than 20 years ago, Tipping and Ketterer (32) proposed that soluble binding proteins may stimulate cytoplasmic transport of their ligands by increasing their aqueous solubility. FABP has been shown to stimulate intracellular fatty acid mobility in cultured cells (18, 20, 22, 23), perfused rat liver (16,19), and artificial cytoplasm (29, 38), whereas knock out of the gene for heart FABP was found to greatly impair cardiac fatty acid metabolism (3). Weisiger and co-workers (15, 16, 20) have shown that the rate of fatty acid diffusion within liver cells is directly proportional to the concentration of liver FABP, and they have proposed that soluble binding proteins act as an aqueous carrier system that reduces binding of fatty acids and other molecules to immobile cytoplasmic membranes (20, 37). On the other hand, Zucker and co-workers (41, 43) found a decreased rate of fatty acid transfer between vesicles in the presence of a membrane-inactive binding protein (liver FABP), although a membrane-active binding protein (intestinal FABP) stimulated transfer under similar conditions (41). Certain tissues such as intestinal mucosa contain both membrane-active and -inactive forms of FABP (31). Collectively, these findings suggest that the two classes of binding proteins may have distinct functions.
To address this possibility, we modeled the transfer of long-chain fatty acids between two membranes as a function of membrane separation and binding protein concentration. Our results indicate that the steady-state transfer rate is determined by two sequential steps: dissociation of the fatty acid from the donor membrane and diffusion of the fatty acid across the aqueous layer separating the membranes. For membrane separation distances less than the mean diffusional excursion of the fatty acid into the water layer (dm), dissociation of fatty acid from the donor membrane is rate determining. Under these conditions, membrane-active binding proteins stimulate the steady-state flux, whereas membrane-inactive binding proteins have no effect. In contrast, for membrane separations much greater than dm, membrane-inactive binding proteins are more effective in stimulating the flux than membrane-active ones. Thus membrane-active binding proteins appear specialized to stimulate fatty acid transport across shorter membrane separations, whereas membrane-inactive binding proteins appear to facilitate diffusion over longer distances.
The model consists of two semi-infinite planar membranes separated by an aqueous solution containing soluble binding protein (Fig.1). We assume rapid equilibrium between the protein-bound and unbound forms of the aqueous fatty acid and that the protein is in molar excess to the fatty acid such that binding sites are never significantly saturated. We further assume that the fatty acid concentration in the donor membrane remains constant (such as might be found in the plasma membrane of a liver cell during fatty acid uptake), whereas that in the acceptor membrane equals zero (such as might exist in the presence of rapid metabolism). In initial analyses, we also assume that the binding protein is not membrane active (although we later consider the effects of membrane-active forms). Thus soluble binding protein with concentration P and affinityK a binds to unbound fatty acid while it is in the aqueous phase but does not directly interact with membrane-bound fatty acid. If a difference exists between the fatty acid concentrations in the two membranes (ΔC), then net transfer of fatty acid from the membrane with the higher concentration (Ld) to the membrane with the lower concentration will occur by aqueous diffusion of both the unbound and bound forms at a rate determined by their respective diffusion constants (Dl and Dp). Intermembrane exchange of aqueous fatty acid is governed by the unidirectional permeabilities of the membrane-water interface [for dissociation (Pmw)] and the water-membrane interface [for binding (Pwm)]. Symbols are further defined in Table 1.
Implementation of model.
We used two different approaches to calculate fatty acid fluxes between the membranes. First, we derived the governing equation that applies to all membrane-inactive binding proteins and all ligands (see ). We then used this equation to explore the dependence of the flux on binding protein concentration and membrane separation. This approach has the advantage of simplicity and general applicability. Because it assumes steady state, however, it cannot be applied to situations in which steady-state diffusion gradients have not yet become established. Most current methods for measuring intracellular transport are not steady-state methods. Therefore, we also modeled transport using numerical simulation, a method that does not assume steady state.
Numerical simulation by finite differences has long been used to predict the time-dependent behavior of complex systems (25) and has recently been extended to understanding intracellular dynamics (8, 24). Assumptions were identical to those for our analytical model outlined above except that steady state was not assumed. The water layer separating the two membranes was divided into a large number of sublayers (typically 32) within which the bound and unbound fatty acid concentrations were assumed to be uniform (i.e., each sublayer was treated as a compartment). Exchange of protein-bound and unbound fatty acid between adjacent sublayers was governed by Fick's law of diffusion (6), whereas exchange with the membranes was governed by the permeability of the membrane water interface for movement of unbound fatty acid into (Pwm) and out of (Pmw) the membrane. At the start of the simulation, the fatty acid was considered to be in the donor membrane only. For each time increment (Δt), a new concentration of bound and unbound fatty acid in each sublayer was calculated by adjusting the existing concentrations of bound and unbound fatty acid for the amount that had diffused into and out of that sublayer during the time interval. For the water layers in contact with the donor or acceptor membrane, the unbound fatty acid concentration was further adjusted for fatty acid dissociating from and rebinding to the membrane. To avoid integration errors, the Δt for each calculation cycle was kept small enough that the concentration adjustments were always minor compared with the total concentration present. Similarly, the number of sublayers used was large enough that the concentration differences between adjacent sublayers were always a small fraction of the total. A computer program to perform these simulations using Microsoft Excel (Microsoft, Redland, WA) is available from the corresponding author.
Selection of model parameters.
Whenever possible, results are presented in general terms that apply to all binding proteins and ligands. Thus binding protein concentrations are expressed relative to the concentration that binds 50% of the total aqueous amphipath [i.e., as a ratio to the affinity constant (K d)], and membrane separations are expressed relative to the mean diffusional excursion of the unbound molecule into the water layer in the absence of binding protein, dm, as defined in the . Specific parameter values were required only for the numerical simulations (Table 1). Membrane permeabilities were obtained from earlier in vitro studies of fatty acid diffusion to and from a water-lipid interface (39, 40), diffusion coefficients for FABP and fatty acid were derived from published estimates (40), and binding protein concentrations were assumed to be 10,000 times K d [based on a typical concentration of cytosolic FABP of 0.1 mM (2) and a K d of 10−8 M (26)].
Validation of the steady-state model.
The equation governing intermembrane fatty acid transfer in the presence and absence of membrane-inactive binding proteins was derived from first principles as detailed in the . This general solution predicts the steady-state transport flux between the membranes as a function of distance, diffusion constants for bound and unbound fatty acid, membrane permeabilities, and the concentration and affinity of the binding protein (Eq. EA10 ). To test the validity of this solution, we compared the predicted flux with that predicted by the numerical model. The predictions of the two models agreed within 0.1% for all conditions tested. The mathematical solution will be presented later after certain model predictions have been validated and their implications considered. Our intent is to develop an intuitive understanding of the transport process before focusing on the specific equations.
Effect of fatty acid concentration.
We refer to the membrane with the larger fatty acid concentration as the “donor” membrane, whereas that with the lesser concentration is the “acceptor” membrane. As expected, the steady-state flux predicted by both models was proportional to the difference between the fatty acid concentrations in the donor and acceptor membranes, ΔC, and was directed into the membrane with the lower concentration (seeEq. EA10 ).
Effect of distance on the flux in the absence of binding proteins.
For any given value of ΔC, the net flux from the donor to the acceptor membrane decreases as the membrane separation increases (Fig.2 A). Note that the flux approaches a maximum value (J max) as the membrane separation approaches zero. Our analyses further show thatJ max is unaffected by changes in the aqueous diffusion constant of the fatty acid but is proportional to the rate of dissociation of the fatty acid from the membrane (see ). As the membrane separation increases, diffusion gradients begin to form (Fig. 2 B) and the flux begins to decline (Fig. 2 A), eventually becoming proportional to the aqueous diffusion constant of the fatty acid and inversely proportional to the membrane separation. Thus dissociation of fatty acid from the membrane is the rate-limiting transport step for smaller membrane separations, whereas diffusion of unbound fatty acid across the water layer is the rate-limiting transport step for larger separations. An important distance is the membrane separation for which dissociation and diffusion are equally limiting. This distance, dm, may be thought of as the mean distance that unbound fatty acid is able to randomly diffuse into the water layer before it rebinds to the donor membrane (i.e., the mean diffusional excursion). Although this distance is unpredictable for any given fatty acid molecule, a mean value for dm can be determined by considering a large population of molecules, as shown in the (Eq. EA14 ).
Effect soluble binding protein on fatty acid flux.
We next investigated the effect of a membrane-inactive binding protein dissolved in the aqueous phase between the membranes. As shown in Fig.3, binding proteins allowed the flux to remain at J max for much larger membrane separations than in the absence of binding protein. For example, at the highest binding protein concentration examined (10,000 times theK d), the flux remained atJ max for a membrane separation ∼4 log orders greater than in the absence of binding protein. Equation EA10 provides the mathematical basis for this observation. These findings indicate that under steady-state conditions, binding proteins can extend the distance over which fatty acids are able to diffuse without formation of substantial concentration gradients.
Effect of distance on protein-stimulated flux.
As shown in Fig. 4, membrane-inactive binding proteins stimulate the diffusional flux only for membrane separations that are larger than dm (bottomcurves). For shorter separations (d≪dm), the fatty acid flux is independent of binding protein concentration (topcurve) and equal to J max. The largest stimulation of the flux occurred for the largest membrane separation tested (10,000 times dm) and was ∼2,000-fold. From this, we conclude that an important limitation of membrane-inactive binding proteins is that they can only stimulate the flux toJ max, after which further increases in binding protein concentration have no effect. They cannot increaseJ max because they have no effect on the rate of dissociation of fatty acid from the donor membrane.
We were unable to directly model the effects of membrane-active binding proteins in this study. To properly address this issue, it is necessary to know the affinity and duration of the membrane-protein interaction and how this interaction affects the rate of transfer of fatty acids to and from the membrane. These data are not yet available. In thediscussion, we propose that membrane-active binding proteins may catalyze both limiting steps in intermembrane transfer by not only increasing dm, but also by increasing the rate of dissociation of fatty acids from the donor membrane.
Validation of numerical model.
A numerical model was used to assess the effects of binding proteins on the transport flux during the pre-steady state (where the above outlined mathematical model does not apply). As previously noted, steady-state fluxes predicted by the numerical model agreed with those predicted by the analytical model under comparable conditions to within 0.1%. The numerical model was further tested according to three criteria of a robust model: steady-state results should be independent of starting conditions, the specific value for Δt, and the number of sublayers (n). The model passed these tests. Allowing the simulations to run for double the normal time did not result in any detectible change in the steady-state flux. Doubling the number of sublayers or dividing Δt by two had no effect on the fluxes or gradients predicted by the model, indicating that we had chosen sufficiently small values for these parameters. These findings increase our confidence that the numerical model is valid.
Effect of time on stimulation by binding proteins.
To determine the effect of time on protein-mediated fatty acid diffusion, we used the numerical model to calculate the flux into the acceptor membrane as a function of time, binding protein concentration, and membrane separation using parameter values outlined in Table 1. These values were chosen to approximate conditions for fatty acid transport through liver cytoplasm (see Table 1, legend). At time 0, fatty acid is considered to be solely within the donor membrane. Subsequent movement of fatty acid across the aqueous phase and into the acceptor membrane is determined by its rate of dissociation from the membrane, diffusion through the water phase, and binding to the acceptor membrane. The results are shown in Fig.5.
For each membrane separation studied, the flux into the acceptor membrane is initially smaller with binding protein (solid lines) but eventually rises to equal or exceed the flux without binding protein (broken lines). It is noteworthy that the flux starts at zero and rises to steady-state levels only after a time lag. The time lag is substantially longer for large membrane separations than for shorter ones, reflecting the greater time required to establish diffusional gradients. The time lag is also substantially larger in the presence of binding protein than in its absence, reflecting the additional time required to load the binding protein with fatty acid and the slower diffusion of protein-bound fatty acid. Thus binding proteins actually slow the delivery of fatty acid to the acceptor membrane during the pre-steady state. It follows that if the flux into the acceptor membrane is measured during this lag period, binding proteins will slow intermembrane ligand transfer. In contrast, binding proteins are able to stimulate ligand flux once steady state has become established.
Membrane-active binding proteins would have shorter lag periods than shown in Fig. 5 if faster dissociation of the fatty acid from the membrane results in more rapid loading of the FABP. If so, it might explain why a membrane-active FABP was able to catalyze intermembrane transfer under conditions in which a membrane-inactive FABP was ineffective (41).
Solution for steady-state conditions.
We now consider the mathematical basis for the steady-state results presented earlier. The full derivation of these equations is provided in the . From Eq. EA15 , the net flux from the donor to the acceptor membrane at steady state is described by Equation 1where d is the distance separating the two membranes and d′m is the mean diffusional excursion of the fatty acid into the aqueous solution before it rebinds to the donor membrane. Notice that for sufficiently small values of d, the ratio d/ d′m becomes insignificant, and this equation simplifies to J = J max. Thus this equation explains why J approachesJ max as d approaches zero. The prime has been added to dm to indicate that it may have been increased by binding proteins. In contrast, dm (which was used to calibrate the membrane separations in Figs. 2-5) applies to the absence of binding proteins only.
From Eq. EA12 , J max is described by the expression Equation 2where Ld is the concentration of ligand in the donor membrane, and Pmw is the permeability of the membrane-water interface (which determines the rate of dissociation of the fatty acid from the membrane). Thus J max is equal to one-half the rate of dissociation of the fatty acid from the donor membrane: when the flux equals J max, dissociation from the membrane is rate limiting.
On the basis of Eq. EA13 , d′m may be defined as Equation 3Note that in the absence of binding protein, the right-most term equals zero, and this equation simplifies to dm = 2Dl/Pwm, which reflects the mean diffusional excursion of the fatty acid in the absence of binding protein. The right-most term in Eq. 3 is the protein-dependent portion and represents the additional mean diffusional excursion due to protein binding.
Preliminary model extension to include intermediate membranes.
Our model greatly simplifies a complex process. In particular, fatty acids may bind to one or more intermediate membranes before reaching the acceptor membrane. Surprisingly, however, binding to intermediate membranes does not affect our conclusions if we make certain assumptions. Consider what happens when we add any number of intermediate membranes to the model in Fig. 1. The intermediate membranes must not metabolize the fatty acid themselves, interfere with diffusion (e.g., by increasing the tortuosity of the path), or contribute to the diffusional flux either through lateral diffusion of bound fatty acid or their own movement. Under these conditions, we can demonstrate that the starting assumptions in the are unaffected and all resulting equations therefore remain valid.
The rationale for this conclusion is as follows. The intermediate membranes are neither a source or a sink for the fatty acids. Because they are also immobile, they will equilibrate with the local concentration of fatty acid for every value of x from 0 to d. Once they have equilibrated, they have no effect on the local fatty acid concentration because no net binding or dissociation of fatty acid occurs. Because the intermediate membranes affect neither the fatty acid concentrations nor the diffusional path, they have no effect on the diffusional flux.
On the other hand, intermediate membranes should add to the time lags in Fig. 5. In addition to loading the dissolved binding protein, the fatty acid dissociating from the donor membrane must also load the intermediate membranes before steady state can be achieved. This will prolong the time lag before steady state is established.
To gain insight into the functions of soluble binding proteins, we modeled the transfer of fatty acids between two planar membranes. We focused on the functions of membrane-inactive binding proteins (i.e., binding proteins that interact with soluble fatty acids but do not interact directly with membrane-bound fatty acids) because most prior studies of the transport functions of FABP have used membrane-inactive forms. Our data suggest that under steady-state conditions, membrane-inactive binding proteins can greatly increase fatty acid fluxes between the membranes but only when the membrane separation is greater than the critical distance dm. For membrane separations less than this value, the flux is already maximal and is determined by the rate of dissociation of the fatty acid from the donor membrane. The only way to increase the flux aboveJ max would be to increase the rate of dissociation of the fatty acid from the membrane.
Storch and Thumser (30) have shown that membrane-active forms of FABP (e.g., intestinal-, heart-, and adipocyte-FABP) transfer fatty acids to acceptor membranes during brief collisional interactions with the bilayer. Stimulation of dissociation of fatty acids from donor membranes was also demonstrated (31). According toEq. EA12 , however, any increase in the rate of dissociation (i.e., in Pmw) will also increaseJ max. We therefore propose that a major function of membrane-active binding proteins is to catalyze dissociation of fatty acids from membranes during intermembrane transport, thus increasing J max. Although the rate of dissociation of fatty acids from biological membranes is not known, the half-time for dissociation of long-chain fatty acids from large phospholipid-cholesterol vesicles is in the range of 1 s (14). When the donor and acceptor membranes are closely spaced, this may be slow enough to limit the flux between the membranes.
Viewed in this context, the functions of binding proteins come into sharper focus. Membrane-active binding proteins are most effective in catalyzing the intermembrane flux when the membrane separation is sufficiently small (d≪dm) that the transport rate is limited by the rate of dissociation of the fatty acid from the donor membrane. Thus they would be expected in cells with large fatty acid fluxes across short membrane separations. For example, the cardiac myocyte typically experiences large fatty acids fluxes across the relatively short distance between the sarcoplasmic reticulum and the outer mitochondrial membrane (33). Not surprisingly, heart FABP is membrane-active (10). In contrast, membrane-inactive binding proteins are most effective when the membrane separation is large enough (d≫dm) that diffusion of the fatty acid across the aqueous layer is rate limiting. Thus they would be expected in cells with larger distances between the donor and acceptor membranes (e.g., liver-FABP in hepatocytes).
This simple interpretation ignores the fact that membrane-active binding proteins seem able to perform all of the functions of their membrane-inactive cousins. Not only can they catalyze dissociation of the fatty acid from the membrane, but they can also increase the concentration of fatty acid in the soluble (i.e., diffusible) phase just as membrane-inactive binding proteins do. Given this, why do cells need membrane-inactive binding proteins at all? The simplest answer is that, in most cases, they don't. Only one form of FABP has been clearly shown to be membrane inactive (liver FABP), and this form has been found only in liver and intestinal mucosa. Most cells appear to transport fatty acids efficiently without a membrane-inactive binding protein.
However, the tendency of membrane-active binding proteins to interact with membranes could slow diffusion under some circumstances. For example, if the cell cytoplasm is rich in membranes, a significant fraction of a membrane-active FABP might be immobilized by membrane interactions at any given time and thus unable to contribute to the diffusional flux. Under these circumstances, membrane-active forms of FABP would be less effective than membrane-inactive forms. This might explain why liver and intestinal epithelial cells both contain membrane-inactive FABPs, because both contain an extensive membranous endoplasmic reticulum. Studies comparing rates of diffusion of membrane-active and -inactive forms of FABP within membrane-rich cells are needed to test this hypothesis.
Weisiger and co-workers (20, 37) previously attributed the ability of binding proteins to catalyze intracellular transport of their ligands to an increase in the fraction of fatty acids in the soluble (and therefore mobile) pool relative to that in the less mobile membrane-bound pool. Although this interpretation remains valid, it is less complete than the current view because it does not consider the rate of dissociation of the fatty acid from the membranes. Other early models of binding protein functions share this limitation (21,32, 34). We are aware of only two models that explicitly include the rate of dissociation of fatty acids from membranes, and each has significant restrictions. Bass and Pond (1) developed a detailed analytical model of binding protein function that included the rates of fatty acid dissociation from the membrane and binding protein. However, they applied it to transfer of fatty acid to and from a single membrane only rather than between membranes (39). Vork and co-workers (35) used a numerical model similar to that used here to show that membrane-active binding proteins were more effective than membrane-inactive ones at stimulating fatty acid transfer between two membranes. Because this study used only numerical methods, it is difficult to generalize these results to conditions other than those studied.
The current study introduces the concept ofJ max, which is equal to one-half the rate of dissociation of the fatty acid from the donor membrane. The reason this rate must be divided by two follows from the fact that diffusion is a random process. Once a fatty acid molecule has dissociated from the membrane, it is equally likely to move randomly toward the donor membrane (resulting in rebinding) or to move further away from the membrane. If an acceptor membrane is sufficiently close, it can capture, at most, the half of the fatty acid molecules that moves away from the donor membrane. It cannot capture the half that randomly moves back onto the donor membrane no matter how small the value of d. On the other hand, if the acceptor membrane is sufficiently distant, the random path followed by the unbound fatty acid molecule is more likely to encounter the donor membrane before it reaches the acceptor membrane, and the probability of transfer is <50%.
The critical distance, dm, determines how close the membranes must be for the flux to reach half ofJ max. Thus dm is a statistical measure of the mean distance that the fatty acid molecule can penetrate into the aqueous layer by random diffusion before it rebinds to the donor membrane, a distance that we call the mean diffusional excursion. Binding proteins increase the excursion by decreasing the rate of rebinding of the fatty acid to the donor membrane much more than they reduce the rate of diffusion [dm is determined by the ratio of these rate constants (see Eq. EA11 )]. We refer to dm in the presence of binding proteins as d′m. This distinction is needed solely because we used dm to calibrate the membrane separations in the figures. The significance remains the same. Transport is dissociation limited for d≪d′m, and diffusion limited for d≫d′m.
It is noteworthy that the value of d′m depends not only on the binding protein concentration but also on its affinity and diffusion constant (see Eq. 3 ). Smaller proteins have greater diffusion constants than large ones. Cytosolic fatty acid binding proteins, which typically have molecular masses around 14 kDa, are among the smallest proteins with stable structures and thus appear to have been optimized for rapid diffusion. The binding affinity of these proteins for long-chain fatty acids, ∼108M−1 (26), is also quite large. These data suggest that soluble fatty acid binding proteins are very well suited to stimulating cytoplasmic diffusion of fatty acids.
It is not possible to calculate a numerical value for dmunless the value of Pwm (which determines the probability of the fatty acid rebinding to the donor membrane) is known. From the value for Pwm, measured for palmitate using an oil-water interface (39), Eq. EA14 predicts a value for dm of ∼0.71 μm. If this value applies to biological membranes, it would suggest that diffusion of fatty acids across living cells (typical diameter 5–20 μm) would benefit from the presence of membrane-inactive binding proteins, whereas transfer between closely adjacent membranes would not benefit unless the binding protein were membrane active. Studies measuring the value of Pwm for biological membranes are needed to provide more precise estimates for dm.
We defined J max using a zero concentration of fatty acid in the acceptor membrane because it simplifies intuitive understanding of this constant. However, it is not necessary to make this assumption. If La is not zero, thenJ max is simply half of the net rate of fatty acid dissociation (e.g., the rate of dissociation of fatty acid from the donor membrane minus the rate of dissociation from the acceptor membrane; see Eq. EA11 ). Thus dissociation can be rate limiting even when the difference between the fatty acid concentrations in the donor and acceptor membranes is small.
The results of our studies may also explain apparently paradoxical findings regarding the effect of soluble binding proteins on fatty acid fluxes. A number of investigators have employed the technique of fluorescence recovery after photobleaching (FRAP) to examine the influence of soluble binding proteins on the fatty acid flux. These studies have consistently shown that the membrane-inactive binding proteins (e.g., bovine serum albumin and liver FABP) enhance the rate of fatty acid diffusion in model cytoplasm (17, 38) as well as in permeabilized (17) and intact cells (15,18, 23). In contrast, analyses of the effect of these same binding proteins on fatty acid diffusion between membrane vesicles have demonstrated a strong inverse correlation between binding protein concentration and transfer rate (41). On the basis of the modeling data presented in Fig. 5, it appears likely that these divergent results reflect differences in the conditions used for these experiments. In FRAP methods, the donor membranes outside the bleach site are already in equilibrium with protein-bound fatty acid at the start of the measurement, so the lag phase is relatively short. In contrast, in the intermembrane transfer studies, the fatty acid starts on the donor membrane only. It is likely that the reduced rates of transport observed in the latter studies reflect the time lag required to load the protein, causing most data to be gathered during the lag phase.
This study is an example of the growing field of simulating complex biological processes “in silico” (8, 24). Whereas this approach offers great experimental freedom, the degree to which our results can be extrapolated to physiological conditions remains to be determined. Cytoplasmic transport is clearly more complicated than depicted by this model. For example, cytoplasmic membranes are not planar, and transport of fatty acids from one location to another within a cell may involve multiple cycles of membrane binding and dissociation. Additional studies will be needed to address the importance of geometry. Whereas the dissociation of fatty acids from different forms of FABP seems fast enough to justify our assumption of binding equilibrium in the bulk cytoplasm (27, 28), slow binding equilibrium near membranes might limit the ability of membrane-inactive binding proteins to catalyze diffusional fluxes (1, 39). Nevertheless, our results seem to fit well with prior observations of cytoplasmic fatty acid transport, and the general insights gained using this simple model seem likely to apply to more complex systems.
In summary, different forms of FABP may have evolved to maximize intracellular transport according to the specific conditions found within each cell. Potentially important determinants include the mean distance across which diffusion must occur and the density of cytoplasmic membranes present. Cells with high membrane densities and longer intracellular transport distances would tend to use membrane-inactive forms to catalyze diffusion, whereas cells with fewer membranes and shorter transport requirements would tend to use membrane-active forms to catalyze dissociation. Cells with both requirements would benefit from the presence of both types of FABP.
This study was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-32898 (to R. A. Weisiger) and DK-51679 (to S. D. Zucker) and by Liver Center Grant DK-26743.
First published September 21, 2001;10.1152/ajpgi.00238.2001
Address for reprint requests and other correspondence: R. A. Weisiger, Div. of Gastroenterology and the Liver Center, 357 Sciences Bldg., Univ. of California, San Francisco, San Francisco, CA 94143–0538 (E-mail:).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
- Copyright © 2002 the American Physiological Society
Our intent is to derive an exact solution for the transport flux from the donor to acceptor membrane for the model shown in Fig. 1. We make three assumptions to simplify our task. First, we assume binding equilibrium between protein-bound and unbound fatty acid. Second, we assume that the binding sites on the protein are never significantly saturated with fatty acid. Third, we assume that the binding protein interacts only with aqueous fatty acid (i.e., is membrane inactive). The main text further assumes that the fatty acid concentration in the acceptor membrane is zero. However, we avoid that assumption in the to obtain a more general solution.
The net steady-state flux (J) from the membrane with the higher concentration (donor) to the membrane with the lower concentration (acceptor) may then be defined by three simultaneous equations, which must be equivalent at steady state:
1) The difference between the rate of fatty acid dissociation from and rebinding to the donor membrane can be expressed as Equation A1where Ld is the fatty acid concentration in the donor membrane, Pmw is the permeability of the membrane-water interface that controls the rate of dissociation of the fatty acid, Lui is the concentration of unbound fatty acid in the water layer closest to the donor membrane, and Pwmis the permeability of the water-membrane interface for rebinding of the fatty acid.
2) The difference between the rate of fatty acid dissociation from and rebinding to the acceptor membrane is described by Equation A2where Luf is the unbound fatty acid concentration in the aqueous layer closest to the acceptor membrane and Lais the fatty acid concentration in the acceptor membrane.
3) The sum of the diffusional fluxes of protein-bound and unbound fatty acid across the aqueous layer is determined by Fick's law of diffusion (6), which states that the diffusional flux for any dissolved substance is the product of the concentration difference and the diffusion constant divided by the distance over which diffusion occurs Equation A3Here Dl is the diffusion constant of the unbound fatty acid, Dp is the diffusion constant of the protein-bound fatty acid, d is the distance separating the two membranes, and Lbi and Lbf are the bound fatty acid concentrations in the initial and final aqueous layers, respectively.
From the law of mass action, we know that the equilibrium concentration of unbound fatty acid in each sublayer is equal to the total fatty acid concentration (Lu + Lb) divided by (1 + K a P), where P is the binding protein concentration. Thus the unbound fatty acid concentrations in the sublayers adjacent to the donor and acceptor membranes are Equation A4 Equation A5To solve for J, we need to eliminate the unknown quantities Lui, Luf, Lbi, and Lbf from these equations. To do this, we first solveEq. EA1 for Lui, Eq. EA2 for Luf, Eq. EA4 for Lbi, and Eq.EA5 for Lbf. Equation A6 Equation A7 Equation A8 Equation A9After Eq. EA6 is substituted into Eq. EA8 and Eq. EA7 is substituted into Eq. EA9 , the resulting expressions for Lui, Luf,Lbi, and Lbf are substituted into Eq.EA3 . We also replace the membrane concentration difference Ld − La with ΔC. After simplification, we obtain the general solution for the steady-state J as a function of known parameter values Equation A10Note that for sufficiently small membrane separations d, theright-hand term in the denominator in Eq. EA10 becomes very small compared with two and may therefore be ignored. Thus, for very small d, J approaches a maximum value defined by Equation A11If we assume that the concentration in the acceptor membrane is always zero (as in the main text), then Eq. EA11 simplifies to one-half the rate of dissociation of fatty acid from the donor membrane Equation A12A critical value is dm, the membrane separation for which the J is one-half of J max. At this distance, dissociation of fatty acid from the membrane and diffusion of fatty acid across the water layer are equally limiting. The distance dm reflects the mean diffusional excursion of the fatty acid into the water phase (see text). When binding proteins are present, dm is increased and we use the symbol d′m to reflect this fact. The expression for d′m is found by setting the two terms in the denominator of Eq. EA10 equal and solving for d. Equation A13This equation shows how the mean diffusional excursion of the fatty acid increases with the binding protein concentration. The two terms to the right of the equal sign represent the components of d′m that are due to diffusion of unbound and bound fatty acid, respectively. In the absence of binding protein (P = 0), this equation simplifies to Equation A14Substituting Eqs. EA11 and EA13 intoEq. EA10 gives the fundamental equation governing cytoplasmic transport, which is reproduced as Eq. 1 in the text. Equation A15This completes our derivation.