Michael Grabe* and George Oster - nature.berkeley.edu

329

J. Gen. Physiol.

© The Rockefeller University Press

•

0022-1295/2001/04/329/15 $5.00Volume 117 April 2001 329–343http://www.jgp.org/cgi/content/full/117/4/329

Regulation of Organelle Acidity

✪

Michael Grabe*

and

George Oster

‡§

From the *Department of Physics,

‡

Department of Molecular and Cellular Biology, and

§

College of Natural Resources, University ofCalifornia, Berkeley, Berkeley, California 94720

abstract

Intracellular organelles have characteristic pH ranges that are set and maintained by a balance be-tween ion pumps, leaks, and internal ionic equilibria. Previously, a thermodynamic study by Rybak et al. (Rybak,

S., F. Lanni, and R. Murphy. 1997.

Biophys. J.

73:674–687) identified the key elements involved in pH regulation;however, recent experiments show that cellular compartments are not in thermodynamic equilibrium. We presenthere a nonequilibrium model of lumenal acidification based on the interplay of ion pumps and channels, thephysical properties of the lumenal matrix, and the organelle geometry. The model successfully predicts experi-mentally measured steady-state and transient pH values and membrane potentials. We conclude that morphologi-cal differences among organelles are insufficient to explain the wide range of pHs present in the cell. Using sensi-tivity analysis, we quantified the influence of pH regulatory elements on the dynamics of acidification. We foundthat V-ATPase proton pump and proton leak densities are the two parameters that most strongly influence restingpH. Additionally, we modeled the pH response of the Golgi complex to varying external solutions, and our find-ings suggest that the membrane is permeable to more than one dominant counter ion. From this data, we deter-

mined a Golgi complex proton permeability of 8.1

3

10

2

6

cm/s. Furthermore, we analyzed the early-to-late transi-tion in the endosomal pathway where Na,K-ATPases have been shown to limit acidification by an entire pH unit.Our model supports the role of the Na,K-ATPase in regulating endosomal pH by affecting the membrane poten-tial. However, experimental data can only be reproduced by (1) positing the existence of a hypothetical voltage-gated chloride channel or (2) that newly formed vesicles have especially high potassium concentrations and smallchloride conductance.

key words:

pH regulation • V-ATPase • proton leak • membrane potential

I N T R O D U C T I O N

Intracellular organelles have characteristic lumenalpHs suited to their biochemical function. Most of theorganelles along the endocytic and secretory pathways,as well as lysosomes, maintain acidic interiors throughthe action of a primary electrogenic proton pump, theV-ATPase. From Fig. 1 A we see the diverse range ofpHs present in the cell. It is our goal to understand themechanisms that enable organelles to establish andmaintain these lumenal pHs. In addressing this prob-lem, we hope to elucidate the connection between or-ganelle pH, morphology, and lumenal contents. Wehave constructed a model of acidification based uponspecific membrane ion pumps and leaks and the inter-nal ionic equilibria characteristic of the lumenal matrixmaterial. Using the model, we have explored the mech-anisms of acidification in specific organelles and de-duced some general features of pH regulation.

Proton concentration plays a fundamental role inmany cellular processes. These roles can be quite dif-ferent. For instance, osteoclasts use low pH to dissolve

bone, whereas neural synaptic vesicles utilize the pHgradient across the bilayer to drive the secondary trans-port of neurotransmitters (Moriyama and Futai, 1990;Chatterjee et al., 1992). Interestingly, organelles alongboth the secretory and endocytic pathways experiencea gradient of decreasing pH (Cain et al., 1989; Fuchs etal., 1989; Wu et al., 2000). In endosomes, acidificationis required for the proper sorting of receptors fromligands. An excellent review of these systems can befound in Futai et al. (2000).

Many of the pH regulatory elements appear in all cel-lular organelles. For instance, proton leaks have beenidentified in both the exocytic and secretory pathways,whereas V-ATPases appear to be integral componentsof nearly all organelles (Van Dyke and Belcher, 1994;Kim et al., 1996; Schapiro and Grinstein, 2000; Wu etal., 2000). Endosomes and Golgi complex are perme-able to counter ions such as chloride and potassium,which can affect pH by altering the membrane poten-tial (Van Dyke and Belcher, 1994; Schapiro and Grin-stein, 2000). Although many regulatory elements arepresent in most, if not all, organelles, other elementsappear to be restricted to particular organelles; e.g.,the Na,K-ATPase that limits acidification in the earlypart of the endosomal pathway (Cain et al., 1989; Fuchset al., 1989; Teter et al., 1998). Fig. 1 B lists the major el-

Address correspondence to George Oster, 201 Wellman Hall, Univer-sity of California, Berkeley, Berkeley, CA 94720-3112. Fax: (510) 642-7428; E-mail: [email protected]

✪

The online version of this article contains supplemental material.

330


ements that we have considered in our study. Many ofthe seminal experiments from which this list was com-piled will be discussed as we construct the model; foran alternative discussion see Rybak et al. (1997).

Motivation for our model comes from the large num-ber of uncertainties and interrelated effects involved inpH regulation. It is often difficult to extract useful in-formation from even the most precise experimentswhen the systems involved are as complex as the oneswe have examined. Thus, a model based on the phys-ics of membrane biology that employs realistic de-scriptions of the relevant bioenergetic proteins is use-ful in precisely interpreting experimental data. Sinceour model is kinetic, it is able to explain the entirepH curve measured during acidification experiments.

Thus, in addition to fitting steady-state data, our modelcan fit initial slopes and transients. From these fits, wedetermine the total numbers of pumps present in aparticular experiment. In principle, using this analysisto compare different organelles can generate predic-tions about pump sorting events. While such biochemi-cal information may be present in the experimentaldata, it requires a comprehensive model to quantify it.Sorting events, and hence their quantification, are criti-cal to understanding pH regulation.

M A T E R I A L S A N D M E T H O D S

A Model for Organelle Acidification

The problem of pH regulation largely reduces to regulating theactivity of the vacuolar proton pumps (the V-ATPases), which arethe primary acidifying agent for most cellular organelles. As an or-ganelle acidifies, a proton motive force (pmf)

1

is built up that op-poses further acidification. The transmembrane pmf (

Dm

) acrossan organelle bilayer has two components: (1) the membrane po-tential (

Dc

), and (2) the proton concentration gradient (

D

pH):

(1)

where k

B

is Boltzmann’s constant, T is the absolute temperature,and e the electronic charge. However, the varying concentrationsof all other ions also affect the membrane potential, so that pre-dicting

DC

requires tracking the movement of the dominantcounter ions, the membrane capacitance, as well as the bufferingand Donnan properties of the lumenal matrix.

By their effect on the membrane potential, the proton pump-ing ability of the V-ATPase can be influenced by other ionpumps, such as the Na,K-ATPases as well as chloride and/or po-tassium channels. In addition, proton leaks and can dissipate gra-dients that are built up by the proton pump. Moreover, organelleshape and size are important parameters in determining mem-brane capacitance and, thus, acidification rates. Matters are fur-ther complicated by the osmotically driven movement of water inresponse to changing ion concentrations. To facilitate discussionof the various factors influencing acidification, we shall divideour discussion into three general categories: pumps, channels,and organelle shape and contents.

Ion Pumps

For our purposes, an ion pump can be characterized by its “per-formance surface” giving the average rate of a single pump as afunction of the transmembrane ion gradient and electrical po-tential, denoted J(

D

pH,

DC

). To measure this surface, a compre-hensive series of experiments must be performed. However,since these experiments have not yet been carried out in suffi-cient detail, we shall rely on models for each pump.

We constructed a mechanochemical model of the V-ATPasethat predicts the proton flux as a function of environmental con-ditions (Grabe et al., 2000). By calibrating the model from wholevacuole patch-clamp experiments the proton pumping perfor-mance surface, J

H

(

D

pH,

DC

), was computed (Fig. 2 A).A similar mechanistic molecular model of the Na,K-ATPase

does not yet exist. However, for our purposes, an accurate kineticdescription will suffice. These enzymes generally localize to the

pmf ∆µ ∆ψ 2.3kBT

e----------∆pH,–=≡

Figure 1. (A) A prototypical mammalian cell showing the majororganelles that exhibit distinct lumenal pHs along with typical pHvalues. These compartments tend to be in the secretory or en-docytic pathways. Arrows indicate the progression along thesepathways. (B) Cartoon of an organelle illustrating the principle el-ements influencing organelle pH regulation. (1) V-ATPase activitycan be directly regulated, e.g., by dissociation of V1; (2) passiveproton leaks; (3) buffering by the lumenal polyelectrolyte matrix;(4) chloride channels; (5) potassium channels; (6) sodium chan-nels; and (7) the Na,K-ATPase.

1

Abbreviations used in this paper:

MVB, multivesicular bodies;

pmf, pro-ton motive force; RRC, receptor recycling compartments.

331

Grabe and Oster

plasma membrane of cells where high turnover rates are neededto maintain low cytoplasmic sodium levels. Many kinetic modelsdescribe the pump only under this limited set of environmentalconditions and generally neglect reverse reactions necessary forpreserving detailed balance (Hartmann and Verkman, 1990; Sa-gar and Rakowski, 1994). However, the Na,K-ATPases found in or-ganelles may not operate over a much greater set of conditions.Therefore, we shall use the more general model of Apell andLauger, which obeys microscopic reversibility (Lauger and Apell,

1986). We have combined two extensions of this model that takeinto account most of the known experimental data on the potas-sium and sodium portions of the pump cycle (Heyse et al., 1994;Sokolov et al., 1998). From this combined model, the sodium flux(J

Na

) and potassium flux (J

K

) for an individual ATPase were deter-mined. We have plotted the response surface for the potassiumportion of the flux in Fig. 2 B. Accurate performance surfacessuch as this are sufficient to predict the Na,K-ATPase’s response totransmembrane ionic and potential differences. A description ofboth pump models along with the Matlab™ code used to com-pute them is given in

Online

Supplemental Material

.The performance surfaces characterize the average behavior

of individual pumps. To compute acidification rates, we mustknow the number of active pumps. This can be estimated fromelectron microscopy and biochemical assays (e.g., mAb tagging).However, these estimates are difficult and often inaccurate sinceonly the total number of pumps, not the active fraction, can bededuced from microscopy. Therefore, we shall estimate the num-ber of active pumps in a given organelle by fitting experimentalacidification curves or by matching steady-state pH values.

Before continuing, it is natural to ask how heavily our resultsdepend upon the exact shape of the performance surfaces inFig. 2. To address this, all simulations have been computed withsimpler pump models assuming that the pump flux decreases lin-early with increasing electrochemical gradient. Most of the datawe fit with the nonlinear performance surface can be fit witha linear pump model. However, this necessitates significantchanges in key parameters, such as the number of active pumps.Thus, for the systems examined, the exact dependence of thepumps on environmental conditions is not critical. What is im-portant is that the pumps provide a source of ion flux, and pHregulation results from the interplay of this source with otherregulatory elements to be discussed. Why then use the nonlinearpumping surfaces in Fig. 2? First, there are situations where thepump mechanism is crucial to pH regulation, such as hypera-cidic organelles whose pH is below 2. Second, it is as easy to com-pute the regulation curves using the nonlinear pump model aswith the linear model, and it does not obfuscate the role thatpumps play in the overall scheme.

Channels: Chloride, Potassium, and Proton Leak

Intact bilayers are somewhat permeable to protons, but relativelyimpermeable to other ions. Ion-specific channels allow an or-ganelle to equilibrate specific ions between the lumen and cyto-plasm. Movement of these ions through the channel is driven bythe transmembrane concentration difference and the membranepotential. The simplest model for this movement assumes thatthe ion and the channel do not interact during transport. In thislimit, the permeability coefficient is the only relevant parameterdetermining the dynamics of transport. In the presence of amembrane potential, the diffusion flux of ions can be describedby Eq. 2 (Hille, 1992; Weiss, 1996):

(2)

where i denotes the ionic species (Cl

2

, K

1

, or H

1

), P

i

is the per-meability of the membrane to ion i, S is the surface area of thecompartment, C

i

is the concentration of the ion in the cytoplasm(C) or lumen (L), z

i

is the valance of the ion, and U is the re-duced membrane potential, U

5

C

F/(RT). F, R, and T havetheir usual meanings.

Proton movement across membranes is not well understood.Protons are permeant to intact bilayers, so it is possible that theirtransport is not channel mediated. It may involve a more compli-cated mechanism using water wires or the partitioning of weak

Ji Pi SziU Ci[ ] L Ci[ ] Ce

ziU––( )

1 eziU–

–---------------------------------------------------------- ,⋅ ⋅=

Figure 2. Performance surfaces. (A) The proton pumping rate[H1/s] for a single V-ATPase is plotted as a function of lumenalpH and membrane potential across the bilayer. The bulk cytoplas-mic pH is 7.4, but the proton concentration at the membrane ismodified by a 250-mV surface potential. The free energy of ATPhydrolysis is 21 kBT. In principle, the pumping surface can be mea-sured empirically. However, in the absence of such measurements,we have computed the surface based on the mechanochemicalmodel for the V-ATPase as described in Grabe et al. (2000). Theparameters used in the computations are given in Table S1 (seeOnline Supplemental Material available at http://www.jgp.org/cgi/content/full/117/4/329/DC1). (B) The pumping rate for a sin-gle Na,K-ATPase plotted as a function of lumenal potassium con-centration and membrane potential. Bulk cytoplasmic potassiumand sodium concentrations are maintained constant at 140 and 20mM, respectively. Membrane values of these concentrations arealso modified by a 250-mV surface potential as in A. The free en-ergy of ATP hydrolysis is 21 kBT, and the lumenal sodium is fixed at145 mM. The pumping profile was computed from the compositemodel found in Online Supplemental Material. See Table S2 (avail-able at http://www.jgp.org/cgi/content/full/117/4/329/DC1) fora complete list of all parameters.

332


acids into the bilayer (Paula et al., 1996; Hill et al., 1999). Addi-tionally, voltage-gated proton channels exist in the plasma mem-brane of certain cell lines, where they are responsible for extrud-ing acid from the cytoplasm (DeCoursey and Cherny, 2000).Whether such channels are functional in organelles is notknown. If transport is mediated by voltage-gated channels, themathematical description above may be inappropriate for theproton flux. Lacking a concrete description of the passive protonflux, we continue with the simple treatment of Eq. 2. As we shallsee, this model fits experimental data quite well.

Net proton movement across the membrane is a competitionbetween V-ATPase pumping and channel-mediated leaking. Ingeneral, high physiological concentrations of counter ions leadto counter ion movement that is much faster than either of thesetwo processes. This follows from Eq. 2 since the flux is propor-tional to the difference in ionic concentrations. Thus, no newtime scale is introduced into the problem by incorporating chan-nels; the dynamics of acidification will primarily be determinedby the proton movement.

We deduce proton permeabilities from data associated witheach of the experiments examined. The permeability value forchloride, listed in Table I, is taken from measurements on non-voltage-activated channels residing in the plasma membrane. Weshall assume that the permeabilities of the other counter ionshave this same value. The rate of acidification is unaffected byvariations in the permeabilities as long as the permeability doesnot become so small as to place counter ion movement on thesame time scale with proton movement.

Organelle Properties: Buffering Capacity, Donnan Potential, Osmotic Effects, and Shape

The lumenal matrix of an organelle is composed of a polymericnetwork of proteins. These proteins tend to be negativelycharged, and can buffer changes in pH. The negative charge ofthe matrix is balanced by mobile counter ions which generate aDonnan potential. The volume and surface area of the organellealso play an important role in determining concentrations, mem-brane potential, and membrane fluxes. These parameters areamong the most difficult to quantify primarily because of thelarge variations in organelle size and shape. Electron micro-graphs show that even vesicles from purified populations exhibita broad distribution of volumes and surface areas (Belcher et al.,1987). For the organelles studied here, we have tried to find themost current measurements of these two parameters; a list isgiven in Table I. When values were uncertain, these parameterswere varied to determine the best fit to experimental data.

The buffering capacity,

b

(units: mol/pH), is the degree withwhich the lumenal matrix is able to bind protons from solution.Since the matrix composition is quite complex, the buffering ca-pacity can only be determined experimentally by biochemicalanalysis. At a particular lumenal pH, this analysis determines theconstant of proportionality between the change in proton con-centration of the lumen and the change in pH,

D

[H

1

]

5

b ?D

pH. In general, this relation is not linear since

b

depends on thelumenal pH. For pH values near 7, the buffering capacities forthe ER and Golgi complex are found to be approximately con-stant, varying between 6–40 mM/pH unit. Similar experiments

T A B L E I

Parameters of Interest

Description Units Symbol Value Reference

Energy of ATP hydrolysis

D

G

ATP

21 k

B

T Rybak et al., 1997

Cytoplasmic pH* pH

C

7.0–7.5 Roos and Boron, 1981;Wu et al., 2000

Cytoplasmic sodium concentration mM Na

c

1

15 Alberts et al., 1994

Cytoplasmic potassium concentration mM K

c

1


Cytoplasmic chloride concentration mM Cl

c

2


Cytoplasmic ATP concentration mM ATP

c

5 Lauger and Apell, 1986

Cytoplasmic ADP concentration mM ADP

c

0.1 Lauger and Apell, 1986

Cytoplasmic phosphate concentration mM P

c

5 Lauger and Apell, 1986

Cytoplasmic calcium concentration M Ca

c2

1

10

2

7

Alberts et al., 1994

Extracellular sodium concentration mM Na

e

1

145 Alberts et al., 1994; Rybak et al., 1997

Extracellular potassium concentration mM Ke

1


Extracellular chloride concentration mM Cl

e

2


Extracellular calcium concentration mM Ca

e2

1

2.5-5 Alberts et al., 1994

Extracellular phosphate concentration mM P

e

1 Vander et al., 1985

Bilayer capacitance

m

F/cm

2

C

0

1 Israelachvili, 1992

Golgi complex buffering capacity mM/pH

b

G

10, 40 Farinas and Verkman, 1999; Wu et al., 2000

ER buffering capacity mM/pH ER 6 Wu et al., 2000

Endosome buffering capacity mM/pH EN 50 Rybak et al., 1997

Chloride permeability cm/s P

Cl

1.2

3

10

2

5

Hartmann and Verkman, 1990

Proton permeability cm/s P

p

4.8–0.67

3

10

2

3

Van Dyke and Belcher, 1994

Golgi complex volume cm

3

V

G

6

3

10

2

12

Ladinsky et al., 1999

Endosome volumes cm

3

V

EN

88–0.7

3

10

2

15


Golgi complex surface area cm

2

S

G

8

3

10

2

6

Ladinsky et al., 1999

Endosome surface area cm

2

S

EN

1–0.06

3

10

2

8


Names and values of relevant physiological parameters.*We occasionally refer to pH

C

5

7.4 as a typical cytoplasmic pH. This value may appear high in relation to older literature (Roos and Boron, 1981), but itis in line with many of the most recent fluorescence experiments (Wu et al., 2000).

333

Grabe and Oster

have been performed on endosomes and lysosomes in which thebuffering capacity did not appear to be constant (see Figure S5;Van Dyke, 1993; Van Dyke and Belcher, 1994). For each simula-tion, the buffering capacity was determined from experimentallymeasured values. This is discussed in more detail in

Online

Supple-mental Material

and the review by Roos and Boron (1981).The membrane potential affects the flow of ions across lipid

membranes and biases the distributions of those ions at steadystate. Electroneutrality requires that no net charge exists in anysmall volume; the membrane potential arises from the micro-scopic deviation from electroneutrality at a lipid boundary. Physi-ological models generally exploit the concept of electroneutral-ity to solve for the membrane potential without detailed informa-tion about the electrical makeup of the cell. This requires thatthe ionic currents crossing the membrane sum to zero at alltimes. This constraint results in a single algebraic equation whoseroot gives the membrane potential. Additionally, Hodgkin-Hux-ley type models relate the electrical activity of the cell to themovement of ions across the membrane by a differential equa-tion for the time dependence of the potential (Hodgkin andHuxley, 1952). Both of these approaches ignore two importantfeatures that strongly influence the membrane potential: (1)fixed lumenal charges, and (2) charged lipid headgroups in thebilayer. To include these elements, a physical model of the mem-brane potential in terms of ionic charge distributions is required.

Poisson-Boltzmann models (more specifically, Gouy-Chapmanmethods) provide one approach for determining the electricalprofile and ion concentrations near the lipid bilayer. To accu-rately determine these profiles all charged solutes must be in-cluded in the calculation (McLaughlin et al., 1981). This is be-yond the scope of the present model, since we track only thedominant counter ion concentrations. Motivated by Rybak et al.(1997), we give an explicit form for the membrane potentialacross the bilayer in terms of the excess charge inside the or-ganelle. We assume that the net charge localizes to the lumenalleaflet, so that we can treat the membrane as a parallel plate ca-pacitor. This is valid since the radius of curvature of organellesurfaces is quite small compared with their thickness. The poten-tial drop across the bilayer is written as Eq. 3:

(3)

where A is the surface area of the membrane, C

0

is the capaci-tance per unit area of the membrane (C

0

?

A is the total capaci-tance of the membrane), V is the volume of the organelle, F isFaraday’s constant, and the numbered terms giving the concen-trations of charged particles are as follows. (1) Sum of ions (chlo-ride, potassium, sodium, and calcium) weighted by their stoichio-metric coefficient n (e.g., n 5 1 for monovalent and 2 for diva-lent ions). (2) Total amount of buffered and free protons in thelumen. b is the buffering capacity. We assume that protons donot contribute to the membrane potential when the lumenal pHis equal to the cytoplasmic pH. When the buffering capacity isnot constant, this term must be integrated to give the total num-ber of lumenal protons contributing to the membrane potential.(3) Molar concentration of all impermeant charges. This termprimarily represents fixed negative protein charges trapped inthe lumen; however, fixed cations (such as calcium), which keepthe lumenal matrix condensed, may reduce the magnitude ofthis term (Verdugo, 1991). Experimental estimates of these con-

∆ψ FC0------ V

A---

ν i cations[ ] i⋅i

∑ ν i anions[ ] i⋅i

∑–

1

β pHC pHL–( )⋅

2

B

3–+

,

⋅ ⋅=

{

centrations vary by .100 mM from organelle to organelle. Someof these values are listed in Table I. In each simulation, this pa-rameter is estimated from resting pH values in the absence ofproton pumping or from best fits as a variable parameter.

Changes in the membrane potential, according to Eq. 3, comeabout as a direct consequence of changes in the lumenal concen-trations of those ions being tracked. The constant charge densityterm (B) sets the zero point of the membrane potential and biasesthe lumenal proton concentration as well as the ionic concentra-tions in term 1. Large values of B have been implicated in theacidic pH that lysosomes maintain in the absence of proton pumps(Moriyama et al., 1992). A recent study using a similar model forthe membrane potential has shown good agreement with experi-mentally predicted membrane potentials (Endresen et al., 2000).

To account for the effects of surface charge, we include a phe-nomenological potential difference between the bulk cytoplasmand the organelle’s outer leaflet, DCC,0, and the bulk lumen andthe inner leaflet, DCL,0 (Hille, 1992). These potentials modify allionic concentrations at these surfaces by a Boltzmann factor:

(4)

where the surface concentrations are denoted by a zero sub-script. Given the typical surface potentials measured for negativephospholipid bilayers bathed in frog Ringer’s solution, this the-ory predicts Ca21 surface concentrations to be 20–104 timeshigher than bulk values (Hille, 1992). These modified concentra-tions are the values that must be used for computing current flowthrough membrane channels and pumps. For example, surfaceconcentrations must replace the bulk values in Eq. 2 to solve forthe instantaneous flow of ions across the membrane. In general,as the lumen is acidified and ionic concentrations change, DCL,0

will also change. We assume that the change in net ionic concen-trations is small enough to ignore this effect.

The lipid compositions of cellular membranes are inherentlyasymmetric, with the cytoplasmic leaflet tending to be more neg-atively charged due to the preponderance of acidic lipids (Roth-man and Lenard, 1977). The surface potential depends stronglyon this composition. Vesicles containing 33% acidic lipid in 100mM monovalent salt concentrations have surface potentialsabout 250 mV (McLaughlin, 1989). The most relevant physio-logical measurements of surface potentials have been performedon the plasma membrane. Cahalan and Hall (1982) observed acytoplasmic surface potential of 230 mV in frog nerve cells.Lacking specific information about the lipid composition of theorganelles studied, we assign reasonable, yet arbitrary, values ofDCC,0 5 250 mV and DCL,0 5 0 mV. We retain these valuesthroughout our study and examine the sensitivity of our modelto these parameters in Table V. For cytoplasmic surface poten-tials in the range of 230 to 250 mV, we expect surface concen-trations of divalent and monovalent cations to be z5–50 timeshigher than bulk concentrations.

Although it is generally accepted that surface charges influ-ence ion concentrations near the membrane, this effect on theoperation of pumps and channels is not well characterized. Thecharges on the transporter itself could affect the local ion con-centration much more than the nearby lipids. Additionally, theV-ATPase, like many transporters, is quite large, and the ionbinding sites may be more than a Debye length away from theclosest membrane lipids. Thus, electric fields generated at themembrane will be screened at the transporter, and ionic concen-trations may not be very different from the bulk solution. Theseissues can be answered only by experiments. For the model trans-

Ci[ ] L,0 Ci[ ] L expz– iF∆ΨL,0

RT-------------------------

⋅=

Ci[ ] C,0 Ci[ ] C expz– iF∆ΨC,0

RT--------------------------

,⋅=

334 Regulation of Organelle Acidity

porters used here, we discuss how changes in surface potential af-fect flux rates in the Online Supplemental Material . There, we showthat when the surface potential is zero, predicted permeabilitycoefficients increase by a factor of 2–3, and proton pumpingrates decrease by less than a factor of 2. Nevertheless, the overalltheme of our discussion remains unchanged.

Experimental measurements of the membrane potential arebetween the bulk cytoplasm and the bulk lumen. Using the modeljust outlined, this total potential difference is given by Eq. 5:

(5)

All reported membrane potentials refer to the total membranepotential. Using the definitions in Eq. 5, DCTotal . 0 will drive an-ions into the lumen and cations out.

Differences in ion concentrations across the membrane giverise to an osmotic pressure that drives water across the mem-brane. This effect will change the total volume of the organelleand may influence the lumenal pH. The flux of water into the lu-men can be written as Eq. 6:

(6)

where FC is the osmolarity of the cytoplasm or external solutionand PW is the permeability. The water permeability of endosomesisolated from rat kidney is PW < 0.052 cm/s (Lencer et al., 1990).For ionic species i, fi is the osmotic coefficient, ai is the solute ac-tivity, and [Ci]L is the lumenal concentration. The osmolarity ofthe cytoplasm is constant with a typical value of FC < 291 mM.We assume that vesicles are initially in osmotic equilibrium. Thisallows us to assign an osmolarity for those lumenal solutes thatare impermeable to the membrane. The osmotic coefficient ofthe charged solutes is taken as 0.73.

Numerical Solutions

With Eqs. 1–6 we can construct a model organelle and write a setof balanced equations that keep track of the movement of ionsacross its membrane. A typical set of equations are as follows:

(7)

The fluxes, JH1, JNa

1, JK1, and the buffer curve, b, are imported

numerical functions from the pump performance surfaces andthe titration data, respectively. As always, brackets denote molarquantities, whereas stand alone symbols represent number ofions. Time varying variables refer to lumenal quantities, all of

Ψ∆ Total ∆Ψ ∆ΨC,0 ∆ΨL,0–( )+=

JW PW Sφi

ai---- Ci[ ] L ΦC–⋅

i∑

,⋅ ⋅=

a( ) Hd 1

td----------- N

H1 JH1 pHL Ψ,( ) P

H1– S U H1[ ] L H1[ ] C e U–⋅–(1 e U––

-----------------------------------------------------------⋅ ⋅ ⋅=

b( ) Kd 1

td---------- 2– NNa, K JNa, K⋅ NaL,K L,Ψ( )⋅ –=

PK1 S U K1[ ] L K1[ ] C e U–⋅–( )

1 e U––----------------------------------------------------------⋅ ⋅

c( ) Nad 1

td-------------- 13 NNa, K JNa, K⋅ NaL,K L,Ψ( )⋅=

d( ) Cld 2

td------------ P

Cl2S U Cl2[ ] L Cl2[ ] C e U+⋅–( )

1 e U+–--------------------------------------------------------------⋅ ⋅=

e( ) Ψ FC0------ V

S---⋅=

Na1[ ] L K1[ ] L Cl2[ ] L β pHL( ) pHC pHL–( )⋅ B–+–+{ }

f( ) pHdtd

----------- 1β pHL( )--------------------–

H1[ ]dtd

---------------⋅=

which are coupled by the membrane potential. The equationswere solved using Berkeley Madonna™ with the Rosenbrock al-gorithm for stiff differential equations (Press, 1997). All simula-tions start with an initial membrane potential of zero, and nega-tive Donnan particles are masked by an equal concentration oflumenal cations. Initial lumenal concentrations of counter ionsdo not affect steady-state values since they are allowed to diffuseuntil their Nernst potential is zero. In the next section we solvethese equations for various experimental configurations.

Online Supplemental Materials

Detailed calculations and further discussion of specific topics areavailable online at http://www.jgp.org/cgi/content/full/117/4/329/DC1. These include all of the following: V-ATPase and Na,K-ATPase pumping profiles, buffering capacity, and surface poten-tials. This site also contains the computer program used to com-pute organelle acidification rates. It is easy to use, and can betailored by the user for particular experimental situations; itrequires the commercial software Berkeley Madonna™. TheMatlab™ code used to compute the pumping profiles in Fig. 2can be downloaded.

R E S U L T S

Using the model outlined above, we examined experi-ments performed on reconstituted organelles and onintact cells. Reconstitution experiments make it possi-ble to measure ATP-dependent acidification of or-ganelles that have been incubated in solutions lackingATP. These experiments give important information onthe kinetics of acidification. However, it is not clear thatthe process of reconstitution does not drastically alterthe biochemical makeup of the organelle. Experimentsperformed on intact cells can also provide kinetic in-formation, and they are much less invasive than recon-stitution experiments. In this section we show that ourmodel is in quantitative agreement with reconstitutionexperiments, and highlight areas of pH regulation thatour model can help elucidate. We also examine data re-corded from intact cells to address several outstandingquestions in pH regulation.

Calibration: The Model Accurately Describes the Acidification of Organelles

It is necessary to confirm the ability of our model to de-scribe acidification before attempting to draw generalconclusions from ambiguous data. We begin by analyz-ing the ATP-dependent acidification of rat kidney endo-somes recorded by Shi et al. (1991). Endosomes were re-constituted in a solution devoid of ATP, effectively turn-ing off the V-ATPase. Next, ATP was reintroduced intothe external solution and the subsequent acidificationof single endosomes was measured. These experimentsform a particularly good benchmark because externalvariables were tightly controlled in the reconstituted me-dium and the acidification measurements were per-formed on single endosomes rather than a population.When modeling a single organelle, model parameters

George Oster

)

George Oster

.

335 Grabe and Oster

such as buffering capacity and pump numbers can be in-terpreted as literal properties of the organelle ratherthan average properties of a population of organelles.

In Fig. 3 (A and B), we modeled the acidification ofan endosome using a constant (Fig. 3 A) and a variable(Fig. 3 B) buffering capacity. The simulation is of aspherical vesicle 163 nm in diameter containing protonpumps and leaks, potassium channels, and trappednegative charges. The experiment (solid line) was per-formed in a solution of 100 mM potassium, and we as-sume that potassium is permeant to the membrane.The diameter, number of proton pumps, and protonpermeability were varied to give the best fit to the data.At time zero, the lumen was acidified from its initial pH

of z7.4 by turning on the ATP-dependent V-ATPase. Inboth simulations, the final membrane potentials werez15 mV, the energetic equivalent of DpH < 0.3. Thisjustifies the authors’ assertion that the membrane po-tential is not appreciable when potassium is free to dif-fuse across the bilayer.

We investigated two parameters that affect the shape ofthe acidification curves: buffering capacity and osmoticfluid flow. In Fig. 3 A, we used a constant lumenal buffer-ing capacity of 40 mM/pH unit, as reported by Shi et al.(1991). The buffering curve used in Fig. 3 B was adaptedfrom endosomal data recorded by Van Dyke and Belcher(1994; see Figure S5 available at http://www.jgp.org/cgi/content/full/117/4/329/DC1). At the same vol-ume, the variable buffering capacity gives a noticeablybetter fit to the data, indicating the importance of buffer-ing in describing pH dynamics. Water channels were alsoincluded in the simulation to allow volume change whilethe surface area remained constant. We assumed that thespherical vesicle was initially in osmotic equilibrium andthat the osmolarity of the external medium was 291 mM,a typical cytoplasmic value. From the shapes in Fig. 3 A,we see that endosomal shrinking occurs as buffered pro-tons expel osmotically active potassium ions. The modelpredicts an 11% decrease in volume, which is in agree-ment with the authors’ estimate of a decrease of less than15%. Interestingly, the change in volume during acidifi-cation has no effect on the shape of the curve in Fig. 3 A:the curves with and without water exchange are virtuallyindistinguishable.

The goodness of the fits, coupled with the constraintsimposed by the experiments, indicate that the modelcan accurately represent the physical situation. Thebuffering capacity plays a dominant role in the dynam-ics of acidification, and our ability to extract reliablenumbers from fits depends upon the fidelity with whichthe buffering curves are known. We predict shapechanges due to osmotic shrinking, but this effect has lit-tle influence over the shape of the endosomal acidifica-tion curves. Finally, the predicted steady-state pmf’s arerelatively small, indicating that the V-ATPase proton

Figure 3. (Top) Modeling the ATP-dependent acidification of asingle endocytic vesicle from rat kidney. The model (dashed) iscompared against the experiments (solid) of Shi et al. (1991). Sim-ulations include the effects of V-ATPases, proton leaks, passive po-tassium channels, Donnan equilibrium, and buffering capacity. Ta-ble II lists all parameters. (A) Acidification with a constant buffer-ing capacity of 40 mM/pH, as reported by Shi et al. (1991). Thevesicle is initially in osmotic equilibrium with an external solutionhaving an osmolarity of 291 mM. The efflux of potassium uponacidification results in a decrease of osmotically active particles;thus, shrinking occurs. Cross-sectional areas of the initial sphericalvesicle and an ellipsoid corresponding to the final volume of thevesicle are drawn to scale. The osmotic activity of the protons andthe potassium ions is given in Table I. The shapes of the acidifica-tion curves with and without water flow are indistinguishable atthis scale.(B) An approximation to the buffering data reported byVan Dyke and Belcher for MVB endosomes was used to modelthe acidification curve in A. This buffering curve can be seen inFigure S5 (available at http://www.jgp.org/cgi/content/full/117/4/329/DC1). The lumenal buffering capacity has a noticeable in-fluence on the models ability to fit the data.

T A B L E I I

Endosome Model Parameters I

Parameter Units Panels A and B

H1 permeability cm/s 4.1 3 1023, 3.4 3 1023

Surface cm2 8.35 3 10210

Volume cm3 2.27 3 10215

No. of V-ATPases 16

External potassium mM 100

External pH 7.5

Donnan particles mM 100

Buffering capacity mM/pH see figure legend

List of parameters used in the simulations in Fig. 3. In column three, leftvalues refer to panel A and right values to panel B.


pump is not working close to thermodynamic equilib-rium, which corresponds to Dpmf < 4.5 pH units(Grabe et al., 2000). This suggests that the steady-statepH in endosomes, lacking Na,K-ATPases, is the resultof a balance between proton pumping and leaking.

The Model Provides a Consistent Theory of pH Gradient and Membrane Potential

A satisfactory theory of pH regulation requires an un-derstanding of both the electrical- and concentration-dependent portions of the pmf across the organelle bi-layer. Here, we model experiments that monitor mem-brane potential in addition to acidification. A similarclass of experiments to those of Shi et al. (1991) in-volves measuring the fluorescence from whole popula-tions of a specific organelle. One acidification curve ismeasured for the entire population, and it is assumedthat this curve represents the dynamics of an averagevesicle in that population. Van Dyke and Belcher(1994) have performed a number of experiments onthree distinct endosomal populations extracted fromnormal rat kidney cells. We modeled their acidificationdata from the receptor recycling compartments (RRC)and multivesicular bodies (MVB) where they report av-erage values for the surface area, volume, and pumpleak rate for each of these populations. In addition,they show that the buffering power of the lumenal ma-trix depends strongly upon the value of the internal pH

(see Online Supplemental Material available at http://www.jgp.org/cgi/content/full/117/4/329/DC1).

Fig. 4 (A and B) shows the acidification of vesicles inthe presence of chloride and potassium channels, fixednegative lumenal proteins, and proton pumps andleaks. For this calculation, we used the buffering curvesfrom Figure S5 (available at http://www.jgp.org/cgi/content/full/117/4/329/DC1); other parameters forthese simulations and the percent deviation from mea-sured values are given in Table III. As in the Verkmanexperiments, large proton permeabilities, on the orderof 1023 cm/s, suggest that the ratio of the leak rate tothe pump rate is the major determinant of steady-statepH. Table III shows that the model parameters arecomparable to the average experimental values. TheRRC parameters are consistently closer than the MVBparameters to the measured values. However, the mostinaccurate MVB parameter (volume) is only a factor of1.7 larger than the reported value.

To investigate the role of counter ions, Van Dyke andBelcher (1994) varied the external chloride concentra-tion and measured the resulting steady-state membranepotential. When the membrane is permeable to chlo-ride, Eq. 2 predicts that changes in the external chlo-ride concentration will affect the steady-state distribu-tion of the ion across the membrane. Subsequently, themembrane potential will change according to Eq. 3.Fig. 4 (C and D) shows that the model describes themembrane potential’s dependence on the external

Figure 4. (Top) Modeling the ATP-dependentacidification of vesicles in rat liver from two differ-ent stages of endocytosis. Model predictions areshown in solid lines and the experimental datapoints were provided by Van Dyke and Belcher(1994). Controls include the following: V-ATP-ases, proton leaks, passive chloride channels, pas-sive potassium channels, Donnan equilibrium,and buffering capacity. See Table III for all param-eters. (A) The acidification of the receptor recy-cling compartment (RRC). The buffering capac-ity is extrapolated from experimental valuesdetermined by Van Dyke and Belcher. (B) Acidifi-cation of the much larger multivesicular bodies(MVB). Fitting the data required a pump densityon the MVB vesicles approximately eight timesgreater than the RRC vesicles. (Bottom) Steady-state membrane potential as a function of exter-nal chloride concentration. The presence of chlo-ride enhances acidification by allowing chlorideto enter the vesicle to reduce the membrane po-tential against which the pump must operate.Both C and D show that increasing the externalchloride concentration is an effective way to re-duce the resting membrane potential. (C) For theRRC population, the dependence of the mem-brane potential on the external chloride concen-

tration matches experimental observations with the same parameters for the simulations in A. (D) A fivefold decrease in the number ofproton pumps relative to B is required to describe the MVB population’s dependence on the external chloride concentration.

337 Grabe and Oster

chloride concentration quite well. As the chloride con-centration is increased, a greater lumenal concentra-tion of chloride counter balances the presence of pro-tons, thus reducing the membrane potential.

One set of parameters described both sets of RRCdata, yet the MVB data required an eightfold decreasein the number of proton pumps to describe the depen-dence of membrane potential on external chlorideconcentration. Whether such variations are to be ex-pected from trial to trial is not clear. Interestingly, pan-els A, C, and D of Fig. 4 have very similar pump den-sities (z50 pumps/mm2) indicating that no activesorting of the proton pump occurs between these pop-ulations. However, the MVB simulation in Fig. 4 B re-quired a pump density of 270 pumps/mm2. More con-sistent fits that agree over more data sets are requiredbefore we can make any claim about sorting events. Fig.4 (C and D) shows that our model quantitatively de-scribes the dependence of the membrane potential onexternal ion concentrations. The ability to fit both themembrane potential and acidification data with thesame set of parameters in Fig. 4 (A and C) suggests thatthe model provides a consistent theory of both compo-nents of the pmf.

Morphological Changes Alone Cannot Account for the Decrease in pH along the Secretory Pathway

In the absence of proton leaks, the steady-state pH of amodel vesicle depends strongly on its shape and size(Rybak et al., 1997). This dependence can be seenfrom Eq. 3 in which the membrane potential is propor-tional to the ratio of the volume to the surface area.When this ratio is small, we expect to see a reduction inmembrane potential and lower steady-state pH values.Can this effect alone explain the decrease in pH alongthe secretory pathway?

We have not been able to find data to directly modelthis question. However, we note that many experimentsperformed on intact cells have shown that significant

proton leakage occurs in many major organelles (Fari-nas and Verkman, 1999; Schapiro and Grinstein, 2000;Wu et al., 2000). Therefore, when revisiting the analy-ses of Rybak et al. (1997) we included a proton leakwhen exploring the effects of vesicle shape and size.

In Fig. 5, we plotted the steady-state pH of a nonequi-librium system containing a constant density of protonleaks and proton pumps. The final pH is plotted as afunction of surface area and the logarithm of volume.In the bottom panel of Fig. 5, numbers indicate experi-mentally measured surface areas and volumes of sev-eral different types of organelles. In the absence (Fig.5, top) and presence (Fig. 5, bottom panel) of chloridechannels, there is very little change in the final pH asthe shape changes from small spherical endosomes andsecretory granules (near population 1) to large floppyGolgi complex (population 3). Both panels exhibit pHchanges of one unit over the range of physiologicalshapes, as also reported by Rybak et al. (1997). How-ever, this effect would only explain pH variations in thecell if small vesicles and endosomes had much moresurface area than is seen in electron micrographs.

The model predicts when proton leaks are present,changes in volume and surface area alone cannot ac-count for the wide range of pHs seen in cellular or-ganelles. This suggests that some other mechanism,

T A B L E I I I

Endosome Model Parameters II

Parameter Units Panels A and C RRC Panels B and D MVB

H1 permeability cm/s 0.60 3 1023 210 2.0 3 1023 258

Surface cm2 0.45 3 1029 224 13.6 3 1029 32

Volume cm3 0.07 3 10214 0 15 3 10214 70

Donnan particles mM 7 – 9 –

Buffering capacity mM/pH see Figure S5

List of parameters used in the simulations in Fig. 4. Simulations in panels Aand B were performed with 10 mM bulk chloride and potassium, whereas thepotassium was 140 mM in panels C and D. The number of V-ATPase protonpumps in panels A and C was 2, whereas in panels B and D had 354 and 70pumps, respectively. The columns RRC and MVB are the percent differencesbetween model parameters and the average values recorded by Van Dyke andBelcher (1994). The external pH was held fixed at 7.0 for all simulations.

Figure 5. The effect of surface area and volume on steady-statepH. Simulations include V-ATPases, proton leaks, chloride chan-nels, Donnan equilibrium (120 mM), and buffering capacity (10mM/pH unit). The density of proton pumps and the proton per-meability were held constant at 109 pumps/cm2 and 1024 cm/s, re-spectively. Constant steady-state pH is plotted as a function of thesurface area and volume in the absence (top) or presence (bottom)of 5 mM chloride (140 mM of impermeant potassium is assumed tobe present internally). The bulk cytoplasmic pH is 7.4. Even at lowcounter ion concentrations, the effect of changes in the vesicleshape on the steady-state pH is small. The bottom panel shows high-lighted regions corresponding to reported surface areas and vol-umes for specific organelles: (1) normal rat kidney Golgi complex(Ladinsky et al., 1999); (2) rat kidney endosomes (Shi et al., 1991);and (3) rat liver endosomes (Van Dyke and Belcher, 1994).


such as the sorting of proton pumps or proton leaks, isthe dominant factor in regulating lumenal pH.

Permeability to Multiple Counter Ions Decreases Changes in Golgi Complex pH

Our goal was to apply the model to the data recordedfrom intact cells to predict Golgi complex proton per-meability values and explore the role of counter ions inregulating Golgi complex pH. Through the biochemi-cal manipulation of intact and membrane permeabi-lized cells, Wu et al. (2000) were able to affect intracel-lular conditions and record subsequent changes inorganelle pHs. Thus, they are able to carry out ex-periments similar to the reconstitution studies dis-cussed above. We modeled two different experimentsaimed at understanding proton leak and counter ionregulation. First, in the presence of bafilomycin, a po-tent inhibitor of V-ATPase, the cytoplasm was acidloaded by externally pulsing membrane permeableNH4 under sodium-free conditions. The absence of so-dium inhibits Na1/H1 exchange across the plasmamembrane thereby keeping the cytoplasm acidic. AfterNH4 washout, sodium was reintroduced, and the alka-linization of the cytoplasm and Golgi complex was mea-sured over time. Since these experiments lack func-tional proton pumps, the proton permeability can bedetermined from the rate of alkalinization. Second, therole of counter ions was investigated in membrane per-meabilized cells in Ringer’s solution. The pH of theGolgi complex was recorded as the external solutionwas replaced with a chloride-free Ringer’s solution.

The experimental information available is too sparseto let us address the dynamic nature of most organelles.Therefore, in the following analysis, we ignore the dy-namic nature of the endomembranes. For example, weassume constant volumes, surface areas, and numbersof active pumps. We also ignore the biochemical differ-ences between the trans, medial, and cis portions of theGolgi complex. This simplification is dictated by thefact that pH-sensitive dyes localized to the Golgi com-plex are spread throughout the entire network, so thatthe experimental recordings that we model are alreadyaveraged over the entire compartment.

In Fig. 6, we examined both the counter ion regula-tion (main graph) and proton leak (inset) experiments.The determination of the Golgi complex proton perme-ability came from the experiment in the inset. Since pro-ton pumping was experimentally inhibited, the simula-tion lacks V-ATPases. At time zero, the cytoplasmic pHbegins to alkalinize from the reintroduction of sodiumto the external solution (Fig. 6, black circles). The Golgicomplex pH rises in response to the change in protongradient across the Golgi complex membrane (Fig. 6,open squares). Modeling the Golgi complex pH, we areable to predict the rate of alkalization of the Golgi com-

plex in response to the increase in cytoplasmic pH (Fig.6, solid line). The Golgi complex leak is modeled well bya simple passive channel. Using the volume and surfacearea measured by Ladinsky et al. (1999), we predict aproton permeability of 8.1 3 1026 cm/s. This value forthe proton permeability is used in the main figure to ex-amine chloride counter ion effects.

The external chloride analysis discussed above dem-onstrated that cytoplasmic counter ion concentrationscan have dramatic effects on the membrane potential

Figure 6. Counter ion regulation of Golgi complex pH in livingHeLa cells. Wu et al. (2000) have shown that the steady-state pH ofthe Golgi complex is not affected by removal of chloride from thecytoplasmic solution (solid circles). These simulations includeV-ATPases, proton leaks, passive chloride channels, passive potas-sium channels, Donnan equilibrium, and buffering (see Table IV forparameter values). At time zero, the external chloride is removed.Three theoretical curves are shown with increasing counter ionconcentration. When the bulk potassium concentration is 1 mM(dotted line), the lumenal pH rises half a pH unit in response tothe removal of chloride. The presence of as little as 14 mM counterion concentration (dashed line) significantly suppresses pHchanges because of chloride removal, whereas the pH change inthe presence of 140 mM counter ion concentration (solid line) isimperceptible. This last case is most likely if the membrane is per-meable to potassium as well as chloride. (Inset) Determination ofthe passive leak of protons out of bafilomycin-inhibited Golgi com-plex. The cytoplasm has been preacidified in sodium-free condi-tions that inhibit plasma membrane Na1/H1 exchangers. At timezero, sodium is reintroduced, and the cytoplasmic pH quickly risesdue to the exchangers. This experimental curve (circles) is de-scribed by a single exponential (solid line). The Golgi complex re-sponse to the rise in cytoplasmic pH (solid line) is then fit, usingthe model, to the experimentally measured data (solid circles). Weestimate the permeability constant of the Golgi complex to be P 58.1 3 1026 cm/s. The Golgi complex volume and surface area areestimated from the detailed EM tomography experiments per-formed by Ladinsky et al. (1999). These measurements were per-formed on a different cell type, and they may not be representativeof the Golgi complex from HeLa cells. Since the characteristic timescale for a passive channel is given by the quantity P3(surfacearea), errors in the estimate of the surface area lead to uncertain-ties in our prediction of the permeability constant.

339 Grabe and Oster

and, consequently, on the lumenal pH. The main graphof Fig. 6 shows the resting pH of a Golgi complex in aplasma membrane compromised cell. At time zero, theexternal solution is replaced by a chloride-free Ringer’ssolution, effectively decreasing the cytoplasmic chloridefrom 20 to 0 mM. From our analysis in Fig. 4 (C and D),we expect the Golgi complex membrane potential to in-crease. After this increase in pmf, the V-ATPase pump-ing rate should slow, causing the lumen to alkalinize,yet experimentally the pH remains unchanged (Fig. 6,black circles). This can be explained two ways. TheGolgi complex is either impermeable to chloride orpermeable to chloride and at least one other dominantcounter ion. Using the model, we explored the latterpossibility. Three separate simulations with differinglevels of cytoplasmic counter ion concentration were fitto the data. In the presence of only 1 mM additionalcounter ion, we predict a noticeable alkalization ofthe Golgi complex, DpH . 0.5 (Fig. 6, dotted line).The presence of a 14-mM cytoplasmic concentrationcounter ion (Fig. 6, dashed line) strongly suppresseschanges in the lumenal pH upon removal of chloride.However, a counter ion concentration of at least 140mM (Fig. 6, solid line) is required to eliminate any no-ticeable change in the Golgi complex pH.

Our analysis suggests that if the Golgi complex mem-brane is permeable to chloride ions, then it is also per-meable to potassium. This conclusion is consistent withrecent Golgi complex experiments (Schapiro and Grin-stein, 2000). The fidelity with which the model repro-duces the experimental data suggests that our treat-ment of the regulatory elements present in the Golgicomplex is accurate. We predict the proton permeabil-ity of the Golgi complex to be 8.1 3 1026 cm/s. How-ever, this value depends upon many organelle charac-teristics that are difficult to quantify, such as volume,surface area, and bilayer surface potential.

Can the Na,K-ATPase Limit Acidification in the Early Endosome through Increasing the Membrane Potential?

Endosomes maintain acidic internal pHs to ensureproper sorting of receptors and ligands during endocy-tosis. Two subpopulations of endosomes can be identi-fied based on their function and capacity for acidifica-tion. Early endosomes are responsible for the rapidrecycling of receptors, whereas late endosomes arerequired for transporting cargo on to lysosomes. Lateendosomes maintain more acidic lumen than early en-dosomes. The use of ouabain, an inhibitor of Na,K-ATPase, enhances the acidification of early endosomesboth in vivo and in vitro (Cain et al., 1989; Fuchs et al.,1989). The electrogenic Na,K-ATPase is thought to in-hibit acidification by increasing the membrane poten-tial across the bilayer. A greater membrane potential,and hence greater pmf, decreases the proton pumping

rate of the V-ATPase, which results in a more alkalinelumen. Rybak et al. (1997) have attempted to quantifythis hypothesis by computing equilibrium solutions ofvesicles containing V-ATPases, Na,K-ATPases, and manyof the regulatory elements considered here. We takethis analysis one step further and use our kinetic modelto simulate the dynamics of acidification in an attemptto shed more light on the feasibility of this proposedmechanism. It should be noted that some endosomesappear to be insensitive to ouabain (Teter et al., 1998).Thus, the mechanism we now examine may not be auniversal property of all cell lines.

Cain et al. (1989) have measured the acidification ofendosomes in single intact cells both in the presence andabsence of ouabain. Initially, cells are bathed in a me-dium containing fluorescent dye at temperatures prohib-itive to endocytosis. The bath is quickly warmed to let en-docytosis begin. The total integrated fluorescence of thedye taken up by the newly formed endosomes is mea-sured, and the acidification curves are determined.These acidification curves are averaged recordings fromthe entire endosomal population as it undergoes allstages of endocytosis. The early measurements are pre-sumably dominated by those endosomes undergoingearly endocytosis, and it is in this early portion in whichthe effects of ouabain are particularly acute.

In Fig. 7, we modeled the initial acidification of theentire endosomal population with a single averagespherical vesicle. All simulations included buffering ca-pacity, chloride and potassium channels, V-ATPases,and proton leak. The initial pH of the vesicle was thatof the bathing solution, and all internal ion concentra-tions were assigned standard extracellular values fromTable I. The middle dashed curve is the simulation inthe presence of Na,K-ATPases, whereas the bottomcurve is the same simulation in the absence of Na,K-ATPases. We sought a single set of parameters thatcould simulate the dynamics of acidification both in thepresence (Fig. 7, circles) and absence of ouabain (Fig.7, diamonds) simply by turning off the Na,K-ATPase.From the two lower curves, we see that our modelcould not fit both sets of data with this constraint. Inthe presence of chloride and potassium counter ionconductance, it is very difficult for the Na,K-ATPase toincrease the membrane potential enough to limit acidi-fication by more than half a pH unit.

From our studies, the Na,K-ATPase is able to limitacidification by a whole pH unit only (1) when the en-domembrane no longer conducts chloride above a cer-tain critical membrane potential, or (2) when the ini-tial internal potassium concentration is far greater thantypical external values and there is no chloride conduc-tance. To explore the former hypothesis, we replacedthe passive chloride channel, described by Eq. 2, with avoltage-gated channel that has zero conductance when


the total membrane potential exceeds 83 mV. In thepresence of such a voltage-gated chloride channel, themiddle dashed line becomes the top solid curve (Fig.7), whereas the simulation in the absence of Na,K-ATP-ases remains unchanged from the bottom curve. With-out Na,K-ATPases, the steady-state membrane potentialis below the switching voltage of the activated chloridechannels, and the chloride counter ions mask themovement of protons into the lumen. The addition ofNa,K-ATPases pushes the membrane potential abovethe switching voltage of the chloride channel, andthe chloride ions no longer act as counter ions. As thepumps continue to push protons into the lumen, themembrane potential quickly increases. This adverselyaffects the V-ATPase pumping rate resulting in a muchmore alkaline lumen.

From the two solid curves in Fig. 7, we see that thepresence of a voltage-gated chloride channel, first pro-posed in this context by Rybak et al. (1997), greatly in-creases the ability of the Na,K-ATPase to affect lumenalpH. ClC-5 is a chloride channel that localizes to endo-somes in proximal tubule cells; it is a likely candidatefor the chloride conductance in the experiments exam-

ined here (Günther et al., 1998). This protein belongsto the ClC family of chloride channels that exhibitwidely varying current-voltage dependencies (Madukeet al., 2000). For inside positive voltages, ClC-5 chan-nels carry a small inward flux of chloride; however, thisflux does not vanish for total membrane potentials .83mV as we hypothesize above (Friedrick et al., 1999).Without such a gated chloride channel, or the pres-ence of a high initial potassium concentration in theendosome coupled with no chloride conductance, ourmodel predicts that the Na,K-ATPase can alkalinize thelumen by half of a pH unit, at most.

Lumenal pH Is Most Sensitive to Changesin Pump and Leak Densities

We have reviewed many experiments that attempt toaddress the importance of particular regulatory ele-ments in establishing and maintaining pH. Our modelincludes many of these elements, and it is important todetermine the relative sensitivity of each of these regu-latory factors. In principle, such analysis will help ex-perimenters determine which parameters are most crit-ical to the type of questions they want to answer. Wechose to revisit the Golgi complex simulations in Fig. 6to carry out this sensitivity analysis.

The acidification of a model organelle was simulatedfrom an initial lumenal pH of 7.4 until steady state wasreached. All relevant parameters were varied 610%from the original values given in Table IV. As in Fig. 5,the proton pump and leak densities remain constantwhen the surface area is changed. The difference be-tween the curves generated with and without variationwere characterized in two ways: (1) the percentagechange in the total pH gradient across the membrane(%DpH), and (2) the percent change in the half-time toacidification (%t). The list of varied parameters is givenin Table V, along with the values for %DpH and %t.

From the top two entries in Table V, we see that themost effective ways to change the steady-state pH are tovary the density of active proton pumps or proton leaks

Figure 7. Endosomal acidification in the selective presence (2ouabain) or absence (1 ouabain) of Na,K-ATPase. We model theacidification of a single 163-nm-diam spherical endosome in thepresence (top curves) or absence (bottom line) of 350 Na,K-ATP-ases. All three simulations include the following: 16 V-ATPases, aproton leak with a permeability of 8.24 3 1025 cm/s, chloridechannels, passive potassium channels, 140 mM of Donnan parti-cles, and 40 mM/(pH unit) buffering capacity. The initial lumenalconcentrations of sodium and potassium were assumed equal tostandard extracellular values (Table I). A significant increase in lu-menal pH is achieved by incorporating Na,K-ATPases into the vesi-cle (dashed curve). However, the addition of a voltage-gated chlo-ride channel, which is inactivated above 133 mV, is necessary to ob-tain the experimentally measured full pH difference (top curve).Data points have been adapted from Cain et al. (1989). The modelendosome begins acidifying at a single instance, hence, a kink inthe pH is present at time zero. The experimental points aresmoother since only a small fraction of the what will be the entireendosomal population has begun acidifying near time zero. Wehave shaded the graph at later times to highlight the portion ofthe curves best modeled by a single endosome.

T A B L E I V

Golgi Complex Parameter Value

Parameter Units Main figure Inset

H1 permeability cm/s 8.1 3 1026 8.1 3 1026

Surface cm2 8.0 3 1026 8.0 3 1026

Volume cm3 6.0 3 10212 6.0 3 10212

No. of V-ATPases 7.0 3 102 0

Cytoplasmic pH 7.4 see caption

Cytoplasmic chloride mM see caption 20

Cytoplasmic potassium mM 1, 14, 140 140

Donnan particles mM 1, 15, 160 140

Buffering capacity mM/pH 10 10

List of parameters used in the simulations in Fig. 6.

341 Grabe and Oster

(permeability). Thus, the principle determinant of thesteady-state Golgi complex pH is the ratio of the protonpump rate to the proton leak rate. The Donnan parti-cles, surface potential, and external potassium concen-tration influence the pH gradient indirectly by alteringthe membrane potential, as discussed in the analysis ofFig. 4 (C and D). The chloride concentration acts in asimilar manner, but with a smaller effect due to reducedchanges in concentration. The pH gradient is least sensi-tive to the last three parameters: buffering capacity, vol-ume, and surface area. However, these elements dramat-ically influence the transient pH dynamics. Thus, the rel-ative importance of a particular regulatory element to anexperiment depends on whether transient or steady-state measurements are being made. For instance,experiments that attempt to determine the proton per-meability of a specific organelle, as in Fig. 6, require ac-curate knowledge of the volume, surface area, and buff-ering capacity to meaningfully model the transient data.

It is important to realize that this sensitivity analysispertains to the acidification of the Golgi complex.Many of the trends in Table V will persist for descrip-tions of different organelles; however, the values willsurely be different. For example, changes in the vol-ume of endosomes had no noticeable effect on theacidification curve, yet comparable percent changes inthe Golgi complex volume are quite noticeable. In ad-dition, this analysis is only valid for determining the rel-ative importance of each parameter near the target pHof the Golgi complex.

Sensitivity analysis confirms what has been a generaltheme throughout this paper: steady-state pH is mostdependent upon proton pump and leak densities.However, transient data is most sensitive to volume, sur-face area, buffering capacity, and proton pump density.

The model provides a method to explore differentmechanisms in comparison with experimental data fo-cusing on the most sensitive parameters.

D I S C U S S I O N

We have constructed a quantitative model for organelleacidification that can be tailored to address many of theorganelles along the secretory and endosomal pathways.Thus, the model can provide a general framework forunderstanding acidification. By using physically accuratedescriptions of but a handful of pH regulatory elements,we have been able to successfully model the acidificationof several organelles. The close attention paid to the bio-chemical makeup of the systems studied eliminatedsome of the extreme pH and membrane potential valuespredicted in the earlier work of Rybak et al. (1997). Wehave presented the first estimates of Golgi complex pro-ton permeability, z1025 cm/s, as well as the first predic-tions of the active ATPase pump numbers for the Golgicomplex and endosome. By modeling experimentaldata, we were able to make hypotheses about the regula-tory elements present in specific organelles. Our modelsuggests that the Golgi complex is permeable to at leasttwo dominant counter ions. In the endosome, we con-clude that it is possible for the Na,K-ATPase to limit lu-menal acidification by increasing the membrane poten-tial. However, the presence of voltage-gated chloridechannels or extreme ionic conditions are required to ex-plain the high degree with which the Na,K-ATPase hasbeen shown to limit acidification.

The greatest asset of our model lies in its ability toquantify and compare experimental data; however, agood deal of our analysis has been spent justifyingthe accuracy of our approach. We have not addressedwhat biochemical differences are required to elicit pHchanges between organelles. Secretory granules thatbud from the Golgi complex at pH < 6.4 eventuallyachieve lumenal pHs of z5.4. Table V suggests thatchanges in the proton pump and leak density are mosteffective in bringing about this 1-pH unit drop. Calcula-tions were performed on a 200-nm-diam vesicle withproperties similar to the Golgi complex simulation ofFig. 6. A 10-fold increase in the proton pump density ora 10-fold decrease in the proton permeability provedsufficient to describe this drop in lumenal pH. Ulti-mately, we hope that analyses such as these, in conjunc-tion with experimental data, will help elucidate the na-ture of pH regulation in the cell.

It is interesting to note the discrepancy in protonpermeabilities between reconstitution and intact cellexperiments. Proton permeability values deduced fromreconstitution experiments are z100 times higher thanvalues deduced from data recorded from intact cells. Itis possible that the process of reconstitution artificiallyincreases the proton permeability of the membrane.

T A B L E V

Sensitivity Analysis

Parameter Percent change DpH Percent change t

H1 permeability 27.5/8.3 20.2/0.4

Number of V-ATPases 7.5/28.3 28.8/10.9

Cytoplasmic potassium 23.5/3.9 1.4/21.5

Donnan particles 3.4/23.6 21.2/1.3

Cytoplasmic surface potential 1.5/21.4 24.2/5.1

Cytoplasmic chloride 0.5/20.5 20.2/0.2

Buffering capacity 20.3/0.3 10/210

Volume 0.0/0.0 10/210

Surface area 0.0/0.0 29.0/11

Sensitivity analysis of lumenal pH and time to acidification. The parametervalues describing the acidification of the Golgi complex (Table IV) havebeen individually varied 610%, and the resulting percent change in thelumenal pH gradient and time constant are given. Percentages on the leftside of the slash correspond to the positive variation in the parameter.Experimental quantities are listed in order of decreasing influence onsteady-state pH. The external potassium and chloride concentrations were140 and 20 mM, respectively.


Extending the model to describe additional or-ganelles can be accomplished easily by including thespecialized proteins that make these systems unique;e.g., synaptic vesicles that use the proton gradient todrive the uptake of neurotransmitter. This is the ap-proach we used in modeling the early-to-late endosometransition, which required a detailed model of the Na,K-ATPase. Some of the most acidic organelles, such as lyso-somes and the central vacuoles of plant cells, have notbeen discussed. Lemon vacuoles have been shown toacidify to pH 2.5, whereas giant blood cells of the ma-rine Ascidian have single large vacuoles with lumenalpHs below 1 (Muller et al., 1996; Futai et al., 1998). It isnot clear how the V-ATPase could be solely responsiblefor creating such acidic environments since it is gener-ally thought to be thermodynamically limited to a maxi-mum pH difference of 4.5 units. It is possible that an-other enzyme contributes to acidification of these sys-tems or that other regulatory elements conspire tocreate a very negative membrane potential that helps todrive protons into the lumen. Motivated by the presenceof V-ATPases in lemon vacuoles, we previously proposeda mechanism by which the V-ATPase proton pumpcould “change gears” and pump below pH 1 (Grabe etal., 2000). In Online Supplemental Material (see http://www.jgp.org/cgi/content/full/117/4/329/DC1), we de-scribe the molecular mechanism of this gear change andinclude a calculation confirming that this hypotheticalscheme can indeed produce large pH gradients.

The authors thank T. Machen, H.P. Moore, G. Chandy, M. Wu, D.Hilgemann, and A. Tobin for many thoughtful discussions andcareful reading of the manuscript, and R. Murphy for his helpfuldiscussions. Additionally, the construction of the Na,K-ATPasemodel would have been incomplete without the guidance of H.-J.Apell and R.F. Rakowski. Finally, we thank R. Van Dyke for her valu-able correspondence and for providing us with her original data.

This work was supported by National Science Foundationgrant DMS 9220719 (to G. Oster).

Submitted: 24 August 2000Revised: 14 February 2001Accepted: 15 February 2001

R E F E R E N C E S

Alberts, B., D. Bray, J. Lewis, M. Raff, K. Roberts, and J. Watson.1994. Molecular Biology of the Cell. 3rd ed. Garland Press, NewYork, NY. 1294 pp.

Belcher, J.D., R.L. Hamilton, S.E. Brady, C.A. Hornick, S. Jaeckle,W.J. Schneider, and R.J. Havel. 1987. Isolation and characteriza-tion of three endosomal fractions from the liver of estradiol-treated rats. Proc. Natl. Acad. Sci. USA. 84:6785–6789.

Cahalan, M.D., and J.E. Hall. 1982. Alamethicin channels incorpo-rated into frog node of ranvier: calcium-induced inactivation andmembrane surface charges. J. Gen. Physiol. 79:411–436

Cain, C.C., D.M. Sipe, and R.F. Murphy. 1989. Regulation of en-docytic pH by the Na1,K1-ATPase in living cells. Proc. Natl. Acad.Sci. USA. 86:544–548.

Chatterjee, D., M. Chakraborty, M. Leit, L. Neff, S. Jamsa-Kello-kumpu, R. Fuchs, M. Bartkiewicz, N. Hernando, and R. Baron.1992. The osteoclast proton pump differs in its pharmacology

and catalytic subunits from other vacuolar proton ATPases. J.Exp. Biol. 172:193–204.

DeCoursey, T.E., and V.V. Cherny. 2000. Common themes andproblems of bioenergetics and voltage-gated proton channels.Biochim. Biophys. Acta. 1458:104–119.

Endresen, L.P., K. Hall, J.S. Hoye, and J. Myrheim. 2000. A theory forthe membrane potential of living cells. Eur. Biophys. J. 29:90–103.

Farinas, J., and A.S. Verkman. 1999. Receptor-mediated targetingof fluorescent probes in living cells. J. Biol. Chem. 274:7603–7606.

Friedrick, T., T. Breiderhoff, T. Jentsch. 1999. Mutational analysisdemonstrates that ClC-4 and ClC-5 directly mediate plasma mem-brane currents. J. Biol. Chem. 274: 896–902

Fuchs, R., S. Schmid, and I. Mellman. 1989. A possible role for so-dium, potassium-ATPase in regulating ATP-dependent endo-some acidification. Proc. Natl. Acad. Sci. USA. 86:539–543.

Futai, M., T. Oka, Y. Moriyama, and Y. Wada. 1998. Diverse roles ofsingle membrane organelles: factors establishing the acid lume-nal pH. J. Biochem. 124:259–267.

Futai, M., T. Oka, G.-H. Sun-Wada, Y. Moriyama, H. Kanazawa, andY. Wada. 2000. Luminal acidification of diverse organelles byV-ATPase in animal cells. J. Exp. Biol. 203:107–116.

Grabe, M., H. Wang, and G. Oster. 2000. The mechanochemistry ofV-ATPase proton pumps. Biophys. J. 78:2798–2813.

Günther, W., A. Luechow, F. Cluzeaud, A. Vandewalle, T. Jentsch.1998. CIC-5, the chloride channel mutated in Dent’s disease,colocalizes with the proton pump in endocytotically active kidneycells. Proc. Natl. Acad. Sci. USA. 95:8075–8080.

Hartmann, T., and A.S. Verkman. 1990. Model of ion transport reg-ulation in chloride-secreting airway epithelial cells. Integrateddescription of electrical, chemical, and fluorescence measure-ments. Biophys. J. 58:391–401.

Heyse, S., I. Wuddel, H.-J. Apell, and W. Stuermer. 1994. Partial re-actions of the Na,K-ATPase: determination of rate constants. J.Gen. Physiol. 104:197–240.

Hill, W.G., R.L. Rivers, and M.L. Zeidel. 1999. Role of leaflet asym-metry in the permeability of model membranes to protons, sol-utes, and gases. J. Gen. Physiol. 114:405–414.

Hille, B. 1992. Ionic Channels of Excitable Membranes. 2nd ed.Sinauer Associates, Inc., Sunderland, MA. 607 pp.

Hodgkin, A.L., and A.F. Huxley. 1952. A quantitative description ofmembrane current and its application to conduction and excita-tion in nerve. J. Physiol. 117:500–544.

Israelachvili, J.N. 1992. Intermolecular and Surface Forces. 2nd ed.Academic Press Inc., London. 450 pp.

Kim, J.H., C.A. Lingwood, D.B. Williams, W. Furuya, M.F. Manol-son, and S. Grinstein. 1996. Dynamic measurement of the pH ofthe Golgi complex in living cells using retrograde transport ofthe verotoxin receptor. J. Cell Biol. 134:1387–1399.

Ladinsky, M.S., D.N. Mastronarde, J.R. McIntosh, K.E. Howell, andL.A. Staehelin. 1999. Golgi structure in three dimensions: func-tional insights from the normal rat kidney cell. J. Cell Biol. 144:1135–1149.

Lauger, P., and H.-J. Apell. 1986. A microscopic model for the cur-rent-voltage behaviour of the Na,K-pump. Eur. Biophys. J. 13:309–321.

Lencer, W.I., P. Weyer, A.S. Verkman, D.A. Ausiello, and D. Brown.1990. FITC-dextran as a probe for endosome function and local-ization in kidney. Am. J. Physiol. 258:C309–C317.

Maduke, M., C. Miller, and J. Mindell. 2000. A decade of CLC chlo-ride channels: structure, mechanism, and many unsettled ques-tions. Annu. Rev. Biophys. Biomol. Struct. 29:411–438

McLaughlin, S. 1989. The electrostatic properties of membranes.Annu. Rev. Biophys. Biophys. Chem. 18:113–136.

McLaughlin, S., N. Mulrine, T. Gresalfi, G. Vaio, and A. McLaugh-lin. 1981. Adsorption of divalent cations to bilayer membranes

343 Grabe and Oster

containing phosphatidylserine. J. Gen. Physiol. 77:445–473.Moriyama, Y., and M. Futai. 1990. Proton-ATPase, a primary pump

for accumulation of neurotransmitters, is a major constituent ofbrain synaptic vesicles. Biochem. Biophys. Res. Comm. 173:443–448.

Moriyama, Y., M. Maeda, and M. Futai. 1992. Involvement of a non-proton pump factor (possibly Donnan-type equilibrium) inmaintenance of an acidic pH in lysosomes. FEBS Lett. 302:18–20.

Muller, M., U. Irkens-Kiesecker, B. Rubinstein, and L. Taiz. 1996.On the mechanism of hyperacidification in lemon. J. Biol. Chem.271:1916–1924.

Paula, S., A.G. Volkov, A.N. Van Hoek, T.H. Haines, and D.W.Deamer. 1996. Permeation of protons, potassium ions, and smallpolar molecules through phospholipid bilayers as a function ofmembrane thickness. Biophys. J. 70:339–348.

Press, W.H. 1997. Numerical Recipes in C: The Art of Scientific Com-puting. 2nd ed. Cambridge University Press, New York, NY. 994 pp.

Roos, A., and W.F. Boron. 1981. Intracellular pH. Physiol. Rev. 61:296–434.

Rothman, J.E., and J. Lenard. 1977. Membrane asymmetry. Science.195:743–753.

Rybak, S., F. Lanni, and R. Murphy. 1997. Theoretical consider-ations on the role of membrane potential in the regulation of en-dosomal pH. Biophys. J. 73:674–687.

Sagar, A., and R.F. Rakowski. 1994. Access channel model for thevoltage dependence of the forward-running Na1/K1 pump. J.Gen. Physiol. 103:869–894.

Schapiro, F., and S. Grinstein. 2000. Determinants of the pH of theGolgi complex. J. Biol. Chem. 275:21025–21032.

Shi, L., K. Fushimi, H. Bae, and A.S. Verkman. 1991. Heterogeneityon ATP-dependent acidification in endocytic vesicles from kid-ney proximal tubule. Biophys. J. 59:1208–1217.

Sokolov, V.S., H.-J. Apell, J.E.T. Corrie, and D.R. Trentham. 1998.Fast transient currents in Na,K-ATPase induced by ATP concen-tration jumps from the P3-(1-(39,59-dimethoxyphenyl)-2-phenyl-2-oxo)ethyl ester of ATP. Biophys. J. 74:2285–2298.

Teter, K., G. Chandy, B. Quinones, K. Pereyra, T. Machen, and H.-P.Moore. 1998. Cellubrevin-targeted fluorescence uncovers hetero-geneity in the recycling endosomes. J. Biol. Chem. 273:19625–19633.

Van Dyke, R.W. 1993. Acidification of rat liver lysosomes: quantita-tion and comparison with endosomes. Am. J. Physiol. 265:C901–C917.

Van Dyke, R.W., and J.D. Belcher. 1994. Acidification of three typesof liver endocytic vesicles: similarities and differences. Am. J. Phys-iol. 266:C81–C94.

Vander, A.J., J.H. Sherman, and D.S. Luciano. 1985. Human Physi-ology: The Mechanisms of Body Function. 4th ed. McGraw-HillInc., New York. 715 pp.

Verdugo, P. 1991. Mucin exocytosis. Am. Rev. Respir. Dis. 144:533–537.Weiss, T.F. 1996. Cellular Biophysics. MIT Press, Cambridge, MA.

693 pp.Wu, M., J. Llopis, S. Adams, J. McCaffery, T. Machen, H.P. Moore,

and R. Tsien. 2000. Using targeted avidin and membrane per-meant fluorescent biotin to study secretory pathway organellepH regulation. Chem. Biol. 7:197–209.

Online Supplemental Material for

J. Gen. Physiol. Vol. 117 No. 4 p. 329, Grabe and Oster

Detailed Calculations and Discussion

Michael Grabe and George Oster

V-ATPase Model

We have constructed a mechanochemical model for the V-ATPase that predicts proton pumping rates over a widerange of environmental conditions. We have used this model to determine the acidification of organelles. The con-ventional paradigm for active transmembrane ion transport is the “alternating access” mechanism: ions are boundtightly on the low concentration side of the membrane and a conformational change exposes them to the high con-centration side and weakens their binding affinity so that they dissociate. The pump then resets its conformation torepeat the cycle (Eisenberg and Hill, 1985; Alberts et al., 1994). The energy to drive the cycle is supplied by nucle-otide hydrolysis or an ion gradient. The Na-K-ATPase conforms to this traditional view of alternating access (see be-low); however, the conformational change in the V-ATPase is a simple rotation. In our computations here, we as-sume that the V-ATPase is working under normal operating conditions and that ATP concentrations are sufficientlyhigh that hydrolysis is not rate limiting.

By analogy with the F-ATPases, the V-ATPase structure is conventionally divided into a counter-rotating “stator”and “rotor” (Boekema et al., 1997; Finbow and Harrison, 1997; Forgac, 2000). Hydrolysis of ATP in the V1-solubleheadpiece provides the torque that rotates the membrane-bound section Vo. Two models have been suggested forthe rotor–stator assemblies: a two-channel model and a one-channel model (Vik and Antonio, 1994; Junge et al.,1997; Elston et al., 1998; Grabe et al., 2000). They differ in the path that the protons take through the enzyme andin the degree to which protons bound to the rotor section communicate with the cytoplasm. There is significant de-bate about the true structure of the channel in the V- and F-ATPase enzymes. We have shown that both channels be-have similarly within the context of our mechanochemical model (Dimroth et al., 1999; Grabe et al., 2000). Morerecently, experiments on sodium V-ATPases strongly support the one channel model (Murata et al., 2000). There-

1 J. Gen. Physiol. © The Rockefeller University Press • 0022-1295Volume 117 April 2001http://www.jgp.org/cgi/content/full/117/4/329/DC1

HydrophobicInterface

Lumen

Cytoplasm

ROTATION

H+ H+

H+

His743Glu789Lys593

e H

OutputChannel

H+

Stator

Cytoplasm

H+

Rotor

PolarStrip

Lumen

PolarStrip

R227Rotor

Figure S1. Perspective and face-on views of the rotor–stator assem-bly showing the paths protons follow in moving from the cytoplasm(bottom) into the lumen (top). Torque supplied by V1 from ATP hy-drolysis turns the rotor to the right. A single half channel penetratesthe stator to the level of the rotor sites. A horizontal polar strip con-nects the right side of the channel and the cytoplasm to allow thepassage of an unprotonated site, but protons are blocked from leak-ing by the stator charge. All other parts of the rotor–stator interfaceare hydrophobic. Cytoplasmic protons on the left bind to a rotor site,largely neutralizing it. Rotation carries the protonated site across thehydrophobic interface where the site enters the output (acidic)channel (from the left). The stator charges force the site to relin-quish its proton into the lumen. The unprotonated site exits the in-terface to the right and is again in equilibrium with the cytoplasmicreservoir. Note that the sizes of the rotor and stator are such that tworotor sites cannot fit in the rotor–stator interface at once.

fore, in Fig. S1 we present the putative structure for the one-channel model that was used to compute the V-ATPaseperformance surface in Fig. 2 A of the text.

The structure in Fig. S1 must possess a few key properties to be a physically viable model. The interface betweenrotor and stator must present a hydrophobic barrier to prevent proton leakage between the reservoirs. When not inthe stator interface, the six rotor acidic sites exchange protons with the cytoplasm. Finally, the output (acidic) chan-nel extends into the stator to the level of the rotor proton binding sites to allow exchange with the lumen.

The one-channel proton pump model works according to the following sequence of events (shown in Fig. S2 A).Rotor sites are in equilibrium with the basic reservoir. Until a rotor site is protonated, the torque generated by V1

cannot force it over the hydrophobic barrier. However, once neutralized by protonation, the torque from V1 drivesthe site across the barrier into the stator output channel. A protonated site that rotates out of the channel into thepolar strip interacts electrostatically with the stator charge, reducing its pKa and forcing it to relinquish its proton tothe acidic reservoir. Further rotation carries the rotor site out of the stator interface, where it quickly equilibrateswith the basic reservoir. The full mathematical formulation is contained in the original paper (Grabe et al., 2000).

The pumping cycle in Fig. S2 A does not always culminate in the proton exiting through the output channel.Rather, the protonated site “slips” past the stator charge and reequilibrates with the input reservoir. Fig. S2 B out-

2 Grane and Oster • Online Supplemental Material

A

B

E

F

JS1 JS2

JSLIP =JS1+JS2

Jin

JOUT

Stator

koffonk

Figure S2. Mechanistic description of atypical pumping event and slip. (A) A typi-cal sequence of events following a site as itpasses through the rotor–stator interface.An empty site in the basic channel is re-flected by the dielectric boundary until aproton binds, neutralizing the site (1 → 2).The V1 motor rotates a protonated site outof the membrane, across the hydrophobicinterface, and into the acidic channel (2 →3). In the channel, the site has a high prob-ability of staying protonated since kon islarge. However, when the protonated siterotates close to the stator charge, the rotorsite pKa decreases so that koff increases untilthe proton is relinquished to the acidic res-ervoir (3 → 4). The empty site is thendriven through the hydrophilic strip, pastthe stator charge, and into the basic reser-voir (4 → 5). The torque from V1 can thenrotate the site through an entire revolutionback to position (1) where the cycle re-peats. (B) The possible pathways for pro-tons passing through the stator. A fractionof the protons that rotate into the outputchannel slip past the stator charge and dis-sociate back into the basic reservoir. Theproton flux (JIN) traverses the hydrophobicinterface from the left. A fraction (JOUT)dissociates into the output channel, and afraction (JS1) slips past the stator chargeand leaves the input channel; thus JS1 � JIN

� JOUT. Actually, a site traversing the outputchannel will bind and dissociate a proton�104–105�, so the proton that slips backinto the basic reservoir is unlikely to be thesame one that initially entered on the rotorsite (Grabe et al., 2000). Additionally, theback and forth diffusive motion of the rotormay shuttle protons from the output chan-nel to the basic reservoir, creating a secondcontribution to the slip flux (JS2). Thus, thetotal slip flux is JSLIP � JS1 � JS2. This secondcomponent of the slip flux is only impor-tant when the V1 motor is nearly at stall; i.e.,when the rotation rate of Vo is small.

lines various slip mechanisms. Slip can have a profound influence upon acidification since it limits the pH that a ves-icle can attain. As the pmf across the membrane increases, the enzyme is more prone to slip. The degree of slip canonly be determined experimentally from measuring ATP hydrolysis rates and proton pumping. The surface in Fig.S2 A is partially calibrated from patch-clamp experiments and largely free of slip over the range of plotted pmf’s. Ex-cellent reviews of this enzyme can be found in the February special edition of the J. Bioenerg. Biomembr. (2000).

The “Gear Change” Model for Hyperacidification

It is generally accepted that the V1 subunit has three ATP hydrolyzing sites, while the Vo rotor has six proton bindingsites on each of its proteolipid subunits (c, c�, and c�) that span the lipid bilayer. Under optimal conditions (i.e., noslip), every rotation of the enzyme transports six protons at the expense of three ATP’s, corresponding to a couplingratio of two. If the six proton binding sites have different pKa’s (for instance, three with low and three with high val-ues), exposure to a very acidic lumenal environment would permanently titrate the high pKa rotor sites. Then thenumber of active rotor sites would be reduced from six to three, thus decreasing the coupling ratio from two to one.With this coupling ratio, the energy of each ATP would transport one proton, and the maximum achievable pH gra-dient would double from 4.5 to 9. This may help account for the phenomenon of very acidic organelles. We have re-visited the V-ATPase model above, assuming that the six proton binding sites are not equivalent. The sites are as-sumed to alternate between tightly binding, pKa � 7.3, and weakly binding, pKa � 3.8. We also assume that thetightly binding sites hold the proton much closer to the proton-accepting amino acid, creating a much smaller di-pole electric field. The existence of more than one proteolipid subunit in the V-ATPase could explain this heteroge-neity in proton binding sites. Although all three subunits (c, c�, and c�) are homologous, subtle differences in theirproton binding carboxyl groups could account for the variations in proton binding affinities (Forgac, 1999).

In Fig. S3, the proton flux is plotted as a function of lumenal pH. The list of simulation parameters are given inTable SI. At modest lumenal pH values, pH � 6–7, all sites participate in proton transport; however, as lumenal pHdrops (pH � 3.5–6), the high pKa sites no longer deliver protons to the lumen. On average, these sites do no workagainst the pH gradient, thereby enabling the low pKa sites to use a larger fraction of the energy of ATP hydrolysisand continue pumping protons. This proposed ability of the V-ATPase to sense and adjust to the absolute pH of thelumen is theoretically interesting. In more alkaline organelles, the V-ATPase works far from thermodynamic stall,where all six proton binding sites would participate in pumping in order to counter the high leak rate of protons.However, in the very acidic organelles, where proton permeabilities are likely to be small, the overall rate of protonpumping should be less important than the size of the pH gradient the enzyme can create. Therefore, as an or-ganelle acidifies, the loss of some of the proton transporting sites is more than made up for by the enzyme’s abilityto continue working to produce a low lumenal pH.

These calculations are carried out in program files: v_pump.m (main program); cal_2.m; vel_a.m; v_param.m;barrier.m; delc_cf.m; pot.m; pot2.m.

3 http://www.jgp.org/cgi/content/full/117/4/329/DC1

1 2 3 4 5 6 7 8-40

-20

0

20

40

60

80

Lumenal pH

Pro

ton

flux

[H+

/sec

]

high pK flux

low pK flux

total flux

Figure S3. The proton flux of a single V-ATPase in the absenceof membrane potential is plotted as a function of lumenal pH. Itis generally accepted that the rotor subunit of the V-ATPase iscomposed of six proton binding sites. Here we explore the possi-bility that these sites have alternating high and low pKa’s. At highpH values, all sites participate in proton transport from the cyto-plasm to the lumen. As the pH drops, the high pKa sites begin toleak protons, and then eventually stop participating in transport.This effectively lowers the number of protons transported to ATPsconsumed per revolution; e.g., the coupling ratio. Lowering thecoupling ratio makes it possible to achieve very acidic lumenalpHs. See Table SI for a list of parameters.

T A B L E S I

V-ATPase Model Parameters

Parameter Fig. 2 A Fig. S3

Screening length (1/

�

)

0.67 nm 0.67 nm

Rotational diffusion constant of rotor (D

r

) 10

4

s

-1

10

4

s

�

1

Proton diffusion constant (D

p

) 9.3

�

10

9

nm

2

s

�

1

9.3

�

10

9

nm

2

s

�

1

Dielectric constant of cytoplasm (

c

) 80 80

Dielectric constant of stator (

s

) 4.0 4.0

Proton conductivity of lumenal channel 2.2

�

10

10

M

�

1

s

�

1

2.2

�

10

10

M

�

1

s

�

1

Proton conductivity of cytoplasmic channel 2.2

�

10

10

M

�

1

s

�

1

2.2

�

10

10

M

�

1

s

�

1

Radius of rotor (R) 3 nm 3 nm

Bilayer viscosity (

) 10 poise 10 poise

Zero load rotation rate of V

1

20 Hz 20 Hz

Free energy of ATP hydrolysis (

�

G

ATP

) 21 k

B

T 21 k

B

T

Central stator charge (elementary charge) 1.4 0.14

Rotor pK

a

5.4 8.1, 5.65

Radial position of central stator charge 3.54 nm 3.54 nm

Dipole length of protonated site 0.22 nm 0.16 nm, 0.39 nm

Cytoplasmic pH (pH

C

) See caption 7.4

Width of polar strip 0.79 nm 0.90 nm

Width of acidic channel 1.18 nm 1.43 nm

Width of hydrophobic interface 1.18 nm 2.5 nm

Parameters used in creating the V-ATPase pumping profile in Fig. 2 A (text) and the pumping curve in Fig. S3. In the right column, multiple entries referto the value for the high and low rotor pK site, respectively.

Na-K ATPase Model

We have attempted to construct a more general model of the Na-K-ATPase pumping rate over a large range of envi-ronmental conditions. This was accomplished by simulating the detailed reaction schemes devised by Heyse et al.(1994) and Sokolov et al. (1998) while constraining the current to obey the voltage dependence measured by Sagarand Rakowski (1994). The model presented here is an amalgam of the most recent Apell model (Heyse et al.,1994), which focuses on sodium transport, with an older model that includes potassium transport. The kineticscheme can be found in Fig. S4 A. The primary pumping cycle of this model, indicated by solid lines, closely resem-bles the standard Post-Albers scheme (Heyse et al., 1994). In addition, there are three escapements that involve in-complete pumping cycles. These closed loops are represented by dashed lines and are essential for describing thepump’s behavior under extreme conditions such as low extracellular potassium concentrations.

Steady state values were computed by constructing the kinetic equations corresponding to the reaction schemein Fig. S4 A and simulating their time evolution until steady state was reached. Alternatively, setting these equationsequal to zero results in a set of linear coupled equations that can be solved algebraically to find the steady state val-ues. For ease of implementation, we chose the former. The rate of ion transport was determined in the usual way bysumming the forward and backward jump rates between those states that correspond to ion movement from theenzyme to the bulk cytoplasm (or the extracellular space).

To accurately describe pumping under physiological conditions, model parameters have been varied to obtainbest fits to the current–voltage data measured by Sagar and Rakowski (1994). Four reverse rates along the potas-sium portion of the reaction cycle have been adjusted. Additionally, the reverse rates of four more parameters weremodified to ensure detailed balance around each of the independent pump cycles. These parameters are indicatedin Table SII (bold). To simulate the data from Sagar and Rakowski (1994), additional assumptions about cytoplas-mic concentrations and the free energy of ATP must be made. For all simulations, the cytoplasmic values for ATP,ADP, and Pi can be found in Table I of the main article text. Sodium and potassium values can be found in Fig. S4.

Given an initial set of model parameters, the steady state current as a function of membrane potential was com-puted for each of the extracellular potassium concentrations examined by Sagar and Rakowski (1994). For each po-tassium concentration, the membrane potential was discretized and the reaction scheme simulated to determinethe steady state current. This family of curves was then scaled by a common parameter to find the best simultaneousfit to all of the experimental data. Such a scale factor is required to normalize the single pump current to the exper-imentally measured current, which is the result of many pumps. This root mean square fitness score was then re-



A

Ext

ra-c

ellu

lar

Cyt

opla

sm

h

(Na3)E1-P

P-E2

P-E2(K)

P-E2(K2)E

2(K2)ATP-E

2(K2)K2E

1K2E

1-ATP

E1

Na3E1-ATP

E1-ATP

KE1-ATP

Na2E1-ATP

NaE1-ATP

Na3E1

Na2E1

NaE1

K1E1

P-E2(Na)

P-E2(Na2)

E2(Na

2)

P-E2(Na3)

l

r

d

p

g3

qa

e

n3

a

a

a

a

a

g1

g2

m1

m2sk

n3

ADP

Pi

Pi

ATPATP

ATP

ATP

ATP

ATP

ATP

Pi

K2E1

E1

Na2E1

NaE1

K1E1

A

K2E1-ATP

E1-ATP

KE1-ATP

Na2E1-ATP

NaE1-ATP

B

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

Ψ [mV]

Pum

p cu

rren

t [µ

A/c

m2 ]

10 mM

[K+] = 0.652 mM

5 mM

2.5 mM

1.25 mM

P

in out

P

in out

in out

in out

Figure S4. Na-K-ATPase reaction scheme and current volt-age fit. (A) This reaction scheme combines aspects of two pre-vious models published by Heyse et al. (1994) and Sokolov etal. (1998). The basic tenants of this model follow the acceptedPost-Albers description that involves the pump switching be-tween an inward and outward facing conformation, E1 and E2.Four cartoon diagrams have been included to illustrate thephysical state of the enzyme. Chemical states within grayed re-gions are assumed to interconvert on a very fast time scale andare mathematically treated as one state. Please refer to theoriginal papers for details concerning the construction of thedifferential equations from this reaction scheme. (B) The ki-netic scheme in A is simulated using the model parameters inTable SII. The total number of pumps giving rise to the mea-sured current is unknown; thus, an arbitrary scale factor hasbeen used to scale the single pump current (solid line) to theexperimentally measured data (�). Cytoplasmic sodium andpotassium concentrations are held constant at 20 and 120mM, respectively. Lumenal sodium concentration is initially120 mM. Data is adapted from Figure 5 A of Sagar and Ra-kowski (1994).

T A B L E S I I

Na-K-ATPase Model Parameters

ParameterForward value/

equilibrium constant Reverse valueDielectriccoefficient

n

3

2

�

10

4

M

�

1

s

�

1

80 s

�

1

�

�

0.25 Sokolov et al. (1998)

a

3.5

�

10

6

M

�

1

s

�

1

2.76

�

10

�

1

s

�

1

* 0 Sokolov et al. (1998)

p 600 s

�

1

1.5

�

10

6

M

�

1

s

�

1

0 Sokolov et al. (1998)

l 25 s

�

1

2.75 s

�

1

�


g

3

1.4

�

10

3

s

�

1

1.4

�

10

4

M

�

1

s

�

1

�

0

�


g

2

5

�

10

3

s

�

1

3.25

�

10

3

M

�

1

s

�

1

�

1

�


g

1

7

�

10

2

s

�

1

7.77

�

10

3

M

�

1

s

�

1

�

2

� 0.1 Sokolov et al. (1998)

r 0.023 s�1 3.10 � 103 M�1 s�1* � � 1.8 Sokolov et al. (1998)

d 1.4 s�1 2.45 M�1 s�1* 0 Sokolov et al. (1998)

e 50 s�1 2.5 � 102 s�1 0 Sokolov et al. (1998)

k 22 s�1 8.71 s�1* 0 Heyse et al. (1994)

h 0.1 s�1 102 s�1 0 Heyse et al. (1994)

s 5 � 105 M�1 s�1 9.95 � 10 s�1* 0 Heyse et al. (1994)

q 105 s�1 7.58 � 105 M�1 s�1* 0 Heyse et al. (1994)

m1 3.4 � 104 M�1 s�1 1.14 � 102 s�1* 1� 0.1 Heyse et al. (1994)

m2 5 � 106 M�1 s�1 3.83 � 106 s�1* 2� 0.1 Heyse et al. (1994)

KNa 3 � 10�3 M — 0 Sokolov et al. (1998)

KK 1.2 � 10�2 M — 0 Heyse et al. (1994)

Parameters used for the Na-K-ATPase pumping profile. *Parameters differ from the values presented in the references. The kinetic scheme presented inFig. S4 A has four closed loops. Detailed balance has been enforced around each of these loops by constraining the reverse rates of the parameters set offin bold. The remaining four * parameters correspond to reverse rates along the potassium pumping portion of the cycle. These parameters have beendetermined from fitting current voltage data (Fig. S4 B).

corded. A Nelder-Mead algorithm was used to search through the space of model parameters and find the set thatminimized the fitness score (Press, 1997). These parameters are presented in Table SII, and the fit to the data ispresented in Figure S4 B. In all figures, ionic flux from the cytoplasm to the extra-cellular/lumenal space is definedto be positive, and the membrane potential is defined relative to the lumen. Thus, a positive membrane potentialwill induce positive charges to flow in the negative direction.

This calibrated model has been used to compute the sodium and potassium rates over a wide range of external/lumenal concentrations and membrane potentials. To illustrate what a slice of this surface looks like, the sodiumrate is plotted as a function of membrane potential and lumenal potassium concentration in Figure 2B. It is this re-sponse surface combined with the corresponding surface for the potassium flux that are used to simulate the effectof incorporating Na-K-ATPases in model vesicles. When creating these surfaces, cytoplasmic concentrations of allspecies are assumed constant. Values can be found in Table I of the main article text.

These computations are given in program files: nak_pump.m (main program); delta.m; nak_param.m.

Buffering Capacity

When a proton enters the lumen or the cytoplasm, it will equilibrate with the lumenal matrix. A bound proton isthen no longer free to contribute to the pH in that compartment. The ability of the matrix to bind free protons isreferred to as its buffering power. The standard definition of buffering power originally given by Michaelis is (Roosand Boron, 1981):

(1)

where �[B] is the amount of strong base added to the system. Since �[B] � �[H�], the relation between change inpH and change in proton concentration is determined. Some experiments suggest that the buffering capacity of someorganelles is nearly constant in the pH range 6–8 (Farinas and Verkman, 1999; Wu et al., 2000). However, Van Dykeand Belcher (1994) find that the buffering capacity of endosomes depends strongly upon the lumenal pH (Fig. S5).

In the simplest case, there is only one kind of proton accepting site with some intrinsic pKa. Once the internal pHgoes below this pKa, the buffering ability of the matrix quickly diminishes. From the definition of pH, one then ob-tains a buffering capacity, � 10�pH. In reality, the matrix is composed of many titratable sites with a distribution ofpKa’s. With this interpretation, the jagged variation in the buffer capacity found by Van Dyke and Belcher (1994)seems more reasonable.

A sensitivity analysis shows that the buffering capacity is one of the most dominant factors in determining theshape of the acidification curves. Originally, it was our goal to try to recreate the pumping performance of the V-ATPase from the experimentally measured curves in Figs. 3 and 4 (text). This would serve as a calibration for theproton pump. However, uncertainties in the volume and buffering capacity made it impossible to extract depend-able information about the pump.

β ∆ B[ ]∆pH------------,=


6 7

30

40

50

60

70

80

Lumenal pH

Buf

ferin

g ca

paci

ty [m

M/p

H]

6.5 7.5

RRCMVB

Figure S5. Endosomal buffering capacity as a function of pH.The buffering capacities of RRC (�) and MVB (�) vesicleshave a strong dependence on lumenal pH (data reproducedfrom Van Dyke and Belcher, 1994). For each of the two vesiclepopulations, the true buffering curve was approximated by alinear interpolation of the buffering data (solid lines). Thesecurves were used to simulate the acidification of the respectivevesicle populations in Fig. 4 (text). In addition, the effect onacidification of a constant versus nonconstant buffering curve(dashed line) is explored in Fig. 3, A and B (text).

Surface Potentials

Here we address how our description of pH regulation is affected by negatively charged lipids on the cytoplasmicleaflet of organelles. First consider an organelle permeable to ions, but with no surface charge or pumps. At equi-librium, only a Donnan potential, ��, will exist across the organelle due to the impermeable macro-ions. If a nega-tive surface potential is imposed on the organelle, ��, the surface values of positive (negative) ions in solutionwill increase (decrease) by a Boltzmann factor (see Eq. 4 in text). The concomitant flow of ions quickly builds upan opposing membrane potential, �� → �� . Changes in lumenal concentrations are marginal since very fewions are required to produce a sizable membrane potential. Thus, the addition of a surface potential shifts the bal-ance of the (equilibrium) electrochemical potential from a chemical to an electrical gradient, while the magnituderemains unchanged. However, steady state and transient dynamics do depend upon surface potentials since chan-nel and pump activities depend on the relative magnitudes of the concentration and electrical components of theelectrochemical potential.

Consider the determination of Golgi proton permeability in Fig. 6, inset (text). In the absence of surface charge,membrane potentials are very small and an analytic analysis is possible. To see how this simulation depends on sur-face potential, we must consider our model for proton flux:

(2)

When the reduced surface potential, ��, is added [H�]C → [H�] C � exp(��) and U → U � ��, as discussedabove. Because of the exponential Boltzmann factor, the membrane potential is no longer negligible. The protonflux is:

(3)

where the approximation U � �� has been made. When �� 50 mV, the cofactor in the last equation�2.27. When the simulation is rerun without the surface potential, the best fit permeability is exactly 2.27� largerthan the one reported in Table IV (main article text).

From this example, we see that proton fluxes can be two to three times smaller in the absence of surface charge.This makes the pH simulations in Figs. 3 and 4 (text) undershoot the present trajectories by �0.2–0.3 pH units, ifthe proton pump remains unaffected by the changes in membrane potential and surface concentrations. However,in our model for the proton pump, the rate-limiting step near cytoplasmic pH values is the capture of a protonfrom the cytoplasm. Removal of the �50-mV surface potential lowers the surface concentration of protons by a fac-tor of 10. This decreases the pumping rate by �1.7� when the lumen is near neutral pH. Examination of the so-dium performance surface (Fig. 2 B, text) in the absence of a surface potential reveals a less dramatic influence onthe dynamics (�6 Na�/s). Thus, only the exact pump numbers and permeability values change in the absence ofsurface potentials, and the nature of our presentation is unaffected.

r e f e r e n c e s

Alberts, B., D. Bray, J. Lewis, M. Raff, K. Roberts, and J. Watson. 1994. Molecular Biology of the Cell. 3rd ed. Garland Press, New York, NY.1294 pp.

Boekema, E., T. Ubbink-Kok, J. Lolkema, A. Brisson, and W. Konings. 1997. Visualization of a peripheral stalk in V-type ATPase: evidence forthe stator structure essential to rotational catalysis. Proc. Natl. Acad. Sci. USA. 94:14291–14293.

Dimroth, P., H. Wang, M. Grabe, and G. Oster. 1999. Energy transduction in the sodium F-ATPase of Propionigenium modestum. Proc. Natl.Acad. Sci. USA. 96:4924–4929.

Eisenberg, E., and T. Hill. 1985. Muscle contraction and free energy transduction in biological systems. Science. 227:999–1006.Elston, T., H. Wang, and G. Oster. 1998. Energy transduction in ATP synthase. Nature. 391:510–514.Farinas, J., and A.S. Verkman. 1999. Receptor-mediated targeting of fluorescent probes in living cells. J. Biol. Chem. 274:7603–7606.Finbow, M., and M. Harrison. 1997. The vacuolar H�-ATPase: a universal proton pump of eukaryotes. Biochem. J. 324:697–712.Forgac, M. 1999. Structure and properties of the vacuolar (H�)-ATPases. J. Biol. Chem. 274:12951–12954.Forgac, M. 2000. Structure, mechanism and regulation of the clathrin-coated vesicle and yeast vacuolar H�-ATPases. J. Exp. Biol. 203:71–80.

Jφ 0= P S U H�[ ] L H�[ ] C– e U–⋅( )1 e U––

------------------------------------------------------⋅ ⋅ ⋅= P S H�[ ] L H�[ ] C– e U–⋅( ).⋅ ⋅≈

J∆φ 0≠ P S U ∆Φ+( ) H�[ ] L H�[ ] C– e U ∆+ Φ( )–⋅{ }1 e U ∆+ Φ( )––

--------------------------------------------------------------------⋅ ⋅ ⋅=

P S ∆Φ H�[ ] L H�[ ] C– e U–⋅( )1 e ∆Φ––

------------------------------------------------------⋅ ⋅ ⋅≈

J∆φ 0=∆Φ

1 e ∆Φ––-------------------

,⋅=


Grabe, M., H. Wang, and G. Oster. 2000. The mechanochemistry of V-ATPase proton pumps. Biophys. J. 78:2798–2813.Heyse, S., I. Wuddel, H.-J. Apell, and W. Stuermer. 1994. Partial reactions of the Na,K-ATPase: determination of rate constants. J. Gen. Phys-

iol. 104:197–240.Junge, W., H. Lill, and S. Engelbrecht. 1997. ATP synthase: an electro-chemical transducer with rotatory mechanics. Trends Biochem. Sci. 22:

420–423.Murata, T., K. Igarashi, Y. Kakinuma, and I. Yamato. 2000. Na� binding of V-type Na�-ATPase in Enterococcus hirae. J. Biol. Chem. 275:13415–

13419.Press, W.H. 1997. Numerical Recipes in C: The Art of Scientific Computing. 2nd ed. Cambridge University Press, New York, NY. 994 pp.Roos, A., and W.F. Boron. 1981. Intracellular pH. Physiol. Rev. 61:296–434.Sagar, A., and R.F. Rakowski. 1994. Access channel model for the voltage dependence of the forward-running Na�/K� pump. J. Gen. Physiol.

103:869–894.Sokolov, V.S., H.-J. Apell, J.E.T. Corrie, and D.R. Trentham. 1998. Fast transient currents in Na,K-ATPase induced by ATP concentration

jumps from the P3-(1-(3�,5�-dimethoxyphenyl)-2-phenyl-2-oxo)ethyl ester of ATP. Biophys. J. 74:2285–2298.Van Dyke, R.W., and J.D. Belcher. 1994. Acidification of three types of liver endocytic vesicles: similarities and differences. Am. J. Physiol. 266:

C81–C94.Vik, S.B., and B.J. Antonio. 1994. A mechanism of proton translocation by F1F0 ATP synthases suggested by double mutants of the a sub-

unit. J. Biol. Chem. 269:30364–30369.Wu, M., J. Llopis, S. Adams, J. McCaffery, T. Machen, H.P. Moore, and R. Tsien. 2000. Using targeted avidin and membrane permeant fluo-

rescent biotin to study secretory pathway organelle pH regulation. Chem. Biol. 7:197–209.


1

Running Supplemental Programs

Programs for use with:J. Gen Physiol. Vol. 117 No. 4 p. 329, M. Grabe and G. OsterRegulation of Organelle Acidity

Download FilesThere are several files included in the supp_prog.zip file that you should have downloaded fromhttp://www.jgp.org/cgi/content/full/114/4/329/DC1

1. This Read Me file.2. pH_regulation.mmd.3. v_pump.m; cal_2.m; vel_a.m; v_param.m; barrier.m; delc_cf.m; pot.m; pot2.m.4. nak_pump.m; delta.m; nak_param.m.

The above program files require two pieces of commercial software to run. Item 2 , a Berkeley MadonnaTM

program file, was used to compute the curves in Figure 3B of the manuscript. Items 3 and 4 are Matlab™program files used to compute the ion pumping surfaces in Figure 2. Below outlines how to go aboutgetting and running these software packages.

http://www.jgp.org/cgi/content/full/117/4/329/DC1

2

Berkeley MadonnaTM

InstallationYou can download a Macintosh or PC version of Berkeley MadonnaTM from:http://www.berkeleymadonna.com

Once you have downloaded the program you will have to install the Java Runtime Environment™ (JRE)on your machine in order to use the Visual Interface. Java™ is only required for the visual interface; ifyou can work from the Equation window you do not need Java™. From the link above, sources for boththe Macintosh and PC versions of Java can be reached. You must purchase a registration number in orderto use all of the features of Berkeley MadonnaTM, such as saving and printing files. Upon purchase, asystem ID will be sent to you and no further installation will be required.

RunningAfter installation, you should be able to click on and open the file pH_regulation.mmd. If this fails, tryopening the file from inside Berkeley MadonnaTM. Upon opening the file a Notes file should appear thatdescribes some basics and how to get started. If this file does not appear, choose Notes from under theEdit or Model menu. Having a copy of the manuscript will help keep track of the symbols andequations.

http://www.berkeleymadonna.com

3

MATLAB™

InstallationMatlab™ must be installed on the machine that you are using before you can run the programs in items 3and 4 above. This program is not recommended for novices. Additional information can be obtained fromhttp://www.mathworks.com

RunningCollect all files from Item 3 or Item 4 into one folder. Start the Matlab™ program from this directory. Thisis usually done by typing ‘matlab’ at the command prompt in UNIX. On the PC version, this step is morecomplicated and involves setting the Matlab™ path by hand. Once Matlab™ has started, type v_pump ornak_pump in order to run the respective codes. For each model, these are the main functions and allother functions are called from within the program. If any of the files are missing an error will occur.Otherwise, the command prompt will return in a few minutes when the computation is complete. Keyparameters can be found and manipulated before each run in the files nak_param.m and v_param.m.These files should be opened in a text editor.

V-ATPase model

When the code has finished running, an array, v_flux, will be in the Matlab™ command window. Thearray contains information on the proton pumping rate of the enzyme under different membranepotentials and pH gradients. The key entries are:

v_flux(i=1:N, j=1:M, 1) = Ψ across the enzyme [mV]v_flux(i=1:N, j=1:M, 2) = Lumenal pHv_flux(i=1:N, j=1:M, 3) = Proton flux into the lumen [H+/sec]

In this case, the flux is computed for N different values of Ψ and M different values of lumenal pH.

Na,K-ATPase model

When the code has finished running, several important arrays will be in the Matlab™ command window.The arrays contains information on the ion pumping rate of the enzyme under different membranepotentials and ion gradients. The key elements are:

kflux_psi(i=1:N, j=1:M, k=1:O) = Potassium flux into the lumen [K+/sec]naflux_psi(i=1:N, j=1:M, k=1:O) = Sodium flux into the lumen [Na+/sec]psilist(i=1:N, j=1:M, k=1:O) = Ψ across the enzyme [mV]klist(i=1:N, j=1:M, k=1:O) = Lumenal potassium concentration [Molar]nalist(i=1:N, j=1:M, k=1:O) = Lumenal sodium concentration [Molar]

In this case, the flux is computed for N different concentrations of lumenal potassium, M differentconcentrations of lumenal sodium, and O different membrane potentials.

Reading the Detailed Calculations and Discussion included with the Supplemental Material will behelpful in understanding this code.

These instructions were written solely by the author of this article, with the exception of possible file name changesmade by The Rockefeller University Press. Any suggested software downloads and purchases do not imply endorsementof the mentioned products by The Journal of General Physiology, The Rockefeller University Press, or HighWire Press, norare these organizations liable for any deleterious effects experienced as a result of downloading or running theapplications.

http://www.mathworks.com

Michael Grabe* and George Oster - nature.berkeley.edu

Documents

Transcript of Michael Grabe* and George Oster - nature.berkeley.edu