The logic of ionic homeostasis: Cations are for voltage, but not for volume
[['The transmembrane movements of ions during neuronal activity are inevitably associated with changes in ionic concentrations , both intracellularly and extracellularly .', 'This activity-dependent redistribution of ions can be osmotically imbalanced and consequently can lead to changes in volume of the cells and of the extracellular space ( ECS ) .', 'It is usually assumed that three principal ions–Na+ , K+ , and Cl-—are responsible for the link between transmembrane conductance , voltage and volume alterations associated with neuronal activity .', 'Among them , extracellular K+ experiences the largest relative activity-dependent changes .', 'The increase of the extracellular K+ concentration ( [K+]o ) during neural activity is the easiest to detect , and was recorded first [1 , 2] , using ion-selective microelectrodes [3] .', 'Soon it was discovered that the increased [K+]o induced by electrical stimulation was associated with a 50% reduction of ECS volume ( [4] in the cortex of cat ) .', 'However , it also was shown that the relationship between ions , voltage and volume was not simple , since in some other layers of the cortex , shrinkage of ECS was not detected in spite of considerable elevation of [K+]o [4] .', 'The correlation between the increase of [K+]o and decrease of ECS volume was found in different parts of the nervous system under various conditions ( honey bee eye , light stimulation—[5]; optic nerve , electrical stimulation—[6]; spinal cord , electrical stimulation—[7]; cortex during spreading depression—[8] ) .', 'However , when extracellular concentrations of Na+ and Cl- ( [Na+]o and [Cl-]o ) were measured to obtain a full picture of the anticipated osmotic imbalance of ions , the results were sometimes confusing .', 'For instance , it was shown that the decrease of ECS volume evoked by electrical stimulation was indeed accompanied by a decrease of [Na+]o , since Na+ enters cells during stimulation , but this decrease had the same amplitude as the [K+]o increase , and [Cl-]o started to change only after the stimulation , during so called “self-sustained neuronal afterdischarges” [9] .', 'During spreading depression , when ECS was reduced to one fourth of its original volume [8] , a large increase of [K+]o was indeed exceeded by even larger decrease of [Na+]o [10 , 11] .', 'But the decrease of [Cl-]o was markedly larger ( by 12–31 mM ) than needed for electrical compensation of extracellular cation deficiency .', 'As discussed further below , these changes still have not been fully explained with a mechanistic model .', 'The vertebrate retina presents a case of special interest in this respect because it consists of cells that respond to light differently–with predominantly depolarization in proximal layers ( ganglion cells , amacrine cells; see S1 Text ) , but with hyperpolarization in distal layers ( photoreceptors , horizontal cells ) .', 'Accordingly , during illumination the ECS volume decreases in proximal retina , but increases in distal retina [12 , 13] .', 'In proximal retina , similarly to brain , ECS shrinkage is associated with a [K+]o increase and also with a larger [Na+]o decrease and with a compensating [Cl-]o decrease [14] .', 'In distal retina the changes are reversed: ECS expansion is associated with a [K+]o decrease and with a larger [Na+]o increase [14] .', 'However , [Cl-]o still decreases in the outer retina when it is expected to increase in order to compensate for total ECS cation excess .', 'It should be noted that although measuring extracellular ion concentrations with ion-selective microelectrodes is the best available method to obtain data on ionic redistribution during neuronal activity , these measurements are technically difficult , and results must be interpreted with caution .', 'In the case of [Na+]o and [Cl-]o changes , the measured voltage changes of the ion-selective electrode are small ( except in spreading depression ) and can be partly compromised by possible electrical artifact arising from a combination of changes in field potential and the huge resistance of the ion-selective microelectrode [14 , 15] .', 'Also , the sensors are not absolutely selective; for instance , some Na+-selective sensors are influenced by changes in extracellular Ca2+ [9] , and some Cl--selective sensors respond to pH fluctuations [14] .', 'Most importantly , the changes of extracellular ions and volume evoked by neuronal activity often stimulate reactions of ever present nearby glial elements , which in turn can alter those ionic changes and affect ECS volume [16 , 17] .', 'Nevertheless , it is possible to conclude that measurements with ion-selective microelectrodes revealed a certain pattern of related changes in voltage , volume and distribution of the principal ions ( K+ , Na+ , and Cl- ) that more or less consistently ( within the limits of the method ) repeats itself in various parts of the nervous system .', 'Neuronal excitation , which is in most cases associated with depolarization due to an increase of Na+ conductance , results in Na+ influx responsible for the [Na+]o decrease .', 'This Na+ influx is electrically compensated partly by outward K+ flux and partly by inward Cl- flux , which leads to an increase of [K+]o and a decrease of [Cl-]o .', 'The consequent transmembrane transfer of NaCl into the cells evokes osmotically obliged movement of water , decreases ECS volume , and increases the cell volume .', 'When the neuronal activity is associated with hyperpolarization ( like light-induced responses of photoreceptors and horizontal cells in the distal part of the vertebrate retina ) the changes of ion concentrations and volumes have opposite signs .', 'However , the reasons for the voltage-dependent , osmotically imbalanced redistribution of the main ions that lead to volume changes are not entirely clear .', 'Movement of a very small amount of ions is enough for changes of the membrane potential , and that amount has no practical effect on ionic concentration , so ion and volume changes must be more complex than expected from membrane potential considerations alone .', 'The ionic concentrations are changed when the precise balance between influxes and effluxes through membrane passive and active transport systems that existed at rest is temporarily disturbed during activity .', 'For instance , the light-induced decrease of [K+]o in the distal retina is a result of a temporal inequity between the passive K+ leak out of the photoreceptors , which is quickly reduced by the hyperpolarization , and the active K+ pumping into the cell by Na+/K+-ATPase , which needs time for adjustment [18] .', 'It is natural to assume that the osmotically imbalanced ionic changes that lead to changes in volume could be a result of the unequal exchange of Na+ for K+ by the Na+/K+-ATPase ( 3 Na+ out of a cell for 2 K+ in ) .', 'Because of this , the decrease of [Na+]o during neuronal depolarization could be expected to be larger than the increase of [K+]o .', 'That cation imbalance must be electrically compensated by the extracellular decrease in an anion ( most probably by Cl- ) concentration , further diminishing the extracellular osmolarity compared to the intracellular .', 'On the other hand , the passive transmembrane Cl- flux itself is directly affected by changes in membrane potential–attracted into the cell by depolarization and repelled out of the cell by hyperpolarization .', 'In this case , appropriate Na+ and/or K+ flux is needed for electroneutrality , creating osmotically active NaCl/KCl transfer .', 'Thus , there is currently no simple and single explanation for the link between voltage , volume and ions in the nervous system .', 'Is unequal exchange of Na+ for K+ by the Na+/K+-ATPase responsible for osmotically imbalanced redistribution of ions ?', 'Or is ion redistribution the consequence of the direct influence of the changing membrane potential on Cl- flux ?', 'Maybe both factors play their roles; in this case , what is the contribution of each ?', 'Here we will use computational modeling to provide answers for those questions .', 'Numerous computational models that aim to understand the interrelations among ions , voltage and volume have been developed recently [19–25] , ( for review see [26 , 27] ) .', 'But most of them are based on modifications of the Goldman equation , and the limitations of this approach ( particularly for a dynamic , changing system ) have been well described [27] .', 'Alternatively , a much more attractive “charge-difference” method for calculations of simultaneously changing ionic concentrations , membrane potential , and cell volume was introduced [28] .', 'In this work a charge-difference computational model was used for further understanding of the link between ions , voltage , and volume .', 'The list of new and improved features that make our program different from that published earlier is presented in the Methods .', 'The focus was on the specific roles for cations and anions , which is probably the most important feature of the work distinguishing it from current literature , where Na+ , K+ , and Cl- were usually treated together .', 'Special attention was directed to the energy requirement to support voltage and volume changes .', 'It also was demonstrated that , contrary to intuitive assumptions , changes of the membrane potential do not necessarily lead to changes in volume , and changes of volume can have no effect on the cation concentration and the membrane potential .', 'Additionally , both Donnan and Double Donnan equilibrium were reexamined .', 'Since the water permeability of the membrane is critically important for Donnan equilibrium , some simulations with various values of this parameter were performed .', 'Our program is the only one that is capable of such calculations .', 'It should be noted that bicarbonate ion , which is probably the second most important anion after Cl- , was omitted here .', 'Although HCO3- can move across cell membranes through numerous GABA and glycine channels and by certain cotransporters and exchangers , it is involved in the fundamentally important CO2/HCO3- buffering system , and its concentration is mostly determined by highly diffusible CO2 .', 'Accordingly , it is tightly linked to energy metabolism , generation and evacuation of acidic metabolic wastes and other processes that require separate study , and it is beyond the scope of this paper .', 'The software is offered to share ( https://sites . northwestern . edu/ralcomputational/ ) and significant efforts were directed toward making it flexible and user friendly .'], ['The model calculates membrane potential ( Em ) and cell volume ( Vol ) depending on extra- and intracellular concentrations of K+ , Na+ , and Cl- , their transmembrane conductances , and the activities of the Na+/K+ ATPase , Na+ , K+ , 2Cl--cotransporter and K+ , Cl--cotransporter .', 'These two cation-Cl- cotransporters were included in the model because they directly link Cl- with cations and are prevalent in the nervous system .', 'We also take into account the concentration ( [An-]i ) and mean charge valence ( z ) of intracellular membrane-impermeant osmolytes , which comprise a substantial amount of the intracellular anions .', 'In this respect our model is similar to the most advanced and recent models [23–25 , 28] .', 'Additionally , our model has some capabilities which previous models do not have .', 'First , the calculations can be performed with different values of transmembrane water permeability .', 'In existing models on volume regulation very high water permeability is accepted by default , so water instantly follows the ions and other osmolytes , changing the cell volume , but preventing any osmotic difference between the internal and external solutions .', 'In most of our simulations the water permeability was also assumed to be large in order to focus on other aspects of ion-dependent volume-voltage regulation .', 'But in one series of simulations we used the widest possible range of water permeability ( from infinity to zero ) to investigate its consequences for the cell volume and internal osmolarity .', 'In the extreme case of zero water permeability we can simulate the development of Donnan equilibrium .', 'Second , electrically neutral impermeant osmolytes ( like glucose ) can be added to extracellular fluid to simulate Double Donnan effects .', 'Third , to simulate certain experimental environments , the model can perform calculations in conditions where osmolyte concentrations change with time .', 'In this part of the work , we modeled the buildup of four different substances:', 'a ) external electrically neutral impermeant osmolytes ,', 'b ) extracellular NaCl ,', 'c ) internal electrically neutral impermeant osmolytes , and', 'd ) internal electrically charged impermeant osmolytes with the addition of an appropriate amount of Na+ to maintain electroneutrality .', 'Besides , the model permits various stoichiometries of the Na+/K+-ATPase , which has been done [24] or can be done [23] in previous work .', 'In most simulations a normal stoichiometry of 3 Na+ to 2 K+ was used , but we also examined a hypothetical case when an electroneutral pump ( 3 Na+:3 K+ ) was responsible for redistribution of the cations , and some other more exotic stoichiometries were also tested .', 'With the exception of mentioned above buildups of external NaCl and electrically neutral impermeant osmolytes , all extracellular concentrations during all other simulations were assumed to be constant ( as for an isolated cell in a Petri dish ) .', 'The parameters are expressed in an easily appreciable physical form and the values are biophysically realistic as far as possible .', 'The conductance of ions is expressed in ions/ ( sec*V ) and can be converted to the usual electrophysiological measure of conductance in Siemens .', 'For instance , the value of 2*1010 ions/ ( sec*V ) for total ionic conductance , which was often used in this work , is equal to 3 . 2*10−9 coulombs/ ( sec*V ) , i . e . 3 . 2*10−9 Siemens ( or 3 . 2 nS ) .', 'This corresponds to an input resistance of 312 . 5 MΩ , a reasonable value for a small sized neuron ( for instance , the input resistance of a starburst amacrine cell in the rabbit retina is in the range of 200–250 MΩ [29] .', 'Similarly , the basis for many parameters that were chosen here is experimental work performed mostly on vertebrate retina .', 'For example , a relatively low ratio of gK/gNa is characteristic of the photoreceptors [30 , 31] , which respond to light with a decrease of gNa [32] , as in the Results: “Conductance of Cl- and cell volume . ” .', 'The transporter activities are presented in cycles/sec for better comparison to fluxes through the conductances , since the number of cycles/sec is proportional ( and in some cases–equal ) to the number of ions transferred per second .', 'The same principles for calculations were used here as in the “charge-difference” method of Fraser and Huang [27 , 28] .', 'There was no attempt to derive an equation that describes Em and Vol from ionic concentrations and conductances and activities of transporters .', 'Instead , our program 1 ) counts transmembrane fluxes through the channels and transporters for each ion during a short time period , when ( important ! ) the conditions are assumed to be unchanged , 2 ) calculates the resulting changes in intracellular ionic concentrations , osmolarity and electrical charge at the end of this time period , and 3 ) makes appropriate adjustments of the intracellular concentrations , Em and Vol .', 'These three steps in the calculation are described in the sections below ( e . g . “The 1st set of calculations” ) .', 'This cycle repeats over and over again .', 'If a balanced ( resting ) state was reached , the combined fluxes through all appropriate channels and transporters for each ion ( K+ , Na+ , and Cl- ) would equal 0 and ionic concentrations , Em and Vol would not change .', 'When the conditions ( a conductance or activity of a transporter ) change , the balance will be disturbed , and a new balanced state will be found with potentially new values for Em and Vol , as well as for ionic concentrations .', 'Consequently , at the end of the calculation the activities of the transporters that depend on ionic concentrations can be different from their initial values .', 'The discretization of a continuous process , which is the core of this method , is an apparent idealization , but it does not affect the precision of results when the resting state is found .', 'And during the dynamic phase , any desired level of accuracy can be achieved by choosing an appropriately short discrete time step .', 'In this work the duration of the time step was 0 . 1 , 0 . 5 , or 1 . 0 millisecond .', 'After choosing an initial set of concentrations and pump rates , transmembrane fluxes for each ion ( in ions/sec ) through each transport system are calculated .', 'Inward fluxes are assumed to have a positive sign , and outward fluxes are considered to be negative , in accordance with the way they affect intracellular ionic concentrations .', 'First , fluxes are added separately for each ion and the sums are multiplied by the time step to produce the amount of ions that were moved in or out of the cell .', 'Then the amounts are converted into concentrations .', 'Buildups of extra- or intracellular osmolytes defined by the investigator are also taken into account .', 'Buildups are distinguished from fluxes because these are exogenous substances that are added de novo to one side of the membrane at a specified rate .', 'They then can affect the concentrations of substances , but never affect electrical charge , since they are either electrically neutral substances or an electrically balanced combination of cations and anions .', 'The concentrations at the beginning of the current time step are marked with index b and the concentrations at the end of the time step ( yet before possible adjustments for volume changes ) are marked with the index e .', 'To obtain the changes in total intracellular electrical charge , the change in the amount of Cl- is subtracted from the change in the amount of cations and the result is multiplied by the charge of one cation .', 'dNa= ( NaFc+NaFp+NaFnkc ) *st;', '( 12 )', 'dK= ( KFc+KFp+KFnkc+KFkc ) *st;', '( 13 )', 'dCl= ( ClFc+ClFpc+ClFnkc+ClFkc ) *st;', '( 14 )', '[osm]o , e=[osm]o , b+bOso*st;', '( 15 )', '[Na+]o , e=[Na+]o , b+bNaCl*st;', '( 16 )', '[Cl‑]o , e=[Cl‑]o , b+bNaCl*st;', '( 17 )', '[osm]i , e=[osm]i , b+bOsi*st;', '( 18 )', '[An‑]i , e=[An‑]i , b+bAn*st;', '( 19 )', '[Na+]i , e=[Na+]i , b+dNa/ ( vole*L ) +bAn*st* ( ‑z ) ;', '( 20 )', '[K+]i , e=[K+]i , b+dK/ ( vole*L ) ;', '( 21 )', '[Cl‑]i , e=[Cl‑]i , b+dCl/ ( vole*L ) ;', '( 22 )', 'dQ= ( dNa+dK–dCl ) *e;', '( 23 )', 'where dNa , dK , and dCl are changes in the intracellular amount of respective ions; st is the duration of the discrete time step , [osm]o and [osm]i are extra- and intracellular concentrations of impermeant neutral osmolytes; [An-]i is the concentration of internal impermeant anion; z is the mean valence of An; bOso , bOsi , bNaCl , and bAn are “buildups” of extra- and intracellular impermeant neutral osmolytes , external NaCl and of internal impermeant anion , respectively; vole is the cell volume at the beginning of the time step; L is Avogadro’s number ( 6 . 02*1023 mol-1 ) ; dQ is the change in internal electrical charge ( in coulombs ) ; and e is the electrical charge of one cation ( 1 . 6*10−19 coulomb ) .', '“Buildups” are inputs to the model to allow gradual changes in applied concentrations over some period of time .', 'Notes: All fluxes are added to each other algebraically .', 'For instance , KFp and KFnkc are positive and increase the amount of intracellular K+ , but KFc and KFkc are negative and decrease it ( Eq 13 ) .', 'Addition of Cl- ( dCl ) increases the intracellular osmolarity , but it decreases the total intracellular electrical charge; so dCl is subtracted from total charge ( Eq 23 ) .', 'The buildup of internal impermeant anion ( bAn ) is associated for electroneutrality with buildup of internal Na+ , which must be multiplied by the mean valence of An- ( -z , Eq 20 ) .', 'The changes in the ionic concentrations obtained in the 2nd set of calculations can affect the intracellular osmolarity .', 'Buildup of extra- or intracellular osmolytes , if defined by the investigator , will result in an additional imbalance in osmolarity .', 'In the 3rd set of calculations , water is allowed to move across the cell membrane to restore osmotic equilibrium by changing the cell volume .', 'The calculations can be performed with different transmembrane water permeability , i . e . with different rates of osmotically driven adjustment of the cell volume .', 'This is an important difference of our program from previous models concerning volume regulation .', 'The change in Em is found from the change in intracellular charge ( dQ ) divided by the cell membrane capacitance', '( c ) .', 'The concentrations , Em and vol at the end of the current time step , but before the final volume adjustments , are marked with index e and after it with the index f .', 'Em , f=Em , e+dQ/c;', '( 24 )', 'osV= ( [Na+]i , e+[K+]i , e+[Cl‑]ie+[An‑]i , e+[osm]i ) / ( [Na+]o , e+[K+]o , e+[Cl‑]o , e+[osm]o ) ;', '( 25 )', 'VoR=1/tau;', '( 26 )', 'chV=1‑ ( 1‑osV ) *VoR*st;', '( 27 )', 'volf=vole*chV;', '( 28 )', '[K+]i , f=[K+]i , e/chV;', '( 29 )', '[Na+]i , f=[Na+]i , e/chV;', '( 30 )', '[Cl‑]i , f=[Cl‑]i , e/chV;', '( 31 )', '[An‑]i , f=[An‑]i , e/chV;', '( 32 )', '[osm]i , f=[osm]i , e/chV;', '( 33 )', 'where tau is the time constant of exponential changes of the cell volume in response to a sudden change in osmolarity , VoR is a constant inversely related to tau , but VoR = 1/st if tau < st and VoR = 0 if tau > 108 sec ( more than 3 years ) , osV is a coefficient of osmosis driven volume changes assuming instant transmembrane water movement , chV is a coefficient of osmosis driven volume changes corrected for limited water permeability of the membrane .', 'Notes: The calculations in this part are straightforward , but the section concerning the water permeability and the time constant of volume changes ( Eqs 26 and 27 ) should be explained .', 'Imbalance between intra- and extracellular osmolarity will induce transmembrane movement of water that changes the cell volume and restores osmotic equilibrium .', 'If membrane water permeability is assumed to be infinite , the water shift and volume changes would happen instantly .', 'In reality , the water permeability is high , but not infinite .', 'It is also different in different cell types .', 'The higher the water permeability , the faster the volume change .', 'The water flux , and consequently , the speed of volume changes also depends on the osmotic gradient , and both the flux and the speed decrease with time as the gradient diminishes .', 'This is similar to the well described discharge of a capacitor in a simple RC circuit .', 'In our case the cell volume is analogous to the voltage of the RC circuit , and membrane water resistance ( which is the inverse of water permeability ) –to the electrical resistance .', 'Thus , the dynamics of cell volume changes evoked by a sudden shift in osmolarity can be described , by analogy with an RC circuit , with the following equation of exponential decay:', 'vol ( t ) =vol1+ ( vol2–vol1 ) * ( 1–e−t/tau )', '( 34 )', 'where vol ( t ) is the cell volume in time t , vol1 is the initial cell volume , vol2 is the final cell volume with osmolarity equilibrated , and tau is a time constant , which is inversely proportional to the membrane water permeability .', 'Thus , tau is a convenient measure of water permeability , and if water permeability is such that the time constant of the volume change is 1 minute , it means that after 1 minute the cell experiences 63 . 2% of the expected volume change .', 'Twofold smaller water permeability will correspond to twofold larger time constant of the volume changes , and the cell will need 2 minutes for 63 . 2% of the volume change .', 'Eq 25 determines the coefficient osV , by which the cell volume and consequently intracellular concentrations must be corrected in order to return to osmotic equilibrium .', 'But due to limited water permeability another coefficient ( chV ) , which is a fraction of osV and determined by Eq 27 , is used later for correction of the cell volume and the intracellular concentrations .', 'The smaller tau , the larger chV ( and closer to osV ) .', 'For practical purposes , in Eq 27 we used VoR , which is the inverse of tau .', 'It permits a pair of convenient conditions .', 'If tau determined by the investigator is 0 or just shorter than the time step ( st ) , VoR is assumed to be equal to 1/st; as a result osV = chV , i . e . water instantly corrects the cell volume and osmolarity .', 'If extremely low water permeability with tau > 108 sec is chosen , VoR is assumed to be equal to 0; as a result osV = 1 , i . e . water does not move across the membrane at all as in simulation of Donnan equilibrium .', 'After completing the 3rd set of calculations , the program moves forward in time by one time step and repeats all the cycles .', 'Our modeled cell is assumed to have an effective volume of 7 . 5*10−13 L , equivalent to a cube 10x10x10 μm , with 25% of the volume occupied by organelles ( like in rat rod photoreceptors [38] ) .', 'All calculations are done with respect to this effective cell volume of water containing ions and other osmolytes .', 'Since the surface area of a cube 10x10x10 μm is equal to 6*10−6 cm2 , the total membrane capacitance of the cell is 1 . 2*10−11 F ( given a specific capacitance for a neuronal membrane of 2 μF/cm2 ) , and remains the same in all calculations .', 'During volume changes cells usually alter their shape but keep the same surface area ( for a recent reference see [39] ) .', 'We specifically pointed out that our model cell is a cube , which permits an increase in cell volume by almost 40% by turning it into a sphere without any changes to surface area , and consequentially with no change in capacitance .', 'During our simulations the volume increases were usually within this range , except in the catastrophic occasion of a swelling cell whose membrane was permeable to Na+ , Cl- and water , but lacked Na+/K+-ATPase ( Figs 1D and 2C ) .', 'Nothing prevents the cell from keeping the same surface area when the volume decreases .', 'The initial conditions and manipulations of the examined system ( concentrations , conductances , transporter activities ) varied from simulation to simulation , and their exact values are listed in the figure legends .', 'It should be noted that initial conditions that describe the starting point of the system before manipulations can refer to equilibrium states ( Figs 1–5 ) or nonequilibrium but steady states with stable concentrations and Em ( Figs 6–10 ) .', 'In the second case ionic conductances and the rate of the Na+/K+-ATPase were chosen for reasons explained in the text , but concentrations and Em that are given as initial conditions in the figure legends were the results of preparatory calculations that led to the state of the system at t = 0 .', 'Preparatory calculations were also necessary to determine initial conditions in some equilibrium states .', 'One particular set of conditions includes 6 mOsm external impermeant osmolyte and a valence of -1 . 5 for internal impermeant anions , which exemplifies an osmolarity-charge asymmetry , i . e . conditions when the quantity of equivalent charge is not the same as the quantity of osmotically active molecules .', 'Since such asymmetry is expected to be common in real cells , this set is called the “realistic” conditions and often will be compared to “simplified” conditions , which are symmetrical in osmolarity-charge respect , when the valence of internal impermeant anions is equal to -1 and there are no other external osmolytes besides NaCl and KCl .'], ['The classical example of a system that generates a considerable transmembrane potential without spending energy is the one based on Donnan equilibrium .', 'The conditions which can lead to Donnan equilibrium—unequal distribution of Cl- across the cell membrane and selective permeability of the membrane to cation ( s ) and to Cl- , but not to other intracellular anions–are typical for living cells , including neurons .', 'Those impermeant “other intracellular anions” consist of a diverse group of large and small molecules which contribute noticeably to the voltage-volume regulation , and they are addressed specifically later .', 'For the current simulation of Donnan equilibrium let us just assume that the impermeant anions ( [An-]i ) account for most of the internal anion concentration ( 135 mM ) and have a mean valence = -1 . At the beginning ( before time = 0 ) the membrane is permeable to nothing , the membrane potential is 0 mV , and for the osmotic and electrical balance the extracellular Cl- concentration ( [Cl-]o ) is 150 mM , the intracellular Cl- concentration ( [Cl-]i ) is 15 mM , and cation concentrations ( in this case Na+ ) are 150 mM , both inside and outside .', 'The membrane remains impermeable to water , so the cell volume does not change .', 'The results of the simulations are presented in Fig 1A and 1B .', 'At time = 0 the cell begins to be permeable to Cl- and to Na+ .', 'In the simulation , gNa and gCl are set to be equal .', 'Cl- immediately starts to move into the cell because [Cl-]o is 10 times larger than [Cl-]i .', 'Rapidly the influx of Cl- brings negative charge into the cell and hyperpolarizes the membrane .', 'This creates an electrical driving force for Na+ influx , and the entering Na+ partly neutralizes intracellular negativity .', 'As a result , Cl- continues to enter because its strong inward concentration-dependent driving force exceeds its outward electrical driving force .', 'Na+ also continues to enter because its inward electrical driving force exceeds its weak outward concentration-dependent driving force .', 'In 20 minutes both [Na+]i and [Cl-]i are increased by 81 . 8 mM and practically stabilized .', 'At these concentrations ( 231 . 8 mM for [Na+]i and 96 . 8 mM for [Cl-]i ) their Nernst potentials are both equal to the membrane potential Em = - 11 . 6 mV ( Fig 1B ) , which means that the concentration-driven fluxes are counter-balanced by electrically-driven fluxes for both ions and the system has reached a stable equilibrium .', 'This is Donnan equilibrium , and as expected [Na+]i * [Cl-]i = [Na+]o * [Cl-]o .', 'Indeed , 231 . 8 * 96 . 8 = 22438 . 24 ≈ 22500 = 150 * 150 .', 'Additional time is required to achieve a more accurate fit between the results of the calculation and the Donnan expectations .', 'With a chosen accuracy of 15 significant digits , the concentrations are finally stabilized after about 90 min at 231 . 986477689649 mM for [Na+]i and 96 . 9884116601443 mM for [Cl-]i; their product is 22499 . 9999977506 .', 'Special calculations were not performed for validating the model but results like these ( as well as some others that will be presented later ) demonstrate the model’s validity .', 'It is also worth mentioning that the initial change of Em , which looks instantaneous in Fig 1B , in reality decreases exponentially with a time constant of 3 . 7 ms ( see S1 Fig ) , and this is exactly what is expected in our cell , which was set to have an input resistance of 312 . 5 MΩ and a membrane capacitance of 12 pF .', 'Increasing both [Na+]i and [Cl-]i , of course , increases the total intracellular osmolarity .', 'In this example the intracellular osmolarity increased from 300 to 463 . 6 mOsm ( [Na+]i = 231 . 8 mM , [Cl-]i = 96 . 8 mM , and [An-]i = 135 mM ) , creating a strong osmotic gradient .', 'Thus , another fundamental condition that is necessary for Donnan equilibrium is the prevention of transmembrane movement of osmotically obliged water .', 'Otherwise the water enters the cell following ions , increasing the cell volume and diluting ionic concentrations .', 'Exactly that happened when the same simulations as for Fig 1A and 1B were repeated , but under the assumption that the membrane was highly water-permeable , and water instantly compensated for the potential osmotic imbalance associated with ionic transfer ( Fig 1C and 1D ) .', 'As in the previous simulation , opening of Cl- and Na+ conductances permits both ions to enter the cell ( Cl- due to the concentration gradient , and Na+ because of the intracellular negat
['Introduction', 'Methods', 'Results', 'Discussion']
['Neuronal activity is associated with transmembrane ionic redistribution , which can lead to an osmotic imbalance .', 'Accordingly , activity-dependent changes of the membrane potential are sometimes accompanied by changes in intracellular and/or extracellular volume .', 'Experimental data that include distributions of ions and volume during neuronal activity are rare and rather inconsistent partly due to the technical difficulty of performing such measurements .', 'However , progress in understanding the interrelations among ions , voltage and volume has been achieved recently by computational modelling , particularly “charge-difference” modelling .', 'In this work a charge-difference computational model was used for further understanding of the specific roles for cations and anions .', 'Our simulations show that without anion conductances the transmembrane movements of cations are always osmotically balanced , regardless of the stoichiometry of the pump or the ratio of Na+ and K+ conductances .', 'Yet any changes in cation conductance or pump activity are associated with changes of the membrane potential , even when a hypothetically electroneutral pump is used in calculations and K+ and Na+ conductances are equal .', 'On the other hand , when a Cl- conductance is present , the only way to keep the Cl-equilibrium potential in accordance with the changed membrane potential is to adjust cell volume .', 'Importantly , this voltage-evoked Cl--dependent volume change does not affect intracellular cation concentrations or the amount of energy that is necessary to support the system .', 'Taking other factors into consideration ( i . e . the presence of internal impermeant poly-anions , the activity of cation-Cl- cotransporters , and the buildup of intra- and extracellular osmolytes , both charged and electroneutral ) adds complexity , but does not change the main principles .']
['We have developed software that calculates membrane potential and cell volume that result from redistribution of principal ions ( K+ , Na+ , and Cl- ) during normal cellular activity and experimental manipulations .', 'Calculations in the model are done by an iterative charge-difference method that makes few assumptions about governing equations .', 'Most of the features that were considered to be important for volume and voltage regulation were incorporated in the model , including the unique capability to perform calculations with different values of transmembrane water permeability .', 'We have used the program to reexamine interactions between ionic fluxes , membrane potential , and cell volume and found that there was a previously unappreciated difference in the way that the distribution of cations and anions affect the cell .', 'Na+ and K+ , which are distributed unevenly across the membrane by the Na+/K+-ATPase , are primarily responsible for the membrane potential , but , contrary to popular belief , do not directly participate in volume regulation .', 'On the other hand , the Cl- conductance determines the extent of volume changes , because Cl- has to follow the changes of membrane potential , which inevitably leads to changes in cell volume .', 'The software is available to download and use for other investigations .']
['medicine and health sciences', 'membrane potential', 'ions', 'anions', 'permeability', 'electrophysiology', 'materials science', 'cations', 'cellular structures and organelles', 'physical chemistry', 'chemistry', 'cell membranes', 'hyperpolarization', 'intracellular membranes', 'cell biology', 'physiology', 'biology and life sciences', 'physical sciences', 'material properties']