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Distribution of the Na/K Pumps’ Turnover Rates As a Function of Membrane Potential, Temperature, and Ion Concentration Gradients and Effect of Fluctuations Feiran Huang, David Rabson, and Wei Chen* Department of Physics, UniVersity of South Florida, Tampa, Florida 33620 ReceiVed: June 19, 2008; ReVised Manuscript ReceiVed: April 7, 2009
Because of structural independence of the Na/K pump molecules, the pumping rates of individual pumps may not be the same, instead showing some sort of distribution. Detailed information about the distribution has not previously been reported. The pumping rate of Na/K pumps depends on many parameters, such as membrane potential, temperature, and ion concentration gradients across the cell membrane. Fluctuation of any of the variables will change the pumping rate, resulting in a distribution. On the basis of a simplified six-state model, a steady-state pumping flux and therefore the pumping rate were obtained. Parameters were determined based on previous experimental results on amphibian skeletal muscle and theoretical work. Gaussian fluctuations of all the variables were considered to determine the changes in the pumping rate. These variable fluctuations may be totally independent or correlated to each other. The results showed that the pumping rates of the Na/K pumps are distributed in an asymmetric profile, which has a higher probability at the lower pumping rate. We present a model distribution of pumping rates as a function of temperature, membrane potential, and ion concentration. Introduction In contrast to ion channels, most of which are in a closed state at the membrane resting potential, many electrogenic pumps, such as the Na/K pump, run in a wide range of membrane potentials. The pumping rates of individual pumps may not be the same, and their pumping pace may be randomly distributed. Because only 3 Na and 2 K ions are transported in each pumping cycle, one has always to measure large numbers of pumps. Consequently, the pump currents show only outward currents due to the cancellation of the inward K and outward Na currents from different pumps. Recently, we developed a technique of synchronization modulation to electrically activate the pumping rate of Na/K pumps1-5 on amphibian skeletal muscle fibers and mammalian cardiomyocytes. The first step in this technique is to use an oscillating electric field to synchronize the pump molecules to work at the same pumping rate and phase. As a result, the synchronized pumps show separated inward and outward current components. The magnitude ratio of the outward over inward currents exhibits 3:2 reflecting the stoichiometric number. In our experiments, we found that with different oscillating frequency, the magnitudes of the synchronized pump currents differ significantly. Because any oscillating electric field with a fixed frequency can only synchronize the pumps with a turnover rate comparable to the field frequency, the results indicated that the pumping rate of individual pumps may not be the same and have some kind of distribution. To date, very little is known about this distribution of the Na/K pumping rate because the traditional pump-current measurements obtained a summation of the currents from all the pumps. Gadsby et al.6,7 studied the functions of Na/K pumps and estimated a 50 Hz turnover rate of the pumps in the depolarized regime. Clearly, this estimated rate is only a mean * To whom correspondence should be addressed. E-mail: weichen@ cas.usf.edu. Phone: 813-974-5038. Fax: 813-974-5831. Website: http:// uweb.cas.usf.edu/∼chen/labwst.htm.
of all the measured pump molecules. However, in order to accelerate the pump functions, we have to synchronize as much as possible the pump molecules. To do so, we need to understand their pumping-rate distribution. In this paper, we would like to investigate the possible distribution of the pumping rate. Functions of each pump molecule depend on many factors, such as membrane potential, temperature, and ion-concentration gradient around the pump. In most of the studies, the pump currents were measured by using a voltage clamp in a thermally controlled situation. In addition, ion channels in the cell membrane were maximally blocked in the measurements. In a natural situation without voltage clamp, channel blocker, and thermal control, the environmental parameters for each individual pump may differ significantly. For example, ion channel opening, especially in excitable cells, generates a large transient channel current that may significantly affect the “local” ion concentrations, and dramatically influence the local membrane potential and temperature. All these will inevitably affect pump molecules nearby. In addition, when we apply an external electric field to cells and tissues, due to differences in cell physical parameters, such as the size, shape, and curvature of the cell membrane, and the location of the cells, pump molecules at different locations may see significantly different membrane potentials. For generality, we employed a six-state model of a carriermediated ion-exchanger to represent the Na/K pumping loop simplified from the Post-Albers model8,9 in which ion A or sodium extrusion and ion B or potassium influx occur sequentially in the loop.11-15 The model is summarized in Figure 1. E1 and E2 represent the two states of pump molecule. E1mA represents the pump molecules at E1 state binding a number m of ion A and E2nB represents the pump molecules at state E2 binding with n of ion B. The figure depicts the ion A-transport limb at the top, which consists of the electrogenic binding and unbinding steps in the two opposite narrow access channels, separated by a nonelectrogenic step representing the channel
10.1021/jp8054153 CCC: $40.75 2009 American Chemical Society Published on Web 05/15/2009
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J. Phys. Chem. B, Vol. 113, No. 23, 2009 8097 terms that can be determined by the diagram model.31 Because of rapid equilibrium for the binding and unbinding steps, these reactions can be represented by the corresponding dissociation constants as
Figure 1. An asymmetric six-state model of the Na/K pumping loop simplified from the Post-Albers model. E1 and E2 represent the two states of pump molecule. E1mA represents the pump molecules at E1 state binding m of ion A and E2nB represents the pump molecules at state E2 binding with n of ion B. The intracellular and extracellular ion concentrations are represented by ci and co, respectively, with subscripts representing ions A and B. Four voltage-dependent steps of binding and unbinding represent processes dependent upon the membrane potential. The sodium-transport limb on the top represents the electrogenic binding and unbinding steps in the two opposite narrow access channels, separated by a nonelectrogenic step representing the channel occlusion and deocclusion, protein conformational changes, and phosphorylation, with forward and backward rate- coefficients R1 and β1, respectively. Similarly, the bottom represents the binding and unbinding processes of K ions in the access channels. The in-between is the protein’s conformational change, with forward and backward ratecoefficients R2 and β2. Only the binding and unbinding steps are electrogenic.
occlusion and deocclusion, protein conformational changes, and phosphorylation. Similarly, the bottom consists of the binding and unbinding processes of ion B or K in the access channels. The in-between is the protein’s conformational change. Only the binding and unbinding steps are electrogenic. The experimental studies of the I-V curve of Na/K pumps showed a sigmoidal form with a shallow slope, saturation behavior, and possibly a negative slope at high bias.19-23 In addition, the temperature dependence of the pump molecules has been studied, and the temperature coefficient, Q10, defined as the rate of increase in pump current or flux over a 10 °C increase in temperature, ranges from 2 to 3.24-28 Furthermore, experimental studies29 have shown that ion concentration changes may shift the pumps’ I-V curve and alter its shape but keep the sigmoidal trends. All these results provided experimental bases that allow us to study the distribution profile of the pumping rate as a function of these parameters. Results from these works were used as criteria to determine the necessary parameters in the calculation. Finally, effects of fluctuations of each variable on the distribution profile of the pumping rates were studied. Simulation Method 1. Expressions for the Pumping Rate As a Function of the Membrane Potential, Temperature, and the Ion Concentrations. The six-state model can be solved for the steadystate flux rate φ as ref 23, which is the flux of the pump molecules in the pumping cycle, or the speed of the pumping cycle 6
φ)
6
∏ ai - ∏ βi 1
1
Σ
(1)
where Ri and βi are the secondary forward and backward reaction rates for each individual step. Σ is a sum of positive
i KmA
)
(KAi )m
o KmA
)
(KAo )m
o KnB
)
(KBo )n
i KnB
)
(KBi )n
)
CE1(CAi )m
)
CE2(CAo )m
)
CE2(CBo )n
)
CE1(CBi )n
i ) kmA (CAi )m
CE1mA
o ) kmA (CAo )m
CE2mA CE2nB CE1nB
(2) )
o knB (CBo )n
i ) knB (CBi )n
The superscripts i and o represent the cytoplamic side and extracellular side of the cell membrane, respectively. The letters m and n mean the protein binding of m of ion A and n of ion B. The pumping flux can be simplified as
φ ) CET
[
i knB i kmA
R1R2 -
(R2 + β2) +
i knB i kmA
o knB o kmA
][
β1β2 /
(R1 + R2) +
i o knB knB i o kmA kmA
o knB
o kmA
(β1 + β2) + o knB o kmA
where i kmA
o kmA
o knB
i KmA
(R1 + β1) + i knB i kmA
o knB R1 +
o i i knB β1 + knB R2 + knBβ 2
() () () ()
]
i 1 βA ) i m ) i m i (CA) (CA) RA
) )
i knB )
o KmA (CAo )m o KnB (CBo )n i KnB (CBi )n
o 1 RA ) i m o (CA) βA o 1 βB ) o n o (CB) RB
)
(3)
i 1 RB (CBi )n βBi
The pumping flux is a function of the concentrations of ions A and B, dissociation constants of the binding and unbinding steps, and the reaction rates for other steps. According to the theory of absolute reaction rates, the reaction rates are proportional to an exponential of the ratio of the activation energy, which consists of a voltage-independent intrinsic term and a voltage-dependent term, over the thermal energy, kT. The intrinsic term is a constant. The voltagedependent term is proportional to the membrane potential, V. We now take the Na/K pump as an example and introduce apportionment factors a, b, r, and h. The partial membrane potentials aV and bV affect ion binding and unbinding access channels, respectively. The rest of partial membrane potential rV represents the dielectric constant on the liganding residues.15,23 Specifically, hrV affects the reaction rate in the forward direction, while (1 - h)rV affects the reverse one
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i RAi ) RA0 e{3a+[(3+Z)hr] /2}V / kT i βAi ) βA0 e-{3a+[(3+Z)(1-h)r] / 2}V / kT o {3b+[(3+Z)hr] / 2}V / kT RAo ) RA0 e o -{3b+[(3+Z)(1-hr] / 2}V / kT βAo ) βA0 e o {2b+[(2+Z)hr / 2}(-V) / kT RBo ) RB0 e
(4)
o -{2b+ (2+Z)(1-h)r] / 2}(-V) / kT βBo ) βB0 e [
i RBi ) RB0 e{2a+[(2+Z)hr] / 2}(-V) / kT i βBi ) βB0 e-{2a+[(2+Z)(1-h)r] / 2}(-V) / kT
Let z) -2 be the liganding residues,2 and take a ratio of 3:2 for the Na and K ion transports. Inserting R, β into k yields
1 ) (CAi )mXAi e[6a+(r / 2)]V / kT i kmA 1 ) (CAo )mXAo e-[6b+(r / 2)]V / kT o kmA 1 o knB ) o n XBoe4bV / kT (CB) 1 i knB ) i n XBi e-4aV / kT (CB)
(5)
where the four voltage independent terms X are expressed in general as
X ) xeE / kT
(6)
E is an intrinsic energy related and voltage-independent term. On the basis of previous experimental results7,20,23 showing that the pumps’ stoichiometric numbers, extruding 3 Na and pumping in 2 K ions in each cycle, remain unchanged over a wide range of membrane potentials, the state flux, φA, is linearly related to the ion flux φ, the transmembrane current I, and the transporter turnover rate.
I ) ckφ
(7)
These equations yield the pump currents I, as a function of the membrane potential, V, temperature, T, and the ion concentrations. The parameters are determined by fitting the experimental data. 2. Determination of the Parameters in the Expression. In previous studies,10 we fixed temperature, the ion concentrations, and the intrinsic-energy terms and focused only on the voltagedependent parameters, a, b, r, and h where a + b + r e 1 and h < 1. In this paper, a, b, and r were fitted by using the values determined in the previous study10 as trial. Because a and b are mainly involved in the ion movements in the binding and unbinding access channels, respectively, we have aA ) aB ) a, and bA ) bB ) b for the two ion-transport limbs. R1, R2, β1, and β2 are voltage independent and were selected in the range of previous study and set to be flexible parameters. In order to study the temperature dependence, all the intrinsic energy related terms, XAi, XAo, XBo, and XBi are needed. All above parameters as well as ck were fitted to the experimental data of Figure 25 with the Levenberg-Marquardt nonlinear least-squares algorithm. In order to estimate uncertainties in the fits of these two parameters, we used the bootstrap method,30 as explained below. One thousand sets of hypothetical data were generated by computer, and a, b, and c are fitted while other parameters are fixed. In addition to the membrane potential, many other factors affect the pump flux. Experimental results have shown that the Na/K pumps have Q10 values from 2 to 3 at room temperature
Figure 2. Experimental data (circles) of pump current fitted with the six-state model (line). The Levenberg-Marquardt nonlinear leastsquares algorithm was used to fit the data. On thousand sets of hypothetical data (gray dots) were generated and fitted in order to estimate the standard deviations of the fitted parameters.
297 K under different experimental conditions.24-28 The temperature dependence is sensitive to the intrinsic-energy parameters E as well as the voltage-dependent parameters. EAi, EAo, EBo, EBi were determined with the fitted intrinsic energy terms, XAi, XAo, XBo, and XBi while the pump flux (i) remains a sigmoidal shaped I-V curve having all the features agreeing with the experimental results,6,19-21 and (ii) increases about 3 times when temperature increases from 292 to 302 K, a typical laboratory temperature range. Because of the fixed stoichiometric numbers, we can represent the pump flux as a function of temperature and the membrane potential. The value at a membrane potential of -40 mV and temperature of 297 K was set to be 50 Hz based on the experimental estimate.6,7 Moreover, experimental results have also shown that an increment in the internal Na (A) ion concentration or decrement in the internal K (B) ion concentration changes the magnitude of the pump currents but does not affect the trend of the I-V curve. As the ion concentrations vary, the I-V curve may scale up or down, or shift left or right, without loss of sigmoidal trends.29 Therefore, another criterion to verify the fitted parameters is to show shifting and scaling of the I-V curve without an effect on sigmoidal characteristics. 3. Pumping Rate Distribution As a Function of Fluctuations in the Constituent Variables. Change in any variable will affect the pumping rate. If the change of the variable follows some sort of distribution, the pumping rate will exhibit a corresponding distribution profile. In order to study the distribution profile of the pumping rates as functions of the variable changes, we assumed that each variable is fluctuating following a Gaussian distribution
M(x) ) M0
1
√2πδx ∆x ) 2.355δx
e-(x
- x0)2/2δx2
(8)
where ∆x is the full width at half-maximum (fwhm) with respect to the peak value, or the nominal value. On the basis of laboratory environment and instrument performance, we estimated the ranges of the changes in the membrane potential and temperature as ∆T ) 2.5 K at T0 ) 297 K, and ∆V ) 5 mV at V0 ) -90 mV. Under physiological conditions, the intracellular ion concentrations are more sensitive than those in the extracellular fluid to environmental changes. Therefore, we assumed that the extracellular ion concentrations remain constants but the intracellular ion concentrations change to the extents ∆cAi ) 5 mM at cAi ) 50 mM, and ∆cBi ) 5 mM at cBi ) 33 mM, which are about 10-15% variations.
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When involving fluctuations of multiple variables, there are two ways to consider these variables. First, fluctuation of each variable can be expressed as a Gaussian function with specific characteristics. These variables fluctuate independently, since the fluctuations are induced by the environment. We started from an assumption that only the membrane potential, V, fluctuates. Then, temperature fluctuation was added, and finally, all the variables were set to fluctuate. Second, if the fluctuations of these variables are induced intrinsically due to ion traffic across the cell membrane, the corresponding changes will be correlated. For example, when electric stimulation or a membrane action potential opens ion channels in the cell membrane, channel currents hundreds or thousands of times larger than the pump currents reduce local ion concentration, and therefore decrease the membrane potential. In addition, large channel currents may induce thermal effects, resulting in an increase in local temperature around the pump molecules. Therefore, although the fluctuation of each variable can still be expressed by a Gaussian distribution, directions of the changes are correlated. In our calculation the variables are completely linearly correlated, through the expression
x2 ) µ2 +
σ2 (x - µ1) σ1 1
(9)
where x1 is the variable whose distribution is determined, and any other variable, x2 is correlated to x1; µi is the mean of xi and σi its standard deviation. The peak value of pumping rate in the presence of fluctuations, φp ) φ(∆T, ∆V, ∆cAi, ∆cBi), may be changed from the reference value defined as that in the absence of fluctuations, φ0 ) φ(T0, V0, cA0i, cB0i). In addition, the full-width at 10% of maximum may also change. 4. Computation. Originlab 7.5 was used for the initial fittings. Several MATLAB programs implement the curve-fitting and distribution calculations. Two subroutines31 were used for determining error of a, b, and ck. The Gaussian distribution functions become discrete in the computation. Step sizes were selected carefully in order to reduce computational time while retaining good resolution. We computed the distribution density, N(φ), as a histogram. Results 1. Fitting of Parameters to Experimental Results. In order to estimate values for the parameters in the model, we used a nonlinear least-squares fitting algorithm, employing the bootstrap method for the purpose of estimating uncertainties. In an initial fit, values were obtained for R1, R2, β1, β2, XAi, XAo, XBo, XBi, a, b, and ck; of these, R1 ) 390, R2 ) 208, β1 ) 55, β2 ) 77, XAi, ) 0.01, XAo ) 1 × 10-6, XBo ) 0.5, XBi ) 0.02 were deemed reliable and so fixed. To estimate errors in a, b, and c, we followed the bootstrap method; measuring the deviations of the data from the initial best-fit curve, we generated 1000 hypothetical data sets with the same overall χ2. Each of these was fitted in a three-parameter model. The 1000 fits yielded a ) 0.42 ( 0.04, b ) 0.12 ( 0.01, ck ) 0.060 ( 0.006. Figure 2 shows the fitted I-V curve (line), experimental data (circle), and hypothetical data sets (gray bars). The ion concentrations cAo ) 120 mM, cBo ) 4 mM, cAi ) 50 mM, and cBi ) 33 mM were used in the fitting as well as in the experiment. This I-V curve fits the experimental data on amphibian skeletal muscle fibers well and also behaves similarly to another I-V curve experimentally measured from the Na/K pumps.21 The pumping flux is expressed as the pumps’ turnover
Figure 3. Calculated pumping rate as a function of temperature at a membrane potential of -90 mV in the range of 292 to 302 K where Q10 is 3.1.
rate, and the value at a membrane potential of -40 mV is scaled to 50 Hz. In this representation, the turnover rate reduces to 7.6 Hz at the membrane resting potential -90 mV; the turnover rate is lower than 5 Hz at -120 mV as the membrane is hyperpolarized. On the other hand, the turnover rate increases and reaches a plateau at ∼110 Hz at ∼10 mV as the membrane is depolarized. The turnover rate goes down if the membrane is further depolarized. Previous experimental results showed that the pump currents or turnover rates increase with temperature. Figure 3 shows the calculated turnover rate as a function of temperature from 292 to 302 K at a membrane potential of -90 mV while EAi ) 38 kJ, EAo ) 2 kJ, EBo ) 2 kJ, EBi ) 38 kJ. Turnover rate monotonically increases from ∼5 to ∼13 Hz agreeing with experimental estimates where Q10 is 3. Figure 4 is a threedimensional plot of the turnover rate against temperature and the membrane potential. As temperature changes, the I-V curve keeps the sigmoidal shape. When the membrane potential is lower than -100 mV, the slope of the I-T curve becomes very shallow, and the maximum turnover rate is lower than 10 Hz; as the membrane is depolarized to 0 mV, the I-T curve has a steep slope, and the turnover rate changes from ∼100 to ∼120 Hz. On the other hand, at lower temperature (292 K) the I-V curve has a shallow slope and a lower maximum value of ∼100 Hz. As temperature increases to 302 K, the I-V curve has a steep slope and a higher maximum of ∼120 Hz. Figure 5 shows three I-V curves calculated at a temperature of 297 K from different internal ion concentrations of B, cBi ) 15, 30, and 45 mM while the others remain the same: cAo ) 120 mM, cBo ) 4 mM, cAi ) 50 mM. Reducing the internal ion B concentration from 30 to 15 mM, the I-V curve scales up, and while it increases to 45 mM the I-V curve scales down and shifts slightly to the right. These results are qualitatively consistent with experimental results.29 2. Pumping-Rate Distribution Induced by Variable Fluctuations. The upper panel of Figure 6 shows the distribution profile of the pumping rates due to fluctuation of only the membrane potential, V. Temperature T and the ion concentrations remain unchanged. The location of the peak, φp ) 7.4 Hz, is 0.2 Hz lower than the value of φ0 ) 7.6 Hz without fluctuations. In addition, the distribution profile is slightly asymmetric. This shift and asymmetry originate from nonlinearity of the I-V curves. The width of the profile defined as the full-width at 10% of maximum is 3 Hz. If fluctuation of temperature, independently from V, is added, the distribution of the pumping rate shifts more to lower values, and the
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Figure 4. A three-dimensional plot of the pumping rate versus temperature and membrane potential.
Similarly, we plot the distribution of the pumping rate at different nominal membrane potentials in Figure 9. All of the parameters are identical to those used in the plot shown in Figure 7 except that fluctuations of the three variables are correlated. The absolute width of the profile is 6.8 Hz at -110 mV and 62 Hz at -40 mV in the depolarized regime. Clearly, the widths of the distributions become greater than those when the variables fluctuated independently. This result indicates that the intrinsic fluctuations induced by the same source significantly influence the pumping rate, generating a much broader distribution of pumping rates. Discussion Figure 5. Calculated pumping rates as a function of the membrane potential with cAB equal to 15 (dot), 30 (dash), and 45 mM (solid) at 297 K. Reducing internal B concentration causes the I-V curve to scale and shift slightly to the left.
distribution profile becomes broader, as shown in the middle panel of Figure 6, where φp ) 7.2 Hz shifts 0.4 Hz down. The width of the profile 4.6 Hz. Furthermore, the lower panel of Figure 6 shows the distribution of the pumping rate as a function of uncorrelated fluctuations of the membrane potential, temperature, and ion concentrations. The peak still shifts to 7.2 Hz, close to that in the middle panel, but the width of the profile is 7.6 Hz. The distributions of the pumping rate at the membrane potentials of -40, -90, and -110 mV with all three variables allowed to fluctuate were calculated. The results are shown in Figure 7. The absolute width of the profile gets narrower to 3.5 Hz as the membrane is hyperpolarized, peaking at 2.2 Hz at -110 mV. On the other hand, it gets broader as 31 Hz in the depolarized regime peaked at 48 Hz at -40 mV. In Figure 8, we calculated the distribution of the pumping rate when all the variables, T, V, cAA, and cAB, fluctuate in a correlated manner. Each variable fluctuates following a Gaussian distribution, but the fluctuation directions are correlated. If the ion concentration gradient goes down due to the ion traffic through ion channels, the membrane potential also decreases while temperature goes up. Similarly, if the ion concentration gradient goes up, the membrane potential also goes up, and the temperature goes down. The pumping-rate distribution becomes significantly broader than in the case of independent fluctuations shown in Figures 6 and 7. The width of the profile is 14.5 Hz.
By using an asymmetric six-state model, we obtained an explicit expression for the Na/K pumping rate with multiple parameters in the expression. The pumping rate is a function of the membrane potential, temperature, and the ion concentration gradient across the cell membrane. Experimental results have shown many features of the pumping rate such as sigmoidal shaped I-V curve,19-23 I-V curve shifting in response to ion concentration changes,29 Q10 value,24-28 and the measured pumping rate at a specific membrane potential.6,7 On the basis of these experimental results, we fitted the values of the parameters in the model. The model then can be used to predict some results which are difficult to obtain experimentally. On the basis of the Boltzmann theory, fluctuation of the pumps’ turnover rate is caused by fluctuations of the variables that determine the pumping rate, including temperature, membrane potential and ion concentrations. Fluctuation of each variable can be expressed as an individual Gaussian function. Fluctuation of these variables may be totally independent, or correlated to each other depending on the source of the fluctuations. The detailed fluctuation values in local temperature, membrane potential, and ion concentration around individual pump molecules are not available. In this paper, we estimate the magnitude of fluctuation in these parameters in order to get a general picture of the probability distribution of the pumping rate. The probability distribution of pumping rates obtained in this study shows a dependence on both the number of variables allowed to fluctuate and the relationship among the fluctuations. When only one variable fluctuates, the profile of pumping-rate
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Figure 7. Distributions of the pumping rate at peak membrane potentials, V0, of -40, -90, and -110 mV. The fluctuations of the membrane potential, temperature, and ion concentrations are all the same as those in the lower panel of Figure 6. The width of profile gets narrower as the nominal membrane potential goes down.
Figure 8. Distribution of the pumping rate as T, V, cAA, and cAB fluctuate in a correlated manner. Each variable fluctuates following a Gaussian distribution, but the fluctuation directions are correlated. If the ion concentration gradient goes down due to the ion traffic through ion channels, the membrane potential also decreases while temperature goes up. Similarly, if the ion concentration gradient goes up, the membrane potential also goes up, and the temperature goes down. The pumping-rate distribution becomes significantly broader. The width of profile is 14.5 Hz.
Figure 6. (Upper panel) Pumping rate distribution generated only from the membrane potential, V, fluctuating at -90 mV as a Gaussian function. The peak φp ) 7.4 Hz shifts 0.2 Hz away from φ0 ) 7.6 Hz toward lower rate. The width of profile is 3 Hz. (Middle panel) Distribution of the pumping rate as both the membrane potential, V, and temperature, T, undergo independent Gaussian fluctuations at -90 mV and 297 K. The distribution becomes more asymmetric. The peak φp ) 7.2 Hz shifts down 0.4 Hz. The width of profile is 4.6 Hz. (Lower panel) Distribution of the pumping rate as temperature, membrane potential, and intracellular ion concentrations, T, V, cAA, and cAB all fluctuate independently. The distribution stays asymmetric and covers a broader range. The peak at φp ) 7.2 Hz has shifted down 0.4 Hz from φ0 ) 7.6 Hz. The width of profile is 7.6 Hz. AU means arbitrary unit.
distribution shows little asymmetry. When more variables are fluctuating, the resultant pumping rate distribution becomes broader and more asymmetric. Permitting all the variables to fluctuate shifts the peak of the distribution from 7.6 to 7.2 Hz. The width of the profile is 7.6 Hz. When the changes in individual variables are induced by the same source so that the fluctuations are no longer independent but correlated, the same fluctuations for the individual variables
Figure 9. Distributions of the pumping rate at peak membrane potentials, V0, of -40, -90, and -110 mV. The distribution widths of the membrane potential, temperature, internal ion concentrations are all the same as those in Figure 7 but correlated. The profiles are broader than those shown in the lower panel of Figure 6.
generate a broader asymmetric profile of pumping rates. On the basis of the same variable fluctuations, the peak pumping rate shifts to 6.3 Hz, and the full-width at 10% of maximum is broadened to 14.5 Hz. The profile of the distribution of pumping rates is not symmetric, though individual variables are symmetrically fluctuating as Gaussian functions. This result is not unexpected. On the basis of the Boltzmann distribution, the less the activation energy involved, the larger the probability of the reaction state. The results show that the pumping rates of the Na/K pumps are distributed in an asymmetric profile which has a higher probability at the lower pumping rate.
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In this study, we only assumed the changes of ∆T ) 2.5 K, ∆V ) 5 mV, and 5 mM in ionic concentration difference. It is necessary to point out that we are studying the probability of the pumping rate of each individual pump in a realistic situation. Therefore, we should not consider the situations under voltageclamp and temperature control, as well as the ion channels maximally blocked. In the real situation, especially in excitable cells, the environment of each pump may differ significantly. Spatially, the pumps close to the ion channels may suffer dramatic change in the ion concentration and the membrane potential, while those away from the channels may experience less fluctuation in the environments. In addition, when the external electric field is applied to the cells and tissues without using a voltage clamp, pump molecules at different membrane locations may experience significant difference in the fieldinduced membrane potential. This difference may easily be much larger than 5 mV. This study only provides a general picture of the pumping rate distribution as function of membrane potential, temperature, and ionic concentrations. The greater the number of variable fluctuations involved, the broader the pumping rate distribution is, and the distribution shows more asymmetry. The correlated variable fluctuations generate a much broader asymmetric distribution. Acknowledgment. This work is partially supported by National Institutes of Health Grant 2NIGM50785 and National Science Foundation Grant PHY-0515787. References and Notes (1) Chen, (2) Chen, 39, 331. (3) Chen, (4) Chen, (5) Chen, 40, 347.
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