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Human HeLa cells transfected with mouse Cx45 and rat RIN cells transfected with chicken Cx45 were used to study the electrical and permeability properties of Cx45 gap junction hemichannels. With no extracellular Ca2+, whole-cell recording revealed currents arising from hemichannels in both transfected cell lines. Multichannel currents showed a time-dependent activation or deactivation sensitive to voltage, Vm. These currents did not occur in nontransfected cells. The hemichannel currents were inhibited by raising extracellular Ca2+ or by acidification with CO2. The unitary conductance exhibited Vm dependence (i.e., γhc,main increased/decreased with hyperpolarization/depolarization). Extrapolation to Vm = 0 mV led to a γhc,main of 57 pS, roughly twice the conductance of an intact Cx45 gap junction channel. The open channel probability, Po, was Vm-dependent, declining at negative Vm (Po < 0.11, Vm < −50 mV), and increasing at positive Vm (Po ∼0.76, Vm > 50 mV). Moreover, Cx45 nonjunctional hemichannels appeared to mediate lucifer yellow (LY) and propidium iodide (PI) dye uptake from the external solution when extracellular Ca2+ level was reduced. Dye uptake was directly proportional to the number of functioning hemichannels. No significant dye uptake was detected in nontransfected cells. Cx45 transfected HeLa and RIN cells also allowed dye to leak out when preloaded with LY and then incubated in Ca2+-free external solution, whereas little or no dye leakage was observed when these cells were incubated with 2 mM external Ca2+. Intact Cx45 gap junction channels allowed passage of either LY or PI dye, but their respective flux rates were different. Comparison of LY diffusion through Cx45 hemichannels and intact gap junction channels revealed that the former is more permeable, suggesting that gap junction channel pores exhibit more allosterical restriction to the dye molecules than the unopposed hemichannel. The data demonstrate the opening of Cx45 nonjunctional hemichannels in vertebrate cells when the external Ca2+ concentration is reduced.

## Introduction

Intercellular communication is mediated via conduction and/or diffusion through gap junction channels. Gap junctions constitute assemblies of intercellular channels. Each channel consists of two hemichannels (connexons) composed of six transmembrane proteins (connexins). Each intercellular channel provides an aqueous pathway for the passage of intracellular ions and small molecules. It has been demonstrated that gap junction channels are permeable to ions, fluorescent dyes, and physiologically active molecules including amino acids, second messengers, and small peptides (Bevans et al., 1998; Saez et al., 1989; Vaney et al., 1998). To date, at least 19 connexins have been identified in vertebrate cells encoded by a multigene family (Bruzzone et al., 1996; Beyer and Willecke, 2000). An interesting aspect of the various gap junctions is their apparent differences in terms of perm-selectivity (molecular permeability and ion selectivity). It has been shown that Cx40, Cx43, and Cx37 gap junction channels each have a unique selectivity pattern for cations and anions (Beblo and Veenstra, 1997; Wang and Veenstra, 1997). Moreover, the permeability to negatively charged dyes of each these gap junction channels appears to be different (Veenstra et al., 1995).

Oligomerization of connexins into hemichannels occurs before entry into the Golgi apparatus (Evans et al., 1999). Docking of two hemichannels to form a gap junction channel involves noncovalent interactions (Foote et al., 1998) and results in the formation of leak-free intercellular channel (Bukauskas et al., 1995a). Hemichannels in nonjunctional membrane areas can gate open under appropriate conditions. Evidence for the existence of nonjunctional hemichannels was first provided by electrical studies on horizontal cells of fish retina (De Vries and Schwartz, 1992; Malchow et al., 1993). Similar observations were subsequently made in Xenopus oocytes exogenously expressing known connexins. Injection of mRNA for rat Cx46 or chicken Cx56 induced currents attributable to hemichannels (Ebihara and Steiner, 1993; Ebihara et al., 1995). Later, it was shown that Cx38, the intrinsic connexin of Xenopus oocytes, also forms hemichannels (Ebihara, 1996). It also has been demonstrated that injected Xenopus oocytes are suitable to study single hemichannels (Trexler et al., 1996).

Under pathophysiological conditions, like ischemia or other forms of metabolic insult, rapid disturbances in ionic homeostasis contribute to cellular injury and death (Wilde and Aksnes, 1995). The physiological role of nonjunctional hemichannels remains to be identified, but the hemichannel activity has been reported to be controlled by the extracellular Ca2+ (Ebihara and Steiner, 1993; Pfahnl and Dahl, 1999; Valiunas and Weingart, 2000), the metabolic state (John et al., 1999), the membrane potential (De Vries and Schwartz, 1992; Ebihara et al., 1995; Trexler et al., 1996; Valiunas and Weingart, 2000), and Ca2+-dependent volume regulation (Quist et al., 2000). Conceivably, nonjunctional hemichannels also may be involved in the pathogenesis of ionic disturbances during myocardial ischemia and hypoxia. However, little is known about the properties of cardiac nonjunctional hemichannels.

This study was concentrated on the properties of Cx45 hemichannels. Cx45 is one of the three major connexins expressed in heart. It has been reported to be widely distributed in myocytes of different types of cardiac tissues (working, conductive and nodal; Kanter et al., 1993; Davis et al., 1994; Saffitz et al., 1994; Alcolea et al., 1999; Coppen et al., 1999). Cx45 gap junction channels possess relatively small unitary conductances for gap junction channels (∼26 pS), but still are able to pass some dyes (Veenstra et al., 1994a).

The aim of this study was to investigate the electrical and perm-selectivity properties of hemichannels in vertebrate cells. We used human HeLa cells transfected with mouse Cx45 and rat islet tumor (RIN)* cells transfected with chicken Cx45. The main emphasis was on Cx45 hemichannel conductance, gating and permeability and comparison to that of the intact gap junction channel. The study illustrates for the first time that Cx45 is capable of forming operational hemichannels, which are voltage, pH and calcium gated. Moreover, we provide the first characterization of the biophysical properties of Cx45 hemichannels.

## Materials And Methods

### Cells and Culture Conditions

Experiments were performed on human HeLa cells transfected with cDNA coding for mouse Cx45 and rat islet tumor (RIN)* cells transfected with cDNA coding for chicken Cx45. HeLa cells and RIN cells were grown in DME medium and RPMI 1640 medium, respectively, supplemented with 10% FCS, 100 μg/ml streptomycin, and 100 U/ml penicillin. Transfected HeLa cells and RIN cells were selected using 0.5–1 μM puromycin (Sigma-Aldrich) and 0.4 mg/ml G418 (Geneticin; Life Technologies), respectively. The cells were passaged weekly, diluted 1:10, and kept at 37°C in a CO2 incubator (5% CO2, 95% ambient air). To perform experiments, the cells were harvested and seeded onto sterile glass coverslips placed in multiwell culture dishes (∼104 cells/cm2). Electrophysiological experiments were performed on cells cultured for 1–3 d.

### Solutions and Pipettes

During experiments, the cells were superfused with bath solution containing the following (in mM): 120 potassium aspartate, 10 NaCl, 2 CaCl2, 5 HEPES, pH 7.4, 5 glucose, and 2 mM CsCl, BaCl2, and TEA+ Cl were added. For the Ca2+-free (0 Ca2+) bath solution, CaCl2 was omitted. The patch pipettes were filled with solution containing the following (in mM): 120 potassium aspartate, 10 NaCl, 3 MgATP, 5 HEPES, pH 7.2, and 10 EGTA (pCa ∼8); filtered through 0.22-μm pores. In perforated patch experiments, the pipette solution contained 30–50 μM β-escin (Fan and Palade, 1998).

### Electrical Measurements

Glass coverslips with adherent cells were transferred to an experimental chamber perfused with bath solution at room temperature (∼22°C). The chamber was mounted on the stage of an inverted microscope (Olympus IMT2). Patch pipettes were pulled from glass capillaries (code 7052; A-M Systems) with a horizontal puller (Sutter lnstruments). When filled, the resistance of the pipettes measured 1–2 MΩ.

Experiments were performed on single cells using the whole-cell voltage-clamp technique (Valiunas and Weingart, 2000). A selected cell was attached to a patch pipette connected to a micromanipulator (model WR-88; Narishige Scientific Instrument) and an amplifier (model Axopatch 200; Axon Instruments, Inc.). This method permitted control of the membrane potential (Vm), and allowed measurement of the associated membrane current (Im).

Dual whole-cell patch clamp was used in experiments with cell pairs. It allowed to control the membrane potential of both cells and to measure the currents (Bukauskas et al., 1995b; Brink et al., 1996). Initially, the membrane potential of cell 1 and cell 2 was clamped to the same value, V1 = V2. V2 was then changed to establish a transjunctional voltage, Vj = V2V1. Currents recorded from cell 2 represent the sum of two components, the junctional current (Ij) and the membrane current of cell 2 (Im,2); the current obtained from cell 1 corresponds to Ij.

### Dye-uptake and Dye-injection Studies

Nontransfected (NT) and transfected HeLa or RIN cells were incubated in the bath solution (with 2 mM Ca2+ or Ca2+ free) containing 1 mg/ml lucifer yellow (LY; Molecular Probes) or 2 mg/ml propidium iodide (PI; Molecular Probes) for 30 min at room temperature (22°C).

The dye transfer through gap junctions was investigated using cell pairs. Dyes were dissolved in pipette solution to reach a concentration 0.1% for LY and 0.2% for PI. In each cell pair examined, the whole-cell recording mode was established on one cell with a dye-filled pipette. The second cell was patched either in the perforated-patch mode or a gigaohm seal was established with the pipette leaving the patch intact. This allowed the dye to spread into the recipient cell. Once dye had spread, the whole-cell recording mode also was established in the recipient cell and the gap junction conductance (gj) was measured. This procedure and the perforated patch method allowed the dye to spread to the neighboring cell without loss of dye caused by patch pipette dialysis and to measure simultaneously gj.

Fluorescent dye uptake and cell-to-cell spread was imaged using a 12-bit 64,000 pixel grayscale digital CCD-camera (model LYNXX 2000T; Spectra Source Instruments).

### Signal Recording and Analysis

Voltage and current signals were recorded on chart paper (model Gould RS 2400; Gould Instruments) and videotape (pulse-code modulated DR-384; Neuro Data Instruments). For off-line analysis, the current signals were filtered at 1 kHz, digitized with a 12-bit A/D-converter (model DT21EZ; Data Translation) and stored with a personal computer. Data acquisition and analysis were performed with custom-made software (Brink et al., 1996; Krisciukaitis, 1997). Curve fitting and statistical analyses were done with SigmaPlot and SigmaStat, respectively (Jandel Scientific). The results are presented as means ± SEM.

## Results

### Hemichannel Currents in Transfected Cells: Modulation by Voltage and External Ca2+

Recordings from HeLa and RIN cells expressing mouse Cx45 and chicken Cx45, respectively, showed slowly activating currents in Ca2+-free bath solution. These currents were absent in cells not expressing Cx45, suggesting they were mediated by Cx45 hemichannels.

Fig. 1 A illustrates the voltage protocol used to activate hemichannel currents. A family of Im traces were elicited by depolarizing (outward currents) and hyperpolarizing pulses (inward currents). The records were obtained from a Cx45-HeLa cell superfused with Ca2+-free solution. After establishing of the whole-cell recording conditions, the membrane potential (Vm) was clamped to 0 mV. Bipolar voltage pulses of 10-s duration were then delivered to alter Vm in steps of ±5, ±10, and then to ±110 mV using increments of 20 mV. The associated membrane currents (Im) increased proportionally with Vm and showed a voltage- and time-dependent activation or deactivation. In the absence of Ca2+, depolarization produced slowly activating outward currents (Fig. 1 A), whereas hyperpolarization induced inward currents which deactivated with time.

Perfusion with 2 mM Ca2+ closed the hemichannels in a time-dependent manner. In the presence of 2 mM extracellular Ca2+, the bipolar pulse protocol applied to the same cell elicited small outward and inward currents. These currents increased with depolarization and hyperpolarization, but showed no slow time dependent changes (Fig. 1 B). Thus, the hemichannel activity was suppressed completely in the presence of 2 mM external Ca2+.

These experiments show that membrane currents can be activated in Cx45 expressing cells by lowering Ca2+ in the external solution. Similar observations were made in Xenopus oocytes (Ebihara et al., 1995; Trexler et al., 1996) and HeLa cells (Valiunas and Weingart, 2000) expressing other connexins. Beside the hemichannels, which opened at 0 mV by lowering of Ca2+, positive voltages cause further activation of hemichannels. In contrast, negative Vm closed the hemichannels previously opened by lowering Ca2+ and depolarization.

The signals obtained in Ca2+-free external solution (Fig. 1 A) were analyzed to obtain the relationships Im = f(Vm). The amplitudes of Im were determined at the beginning (Im,inst; inst, instananeous) and end of each pulse (Im,ss; ss, steady state). To distinguish between capacitive and ionic currents at the beginning of each record, the signals were displayed at fast time resolution. At large negative voltages, Im,inst was determined by fitting the inward current with an exponential. As shown in Fig. 1 C, Im,inst = f(Vm) (○) was slightly nonlinear over the voltage range examined (i.e., ±110 mV). The Im,inst was ∼20% larger at −110 mV in comparison to the current at +110 mV. In contrast, Im,ss = f(Vm) (•) deviated strongly from linearity. It was shallower at negative Vm and steeper at positive Vm. Fig. 1 D shows the Im = f(Vm) relationship recorded in the presence of 2 mM extracellular Ca2+. The instantaneous (Fig. 1 D, ○) and steady-state currents (Fig. 1 D, •) were virtually linear and superimposable. The analysis yielded a uniform slope conductance of 0.6 nS.

The presence 2 mM Cs+, Ba2+, or TEA+ in the extracellular and intracellular solution does not block the novel current seen in Ca2+-free solution, suggesting that the Ca2+-sensitive Im reflects current through hemichannels, Ihc (hc, hemichannel). Similar results were obtained from RIN cells transfected with chicken Cx45 (unpublished data).

### Voltage Dependence of Hemichannel Currents

The relationship between Vm and ghc was studied in Ca2+-free solution using the bipolar pulse protocol. The signals in Fig. 1 A and others were analyzed to determine the voltage dependence of Ihc. Fig. 1 E summarizes the data gathered from five cells. The normalized steady-state conductance (ghc,ss) values were calculated from the ratios Ihc,ss/Vm, normalized with respect to the value at Vm = +5 mV, were averaged, and plotted versus Vm. The ghc,ss decreased in a sigmoidal manner to ghc,min when Vm was made negative and increased to ghc,max at positive voltages. The smooth curve represents the best fit of data to the Boltzmann equation:

$g_{hc,ss}=\frac{g_{hc,\mathrm{max}}{-}g_{hc,\mathrm{min}}}{1+e^{\left[A{\cdot}\left(V_{m}{-}V_{m,0}\right)\right]}}+g_{hc,\mathrm{min}}\mathrm{,}$

where ghc,max and ghc,min are the maximal and minimal conductance at large positive and negative Vm, respectively. Vm,0 corresponds to Vm at which ghc,ss is half maximally inactivated. A is a constant which expresses gating charge, zq(kT)−1, where z is the equivalent number of unitary positive charges, q moving through the applied electric field, and k and T are the Boltzmann constant and the temperature in Kelvin, respectively. The analysis yielded the following values: Vm,0 = 11.1 mV, ghc,max = 2.34, ghc,min = 0.21, and z = 1.7. The relationship in Fig. 1 E resembles the negative limb of the function gj,ss = f(Vj) of gap junctions studied in cell pairs (Valiunas et al., 2000).

The value of ghc,inst determined at the beginning of a depolarizing pulse reflects the sum of the conductances of the hemichannels opened by the reduction of the extracellular Ca2+. In the case of Cx45-HeLa cells, ghc,inst averaged 6.54 ± 1.1 nS (n = 19); in the case of Cx45-RIN cells, ghc,inst averaged 1.72 ± 0.52 nS (n = 8). This suggests that the level of connexin expression is ∼3.8-fold larger in HeLa cells than in RIN cells.

Assuming that endogenous membrane channels are involved, the value of ghc,inst has to be corrected by subtraction of membrane leak conductance (∼0.6 nS, Fig. 1 D). Hence, the corrected ghc,inst for the Cx45-HeLa cells would be ∼5.94 nS.

### Acidification Affects Macroscopic Hemichannel Currents

Exposure to 100% CO2 reduces the gap junction conductance (gj) via lowering of the pHi (Bukauskas and Peracchia, 1997). Trexler et al. (1999) have shown that this intervention reduces the macroscopic currents through Cx46 hemichannels in Xenopus oocytes. The effects of acidification on macroscopic hemichannel currents were studied in Cx45-HeLa cells. Fig. 2 illustrates the effects of 100% CO2 on Im (Ihc). Hemichannel currents were activated and deactivated by bipolar pulses (±50 mV, 10-s duration each) starting from a holding potential of 0 mV. The bipolar pulses were repeated every 10 s. After a 2-min control, perfusion with bath solution equilibrated with 100% CO2 was started. This led to a gradual decrease in Ihc (Fig. 2, continuous current trace and insets). After ∼160 s, the slowly activating/deactivating outward/inward Ihc were completely abolished. Depolarization and hyperpolarization exhibited small constant currents similar to those seen in NT HeLa cells (Valiunas and Weingart, 2000) or in Cx45-HeLa cells in the presence of 2 mM external Ca2+ (Fig. 1 B). Washout of CO2 led to a gradually recovery of Ihc; 13.5 min later the recovery was 75% complete. Similar results were obtained in five other cells.

### Multichannel versus Single-channel Currents

Transfected HeLa cells and RIN cells often showed Ihc signals from tens of hemichannels. They were used to examine multichannel currents. Fig. 3 compares multichannel and single-channel currents through hemichannels. The cell was superfused with Ca2+-free solution. Fig. 3 A shows multichannel currents elicited by biphasic pulse (±50 mV; top trace). Depolarization gave rise to a time-dependent increase in Ihc exhibiting discrete current steps. About 9–10 hemichannels were involved during this episode. Hyperpolarization led to a large Ihc,inst which decreased rapidly to a level attributable to endogenous channels. With this time resolution no discrete current steps could be resolved.

Fig. 3 B shows a current signal displayed at higher magnification, evoked by depolarization of 30 mV (top trace). After a capacitive spike, Ihc revealed discrete steps indicating the sequential opening and closing of hemichannels (channel opening: upward deflections). The smooth curve on the right-hand side represents an all-point histogram of the current trace. It suggests the involvement of three to four hemichannels. The distances between the peaks were 1.53, 1.48, and 1.54 pA. Hence, the current steps were of comparable amplitudes and correspond to conductance steps of ∼49–51 pS.

Deactivation of Ihc associated with hyperpolarization appears to be much faster than activation. Fig. 3 C shows a current signal displayed at high magnification induced by hyperpolarization from 50 to −50 mV (Fig. 3 C, top trace). At the beginning of the episode, Vm was at 50 mV. The outward Ihc showed discrete steps indicative of hemichannels. The following hyperpolarization to −50 mV led to an inward Ihc that deactivated quickly. Discrete steps indicative of sequential closure of hemichannels could now be resolved (channel closure: upward deflections). The analysis of the current peaks yielded conductance steps of 65–73 pS.

Hence, the time-dependent increase/decrease of Ihc associated with depolarization/hyperpolarization shown in Fig. 3 (A–C) is caused by activation/deactivation of hemichannels, whose single-channel conductance for inward current is somewhat higher than for outward current.

### Single Hemichannel Activity in the Cell-attached Mode

The data presented so far were obtained in the whole-cell recording mode. The advantage of this method is that it allows us study of multichannel and single-channel currents. However, data also have been collected in the cell-attached patch mode, a method suitable to analyze single-channel currents. At the end of an experiment, the membrane patch was disrupted and Vm of ∼0 mV was recorded, so the voltage across the membrane patch, V essentially equals −Vpip.

Fig. 3 (D–F) illustrate records of one operational channel obtained from cells at different voltages. The current histograms in all records show the following conductive states: O (hemichannel open), C (hemichannel closed), I = 0 (zero current level; i.e., Vpip = 0). The hemichannel closed state (C) also includes the sum of the patch leak current and membrane current. To be sure that the closed state for the hemichannel does not include any hemichannel conductance, the membrane patch conductance was investigated in cell-attached mode. It yielded a conductance of 43 ± 7 pS (n = 8, no open hemichannel observed). Application of CO2 reduced it to 27 ± 5 pS (presumably by closing some membrane channels); i.e., CO2 reduced leak/membrane current by 16 pS, which is less than hemichannel unitary conductance. In records presented here, the membrane leak conductance varies from 18 to 26 pS. Such an explanation is supported by current records exhibiting a residual state (see Fig. 3 F), which is different from the closed state, illustrated in Fig. 3 (D and E).

In Fig. 3 D, Vpip was stepped from 0 to 70 and then to −70 mV (i.e., V was made −70 and +70 mV, respectively). Fig. 3 E demonstrates a single channel at ±90 mV. In both cases, the channel opened when a positive V was applied and started to flicker spending most time in the main state. After inversion to a negative V, the channel was still apparent, but it now spent less time in the main state. The channel closure at negative voltage was voltage-dependent. At larger voltages, the channels spent less time in the main state (Fig. 3, D and E). The current histograms yielded the following conductances: 52/63 pS (positive/negative V) for V = 70 mV and 50/66 pS (positive/negative V) for V = 90 mV. In Fig. 3 F, Vpip was stepped from 0 to −70 mV. Ihc exhibited two prominent levels corresponding to Ihc,main (bottom dotted line) and Ihc,residual (middle dotted line). The current histogram shows that the Ihc,residual level is different from the closed state current level. The analysis yielded the following conductances: γhc,main = 62 pS; and γhc,residual = 15 pS. A residual state is the property of gap junction channels and hemichannels (Bukauskas et al., 1995a; Trexler et al., 1996; Valiunas et al., 1997; Valiunas and Weingart, 2000; Oh et al., 2000).

### Voltage Dependence of Single Hemichannel Conductance

The relationship between single hemichannel conductance and Vm was investigated in Cx45-HeLa cells and Cx45-RIN cells. For this purpose, γhc,main was determined using voltage pulses of different amplitude and either polarity. The experiments were performed in Ca2+-free bath solution. The data were sampled, averaged, and plotted versus Vm. Fig. 4 summarizes the results from four Cx45-HeLa cells. Similar results were obtained from Cx45-RIN cells (unpublished data). Over the voltage range examined, i.e., −90 to 90 mV, γhc,main was dependent on Vm. It increased with hyperpolarization and decreased with depolarization. The solid curve represents the best fit of the exponential bellow to the data:

$\mathrm{{\gamma}}_{hc,\mathrm{main}}=\mathrm{{\Gamma}}_{H}{\cdot}e^{{-}\frac{V}{V_{H}}}\mathrm{,}$

where ΓH corresponds to the conductance at Vm = 0 mV, and VH is the decay constant at which γhc,main declines to e−1 (Vogel and Weingart, 1998). The analysis yielded the following values: ΓH = 57 pS and VH = −615 mV. A gap junction channel consists of two hemichannels in series. Hence, the conductance of a gap junction channel (γj,main) can be calculated from the conductance of its hemichannels (γhc1, γhc2):

$\mathrm{{\gamma}}_{j,\mathrm{main}}=\frac{\mathrm{{\gamma}}_{hc1,\mathrm{main}}{\cdot}\mathrm{{\gamma}}_{hc2,\mathrm{main}}}{\mathrm{{\gamma}}_{hc1,\mathrm{main}}+\mathrm{{\gamma}}_{hc2,\mathrm{main}}}\mathrm{,}$
(1)

Using the values extrapolated to V = 0 mV, γj,main was 28.5 pS. For comparison, using the same experimental conditions (22°C, pipette solution potassium aspartate [120 mM]).

γj,main determined in pairs of Cx45-HeLa cells was 25 pS (unpublished data). A similar result (26 pS) for Cx45 was reported earlier (Veenstra et al., 1994a). These values are in good agreement with γj,main calculated from the hemichannels data.

### Open-state Probability

Cx45-HeLa cells with a single operational hemichannel also were used to study channel kinetics. The analysis of long records allowed us to explore the open-state probability (Po) at steady state. The protocol involved the establishment of Vm gradients of different amplitude (±50, ±70, and ±90 mV) and duration (8–10 s). Current traces with one operational single hemichannel were analyzed for the dwell times in the main state. To determine Po, the time a channel spent in the main state was measured and expressed as fraction of record duration. Po was maximal (0.7–0.76) at positive Vm (Vm > 50 mV). It decreased to 0.06–0.11 when Vm was negative (Vm < −50 mV). This suggests that Cx45 hemichannels close in a Vm-dependent manner at negative voltage and remain open at positive Vm.

### Dye Uptake Mediated by Cx45 Hemichannels

To examine the permeability properties of Cx45 hemichannels to larger molecules NT and transfected HeLa and RIN cells were incubated in bath solution (with or without 2 mM Ca2+) containing 1 mg/ml LY or 2 mg/ml PI for 30 min at room temperature (22°C). Significant dye uptake occurred in the absence of external Ca2+. Fig. 5 (A and C) illustrates such an experiment. The left-hand panels show phase-contrast micrographs of Cx45-HeLa and Cx45-RIN cells, respectively. Epifluorescent micrographs taken 30 min after incubation in Ca2+-free bath solution containing LY showed dye uptake in Cx45-HeLa cells (Fig. 5 A) and Cx45-RIN cells (Fig. 5 C). No significant dye uptake was detected in this example in the presence of extracellular Ca2+ (Fig. 5, B and D). In control experiments, NT HeLa and RIN cells usually showed very little dye uptake in the presence or absence of Ca2+.

Fig. 5 E summarizes the data obtained from Cx45-HeLa cells and NT HeLa cells. 85% of the Cx45-HeLa cells (n = 650) showed dye uptake in Ca2+-free external solution. In the presence of 2 mM extracellular Ca2+, only 1.4% of the Cx45-HeLa cells (n = 517) showed some loading with LY. 9.4% of the NT HeLa cells (n = 583) showed similar low level dye uptake when bathed in Ca2+-free solution. The large difference in dye uptake between transfected and NT cells in the absence of Ca2+ suggests that uptake occurred through Cx45 hemichannels. The observation that dye uptake by NT HeLa cells in the absence of Ca2+ was larger than dye uptake by Cx45-HeLa cells in the presence of Ca2+ may be due to an endogenous connexin. The ratio of intensity between Cx45-HeLa cells and NT HeLa cells was large, i.e., ICx45/INT = 16.3, suggesting that expression level of endogenous connexins in NT HeLa cells is low (Eckert et al., 1993).

### Induction of Dye Leak by Reduction of Extracellular Ca2+

The rate of dye uptake is difficult to measure with dye in the bath. Hence, Cx45-HeLa cells that had been loaded with LY were washed with Ca2+-free extracellular solution without dye and time-dependent dye unloading observed. Fig. 6 illustrates such experiments. In Fig. 6 A, the top panel shows a phase-contrast micrographs of several Cx45-HeLa cells. The epifluorescent micrographs (Fig. 6 A, middle and bottom panels), taken at 0, 30, 60, and 120 min after incubation in Ca2+-free solution, show a decrease in cell fluorescence with time. Time zero (min) indicates the beginning of washout. The reduction of fluorescence intensity indicates a loss of dye into Ca2+-free extracellular solution. In contrast, when LY preloaded cells were washed and bathed with external solution containing 2 mM Ca2+, a marginal decrease in intensity was observed. Fig. 6 B shows epifluorescent images 0, 30, 60, and 120 min after preloading and after incubation in extracellular solution containing 2 mM Ca2+. Similar results were obtained for Cx45-RIN cells (unpublished data).

Fig. 6 (C and D) summarizes the data on dye unloading from Cx45-HeLa and Cx45-RIN cells. The fluorescence intensity was normalized, averaged, and plotted versus time. Fig. 6 C shows the results from eight Cx45-HeLa cells obtained in Ca2+-free external solution (•). Over the time examined (i.e., >2 h), the intensity decreased as an exponential (smooth curve) with a time constant (τ) of 33 min to a minimum (Imin) corresponding to 0.08. When LY preloaded cells were incubated in Ca2+-containing solution, a marginal drop of fluorescence was observed (Fig. 6 C, ○). Fitting of the data to an exponential yielded a τ of 72 min and a Imin of 0.85. Fig. 6 D illustrates similar results obtained with Cx45-RIN cells incubated in Ca2+-containing (○) or Ca2+-free (•) solution.

### Concentration Dependence of Dye Uptake

A series of experiments was done to investigate the dependency of dye uptake on extracellular concentration. Cx45-HeLa cells were incubated for 30 min in external Ca2+-free solution containing 0.5, 1.0, or 2 mM LY. After dye uptake, the cells were washed in external solution containing 2 mM Ca2+. Fluorescence intensity at each dye concentration was measured in randomly picked cells (for 0.5 mM, n = 135 cells; for 1.0 mM, n = 218; and for 2 mM, n = 274), averaged, and plotted versus concentration (Fig. 7). The fluorescence intensity increased linearly (r2 = 0.995) over the concentration range investigated, showing no saturation at these concentrations of dye.

### Dye Uptake versus Conductance

Another series of experiments was performed to investigate the correlation between dye uptake and the conductance due to functioning hemichannels. Cx45-HeLa cells were incubated in Ca2+-free bath solution containing 1 mg/ml LY for 10 min at room temperature (22°C). Immediately after dye uptake, cells were washed with Ca2+-free extracellular solution without dye and fluorescent images of the cells were taken. The whole-cell recording conditions were established on each of the same cells and macroscopic hemichannel currents were measured (Fig. 1 A). For each cell, ghc,inst and fluorescence intensity values were determined and later normalized with respect to the maximum value in the pool. Fig. 7 B shows a plot of normalized fluorescence intensity plot versus normalized instantaneous hemichannel conductance. The fluorescence intensity was directly proportional to the ghc,inst. The solid line corresponds to a first order regression with r2=0.86. Correlation of electrophysiological data and dye uptake demonstrates that individual cells take up dye in proportion to the number of open hemichannels.

### Gap Junction Channels versus Hemichannels

The question arises whether the permeability of gap junction channels is the same as that of hemichannels and how these properties correlate with macroscopic conductances (Trexler et al., 1996). Does docking of hemichannels change the properties of the pore? Dye transfer through gap junctions was investigated using Cx45-HeLa cell pairs. LY, a negatively charged dye, was injected into one cell of a pair via patch pipette, and gj was measured in conjunction with dye flux. Fig. 8 A illustrates an experiment demonstrating cell-to-cell spread of LY in Cx45-Hela cell pairs. The pipette containing 2 mM LY was attached in the whole-cell configuration to the left-handed cell (Fig. 8 A, cell 1; see top left-hand panel). The right-handed cell (cell 2) was patched using the perforated patch configuration. This allowed the dye spread to cell 2 without loss caused by cell dialysis via the pipette and simultaneous measurement of gj. The epifluorescent micrographs taken at 1, 5, 10, and 30 min after dye injection in cell 1 demonstrate an increase in fluorescence in the recipient cell (Fig. 8 A). To measure gj during dye flux, short 10-mV bipolar pulses were used to generate a transjunctional current, Ij. Fig. 8 B (left) shows the Ij at different time (5, 15, and 30 min) of the LY spread. The right panel shows a family of the voltage-dependent gap junction currents generated by series of voltage pulses (from ±10 to ±150 mV) at the 30-min time mark. The slightly asymmetric current profiles could be related to different patch configuration (cell 1, whole cell; and cell 2, perforated) and/or the influence of negatively charged LY on the voltage gate (sensor). The total gj measured at 30 min was ∼17 nS.

Fig. 8 C summarizes data gained from four dye transfer experiments. It shows a plot of fluorescence intensity versus time for injected cells (•) and moderately coupled (10–12 nS) recipient cells (O). The smooth lines correspond to the best fit of the exponential to the data. The intensity of injected cells increased to a steady state with a time constant (τ) of 5.5 min. The recipient cells filled with LY with a τ of 244.3 min. In comparison with unapposed hemichannels, the dye transfer through gap junction channels takes more time.

### Estimation of Intracellular Dye Concentration

Fluorescence intensity for injected cells reaches steady state at ∼56,000 intensity units (Fig. 8 C). It is assumed that at the steady state the dye concentration within the injected cell is ∼2 mM, i.e., reaches LY concentration of the patch pipette. The fluorescent intensity measured from Cx45-HeLa cells with a 30-min 2-mM LY load time in Ca2+-free solution is ∼6,800 (Fig. 7 A). This intensity corresponds to ∼0.24 mM of LY. The data shown indicate that the total intracellular LY concentration in Cx45-HeLa cells was ∼0.24 mM after 30 min of dye uptake in Ca2+-free solution.

### Positively Charged Dye Diffusion through Hemichannels and Gap Junction Channels

The molecules that can pass through gap junctions display a size limitation of ∼1 kD (Schwarzmann et al., 1981) and junctions formed by some connexins exhibit a charge dependence (Cao et al., 1998). The dye uptake experiments described above were done using LY, which has two negative charges (MW = 457). To test whether positive charged molecules can pass through Cx45 hemichannels, 2 mg/ml propidium iodide (two positive charges, MW = 668) was used in dye uptake assays. The Cx45-HeLa and NT HeLa cells were incubated in bath solution (with 2 mM Ca2+ or Ca2+ free) containing PI for 30 min at room temperature (22°C). Only Cx45-transfected cells bathed in Ca2+-free external solution sequestered PI. Fig. 9 A illustrates such an experiment. The left-hand panel shows phase-contrast micrographs of Cx45-HeLa. No significant dye uptake was detected in the presence of extracellular Ca2+ in Cx45-HeLa cells or nontransfected HeLa cells.

Fig. 9 B summarizes results with Cx45-HeLa cells and NT HeLa cells. Dye uptake in Ca2+-free external solution occurred in 63% of the Cx45-HeLa cells (n = 849) showed. In the presence of 2 mM extracellular Ca2+, only 1.2% of the Cx45-HeLa cells (n = 1,247) were loaded with PI; 2.3% of the NT HeLa cells (n = 973) showed PI uptake when bathed in Ca2+-free solution. Comparison of LY data with that of PI for Cx45-HeLa cells in Ca2+-free external solution shows a difference (63% for PI versus 85% for LY). Further, PI loaded Cx45-HeLa cells bathed in Ca2+-free solution revealed no significant dye leak with the time in contrast to LY experiments. This could be explained by the binding of PI to DNA, negatively charged proteins, and lipids.

The PI transfer through intact gap junction channels was investigated using Cx45-HeLa cell pairs. The experimental procedure was the same as used for LY experiments. In each cell pair examined, the whole-cell recording mode was established on one cell with a dye-filled pipette. For the second cell, a gigaohm seal was established with the pipette leaving the patch intact. Dye spread into the recipient cell was recorded, then the whole-cell recording mode was established in the recipient cell and gap junction conductance (gj) was measured. Although Cx45-HeLa cell pairs showed detectable PI coupling, PI diffusion was sufficiently low that it was not easily detected for a number of minutes after dye injection.

Fig. 10 shows the data from PI transfer experiments. The epifluorescent micrographs taken at 1, 30, and 60 min after PI injection in cell 1 demonstrate dye spread into the recipient cell 2 (Fig. 10 A). Fig. 10 B summarizes PI transfer data from Cx45 HeLa cell pairs (n = 6, gj = 6–9 nS). It shows a plot of fluorescence intensity versus time for injected cells (Fig. 10 B, •) and the recipient cells (Fig. 10 B, ○). The inset in Fig. 10 B shows the fluorescence intensity increase in cell 2 versus time on an expanded scale. The intensity of fluorescence in the recipient cell increased very slowly over a period of 60 min, suggesting either that PI diffusion through Cx45 gap junction channels is very slow and/or PI binding is so strong that the free concentration of dye is very low and only a small amount is able to diffuse to the adjacent cell.

The data show that the quantitative analysis of PI diffusion through gap junction is complicated by PI binding effects. Quantification of the flux rates becomes very difficult. The results indicate that PI can traverse gap junctions but it is not possible to determine the flux/channel.

## Discussion

Hemichannel currents have been measured in Xenopus oocytes expressing identified connexins (for multichannel currents, rat Cx46, chicken Cx56, and Xenopus Cx38 [Ebihara and Steiner, 1993; Ebihara et al., 1995; Ebihara, 1996]; for single-channel currents, rat Cx46 [Trexler et al., 1996]). Recently, hemichannel currents also were recorded from human HeLa cells expressing mouse connexins Cx30, Cx46, and Cx50 (Valiunas and Weingart, 2000).

The aim of the present study has been to examine the conductance and perm-selectivity of mouse Cx45 and chicken Cx45 hemichannels in vertebrate cells and compare these properties to those of the intact gap junction channels. The electrophysiological studies in combination with dye uptake and leakage assays in the present study provide evidence for the opening of nonjunctional Cx45 hemichannels when external Ca2+ concentration is reduced.

### Evidence for Cx45 Hemichannels

The combination of several experimental features allows the identification of Cx45 hemichannel currents. Removal of extracellular Ca2+ in conjunction with depolarization of Vm induced a novel current, Ihc (apparent in multichannel and single-channel signals) in Cx45 transfected HeLa and RIN cells, but not in nontransfected cells. This suggests that Ihc is carried by channels introduced by transfection, possibly Cx45 hemichannels. Ihc disappeared when the external Ca2+ was restored. This is consistent with the observation that cytosolic Ca2+ impairs gap junctions via chemical gating (Weingart and Bukauskas, 1995). Ihc in Cx45 transfected HeLa and RIN cells resembles the currents seen in Xenopus oocytes expressing connexins and attributed to hemichannels (Ebihara and Steiner, 1993; Ebihara et al., 1995; Trexler et al., 1996). Intracellular acidification with CO2 caused a reversible decrease of Ihc in Cx45-HeLa cells (Fig. 2 A), showing that chemical gating mediated by pH changes is also a feature of Cx45 hemichannels—yet another property it shares with Cx46 hemichannels in Xenopus oocytes (Trexler et al., 1999). Both the pH and Ca2+ sensitivity of Ihc are consistent with the modulation of gap junction channel gating by pH and Ca2+ (Weingart and Bukauskas, 1995; Bukauskas and Peracchia, 1997; Francis et al., 1999). Moreover, Cx45-HeLa, and Cx45-RIN cells showed significant dye uptake in the absence of external Ca2+. However, no significant dye uptake was detected in the presence of normal (2 mM) extracellular Ca2+. No significant dye uptake was detected in nontransfected cells, in the presence or absence of Ca2+. Thus dye transfer occurred through Cx45 hemichannels. This is consistent with studies on Cx43 (Li et al., 1996; John et al., 1999; Quist et al., 2000)

Other types of membrane channels have been identified in HeLa (two K+ and a Cl channel) and RIN (BK channels) cells. These are easily distinguished from Cx45 hemichannels based on their Ca2+ and/or voltage sensitivity, and unitary conductance (Sauve et al., 1983, 1986; Diaz et al., 1993; Diaz and Sepulveda, 1995; Li et al., 1999).

### Multichannel and Single-channel Properties

The currents of Cx45 hemichannels showed a time- and voltage-dependent deactivation or activation. Ihc was present at depolarized Vm. It decreased and disappeared with hyperpolarization. Gating of Cx45 hemichannels with negative Vm is expected and is consistent to the findings on other connexins (Trexler et al., 1996; Valiunas and Weingart, 2000). The slow current activation of Cx45 with positive Vm is also a typical characteristic of hemichannels (Ebihara and Steiner, 1993; Ebihara et al., 1995; Trexler et al., 1996; Valiunas and Weingart, 2000). The decline of Cx45 currents with hyperpolarization is another feature typical of hemichannels (Ebihara and Steiner, 1993; Ebihara et al., 1995).

In the present study, a key finding was that ghc,ss = f(Vm) of Cx45 hemichannels is S-shaped and shows no deactivation at positive Vm (Fig. 1, A and E). This suggests that voltage gating of these hemichannels is governed by negative voltage. For comparison, experiments on heterotypic gap junctions also suggested that Cx45 connexons are gated by negative Vj (Steiner and Ebihara, 1996; Valiunas et al., 2000). Earlier it was proposed that hemichannels may have two different mechanisms of voltage gating: (1) gating due to the transmembrane voltage (Vm-gating), which affects properties of cell membranes; and (2) gating, reflecting transjunctional voltage gating (Vj-gating) of gap junction channels (Trexler et al., 1996; Oh et al., 2000; Valiunas and Weingart, 2000). Hemichannel gating was studied in detail by Oh et al. (2000) in Xenopus oocytes expressing chimeric connexins. The authors conclude that the mechanism of Vj dependence of intercellular channels is conserved in conductive hemichannels and suggested that Vj gating results from conformational changes in individual connexin subunits.

It is interesting to note that the ghc,ss at Vm = 0 (Fig. 1 E) is ∼43% of the maximum conductance at steady state for positive voltages. The time-dependent increase in hemichannel current with positive voltage steps suggests that either channel insertion occurs, or the open probability of the individual channels is increasing, or a combination of the two occur. It is probable that the Po of Cx45 for gap junction channels at Vj = 0 is <1, making a combination effect most likely. Studies on the gating properties of heterotypic gap junction channels involving Cx45 connexons (i.e., Cx40-Cx45 and Cx43-Cx45; Valiunas et al., 2000; Steiner and Ebihara, 1996) are consistent with this idea: the steady-state conductance increases when the cytoplasmic side of Cx45 is positive. The Vm,0 of Cx45 is in the positive voltage range (Vm,0 = 11 mV), implying that some gap junction channels involving Cx45 connexons may be closed at Vj = 0. The single-channel data for Cx45 collected in this study allowed us to estimate of open probability at various Vm. The Po ranged from 0.06 to 0.76 over a 100-mV range and yielded an estimated value of ∼0.4 at Vm = 0 mV.

At the single-channel level, Ihc of Cx45 exhibited two major single-channel conductances: a main state and a residual state with short-lived substates seen occasionally (Fig. 3 F). Such a behavior was observed for Cx30 and Cx50 hemichannels in HeLa cells (Valiunas and Weingart, 2000), for Cx46 (Trexler et al., 1996) and for chimeric Cx32*Cx43 (Oh et al., 2000) hemichannels in Xenopus oocytes and resembles gap junction channels properties (Bukauskas et al., 1995a; Valiunas et al., 1999).

The measurements yielded a γhc,main = 57 pS for Cx45-HeLa hemichannels which can be estimated from γj,main by means of Eq. 1. In comparison, the conductance of Cx45-HeLa gap junction channels (γj,main) was found to be 25 pS under similar ionic conditions (unpublished data). The analogous observations for Cx46 hemichannel and intact channel conductances were obtained in Xenopus oocytes (Trexler et al., 2000).

The unitary conductance of Cx45 hemichannels revealed a distinct Vm dependency. The γhc,main increased with hyperpolarization and decreased with depolarization (Fig. 4). This feature has been observed earlier for other hemichannels in HeLa cells (Valiunas and Weingart, 2000) and for Cx46 hemichannels expressed in Xenopus oocytes (Trexler et al., 1996; Pfahnl and Dahl, 1998).

Electrophysiological studies of single and multichannel currents demonstrate the following: (1) Cx45 hemichannels are voltage-, pH-, and Ca2+-gated; and (2) docking does not interfere with the polarity of gating of Cx45 gap junction channels.

### Permeability Properties

#### Dye Uptake and Leak

Dye uptake from the external bath solution with reduced extracellular Ca2+ was used to assay for the opening of hemichannels. There is a strong correlation between dye uptake and Cx45 expression in the cell culture system. Cx45-transfected HeLa or RIN cells were able to sequester dye when extracellular Ca2+ was reduced. There was no significant dye uptake observed for NT HeLa and RIN cells (only 9.4% of NT HeLa cells took up LY versus 85% of Cx45-Hela cells). LY uptake was detected only in a small fraction of transfected cells (1.4%) in the presence of extracellular Ca2+, demonstrating that extracellular Ca2+ plays a significant role in regulation of nonjunctional hemichannels (Ebihara and Steiner, 1993; Pfahnl and Dahl, 1999; Zampighi et al., 1999; Quist et al., 2000). Correlation of electrophysiological data and dye uptake (Fig. 7 B) demonstrated that individual cell dye uptake was in proportion to the number of open hemichannels.

The measurement of the leak of LY from preloaded cells was the principle approach in assessing hemichannel permeability properties. The fluorescent intensity of LY preloaded Cx45-transfected HeLa and RIN cells decreased markedly when incubated in Ca2+-free external solution (Fig. 6). However, when LY preloaded cells were placed in solution containing 2 mM extracellular Ca2+, only marginal fluorescence intensity loss was observed (∼15%). The 15% loss could arise from the following: photo-bleaching effect, a small fraction of open hemichannels, or nonspecific membrane leak or efflux involving other membrane channels.

Cx45-transfected cells took up PI dye in Ca2+-free external solution, suggesting that positively charged PI can pass through Cx45 hemichannels. However, in comparison to LY fewer Cx45-transfected cells absorbed PI (63% vs. 85%). This suggests that PI is less permeant than LY. Three possibilities are as follows: (1) the PI molecules are bigger than LY molecules (molecular weight of 668 vs. 457) and diffuse slower; (2) Cx45 hemichannels are selective for negative over positive charge; and (3) there is low affinity binding of PI to the hemichannel itself. Whole-cell currents obtained in Ca2+-free solution from PI preloaded cells or those obtained when PI was in the patch pipette were not significantly different from nonloaded cells suggesting that if binding occurs it does not interfere with the conductive path of the hemichannel.

No detectable PI leak was observed from loaded cells even in Ca2+-free solution. PI is known to localize in the nucleus and is able to bind DNA, thus reducing the free dye concentration dramatically.

#### Dye Diffusion through Hemichannels versus Gap Junction Channels

Dye permeation through hemichannels was qualitatively like that of gap junction channels, but the flux of dye through the hemichannel was greater than predicted based on comparison of unitary conductance for the intact gap junction channels and hemichannels. The time constant of fluorescent intensity decay (τ = 33 min for Cx45-HeLa cells) obtained from dye leak experiments was 7.4-fold less than for the Cx45 gap junction channels (33 vs. 244 min). It is possible to determine the number of LY molecules passing per channel per second by taking into account the concentrations of LY within the cells. The intracellular concentration of LY in the cells exposed to 2 mM LY bathing media was 0.24 mM. The average whole cell conductance due to hemichannels was ghc = 5.94 nS, which translates into ∼104 hemichannels being open at any instance in time. The initial slope of the fluorescence intensity loss (Fig. 6 C) yields a 40% reduction in the first 20 min or an efflux of 0.09 mM. The number of molecules (MN) that efflux can be determined by: MN = Vc · ΔCi · NA, where Vc is cell volume, ΔCi is an efflux concentration, and NA is Avogadro's number. Assuming a cell volume of 1 pl, ∼6.02 × 107 molecules have diffused out of the cell via the hemichannel pathway; i.e., ∼434 molecules pass through each hemichannel per second (6.02 × 107/104/1,200 s = 434 molecules/channel/s). The same calculation can be made for the intact gap junction channels. In this case, the concentration in the recipient cell after a 20-min exposure can be determined from Fig. 8 C. The injected cell concentration was 2 mM and the recipient cell at the 20-min mark was 0.17 mM. The average number of open channels (gj = 10 nS) was ∼400. This yields a value of 213 molecules/channel/s. Comparison of the intact channels and hemichannels permeability to LY requires normalizing the concentrations. In this case, the hemichannel concentration can be normalized to the 2 mM, concentration obtained in the pairs data. By normalizing LY concentration for the gap junction channel data and hemichannel data (about eightfold concentration increase from 0.24 to 2 mM) the difference in LY permeability can be determined. The normalized ratio for hemichannel flux to that of the gap junction channels is 3,475/213, which implies that hemichannels are 16-fold more permeable to LY than the intact gap junction channels.

There are several factors which may contribute to such a difference. In the absence of an electrical force, the ions might move independently by a random walk, for which an averaged time of d2/2D is required to diffuse a distance d (Hille, 1992), where D is diffusion coefficient. Assuming that for the intact gap junction channel distance is >2d, diffusion time through hemichannel should be at least four times less in comparison to gap junction channel. The hemichannel/gap junction channel flux ratio of 16 is consistent with a path length for LY in the hemichannel being about one fourth the length of the path in the gap junction channel. The difference between the conductance ratio (γhc,mainj,main = 2.28) and the LY flux ratio implies that the rate-limiting barriers/pore lengths are much more restrictive for LY than to monovalent ions in the intact gap junction channel relative to the hemichannel. It is an indication that the pore structure of the hemichannel is not merely half of a gap junction channel, but rather is significantly different than one half of an intact gap junction channel and/or the pore structure of the hemichannel is significantly different than its construct in the intact gap junction channel. For example, the hemichannel may lose the selectivity filter for LY (big charged molecules), which even in intact channel does not affect monovalent ions mobility when electrical force is applied.

Other factors that could contribute to the high hemichannel/gap junction channel permeability difference are long pore effects, such as flux coupling, flux saturation or the overcoming electrostatic barriers would result in less efficient flux of solutes (Hille, 1992). The LY binding (nonspecific, low affinity) could also affect the free pool of LY and therefore affect the hemichannel/gap junction channel permeability ratio. However, the time course for the binding of LY in the millimolar concentration range is on the order of hours (Brink and Ramanan, 1985; Ramanan and Brink, 1990) reducing the effective diffusion coefficient by a factor of 10 over a 6-h time frame, and it unlikely is an explanation for the high hemichannel/gap junction channel flux ratio.

Another possible factor which also could influence hemichannel/gap junction channel flux ratio and should be consider is the open channel probability. Under conditions where the open probability of the dye permeable channel conductance state is reduced, the time required for dye transfer could be increased. If channel substates are not permeable or less permeable to dye the flux would depend entirely upon the cumulative open time of the main state (Veenstra et al., 1994b). If open channel probability is reduced because the majority of channels are not in closed state, but in substate, the number of open channels may be overestimated from macroscopic conductance regarding to dye flux. The exact quantitatively evaluation of hemichannels and gap junction channels permeability needs further investigation.

In the case of positively charged PI, such a comparison of hemichannels and intact gap junction channels perm-selectivity is dubious because of PI binding to DNA or other negatively charged intracellular sites. Without a quantitative measure of binding (Brink and Ramanan, 1985) estimation of PI perm-selectivity is impossible. Only the qualitative statement that PI diffuses through Cx45 hemichannels or intact gap junction channels can be made.

### Biological Role

This study shows that Cx45 hemichannels permit transfer of small ions as well as molecules of considerable size. This may be relevant for tissues when strong intercellular signaling and/or metabolic cooperation is involved. It is known that ischemia, hypoxia and/or other forms of metabolic inhibition contribute to cellular injury and death (Wilde and Aksnes, 1995). Recently, John et al. (1999) showed that opening of Cx43 nonjunctional hemichannels was induced by metabolic inhibition, indicating that nonjunctional hemichannels may be involved in the pathogenesis of ionic disturbances. Hemichannel activity is regulated by extracellular Ca2+ (Ebihara and Steiner, 1993; Pfahnl and Dahl, 1999; Valiunas and Weingart, 2000) and by membrane depolarization (De Vries and Schwartz, 1992; Ebihara et al., 1995; Trexler et al., 1996; Valiunas and Weingart, 2000). In this study, Cx45 hemichannels have been shown to open when extracellular Ca2+ is reduced. Membrane depolarization also is able to trigger Cx45 hemichannel opening. The large pore size of the open hemichannel may allow the flow of ions down their concentration gradients. This would result in membrane depolarization and may be a potential arrythmogenic factor in the heart. Thus, the prevalence of Cx45 expression in the heart conductive system (Coppen et al., 1999) is consistent with this view. Reports of the putative role of hemichannels in calcium-dependent isosmotic volume regulation (Quist et al., 2000) also suggest that nonjunctional hemichannels may be physiological relevant.

## Acknowledgments

The author acknowledges the expert technical assistance of Laima Valiuniene. The author would like to thank Drs. P.R. Brink and R. Weingart for advice and support, and Drs. T.W. White and R.T. Mathias for their critical comments on the manuscript. The Cx45-HeLa cells were provided by Dr. K. Willecke; parental RIN cells and Cx45-RIN cells were provided by Dr. T. Steinberg.

This work was supported by a National Institutes of Health grant (GM55263 to P.R. Brink).

V. Valiunas is on leave from Institute for Biomedical Research, Kaunas Medical University, Eiveniu 4 LT-3007 Kaunas, Lithuania.

*

Abbreviations used in this paper: NT, nontransfected; LY, lucifer yellow; PI, propidium iodide; Po, open state probability; RIN, rat islet tumor.

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text:
# RStudio internship week 2

I think I can make it through the summer

The main topics and events of last week were:

1. Much git.
2. Metaprogramming and non-standard evaluation (NSE) in R
3. Four 1-hour workshops by Allison Hill on the summer-of-blogdown
• moving things over from jekyll will take some time

So many of the random things I’ve tinkered with in the past have come front and center. As an educator, I know seeing these things again make learning and understanding them easier. You build on your previous knowledge to solidify, fix, and fill in gaps in your mental model. The process repeats until you get an understanding about a topic.

For me, I’m getting a better foundation to how NSE works and how it all plays together within the Tidyverse.

### Git

I got some things merged!

The pull-request that was broken and merged on my first day finally was fixed and merged. I also got to work with the lintr package and merged it into grader too.

This week was probably the first time I’ve used git amend in a long time (if ever?). I’ve typically always just made the commit, and run git rebase -i to squash and/or amend my commits. I can see why common operations like making changes to the previous commit would have a shortcut. I typically don’t use these features because it’s another thing to teach, and understanding git rebase -i is more general than git commit --amend.

What --amend allows you to do is replace the previous git commit with another one. You can fix the commit message, or add/fix file you missed. Theses are all ways to make the commit history cleaner.

The recipe looks like this:

git add <file>
git commit --amend

<Fix/modify the commit message>

git push -f origin master

The last line does a -f force push, because the commit is actually different from the one before you --amended.

### R package versions

There’s a convention about version numbering adding a .9000+ after the patch (e.g., v0.1.4.9000) to show a a development version number. You can couple this with the DESCRIPTION file by forcing a particular version to make sure you and the team all have access to the same development features.

I came across this in grader that has a Imports for learnr (>= 0.9.2.9001).

### grader progress

We’re probably going to change the name of the package to gradethis because the package gradeR (note capital R) was submitted to CRAN right as I started.

Here’s what I learned about the library I’m working on this week:

• learnr set’s the checker function
• In the knitr chunk exercise.checker specifies the checker function
• tutorial_options(exercise.timelimit = 60, exercise.checker = grade_learnr)
• In this example, grade_learnr is the main entrypoint from learnr to grader and my work starts with this function.
• checker is called on line 129 in exercise.R in learnr
• the checker function (i.e., grade_learnr) returns a value depending on what is passed into it
• if missing: returns a list() with message, correct, type, and location keys
• if error: graded object (named checked_result of class grader_graded) with correct and message
• or evaluated code

There’s a bunch of stuff within the exercise.R function in learnr that captures information from shiny, sets up the knitr environment, and inserts the output and results into the correct place in the DOM. That’ll be a separate writeup when I leave the grader world.

For the next week or so the goal is to update the check_result API, which got me down the rabbit hole of non-standard evaluation in R (I’ll talk about it in a separate set of posts).

### Non-standard Evaluation (NSE)

I gave a talk about writing functions in R which touched on NSE but it was pretty superficial. Since NSE is so crucial to grader I’ll write a series of posts about this topic and eventually turn it into a talk.

In the meantime, here are the materials (in no particular order) I’ll be reading to:

### Misc

Other random things I’ve discovered this week

#### dplyr::count vs dplyr::tally

From the docs:

tally() is a convenient wrapper for summarise that will either call n() or sum(n) depending on whether you’re tallying for the first time, or re-tallying. count() is similar but calls group_by() before and ungroup() after. If the data is already grouped, count() adds an additional group that is removed afterwards.

dplyr::count can count observations with 0 counts (useful for group_by operations) with the .drop argument

#### pryr::standarise_call

Manipulating the function call is black magic NSE voodoo. This is the stuff that is happening within grader that gets student code, solution code, learnr and grader arguments that are all passed into grade_learnr.

# code below was copy/pasted from the console

x + y
}

# pass in part of an expression

# on the fly add more parameters!
call\$y <- 10

# evaluate the thing
eval(call)

## [1] 13

This all uses the global environment, but grader will be doing this type of thing with separate environments for each exercise that will be checked.

Also, this is all how match.call works in base R.

#### checkmate package

There is a package called checkmate that is unit testing (e.g., testthat) on steroids. It allows you to more specific type and argument checking in R. I haven’t work with the package personally yet, but it does seem to be like type hints in Python and allows more specific checks into what objects in R contain.

#### Credentials

One of the coolest things about being an intern at RStudio is being on the slack channel! I try to keep my questions reserved but one of the things that have always bothered me was how store and access credentials for R. Putting in API keys in .Renviron are common practice, but I piggy-backed on another intern’s question by asking about storing passwords more securely than in a plain text file.

I’ve used the rstudioapi, secret, and getPass libraries before, but as Raymond Hettinger always says: There must be a better way.

The resource I was given was to look at how database credentials are stored: https://db.rstudio.com/best-practices/managing-credentials/

### Summer of blog down

Lastly, the great Allison Hill hosted a series of blogdown workshops for people who were interested, summer-of-blogdown. It was a total of 4 days and we covered the basics of blogdown, how to pick (the academic) themes, deploying it on netlify, and best ways to maintain the site.

I didn’t realize how amazingly flexible the academic theme was until this workshop. I’ll be sure to move my own website over to blogdown + academic one of these days.

I’m currently trying to find out how to save urls in a common location so they can be maintained in one place and be used in links throughout the site. The ongoing search for how to use variables in an md document (tl;dr: you can’t, but you might still be able to do what I want): https://discourse.gohugo.io/t/variables-in-markdown/7113/12. It almost seemed that site variables were going to be the way to go, but that ended up in a dead end.

What I’m most excited about is the ability to write posts in Rmd and jupyter notebooks for R and Python posts.

Things I’ve learned the 4 days:

• Each folder in content is a “section” and each “section” has a “page”
• /content/home/ contain widgets
• Learn x in y minutes website for TOML
• From Greg Wilson: Put a LICENSE and CITATION + orcid on your website. Make the librarians happy.
##### Daniel Chen
###### PhD Student

My research interests include trying anything to graduate.

format: plain


text:
# Basic Electrical Engineering – Formulas and Equations

The branch of engineering that deals with the study of design and implementation of various electrical devices and systems used in our everyday life as well as generation, transmission and distribution of electrical power is popularly known as Electrical Engineering.

Electrical engineering primarily convers the study of electric circuits, power systems, electrical machines, power electronics, control system, and many more. Electrical engineering uses formulae and mathematical equations to explain and prove the truth of concepts. These formulae and equations are very useful in understanding the behavior of electrical systems and helps to perform the different calculations in practice.

## Electric Charge

The subatomic property of substances by virtue of which the substances show the electrical behavior is called electric charge. The electric charge is denoted by Q (or q), and is measured in Coulombs (C). The electric charge is carried two elementary atomic particles namely electrons and protons, where an electron carries a negative charge while a proton carries a positive charge.

$$\mathrm{Charge\: on\: an\: electron, \mathit{e} \, =\, -1.6 \times 10^{-19} C}$$

The electric charge is a quantized quantity, which means the electric charge always exists as the integral multiple of elementary charge (e), i.e.,

$$\mathrm{Q = ne;\:\:\: where, n = 0,\, 1,\, 2,\, 3,\, \cdot \cdot \cdot}$$

## Voltage

Voltage, also called potential difference, is defined as the difference of electric potentials of two points in an electrical circuit. It is measured in Volt (V). It may also be defined as the amount of work done required to move a unit charge from one point to another in an electric circuit, i.e.,

$$\mathrm{Voltage, \: V\, =\, \frac{Work\: done\: \left ( W \right )}{Charge\: \left ( Q \right )}}$$

## Electric Current

The directed flow of electric charge, more specifically electrons, through a conductor is known as electric current. Electric current is denoted by I (or i), and is measured in Ampere (A). Electrical current may also be defined as the time rate of change of charge, i.e.,

$$\mathrm{I\, =\, \frac{Q}{t}}$$

In differential form,

$$\mathrm{i\, =\, \frac{dq}{dt}}$$

## Electrical Resistance

Electrical resistance is the measure of opposition offered by a substance in the flow of electric current. It is denoted by R and is measured in Ohms (Ω).

$$\mathrm{R\, =\, \frac{\rho l}{a}}$$

Where, ρ is a constant called resistivity or specific resistance of the material. The resistivity is defined as the property of material by which it opposes the flow of current through it.

## Ohm’s Law

Ohm’s law is a fundamental law related to electrical circuit. It states that the voltage across a conductor is directly proportional to the current flowing through it provided the physical conditions remain constant. It gives the relationship between voltage, current and resistance of a conductor as,

$$\mathrm{V \, =\, IR}$$

## Conductance

The measure of ease that the material offers in the path of current is called conductance, and it is given by as reciprocal of electrical resistance, i.e.

$$\mathrm{Conductance,\, G \, =\, \frac{I}{R}\, =\, \frac{a}{\rho l}\, =\,\frac{\sigma a}{l}}$$

Where, σ is the conductivity of the material, and is given by,

$$\mathrm{Conductivity,\, \sigma \, =\, \frac{1}{\rho }}$$

## Electric Power

The rate of doing work in an electric circuit is called electric power. It is denoted by P and measured in watt (W).

$$\mathrm{P\, \sigma \, =\, \frac{dW}{dt }}$$

In DC circuits,

$$\mathrm{Power,P\, =\,VI\, = \, I^{2}R\, =\,\frac{V^{2}}{R}}$$

In single-phase AC circuits,

$$\mathrm{Active\: power, P = VI\: cos\, \phi }$$

$$\mathrm{Reactive\: power, Q = VI\: sin\, \phi }$$

$$\mathrm{Apparent\: power, S = VI}$$

Where, active power is measured in watts (W), reactive power in volt-ampere reactive (VAr), and apparent power in volt-ampere (VA).

In three-phase AC circuits,

$$\mathrm{Active\: power,P\, =\, 3V_{p}I_{p}\, cos\, \phi \, = \, \, =\, \sqrt{3}V_{L}I_{L}\, cos\, \phi }$$

$$\mathrm{Reactive\: power,Q\, =\, 3V_{p}I_{p}\, sin\, \phi \, = \, \, =\, \sqrt{3}V_{L}I_{L}\, sin\, \phi }$$

$$\mathrm{Apparent\: power,S\, =\, 3V_{p}I_{p}\, = \, \, =\, \sqrt{3}V_{L}I_{L}}$$

## Power Factor

Power factor is measure of utilization of electric power in an ac electric circuit. It provides information about the part of total power utilized by load in an electrical system. It is given by the ratio of active power to the apparent power, i.e.,

$$\mathrm{Power\: factor,cos\, \phi \, =\, \frac{Active\: power (P)}{Apparent\: power (S)}}$$

The power factor of an electrical load varies from -1 to 1. For resistive load it is unity (1), for inductive load it is lagging, and for capacitive load it is leading.

## Frequency and Time Period

The number of cycle that an alternating quantity completes in one second is called frequency of the quantity. It is denoted by f and is measured in Hertz (Hz).

$$\mathrm{f \, =\, \frac{No.\: of\: cycles}{time}}$$

The time required by an alternating quantity to complete one cycle is called its time period. It is denoted by T, and is measured in seconds (s).

The frequency of an alternating quantity is inversely proportional to its time period, i.e.,

$$\mathrm{f \propto \frac{1}{T}}$$

## Wavelength

For an alternating quantity, the distance between two consecutive crests in the adjacent cycles of a wave is called wavelength of the signal. It is denoted by Greek letter Lambda (λ).

$$\mathrm{\lambda \, =\, \frac{ u }{f}}$$

Where, v is the velocity of wave and f is the frequency

## Capacitance

The property of a substance of storing electric charge in the form of electrostatic field is called capacitance of the substance. The circuit element used to introduce the capacitance effect in an electric circuit is called capacitor. The capacitance of capacitor is denoted by C, and measured in Farad (F).

The capacitance of a capacitor is given by,

$$\mathrm{C \, =\, \frac{Q }{V}}$$

Where, Q is the electric charge accumulated on each plate of the capacitor, and V is the voltage across the plates of the capacitor.

We may also express the capacitance of a capacitor in terms of its physical dimensions as,

$$\mathrm{C \, =\, \frac{\epsilon A }{d}}$$

Where, ε is the permittivity of medium between plates, A is the area of cross-section of capacitor plate, and d is the distance between plates of the capacitor.

## Inductance

Inductance is a property of materials by which they store electrical energy in the form of magnetic field. The element is known as inductor. The inductance is denoted by L and measured in Henry (H). It is given by the ratio of magnetic flux linkage and current as,

$$\mathrm{L \, =\, \frac{N \phi }{I}}$$

Where, N is the number of turn in inductor coil, and ϕ is the magnetic flux.

## Electric Field Intensity

The space around an electrically charge body in which a test charge experiences a force of attraction or repulsion is known as electric field. The strength of force acting on the charge placed in the field is called electric field intensity. It is denoted by E and is measured in Newton per coulomb (N/C).

$$\mathrm{E \, =\, \frac{F}{Q}}$$

## Coulomb’s Law

The Coulomb’s law is a fundamental law of electrostatics, which states that the electrostatic force acting between two charges is directly proportional to the product of magnitude of charges and inversely proportional to square of the distance between charges, i.e.,

$$\mathrm{F \propto \frac{Q_{1}Q_{2}}{d^{2}}}$$

$$\mathrm{\Rightarrow F \, =\, k \frac{Q_{1}Q_{2}}{d^{2}} \, =\,\frac{1}{4\pi \epsilon }\frac{Q_{1}Q_{2}}{d^{2}} }$$

Where, Q1 and Q2 are static charges, and d is the distances between these charges

## Gauss’s Law for Electric Flux

The Gauss’s law of electrostatics gives the value of electric flux, which is,

$$\mathrm{\phi _{e}\, =\, \frac{Q}{\epsilon } }$$

## EMF Equation of DC Generator

The emf generated by a DC generator is given by,

$$\mathrm{E_{g}\, =\, \frac{NP\phi Z}{60A } }$$

Where, N is the speed of rotation of armature, P is the number of field poles, ϕ is the magnetic flux per poles, Z is the number of armature conductors and A is the no. of parallel paths.

## Back EMF of DC Motor

The emf induced in a dc motor due to electromagnetic induction is called back emf or counter emf. It is given by,

$$\mathrm{E_{b}\, =\, \frac{NP\phi Z}{60A } }$$

## EMF Equation of Transformer

The expression that gives the value of emf induced in windings of a transformer is called emf equation. It is given by,

$$\mathrm{E\, =\,4.44\, f\phi _{m}N }$$

Where, f is the frequency of ac supply, ϕm is the maximum flux and N is the number of turns in winding.

## Hysteresis Loss

The power loss in iron core of electrical machines (motor, generator, transformer, etc.) due to magnetic reversal is called hysteresis loss, and it is calculated by,

$$\mathrm{P_{h}\, =\,\eta\, B_{m}^{1.6}\, fV }$$

Where, η is the hysteresis coefficient, Bm is maximum flux density in core, V is volume of the core, and f is the frequency of magnetic reversal.

## Eddy-Current Loss

The power loss occurs due to eddy currents induced in iron core of machines is called eddy current loss, and is given by,

$$\mathrm{P_{e}\, =\,k_{e}\, B_{m}^{2}\, f^{2}t^{2}\, V }$$

Where, ke is a constant, and t is the thickness of each lamination of core.

## Transformer Turns Ratio and Transformation Ratio

Turns ratio of a transformer is defined as the ratio of number of turns in primary winding to the number of turns in secondary winding, i.e.

$$\mathrm{Turns \: ratio, a\, =\,\frac{N_{1}}{N_{2}}\,=\,\frac{E_{1}}{E_{2}}\,=\,\frac{V_{1}}{V_{2}}\,=\,\frac{I_{2}}{I_{1}}}$$

Transformation ratio of a transformer is defined as the ratio of output voltage to the input voltage, i.e.,

$$\mathrm{Transformation \: ratio\, =\,\frac{V_{2}}{V_{1}}\,=\,\frac{E_{2}}{E_{1}}\,=\,\frac{N_{2}}{N_{1}}\,=\,\frac{I_{1}}{I_{2}}\,=\,\frac{1}{a}}$$

## Synchronous Speed

In rotating electric machines such as motors and generators, the magnetic field rotates at a constant speed, which is called synchronous speed.

$$\mathrm{N_{s}\, =\, \frac{120\, f}{P}}$$

Where, f is the supply frequency, and P is the field poles in the machine.

## Efficiency

For an electrical machine, the ratio of output power to the input power is known as efficiency of the machine, i.e.,

$$\mathrm{Efficiency,\eta \, =\, \frac{Output\: power\left ( P_{o} \right )}{Input\: power\left ( P_{i} \right )}}$$

## EMF Equation of 3-Phase Alternator

The EMF equation of a 3-phase alternator gives the magnitude of generated EMF. Generated EMF per phase is,

$$\mathrm{E_{ph} \, =\,2.22k_{p}k_{d}f\phi Z\, =\, 4.44k_{p}k_{d}f\phi T }$$

Where, kp and kd are pitch factor and distribution factor of armature winding respectively, Z is the number of conductors per phase, and T is the number of turns per phase.

## Electrical Impedance

In AC circuits, the combined opposition offered by the resistance, inductance and (or) capacitance in the flow of current is called impedance. It is denoted by Z and measured in Ohms (Ω).

$$\mathrm{Z \, =\,R+jX}$$

Where, X is the reactance, which is opposition offered by inductor or capacitor.

For inductor,

$$\mathrm{Inductive\: reactance,\, X_{L} \, =\,\omega L\, =\, 2\pi fL}$$

For capacitor,

$$\mathrm{Capacitive\: reactance,\, X_{C} \, =\,\frac{1}{\omega C}\, =\, \frac{1}{2\pi fC}}$$

## Conclusion

In this article, we listed all the important formulae and equations of basic electrical engineering. Also, we defined each quantity for your reference. Every electrical engineer must know all these formulae because they are frequently used in different calculations.


text:
## On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $$\mathbb{R}^N$$.(English)Zbl 0935.35044

The author applies a “monotonicity trick” introduced by Struwe in order to derive an existence result for a large class of functionals having a mountain-pass geometry. The abstract theorem establishes, essentially, the existence of a bounded Palais-Smale sequence at the mountain-pass level. This result is then applied to deduce the existence of a positive solution $$u\in H^1({\mathbb R}^N)$$ to the problem $$-\Delta u+Ku=f(x,u)$$, where $$K$$ is a positive constant, provided that the energy functional associated to the above problem has a mountain-pass geometry. The nonlinearity $$f$$ is assumed to satisfy the following conditions: (i) $$f(x,u)u^{-1}\rightarrow a\in (0,+\infty ]$$ as $$u\rightarrow +\infty$$; and (ii) the mapping $$[0,+\infty)\ni u\mapsto f(x,u)u^{-1}$$ is non-decreasing, a.e. $$x\in{\mathbb R}^N$$.
The paper gives a new and interesting perspective in the critical point theory and its applications to the study of variational problems.

### MSC:

 35J60 Nonlinear elliptic equations 35A15 Variational methods applied to PDEs 49J35 Existence of solutions for minimax problems 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text:

### References:

 [1] Schechter, Diff. Int. Eqns 3 pp 889– (1990) [2] Struwe, Variational methods (1996) [3] DOI: 10.1007/BF01205672 · Zbl 0618.35111 [4] Brezis, Analyse fonctionnelle (1983) [5] Berestycki, Arch. Ration. Mech. Analysis 82 pp 313– (1983) [6] DOI: 10.1016/0362-546X(83)90115-3 · Zbl 0522.58012 [7] DOI: 10.1002/cpa.3160450908 · Zbl 0801.35026 [8] DOI: 10.1007/BF01053458 · Zbl 0694.76012 [9] DOI: 10.1016/0022-1236(73)90051-7 · Zbl 0273.49063 [10] Ambrosetti, Homoclinics for second order convervative systems. Partial differential equations and related subjects (1992) [11] DOI: 10.1007/s000330050128 · Zbl 0916.35036 [12] Ambrosetti, Atti Ace. Naz. Lincei 52 pp 660– (1972) [13] Tarantello, C. R. Acad. Sci. Paris 313 pp 441– (1991) [14] DOI: 10.1002/(SICI)1099-1476(19961125)19:17<1397::AID-MMA833>3.0.CO;2-B · Zbl 0862.35123 [15] Lions, Ann. Inst. H. Poincaré Analyse non linéaire 1 pp 109– (1984) · Zbl 0541.49009 [16] DOI: 10.1016/S0362-546X(96)00021-1 · Zbl 0877.35091 [17] DOI: 10.1006/jdeq.1994.1095 · Zbl 0804.35033 [18] DOI: 10.1155/S1073792895000079 · Zbl 0835.58029 [19] DOI: 10.1017/CBO9780511551703 [20] DOI: 10.1016/0022-247X(74)90025-0 · Zbl 0286.49015 [21] DOI: 10.1002/cpa.3160451002 · Zbl 0785.35029 [22] Cerami, Rend. Acad. Sci. Let. 1st. Lombardo 112 pp 332– (1978) [23] DOI: 10.1090/S0002-9939-1993-1145940-X [24] DOI: 10.1112/plms/s3-57.3.511 · Zbl 0673.35005 [25] DOI: 10.1112/plms/s3-45.1.169 · Zbl 0505.35010 [26] Struwe, C. R. Acad. Sci. Pans 317 pp 677– (1993) [27] DOI: 10.1007/BF02585433 · Zbl 0719.58032 [28] DOI: 10.1007/BF02392272 · Zbl 0646.53005 [29] DOI: 10.1007/BF02571356 · Zbl 0707.58022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
## Essential University Physics: Volume 1 (4th Edition) Clone

b) $5.5 \ km/s$
We use the equation for escape speed to find: $V_{esc} =\sqrt{\frac{2GM}{r}}$ $V_{esc} =\sqrt{\frac{2(6.67\times10^{-11})(5.97\times10^{24})}{6.37\times10^6+(20.2\times10^6)}}=5470\ m/s = 5.5 \ km/s$


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# Thread: What Do These Question Mean?

1. ## What Do These Question Mean?

Through these problems, express the given limit as a definite integral over the indicated interval [a,b]. Assuming that [xi-1,x] denotes the ith subinterval of a subdivision of [a,b] into n subintervals, all with the same length Δx = (b-a/n, and that mi =1/2(xi-1 + xi is the midpoint of the ith subinterval.
qqqqqqqqqqn
qqqlimqqqqqΣ (x^3-3xi^2 +1) Δx over [0,3]
qqqqn->∞qi=1

qqqqqqqqqqn
qqqlimqqqqqΣ (25-xi^2)^0.5 Δx over [0,5]
qqqn->∞qqi=1

Please show work and explain, I am very lost . Ignore the q's in the formulas.

2. Have you heard of Riemann Sums?

3. Wow. Maybe it's just my browser, but your "equations" are displaying as mostly long strings of the letter "q".

I see a lot of q's and n's too

5. That's because most browsers do not respect "spaces" at beginnings of lines. In order to maintain the spacing, he put "q" at the beginnings of lines. Any way, LaTex is better. They are:

$\sum_{i= 1}^n (x^3-3x_i^2 +1) \Delta x$ over [0, 3]
and
$\sum_{i+ 1}^n \sqrt{25- x_i^2} \Delta x$ over [0, 5]

Billy Maize, obviously whoever gave you these problems expects you to know about the "Riemann sums" chengbin refers too. You can hardly expect a tutorial here but the basic idea is to divide the x-axis into a number of intervals, use the height of the function at each point to construct rectangles, then add the areas of the rectangle to approximate the integral.

Taking the limit, as the number of intervals goes to infinity and $\Delta x$ goes to 0, gives the exact integral. What you want to write for your answer is something like $\int_a^b f(x) dx$. Now what do you think a, b, and f(x) is for each of those? (I'll give you this hint- $\Delta x$ becomes dx!)

6. Thank you halls!


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# Find the sum of all Truncatable primes below N in Python

Suppose we have a given integer N; we have to find the sum of all Truncatable primes less than N. As we know the truncatable prime is a number which is left-truncatable prime (if the leading "left" digit is successively removed, then all resulting numbers are treated as prime) as well as right-truncatable prime (if the last "right" digit is successively removed, then all the resulting numbers are treated as prime). An example of truncatable prime is 9137 as this is lefttruncatable prime because 9137, 137, 37 and 7 are primes. Hence 9137 is a truncatable prime.

So, if the input is like N = 55, then the output will be 130 as (2 + 3 + 5 + 7 + 23 + 37 + 53) =

To solve this, we will follow these steps −

• N := 1000005

• prime := a list of size N and fill with True

• Define a function sieve() . This will take

• prime[1] := False, prime[0] := False

• for i in range 2 to N, do

• if prime[i] is same as True, then

• for j in range i * 2 to N, update in each step by i, do

• prime[j] := False

• From the main method, do the following −

• sum := 0

• for i in range 2 to n, do

• current := i

• f := True

• while current is non-zero, do

• if prime[current] is False, then

• f := False

• come out from the loop

• current := current / 10

• current := i

• power := 10

• while quotient of (current / power) is non-zero, do

• if prime[current mod power] is False, then

• f := False

• come out from the loop

• power := power * 10

• if f is True, then

• sum := sum + i

• return sum

## Example

Let us see the following implementation to get better understanding −

Live Demo

N = 1000005
prime = [True for i in range(N)]
def sieve():
prime[1] = False
prime[0] = False
for i in range(2, N):
if (prime[i]==True):
for j in range(i * 2, N, i):
prime[j] = False
def get_total_of_trunc_primes(n):
sum = 0
for i in range(2, n):
current = i
f = True
while (current):
if (prime[current] == False):
f = False
break
current //= 10
current = i
power = 10
while (current // power):
if (prime[current % power] == False):
f = False
break
power *= 10
if f:
sum += i
return sum
n = 55
sieve()
print(get_total_of_trunc_primes(n))

## Input

55

## Output

130