2416:
where subscripts 'i' and 'o' denote the intra- and extracellular compartments, respectively. Intuitively one may understand these limits as follows: if an ion is only found outside a cell, then the flux is Ohmic (proportional to voltage) when the voltage causes the ion to flow into the cell, but no
1018:
1678:
367:
2335:
2100:
1941:
Since one of the assumptions of the GHK flux equation is that the ions move independently of each other, the total flow of ions across the membrane is simply equal to the sum of two oppositely directed fluxes. Each flux approaches an
1931:
1180:
2471:
scaled by the surface area). As the ratio between the two concentrations increases, so does the difference between the two slopes, meaning that the current is larger in one direction than the other, given an equal
2411:
2176:
1427:
728:
1380:
606:
550:
128:
57:
and the concentrations of the ion inside and outside of the cell. Since both the voltage and the concentration gradients influence the movement of ions, this process is a simplified version of
1799:
682:
1321:
1249:
2187:
1952:
709:
1419:
1216:
642:
1052:
2446:
394:
35:
1814:
1060:
2540:
2467:. Thus, if the two concentrations are identical, the slope will be identical (and constant) throughout the voltage range (corresponding to
2587:
1673:{\displaystyle \Phi _{S}=0=P_{S}{\frac {z_{S}^{2}F^{2}}{RT}}{\frac {V_{m}(_{i}-_{o}\exp(-z_{S}V_{m}F/RT))}{1-\exp(-z_{S}V_{m}F/RT)}}}
2476:
of opposite signs. This is contrary to the result obtained if using Ohm's law scaled by the surface area, and the effect is called
2341:
2106:
1013:{\displaystyle \lim _{V_{m}\rightarrow 0}\Phi _{S}=P_{S}{\frac {z_{S}^{2}F^{2}}{RT}}{\frac {_{i}-_{o}\exp(-z_{S}V_{m}F/RT))]'}{'}}}
362:{\displaystyle \Phi _{S}=P_{S}z_{S}^{2}{\frac {V_{m}F^{2}}{RT}}{\frac {_{i}-_{o}\exp(-z_{S}V_{m}F/RT)}{1-\exp(-z_{S}V_{m}F/RT)}}}
1329:
555:
499:
493:
is shown to be contained in the GHK flux equation (Flax 2008). The proof is replicated from the reference (Flax 2008) here.
2597:
1689:
62:
2459:. It is evident that the ratio between the two asymptotes is merely the ratio between the two concentrations of S,
2417:
voltage could cause the ion to flow out of the cell, since there are no ions inside the cell in the first place.
719:
2508:
647:
496:
We wish to show that when the flux is zero, the transmembrane potential is not zero. Formally it is written
65:
and the GHK flux equation is a solution to the Nernst–Planck equation with the assumptions listed below.
104:
The electrical field is constant so that the transmembrane potential varies linearly across the membrane
2330:{\displaystyle \Phi _{S|o\to i}=P_{S}z_{S}^{2}{\frac {V_{m}F^{2}}{RT}}_{o}\ {\mbox{for}}\ V_{m}\ll \;0}
2095:{\displaystyle \Phi _{S|i\to o}=P_{S}z_{S}^{2}{\frac {V_{m}F^{2}}{RT}}_{i}\ {\mbox{for}}\ V_{m}\gg \;0}
1254:
1221:
461:
687:
2592:
78:
2572:
1391:
712:
490:
82:
54:
97:
Several assumptions are made in deriving the GHK flux equation (Hille 2001, p. 445) :
2545:
2550:
Goldman-Hodgkin-Katz
Cochlear Hair Cell Models – a Foundation for Nonlinear Cochlear Mechanics
2536:
2484:
1188:
614:
2562:
2496:
441:
74:
31:
2431:
379:
2567:
1805:
400:
is the current density (flux) outward through the membrane carried by ion S, measured in
107:
The ions access the membrane instantaneously from the intra- and extracellular solutions
1026:
2581:
2495:
is large and/or when one or both of the concentrations change considerably during an
50:
608:, which states that when the transmembrane potential is zero, the flux is not zero.
2528:
451:
86:
2468:
17:
1926:{\displaystyle V_{m}=-{\frac {RT}{z_{S}F}}\ln \left({\frac {_{i}}{_{o}}}\right)}
113:
The movement of ions is affected by both concentration and voltage differences
2477:
1943:
45:(or GHK flux equation or GHK current density equation) describes the ionic
1175:{\displaystyle \lim _{V_{m}\rightarrow 0}\Phi _{S}=P_{S}z_{S}F(_{i}-_{o})}
474:
is the intracellular concentration of ion S, measured in mol·m or mmol·l
2500:
1946:
as the membrane potential diverges from zero. These asymptotes are
465:
401:
2473:
432:
46:
2406:{\displaystyle \Phi _{S|o\to i}=0\;{\mbox{for}}\ V_{m}\gg \;0}
2171:{\displaystyle \Phi _{S|i\to o}=0\;{\mbox{for}}\ V_{m}\ll \;0}
480:
is the extracellular concentration of ion S, measured in mol·m
122:
The GHK flux equation for an ion S (Hille 2001, p. 445):
413:
is the permeability of the membrane for ion S measured in m·s
2427:
constant, the equation yields a straight line when plotting
2535:, 3rd ed., Sinauer Associates, Sunderland, Massachusetts.
1054:
represents the differential of f and the result is :
2511:
cycle can change 100-fold or more, and the ratio between
1375:{\displaystyle \lim _{\Phi _{S}\rightarrow 0}V_{m}\neq 0}
601:{\displaystyle \lim _{V_{m}\rightarrow 0}\Phi _{S}\neq 0}
545:{\displaystyle \lim _{\Phi _{S}\rightarrow 0}V_{m}\neq 0}
611:
However, due to the form of the GHK flux equation when
2377:
2301:
2281:
2142:
2066:
2046:
1900:
1878:
1720:
1697:
1542:
1519:
1288:
1265:
1153:
1130:
863:
840:
234:
211:
2499:. The most common example is probably intracellular
2434:
2344:
2190:
2109:
1955:
1817:
1692:
1430:
1394:
1332:
1257:
1224:
1191:
1063:
1029:
731:
690:
650:
617:
558:
502:
382:
131:
61:. Electrodiffusion is most accurately defined by the
1385:which is the definition of the reversal potential.
1185:It is evident from the previous equation that when
27:
Expression of the ionic flux across a cell membrane
2440:
2405:
2329:
2170:
2094:
1925:
1793:
1672:
1413:
1374:
1315:
1243:
1210:
1174:
1046:
1012:
703:
676:
636:
600:
544:
388:
361:
1421:we can also obtain the reversal potential :
1794:{\displaystyle _{i}-_{o}\exp(-z_{S}V_{m}F/RT)=0}
1334:
1065:
733:
560:
504:
8:
2552:, Conference proceedings: Interspeech 2008.
2399:
2375:
2323:
2164:
2140:
2088:
2433:
2390:
2376:
2353:
2349:
2343:
2314:
2300:
2291:
2280:
2260:
2250:
2243:
2237:
2232:
2222:
2199:
2195:
2189:
2155:
2141:
2118:
2114:
2108:
2079:
2065:
2056:
2045:
2025:
2015:
2008:
2002:
1997:
1987:
1964:
1960:
1954:
1910:
1899:
1888:
1877:
1871:
1849:
1834:
1822:
1816:
1771:
1762:
1752:
1730:
1719:
1707:
1696:
1691:
1653:
1644:
1634:
1593:
1584:
1574:
1552:
1541:
1529:
1518:
1506:
1499:
1482:
1472:
1467:
1460:
1454:
1435:
1429:
1399:
1393:
1360:
1342:
1337:
1331:
1298:
1287:
1275:
1264:
1256:
1229:
1223:
1196:
1190:
1163:
1152:
1140:
1129:
1114:
1104:
1091:
1073:
1068:
1062:
1028:
985:
976:
966:
914:
905:
895:
873:
862:
850:
839:
827:
817:
800:
790:
785:
778:
772:
759:
741:
736:
730:
691:
689:
664:
655:
649:
622:
616:
586:
568:
563:
557:
530:
512:
507:
501:
485:Implicit definition of reversal potential
381:
342:
333:
323:
285:
276:
266:
244:
233:
221:
210:
204:
187:
177:
170:
164:
159:
149:
136:
130:
2483:The GHK flux equation is mostly used by
677:{\displaystyle \Phi _{S}={\frac {0}{0}}}
101:The membrane is a homogeneous substance
36:Goldman–Hodgkin–Katz voltage equation
7:
722:to find the solution for the limit:
684:. This is a problem as the value of
2533:Ion channels of excitable membranes
2435:
2346:
2192:
2111:
1957:
1432:
1396:
1339:
1226:
1088:
756:
652:
583:
509:
444:, equal to 96,485 C·mol or J·V·mol
431:is the transmembrane potential in
383:
133:
81:, and the English Nobel laureates
43:Goldman–Hodgkin–Katz flux equation
25:
1316:{\displaystyle (_{i}-_{o})\neq 0}
110:The permeant ions do not interact
1244:{\displaystyle \Phi _{S}\neq 0}
552:which is equivalent to writing
2361:
2354:
2288:
2277:
2207:
2200:
2126:
2119:
2053:
2042:
1972:
1965:
1907:
1896:
1885:
1874:
1782:
1742:
1727:
1716:
1704:
1693:
1664:
1624:
1607:
1604:
1564:
1549:
1538:
1526:
1515:
1512:
1348:
1304:
1295:
1284:
1272:
1261:
1258:
1169:
1160:
1149:
1137:
1126:
1123:
1079:
1037:
1030:
1000:
996:
956:
941:
932:
928:
925:
885:
870:
859:
847:
836:
833:
820:
747:
704:{\displaystyle {\frac {0}{0}}}
574:
518:
353:
313:
296:
256:
241:
230:
218:
207:
1:
2548:and Holmes, W.Harvey (2008).
468:(= degrees Celsius + 273.15)
1414:{\displaystyle \Phi _{S}=0}
2614:
2519:can reach 20,000 or more.
29:
2588:Electrochemical equations
2420:Keeping all terms except
2509:cardiac action potential
1683:which reduces to :
454:, equal to 8.314 J·K·mol
30:Not to be confused with
2487:when the ratio between
1211:{\displaystyle V_{m}=0}
637:{\displaystyle V_{m}=0}
422:is the valence of ion S
89:derived this equation.
55:transmembrane potential
2442:
2407:
2331:
2172:
2096:
1927:
1795:
1674:
1415:
1376:
1317:
1245:
1212:
1176:
1048:
1014:
705:
678:
638:
602:
546:
404:per square meter (A·m)
390:
363:
63:Nernst–Planck equation
2443:
2441:{\displaystyle \Phi }
2408:
2332:
2173:
2097:
1928:
1796:
1675:
1416:
1377:
1318:
1246:
1213:
1177:
1049:
1015:
706:
679:
639:
603:
547:
391:
389:{\displaystyle \Phi }
364:
53:as a function of the
2485:electrophysiologists
2432:
2342:
2188:
2107:
1953:
1815:
1690:
1428:
1392:
1330:
1255:
1222:
1189:
1061:
1027:
729:
688:
648:
615:
556:
500:
462:absolute temperature
380:
129:
2598:Bioelectrochemistry
2242:
2007:
1477:
795:
169:
79:Columbia University
2573:Reversal potential
2438:
2403:
2381:
2327:
2305:
2285:
2228:
2168:
2146:
2092:
2070:
2050:
1993:
1923:
1904:
1882:
1791:
1724:
1701:
1670:
1546:
1523:
1463:
1411:
1372:
1355:
1313:
1292:
1269:
1241:
1208:
1172:
1157:
1134:
1086:
1044:
1010:
867:
844:
781:
754:
701:
674:
634:
598:
581:
542:
525:
491:reversal potential
386:
359:
238:
215:
155:
83:Alan Lloyd Hodgkin
2541:978-0-87893-321-1
2507:, which during a
2385:
2380:
2309:
2304:
2299:
2284:
2275:
2150:
2145:
2074:
2069:
2064:
2049:
2040:
1917:
1903:
1881:
1859:
1804:and produces the
1723:
1700:
1668:
1545:
1522:
1497:
1333:
1291:
1268:
1156:
1133:
1064:
1047:{\displaystyle '}
1008:
866:
843:
815:
732:
699:
672:
559:
503:
357:
237:
214:
202:
18:GHK flux equation
16:(Redirected from
2605:
2563:Goldman equation
2497:action potential
2447:
2445:
2444:
2439:
2412:
2410:
2409:
2404:
2395:
2394:
2383:
2382:
2378:
2368:
2367:
2357:
2336:
2334:
2333:
2328:
2319:
2318:
2307:
2306:
2302:
2297:
2296:
2295:
2286:
2282:
2276:
2274:
2266:
2265:
2264:
2255:
2254:
2244:
2241:
2236:
2227:
2226:
2214:
2213:
2203:
2177:
2175:
2174:
2169:
2160:
2159:
2148:
2147:
2143:
2133:
2132:
2122:
2101:
2099:
2098:
2093:
2084:
2083:
2072:
2071:
2067:
2062:
2061:
2060:
2051:
2047:
2041:
2039:
2031:
2030:
2029:
2020:
2019:
2009:
2006:
2001:
1992:
1991:
1979:
1978:
1968:
1944:asymptotic value
1932:
1930:
1929:
1924:
1922:
1918:
1916:
1915:
1914:
1905:
1901:
1894:
1893:
1892:
1883:
1879:
1872:
1860:
1858:
1854:
1853:
1843:
1835:
1827:
1826:
1800:
1798:
1797:
1792:
1775:
1767:
1766:
1757:
1756:
1735:
1734:
1725:
1721:
1712:
1711:
1702:
1698:
1679:
1677:
1676:
1671:
1669:
1667:
1657:
1649:
1648:
1639:
1638:
1610:
1597:
1589:
1588:
1579:
1578:
1557:
1556:
1547:
1543:
1534:
1533:
1524:
1520:
1511:
1510:
1500:
1498:
1496:
1488:
1487:
1486:
1476:
1471:
1461:
1459:
1458:
1440:
1439:
1420:
1418:
1417:
1412:
1404:
1403:
1381:
1379:
1378:
1373:
1365:
1364:
1354:
1347:
1346:
1322:
1320:
1319:
1314:
1303:
1302:
1293:
1289:
1280:
1279:
1270:
1266:
1250:
1248:
1247:
1242:
1234:
1233:
1217:
1215:
1214:
1209:
1201:
1200:
1181:
1179:
1178:
1173:
1168:
1167:
1158:
1154:
1145:
1144:
1135:
1131:
1119:
1118:
1109:
1108:
1096:
1095:
1085:
1078:
1077:
1053:
1051:
1050:
1045:
1043:
1019:
1017:
1016:
1011:
1009:
1007:
1006:
989:
981:
980:
971:
970:
939:
938:
918:
910:
909:
900:
899:
878:
877:
868:
864:
855:
854:
845:
841:
832:
831:
818:
816:
814:
806:
805:
804:
794:
789:
779:
777:
776:
764:
763:
753:
746:
745:
720:l'Hôpital's rule
710:
708:
707:
702:
700:
692:
683:
681:
680:
675:
673:
665:
660:
659:
643:
641:
640:
635:
627:
626:
607:
605:
604:
599:
591:
590:
580:
573:
572:
551:
549:
548:
543:
535:
534:
524:
517:
516:
442:Faraday constant
395:
393:
392:
387:
368:
366:
365:
360:
358:
356:
346:
338:
337:
328:
327:
299:
289:
281:
280:
271:
270:
249:
248:
239:
235:
226:
225:
216:
212:
205:
203:
201:
193:
192:
191:
182:
181:
171:
168:
163:
154:
153:
141:
140:
75:David E. Goldman
59:electrodiffusion
32:Goldman equation
21:
2613:
2612:
2608:
2607:
2606:
2604:
2603:
2602:
2578:
2577:
2568:Nernst equation
2559:
2525:
2518:
2514:
2506:
2494:
2490:
2466:
2462:
2458:
2451:
2430:
2429:
2426:
2386:
2345:
2340:
2339:
2310:
2287:
2267:
2256:
2246:
2245:
2218:
2191:
2186:
2185:
2151:
2110:
2105:
2104:
2075:
2052:
2032:
2021:
2011:
2010:
1983:
1956:
1951:
1950:
1939:
1906:
1895:
1884:
1873:
1867:
1845:
1844:
1836:
1818:
1813:
1812:
1806:Nernst equation
1758:
1748:
1726:
1703:
1688:
1687:
1640:
1630:
1611:
1580:
1570:
1548:
1525:
1502:
1501:
1489:
1478:
1462:
1450:
1431:
1426:
1425:
1395:
1390:
1389:
1356:
1338:
1328:
1327:
1294:
1271:
1253:
1252:
1225:
1220:
1219:
1192:
1187:
1186:
1159:
1136:
1110:
1100:
1087:
1069:
1059:
1058:
1036:
1025:
1024:
999:
972:
962:
940:
931:
901:
891:
869:
846:
823:
819:
807:
796:
780:
768:
755:
737:
727:
726:
686:
685:
651:
646:
645:
618:
613:
612:
582:
564:
554:
553:
526:
508:
498:
497:
487:
479:
473:
430:
421:
412:
399:
378:
377:
329:
319:
300:
272:
262:
240:
217:
206:
194:
183:
173:
172:
145:
132:
127:
126:
120:
95:
71:
39:
28:
23:
22:
15:
12:
11:
5:
2611:
2609:
2601:
2600:
2595:
2590:
2580:
2579:
2576:
2575:
2570:
2565:
2558:
2555:
2554:
2553:
2543:
2524:
2521:
2516:
2512:
2504:
2492:
2488:
2464:
2460:
2456:
2449:
2437:
2424:
2414:
2413:
2402:
2398:
2393:
2389:
2374:
2371:
2366:
2363:
2360:
2356:
2352:
2348:
2337:
2326:
2322:
2317:
2313:
2294:
2290:
2279:
2273:
2270:
2263:
2259:
2253:
2249:
2240:
2235:
2231:
2225:
2221:
2217:
2212:
2209:
2206:
2202:
2198:
2194:
2179:
2178:
2167:
2163:
2158:
2154:
2139:
2136:
2131:
2128:
2125:
2121:
2117:
2113:
2102:
2091:
2087:
2082:
2078:
2059:
2055:
2044:
2038:
2035:
2028:
2024:
2018:
2014:
2005:
2000:
1996:
1990:
1986:
1982:
1977:
1974:
1971:
1967:
1963:
1959:
1938:
1935:
1934:
1933:
1921:
1913:
1909:
1898:
1891:
1887:
1876:
1870:
1866:
1863:
1857:
1852:
1848:
1842:
1839:
1833:
1830:
1825:
1821:
1802:
1801:
1790:
1787:
1784:
1781:
1778:
1774:
1770:
1765:
1761:
1755:
1751:
1747:
1744:
1741:
1738:
1733:
1729:
1718:
1715:
1710:
1706:
1695:
1681:
1680:
1666:
1663:
1660:
1656:
1652:
1647:
1643:
1637:
1633:
1629:
1626:
1623:
1620:
1617:
1614:
1609:
1606:
1603:
1600:
1596:
1592:
1587:
1583:
1577:
1573:
1569:
1566:
1563:
1560:
1555:
1551:
1540:
1537:
1532:
1528:
1517:
1514:
1509:
1505:
1495:
1492:
1485:
1481:
1475:
1470:
1466:
1457:
1453:
1449:
1446:
1443:
1438:
1434:
1410:
1407:
1402:
1398:
1383:
1382:
1371:
1368:
1363:
1359:
1353:
1350:
1345:
1341:
1336:
1312:
1309:
1306:
1301:
1297:
1286:
1283:
1278:
1274:
1263:
1260:
1240:
1237:
1232:
1228:
1207:
1204:
1199:
1195:
1183:
1182:
1171:
1166:
1162:
1151:
1148:
1143:
1139:
1128:
1125:
1122:
1117:
1113:
1107:
1103:
1099:
1094:
1090:
1084:
1081:
1076:
1072:
1067:
1042:
1039:
1035:
1032:
1021:
1020:
1005:
1002:
998:
995:
992:
988:
984:
979:
975:
969:
965:
961:
958:
955:
952:
949:
946:
943:
937:
934:
930:
927:
924:
921:
917:
913:
908:
904:
898:
894:
890:
887:
884:
881:
876:
872:
861:
858:
853:
849:
838:
835:
830:
826:
822:
813:
810:
803:
799:
793:
788:
784:
775:
771:
767:
762:
758:
752:
749:
744:
740:
735:
698:
695:
671:
668:
663:
658:
654:
633:
630:
625:
621:
597:
594:
589:
585:
579:
576:
571:
567:
562:
541:
538:
533:
529:
523:
520:
515:
511:
506:
486:
483:
482:
481:
477:
475:
471:
469:
464:, measured in
455:
445:
435:
428:
423:
419:
414:
410:
405:
397:
385:
370:
369:
355:
352:
349:
345:
341:
336:
332:
326:
322:
318:
315:
312:
309:
306:
303:
298:
295:
292:
288:
284:
279:
275:
269:
265:
261:
258:
255:
252:
247:
243:
232:
229:
224:
220:
209:
200:
197:
190:
186:
180:
176:
167:
162:
158:
152:
148:
144:
139:
135:
119:
116:
115:
114:
111:
108:
105:
102:
94:
91:
70:
67:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2610:
2599:
2596:
2594:
2591:
2589:
2586:
2585:
2583:
2574:
2571:
2569:
2566:
2564:
2561:
2560:
2556:
2551:
2547:
2546:Flax, Matt R.
2544:
2542:
2538:
2534:
2530:
2529:Hille, Bertil
2527:
2526:
2522:
2520:
2510:
2502:
2498:
2486:
2481:
2479:
2478:rectification
2475:
2474:driving force
2470:
2455:
2448:
2423:
2418:
2400:
2396:
2391:
2387:
2372:
2369:
2364:
2358:
2350:
2338:
2324:
2320:
2315:
2311:
2292:
2271:
2268:
2261:
2257:
2251:
2247:
2238:
2233:
2229:
2223:
2219:
2215:
2210:
2204:
2196:
2184:
2183:
2182:
2165:
2161:
2156:
2152:
2137:
2134:
2129:
2123:
2115:
2103:
2089:
2085:
2080:
2076:
2057:
2036:
2033:
2026:
2022:
2016:
2012:
2003:
1998:
1994:
1988:
1984:
1980:
1975:
1969:
1961:
1949:
1948:
1947:
1945:
1937:Rectification
1936:
1919:
1911:
1889:
1868:
1864:
1861:
1855:
1850:
1846:
1840:
1837:
1831:
1828:
1823:
1819:
1811:
1810:
1809:
1807:
1788:
1785:
1779:
1776:
1772:
1768:
1763:
1759:
1753:
1749:
1745:
1739:
1736:
1731:
1713:
1708:
1686:
1685:
1684:
1661:
1658:
1654:
1650:
1645:
1641:
1635:
1631:
1627:
1621:
1618:
1615:
1612:
1601:
1598:
1594:
1590:
1585:
1581:
1575:
1571:
1567:
1561:
1558:
1553:
1535:
1530:
1507:
1503:
1493:
1490:
1483:
1479:
1473:
1468:
1464:
1455:
1451:
1447:
1444:
1441:
1436:
1424:
1423:
1422:
1408:
1405:
1400:
1386:
1369:
1366:
1361:
1357:
1351:
1343:
1326:
1325:
1324:
1310:
1307:
1299:
1281:
1276:
1238:
1235:
1230:
1205:
1202:
1197:
1193:
1164:
1146:
1141:
1120:
1115:
1111:
1105:
1101:
1097:
1092:
1082:
1074:
1070:
1057:
1056:
1055:
1040:
1033:
1003:
993:
990:
986:
982:
977:
973:
967:
963:
959:
953:
950:
947:
944:
935:
922:
919:
915:
911:
906:
902:
896:
892:
888:
882:
879:
874:
856:
851:
828:
824:
811:
808:
801:
797:
791:
786:
782:
773:
769:
765:
760:
750:
742:
738:
725:
724:
723:
721:
716:
714:
713:indeterminate
696:
693:
669:
666:
661:
656:
631:
628:
623:
619:
609:
595:
592:
587:
577:
569:
565:
539:
536:
531:
527:
521:
513:
494:
492:
484:
476:
470:
467:
463:
459:
456:
453:
449:
446:
443:
439:
436:
434:
427:
424:
418:
415:
409:
406:
403:
396:
375:
374:
373:
350:
347:
343:
339:
334:
330:
324:
320:
316:
310:
307:
304:
301:
293:
290:
286:
282:
277:
273:
267:
263:
259:
253:
250:
245:
227:
222:
198:
195:
188:
184:
178:
174:
165:
160:
156:
150:
146:
142:
137:
125:
124:
123:
117:
112:
109:
106:
103:
100:
99:
98:
92:
90:
88:
84:
80:
76:
73:The American
68:
66:
64:
60:
56:
52:
51:cell membrane
48:
44:
37:
33:
19:
2549:
2532:
2482:
2453:
2428:
2421:
2419:
2415:
2180:
1940:
1803:
1682:
1387:
1384:
1184:
1022:
717:
610:
495:
488:
457:
452:gas constant
447:
437:
425:
416:
407:
376:
371:
121:
96:
87:Bernard Katz
72:
58:
42:
40:
1388:By setting
718:We turn to
93:Assumptions
2593:Biophysics
2582:Categories
2523:References
1323:and thus
2469:Ohm's law
2436:Φ
2397:≫
2362:→
2347:Φ
2321:≪
2208:→
2193:Φ
2162:≪
2127:→
2112:Φ
2086:≫
1973:→
1958:Φ
1865:
1832:−
1746:−
1740:
1714:−
1628:−
1622:
1616:−
1568:−
1562:
1536:−
1433:Φ
1397:Φ
1367:≠
1349:→
1340:Φ
1308:≠
1282:−
1236:≠
1227:Φ
1147:−
1089:Φ
1080:→
960:−
954:
948:−
889:−
883:
857:−
757:Φ
748:→
653:Φ
593:≠
584:Φ
575:→
537:≠
519:→
510:Φ
384:Φ
317:−
311:
305:−
260:−
254:
228:−
134:Φ
49:across a
2557:See also
2531:(2001).
2452:against
1808: :
1041:′
1004:′
936:′
118:Equation
2501:calcium
466:kelvins
460:is the
450:is the
440:is the
402:amperes
2539:
2384:
2308:
2298:
2149:
2073:
2063:
1023:where
372:where
69:Origin
433:volts
2537:ISBN
2515:and
2491:and
2463:and
2181:and
489:The
85:and
47:flux
41:The
2379:for
2303:for
2144:for
2068:for
1737:exp
1619:exp
1559:exp
1335:lim
1251:if
1066:lim
951:exp
880:exp
734:lim
711:is
561:lim
505:lim
308:exp
251:exp
77:of
34:or
2584::
2503:,
2480:.
1862:ln
1218:,
715:.
644:,
2517:i
2513:o
2505:i
2493:o
2489:i
2465:o
2461:i
2457:m
2454:V
2450:S
2425:m
2422:V
2401:0
2392:m
2388:V
2373:0
2370:=
2365:i
2359:o
2355:|
2351:S
2325:0
2316:m
2312:V
2293:o
2289:]
2283:S
2278:[
2272:T
2269:R
2262:2
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2224:S
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2216:=
2211:i
2205:o
2201:|
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2166:0
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2153:V
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2135:=
2130:o
2124:i
2120:|
2116:S
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2081:m
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2058:i
2054:]
2048:S
2043:[
2037:T
2034:R
2027:2
2023:F
2017:m
2013:V
2004:2
1999:S
1995:z
1989:S
1985:P
1981:=
1976:o
1970:i
1966:|
1962:S
1920:)
1912:o
1908:]
1902:S
1897:[
1890:i
1886:]
1880:S
1875:[
1869:(
1856:F
1851:S
1847:z
1841:T
1838:R
1829:=
1824:m
1820:V
1789:0
1786:=
1783:)
1780:T
1777:R
1773:/
1769:F
1764:m
1760:V
1754:S
1750:z
1743:(
1732:o
1728:]
1722:S
1717:[
1709:i
1705:]
1699:S
1694:[
1665:)
1662:T
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1655:/
1651:F
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1625:(
1613:1
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1602:T
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1595:/
1591:F
1586:m
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1572:z
1565:(
1554:o
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1544:S
1539:[
1531:i
1527:]
1521:S
1516:[
1513:(
1508:m
1504:V
1494:T
1491:R
1484:2
1480:F
1474:2
1469:S
1465:z
1456:S
1452:P
1448:=
1445:0
1442:=
1437:S
1409:0
1406:=
1401:S
1370:0
1362:m
1358:V
1352:0
1344:S
1311:0
1305:)
1300:o
1296:]
1290:S
1285:[
1277:i
1273:]
1267:S
1262:[
1259:(
1239:0
1231:S
1206:0
1203:=
1198:m
1194:V
1170:)
1165:o
1161:]
1155:S
1150:[
1142:i
1138:]
1132:S
1127:[
1124:(
1121:F
1116:S
1112:z
1106:S
1102:P
1098:=
1093:S
1083:0
1075:m
1071:V
1038:]
1034:f
1031:[
1001:]
997:)
994:T
991:R
987:/
983:F
978:m
974:V
968:S
964:z
957:(
945:1
942:[
933:]
929:)
926:)
923:T
920:R
916:/
912:F
907:m
903:V
897:S
893:z
886:(
875:o
871:]
865:S
860:[
852:i
848:]
842:S
837:[
834:(
829:m
825:V
821:[
812:T
809:R
802:2
798:F
792:2
787:S
783:z
774:S
770:P
766:=
761:S
751:0
743:m
739:V
697:0
694:0
670:0
667:0
662:=
657:S
632:0
629:=
624:m
620:V
596:0
588:S
578:0
570:m
566:V
540:0
532:m
528:V
522:0
514:S
478:o
472:i
458:T
448:R
438:F
429:m
426:V
420:S
417:z
411:S
408:P
398:S
354:)
351:T
348:R
344:/
340:F
335:m
331:V
325:S
321:z
314:(
302:1
297:)
294:T
291:R
287:/
283:F
278:m
274:V
268:S
264:z
257:(
246:o
242:]
236:S
231:[
223:i
219:]
213:S
208:[
199:T
196:R
189:2
185:F
179:m
175:V
166:2
161:S
157:z
151:S
147:P
143:=
138:S
38:.
20:)
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