2405:
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
1007:
1667:
356:
2324:
2089:
1930:
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
1920:
1169:
2460:
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
2400:
2165:
1416:
717:
1369:
595:
539:
117:
46:
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
1788:
671:
1310:
1238:
2176:
1941:
698:
1408:
1205:
631:
1041:
2435:
383:
24:
1803:
1049:
2529:
2456:. Thus, if the two concentrations are identical, the slope will be identical (and constant) throughout the voltage range (corresponding to
2576:
1662:{\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)}}}
2465:
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
2330:
2095:
1002:{\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))]'}{'}}}
351:{\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)}}}
1318:
544:
488:
482:
is shown to be contained in the GHK flux equation (Flax 2008). The proof is replicated from the reference (Flax 2008) here.
2586:
1678:
51:
2448:. It is evident that the ratio between the two asymptotes is merely the ratio between the two concentrations of S,
2406:
voltage could cause the ion to flow out of the cell, since there are no ions inside the cell in the first place.
708:
2497:
636:
485:
We wish to show that when the flux is zero, the transmembrane potential is not zero. Formally it is written
54:
and the GHK flux equation is a solution to the Nernst–Planck equation with the assumptions listed below.
93:
The electrical field is constant so that the transmembrane potential varies linearly across the membrane
2319:{\displaystyle \Phi _{S|o\to i}=P_{S}z_{S}^{2}{\frac {V_{m}F^{2}}{RT}}_{o}\ {\mbox{for}}\ V_{m}\ll \;0}
2084:{\displaystyle \Phi _{S|i\to o}=P_{S}z_{S}^{2}{\frac {V_{m}F^{2}}{RT}}_{i}\ {\mbox{for}}\ V_{m}\gg \;0}
1243:
1210:
450:
676:
2581:
67:
2561:
1380:
701:
479:
71:
43:
86:
Several assumptions are made in deriving the GHK flux equation (Hille 2001, p. 445) :
2534:
2539:
Goldman-Hodgkin-Katz
Cochlear Hair Cell Models – a Foundation for Nonlinear Cochlear Mechanics
2525:
2473:
1177:
603:
2551:
2485:
430:
63:
20:
2420:
368:
2556:
1794:
389:
is the current density (flux) outward through the membrane carried by ion S, measured in
96:
The ions access the membrane instantaneously from the intra- and extracellular solutions
1015:
2570:
2484:
is large and/or when one or both of the concentrations change considerably during an
39:
597:, which states that when the transmembrane potential is zero, the flux is not zero.
2517:
440:
75:
2457:
1915:{\displaystyle V_{m}=-{\frac {RT}{z_{S}F}}\ln \left({\frac {_{i}}{_{o}}}\right)}
102:
The movement of ions is affected by both concentration and voltage differences
2466:
1932:
34:(or GHK flux equation or GHK current density equation) describes the ionic
1164:{\displaystyle \lim _{V_{m}\rightarrow 0}\Phi _{S}=P_{S}z_{S}F(_{i}-_{o})}
463:
is the intracellular concentration of ion S, measured in mol·m or mmol·l
2489:
1935:
as the membrane potential diverges from zero. These asymptotes are
454:
390:
2462:
421:
35:
2395:{\displaystyle \Phi _{S|o\to i}=0\;{\mbox{for}}\ V_{m}\gg \;0}
2160:{\displaystyle \Phi _{S|i\to o}=0\;{\mbox{for}}\ V_{m}\ll \;0}
469:
is the extracellular concentration of ion S, measured in mol·m
111:
The GHK flux equation for an ion S (Hille 2001, p. 445):
402:
is the permeability of the membrane for ion S measured in m·s
2416:
constant, the equation yields a straight line when plotting
2524:, 3rd ed., Sinauer Associates, Sunderland, Massachusetts.
1043:
represents the differential of f and the result is :
2500:
cycle can change 100-fold or more, and the ratio between
1364:{\displaystyle \lim _{\Phi _{S}\rightarrow 0}V_{m}\neq 0}
590:{\displaystyle \lim _{V_{m}\rightarrow 0}\Phi _{S}\neq 0}
534:{\displaystyle \lim _{\Phi _{S}\rightarrow 0}V_{m}\neq 0}
600:
However, due to the form of the GHK flux equation when
2366:
2290:
2270:
2131:
2055:
2035:
1889:
1867:
1709:
1686:
1531:
1508:
1277:
1254:
1142:
1119:
852:
829:
223:
200:
2488:. The most common example is probably intracellular
2423:
2333:
2179:
2098:
1944:
1806:
1681:
1419:
1383:
1321:
1246:
1213:
1180:
1052:
1018:
720:
679:
639:
606:
547:
491:
371:
120:
50:. Electrodiffusion is most accurately defined by the
1374:which is the definition of the reversal potential.
1174:It is evident from the previous equation that when
16:
Expression of the ionic flux across a cell membrane
2429:
2394:
2318:
2159:
2083:
1914:
1782:
1661:
1402:
1363:
1304:
1232:
1199:
1163:
1035:
1001:
692:
665:
625:
589:
533:
377:
350:
1410:we can also obtain the reversal potential :
1783:{\displaystyle _{i}-_{o}\exp(-z_{S}V_{m}F/RT)=0}
1323:
1054:
722:
549:
493:
8:
2541:, Conference proceedings: Interspeech 2008.
2388:
2364:
2312:
2153:
2129:
2077:
2422:
2379:
2365:
2342:
2338:
2332:
2303:
2289:
2280:
2269:
2249:
2239:
2232:
2226:
2221:
2211:
2188:
2184:
2178:
2144:
2130:
2107:
2103:
2097:
2068:
2054:
2045:
2034:
2014:
2004:
1997:
1991:
1986:
1976:
1953:
1949:
1943:
1899:
1888:
1877:
1866:
1860:
1838:
1823:
1811:
1805:
1760:
1751:
1741:
1719:
1708:
1696:
1685:
1680:
1642:
1633:
1623:
1582:
1573:
1563:
1541:
1530:
1518:
1507:
1495:
1488:
1471:
1461:
1456:
1449:
1443:
1424:
1418:
1388:
1382:
1349:
1331:
1326:
1320:
1287:
1276:
1264:
1253:
1245:
1218:
1212:
1185:
1179:
1152:
1141:
1129:
1118:
1103:
1093:
1080:
1062:
1057:
1051:
1017:
974:
965:
955:
903:
894:
884:
862:
851:
839:
828:
816:
806:
789:
779:
774:
767:
761:
748:
730:
725:
719:
680:
678:
653:
644:
638:
611:
605:
575:
557:
552:
546:
519:
501:
496:
490:
474:Implicit definition of reversal potential
370:
331:
322:
312:
274:
265:
255:
233:
222:
210:
199:
193:
176:
166:
159:
153:
148:
138:
125:
119:
2472:The GHK flux equation is mostly used by
666:{\displaystyle \Phi _{S}={\frac {0}{0}}}
90:The membrane is a homogeneous substance
25:Goldman–Hodgkin–Katz voltage equation
7:
711:to find the solution for the limit:
673:. This is a problem as the value of
2522:Ion channels of excitable membranes
2424:
2335:
2181:
2100:
1946:
1421:
1385:
1328:
1215:
1077:
745:
641:
572:
498:
433:, equal to 96,485 C·mol or J·V·mol
420:is the transmembrane potential in
372:
122:
70:, and the English Nobel laureates
32:Goldman–Hodgkin–Katz flux equation
14:
1305:{\displaystyle (_{i}-_{o})\neq 0}
99:The permeant ions do not interact
1233:{\displaystyle \Phi _{S}\neq 0}
541:which is equivalent to writing
2350:
2343:
2277:
2266:
2196:
2189:
2115:
2108:
2042:
2031:
1961:
1954:
1896:
1885:
1874:
1863:
1771:
1731:
1716:
1705:
1693:
1682:
1653:
1613:
1596:
1593:
1553:
1538:
1527:
1515:
1504:
1501:
1337:
1293:
1284:
1273:
1261:
1250:
1247:
1158:
1149:
1138:
1126:
1115:
1112:
1068:
1026:
1019:
989:
985:
945:
930:
921:
917:
914:
874:
859:
848:
836:
825:
822:
809:
736:
693:{\displaystyle {\frac {0}{0}}}
563:
507:
342:
302:
285:
245:
230:
219:
207:
196:
1:
2537:and Holmes, W.Harvey (2008).
457:(= degrees Celsius + 273.15)
1403:{\displaystyle \Phi _{S}=0}
2603:
2508:can reach 20,000 or more.
18:
2577:Electrochemical equations
2409:Keeping all terms except
2498:cardiac action potential
1672:which reduces to :
443:, equal to 8.314 J·K·mol
19:Not to be confused with
2476:when the ratio between
1200:{\displaystyle V_{m}=0}
626:{\displaystyle V_{m}=0}
411:is the valence of ion S
78:derived this equation.
44:transmembrane potential
2431:
2396:
2320:
2161:
2085:
1916:
1784:
1663:
1404:
1365:
1306:
1234:
1201:
1165:
1037:
1003:
694:
667:
627:
591:
535:
393:per square meter (A·m)
379:
352:
52:Nernst–Planck equation
2432:
2430:{\displaystyle \Phi }
2397:
2321:
2162:
2086:
1917:
1785:
1664:
1405:
1366:
1307:
1235:
1202:
1166:
1038:
1004:
695:
668:
628:
592:
536:
380:
378:{\displaystyle \Phi }
353:
42:as a function of the
2474:electrophysiologists
2421:
2331:
2177:
2096:
1942:
1804:
1679:
1417:
1381:
1319:
1244:
1211:
1178:
1050:
1016:
718:
677:
637:
604:
545:
489:
451:absolute temperature
369:
118:
2587:Bioelectrochemistry
2231:
1996:
1466:
784:
158:
68:Columbia University
2562:Reversal potential
2427:
2392:
2370:
2316:
2294:
2274:
2217:
2157:
2135:
2081:
2059:
2039:
1982:
1912:
1893:
1871:
1780:
1713:
1690:
1659:
1535:
1512:
1452:
1400:
1361:
1344:
1302:
1281:
1258:
1230:
1197:
1161:
1146:
1123:
1075:
1033:
999:
856:
833:
770:
743:
690:
663:
623:
587:
570:
531:
514:
480:reversal potential
375:
348:
227:
204:
144:
72:Alan Lloyd Hodgkin
2530:978-0-87893-321-1
2496:, which during a
2374:
2369:
2298:
2293:
2288:
2273:
2264:
2139:
2134:
2063:
2058:
2053:
2038:
2029:
1906:
1892:
1870:
1848:
1793:and produces the
1712:
1689:
1657:
1534:
1511:
1486:
1322:
1280:
1257:
1145:
1122:
1053:
1036:{\displaystyle '}
997:
855:
832:
804:
721:
688:
661:
548:
492:
346:
226:
203:
191:
2594:
2552:Goldman equation
2486:action potential
2436:
2434:
2433:
2428:
2401:
2399:
2398:
2393:
2384:
2383:
2372:
2371:
2367:
2357:
2356:
2346:
2325:
2323:
2322:
2317:
2308:
2307:
2296:
2295:
2291:
2286:
2285:
2284:
2275:
2271:
2265:
2263:
2255:
2254:
2253:
2244:
2243:
2233:
2230:
2225:
2216:
2215:
2203:
2202:
2192:
2166:
2164:
2163:
2158:
2149:
2148:
2137:
2136:
2132:
2122:
2121:
2111:
2090:
2088:
2087:
2082:
2073:
2072:
2061:
2060:
2056:
2051:
2050:
2049:
2040:
2036:
2030:
2028:
2020:
2019:
2018:
2009:
2008:
1998:
1995:
1990:
1981:
1980:
1968:
1967:
1957:
1933:asymptotic value
1921:
1919:
1918:
1913:
1911:
1907:
1905:
1904:
1903:
1894:
1890:
1883:
1882:
1881:
1872:
1868:
1861:
1849:
1847:
1843:
1842:
1832:
1824:
1816:
1815:
1789:
1787:
1786:
1781:
1764:
1756:
1755:
1746:
1745:
1724:
1723:
1714:
1710:
1701:
1700:
1691:
1687:
1668:
1666:
1665:
1660:
1658:
1656:
1646:
1638:
1637:
1628:
1627:
1599:
1586:
1578:
1577:
1568:
1567:
1546:
1545:
1536:
1532:
1523:
1522:
1513:
1509:
1500:
1499:
1489:
1487:
1485:
1477:
1476:
1475:
1465:
1460:
1450:
1448:
1447:
1429:
1428:
1409:
1407:
1406:
1401:
1393:
1392:
1370:
1368:
1367:
1362:
1354:
1353:
1343:
1336:
1335:
1311:
1309:
1308:
1303:
1292:
1291:
1282:
1278:
1269:
1268:
1259:
1255:
1239:
1237:
1236:
1231:
1223:
1222:
1206:
1204:
1203:
1198:
1190:
1189:
1170:
1168:
1167:
1162:
1157:
1156:
1147:
1143:
1134:
1133:
1124:
1120:
1108:
1107:
1098:
1097:
1085:
1084:
1074:
1067:
1066:
1042:
1040:
1039:
1034:
1032:
1008:
1006:
1005:
1000:
998:
996:
995:
978:
970:
969:
960:
959:
928:
927:
907:
899:
898:
889:
888:
867:
866:
857:
853:
844:
843:
834:
830:
821:
820:
807:
805:
803:
795:
794:
793:
783:
778:
768:
766:
765:
753:
752:
742:
735:
734:
709:l'Hôpital's rule
699:
697:
696:
691:
689:
681:
672:
670:
669:
664:
662:
654:
649:
648:
632:
630:
629:
624:
616:
615:
596:
594:
593:
588:
580:
579:
569:
562:
561:
540:
538:
537:
532:
524:
523:
513:
506:
505:
431:Faraday constant
384:
382:
381:
376:
357:
355:
354:
349:
347:
345:
335:
327:
326:
317:
316:
288:
278:
270:
269:
260:
259:
238:
237:
228:
224:
215:
214:
205:
201:
194:
192:
190:
182:
181:
180:
171:
170:
160:
157:
152:
143:
142:
130:
129:
64:David E. Goldman
48:electrodiffusion
21:Goldman equation
2602:
2601:
2597:
2596:
2595:
2593:
2592:
2591:
2567:
2566:
2557:Nernst equation
2548:
2514:
2507:
2503:
2495:
2483:
2479:
2455:
2451:
2447:
2440:
2419:
2418:
2415:
2375:
2334:
2329:
2328:
2299:
2276:
2256:
2245:
2235:
2234:
2207:
2180:
2175:
2174:
2140:
2099:
2094:
2093:
2064:
2041:
2021:
2010:
2000:
1999:
1972:
1945:
1940:
1939:
1928:
1895:
1884:
1873:
1862:
1856:
1834:
1833:
1825:
1807:
1802:
1801:
1795:Nernst equation
1747:
1737:
1715:
1692:
1677:
1676:
1629:
1619:
1600:
1569:
1559:
1537:
1514:
1491:
1490:
1478:
1467:
1451:
1439:
1420:
1415:
1414:
1384:
1379:
1378:
1345:
1327:
1317:
1316:
1283:
1260:
1242:
1241:
1214:
1209:
1208:
1181:
1176:
1175:
1148:
1125:
1099:
1089:
1076:
1058:
1048:
1047:
1025:
1014:
1013:
988:
961:
951:
929:
920:
890:
880:
858:
835:
812:
808:
796:
785:
769:
757:
744:
726:
716:
715:
675:
674:
640:
635:
634:
607:
602:
601:
571:
553:
543:
542:
515:
497:
487:
486:
476:
468:
462:
419:
410:
401:
388:
367:
366:
318:
308:
289:
261:
251:
229:
206:
195:
183:
172:
162:
161:
134:
121:
116:
115:
109:
84:
60:
28:
17:
12:
11:
5:
2600:
2598:
2590:
2589:
2584:
2579:
2569:
2568:
2565:
2564:
2559:
2554:
2547:
2544:
2543:
2542:
2532:
2513:
2510:
2505:
2501:
2493:
2481:
2477:
2453:
2449:
2445:
2438:
2426:
2413:
2403:
2402:
2391:
2387:
2382:
2378:
2363:
2360:
2355:
2352:
2349:
2345:
2341:
2337:
2326:
2315:
2311:
2306:
2302:
2283:
2279:
2268:
2262:
2259:
2252:
2248:
2242:
2238:
2229:
2224:
2220:
2214:
2210:
2206:
2201:
2198:
2195:
2191:
2187:
2183:
2168:
2167:
2156:
2152:
2147:
2143:
2128:
2125:
2120:
2117:
2114:
2110:
2106:
2102:
2091:
2080:
2076:
2071:
2067:
2048:
2044:
2033:
2027:
2024:
2017:
2013:
2007:
2003:
1994:
1989:
1985:
1979:
1975:
1971:
1966:
1963:
1960:
1956:
1952:
1948:
1927:
1924:
1923:
1922:
1910:
1902:
1898:
1887:
1880:
1876:
1865:
1859:
1855:
1852:
1846:
1841:
1837:
1831:
1828:
1822:
1819:
1814:
1810:
1791:
1790:
1779:
1776:
1773:
1770:
1767:
1763:
1759:
1754:
1750:
1744:
1740:
1736:
1733:
1730:
1727:
1722:
1718:
1707:
1704:
1699:
1695:
1684:
1670:
1669:
1655:
1652:
1649:
1645:
1641:
1636:
1632:
1626:
1622:
1618:
1615:
1612:
1609:
1606:
1603:
1598:
1595:
1592:
1589:
1585:
1581:
1576:
1572:
1566:
1562:
1558:
1555:
1552:
1549:
1544:
1540:
1529:
1526:
1521:
1517:
1506:
1503:
1498:
1494:
1484:
1481:
1474:
1470:
1464:
1459:
1455:
1446:
1442:
1438:
1435:
1432:
1427:
1423:
1399:
1396:
1391:
1387:
1372:
1371:
1360:
1357:
1352:
1348:
1342:
1339:
1334:
1330:
1325:
1301:
1298:
1295:
1290:
1286:
1275:
1272:
1267:
1263:
1252:
1249:
1229:
1226:
1221:
1217:
1196:
1193:
1188:
1184:
1172:
1171:
1160:
1155:
1151:
1140:
1137:
1132:
1128:
1117:
1114:
1111:
1106:
1102:
1096:
1092:
1088:
1083:
1079:
1073:
1070:
1065:
1061:
1056:
1031:
1028:
1024:
1021:
1010:
1009:
994:
991:
987:
984:
981:
977:
973:
968:
964:
958:
954:
950:
947:
944:
941:
938:
935:
932:
926:
923:
919:
916:
913:
910:
906:
902:
897:
893:
887:
883:
879:
876:
873:
870:
865:
861:
850:
847:
842:
838:
827:
824:
819:
815:
811:
802:
799:
792:
788:
782:
777:
773:
764:
760:
756:
751:
747:
741:
738:
733:
729:
724:
687:
684:
660:
657:
652:
647:
643:
622:
619:
614:
610:
586:
583:
578:
574:
568:
565:
560:
556:
551:
530:
527:
522:
518:
512:
509:
504:
500:
495:
475:
472:
471:
470:
466:
464:
460:
458:
453:, measured in
444:
434:
424:
417:
412:
408:
403:
399:
394:
386:
374:
359:
358:
344:
341:
338:
334:
330:
325:
321:
315:
311:
307:
304:
301:
298:
295:
292:
287:
284:
281:
277:
273:
268:
264:
258:
254:
250:
247:
244:
241:
236:
232:
221:
218:
213:
209:
198:
189:
186:
179:
175:
169:
165:
156:
151:
147:
141:
137:
133:
128:
124:
108:
105:
104:
103:
100:
97:
94:
91:
83:
80:
59:
56:
15:
13:
10:
9:
6:
4:
3:
2:
2599:
2588:
2585:
2583:
2580:
2578:
2575:
2574:
2572:
2563:
2560:
2558:
2555:
2553:
2550:
2549:
2545:
2540:
2536:
2535:Flax, Matt R.
2533:
2531:
2527:
2523:
2519:
2518:Hille, Bertil
2516:
2515:
2511:
2509:
2499:
2491:
2487:
2475:
2470:
2468:
2467:rectification
2464:
2463:driving force
2459:
2444:
2437:
2412:
2407:
2389:
2385:
2380:
2376:
2361:
2358:
2353:
2347:
2339:
2327:
2313:
2309:
2304:
2300:
2281:
2260:
2257:
2250:
2246:
2240:
2236:
2227:
2222:
2218:
2212:
2208:
2204:
2199:
2193:
2185:
2173:
2172:
2171:
2154:
2150:
2145:
2141:
2126:
2123:
2118:
2112:
2104:
2092:
2078:
2074:
2069:
2065:
2046:
2025:
2022:
2015:
2011:
2005:
2001:
1992:
1987:
1983:
1977:
1973:
1969:
1964:
1958:
1950:
1938:
1937:
1936:
1934:
1926:Rectification
1925:
1908:
1900:
1878:
1857:
1853:
1850:
1844:
1839:
1835:
1829:
1826:
1820:
1817:
1812:
1808:
1800:
1799:
1798:
1796:
1777:
1774:
1768:
1765:
1761:
1757:
1752:
1748:
1742:
1738:
1734:
1728:
1725:
1720:
1702:
1697:
1675:
1674:
1673:
1650:
1647:
1643:
1639:
1634:
1630:
1624:
1620:
1616:
1610:
1607:
1604:
1601:
1590:
1587:
1583:
1579:
1574:
1570:
1564:
1560:
1556:
1550:
1547:
1542:
1524:
1519:
1496:
1492:
1482:
1479:
1472:
1468:
1462:
1457:
1453:
1444:
1440:
1436:
1433:
1430:
1425:
1413:
1412:
1411:
1397:
1394:
1389:
1375:
1358:
1355:
1350:
1346:
1340:
1332:
1315:
1314:
1313:
1299:
1296:
1288:
1270:
1265:
1227:
1224:
1219:
1194:
1191:
1186:
1182:
1153:
1135:
1130:
1109:
1104:
1100:
1094:
1090:
1086:
1081:
1071:
1063:
1059:
1046:
1045:
1044:
1029:
1022:
992:
982:
979:
975:
971:
966:
962:
956:
952:
948:
942:
939:
936:
933:
924:
911:
908:
904:
900:
895:
891:
885:
881:
877:
871:
868:
863:
845:
840:
817:
813:
800:
797:
790:
786:
780:
775:
771:
762:
758:
754:
749:
739:
731:
727:
714:
713:
712:
710:
705:
703:
702:indeterminate
685:
682:
658:
655:
650:
645:
620:
617:
612:
608:
598:
584:
581:
576:
566:
558:
554:
528:
525:
520:
516:
510:
502:
483:
481:
473:
465:
459:
456:
452:
448:
445:
442:
438:
435:
432:
428:
425:
423:
416:
413:
407:
404:
398:
395:
392:
385:
364:
363:
362:
339:
336:
332:
328:
323:
319:
313:
309:
305:
299:
296:
293:
290:
282:
279:
275:
271:
266:
262:
256:
252:
248:
242:
239:
234:
216:
211:
187:
184:
177:
173:
167:
163:
154:
149:
145:
139:
135:
131:
126:
114:
113:
112:
106:
101:
98:
95:
92:
89:
88:
87:
81:
79:
77:
73:
69:
65:
62:The American
57:
55:
53:
49:
45:
41:
40:cell membrane
37:
33:
26:
22:
2538:
2521:
2471:
2442:
2417:
2410:
2408:
2404:
2169:
1929:
1792:
1671:
1376:
1373:
1173:
1011:
706:
599:
484:
477:
446:
441:gas constant
436:
426:
414:
405:
396:
365:
360:
110:
85:
76:Bernard Katz
61:
47:
31:
29:
1377:By setting
707:We turn to
82:Assumptions
2582:Biophysics
2571:Categories
2512:References
1312:and thus
2458:Ohm's law
2425:Φ
2386:≫
2351:→
2336:Φ
2310:≪
2197:→
2182:Φ
2151:≪
2116:→
2101:Φ
2075:≫
1962:→
1947:Φ
1854:
1821:−
1735:−
1729:
1703:−
1617:−
1611:
1605:−
1557:−
1551:
1525:−
1422:Φ
1386:Φ
1356:≠
1338:→
1329:Φ
1297:≠
1271:−
1225:≠
1216:Φ
1136:−
1078:Φ
1069:→
949:−
943:
937:−
878:−
872:
846:−
746:Φ
737:→
642:Φ
582:≠
573:Φ
564:→
526:≠
508:→
499:Φ
373:Φ
306:−
300:
294:−
249:−
243:
217:−
123:Φ
38:across a
2546:See also
2520:(2001).
2441:against
1797: :
1030:′
993:′
925:′
107:Equation
2490:calcium
455:kelvins
449:is the
439:is the
429:is the
391:amperes
2528:
2373:
2297:
2287:
2138:
2062:
2052:
1012:where
361:where
58:Origin
422:volts
2526:ISBN
2504:and
2480:and
2452:and
2170:and
478:The
74:and
36:flux
30:The
2368:for
2292:for
2133:for
2057:for
1726:exp
1608:exp
1548:exp
1324:lim
1240:if
1055:lim
940:exp
869:exp
723:lim
700:is
550:lim
494:lim
297:exp
240:exp
66:of
23:or
2573::
2492:,
2469:.
1851:ln
1207:,
704:.
633:,
2506:i
2502:o
2494:i
2482:o
2478:i
2454:o
2450:i
2446:m
2443:V
2439:S
2414:m
2411:V
2390:0
2381:m
2377:V
2362:0
2359:=
2354:i
2348:o
2344:|
2340:S
2314:0
2305:m
2301:V
2282:o
2278:]
2272:S
2267:[
2261:T
2258:R
2251:2
2247:F
2241:m
2237:V
2228:2
2223:S
2219:z
2213:S
2209:P
2205:=
2200:i
2194:o
2190:|
2186:S
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2146:m
2142:V
2127:0
2124:=
2119:o
2113:i
2109:|
2105:S
2079:0
2070:m
2066:V
2047:i
2043:]
2037:S
2032:[
2026:T
2023:R
2016:2
2012:F
2006:m
2002:V
1993:2
1988:S
1984:z
1978:S
1974:P
1970:=
1965:o
1959:i
1955:|
1951:S
1909:)
1901:o
1897:]
1891:S
1886:[
1879:i
1875:]
1869:S
1864:[
1858:(
1845:F
1840:S
1836:z
1830:T
1827:R
1818:=
1813:m
1809:V
1778:0
1775:=
1772:)
1769:T
1766:R
1762:/
1758:F
1753:m
1749:V
1743:S
1739:z
1732:(
1721:o
1717:]
1711:S
1706:[
1698:i
1694:]
1688:S
1683:[
1654:)
1651:T
1648:R
1644:/
1640:F
1635:m
1631:V
1625:S
1621:z
1614:(
1602:1
1597:)
1594:)
1591:T
1588:R
1584:/
1580:F
1575:m
1571:V
1565:S
1561:z
1554:(
1543:o
1539:]
1533:S
1528:[
1520:i
1516:]
1510:S
1505:[
1502:(
1497:m
1493:V
1483:T
1480:R
1473:2
1469:F
1463:2
1458:S
1454:z
1445:S
1441:P
1437:=
1434:0
1431:=
1426:S
1398:0
1395:=
1390:S
1359:0
1351:m
1347:V
1341:0
1333:S
1300:0
1294:)
1289:o
1285:]
1279:S
1274:[
1266:i
1262:]
1256:S
1251:[
1248:(
1228:0
1220:S
1195:0
1192:=
1187:m
1183:V
1159:)
1154:o
1150:]
1144:S
1139:[
1131:i
1127:]
1121:S
1116:[
1113:(
1110:F
1105:S
1101:z
1095:S
1091:P
1087:=
1082:S
1072:0
1064:m
1060:V
1027:]
1023:f
1020:[
990:]
986:)
983:T
980:R
976:/
972:F
967:m
963:V
957:S
953:z
946:(
934:1
931:[
922:]
918:)
915:)
912:T
909:R
905:/
901:F
896:m
892:V
886:S
882:z
875:(
864:o
860:]
854:S
849:[
841:i
837:]
831:S
826:[
823:(
818:m
814:V
810:[
801:T
798:R
791:2
787:F
781:2
776:S
772:z
763:S
759:P
755:=
750:S
740:0
732:m
728:V
686:0
683:0
659:0
656:0
651:=
646:S
621:0
618:=
613:m
609:V
585:0
577:S
567:0
559:m
555:V
529:0
521:m
517:V
511:0
503:S
467:o
461:i
447:T
437:R
427:F
418:m
415:V
409:S
406:z
400:S
397:P
387:S
343:)
340:T
337:R
333:/
329:F
324:m
320:V
314:S
310:z
303:(
291:1
286:)
283:T
280:R
276:/
272:F
267:m
263:V
257:S
253:z
246:(
235:o
231:]
225:S
220:[
212:i
208:]
202:S
197:[
188:T
185:R
178:2
174:F
168:m
164:V
155:2
150:S
146:z
140:S
136:P
132:=
127:S
27:.
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