1927:
One of the main problems of chemical reaction network theory is the connection between network structure and properties of dynamics. This connection is important even for linear systems, for example, the simple cycle with equal interaction weights has the slowest decay of the oscillations among all
2576:
The quasi steady state approximation or QSS (some of the species, very often these are some of intermediates or radicals, exist in relatively small amounts; they reach quickly their QSS concentrations, and then follow, as dependent quantities, the dynamics of these other species remaining close to
1804:
Results regarding stable periodic solutions attempt to rule out "unusual" behaviour. If a given chemical reaction network admits a stable periodic solution, then some initial conditions will converge to an infinite cycle of oscillating reactant concentrations. For some parameter values it may even
2568:
Modelling of large reaction networks meets various difficulties: the models include too many unknown parameters and high dimension makes the modelling computationally expensive. The model reduction methods were developed together with the first theories of complex chemical reactions. Three simple
930:
147:. The paper of R. Aris in this journal was communicated to the journal by C. Truesdell. It opened the series of papers of other authors (which were communicated already by R. Aris). The well known papers of this series are the works of Frederick J. Krambeck, Roy Jackson,
329:
1145:
For physical reasons, it is usually assumed that reactant concentrations cannot be negative, and that each reaction only takes place if all its reactants are present, i.e. all have non-zero concentration. For mathematical reasons, it is usually assumed that
775:
1795:
in population dynamics can go extinct for some (or all) initial conditions. Similar questions are of interests to chemists and biochemists, i.e. if a given reactant was present to start with, can it ever be completely used up?
3042:
786:
2051:
1237:
These results relate to whether a chemical reaction network can produce significantly different behaviour depending on the initial concentrations of its constituent reactants. This has applications in e.g. modelling
3016:
M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems. Chemical
Engineering Science. 1987 31, 42(10), 2229-2268.
2486:
155:
and others, published in the 1970s. In his second "prolegomena" paper, R. Aris mentioned the work of N.Z. Shapiro, L.S. Shapley (1965), where an important part of his scientific program was realized.
2349:
2155:
1931:
For nonlinear systems, many connections between structure and dynamics have been discovered. First of all, these are results about stability. For some classes of networks, explicit construction of
1763:
are the intermediates on the surface (adatoms, adsorbed molecules or radicals). This system may have two stable steady states of the surface for the same concentrations of the gaseous components.
1190:), and that increasing the concentration of a reactant increases the rate of any reactions that use it up. This second assumption is compatible with all physically reasonable kinetics, including
203:
198:
2573:
The quasi-equilibrium (or pseudo-equilibrium, or partial equilibrium) approximation (a fraction of reactions approach their equilibrium fast enough and, after that, remain almost equilibrated).
1242:
switches—a high concentration of a key chemical at steady state could represent a biological process being "switched on" whereas a low concentration would represent being "switched off".
2550:
2584:
or bottleneck is a relatively small part of the reaction network, in the simplest cases it is a single reaction, which rate is a good approximation to the reaction rate of the whole network.
513:
Mathematical modelling of chemical reaction networks usually focuses on what happens to the concentrations of the various chemicals involved as time passes. Following the example above, let
1901:
1504:
1347:
1438:
989:. The number of molecules of each reactant used up each time a reaction occurs is constant, as is the number of molecules produced of each product. These numbers are referred to as the
3026:
1775:
models) tend to be subject to random background noise, an unstable steady state solution is unlikely to be observed in practice. Instead of them, stable oscillations or other types of
2671:
F.J. Krambeck, The mathematical structure of chemical kinetics in homogeneous single-phase systems, Archive for
Rational Mechanics and Analysis, 1970, Volume 38, Issue 5, pp 317-347,
1045:
2716:
R. Aris, Prolegomena to the rational analysis of systems of chemical reactions II. Some addenda, Archive for
Rational Mechanics and Analysis, 1968, Volume 27, Issue 5, pp 356-364
987:
1565:
2279:
2249:
1673:
626:
496:
460:
2382:
1761:
1739:
1717:
1641:
1619:
1587:
587:
551:
413:
377:
1695:
670:
2181:
1229:
As chemical reaction network theory is a diverse and well-established area of research, there is a significant variety of results. Some key areas are outlined below.
1094:
1074:
2212:
2078:
1173:
1127:
2815:
2899:
2662:
R. Aris, Prolegomena to the rational analysis of systems of chemical reactions, Archive for
Rational Mechanics and Analysis, 1965, Volume 19, Issue 2, pp 81-99.
1817:. The simplest catalytic oscillator (nonlinear self-oscillations without autocatalysis) can be produced from the catalytic trigger by adding a "buffer" step.
681:
2699:
143:
925:{\displaystyle {\dot {x}}\equiv {\frac {dx}{dt}}=\left({\begin{array}{c}{\frac {da}{dt}}\\{\frac {db}{dt}}\\{\frac {dc}{dt}}\\\vdots \end{array}}\right).}
2787:
2084:-th component. The theorem about systems without interactions between different components states that if a network consists of reactions of the form
1953:
96:
Three eras of chemical dynamics can be revealed in the flux of research and publications. These eras may be associated with leaders: the first is the
119:
was searching for the general law of chemical reaction related to specific chemical properties. The term "chemical dynamics" belongs to van’t Hoff.
3043:
Model reduction in chemical dynamics: slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph.
347:
form a reaction network. The reactions are represented by the arrows. The reactants appear to the left of the arrows, in this example they are
122:
The
Semenov-Hinshelwood focus was an explanation of critical phenomena observed in many chemical systems, in particular in flames. A concept
1813:
behaviour. While stable periodic solutions are unusual in real-world chemical reaction networks, well-known examples exist, such as the
1935:
is possible without apriori assumptions about special relations between rate constants. Two results of this type are well known: the
1814:
1523:
628:, and so on. Since all of these concentrations will not in general remain constant, they can be written as a function of time e.g.
2977:
V.I. Bykov, G.S. Yablonskii, V.F. Kim, "On the simple model of kinetic self-oscillations in catalytic reaction of CO oxidation",
3073:
2852:
2393:
939:
116:
97:
158:
Since then, the chemical reaction network theory has been further developed by a large number of researchers internationally.
2284:
2087:
324:{\displaystyle {\begin{aligned}{\ce {{2H2}+ O2}}&{\ce {-> 2H2O}}\\{\ce {{C}+ O2}}&{\ce {-> CO2}}\end{aligned}}}
45:
systems. Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in
2577:
the QSS). The QSS is defined as the steady state under the condition that the concentrations of other species do not change.
1195:
2916:
I. Otero-Muras, J. R. Banga and A. A. Alonso, "Characterizing multistationarity regimes in biochemical reaction networks",
1771:
Stability determines whether a given steady state solution is likely to be observed in reality. Since real systems (unlike
1202:
kinetics. Sometimes further assumptions are made about reaction rates, e.g. that all reactions obey mass action kinetics.
2869:
1199:
2638:
2588:
The quasi-equilibrium approximation and the quasi steady state methods were developed further into the methods of slow
2736:
2725:
N.Z. Shapiro, L.S. Shapley, Mass action law and the Gibbs free energy function, SIAM J. Appl. Math. 16 (1965) 353–375.
2491:
993:
of the reaction, and the difference between the two (i.e. the overall number of molecules used up or produced) is the
2766:
M. Mincheva and D. Siegel, "Nonnegativity and positiveness of solutions to mass action reaction–diffusion systems",
1946:
The deficiency zero theorem gives sufficient conditions for the existence of the
Lyapunov function in the classical
2897:
E. Feliu, M. Knudsen and C. Wiuf., "Signaling cascades: Consequences of varying substrate and phosphatase levels",
2626:
1825:
1461:
1261:
1182:
It is also commonly assumed that no reaction features the same chemical as both a reactant and a product (i.e. no
179:
2749:
H. Kunze and D. Siegel, "Monotonicity properties of chemical reactions with a single initial bimolecular step",
148:
2614:Ăśber simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme
1370:
105:
90:
2784:
1218:
1003:
3007:
B.L. Clarke, Theorems on chemical network stability. The
Journal of Chemical Physics. 1975, 62(3), 773-775.
69:
Dynamical properties of reaction networks were studied in chemistry and physics after the invention of the
1130:
1051:
58:
2593:
2581:
1947:
506:
and neither of the products are used in the reactions, the set of reactants and the set of products are
50:
1246:
1806:
1772:
1054:
948:
129:
Aris’ activity was concentrated on the detailed systematization of mathematical ideas and approaches.
126:
elaborated by these researchers influenced many sciences, especially nuclear physics and engineering.
1788:
936:
503:
34:
3027:
Thermodynamic function analogue for reactions proceeding without interaction of various substances
1532:
2954:
2934:
2873:
2596:. The methods of limiting steps gave rise to many methods of the analysis of the reaction graph.
2589:
2254:
2221:
1191:
138:
78:
70:
38:
2354:
1646:
596:
469:
430:
1744:
1722:
1700:
1624:
1592:
1570:
631:
560:
524:
386:
350:
1932:
1678:
1134:
997:. This means that the equation representing the chemical reaction network can be rewritten as
183:
171:
2850:
M. Domijan and M. Kirkilionis, "Bistability and oscillations in chemical reaction networks",
2160:
1079:
1059:
2385:
942:
101:
82:
74:
54:
2190:
2056:
112:
era. The "eras" may be distinguished based on the main focuses of the scientific leaders:
2791:
1176:
1149:
1103:
152:
134:
109:
2937:, T.A. Akramov, "Multiplicity of the Steady State in Heterogeneous Catalytic Reactions",
17:
770:{\displaystyle x(t)=\left({\begin{array}{c}a(t)\\b(t)\\c(t)\\\vdots \end{array}}\right)}
499:
123:
3067:
1250:
1187:
990:
518:
507:
1810:
1206:
46:
2959:"The simplest catalytic mechanism permitting several steady states of the surface"
133:
The mathematical discipline "chemical reaction network theory" was originated by
2046:{\displaystyle G(c)=\sum _{i}c_{i}\left(\ln {\frac {c_{i}}{c_{i}^{*}}}-1\right)}
1210:
1919:
where (BZ) is an intermediate that does not participate in the main reaction.
187:
81:(1901), development of the quantitative theory of chemical chain reactions by
1697:
is the "adsorption place" on the surface of the solid catalyst (for example,
1776:
1183:
86:
42:
2833:
G. Craciun and C. Pantea, "Identifiability of chemical reaction networks",
2281:
are non-negative integers) and allows the stoichiometric conservation law
2807:
2627:
Some
Problems Relating to Chain Reactions and to the Theory of Combustion
1214:
380:
2958:
2613:
1941:
theorem about systems without interactions between different components
1792:
1239:
1133:
where each output value represents a reaction rate, referred to as the
175:
2813:
properties, injectivity and stability in chemical reaction systems",
424:
416:
833:
2995:
2878:"Extended detailed balance for systems with irreversible reactions"
2650:
2877:
2734:
P. Érdi and J. Tóth, "Mathematical models of chemical reactions",
463:
427:). The products appear to the right of the arrows, here they are
3058:
2994:
A.N. Gorban, N. Jarman, E. Steur, C. van
Leeuwen, I.Yu. Tyukin,
2653:, Mathematical Modelling of Natural Phenomena 10(5) (2015), 1–5.
137:, a famous expert in chemical engineering, with the support of
2680:
F. J. M. Horn and R. Jackson, "General Mass Action
Kinetics",
2697:
M. Feinberg, "Complex balancing in general kinetic systems",
2998:
Math. Model. Nat. Phenom. Vol. 10, No. 3, 2015, pp. 212–231.
2616:, Monatshefte fĂĽr Chemie / Chemical Monthly 32(8), 849--906.
2481:{\displaystyle \sum _{i}m_{i}|c_{i}^{1}(t)-c_{i}^{2}(t)|}
1661:
1607:
1547:
1277:
611:
575:
539:
484:
445:
401:
365:
313:
293:
263:
239:
222:
73:. The essential steps in this study were introduction of
3045:
Current Opinion in Chemical Engineering 2018 21C, 48-59.
3029:, Chemical Engineering Science, 1986 41(11), 2739-2745.
3059:
Specialist wiki on the mathematics of reaction networks
2344:{\displaystyle M(c)=\sum _{i}m_{i}c_{i}={\text{const}}}
2150:{\displaystyle n_{k}A_{i}\to \sum _{j}\beta _{kj}A_{j}}
1076:
represents the net stoichiometry of a reaction, and so
2516:
1862:
1407:
1312:
705:
2494:
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1956:
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951:
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353:
201:
57:
due to the interesting problems that arise from the
675:These variables can then be combined into a vector
2544:
2480:
2376:
2343:
2273:
2243:
2206:
2175:
2149:
2072:
2045:
1895:
1755:
1733:
1711:
1689:
1667:
1635:
1613:
1581:
1559:
1498:
1432:
1341:
1253:that allows multiplicity of steady states (1976):
1167:
1121:
1088:
1068:
1039:
981:
924:
769:
664:
620:
581:
545:
490:
454:
407:
371:
323:
166:A chemical reaction network (often abbreviated to
1870:
1869:
1852:
1851:
1415:
1414:
1397:
1396:
1320:
1319:
1302:
1301:
141:, the founder and editor-in-chief of the journal
2545:{\displaystyle c^{1}(t)\;{\mbox{and}}\;c^{2}(t)}
1928:linear systems with the same number of states.
3037:
3035:
1896:{\displaystyle {\ce {{B}+ Z <=> (BZ)}}}
1499:{\displaystyle {\ce {{AZ}+BZ -> {AB}+2Z}}}
1342:{\displaystyle {\ce {{A2}+2Z <=> 2AZ}}}
780:and their evolution with time can be written
8:
2783:P. De Leenheer, D. Angeli and E. D. Sontag,
502:). In this example, since the reactions are
3025:A.N. Gorban, V.I. Bykov, G.S. Yablonskii,
1249:is the simplest catalytic reaction without
144:Archive for Rational Mechanics and Analysis
2522:
2514:
1923:Network structure and dynamical properties
1433:{\displaystyle {\ce {{B}+Z <=> BZ}}}
2639:Chemical Kinetics in the Past Few Decades
2527:
2515:
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2493:
2473:
2458:
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2431:
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2401:
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2256:
2229:
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2141:
2128:
2118:
2105:
2095:
2089:
2064:
2058:
2024:
2019:
2009:
2003:
1986:
1976:
1955:
1881:
1871:
1864:
1863:
1861:
1853:
1846:
1844:
1843:
1841:
1830:
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1827:
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1303:
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1081:
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1005:
953:
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2806:M. Banaji, P. Donnell and S. Baigent, "
2605:
1845:
1800:Existence of stable periodic solutions
1390:
1295:
1221:concentration of reactants, and so on.
1040:{\displaystyle {\dot {x}}=\Gamma V(x)}
77:for the complex chemical reactions by
53:. It has also attracted interest from
2996:Leaders do not Look Back, or do They?
2796:J. Math. Chem.', 41(3):295–314, 2007.
2785:"Monotone chemical reaction networks"
7:
1819:
1455:
1364:
1255:
192:
182:the set of reactants), and a set of
108:era and the third is definitely the
2560:) monotonically decreases in time.
85:(1934), development of kinetics of
1083:
1063:
1022:
100:era, the second may be called the
25:
2651:Three Waves of Chemical Dynamics
2569:basic ideas have been invented:
31:Chemical reaction network theory
2953:V.I. Bykov, V.I. Elokhin, G.S.
2933:M.G. Slin'ko, V.I. Bykov, G.S.
982:{\displaystyle {\dot {x}}=f(x)}
945:, commonly written in the form
593:represent the concentration of
557:represent the concentration of
2539:
2533:
2511:
2505:
2474:
2470:
2464:
2443:
2437:
2418:
2297:
2291:
2111:
1966:
1960:
1888:
1882:
1872:
1847:
1815:Belousov–Zhabotinsky reactions
1477:
1417:
1392:
1322:
1297:
1162:
1156:
1116:
1110:
1034:
1028:
976:
970:
749:
743:
733:
727:
717:
711:
694:
688:
659:
653:
644:
638:
300:
246:
1:
2637:Hinshelwood's Nobel Lecture
1787:Persistence has its roots in
1560:{\displaystyle {\ce {A2, B}}}
2882:Chemical Engineering Science
2649:A.N. Gorban, G.S. Yablonsky
2187:is the number of reactions,
2080:is the concentration of the
41:the behaviour of real-world
2737:Manchester University Press
2274:{\displaystyle \beta _{kj}}
2244:{\displaystyle n_{k}\geq 1}
1668:{\displaystyle {\ce {CO2}}}
1177:continuously differentiable
621:{\displaystyle {\ce {H2O}}}
491:{\displaystyle {\ce {CO2}}}
455:{\displaystyle {\ce {H2O}}}
186:. For example, the pair of
178:, a set of products (often
27:Area of applied mathematics
3090:
2963:React. Kinet. Catal. Lett.
2700:Arch. Rational Mech. Anal.
2625:Semyonov's Nobel Lecture
2377:{\displaystyle m_{i}>0}
1767:Stability of steady states
1756:{\displaystyle {\ce {BZ}}}
1734:{\displaystyle {\ce {AZ}}}
1712:{\displaystyle {\ce {Pt}}}
1636:{\displaystyle {\ce {CO}}}
1614:{\displaystyle {\ce {O2}}}
1582:{\displaystyle {\ce {AB}}}
1205:Other assumptions include
582:{\displaystyle {\ce {O2}}}
546:{\displaystyle {\ce {H2}}}
408:{\displaystyle {\ce {O2}}}
372:{\displaystyle {\ce {H2}}}
149:Friedrich Josef Maria Horn
93:, and many other results.
1690:{\displaystyle {\ce {Z}}}
1050:Here, each column of the
665:{\displaystyle a(t),b(t)}
18:Chemical reaction network
1589:are gases (for example,
1526:of catalytic oxidation.
935:This is an example of a
553:in the surrounding air,
91:Cyril Norman Hinshelwood
2612:Wegscheider, R. (1901)
2176:{\displaystyle k\leq r}
1937:deficiency zero theorem
1233:Number of steady states
1089:{\displaystyle \Gamma }
1069:{\displaystyle \Gamma }
59:mathematical structures
3074:Mathematical chemistry
2682:Archive Rational Mech.
2546:
2488:between two solutions
2482:
2378:
2345:
2275:
2245:
2208:
2177:
2151:
2074:
2047:
1897:
1757:
1735:
1713:
1691:
1669:
1637:
1615:
1583:
1561:
1522:This is the classical
1500:
1434:
1343:
1169:
1131:vector-valued function
1123:
1090:
1070:
1041:
983:
926:
771:
666:
622:
583:
547:
492:
456:
409:
373:
325:
2939:Dokl. Akad. Nauk SSSR
2594:singular perturbation
2547:
2483:
2384:), then the weighted
2379:
2346:
2276:
2246:
2209:
2207:{\displaystyle A_{i}}
2178:
2152:
2075:
2073:{\displaystyle c_{i}}
2048:
1898:
1758:
1736:
1714:
1692:
1670:
1638:
1616:
1584:
1562:
1501:
1435:
1344:
1170:
1124:
1091:
1071:
1042:
984:
927:
772:
667:
623:
584:
548:
493:
457:
410:
374:
326:
51:theoretical chemistry
2985:(3) (1978), 637–639.
2968:(2) (1976), 191–198.
2900:Adv. Exp. Med. Biol.
2824:(6):1523–1547, 2007.
2492:
2394:
2355:
2285:
2255:
2222:
2191:
2161:
2088:
2057:
1954:
1826:
1745:
1723:
1701:
1679:
1647:
1625:
1593:
1571:
1533:
1524:adsorption mechanism
1462:
1371:
1262:
1168:{\displaystyle V(x)}
1150:
1122:{\displaystyle V(x)}
1104:
1098:stoichiometry matrix
1080:
1060:
1004:
949:
787:
682:
632:
597:
561:
525:
470:
431:
387:
351:
199:
2590:invariant manifolds
2463:
2436:
2029:
1858:
1791:. A non-persistent
1789:population dynamics
1663:
1609:
1549:
1403:
1308:
1279:
613:
577:
541:
486:
447:
403:
367:
315:
295:
265:
241:
224:
55:pure mathematicians
35:applied mathematics
2860:(4):467–501, 2009.
2790:2014-08-12 at the
2757:(4):339–344, 2002.
2592:and computational
2542:
2520:
2478:
2449:
2422:
2406:
2374:
2341:
2312:
2271:
2241:
2204:
2173:
2147:
2123:
2070:
2043:
2015:
1981:
1933:Lyapunov functions
1893:
1877:
1753:
1731:
1709:
1687:
1665:
1651:
1633:
1611:
1597:
1579:
1557:
1537:
1496:
1430:
1422:
1339:
1327:
1267:
1165:
1141:Common assumptions
1119:
1086:
1066:
1037:
979:
922:
913:
767:
761:
662:
618:
601:
579:
565:
543:
529:
488:
474:
452:
435:
405:
391:
369:
355:
321:
319:
303:
283:
253:
229:
212:
139:Clifford Truesdell
79:Rudolf Wegscheider
71:law of mass action
2903:(Adv Syst Biol),
2888::5388–5399, 2011.
2774::1135–1145, 2007.
2519:
2397:
2339:
2303:
2214:is the symbol of
2114:
2030:
1972:
1917:
1916:
1887:
1879:
1840:
1833:
1751:
1729:
1707:
1685:
1654:
1631:
1600:
1577:
1555:
1540:
1520:
1519:
1494:
1483:
1476:
1469:
1454:
1453:
1428:
1424:
1385:
1378:
1363:
1362:
1337:
1329:
1290:
1270:
1247:catalytic trigger
1245:For example, the
1219:spatially uniform
1016:
995:net stoichiometry
961:
902:
878:
854:
823:
799:
616:
604:
568:
532:
477:
450:
438:
394:
358:
345:
344:
306:
286:
278:
268:
256:
232:
215:
37:that attempts to
16:(Redirected from
3081:
3046:
3039:
3030:
3023:
3017:
3014:
3008:
3005:
2999:
2992:
2986:
2975:
2969:
2951:
2945:
2944:(4) (1976), 876.
2931:
2925:
2924:(7):e39194,2012.
2914:
2908:
2895:
2889:
2867:
2861:
2848:
2842:
2831:
2825:
2804:
2798:
2781:
2775:
2764:
2758:
2747:
2741:
2732:
2726:
2723:
2717:
2714:
2708:
2695:
2689:
2678:
2672:
2669:
2663:
2660:
2654:
2647:
2641:
2635:
2629:
2623:
2617:
2610:
2551:
2549:
2548:
2543:
2532:
2531:
2521:
2517:
2504:
2503:
2487:
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2484:
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2337:
2332:
2331:
2322:
2321:
2311:
2280:
2278:
2277:
2272:
2270:
2269:
2250:
2248:
2247:
2242:
2234:
2233:
2213:
2211:
2210:
2205:
2203:
2202:
2182:
2180:
2179:
2174:
2156:
2154:
2153:
2148:
2146:
2145:
2136:
2135:
2122:
2110:
2109:
2100:
2099:
2079:
2077:
2076:
2071:
2069:
2068:
2052:
2050:
2049:
2044:
2042:
2038:
2031:
2028:
2023:
2014:
2013:
2004:
1991:
1990:
1980:
1911:
1902:
1900:
1899:
1894:
1892:
1891:
1885:
1880:
1878:
1876:
1875:
1868:
1860:
1859:
1857:
1850:
1842:
1838:
1834:
1831:
1820:
1762:
1760:
1759:
1754:
1752:
1749:
1740:
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1737:
1732:
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1727:
1718:
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1705:
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1683:
1674:
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1666:
1664:
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1642:
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1634:
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1620:
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1610:
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1586:
1585:
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1578:
1575:
1566:
1564:
1563:
1558:
1556:
1553:
1548:
1545:
1538:
1514:
1505:
1503:
1502:
1497:
1495:
1492:
1484:
1481:
1474:
1470:
1467:
1456:
1448:
1439:
1437:
1436:
1431:
1429:
1426:
1425:
1423:
1421:
1420:
1413:
1405:
1404:
1402:
1395:
1387:
1383:
1379:
1376:
1365:
1357:
1348:
1346:
1345:
1340:
1338:
1335:
1330:
1328:
1326:
1325:
1318:
1310:
1309:
1307:
1300:
1292:
1288:
1280:
1278:
1275:
1268:
1256:
1225:Types of results
1196:Michaelis–Menten
1174:
1172:
1171:
1166:
1128:
1126:
1125:
1120:
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1046:
1044:
1043:
1038:
1018:
1017:
1009:
988:
986:
985:
980:
963:
962:
954:
943:dynamical system
931:
929:
928:
923:
918:
914:
903:
901:
893:
885:
879:
877:
869:
861:
855:
853:
845:
837:
824:
822:
814:
806:
801:
800:
792:
776:
774:
773:
768:
766:
762:
671:
669:
668:
663:
627:
625:
624:
619:
617:
614:
612:
609:
602:
592:
588:
586:
585:
580:
578:
576:
573:
566:
556:
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550:
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542:
540:
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436:
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378:
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370:
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304:
296:
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279:
276:
269:
266:
264:
261:
254:
242:
240:
237:
230:
225:
223:
220:
213:
193:
83:Nikolay Semyonov
75:detailed balance
21:
3089:
3088:
3084:
3083:
3082:
3080:
3079:
3078:
3064:
3063:
3055:
3050:
3049:
3040:
3033:
3024:
3020:
3015:
3011:
3006:
3002:
2993:
2989:
2979:Doklady AN USSR
2976:
2972:
2952:
2948:
2932:
2928:
2915:
2911:
2896:
2892:
2868:
2864:
2849:
2845:
2832:
2828:
2805:
2801:
2792:Wayback Machine
2782:
2778:
2765:
2761:
2748:
2744:
2733:
2729:
2724:
2720:
2715:
2711:
2707::187–194, 1972.
2696:
2692:
2679:
2675:
2670:
2666:
2661:
2657:
2648:
2644:
2636:
2632:
2624:
2620:
2611:
2607:
2602:
2566:
2564:Model reduction
2523:
2495:
2490:
2489:
2407:
2392:
2391:
2358:
2353:
2352:
2323:
2313:
2283:
2282:
2258:
2253:
2252:
2225:
2220:
2219:
2194:
2189:
2188:
2159:
2158:
2137:
2124:
2101:
2091:
2086:
2085:
2060:
2055:
2054:
2005:
1996:
1992:
1982:
1952:
1951:
1925:
1909:
1824:
1823:
1802:
1785:
1769:
1743:
1742:
1721:
1720:
1699:
1698:
1677:
1676:
1645:
1644:
1623:
1622:
1591:
1590:
1569:
1568:
1531:
1530:
1512:
1460:
1459:
1446:
1369:
1368:
1355:
1260:
1259:
1235:
1227:
1148:
1147:
1143:
1102:
1101:
1078:
1077:
1058:
1057:
1002:
1001:
947:
946:
912:
911:
905:
904:
894:
886:
881:
880:
870:
862:
857:
856:
846:
838:
828:
815:
807:
785:
784:
760:
759:
753:
752:
737:
736:
721:
720:
700:
680:
679:
630:
629:
595:
594:
590:
559:
558:
554:
523:
522:
514:
468:
467:
429:
428:
420:
385:
384:
349:
348:
337:
318:
317:
297:
271:
270:
243:
197:
196:
164:
153:Martin Feinberg
135:Rutherford Aris
124:chain reactions
67:
28:
23:
22:
15:
12:
11:
5:
3087:
3085:
3077:
3076:
3066:
3065:
3062:
3061:
3054:
3053:External links
3051:
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3047:
3031:
3018:
3009:
3000:
2987:
2970:
2946:
2926:
2909:
2890:
2862:
2853:J. Math. Biol.
2843:
2835:J. Math. Chem.
2826:
2818:J. Appl. Math.
2799:
2776:
2768:J. Math. Chem.
2759:
2751:J. Math. Chem.
2742:
2727:
2718:
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2673:
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2618:
2604:
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2574:
2565:
2562:
2552:with the same
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2299:
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2240:
2237:
2232:
2228:
2218:th component,
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1223:
1164:
1161:
1158:
1155:
1142:
1139:
1118:
1115:
1112:
1109:
1096:is called the
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1065:
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1036:
1033:
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1024:
1021:
1015:
1012:
978:
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608:
572:
536:
517:represent the
500:carbon dioxide
481:
442:
398:
362:
343:
342:
333:
331:
310:
302:
298:
290:
282:
273:
272:
260:
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205:
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170:) comprises a
163:
160:
131:
130:
127:
120:
66:
63:
33:is an area of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3086:
3075:
3072:
3071:
3069:
3060:
3057:
3056:
3052:
3044:
3038:
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3032:
3028:
3022:
3019:
3013:
3010:
3004:
3001:
2997:
2991:
2988:
2984:
2981:(Chemistry)
2980:
2974:
2971:
2967:
2964:
2960:
2956:
2950:
2947:
2943:
2940:
2936:
2930:
2927:
2923:
2919:
2913:
2910:
2907::81–94, 2012.
2906:
2902:
2901:
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2891:
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2859:
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2854:
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2840:
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2827:
2823:
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2810:
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2800:
2797:
2793:
2789:
2786:
2780:
2777:
2773:
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2763:
2760:
2756:
2752:
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2743:
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2738:
2731:
2728:
2722:
2719:
2713:
2710:
2706:
2702:
2701:
2694:
2691:
2687:
2683:
2677:
2674:
2668:
2665:
2659:
2656:
2652:
2646:
2643:
2640:
2634:
2631:
2628:
2622:
2619:
2615:
2609:
2606:
2599:
2597:
2595:
2591:
2583:
2582:limiting step
2579:
2575:
2572:
2571:
2570:
2563:
2561:
2559:
2555:
2536:
2528:
2524:
2508:
2500:
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2467:
2459:
2454:
2450:
2446:
2440:
2432:
2427:
2423:
2412:
2408:
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2238:
2235:
2230:
2226:
2217:
2199:
2195:
2186:
2170:
2167:
2164:
2142:
2138:
2132:
2129:
2125:
2119:
2115:
2106:
2102:
2096:
2092:
2083:
2065:
2061:
2039:
2035:
2032:
2025:
2020:
2016:
2010:
2006:
2000:
1997:
1993:
1987:
1983:
1977:
1973:
1969:
1963:
1957:
1949:
1944:
1942:
1938:
1934:
1929:
1922:
1920:
1913:
1906:
1904:
1865:
1854:
1835:
1822:
1821:
1818:
1816:
1812:
1808:
1807:quasiperiodic
1799:
1797:
1794:
1790:
1782:
1780:
1778:
1774:
1773:deterministic
1766:
1764:
1656:
1602:
1550:
1542:
1527:
1525:
1516:
1509:
1507:
1488:
1485:
1471:
1458:
1457:
1450:
1443:
1441:
1410:
1399:
1380:
1367:
1366:
1359:
1352:
1350:
1331:
1315:
1304:
1284:
1281:
1272:
1258:
1257:
1254:
1252:
1251:autocatalysis
1248:
1243:
1241:
1232:
1230:
1224:
1222:
1220:
1216:
1212:
1208:
1203:
1201:
1197:
1193:
1189:
1188:autocatalysis
1185:
1180:
1178:
1159:
1153:
1140:
1138:
1136:
1132:
1113:
1107:
1099:
1056:
1053:
1031:
1025:
1019:
1013:
1010:
1000:
999:
998:
996:
992:
991:stoichiometry
973:
967:
964:
958:
955:
944:
941:
938:
919:
915:
908:
898:
895:
890:
887:
874:
871:
866:
863:
850:
847:
842:
839:
829:
825:
819:
816:
811:
808:
802:
796:
793:
783:
782:
781:
763:
756:
746:
740:
730:
724:
714:
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691:
685:
678:
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534:
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519:concentration
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125:
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118:
115:
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113:
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99:
94:
92:
89:reactions by
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80:
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72:
64:
62:
60:
56:
52:
48:
44:
40:
36:
32:
19:
3041:A.N.Gorban,
3021:
3012:
3003:
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2973:
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2962:
2949:
2941:
2938:
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2904:
2898:
2893:
2885:
2881:
2870:A. N. Gorban
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2704:
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2693:
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2681:
2676:
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2658:
2645:
2633:
2621:
2608:
2587:
2567:
2557:
2553:
2386:
2215:
2184:
2081:
1945:
1940:
1936:
1930:
1926:
1918:
1907:
1803:
1786:
1779:may appear.
1770:
1528:
1521:
1510:
1444:
1353:
1244:
1236:
1228:
1207:mass balance
1204:
1181:
1144:
1097:
1049:
994:
934:
779:
674:
512:
504:irreversible
346:
335:
180:intersecting
167:
165:
157:
142:
132:
95:
68:
47:biochemistry
30:
29:
2351:(where all
1948:free energy
1783:Persistence
1213:, constant
1211:temperature
1209:, constant
1192:mass action
106:Hinshelwood
98:van 't Hoff
2955:Yablonskii
2935:Yablonskii
2872:and G. S.
2688::81, 1972.
2600:References
1910:reaction 5
1777:attractors
1513:reaction 4
1447:reaction 3
1356:reaction 2
1240:biological
940:autonomous
937:continuous
338:reaction 1
190:reactions
188:combustion
117:van’t Hoff
61:involved.
2874:Yablonsky
2841::1, 2008.
2447:−
2399:∑
2305:∑
2260:β
2236:≥
2168:≤
2126:β
2116:∑
2112:→
2033:−
2026:∗
2001:
1974:∑
1873:⇀
1866:−
1855:−
1848:↽
1478:⟶
1418:⇀
1411:−
1400:−
1393:↽
1323:⇀
1316:−
1305:−
1298:↽
1184:catalysis
1084:Γ
1064:Γ
1023:Γ
1014:˙
959:˙
909:⋮
803:≡
797:˙
757:⋮
301:⟶
247:⟶
184:reactions
176:reactants
87:catalytic
3068:Category
2918:PLoS ONE
2788:Archived
2389:distance
2183:, where
2053:, where
1939:and the
1805:exhibit
1215:pressure
1135:kinetics
1052:constant
508:disjoint
381:hydrogen
162:Overview
43:chemical
2740:, 1989.
1811:chaotic
1793:species
672:, etc.
102:Semenov
65:History
2811:matrix
2251:, and
1529:Here,
1055:matrix
466:) and
425:carbon
419:) and
417:oxygen
2338:const
2157:(for
1950:form
1129:is a
464:water
39:model
2816:SIAM
2580:The
2369:>
1741:and
1643:and
1567:and
1200:Hill
1198:and
110:Aris
49:and
2983:242
2942:226
2905:736
2518:and
1809:or
1719:),
1675:),
1186:or
1175:is
521:of
383:),
174:of
172:set
168:CRN
3070::
3034:^
2961:,
2957:,
2886:66
2884:,
2880:,
2876:,
2858:59
2856:,
2839:44
2837:,
2822:67
2820:,
2794:,
2772:42
2770:,
2755:31
2753:,
2705:49
2703:,
2686:47
2684:,
1998:ln
1943:.
1886:BZ
1750:BZ
1728:AZ
1706:Pt
1653:CO
1630:CO
1621:,
1576:AB
1482:AB
1475:BZ
1468:AZ
1427:BZ
1336:AZ
1217:,
1194:,
1179:.
1137:.
1100:.
589:,
510:.
476:CO
305:CO
151:,
2966:4
2922:7
2920:,
2809:P
2558:c
2556:(
2554:M
2540:)
2537:t
2534:(
2529:2
2525:c
2512:)
2509:t
2506:(
2501:1
2497:c
2475:|
2471:)
2468:t
2465:(
2460:2
2455:i
2451:c
2444:)
2441:t
2438:(
2433:1
2428:i
2424:c
2419:|
2413:i
2409:m
2403:i
2387:L
2372:0
2364:i
2360:m
2334:=
2329:i
2325:c
2319:i
2315:m
2309:i
2301:=
2298:)
2295:c
2292:(
2289:M
2267:j
2264:k
2239:1
2231:k
2227:n
2216:i
2200:i
2196:A
2185:r
2171:r
2165:k
2143:j
2139:A
2133:j
2130:k
2120:j
2107:i
2103:A
2097:k
2093:n
2082:i
2066:i
2062:c
2040:)
2036:1
2021:i
2017:c
2011:i
2007:c
1994:(
1988:i
1984:c
1978:i
1970:=
1967:)
1964:c
1961:(
1958:G
1912:)
1908:(
1889:)
1883:(
1839:Z
1836:+
1832:B
1684:Z
1657:2
1603:2
1599:O
1554:B
1551:,
1543:2
1539:A
1515:)
1511:(
1493:Z
1489:2
1486:+
1472:+
1449:)
1445:(
1384:Z
1381:+
1377:B
1358:)
1354:(
1332:2
1289:Z
1285:2
1282:+
1273:2
1269:A
1163:)
1160:x
1157:(
1154:V
1117:)
1114:x
1111:(
1108:V
1035:)
1032:x
1029:(
1026:V
1020:=
1011:x
977:)
974:x
971:(
968:f
965:=
956:x
920:.
916:)
899:t
896:d
891:c
888:d
875:t
872:d
867:b
864:d
851:t
848:d
843:a
840:d
830:(
826:=
820:t
817:d
812:x
809:d
794:x
764:)
750:)
747:t
744:(
741:c
734:)
731:t
728:(
725:b
718:)
715:t
712:(
709:a
702:(
698:=
695:)
692:t
689:(
686:x
660:)
657:t
654:(
651:b
648:,
645:)
642:t
639:(
636:a
615:O
607:2
603:H
591:c
571:2
567:O
555:b
535:2
531:H
515:a
498:(
480:2
462:(
449:O
441:2
437:H
423:(
421:C
415:(
397:2
393:O
379:(
361:2
357:H
340:)
336:(
309:2
289:2
285:O
281:+
277:C
267:O
259:2
255:H
250:2
235:2
231:O
227:+
218:2
214:H
209:2
104:–
20:)
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