1916:
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
2565:
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
1793:
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
2557:
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
919:
136:. 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,
318:
1134:
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
764:
1784:
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?
3031:
775:
2040:
1226:
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
3005:
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.
2475:
144:
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.
2338:
2144:
1920:
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
1752:
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.
1179:), 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
192:
187:
2562:
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).
1231:
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".
2539:
2573:
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.
502:
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
1890:
1493:
1336:
1427:
978:. 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
3015:
1764:
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
2660:
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,
1034:
2705:
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
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402:
366:
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659:
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1218:
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.
1083:
1063:
2201:
2067:
1162:
1116:
2651:
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.
1806:. The simplest catalytic oscillator (nonlinear self-oscillations without autocatalysis) can be produced from the catalytic trigger by adding a "buffer" step.
670:
914:{\displaystyle {\dot {x}}\equiv {\frac {dx}{dt}}=\left({\begin{array}{c}{\frac {da}{dt}}\\{\frac {db}{dt}}\\{\frac {dc}{dt}}\\\vdots \end{array}}\right).}
1235:
2776:
2073:-th component. The theorem about systems without interactions between different components states that if a network consists of reactions of the form
1942:
85:
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
108:
was searching for the general law of chemical reaction related to specific chemical properties. The term "chemical dynamics" belongs to van’t Hoff.
3032:
Model reduction in chemical dynamics: slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph.
2804:
336:
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
2888:
111:
The
Semenov-Hinshelwood focus was an explanation of critical phenomena observed in many chemical systems, in particular in flames. A concept
2688:
132:
1802:
behaviour. While stable periodic solutions are unusual in real-world chemical reaction networks, well-known examples exist, such as the
1924:
is possible without apriori assumptions about special relations between rate constants. Two results of this type are well known: the
617:, 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.
2966:
V.I. Bykov, G.S. Yablonskii, V.F. Kim, "On the simple model of kinetic self-oscillations in catalytic reaction of CO oxidation",
2382:
147:
Since then, the chemical reaction network theory has been further developed by a large number of researchers internationally.
2273:
2076:
313:{\displaystyle {\begin{aligned}{\ce {{2H2}+ O2}}&{\ce {-> 2H2O}}\\{\ce {{C}+ O2}}&{\ce {-> CO2}}\end{aligned}}}
34:
systems. Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in
2566:
the QSS). The QSS is defined as the steady state under the condition that the concentrations of other species do not change.
2905:
I. Otero-Muras, J. R. Banga and A. A. Alonso, "Characterizing multistationarity regimes in biochemical reaction networks",
1760:
Stability determines whether a given steady state solution is likely to be observed in reality. Since real systems (unlike
1191:
kinetics. Sometimes further assumptions are made about reaction rates, e.g. that all reactions obey mass action kinetics.
2627:
2577:
The quasi-equilibrium approximation and the quasi steady state methods were developed further into the methods of slow
2714:
N.Z. Shapiro, L.S. Shapley, Mass action law and the Gibbs free energy function, SIAM J. Appl. Math. 16 (1965) 353–375.
2480:
982:
of the reaction, and the difference between the two (i.e. the overall number of molecules used up or produced) is the
1803:
1512:
2755:
M. Mincheva and D. Siegel, "Nonnegativity and positiveness of solutions to mass action reaction–diffusion systems",
1935:
The deficiency zero theorem gives sufficient conditions for the existence of the
Lyapunov function in the classical
2886:
E. Feliu, M. Knudsen and C. Wiuf., "Signaling cascades: Consequences of varying substrate and phosphatase levels",
2615:
1814:
1450:
1250:
1171:
It is also commonly assumed that no reaction features the same chemical as both a reactant and a product (i.e. no
3062:
2841:
928:
105:
86:
2738:
H. Kunze and D. Siegel, "Monotonicity properties of chemical reactions with a single initial bimolecular step",
2603:Ăśber simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme
1359:
2773:
1184:
992:
2996:
B.L. Clarke, Theorems on chemical network stability. The
Journal of Chemical Physics. 1975, 62(3), 773-775.
2858:
1188:
58:
Dynamical properties of reaction networks were studied in chemistry and physics after the invention of the
2725:
495:
and neither of the products are used in the reactions, the set of reactants and the set of products are
937:
168:
118:
Aris’ activity was concentrated on the detailed systematization of mathematical ideas and approaches.
115:
elaborated by these researchers influenced many sciences, especially nuclear physics and engineering.
94:
79:
3016:
Thermodynamic function analogue for reactions proceeding without interaction of various substances
1521:
2585:. The methods of limiting steps gave rise to many methods of the analysis of the reaction graph.
2243:
2210:
1207:
2343:
1635:
585:
458:
419:
1733:
1711:
1689:
1613:
1581:
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1119:
1040:
620:
549:
513:
375:
339:
47:
1667:
986:. This means that the equation representing the chemical reaction network can be rewritten as
2839:
M. Domijan and M. Kirkilionis, "Bistability and oscillations in chemical reaction networks",
2582:
2570:
2149:
1936:
1068:
1048:
39:
1799:
2179:
2045:
1795:
1761:
1043:
101:
era. The "eras" may be distinguished based on the main focuses of the scientific leaders:
1138:
1092:
8:
2926:, T.A. Akramov, "Multiplicity of the Steady State in Heterogeneous Catalytic Reactions",
1777:
925:
492:
23:
759:{\displaystyle x(t)=\left({\begin{array}{c}a(t)\\b(t)\\c(t)\\\vdots \end{array}}\right)}
2943:
2923:
2862:
2578:
1180:
127:
67:
59:
27:
1921:
1123:
172:
160:
2374:
931:
90:
71:
63:
43:
2948:"The simplest catalytic mechanism permitting several steady states of the surface"
2780:
1165:
141:
123:
122:
The mathematical discipline "chemical reaction network theory" was originated by
98:
2035:{\displaystyle G(c)=\sum _{i}c_{i}\left(\ln {\frac {c_{i}}{c_{i}^{*}}}-1\right)}
488:
112:
1908:
where (BZ) is an intermediate that does not participate in the main reaction.
70:(1901), development of the quantitative theory of chemical chain reactions by
3056:
1686:
is the "adsorption place" on the surface of the solid catalyst (for example,
1239:
1176:
979:
507:
496:
2822:
G. Craciun and C. Pantea, "Identifiability of chemical reaction networks",
1195:
35:
2270:
are non-negative integers) and allows the stoichiometric conservation law
2616:
Some
Problems Relating to Chain Reactions and to the Theory of Combustion
1199:
2947:
2602:
1930:
theorem about systems without interactions between different components
1122:
where each output value represents a reaction rate, referred to as the
176:
2802:
properties, injectivity and stability in chemical reaction systems",
1765:
1172:
75:
31:
822:
2984:
2867:"Extended detailed balance for systems with irreversible reactions"
2796:
2639:
1203:
369:
2866:
2723:
P. Érdi and J. Tóth, "Mathematical models of chemical reactions",
1781:
1228:
416:). The products appear to the right of the arrows, here they are
164:
3047:
2983:
A.N. Gorban, N. Jarman, E. Steur, C. van
Leeuwen, I.Yu. Tyukin,
2642:, Mathematical Modelling of Natural Phenomena 10(5) (2015), 1–5.
413:
405:
126:, a famous expert in chemical engineering, with the support of
2669:
F. J. M. Horn and R. Jackson, "General Mass Action
Kinetics",
2686:
M. Feinberg, "Complex balancing in general kinetic systems",
1911:
452:
2987:
Math. Model. Nat. Phenom. Vol. 10, No. 3, 2015, pp. 212–231.
2605:, Monatshefte fĂĽr Chemie / Chemical Monthly 32(8), 849--906.
2470:{\displaystyle \sum _{i}m_{i}|c_{i}^{1}(t)-c_{i}^{2}(t)|}
1788:
1650:
1596:
1536:
1266:
600:
564:
528:
473:
434:
390:
354:
302:
282:
252:
228:
211:
62:. The essential steps in this study were introduction of
3034:
Current Opinion in Chemical Engineering 2018 21C, 48-59.
3018:, Chemical Engineering Science, 1986 41(11), 2739-2745.
3048:
Specialist wiki on the mathematics of reaction networks
2333:{\displaystyle M(c)=\sum _{i}m_{i}c_{i}={\text{const}}}
2139:{\displaystyle n_{k}A_{i}\to \sum _{j}\beta _{kj}A_{j}}
1065:
represents the net stoichiometry of a reaction, and so
2505:
1851:
1396:
1301:
694:
2483:
2385:
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378:
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46:
due to the interesting problems that arise from the
664:These variables can then be combined into a vector
2533:
2469:
2365:
2332:
2262:
2232:
2195:
2164:
2138:
2061:
2034:
1884:
1744:
1722:
1700:
1678:
1656:
1624:
1602:
1570:
1548:
1487:
1421:
1330:
1242:that allows multiplicity of steady states (1976):
1156:
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1077:
1057:
1028:
970:
913:
758:
653:
609:
570:
534:
479:
443:
396:
360:
312:
155:A chemical reaction network (often abbreviated to
1859:
1858:
1841:
1840:
1404:
1403:
1386:
1385:
1309:
1308:
1291:
1290:
130:, the founder and editor-in-chief of the journal
3054:
2534:{\displaystyle c^{1}(t)\;{\mbox{and}}\;c^{2}(t)}
1917:linear systems with the same number of states.
3026:
3024:
1885:{\displaystyle {\ce {{B}+ Z <=> (BZ)}}}
1755:
1488:{\displaystyle {\ce {{AZ}+BZ -> {AB}+2Z}}}
1331:{\displaystyle {\ce {{A2}+2Z <=> 2AZ}}}
769:and their evolution with time can be written
2772:P. De Leenheer, D. Angeli and E. D. Sontag,
491:). In this example, since the reactions are
3014:A.N. Gorban, V.I. Bykov, G.S. Yablonskii,
1238:is the simplest catalytic reaction without
133:Archive for Rational Mechanics and Analysis
3021:
2511:
2503:
1912:Network structure and dynamical properties
1422:{\displaystyle {\ce {{B}+Z <=> BZ}}}
1221:
2628:Chemical Kinetics in the Past Few Decades
1480:
1323:
1276:
241:
200:
2795:M. Banaji, P. Donnell and S. Baigent, "
1834:
1789:Existence of stable periodic solutions
1379:
1284:
1210:concentration of reactants, and so on.
3055:
1029:{\displaystyle {\dot {x}}=\Gamma V(x)}
66:for the complex chemical reactions by
42:. It has also attracted interest from
2985:Leaders do not Look Back, or do They?
2785:J. Math. Chem.', 41(3):295–314, 2007.
2774:"Monotone chemical reaction networks"
1129:
1898:
1808:
1501:
1444:
1435:
1353:
1344:
1244:
326:
181:
171:the set of reactants), and a set of
97:era and the third is definitely the
2549:) monotonically decreases in time.
1213:
74:(1934), development of kinetics of
13:
2552:
1072:
1052:
1011:
89:era, the second may be called the
14:
3074:
3041:
2640:Three Waves of Chemical Dynamics
2558:basic ideas have been invented:
20:Chemical reaction network theory
3008:
2999:
2990:
2977:
2960:
2942:V.I. Bykov, V.I. Elokhin, G.S.
2936:
2922:M.G. Slin'ko, V.I. Bykov, G.S.
2916:
2899:
2880:
2852:
2833:
2816:
2789:
2766:
2749:
2732:
2717:
2708:
971:{\displaystyle {\dot {x}}=f(x)}
934:, commonly written in the form
582:represent the concentration of
546:represent the concentration of
2699:
2680:
2663:
2654:
2645:
2632:
2620:
2608:
2595:
2528:
2522:
2500:
2494:
2463:
2459:
2453:
2432:
2426:
2407:
2286:
2280:
2100:
1955:
1949:
1877:
1871:
1861:
1836:
1804:Belousov–Zhabotinsky reactions
1771:
1466:
1406:
1381:
1311:
1286:
1151:
1145:
1105:
1099:
1023:
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648:
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289:
235:
1:
2626:Hinshelwood's Nobel Lecture
2588:
1776:Persistence has its roots in
1549:{\displaystyle {\ce {A2, B}}}
2871:Chemical Engineering Science
2638:A.N. Gorban, G.S. Yablonsky
2176:is the number of reactions,
2069:is the concentration of the
30:the behaviour of real-world
7:
2726:Manchester University Press
2263:{\displaystyle \beta _{kj}}
2233:{\displaystyle n_{k}\geq 1}
1657:{\displaystyle {\ce {CO2}}}
1166:continuously differentiable
610:{\displaystyle {\ce {H2O}}}
480:{\displaystyle {\ce {CO2}}}
444:{\displaystyle {\ce {H2O}}}
175:. For example, the pair of
167:, a set of products (often
150:
16:Area of applied mathematics
10:
3079:
2952:React. Kinet. Catal. Lett.
2689:Arch. Rational Mech. Anal.
2614:Semyonov's Nobel Lecture
2366:{\displaystyle m_{i}>0}
1756:Stability of steady states
1745:{\displaystyle {\ce {BZ}}}
1723:{\displaystyle {\ce {AZ}}}
1701:{\displaystyle {\ce {Pt}}}
1625:{\displaystyle {\ce {CO}}}
1603:{\displaystyle {\ce {O2}}}
1571:{\displaystyle {\ce {AB}}}
1194:Other assumptions include
571:{\displaystyle {\ce {O2}}}
535:{\displaystyle {\ce {H2}}}
397:{\displaystyle {\ce {O2}}}
361:{\displaystyle {\ce {H2}}}
138:Friedrich Josef Maria Horn
82:, and many other results.
53:
1679:{\displaystyle {\ce {Z}}}
1039:Here, each column of the
654:{\displaystyle a(t),b(t)}
1578:are gases (for example,
1515:of catalytic oxidation.
924:This is an example of a
542:in the surrounding air,
80:Cyril Norman Hinshelwood
2601:Wegscheider, R. (1901)
2165:{\displaystyle k\leq r}
1926:deficiency zero theorem
1222:Number of steady states
1078:{\displaystyle \Gamma }
1058:{\displaystyle \Gamma }
48:mathematical structures
3063:Mathematical chemistry
2671:Archive Rational Mech.
2535:
2477:between two solutions
2471:
2367:
2334:
2264:
2234:
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2140:
2063:
2036:
1886:
1746:
1724:
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1658:
1626:
1604:
1572:
1550:
1511:This is the classical
1489:
1423:
1332:
1158:
1120:vector-valued function
1112:
1079:
1059:
1030:
972:
915:
760:
655:
611:
572:
536:
481:
445:
398:
362:
314:
2928:Dokl. Akad. Nauk SSSR
2583:singular perturbation
2536:
2472:
2373:), then the weighted
2368:
2335:
2265:
2235:
2198:
2196:{\displaystyle A_{i}}
2167:
2141:
2064:
2062:{\displaystyle c_{i}}
2037:
1887:
1747:
1725:
1703:
1681:
1659:
1627:
1605:
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1551:
1490:
1424:
1333:
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1113:
1080:
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1031:
973:
916:
761:
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612:
573:
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482:
446:
399:
363:
315:
40:theoretical chemistry
2974:(3) (1978), 637–639.
2957:(2) (1976), 191–198.
2889:Adv. Exp. Med. Biol.
2813:(6):1523–1547, 2007.
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2383:
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1943:
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1614:
1582:
1560:
1522:
1513:adsorption mechanism
1451:
1360:
1251:
1157:{\displaystyle V(x)}
1139:
1111:{\displaystyle V(x)}
1093:
1087:stoichiometry matrix
1069:
1049:
993:
938:
776:
671:
621:
586:
550:
514:
459:
420:
376:
340:
188:
2579:invariant manifolds
2452:
2425:
2018:
1847:
1780:. A non-persistent
1778:population dynamics
1652:
1598:
1538:
1392:
1297:
1268:
602:
566:
530:
475:
436:
392:
356:
304:
284:
254:
230:
213:
44:pure mathematicians
24:applied mathematics
2849:(4):467–501, 2009.
2779:2014-08-12 at the
2746:(4):339–344, 2002.
2581:and computational
2531:
2509:
2467:
2438:
2411:
2395:
2363:
2330:
2301:
2260:
2230:
2193:
2162:
2136:
2112:
2059:
2032:
2004:
1970:
1922:Lyapunov functions
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1866:
1742:
1720:
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1600:
1586:
1568:
1546:
1526:
1485:
1419:
1411:
1328:
1316:
1256:
1154:
1130:Common assumptions
1108:
1075:
1055:
1026:
968:
911:
902:
756:
750:
651:
607:
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568:
554:
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310:
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292:
272:
242:
218:
201:
128:Clifford Truesdell
68:Rudolf Wegscheider
60:law of mass action
2892:(Adv Syst Biol),
2877::5388–5399, 2011.
2763::1135–1145, 2007.
2508:
2386:
2328:
2292:
2203:is the symbol of
2103:
2019:
1961:
1906:
1905:
1876:
1868:
1829:
1822:
1740:
1718:
1696:
1674:
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1620:
1589:
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1544:
1529:
1509:
1508:
1483:
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1465:
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1443:
1442:
1417:
1413:
1374:
1367:
1352:
1351:
1326:
1318:
1279:
1259:
1236:catalytic trigger
1234:For example, the
1208:spatially uniform
1005:
984:net stoichiometry
950:
891:
867:
843:
812:
788:
605:
593:
557:
521:
466:
439:
427:
383:
347:
334:
333:
295:
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267:
257:
245:
221:
204:
26:that attempts to
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3035:
3028:
3019:
3012:
3006:
3003:
2997:
2994:
2988:
2981:
2975:
2964:
2958:
2940:
2934:
2933:(4) (1976), 876.
2920:
2914:
2913:(7):e39194,2012.
2903:
2897:
2884:
2878:
2856:
2850:
2837:
2831:
2820:
2814:
2793:
2787:
2770:
2764:
2753:
2747:
2736:
2730:
2721:
2715:
2712:
2706:
2703:
2697:
2684:
2678:
2667:
2661:
2658:
2652:
2649:
2643:
2636:
2630:
2624:
2618:
2612:
2606:
2599:
2540:
2538:
2537:
2532:
2521:
2520:
2510:
2506:
2493:
2492:
2476:
2474:
2473:
2468:
2466:
2451:
2446:
2424:
2419:
2410:
2405:
2404:
2394:
2372:
2370:
2369:
2364:
2356:
2355:
2339:
2337:
2336:
2331:
2329:
2326:
2321:
2320:
2311:
2310:
2300:
2269:
2267:
2266:
2261:
2259:
2258:
2239:
2237:
2236:
2231:
2223:
2222:
2202:
2200:
2199:
2194:
2192:
2191:
2171:
2169:
2168:
2163:
2145:
2143:
2142:
2137:
2135:
2134:
2125:
2124:
2111:
2099:
2098:
2089:
2088:
2068:
2066:
2065:
2060:
2058:
2057:
2041:
2039:
2038:
2033:
2031:
2027:
2020:
2017:
2012:
2003:
2002:
1993:
1980:
1979:
1969:
1900:
1891:
1889:
1888:
1883:
1881:
1880:
1874:
1869:
1867:
1865:
1864:
1857:
1849:
1848:
1846:
1839:
1831:
1827:
1823:
1820:
1809:
1751:
1749:
1748:
1743:
1741:
1738:
1729:
1727:
1726:
1721:
1719:
1716:
1707:
1705:
1704:
1699:
1697:
1694:
1685:
1683:
1682:
1677:
1675:
1672:
1663:
1661:
1660:
1655:
1653:
1651:
1648:
1641:
1631:
1629:
1628:
1623:
1621:
1618:
1609:
1607:
1606:
1601:
1599:
1597:
1594:
1587:
1577:
1575:
1574:
1569:
1567:
1564:
1555:
1553:
1552:
1547:
1545:
1542:
1537:
1534:
1527:
1503:
1494:
1492:
1491:
1486:
1484:
1481:
1473:
1470:
1463:
1459:
1456:
1445:
1437:
1428:
1426:
1425:
1420:
1418:
1415:
1414:
1412:
1410:
1409:
1402:
1394:
1393:
1391:
1384:
1376:
1372:
1368:
1365:
1354:
1346:
1337:
1335:
1334:
1329:
1327:
1324:
1319:
1317:
1315:
1314:
1307:
1299:
1298:
1296:
1289:
1281:
1277:
1269:
1267:
1264:
1257:
1245:
1214:Types of results
1185:Michaelis–Menten
1163:
1161:
1160:
1155:
1117:
1115:
1114:
1109:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1035:
1033:
1032:
1027:
1007:
1006:
998:
977:
975:
974:
969:
952:
951:
943:
932:dynamical system
920:
918:
917:
912:
907:
903:
892:
890:
882:
874:
868:
866:
858:
850:
844:
842:
834:
826:
813:
811:
803:
795:
790:
789:
781:
765:
763:
762:
757:
755:
751:
660:
658:
657:
652:
616:
614:
613:
608:
606:
603:
601:
598:
591:
581:
577:
575:
574:
569:
567:
565:
562:
555:
545:
541:
539:
538:
533:
531:
529:
526:
519:
505:
486:
484:
483:
478:
476:
474:
471:
464:
450:
448:
447:
442:
440:
437:
435:
432:
425:
411:
403:
401:
400:
395:
393:
391:
388:
381:
367:
365:
364:
359:
357:
355:
352:
345:
328:
319:
317:
316:
311:
309:
305:
303:
300:
293:
285:
283:
280:
273:
268:
265:
258:
255:
253:
250:
243:
231:
229:
226:
219:
214:
212:
209:
202:
182:
72:Nikolay Semyonov
64:detailed balance
3078:
3077:
3073:
3072:
3071:
3069:
3068:
3067:
3053:
3052:
3044:
3039:
3038:
3029:
3022:
3013:
3009:
3004:
3000:
2995:
2991:
2982:
2978:
2968:Doklady AN USSR
2965:
2961:
2941:
2937:
2921:
2917:
2904:
2900:
2885:
2881:
2857:
2853:
2838:
2834:
2821:
2817:
2794:
2790:
2781:Wayback Machine
2771:
2767:
2754:
2750:
2737:
2733:
2722:
2718:
2713:
2709:
2704:
2700:
2696::187–194, 1972.
2685:
2681:
2668:
2664:
2659:
2655:
2650:
2646:
2637:
2633:
2625:
2621:
2613:
2609:
2600:
2596:
2591:
2555:
2553:Model reduction
2516:
2512:
2504:
2488:
2484:
2482:
2479:
2478:
2462:
2447:
2442:
2420:
2415:
2406:
2400:
2396:
2390:
2384:
2381:
2380:
2351:
2347:
2345:
2342:
2341:
2325:
2316:
2312:
2306:
2302:
2296:
2275:
2272:
2271:
2251:
2247:
2245:
2242:
2241:
2218:
2214:
2212:
2209:
2208:
2187:
2183:
2181:
2178:
2177:
2151:
2148:
2147:
2130:
2126:
2117:
2113:
2107:
2094:
2090:
2084:
2080:
2078:
2075:
2074:
2053:
2049:
2047:
2044:
2043:
2013:
2008:
1998:
1994:
1992:
1985:
1981:
1975:
1971:
1965:
1944:
1941:
1940:
1914:
1870:
1860:
1853:
1852:
1850:
1842:
1835:
1833:
1832:
1830:
1819:
1818:
1816:
1813:
1812:
1791:
1774:
1758:
1737:
1735:
1732:
1731:
1715:
1713:
1710:
1709:
1693:
1691:
1688:
1687:
1671:
1669:
1666:
1665:
1649:
1644:
1639:
1637:
1634:
1633:
1617:
1615:
1612:
1611:
1595:
1590:
1585:
1583:
1580:
1579:
1563:
1561:
1558:
1557:
1535:
1530:
1525:
1523:
1520:
1519:
1469:
1455:
1454:
1452:
1449:
1448:
1405:
1398:
1397:
1395:
1387:
1380:
1378:
1377:
1375:
1364:
1363:
1361:
1358:
1357:
1310:
1303:
1302:
1300:
1292:
1285:
1283:
1282:
1280:
1265:
1260:
1255:
1254:
1252:
1249:
1248:
1224:
1216:
1140:
1137:
1136:
1132:
1094:
1091:
1090:
1070:
1067:
1066:
1050:
1047:
1046:
997:
996:
994:
991:
990:
942:
941:
939:
936:
935:
901:
900:
894:
893:
883:
875:
873:
870:
869:
859:
851:
849:
846:
845:
835:
827:
825:
821:
817:
804:
796:
794:
780:
779:
777:
774:
773:
749:
748:
742:
741:
726:
725:
710:
709:
693:
689:
672:
669:
668:
622:
619:
618:
599:
594:
589:
587:
584:
583:
579:
563:
558:
553:
551:
548:
547:
543:
527:
522:
517:
515:
512:
511:
503:
472:
467:
462:
460:
457:
456:
433:
428:
423:
421:
418:
417:
409:
389:
384:
379:
377:
374:
373:
353:
348:
343:
341:
338:
337:
307:
306:
301:
296:
288:
286:
281:
276:
264:
263:
260:
259:
251:
246:
234:
232:
227:
222:
210:
205:
196:
195:
191:
189:
186:
185:
153:
142:Martin Feinberg
124:Rutherford Aris
113:chain reactions
56:
17:
12:
11:
5:
3076:
3066:
3065:
3051:
3050:
3043:
3042:External links
3040:
3037:
3036:
3020:
3007:
2998:
2989:
2976:
2959:
2935:
2915:
2898:
2879:
2851:
2842:J. Math. Biol.
2832:
2824:J. Math. Chem.
2815:
2807:J. Appl. Math.
2788:
2765:
2757:J. Math. Chem.
2748:
2740:J. Math. Chem.
2731:
2716:
2707:
2698:
2679:
2662:
2653:
2644:
2631:
2619:
2607:
2593:
2592:
2590:
2587:
2575:
2574:
2567:
2563:
2554:
2551:
2541:with the same
2530:
2527:
2524:
2519:
2515:
2502:
2499:
2496:
2491:
2487:
2465:
2461:
2458:
2455:
2450:
2445:
2441:
2437:
2434:
2431:
2428:
2423:
2418:
2414:
2409:
2403:
2399:
2393:
2389:
2362:
2359:
2354:
2350:
2324:
2319:
2315:
2309:
2305:
2299:
2295:
2291:
2288:
2285:
2282:
2279:
2257:
2254:
2250:
2229:
2226:
2221:
2217:
2207:th component,
2190:
2186:
2161:
2158:
2155:
2133:
2129:
2123:
2120:
2116:
2110:
2106:
2102:
2097:
2093:
2087:
2083:
2056:
2052:
2030:
2026:
2023:
2016:
2011:
2007:
2001:
1997:
1991:
1988:
1984:
1978:
1974:
1968:
1964:
1960:
1957:
1954:
1951:
1948:
1913:
1910:
1904:
1903:
1894:
1892:
1879:
1873:
1863:
1856:
1845:
1838:
1826:
1790:
1787:
1773:
1770:
1757:
1754:
1647:
1593:
1541:
1533:
1507:
1506:
1497:
1495:
1479:
1476:
1468:
1462:
1441:
1440:
1431:
1429:
1408:
1401:
1390:
1383:
1371:
1350:
1349:
1340:
1338:
1322:
1313:
1306:
1295:
1288:
1275:
1272:
1263:
1223:
1220:
1215:
1212:
1153:
1150:
1147:
1144:
1131:
1128:
1107:
1104:
1101:
1098:
1085:is called the
1074:
1054:
1037:
1036:
1025:
1022:
1019:
1016:
1013:
1010:
1004:
1001:
967:
964:
961:
958:
955:
949:
946:
922:
921:
910:
906:
899:
896:
895:
889:
886:
881:
878:
872:
871:
865:
862:
857:
854:
848:
847:
841:
838:
833:
830:
824:
823:
820:
816:
810:
807:
802:
799:
793:
787:
784:
767:
766:
754:
747:
744:
743:
740:
737:
734:
731:
728:
727:
724:
721:
718:
715:
712:
711:
708:
705:
702:
699:
696:
695:
692:
688:
685:
682:
679:
676:
650:
647:
644:
641:
638:
635:
632:
629:
626:
597:
561:
525:
506:represent the
489:carbon dioxide
470:
431:
387:
351:
332:
331:
322:
320:
299:
291:
287:
279:
271:
262:
261:
249:
240:
237:
233:
225:
217:
208:
199:
194:
193:
159:) comprises a
152:
149:
120:
119:
116:
109:
55:
52:
22:is an area of
15:
9:
6:
4:
3:
2:
3075:
3064:
3061:
3060:
3058:
3049:
3046:
3045:
3033:
3027:
3025:
3017:
3011:
3002:
2993:
2986:
2980:
2973:
2970:(Chemistry)
2969:
2963:
2956:
2953:
2949:
2945:
2939:
2932:
2929:
2925:
2919:
2912:
2908:
2902:
2896::81–94, 2012.
2895:
2891:
2890:
2883:
2876:
2872:
2868:
2864:
2860:
2855:
2848:
2844:
2843:
2836:
2829:
2825:
2819:
2812:
2808:
2806:
2801:
2799:
2792:
2786:
2782:
2778:
2775:
2769:
2762:
2758:
2752:
2745:
2741:
2735:
2728:
2727:
2720:
2711:
2702:
2695:
2691:
2690:
2683:
2676:
2672:
2666:
2657:
2648:
2641:
2635:
2629:
2623:
2617:
2611:
2604:
2598:
2594:
2586:
2584:
2580:
2572:
2571:limiting step
2568:
2564:
2561:
2560:
2559:
2550:
2548:
2544:
2525:
2517:
2513:
2497:
2489:
2485:
2456:
2448:
2443:
2439:
2435:
2429:
2421:
2416:
2412:
2401:
2397:
2391:
2387:
2379:
2377:
2360:
2357:
2352:
2348:
2322:
2317:
2313:
2307:
2303:
2297:
2293:
2289:
2283:
2277:
2255:
2252:
2248:
2227:
2224:
2219:
2215:
2206:
2188:
2184:
2175:
2159:
2156:
2153:
2131:
2127:
2121:
2118:
2114:
2108:
2104:
2095:
2091:
2085:
2081:
2072:
2054:
2050:
2028:
2024:
2021:
2014:
2009:
2005:
1999:
1995:
1989:
1986:
1982:
1976:
1972:
1966:
1962:
1958:
1952:
1946:
1938:
1933:
1931:
1927:
1923:
1918:
1909:
1902:
1895:
1893:
1854:
1843:
1824:
1811:
1810:
1807:
1805:
1801:
1797:
1796:quasiperiodic
1786:
1783:
1779:
1769:
1767:
1763:
1762:deterministic
1753:
1645:
1591:
1539:
1531:
1516:
1514:
1505:
1498:
1496:
1477:
1474:
1460:
1447:
1446:
1439:
1432:
1430:
1399:
1388:
1369:
1356:
1355:
1348:
1341:
1339:
1320:
1304:
1293:
1273:
1270:
1261:
1247:
1246:
1243:
1241:
1240:autocatalysis
1237:
1232:
1230:
1219:
1211:
1209:
1205:
1201:
1197:
1192:
1190:
1186:
1182:
1178:
1177:autocatalysis
1174:
1169:
1167:
1148:
1142:
1127:
1125:
1121:
1102:
1096:
1088:
1045:
1042:
1020:
1014:
1008:
1002:
999:
989:
988:
987:
985:
981:
980:stoichiometry
962:
956:
953:
947:
944:
933:
930:
927:
908:
904:
897:
887:
884:
879:
876:
863:
860:
855:
852:
839:
836:
831:
828:
818:
814:
808:
805:
800:
797:
791:
785:
782:
772:
771:
770:
752:
745:
735:
729:
719:
713:
703:
697:
690:
686:
680:
674:
667:
666:
665:
662:
645:
639:
636:
630:
624:
595:
559:
523:
509:
508:concentration
500:
498:
494:
490:
468:
454:
429:
415:
407:
385:
371:
349:
330:
323:
321:
297:
277:
269:
247:
238:
223:
215:
206:
197:
184:
183:
180:
178:
174:
170:
166:
162:
158:
148:
145:
143:
139:
135:
134:
129:
125:
117:
114:
110:
107:
104:
103:
102:
100:
96:
92:
88:
83:
81:
78:reactions by
77:
73:
69:
65:
61:
51:
49:
45:
41:
37:
33:
29:
25:
21:
3030:A.N.Gorban,
3010:
3001:
2992:
2979:
2971:
2967:
2962:
2954:
2951:
2938:
2930:
2927:
2918:
2910:
2906:
2901:
2893:
2887:
2882:
2874:
2870:
2859:A. N. Gorban
2854:
2846:
2840:
2835:
2827:
2823:
2818:
2810:
2803:
2797:
2791:
2784:
2768:
2760:
2756:
2751:
2743:
2739:
2734:
2724:
2719:
2710:
2701:
2693:
2687:
2682:
2674:
2670:
2665:
2656:
2647:
2634:
2622:
2610:
2597:
2576:
2556:
2546:
2542:
2375:
2204:
2173:
2070:
1934:
1929:
1925:
1919:
1915:
1907:
1896:
1792:
1775:
1768:may appear.
1759:
1517:
1510:
1499:
1433:
1342:
1233:
1225:
1217:
1196:mass balance
1193:
1170:
1133:
1086:
1038:
983:
923:
768:
663:
501:
493:irreversible
335:
324:
169:intersecting
156:
154:
146:
131:
121:
84:
57:
36:biochemistry
19:
18:
2340:(where all
1937:free energy
1772:Persistence
1202:, constant
1200:temperature
1198:, constant
1181:mass action
95:Hinshelwood
87:van 't Hoff
2944:Yablonskii
2924:Yablonskii
2861:and G. S.
2677::81, 1972.
2589:References
1899:reaction 5
1766:attractors
1502:reaction 4
1436:reaction 3
1345:reaction 2
1229:biological
929:autonomous
926:continuous
327:reaction 1
179:reactions
177:combustion
106:van’t Hoff
50:involved.
2863:Yablonsky
2830::1, 2008.
2436:−
2388:∑
2294:∑
2249:β
2225:≥
2157:≤
2115:β
2105:∑
2101:→
2022:−
2015:∗
1990:
1963:∑
1862:⇀
1855:−
1844:−
1837:↽
1467:⟶
1407:⇀
1400:−
1389:−
1382:↽
1312:⇀
1305:−
1294:−
1287:↽
1173:catalysis
1073:Γ
1053:Γ
1012:Γ
1003:˙
948:˙
898:⋮
792:≡
786:˙
746:⋮
290:⟶
236:⟶
173:reactions
165:reactants
76:catalytic
3057:Category
2907:PLoS ONE
2777:Archived
2378:distance
2172:, where
2042:, where
1928:and the
1794:exhibit
1204:pressure
1124:kinetics
1041:constant
497:disjoint
370:hydrogen
151:Overview
32:chemical
2729:, 1989.
1800:chaotic
1782:species
661:, etc.
91:Semenov
54:History
2800:matrix
2240:, and
1518:Here,
1044:matrix
455:) and
414:carbon
408:) and
406:oxygen
2327:const
2146:(for
1939:form
1118:is a
453:water
28:model
2805:SIAM
2569:The
2358:>
1730:and
1632:and
1556:and
1189:Hill
1187:and
99:Aris
38:and
2972:242
2931:226
2894:736
2507:and
1798:or
1708:),
1664:),
1175:or
1164:is
510:of
372:),
163:of
161:set
157:CRN
3059::
3023:^
2950:,
2946:,
2875:66
2873:,
2869:,
2865:,
2847:59
2845:,
2828:44
2826:,
2811:67
2809:,
2783:,
2761:42
2759:,
2744:31
2742:,
2694:49
2692:,
2675:47
2673:,
1987:ln
1932:.
1875:BZ
1739:BZ
1717:AZ
1695:Pt
1642:CO
1619:CO
1610:,
1565:AB
1471:AB
1464:BZ
1457:AZ
1416:BZ
1325:AZ
1206:,
1183:,
1168:.
1126:.
1089:.
578:,
499:.
465:CO
294:CO
140:,
2955:4
2911:7
2909:,
2798:P
2547:c
2545:(
2543:M
2529:)
2526:t
2523:(
2518:2
2514:c
2501:)
2498:t
2495:(
2490:1
2486:c
2464:|
2460:)
2457:t
2454:(
2449:2
2444:i
2440:c
2433:)
2430:t
2427:(
2422:1
2417:i
2413:c
2408:|
2402:i
2398:m
2392:i
2376:L
2361:0
2353:i
2349:m
2323:=
2318:i
2314:c
2308:i
2304:m
2298:i
2290:=
2287:)
2284:c
2281:(
2278:M
2256:j
2253:k
2228:1
2220:k
2216:n
2205:i
2189:i
2185:A
2174:r
2160:r
2154:k
2132:j
2128:A
2122:j
2119:k
2109:j
2096:i
2092:A
2086:k
2082:n
2071:i
2055:i
2051:c
2029:)
2025:1
2010:i
2006:c
2000:i
1996:c
1983:(
1977:i
1973:c
1967:i
1959:=
1956:)
1953:c
1950:(
1947:G
1901:)
1897:(
1878:)
1872:(
1828:Z
1825:+
1821:B
1673:Z
1646:2
1592:2
1588:O
1543:B
1540:,
1532:2
1528:A
1504:)
1500:(
1482:Z
1478:2
1475:+
1461:+
1438:)
1434:(
1373:Z
1370:+
1366:B
1347:)
1343:(
1321:2
1278:Z
1274:2
1271:+
1262:2
1258:A
1152:)
1149:x
1146:(
1143:V
1106:)
1103:x
1100:(
1097:V
1024:)
1021:x
1018:(
1015:V
1009:=
1000:x
966:)
963:x
960:(
957:f
954:=
945:x
909:.
905:)
888:t
885:d
880:c
877:d
864:t
861:d
856:b
853:d
840:t
837:d
832:a
829:d
819:(
815:=
809:t
806:d
801:x
798:d
783:x
753:)
739:)
736:t
733:(
730:c
723:)
720:t
717:(
714:b
707:)
704:t
701:(
698:a
691:(
687:=
684:)
681:t
678:(
675:x
649:)
646:t
643:(
640:b
637:,
634:)
631:t
628:(
625:a
604:O
596:2
592:H
580:c
560:2
556:O
544:b
524:2
520:H
504:a
487:(
469:2
451:(
438:O
430:2
426:H
412:(
410:C
404:(
386:2
382:O
368:(
350:2
346:H
329:)
325:(
298:2
278:2
274:O
270:+
266:C
256:O
248:2
244:H
239:2
224:2
220:O
216:+
207:2
203:H
198:2
93:–
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