Detailed balance and complex balance: modern history of 130 year old laws

The history of detailed and semidetailed balanceThe Michaelis–Menten–Stueckelberg theorem

Let some equilibria go to zeroSummary

Detailed balance and complex balance:modern history of 130 year old laws

Alexander N. Gorban

Department of MathematicsUniversity of Leicester, UK

CRNT Workshop, Portsmouth, 2014

Alexander N. Gorban Detailed and complex balance



Outline

1 The history of detailed and semidetailed balanceBoltzmann’s discoveryTimeline of the main events

2 The Michaelis–Menten–Stueckelberg theoremAsymptotic assumptionsMichaelis–Menten versus Briggs–HaldaneThermodynamics of fast equilibriaGeneralized mass action law

3 Let some equilibria go to zero




Boltzmann’s discoveryTimeline of the main events

Outline








Boltzmann’s detailed balance (1872)

=

At equilibrium





Why detailed balance?

T is time reversal, P is space reflection

1 Dynamics is T - and P-invariant.2 Equilibrium is T - and P-invariant.3 PT transforms v + w → v ′ + w ′ into reverse collision

v ′ + w ′ → v + w .4 At equilibrium, the rate of v + w → v ′ + w ′ is equal to the

rate of v ′ + w ′ → v + w .

1⇒ 2 without spontaneous symmetry breaking; 2&3⇒ 4





Boltzmann’s cyclic balance (1887)(or semi-detailed balance or complex balance)

=

At equilibrium

.

.

.

.

.

.





Why cyclic balance?

v + w is a complex Θ.

At equilibrium, kinetics can be decomposed into a systemsof cycles Θ1 → Θn (n = 2, . . .) in such a way that eachcycle is equilibrated (in each cycle and for any complexcomplex the input rate is equal to the output rate).

This cyclic decomposition proves cyclic balance.





Outline








From Boltzmann to Wegscheider

In 1872 Boltzmann proved H-theorem on the basis ofdetailed balance;

In 1887 Lorentz stated the nonexistence of inversecollisions for polyatomic molecules and impossibility ofdetailed balance for them;

In the same issue of the “Sitzungsberichte der KaiserlichenAkademie” (1887) Boltzmann proposed the “cyclicbalance” condition and proved H-theorem for systems withcyclic balance;

In 1901 Wegscheider formulated detailed balance forchemical kinetics;





From Einstein to Tolman

In 1916 Einstein used detailed balance as the backgroundfor his quantum theory of radiation (Nobel prize);

In 1925 Lewis published in PNAS “A new principle ofequilibrium" with foundation of detailed balance;

In 1932 Onsager used detailed balance in his work aboutreciprocal relations (Nobel prize);

In 1938 Tolman published “The Principles of StatisticalMechanics” with analysis of Boltzmann’s detailed balance;





From Heitler to Cercignani

In 1944 Heitler analyzed Boltzmann’s cyclic balance and in1951 Coester proved Heitlers hypothesis about cyclicbalance (the “Heitler-Coestler theorem of semi-detailedbalance”);

In 1952 Stueckelberg demonstrated that cyclic balanceholds for Markov microkinetics under condition thatintermediates are in fast equilibrium and in small amount;

In 1972 Horn and Jackson rediscovered cyclic (complex)balance for chemical kinetics.

In 1981 Cercignani and Lampis demonstrated that theLorenz arguments are wrong and the new Boltzmannrelations are not needed, detailed balance should hold!




Asymptotic assumptionsMichaelis–Menten versus Briggs–HaldaneThermodynamics of fast equilibriaGeneralized mass action law

Outline








2n-tail scheme with intermediates —a starting point in the Stueckelberg asymptotic





Reversible, fast and small

Fast equilibria

Small amounts





Vocabulary

A1, . . . , An – components;B1, . . . , Bm – intermediate compounds;νji ≥ 0 (i = 1, . . . , n, j = 1, . . . , m) – stoichiometriccoefficients;Θj =

∑i νjiAi – complexes;

Θj � Bj – fast equilibria;Bi → Bj – intermediates’ transitions;ci ≥ 0 – concentration of Ai ;ςj ≥ 0 – concentration of Bj .





Michaelis–Menten trick (1913)

1 Equilibria are defined by thermodynamics;2 We do not know kinetics of nonlinear reactions Θ j � Bj but

we know their equilibria;3 If the reactions Θj � Bj are fast then we do not need

kinetics for these reactions and thermodynamics gives theequilibria;

4 If all ςj are small then only linear reactions with Bj shouldbe taken into account and their rates are linear in ς j .

Behind this trick we can imagine a general system of kineticequations which has a Lyapunov function (free energy) andincludes some small parameters.





Outline








Briggs–Haldane (1925)

The famous Michelis–Menten formula was produced byBriggs & Haldane (1925).

Briggs and Haldane considered the simplest enzymereaction S + E � SE → P + E .

They assumed mass action law kinetics for all elementaryreactions.

They mentioned that the total concentration of enzyme([E ] + [SE ]) is “negligibly small” compared with theconcentration of substrate [S].

After that they produced the now famous‘Michaelis–Menten’ formula for the rate of thebrutto–reaction S + E � P + E .





Michaelis–Menten (1913)

Michaelis & Menten considered the fully reversiblemechanisms like S + E � B1 � B2 � P + E .

They did not assume any kinetic formula for non-linearreactions.

They assumed that equilibria S + E � B1 and B2 � P + Eare fast, and concentrations of compounds B1,2 are small.

They produced mass action law kinetics for thebrutto–reaction S + E � P + E .





Outline








Free energy

The basic hypothesis is that the compounds are the smalladmixtures to the system,

That is, the amount of compounds Bj is much smaller thanamount of initial components Ai .

Take the energy of their interaction with Ai in the linearapproximation in concentrations of Bj ,

Use the perfect entropy for Bi .

Under these standard assumptions for a small admixturesget the free energy:

F = Vf (c, T ) + VRTq∑

j=1

ςj

(uj(c, T )

RT+ ln ςj − 1

)





Standard equilibrium

The standard equilibrium concentration for Bj is by definition:

ς∗j (c, T ) = exp(−uj(c, T )

RT

)

It is defined because of Boltzmann distribution ( 1Z exp(−u/RT ))

with unite normalization factor (statistical sum) Z = 1. With thisnotation

F = Vf (c, T ) + VRTq∑

j=1

ςj

(ln

(ςj

ς∗j (c, T )

)− 1

)





Fast equilibria

Use the formula for free energy to define the fast equilibriaΘj � Bj .

Find the concentration of Bi , ςi as a function of theconcentrations ci of Ai .

Such an equilibrium is the minimizer of the free energy onthe straight line parameterized by a: ci = c0

i − aνji , ςj = a.

ςj = ς∗j (c, T ) exp(∑

i νjiμi(c, T )

RT

)+ o(ς)

where μi = ∂f (c,T )∂ci

is the chemical potential of Ai .





Outline








Transformation of compounds

Linear kinetics of compounds is just a Markov chain (masterequation)

dςj

dt=∑l , l �=j

(κjl ςl − κljςj

)It should be in agreement with thermodynamics: the standardequilibrium should be a kinetic equilibrium:

∑l , l �=j

κjl ς∗l =

∑l , l �=j

κlj ς∗j





Quasiequilibrium elimination of compounds

In the quasiequilibrium approximation the rate of thebrutto–process Θj → Θl is

wlj = κlj ςj

In the same approximation

wlj = κlj ς∗j (c, T ) exp

(∑i νjiμi(c, T )

RT

)= ϕlj(c, T )Ωlj(c, t)

where ϕlj is the kinetic factor and Ωlj is the Boltzmann factor

Ωlj = exp(∑

i νjiμi(c, T )

RT

), ϕlj = κlj ς

∗j (c, T )





Complex balance

The complex balance condition appears as a trivial re-writing ofthe condition that the Boltzmann equilibrium of compounds ς ∗

should be an equilibrium of Markov kinetics:

∑l

ϕlj =∑

l

ϕjl

This condition is sufficient for the thermodynamic behavior ofthe system because the perfect entropy is the Lyapunovfunction for Markov kinetics.





Restart of the formalism: GMAL+complex balance

Now, we can:

“Forget” about the compounds (temporarily, at least);

Start from the formulas for wlj , the Boltzmann factor Ωlj andthe system of non-negative kinetic factors ϕjl ;

Kinetic factors should satisfy the complex balancecondition.

This asymptotic representation of the kinetics that goes throughintermediates, which (1) are in small amounts and (2) persist infast equilibria with the components is the

Michaelis–Menten–Stuekelberg theorem.




A simple cycle with detailed balance

Consider a simple cycle

A1k1�

k−1

A2k2�

k−2

A3k3�

k−3

A1

with equilibrium ceq = (ceq1 , ceq

2 , ceq3 ) and detailed balance:

kiceqi = k−i c

eqi+1

(here, c3+1 = c1). The perfect free energy

F =∑

i

RTVci

(ln(

ci

ceqi

)− 1)

.




The classical thermodynamics has no limit but thekinetic equations have

Let ceq1 → 0 for given ceq

2,3 > 0 and fixed k1, k±2, k−3 > 0.

The limit kinetic system exists

A1k1→A2

k2�k−2

A3 ←k−3

A1 .

The limit free energy does not exist.

Two questions arise:1 Which system appear as the limits of systems with detailed

balance when some reaction rate constants tend to zero?2 Do these systems have Lyapunov functions?




The extended principle of detailed balance

A system which obey the generalized mass action law is thelimits of the systems with detailed balance when some of thereaction rate constants tend to zero if and only if:

The principle of detailed balance is valid for thereversible part. (This means that for the set of all reversiblereactions there exists a positive equilibrium where all theelementary reactions are equilibrated by their reverse reactions.)

The convex hull of the stoichiometric vectors of theirreversible reactions has empty intersection with thelinear span of the stoichiometric vectors of thereversible reactions. (Physically, this means that theirreversible reactions cannot be included in oriented cyclicpathways.)

The Lyapunov functions have been constructed too.Alexander N. Gorban Detailed and complex balance



Example: H2+O2 system

Consider a standard mechanism with 8 components H2, O2,OH, H2O, H, O, HO2, and H2O2 and 20 elementary reactions.Some of them are “almost irreversible”. Consider theirirreversible limits.Let reactions H2 + O2 � 2OH, H2 + OH � H2O + H, H2 + O �OH + H, H2O2 � 2OH, H + H2O2 � H2 + HO2, OH + H2O2 �H2O + HO2 be reversible. Find the maximal sets of reactionswhich may be irreversible.There are two maximal sets, the following one and the set ofreverse reactions:OH + HO2 → O2 + H2O, H + HO2 → 2OH, O + HO2 → O2 +OH, 2H→ H2, OH + H→ H2O, H + O→ OH, 2O→ O2, H +HO2 → H2 + O2, 2HO2 → O2 + H2O2.




Main message 1

Boltzman introduced detailed (1872) and complex (1887)balance:

=

At equilibrium

;

=

At equilibrium

.

.

.

.

.

.




Main message 2

The generalized mass action law with the complex balancecondition appear in the limit

Fast equilibria

Small amounts




Main message 3

There exists extended detailed balance when some reactionsbecome irreversible. It consists of two parts:

The principle of detailed balance is valid for the reversiblepart.

The convex hull of the stoichiometric vectors of theirreversible reactions has empty intersection with the linearspan of the stoichiometric vectors of the reversiblereactions.




References: some history

Boltzmann, L. Weitere Studien über dasWärmegleichgewicht unter Gasmolekülen. Sitzungsber.keis. Akad. Wiss. 1872, 66, 275–370. Translation: InKinetic Theory of Gases; Imperial Colledge Press: London,UK, 2003.

Boltzmann, L. Neuer Beweis zweier Sätze über dasWärmegleichgewicht unter mehratomigen Gasmolekülen.Sitzungsberichte der Kaiserlichen Akademie derWissenschaften in Wien. 1887, 95, 153–164.

Wegscheider, R. Über simultane Gleichgewichte und dieBeziehungen zwischen Thermodynamik undReactionskinetik homogener Systeme, Monatshefte fürChemie / Chemical Monthly 32(8) (1901), 849–906.




References: some history

Lewis, G.N. A new principle of equilibrium, PNAS March 1,1925 vol. 11 no. 3 179-183.Coester, F. Principle of detailed balance. Phys. Rev. 1951,84, 1259.Stueckelberg, E.C.G. Theoreme H et unitarite de S. Helv.Phys. Acta 1952, 25, 577–580.

Horn, F.; Jackson, R. General mass action kinetics. Arch.Ration. Mech. Anal. 1972, 47, 81–116.

Feinberg, M. Complex balancing in general kineticsystems. Arch. Ration. Mech. Anal. 1972, 49, 187–194.Yablonskii, G.S.; Bykov, V.I.; Gorban, A.N.; Elokhin, V.I.Kinetic Models of Catalytic Reactions; Elsevier:Amsterdam, The Netherlands, 1991.




References

Gorban, A.N.; Shahzad, M. The Michaelis – Menten –Stueckelberg Theorem. Entropy 2011, 13, 966–1019.Corrected postprint: arXiv:1008.3296 [physics.chem-ph]

A.N. Gorban, G.S. Yablonskii, Extended detailed balancefor systems with irreversible reactions, Chem. Eng. Sci. 662011 5388–5399; arXiv:1101.5280 [cond-mat.stat-mech].

D. Grigoriev, P.D. Milman, Nash resolution for binomialvarieties as Euclidean division. A priori termination bound,polynomial complexity in essential dimension 2, Advancesin Mathematics, 231 (6) 2012, 3389–3428.




References

A.N. Gorban, E.M. Mirkes, G.S. Yablonskii,Thermodynamics in the Limit of Irreversible Reactions,Physica A 392 2013 1318–1335, arXiv:1207.2507[cond-mat.stat-mech]

A. N. Gorban, General H-theorem and entropies thatviolate the second law, Entropy 2014, 16(5), 2408–2432,Extended postprint arXiv:1212.6767 [cond-mat.stat-mech].

Gorban, A.N., Sargsyan, H.P., and Wahab, H.A.Quasichemical Models of Multicomponent NonlinearDiffusion, Mathematical Modelling of Natural Phenomena,Volume 6 / Issue 05, 2011, 184-262.


Education

Detailed balance and complex balance: modern history of 130 year old laws