Mesoscopic Kinetic Basis of Macroscopic Chemical Thermodynamics: A Mathematical Theory

0.09 MB·

by  Hao Ge

Mesoscopic Kinetic Basis of Macroscopic
Chemical Thermodynamics: A Mathematical
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0 Theory
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Hao Ge1∗and Hong Qian2†
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1 1Beijing International Center for Mathematical Research (BICMR)

Biodynamic Optical Imaging Center (BIOPIC)
]
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Peking University, Beijing 100871, P.R.C.
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m 2Department of Applied Mathematics, University of Washington
e Seattle, WA 98195-3925, U.S.A
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c January 14, 2016
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Abstract

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v From a mathematical model that describes a complex chemical kinetic
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system of N species and M elementrary reactions in a rapidly stirred ves-
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1 sel of size V as a Markov process, we show that a macroscopic chemical
3 thermodynamics emerges as V → ∞. The theory is applicable to linear
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and nonlinear reactions, closed systems reaching chemical equilibrium, or
.
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open, driven systems approaching to nonequilibrium steady states. A gen-
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6 eralized mesoscopic free energy gives rise to a macroscopic chemical en-
1 ergy function ϕss(x) where x = (x ,··· ,x ) are the concentrations of
1 N
v: theN chemical species. Themacroscopic chemicaldynamics x(t)satisfies
Xi two emergent laws: (1) (d/dt)ϕss[x(t)] ≤ 0; and (2) (d/dt)ϕss[x(t)] =
cmf(x) − σ(x) where entropy production rate σ ≥ 0 represents the sink
r
a for the chemical energy, and chemical motive force cmf ≥ 0 is non-zero
if the system is driven under a sustained nonequilibrium chemostat. For
systems with detailed balance cmf = 0; and if one assumes the law of
mass action, ϕss(x) is precisely the Gibbs’ function N x µo + lnx
i=1 i i i
P (cid:2) (cid:3)
∗[email protected]
†[email protected]
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for ideal solutions. For aclass of kinetic systems called complex balanced,
whichincludemanynonlinear systemsaswellasmanysimpleopen,driven
chemical systems, the ϕss(x), with global minimum at x∗, has the generic
form N x ln(x /x∗)−x +x∗ ,whichhasbeenknowninchemicalki-
i=1 i i i i i
neticlPiterature(cid:2). Macroscopicemerg(cid:3)ent“laws”areindependentofthedetails
of the underlying kinetics. This theory provides a concrete example from
chemistry showing how a dynamic macroscopic law can emerge from the
kinetics atalevelbelow.
Ever since the work of J. W. Gibbs and the influential treatise of Lewis and
Randall [1], chemical thermodynamics has been one of the most important the-
oretical cornerstones of chemical science. While temperature is one of the key
concepts in the theory, its origin resides in the mechanical movement of atoms
and molecules, as already clearly articulated by L. Boltzmann in his mechanical
theory of heat [2]. The notion of chemical potential, however, can not be under-
stoodfrom Newtonianmechanics. Thisisbest illustratedthroughits definitionin
Gibbs’equation:
n
dU = TdS −pdV + dµ N , (1)
i i
Xi=1
in which
∂U ∂U ∂U
T = , p = − , µ = . (2)
i
(cid:18)∂S(cid:19) (cid:18)∂V (cid:19) (cid:18)∂N (cid:19)
V,{Ni} S,{Ni} i S,V,{Nj,j6=i}
In graduate texts on Newtonian mechanics, there is a demonstration of that [3]
∂U/∂S is the mean kinetic energy if one identifies the S ≡ k lnΩ as the phase
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volume of a Hamiltonian dynamics; and −∂U/∂V as the momentum transfer on
the wall of a box that contains gas particles. While these demonstrations are not
general,theyhaveprovideddefinitivemechanicalinterpretationsofthetwoemer-
gent thermodynamicquantities. On the contrary, there is no mechanical interpre-
tationforthe∂U/∂N in(2). Itiswidelyfeltthata“mechanistic”interpretationof
the chemical potentialµ has to bea probabilisticone. We hasten to mentionthat
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a living organism is sustained as a nonequilibriumsystem neither by a difference
in T nor p, rather it is a phenomenon driven by ∆µ. Therefore, a fundamental
understanding of the irreversibility of life requires a deeper understanding of µ,
which isstilllackingarigoroussub-macroscopicfoundation.
Thedescriptionofthe mathematicalmodel. We report inthisnotearecently
discovered mathematical result that provides the chemical potential µ, as a emer-
gent macroscopic quantity, an origin in a mesoscopic description of chemical ki-
netics. The mesoscopic description of a chemical kinetics is based on a system
of elementary chemical reactions with arbitrary complexity; reactions occur one
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at a time in a stochastic fashion, as now clearly demonstrated in single-molecule
studies[4]. ThegeneralsettinghasN chemicalspeciesandM reactionsinafixed
volumeofV [5]:
ν+X +ν+X +···ν+ X FGGkGG+GGBGℓG ν−X +ν−X +···ν− X , (3)
ℓ1 1 ℓ2 2 ℓN N ℓ1 1 ℓ2 2 ℓN N
k−ℓ
in which1 ≤ ℓ ≤ M. ν = (ν− −ν+) are thestoichiometriccoefficientsrelating
ij ij ij
species to reactions. In a reaction vessel with rapidly stirred chemical solutions,
thenumbersofspecies iat timetisdenotedby n (t). Ourtheory assumesthat
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(i) Each reaction is microscopically reversible, with forward rate r (n(t))
+ℓ
andbackwardrater (n(t))wheren = (n ,··· ,n )denotingthecopynumbers
−ℓ 1 N
of all the species. Both r are mathematically non-negative; but their depen-
±ℓ
dences on n are essentailly arbitrary (with some minor mathematical assump-
tions).
(ii) As an elementrary reaction, each occurs as a Markov process with expo-
nentialwaitingtimefollowingthedistributionr(n)e−r(n)t.
(iii) The mesoscopic rate r (n) is the number of occurrences of the ℓth for-
+ℓ
wardreactionperunittimeinthevolumeV. Therefore,foramacroscopicsystem
with extremely large n and V: (x ,··· ,x ) ≡ x = n/V are the concentrations,
1 N
and R (x) = r (Vx)/V are theconcentration-based rates ofthereactions.
±ℓ ±ℓ
(iv) The chemical reaction system can be either closed or open. A closed
system has no exchange of matter with its surrounding; an open system can ex-
change various chemical species with its surrounding, which are kept at constant
concentrations. If all the “externally buffered” species are themselves at a chem-
ical equilibrium, the situation is like a dialysis system, which ultimately reach
a chemical equilibrium both within and with its surrounding. If, however, there
are at least two species that are sustained at a nonequilibrium condition, then the
system eventually settles in a nonequilibrium state with stationary concentration
fluctuations. This last scenario is the biochemical kinetic setup for modeling a
livingcells underacontinuouslychemostat.
Withtheserathergeneralassumptions,T.G.Kurtzhasshownin1972[6]that
in the limit of V → ∞, the mesoscopic stochastic description of the system of
chemical reactions becomes the following set of rate equations for the macro-
scopickinetics:
M
dx (t)
i = ν R (x)−R (x) , (4)
ℓi +ℓ −ℓ
dt
X (cid:16) (cid:17)
ℓ=1
1 ≤ i ≤ N. Kurtz’s theorem paves the way for a unified mathematical theory of
chemical kineticsinarapidlystirred vesselofbothsmallandlarge size.
BasedonsuchMarkovprocesses,amesoscopicstatistical(orstochastic)ther-
modynamic theory has been developed in recent years in the field of nonequi-
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librium statistical physics. The most celebrated results from this theory is the
Jarzynski-Crooks equality [7, 8]. The mesoscopic theory also, for the first time,
demonstrated a free energy blanace equation: One can introduce a generalized,
nonequilibriumfreeenergyF(meso)foranychemicalreactionsystemsatthemeso-
scopic level. Then it can be mathematically shown that this F(meso) satisfied an
instantaneousbalancerelation[9, 10, 11]:
dF(meso)
= E −e , (5)
in p
dt
in which both E and e are non-negative, and dF(meso)/dt is never positive.
in p
When applied to the chemomechanics of a single ATPase motor protein, the E
in
istheamountofchemicalenergyinputperunittime,e.g.,ATPhydrolysis,ande
p
istheminimalamountofheatdissipation[12]. Ifthemotorisactuallyperforming
mechanical work against an external elastic force fext and moving with velocity
v, thenbothE ande containsthemechanicalpowerf(ext)v.
in p
Foraclosedchemicalreactionsystem,oranopensystemincontactwithasin-
gleexternalchemicalpotentialµext,itcanbeshownthatE = 0. Inthiscase,the
in
F(meso) isindeedthefree energy ofthemolecularsystem[13, 14]. Actually,with
the help of detailed balance, the stochastic Markov theory and Gibbs’ canonical
and grand canonical emsembletheories areprovenequivalent.
The kinetic theory of macroscopic chemical (free) energy. We summarize
the new mathematical results [15, 16]. So far, the equation is only establised in
the mesoscopic setting. So all three non-negative quantities, entropy production
e , energy input E , and free energy dissipation −dF(meso)/dt are functions of
p in
the volume parameter V. Furthermore, they are functions of the probability dis-
tribution of the number of each species at time t, p (n,t), which itself is also a
V
functionofV.
Now if we perform the limit of V → ∞, Kurtz’s theorem tells us that x(t) =
n(t)/V is the solution to the nonlinear rate equation (4). Furthermore, the prob-
ability theory also suggests that for a wide class of models, p (Vx,t) can be
V
written as exp −Vϕ(x,t) , and in thestationary statea ϕss(x) emerges, which
satisfies thefol(cid:0)lowingequat(cid:1)ion,derivedindependentlyby Kurtzin 1978[17]and
G. Hu in1986[18]
M
ν ν
R (x) 1−e ℓ·∇xϕss(x) +R (x) 1−e− ℓ·∇xϕss(x) = 0, (6)
+ℓ −ℓ
Xℓ=1 h i h i
where ν = (ν ,··· ,ν ). We have shown that in the “macroscopic limit”, as
1 M
intensivequantities
V−1F(meso) → ϕss(x), (7a)
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dϕss(x)
= cmf(x)−σ(x), (7b)
dt
M R (x)
V−1e → σ(x) = R (x)−R (x) ln +ℓ , (7c)
p +ℓ −ℓ (cid:18)R (x)(cid:19)
Xℓ=1 (cid:16) (cid:17) −ℓ
M R (x) ν
V−1E → cmf(x) = R (x)−R (x) ln +ℓ e ·∇xϕss(x) . (7d)
in +ℓ −ℓ (cid:18)R (x) (cid:19)
Xℓ=1 (cid:16) (cid:17) −ℓ
Amacroscopicchemicalenergyfunctionϕss(x)emerges. Forthestationarystate
with large but finite V, the probability distribution for concentration fluctuations
attainsa universalexpression
Ω(x)e−Vϕss(x)
f(x|V) = , where Ξ(V) = Ω(x)e−Vϕss(x)dx, (8)
Ξ(V) Z
inwhichΩ(x)isanalogoustothe“degeneracy”inapartitionfunctioncalculation.
As a part of the mathematical theory of large deviations in a probability distribu-
tion, Eq. 8 and alike have been considered as the mathematical foundation of
equilibrium statistical thermodynamics [19]. Our present work is an application
ofadynamicversionofsuch mathematics.
What is the mechanistic force corresponding to the emergent energy function
ϕss(x)? Morespecifically,howdoesthis“force”affectthemacroscopickinetics?
First, one needs to consciously recognize the vast separation of timescales in the
mesoscopic and macroscopic kinetics. That is why the dynamics of the latter is
partially dictated by the stationary behavior of the former, in the form of ϕss(x).
It is clear that ϕss(x) is a consequence of a global, infinitely long time behavior
ofthemesoscopicsystem.
Eq. 7b isamacroscopicchemical(free) energy balanceequation,in whichan
theemergentchemicalmotiveforce(cmf)characterizestheforcetheenvironment
puts upon the kinetic system, and entropy production rate σ(x) characterized the
amountoffree energy thatis dissipatedfrom thesystem.
Detailed balance and chemical equilibrium. If this global dynamic conse-
quence is in completeconsistencywith thelocal kineticln R (x)/R (x) , we
+ℓ −ℓ
say thechemicalkineticsisin “equilibrium”: (cid:2) (cid:3)
R (x)
ln(cid:18)R+ℓ(x)(cid:19) = exp −νℓ ·∇xϕexq(x) , ∀x. (9)
−ℓ (cid:16) (cid:17)
Thencmf(x) = 0forallx,andviceversa. Eq. (9)formalizesthefundamentalin-
sightsofG.N.Lewisontheimportantaceofdetailedbalanceinchemicalkinetics
[20]. Inthiscasethereisachemicalequilibriumbetweenthelocalkineticsandits
environmentthatiscreatedbytheotherreactionsinthesamekineticsystem. Ifwe
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furtherassumetheLawofMassAction,thenitcanbeprovenmathematicallythat
ϕ(x) is actually the Gibbs function G(x)/k T = N x µo +k T lnx , fur-
B i=1 i i B i
thermoreEq. 9isequivalenttoR+ℓ(xeq) = R−ℓ(xePq),∀ℓ,w(cid:0)herexeq istheu(cid:1)nique
minimumofG(x). Then Eq. 7b becomes ddGt = −σ(x) ≤ 0; and νℓ ·∇xG(x) =
∆µ (x) = k T ln R (x)/R (x) ,∆µo = N ν µo = k T ln(k /k ).
ℓ B −ℓ +ℓ ℓ i=1 ℓi i B −ℓ +ℓ
WhenEq. 9isn(cid:2)othold,whichis(cid:3)equivalenPttosaythatcmf 6= 0,thentheopen,
driven kineticsystemeventuallysettles into a nonequilibriumsteady state xss (or
somemorecomplexbeheviorslikeoscillations,)withcmf(xss) = σ(xss) > 0.
Kinetics with complex balance. The emergence of ϕss(x) given in Eq. 6 is
highly abstract. One example of this, thanks to the recent work of Anderson et.
al [21], is when the kinetic system is complex balanced, a notion introduced by
Horn and Jackson in 1972[22]. Thisis a class ofmodelswhich contains detailed
balance, all unimolecular reaction networks, as well as many open, driven, non-
linear chemical systems. In this case, it can be shown that the kinetics equation
(4)has auniquesteady statexss and
N
x
ϕss(x) = x ln i −x +xss, (10)
i (cid:18)xss(cid:19) i i
Xi=1 i
whichisasolutionto(6),andthelimitof−V−1lnpss(Vx)whenV tendsinfinity.
V
We note that for this large class of chemical kinetics, linear and nonlinear,
closedandopen,equilibriumandnonequilibirum,thereisageneric,universalex-
pression for the chemical energy ϕss(x), Eq. 10, which has aleady been in the
chemical literature. This illustrates the important idea that macroscopic emer-
gent behavior, such as thermodynamics,should be independent of the underlying
detailsofthekinetics.
Discussion. In summary, macroscopic chemical thermodynamics can have
a rigous mesoscopic, statistical, reaction kinetic foundation. Chemical free en-
ergy, which is a generalization of Gibbs’ equilibrium free energy, is an emergent
quantity in the macroscopic limit. It actually has a free energy balance equation
which is different from Newtonian mechanical energy conservation as well as
Helmholtz-Boltzmann’sderivationoftheFirstLawofThermodynamicsbasedon
their mechanical theory of heat. This free energy balance equation is applicable
to closed and driven chemical reaction kinetic systems under isothermal condi-
tions. This chemical theory of reaction kinetics also provides a concrete example
forP.W.Anderson’sstructureofscientificlaws[23]: Macroscopiclawsareemer-
gent behaviors from the dynamics of a level below; such laws are insensitive to
a large extend the details of the dynamics. It also provides us a tantalizing pos-
sibility for high-energy theoretical physics: The universe is made of particles of
many kinds and types; the laws that governs their creation, annihilation, trans-
formation, and interactions are the fundamental theory of quantum world. Yet,
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at the macroscopic, cosmological scale, one expects emergent laws that govern
its macroscopic dynamics in terms of a “mysterious” force. Such a force should
satisfy an equation like our Eq. 6. Currently, there is an active research program
in mathematicsthat studies this typeof nonlinearequations [24] that includes (6)
as well as Ricci flow,themathematicalstructureofEinstein’sspace-time.
Perhaps, chemical science, in addition to providing the livingworld with tan-
gible materials and useful energy, can also offer some fundamental insights on
howtheuniverseworks.
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