Nonequilibrium Thermodynamics of Closed and Open · PDF fileNonequilibrium Thermodynamics ......
Transcript of Nonequilibrium Thermodynamics of Closed and Open · PDF fileNonequilibrium Thermodynamics ......
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Nonequilibrium Thermodynamics Nonequilibrium Thermodynamics of Closed and Open Chemical Networksof Closed and Open Chemical Networks
Matteo Polettini and Massimiliano Esposito
Irreversible thermodynamics of open chemical networks I: Emergent cycles and broken conservation laws
J. Chem. Phys. 141, 024117 (2014)
Singapore, May 4, 2015
IntroductionIntroduction
Metabolic network reconstruction (Palsson, Thiele, Bonarius, Stephanopoulos, …):
Steady-state CN + Kirchhoff's law (Flux Balance Analysis & Metabolic Flux Analysis)
Energy balance analysis (Beard, Qian, …): Avoiding thermodynamically infeasible cycles
Elementary flux mode (Schuster, …):
Bottom up versus Top down
Our key assumptions:
- Elementary (reversible) reactions - Mass action kinetics
- Use of chemostats instead of fixed fluxes
Reminder: Chemical thermodynamics of a single reactionReminder: Chemical thermodynamics of a single reaction
Equilibrium:
Chemical potential:
Mass action law:
Entropy production:
Gibbs free energy:
stoichiometric matrix
Mass action law kinetics (elementary reactions)
Closed chemical networkClosed chemical network
where
Steady state: Equilibrium:
Example:
Kinetic constantsConcentrations
Conserved quantity
Null left eigenvectors of(cokernel)
Steady stateNull right eigenvectors of
(kernel)
rank-nullity theorem:
Species
Conservation laws (dim cokernel)
Reactions Cycles (dim kernel)
4 vector spaces: image (column vectors) and its orthogonal cokernel (left nul space), coimage (raw vectors) and its orthogonal kernel (right null space)
cycle current:
Mathematical properties of the stoichiometric matrix
Class of equilibrium states
Thermodynamic forces:
Chemical potential:
Equilibrium:
Entropy production:
Thermodynamics of closed CNThermodynamics of closed CN
Thermodynamic network independence: do not depend on
do depend on
Shannon entropy:
Lyapunov function
A closed network always relaxes to equilibrium
Open chemical networksOpen chemical networks
Variable species Chemostats External fluxes:
New network with effective rates
Example:
Cycles of the closed network belong to the kernel of
The reverse is not true, since there might exist vectors such that
Emergent cyclesEmergent cycles
At steady state:
Chemostating cannot decrease the number of cycles
Circulation of the force along cycles of the closed network vanishes:
Circulation of the force along emergent cycles yields nonnull (De Donder) affinities
Thermodynamics of open CNThermodynamics of open CN
New forces:
where
where
Entropy production:
Analogue of Hill and Schnakenberg decomposition
Steady state:
only depends on flux and affinities along emergent cycles
Steady-state EP:
Broken conservation laws and symmetriesBroken conservation laws and symmetries
if is a conservation law for is a conservation law for and is a conservation law of .
Chemostatting cannot increase the number of conservation laws
The converse is not always true
Balance of conserved quantities:
Mass conservation
# of independentaffinities
# of broken symmetries
Number of chemostats
Chemostatting: # of cycles cannot decrease and # of conservation laws cannot increase
rank-nullity theorem:
When chemostatting: fixedindependent
emergent cycles
It takes at least two chemostats to generate a nonequilibrium current
mass conservation law is always broken as the first chemostat is fixed
SummarySummary
Assumptions:
Elementary reactions satisfying mass action law + Chemostating by fixing concentrations
Closed CN:
A closed network always relaxes to an equilibrium steady-state where EP is zero.
Open CN:
Affinities are zero along cycles of the closed CN and non-zero only along emerging cycles:
Steady-state EP of an open CN is the sum of flux-affinity products over emerging cycles (analogue of the Schnakenberg decomposition for CN)
When a chemostat is created, either a symmetry is broken or a new cycle emerges:
# of affinities = # of chemostats - # of broken symmetries
Perspectives:
Coarse-graining, Aggregation dynamics, Oscillations, Chemical master equation, ...