Funding agency:

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

Example:

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

Emergent cycles

Example:

Old cycle

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:

Example:

Affinities Cycle currents

EPSteady-state

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

Example:

Two broken One created

A chemical network cannot be represented as a graph

Cycles are not cycles on a graph (hypergraphs)

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, ...