Open Markov Processes: A Compositional Perspective on Non-Equilibrium Steady States in Biology

@article{Pollard2016OpenMP,
  title={Open Markov Processes: A Compositional Perspective on Non-Equilibrium Steady States in Biology},
  author={Blake S. Pollard},
  journal={Entropy},
  year={2016},
  volume={18},
  pages={140}
}
In recent work, Baez, Fong and the author introduced a framework for describing Markov processes equipped with a detailed balanced equilibrium as open systems of a certain type. These `open Markov processes' serve as the building blocks for more complicated processes. In this paper, we describe the potential application of this framework in the modeling of biological systems as open systems maintained away from equilibrium. We show that non-equilibrium steady states emerge in open systems of… 

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References

SHOWING 1-10 OF 30 REFERENCES

A compositional framework for Markov processes

It is proved that black boxing gives a symmetric monoidal dagger functor sending open detailed balanced Markov processes to open circuits made of linear resistors, and described how to “black box” an open Markov process.

Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws.

The steady state entropy production rate is decompose in terms of fundamental fluxes and affinities in the spirit of Schnakenberg's theory of network thermodynamics, paving the way for the forthcoming treatment of the linear regime, of efficiency and tight coupling, of free energy transduction, and of thermodynamic constraints for network reconstruction.

The Chemical Master Equation Approach to Nonequilibrium Steady-State of Open Biochemical Systems: Linear Single-Molecule Enzyme Kinetics and Nonlinear Biochemical Reaction Networks

The stochastic, chemical master equation is developed as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment and it is suggested that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting.

Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium.

Open-system nonequilibrium steady state: statistical thermodynamics, fluctuations, and chemical oscillations.

  • H. Qian
  • Physics, Chemistry
    The journal of physical chemistry. B
  • 2006
A nonequilibrium statistical thermodynamic theory based on stochastic kinetics is introduced, mainly through a series of examples: single-molecule enzyme kinetics, nonlinear chemical oscillation, molecular motor, biochemical switch, and specificity amplification.

Network theory of microscopic and macroscopic behavior of master equation systems

A general microscopic and macroscopic theory is developed for systems which are governed by a (linear) master equation. The theory is based on a network representation of the master equation, and the

Inadequacy of entropy and entropy derivatives in characterizing the steady state

In bistable systems the transition kinetics between the two locally stable states can be altered without changing the behavior in the immediate vicinity of the two favored steady states. It follows

Thermodynamic network analysis of biological systems

1. Introduction.- 2. Models.- 2.1 The Purpose and Nature of Models.- 2.2 Enzyme-Catalyzed Reactions and the Michaelis-Menten Kinetics.- 2.3 Transport Across Membranes: A Black-Box Approach.- 2.4

Stability and entropy production in electrical circuits

A number of the theorems expounded by Prigogine, Glansdorff and their collaborators are translated into electrical circuit terminology and their validity and significance discussed. The simultaneous

Non-equilibrium entropy and irreversibility

1. Introduction and Summary.- 2. Dynamics and Work.- 3. Information Entropy.- 3.a Entropy and relative entropy.- 3.b Gibbs states.- 3.c Entropy-increasing processes.- 4. Heat Baths.- 5. Reversible