135 research outputs found
Open Markov processes: A compositional perspective on non-equilibrium steady states in biology
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 this type, even when the rates of the
underlying process are such that a detailed balanced equilibrium is permitted.
It is shown that these non-equilibrium steady states minimize a quadratic form
which we call `dissipation.' In some circumstances, the dissipation is
approximately equal to the rate of change of relative entropy plus a correction
term. On the other hand, Prigogine's principle of minimum entropy production
generally fails for non-equilibrium steady states. We use a simple model of
membrane transport to illustrate these concepts
Relative Entropy in Biological Systems
In this paper we review various information-theoretic characterizations of
the approach to equilibrium in biological systems. The replicator equation,
evolutionary game theory, Markov processes and chemical reaction networks all
describe the dynamics of a population or probability distribution. Under
suitable assumptions, the distribution will approach an equilibrium with the
passage of time. Relative entropy - that is, the Kullback--Leibler divergence,
or various generalizations of this - provides a quantitative measure of how far
from equilibrium the system is. We explain various theorems that give
conditions under which relative entropy is nonincreasing. In biochemical
applications these results can be seen as versions of the Second Law of
Thermodynamics, stating that free energy can never increase with the passage of
time. In ecological applications, they make precise the notion that a
population gains information from its environment as it approaches equilibrium.Comment: 20 page
Open Markov Processes and Reaction Networks
We define the concept of an `open' Markov process, a continuous-time Markov
chain equipped with specified boundary states through which probability can
flow in and out of the system. External couplings which fix the probabilities
of boundary states induce non-equilibrium steady states characterized by
non-zero probability currents flowing through the system. We show that these
non-equilibrium steady states minimize a quadratic form which we call
`dissipation.' This is closely related to Prigogine's principle of minimum
entropy production. We bound the rate of change of the entropy of a driven
non-equilibrium steady state relative to the underlying equilibrium state in
terms of the flow of probability through the boundary of the process.
We then consider open Markov processes as morphisms in a symmetric monoidal
category by splitting up their boundary states into certain sets of `inputs'
and `outputs.' Composition corresponds to gluing the outputs of one such open
Markov process onto the inputs of another so that the probability flowing out
of the first process is equal to the probability flowing into the second. We
construct a `black-box' functor characterizing the behavior of an open Markov
process in terms of the space of possible steady state probabilities and
probability currents along the boundary. The fact that this is a functor means
that the behavior of a composite open Markov process can be computed by
composing the behaviors of the open Markov processes from which it is composed.
We prove a similar black-boxing theorem for reaction networks whose dynamics
are given by the non-linear rate equation. Along the way we describe a more
general category of open dynamical systems where composition corresponds to
gluing together open dynamical systems.Comment: 140 pages, University of California Riverside PhD Dissertatio
Network Models
Networks can be combined in various ways, such as overlaying one on top of
another or setting two side by side. We introduce "network models" to encode
these ways of combining networks. Different network models describe different
kinds of networks. We show that each network model gives rise to an operad,
whose operations are ways of assembling a network of the given kind from
smaller parts. Such operads, and their algebras, can serve as tools for
designing networks. Technically, a network model is a lax symmetric monoidal
functor from the free symmetric monoidal category on some set to
, and the construction of the corresponding operad proceeds via a
symmetric monoidal version of the Grothendieck construction.Comment: 46 page
- …