7 research outputs found
Polyhedral geometry and combinatorics of an autocatalytic ecosystem
Developing a mathematical understanding of autocatalysis in reaction networks
has both theoretical and practical implications. We review definitions of
autocatalytic networks and prove some properties for minimal autocatalytic
subnetworks (MASs). We show that it is possible to classify MASs in equivalence
classes, and develop mathematical results about their behavior. We also provide
linear-programming algorithms to exhaustively enumerate them and a scheme to
visualize their polyhedral geometry and combinatorics. We then define cluster
chemical reaction networks, a framework for coarse-graining real chemical
reactions with positive integer conservation laws. We find that the size of the
list of minimal autocatalytic subnetworks in a maximally connected cluster
chemical reaction network with one conservation law grows exponentially in the
number of species. We end our discussion with open questions concerning an
ecosystem of autocatalytic subnetworks and multidisciplinary opportunities for
future investigation.Comment: 36 pages, 17 figures, 7 table
Universal motifs and the diversity of autocatalytic systems
Autocatalysis is essential for the origin of life and chemical evolution. However, the lack of a unified framework so far prevents a systematic study of autocatalysis. Here, we derive, from basic principles, general stoichiometric conditions for catalysis and autocatalysis in chemical reaction networks. This allows for a classification of minimal autocatalytic motifs called cores. While all known autocatalytic systems indeed contain minimal motifs, the classification also reveals hitherto unidentified motifs.We further examine conditions for kinetic viability of such networks, which depends on the autocatalytic motifs they contain and is notably increased by internal catalytic cycles. Finally, we show how this framework extends the range of conceivable autocatalytic systems, by applying our stoichiometric and kinetic analysis to autocatalysis emerging from coupled compartments. The unified approach to autocatalysis presented in this work lays a foundation toward the building of a systems-level theory of chemical evolution
Intermediates, Catalysts, Persistence, and Boundary Steady States
For dynamical systems arising from chemical reaction networks, persistence is
the property that each species concentration remains positively bounded away
from zero, as long as species concentrations were all positive in the
beginning. We describe two graphical procedures for simplifying reaction
networks without breaking known necessary or sufficient conditions for
persistence, by iteratively removing so-called intermediates and catalysts from
the network. The procedures are easy to apply and, in many cases, lead to
highly simplified network structures, such as monomolecular networks. For
specific classes of reaction networks, we show that these conditions for
persistence are equivalent to one another. Furthermore, they can also be
characterized by easily checkable strong connectivity properties of a related
graph. In particular, this is the case for (conservative) monomolecular
networks, as well as cascades of a large class of post-translational
modification systems (of which the MAPK cascade and the -site futile cycle
are prominent examples). Since one of the aforementioned sufficient conditions
for persistence precludes the existence of boundary steady states, our method
also provides a graphical tool to check for that.Comment: The main result was made more general through a slightly different
approach. Accepted for publication in the Journal of Mathematical Biolog
Computational Complexity of Atomic Chemical Reaction Networks
Informally, a chemical reaction network is "atomic" if each reaction may be
interpreted as the rearrangement of indivisible units of matter. There are
several reasonable definitions formalizing this idea. We investigate the
computational complexity of deciding whether a given network is atomic
according to each of these definitions.
Our first definition, primitive atomic, which requires each reaction to
preserve the total number of atoms, is to shown to be equivalent to mass
conservation. Since it is known that it can be decided in polynomial time
whether a given chemical reaction network is mass-conserving, the equivalence
gives an efficient algorithm to decide primitive atomicity.
Another definition, subset atomic, further requires that all atoms are
species. We show that deciding whether a given network is subset atomic is in
, and the problem "is a network subset atomic with respect to a
given atom set" is strongly -.
A third definition, reachably atomic, studied by Adleman, Gopalkrishnan et
al., further requires that each species has a sequence of reactions splitting
it into its constituent atoms. We show that there is a to decide whether a given network is reachably atomic, improving
upon the result of Adleman et al. that the problem is . We
show that the reachability problem for reachably atomic networks is
-.
Finally, we demonstrate equivalence relationships between our definitions and
some special cases of another existing definition of atomicity due to Gnacadja
Unstable Cores are the source of instability in chemical reaction networks
In biochemical networks, complex dynamical features such as superlinear
growth and oscillations are classically considered a consequence of
autocatalysis. For the large class of parameter-rich kinetic models, which
includes Generalized Mass Action kinetics and Michaelis-Menten kinetics, we
show that certain submatrices of the stoichiometric matrix, so-called unstable
cores, are sufficient for a reaction network to admit instability and
potentially give rise to such complex dynamical behavior. The determinant of
the submatrix distinguishes unstable-positive feedbacks, with a single
real-positive eigenvalue, and unstable-negative feedbacks without real-positive
eigenvalues. Autocatalytic cores turn out to be exactly the unstable-positive
feedbacks that are Metzler matrices. Thus there are sources of dynamical
instability in chemical networks that are unrelated to autocatalysis. We use
such intuition to design non-autocatalytic biochemical networks with
superlinear growth and oscillations.Comment: 47 pages. Main text pp 1-14, Supplementary Information pp 15-47. 8
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