33 research outputs found
Inferring Biologically Relevant Models: Nested Canalyzing Functions
Inferring dynamic biochemical networks is one of the main challenges in
systems biology. Given experimental data, the objective is to identify the
rules of interaction among the different entities of the network. However, the
number of possible models fitting the available data is huge and identifying a
biologically relevant model is of great interest. Nested canalyzing functions,
where variables in a given order dominate the function, have recently been
proposed as a framework for modeling gene regulatory networks. Previously we
described this class of functions as an algebraic toric variety. In this paper,
we present an algorithm that identifies all nested canalyzing models that fit
the given data. We demonstrate our methods using a well-known Boolean model of
the cell cycle in budding yeast
Generic Cohen-Macaulay monomial ideals
Given a simplicial complex, it is easy to construct a generic deformation of
its Stanley-Reisner ideal. The main question under investigation in this paper
is how to characterize the simplicial complexes such that their Stanley-Reisner
ideals have Cohen-Macaulay generic deformations. Algorithms are presented to
construct such deformations for matroid complexes, shifted complexes, and tree
complexes.Comment: 18 pages, 8 figure
Reverse-engineering of polynomial dynamical systems
Multivariate polynomial dynamical systems over finite fields have been
studied in several contexts, including engineering and mathematical biology. An
important problem is to construct models of such systems from a partial
specification of dynamic properties, e.g., from a collection of state
transition measurements. Here, we consider static models, which are directed
graphs that represent the causal relationships between system variables,
so-called wiring diagrams. This paper contains an algorithm which computes all
possible minimal wiring diagrams for a given set of state transition
measurements. The paper also contains several statistical measures for model
selection. The algorithm uses primary decomposition of monomial ideals as the
principal tool. An application to the reverse-engineering of a gene regulatory
network is included. The algorithm and the statistical measures are implemented
in Macaulay2 and are available from the authors
Discrete Cubical and Path Homologies of Graphs
In this paper we study and compare two homology theories for (simple and
undirected) graphs. The first, which was developed by Barcelo, Caprano, and
White, is based on graph maps from hypercubes to the graph. The second theory
was developed by Grigor'yan, Lin, Muranov, and Yau, and is based on paths in
the graph. Results in both settings imply that the respective homology groups
are isomorphic in homological dimension one. We show that, for several infinite
classes of graphs, the two theories lead to isomorphic homology groups in all
dimensions. However, we provide an example for which the homology groups of the
two theories are not isomorphic at least in dimensions two and three. We
establish a natural map from the cubical to the path homology groups which is
an isomorphism in dimension one and surjective in dimension two. Again our
example shows that in general the map is not surjective in dimension three and
not injective in dimension two. In the process we develop tools to compute the
homology groups for both theories in all dimensions
On the vanishing of discrete singular cubical homology for graphs
We prove that if G is a graph without 3-cycles and 4-cycles, then the
discrete cubical homology of G is trivial in dimension d, for all d\ge 2. We
also construct a sequence { G_d } of graphs such that this homology is
non-trivial in dimension d for d\ge 1. Finally, we show that the discrete
cubical homology induced by certain coverings of G equals the ordinary singular
homology of a 2-dimensional cell complex built from G, although in general it
differs from the discrete cubical homology of the graph as a whole.Comment: Minor changes, background information adde