Conjunctive Bayesian networks

Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. by allowing the occurrence of an event to depend on more than one predecessor event. The present paper studies the statistical and algebraic properties of CBNs. We determine the maximum likelihood parameters and present a combinatorial solution to the model selection problem. Our method performs well on two datasets where the events are HIV mutations associated with drug resistance. Concluding with a study of the algebraic properties of CBNs, we show that CBNs are toric varieties after a coordinate transformation and that their ideals possess a quadratic Gr\"{o}bner basis.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6133 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On the stability and number of steady states of chemical reaction networks

Chemical reaction networks model the interactions of chemical substances. An important aim is to understand the long-term behaviors of these chemical reactions, which are called steady states. Interesting behaviors have implications in biology, such as in cell-signaling or regulatory processes. This motivates the need to predict if and when a network can exhibit interesting long-term behavior. Exact values of the physical parameters that describe these networks are often unknown. Because of this, the goal is to prove results based on a network’s structure alone, independent of any specific numerical values. This dissertation harnesses theoretical mathematical techniques to investigate the number and stability of steady states of a chemical reaction network. In this dissertation, we answer the following: Does a given chemical reaction network have the capacity for Hopf bifurcations (an important unstable steady state)? How many steady states can it have? Our first contribution is a novel procedure for constructing a Hopf bifurcation of a chemical reaction network. This algorithm gives an easy-to-check condition for the existence of a Hopf bifurcation and explicitly constructs one if it exists. Our second set of contributions are new upper bounds on the number of steady states of a chemical reaction network. These new numerical invariants are both quick to compute and are surprisingly good bounds on the number of steady states. As the main application of our new tools, we analyze an important biological cell-signaling network called the Extracellular signal Regulated Kinase (ERK) network. Malfunctions in the ERK network are linked with human diseases, including cancers and developmental abnormalities, making it crucial to understand the ERK network’s long-term behavior. We show how the ERK network has the capacity for different dynamic regimes, including multiple steady states, two stable steady states, simple Hopf bifurcations, and a unique, stable steady state. Applying our new tools, we directly relate each dynamic behavior to the network structure, specifically the presence of certain species or reactions

Groebner bases of symmetric ideals

In this article we present two new algorithms to compute the Groebner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in SINGULAR (cf. [DGPS12]). The first and major algorithm is most performant over finite fields whereas the second algorithm is a probabilistic modification of the modular computation of Groebner bases based on the articles by Arnold (cf. [A03]), Idrees, Pfister, Steidel (cf. [IPS11]) and Noro, Yokoyama (cf. [NY12], [Y12]). In fact, the first algorithm that mainly uses the given symmetry, improves the necessary modular calculations in positive characteristic in the second algorithm. Particularly, we could, for the first time even though probabilistic, compute the Groebner basis of the famous ideal of cyclic 9-roots (cf. [BF91]) over the rationals with SINGULAR.Comment: 17 page

Solving Equations Using Khovanskii Bases

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Pl\"ucker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.Comment: 25 pages, 1 figure, 2 table

Identifiable reparametrizations of linear compartment models

Identifiability concerns finding which unknown parameters of a model can be quantified from given input-output data. Many linear ODE models, used in systems biology and pharmacokinetics, are unidentifiable, which means that parameters can take on an infinite number of values and yet yield the same input-output data. We use commutative algebra and graph theory to study a particular class of unidentifiable models and find conditions to obtain identifiable scaling reparametrizations of these models. Our main result is that the existence of an identifiable scaling reparametrization is equivalent to the existence of a scaling reparametrization by monomial functions. We also provide partial results beginning to classify graphs which possess an identifiable scaling reparametrization.Comment: 5 figure

Symmetry in Multivariate Ideal Interpolation

An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes,while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Groebner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group

Cohomological Aspects of Hamiltonian Group Actions and Toric Varieties

[no abstract available