Papers by Brandilyn Stigler
Bioinformatics
Motivation While there are software packages that analyze Boolean, ternary, or other multi-state ... more Motivation While there are software packages that analyze Boolean, ternary, or other multi-state models, none compute the complete state space of function-based models over any finite set. Results: We propose Cyclone, a simple light-weight software package which simulates the complete state space for a finite dynamical system over any finite set. Availability and implementation Source code is freely available at https://github.com/discretedynamics/cyclone under the Apache-2.0 license.
Automatica
Boolean functions can be represented in many ways including logical forms, truth tables, and poly... more Boolean functions can be represented in many ways including logical forms, truth tables, and polynomials. Additionally, Boolean functions have different canonical representations such as minimal disjunctive normal forms. Other canonical representation is based on the polynomial representation of Boolean functions where they can be written as a nested product of canalizing layers and a polynomial that contains the noncanalizing variables. In this paper we study the problem of identifying the canalizing layers format of Boolean functions. First, we show that the problem of finding the canalizing layers is NP-hard. Second, we present several algorithms for finding the canalizing layers of a Boolean function, discuss their complexities, and compare their performances. Third, we show applications where the computation of canalizing layers can be used for finding a disjunctive normal form of a nested canalizing function. Another application deals with the reverse engineering of Boolean networks with a prescribed layering format. Finally, implementations of our algorithms in Python and in the computer algebra system Macaulay2 are available at https://github.com/ckadelka/BooleanCanalization.
arXiv (Cornell University), Aug 4, 2022
Over the past several decades, algebraic geometry has provided innovative approaches to biologica... more Over the past several decades, algebraic geometry has provided innovative approaches to biological experimental design that resolved theoretical questions and improved computational efficiency. However, guaranteeing uniqueness and perfect recovery of models are still open problems. In this work we study the problem of uniqueness of wiring diagrams. We use as a modeling framework polynomial dynamical systems and utilize the correspondence between simplicial complexes and square-free monomial ideals from Stanley-Reisner theory to develop theory and construct an algorithm for identifying input data sets V ⊂ F n p that are guaranteed to correspond to a unique minimal wiring diagram regardless of the experimental output. We apply the results on a tumor-suppression network mediated by epidermal derived growth factor receptor and demonstrate how careful experimental design decisions can lead to a unique minimal wiring diagram identification. One of the insights of the theoretical work is the connection between the uniqueness of a wiring diagram for a given V ⊂ F n p and the uniqueness of the reduced Gröbner basis of the polynomial ideal I(V) ⊂ Fp[x1,. .. , xn]. We discuss existing results and introduce a new necessary condition on the points in V for uniqueness of the reduced Gröbner basis of I(V). These results also point to the importance of the relative proximity of the experimental input points on the number of minimal wiring diagrams, which we then study computationally. We find that there is a concrete heuristic way to generate data that tends to result in fewer minimal wiring diagrams.
arXiv: Molecular Networks, 2008
The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene s... more The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models have been proposed to capture this phenomenon. While most are based on complex mathematical formulations, it has been suggested that for other gene systems network topology is sufficient to produce the desired dynamical behavior. We present a Boolean network as a discrete model for the lac operon. We in- clude the two main glucose control mechanisms of catabolite repression and inducer exclusion in the model and show that it exhibits bistability. Further we present a reduced model which shows that lac mRNA and lactose form the core of the lac operon, and that this reduced model also exhibits the same dynamics. This work corroborates the claim that the key to dynamical properties is the topology of th...
In the field of algebraic systems biology, the number of minimal polynomial models constructed us... more In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Grobner bases for the ideal of the data points. While the theory of Grobner bases is extensive, what is missing is a closed form for their number for a given ideal. This work contributes connections between the geometry of data points and the number of Grobner bases associated to small data sets. Furthermore we improve an existing upper bound for the number of Grobner bases specialized for data over a finite field.
arXiv: Algebraic Geometry, 2019
In the field of algebraic systems biology, the number of minimal polynomial models constructed us... more In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Grobner bases for the ideal of the data points. While the theory of Grobner bases is extensive, what is missing is a closed form for their number for a given ideal. This work contributes connections between the geometry of data points and the number of Grobner bases associated to small data sets. Furthermore we improve an existing upper bound for the number of Grobner bases specialized for data over a finite field.
Advances in Applied Mathematics, 2022
Design of experiments and model selection, though essential steps in data science, are usually vi... more Design of experiments and model selection, though essential steps in data science, are usually viewed as unrelated processes in the study and analysis of biological networks. Not accounting for their inter-relatedness has the potential to introduce bias and increase the risk of missing salient features in the modeling process. We propose a data-driven computational framework to unify experimental design and model selection for discrete data sets and minimal polynomial models. We use a special affine transformation, called a linear shift, to provide both the data sets and the polynomial terms that form a basis for a model. This framework enables us to address two important questions that arise in biological data science research: finding the data which identify a set of known interactions and finding identifiable interactions given a set of data. We present the theoretical foundation for a web-accessible database. As an example, we apply this methodology to a previously constructed pharmacodynamic model of epidermal derived growth factor receptor (EGFR) signaling.
Bulletin of Mathematical Biology, 2019
Applications of Gröbner bases, such as reverse engineering of gene regulatory networks and combin... more Applications of Gröbner bases, such as reverse engineering of gene regulatory networks and combinatorial encoding of receptive fields, consider data sets whose ideals of points have unique reduced Gröbner bases. The significance is that uniqueness provides a canonical representation of the input data. In this work, we identify geometric properties of input data that result in a unique reduced Gröbner basis. We show that if the data form a staircase or a so-called linear shift of a staircase, the ideal of the points has a unique reduced Gröbner basis. These results serve to minimize computational effort in using Gröbner bases and are a stepping stone for developing algorithms to generate such data sets.
Journal of Algebra and Its Applications, 2019
In the context of modeling biological systems, it is of interest to generate ideals of points wit... more In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Gröbner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this property. Moreover, we provide methodologies for constructing such ideals. We then relax the condition of uniqueness. The second and most relevant topic discussed here is to consider and identify pairs of ideals with the same number of reduced Gröbner bases, that is, with the same cardinality of their associated Gröbner fan.
IFAC Proceedings Volumes, 2009
This paper gives a review of tools for the system identification of dynamic models for gene regul... more This paper gives a review of tools for the system identification of dynamic models for gene regulatory networks, using the modeling framework of polynomial dynamical systems over finite fields.
ACM Communications in Computer Algebra, 2006
A contemporary and exciting application of Gröbner bases is their use in computational biology, p... more A contemporary and exciting application of Gröbner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Gröbner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Gröbner bases, to date none has been specifically designed to cope with large numbers of variables and few distinct data points. In this paper, we present an algorithm for computing Gröbner bases of zero-dimensional ideals that is optimized for the ca...
Proceedings of the 2007 international symposium on Symbolic and algebraic computation, 2007
Polynomial dynamical systems (PDSs) have been used suc- cessfully as a framework for the reconstr... more Polynomial dynamical systems (PDSs) have been used suc- cessfully as a framework for the reconstruction, or reverse engineering, of biochemical networks from experimental data. Within this modeling space, a particular PDS is chosen by way of a Grobner basis, and using different monomial orders may result in different polynomial models. In this paper, we present a systematic method for selecting
Mathematical Concepts and Methods in Modern Biology, 2013
A contemporary and exciting application of Grobner bases is their use in computational biology, p... more A contemporary and exciting application of Grobner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Grobner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Grobner bases, to date none has been specifically designed to cope with large numbers of variables and few data points. In this paper, we present an algorithm for computing Grobner bases of zero-dimensional ideals that is optimized for the case when t...
Theoretical Computer Science, 2011
Boolean networks have long been used as models of molecular networks and play an increasingly imp... more Boolean networks have long been used as models of molecular networks and play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.
Journal of Theoretical Biology, 2004
This paper proposes a new method to reverse engineer gene regulatory networks from experimental d... more This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is time-discrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in which variables have only two possible states. The use of a larger number of possible states allows a finer discretization of experimental data and more than one possible mode of action for the variables, depending on threshold values. Furthermore, with a suitable choice of state set, one can employ powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to meso-scale networks rather than networks on a genome-wide scale. The complexity of the algorithm is quadratic in the number of variables and cubic in the number of time points. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster.
Journal of Computational Biology, 2011
The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene s... more The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models have been proposed to capture this phenomenon. While most are based on complex mathematical formulations, it has been suggested that for other gene systems network topology is sufficient to produce the desired dynamical behavior. We present a Boolean network as a discrete model for the lac operon. We include the two main glucose control mechanisms of catabolite repression and inducer exclusion in the model and show that it exhibits bistability. Further we present a reduced model which shows that lac mRNA and lactose form the core of the lac operon, and that this reduced model also exhibits the same dynamics. This work corroborates the claim that the key to dynamical properties is the topology of the network and signs of interactions.
Annals of the New York Academy of Sciences, 2007
We consider the problem of reverse-engineering dynamic models of biochemical networks from experi... more We consider the problem of reverse-engineering dynamic models of biochemical networks from experimental data using polynomial dynamical systems. In earlier work, we developed an algorithm to identify minimal wiring diagrams, that is, directed graphs that represent the causal relationships between network variables. Here we extend this algorithm to identify a most likely dynamic model from the set of all possible dynamic models that fit the data over a fixed wiring diagram. To illustrate its performance, the method is applied to simulated time-course data from a published gene regulatory network in the fruitfly Drosophila melanogaster.
We present an algorithm for computing Gröbner bases of vanishing ideals of points that is optimiz... more We present an algorithm for computing Gröbner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Möller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Möller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Gröbner bases to the problem of network reconstruction in molecular biology.
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Papers by Brandilyn Stigler