Joint work with Dana L. Nettles³, Kimberly Trabbic Carlson³, Ashutosh Chilkoti³, Lori A. Setton^3,4, Mansoor A. Haider^1,2

Elastin-like polypeptide (ELP) hydrogels are a class of biomaterials that have potential utility as a biocompatible scaffold for filling defects due to osteoarthritis and for regenerating cartilage. Because of the facility to genetically engineer elastin sequence, there are almost endless possible configurations of ELPs and conformations of the networks formed after crosslinking. ELP biomaterial function will exhibit a complex dependence on these polymer characteristics that impacts properties expected to affect cartilage regeneration, such as mechanical load support. These complex structure-function relationships for crosslinked ELP hydrogels are not well described. A method for predicting the mechanical properties of ELP hydrogels was developed based on structural properties and Supervised Artificial Neural Network (ANN) modeling. The ANN Model used concentration, molecular weight, crosslink density, and sample number to predict the dynamic shear modulus and loss angle of the hydrogels. The ANN was implemented in a custom compiled code based on the Scaled Conjugate Gradient minimization algorithm and a Monte Carlo Method was used to expand the dataset. The ANN was trained using a varying subsets of the full dataset (22 formulations), with the complementary subset used for validation. Trained networks demonstrated excellent accuracy in prediction of hydrogel dynamic shear modulus at physiological temperature, based on polymer design and predictions were robust with respect to statistical variations. The results are used to show the validity of an intermediate screening process using ANNs to obtain the optimal mechanical properties for the ELP.

¹ Biomathematics Graduate Program, North Carolina State University, ² Department of Mathematics, North Carolina State University, ³ Department of Biomedical Engineering, Duke University, ⁴ Department of Surgery, Duke Medical Center

Molecular phylogenetics is concerned with inferring evolutionary relationships (phylogenetic trees) from biological sequences (such as aligned DNA sequences for a gene shared by a collection of species). The probabilistic models of sequence evolution that underly statistical approaches in this field exhibit a rich algebraic structure.

After an introduction to the inference problem and phylogenetic models, this talk will survey some of the highlights of current algebraic understanding. Results on the important statistical issue of identifiability of phylogenetic models will be emphasized, as the algebraic viewpoint has been crucial to obtaining such results.

The relationship between the shape of a fitness landscape and the underlying gene interactions, or epistasis, has been extensively studied in the two-locus case. Epistasis has been linked to biological important properties such as the advantage of sex. Gene interactions among multiple loci are usually reduced to two-way interactions. Here, we present a geometric theory of shapes of fitness landscapes for multiple loci. We investigate the dynamics of evolving populations on fitness landscapes and the predictive power of the geometric shape for the speed of adaptation. Finally, we discuss applications to fitness data from viruses and bacteria.

Many statistical models of evolution can be viewed as algebraic varieties. The generators of the ideal associated to a model and a phylogenetic tree are called invariants. The invariants of an statistical model of evolution should allow to determine what is the tree formed by a set of living species.

We will present a method of phylogenetic inference based on invariants and we will discuss why algebraic geometry should be considered as a powerful tool for phylogenetic reconstruction. The performance of the method has been studied for quartet trees and the Kimura 3-parameter model and it will be compared to widely known phylogenetic reconstruction methods such as Maximum likelihood estimate and Neighbor-Joining.

Chemical reaction network models give rise to polynomial dynamical systems that are usually high dimensional, nonlinear, and have many unknown parameters. Due to the presence of these unknown parameters (such as reaction rate constants) direct numerical simulation of the chemical dynamics is practically impossible. On the other hand, we will show that important properties of these systems are determined only by the network structure, and do not depend on the unknown parameters. Also, we will show how some of these results can be generalized to systems of polynomial equations that are not necessarily derived from chemical kinetics. In particular, we will point out connections with classical problems in algebraic geometry, such as the real Jacobian conjecture. This talk describes joint work with Martin Feinberg, and can be regarded as a continuation of his earlier talk.

We work in the space of n-by-n real symmetric matrices with the Frobenius inner product. Consider the following problem:

Problem: Positive semi-definite approximation. Given an n-by-n real symmetric matrix A, find the positive semi-definite matrix which is closest to A.

I discuss the following differential equation in the space of symmetric matrices: X^′ = (A-X)X² + X²(A-X) .

The corresponding flow preserves inertia. In particular, if the initial value X(0)=M is a positive definite matrix then X(t) is positive definite for all t>0. I show that the distance between A and X(t) decreases as t increases. I also show that if A has distinct nonzero eigenvalues (which is a generic condition) then the solution X(t) converges to the positive semi-definite matrix which is closest to A.

The Behrens-Fisher problem concerns testing the statistical hypothesis of equality of the means of two normal populations with possibly different variances. This problem furnishes one of the simplest statistical models for which the likelihood equations may have more than one real solution. In fact, with probability one, the equations have either one or three real solutions. Using the cubic discriminant, we study the large-sample probability of one versus three solutions.

Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff's theorem hold for semi-algebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point. At boundary points or singularities, the tangent cone need not be a linear space such that non-standard limiting distributions may arise. Besides the well-known mixtures of chi-square distributions, such non-standard limits are shown to include the distributions of minima of chi-square random variables. Via algebraic tangent cones, connections to eigenvalues of Wishart matrices are found in factor analysis.

We introduce new methods for phylogenetic tree construction by using machine learning to optimize the power of phylogenetic invariants. Phylogenetic invariants are polynomials in the joint probabilities which vanish under a model of evolution on a phylogenetic tree. We give algorithms for selecting a good set of invariants and for learning a metric on this set of invariants which optimally distinguishes the different models. Our learning algorithms involve semidefinite programming on data simulated over a wide range of parameters. Simulations on trees with four leaves under the Jukes-Cantor and Kimura 3-parameter models show that our method improves on other uses of invariants and is competitive with neighbor-joining. Our main biological result is that the trained invariants can perform substantially better than neighbor joining on quartet trees with short interior edges.

This is joint work with Yuan Yao (Stanford).

In nature there are millions of distinct networks of chemical reactions that might present themselves for study at one time or another. Written at the level of elementary reactions taken with classical mass action kinetics, each new network gives rise to its own (usually large) system of polynomial equations for the species concentrations. In this way, chemistry presents a huge and bewildering array of polynomial systems, each determined in a precise way by the underlying network up to parameter values (e.g., rate constants). Polynomial systems in general, even simple ones, are known to be rich sources of interesting and sometimes wild dynamical behavior. It would appear, then, that chemistry too should be a rich source of dynamical exotica.

Yet there is a remarkable amount of stability in chemistry. Indeed, chemists and chemical engineers generally expect homogeneous isothermal reactors, even complex ones, to admit precisely one (globally attractive) equilibrium. Although this tacit doctrine is supported by a long observational record, there are certainly instances of homogeneous isothermal reactors that give rise, for example, to multiple equilibria. The vast landscape of chemical reaction networks, then, appears to have wide regions of intrinsic stability (regardless of parameter values) punctuated by far smaller regions in which instability might be extant (for at least certain parameter values).

In this talk, I will present some recent joint work with Gheorghe Craciun that goes a long way toward explaining this landscape — in particular, toward explaining how biological chemistry "escapes" the stability doctrine to (literally) "make life interesting." A subsequent talk by Craciun will emphasize more mathematical detail.

The past decade has seen considerable interest in the reformulation of statistical models and methods for the analysis of contingency tables using the language and results of algebraic and polyhedral geometry. But as algebraic statistics has developed, new ideas have emerged that have changed how we view a number of statistical problems. This talk reviews some of these recent advances and suggests some challenges for collaborative research, especially those involving large scale databases.

A subspace arrangement is a union of a finite number of subspaces of a vector space. We will discuss the importance of subspace arrangements first as mathematical objects and now as a popular class of models for engineering.

We will then introduce some of new theoretical results that were motivated from practice. Using these results we will address the computational issue about how to extract subspace arrangements from noisy or corrupted data.

Finally we will turn to the importance of subspace arrangements by briefly discussing the connections to sparse representations, manifold learning, etc...

A coupled cell system is a collection of interacting dynamical systems. Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: How much of the qualitative dynamics observed in coupled cells is the product of network architecture and how much depends on the specific equations? Speficially we study the structure of synchrony-breaking bifurcations in these systems.

The ideas will be illustrated through a series of examples and theorems. One example shows how a frequency filter / amplifier can be built from a small three-cell feed forward network; and a second illustrates patterns of synchrony in lattice dynamical systems. One theorem gives necessary and sufficient conditions for synchrony in terms of network architecture; and a second shows that synchronous dynamics may itself be viewed as dynamics in a coupled cell system through a quotient construction.

Regular patterns appear all around us: from vast geological formations to the ripples in a vibrating coffee cup, from the gaits of trotting horses to tongues of flames, and even in visual hallucinations. The mathematical notion of symmetry is a key to understanding how and why these patterns form. In this lecture Professor Golubitsky will show some of these fascinating patterns and explain how mathematical symmetry enters the picture.

Mass-action kinetics is a powerful tool to describe events created by collission of molecules or individuals in a well-mixed environment giving them locally the same probability to meet each other. Moreover this probability is only dependent on the concentration of the mutual partners. Mass action systems can be found in chemistry, cell biology, but also game theory and economics. Mathematically this gives rise to dynamical systems of a special type, more specific of polynomial type. I will give an overview how this property can be used to determine different types of bifurcations, for example the ocurrence of bistability, or oscillations via a Hopf bifurcation. All tools will be borrowing methods from algebraic geometry. Finally I will give an outlook what usually goes wrong in the modelling part while using mass-action kinetics if biochemical or cellular molecular events are considered. Finally the talk ends with a fresh look on mass-action kinetics applied to a spatial setting.

Time-discrete dynamical systems with a finite state space have been used as models of biological systems since the use/invention of cellular automata by von Neumann in his attempt to model a self-replicating organism in the 1950s. More recently, they have appeared as models of a variety of biological systems, from gene regulatory networks to large-scale epidemiological networks. This talk will focus on theoretical and computational results about polynomial dynamical systems using tools from computational algebra and algebraic geometry.

Exponential families underpin numerous models of statistics and information geometry that have significant applications. For a standard full exponential family, or its canonically convex subfamily, if the corresponding likelihood function from a sample has a maximizer t* then, by the maximum likelihood principle, the data are judged to be generated by the probability measure P* from the family that is parameterized by t*. Since the likelihood depends on data only through their mean, in this way the mean is mapped to P*. In a joint work with Imre Csiszar, Budapest, we study an extension of this mapping, the generalized maximum likelihood estimator. It assigns to each point of the space at which the likelihood function is bounded above, a probability measure from the closure of the family in variation distance. A detailed description, complete characterization of domain and range, and additional results will be presented, not imposing any regularity assumptions.

We investigate the polyhedral geometry of conditional probability and undirected graphical models, developing new statistical procedures called convex rank tests. The polytope associated to an undirected graphical conditional independence model is the graph associahedron. The convex rank test defined by the dual semigraphoid to the n-cycle graphical model is applied to microarray data analysis to detect periodic gene expression.

We address the problem of studying the toric ideals of phylogenetic invariants for a general group-based model on an arbitrary claw tree. We focus on the group ₂ and choose a natural recursive approach that extends to other groups. The study of the lattice associated with each phylogenetic ideal produces a list of circuits that generate the corresponding lattice basis ideal. In addition, we describe explicitly a quadratic lexicographic Gröbner basis of the toric ideal of invariants for the claw tree on an arbitrary number of leaves. Combined with a result of Sturmfels and Sullivant, this implies that the phylogenetic ideal of every tree for the group ₂ has a quadratic Gröbner basis. Hence, the coordinate ring of the toric variety is a Koszul algebra.

This is joint work with Julia Chifman, University of Kentucky.

Recent presentations of Information Geometry (IG), e.g. Amari and Nagaoka (2000), consider general statistical models and general sample spaces. However, the seminal discussion by Cenkov (transl. 1982) is based on finite sample spaces, as it is in Algebraic Statistics (AS).

This talk will first review basic IG from the point of view of AS. In the second part, it discusses the issue of computation IG quantities and presents a few examples.

In the 1920's the geneticist Sewall Wright introduced a class of Gaussian statistical models represented by graphs containing directed and bi-directed edges, known as path diagrams. These models have been used extensively in psychometrics and econometrics where they are called structural equation models.

I will first describe the subclass of bow-free acyclic path diagrams, which have desirable statistical properties. I will then characterize a subclass of models that are characterized by their Markov properties. Lastly I will outline recent work aimed at characterizing non-Markovian constraints that may arise.

(This is joint work with Mathias Drton, Michael Eichler and Masashi Miyamura.)

Statistical disclosure limitation applies statistical tools to the problems of limiting sensitive information releases about individuals and groups that are part of statistical databases while allowing for proper statistical inference. The limited releases can be in a form of arbitrary collections of marginal and conditional distributions, and odds ratios for contingency tables. Given this information, we discuss how tools from algebraic geometry can be used to give both complete and incomplete characterization of discrete distributions for contingency tables. These problems also lead to linear and non-linear integer optimization formulations. We discuss some practical implication, and challenges, of using algebraic statistics for data privacy and confidentiality problems.

This talk introduces five or six mathematical problems whose solution would likely be a significant contribution to the emerging interactions between algebraic geometry, statistics, and computational biology.

It is known that for general distributions, there is no finite list of conditional independence axioms that can be used to deduce all implications among a collection of conditional independence statements. We show the same result holds among the class of Gaussian random variables by exhibiting, for each n>3, a collection of n independence statements on n random variables, which, in the Gaussian case imply that X_1 is independent of X_2, but such that no subset implies that X_1 is independent of X_2. The proof depends on the fact that conditional independence models for Gaussian random variables are algebraic varieties in the cone of positive definite matrices and makes use of binomial primary decomposition.

Joint work with Reinhard Steffens.

Linkages are graphs whose edges are rigid bars, and they arise as a natural model in many applications in computational geometry, molecular biology and robotics. Studying linkages naturally leads to a variety of questions in real algebraic geometry, such as:

1. Given a rigid graph with prescribed edge lengths, how many embeddings are there?
2. Given a 1-degree-of-freedom linkage, how can one characterize and compute the trajectory of the vertices?

From the real algebraic point of view, these questions are questions of specially-structured real algebraic varieties. On the poster we exhibit some techniques from sparse elimination theory to analyze these problems. In particular, we show that certain bounds (e.g. for Henneberg-type graphs) naturally arise from mixed volumes and Bernstein's theorem.

I will discuss geometric methods of investigating phylogenetic trees. In a joint project with Weronika Buczynska we investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. They have Gorenstein terminal singularities and are Fano varieties. Moreover any two such varieties which are of the same dimension are deformation equivalent, that is, they are in the same connected component of the Hilbert scheme of the projective space whose coordinates are indexed by subsets of their leaves.

Symbioses of grasses and fungal endophytes constitute an interesting model for evolution of mutualism and parasitism. Grasses of all subfamilies can harbor systemic infections by fungi of the family Clavicipitaceae. Subfamily Poöideae is specifically associated with epichloë endophytes (species of Epichloë and their asexual derivatives, the Neotyphodium species) in intimate symbioses often characterized by highly efficient vertical transmission in seeds, and bioprotective benefits conferred by the symbionts to their hosts. These remarkable symbioses have been identified in most grass tribes spanning the taxonomic range of the subfamily. Here we examine the possibility of codivergence in the phylogenetic histories of Poöideae and epichloë. We introduce a method of analysis to detect significant codivergence even in the absence of strict cospeciation, and to address problems in previously developed methods. Relative ages of corresponding cladogenesis events were determined from ultrametric maximum likelihood H (host) and P (parasite = symbiont) trees by an algorithm called MRCALink (most recent common ancestor link), an improvement over previous methods that greatly weight deep over shallow H and P node pairs. We then compared the inferred correspondence of MRCA ages in the H and P trees to the spaces of trees estimated from 10,000 randomly generated H and P tree pairs. Analysis of the complete dataset, which included a broad host-range species and some likely host transfers (jumps), did not indicate significant codivergence. However, when likely host jumps were removed the analysis indicated highly significant codivergence. Interestingly, early cladogenesis events in the Poöideae corresponded to early cladogenesis events in epichloë, suggesting concomitant origins of the Poöideae and this unusual symbiotic system.

This is joint work with C. L. Schardl, K. D. Craven, A. Lindstrom, and A. Stromberg.

Genes play a complicated role in how likely one is to get a certain disease. Biologists would like to model how one's genotype affects their likelihood of illness. We propose a new classification of two-locus disease models, where each model corresponds to an induced subdivision of a point configuration (basically a picture of connected dots). Our models reflect epistasis, or gene interaction. This work is joint with Ingileif Hallgrimsdottir. For more information, see our preprint at arXiv:q-bio.QM/0612044.

Latent class models have been used to explain the heterogeneity of the observed relationship among a set of categorical variables and have received more and more attention as a powerful methodology for analyzing discrete data. The central goal of our work is to study the existence and computation of maximum likelihood estimates (MLEs) for these models, which are cardinal for assessment of goodness of fit and model selection. Our study is at the interface between the fields of algebraic statistics and machine learning.



Traditionally, the expectation maximization (EM) algorithm has been applied to compute the MLEs of a latent class model. However, the solutions provided by the EM correspond to local maxima only, so, although we are able to compute them effectively, we still lack methods for assessing uniqueness and existence of the MLEs. Another interesting problem in statistics is the identifiability of the model. When a model is unidentifiable, it is necessary to adjust the number of degrees of freedom in order to apply correctly goodness-of-fit tests. In our work, we show that both the existence and identifiability problems are closely related to the geometric properties of the latent class models. Therefore, studying the algebraic varieties and ideals arising from these models is particularly relevant to our problem. We include a number of examples as a way of opening a discussion on a general method for addressing both MLE existence and identifiability in latent class models.