Research
In my research, I am interested in applications of theoretical mathematics, including graph theory, combinatorics, and algebraic geoemtry, to problems from biology and physics. Often this work is categorized as being in the field of algebraic statistics. I love this work because it allows me to use several different areas of math while also having underlying motivation from the real world for the problems that I work on. Almost every project I do involves some amount of code for testing examples in Python/Sage, Mathematica, and/or Macaulay2.
Students interested in working with me should be comfortable writing proofs, i.e. have taken MATH 3400 “Set Theory and Logic.” We will learn anything else we need along the way!
Publications
Published in Preprint., 2024
An important problem in biological modeling is choosing the right model. Given experimental data, one is supposed to find the best mathematical representation to describe the real-world phenomena. However, there may not be a unique model representing that real-world phenomena. Two distinct models could yield the same exact dynamics. In this case, these models are called indistinguishable. In this work, we consider the indistinguishability problem for linear compartmental models, which are used in many areas, such as pharmacokinetics, physiology, cell biology, toxicology, and ecology. We exhibit sufficient conditions for indistinguishability for models with a certain graph structure: paths from input to output with “detours”. The benefit of applying our results is that indistinguishability can be proven using only the graph structure of the models, without the use of any symbolic computation. This can be very helpful for medium-to-large sized linear compartmental models. These are the first sufficient conditions for indistinguishability of linear compartmental models based on graph structure alone, as previously only necessary conditions for indistinguishability of linear compartmental models existed based on graph structure alone. We prove our results by showing that the indistinguishable models are the same up to a renaming of parameters, which we call permutation indistinguishability.
Recommended citation: Bortner, C. & Meshkat, N. (2024). "Graph-Based Sufficient Conditions for Indistinguishability of Linear Compartmental Models." Preprint. https://arxiv.org/abs/2309.10861
Published in Advances in Applied Mathematics, 2023
A foundational question in the theory of linear compartmental models is how to assess whether a model is structurally identifiable – that is, whether parameter values can be inferred from noiseless data – directly from the combinatorics of the model. Our main result completely answers this question for models (with one input and one output) in which the underlying graph is a bidirectional tree; moreover, identifiability of such models can be verified visually. Models of this structure include two families of models often appearing in biological applications: catenary and mammillary models. Our analysis of such models is enabled by two supporting results, which are significant in their own right. One result gives the first general formula for the coefficients of input-output equations (certain equations that can be used to determine identifiability) that allows for input and output to be in distinct compartments. In another supporting result, we prove that identifiability is preserved when a model is enlarged and altered in specific ways involving adding a new compartment with a bidirected edge to an existing compartment.
Recommended citation: Bortner, C., Gross, E., Meshkat, N., Shiu, A., & Sullivant, S. (May 2023). "Identifiability of linear compartmental tree models and a general formula for the input-output equations." Advances in Applied Mathematics. 146. https://www.sciencedirect.com/science/article/pii/S0196885823000088?casa_token=wx3f6jO9YAkAAAAA:BNh1bxkOuQJt1u_M-okXZhcyFp9A0NqzYz5wxE6PLscKRKVdTBnUdeEHqXxz3o9suzCyGgJjgT8
Published in Journal of Symbolic Computation., 2022
We consider the identifiability problem for the parameters of series-parallel LCR circuit networks. We prove that for networks with only two classes of components (inductor-capacitor (LC), inductor-resistor (LR), and capacitor-resistor (RC)), the parameters are identifiable if and only if the number of non-monic coefficients of the constitutive equations equals the number of parameters. The notion of the “type” of the constitutive equations plays a key role in the identifiability of LC, LR, and RC networks. We also investigate the general series-parallel LCR circuits (with all three classes of components), and classify the types of constitutive equations that can arise, showing that there are 22 different types. However, we produce an example that shows that the basic notion of type that works to classify identifiability of two class networks is not sufficient to classify the identifiability of general series-parallel LCR circuits.
Recommended citation: Bortner, C. & Sullivant, S. (Mar. 2022). "Structural Identifiability of Series-Parallel LCR Systems." Journal of Symbolic Computation. 112 (pp 79-104). https://www.sciencedirect.com/science/article/abs/pii/S0747717122000025
Published in Bulletin of Mathematical Biology, 2022
We introduce a class of linear compartmental models called identifiable path/cycle models which have the property that all of the monomial functions of parameters associated to the directed cycles and paths from input compartments to output compartments are identifiable and give sufficient conditions to obtain an identifiable path/cycle model. Removing leaks, we then show how one can obtain a locally identifiable model from an identifiable path/cycle model. These identifiable path/cycle models yield the only identifiable models with certain conditions on their graph structure and thus we provide necessary and sufficient conditions for identifiable models with certain graph properties. A sufficient condition based on the graph structure of the model is also provided so that one can test if a model is an identifiable path/cycle model by examining the graph itself. We also provide some necessary conditions for identifiability based on graph structure. Our proofs use algebraic and combinatorial techniques.
Recommended citation: Bortner, C. & Meshkat, N. (Mar. 2022). "Identifiable Paths and Cycles in Linear Compartmental Models." Bulletin of Mathematical Biology. 84 (53). https://link.springer.com/article/10.1007/s11538-022-01007-5#:~:text=We%20introduce%20a%20class%20of,sufficient%20conditions%20to%20obtain%20an
Published in The Australasian Journal of Combinatorics, 2019
The set splittability problem is the following: given a finite collection of finite sets, does there exist a single set that contains exactly half the elements from each set in the collection? (If a set has odd size, we allow the floor or ceiling.) It is natural to study the set splittability problem in the context of combinatorial discrepancy theory and its applications, since a collection is splittable if and only if it has discrepancy at most 1. We introduce a natural generalization of the splittability problem called the p-splittability problem, where we replace the fraction 1 2 in the definition with an arbitrary fraction p ∈ (0, 1). We first show that the p-splittability problem is NP-complete. We then give several criteria for p-splittability, including a complete characterization of p-splittability for three or fewer sets (p arbitrary), and for four or fewer sets (p = 1 2 ). Finally we prove the asymptotic prevalence of splittability over unsplittability in an appropriate sense.
Recommended citation: Bernstein, P., Bortner, C., Coskey, C., Li, S., & Simnpson, C. (Oct. 2019). "The Set Splittability Problem" The Australasian Journal of Combinatorics. 85 (pp 190-209). https://ajc.maths.uq.edu.au/pdf/75/ajc_v75_p190.pdf
