Mathematics Research

Systems Biology

A rising field of interest is that of cancer systems biology, which often requires multidisciplinary collaboration to provide a wholistic understanding of a system's complexities. Rather than relying strictly on traditional experimental approaches in vivo or in vitro, this field uses in silico models based on available clinical data as well as literature. Notably, these models are often much less complex than the actual biological system due to the computational demand of high-dimensional systems. Researchers in phenotype control theory are primarily concerned with identifying key markers of the system that aid in understanding the various functions of cells and their molecular mechanisms. These models can perform simulations in silico to help predict outcomes and optimal targets for therapy intervention strategies. They can also be used to simulate, validate, and optimize existing hypotheses. Recent efforts have allowed modelers to expand to multiscale models, which is an important challenge because cancer growth occurs over varying time scales, depending on the hierarchy of spatial scales. Inherently, implementing multiple scales considerably increases the model’s complexity [1,2]. 

Populations of cells, multiple scales (time and size), and highly nonlinear dynamics all contribute to a significant burden for in silico models to overcome. There are a wide range of mathematical tools available to implement these complex models. Models are traditionally classified based on the time and population of gene products. For instance, there are techniques for continuous population with continuous time such as ordinary differential equations, discrete population with continuous time such as the Gillespie formulation, and discrete population with discrete time such as BNs, logical models, local models, and also their related stochastic counterparts [3]. These models are typically multiscale integrating interactions at different scales, making it possible to simulate clinically relevant spatiotemporal scales, and at the same time simulate the effect of molecular drugs on tumor progression. Even though a multiscale model would likely provide more realistic simulations, there are currently no control methods that apply directly to multiscale models . Thus, the implementation of multistate models (eg. Boolean Networks) allows for a mechanistic approach that eliminates the need for tedious parameter fitting. Such models have a well-studied and effective track record for capturing various biological system dynamics [3]. 

Cancer Systems Biology

In silico models are commonly used in cancer research for the discovery of general principles and novel hypotheses that can guide the development of new treatments. It is eventually possible that, when combined with cancer data and modern control techniques, in silico models will predict clinically relevant endpoints and find optimal control interventions to stop cancer progression. Despite their potential, concrete examples of predictive models of cancer progression are scarce due to limited data availability. One reason is that most models have focused on single–cell type dynamics, ignoring the interactions between cancer cells and their local microenvironment. Indeed, it has been demonstrated that the local microenvironment has a critical effect on the behavior of cancer cells. Consequently, ignoring the effect of cells and signals of the local microenvironment can generate confounding conclusions.

Stochastic Modeling

Discrete models of gene regulatory networks can involve stochastic processes depending on the update schedules chosen. Synchronous update schedules produce deterministic dynamics, wherein nodes are all updated simultaneously. On the other hand, asynchronous update schedules produce stochastic dynamics, wherein a randomly selected node is updated at each time step. In the SDDS framework, one propensity is used when the update positively impacts the variable in the sense that the variable increases its value from OFF to ON. Another propensity is used when the update negatively affects the variable in the sense that the variable decreases its value from ON to OFF. The SDDS framework [3] incorporates Markov chain tools to study long-term dynamics of Boolean networks. It also uses parameters based on designated propensities to model node (and pathway) signal activation and deactivation, also referred to as degradation. 

Phenotype Control

Phenotype control theory is primarily concerned with identifying key markers of the system that aid in understanding the various functions of cells and their molecular mechanisms. In a biological sense, phenotypes represent the observable traits of an organism. In a similar fashion, we associate a phenotype with group of attractors where a subset of the system's variables have a shared state. These shared states are then used as biomarkers or identifiers. Thus, phenotype control is the ability to drive the system to a predetermined phenotype from any initial state. This is different than classical control because objectives are related to the attractors of highly nonlinear systems, and it uses open-loop interventions which are not adjusted throughout the time-course. 

Publications

1. Phenotype control techniques for Boolean gene regulatory networks. Daniel Plaugher and David Murrugarra. Bulletin of Mathematical Biology, 85:89, 2023. https://rdcu.be/dkN6A

2. Investigating the effect of changes in model parameters on optimal control policies, time to absorption, and mixing times. Kathleen Johnson,  Daniel Plaugher,  David Murrugarra. Letters in Biomathematics 2023; 10:193–206. doi: 10.30707/. https://www.biorxiv.org/content/10.1101/2023.01.23.525286v1 

3. An Integrated Computational Pipeline to Construct Patient-Specific Cancer Models (2022). Daniel Plaugher, Theses and Dissertations--Mathematics. 93. https://uknowledge.uky.edu/math_etds/93

4. Uncovering potential interventions for pancreatic cancer patients via mathematical modeling. Daniel Plaugher, Boris Aguilar, David Murrugarra. Journal of Theoretical Biology, 548, 111197, 2022. https://www.sciencedirect.com/science/article/abs/pii/S0022519322001953

5. Modeling the Pancreatic Cancer Microenvironment in Search of Control Targets. Daniel Plaugher and David Murrugarra. Bulletin of Mathematical Biology, 83, (11):115, 2021.  https://www.biorxiv.org/content/10.1101/2021.05.04.442611v2 

Resources

1. Boris Aguilar, David L Gibbs, David J Reiss, Mark McConnell, Samuel A Danziger, Andrew Dervan, Matthew Trotter, Douglas Bassett, Robert Hershberg, Alexander V Ratushny,and Ilya Shmulevich.  A generalizable data-driven multicellular model of pancreatic ductal adenocarcinoma.Gigascience, 9(7), 07 2020.

2. Thomas S. Deisboeck, Zhihui Wang, Paul Macklin, and Vittorio Cristini.   Multiscale cancer  modeling. Annual Review of Biomedical Engineering,  13(1):127–155,  2011.   PMID:21529163.

3. D. Murrugarra and B. Aguilar. Modeling the stochastic nature of gene regulation with Boolean networks. Algebraic and Combinatorial Computational Biology, edited by Raina Robeva and Matthew Macauley, Academic Press, 2018.