A Nonparametric Swiss Army Knife for Medicine

By Matthew R. Francis

The complexity of living things is frequently humbling for mathematicians. Even a single cell contains a plethora of processes and complicated interactions that tractable mathematical models cannot easily describe. Researchers have applied nonlinear dynamics, mechanical analogs, and numerous other techniques to understand biological systems, but the tradeoffs of modeling often err on the side of reductionism.

For this reason, Heather Harrington of the University of Oxford and her collaborators are turning to global mathematical methods and drawing on experimental data to identify the best techniques. Harrington described several of these methods during her invited talk at the 2021 SIAM Conference on Applications of Dynamical Systems, which took place virtually earlier this year.

“The way that we look at dynamical systems is usually in a small region of the parameter space,” Harrington said. This approach is helpful if one knows a lot about the model and its parameters, but it can be hard to extract detailed predictions from the model if the parameters in question range over large values. “In biology, we often don’t know if the system is very close to a value in parameter space because the variables or parameters are difficult to measure or the data is too messy,” she added.

In particular, parameters may vary significantly between healthy and diseased systems — or even between two healthy states within the same system. “Sometimes you can look at this as changes in parameters with bifurcation theory,” Harrington said. “But if you want to look at what’s possible in the system more globally, studying the structure of the equations algebraically can be helpful.”

She and her colleagues have developed nonparametric techniques that use algebraic geometry, topology, and other mathematical fields to study problems in biology and medicine. They apply whichever mathematical methods are most enlightening to examine signaling in both healthy and cancerous cells, as well as blood vessel growth in tumors. 

“What I’m looking at really is application driven,” Harrington said. “If the data that we have is spatio-temporal, then the models I want to look at are too. It’s really about finding the appropriate math for the problem.”

She Wnt Thataway

Textbook illustrations of cells generally make them look simple, with spheres and bean-like shapes that sit peacefully in jelly. But real cells are phenomenally complicated; organelles move around, proteins and RNA carry instructions to the various parts, and membranes allow some chemicals to pass through while repelling others. All of these processes maintain a cell’s functions, manage when it divides, and determine when it dies.

One such set of processes is the Wnt signaling pathway,¹ which passes signals through the membranes that govern actions like cell growth and movement. These chemical pathways are important for embryonic development and are disrupted in cancer. They also evolved early in the timeline of life on Earth, so experiments that involve Wnt in one organism—such as a fruit fly or Xenopus frog—can act as experimental proxies for Wnt in humans. “The Wnt pathway experiences dysfunction in 90 percent of colorectal cancer,” Harrington said, noting that colorectal cancer is among the most common cancers in men and women. “So it’s really important to understand the underlying molecular mechanisms.”

Researchers have proposed multiple models for the processes that shuttle proteins around cells and the degradation of the protein \(\beta\)-catenin, which mediates the Wnt signaling pathway. Each model involves multiple coupled nonlinear differential equations, where the equations govern the amounts of particular protein species in the pathway in terms of their chemical reaction, formation, and destruction. Harrington and her collaborators incorporated features from these existing models to build a “shuttle model” that characterizes the entire pathway and has 19 species of proteins and 31 parameters [2].

In general terms, these models consist of 19 polynomial differential equations, one for each species \(x_i \in \{x\}\):

\[\dot{x}=YA_k\Psi(x).\]

Here, \(Y\) is a matrix of integers that reveal the number of occurrences for each reaction (the stoichiometric parameters, in chemical terms); \(A_k\) is a matrix containing the parameters that govern reaction, growth, and decay rates; and \(\Psi(x)\) includes the dynamics of the species concentrations. Writing out all 19 equations for the shuttle model would take up an unreasonable amount of space; fortunately, these equations are simplified by the fact that not all species interact with each other. The following two equations demonstrate how they look in practice:

\[\dot{x}_1=k_1x_1+k_2x_2,\]\[\dot{x}_2=k_1x_1-(k_2+k_{26})x_2+k_{27}x_3-k_3x_2x_4+(d_4+k_5)x_{14}.\]

Rather than use the matrix indices, the nonzero parameters are labeled \(k_{\alpha}\).

As with other dynamical systems, the fixed points of the shuttle model (where \(\dot{x}_1=0\)) characterize important behaviors like cell death. However, the number of equations and parameters make extraction of these behaviors extremely complicated, apart from special parameter choices.

To circumvent this issue, Harrington and her colleagues employed chemical reaction network theory (CRNT): a mathematical framework in applied algebraic geometry that identifies equations’ qualitative features without having to establish parameter values. At the fixed points, the differential equations become polynomial equations for which only real-valued positive protein amounts are biologically realistic. Harrington combined CRNT, numerical algebraic geometry, and matroid theory to identify stable states for the Wnt pathway in the shuttle model and design future experiments for model testing.

Despite this method’s complexity, Harrington does not pretend that it corresponds perfectly to real-world processes in cells. “A biologist abstracts a signaling network from the messy molecular interactions inside a cell,” she said. “But it’s not a true network in a mathematical sense. One of the assumptions we make when looking at these systems using chemical reaction networks is that the rate at which a reaction proceeds is proportional to the product of the species’ concentration.”

Though Harrington’s earlier work attempted to analyze a single pathway at steady state, her team is becoming increasingly interested in joining multiple pathways and studying them with transient time-course data [1]. New mathematics and methods are necessary if scientists hope to better represent and understand the true biology.

Figure 1. The use of topological methods to analyze a real biological system into something tractable requires several steps, shown here for the growth of blood vessels in a tumor. Figure courtesy of [3].

Topologically Speaking

The aforementioned dynamical models that describe Wnt only involve changes in time. However, many important biological applications require consideration of variations in space as well. For example, researchers can map angiogenesis (the growth of blood vessels) in tumors, which offers insight as to how cancers receive nutrients and how the immune system fights back.

Angiogenesis in cancer has intrigued Harrington ever since she conducted a summer research project as an undergraduate student at the University of Massachusetts. “We were running simulations on the big computing cluster at UMass, tracking the angiogenesis and growth of new vasculature towards a tumor,” she said. Harrington then described how her group judged the results of their simulations by visual inspection rather than with a more mathematically rigorous method. “I thought that there must be a better way to quantify this,” she added.

Yet at the time, the simulations taxed the available computers, thereby limiting the kinds of simulations that Harrington desired. Thankfully, improvements in algorithms and computers over the following decades have moved these seemingly impossible problems into the realm of solvability.

Real-world angiogenesis data tends to be very noisy, which complicates the analysis of spatial models and data. Harrington’s team addressed this issue by adapting topological data analysis (TDA)—specifically the persistent homology algorithm—to create a TDA pipeline that analyzes angiogenesis (see Figure 1). More recently, she showcased a statistical topology technique to determine diferent spatial patterns of immune cells that infiltrate a tumor.

Persistent homology allows researchers to simultaneously examine the topology of a set of data points on multiple scales so that the technique can handle the fuzziness of experimental data. In fact, fuzzy data that also contained outliers helped motivate the development of multi-parameter persistent homology. “Persistent homology landscapes are a statistical way to compare when you have more than one parameter,” Harrington said. She used the example of the density of immune cells relative to regions where the tumor is deprived of oxygen (hypoxia). She and her collaborators were even able to distinguish between different types of immune cells with these methods [4].

Both the algebraic and topological approaches have their advantages, but Harrington finds that their power comes through in flexibility. “I’m not using one approach and trying to make everything fit into that,” she said. “As we go from a couple of molecular species to groups of cells or tissues, the problems become spatio-temporal and the mathematical approaches need to change — especially when we want to include real data with noise.”

Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.