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The Mathematics of Synthetic Biology

By Lina Sorg

Synthetic biology is a relatively new, interdisciplinary field that combines mathematical modeling, engineering principles, and molecular biology methods to create novel biological systems or entities. These entities can include enzymes, cells, or genetic circuits. The field’s focus on system creation—rather than the study of existing methods—separates it from systems biology, and its broader, systems-level approach sets it apart from genetics (a related, foundational field). During a minisymposium at the 2017 SIAM Annual Meeting, taking place in Pittsburgh, Pa., this week, Laurie Heyer of Davidson College spoke about synthetic biology’s reliance on mathematical models.  

“Synthetic biology is briefly described as the design, construction, modeling, and implementation of biological devices and systems at the molecular level,” Heyer said. The field finds application in energy, the environment, medicine, and technology. While synthetic biologists must think like engineers and embrace abstraction and modularity during the design phase, mathematical modeling is particularly important during the testing phases; researchers can model and predict an apparatus’ performance much more rapidly than they can build it. Modeling helps researchers understand biological processes, while biological systems help researchers solve mathematical problems. 

As part of a research collaboration with Missouri Western State University, Heyer’s undergraduate group is focusing on the connection between mathematics and biology. They built a system that uses “programmed evolution” to tackle a specific biological problem: the creation of theophylline, a methylxanthine drug to treat respiratory conditions such as asthma. “Programmed evolution involves designing evolution to proceed in the way we want so we can produce a small molecule,” Heyer said. The technique is ideally meant to produce this molecule at an optimal rate and prevent subsequent evolution away from the optimal solution; such behavior is unnatural when compared to typical evolutionary patterns. In short, Heyer’s team wants bacteria to produce theophylline indefinitely without further evolution. 

Heyer introduced three modules: the combinatorics module, the fitness module, and the biosensor module. The combinatorics module processes the numerous gene variations, the fitness module forces the cells to behave a certain way (and kills those that don’t function according to plan), and the biosensor module determines if things are going as intended. A Riboswitch, a section of a messenger RNA molecule that binds small target molecules, drives evolution in the fitness module, and theophylline becomes necessary to cells if they want to survive. Heyer is specifically interested in the design of RNA molecules that make up the Riboswitch. As a result, her uniue Riboswitch is well-characterized and useful for a variety of applications. 

The combinatorics module processes the numerous gene variations; the fitness module forces the cells to behave a certain way, thus driving evolution; and the biosensor module determines if things are going as intended. Image credit: Laurie Heyer, AN17 presentation.

Heyer then introduced her evolution target: the conversion of caffeine to theophylline with use of demethylase, which removes the offending methyl group. “When we feed our bacteria coffee, they produce asthma medication,” she quipped. “After a period of time, we see evolution.” Her group delivers the caffeine to cells via a caffeine-soaked disc placed in the center of a petri dish. Heyer used agent-based diffusion models, which are readily accessible to her undergraduate students, to understand the stimulant’s dispersion. “You don’t need to know as many parameters,” she said. “They’re a much less complicated way to get what you’d essentially have to use stochastic differential equations to model.” Her agent-based models revealed diffusion via random motion. Heyer also observed that caffeine greatly reduces cell density, and is deadly to cells at peak radius; this explains the presence of a small, unpopulated space immediately surrounding the caffeine-soaked disk.

Some cells evolved to be especially adept at converting caffeine to theophylline, allowing less-able cells to simply live off of the converters’ theophylline diffusion. Heyer dubbed these two different types “makers” and “moochers,” and modeled their respective interactions. “We do have serious concerns based on these models that moochers could dominate our population,” she said, despite a chance that makers could come out on top. These dynamics reveal critical information about how to best control the rates of caffeine conversion. 

Caffeine diffusion model. A caffeine-soaked disc placed in the center of a petri dish delivers caffeine to bacteria cells, which then convert it to theophylline. Image credit: Laurie Heyer, AN17 presentation.

Heyer also discussed the possible creation of a Riboswitch—produced by cells—able to produce substances other than theophylline. She began by mutating the theophylline Riboswitch in a few specific places, then tackled mathematical questions of how to mutate a subset of sites. Heyer and her team were fairly certain they had to change a few certain sites, so they arranged for those sites to appear together in a set as many times as possible and rearranged different combinations so pairs were seen at least once. 

Heyer ended her talk with a brief discussion of a new complicated combinatorics model called phase-assisted continued evolution (PACE). She created an agent-based model to observe PACE’s dynamics; of PACE; agent-based models are particularly good for students without much programming experience. Through an extensive discussion of both mathematical and computational methods, Heyer demonstrated modeling’s increasing relevance and significance in the continued growth of synthetic biology.

Click here for more coverage of the 2017 SIAM Annual Meeting.

 Lina Sorg is the associate editor of SIAM News.