Interested in learning how mathematics is used in biology? Then read November's *Monthly*, featuring a special issue in Mathematical Biology. Edited by guest editors Elizabeth Allman, Fred Adler, and Lisette de Pillis, this issue's seven articles showcase the interactions of mathematics and biology. How do animals determine territorial boundaries? How are cancer dynamics modeled? What kind of `Frankenstein' might be created in a petrie dish using synthetic biology? All this, and more, in the latest issue of the Monthly. — *Elizabeth Allman*

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## Table of Contents

### A Letter from the Guest Editors

Fred Adler, Elizabeth S. Allman, and Lisette G. de Pillis

### A Mathematical Approach to Territorial Pattern Formation

Jonathan R. Potts and Mark A. Lewis

Territorial behavior is widespread in the animal kingdom, with creatures seeking to gain parts of space for their exclusive use. It arises through a complicated interplay of many different behavioral features. Extracting and quantifying the processes that give rise to territorial patterns requires both mathematical models of movement and interaction mechanisms, together with statistical techniques for rigorously extracting parameters from data. Here, we give a brisk, pedagogical overview of the techniques so far developed to tackle the problem of territory formation. We give some examples of what has already been achieved using these techniques, together with pointers to where we believe the future lies in this area of study. This progress is a single example of a major aim for 21st century science: to construct quantitatively predictive theory for ecological systems.

### Tracing Evolutionary Links between Species

Mike Steel

The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's *Origin of Species*. Ever since, biologists have tried to piece together parts of this ‘tree of life’ based on what we can observe today: fossils and the evolutionary signal that is present in the genomes and phenotypes of different organisms. Mathematics has played a key role in helping transform genetic data into phylogenetic (evolutionary) trees and networks. Here, I will explain some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability, and algebra.

### Dynamics in Genetic Networks

Roderick Edwards and Leon Glass

Living organisms contain thousands of genes. These genes contain the genetic code for protein molecules required for the functioning of the organism. During the development of the organisms, genes are activated and deactivated at different times and in different places. Molecular biologists have developed remarkable techniques for determining the expression patterns of genes and the mechanisms regulating their expression. Mathematical modeling of these networks proves a formidable task. We show how logical circuits and piecewise linear equations are being used to meet this challenge. The mathematics not only can successfully model gene control in complex organisms but is also posing new questions for mathematical analysis.

### Computing Flows around Microorganisms: Slender-Body Theory and Beyond

Hoa Nguyen, Ricardo Cortez, and Lisa Fauci

We present the mathematical framework that governs the interaction of a forcegenerating microorganism with a surrounding viscous fluid. We review slender-body theories that have been used to study flagellar motility, along with the method of regularized Stokeslets. We investigate the role of a dinoflagellate transverse flagellum as well as the flow structures near a choanoflagellate.

### Mathematical Neuroscience

David Terman

The Hodgkin–Huxley equations model the propagation of electrical signals in a nerve axon. Here we give examples of mathematics inspired by this model. The presentation is organized chronologically, beginning with a brief introduction to the underlying biology and a description of the model. Mathematical issues related to both single cell dynamics and large neuronal networks are discussed, along with applications to neurological disease.

### Mathematical Oncology: Using Mathematics to Enable Cancer Discoveries

Trachette Jackson, Natalia Komarova, and Kristin Swanson

Mathematical and computational modeling approaches have been applied to every aspect of tumor growth from mutation acquisition and tumorigenesis to metastasis and treatment response. In this article, we discuss some of the current mathematical trends in the field and the exciting applications and challenges that lie ahead. In particular, we focus on mathematical approaches that are able to address critical questions associated with tumor initiation; angiogenesis and vascular tumor growth; and the new frontier of computer-aided, patient-specific cancer evaluation and treatment.

### Synthetic Biology: A New Frontier

Laurie J. Heyer and Jeffrey L. Poet

Synthetic biology is a new field that combines engineering principles, mathematical modeling, and molecular biology techniques to design and construct novel biological parts, devices, and systems with applications in energy, medicine, environmental science, and technology. We discuss examples of mathematical models aiding in biological investigations, biology aiding in mathematical investigations, and the two fields working together in synthetic biology to attack some of the world's biggest problems. Opportunities abound for mathematicians to contribute to this budding field.