Viral Packaging

Print or copy the triangle template onto the cardstock (the two strips of 20 triangles each should fit onto one standard 8.5 x 11-inch [A4] sheet). Only one strip is required for each model.

Cut out the individual equilateral triangles from the template. Experiment with different ways of taping up to 20 of the triangles together so that they completely enclose the yarn and cotton balls.

Viruses are composed of nucleic acid genomes and interior proteins that are surrounded by a protective protein shell called a capsid.

Instead of making the capsid out of one giant protein, viruses typically utilize many identical copies of the same protein that combine together to form this outer shell. This way, the virus can be economical, using one gene repetitively to make many small proteins instead of devoting a large portion of its genome to making a large protein coat.  

When making your paper container, you may have found a shape that uses several triangles to enclose the yarn and cotton balls, which represent a virus's nucleic acid and interior proteins, respectively. The majority of viruses are composed of triangular protein sub-units that associate to form an icosahedron—a 20-sided shape. This shape helps the virus to minimize its surface-area-to-volume ratio, which allows it to carry the most genetic material and internal proteins inside a given protein shell.

One way to categorize viruses is by whether or not they are surrounded by a membrane, called a viral envelope. A large number of viruses that infect humans have envelopes, including HIV, the virus that causes AIDS. You can place your capsid into the plastic bag to model these enveloped viruses.

Coronaviruses, such as the one that causes COVID-19, also have envelopes. Unlike HIV, they have helical interiors, which is another common shape for viruses.

Icosahedrons have three axes of rotational symmetry—two-fold (180°), three-fold (120°) and five-fold (72°). At the two-fold axis of symmetry, the shape will look the same if it’s rotated on this axis 180°. Can you find all three axes of symmetry?

This model can also be used to illustrate the concept of gene therapy. Gene therapy seeks to treat diseases caused by defective or deficient proteins by introducing genetic material as medicine. Instead of manufacturing and injecting a functional version of the protein, gene therapists modify viruses so that their genomes contain a copy of the gene that encodes for the correct form of the diseased protein. Viral genes are removed from the genome, so that when the virus—now a viral vector—enters the target cell, it produces correct copies of the therapeutic protein instead of creating new virions (complete virus particles).

Gene therapy has been used in clinical trials to treat a wide range of diseases, including HIV, hemophilia, cancer, and “bubble-boy disease” (Severe Combined Immune Deficiency).

If you are a teacher making these models with your class, we recommend one model for every 1–2 students.

This Science Snack is part of a collection that showcases female mathematicians and math educators whose work aids or expands our understanding of the phenomena explored in each Snack.

Source: Wikimedia Commons

Every cell in your body has two yards (six feet) of DNA inside. How does your body pack in all that DNA? Mariel Vázquez (pictured above), a mathematical biologist, studies the shape of DNA. Using a type of math called topology, she simplifies DNA into basic shapes like curves, ribbons, and lines connecting dots to better understand how DNA knots and coils. Vázquez grew up in Mexico and studied at the National Autonomous University of Mexico (UNAM) and UC Berkeley. In our Science Snack Viral Packaging, you can see if you can fit nucleic acids and proteins into a small package.