Cells actually like Trigonometry

One of our first childhood lessons is that fitting a square peg in a round hole doesn’t work. Shape matters. For cells, the shape of objects also matters, whether these objects are cellular debris, viruses, bacteria or biological signaling complexes. All cells move molecules or whole objects from the outside to the inside through multiple biological mechanisms. Viruses, bacteria and single celled parasites actively seek out and invade cells. Some defensive cell types – phagocytes - bind objects and can actively engulf them for destruction or use the object’s information to generate protective antibodies. Just how this process works has recently been elegantly described using basic principles of thermodynamics, trigonometry and calculus, but cells don’t need advanced degrees to make it work.

In the May 31, 2016 edition of PNAS, two British scientists, David M. Richards and Robert G. Enders, present their mathematical interpretation of how different biologically relevant shapes can affect this internal movement - endocytosis - with implications for human disease, cellular responses and future drug development.

From their modeling, they have figured out the rates of cell entry for particles of all sizes and shapes. They have defined biological shapes that roughly fall into several discrete categories: spheres, ellipses, cylinders and hourglass-shaped forms. Further, they break down entry into receptor-driven or actin-driven mechanisms, determining which shape moves the fastest inward and from which orientation.

Think of pushing your finger into a full balloon. The balloon curves around your finger and to gain “entry” the balloon must “pop”. Just like the balloon, the outer coating of cells – the plasma membrane – must curve and deform to accommodate object endocytosis, but must also be self-healing to maintain cellular integrity.

This membrane curvature is the defining cellular feature portion of endocytosis and is governed by the thermodynamic principles of free energy and entropy. The rate of entry is determined directly by the object and is governed by the principles of trigonometry; the radius of the object and the angle of attachment, and calculus; the function of time over distance.

As expected, the rounder the object, the faster and easier the entry. The fatter the ellipse, the less likely it is to be internalized, unless it attaches point first. Think a torpedo or a surfboard sinking into quicksand. These may seem like somewhat obvious conclusions, but their model is the first to understand why non-spherical objects can enter cells quickly if they attach at an angle to flat surfaces. Many human bacterial pathogens exploit this principle through long filamentous structures, which can avoid endocytosis through shape alone. The strange shapes of other pathogens, such as the contorted shape of Ebola virus, are highly evolved patterns to gain rapid cell entry for infection and rapid propagation. Have humans also evolved different shapes to influence intracellular signaling or nutrient supply?

From microscopy studies, signaling proteins or critical cellular components are packaged in spherical vesicles, like neuronal synaptic vesicles or lipoprotein particles. However, this research seems to suggest that we need to take a closer look at some human transport diseases and visualize in 3-D what each particle actually looks like. Besides just genetic changes, some diseases, like hypercholesterolemia may also be caused by less than spherical lipoprotein particles which would have reduced receptor-driven endocytosis to target cells.

On the other hand, like some pathogens, we can exploit shape to our advantage by designing drug delivery systems using highly favorable nanoparticle shapes and sizes. For increased drug life, rod shaped cylinders will prolong blood levels by reduced endocytosis. While small spherical particles are rapidly endocytosed, releasing drugs to needy cells quickly.

So next time your teen whines about math, just remind them that their cells are already math experts. They just need to catch up.

Reading:

Target shape dependence in a simple model of receptor-mediated endocytosis and phagocytosis. Richards DM, Endres RG. Proc Natl Acad Sci U S A. 2016 May 31;113(22):6113-8. doi: 10.1073/pnas.1521974113