The way immune cells pick friends from foes can be described by a classic math puzzle known as the “narrow escape problem.”
That’s a key finding arising from an international collaboration between biologists, immunologists and mathematicians, published in the journal Proceedings of the National Academy of Sciences.
The narrow escape problem is a framework often applied in cellular biology.
It posits randomly moving particles trapped in a space with only a tiny exit, and calculates the average time required for each one to escape.
“This is a new application for some familiar equations,” says co-author Justin Tzou from Macquarie University’s Department of Mathematics and Statistics.
Tzou worked with colleagues at the universities of Oxford and Cambridge in the UK, the University of British Columbia in Canada, and the University of Skövde in Sweden to analyze how potential pathogens are probed by T cells, which identify and attack invaders.
The researchers discovered that the equations used in the narrow escape problem play a key role in determining whether an immune response is triggered.
The narrow escape problem turns out to be a close cousin of the situation with T cell receptors,” Justin says. “It is about determining how long a diffusing particle remains in a certain region before escaping.”
Unlike most cells, which have relatively smooth surfaces, T cells are covered in ruffles, bumps and other protrusions.
Scientists have known for a long time that T cell receptor molecules sit on the surface of the cells to recognize enemies and trigger a hostile response.
The receptors contain molecular patterns that mirror those found on the surfaces of bacteria, tumors, and other dangerous interlopers.
But exactly how the process of recognition and triggering works – and particularly how it works so quickly and accurately – has been a mystery.
The study of the diffusive motion of ions or molecules in confined biological microdomains requires the derivation of the explicit dependence of quantities, such as the decay rate of the population or the forward chemical reaction rate constant on the geometry of the domain.
Here, we obtain this explicit dependence for a model of a Brownian particle (ion, molecule, or protein) confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. We call the calculation of the mean escape time the narrow escape problem.
This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.
Here, we present asymptotic formulas for the mean escape time in several cases, including regular domains in two and three dimensions and in some singular domains in two dimensions.
The mean escape time comes up in many applications, because it represents the mean time it takes for a molecule to hit a target binding site. We present several applications in cellular biology: calcium decay in dendritic spines, a Markov model of multicomponent chemical reactions in microdomains, dynamics of receptor diffusion on the surface of neurons, and vesicle trafficking inside a cell.
More information: Ricardo A. Fernandes et al. A cell topography-based mechanism for ligand discrimination by the T cell receptor, Proceedings of the National Academy of Sciences (2019). DOI: 10.1073/pnas.1817255116
Journal information: Proceedings of the National Academy of Sciences
Provided by Macquarie University