Cheers to Applied Mathematics !
Cheers to Applied Mathematics !
There is a nice discussion about this in Calculus and Analytic Geometry, G. B. Thomas.
Also, Prof. Steven Strogatz discusses the brachistochrone here:
The following is one of the greatest triumphs of pure mathematics — its applications to the glamorous world of IT, finance, science and engineering! Hats off to Prof. Yves Meyer, winner of Abel Prize, 2017:
Reproduced from newspaper, DNA, print edition, Mumbai, Sunday, Mar 5, 2017, (Section on Health):
Biomedical solutions via math:
(A new model combines mathematics with biology to set the stage for cancer cure and other diseases):
How do our genes give rise to proteins, proteins to cells, and cells to tissues and organs? What makes a cluster of cells become a liver or a muscle? The incredible complexity of these biological systems drives the work of biomedical scientists. But, two mathematicians have introduced a new way of thinking that may help set the stage for better understanding of our bodies and other living things.
The pair from University of Michigan Medical School and University of California, Berkeley talk of using math to understand how generic information and interactions between cells give rise to the actual function of a particular type of tissue. While the duo admit that it’s a highly idealized framework, which does not take into account every detail of this process, that’s what’s needed. By stepping back and making a simplified model based on mathematics, they hope to create a basis for scientists to understand the changes that happen over time within and between cells to make living tissues possible. It could also help with understanding of how diseases such as cancer can arise when things don’t go as planned.
Turning to Turing’s machine:
U-M Medical School Assistant Professor of Computational Medicine, Indika Rajapakse and Berkeley Professor Emeritus, Stephen Smale have worked on the concepts for several years. “All the time, this process is happening in our bodies, as cells are dying and arising, and yet, they keep the function of the tissue going,” says Rajapakse. “We need to use beautiful mathematics and beautiful biology together to understand the beauty of a tissue.”
For the new work, they even hearken back to the work of Alan Turing, the pioneering Btitish mathematician famous for his “Turing machine” computer that cracked the codes during World War II.
Toward the end of his life, Turing began looking at the mathematical underpinnings of morphogenesis — the process that allows natural patterns such as a zebra’s stripes to develop as a living thing grows from an embryo to an adult.
“Our approach adapts Turing’s technique, combining genome dynamics within the cell and the diffusion dynamics between cells,” says Rajapakse, who leads the U-M 4D —- Genome Lab in the Department of Computational Medicine and Bio-Informatics.
His team of biologists and engineers conduct experiments that capture human genome dynamics to three dimensions using bio-chemical methods and high resolution imaging.
Bringing math and the genome together
Smale, who retired from Berkeley, but is still active in research, is considered a pioneer of modelling dynamic systems. Several years ago, Rajapakse approached him during a visit to U-M, where Smale earned his undergraduate and graduate degrees. They began exploring how to study the human genome — the set of genes in an organism’s DNA — as a dynamic system.
They based their work on the idea that while the genes of an organism remain the same throughout life, how cells use them does not.
Last spring, they published a paper that lays a mathematical foundation for gene regulation — the process that governs how often and when genes get “read” by cells in order to make proteins.
Instead of the nodes of those networks being static, as Turing assumed, the new work sees them as dynamic systems. The genes may be “hard-wired” into the cell, but how they are expressed depends on factors such as epigenetic tags added as a result of environmental factors, and more.
As a result of his work with Smale, Rajapakse now has funding from the Defense Advanced Research Projects Agency (DARPA), to keep exploring the issue of emergence of function — including what happens when the process changes.
Cancer, for instance, arises from a cell development and proliferation cycle gone awry. And the process by which induced pluripotent stem cells are made in a lab —- essentially turning back the clock on a cell type so that it regains the ability to become other cell types — is another example.
Rajapakse aims to use data from real world genome and cell biology experiments in his lab to inform future work, focused on cancer and cell reprogramming.
He’s also organizing a gathering of mathematicians from around the world to look at computational biology and the genome this summer in Barcelona.
Thanks to DNA, Prof. Stephen Smale and Prof. Indika Rajapakse; this, according to me, is one of the several applications of math.