Michaelis-Menten kinetics

This year is the one hundred anniversary of the Michaelis-Menten equation, which was published in 1913 by German born biochemist Leonor Michaelis and Canadian physician Maud Menten. Menten was one of the first women to obtain a medical degree in Canada and travelled to Berlin to work with Michaelis because women were forbidden from doing research in Canada. After spending a few years in Europe she returned to the US to obtain a PhD from the University of Chicago and spent most of her career at the University of Pittsburgh. Michaelis also eventually moved to the US and had positions at Johns Hopkins University and the Rockefeller University.

The Michaelis-Menten equation is one of the first applications of mathematics to biochemistry and perhaps the most important. These days people, including myself, throw the term Michaelis-Menten around to generally mean any function of the form

f(x)= \frac {Vx}{K+x}

although its original derivation was to specify the rate of an enzymatic reaction.  In 1903, it had been discovered that enzymes, which catalyze reactions, work by binding to a substrate. Michaelis took up this line of research and Menten joined him. They focused on the enzyme invertase, which catalyzes the breaking down (i.e. hydrolysis) of the substrate sucrose (i.e. table sugar) into the simple sugars fructose and glucose. They modelled this reaction as

E + S \overset{k_f}{\underset{k_r}{\rightleftharpoons}} ES \overset{k_c}{\rightarrow }E +P

where the enzyme E binds to a substrate S to form a complex ES which releases the enzyme and forms a product P. The goal is to calculate the rate of the appearance of P.

Continue reading

Talk in Taiwan

I’m currently at the National Center for Theoretical Sciences, Math Division, on the campus of the National Tsing Hua University, Hsinchu for the 2013 Conference on Mathematical Physiology.  The NCTS is perhaps the best run institution I’ve ever visited. They have made my stay extremely comfortable and convenient.

Here are the slides for my talk on Correlations, Fluctuations, and Finite Size Effects in Neural Networks.  Here is a list of references that go with the talk

E. Hildebrand, M.A. Buice, and C.C. Chow, `Kinetic theory of coupled oscillators,’ Physical Review Letters 98 , 054101 (2007) [PRL Online] [PDF]

M.A. Buice and C.C. Chow, `Correlations, fluctuations and stability of a finite-size network of coupled oscillators’. Phys. Rev. E 76 031118 (2007) [PDF]

M.A. Buice, J.D. Cowan, and C.C. Chow, ‘Systematic Fluctuation Expansion for Neural Network Activity Equations’, Neural Comp., 22:377-426 (2010) [PDF]

C.C. Chow and M.A. Buice, ‘Path integral methods for stochastic differential equations’, arXiv:1009.5966 (2010).

M.A. Buice and C.C. Chow, `Effective stochastic behavior in dynamical systems with incomplete incomplete information.’ Phys. Rev. E 84:051120 (2011).

MA Buice and CC Chow. Dynamic finite size effects in spiking neural networks. PLoS Comp Bio 9:e1002872 (2013).

MA Buice and CC Chow. Generalized activity equations for spiking neural networks. Front. Comput. Neurosci. 7:162. doi: 10.3389/fncom.2013.00162, arXiv:1310.6934.

Here is the link to relevant posts on the topic.