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What are some of the original examples of uses of differential equations for modeling and analyzing chemical reactions, particularly those relevant to biochemistry, involving proteins and enzymes? Michaelis and Menten's work in 1910s is one example.
What are the earlier examples or antecedents to this work? (references to good historical accounts of this modeling endeavor would also be helpful).
An early result in chemical kinetics was Waage and Guldberg's law of mass action, that relates the speed of a chemical reaction to the quantity of the reacting substances. The field came of age in 1884 with van'tHoff's "Etudes de dynamique chimique", which is filled with differential equations (see the English 1895 translation), he was awarded the first Nobel Prize in Chemistry in 1901. See Wikipedia's History of Chemical Kinetics.
The historical context of the Michaelis-Menten 1913 paper Die Kinetik der Invertinwirkung (Biochem. Z. 49, 333–369) is described in The Original Michaelis Constant Johnson and Goody , who translated it into English. According to them, their chief precursor was Victor Henri in his Lois Générales de l’Action des Diastases (1903):
"Perhaps the unsung hero of the early history of enzymology is Victor Henri, who first derived an equation predicting the relationship between rate and substrate concentration based upon a rational model involving the formation of a catalytic enzyme−substrate complex. However, as Michaelis and Menten point out, Henri made two crucial mistakes, which prevented him from confirming the predicted relationship between rate and substrate concentration. He failed to account for the slow mutarotation of the products of the reaction (equilibration of the α and β anomers of glucose), and he neglected to control pH. Thus, errors in his data precluded an accurate test of the theory. Otherwise, we would probably be writing about the Henri equation."
A more distant inspiration was the Henderson-Hasselbalch equation for pH dependence on acidity derived by Henderson in 1908 and turned into the logarithmic form that Michaelis and Menten imitated by Hasselbalch.
A celebrated textbook on enzyme kinetics is
Fundamentals of Enzyme Kinetics by Athel Cornish-Bowden, Fourth Edition, 2012, Wiley-VCH, Weinheim.
As a sample, one can download the first chapter for free, which intersperses historical references with a discussion of the basic principles of enzyme kinetics.
From that chapter, I think this paragraph is most relevant to the question:
Chemical kinetics as a science began in the middle of the 19th century, when Wilhelmy was apparently the first to recognize that the rate at which a chemical reaction proceeds follows definite laws, but although his work paved the way for the law of mass action of Waage and Guldberg, it attracted little attention until it was taken up by Ostwald towards the end of the century, as discussed by Laidler. Wilhelmy realized that chemical rates depended on the concentrations of the reactants, but before considering some examples we need to examine how chemical reactions can be classified.
L. F. Wilhelmy (1850) “Über das Gesetz, nach welchem die Einwirkung der Säuren auf Rohrzucker stattfindet” Poggendorff’s Annalen der Physik und Chemie 81, 413–433, 499–526, https://doi.org/10.1002/andp.18501571203
P. Waage and C. M. Guldberg (1864) “Studier over Affiniteten” Forhandlinger: Videnskabs-Selskabet i Christiana, 35–40, 111-120. There is an English translation by H. I. Abrash at http://chimie.scola.ac-paris.fr/sitedechimie/hist_chi/text_origin/guldberg_waage/Concerning-Affinity.htm [The link is dead when I'm posting this, this one works: http://www.chemteam.info/Chem-History/Concerning-Affinity.html ]
K. J. Laidler (1993) The World of Physical Chemistry, pages 232–289, Oxford University Press, Oxford
Adding to Bence Mélykúti's response, you can find a discussion of the relative contributions of Michaelis and Menten on the one hand and Henri on the other in a paper by Deichmann et al. "Commemorating the 1913 Michaelis-Menten paper Die Kinetik der Invertinwirkung: three perspectives" FEBS J. 281 (2014) 435-463. The "three perspectives" have different authors because there wasn't full agreement between the four authors of the whole paper about the amount of credit to be assigned, but the editor of the journal wanted one paper, not three.
