| Summary: | <p>We describe an approach to formulating the kinetic master
equations of the time evolution of NMR signals in reacting (bio)chemical
systems. Special focus is given to studies that employ signal enhancement
(hyperpolarization) methods such as dissolution dynamic nuclear polarization
(dDNP) and involving nuclear spin-bearing solutes that undergo reactions
mediated by enzymes and membrane transport proteins. We extend the work
given in a recent presentation on this topic (Kuchel and
Shishmarev, 2020) to now include enzymes with two or more substrates and
various enzyme reaction mechanisms as classified by Cleland, with particular
reference to non-first-order processes. Using this approach, we can address some pressing questions in the field from a theoretical standpoint. For
example, why does binding of a hyperpolarized substrate to an enzyme <i>not</i> cause
an appreciable loss of the signal from the substrate or product? Why does
the concentration of an unlabelled pool of substrate, for example <span class="inline-formula"><sup>12</sup></span>C
lactate, cause an increase in the rate of exchange of the <span class="inline-formula"><sup>13</sup></span>C-labelled pool? To what extent is the equilibrium position of the reaction perturbed
during administration of the substrate? The formalism gives a full
mechanistic understanding of the time courses derived and is of relevance to
ongoing clinical trials using these techniques.</p>
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