Multidimensional inflection points

Caplan and Essig (1989) provided a simple model of active ion transport, having properties consistent with a multidimensional inflection point when one of the variables was the electrical potential difference across the membrane. A multiple inflection point may not be unique other conditions may exist where flows Ji and J2 simultaneously pass through an inflection point of on variation of Xi at constant X2, and vice versa. It is frequently not possible to vary both forces independently in biological systems. However, if Xi can be controlled experimentally along a proper pathway, while X2 is kept constant, the response of the flows to change in Xi will permit a thermodynamic characterization of the system. 

Under sufficient conditions, an enzyme mechanism exhibits a multidimensional inflection point around which a set of linear flow-force equations may be valid over an extended range outside of equilibrium. Enzyme catalyzed reactions obey approximately the Michaelis-Menten rate equation, which can show a high degree of linearity in the chemical affinity for certain values of substrate concentrations. 

Sheppard, A.N., Acevedo, O. Multidimensional exploration of valley-ridge inflection points on potential energy surfaces. J. Am. Chem. Soc. 2009,131, 2530 40. 

Figure 3. Stationary points and inflection surfaces in a multidimensional potential energy surface.

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