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What is the optimum temperature, temperature stability and pH stability of beta-galactosidase from E. coli?  Is the enzyme inhibited by either glucose or galactose?”

We don't have a profile of activity vs temperature, but I'd be surprised if the activity at 37°C is much below the maximum, since that's the temperature where everyone uses it.

As for temperature stability, it depends on what buffer it is stored in.  In our storage buffer, we've left it at 37°C for 2-3 days without significant loss of activity.  According to Wallenfels and Weil (1972), 50% of the activity remains after 40 minutes at 55°C when Mg and mercaptoethanol are present; without these, the activity is lost within 10 minutes.

pH Stability:  According to the same review, the enzyme is stable at 40°C between pH 6 and 8.  They say the stability decreases sharply below pH 6 and more gradually above pH 8.

As for inhibition, Kuby and Lardy (1952) reported that galactose is a competitive inhibitor with a Ki of 9.4 x 10-3, whereas glucose is a non-competitive inhibitor that doesn't have much effect on activity at concentrations below 10 mM.
 

Wallenfels and Weil (1972) The Enzymes, vol. 7; P.D. Boyer, Ed. Academic Press, NY; pp 617-663.

Kuby and Lardy (1952) Journal of the American Chemical Society, vol. 75; pp. 890-896.

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