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I understand that an enzyme lowers the activation energy, allowing the reaction to run faster, however I am not sure, how it may be favorable during the addition of heat energy. Wouldn't that lower the activity of the enzyme since the enzyme is sensitive to heat?

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Enzymes have two benefits over raising the temperature:

First, reducing the activation energy by reusable enzymes is far more energy-efficient, apart from the fact that the required temperature would vary for different reactions.

Second, enzymes are able to be controlled by allosteric mechanisms, which facilitates metabolic control by feedback loops, feedforward motivs and receptor-mediated processes. In the example of glycolysis this enables the cell to fine-control its supply with energy equivalents (see figure from Wikimedia Commons below). This mechanism may be the most important function of enzymes.

enter image description here

Reference

  • William C. Plaxton: Principles of Metabolic Control. In Kenneth B. Storey (ed.): Functional Metabolism: Regulation and Adaptation, John Wiley & Sons, Inc. 2004
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The enzyme does only one thing: It lowers the activation energy which is necessary for the specific reaction to happen. It does not change the speed of the reaction, nor does it change the rection enthalpy or the quilibrium of the reaction. See the figure for clarification (from here):

enter image description here

Without the enzyme the Ea (activation energy) would be so high, that the reaction would be possible in the cell. Either because this high amount of energy is not available or because the amount of energy would destroy the cell or the reaction. So the lowering of Ea allows this specific reaction to take place.

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By reactions there is an energy barrier called activation energy. Your substrate molecules have kinetic energy (they move, rotate, etc...), so when they collide properly, their kinetic energy can be used to overcome the activation energy and so the reaction takes place. According to the Arrhenius equation the reaction rate depends on the temperature, which is evident, because the kinetic energy of your molecules is higher by higher temperature. Enzymes lower the activation energy, because they help the substrates to collide with the proper orientation.

Enzymes are mostly composed of amino acid chains. These chains must have proper folding. Increasing (or decreasing) the temperature can disrupt this folding and make the enzyme useless. This inactivation can be reversible or irreversible depending on the enzyme type, the temperature and other environmental factors e.g. pressure, osmotic pressure, etc...

To answer your question, every enzyme have an optimum working temperature. At this temperature the inactivation is slow enough to be compensated by the increased reaction rate.

optimum temperature

  • figure 1 - optimum temperature of a single enzyme - ref

optimum temperatures of multiple enzymes

  • figure 2 - optimum temperatures of multiple enzymes - ref
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