Chapter 4, Problem 23Q

Summary Introduction

To explain: How the features described in the text accurately represent the Michaelis-Menten equation.

Introduction: Enzymes are the protein molecules that catalyze the rate of reaction at different substrate concentration. Enzymes are biological protein catalysts which alter the speed of the reaction in the biological system. The chemical reactions which are catalyzed by these enzymes are studied under the enzyme kinetics. The reaction rate is measured based on the rate at which the substrate of the enzymes combines with the enzyme and forms the products.

Explanation of Solution

The Michaelis-Menten equation is used for enzyme kinetics reaction. It is given as follows,

$\text{The rate of enzyme reaction }\left(\text{V}\right)\text{ =}\frac{{\text{V}}_{\text{max}}\left[\text{S}\right]}{\left({\text{K}}_{\text{M}}\text{+}\left[\text{S}\right]\right)}\mathrm{............}\text{(1)}$$\begin{array}{l}\text{Where, V}\text{=}\text{velocity}\text{(molar/times)}\\ \text{ Vmax}\text{=}\text{maximum velocity}\\ \left[\text{S}\right]\text{=}\text{substrate}\text{concentration }\\ \text{ K}\text{=}\text{substrate}\text{concentration at}\text{half}\text{of}\text{Vmax}\end{array}$

Refer to Fig.4-35, “An enzyme’s performance depends on how rapidly it can process its substrate”, in the text book. This figure shows a hyperbolic relationship of enzymes at high substrate concentration. In this figure, the rate of enzyme reaction (V) increases with an increase in the substrate concentration to reach the maximum value (V_max). At this point, the enzymes become saturated and the rate becomes limited. K_M indicates the Michaelis rate constant and the substrate concentration on half of the maximum value (V_max). Hence, the descriptions given in the text book represents the Michaelis-Menten equation.

Conclusion

The descriptions given in the text book represents the Michaelis-Menten equation.

(A)

Summary Introduction

To determine: How the Michaelis-Menten equation from the given substrate concentrations is simplified if [S] is much smaller than the K_M.

Introduction: The Michaelis–Menten equation is generally used for enzyme kinetics reaction.

Explanation of Solution

Option (A) is given as, “[S] is much smaller than the K_M". When [S] << K_M, the term ([S] +K_M) nearly equal to the term K_M.

So the equation (1) can be simplified as follows,

$\text{The rate of enzyme reaction ( V)}\text{=}\frac{\text{Vmax [S]}}{\text{ K}}$

Therefore, the rate of enzyme reaction is directly proportional to the substrate concentration.

Conclusion

When [S] is much smaller than the K_M, then the rate is directly proportional to the substrate concentration.

(B)

Summary Introduction

To determine: How the Michaelis-Menten equation from the given substrate concentrations be simplified if [S] equals the K_M.

Introduction: K_M is the Michaelis-Menten constant. Vmax is the maximum velocity with which the enzyme catalyzes the reaction.

Explanation of Solution

Option (B) is given as, “[S] equals the K_M”.

So, Equation (1) can be simplified as follows,

$\begin{array}{l}\text{The rate of enzyme reaction ( V)}\text{=}\frac{\text{Vmax K}}{\text{ K}\text{+}\text{K}}\\ \text{=1/2Vmax}\\ \end{array}$

Therefore, the rate of the reaction is half of the maximal rate.

Conclusion

If [S] equals the K_M, then the rate of enzyme reaction is half of the maximal rate.

(C)

Summary Introduction

To determine: How the Michaelis-Menten equation from the given substrate concentrations would be simplified if [S] is much smaller than the K_M.

Introduction: K_M is the Michaelis-Menten constant and [S] is the substrate concentration.

Explanation of Solution

Option (C) is given as, “[S] is much smaller than the K_M”. When [S] >> K_M, the term ([S] +K_M) nearly equal to the term [S].

So, Equation (1) can be simplified as follows,

$\begin{array}{l}\text{The rate of enzyme reaction ( V)}\text{=}\frac{\text{Vmax [S] }}{\text{ [S]}}\\ \text{= Vmax}\\ \end{array}$

Therefore, the maximum rate of the reaction occurs at its maximal rate.

Conclusion

If [S] is much smaller than the K_M, the maximum rate of the reaction occurs at its maximal rate.