Alfisol, Item 003:

LMMpro, version 2.0 The Langmuir Optimization Program plus The Michaelis-Menten Optimization Program

What does LMMpro do? List of Published Articles Using LMMpro History of Linear Regression Methods Linear Regressions Deriving Linear Regression Methods

NLLS Regressions v-NLLS Optimization n-NLLS Optimization Axes Conversion Factor (ACF) Types of Errors

Correlation Coefficient (r2) Correlation Coefficient (η2) Correlation Coefficient (η*2) Do's & Don't's References Cited

v-NLLS Optimization

v-NLLS stands for vertical nonlinear least squares. This regression method optimizes the parameters of the equation without converting the equation into another form or shape. The best fit is that equation that yields the smallest error. The error of each datum point is defined as the vertical difference between the predicted value and the actual value. The predicted value is the value calculated for a given x_i, and the actual value is the value measured for a given x_i.

For the Langmuir Equation, the optimization is as follows:

1. Let Σ ε_i² = Σ error² = Σ (measured - predicted)²
2. Substitute the Langmuir Equation for predicted values:
   

   Σ εi2 = Σ [ Γi - Γmax K ci 1 + K ci ]2

3. Expand the square term:
   

   Σ εi2 = Σ [ Γmax K ci 1 + K ci ]2 - Σ [ 2 Γi Γmax K ci 1 + K ci ] + Σ Γi2

4. To optimize Γ_max, set the first derivative equal to zero and solve:
   

   d(Σ εi2) dΓmax = 0 = 2 Γmax Σ [ K ci 1 + K ci ]2 - Σ [ 2 Γi K ci 1 + K ci ] + 0

5. Solving Equation [4] for Γ_max yields:
   

   Σ [ Γi K ci 1 + K ci ] Γmax = Σ [ K ci 1 + K ci ]2

6. If K is known (or fixed by the user), then use Equation [5] above to get the optimized Γ_max value.
7. If K is not known, guess K values, and then use Equation [5] to get the corresponding best Γ_max value. Finally, use Equation [2] to determine the error that corresponds to the guesses made. Repeat the process by incrementally increasing or decreasing the K value guesses until the minimum error condition is found.
8. If Γ_max is known (or fixed by the user), then Equation [5] is not needed. The best value of K, however, is still determined with Equation [2] and an iteration loop to find the condition with the minimum error.

For the Michaelis-Menten Equation, the optimization is as follows:

1. Let Σ ε_i² = Σ error² = Σ (measured - predicted)²
2. Substitute the Michaelis-Menten Equation for predicted values:
   

   Σ εi2 = Σ [ vi - Vmax Si KM + Si ]2

3. Expand the square term:
   

   Σ εi2 = Σ [ Vmax Si KM + Si ]2 - Σ [ 2 vi Vmax Si KM + Si ] + Σ vi2

4. To optimize V_max, set the first derivative equal to zero and solve:
   

   d(Σ εi2) dVmax = 0 = 2 Vmax Σ [ Si KM + Si ]2 - Σ [ 2 vi Si KM + Si ] + 0

5. Solving Equation [4] for V_max yields:
   

   Σ [ vi Si KM + Si ] Vmax = Σ [ Si KM + Si ]2

6. If K_M is known (or fixed by the user), then use Equation [5] above to get the optimized V_max value.
7. If K_M is not known, guess K_M values, and then use Equation [5] to get the corresponding best V_max value. Finally, use Equation [2] to determine the error that corresponds to the guesses made. Repeat the process by incrementally increasing or decreasing the K_M value guesses until the minimum error condition is found.
8. If V_max is known (or fixed by the user), then Equation [5] is not needed. The best value of K_M, however, is still determined with Equation [2] and an iteration loop to find the condition with the minimum error.

Note that the v-NLLS regression will optimize the parameters assuming that minimizing the vertical error yields the best results. This method does have some bias in favor of any region of the curve where the vertical changes are most pronounced. In other words, for a parabolic equation such as the Langmuir Equation or the Michaelis-Menten Equation, the v-NLLS regression has some bias toward the lower left region of the graph.

Also note that the v-NLLS regression does not result in an optimized curve with the data evenly distributed above and below the curve. That is, the sum of the errors above the curve is not the same value as the sum of the errors below the curve. Least squares regressions will only balance the data around the curve if the function has a constant term. For example, in y = f(x) + b, b is the constant term. The Langmuir Equation and the Michaelis-Menten Equation are lacking the contant term (b).