Nonlinear Regression

Nonlinear Regression

Example: Puromycin

• Puromycin is an antibiotic used in bio-research to select cells modified by genetic engineering
• The Michaelis-Menten model for enzyme kinetics relates the initial velocity of an enzymatic reaction to the substrate concentration x by the equation
• At high concentrations, the velocity is essentially constant (θ1), while at low concentrations it is a linear function of the concentration

Puromycin Kinetics

The model:

Model Linearization

• The initial model reads
• This can be rearranged to
• Renaming the variables yields a linear model

Linearized Model: Fit

Regression Line
b1 = 0.0051072
b2 = 0.00024722

Regression from Linearized Model

Regression from linearized model
q1 = 195.8
q2 = 0.048407

Nonlinear Regression

• The objective function of non-linear regression reads where y is the vector of responses, x is the vector of observations, θ is the vector of parameters and f is the nonlinear model function
• In the 2-parameter case, we can plot S(θ) as a function of the parameter values to look for a minimum

Objective Function S(θ)

Minimum
Estimated value of q from linearization

Minimization of S(θ)

• If we linearize the model with respect to the parameters, we get where J(θ0) is the Jacobian matrix of f with respect to θ, evaluated at θ0
• In this case, the residuals are
• If we now define the objective function in terms of the residuals, we can apply the Gauss-Newton method

Analogy: