Protein-Ligand Biosensor Data Fitting

1.1 Simple one to one model:

$A + L \overset{k_{on}}{\underset{k_{off}}\rightleftarrows} AL$

A: the protein to be analyzed.
L: Ligand immobilized on the pin
AL: Complex formed on the pin

Below is the rate equation:

${d[AL] \over dt} = k_{on} [A] [L] - k_{off} [AL]$

other forms:
- ${dy \over dt} = k_{on} [conc] (R_{max} - R) - k_{off} y$
- ${dy \over dt} = k_{on} [conc] R_{max} - (k_{on} [conc] + k_{off}) y$
- ${dy \over dt} = k_{on} C R_{max} - (k_{on} C + k_{off}) R$
Parameters:
- Response (nm): $R = [AL] = y$
- Ligand on the sensor: $[L] = R_{max} - R$
- Conc. of the analytes: $C = [conc] = [A]$
- $R_{max} = [AL] + [L]$

1.2 Dimerization model

1.2.1 Dimerization model in equilibration

On sensor tip:

$A + B \overset{Kd_1}{\rightleftarrows} AB \;\;$ $\Rightarrow Response (nm) = \frac{R_{max}*[A]}{Kd_1+[A]}$

In solution :

$A + A \overset{Kd_2}{\rightleftarrows} AA$
$Kd_2=\frac{[A]*[A]}{[AA]}=\frac{[A]^2}{[AA]}\Rightarrow [AA]=\frac{[A]^2}{Kd_2}$ & $[At] = [A] + 2[AA]$
$\Rightarrow [At]=[A]+2*\frac{[A]^2}{Kd_2} \Rightarrow 2[A]^2 + Kd_2 * [A] -Kd_2*[At] = 0$
$\Rightarrow [A] = \frac{-Kd_2+\sqrt{Kd_2^2+8Kd_2*[At]}}{4}$

Combine above equations together, assumming $Kd_1 = Kd_2 = Kd$ :

R e s p o n s e (n m) = \frac{R_{m a x} * (- K d + \sqrt{K d^{2} + 8 K d * [A t]})}{3 K d + \sqrt{K d^{2} + 8 K d * [A t]}}

$Response(nm)=\frac{R_{max}*(-Kd+\sqrt[]{Kd^2+8Kd*[At]})}{3Kd+\sqrt[]{Kd^2+8Kd*[At]}}$

1.2.2 Kinetic model of dimerization

D1D2 dimerization in solution:

$A + A \overset{Kd_2}{\rightleftarrows} AA$
$\Rightarrow [A] = \frac{-Kd_2+\sqrt{Kd_2^2+8Kd_2*[At]}}{4}$
Where: $[A]$ is the D1D2 monomer contration.

On sensor tip:

$A + B \overset{k_{on}}{\underset{k_{off}}\rightleftarrows} AB \; (Kd_1)$
$\frac{d[AB]}{dt} = k_{on} [A] [B] - k_{off} [AL]$

$\Rightarrow$ $\frac{dy}{dt} = k_{on} [A] (R_{max} - R) - k_{off} * R$ ( Called “simple 1:1 model” )

$\Rightarrow$ $\frac{dy}{dt} = k_{on} \frac{-Kd_2+\sqrt{Kd_2^2+8Kd_2*[At]}}{4} (R_{max} - R) - k_{off} * R$
Where $Kd_{2} = \frac{k_{off}}{k_{on}}$ , assuming $Kd_1 = Kd_2$ .

2. Fiting algorithm using numerical intergration

2.1 Input data:

xdata: time (sec)
ydata: response (nm)

2.2 Procedure:

rate equation: $\frac{dy}{dt}$
calculated response or f(xdata, params) = cumsum (dy*dt)
Error func: resid(params) = calculated response f(xdata, params) - measured response (ydata)
- should take at least one (possibley length N vector ) argument
- and returns M floating numbers
- it must not return NA, NaN or fitting might fail.
sum square of the errors: sum(resid ** 2)
optimization, minimize:
- nls in R or scipy.optimize.leastsq in python
- minipack.lm::nlsLM, LMA, levenberg-Maquardt algorithm
- GenSA, simulatd annealing
- DEoption

2.3 Minimization:

2.3.1 nls.lm from minpack.lm:

minimize the sum square of the vector returned by gthe function fn
use a modification of the levenberg-marquardt algorithm.
usage: nls.lm(par, lower = NULL, upper = NULL, fn, jac = NULL, control = nls.lm.control(), …)
- par: a list of numeric vector with initial estimates.
- fn: a function that returns a vector of residuals, the sum squae of which is to be minimized.
- fn: the first argument of fn must be par.
- jac: a function to return the Jacobian for the fn function.
- control: see nls.lm.control
- …: further arguments to be passed to fn and jac
both functions fn and jac (if provided) must return numeric vectors.
the length of the vector returned by fn must not be lower than the length of par.

2.3.2 GenSA, Generalized Simulated Annealing:

searches for global minimum of a very complex non-linear objetive function with a very large number of optima.
Usage: GenSA(pan, fn, lower, upper, control=list(), …)
- par: initial values for the compnents to be optimized.
- par: Default is NULL, in which case, default values will be generated automatically.
- fn: a function to be minimized.
- fn: the first argument of fn should be the vector of parameters over which minimization is to take place.
- fn: fn should return a scalar result.
- lower: vector with length of par. Lower bounds for components.
- upper: vector with length of par. Upper bounds for components.
- …: allows the user to pass additinal arguments to the function fn.
- control: control parameters, including temperature, maxit, etc.

Protein-Ligand Biosensor Data Fitting

Ren-Huai Huang

2016-02-24

1. Model equation:

1.1 Simple one to one model:

$R_{max} = [AL] + [L]$

1.2 Dimerization model

1.2.1 Dimerization model in equilibration

1.2.2 Kinetic model of dimerization

2. Fiting algorithm using numerical intergration

2.1 Input data:

2.2 Procedure:

2.3 Minimization:

2.3.1 nls.lm from minpack.lm:

2.3.2 GenSA, Generalized Simulated Annealing:

Protein-Ligand Biosensor Data Fitting

Ren-Huai Huang

2016-02-24

1. Model equation:

1.1 Simple one to one model:

Rmax=[AL]+[L]Rmax=[AL]+[L]R_{max} = [AL] + [L]

1.2 Dimerization model

1.2.1 Dimerization model in equilibration

1.2.2 Kinetic model of dimerization

2. Fiting algorithm using numerical intergration

2.1 Input data:

2.2 Procedure:

2.3 Minimization:

2.3.1 nls.lm from minpack.lm:

2.3.2 GenSA, Generalized Simulated Annealing:

$R_{max} = [AL] + [L]$