In huangrenhuai/pldfit: SensorGram kinetic data fitting, especialy for BLI and SPR data

1. Model equation:

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 $Kd_1 = Kd_2 = Kd$:
$Response(nm)=\frac{R_{max}*(-Kd+\sqrt[]{Kd^2+8Kd*[At]})}{3Kd+\sqrt[]{Kd^2+8Kd*[At]}}$

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.

huangrenhuai/pldfit documentation built on May 17, 2019, 9:10 p.m.