Simple One to One Binding: Simulation

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: $C = [conc] = [A]$
    

  • $R_{max} = [AL] + [L]$

Simulation & Plot

# simulate the data use the following parameters
par = list(kon  = 2e2, koff = 1e-2, rmax = 1)
datsim = list()
datsim$concs <- 1e-5 * (2^(0:5));
datsim$xdata = seq(0, 300, length.out = 1501); # time 
datsim$t2   = 150
datsim$ydata = NULL

# simulation
xySimulated <- dfsimple1to1(par = par, dat = datsim, noise = 0.01) 

# plot the simulation
# ySimulated$Time = time; 
xy <-reshape2::melt(data = xySimulated, 
                     id.vars = "Time", 
                     measure.vars = rev(1:6), 
                     variable.name = "Conc")

g <- ggplot()  + xlab("Time (sec)") + ylab("Response (nm)") +
    labs(linetype= 'title') + 
    ylim(-0.025,1) + 
    theme_classic() + 
    theme(legend.position=c(0.9, 0.65),
          legend.text=element_text(size = rel(1)),
          legend.key.size=unit(0.9,"line"));
g <- g + geom_line(data = xy, aes(x = Time, y = value, color = Conc));
print(g)

Fitting

# init
initPar_test = list(kon =1, koff = 1, rmax = 1) 
dat_test <- datsim
dat_test$t2 <- 149.9 # t2 is the beginning of the diassociation. 
dat_test$lowerBound = list(kon =1e-04, koff=1e-04, rmax = 0.01);
dat_test$upperBound = list(kon =1e04, koff=1e04, rmax = 10);
dat_test$datF       = within(xySimulated, rm("Time")); 

# Fit
fit <- kinfit(par = initPar_test, dat = dat_test, model = "simple1to1")
names(fit)

## [1] "par"      "hessian"  "fvec"     "info"     "message"  "diag"    
## [7] "niter"    "rsstrace" "deviance"

par # simulation parameters

## $kon
## [1] 200
## 
## $koff
## [1] 0.01
## 
## $rmax
## [1] 1

fit$par # paramenters after fitting

## $kon
## [1] 203.0149
## 
## $koff
## [1] 0.009974569
## 
## $rmax
## [1] 0.9894468

cbind(simulation= par, init = initPar_test, fitting = fit$par)

##      simulation init fitting    
## kon  200        1    203.0149   
## koff 0.01       1    0.009974569
## rmax 1          1    0.9894468

#prodict and plot
predFit <- dfsimple1to1(par = fit$par, dat = dat_test, noise = 0)
predFit <- reshape2::melt(predFit, id.vars = "Time")
g + geom_line(data=predFit, aes(x = Time, y = value, group = variable) )