non-linear optimisation with constraints - r

Question

I try to find the initial values of the system of differential equations. My parameter estimation returns negative values for some of the initial values, but all of them have to be bigger than or equal 0. On top of that I would like to specify that initial conditions should be as follow:
A>0
B>0<1
C>0<1
D>0<1
E=0
F=0
G>0<5
H>0<100
All results obtained should be positive (concentrations).
How can I introduce constraints? I found some info about how to use it while using optim, but I can't find any info relevant to my problem.
Any help will be appreciated.
Malgosia
library(ggplot2)
library(reshape2)
library(deSolve)
library(minpack.lm)
time<-seq(0, 5, by=1)
P4=c(0.018,0.028,0.201,0.888,0.934,2.044)
E2=c(0.355,0.28,0.665,0.995,0.934)
FSH=c(0.408,0.226,0.126,0.224,0.123)
signal<-as.data.frame(list(time=time,P4))
input<-approxfun(P4,rule=2)
df<-data.frame(time,E2,FSH)
df
names(df)=c("time","E2","FSH")
#plot data
tmp=melt(df,id.vars=c("time"),variable.name="species",value.name="conc")
ggplot(data=tmp,aes(x=time,y=conc,color=species))+geom_point(size=4)
#rate function
rxnrate=function(t,c,parms){
#c is the concentration of species
#derivatives dc/dt are computed below
P4<-input(t)
r=rep(0,length(c))
r[1]<-12.84*1/(1+(P4/5)^5)*((c["G"]/3)^10)/(1+(c["G"]/3)^10)-2.14*c["A"];
r[2]<-75*(((c["A"]/5)^10)/(1+(c["A"]/5)^10))-8.56*c["B"];
r[3]<-12.84*(1/(1+(c["H"]/2)^2))*(1/(1+(c["G"]/10)^10))+1*(c["A"]/1)^1)/(1+(c["A"]/1)^1))-2.14*c["C"];
r[4]<-0.0107*c["C"]+0.321*c["C"]*c["D"]- 0.749*c["D"];
r[5]<-0.749*c["D"]- 0.749*c["E"]+0.214*c["C"]*c["E"]^2;
r[6]<-0.749*c["E"]-0.749*c["F"]+0.214*c["B"]*c["F"]^2;
r[7]<-0.0107 + 2.14*c["E"] + 10.7*c["F"]-1.07*c["G"];
r[8]<-0.0107+3*c["E"]+ 3.21*c["F"]+3.21*c["G"]-1.07*c["H"];
return(list(r))
}
cinit<-c(A=0.3947,B=0.40727,C=0.408,D=0.17828,E=0,F=0.05,G=0.355,H=0.9);
t=df$time;
out=ode(y=cinit,times=t,func=rxnrate)
head(out)
plot(out)
ssq=function(myparms){
#initial concentration
cinit=c(A=myparms[1],B=myparms[2],C=myparms[3],D=myparms[4],E=myparms[5],F=myparms[6],G=myparms[7],H=myparms[8])
cinit=c(A=unname(myparms[1]),B=unname(myparms[2]),C=unname(myparms[3]),D=unname(myparms[4]),E=unname(myparms[5]),F=unname(myparms[6]),G=unname(myparms[7]),H=unname(myparms[8]))
print(cinit)
#time points for which conc is reported
#include the points where data is available
t=c(seq(0,5,1),df$time)
t=sort(unique(t))
#parameters from the parameters estimation
#solve ODE for a given set of parameters
out=ode(y=cinit,times=t,func=rxnrate)
#Filter data that contains time points
outdf=data.frame(out)
outdf=outdf[outdf$time%in% df$time,]
#Evaluate predicted vs experimental residual
preddf=melt(outdf,id.var="time",variable.name="species",value.name="conc")
expdf=melt(df,id.var="time",variable.name="species",value.name="conc")
ssqres=preddf$conc-expdf$conc
return(ssqres)
}
# parameter fitting using levenberg marquart
#initial guess for parameters
myparms=c(A=0.6947,B=0.4072,C=0.408,D=0.2,E=0,F=0,G=0.355,H=0.9)
#fitting
fitval=nls.lm(par=myparms,fn=ssq)
#summary of fit
summary(fitval)