#########################################################################################
### R code for PCOMPBIOL-D-12-01525R1 / 10.1371/journal.pcbi.1002900: ###################
### C. Magnus. Virus Neutralisation: New Insights from Kinetic Neutralisation Curves. ###
#########################################################################################


##### 0. Definition of spike number distribution, eta #####

ETA_vars<-function(a,b,s.max){
	
	# a		 	1st parameter of Beta-distribution
	# b		 	2nd parameter of Beta-distribution
	# s.max		maximal number of spikes
	
	if(sum(c(a,b)<=0)!=0){ETA_vars<-rep(1/(s.max+1),s.max+1)}
	else{
		if(sum(c(a,b)>=1)==2){ETA_vars<-dbeta(seq(from=0,to=1,by=1/s.max),a,b)/(sum(dbeta(seq(from=0,to=1,by=1/s.max),a,b)))}
		else{v<-dbeta(seq(from=0,to=1,by=1/s.max),a,b)
	              	if(v[1]==Inf){v[1]<-ceiling(v[2])}
	              	if(v[s.max+1]==Inf){v[s.max+1]<-ceiling(v[s.max])}
	               ETA_vars<-v/(sum(v))
	              	}
		}
	}
	
meandos<-function(dos,s.max)sum(dos*0:s.max)

vardos<-function(dos,s.max)sum(dos*(0:s.max-sum(dos*0:s.max))^2)

rssab<-function(X,ew,v,s.max){
	dos<-ETA_vars(X[1],X[2],s.max)
	(sum(dos*0:s.max)-ew)^2+(sum(dos*(0:s.max-sum(dos*0:s.max))^2)-v)^2
	
	}

fitab_m_var<-function(ew,v,s.max,start){
	optim(start,rssab,ew=ew,v=v,s.max=s.max,)
	}
	
dos_m_var<-function(ew,v,s.max,start){
	bla<-fitab_m_var(ew,v,s.max,start)
	ETA_vars(bla$par[1],bla$par[2],s.max)
	
	}


# HIV specific trimer number distribution, described in the article
	
dos1449<-dos_m_var(14,49,100,c(5,2))


# equally distributed trimer numbers, used in "Other viral systems", especially for Figure 6

dos100<-c(rep(0,10),1)

dos10218<-c(rep(0,2),rep(1/17,17))

dos360<-c(rep(0,36),1)

dos36072<-rep(1/73,73)







######### 1. Models for three epitopes per trimer only #########



##### elementary reaction model

#### definition of the ODE - has to be uncommented when used

# TRAB<-function(t,y,parms){
	
# with(as.list(c(parms,y)),{
# dTr 	<- k1off*TrAb-k1on*Tr*Ab
# dAb	<- k1off*TrAb + k2off*TrAb2 + k3off*TrAb3 - (k1on*Tr*Ab + k2on*TrAb*Ab + k3on*TrAb2*Ab)
# dTrAb<- k1on*Tr*Ab + k2off*TrAb2 -(k1off*TrAb + k2on*TrAb*Ab)
# dTrAb2<-k2on*TrAb*Ab + k3off*TrAb3 - (k2off*TrAb2 + k3on*TrAb2*Ab)
# dTrAb3<-k3on*TrAb2*Ab - k3off*TrAb3

# der<-c(dTr,dAb,dTrAb,dTrAb2,dTrAb3)
# list(der)

# }
# )	

# }

##### stoichiometric reaction model 

### definition of the ODE - has to be commented-out when the elementary reaction model is used

TRAB<-function(t,y,parms){
	
with(as.list(c(parms,y)),{
dTr 	<- k1off*TrAb-k1on*Tr*Ab^3
dAb	<- k1off*TrAb + k2off*TrAb2 + k3off*TrAb3 - (k1on*Tr*Ab^3 + k2on*TrAb*Ab^2 + k3on*TrAb2*Ab)
dTrAb<- k1on*Tr*Ab^3 + k2off*TrAb2 -(k1off*TrAb + k2on*TrAb*Ab^2)
dTrAb2<-k2on*TrAb*Ab^2 + k3off*TrAb3 - (k2off*TrAb2 + k3on*TrAb2*Ab)
dTrAb3<-k3on*TrAb2*Ab - k3off*TrAb3

der<-c(dTr,dAb,dTrAb,dTrAb2,dTrAb3)
list(der)

}
)	

}

abneutra_ODE<-function(dt,parms,ctr,cab,plot){

	require(odesolve)
	
	# dt		time steps, eg dt<-1:10^5
	# parms		kinetic parameters, eg parms<-c(1e+6,1e-5,5e+5,1e-5,1e+5,1e-5)
	# ctr 		trimer concentration
	# cab		antibody concentration
	# plot		TRUE or FALSE if a plot should be created or not
	
	names(parms)<-c("k1on","k1off","k2on","k2off","k3on","k3off")
	
	inits<-c(Tr=ctr,Ab=cab,TrAb=0,TrAb2=0,TrAb3=0)
	

	simulation<-lsoda(inits,dt,TRAB,parms=parms)
	
		
	
	if(plot){# plotting the results
	
		
	
		plot(dt,simulation[,"Tr"],type="l",col="blue",ylim=c(0,max(simulation[,c("Ab","Tr","TrAb","TrAb2","TrAb3")])),xlab="time",ylab="product concentrations")
		lines(dt,simulation[,"Ab"],type="l",col="black",lwd=2)
		lines(dt,simulation[,"TrAb"],type="l",col="red",lwd=2)
		lines(dt,simulation[,"TrAb2"],type="l",col="green",lwd=2)
		lines(dt,simulation[,"TrAb3"],type="l",col="magenta",lwd=2)
	
		legend(dt[10],max(simulation[,c("Ab","Tr","TrAb","TrAb2","TrAb3")]),legend=c("Tr","Ab","TrAb","TrAb2","TrAb3"),col=c("blue","black","red","green","magenta"),bty="n",lty=rep(1,5))

	}
	
	return(simulation)

}


inf.virions.tit<-function(eta,TT,N,f0,f1,f2,f3,relativePI){
	# eta:			trimer number distribution, the last entry in this vector MUST be >0
	# TT:			stoichiometry of entry, must be >=1
	# N:			stoichiometry of neutralization
	# fi:			fraction of trimers with i abs
	# relativePI:	relative percent infectivity when TRUE. Not normalised with sum(eta[(TT:s.max)+1]) if FALSE. If the V/Vo is calculated by dividing with the total number of virions, relativeI=FALSE. If Vo is meant to be the number of infective virions (for example in infectivity experiments), this parameter is TRUE.
	
	s.max<-length(eta)-1
	
	if(s.max<TT){ 	# if there are less than TT spikes, the virion is not infectious
		out<-0
		relativePI<-FALSE
	}
	
	else{
		out<-0
	
		for(s in TT:s.max){
			zwischen<-0
			if(N==1){
				
				zwischen<-sum(dbinom(TT:s,s,f0))
			}
			
			
			if(N==2){
				for(j in TT:s){
					for(y0 in 0:j){
						y1<-j-y0
						zwischen<-zwischen+dmultinom(c(y0,y1,s-y0-y1),s,c(f0,f1,f2+f3))
					}
					}
				
			}
			
			if(N==3){
				zwischen<-sum(dbinom(0:(s-TT),s,f3))
			}
					
			out<-out+zwischen*eta[s+1]
		}
	
	
	}
	
	if(relativePI){
	out/(sum(eta[(TT:s.max)+1]))
	}
	else{out}
	
}


inf.virions.tit.vN<-function(eta,TT,v.N,f0,f1,f2,f3,relativePI)sapply(v.N,inf.virions.tit,eta=eta,TT=TT,f0=f0,f1=f1,f2=f2,f3=f3,relativePI=relativePI)
inf.virions.tit.vN.vTT<-function(eta,v.TT,v.N,f0,f1,f2,f3,relativePI)sapply(v.TT,inf.virions.tit.vN,eta=eta,v.N=v.N,f0=f0,f1=f1,f2=f2,f3=f3,relativePI=relativePI)


kin.neut.curve.theo.b<-function(dt,show.dt,parms,ctr,cab,N,TT,eta,relativePI){ 
	# dt: 			times at which the simulation runs
	# show.dt:		time steps at which data is needed
	# relativePI:	relative percent infectivity when TRUE. Not normalised with sum(eta[(TT:s.max)+1]) if FALSE. If the V/Vo is calculated by dividing with the total number of virions, relativePI=FALSE. If Vo is meant to be the number of infective virions (for example in infectivity experiments), this parameter is TRUE.
	# dt: 			vector with time point at which the ODE shall be solved, dt<-1:10^5	
	
	zwischen<-abneutra_ODE(dt,parms,ctr,cab,FALSE) # time dependent concentrations, 1st step in the model!
	
	# negative concentrations are set to 0
	zwischen[which(zwischen<0)]<-0
	
	pos.dt<-show.dt

	out<-matrix(NA,nrow=length(pos.dt),ncol=2,dimnames=list(1:length(pos.dt),c("time","perc.inf")))
	
	# 2nd step in the model
	k<-0
	for(i in pos.dt){
		out[k+1,"perc.inf"]<-as.vector(inf.virions.tit(eta,TT,N,zwischen[i,"Tr"]/(zwischen[i,"Tr"]+zwischen[i,"TrAb"]+zwischen[i,"TrAb2"]+zwischen[i,"TrAb3"]),zwischen[i,"TrAb"]/(zwischen[i,"Tr"]+zwischen[i,"TrAb"]+zwischen[i,"TrAb2"]+zwischen[i,"TrAb3"]),zwischen[i,"TrAb2"]/(zwischen[i,"Tr"]+zwischen[i,"TrAb"]+zwischen[i,"TrAb2"]+zwischen[i,"TrAb3"]),zwischen[i,"TrAb3"]/(zwischen[i,"Tr"]+zwischen[i,"TrAb"]+zwischen[i,"TrAb2"]+zwischen[i,"TrAb3"]),relativePI))

		out[,"time"]<-pos.dt
		k<-k+1
	}
	out	
}


#### example for how to call this function ####
# dt<-1:10^6
# show.dt<-c(1,(1:100)*10^4)
# parms<-c(10^6,10^(-5),10^3,10^(-4),1,10^(-3))
# ctr<-4*10^(-5)
# cab<-4.2*10^(-4)
# N<-1
# TT<-8
# eta<-dos1449
# relativePI<-TRUE
# kin.neut.curve.theo.b(dt,show.dt,parms,ctr,cab,N,TT,eta,relativePI)->Example
# plot(Example[,"time"],log(Example[,"perc.inf"],10),xlab="time",ylab="log(Vt/V0)",type="l")





###################### data fitting ######################

### data extracted from McLain, L, Dimmock, N (1994). Single-hit and multi-hit kinetics of immunoblobullin-G neutralization of human immunodeficiency virus type-1 by monoclonal antibodies. J gen Virol 75: 1457-1460, Figure 2
# the percent infectivity is displayed as log(PI); PI=V_t/V_0

Tmcla<-data.frame(matrix(c(300, 600, 900, 1860, 2700, 3660, 5400, 7200, -0.109321, -0.195690, -0.350959, -0.460785, -0.582048, -0.680390, -0.825324, -1.010500), nrow=8, ncol=2, dimnames=list(1:8,c("Time","logPI"))))
Tmclb<-data.frame(matrix(c(360, 660, 960, 1860, 2700, 3660, 5400, 7200, 0.00647773, -0.13034800, -0.30299300, -0.46268200, -0.61046300, -0.74030300, -1.08962000, -1.08661000), nrow=8, ncol=2, dimnames=list(1:8,c("Time","logPI"))))
Tmclc<-data.frame(matrix(c(300, 660, 900, 1800, 2760, 3660, 5400, 7200, 0.00670391, 0.01340780, -0.00670391, -0.14078200, -0.34860300, -0.55642500, -0.87150800, -1.09944000), nrow=8, ncol=2, dimnames=list(1:8,c("Time","logPI"))))

# definition of the residual sum of squares


rss.knc<-function(parms,data,ctr,cab,N,TT,eta){
	# parms:		binding and dissociation parameters
	# data:			matrix with two columns: Time, logPI, where Time is measured in SECONDS
	# ctr:			concentration of trimers
	# cab:			concentration of antibodies
	# N:			stoichiometry of neutralisation
	# TT:			stoichiometry of entry
	# eta:			trimer number distribution
	trafo<-function(x){x}
	
	dt<-seq(0,max(data$Time),by=1)
	
	sum((trafo(10^(data$logPI))-trafo(kin.neut.curve.theo.b(dt,data$Time,parms,ctr,cab,N,TT,eta,TRUE)[,"perc.inf"]))^2)
	
}


# optimize residual sum of square

knc.optim<-function(start,data,ctr,cab,N,TT,eta){
	
	# start:		starting value: log10 of the constants

 	zw<-optim(start,function(parms) rss.knc(10^parms,data,ctr,cab,N,TT,eta),lower=c(0,-10,0,-10,0,-10),method="L-BFGS-B",control=list(factr=1e06))
 	
 	matrix(c(zw$par,zw$value,zw$convergence),nrow=1,ncol=8,dimnames=list("",c("k1on","k1off","k2on","k2off","k3on","k3off","rss","conv")))
 
 }
 
 
 #### example for calling this function
 # start<-c(8,-2,5,-3,0,-4)
 # data<-Tmcla
 # ctr<-4*10^(-5)
 # cab<-4.2*10^(-4)
 # N<-1
 # TT<-2
 # eta<-dos1449
 # knc.optim(start,data,ctr,cab,N,TT,eta)