multiple step decay

A -k1-> B -k2-> C -k3-> D -k4->E -k5-> F
A'= -k1A; d (lnA)= d(-k1t) ; A=A0 e^(-k1t)
B'=  k1A - k2B
 B'+k2B=k1A=A0 k1 e^(-k1t);令B(t)=x(t)e^(-k1t)+y(t)e^(-k2t);
> B'(t)+k2B(t) = (k2-k1)x(t)e^(-k1t) + x'(t)e^(-k1t) + y'(t)e^(-k2t) > (k2-k1)x(t)+x'(t) = A0k1 & y'(t)=0 ; y(t)=y0
if k2  = k1 then x'(t) =A0k1; x(t)= A0k1t + x0
>B(t)= (A0k1t + x0)e^(-k1t)+y0e^(-k1t)= A0k1t e^(-k1t)+ (x0+y0)e^(-k1t); B0= (x0+y0)
>B(t)= B0e^(-k1t) + A0k1t e^(-k1t)
if k2 != k1 then x(t) = A0 k1/(k2-k1)
>B(t)= A0 k1/(k2-k1) e^(-k1t)+y0e^(-k2t); B0=  A0 k1/(k2-k1) +y0
>B(t)= A0 k1/(k2-k1) e^(-k1t)+ (B0-A0 k1/(k2-k1))e^(-k2t)
>B(t)= B0e^(-k2t) + A0k1( e^(-k1t)/(k2-k1) + e^(-k2t)/(k1-k2) )

C'=  k2B - k3C

if k3 != k2 != k1
C(t) =  C0e^(-k3t) + B0k2( e^(-k2t)/(k3-k2) + e^(-k3t)/(k2-k3) ) + A0k1k2 ( e^(-k1t)/[(k3-k1)(k2-k1)] + e^(-k2t)/[(k3-k2)(k1-k2)] + e^(-k3t)/[(k1-k3)(k2-k3)] )



IF k1=k2=k3=.........=kn=k AND B0=C0=D0=E0=F0=.........=0 let A1=B, A2=C,.....,An
A=A0 e^(-kt)
A1=B=A0 kt e^(-kt)
A2=C=A0 [ (kt)^2 /2! ] e^(-kt)
A3=D=A0 [ (kt)^3 /3! ] e^(-kt)
A4=E=A0 [ (kt)^4 /4! ] e^(-kt)
A5=F=A0 [ (kt)^5 /5! ] e^(-kt)
An=..=A0 [ (kt)^n /n! ] e^(-kt)
All summation: A0  e^(kt) e^(-kt) = A0

for each An, peak An occured when (An)'=0; (kt/n)=1;  tmax = n/k
[n - (n-1)]/[(tmax)n-(tmax)n-1 ] = k ; So The peak wave moves at speed k(let x=kt),  (x^n/n!)e^(-x) dx = d[(-x^n/n!) e^(-x)] + (x^(n-1)/(n-1)!)e^(-x) dx,
(-x^n/n!) e^(-x) will be 0(when x=infinite), 0(when x=0,n>0), -1(when x=n=0)

intergration An from x=0 to infinite:
A0[ (kt)^n /n! ]e^(-kt) dt = A0/k[ (kt)^n /n! ] e^(-kt) d(kt) = A0/k (x^n/n!)e^(-x) dx =A0/k[0-(-1)]=A0/k

An= An-1 x ( kt / n )
for each time frame t (and kt), peak A occurred at nmax=kt (where kt/n=1;An=An-1)
at nmax, Anmax-2/Anmax-1= (kt-1)/kt; Anmax+1/Anmax= kt/(kt+1);when kt large enough ->