Two species and move in an electrostatic, centrifugal, or velocity field and mix according to the cyclic reaction system: . All reaction rate constants are assumed to be equal to .The two governing equations for the concentrations of the species are:=--+,where } are two permutations of .The boundary and initial conditions are (x,t)=0 and (x,0)=δ(x) with =1 and =0.All components are assumed to be sufficiently similar to have identical diffusion coefficients The velocities of species and are such as =-=v.First, expressions for the spatial moments (i.e., for are obtained. Then, we compute the reduced first moment, as well as the second, third and fourth central moments (, and ) .These spatial moments are used to construct the concentration profile using the following truncated series of the Gram-Charlier expansion:(x,t)=1+8-126+124-34-12+3,where =.Here, we plot the concentration profiles of the species and for the case where =t=0.96 and . The concentrations of the species and are shown as blue and red peaks, respectively. The green peak corresponds to (i.e., the concentration of the combined species). Finally, our generated concentration distributions are benchmarked against their counterparts from Killalea and McCoy (1980).
A
1
A
2
A
1
k
e
⇆
k
e
A
2
k
e
c
i
A
i
∂
c
i
∂t
D
i
2
∂
c
i
∂
2
x
v
i
∂
c
i
∂x
k
e
c
i
k
e
c
j
{i,j
{1,2}
lim
x±∞
c
j
c
j
f
j
f
1
f
2
(=).
D
1
D
2
A
1
A
2
v
1
v
2
m
kj
k∈{1,2,3,4}andj∈{1,2}
μ
1j
μ
2j
μ
3j
μ
4j
c
j
m
0j
-
2
X
j
μ
3j
3
X
j
X
j
1.5
2
μ
2j
μ
4j
2
()
μ
2j
4
X
j
2
X
j
2π
μ
2j
X
j
x-
μ
1j
2
μ
2j
A
1
A
2
*
t
k
e
β=2v=10
k
e
D
2
A
1
A
2
2
∑
i=1
c
i
In[]:=
Eliminate[{D[[t],{t,2}]+d1D[[t],{t,1}]-k2D[[t],{t,1}]==0,D[[t],{t,1}]+d1[t]-k2[t]==0,D[[t],{t,1}]+d2[t]-k1[t]==0},{D[[t],{t,1}],[t]}]
c
1
c
1
c
2
c
1
c
1
c
2
c
2
c
2
c
1
c
2
c
2
Out[]=
′′
c
1
c
1
c
1
′
c
1
′
c
1
In[]:=
roots=Solve[r^2+(d1+d2)r+(d1d2-k1k2)==0,r]
Out[]=
r-d1-d2--2d1d2++4k1k2,r-d1-d2+-2d1d2++4k1k2
1
2
2
d1
2
d2
1
2
2
d1
2
d2
In[]:=
r1=roots[[1]][[1,2]]
Out[]=
1
2
2
d1
2
d2
In[]:=
r2=roots[[2]][[1,2]]
Out[]=
1
2
2
d1
2
d2
In[]:=
c
1
In[]:=
c
2
c
1
c
1
In[]:=
ConstInteg=Solve[{[0]==f1,'[0]==f2k2-d1f1},{C11,C22}]
c
1
c
1
Out[]=
C11-,C22-
-d1f1+d2f1+2f2k2-f1-2d1d2++4k1k2
2
d1
2
d2
2-2d1d2++4k1k2
2
d1
2
d2
d1f1-d2f1-2f2k2-f1-2d1d2++4k1k2
2
d1
2
d2
2-2d1d2++4k1k2
2
d1
2
d2
In[]:=
C11=ConstInteg[[1]][[1,2]]
Out[]=
-
-d1f1+d2f1+2f2k2-f1-2d1d2++4k1k2
2
d1
2
d2
2-2d1d2++4k1k2
2
d1
2
d2
In[]:=
C22=ConstInteg[[1]][[2,2]]
Out[]=
-
d1f1-d2f1-2f2k2-f1-2d1d2++4k1k2
2
d1
2
d2
2-2d1d2++4k1k2
2
d1
2
d2
In[]:=
d1=D1k^2+v1kI+k1;
In[]:=
d2=D2k^2+v2kI+k2;
In[]:=
k1=k2;D1=D2;v1=v;v2=-v;f1=1;f2=0;
Spatial moments for species 1 and 2
In[]:=
m[0,1]=FullSimplify[Limit[[t],k0],k2>0]//Expand
c
1
Out[]=
1
2
1
2
-2k2t
In[]:=
m[0,2]=FullSimplify[Limit[[t],k0],k2>0]//Expand
c
2
Out[]=
1
2
1
2
-2k2t
In[]:=
m[1,1]=FullSimplify[ILimit[D[[t],{k,1}],k0],k2>0]//Expand
c
1
Out[]=
v
2k2
-2k2t
2k2
In[]:=
m[1,2]=FullSimplify[ILimit[D[[t],{k,1}],k0],k2>0]//Expand
c
2
Out[]=
0
In[]:=
m[2,1]=FullSimplify[I^2Limit[D[[t],{k,2}],k0],k2>0]//Expand
c
1
Out[]=
2D2tCosh[k2t]+tSinh[k2t]
-k2t
-k2t
2
v
k2
In[]:=
m[2,2]=FullSimplify[I^2Limit[D[[t],{k,2}],k0],k2>0]//Expand
c
2
Out[]=
-k2t
2
v
k2
-k2t
-k2t
2
v
2
k2
In[]:=
m[3,1]=FullSimplify[I^3Limit[D[[t],{k,3}],k0],k2>0]//Expand
c
1
Out[]=
3tCosh[k2t]
-k2t
3
v
2
k2
6D2tvSinh[k2t]
-k2t
k2
3Sinh[k2t]
-k2t
3
v
3
k2
In[]:=
m[3,2]=FullSimplify[I^3Limit[D[[t],{k,3}],k0],k2>0]//Expand
c
2
Out[]=
0
In[]:=
m[4,1]=FullSimplify[I^4Limit[D[[t],{k,4}],k0],k2>0]//Expand
c
1
Out[]=
12Cosh[k2t]++-
2
D2
-k2t
2
t
3Cosh[k2t]
-k2t
2
t
4
v
2
k2
12D2Sinh[k2t]
-k2t
2
t
2
v
k2
3tSinh[k2t]
-k2t
4
v
3
k2
In[]:=
m[4,2]=FullSimplify[I^4Limit[D[[t],{k,4}],k0],k2>0]//Expand
c
2
Out[]=
12D2Cosh[k2t]
-k2t
2
t
2
v
k2
9tCosh[k2t]
-k2t
4
v
3
k2
2
D2
-k2t
2
t
12D2tSinh[k2t]
-k2t
2
v
2
k2
9Sinh[k2t]
-k2t
4
v
4
k2
3Sinh[k2t]
-k2t
2
t
4
v
2
k2
Reduced moments for species 1 and 2
In[]:=
red[i_,j_]:=m[i,j]/m[0,j]
Central moments for species 1 and 2
In[]:=
cen[i_,j_]:=Sum[Binomial[i,k]red[i-k,j](-red[1,j])^k,{k,0,i}]
Concentration profile using the Gram-Charlier expansion
In[]:=
c1=m[0,j]Exp[-(X)^2](1+cen[3,j](8(X)^3-12(X))/(6(2cen[2,j])^1.5)+1/24(cen[4,j]/cen[2,j]^2-3)(4(X)^4-12(X)^2+3))
2πcen[2,j]
a/.j1;In[]:=
a=Sqrt[D2t]
Out[]=
D2t
In[]:=
sol=First[Solve[{v-5Sqrt[D2k2],tk20.96},{v,k2}]]
Out[]=
v-4.89898,k2
D2
t
0.96
t
In[]:=
X[j_]:=(x-red[1,j])/Sqrt[2cen[2,j]];
In[]:=
X1=X[j]/.j1;
In[]:=
c1/.sol/.XX1/.xxoSqrt[D2/k2]/.sol//Expand//FullSimplify
Out[]=
1
3.5
(D2t)
--0.084261xoxo
0.627135t
D2
t
D2t
D2
t
3.
(D2t)
3
xo
3
D2
D2
t
3
t
3
xo
2.5
(D2t)
2
xo
4
xo
In[]:=
c1p=(E^(((-0.6271351215135569)-0.08426099869750073*xo)*xo)*((-0.005193249821722638+0.003941118485202517*xo^2)+0.0006399962877921872*(-3.9531289210574077*xo+1.*xo^3)+0.0003530154487780164*(23.744273843440205*xo+1.*xo^3)+(0.02022875903003083+0.0020417465304259333*xo^2+0.00004299450351458052*xo^4)))
Out[]=
(-0.627135-0.084261xo)xo
2
xo
4
xo
3
xo
3
xo
Concentration profile for species 1 plotted in blue
In[]:=
p1=Plot[c1p,{xo,-15,15},PlotRangeAll,PlotStyleDarker@Blue,PlotRange{{-15,15},{0,0.15}}];
In[]:=
c2=m[0,j]Exp[-(X)^2](1+cen[3,j](8(X)^3-12(X))/(6(2cen[2,j])^1.5)+1/24(cen[4,j]/cen[2,j]^2-3)(4(X)^4-12(X)^2+3))
2πcen[2,j]
a/.j2;In[]:=
X2=X[j]/.j2;
In[]:=
c2p=c2/.sol/.XX2/.xxoSqrt[D2/k2]/.sol//FullSimplify
Concentration profile for species 2 plotted in red
Overall concentration profile plotted in green
Comparison of our concentration distributions with Fig. 2 of Ref. 1
Effect of neglecting the fourth moment on the total concentration profile
In[]:=
c2 = m[0, j] Exp[-(X)^2] (1 + cen[3, j] (8 (X)^3 - 12 (X))/(6 (2 cen[2, j])^1.5))/Sqrt[2 π cen[2, j]] a /. j 2;
X2 = X[j] /. j 2;
c2p = c2 /. sol /. X X2 /. x xo Sqrt[D2/k2] /. sol // FullSimplify
X2 = X[j] /. j 2;
c2p = c2 /. sol /. X X2 /. x xo Sqrt[D2/k2] /. sol // FullSimplify
Comparison of our concentration distributions with Fig. 3 of Ref. 1
References
References
[1] Killalea, M.K. and McCoy, B.J. (1980), Concentration distribution and spatial moments of moving macromolecules undergoing isomerization. Biopolymers, 19: 1875-1886. https://doi.org/10.1002/bip.1980.360191014
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Concentration distribution of moving macromolecules undergoing isomerization
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, June 9, 2025
https://community.wolfram.com/groups/-/m/t/3476080
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, June 9, 2025
https://community.wolfram.com/groups/-/m/t/3476080