### 统计代写|R代写project|Host Cell Proteins with Cross Diffusion

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## 统计代写|R代写project|Keywords

Abstract A model is developed for the production of two proteins within a host cell that diffuse outward through the cell membrane with cross diffusion added for the interaction of the proteins during diffusion.

The numerical and graphical output for the three protein model is displayed with standard R utilities. The formation of the second and third proteins within the host cell can result in the transmission of the resulting virions to other host cells, and thus is the basis for virus transmission. The transmission can be moderated by cross diffusion of the two proteins produced in the host cell.

Keywords host cell protein production – viral genetic material (VGM) transport . VGM mutation – mathematical model – partial differential equation (PDE). initial condition (IC) – boundary condition $(\mathrm{BC})$ – ordinary differential equation (ODE) · cross diffusion · R coding – method of lines (MOL)

## 统计代写|R代写project|R routines for an ODE/PDE model with cross diffusion

Eqs. (1.1), (1.2) are extended to three proteins, one for entry into the host cell, denoted with a subscript 1 , and two proteins produced in the cell that diffuse outward across the cell membrane, with subscripts 2,3 . Eqs. (1.1), the ODE/PDE model for component 1 , are restated here (so that the following development is self contained).

$D_{V 1} \frac{\partial V_{1}\left(x=x_{u t}, t\right)}{\partial x}=k_{1 u}\left(V_{1 s}(t)-V_{1}\left(x=x_{u}, t\right)\right)$
$-D_{V 1} \frac{\partial V_{1}\left(x=x_{l}, t\right)}{\partial x}=k_{1 l}\left(C_{1}(t)-V_{1}\left(x=x_{l}, t\right)\right)$
$V_{1}(x, t=0)=V_{10}(x)$
$\frac{d C_{1}(t)}{d t}=-k_{1 l}\left(C_{1}(t)-V_{1}\left(x=x_{1}, t\right)\right)$
$C_{1}(t=0)=C_{10}$
Eqs. (3.1) define $V_{1}(x, t), C_{1}(t)$ as dependent variables.
A detailed explanation of these equations is given in Chapter 1 .
Eqs. (1.2), the ODE/PDE model for component 2, are restated here with cross diffusion added to eq. (1.2-1)
$$\frac{\partial V_{2}(x, t)}{\partial t}=D_{V 2} \frac{\partial^{2} V_{2}(x, t)}{\partial x^{2}}-\chi_{2} \frac{\partial^{2}\left(V_{2}(x, t) V_{3}(x, t)\right)}{\partial x^{2}}$$
$D_{V 2} \frac{\partial V_{2}\left(x=x_{u}, t\right)}{\partial x}=k_{2 u}\left(V_{2 a}-V_{2}\left(x=x_{u}, t\right)\right)$
$D_{V 2} \frac{\partial V_{2}\left(x=x_{l}, t\right)}{\partial x}=-k_{2 l}\left(C_{2}(t)-V_{2}\left(x=x_{l}, t\right)\right)$
$V_{2}(x, t=0)=V_{20}(x)$
$\frac{d C_{2}(t)}{d t}=-k_{2 l}\left(C_{2}(t)-V_{2}\left(x=x_{l}, t\right)\right)+k_{r 2} C_{1}^{n_{2}}$
$C_{2}(t=0)=C_{20}$
Eqs. (3.2), the ODE/PDE model for component 2, are restated here for component 3 (with the cross diffusion included in eq. (3.3-1)).
$$\frac{\partial V_{3}(x, t)}{\partial t}=D_{V 3} \frac{\partial^{2} V_{3}(x, t)}{\partial x^{2}}-\chi_{3} \frac{\partial^{2}\left(V_{2}(x, t) V_{3}(x, t)\right)}{\partial x^{2}}$$
$D_{V 3} \frac{\partial V_{3}\left(x=x_{u}, t\right)}{\partial x}=k_{3 u}\left(V_{3 a}-V_{3}\left(x=x_{u}, t\right)\right)$

$$\begin{gathered} D_{V 3} \frac{\partial V_{3}\left(x=x_{l}, t\right)}{\partial x}=-k_{3 l}\left(C_{3}(t)-V_{3}\left(x=x_{l}, t\right)\right) \ V_{3}(x, t=0)=V_{30}(x) \ \frac{d C_{3}(t)}{d t}=-k_{3 l}\left(C_{3}(t)-V_{3}\left(x=x_{l}, t\right)\right)+k_{r 3} C_{1}^{n_{3}} \ C_{3}(t=0)=C_{30} \end{gathered}$$
Eqs. (3.1), (3.2), (3.3) constitute the ODE/PDE models for component 1 , the protein that enters the cell from the virus, and components 2,3 , the protens produced in the cell that leave the cell to become virions 1 and possibly infect other cells.
The $R$ routines for eqs. (3.1), (3.2), (3.3) are discussed next.

## 统计代写|R代写project|Main program

The main program for three proteins with cross diffusion follows.
#

###### Three ODE, three PDE model

$#$

###### Delete previous workspaces

$\operatorname{rm}($ list $=1 \mathrm{~s}(\mathrm{all}=\mathrm{TRUE}))$
#

###### Access ODE integrator

library (“desolve”);
#

###### Access functions for numerical solution

setwd (“f:/vci/chap $3 “)$;
source (“pdela. $R^{\prime \prime}$ );
source (“dss004. $R^{\prime \prime}$ );
#

###### Parameters

Dv $1=1.0 e-02 ;$
Dv2 $=1 \cdot 0 e-02$;
Dv $3=1,0 e-02 ;$
$\mathrm{V} 1 \mathrm{~s}=1$;
$\mathrm{kml}=0.1$;
$\mathrm{km} 2=0.1$;

$\operatorname{km3} 3=0.1$;
$\mathrm{V} 2 \mathrm{a}=0$;
$\mathrm{V} 3 \mathrm{a}=0$;
kr2 =1;
kr3 =1;
$n 2=1$;
$\mathrm{n} 3=1$;
chi $2=1.0 e-04$;
chi $3=1.0 e-04$;
$\mathrm{V} 10=0$;
$\mathrm{V} 20=0$;
$\mathrm{V} 30=0$;
$\mathrm{C} 10=0$;
C2 $2=0$;
$\mathrm{C} 30=0$;
$#$

###### Spatial grid (in $x$ )

$\mathrm{nx}=21 ; \mathrm{xl}=0 ; \mathrm{xu}=1$;
$\mathrm{x}=\mathrm{seq}($ from $=\mathrm{x} 1, \mathrm{to}=\mathrm{xu}, \mathrm{by}=(\mathrm{xu-xl}) /(n x-1))$;

## 统计代写|R代写project|R routines for an ODE/PDE model with cross diffusion

D在1∂在1(X=X在吨,吨)∂X=ķ1在(在1s(吨)−在1(X=X在,吨))
−D在1∂在1(X=Xl,吨)∂X=ķ1l(C1(吨)−在1(X=Xl,吨))

dC1(吨)d吨=−ķ1l(C1(吨)−在1(X=X1,吨))
C1(吨=0)=C10

∂在2(X,吨)∂吨=D在2∂2在2(X,吨)∂X2−χ2∂2(在2(X,吨)在3(X,吨))∂X2
D在2∂在2(X=X在,吨)∂X=ķ2在(在2一种−在2(X=X在,吨))
D在2∂在2(X=Xl,吨)∂X=−ķ2l(C2(吨)−在2(X=Xl,吨))

dC2(吨)d吨=−ķ2l(C2(吨)−在2(X=Xl,吨))+ķr2C1n2
C2(吨=0)=C20

∂在3(X,吨)∂吨=D在3∂2在3(X,吨)∂X2−χ3∂2(在2(X,吨)在3(X,吨))∂X2
D在3∂在3(X=X在,吨)∂X=ķ3在(在3一种−在3(X=X在,吨))D在3∂在3(X=Xl,吨)∂X=−ķ3l(C3(吨)−在3(X=Xl,吨)) 在3(X,吨=0)=在30(X) dC3(吨)d吨=−ķ3l(C3(吨)−在3(X=Xl,吨))+ķr3C1n3 C3(吨=0)=C30

## 统计代写|R代写project|Main program

#

##

###### 删除以前的工作区

R M⁡(列表=1 s(一种ll=吨R在和))
#

#

###### 数值解法的访问函数

setwd (“f:/vci/chap3“);

#

DV1=1.0和−02;
DV2=1⋅0和−02;
DV3=1,0和−02;

ķ米l=0.1;
ķ米2=0.1;

kr2 =1;
kr3 =1;
n2=1;
n3=1;

C10=0;
C22=0;
C30=0;
##

###### 空间网格（在X )

nX=21;Xl=0;X在=1;
X=s和q(从=X1,吨这=X在,b是=(X在−Xl)/(nX−1));

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