Last updated: 2020-02-01

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Knit directory: Thesis_single_RNA/

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Data generation for one gene under two-state model with different parameters

Three-parameter model for stochastic gene expression assumes that cell can be either in “on” and “off” transcription states which is characterized by Beta distribution.

While the cell is in “on” mode, mRNA is transcribed with \(k_r\) rate, so mRNA expression follows two-state model of gene expression, where

\(x|k_r,p\sim Poisson(k_rp)\) and \(p|k_{on},k_{off}\sim Beta(k_{on},k_{off})\)

Further extension of this model includes four-parameter model, that accounts for degradation rate d, which is a positive value less than one, so now model is \(x|k_r,p,d\sim d*Poisson(k_rp)\)

Four-parameter model can be written as three-parameter model is we scale mRNA expression level by d or \(BetaPoisson(x|k_{on},k_{off},k_r,d) = BetaPoisson(\frac{x}{d}|k_{on},k_{off},k_r)\)

Here is the R code to simulate mRNA copy number for 1 gene in n cells:

generate_one_gene<-function(k_on,k_off,k_r,n){
  p=rbeta(n,k_on,k_off)
  x=rpois(n,p*k_r)
  return(x)}

Different parameters of \(k_{on}\) and \(k_{off}\) lead to different mRNA copy number distribution

I) Small \(k_{on}\) and \(k_{off}\) lead to bimodal distribution (large number of cells have low expression with few cells with high expression):

#Fixed k=100 as in Munsky paper the Fig.2
par(mfrow=c(2,2))
k=c(0.001,0.01,0.1,0.5)
for (i in k){
  x=generate_one_gene(i,i,100,n=10000)
  name=paste('k_on,k_off=',i)
  hist(x,xlim=c(0,200),ylim=c(0,0.06),freq=FALSE,main=name)}

II) Large \(k_{on}\) and \(k_{off}\) lead to Poisson or negative binomial distribution:

par(mfrow=c(2,2))
k=c(2,10,50,100)
for (i in k){
  x=generate_one_gene(i,i,100,n=10000)
  name=paste('k_on,k_off=',i)
  hist(x,xlim=c(0,200),ylim=c(0,0.06),freq=FALSE,main=name)}

III) Large value if \(k_{off}\) and small \(k_{on}\)

par(mfrow=c(1,2))
k=list(c(0.1,1),c(1,10))
for (i in k){
  x=generate_one_gene(i[1],i[2],100,n=10000)
  name=paste("k_on",i[1],'k_off',i[2])
  hist(x,xlim=c(0,200),ylim=c(0,0.1),freq=FALSE,main=name)}

IV) Small value if \(k_{off}\) and big \(k_{on}\)

par(mfrow=c(1,2))
k=list(c(1,0.1),c(10,1))
for (i in k){
  x=generate_one_gene(i[1],i[2],100,n=10000)
  name=paste("k_on",i[1],'k_off',i[2])
  hist(x,xlim=c(0,200),ylim=c(0,0.1),freq=FALSE,main=name)}


sessionInfo()
R version 3.5.0 (2018-04-23)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 17763)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[2] LC_CTYPE=English_United States.1252   
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.5.0 Rcpp_1.0.2      later_0.8.0     digest_0.6.17  
 [5] rprojroot_1.3-2 R6_2.3.0        backports_1.1.2 git2r_0.26.1   
 [9] magrittr_1.5    evaluate_0.11   stringi_1.1.7   promises_1.0.1 
[13] fs_1.3.1        rmarkdown_1.10  tools_3.5.0     stringr_1.3.1  
[17] glue_1.3.0      httpuv_1.5.1    yaml_2.2.0      compiler_3.5.0 
[21] htmltools_0.3.6 knitr_1.20