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Markov chains are probabilistic models which can be used for the modeling of sequences given a probability distribution and then, they are also very useful for the characterization of certain parts of a DNA or protein string given for example, a bias towards the AT or GC content.

For now, I am going to introduce how to build our own Markov Chain of zero order and first order in R programming language. The definition of a zero order Markov Chain relies in that, the current state (or nucleotide) does not depends on the previous state, so there’s no “memory” and every state is untied.

For the first order Markov Chain the case is different because the current state actually depends only on the previous state. Given that points clear, a second order Markov Model will be a model that reflects that the current state only depends on the previous two states before it (This model will be useful for the study of codons, given that they are substrings of 3 nucleotides long).

Example of a Markov Chain of zero order (the current nucleotide is totally independent of the previous nucleotide).

The multinomial model is:
p(A)+p(C)+p(G)+p(T) = 1.0
0.4 +0.1 +0.1 +0.4 = 1.0

Example of the structure of the zero order model:

Example of a Markov Chain of first order (the current nucleotide only depends on the previous nucleotide).

The multinomial model per base is:
p(A|A)+p(C|A)+p(G|A)+p(T|A) = 1.0
p(A|C)+p(C|C)+p(G|C)+p(T|C) = 1.0
p(A|G)+p(C|G)+p(G|G)+p(T|G) = 1.0
p(A|T)+p(C|T)+p(G|T)+p(T|T) = 1.0

So:
0.6 + 0.1 + 0.1 + 0.2  = 1.0
0.1 + 0.5 + 0.3 + 0.1  = 1.0
0.05+ 0.2 + 0.7 + 0.05 = 1.0
0.4 + 0.05+0.05 + 0.5  = 1.0

Example of the structure of the first order model:
Code:
#    Author: Benjamin Tovar
#    Date: 13/April/2012
#
#    Example of a Markov Chain of zero order (the current nucleotide is
#    totally independent of the previous nucleotide).

#    The multinomial model is:
#    p(A)+p(C)+p(G)+p(T) = 1.0
#    0.4 +0.1 +0.1 +0.4  = 1.0

# Define the DNA alphabet that will be used to put names to the objects
alp <- c("A","C","G","T")
# Create the vector that represents the probability distribution of the model
zeroOrder <- c(0.4,0.1,0.1,0.4)
# Put the name of reference of each base
names(zeroOrder) <- alp
# Create a sequence of 1000 bases using this model.
zeroOrderSeq <- sample(alp,1000,rep=T,prob=zeroOrder)

# ***** Study the composition bias of the sequence *****
# We wil use the "seqinr" package.
# For the installation of the package, type:
# install.packages("seqinr")
require("seqinr")

# Count the frequency of each base
# in the sequence using the "count" function
zeroOrderFreq <- count(zeroOrderSeq,1,alphabet=alp,freq=TRUE)

# Count the frequency of dinucleotides
# in the sequence using the "count" function
zeroOrderFreqDin <- count(zeroOrderSeq,2,alphabet=alp,freq=TRUE)

# Now, plot the results in the same plot:
layout(1:2)
barplot(zeroOrderFreq,col=1:4,main="Compositional bias of each nucleotide",
xlab="Base",
ylab="Base proportion")
barplot(zeroOrderFreqDin,col=rainbow(16),
main="Compositional bias of each dinucleotide",
xlab="Base",
ylab="Base proportion")

# &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

#    Example of a Markov Chain of first order (the current nucleotide only
#    depends on the previous nucleotide).

#    The multinomial model per base is:
#    p(A|A)+p(C|A)+p(G|A)+p(T|A) = 1.0
#    p(A|C)+p(C|C)+p(G|C)+p(T|C) = 1.0
#    p(A|G)+p(C|G)+p(G|G)+p(T|G) = 1.0
#    p(A|T)+p(C|T)+p(G|T)+p(T|T) = 1.0

#    So:
#    0.6 + 0.1 + 0.1 + 0.2  = 1.0
#    0.1 + 0.5 + 0.3 + 0.1  = 1.0
#    0.05+ 0.2 + 0.7 + 0.05 = 1.0
#    0.4 + 0.05+0.05 + 0.5  = 1.0

# Create the matrix that will store the probability distribution given
# a certain nucleotide:
firstOrderMat <- matrix(NA,nr=4,nc=4)
# Put names to the 2 dimensions of the matrix
colnames(firstOrderMat) <- rownames(firstOrderMat) <- alp
# Add the probability distribution per base:
firstOrderMat[1,] <- c(0.6,0.1,0.1,0.2)  # Given an A in the 1st position
firstOrderMat[2,] <- c(0.1,0.5,0.3,0.1)  # Given a  C in the 1st position
firstOrderMat[3,] <- c(0.05,0.2,0.7,0.05)# Given a  G in the 1st position
firstOrderMat[4,] <- c(0.4,0.05,0.05,0.5)# Given a  T in the 1st position

# Now we got a matrix
#    > firstOrderMat
#         A    C    G    T
#    A 0.60 0.10 0.10 0.20
#    C 0.10 0.50 0.30 0.10
#    G 0.05 0.20 0.70 0.05
#    T 0.40 0.05 0.05 0.50

# In order to continue, we need an initial probability distribution to know
# which base is the most probable to start up the sequence.
inProb <- c(0.4,0.1,0.1,0.4); names(inProb) <- alp
# So, the sequence will have a 40% to start with an A or a T and 10% with C or G

# Create a function to generate the sequence.
# NOTE: To load the function to the current environment, just copy
# the entire function and paste it inside the R prompt.

generateFirstOrderSeq <- function(lengthSeq,
alphabet,
initialProb,
firstOrderMatrix){
#    lengthSeq = length of the sequence
#    alphabet = alphabet that compounds the sequence
#    initialProb   = initial probability distribution
#    firstOrderMatrix = matrix that stores the probability distribution of a
#                       first order Markov Chain

# Construct the object that stores the sequence
outputSeq <- rep(NA,lengthSeq)
# Which is the first base:
outputSeq  <- sample(alphabet,1,prob=initialProb)
# Let the computer decide:
for(i in 2:length(outputSeq)){
prevNuc <- outputSeq[i-1]
currentProb <- firstOrderMatrix[prevNuc,]
outputSeq[i] <- sample(alp,1,prob=currentProb)
}
cat("** DONE: Sequence computation is complete **\n")
return(outputSeq)
}

# Use the generateFirstOrderSeq function to generate a sequence of 1000 bases
# long
firstOrderSeq <- generateFirstOrderSeq(1000,alp,inProb,firstOrderMat)

# ***** Study the composition bias of the sequence *****
# We wil use the "seqinr" package.
# For the installation of the package, type:
# install.packages("seqinr")
require("seqinr")

# Count the frequency of each base
# in the sequence using the "count" function
firstOrderFreq <- count(firstOrderSeq,1,alphabet=alp,freq=TRUE)

# Count the frequency of dinucleotides
# in the sequence using the "count" function
firstOrderFreqDin <- count(firstOrderSeq,2,alphabet=alp,freq=TRUE)

# Now, plot the results in the same plot:
layout(1:2)
barplot(firstOrderFreq,col=1:4,main="Compositional bias of each nucleotide",
xlab="Base",
ylab="Base proportion")
barplot(firstOrderFreqDin,col=rainbow(16),
main="Compositional bias of each dinucleotide",
xlab="Base",
ylab="Base proportion")

## Lets plot the 4 plots in one window
layout(matrix(1:4,nr=2,nc=2))
# Results from the zero order
barplot(zeroOrderFreq,col=1:4,
main="Compositional bias of each nucleotide\nZero Order Markov Chain",
xlab="Base",
ylab="Base proportion")
barplot(zeroOrderFreqDin,col=rainbow(16),
main="Compositional bias of each dinucleotide\nZero Order Markov Chain",
xlab="Base",
ylab="Base proportion")
# Results from the first order
barplot(firstOrderFreq,col=1:4,
main="Compositional bias of each nucleotide\nFirst Order Markov Chain",
xlab="Base",
ylab="Base proportion")
barplot(firstOrderFreqDin,col=rainbow(16),
main="Compositional bias of each dinucleotide\nFirst Order Markov Chain",
xlab="Base",
ylab="Base proportion")

# end.



Plot of the zero order model:

Plot of the first order model

All plots

Benjamin