TMParameterEstimation: "TMParameterEstimation"

In ChengchenZhao/MethylTransition: Estimated DNA Methylation Parameters and Calculate Methylation Classes after n Cell Cycle(s)

Description

Estimated the parameters that represent the probabilities of three active DNA methylation change types during n cell cycle(s).

Usage

TMParameterEstimation(observation_matrix, iter = 50, cell_cycle = 1)

Arguments

observation_matrix	The transition matrix(5 \times 5) from the original state to the terminal state. original_class1 original_class2 original_class3 original_class4 original_class5 terminal_class1 a1 b1 c1 d1 e1 terminal_class2 a2 b2 c2 d2 e2 terminal_class3 a3 b3 c3 d3 e3 terminal_class4 a4 b4 c4 d4 e4 terminal_class5 a5 b5 c5 d5 e5 # observation_matrix[1,1](a1) is the ratio of original_class1 to terminal_class1,observation_matrix[1,2](b1) is the ratio of original_class2 to terminal_class1 and so on # observation_matrix[2,1](a2) is the ratio of original_class1 to terminal_class2,observation_matrix[2,2](b2) is the ratio of original_class2 to terminal_class2 and so on # The sum of the ratio that original_class1 change to 5 classes should be 1. That is a1+a2+a3+a4+a5=1.
iter	The iteration times of the parameter estimation using the Newton-Raphson method with different initial guesses.
cell_cycle	The cell cycle times from the original state to the terminal state.

Details

The transition matrix of this model describes the changes of DNA methylation during one cell cycle in three steps: passive demethylation by DNA replication, active DNA methylation changes affected by DNA methylation-modifying enzymes and DNA methylation combinations during homologous recombination. For each CpG site in a chromsome, the methylation states are one of these four types: 0-0, 0-1, 1-0, 1-1. The transition matrix after DNA replication would be :

	original(0-0)	original(0-1)	original(1-0)	original(1-1)
after_replication(0-0)	1	a	1-a	0
after_replication(0-1)	0	1-a	0	1-a
after_replication(1-0)	0	0	0	a
after_replication(1-1)	0	0	0	0

among this matrix ,a is the methylation change probability with DNA replication and equal to 0.5. Then the transition matrix after active DNA methylation changes would be:

	after_replication(0-0)	after_replication(0-1)	after_replication(1-0)	after_replication(1-1)
after_enzymemodifying(0-0)	(1-u) \times (1-u)	(1-u-p+u \times p) \times d	d \times (1-u-p+u \times p)	0
after_enzymemodifying(0-1)	u \times (1-u)	(1-u-p+u \times p) \times (1-d)	d \times (u+p-u \times p)	0
after_enzymemodifying(1-0)	u \times (1-u)	(u+p-u \times p) \times d	(1-d) \times (1-u-p+u \times p)	0
after_enzymemodifying(1-1)	u \times u	(u+p-u \times p) \times (1-d)	(1-d) \times (u+p-u \times p)	1

The paramater u described the methylation probablity on CpG site. The paramater d described the de-methylation probablity on 5mCpG site. The paramater p described the methylation probablity on semi-CpG site. Thus the mathlation state change is P = Penzymemodifying \cdot Preplication \cdot original_classes We using t_{i,j} represent the i and j vector of the matrix Penzymemodifying \cdot Preplication Then two chromsomes are combined during homologous recombination. The observed DNA methylation of each CpG site is the combination of the methylation types in both chromsomes. The observed transition matrix would be :

	original_class1(0)	original_class2(1/4)	original_class3(1/2)	original_class4(3/4)	original_class5(1)
terminal_class1(0)	x_{1,1}	x_{1,2}	x_{1,3}	x_{1,4}	x_{1,5}
terminal_class2(1/4)	x_{2,1}	x_{2,2}	x_{2,3}	x_{2,4}	x_{2,5}
terminal_class3(1/2)	x_{3,1}	x_{3,2}	x_{3,3}	x_{3,4}	x_{3,5}
terminal_class4(3/4)	x_{4,1}	x_{4,2}	x_{4,3}	x_{4,4}	x_{4,5}
terminal_class5(1)	x_{5,1}	x_{5,2}	x_{5,3}	x_{5,4}	x_{5,5}

and

x_{1,1}=t_{1,1} \times t_{1,1}

x_{1,1}=t_{{,1},1} \times t_{{,1},1}

x_{1,2}=1/4 \times (t_{1,1} \times t_{1,2}+t_{1,1} \times t_{1,3}+t_{1,2} \times t_{1,1}+t_{1,3} \times t_{1,1})

x_{1,3}=1/6 \times (t_{1,1} \times t_{1,4}+t_{1,2} \times t_{1,2}+t_{1,2} \times t_{1,3}+t_{1,3} \times t_{1,2}+t_{1,3} \times t_{1,3}+t_{1,4} \times t_{1,1})

x_{1,4}=1/4 \times (t_{1,2} \times t_{1,4}+t_{1,3} \times t_{1,4}+t_{1,4} \times t_{1,2}+t_{1,4} \times t_{1,3})

x_{1,5}=t_{1,4} \times t_{1,4}

x_{2,1}=t_{1,1} \times t_{2,1}+t_{1,1} \times t_{3,1}+t_{2,1} \times t_{1,1}+t_{3,1} \times t_{1,1}

x_{2,2}=1/4 \times (t_{1,1} \times t_{2,2}+t_{1,1} \times t_{2,3}+t_{1,2} \times t_{2,1}+t_{1,3} \times t_{2,1}+t_{1,1} \times t_{3,2}+t_{1,1} \times t_{3,3}+t_{1,2} \times t_{3,1}+t_{1,3} \times t_{3,1}+t_{2,1} \times t_{1,2}+t_{2,1} \times t_{1,3}+t_{2,2} \times t_{1,1}+t_{2,3} \times t_{1,1}+t_{3,1} \times t_{1,2}+t_{3,1} \times t_{1,3}+t_{3,2} \times t_{1,1}+t_{3,3} \times t_{1,1})

x_{2,3}=1/6 \times (t_{1,1} \times t_{2,4}+t_{1,2} \times t_{2,2}+t_{1,2} \times t_{2,3}+t_{1,3} \times t_{2,2}+t_{1,3} \times t_{2,3}+t_{1,4} \times t_{2,1}+t_{1,1} \times t_{3,4}+t_{1,2} \times t_{3,2}+t_{1,2} \times t_{3,3}+t_{1,3} \times t_{3,2}+t_{1,3} \times t_{3,3}+t_{1,4} \times t_{3,1}+t_{2,1} \times t_{1,4}+t_{2,2} \times t_{1,2}+t_{2,2} \times t_{1,3}+t_{2,3} \times t_{1,2}+t_{2,3} \times t_{1,3}+t_{2,4} \times t_{1,1}+t_{3,1} \times t_{1,4}+t_{3,2} \times t_{1,2}+t_{3,2} \times t_{1,3}+t_{3,3} \times t_{1,2}+t_{3,3} \times t_{1,3}+t_{3,4} \times t_{1,1})

x_{2,4}=1/4 \times (t_{1,2} \times t_{2,4}+t_{1,3} \times t_{2,4}+t_{1,4} \times t_{2,2}+t_{1,4} \times t_{2,3}+t_{1,2} \times t_{3,4}+t_{1,3} \times t_{3,4}+t_{1,4} \times t_{3,2}+t_{1,4} \times t_{3,3}+t_{2,2} \times t_{1,4}+t_{2,3} \times t_{1,4}+t_{2,4} \times t_{1,2}+t_{2,4} \times t_{1,3}+t_{3,2} \times t_{1,4}+t_{3,3} \times t_{1,4}+t_{3,4} \times t_{1,2}+t_{3,4} \times t_{1,3})

x_{2,5}=t_{1,4} \times t_{2,4}+t_{1,4} \times t_{3,4}+t_{2,4} \times t_{1,4}+t_{3,4} \times t_{1,4}

x_{3,1}=t_{1,1} \times t_{4,1}+t_{2,1} \times t_{2,1}+t_{2,1} \times t_{3,1}+t_{3,1} \times t_{2,1}+t_{3,1} \times t_{3,1}+t_{4,1} \times t_{1,1}

x_{3,2}=1/4 \times (t_{1,1} \times t_{4,2}+t_{1,1} \times t_{4,3}+t_{1,2} \times t_{4,1}+t_{1,3} \times t_{4,1}+t_{2,1} \times t_{2,2}+t_{2,1} \times t_{2,3}+t_{2,2} \times t_{2,1}+t_{2,3} \times t_{2,1}+t_{2,1} \times t_{3,2}+t_{2,1} \times t_{3,3}+t_{2,2} \times t_{3,1}+t_{2,3} \times t_{3,1}+t_{3,1} \times t_{2,2}+t_{3,1} \times t_{2,3}+t_{3,2} \times t_{2,1}+t_{3,3} \times t_{2,1}+t_{3,1} \times t_{3,2}+t_{3,1} \times t_{3,3}+t_{3,2} \times t_{3,1}+t_{3,3} \times t_{3,1}+t_{4,1} \times t_{1,2}+t_{4,1} \times t_{1,3}+t_{4,2} \times t_{1,1}+t_{4,3} \times t_{1,1})

x_{3,3}=1/6 \times (t_{1,1} \times t_{4,4}+t_{1,2} \times t_{4,2}+t_{1,2} \times t_{4,3}+t_{1,3} \times t_{4,2}+t_{1,3} \times t_{4,3}+t_{1,4} \times t_{4,1}+t_{2,1} \times t_{2,4}+t_{2,2} \times t_{2,2}+t_{2,2} \times t_{2,3}+t_{2,3} \times t_{2,2}+t_{2,3} \times t_{2,3}+t_{2,4} \times t_{2,1}+t_{2,1} \times t_{3,4}+t_{2,2} \times t_{3,2}+t_{2,2} \times t_{3,3}+t_{2,3} \times t_{3,2}+t_{2,3} \times t_{3,3}+t_{2,4} \times t_{3,1}+t_{3,1} \times t_{2,4}+t_{3,2} \times t_{2,2}+t_{3,2} \times t_{2,3}+t_{3,3} \times t_{2,2}+t_{3,3} \times t_{2,3}+t_{3,4} \times t_{2,1}+t_{3,1} \times t_{3,4}+t_{3,2} \times t_{3,2}+t_{3,2} \times t_{3,3}+t_{3,3} \times t_{3,2}+t_{3,3} \times t_{3,3}+t_{3,4} \times t_{3,1}+t_{4,1} \times t_{1,4}+t_{4,2} \times t_{1,2}+t_{4,2} \times t_{1,3}+t_{4,3} \times t_{1,2}+t_{4,3} \times t_{1,3}+t_{4,4} \times t_{1,1})

x_{3,4}=1/4 \times (t_{1,2} \times t_{4,4}+t_{1,3} \times t_{4,4}+t_{1,4} \times t_{4,2}+t_{1,4} \times t_{4,3}+t_{2,2} \times t_{2,4}+t_{2,3} \times t_{2,4}+t_{2,4} \times t_{2,2}+t_{2,4} \times t_{2,3}+t_{2,2} \times t_{3,4}+t_{2,3} \times t_{3,4}+t_{2,4} \times t_{3,2}+t_{2,4} \times t_{3,3}+t_{3,2} \times t_{2,4}+t_{3,3} \times t_{2,4}+t_{3,4} \times t_{2,2}+t_{3,4} \times t_{2,3}+t_{3,2} \times t_{3,4}+t_{3,3} \times t_{3,4}+t_{3,4} \times t_{3,2}+t_{3,4} \times t_{3,3}+t_{4,2} \times t_{1,4}+t_{4,3} \times t_{1,4}+t_{4,4} \times t_{1,2}+t_{4,4} \times t_{1,3})

x_{3,5}=t_{1,4} \times t_{4,4}+t_{2,4} \times t_{2,4}+t_{2,4} \times t_{3,4}+t_{3,4} \times t_{2,4}+t_{3,4} \times t_{3,4}+t_{4,4} \times t_{1,4}

x_{4,1}=t_{2,1} \times t_{4,1}+t_{3,1} \times t_{4,1}+t_{4,1} \times t_{2,1}+t_{4,1} \times t_{3,1}

x_{4,2}=1/4 \times (t_{2,1} \times t_{4,2}+t_{2,1} \times t_{4,3}+t_{2,2} \times t_{4,1}+t_{2,3} \times t_{4,1}+t_{3,1} \times t_{4,2}+t_{3,1} \times t_{4,3}+t_{3,2} \times t_{4,1}+t_{3,3} \times t_{4,1}+t_{4,1} \times t_{2,2}+t_{4,1} \times t_{2,3}+t_{4,2} \times t_{2,1}+t_{4,3} \times t_{2,1}+t_{4,1} \times t_{3,2}+t_{4,1} \times t_{3,3}+t_{4,2} \times t_{3,1}+t_{4,3} \times t_{3,1})

x_{4,3}=1/6 \times (t_{2,1} \times t_{4,4}+t_{2,2} \times t_{4,2}+t_{2,2} \times t_{4,3}+t_{2,3} \times t_{4,2}+t_{2,3} \times t_{4,3}+t_{2,4} \times t_{4,1}+t_{3,1} \times t_{4,4}+t_{3,2} \times t_{4,2}+t_{3,2} \times t_{4,3}+t_{3,3} \times t_{4,2}+t_{3,3} \times t_{4,3}+t_{3,4} \times t_{4,1}+t_{4,1} \times t_{2,4}+t_{4,2} \times t_{2,2}+t_{4,2} \times t_{2,3}+t_{4,3} \times t_{2,2}+t_{4,3} \times t_{2,3}+t_{4,4} \times t_{2,1}+t_{4,1} \times t_{3,4}+t_{4,2} \times t_{3,2}+t_{4,2} \times t_{3,3}+t_{4,3} \times t_{3,2}+t_{4,3} \times t_{3,3}+t_{4,4} \times t_{3,1})

x_{4,4}=1/4 \times (t_{2,2} \times t_{4,4}+t_{2,3} \times t_{4,4}+t_{2,4} \times t_{4,2}+t_{2,4} \times t_{4,3}+t_{3,2} \times t_{4,4}+t_{3,3} \times t_{4,4}+t_{3,4} \times t_{4,2}+t_{3,4} \times t_{4,3}+t_{4,2} \times t_{2,4}+t_{4,3} \times t_{2,4}+t_{4,4} \times t_{2,2}+t_{4,4} \times t_{2,3}+t_{4,2} \times t_{3,4}+t_{4,3} \times t_{3,4}+t_{4,4} \times t_{3,2}+t_{4,4} \times t_{3,3})

x_{4,5}=t_{2,4} \times t_{4,4}+t_{3,4} \times t_{4,4}+t_{4,4} \times t_{2,4}+t_{4,4} \times t_{3,4}

x_{5,1}=t_{4,1} \times t_{4,1}

x_{5,2}=1/4 \times (t_{4,1} \times t_{4,2}+t_{4,1} \times t_{4,3}+t_{4,2} \times t_{4,1}+t_{4,3} \times t_{4,1})

x_{5,3}=1/6 \times (t_{4,1} \times t_{4,4}+t_{4,2} \times t_{4,2}+t_{4,2} \times t_{4,3}+t_{4,3} \times t_{4,2}+t_{4,3} \times t_{4,3}+t_{4,4} \times t_{4,1})

x_{5,4}=1/4 \times (t_{4,2} \times t_{4,4}+t_{4,3} \times t_{4,4}+t_{4,4} \times t_{4,2}+t_{4,4} \times t_{4,3})

x_{5,5}=t_{4,4} \times t_{4,4}

The cost function was defined by

f_{cost}=∑_{i=1,j=1}^{n=5} (o_{i,j}-x_{i,j})

and minimized using the Newton-Raphson method.

Value

estimated_parameters The estimated parameters using the maximum likelihood estimation and the Newton-Raphson method.

predicted_matrix The calculated transition matrix using the estimated parameters.

References

Zhao, C. et.al.(2019). A DNA methylation state transition model reveals the programmed epigenetic heterogeneity in pre-implantation embryos. Under revision.

Examples

# Let's start from a transtion matrix
observation_matrix <- matrix(c(0.9388,0.0952,0.0377,0,0.0001,
                               0.0497,0.5873,0.1887,0.0344,0.0149,
                               0.0096,0.2381,0.4151,0.0653,0.0876,
                               0.0019,0.0635,0.3396,0.6151,0.253,
                               0,0.0159,0.0189,0.2852,0.6444),5,5,byrow=T)

# The DNA methylation states change from original state to the terminal state after 1 cell cycle.
TMParameterEstimation(observation_matrix,iter=30,cell_cycle=1)

# The DNA methylation states change from original state to the terminal state after 2 cell cycle.
# TMParameterEstimation(observation_matrix,iter=1,cell_cycle=2)
# if this function was not successful to estimated the function, you may try more iterations with different initial guesses
TMParameterEstimation(observation_matrix,iter=50,cell_cycle=2)

# The DNA methylation states change from original state to the terminal state after 30 cell cycle.
TMParameterEstimation(observation_matrix,iter=50,cell_cycle=30)

ChengchenZhao/MethylTransition documentation built on Nov. 30, 2020, 11:09 a.m.