toggleSwitch: The toggle switch model

In CharlotteJana/pdmpsim: Simulate Piecewise Deterministic Markov Processes

The toggle switch model

Description

This is an example for a piecewise deterministic markov process defined as pdmpModel. It models a gene regulation mechanism that is called toggle switch. This mechanism describes two genes A and B whose gene products are repressors to the other gene, i. e. the gene product of A blocks the gene expression of gene B and vice versa. This model is an example for a PDMP with two discrete and two continous variables. It is included to the package for demonstration purposes and is used in some unit tests and function examples.

Usage

toggleSwitch

Format

An object of class pdmpModel.

Slots

init

There are two continous variables f_A and f_B which describe the concentration of gene products from gene A and gene B, respectivly. Both have initial value 0.5. The two discrete variables d_A and d_B describe the expression state of gene A and gene B and have initial value 1, which means that both genes are not blocked.

discStates

The discrete variables d_A and d_B both have codomain {0, 1} where 0 stands for "Gene A/B is blocked" and 1 stands for "Gene A/B is unblocked".

dynfunc

The dynamic of the continous variable f_A depends on the state of discrete variable d_A. In case d_A = 0 it is given by \frac{df_A}{dt} = -b_A \cdot f_A describing an exponential decay of the concentration of gene product A. If d_A = 1, there is an additional term of linear growth leading to the ODE \frac{df_A}{dt} = -b_A \cdot f_A + a_A. Both formulas can be combined to \frac{df_A}{dt} = -b_A \cdot f_A + a_A \cdot d_A. Accordingly, the dynamic of dB is given as \frac{df_B}{dt} = -b_B \cdot f_B + a_B \cdot d_B.

jumpfunc

There are two jumptypes. If the jump is of type 1, only the value of d_B is changed (from 0 to 1 or 1 to 0, respectivly). All other variables remain unchanged. The second jumptype changes only the value of d_A.

ratefunc

A vector of length two determining the probability of a jump being of type 1 or 2. In this model, a molecule of gene product A can act as repressor and impede the gene expression of gene B. The rates for the first jumptype (which describes a change in the gene expression of gene B) are therefore k_{10A} \cdot fA for the change from the unblocked (d_B = 1) to the blocked (d_B = 0) state and k_{01A} for the change from the blocked to the unblocked state of gene B. The rates for the second jumptype are generated in an analogous way because gene A is repressed by a molecule of gene product B and the second jumptype describes a change in the gene expression of gene A.

parms

There are are a number of parameters that appear in the dynamics and rates of the process, namely b_A, b_B, a_A, a_B, k_{01A}, k_{10A}, k_{01B} and k_{10B}. The values of the parameters in this example are artificial and are not based on real data.

times

The simulations will start at time t = 0 and end at t = 10 with step length 0.01.

See Also

simplePdmp for an easier example of a pdmpModel and pdmpModel for the formal description of the S4 class.

Examples

## the code used to generate this model:

toggleSwitch <- pdmpModel(
    descr = "Toggle Switch with two Promotors",
    parms = list(bA = 0.5, bB = 0.5, aA = 2, aB = 4, 
                 k01A = 0.5, k10A = 2, k01B = 0.3, k10B = 3),
    init = c(fA = 0.5, fB = 0.5, dA = 1.0, dB = 1.0),
    discStates = list(dA = c(0, 1), dB = c(0, 1)),
    times = c(from = 0, to = 100, by = 0.01),
    dynfunc = function(t, x, parms) {
       df <- with(as.list(c(x, parms)), c(-bA*fA + aA*dA, -bB*fB + aB*dB))
       return(c(df, 0, 0))
    },
    ratefunc = function(t, x, parms) {
       return(with(as.list(c(x, parms)), c(switch(dB+1, k01B, k10B*fA),
                                           switch(dA+1, k01A, k10A*fB))))
    },
    jumpfunc = function(t, x, parms, jtype){
       return(with(as.list(c(x, parms)), c(fA, fB, switch(jtype,
                                                          c(dA, 1-dB),
                                                          c(1-dA, dB)))))
    })

## load it and plot a simulation:
data("toggleSwitch")
plot(sim(toggleSwitch))

CharlotteJana/pdmpsim documentation built on Oct. 21, 2024, 4:54 p.m.