User:Tom Adie/CS2

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MRes Case Study 2

R Exercise 2 (ODE Solver)


RK4 solver:

  1. RK4 documentation
  2. Interface: rk4(y, times, func, parms, ...)
  3. Type >library(odesolve) to load the library that includes rk4

Model: Constitutive gene expression

  1. ODE System:
    • d[mRNA]/dt = k_1 - d_1*[mRNA]
    • d[protein]/dt = k_2[mRNA] -d_2*[protein]

Questions:

  • 1. Considering that the half-life for the mRNA and the protein is respectively 3min and 1h. Workout the values of (k_1, d_1, k_2, d_2) so that, at steady state, [mRNA] = 3 and [protein] = 500.

Using the relationship degradation rate = ln2 / half-life we get values of d1 = 0.23 and d2 = 0.012. At steady state, d[mRNA]/dt and d[Protein]/dt are set to 0. Rearranging gives

[mRNA]=k1/d1
k2/d2=[Protein]/[mRNA]

substituting in values of d1, d2 [mRNA] and [Protein] we find k1 = 3*0.23 = 0.69 and k2 = 0.012*500/3 = 2

  • 2. Using the previously found parameter values, plot on the same graph [mRNA](t) and [protein](t) for t=[0, 5h].

Concentrations at time t can be found using a step function as below: 

ODE <- function(k1,d1,k2,d2,mrna,protein,t) {
for (i in 1:t) {
protein <- protein+k2*mrna-d2*protein
mrna <- mrna+k1-d1*mrna
}
cat("        t =",t,"\n")
cat("   [mRNA] =",mrna,"\n")
cat("[Protein] =",protein,"\n")
}
ODE(0.69,0.23,2,0.012,0,0,300)
        t = 300 
   [mRNA] = 3 
[Protein] = 485.8966

this uses the parameters from above to output the values of [mRNA] and [Protein] at time t (where t is in minutes). Saving the values of t, [mRNA] and [Protein] after every step into concatenated lists, then saving those as a data set, would give me something to graph... But this doesn't use rk4!

  • 3. Parameter scanning: Consider parameter d_2 to vary [-20%, 20%] of its nominal value (10 uniformly spread values). Generate on the same graph the 10 [protein](t) trajectories (t=[0, 5h]).

Intro

Links:

Inputs:

  • Wavelength
  • Intensity
  • Stimulation time (pulse length)

Outputs:

  • Viability of cells (bearing in mind YtvA pathway triggers stress response)
  • PoPs
References
  1. One
  2. Two
  3. Three
  4. Four
  5. Five

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