Reference no: EM132660827

Modeling feedback systems

Question 1. Model a negative feedback system that uses trimers for repression. Assume that monomer binds to monomer, then dimers binds to other monomer as the mechanism. You can assume that only monomers degrade and dimer/trimer degradation is negligible. You can assume the total amount of DNA, RNA polymerase, and Ribosomes is constant. Assume some level of basal expression (β) when no repressors are bound, and that transcription is negligible when repressors are bound.
a) Write down reaction list
b) Write down full mass action equations
c) Simplify equations to mRNA/Protein dynamics using QSSA
d) Simplify equations to Protein only dynamics.
e) In plain language, what assumptions are made in parts c) and d)?

Question 2. a) Write down the reaction list for the above positive/negative feedback system. mRNA dynamics description can be neglected if desired. You can assume that monomers degrade and dimer degradation is negligible. You can assume the total amount of DNA, RNA polymerase, and Ribosomes is constant. Assume some level of basal expression (β) when no activator (X) is bound.

b) Simplify the dynamics to mRNA/protein equations, and then to protein only equations for x & y (monomers). mRNA dynamics description can be neglected if desired.

c) Show that by appropriate scaling of parameters, the equations can be non- dimensionalized to the form below. What does the parameter mxy represent?

x^· = 1+ax²/1+x² -y_xx - y_x??xy

y^· = m_xy(1+ax²) - y_yy

d) Simulate the system using ode23 function in MATLAB for two sets of parameters: one that oscillates and one that does not. Make a plot of x vs t. Parameter set [a = 100, g_x = 0.5, g_y = 0.05, g_xy = 0.1, m_xy = 1] can be used as a reference.

e) Make a plot of phase portraits (x vs. y) for same two sets of parameters: one that oscillates and one that does not.

Question 3. Model the lactate biosensor shown above. In the absence of lactate, dimer repressor inhibits pLldR promoter from transcription, in the presence of lactate, dimer derepress and allows for pLldR promoter from transcription/translation, leading to Py production. Assume monomer binds to monomer and dimer represses pLldR promoter activity. You can assume that only monomers degrade and dimer degradation is negligible. You can assume the total amount of DNA, RNA polymerase, and Ribosomes is constant. Assume some level of basal expression

(β) when no repressors are bound, and that transcription is negligible when repressors are bound.
a) Write down reaction list
b) Write down full mass action equations
c) Simplify equations to mRNA/Protein dynamics using QSSA
d) Simplify equations to Protein only dynamics. (leave the lactate term!)
e.) Simulate the Py production by lactate sensor in MATLAB with different concentrations of lactate ( 0, 0.1, 1, 5, 10, 50 mM).
f.) What are some applications for such a sensor?