A gene Y with simple regulation has a time-dependent production rate β(t) and a time-dependent degradation...
A gene Y with simple regulation has a time-dependent production rate β(t) and a time-dependent degradation rate α(t). Solve for its concentration as a function of time.
Homework Answers
Verify by taking the time derivative that the following is correct:
Y(t) = exp (- ʃ α (t’) dt’) [Y(0) + ʃ β (t’) exp (ʃ α (t’’) dt’’) dt’]
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A gene Y with simple regulation has a time-dependent production
rate β(t) and a time-dependent degradation...
Problem 3. Fan-out: Transcription factor X regulates two genes Y1 and Y2. Draw the resulting network,...
Problem 3. Fan-out: Transcription factor X regulates two genes Y1 and Y2. Draw the resulting network, termed a fan- out with two target genes. The activation thresholds for these genes are Ki and K2. The activator X begins to be produced at time t=0 at rate 6, and is degraded/diluted at rate a, and its signal Sx is present throughout. What are the times at which Yn and Y2 reach halfway to their maximal expression? Problem 4. Time-dependent production and...
Suppose output, Y t, is produced using capital, K t, and labor, N t, according to...
Suppose output, Y t, is produced using capital, K t, and labor, N t, according to the production function: Y t = A ⋅ ( K t α N t 1 − α + K t β N t 1 − β )where the parameters satisfy 0 < α < 1, 0 < β < 1 and A > 0. a) (5 pts) Write the production function in “per worker” terms. That is, if we define y t = Y...
Exercise 1: Solow model . Consider an economy whose production function is defined by Y (t)...
Exercise 1: Solow model . Consider an economy whose production function is defined by Y (t) = F (K (t), L (t)) = K (t) 1 − α · L (t) α. with 0 <α <1. In this economy, the population grows at the following rate: L (t) = n + β where n and β are strictly positive constants and k (t) represents capital per capita: k (t) = L (t). Moreover, a constant part of the product is...
1. The time-dependent Schrödinger equation The time-dependent Schrödinger equation is -R2 824(1,t) + V (1,t) (1,t)...
1. The time-dependent Schrödinger equation The time-dependent Schrödinger equation is -R2 824(1,t) + V (1,t) (1,t) = in 2m 0:2 . (a) For V1, t) = 0, show that the wave function (1,t) = A sin (kr - wt) does not satisfy the time- dependent Schrödinger equation. (b) For VI,t) = 0, Show that I, t) = A cos(kr - wt) + i sin (kr - wt) does satisfy this equation. This is a simple demonstration that the wavefunction in...
2. Consider a linear time-invariant system with transfer function H(s)Find the (s + α)(s + β) impulse response, h(t), of the system 2. Consider a linear time-invariant system with transfer f...
2. Consider a linear time-invariant system with transfer function H(s)Find the (s + α)(s + β) impulse response, h(t), of the system
2. Consider a linear time-invariant system with transfer function H(s)Find the (s + α)(s + β) impulse response, h(t), of the system
Consider the following vector field: a(t) where a(t) is an arbitrary time dependent function. (a)...
Consider the following vector field: a(t) where a(t) is an arbitrary time dependent function. (a) Show that the origin is a hyperbolic trajectory. (b) Argue that the graph of y 2 is the global unstable manifold of the origin. What requirements must be made on the function a(t) in order that these conclusions are true?
Consider the following vector field: a(t) where a(t) is an arbitrary time dependent function. (a) Show that the origin is a hyperbolic trajectory. (b) Argue...
(25 pts) Solve for the time-dependent concentration of ammonium chloride based on the following reaction equation....
(25 pts) Solve for the time-dependent concentration of ammonium chloride based on the following reaction equation. Assume that the reaction rate constants and the concentrations of ammonia and hydrogen chloride are constant. Assume an initial concentration of ammonium chloride of 0.5mM. Be sure to solve for the constants. NH3+HCl NH4CI Reaction: a. (5 pts) Write the reaction rate equation for NH2C b. (5 pts) Write the differential equation in our standard form (+A -y = B) dx c. (5 pts)...
8. (4pts) You are interested in the regulation of a gene that regulates cartilage production in...
8. (4pts) You are interested in the regulation of a gene that regulates cartilage production in joints, gene Crt. You find three proteins that regulate gene Crt expression. Protein A, B and C bind the promoter of the Crt gene to increase or decrease transcription of Crt. Proteins A and B both bind to site 1 but cannot bind to site 1 at the same time. Protein C binds to site 2 on the promoter. Transcriptional start site mRNA of...
We have this simple ODE model subject to x(0) = x0 ≥ 0, y(0) = y0...
We have this simple ODE model subject to x(0) = x0 ≥ 0, y(0) =
y0 ≥ 0 (you may choose values of x0 and y0). The constants α, β
> 0.
Question: Find an ODE for y(t) by eliminating x. Solve this ODE
analytically. Plot solutions using Mathematica.
x — ау dx dt dy dt = Вх y
A faulty model rocket moves in the xy-plane (the positive y-direction is vertically upward). The rocket's...
A faulty model rocket moves in the xy-plane (the positive y-direction is vertically upward). The rocket's acceleration has components ax(t)=αt2 and ay(t)=β−γt, where α = 2.50 m/s4, β = 9.00 m/s2, and γ = 1.40 m/s3. At t=0 the rocket is at the origin and has velocity v 0=v0xi^+v0yj^ with v0x = 1.00 m/s and v0y = 7.00 m/s. a. Calculate the velocity vector as a function of time. Express your answer in terms of v0x, v0y, β, γ, and...
Problem 3. Fan-out: Transcription factor X regulates two genes Y1 and Y2. Draw the resulting network, termed a fan- out with two target genes. The activation thresholds for these genes are Ki and K2. The activator X begins to be produced at time t=0 at rate 6, and is degraded/diluted at rate a, and its signal Sx is present throughout. What are the times at which Yn and Y2 reach halfway to their maximal expression? Problem 4. Time-dependent production and...
1. The time-dependent Schrödinger equation The time-dependent Schrödinger equation is -R2 824(1,t) + V (1,t) (1,t) = in 2m 0:2 . (a) For V1, t) = 0, show that the wave function (1,t) = A sin (kr - wt) does not satisfy the time- dependent Schrödinger equation. (b) For VI,t) = 0, Show that I, t) = A cos(kr - wt) + i sin (kr - wt) does satisfy this equation. This is a simple demonstration that the wavefunction in...
2. Consider a linear time-invariant system with transfer function H(s)Find the (s + α)(s + β) impulse response, h(t), of the system
2. Consider a linear time-invariant system with transfer function H(s)Find the (s + α)(s + β) impulse response, h(t), of the system
Consider the following vector field: a(t) where a(t) is an arbitrary time dependent function. (a) Show that the origin is a hyperbolic trajectory. (b) Argue that the graph of y 2 is the global unstable manifold of the origin. What requirements must be made on the function a(t) in order that these conclusions are true?
Consider the following vector field: a(t) where a(t) is an arbitrary time dependent function. (a) Show that the origin is a hyperbolic trajectory. (b) Argue...
(25 pts) Solve for the time-dependent concentration of ammonium chloride based on the following reaction equation. Assume that the reaction rate constants and the concentrations of ammonia and hydrogen chloride are constant. Assume an initial concentration of ammonium chloride of 0.5mM. Be sure to solve for the constants. NH3+HCl NH4CI Reaction: a. (5 pts) Write the reaction rate equation for NH2C b. (5 pts) Write the differential equation in our standard form (+A -y = B) dx c. (5 pts)...
8. (4pts) You are interested in the regulation of a gene that regulates cartilage production in joints, gene Crt. You find three proteins that regulate gene Crt expression. Protein A, B and C bind the promoter of the Crt gene to increase or decrease transcription of Crt. Proteins A and B both bind to site 1 but cannot bind to site 1 at the same time. Protein C binds to site 2 on the promoter. Transcriptional start site mRNA of...
We have this simple ODE model subject to x(0) = x0 ≥ 0, y(0) =
y0 ≥ 0 (you may choose values of x0 and y0). The constants α, β
> 0.
Question: Find an ODE for y(t) by eliminating x. Solve this ODE
analytically. Plot solutions using Mathematica.
x — ау dx dt dy dt = Вх y
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