Glycolysis is a fundamental biochemical process, from which living cells obtain energy by breaking down sugar. In intact yeast cells or muscle extracts, glycolysis can proceed in an oscillatory fashion, with the concentrations of various intermediates increasing and decreasing with a period of several minutes. A simple model of these oscillations is given by the following equations: dx = -x + ay + x*y, dt (1) dy = 6 dt ay – x²y, (2) where x and y are the concentrations of adenosine diphosphate (ADP) and fructose-6-phosphate (F6P), and a, b > 0 are kinetic parameters. 1. The fixed points of a system of ODES are defined as those points, z*, where = 0 (note that bold means it is a vector). Find the fixed point of the model above, (x*, y*), as a function of the kinetic pa- rameters. Note that you can do this by hand, and you should get expressions that depend on a and b. 2. Explore the dynamical behavior of the model using RK4 method or the ode45 built-in function, this means to get x(t) and y(t). As the kinetic parameters are important, you will have to explore the dynamical behavior as these parameters are changed. For this project, the parameter a should be kept at the value a = 0.06. You will need to explore the effect of changing the parameter b in the interval b e [0.1,1.1], at least ten different values of b should be studied. For each of the selected parameter values: (i) use the RK4 method or the ode45 built-in function, and solve the equation for 2 different initial conditions: one close to the fixed point you found in the previous item, and another far from it, but in the region x E [0.1,3.0] and y E [0.1, 3.0]. (i) Plot the time series, x(t) and y(t) vs t, for the 2 different initial conditions, different subplots on same figure. Label each plot according to its initial condition. Choose the total time to be long enough to observe the qualitative behavior clearly. (iii) Generate another figure that plots y vs x for each initial condition in the same figure, also plot the location of the corresponding fixed point (x*, y*).
Glycolysis is a fundamental biochemical process, from which living cells obtain energy by breaking down sugar. In intact yeast cells or muscle extracts, glycolysis can proceed in an oscillatory fashion, with the concentrations of various intermediates increasing and decreasing with a period of several minutes. A simple model of these oscillations is given by the following equations: dx = -x + ay + x*y, dt (1) dy = 6 dt ay – x²y, (2) where x and y are the concentrations of adenosine diphosphate (ADP) and fructose-6-phosphate (F6P), and a, b > 0 are kinetic parameters. 1. The fixed points of a system of ODES are defined as those points, z*, where = 0 (note that bold means it is a vector). Find the fixed point of the model above, (x*, y*), as a function of the kinetic pa- rameters. Note that you can do this by hand, and you should get expressions that depend on a and b. 2. Explore the dynamical behavior of the model using RK4 method or the ode45 built-in function, this means to get x(t) and y(t). As the kinetic parameters are important, you will have to explore the dynamical behavior as these parameters are changed. For this project, the parameter a should be kept at the value a = 0.06. You will need to explore the effect of changing the parameter b in the interval b e [0.1,1.1], at least ten different values of b should be studied. For each of the selected parameter values: (i) use the RK4 method or the ode45 built-in function, and solve the equation for 2 different initial conditions: one close to the fixed point you found in the previous item, and another far from it, but in the region x E [0.1,3.0] and y E [0.1, 3.0]. (i) Plot the time series, x(t) and y(t) vs t, for the 2 different initial conditions, different subplots on same figure. Label each plot according to its initial condition. Choose the total time to be long enough to observe the qualitative behavior clearly. (iii) Generate another figure that plots y vs x for each initial condition in the same figure, also plot the location of the corresponding fixed point (x*, y*).
Computer Networking: A Top-Down Approach (7th Edition)
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![Glycolysis is a fundamental biochemical process, from which living cells obtain energy by breaking down
sugar. In intact yeast cells or muscle extracts, glycolysis can proceed in an oscillatory fashion, with the
concentrations of various intermediates increasing and decreasing with a period of several minutes. A
simple model of these oscillations is given by the following equations:
dx
2
-x + ay + xʻ
(1)
dt
dy
= b – ay
dt
(2)
where x and y are the concentrations of adenosine diphosphate (ADP) and fructose-6-phosphate (F6P),
and a, b > 0 are kinetic parameters.
dz*
1. The fixed points of a system of ODES are defined as those points, z*, where
means it is a vector). Find the fixed point of the model above, (x*, y*), as a function of the kinetic pa-
rameters. Note that you can do this by hand, and you should get expressions that depend on a and b.
= 0 (note that bold
2. Explore the dynamical behavior of the model using RK4 method or the ode45 built-in function, this
means to get x(t) and y(t). As the kinetic parameters are important, you will have to explore the
dynamical behavior as these parameters are changed. For this project, the parameter a should be
kept at the value a =
interval b e [0.1, 1.1], at least ten different values of b should be studied. For each of the selected
parameter values: (i) use the RK4 method or the ode45 built-in function, and solve the equation
for 2 different initial conditions: one close to the fixed point you found in the previous item, and
another far from it, but in the region x E [0.1, 3.0] and y e [0.1,3.0]. (i) Plot the time series, x(t)
and y(t) vs t, for the 2 different initial conditions, different subplots on same figure. Label each plot
according to its initial condition. Choose the total time to be long enough to observe the qualitative
behavior clearly. (iii) Generate another figure that plots y vs x for each initial condition in the same
figure, also plot the location of the corresponding fixed point (x*, y*).
0.06. You will need to explore the effect of changing the parameter b in the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f04e2f-081f-424b-be8b-a6d80cdb27e3%2F8b71b91d-9961-41c9-8f8d-d323f85cc646%2F36ngrqq_processed.png&w=3840&q=75)
Transcribed Image Text:Glycolysis is a fundamental biochemical process, from which living cells obtain energy by breaking down
sugar. In intact yeast cells or muscle extracts, glycolysis can proceed in an oscillatory fashion, with the
concentrations of various intermediates increasing and decreasing with a period of several minutes. A
simple model of these oscillations is given by the following equations:
dx
2
-x + ay + xʻ
(1)
dt
dy
= b – ay
dt
(2)
where x and y are the concentrations of adenosine diphosphate (ADP) and fructose-6-phosphate (F6P),
and a, b > 0 are kinetic parameters.
dz*
1. The fixed points of a system of ODES are defined as those points, z*, where
means it is a vector). Find the fixed point of the model above, (x*, y*), as a function of the kinetic pa-
rameters. Note that you can do this by hand, and you should get expressions that depend on a and b.
= 0 (note that bold
2. Explore the dynamical behavior of the model using RK4 method or the ode45 built-in function, this
means to get x(t) and y(t). As the kinetic parameters are important, you will have to explore the
dynamical behavior as these parameters are changed. For this project, the parameter a should be
kept at the value a =
interval b e [0.1, 1.1], at least ten different values of b should be studied. For each of the selected
parameter values: (i) use the RK4 method or the ode45 built-in function, and solve the equation
for 2 different initial conditions: one close to the fixed point you found in the previous item, and
another far from it, but in the region x E [0.1, 3.0] and y e [0.1,3.0]. (i) Plot the time series, x(t)
and y(t) vs t, for the 2 different initial conditions, different subplots on same figure. Label each plot
according to its initial condition. Choose the total time to be long enough to observe the qualitative
behavior clearly. (iii) Generate another figure that plots y vs x for each initial condition in the same
figure, also plot the location of the corresponding fixed point (x*, y*).
0.06. You will need to explore the effect of changing the parameter b in the
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