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Plotting functions II

Exercise A1: Cell movement - MSD

A freely moving cell in 2D approximately follows the diffusion formula, which 
gives the mean squared distance (MSD) from the origin:

d(t)^2 = 4 * D * t

with D the so-called diffusion constant. Let D = 1 m^2/s. Calculate and plot 
the MSD for the first 100 seconds of movement. 
Note that the unit of d(t)^2 is m^2.
'''

# Your code here

'''
Exercise A2: Cell movement - Fuhrt

A more accurate representation of a 2D moving cell is Fuhrt's formula, 

d(t)^2 = 4 * D * (t - P * (1-e^{-t/P}))

where D and P are the diffusion and persistence constants. For our 
purposes, we assume D = 1 m^2/s and P = 15 s.  

Calculate and plot the MSD for the first 100s of movement. 
Compare with the previous plot. 
'''

# Your code here


'''
Exercise B: The ideal gas law

The ideal gas law is as follows:

p * V = N * kB *T

with p the pressure, V the volume of the gas, N the particle number,
k_B the Boltzmann constant, and T the temperature.

1. Write a function for the volume depending on the pressure and temperature 
(keeping the particle number as a constant).

2. Let N = 1 and k_B = 1.4 * 10^{-23} J/K. Assume 1 atm pressure (101325 Pa). 
Plot the volume between 100 K and 1,000 K. Is the relationship as expected 
from the equation?

3. Let N = 1 and k_B = 1.4 * 10^{-23} J/K. Assume room temperature (about 
300 K). Plot the volume as a function of pressure between 1,000 Pa and 
1,0000 Pa. Is the relationship as expected?
'''

# Your code here


'''
Exercise C: Michaelis-Menten

Remember the Michaelis-Menten equation describing an enzymatic reaction:

v = V_{max}[S] / K_M + [S] 

with V_{max} = k_2 * [E_{tot}] and K_M = k_1+k_2 / k_{-1}. The variables 
here represent the reaction rate constants k_1, k_{-1} and k_2, the total 
enzyme concentration [E_{tot}], the substrate concentration [S] and the 
maximum reaction velocity V_{max}. 

1. Create a function for the Michaelis-Menten equation depending on substrate 
concentration [S] and reaction constant k_1.

2. Compute and plot the reaction velocity as a function of [S] in [0,10] [mmol] 
for k_1 = 2 /s, k_2 = 1 mmol/s, k_{-1} = 1 mmol/s and E_{tot} = 2 mmol. 
Don't forget to properly annotate the axes and give the correct units.

3. Now keep the substrate concentration constant at a value of 4 mmol and plot 
the MM-equation as function of k_1 in [0,5]. 
How does k_1 influence the reaction velocity?
'''

# Your code here