Resources for CMBSGR6300#

Further reading#

Gene input functions#

Step functions as approximations for the input function of a gene

How similar are the dynamics of the accumulation of protein Y in X -> Y when we use:

  1. f(X) = X/(K+X) (Michaelis-Menten)

  2. f(X) = X^2/(K^2+X^2) (Hill cooperativity n = 2)

  3. a step function

as the input function for:

dY/dt = f(X*) - aY

Hey! You can change and run this code yourself.

Hit the rocketship in the corner of the page to select “Live code”! Run the code with these parameters, and then try changing them.

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches


X = np.linspace(.001,1)

# define constants

a = 1 # removal rate
K = 0.33 # dissociation constant
n = 2 # hill coefficient

# INPUT FUNCTION DEFINITIONS

# MM input function
def MM_fxn(X):
    f_X = X / (K + X)
    return f_X

# hill input function
def hill_fxn(X):
    f_X = X**n / (K**n + X**n)
    return f_X

# step input function
def logic_fxn(X):
    f_X = []
    for value in X:
        if value >= K:
            f_X.append(1)
        if value < K:
            f_X.append(0)
    return f_X

# define input function
input1 = MM_fxn(X)
input2 = hill_fxn(X)
input3 = logic_fxn(X)

# plot input fxns
plt.rcParams.update({'font.size': 18})

fig1 = plt.figure(figsize=(6,6))
plt.title('input function')
plt.plot(X/K, input1, 'r.')
plt.plot(X/K, input2, 'g')
plt.plot(X/K, input3, 'b--')
plt.xlabel('X/K')
plt.ylabel('f(X)')

# legend
plt.legend(labels=['MM', 'Hill', 'step'], loc = 'lower right')
print("This is what the different input functions f(X) look like.")
---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In [1], line 2
      1 import numpy as np
----> 2 from scipy.integrate import odeint
      3 import matplotlib.pyplot as plt
      4 import matplotlib.patches as mpatches

ModuleNotFoundError: No module named 'scipy'
# function returning dy/dt
def y_activation_1(Y, t):
    x_star = (1/a)*(1-np.exp(-a*t))
    dy_dt_1 = MM_fxn(x_star) - a*Y
    return dy_dt_1

def y_activation_2(Y, t):
    x_star = (1/a)*(1-np.exp(-a*t))
    dy_dt_2 = hill_fxn(x_star) - a*Y
    return dy_dt_2

x_star_array = []
def y_activation_3(Y, t):
    x_star = (1/a)*(1-np.exp(-a*t))
    if x_star >= K:
        x_star = 1
    if x_star < K:
        x_star = 0
    dy_dt_3 = x_star - a*Y
    return dy_dt_3

# initial condition
y0 = np.zeros(50)

# time points
t = np.linspace(0,5, 100)

# solve ODE
Y1 = odeint(y_activation_1, y0, t)
Y2 = odeint(y_activation_2, y0, t)
Y3 = odeint(y_activation_3, y0, t)

# plot results
fig2 = plt.figure(figsize = (6,6))
plt.title('Y')
plt.plot(a*t, (1/a)*(1-np.exp(-a*t)), 'k', marker = 'X', label = 'X*')
plt.plot(a*t, Y1 / Y1[-1] , 'r.', label = 'MM')
plt.plot(a*t, Y2 / Y2[-1] , 'g', label = 'Hill')
plt.plot(a*t, Y3 / Y3[-1] , 'b--', label = 'step')
plt.xlabel('a*t')
plt.ylabel('X*, Y')
plt.ylim(-.05, 1.05)

# legend
X = mpatches.Patch(color = 'k', label = 'X*')
MM = mpatches.Patch(color = 'r', label = 'MM')
Hill = mpatches.Patch(color = 'g', label = 'Hill')
logic = mpatches.Patch(color = 'b', label = 'step')

plt.legend(handles=[X, MM, Hill, logic], loc = 'lower right')
print('''
This graph reflects Y accumulation over time 
modeled with different gene input functions,
after X* (black line) appears in the cell.
''')
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In [2], line 29
     26 t = np.linspace(0,5, 100)
     28 # solve ODE
---> 29 Y1 = odeint(y_activation_1, y0, t)
     30 Y2 = odeint(y_activation_2, y0, t)
     31 Y3 = odeint(y_activation_3, y0, t)

NameError: name 'odeint' is not defined

Class materials#