Written by G. Gygli.

Contact gudrun.gygli@kit.edu in case of questions.

Made available without any warranty under a CC-By license. This script uses Python 3

In [1]:
# Here, we import all the python packages we need to run this script. 
# in the unlikely case they are not installed on your computer, this might help: 
# https://jakevdp.github.io/blog/2017/12/05/installing-python-packages-from-jupyter/
# accessed 04.10.21
import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import curve_fit

# Michaelis-Menten equation as a function for easy use in the script
def michaelismenten_equation(S0,Km,vmax):
    return vmax*(s/(Km+s)) #v0
# other equations to describe an enzyme reaction can be used by scripting these as functions

Design an initial rate experiment in four steps¶

This script aims to help you design high-quality initial rate experiments. It uses the Michaelis-Menten equation to simulate data for an enzyme with unknown enzyme reaction parameters. Four steps are needed:

1. Estimate the enzyme reaction parameters¶

Input: estimates of $K_\text{m}$ and $v_\text{max}$ and indicate the enzyme concentration concentration you are planning to use.
If you have no estimates to give, try the experiment with the suggested initial values.

2. "Zero-round" experiment¶

Input: estimates from 1. and 5 broadly spaced substrate concentrations.
Output: fake Michaelis-Menten plot for these parameters to obtain an intial first guess of $K_\text{m}$

3. "First-round" experiment¶

Input: estimates from 1. and 10 better spaced substrate concentrations to obtain an intial first guess of $K_\text{m}$.
Output: fake Michaelis-Menten plot for these parameters to obtain a better guess of $K_\text{m}$

4. "Gold-round" experiment¶

Input: estimates from 1. and 10 perfectly spaced substrate concentrations to obtain an intial first guess of $K_\text{m}$.
Output: fake Michaelis-Menten plot for these parameters to obtain $K_\text{m}$ and $v_\text{max}$ values

1. Estimate enzyme reaction parameters¶

Here, you can give an estimate of the enzyme reaction parameters $K_\text{m}$ and $v_\text{max}$. These parameters are needed to model data using the Michaelis-Menten equation, and can be adjusted as you progress with your experiments.

In [2]:
vmax = 100 # units: μM/s, initial value: 100 
Km = 100 # units: μM, initial value: 100
In [3]:
# we will be using a randon noise generator to include some variation in our simulated data
# fixing the seed for the random noise generation to always get the same result. 
# comment this line if you want to always get different data.
np.random.seed(1) # initial value: 1
# noiselevelMM controls how "noisy" your fake data will be, but it has not real meaning for your actual experiment.
noiselevelMM = 1 # initial value: 1

2. "Zero-round" experiment¶

Give the 5 widely spaced concentrations of substrate you want to use for this first-round experiment. 3 Options are given to show you what happens if different S0 are chosen. Uncomment and comment the different lines of code.

  1. S0 spaced widely around $K_\text{m}$
  2. All S0 spaced below $K_\text{m}$
  3. All S0 spaced above $K_\text{m}$
In [4]:
# 1.
S0 = [0,0.01,1,100,1000,1000000] # units: μM, initial values: [0.01,1,100,1000,1000000]

# 2. all S0 below Km
#S0 = [0.01,1,20,50,80] # units: μM, initial values: [0.01,1,20,50,80] 

# 3. all S0 above Km
#S0 = [300,500,1000,5000,10000] # units: μM, initial values: [300,500,1000,5000,10000]

S0 = np.round(S0, 2) #round the substrate concentrations to 2 digits

Now, run all the code below to see what data you can expect to obtain for the $S_\text{0}$, $v_\text{max}$ and $K_\text{m}$ you gave.

This is where the data are simulated and plotted. Depending on how the plot looks, you may want to adjust $S_\text{0}$.

Note that kcat only influences the scale of the y-axis.

Attention:¶

In this zero-round experiment, you will NOT get "nice looking" data, but data that helps you to adjust the substrate concentrations S0 so that you can plan the first-round experiment. Try the different options above (Ln 4) to see how they influence the plot you get.

In [5]:
# some preparations:
v0 = S0.copy() # initialize the array, the values will be overwritten
# fixing the seed for the random noise generation to always get the same result. 
# comment this line if you want to always get different data.
np.random.seed(1) # initial value: 1

#create a figure with satisfactory dimensions and resolution:
fig = plt.figure(figsize=[5,3], dpi=500)

# This code is repeated again and again below. 
# It is not put into a function to enable beginners in Python programming to understand what is happening.

for i,s in enumerate(S0):
    v0[i] = michaelismenten_equation(s,Km,vmax)   
    #adding some artificial, random noise
    noise = noiselevelMM*(np.random.random(1)-0.5)
    data_random = v0[i] + noise
    plt.scatter(s,data_random,c="black")

# add some annotations to the plot
plt.vlines(Km, 0, vmax/2, colors="red", linestyles='dashed')
plt.hlines(vmax/2, S0[-1], Km, colors="red", linestyles='dashed')
plt.text(Km,0,"$K_{m}$",color="red",fontsize=9,backgroundcolor="white",
         bbox=dict(facecolor='none', edgecolor='red', boxstyle='round'))
plt.text(0,vmax/2,"$v_{max}$/2",color="red",fontsize=9,
         bbox=dict(facecolor='none', edgecolor='red', boxstyle='round'))
plt.title("A simulated zero-round experiment", fontsize=14)
plt.ylabel('rates (\u03BC/s)')
plt.xlabel('[$S_{0}$] (μM)')
Out[5]:
Text(0.5, 0, '[$S_{0}$] (μM)')

3. First-round experiment¶

Now that we have a better idea of our Km, we can adjust the S0 to enable a better fit of the data with the Michaelis-Menten equation. The lowest S0 should be at least 10x lower than Km and the highest S0 at least 10x higher than Km, with 6 concentrations below and 4 above Km.

A bad example¶

Here follows an example of how data look if S0 are badly chosen. The initial values given were chosen too closely together.

In [6]:
S0 = [Km/5,Km/4,Km/3,Km/2,Km, Km*2,Km*3] 
# μM, initial values: [Km/5,Km/4,Km/3,Km/2,Km,Km*2,Km*3]

S0 = np.round(S0, 2) #round the substrate concentrations to 2 digits
v0 = S0.copy() #initialize the array, the values will be overwritten

# fixing the seed for the random noise generation to always get the same result. 
# comment this line if you want to always get different data.
np.random.seed(1) # initial value: 1
#create a figure with satisfactory dimensions and resolution:
fig = plt.figure(figsize=[5,3], dpi=500)

for s in S0:
    v0[i] = michaelismenten_equation(s,Km,vmax)   
    #adding some artificial, random noise
    noise = noiselevelMM*(np.random.random(1)-0.5)
    data_random = v0[i] + noise
    plt.scatter(s,data_random,c="black")

# add some annotations to the plot
plt.vlines(Km, 0, vmax/2, colors="red", linestyles='dashed')
plt.hlines(vmax/2, S0[-1], Km, colors="red", linestyles='dashed')
plt.text(Km,0,"$K_{m}$",color="red",fontsize=9,backgroundcolor="white",
         bbox=dict(facecolor='white', edgecolor='red', boxstyle='round'))
plt.text(0,vmax/2,"$v_{max}$/2",color="red",fontsize=9,
         bbox=dict(facecolor='white', edgecolor='red', boxstyle='round'))
plt.title("A simulated bad first-round experiment", fontsize=14)
plt.ylabel('rates (\u03BC/s)')
plt.xlabel('[$S_{0}$] (μM)')
Out[6]:
Text(0.5, 0, '[$S_{0}$] (μM)')

A good example¶

Here follows an example of how data look if S0 are well chosen. You will see that there is still room for improvement though.

In [7]:
S0 = [Km/20,Km/15,Km/5,Km/3,Km/2,Km/1.1, Km*2,Km*9,Km*10,Km*20] 
# μM, initial values: [Km/20,Km/15,Km/5,Km/3,Km/2,Km/1.1, Km*2,Km*9,Km*10,Km*20]

S0 = np.round(S0, 2) #round the substrate concentrations to 2 digits
v0 = S0.copy() #initialize the array, the values will be overwritten

# fixing the seed for the random noise generation to always get the same result. 
# comment this line if you want to always get different data.
np.random.seed(1) # initial value: 1
#create a figure with satisfactory dimensions and resolution:
fig = plt.figure(figsize=[5,3], dpi=500)

for s in S0:
    v0[i] = michaelismenten_equation(s,Km,vmax)   
    #adding some artificial, random noise
    noise = noiselevelMM*(np.random.random(1)-0.5)
    data_random = v0[i] + noise
    plt.scatter(s,data_random,c="black")

# add some annotations to the plot
plt.vlines(Km, 0, vmax/2, colors="red", linestyles='dashed')
plt.hlines(vmax/2, S0[-1], Km, colors="red", linestyles='dashed')
plt.text(Km,0,"$K_{m}$",color="red",fontsize=9,backgroundcolor="white",
         bbox=dict(facecolor='white', edgecolor='red', boxstyle='round'))
plt.text(0,vmax/2,"$v_{max}$/2",color="red",fontsize=9,
         bbox=dict(facecolor='white', edgecolor='red', boxstyle='round'))
plt.title("A simulated good first-round experiment", fontsize=14)
plt.ylabel('rates (\u03BC/s)')
plt.xlabel('[$S_{0}$] (μM)')
Out[7]:
Text(0.5, 0, '[$S_{0}$] (μM)')

4. Gold-round experiment¶

Now that we have a much better idea of our Km and how the data look like, we can adjust the S0 to enable a better fit of the data with the Michaelis-Menten equation. The lowest S0 should be at least 10x lower than Km and the highest S0 at least 10x higher than Km, with 6 concentrations below and 4 above Km.

In [8]:
S0 = [0,Km/20,Km/10,Km/5,Km/2.5,Km/2,Km/1.2,Km*2.5,Km*5,Km*10,Km*20] 
# μM, initial values: [Km/20,Km/10,Km/5,Km/3,Km/2,Km/1.1,Km*2,Km*5,Km*10,Km*20]

S0 = np.round(S0, 2) #round the substrate concentrations to 2 digits
v0 = S0.copy() #initialize the array, the values will be overwritten

# fixing the seed for the random noise generation to always get the same result. 
# comment this line if you want to always get different data.
np.random.seed(1) # initial value: 1
#create a figure with satisfactory dimensions and resolution:
fig = plt.figure(figsize=[5,3], dpi=500)

for s in S0:
    v0[i] = michaelismenten_equation(s,Km,vmax)   
    #adding some artificial, random noise
    noise = noiselevelMM*(np.random.random(1)-0.5)
    data_random = v0[i] + noise
    plt.scatter(s,data_random,c="black")

# add some annotations to the plot
plt.vlines(Km, 0, vmax/2, colors="red", linestyles='dashed')
plt.hlines(vmax/2, S0[-1], Km, colors="red", linestyles='dashed')
plt.text(Km,0,"$K_{m}$",color="red",fontsize=9,backgroundcolor="white",
         bbox=dict(facecolor='white', edgecolor='red', boxstyle='round'))
plt.text(0,vmax/2,"$v_{max}$/2",color="red",fontsize=9,
         bbox=dict(facecolor='white', edgecolor='red', boxstyle='round'))
plt.title("A simulated gold-round experiment", fontsize=14)
plt.ylabel('rates (\u03BC/s)')
plt.xlabel('[$S_{0}$] (μM)')
Out[8]:
Text(0.5, 0, '[$S_{0}$] (μM)')

Bonus¶

Simulated raw data are shown for the gold-round experiment.

In [9]:
# function for linear fit
def linear(x,a,b):
    return x*a+b

The noisier your data, the more datapoints you need to get reliable fits!¶

An example with little noise and 10 datapoints:

In [10]:
# fixing the seed for the random noise generation to always get the same result. 
# comment this line if you want to always get different data.
np.random.seed(1) # initial value: 1

time = np.arange(1, 20, 2) # initial values: (1, 20, 2): Start = 1, Stop = 20, Step Size = 2
# get fake time data, at least 10 datapoints to enable a good fit 
# noiselevelL controls how "noisy" your fake data will be.
noiselevelL = 100 #initial value: 100

#create a figure with satisfactory dimensions and resolution:
fig = plt.figure(figsize=[5,3], dpi=500)

for s in S0:
    v0 = michaelismenten_equation(s,Km,vmax) 
    data = linear(time,v0,0.001)
    #adding some artificial, random noise
    noise = noiselevelL*(np.random.random(len(time))-0.5)
    data_random = data + noise
    plt.scatter(time,data_random)
    popt, pcov = curve_fit(linear, time, data_random)
    rate = popt[0]
    b = popt[1]
    plt.plot(time, linear(time,rate,b))

plt.legend(S0, title="[$S_{0}$] in μM ", loc="upper left", fontsize='small', fancybox=True)   
plt.title("Good quality simulated linear fits of initial rates", fontsize=14)
plt.ylabel('fake abs') # these values are absolutely fake!
plt.xlabel('time (s)')
Out[10]:
Text(0.5, 0, 'time (s)')

But if the noise increases, the fit with only 10 datapoints looks nasty:

In [11]:
# fixing the seed for the random noise generation to always get the same result. 
# comment this line if you want to always get different data.
np.random.seed(1) # initial value: 1


# try this:
time = np.arange(1, 20, 2) # initial values: (1, 20, 2): Start = 1, Stop = 20, Step Size = 2
noiselevelL = 1000 #initial value: 1000

#create a figure with satisfactory dimensions and resolution:
fig = plt.figure(figsize=[5,3], dpi=500)

for s in S0:
    v0 = michaelismenten_equation(s,Km,vmax) 
    data = linear(time,v0,0.001)
    #adding some artificial, random noise
    noise = noiselevelL*(np.random.random(len(time))-0.5)
    data_random = data + noise
    plt.scatter(time,data_random)
    popt, pcov = curve_fit(linear, time, data_random)
    rate = popt[0]
    b = popt[1]
    plt.plot(time, linear(time,rate,b))

plt.legend(S0, title="[$S_{0}$] in μM ", loc="upper left", fontsize='small', fancybox=True)   
plt.title("Bad quality simulated linear fits of initial rates", fontsize=14)
plt.ylabel('fake abs') # these values are absolutely fake!
plt.xlabel('time (s)')
Out[11]:
Text(0.5, 0, 'time (s)')

So it is a good idea to increase the number of datapoints,
i.e. measuring more frequently or increasing the time of the measurement.

In [12]:
# fixing the seed for the random noise generation to always get the same result. 
# comment this line if you want to always get different data.
np.random.seed(1) # initial value: 1

# .... and this:
time = np.arange(1, 200, 2) # initial values: (1, 200, 2): Start = 1, Stop = 200, Step Size = 2
noiselevelL = 1000 #initial value: 1000

#create a figure with satisfactory dimensions and resolution:
fig = plt.figure(figsize=[5,3], dpi=500)

for s in S0:
    v0 = michaelismenten_equation(s,Km,vmax) 
    data = linear(time,v0,0.001)
    #adding some artificial, random noise
    noise = noiselevelL*(np.random.random(len(time))-0.5)
    data_random = data + noise
    plt.scatter(time,data_random)
    popt, pcov = curve_fit(linear, time, data_random)
    rate = popt[0]
    b = popt[1]
    plt.plot(time, linear(time,rate,b))

plt.legend(S0, title="[$S_{0}$] in μM ", loc="upper left", fontsize='small', fancybox=True)   
plt.title("Good quality simulated linear fits of initial rates", fontsize=14)
plt.ylabel('fake abs') # these values are absolutely fake!
plt.xlabel('time (s)')
Out[12]:
Text(0.5, 0, 'time (s)')
In [ ]:
In [ ]: