Markov State Modeling

Last updated: August 31st, 20202020-08-31Project preview

Markov State Modeling Practice

Purinergic Receptor, P2X7, and its gating mechanism based on ATP concentration

Link to the original paper

Fig 1: Markov State Model of P2X7

image

Following parameters in table are corresponding to the parameters listed in Fig 1 (Markov Map): Ignore "distribution"

image

Based on Fig 1, you should be able to construct the ordinary differential equations.

For instance,

$\frac{d[C_1]}{dt} = (L_1\times C_4 + k_1\times C_2) - (3k_2\times A\times C_1 + L_4\times C_1)$

  • In here $L_4$ is not listed in both table and figure but due to the computation, it is necessary to introduce to stabilize the calculation
  • There are two L3 values. You can try either one of them and decide which one works best by comparing the plots from literature given below in Step 4
In [1]:
# Function for integreating ODE and displaying results
import scipy.integrate
from scipy.integrate import odeint
import numpy as np
import matplotlib.pylab as plt
import math
%matplotlib inline

Step 1: define "ordinary differential equations" based on Fig 1

In [2]:
def func(y,t,ATP,t_end):
    C1, C2, C3, C4, Q1, Q2, Q3, Q4 = y
    
    ### List your parameters such as k1, k2, .... 

    ### ATP switch: ATP is off every 40 seconds. 
    ### Using time variable 't', write out resonating ATP stimulation for 40 seconds on and 40 seconds off


    ### ODEs: There should be 8 ordinary differential equations for each state starting from C1, C2, ...
    dC1dt = L1*C4 + k1*C2       - (3*k2*A + L4)*C1
    
    ## list individual ODE in dydt 
    dydt = []
    
    return dydt

Step 2: solve ODE

In [8]:
## Initial state 
y0 = [1,0,0,0,0,0,0,0] ## you need 8 initial states (C1, C2, C3, C4, Q1, Q2, Q3, Q4)

## Time: Ends abount 3 mins after
## Be careful with unit
t_start = 0
t_end   = 120
t_step  = t_end*10
t = scipy.linspace(t_start,t_end,t_step)

## Additional information such as ATP concentration: Test 10, 32, 100 uM.
## Be careful with unit!
## In future, this ATP stimulation would be no longer consistent and we will try. 
ATP = 32*10e-6

## Solve! 
y = scipy.integrate.odeint() # complete this  
<ipython-input-8-247f49d6c430>:9: DeprecationWarning: scipy.linspace is deprecated and will be removed in SciPy 2.0.0, use numpy.linspace instead
  t = scipy.linspace(t_start,t_end,t_step)

Step 3: converting open probability to current

image

  • I : inward current
  • $Q_{1,2,3,4}$ : open probability from calculated "y"... make sure you understand the index corresponding to these dependent variables and properly extract them to calculate the inward current.
  • Tip: if you wonder the structure of "y", type print(np.shape(y)) or print(y) to check what columes and rows indicate
In [9]:
## Parameters
g12 = 1.5e-8
g34 = 4.5e-8
V   = -60e-3
E   = 0

# complete the following expression based on Step3 instruction
I = 

Step 4: Plot and compare them against literature

Fig 2: Plot from literature

image

In [10]:
plt.figure(figsize=(6,2),dpi=150)
plt.tick_params(labelsize=10,direction='in')
plt.plot(t,I)
plt.tight_layout()
In [ ]:
 
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