Fit Gamma factor

This notebook estimates the gamma factor from a set of 5 μs-ALEX smFRET measurements.

What this notebook does?

According to Lee 2005 (PDF, SI PDF), we estimate the $\gamma$-factor from Proximity Ratio (PR) and S values (with background, leakage and direct excitation correction) for a set of 5 μs-ALEX measurements.

The PR and S values are computed by the notebook

usALEX-5samples-PR-leakage-dir-ex-all-ph

which is executed by 8-spots paper analysis.

From Lee 2005 (equation 20), the following linear relation holds:

$$\frac{1}{S} = \Omega + \Sigma \cdot E_{PR}$$

Once $\Omega$ and $\Sigma$ are fitted, we can compute the $\gamma$-factor as (equation 22):

$$\gamma = (\Omega-1)/(\Omega + \Sigma-1)$$$$\beta = \Omega + \Sigma - 1$$

The definition of $\beta$ based on physical parameters is:

$$ \beta = \frac{I_{A_{ex}}\sigma_{A_{ex}}^A}{I_{D_{ex}}\sigma_{D_{ex}}^D}$$

Note that, calling $S_\gamma$ the corrected S, the following relation holds:

$$ S_\gamma = (1 + \beta)^{-1}$$

Import libraries



In [ ]:

    
from __future__ import division
import numpy as np
import pandas as pd
import lmfit
from scipy.stats import linregress

Computation

This notebook read data from the file:



In [ ]:

    
data_file = 'results/usALEX-5samples-PR-leakage-dir-ex-all-ph.csv'



In [ ]:

    
data = pd.read_csv(data_file).set_index('sample')
data



In [ ]:

    
data[['E_gauss_w', 'E_kde_w', 'S_gauss']]



In [ ]:

    
E_ref, S_ref = data.E_gauss_w, data.S_gauss



In [ ]:

    
res = linregress(E_ref, 1/S_ref)
slope, intercept, r_val, p_val, stderr = res

For more info see scipy.stats.linearregress.



In [ ]:

    
Sigma = slope 
Sigma



In [ ]:

    
Omega = intercept
Omega

Pearson correlation coefficient:



In [ ]:

    
r_val

Coefficient of determination $R^2$:



In [ ]:

    
r_val**2

P-value (to test the null hypothesis that the slope is zero):



In [ ]:

    
p_val

Gamma computed from the previous fitted values:



In [ ]:

    
gamma = (Omega - 1)/(Omega + Sigma - 1)
'%.6f' % gamma



In [ ]:

    
with open('results/usALEX - gamma factor - all-ph.csv', 'w') as f:
    f.write('%.6f' % gamma)



In [ ]:

    
beta = Omega + Sigma - 1
'%.6f' % beta



In [ ]:

    
with open('results/usALEX - beta factor - all-ph.csv', 'w') as f:
    f.write('%.6f' % beta)

Fit plot



In [ ]:

    
import matplotlib.pyplot as plt
import seaborn as sns
%matplotlib inline
%config InlineBackend.figure_format='retina'  # for hi-dpi displays

sns.set_style('whitegrid')



In [ ]:

    
x = np.arange(0, 1, 0.01)
plt.plot(E_ref, 1/S_ref, 's', label='dsDNA samples')
plt.plot(x, intercept + slope*x, 'k', label='fit (slope = %.2f)' % slope)
plt.legend(loc=4)
plt.ylim(1, 2)
plt.xlabel('PR')
plt.ylabel('1/SR');