More detailed walkthrough¶

See the simple walkthrough for a quicker introduction to the PyDDM package. Here, we will go into a little more depth.

Tip: Before working through this, go through the simple walkthrough.

Importing the necessary modules¶

We will ned to import numpy, matplotlib, and xarray. We will also import the PyDDM package.

[1]:

%matplotlib inline
import matplotlib
import matplotlib.pyplot as plt

import numpy as np #numerical python used for working with arrays, mathematical operations
import xarray as xr #package for labeling and adding metadata to multi-dimensional arrays

import sys
sys.path.append("../../PyDDM") #must point to the PyDDM folder
import ddm_analysis_and_fitting as ddm

Initiazing DDM_Fit class and fitting our data to a model¶

We initlize the DDM_Fit class by passing it the yaml file containing our analysis parameters.

[13]:

#Note that to initialize the DDM_Fit class we need to pass it the YAML file containing the parameters
# *or* a dictionary with the same parameters.

ddm_fit = ddm.DDM_Fit(ddm_calc.data_yaml)

	Initial guess	Minimum	Maximum
Amplitude	100.0	1.000	1000000.0
Tau	1.0	0.001	10.0
Background	25000.0	0.000	10000000.0
StretchingExp	1.0	0.500	1.1

Loading file C:/Users/rmcgorty/Documents/GitHub/PyDDM/Examples/images_nobin_40x_128x128_8bit_ddmmatrix.nc ...

Based on what was done above in terms of estimating the background, we found that \(B\) should be about 190. We use that estimate of the background to determine \(A(q)\) and the intermediate scattering function, \(f(q,\Delta t)\). We could proceed to then fit the ISF. But here we choose not to. We fit the DDM matrix and have the amplitude and background, \(A\) and \(B\), be fitting parameters. But we can constrain these parameters based on what we found above.

[14]:

#We update the intial guess for the background and the lower and upper bounds
ddm_fit.set_parameter_initial_guess('Background', 190)
ddm_fit.set_parameter_bounds('Background', [10, 220])

Parameter 'Background' set to 190.
Parameter 'Background' lower limit set to 10.
Parameter 'Background' upper limit set to 220.

What to make of the results at high \(q\)¶

So we have tried a couple things to improve the results of our fits for \(q \gtrsim 6 \mu m^{-1}\). We tried: 1. Updating the intial guess for \(\tau\) for each \(q\) based on an expected diffusion coefficient and constraining the bounds of \(\tau\) around that guess. 2. Narrowing the bounds of the stretching exponent to be close to what we would expect for normal diffusive motion (i.e., near 1).

But, these didn’t seem to make the result for \(\tau (q)\) reasonable for the larger \(q\) values. So what is going on?

The horizontal line in the plot above corresponds to the minimum time lag we are measuring (24 ms given our frame rate of 41.7 Hz). So we are not lacking in temporal resolution.

But what about spatial resolution? Even though our pixels are 0.242 \(\mu m\) in size doesn’t mean that we can truly measure dynamics occurring around such small length scales.

The data used here was acquired using a not-so-great microscope with a 40x 0.5NA objective and a 0.3NA condenser. Assuming light of a wavelength of 550 nm, that means our theoretical resolution is \(\frac{1.22 \times 0.550}{0.5+0.3} = 0.763 \mu m\). Note that this is more than 3 times the pixel size.

As described in [Safari 2015], the maximum \(q\) is determined either by (1) the pixel size, (2) the smallest distance a particle can be detected moving between two frames given the frame rate, and (3) the smallest resolvable distance given the optical resolution limit. In the case of (3): \(q_{max} = \frac{2 \pi NA}{\lambda}\). For our data, that would be \(q_{max} = 5.7 \mu m^{-1}\).

So it seems we have found the culprit of the poor fitting at large :math:`q`: our limited optical resolution.

[Safari 2015] Safari, M. S., Vorontsova, M. A., Poling-Skutvik, R., Vekilov, P. G. & Conrad, J. C. Differential dynamic microscopy of weakly scattering and polydisperse protein-rich clusters. Phys. Rev. E 92, 042712 (2015).

[24]:

##################################################################################################
# Plotting the characteristic decay time tau vs wavevector q determined from the last three fits #
##################################################################################################

plt.figure(figsize=(10,10/1.618))
plt.loglog(fit01.q, fit01.parameters.loc['Tau'], 'ro', label="Not updating tau's initial guess")
plt.loglog(fit02.q, fit02.parameters.loc['Tau'], 'bo', label="Updating tau's initial guess")
plt.loglog(fit03.q, fit03.parameters.loc['Tau'], 'go', label="Restricting the stretching exponent")
plt.xlabel("$q$ ($\mu$m$^{-1})$")
plt.ylabel("Decay time, $\\tau$ (s)")
plt.axhline(y = 1.0/fit01.frame_rate)
plt.axvline(x = 5.7)
plt.legend(loc=0)
plt.title("Lines correspond roughly to limits of temporal (horizontal line) and spatial (vertical line) resolution")

[24]:

Text(0.5, 1.0, 'Lines correspond roughly to limits of temporal (horizontal line) and spatial (vertical line) resolution')

Determining the mean squared displacement from the ISF¶

If we assume that the particle displacements are Gaussian, then the ISF can be written as \(f(q,\Delta t) = \exp \left[ (-q^2 / 4) \left< \Delta r^2(\Delta t) \right> \right]\). We can then find the MSD, \(\left< \Delta r^2(\Delta t) \right>\) from the DDM matrix, \(D(q, \Delta t)\), using:

\[\left< \Delta r^2(\Delta t) \right> = \frac{4}{q^2} \ln \left[ \frac{A(q)}{A(q)-D(q,\Delta t)+B(q)} \right]\]

See equation 6 in [Bayles 2017] or equation 7 in [Edera 2017].

We find \(\left< \Delta r^2(\Delta t) \right>\) in this code with the method extract_MSD in the DDM_Fit class. The code will use the ‘Amplitude’ and ‘Background’ parameters of the fit. But, if those parameters are not present (as would be the case if we fit to the ISF instead of the DDM matrix) then the code will use the estimate for \(A(q)\) and \(B\) found using \(\left< | \tilde{I}(q, t) |^2 \right>_t\) as described previously in this document.

[Bayles 2017] Bayles, A. V., Squires, T. M. & Helgeson, M. E. Probe microrheology without particle tracking by differential dynamic microscopy. Rheol Acta 56, 863–869 (2017).

[Edera 2017] Edera, P., Bergamini, D., Trappe, V., Giavazzi, F. & Cerbino, R. Differential dynamic microscopy microrheology of soft materials: A tracking-free determination of the frequency-dependent loss and storage moduli. Phys. Rev. Materials 1, 073804 (2017).

[25]:

###############################################################################################
# Plotting the mean squared displacement as a function of lag time as determined from the ISF #
###############################################################################################

msd, msd_std = ddm_fit.extract_MSD(fit=fit01)
poly_fit_results = np.polyfit(msd.lagtime[1:10], msd[1:10], 1)

fig,ax = plt.subplots()
plt.loglog(msd.lagtime, msd, 'o', color='darkblue')
plt.plot(msd.lagtime, np.polyval(poly_fit_results,msd.lagtime), '-r')
plt.xlabel("Lag time (s)")
plt.ylabel("MSD ($\mu$m$^2$)")
plt.title("MSD vs lag time")

print("Diffusion coefficient: %.5f um^2/s" % (poly_fit_results[0]/4.))

C:\Users\rmcgorty\Anaconda3\lib\site-packages\xarray\core\computation.py:742: RuntimeWarning: invalid value encountered in log
  result_data = func(*input_data)

Diffusion coefficient: 0.66499 um^2/s

[ ]:

PyDDM v0.1.0 documentation

More detailed walkthrough

Contents

More detailed walkthrough¶

Importing the necessary modules¶

Initializing DDM_Analysis class and computing the DDM matrix¶

Digging into the DDM xarray.Dataset¶

Different data selecting methods¶

Determing the A and B parameters to get the ISF¶

Checking the background estimate¶

Additional notes about the background¶

Initiazing DDM_Fit class and fitting our data to a model¶

Our first attemp at fitting the data¶

Our second attemp at fitting the data - updating initial guesses for tau¶

Our third attemp at fitting the data - tightening the bounds on the stretching exponent¶

What to make of the results at high \(q\)¶

Determining the mean squared displacement from the ISF¶