Modeling¶

DeerLab provides a very flexible framework for simulating/modeling signals from a wide range of dipolar EPR spectroscopy experiments. For this, the central task is to use the DeerLab function dipolarmodel to define the model for the signal:

Vmodel = dl.dipolarmodel(t,r,Pmodel,Bmodel,experiment)

The following components need to be provided:

Time range t: This is the range of dipolar evolution times for the desired signal.
Distance range r: This is the range of intra-molecular spin-spin distances where the distribution is defined.
Distribution model Pmodel: Describes the intra-molecular distance distribution in either a parametric form (e.g. a Gaussian distribution) or a non-parametric form (as a histogram).
Background model Bmodel: Describes the dipolar background signal arising from the inter-molecular spin pairs.
Type of experiment experiment: This sets the number of dipolar pathways contributing to the dipolar signal.

For each of these components, a choice needs to be made:

The time range¶

The time range depends the expected range of distances, but is typically beteen about 0.5 and several microseconds. If you are planning to use this model to fit experimental data, the time vector should be the same as the one used in the experiment. Otherwise, to construct a time vector, use the linspace or arange functions from NumPy:

# time from 0.2 µs to 3.0 µs with a resolution of 0.01 µs, 321 points
t = np.linspace(0.2, 3.0, 321)  # start, stop, number of points
t = np.arange(0.2, 3.0, 0.01)   # start, stop, step size

Note that DeerLab’s definition of t depends on the type of dipolar experiment, and it might differ from how the time is conventionally defined in the literature or in experimental acquired data.

For example, for 4-pulse DEER and 5-pulse RIDME, t in DeerLab is defined such that it is zero right at the end of the second observer pulse rather than at the refocusing point of the main dipolar signal (after $\tau_1$ after the second observer pulse). Make sure to check the time definition in the list of experiment models.

The distance range¶

The distance range is a vector that goes from a minimum distance $r_\mathrm{min}$ to a maximum distance $r_\mathrm{max}$ . Any distance distribution is truncated to this range, i.e. it is assumed to be zero outside that range. The lower limit of the distance range is determined by the bandwidth of the pulses, and also by the time increment. Typically, 1.5 nm is a reasonable choice. The upper limit depends on the distances in your sample. The number of points in r is usually set to a certain resolution (typically 0.01-0.05 nm). To construct a distance vector, use the linspace or arange functions from NumPy:

# define distance range from 1.5 nm to 6.5 nm with a resolution of 0.05 nm
r = np.linspace(1.5,6.5,101)
r = np.arange(1.5,6.5,0.05)

The distance distribution model¶

A key aspect of the model is the type of distance distribution. A non-parametric distribution is specified by setting the choice of Pmodel keyword in dipolarmodel to None. In a non-parametric distribution, $P$ is a vector of the same length as r, and each element $P_i$ is a parameter. Basically, the distribution is represented as a histogram. When analyzing data, such distributions are used with methods such as Tikhonov regularization. If there are reasons to believe that the distance distribution has a specific shape (e.g. Gaussian, Rice, random-coil, etc.), or if there is very little information in the data, use a parametric distance distribution model from the list of available models.

The background model¶

Typically, a background model of a homogenous 3D distribution of spins is appropriate. The associated parametric model function is deerlab.bg_hom3d. In some cases, depending on the properties of your sample, other background models might be needed, such as backgrounds arising from distributions of spins in fractal dimensions or when accounting for volume exclusion effects. In such cases, use the associated parametric background models from the list of available models. If there is no inter-molecular background in your sample, or if it is negligible, set the background model to None.

The number of dipolar pathways¶

What distibguishes different dipolar EPR experiments within DeerLab is the number of dipolar pathways and their refocusing times. How many to include depends on the experiment you used to acquire the data and the type of pulses you used. For example, for analyzing a standard 4-pulse DEER signal without 2+1 component at the end, a single pathway suffices. If the 2+1 component (appearing at the right edge of the time trace) is present, then it should be fitted as well, including its counterpart appearing at negative times, making a total of three dipolar pathways. Experiments such as 5-pulse DEER typically require at least two dipolar pathways to be correctly modelled.

Experimental pulse delays¶

The dipolar pathways of a newly constructed dipolar model are initialized at arbitrary refocusing times and fully unconstrained. The refocusing times can be strongly constrained by knowing the experimental pulse sequence delays used to acquire the data. If the experiment used to acquire the data is known, as well as its pulse delays, then it is strongly recommended do so.

DeerLab provides a selection of experimental information generators for some of the most widely employed experimental methods (see the of list of available experiments). These are functions that take the pulse sequence delays, and return an ExperimentInfo object. This can be passed to the dipolarmodel function via the experiment keyword argument, to incorporate the experiment information on the model and constrain some of its parameters. These experimental information generators can also take information on the duration of the longest microwave pulses to more accurately constraint the parameters when using long pulses such as in frequency-swept or shaped microwave pulses.

When using an experimental time axis t and the experiment argument, the model assumes that t is zero at the beginning of the interpulse delay (see the illustrations of the individual experiment models for details). For example, for 4-pulse DEER, this zero point would be immediately after the second pulse. However, experimentally, it is common that the time axis is defined and stored differently. For 4-pulse DEER, a commercial spectrometer usually defines t such that it is zero tau1 after the second pulse. It is important to correct for this offset (also called start time or deadtime):

t0 = 0.4   # Experimental time offset of 400 ns, in μs
# To convert from experimental time axis to DeerLab time axis
t = t + t0 # Shift the experimental time axis to match DeerLab's definition

# To convert from DeerLab time axis to experimental time axis
t = t - t0

Without this shift, an incorrectly defined time vector will result in wrong simulated or fitted signals, and in wrong plots.

Constructing the dipolar model¶

Once all the decisions above have been made, the dipolar model can be constructed using the dipolarmodel function. The models that have an associated parametric function, e.g. bg_hom3d, must be passed directly as inputs to dipolarmodel. In Python, functions can be passed as inputs to other functions. See the details on dipolarmodel for more information.

For example, a model for 4-pulse DEER with a non-parametric distance distribution and a homogenous 3D background can be constructed using

expinfo = dl.ex_4pdeer(tau1=0.5, tau2=5.5, pathways=[1])
Vmodel = dl.dipolarmodel(t, r, Pmodel=None, Bmodel=dl.bg_hom3d, experiment=expinfo)

By default, dipolarmodel assumes a non-parametric distance distribution, a homogenous 3D background and a single pathway. Thus the above is equivalent to

expinfo = dl.ex_4pdeer(tau1=0.5, tau2=5.5, pathways=[1])
Vmodel = dl.dipolarmodel(t, r, experiment=expinfo)

To construct a model for 5-pulse DEER with non-parametric distance distribution and homogenous 3D background, use

expinfo = dl.ex_rev5pdeer(tau1=0.5, tau2=5.5, tau3=0.2, pathways=[1,2])
Vmodel = dl.dipolarmodel(t, r, Pmodel=None, Bmodel=dl.bg_hom3d, experiment=expinfo)

Using and modifying the model¶

To obtain a summary of the constructed model, print it:

>>> print(Vmodel)
Description: Dipolar signal model
Signature: (mod, reftime, conc, P)
Constants: []
Parameter Table:
========= ======= ======= ======= ======== ======== ====== ======================================
Name      Lower   Start   Upper    Type    Frozen   Unit   Description
========= ======= ======= ======= ======== ======== ====== ======================================
mod           0    0.01       1   nonlin     No            Modulation depth
reftime    -inf       0     inf   nonlin     No      μs    Refocusing time
conc       0.01      50   5e+03   nonlin     No      μM    Spin concentration
P             0       0     inf   linear     No     nm⁻¹   Non-parametric distance distribution
========= ======= ======= ======= ======== ======== ====== ======================================

Once defined, the model can be modified, manipulated and expanded freely as any other DeerLab model. Refer to the modeling guide for more details and instructions on model manipulation.