Phasor coordinates from lifetimes

An introduction to the phasor_from_lifetime function.

The phasorpy.lifetime.phasor_from_lifetime() function is used to calculate phasor coordinates as a function of frequency, single or multiple lifetime components, and the pre-exponential amplitudes or fractional intensities of the components.

Import required modules and functions:

import numpy

from phasorpy.lifetime import phasor_from_lifetime
from phasorpy.phasor import phasor_to_polar
from phasorpy.plot import PhasorPlot, plot_phasor, plot_polar_frequency

rng = numpy.random.default_rng(42)  # initialize random number generator

Single-component lifetimes

The phasor coordinates of single-component lifetimes lie on the universal semicircle. For example, 4.0 ns and 1.0 ns at a frequency of 80 MHz:

frequency = 80.0
lifetimes = [4.0, 1.0]

plot_phasor(
    *phasor_from_lifetime(frequency, lifetimes),
    frequency=frequency,
    title='Single-component lifetimes',
)

Multi-component lifetimes

The phasor coordinates of two lifetime components with varying fractional intensities are linear combinations of the coordinates of the pure components:

fractions = numpy.array(
    [[1, 0], [0.25, 0.75], [0.5, 0.5], [0.75, 0.25], [0, 1]]
)

plot_phasor(
    *phasor_from_lifetime(frequency, lifetimes, fractions),
    frequency=frequency,
    linestyle='-',
    title='Multi-component lifetimes',
)

Pre-exponential amplitudes

The phasor coordinates of two lifetime components with varying pre-exponential amplitudes are also located on a line connecting the pure components:

plot_phasor(
    *phasor_from_lifetime(
        frequency, lifetimes, fractions, preexponential=True
    ),
    frequency=frequency,
    linestyle='-',
    title='Pre-exponential amplitudes',
)

Average of lifetime distributions

The average phasor coordinates for broader (larger standard deviation) lifetime distributions lie further inside the universal semicircle compared to narrower (smaller standard deviation) distributions:

frequency = 80.0
lifetime = 4.0
standard_deviations = [0.1, 0.5, 1.0]
samples = 100000

plot = PhasorPlot(
    frequency=frequency, title='Average of lifetime distributions'
)
for sigma in standard_deviations:
    phi = numpy.sqrt(sigma * sigma / lifetime)
    phasor_average = numpy.average(
        phasor_from_lifetime(
            frequency, rng.gamma(lifetime / phi, phi, samples)
        ),
        axis=1,
    )
    plot.plot(*phasor_average, label=f'{sigma=:.1f}')
plot.show()

Lifetime distributions at multiple frequencies

Phasor coordinates can be calculated at once for many frequencies, lifetime components, and their fractions. As an example, random distributions of lifetimes and their fractions are plotted at three frequencies. Here, lifetimes are passed in units of seconds and frequencies in Hz instead of nanoseconds and MHz. This requires specifying the unit_conversion factor accordingly:

samples = 100
lifetimes = [4.0, 2.0, 1.0]

lifetime_distributions = (
    numpy.column_stack(
        [rng.gamma(lifetime / 0.01, 0.01, samples) for lifetime in lifetimes]
    )
    * 1e-9  # seconds
)
fraction_distributions = numpy.column_stack(
    [rng.random(samples) for lifetime in lifetimes]
)

plot_phasor(
    *phasor_from_lifetime(
        frequency=[40e6, 80e6, 160e6],
        lifetime=lifetime_distributions,
        fraction=fraction_distributions,
        unit_conversion=1.0,  # using seconds and Hz
    ),
    marker='.',
    label=['40 MHz', '80 MHz', '160 MHz'],
    title='Lifetime distributions at multiple frequencies',
)

FRET efficiency

The phasor coordinates of a Förster resonance energy transfer (FRET) donor with a single lifetime component of 4.0 ns as a function of FRET efficiency at a frequency of 80 MHz, with some background signal and about 90% of the donors participating in energy transfer, are on a curved trajectory. For comparison, when 100% donors participate in FRET and there is no background signal, the phasor coordinates lie on the universal semicircle:

frequency = 80.0
samples = 25
lifetime = 4.0
efficiency = numpy.linspace(0.0, 1.0, samples)

lifetime_quenched = lifetime * (1.0 - efficiency)

plot = PhasorPlot(frequency=frequency, title='FRET efficiency')
plot.plot(
    *phasor_from_lifetime(frequency, lifetime_quenched),
    color='k',
    marker='.',
    label='100% Donor in FRET',
)
plot.plot(
    *phasor_from_lifetime(
        frequency,
        lifetime=numpy.column_stack(
            [
                numpy.full(samples, lifetime),  # donor-only lifetime
                lifetime_quenched,  # donor lifetime with FRET
                numpy.full(samples, 1e9),  # background with very long lifetime
            ]
        ),
        fraction=[0.1, 0.9, 0.1 / 1e9],
        preexponential=True,  # fraction contains pre-exponential amplitudes
    ),
    linestyle='-',
    label='90% Donor in FRET',
)
plot.show()

Multi-frequency plot

Phase shift and demodulation of multi-component lifetimes can be calculated as a function of the excitation light frequency and fractional intensities:

frequencies = numpy.logspace(-1, 4, 32)
lifetimes = [4.0, 1.0]
fractions = numpy.array([[1, 0], [0.5, 0.5], [0, 1]])

plot_polar_frequency(
    frequencies,
    *phasor_to_polar(*phasor_from_lifetime(frequencies, lifetimes, fractions)),
    title='Multi-frequency plot',
)