Molecular population ($p_i=N_i/N$)

A molecular population is a specified collection of molecules, or the count occupying a selected molecular state. The definition includes the chemical species, compartment, time and state classification. If $N_i$ molecules occupy state $i$ out of $N$ total, its population fraction is $p_i=N_i/N$ and ${\sum }_i{p}_i=1$ for an exhaustive, nonoverlapping classification.

An experimental signal need not equal that fraction. In a simple additive fluorescence model,where $b_i$ is the detected molecular brightness, the fluorescence per molecule in state $i$. If two states are equally populated but one is twice as bright, it contributes two thirds of the signal, despite containing half the molecules. Equal molecular counts are therefore not inferred from equal signal contributions without a response model.

Single-molecule fluorescence microscopy can resolve trajectories or individual emitters, while ensemble methods average over many molecules. Detection, labeling and state-dependent response still determine which molecular population is represented. A conformational population is one important special case, where the classification records molecular shape rather than chemical identity.

Molecular brightness is the detected fluorescence signal contributed by one labelled molecule or fluorescent particle under specified illumination and collection conditions. It is often expressed as detected photons per second per particle, or counts per particle per acquisition interval. It depends on the label, its environment and the apparatus, so it is not an intrinsic universal constant of the protein carrying the label.

For one uniformly observed species in a simple additive model, background-subtracted count rate $F$ and mean particle number $N$ satisfy $F=bN$, where $b$ is the molecular brightness. Ten particles with $b=1000$ counts per second and five particles with $b=2000$ counts per second both yield $F=10,000$ counts per second. Mean intensity alone cannot distinguish the two samples. A particle carrying two independent identical labels can approach twice the molecular brightness of one label, but quenching or incomplete labelling can invalidate that comparison.

Yan Chen and colleagues used the distribution of photon counts as particles diffuse through a tiny illuminated observation volume to estimate both molecular brightness and mean occupancy. This uses fluctuations beyond the average, with the illumination profile included in the measurement model.^[ref]