Many biophysical processes involves diffusing particles in a potential field, especially a harmonic potential. This is the case for a particle trapped in optical or magnetic tweezers, or a particle tethered to DNA.

Two main parameters define such a process: the particle’s diffusion coefficient, *D*, and the harmonic potential spring-constant, *K*. Most of the current methods require calibration of its setup in order to extract these parameters, or having a prior knowledge about one of the parameters in order to extract the other.

We developed a method for extracting both *K* and *D* simultaneously that do not require any prior knowledge or calibration. The method is based on the formulating of the Smoluchowski’s equation for the diffusing particle and its known solution for a harmonic potential.

The Smoluchowski’s equation is a differential equation that describes the time-evolution of the probability to find a diffusing particle in a potential field, given it’s initial position. For a harmonic potential, the solution has a Gaussian shape. The width and the center of the Gaussian function depends on the initial position, the time interval *Δt*, *K* and *D*. During the time, the center of the Gaussian move towards the center of the potential field (where the force iz zero, *x=0*) and the width broadens, until it reaches the Boltzmann steady-state distribution (see animation in figure, click to enlarge).

The Joint Probability Distribution (JPD), represents the probability to find the particle at time *t+Δt*, i.e. *P[x(t+Δt)]*, for any given initial position *x(t)*. Since the initial position, *x(t)*, is distributed according to Boltzmann distribution, the JPD has an ellipse-like shape, where the slope and width depends on the time *Δt*, *K* and *D*.

By plotting the slope of the JPD as a function of time, one can find the relaxation time, *τ*, that depends on the multiplication *K⋅D*. It is similar to the corner frequency, *f _{c}*, that is often calculated in optical or magnetic tweezers experiments.

Nevertheless, from the width of the JPD one can extract both *K* and *D* independently. Instead of plotting the width *σ ^{2}* it is more convenient to plot the width divided by time,

*σ*, as a function of frequency (

^{2}/Δt*f=1/Δt*). In this plot, at high frequencies the curve approaches the value

*2D*(for a single-axis motion). An explanation describing the width calculation could be found here.

From both of these plots (slope and width), 3 frequencies regimes can be identified:

• High frequencies – where the diffusion is dominant (hence we get *σ ^{2}/Δt=2D*, as in normal diffusion).

• Low frequencies – where the harmonic potential is dominant.

• Middle range – where both

*K*and

*D*are dominant. This regime is the relevant for extracting the parameters simultaneously.

More information is available in our paper.

### Related paper:

- M. Lindner, G. Nir, A. Vivante, I. T. Young and Y. Garini,
*Dynamic analysis of a diffusing particle in a trapping potential*, Physical Review E,**87**, 022716 (2013).