Diffusing particles in a harmonic potential field


Illustration of diffusing particle in a harmonic potential field. After a short time, the particle will stay close to its previous position (right), but after a long time, the particle will probably found at the center of the field (left).

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.

Time evolution of probability.

Time evolution of probability.

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).

Joint probability distribution

Joint probability distribution

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, fc, 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 σ2 it is more convenient to plot the width divided by time, σ2/Δt, as a function of frequency (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 σ2/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.

Slope (a) and widht (b) as a function of frequency. Figure taken from Lindner et. al (PRE 2013)

Slope (a) and widht (b) as a function of frequency.
Figure taken from Lindner et. al (PRE 2013)

More information is available in our paper.

Related paper: