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, 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.
More information is available in our 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).