Signal processing for molecular and cellular biological physics: an emerging field

Abstract

Recent advances in our ability to watch the molecular and cellular processes of life in action--such as atomic force microscopy, optical tweezers and Forster fluorescence resonance energy transfer--raise challenges for digital signal processing (DSP) of the resulting experimental data. This article explores the unique properties of such biophysical time series that set them apart from other signals, such as the prevalence of abrupt jumps and steps, multi-modal distributions and autocorrelated noise. It exposes the problems with classical linear DSP algorithms applied to this kind of data, and describes new nonlinear and non-Gaussian algorithms that are able to extract information that is of direct relevance to biological physicists. It is argued that these new methods applied in this context typify the nascent field of biophysical DSP. Practical experimental examples are supplied.

Figure 1.

Simulated traces illustrating the unique characteristics of biophysical time series. (a) Step-like transitions; the molecular system switches between equilibrium states on a time scale faster than the experimental sampling interval. (b) Langevin dynamics and autocorrelated noise; the effect of thermal Gaussian noise and experimental apparatus creates ‘rounding’ of the state switching, and introduces Gaussian stochastic fluctuations around the measured state. (c) Poisson photon count noise; experiments often use light to make recordings and so the random emission of photons causes stochastic fluctuations in the digital signal, the spread of the fluctuations increasing with the intensity of the detected light. In each panel, the horizontal axis is the sample index. (Online version in colour.)

Figure 2.

Extracting discrete states and dwell time distributions of the bacterial flagellar motor. (a) Sequence of EMCCD frames of a bead attached to a bacterial flagellar hook, the estimated centre of the bead over time shown in the yellow, time–position trace. (b) Time–angle signal estimated from the time–position trace in (a). (c) Square magnitude of ECF components averaged over step-smoothed time–position traces, showing the dominant 26-fold periodicity and strong evidence for 11-fold periodicity. This periodicity is related to the number of motor proteins. Horizontal axis is frequency in steps per revolution (d) Estimated dwell times against samples from an exponential model which assumes Poisson motor stepping, and (e) the same estimated dwell times against samples from a power-law model—(d) and (e) show quantile–quantile plots: a perfect model will have all points lying on the dotted diagonal line, and the closer the points to that line, the better the model for the dwell times. The power-law model is visually a much better fit, but additionally the Bayesian information criterion (BIC) that quantifies the quality of the fit is larger for the power-law model, confirming the visual conclusion that the motor stepping is non-Poisson. (Online version in colour.)

Figure 3.

Quantifying F1-ATPase rotation. (a) Time–angle signal obtained by fitting a Gaussian to EMCCD frames of an illuminated bead attached to the molecule. (b) The autocorrelation of the time–angle signal showing clear evidence for Langevin dynamics since the autocorrelation at several of the low-order time delays is statistically significant (blue horizontal lines are the 95% confidence values assuming the null hypothesis of no autocorrelation). Extracting equilibrium states for this autocorrelated time series requires specialized signal processing techniques (see §4). (Online version in colour.)