Inferring pointwise diffusion properties of single trajectories with deep learning

Abstract

To characterize the mechanisms governing the diffusion of particles in biological scenarios, it is essential to accurately determine their diffusive properties. To do so, we propose a machine-learning method to characterize diffusion processes with time-dependent properties at the experimental time resolution. Our approach operates at the single-trajectory level predicting the properties of interest, such as the diffusion coefficient or the anomalous diffusion exponent, at every time step of the trajectory. In this way, changes in the diffusive properties occurring along the trajectory emerge naturally in the prediction and thus allow the characterization without any prior knowledge or assumption about the system. We first benchmark the method on synthetic trajectories simulated under several conditions. We show that our approach can successfully characterize both abrupt and continuous changes in the diffusion coefficient or the anomalous diffusion exponent. Finally, we leverage the method to analyze experiments of single-molecule diffusion of two membrane proteins in living cells: the pathogen-recognition receptor DC-SIGN and the integrin α5β1. The analysis allows us to characterize physical parameters and diffusive states with unprecedented accuracy, shedding new light on the underlying mechanisms.

Figure 1

Heterogeneous trajectories and the STEP pipeline. (A) Examples of trajectories and the corresponding diffusion coefficient D as a function of time for constant D, changes within a discrete set of states with fixed D, continuous and monotonous change of D, and switch between random Ds. (B) Schematic of the pipeline of STEP: an input trajectory is fed to the architecture, which consists of a stack of convolutional layers, a transformer encoder, and a pointwise feedforward layer. The model’s output is the pointwise prediction of the diffusion parameter of interest (in this case D). To see this figure in color, go online.

Figure 2

Time-dependent diffusion properties prediction. (A) 2D histogram of the predicted diffusion coefficient D compared to the ground truth. The mean relative error (MRE) over the whole test set is 0.226. (B) Relative error for the prediction of D as a function of the segment length. (C) 2D histogram of the predicted anomalous diffusion exponent α compared to the ground truth. The MAE over the whole test set is 0.271. (D) MAE for the prediction of α as function of the segment length. (E–H) Jaccard index for the changepoint detection problem as a function of (E) the ratio between consecutive segment Ds, (F) the changepoint position for D, (G) the difference between consecutive segment αs, and (H) the changepoint position for α. For details about the data used in each panel, see Appendix B3 and Table I therein. To see this figure in color, go online.

Figure 3

Continuous changes of diffusion properties. Predictions of the time-dependent diffusion coefficient of two sets of SBM trajectories with $\alpha =0.1$ (blue) and 0.5 (red). The bold continuous lines show the average prediction over 3000 trajectories with STEP and a linear fit of the TA-MSD over a sliding window of 20 points. The thin continuous lines show a few example STEP predictions. The dashed lines indicate the theoretically expected scaling for every α. The lines have been normalized and shifted to compare their slopes easily. To see this figure in color, go online.

Figure 4

Switch between random diffusive states of the pathogen-recognition receptor DC-SIGN. (A) Characteristic features of the ATTM model: an exemplary trajectory undergoing changes of diffusion coefficient; a few examples of the distribution of D with different ${\sigma }_i$, and an example relation between the diffusion coefficient and the dwell time τ for a fixed γ; and the ensemble-average MSD scaling for the ${\alpha }_i={\sigma }_i/\gamma$ that result from each of the previous ${\sigma }_i$ and a fixed γ. (B) Predictions of the diffusion coefficient obtained by applying STEP to simulated ATTM trajectories (dots) and the result of applying the changepoint analysis (black line). (C) Distribution of D obtained through the analysis described in (B), showing the expected power-law behavior at small D. (D) Relation between D and the dwell time τ obtained through the analysis described in (B), showing the expected power-law behavior. (E) Examples of experimental trajectories of DC-SIGN with the corresponding predictions obtained for D (dots) and the changepoint analysis (black line). (F) Histogram of the distribution of D obtained for the experimental trajectories. Inset: power-law fit at small D. (G) 2D histogram of D and α obtained for the experimental trajectories. For details about the data used in each panel, see Appendix B3 and Table I therein. To see this figure in color, go online.

Figure 5

Multi-state diffusion of the integrin $\alpha 5\beta 1$. (A) Examples of experimental trajectories of the integrin α5β1 with the corresponding predictions obtained for D (dots) and the changepoint analysis (black line). (B) 2D histogram of D and α with the respective marginal distributions. (C) Scatter plot of the predictions obtained for D and α at the segment level, color coded according to a clustering analysis performed with a k-means algorithm. (D) Distribution of the turning angle for the four clusters of segments obtained as in (C). (E) Distribution of the confinement radius for the clusters showing restrained diffusion. For details about the data in each panel, see Appendix B3 and Table I therein. To see this figure in color, go online.