Systems-level approach to uncovering diffusive states and their transitions from single-particle trajectories

The stochastic motions of a diffusing particle contain information concerning the particle's interactions with binding partners and with its local environment. However, an accurate determination of the underlying diffusive properties, beyond normal diffusion, has remained challenging when analyzing particle trajectories on an individual basis. Here, we introduce the maximum-likelihood estimator (MLE) for confined diffusion and fractional Brownian motion. We demonstrate that this MLE yields improved estimation over traditional mean-square displacement analyses. We also introduce a model selection scheme (that we call mleBIC) that classifies individual trajectories to a given diffusion mode. We demonstrate the statistical limitations of classification via mleBIC using simulated data. To overcome these limitations, we introduce a version of perturbation expectation-maximization (pEMv2), which simultaneously analyzes a collection of particle trajectories to uncover the system of interactions that give rise to unique normal and/or non-normal diffusive states within the population. We test and evaluate the performance of pEMv2 on various sets of simulated particle trajectories, which transition among several modes of normal and non-normal diffusion, highlighting the key considerations for employing this analysis methodology.

Figure 1

Classification probability via mleBIC of simulated particle trajectories with localization noise for various particle track lengths, undergoing (a) normal diffusion for various underlying diffusivities, (b) confined diffusion for various reduced confinement sizes, and (c) fractional Brownian motion for various anomalous exponents. Each row represents a different particle track length: first row, $N=30$ steps; second row, $N=60$ steps; third row, $N=120$ steps; and last row, $N=240$ steps. The probability of each model was calculated on the basis of the fraction of tracks classified to that model at each point in parameter space, and it is specified by a unique marker and color: immobile (cyan cross), normal diffusion (blue circles), confined diffusion (red squares), and anomalous diffusion (green diamonds). Error bars represent the observed standard deviation.

Figure 2

Empirical probability distributions of the mean covariance matrix elements, $\langle \mathbf{C}(i,j)\rangle$, for $j=i$, $j=i\pm 1$, $j=i\pm 2$, and $j=i\pm 6$ for case 1 (top row), where states 1, 2, 3, and 4 are shown in red (second darkest gray), green (second lightest gray), blue (darkest gray), and cyan (lightest gray), respectively, and for case 2 (bottom row), where states 1 and 2 are shown in green (light gray) and blue (dark gray), respectively. Vertical dashed lines indicate the theoretical values with a color/grey value corresponding to each diffusive state.

Figure 3

ln-probability (base $e$) as a function of the number of diffusive states (model size) for (a) case 1 and (b) case 2, determined by pEMv2 analysis using $f=1$, 2, 4, 6, 9, and 13 off-diagonal covariance matrix elements. The inset key shows the correspondence between color (gray-scale) value and the value of $f$. Each data set consists of five sets of simulated particle tracks (shown as a different curve) for each $f$ [shown as a different color (gray-scale) value]. The inset panel shows a zoomed-in representation near the maximum ln-probability.

Figure 4

Representations of 1500 synthetic particle trajectories for case 1 (top row) and case 2 (bottom row). In the left column, each trajectory is depicted using a color (gray-scale) value, corresponding to the known, simulated diffusive state of the track. In the right column, each trajectory is depicted using a color (grey-scale) value, corresponding to the diffusive state, that yields maximum posterior probability, determined on the basis of pEMv2 using $f=6$ off-diagonal covariance matrix elements. The starting position of each trajectory is on a two-dimensional grid, separated one from another by $1\mu \mathrm{m}$ (top row) and $0.4\mu \mathrm{m}$ (bottom row). In both cases, the scale bar represents $5\mu \mathrm{m}$.

Figure 5

Fraction of trajectories classified into the correct diffusive state as a function of the number of off-diagonal covariance matrix elements used in the pEMv2 analysis of case 1 (top) and case 2 (bottom). Error bars represent the observed standard deviation across five different sets of simulated particle tracks.

Figure 6

Average covariance matrix elements and average taMSD from maximum posterior classification as a function of time lag for (a) states 1 (red, bottom curve), 2 (green, second from bottom curve), 3 (blue, second from top curve), and 4 (cyan, top curve) of case 1, and (b) states 1 (red, bottom curve) and 2 (blue, top curve) of case 2. The $n\mathrm{th}$ covariance matrix element is calculated according to $〈\left[x(i+n+1)-x(i+n)\right]\left[x(i+1)-x\left(i\right)\right]〉$ and the $n\mathrm{th}$ taMSD is calculated according to $〈{\left[x(i+n)-x\left(i\right)\right]}^2〉$ for each time lag, $n\mathrm{\Delta }t$, where $n=0,1,\cdots ,9$. Each data point represents the average over five different sets of simulated particle tracks, where the ensemble taMSD and covariance matrix elements were calculated for each state based on classification by pEMv2 using $f=1$ (left column) and $f=6$ (right column). The solid lines linking the data points are guides to the eye. The error bars correspond to the standard deviation of the mean across five trials. Shown as the dashed curves are the true matrix elements and the true ensemble-averaged taMSD for each state, determined using the known diffusive states of trajectories, while the shaded bands represent its standard deviation.

Figure 7

Performance of pEMv2 on particle tracks that transition between diffusive states given by case 3. ln-probability (base $e$) determined by pEMv2 analysis for five sets of simulated particle tracks (each shown as a different curve) with diffusive states given according to case 3 with (a) track lengths of 15, 30, 60, 90, and 120 steps, and (d) track lengths of 5, 10, 15, 20, and 30 steps created by splitting the 120-step data set in (a). Each simulation set is shown in a different color (grey-scale) value. Average fraction that the classified diffusive states matches the simulated diffusive state for (b) various track lengths and (e) various bin sizes. For the purposes of this comparison, when a bin contains a transition, the “true” diffusive state is chosen to be the state with the highest number of displacements. Average transition rate per track for (c) various track lengths and (f) various bin sizes. (b), (c), (e), and (f) Error bars represent the observed standard deviation across the five data sets.

Figure 8

Average covariance matrix elements and average taMSD from maximum posterior classification as a function of time lag for states 1 (red, bottom curve), 2 (green, middle curve), and 3 (blue, top curve) with a bin size of (a) 120 steps and (b) 10 steps. The $n\mathrm{th}$ covariance matrix element is calculated according to $〈\left[x(i+n+1)-x(i+n)\right]\left[x(i+1)-x\left(i\right)\right]〉$ and the $n\mathrm{th}$ taMSD is calculated according to $〈{\left[x(i+n)-x\left(i\right)\right]}^2〉$ for each time lag, $n\mathrm{\Delta }t$, where $n=0,1,\cdots ,9$. Each data point represents the average over five different sets of simulated particle tracks, analyzed using pEMv2 using $f=6$. The solid lines linking the data points are guides to the eye. The error bars correspond to the standard deviation of the mean across five trials. Shown as the dashed curves are the true matrix elements and the true ensemble-averaged taMSD for each state, determined using the known diffusive states of trajectories, while the shaded bands represent their standard deviations.

Figure 9

Dependence of the optimal bin size on transition rate. (a) Average number of transitions per track vs bin size for various transition rates, $R=\{0,0.36,0.6,1.2,1.8,2.4,3.6\}$ transitions per track [each shown in a different color (gray-scale) value]. (b) Average ln-likelihood value per step vs bin size for various transition rates. (c) Optimal bin size vs the mean number of transitions per track, determined by the maximum ln-likelihood per per step. (d) Average fraction that the classified diffusive state matches the simulated diffusive state as a function of the mean number of transitions per track. Each error bar represents the observed standard deviation across five different sets of simulated particle tracks.