Figure 1.
Simulation algorithm for a 2-state Markov chain.
Figure 2.
Two-state particle trajectories.
(A.) A schematic 2-state particle trajectory consisting of a sequence of observable displacements arising from an underlying state sequence hidden from the observer (B.) Sum of squared displacements (ssd) as a function of time for simulated particle tracks exhibiting purely Brownian motion with a diffusion coefficient , or
, or 2-state motion switching between these two diffusion coefficients with transition probabilities
and
. Each ssd trace is generated from a total of 20 independently simulated tracks, each containing 100 frames sampled at 10 ms intervals. The colored symbols mark the mean±standard deviation of the ssd for each set of tracks, and the solid lines are the best linear fits to the time versus mean ssd data.
Figure 3.
Forward algorithm for calculating log likelihood of parameter values of a 2-state HMM for a given track .
Figure 4.
Algorithm for MCMC maximization of the log likelihood function with respect to the model parameters
.
Figure 5.
Parameter optimization for two-state model.
A typical MCMC parameter optimization for an ensemble of 20 simulated 2-state particle tracks with model parameters ,
,
and
. Each track consists of 1000 frames sampled at 1 ms intervals. (A., B.) HMM parameter values are plotted for an MCMC trajectory that starts with a random initial guess and stochastically evolves in the parameter space according to Algorithm 3 (Fig. 4). The shaded part of the plots indicate the burn-in phase during which the trajectory approaches the log likelihood maxima. (C., D.) Histogram of parameter values from the MCMC trajectory above after excluding the burn-in phase.
and
are in units of
. The gray vertical lines in (D.) mark the values of transition probabilities that were used for simulating the particle tracks. (E., F.) Typical errors and dispersions in maximum likelihood parameter estimates using the stochastic MCMC optimization scheme described in the text. Ten independent particle tracks consisting of 1000 steps each, sampled at 5 ms intervals were simulated with
, different values of
, indicated by the colored dots in the left panel,
and
. These parameter combinations correspond to the first four rows in Table S1. MCMC parameter estimates and 95% coverage intervals of parameter histograms are shown by the corresponding colored crosses that are centered at the maximum likelihood parameter values.
Figure 6.
Comparison between MSD and HMM analysis.
Distribution of values estimated from MSD plots (left side, panels A,C,E) and the distribution of maximum likelihood parameter estimates for a 2-state HMM (right side, panels B,D,F), applied to simulated (top and middle, panels A,B and C,D) and experimental (bottom panels E,F) particle tracks. 20 simulated tracks each containing 1000 frames sampled at 100 frames/s were analyzed for the top and middle examples. The tracks used for the top example (panels A,B) were simulated for a 2-state system with parameters
,
,
and
, and the tracks used for the middle example (panels C,D) were simulated for pure Brownian diffusion with a diffusion coefficient of
. The tracks used for the bottom panels (E,F) are for TS-1/18-labeled LFA-1 in resting T cells, and consist of 75 individual tracks sampled for 4 s at 1000 frames/s [22]. For each track
was calculated for 1/3 of the total length of the track.
values for each set of tracks were binned and plotted as a histogram shown for each plot on the left. The corresponding densities of the distribution of
values were estimated and fitted to the sum of two lognormal distributions (shown in blue and green) as described previously [22].
Figure 7.
Schematic diagram of LFA-1 interactions and experimental conditions.
A schematic diagram showing the putative interaction between LFA-1 and a binding partner (e.g. talin) associated with the actin cytoskeleton, and the pharmacological agents used to perturb the system. cyto D: cytochalasin D; lova: lovastatin; cal-I: calpain inhibitor I. Additionally, PMA was used to activate the cells. See reference [22] for details of treatment conditions.
Table 1.
2-state HMM parameter estimates for LFA-1.
Figure 8.
Forward-backward algorithm for identifying the most likely states of the particle for a given track .
Figure 9.
Segmentation of particle trajectories into the two hidden states.
(A.) A simulated 2-state particle track with 1000 steps sampled at 5ms intervals, and parameters ,
and
, color coded to indicate the particle state (free: blue or bound: red). The state sequence is also depicted in the top bar code in the right panel, and the predicted state sequence, inferred using the track segmentation algorithm (Algorithm 4; Fig. 8), is shown in the bottom bar code. (B.) A selection of LFA-1 trajectories segmented into their two component states. Each enclosing box is a square of side
.
Figure 10.
Relative fractions of time spent in each state.
Classification of LFA-1 trajectories based on (A.) the fraction of total steps when the particle is in the bound state, and (B.) the mean number of transitions per second between the two states, plotted as a function of the overall mobility. The state sequence for each individual trajectory was established using the track segmentation algorithm with the maximum likelihood parameter estimates listed in Table 1. The overall mobility is indicated by values calculated using equation 8 applied to each trajectory.