Detection of Diffusion Heterogeneity in Single Particle Tracking Trajectories Using a Hidden Markov Model with Measurement Noise Propagation

We develop a Bayesian analysis framework to detect heterogeneity in the diffusive behaviour of single particle trajectories on cells, implementing model selection to classify trajectories as either consistent with Brownian motion or with a two-state (diffusion coefficient) switching model. The incorporation of localisation accuracy is essential, as otherwise false detection of switching within a trajectory was observed and diffusion coefficient estimates were inflated. Since our analysis is on a single trajectory basis, we are able to examine heterogeneity between trajectories in a quantitative manner. Applying our method to the lymphocyte function-associated antigen 1 (LFA-1) receptor tagged with latex beads (4 s trajectories at 1000 frames s−1), both intra- and inter-trajectory heterogeneity were detected; 12–26% of trajectories display clear switching between diffusive states dependent on condition, whilst the inter-trajectory variability is highly structured with the diffusion coefficients being related by D 1 = 0.68D 0 − 1.5 × 104 nm2 s−1, suggestive that on these time scales we are detecting switching due to a single process. Further, the inter-trajectory variability of the diffusion coefficient estimates (1.6 × 102 − 2.6 × 105 nm2 s−1) is very much larger than the measurement uncertainty within trajectories, suggesting that LFA-1 aggregation and cytoskeletal interactions are significantly affecting mobility, whilst the timescales of these processes are distinctly different giving rise to inter- and intra-trajectory variability. There is also an ‘immobile’ state (defined as D < 3.0 × 103 nm2 s−1) that is rarely involved in switching, immobility occurring with the highest frequency (47%) under T cell activation (phorbol-12-myristate-13-acetate (PMA) treatment) with enhanced cytoskeletal attachment (calpain inhibition). Such ‘immobile’ states frequently display slow linear drift, potentially reflecting binding to a dynamic actin cortex. Our methods allow significantly more information to be extracted from individual trajectories (ultimately limited by time resolution and time-series length), and allow statistical comparisons between trajectories thereby quantifying inter-trajectory heterogeneity. Such methods will be highly informative for the construction and fitting of molecule mobility models within membranes incorporating aggregation, binding to the cytoskeleton, or traversing membrane microdomains.


One-state diffusion model marginal likelihood calculation
The marginal likelihood is defined π(X|M 1D ) = is an upper incomplete gamma function. Using this gives Approximate one-state diffusion model with measurement noise We consider a trajectory X subject to Gaussian observation error, with fixed localisation accuracy σ 2 . By a result in [1] (also see the section Approximation to the likelihood for one-state diffusion model with measurement noise) an approximation for the likelihood of X given D is The associated posterior is which can be sampled using a Metropolis-Hastings algorithm. We set π(D) = Unif(D; 0, D max ), and use a random walk sampler (RW MCMC) with a symmetric Gaussian proposal, q(D → D ) = N (D ; D, S D ), giving the acceptance probability Thus, any moves outside [0, D max ] are automatically rejected. The value of S D is tuned during the burn-in to ensure that the acceptance rate is approximately 0.25 [2]. The MCMC sampler is also given as pseudocode in S1 Algorithms.

Approximate two-state diffusion model with measurement noise
We now add fixed localisation error to the previous two-state diffusion hidden Markov model. Using the same approximation to the likelihood as the approximate one-state model we can write Letting θ = {D 0 , D 1 , p 01 , p 10 } we can write the posterior as We use the same priors on D 0 , D 1 , p 01 , p 10 and z 1 as in the two-state diffusion model without measurement noise, given in equation (7), main text.
S D0 , S D1 are tuned during the burn-in to ensure an acceptance rate of approximately 0.25. We also impose the condition that D 0 < D 1 , which we enforce after the MCMC run as follows: if the posterior meansD 0 >D 1 then we swap the D 0 , D 1 chains, swap the p 01 , p 10 chains, and swap the 0 and 1 states in the hidden state z throughout the run. This is possible because although state identity switching (0 ↔ 1) is possible because of a permutation symmetry during a run, it isn't observed to occur. The updates for the transition probabilities are Gibbs moves, identical to the two-state model without measurement noise, given by equations (15) and (16), main text. The z update is similar to the other two-state models, the conditional is And again the update is At the endpoints i = 1 and i = N we have Pseudocode for this MCMC sampler is given in S1 Algorithms.
Approximation to the likelihood for one-state diffusion model with measurement noise (This method is mentioned in reference [1].) Consider a 2D trajectory observed with experimental noise with known localisation accuracy σ 2 . Let {U i } N +1 i=1 be the underlying particle position and i=1 be the observed positions. For each time step Which we can write as (summing two Gaussians) shifting the mean So we know that the measured displacement then satisfies X i+1 |X i ∼ X i + N (0, 2D∆t i + 2σ 2 ), which suggests that the likelihood is given by However, this is only true if the displacements are independent, which not the case since the displacements U i+1 − U i and U i − U i−1 both depend on the measurement noise U i − X i at time point i. However, we demonstrate that equation (5) is sufficient for model selection, see Results.
And the final likelihood is π(X|θ) = log e e log e α N (z N =0) + e log e α N (z N =1) .