^{1}

^{*}

^{2}

^{1}

Conceived and designed the experiments: CC. Performed the experiments: CC. Analyzed the data: RD CC. Wrote the paper: RD CC DC. Proposed and developed the model: RD DC.

The authors have declared that no competing interests exist.

The extraction of hidden information from complex trajectories is a continuing problem in single-particle and single-molecule experiments. Particle trajectories are the result of multiple phenomena, and new methods for revealing changes in molecular processes are needed. We have developed a practical technique that is capable of identifying multiple states of diffusion within experimental trajectories. We model single particle tracks for a membrane-associated protein interacting with a homogeneously distributed binding partner and show that, with certain simplifying assumptions, particle trajectories can be regarded as the outcome of a two-state hidden Markov model. Using simulated trajectories, we demonstrate that this model can be used to identify the key biophysical parameters for such a system, namely the diffusion coefficients of the underlying states, and the rates of transition between them. We use a stochastic optimization scheme to compute maximum likelihood estimates of these parameters. We have applied this analysis to single-particle trajectories of the integrin receptor lymphocyte function-associated antigen-1 (LFA-1) on live T cells. Our analysis reveals that the diffusion of LFA-1 is indeed approximately two-state, and is characterized by large changes in cytoskeletal interactions upon cellular activation.

Many important biological processes begin when a target molecule binds to a cell surface receptor protein. This event leads to a series of biochemical reactions involving the receptor and signalling molecules, and ultimately a cellular response. Surface receptors are mobile on the cell surface and their mobility is influenced by their interaction with intracellular proteins. We wish to understand the details of these interactions and how they are affected by cellular activation. An experimental technique called single particle tracking (SPT) uses optical microscopy to study the motion of cell-surface receptors, revealing important details about the organization of the cell membrane. In this paper, we propose a new method of analyzing SPT data to identify reduced receptor mobility as a result of transient binding to intracellular proteins. Using our analysis we are able to reliably differentiate receptor motion when a receptor is freely diffusing on the membrane versus when it is interacting with an intracellular protein. By observing the frequency of transitions between free and bound states, we are able to estimate reaction rates for the interaction. We apply our method to the receptor LFA-1 in T cells and draw conclusions about its interactions with the T cell cytoskeleton.

The lateral mobility of cell-surface proteins plays a critical role in mediating the biological functions of membrane proteins

The analysis of particle trajectories is commonly based on a classification into different modes of motion, such as Brownian, hop diffusion, confined motion or directed diffusion based on fits to their mean squared displacement (MSD) over time

Both FRAP and SPT experiments on adhesion receptors commonly show a large reduction in receptor mobility upon binding with cytoskeletal components. Therefore, receptor motion may involve multiple states (i.e. bound or unbound) that contribute to the diffusion of the receptor in different ways. In a previous study of the T cell integrin receptor, LFA-1, particle trajectories were acquired with a temporal resolution of 1000 frames/s using antibody-conjugated beads

Here, we present a novel analytical framework to identify multiple diffusion states and estimate probabilities of switching between them, from particle trajectories of cell-surface proteins. Transitions between these states represent the binding and unbinding of receptors to cytoskeletal contacts or other intracellular signalling components. We introduce a new model that treats particle trajectories as the outcome of a two-state hidden Markov process, parametrized by diffusion coefficients of the two states and rates of transition between them. We adopt a likelihood maximization strategy to identify model parameters that best describe a set of tracks, thus characterizing the underlying diffusive states and the kinetics of the transitions between them.

This analysis was first tested with a series of simulated trajectories and compared with previous approaches for isolating subpopulations. We show that our analysis achieves a more accurate and informative resolution of the underlying biophysical parameters for a complex trajectory consisting of multiple states of diffusion. We tested the applicability of this analysis to experimental data of LFA-1 particle trajectories, and found that the diffusion of this adhesion molecule can indeed be treated as a two-state process due to its interactions with cytoskeletal binding partners. Our analysis identifies the characteristic diffusion coefficient of LFA-1 in the two states, and reveals the kinetics of switching between them. The use of a likelihood-based approach further allowed us to compare multiple models for given experimental data, and identify the statistically most optimal model that captures the receptor dynamics.

We modeled single particle tracks for a labeled, membrane-associated protein that binds to a uniformly distributed intracellular substrate, such as cytoskeletal binding proteins. This binding is schematically represented by the bimolecular reaction

In this model, the state sequence of the particle during an SPT experiment is regarded as a 2-state Markov chain. The displacement of the particle at each step is the outcome of Brownian diffusion with a diffusion coefficient corresponding to the particle state at that interval. As described in

In an experimental trajectory, only the particle position is recorded and information about the particle state must be inferred from the displacement of the particle between successive frames. Therefore, in our model, a particle trajectory is regarded as the outcome of a 2-state hidden Markov model (HMM)

(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

We first consider a trajectory arising from 2D Brownian diffusion and sampled at fixed time intervals,

The previous equation can be rewritten in the following familiar form

For the 2-state system described above, as the track length increases,

The slope of a linear fit to

The 2-state HMM is characterized by two diffusion coefficients and two transition probabilities. We parametrized the model by the parameter set

The probability of observing a track for a given choice of the parameters

As described in

We then maximized this log likelihood with respect to the four model parameters to calculate their most likely values for a given set of tracks. We used a Markov Chain Monte Carlo (MCMC) algorithm (Algorithm 3;

A typical MCMC parameter optimization for an ensemble of 20 simulated 2-state particle tracks with model parameters

We assessed the MCMC parameter optimization scheme for a range of parameter values, using an ensemble of simulated tracks for each parameter set. The results, summarized in

In

The most commonly used analysis of single particle trajectories is to extract a diffusion coefficient from a linear fit to their mean squared displacement (MSD) over time

Distribution of

To test the applicability of the 2-state HMM described above, we analyzed a set of experimental SPT data for the T cell integrin, LFA-1. LFA-1 is critical for lymphocte adhesion and signaling, and has been previously studied using both SPT

We applied the 2-state HMM analysis to the data set of LFA-1 particle trajectories observed on T cells by Cairo et al.

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

Label | Treatment | ||||||||||||||

Mean | CV | Mean | CV | Mean | CV | Mean | CV | ||||||||

1. | TS1/18 | untreated | 75 | 0.085 | 0.38% | 0.031 | 0.80% | 4.2 | 6.9% | 9.1 | 5.8% | 0.68 | 0.32 | 0.46 | 0.068 |

2. | ICAM-1 | untreated | 38 | 0.081 | 0.43% | 0.015 | 0.91% | 3.9 | 7.7% | 9.8 | 6.3% | 0.72 | 0.28 | 0.40 | 0.062 |

3. | TS1/18 | cyto D | 36 | 0.082 | 0.42% | 0.012 | 1.1% | 3.4 | 8.2% | 12 | 7.1% | 0.78 | 0.22 | 0.27 | 0.067 |

4. | ICAM-1 | cyto D | 48 | 0.088 | 0.36% | 0.019 | 1.2% | 2.5 | 8.1% | 11 | 6.9% | 0.82 | 0.18 | 0.22 | 0.076 |

5. | TS1/18 | PMA | 39 | 0.082 | 0.38% | 0.0038 | 0.78% | 1.0 | 13% | 4.2 | 11% | 0.81 | 0.19 | 0.24 | 0.068 |

6. | ICAM-1 | PMA | 24 | 0.057 | 0.69% | 0.0083 | 1.1% | 19 | 5.2% | 23 | 4.2% | 0.55 | 0.45 | 0.81 | 0.035 |

7. | TS1/18 | lova | 42 | 0.086 | 0.39% | 0.030 | 0.94% | 1.1 | 15% | 3.9 | 13% | 0.78 | 0.22 | 0.28 | 0.074 |

8. | TS1/18 | PMA+lova | 42 | 0.090 | 0.40% | 0.0053 | 0.54% | 1.8 | 9.5% | 3.4 | 8.0% | 0.65 | 0.35 | 0.53 | 0.060 |

9. | TS1/18 | cal-I | 49 | 0.080 | 0.41% | 0.013 | 1.9% | 5.8 | 6.4% | 29 | 4.8% | 0.83 | 0.17 | 0.20 | 0.069 |

10. | TS1/18 | PMA+cal-I | 46 | 0.084 | 0.49% | 0.012 | 0.51% | 5.1 | 6.7% | 4.3 | 5.9% | 0.46 | 0.54 | 1.2 | 0.045 |

We note that for all the experiments analyzed here, the maximum likelihood estimate of

We observed that in untreated cells, ICAM-1 ligation reduces the overall mobility of LFA-1, compared to TS-1/18-labeled LFA-1, as assessed by the

Treating cells with cytochalasin D reduces the lifetime of the bound state, with approximately 40% smaller

PMA-induced activation of T cells lowered

Notably, the combination of ICAM-1 ligation and PMA-induced activation also increases both the transition probabilities,

As previously noted using a

Calpain is a cytosolic protease that cleaves the talin head domain, thus releasing LFA-1 from its cytoskeletal attachment site

The hidden Markov formulation that we used to analyze single particle tracks also allows us to identify the most likely state of the Markov chain at each step along a track. To achieve this, the forward-backward algorithm defines a backward variable

We tested the performance of the segmentation algorithm for simulated trajectories that were previously used to assess the performance of the likelihood maximization algorithm (

(A.) A simulated 2-state particle track with 1000 steps sampled at 5ms intervals, and parameters

We applied the trajectory segmentation algorithm to LFA-1 particle tracks analyzed with a 2-state HMM. A selection of segmented LFA-1 particle tracks is shown in

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

We now address the question of how to determine whether a 2-state model is indeed the best descriptor for the observed data, given one or more alternate models. We compared different models by means of Akaike's information criterion (

To determine whether a 2-state model is sufficient to describe the data, we attempted to further resolve the two states into component “sub-states”. After the intial segmentation of an ensemble of trajectories, we assembled all the displacements ascribed to

In this study, we examined single particle trajectories for a membrane-associated protein that interacts with cytoskeletal binding proteins. Adhesion proteins at the cell membrane regulate a variety of biological phenomena including inflammation and antigen-presentation. Using a hidden Markov formulation to model 2D trajectories of a membrane protein, we outlined a systematic and easily-implemented procedure to parameterize a two-state model of diffusion and binding. Parameter estimates for this model can be used to identify the most probable state at each frame of the trajectory and thus divide it into mobile and immobile fragments. To establish the applicability of this analysis, we rigorously tested it with simulated trajectories for a range of parameter values. The HMM analysis revealed the diffusion coefficients of the individual states and identified transient state changes within single trajectories. Hidden Markov models have been previously used to analyze actomyosin and kinesin-microtubule movement data

Our method expands upon the standard MSD analysis for SPT experiments, and provides previously inaccessible information about hetereogeneous diffusion. We are able to confidently detect the presence of two diffusion coefficients (

We made two key simplifying assumptions: first, that the particle transitions between the two states with first order kinetics, and second, that all transitions occur at the sampling time. First order kinetics are justifiable when there is an excess of binding sites, but without direct experimental data, it is difficult to judge the merit of this assumption. Thus, the transition probabilities reported here must be interpreted with care, as they depend on

The assumption that transitions in the particle state occur on order of the sampling time is more easily justified in light of the relatively low transition probabilities that we observe (less than once every 100 frames). For infrequent transitions relative to the frame rate, the exact transition moment should not significantly alter our analysis. The validity of this assumption must be checked a-posteriori for a given experimental setup, by confirming that the transition probabilities are indeed small (

Our analysis offers some distinct advantages over an MSD-based approach. Firstly, by examining the diffusive behaviour of a particle at each step along a trajectory, heterogeneous diffusion is efficiently resolved. Secondly, unlike the distribution of

Of the two states identified in our analysis, the one with greater mobility (

We note that, the values of diffusion coefficients reported here are influenced by the use of a micron-sized bead to label the protein. The potential effects of a bead on the mobility of a membrane protein are discussed in reference

Our analysis also assumes that the binding partner is homogeneously distributed, such that the transition probabilities have no spatial dependence. In this respect, it differs notably from another class of SPT analysis that has been used to resolve transient spatial confinement of particles

We have tested simulated 2-state trajectories and experimental LFA-1 trajectories using the spatial confinement algorithms described previously

In general, analyzing spatial heterogeneity in mobility with the HMM formulation would require substantially more complex models than the one presented here, as the transition probabilities themselves would vary with the location of the particle. Additional complexity would be introduced by variations in the size of confinement regions. In future studies, we intend to examine modifications to our model that rigorously address these issues. A notable advantage of the present analysis is the lack of any user-tuned parameters, such as a characteristic confinement length (

Finally, the likelihood-based approach that we adopted here is flexible and can be extended to account for other modes of motion. We tested a two-state Brownian model in this work, but the HMM approach could be used to introduce additional states or alternative models of mobility, such as directed motion. This approach has the potential to resolve extremely complex and heterogeneous trajectories. The use of likelihood as a metric for the quality of a model allows for statistically well-defined comparisons between various models, using

Experimental LFA-1 trajectories used were acquired as described in Cairo et al

For a particle undergoing Brownian diffusion in a

In this study we are concerned with single particle tracks of a membrane-associated protein that is imaged at fixed time intervals. Thus,

To simulate trajectories for a particle with 2-state diffusion we first generated a Markov chain

A particle trajectory consists of a sequence of individual displacements, denoted as

To estimate the maximum likelihood parameters of a 2-state HMM for a set of tracks, we used a stochastic Markov Chain Monte Carlo (MCMC) optimization scheme (Algorithm 3;

We define

To compare the effectiveness of different models in describing a set of tracks, we used the Akaike information criterion (

Supporting information text.

(0.12 MB PDF)

Maximum likelihood parameter estimates for simulated 2-state trajectories.

(0.03 MB PDF)

Accuracy of parameter estimates as a function of trajectory length.

(0.05 MB PDF)

Pure Brownian diffusion analyzed with a two-state HMM.

(0.62 MB PDF)

Analysis of segmented trajectories.

(0.04 MB PDF)

We would like to thank Gerda de Vries, Gustavo Carrero and Vishaal Rajani for helpful discussions. RD and DC wish to thank D. Brian Walton for useful discussions.