Probing Membrane Protein Interactions with Their Lipid Raft Environment Using Single-Molecule Tracking and Bayesian Inference Analysis

The statistical properties of membrane protein random walks reveal information on the interactions between the proteins and their environments. These interactions can be included in an overdamped Langevin equation framework where they are injected in either or both the friction field and the potential field. Using a Bayesian inference scheme, both the friction and potential fields acting on the ε-toxin receptor in its lipid raft have been measured. Two types of events were used to probe these interactions. First, active events, the removal of cholesterol and sphingolipid molecules, were used to measure the time evolution of confining potentials and diffusion fields. Second, passive rare events, de-confinement of the receptors from one raft and transition to an adjacent one, were used to measure hopping energies. Lipid interactions with the ε-toxin receptor are found to be an essential source of confinement. ε-toxin receptor confinement is due to both the friction and potential field induced by cholesterol and sphingolipids. Finally, the statistics of hopping energies reveal sub-structures of potentials in the rafts, characterized by small hopping energies, and the difference of solubilization energy between the inner and outer raft area, characterized by higher hopping energies.

: Temporal evolution of the average diffusion in the raft (A) and spring constant (B) after the adjunction of cholesterol oxidase (in black) and of sphingomyelinase (in red).
We also quantified the dispersion of diffusivity values inside the raft after the addition of sphingomyelinase and cholesterol oxidase (Fig. S2).

B.Definition of the hopping energy
Depending on the biological system and on its dynamics there is more than one way to define the hopping energy. Note that biological systems may exist where the hopping energy is not really definable. Here, we deal with confined motion in the membrane with escaping events; hence, the hopping energy is the energy that the random walker needs to escape its confinement.

B.1 1D hopping between two wells
1D hopping between two wells is a well-defined system. We define the hopping energy to be equal to the energy difference between the deepest well and the maximum of the potential energy between the two wells. Figure S3: Hopping in 1D between two wells. E h is the hopping energy.

B.2 1D hopping between a confining well and free motion
We define the hopping energy to be the energy difference between the maximal energy at the border of the domain and the minimal energy inside the confining domain. Figure S4: Hopping in 1D between a confining well and free motion. E h is the hopping energy.

B.3 2D hopping between two wells
2D hopping between two confining wells leads to more issues in the definition of the hopping energy. Because of the multiple ways to access the second well (see Fig. S5), the definition of the hopping energy is not unique. We here define the hopping energy as the energy difference between the maximal potential value on the straight line joining the two confining well minima and the minimum value of the deepest potential well. Figure S5: Hopping energy in 2D between two confining wells. The straight red line joins the two confining wells' minima (here chosen to lie along the x-axis). The hopping energy is defined as the energy difference between the maximum potential value along this line and the lowest potential minimum of the two wells (in this case, the minimum of the well on the right). The curved line shows another possible way to go from one well to the other.

B.4 2D hopping between a confining well and free motion
Here, the definition of the hopping energy is even more ambiguous because there are infinite ways to escape the confining area. We choose to define the hopping energy as the energy difference between the average value of the potential at the limit of the domain and the minimal value of the potential in the confining well (In the simulations the limit of the domains are known, in experimental work different definitions have to be devised depending on the nature of the confinement). This definition ignores the fact that, statistically, the biomolecule will tend to escape the well where the difference of potential is the lowest. Yet, simulations have shown that this definition is useful and it is also applied in Ref. [2] to study the confining potentials and hopping energies of the glycine receptors interacting with the scaffolding molecule gephyrin. Figure S6: Hopping between a confining well and free motion. The hopping energy is defined as the energy difference between the average value of the potential on the green circle and the minimal value of the potential in the well. Here, the domain is circular.

C. Performance of the inference
The large variety of biologically relevant situations prevents an analytical evaluation of all possible outcomes of the inference scheme. Simulations of biomolecule motion in environments mimicking experimental conditions are our main tools to evaluate the behavior of the inference. Here, we focus our simulations on the most relevant situations met experimentally. We also expand partially to conditions that may be found in other experiments.

C.1 Notations and comments for hopping simulations
In the rest of the supportinginformation MAP stands for Maximum A Posteriori and Pdf for Probability density function. Except where indicated otherwise, the double-well potentials used for the simulations in 1D are with E the hopping energy and 0 and and X  the positions of the two well minima. The double-well potentials used for the simulations in 2D are with E the hopping energy, X + and X + the positions of the two well minima, and the spring constant of the harmonic confinement in the y-axis direction. The y-axis confinement will be referred to as lateral confinement. For escaping events from a unique well in both 1D and 2D the confinement potentials are harmonic and symmetrical. Outside the region of confinement the potential value is set to 0 k B T.
We define the confinement factor u as in Ref. [3]. For one confinement well u=Dt/L 2 , with D the average diffusivity, Δt the time between frames and L the characteristic diameter of the well. For two confining wells u=Dt/L 2 with D the maximal value of the average diffusivity in the two wells and L the characteristic diameter of the smaller confining well.
Positioning noise is introduced in the simulations with Gaussian noise. This noise is used to model all the sources of noise found experimentally, i.e. Poisson noise due to the signal and the fluorescent background, camera read-out noise, and noise due to the positioning algorithm and the imperfect modeling of the Point Spread Function.Noise in the potential fields is introduced as a sum of cosine and sinus functions (section C.4).
In simulations with varying diffusivity, the diffusivity takes a common value for |x|>X + and another one for |x|<X + . In simulations with a unique hopping event, trajectories are generated in the same manner as for all the other simulations; only the ones with a unique hopping event are analyzed.
In all the cases where we claim that the bias can be corrected, we mean that by performing simulations in conditions matching experimental ones, a curve or multidimensional surface of the bias evolution with the parameters involved can be extracted and the bias can be corrected by extracting all the inferred parameters from the trajectoryas in Ref. [3].
Finally, each probability density function (Pdf) in the main paper and in the supporting information was obtained from at least 2000 (from 2000 to 10000) numerically generated trajectories.

C.2 Choice of the estimator
In order to extract hopping energies, we chose the maximum of the a posteriori probability (MAP) as the estimator. In previous work (Ref. [4]), we chose (or focused on) the average value estimator (AVE) of the posteriori distribution. In the present case, the MAP outperforms other estimators ( Figure S7). The MAP estimator yields a distribution centered on the theoretical energy used in the simulations (Fig. S7). Biases in the AVE are mostly due to asymmetries in the posteriori distribution favoring higher diffusivities in the mesh squares and concomitantly shifting the hopping energy towards lower values. This phenomenon of drifting of inferred hopping energies towards lower values due to higher diffusivities is often encountered in these kinds of inference [3].
Because we are not using the AVE, we cannot access the error in the hopping energy by sampling the posteriori distribution. Furthermore, we cannot evaluate the Fisher Information [4] due to the large set of possible realizations. Therefore, the error in the hopping energy estimation is derived from the statistics of MAP results for numerical trajectories matching experimental conditions.

C.3 Characteristics of the MAP statistics
We here investigate the tail of the hopping energy MAP statistics (30000 trajectories generated and analyzed with inference). The statistics exhibit a fast decreasing behavior both for low and high hopping energy values (Fig. S8). For high hopping energy values, the statistics evolves as exp(-(E h /0.58) 2 ). Thus, the significant values of the inference are well centered around the maximum of the MAP statistics and there is no long-tail behavior.

C.4 Noise in the potential fields
The biomolecule may experience the combined effect of general potential and local interactions of inferior magnitude leading to a potential with noise structures. The noise structures can also be considered as the consequence of multiple small potential barriers. We simulated trajectories in a double-well potential both in 1D and in 2D with additional sinusoidal terms to model the substructures, with  the standard deviation of the potential noise and constant diffusivity.  Zwanzig [5] who showed that noise in the potential is reflected in the diffusivity. The data in Figs. S9 and S10 show good agreement with the Zwanzig formula for 1D inference. In 2D, a different evolution is shown. To our knowledge, there are no general models describing the evolution of diffusivity with noise in 2D potentials. Note that evolutions tend to be extremely sensitive to the nature of the potential noise.

C.5 Evolution of the inference with a unique hopping event allowed
Experimental recordings often catch a unique hopping event. We study the MAP Pdf of the hopping energy with aunique hopping event limitation in 2D. For various conditions, when the estimator is unbiased with multiple hopping events, it remains unbiased for a unique hopping event. The main consequence of the unique hopping event limitation is the broadening the Pdf.

C.6 Effect of lateral confinement in 2D motion on the inference
Here, we study the effect of lateral confinement on the MAP Pdf of the hopping energy. There are no major effects on the statistics for large sets of lateral confinements. This effect confirms that, for various experimental conditions, 1D and 2D trajectories lead to the same results and that, therefore, 1D simulations may catch most of the information necessary to characterize the inference behavior.

C.7 Effects of diffusivity on the inference of hopping energies between two wells a) Uniform diffusivity in 1Dtrajectories
Certain diffusivity values and experimental conditions leading to a high confinement factor, u, may generate bias on the inference in the case of simple confinement [3]. These effects can be compensated in most cases. Here, we show the effect of the rise of the global diffusivity on the inference for 1D trajectories (Fig. S13). Similar effects of lesser magnitude are observed for 2D trajectories. Interestingly, despite the rise of diffusivity and consequently confinement factor u inside each well,we still obtain good inferences. As the diffusivity rises, the inferred hopping energies decrease. The bias evolution can be fitted by a second-order polynomial and thus compensated (Fig. S14). The drift of hopping energy with diffusivity always shows the same behavior. The drift curve depends on the value of the hopping energy, the level of confinement inside each well, and the distance between the two wells. Whereas confinement was a major issue when dealing with motion inside a unique well,in the double-well case, information can be retrieved for higher confinement factors u than for a unique well confinement. Figure S14: Evolution of the average MAP values with diffusivity. The black dots are the results of the inferences and the red line is aparabolic fit.

b) Varying diffusivities in 1D trajectories
Here, the diffusivitiesare identical in the two wells but are higher in the central zone between the two wells ( ). As the diffusivity in the central hopping zone rises, the inferred value of the hopping energy shifts towards lower values. The shift is deterministic, always lowering the inferred value of the hopping energy with rising central diffusivities. Therefore, this bias can be corrected.

c) Varying diffusivities in 2D trajectories
x  X  Figure S16: Evolution of the MAP Pdf with varying central diffusivity for a 2D double-well potential. D=0.00625 m 2 .s -1 in black, D=0.0125 m 2 .s -1 in red, D=0.05 m 2 .s -1 in green, D=0.075 m 2 .s -1 in blue, D=0.125 m 2 .s -1 in cyan. 2000-point trajectories, 3 k B T theoretical hopping energy (shown by the thick vertical black line), diffusivities in the wells D=0.035 m 2 .s -1 , 400 nm between the two wells, and 25 ms acquisition time.
Here also, the diffusivities are identical in the two wells but are higher in the central zone between the two wells ( ).The change of central diffusivity may have a strong effect on the inference. We here show a unique set of simulation results. Extended simulations, however, have shown that these biases are sensitive to the diffusivity inside the wells, to the distance between the wells, to the hopping energy and to the nature of the lateral confinement. Therefore, when dealing with varying diffusivities, large amounts of simulations must be performed to catch the effect of the different parametersof the environment on the inference and the resulting biases. However, these biases can be compensated.

C.8 Escaping from a Unique Well to Free Motion
Experimentally, none of the toxin receptors exhibited pure escaping behavior from unique wells. Yet, this is an important case that appears in many other biological systems: the biomolecule escapes from a confinement area and then diffuses freely (or under a much more shallow potential). In these cases, the same inference scheme, as the one used for the double well trajectories, is used on the trajectories; however, the two main issues are the definition of the domain size and the definition of the hopping energy (see section A). In the following simulations, the size of the confinement domain and the shape its limits are supposed to be known.

a) 1D trajectories
The hopping energy is defined as the higher of the two energies at the limit of the domain. We have shown, in the main text, the behavior of the inference with the hopping energy assuming that the diffusivity is identical inside and outside the well. Here, we show for a unique hopping energy the evolution of the MAP statistics with a differentexternal diffusivity.
x  X  Figure S17: Evolution of the MAP Pdf with the external diffusivity. D=0.0088 m 2 .s -1 in black, D=0.0175 m 2 .s -1 in red, D=0.07 m 2 .s -1 in green, D=0.105 m 2 .s -1 in blue, D=0.175 m 2 .s -1 in cyan and D=0.35 m 2 .s -1 in yellow. 2000-point trajectories with at least 1000 points inside the well, 3 k B T theoretical hopping energy (shown by the thick vertical black line), diffusivity inside the well D=0.035 m 2 .s -1 , 100 nm radius of the well, and 25ms acquisition time.
In the unique well escaping inference, the external diffusivity does not modify strongly the bias but mainly broadens the Pdf. Here again, bias and evolution are sensitive to the diffusion inside the well, the size of the well and the shape of the potential.

b) 2D trajectories
Here, we show for a unique hopping energy the evolution of the MAP statistics with changing external diffusivity.The evolution is similar to the one for 1D trajectories. The major effect of the rise of external diffusivity is the broadening of the MAP Pdf. Figure S18: Evolution of the MAP Pdf with external diffusivity. D=0.0125 m 2 .s -1 in black, D=0.05 m 2 .s -1 in red, D=0.125 m 2 .s -1 in green, and D=0.25 m 2 .s -1 in blue. 1000-point trajectories with at least 500 points in the well, 5 k B T theoretical hopping energy (shown by the thick vertical black line), diffusivity inside the well D=0.025 m 2 .s -1 , 100 nm radius of the well, and 25 ms acquisition time.

C.9 1D vs. 2D simulations
In many simulation conditions, we noticed that inferences performed on 1D simulated trajectories lead to the same results as inferences performed on 2D trajectories. In these cases, as extensive simulations are needed to evaluate noise and bias on the inferred parameters, 1D simulations which are much faster to compute may be used to capture the essential features of the inference performance.

D. Experimental hopping D.1Identification of the hopping events
In order to eliminate the possibility of fast local motion of the membrane that could be assimilated to hopping, hopping events were only considered as such if locally, in the recorded image, the other receptors did not exhibit any sudden motion or hopping-like events.

D.2Correctingharmonic bias
When performing the inference in the harmonic approximation of the potential, some bias appears during optimization. We mentioned this effect in Ref. [3], showed its evolution with the confinement factor u and how to correct it. Here, we display the inferred values of diffusivity and spring constant as a function of the input values of diffusivity and spring constant when the inference is performed on numerically generated trajectories.

c) Figure 4 with materialization of the trajectory
We show the diffusivity and potential fields of Figure S22 with connected trajectory points to materialize the trajectory.

d) Intermediate structures
In some cases, the inferred potential exhibited a main well and a lateral extension without a second local minimum. We designate this type of potential a confinement well with intermediate structures. An example of these structures is shown in Fig. S23. Figure S23: Interaction potential acting on the ε-toxin receptor inferred from a 4262-point trajectory. On the left, only the trajectory points are superimposed on the image; on the right, the trajectory points are linked to materialize the trajectory.
15 trajectories exhibited this type of structure. Interestingly, these potentials do not show a second minimum, yet there are two well-defined areas of the potential. These potential structures suggest that the larger rafts may be formed by coalescence of smaller rafts that undergo lipid reorganization. The two-well potentials with small hopping energies that we observe may be linked to early stages in the coalescence process, where the lipid organization of the rafts is still similar to that of isolated ones. Experimental trajectories on even longer time scales are necessary to fully confirm that these are indeed coalescence processes. The main difficulty is the related to the local motion of the cell membrane that prevents efficient, reliable use of the inference.