Frame-by-frame rigid-body fitting

In the rigid-body fitting, an ensemble of candidate structures is prepared from experimentally determined structures or structures obtained by MD simulations. The goal of the rigid-body fitting is to select the correct structure and to find its orientation and translation from a given experimental AFM image. To accomplish this, the rigid-body fitting generates a large number of synthetic AFM images with various structures m, 3D rotations ϕ (quaternions are used in this study), translations (dx, dy), and other parameters, and compares them to the experimental AFM images. The parameter set that produces the synthetic AFM image with the highest similarity to the experimental AFM image is considered as the best estimate. Hereafter, we will refer to these synthetic images as pseudo-AFM images.

To generate pseudo-AFM images, we employed a conventional collision-detection method [11,18,19]. This method calculates the height at which a tip perpendicular to the stage collides with a sample molecule. Let the stage be the xy-plane, then the height of the stage (z-coordinate) is aligned with the minimum value of the z-coordinate of the atoms of the molecule. Then, let us describe by dz an offset of the stage from the origin as an unknown parameter. The scaling parameter β for height z is also considered as unknown, if the calibration of the height scale is not perfect. Following Niina et al. [19], we approximate the tip by a hemisphere (with radius r) combined with a circular frustum of a cone. In summary, unknown parameters when generating pseudo-AFM images are structure m, rotation ϕ, and translation (dx, dy), offset dz, scaling parameter β to calibrate the height, and apex radius r of the tip.

The rigid-body fitting measures similarity between the generated pseudo-AFM image and the experimental AFM image. In this approach, the parameters used to generate the pseudo-AFM image with the highest similarity are considered as the best estimate. The similarity measures include pixel-RMSD [22], cross correlation [18,20], cosine similarity [19], and structural similarity index measure (SSIM) [17]. For example, pixel-RMSD is defined as follows, essentially the same as peak signal-to-noise ratio (PSNR): where and are the heights at pixel i of the experimental AFM and the pseudo-AFM images, respectively. N_pixel is the number of pixels in the image. In this case, a smaller pixel-RMSD means a higher similarity.

In general, the calibration for height is not always perfect in AFM data and thus the scaling parameter β for height and the stage offset dz should be treated carefully. In fact, Müller and Engel showed that ion concentration and the affected Debye length lead to different AFM measure heights in biomolecules [24]. Due to this problem, conventional rigid-body fitting studies use cross correlation or cosine similarity as the similarity measure between images. These measures are invariant to the scaling or translations by height. However, these measures are incompatible with the probabilistic modeling of images, and thus cannot be directly used for the likelihood analysis in this study. Considering this context, in the following, we introduce a measure developed by Cossio and Hummer in their analysis of CryoEM images [25], that is invariant to height scaling and translations, to the analysis of AFM images, which enabled us to perform probabilistic modeling of images and extend the conventional rigid-body fitting to multiple frames.

First, we introduce the concept of probability into the rigid-body fitting to extend the analysis to a likelihood-based analysis that allows more flexible modeling. We assume that an experimental AFM image can be generated by adding spatially independent Gaussian noise to a pseudo-AFM image. Then, the probability of observing the experimental AFM image or likelihood, given a set of parameters, can be written as follows (1) where λ is the standard deviation of the Gaussian noise. From the above equation, it is obvious that this likelihood L is maximized for the parameter set with the smallest pixel-RMSD. In principle, the maximum likelihood parameters can be found by exhaustively searching possible parameters, but the large number of possible combinations of the parameters makes it infeasible to search all of them. Therefore, in this study, we reduce the number of parameters by integrating out some of unknown parameters, as was done by Cossio and Hummer in their CryoEM data analyisis [25].

Starting from the same type of equation as Eq 1, Cossio and Hummer analytically integrated out the parameters, dz, β, and λ. The derivation is very general and thus can be applied directly to AFM data, not just CryoEM. The Gaussian integral can be applied to the integration over β and dz. The integration over λ needs some approximation (a saddle-point-type approximation). Then, the marginalized likelihood becomes as follows, (2)

Here, N_pixel is the total number of pixels in the image. C_o, C_c, C_oo, C_cc, C_oc are sums of pixel heights, sums of squares, and inner product of pixel heights, respectively:

This marginalized likelihood depends only on the candidate structure m, 3D rotation ϕ, translation (dx, dy) and tip apex radius r, and now this is more feasible compared to Eq 1.

In this study, using the marginalized likelihood (Eq 2), we conducted maximum likelihood estimates for the candidate structure m, 3D rotation ϕ, and tip apex radius r. For the integration over the 3D rotation ϕ, we used 577 quaternions which are uniformly sampled in the SO (3) group space [26,27]. C_o, C_c, C_oo and C_cc are invariant with respect to translation (dx, dy), thus only C_oc is needed to integrate over (dx, dy). C_oc can be expressed in the form of convolution integrals, thus can be efficiently computed in the inverse space using Fast Fourier Transform (FFT) [27]. In this study, we employed FFT and the computational cost was reduced from to . As for the translation, we just took the slice of the maximum point instead of marginalization. This is important for analyzing AFM movies which contain particles other than the target molecule because marginalization is affected by non-targeted particles.

Hidden Markov modeling extension to multiple frames

To extend the rigid-body fitting described in the previous subsection to treat multiple AFM images as a time-series, we introduce a likelihood for multiple images consisting of temporally consecutive frames (Fig 1C). To define this, we first define state s (s = 1,…,M) by the combination of the candidate structure m and the 3D rotation ϕ, and then introduce the transition probability from state s_t at the t-th frame (t = 1,…,T) to state s_t+1 at the (t+1)-th frame as . We assume that the transitions between states satisfy the Markovian property so that the transitions can be fully characterized by only the transition probabilities. Furthermore, the transition probabilities are assumed to satisfy the detailed balance. Translations (dx, dy) are not included in the definition of the state because, unlike rotation, translations may not affect the conformational dynamics of a molecule because interactions with the stage do not change over translations. Then, the probability of being in state s₁ at the 1st frame and observing image I^ref,1 can be written by p(s₁)L(m₁, ϕ₁, dx₁, dy₁, r) where m₁, ϕ₁, dx₁, dy₁ are the parameter sets at the 1st frame, p(s₁) is the equilibrium probability of s₁. Repeating this, the probability of observing state s₂ and image I^ref,2 in the 2nd frame is written by using the transition probability. By repeating this process up to the Tth frame, and marginalizing over the state s_t for all the frames, the likelihood of AFM time-series images becomes as follows, (3)

Note that if the equilibrium probabilities and transition probabilities are uniform, the likelihood in Eq 3 is equivalent to simply multiplying the likelihoods of the frame-by-frame rigid-body fitting. On the other hand, if the transition probabilities are not uniform, the likelihood is affected by not only the individual likelihood of rigid-body fitting, but also the transition probabilities.

Since the transition probabilities are usually unknown in advance of the analysis of AFM images, they should be estimated from the data using the Baum-Welch algorithm of hidden Markov modeling [23]. In the original Baum-Welch algorithm, the so-called emission, which is the probability of observing an image from a given state, is estimated together from the data. Since the emission is exactly the same as the likelihood L(m, ϕ, dx, dy, r) of the frame-by-frame rigid-body fitting in this case, only the transition probabilities are estimated here. The Baum-Welch algorithm can estimate transition probabilities that reproduce the data with higher likelihood defined in Eq 3 (though it only locally optimizes the likelihood thus it could be trapped a local minimum). In the algorithm, the Forward-Backward algorithm is used for the calculation of the likelihood, which drastically reduces the computational cost from O(2TM^T) to O(2TM²).

An important characteristic of the Baum-Welch algorithm is that the transition probability, initially sets to zero, remains zero even after the algorithm is applied [23]. By imposing these topological constraints as initial conditions, it is possible to introduce a prior knowledge that transitions between certain states completely unlikely happen in a single transition and further reduce computational cost [28]. In the case of AFM experiments, a target molecule is often interacting with the stage, and large rotations between adjacent frames do not likely happen. Thus, we propose to impose a prior knowledge that significantly change its orientations unlikely occur in a single transition. This can be achieved by imposing the transition probabilities to be zero between states with largely different orientations. Throughout this study, we imposed the probability of transitions between states whose quaternion distance is to be zero.

In the calculation of the likelihood of AFM time-series images (Eq 3), we first need to compute the likelihood L(m_t, ϕ_t, dx_t, dy_t, r) of the frame-by-frame rigid-body fitting (Eq 2). The results of the frame-by-frame likelihood can be used to further reduce the number of states used in Eq 3, because the structures and orientations with very small frame-by-frame likelihood (Eq 2) value are also very small even in the likelihood of the time-series (Eq 3). By removing the structures and orientations with very small frame-by-frame likelihoods, we can further reduce computational cost of the hidden Markov modeling.

Once the transition probabilities are estimated by the Baum-Welch algorithm, they can be used to estimate the maximum likelihood states that match the AFM time-series images. This maximum likelihood estimation can be done efficiently with the Viterbi algorithm in the Hidden Markov modeling [23]. The Viterbi algorithm find the maximum likelihood state sequences from the given AFM data by maximizing the likelihood of AFM time-series images over the parameters (sequence of structures, orientations, m₁, ϕ₁,…,m_T, ϕ_T). In this study, the tip radius r is fixed during the analysis instead of optimizing it in order to further reduce the computational cost. In summary, by applying the hidden Markov modeling to AFM time-series images, (i) we first obtain the transition probabilities between states by the Baum-Welch algorithm, (ii) then we obtain the maximum likelihood sequence of structures, rotations with by the Viterbi algorithm.

Twin experiment

To compare the performance of the Hidden Markov modeling method described in the previous subsection with the frame-by-frame rigid-body fitting, we performed twin experiments. In the experiments, the accuracy of the method is assessed by generating pseudo-AFM images and compare the estimated structures and orientations with the ground-truths used to generate the images.

We used the extracellular ligand-binding domains (LBDs) of the medaka fish taste receptor (PDB ID: 5X2M [29]) as a molecular model in the twin experiments (Fig 2A). This is a heterodimer consisting of T1r2a and T1r3. The ligand-binding domains (LBDs) in T1r2a and T1r3 have a ligand-binding pocket. Previous biophysical studies have shown that the T1r2aLBD pocket is characterized by a broad yet discriminating chemical recognition, while the T1r3LBD pocket is rather loosely bound an amino acid [29]. The receptor is an ortholog of human sweet and umami taste receptors, in which each subunit of the heterodimer is known to play different functional roles [30]. In this context, it is important to correctly identify T1r2aLBD and T1r3LBD of this pseudo symmetric structure from AFM images and to characterize the structure and dynamics of each domain.

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Fig 2. Structure of the taste receptor and its Markov state model (MSM).

(A) The extracellular ligand-binding domains and the cysteine-rich domains of the medaka fish taste receptor. (B) MSM constructed from coarse-grained simulation data. (C) Pseudo-AFM data generated by a stochastic simulation with the MSM, which is used for the twin experiments.

https://doi.org/10.1371/journal.pcbi.1010384.g002

To prepare molecular structures used for generating pseudo-AFM images for twin experiment, we performed coarse-grained MD simulations of the taste receptor (without the stage of AFM), sampling all possible structures, and then constructed a Markov state model (MSM) [31–33]. Before starting MD simulations, to clarify the dynamics of the entire extracellular region not only with LBD but also with the downstream cysteine-rich domain, cysteine-rich domains were added to the atomistic structure of 5X2M by homology modeling using the structure of the same protein family with T1rs, class C G protein-coupled receptor (GPCR), the active form of human calcium-sensing receptor extracellular domain (PDB ID: 5K5S [34]) as a template. Then, the subdomains (LB1 and LB2) of LBDs were superimposed on the structure of the inactive form of human calcium-sensing receptor extracellular domain (PDB ID: 5K5T [34]) to create a structure where LBDs are open (called the OO state while the closed state is called the CC state). Also, cysteine-rich domain was also superimposed on the structure of a single protomer of another class C GPCR, the metabotropic receptor (mGluR)3LBD (PDB ID: 2E4U [35]), to create an open structure of the whole dimer (called the R state while the closed state is called the A state). Simple superposition of the LBDs made the interface region of the LBDs sterically crashed, so the structure was superimposed again to the crashed structure by a targeted MD simulation [36] using an all-atom model from the 5X2M structure. In the targeted MD simulation, we did not impose any pulling forces to the interface region to make the regions relaxed according to naturally occurring inter-molecular interactions. Similar calculations were performed for structures with open LBDs (OO state). All obtained structures were then subjected to all-atom MD with positional restraints on their Cα atoms to relax the other regions including side-chains and surrounding solvents. Then, using the obtained structures, Cα-based Karanicolas-Brooks (KS) Go-model [37,38] was created for coarse-grained MD simulations. Based on KB Go-model, we created a quad-basin potential energy by macro-mixing the four states (CCA, CCR, OOR, OOA states, see Fig 2B) obtained by the all-atom modeling, which makes its energy landscape stabilizing these four states. Then, a number of coarse-grained MD simulations were conducted with the created model to sample all possible structures.

All the MD simulations were performed with GENESIS [39,40]. In the all-atom MD simulations, the simulation systems were prepared with the CHARMM-GUI web server [41] using the modeled structures. Missing residues were modelled with MODELLER [42]. CHARMM36 force field [43,44] was used with the TIP3P water model [45]. Electrostatic interactions were treated using the Smooth Particle Mesh Ewald method [46], and covalent bonds including hydrogen atoms were constrained using SHAKE [47] and SETTLE [48]. Temperature was kept at 303.15 K with Langevin dynamics. In the coarse-grained MD simulations, the parameter sets for KB Go-model were prepared by using the MMTSB [49] Go model web server. Quad-basin macro-mixing parameters were created from four single-basin KB Go-model parameters. All the bonds between coarse-grained atoms were constrained using SHAKE, and the temperature was controlled at 200 K by Langevin dynamics.

The obtained MD structures were aligned to the CCA structure, and principal component analysis (PCA [50]) was performed on the Cartesian coordinates of the aligned structures. The 1st PC corresponds to opening-closing structural motions in the entire dimer, and the 2nd PC corresponds to opening-closing motions in LB1 and LB2 of LBDs. We performed the k-means clustering in the space of these 1st and 2nd PCs to define the states in the Markov state model (MSM) with k = 50. Note that, “states” used here, contain only structures (does not contain rotations). By counting the transition between these states from the trajectories, transition probabilities for states (structures) were estimated. Finally, we constructed a MSM of the taste receptor to represent its conformational dynamics.

We used the constructed MSM to generate pseudo-AFM data for twin experiment (see S1 Fig for the computational protocol). To generate pseudo-AFM images, (i) we first conducted stochastic simulations with the MSM to calculate the transitions between the MSM’s discretized states for 100 frames. In the simulation, we first chose one state from the MSM’s discretized fifty states as the initial state according to the equilibrium probabilities. Then, calculate next state according to the transition probabilities of the MSM. By repeating this process, we obtained 100 state sequences. According to the obtained sequence of states, the sequence of corresponding structures was drawn from the centroid of k-means clustering. Then, (ii) 100 frames of AFM images were created from the sequence of structures using a collision-detection method [19]. We assumed the tip by a hemisphere (with the radius of r = 2.5 nm and the half angle of 10 degrees, see S1 Fig for the definition of the half angle) combined with a circular frustum of a cone. When calculating the collision with a tip, the effective radius of the Cα atom for its amino acid residue was taken from the CryoEM study [25]. The pixel resolution is 0.625 nm × 0.625 nm, and the image has 80 pixels along the X-axis and 60 pixels along the Y-axis. For simplicity, we assumed that a molecule is strongly interacted with the stage, so the orientation of the molecule is fixed to a specific angle during the 100 frames. To determine the orientation, a quaternion was randomly selected from 576 quaternions uniformly sampled in SO(3) group [26,27]. Irrespective the chosen orientations, the same transition probabilities between states (structures) were used for the stochastic simulation. Lateral drift of the molecule was simulated as a random walk where each step was sampled by 2D independent Gaussian distribution with the standard deviation of (0.1 nm). (iii) Then, to mimic experimental noise, spatially independent Gaussian noise with a standard deviation of 0.3 nm were added to the generated pseudo-AFM images. The standard deviation of 0.3 nm is a typical noise width of AFM images [19]. The above calculations were repeated 50 times to produce a total of 50 sets of 100-frame AFM data.

We performed the frame-by-frame rigid-body fitting and the maximum likelihood estimation with hidden Markov modeling on the pseudo-AFM images created by the above procedure. We used the same 50 structures with the MSM (i.e., the centroids of k-means clustering) as candidate structures. The same 576 quaternions and AFM image grids were used. RMSD between the ground-truth and estimated structures was used to access the accuracy of the estimation. In order to access the accuracy of not only the structure m but also the 3D rotation ϕ, the RMSD calculation was performed without applying structural alignment.

It is noted again that the same 50 structure with the MSM were used for the analysis. This is mainly due to the requirement that the MSM clusters should be the same as the ground-truth in order to check the accuracy of the reconstructed MSM (relative entropy calculation, described later) in the latter part of our analysis. For checking the effect of this choice, we conducted an additional analysis and calculated RMSDs calculation using different 50 structures obtained by a clustering of MD simulation trajectories.

The actual experimental data often suggest that the shape of a tip is not simple as a circular frustum of a cone assumed in this study, but a more complex one. Therefore, to account for the possibility that the shape of the real tip might be different from the assumed one, we performed maximum likelihood estimations with intentionally different tip apex radii (ground truth used for generating pseudo-AFM images are 2.5 nm while we used 1.5 nm, 1.8 nm, 2.5 nm, 3.2 nm, and 3.5 nm for estimation). We tested how robustly the method estimates the structure and orientation in the situation where there was a discrepancy in the tip apex radius. Furthermore, the hidden Markov modeling can estimate the transition probabilities between states. We reconstructed the transition probabilities between structures by reducing orientations from obtained by the Baum-Welch algorithm. Then, compared the reduced transition probabilities between structures with those of the Markov state model.

Software availability

The Hidden Markov modeling-based analysis method described here is implemented and publicly available as part of the MDToolbox.jl package and can be downloaded from https://github.com/matsunagalab/MDToolbox.jl. In addition, Jupyter notebooks to reproduce the results of this paper are publicly available at https://github.com/matsunagalab/paper_ogane2022.

Twin experiments with identical radii

First, we examined the estimation accuracy when the ground-truth tip radius used to generate the pseudo-AFM data (2.5 nm) and the tip radius used for the maximum likelihood estimation are identical. Fig 3C shows that both methods (the frame-by-frame rigid-body fitting and the hidden Markov modeling) are able to estimate the correct ones from the combinations of 50 candidate structures and 576 angles with almost 100% accuracy, despite the addition of Gaussian noise with a standard deviation of 0.3 nm. Specifically, the frame-by-frame rigid body fitting resulted in zero RMSD (without structural alignment) for 4991 frames out of 50 frames × 100 sets of images. The Hidden Markov modeling resulted in zero RMSD in the same number of frames (4991 frames). The accuracy of this estimation result may be dependent on the noise size; in the current setup, the spatial scale of the average RMSD (~1.1 nm) between the 50 structures and the resolution of the rotations is comparable or larger than the size of Gaussian noise.

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Fig 3. Results of twin experiments.

Histograms of root-mean square deviations (RMSDs) of estimated structures from the ground-truth structures in twin experiments. Note that the RMSDs were computed without structural alignment. Structural estimations were performed with various conditions: with different algorithms (the frame-by-frame rigid-body fitting, the Viterbi algorithm using the ground-truth transition probabilities, estimated probabilities by the Baum-Welch algorithm, and a random matrix), and different tip radii (1.5 nm, 1.8 nm, 2.5 nm that is the ground-truth, 3.2 nm, 3.5 nm).

https://doi.org/10.1371/journal.pcbi.1010384.g003

Twin experiments with discrepant radii

Next, we examined the estimation accuracy when the ground-truth tip radius (2.5 nm) used in generating the AFM data is different from the tip radius used for the maximum likelihood estimation. As described in Methods, the actual experimental probe geometry is more complex than the assumed shape, so we intentionally used a different radius for the estimation to assess how the estimations are robust against such discrepancies.

Fig 3A and 3B and 3D and 3E show the accuracies of the frame-by-frame rigid-body fitting with different tip radii, measured by structural RMSD from the ground-truth structure. When the radius difference is close, the RMSDs are distributed lower than 1.0 nm. However, as the radius difference increases, higher peaks appear at around the RMSD of 5.0 nm, indicating that the estimation accuracy is becoming worse. The structures around the RMSD of 5.0 nm are flipped 180 degrees from the ground-truth structure, indicating that the estimations failed to assign each monomer of the taste receptor to the AFM image correctly. This is because, as the radius difference increases, the likelihood fails to catch the subtle surface differences in the two monomers.

To further investigate the cause of the double peaks in the RMSD distribution, we examined the relation between the orientations and the likelihood values (Eq 2) using the first frame of the first set in the AFM data (shown in Fig 4). Fig 4 shows the likelihood values for each 3D rotation ϕ and candidate structures m. Fig 4B shows that, even when the radius of the tip apex is identical (2.5 nm), the likelihood of the structures flipped about 180 degrees have already higher likelihoods (compared to other angles), but the tip distinguishes the difference in surface geometry of each monomer and the correct angle is chosen as the maximum likelihood. On the other hand, when the radii are different from each other (Fig 4A), the likelihood at the ground-truth angle decreases whereas the likelihoods of the 180-degrees flipped structures become competitive, resulting in the flipped structures to be selected as the maximum likelihood.

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Fig 4. Likelihood distribution over all possible orientations and structures computed using the first single frame of the first set AFM data (defined by Eq 2).

Likelihoods are projected onto the quaternion distance from the ground-truth. The maximum likelihoods are indicated by red arrows. (A) Likelihoods computed with the tip apex radius of 1.8 nm. (B) 2.5 nm (the ground-truth tip radius).

https://doi.org/10.1371/journal.pcbi.1010384.g004

Next, we compared the results of the frame-by-frame rigid-body fitting with those of the hidden Markov modeling (Fig 3). We first used the ground-truth transition probabilities of MSM used for generating pseudo-AFM data. A sequence of maximum likelihood states was computed by the Viterbi algorithm. Fig 3 shows that the RMSDs of the hidden Markov modeling with the ground-truth transition matrix are overall improved compared to the RMSDs of the frame-by-frame rigid-body fitting. In particular, when the probe radius is 1.8 nm, there is a noticeable reduction in the peak around RMSD of 5.0 nm corresponding to 180-degrees flipped structures. This means that hidden Markov modeling enables a robust correct estimation of the orientation of the molecule against the choice of tip radius. As described in Eq 3, the hidden Markov uses all frames to find the maximum likelihood sequence, not just one frame. The use of a transition matrix with rotational constraints gives penalties for large rotational flips. As a result, a fixed orientation throughout all frames is favored as the maximum likelihood sequence by a majority rule; minor orientations are not selected. This mechanism can be considered as the same principle as in the so-called ensemble learning [51]; the variance of estimation becomes large when a single discriminator is used, while the variance becomes small when multiple discriminators are combined. Note, however, when the bias getting larger due to large inconsistencies between tip radii, the majority rule cannot decrease the error as shown in Fig 3A (1.5 nm) and 3E (3.5 nm).

In order to check the sensitivity of the results against noise realizations (Gaussian noise added to the pseudo-AFM images), we conducted the same analyses using different noise realizations (S2 and S3 Figs). The same trend was observed regardless of the noise realization values. Furthermore, to check the sensitivity of the results against noise levels, we conducted the same analyses using the pseudo-AFM images with different noise levels (the standard deviations of Gaussian noise, 0.6, 0.9, and 1.2 nm, shown in S4–S6 Figs). Interestingly, we did not see a significant decrease in RMSDs with these noise levels. Although this may be due in part to the still small noise levels, this result suggests that the tip radius (shown in Fig 3) has a stronger influence on AFM image analysis than the noise level. Moreover, to check the sensitivity of the results against tip radius used for generating pseudo-AFM image, we conducted the same analysis with the tip radius of 3.5 nm, which shows the same tendency as that of the tip radius of 2.5 nm (S7 Fig).

S8 Fig shows the RMSDs from the ground-truth structures calculated using 50 different structures different from those used for generating the pseudo-AFM structures. This corresponds to a more general situation for analyzing HS-AFM data. Although the accuracies of estimations have deteriorated by using the different structures, the overall story that hidden Markov modeling improves estimation accuracy remains unchanged.

To investigate how the majority rule mechanism works in multiple frames, we again examined the relation between orientations and likelihood values, changing the numbers of frames. By imposing a constraint being at a specific 3D angle ϕ in the Viterbi algorithm, we computed the maximum likelihood at each orientation using 1 frame, 10 frames, and 100 frames of the first set of the AFM data. Fig 5B shows the maximum likelihoods at each orientation when both radii are identical (2.5 nm). In this case, the correct orientation has already been selected as the maximum likelihood of the frame-by-frame analysis. As the number of frames increases, the difference between the largest likelihood of the correct orientation and the second largest one increases, indicating that the correct orientation is more likely chosen by using a greater number of frames. Fig 5A shows the results when the probe radius is not identical with the ground truth (2.5 nm and 1.8 nm). When only single frame is used, the maximum likelihood orientation is not correct, a 180 degrees flipped structure is chosen. Interestingly, as the number of frames increases, the relationship switches and the likelihood of the correct orientation begins to be chosen because the other frames have higher likelihoods for the correct orientation. As the number of frames increases, the likelihood of the correct orientation becomes more prominent. In summary, when the likelihood is calculated for only a single frame, it is probable that a wrong orientation is chosen due to the estimation error (variance), but when the number of frames is increased, the correct orientation is robustly chosen by decreasing the variance.

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Fig 5. Maximum likelihood of pseudo-AFM time-series images in each orientation computed using the first single frame, the first 10 frames, the first 100 frames of the first set AFM data (defined by Eq 3).

Note that only the maximum likelihood for each orientation is shown due to computational costs (calculated by the Viterbi algorithm). The maximum likelihoods among all orientations are indicated by red arrows. (A) Likelihoods computed with the tip apex radius of 1.8 nm. (B) 2.5 nm (the ground-truth).

https://doi.org/10.1371/journal.pcbi.1010384.g005

To verify further whether the transition probabilities work to improve estimation accuracy, we computed the maximum likelihood sequence using a random matrix generated by uniform random numbers (normalized over rows) as a transition probability matrix. Interestingly, the estimation accuracy of hidden Markov modeling using the random matrix is almost identical to the results of the frame-by-frame rigid-body fitting. This is due to the fact that the mechanism of the ensemble learning does not work because the transition probabilities can no longer give penalty for large rotational flips. As a result, the likelihood of each frame is independently selected in the maximum likelihood sequence.

Finally, we performed estimations where the transition probabilities were estimated from the data without providing ground-truth transition probabilities in advance. Staring from a random matrix as the initial condition and imposing a rotational constraint, we estimated transition probabilities from the data using the Baum-Welch algorithm, and subsequently obtained the maximum likelihood sequence to verify the accuracy of the estimation. As the constraint on rotation, the probability of transitions between states whose quaternion distance is was imposed to be zero. Fig 3 shows that the estimation accuracy is improved compared with those of the rigid-body fitting, even when the transition probabilities are estimated from the data, although they are not as good as those with the ground-truth transition probabilities. This improvement in accuracy is mainly contributed by rotational constraints on the transition probabilities.

The confidence intervals of the estimated transition probabilities were evaluated by the bootstrap method [52] for the radii of 2.5 and 2.0 nm (S11 and S12 Figs). The results indicate that probabilities can be estimated in the range of 0.1 to 0.2 from the current data set.

Reconstruction of Markov state model

As calculated in the previous subsection, Hidden Markov modeling can estimate transition probabilities from the data using the Baum-Welch algorithm. Estimating transition probabilities and constructing more accurate MSM from experimental data is an important topic for application studies. Here, we examined how accurately our proposed method can estimate the transition probabilities between structures using the same twin experiment data. Again, using a random matrix as the initial condition and imposing a rotation constraint, we estimated transition probabilities from the data using the Baum-Welch algorithm. Then, we marginalized the estimated transition probabilities between states (structures and rotations) to the transition probabilities between structures. After the marginalization, the probabilities were post-processed to suffice the detailed balance. Fig 6 visualizes the transition probabilities between states estimated with various tip radii. The equilibrium probability of each structure (indicated by the area of each node in Fig 6) computed from the detailed balance.

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Fig 6. Reconstruction of MSM from atomic force microscopy data.

(A) Graph representation of the ground-truth MSM used for twin experiments. (B) Reconstructed Markov state model with the tip apex radius of 1.5 nm. (C) Reconstructed MSM with the tip apex radius of 2.5 nm (the ground-truth radius). (D) Accuracies of reconstructed msms evaluated by relative entropies for various tip apex radii.

https://doi.org/10.1371/journal.pcbi.1010384.g006

To access the accuracy of the reconstructed Markov state model, we used the relative entropy between two models [53], where T_ij is the transition probability of the ground-truth model, and p(i) is its equilibrium probability of state i. is the transition probability of a reconstructed model. The relative entropy is zero if the two models are identical and takes increasingly large values the more the two models differ. To avoid singular behavior due to zero divide, we here added a very small value (10⁻¹⁰) uniformly to .

Fig 6 shows that the estimated transition probabilities are almost the same, especially when the probe radii used for estimation are ground truth and identical (2.5 nm). In fact, as in the ground-truth MSM, CCR state is kinetically isolated from other states and the transition to this state is rare. This is true even when the probe radius is 2.0 nm, indicating that our proposed method is robust to the discrepancies in the probe geometry. According to the frame-by-frame rigid-body fitting of real AFM data (actin filament) by Nina et al. [19], the error size in the tip radii estimation from real data looks less than ± 1.0 nm. Thus, the robustness of our method would be applicable to real experimental data in the future study.

On the other hand, when 1.8 nm or 1.5 nm is used as the tip radius, the distinction between structural states becomes blurred, indicating that the kinetic isolation of CCA state is significantly lost at the radius of 1.5 nm. This observation is quantified by the relative entropies between two models which takes the minimum value when two radii are identical.

The relative entropies calculated with various conditions (different noise realizations, noise levels, tip radii for generating pseudo-AFM images) are shown in S13 Fig.