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2.2 Observation types The two proteins, [MATH] and [MATH] , in a homologous pair are associated with a pair of observation sequences [MATH] and [MATH] obtained from experimental data, respectively. An [MATH] th site observation pair, [MATH] , is associated with every aligned site [MATH] in an alignment [MATH] of [MATH]... |
Each site observation, [MATH] and [MATH] , contains amino acid and structural information corresponding to the two [MATH] atoms at aligned site [MATH] belonging to each of the two proteins. A site observation corresponding to a particular protein at aligned site [MATH] [MATH] , is comprised of three different data type... |
2.3 Model structure The sequence of hidden nodes in the HMM is written as [MATH] ). Each hidden node [MATH] in the HMM corresponds to a site observation pair, [MATH] and [MATH] , at an aligned site [MATH] in the alignment [MATH] . Initially we treat the alignment [MATH] as given a priori , but later modify the HMM to m... |
The model is parametrised by [MATH] hidden states. Every hidden node [MATH] corresponding to an aligned site [MATH] takes an integer value from [MATH] to [MATH] for the hidden state at node [MATH] . In turn, each hidden state specifies a distribution over an evolutionary regime pair: [MATH] as a function of evolutionar... |
[MATH] . We return to the specific role of the evolutionary regimes pairs in the next section. The state of [MATH] together with the evolutionary regime pair, [MATH] , and the evolutionary time separating proteins [MATH] and [MATH] [MATH] , specify a distribution over three conditionally independent stochastic processe... |
[EQUATION] The assumption of conditional independence provides computational tractability, allowing us to avoid costly marginalisation when certain combinations of data are missing (e.g. amino acid sequences present, but secondary structures and dihedral angles missing). |
2.4 Stochastic processes: modelling evolutionary dependencies Each evolutionary regime couples together three time-reversible stochastic processes that separately describe the evolution of the three pairs of observation types, as in equation ( ). Each evolutionary regime is intended to capture different features of seq... |
Amino acid evolution As is typical with models of sequence evolution, amino acid evolution, [MATH] , is described by a Continuous-Time Markov Chain (CTMC). Each amino acid CTMC is parametrised in the following way: the exchangeability of amino acids is described by a [MATH] symmetric global exchangeability matrix [MATH... |
Secondary structure evolution Secondary structure evolution, [MATH] , is also described by a CTMC. In our model we use three discrete classes to describe secondary structure at each position: helix (H), sheet (S) and random coil (C). |
The exchangeability of secondary structure classes at a position is described by a [MATH] symmetric global exchangeability matrix [MATH] and a regime-specific set of 3 secondary structure equilibrium frequencies [MATH] . Together they define a regime-specific time-reversible secondary structure rate matrix [MATH] , wit... |
Dihedral angle evolution Central to our model is evolutionary dependence between dihedral angles, [MATH] . Typically, the continuous-time evolution of the continuous-state random variables is modelled by a diffusive process such as the Ornstein–Uhlenbeck (OU) process, as in Challis and Schmidler ( 2012 . However, an OU... |
Topologically, the WN diffusion (see Figure for a pictorial example) can be thought of as the analogue of the OU process on the torus [MATH] . The WN diffusion arises as the wrapping on [MATH] of the following Euclidean diffusion: |
[EQUATION] where [MATH] is the two-dimensional Wiener process, [MATH] is the drift matrix, [MATH] is the stationary mean, [MATH] is the infinitesimal covariance matrix and |
[EQUATION] [MATH] , is a probability density function (pdf) for [MATH] [MATH] stands for the pdf of a bivariate Gaussian [MATH] . The pdf ( ) weights the linear drifts of ( ) such that they become smooth and periodic. |
It is shown in García-Portugués et al. 2017 that the stationary distribution of the WN diffusion is a [MATH] , which has pdf: [EQUATION] |
Despite involving an infinite sum over [MATH] , taking just the first few terms of this sum provides a tractable and accurate approximation to the stationary density for most of the realistic parameter values. |
Maximum Likelihood Estimation (MLE) for diffusions is based on the transition probability density (tpd), which only has a tractable analytical form for very few specific processes. A highly tractable and accurate approximation to the tpd is given for the WN diffusion. This approximation results from weighting the tpd o... |
[EQUATION] with [MATH] [MATH] and [MATH] . The pseudo-tpd provides a good approximation to the true tpd in key circumstances: [MATH] , since it collapses in the Dirac’s delta; ii [MATH] , since it converges to the stationary distribution; iii ) high concentration, since the WN diffusion becomes an OU process. Furthermo... |
A further desirable property of the pseudo-tpd is that it obeys the time-reversibility equation, which in terms of [MATH] is [EQUATION] |
Indeed, the WN diffusion is the unique time-reversible diffusion with the stationary pdf ( ), in the same way the OU is with respect to a Gaussian. Time-reversibility is an assumption of the overall model and many other models of sequence evolution. A benefit of time-reversibility in a pairwise model such as ETDBN is t... |
The likelihood of a dihedral angle observation pair [MATH] , assuming that [MATH] is drawn from the stationary distribution, is given by: |
[EQUATION] [MATH] and [MATH] are constrained to yield a covariance matrix [MATH] . A parametrization that achieves this is [MATH] and [MATH] [MATH] [MATH] and [MATH] are the drift components for the [MATH] and [MATH] dihedral angles, respectively. Dependence (correlation) between the dihedral angles is captured by [MAT... |
The computationally tractable nature of the stationary density and tpd required in ( ) is key to enabling efficient training of and sampling under the model. Specifically, the computation of ( ) involves evaluating [MATH] and [MATH] , which we can work out explicitly. First, [MATH] with [MATH] [MATH] [MATH] and [MATH] ... |
[EQUATION] with [MATH] and [MATH] . This gives a neat interpolation of the stationary and infinitesimal covariance matrices, particularly convenient for efficiently evaluating ( ) at different [MATH] ’s. |
2.5 Evolutionary regimes: modelling shift and drift We now turn to the meaning of the evolutionary regime pairs. Two modes of evolution are modelled: constant evolution and jump evolution . Constant evolution occurs when the evolutionary regime starting in protein [MATH] at aligned site [MATH] [MATH] , is the same as t... |
[MATH] . Conversely, jump evolution occurs when [MATH] . Constant evolution is intended to capture angular drift (changes in dihedral angles localised to a region of the Ramachandran plot), whereas jump evolution is intended to capture angular shift (large changes in dihedral angles, possibly between distant regions of... |
The hidden state at node [MATH] , together with the evolutionary time [MATH] separating proteins [MATH] and [MATH] , specifies a joint distribution over the evolutionary regime pairs: |
[EQUATION] where [EQUATION] and [MATH] and [MATH] [MATH] and [MATH] are model parameters specifying the probability of starting in regime [MATH] or [MATH] , respectively, corresponding to the hidden state specified by node [MATH] [MATH] is a model parameter specifying the jump rate corresponding to the hidden state spe... |
The regime pair jump probabilities have been chosen so that time-reversibility holds, in other words: [EQUATION] The hidden state at node [MATH] , together with a regime pair [MATH] and the evolutionary time [MATH] , specifies the joint likelihood over site observation pairs: |
[EQUATION] In the case of constant evolution, evolution at aligned [MATH] is described in terms of the same evolutionary regime [MATH] . Evolution is considered constant because each observation type is drawn from a single stochastic process specified by [MATH] and [MATH] . Note that the strength of the evolutionary de... |
In the case of jump evolution, the evolutionary processes are, after the evolutionary jump, restarted independently in the stationary distribution of the new evolutionary regime. Thus the site observations [MATH] and [MATH] are assumed to be drawn from the stationary distributions of two separate stochastic processes c... |
[EQUATION] where [MATH] is the set of four regime pairs, [MATH] is given by ( ) and [MATH] is given by ( ). 2.6 Identification of evolutionary motifs encoding jump evolution |
In order to identify aligned sites having potential evolutionary motifs encoding jump evolution, a specific criterion was developed. |
For a particular protein pair, inference was performed under the model conditioned on the amino acid sequence and dihedral angles for both proteins, [MATH] . Homologous sites corresponding to a single hidden state and with evidence of jump evolution ( [MATH] ) at posterior probability [MATH] were identified, that is, t... |
In a second filtering step, amino acid sequences and a single set of dihedral angles corresponding to one of the proteins were used ( [MATH] or [MATH] ) to infer the posterior probability, this time at a lower threshold: [MATH] or [MATH] . This second criterion ensured that the evolutionary motif was identifiable under... |
Only those aligned sites meeting both criteria were selected for further downstream analysis. 2.7 Statistical alignment: modelling insertions and deletions |
Protein sequences can not only undergo amino transitions due to underlying nucleotide mutations in the coding sequence, but also indel events. To account for indels, a modified pairwise TKF92 alignment HMM based on Miklós et al. 2004 was implemented. The TKF92 alignment HMM was augmented with additional evolutionary hi... |
Whilst it is possible to fix the alignment in advance by pre-aligning the sequences using one of the many available alignment methods (Katoh et al. (2002); Edgar (2004)) or using a curated alignment (such as from the HOMSTRAD database), doing so ignores alignment uncertainty. For more detail we refer to the supplementa... |
2.8 Training and test dataset A training dataset of 1200 protein pairs (2400 proteins; 417,870 site observation pairs) and a test dataset of 38 protein pairs (76 proteins; 14125 site observation pairs) were assembled from 1032 protein families in the HOMSTRAD database. |
For each protein family in HOMSTRAD (ranging in size from 2 to 22 homologous proteins each), a phylogenetic tree was inferred from the HOMSTRAD protein family sequence alignment using FastTree (Price et al. 2010 . Each protein family tree was taken and protein pairs selected such that the sum of the branches between pa... |
Dihedral angles were computed from the PDB coordinates of each protein structure using the BioPython.PDB package (Hamelryck and Manderick, 2003 . Furthermore, each protein was taken and the corresponding full length protein structure obtained from the PDB database and the secondary structure annotated at each amino aci... |
2.9 Model training Maximum likelihood estimation of the model parameters, [MATH] , was done using Stochastic Expectation Maximisation (StEM). Forward Filtering Backward Sampling (FFBS) was used to jointly sample alignment configurations ( [MATH] ), hidden node states ( [MATH] ) and evolutionary regimes, ( [MATH] ). The... |
[EQUATION] In the M-step the samples from the previous E-step, were used to update the hidden node parameters ( [MATH] ) using efficient sufficient statistics and COBYLA optimization algorithm (Powell, 1994 in the NLOpt library (Johnson, 2014 was used. |
2.10 Angular distances For benchmarking purposes, the angular cosine distance was used to measure distances between pairs of dihedral angles, [MATH] and [MATH] . It is defined as follows (Downs and Mardia, 2002 |
[EQUATION] The maximum possible distance is [MATH] . It has the property that when [MATH] and [MATH] are near zero it may be approximated by the Euclidean distance – using the small angle approximation for cosine ( [MATH] when [MATH] is near zero): |
[EQUATION] Results and discussion 3.1 Selecting the number of hidden states Models with 8, 16, 32, 48, 64, 80, 96 and 112 hidden states were trained until convergence for three different repetitions using different initial random number seeds. The highest log-likelihood model of the three repetitions for each number of... |
Following that, marginal likelihoods [MATH] i.e. model evidence for each of the 38 protein pairs in the test dataset, were computed under each model by fixing the alignments to the respective HOMSTRAD alignments. The alignments were fixed a priori in order to make computation of the marginal likelihoods computationally... |
The 112 hidden state model had the highest total marginal log-likelihood for protein pairs in the test dataset and predictive accuracy under the homology modelling scenario comparable to the predictive accuracies of the 16, 32, 48, 64, 80 and 96 hidden state models (all were with standard error of the mean, see supplem... |
Although the model with 112 hidden states had the highest total marginal log-likelihoods, the model with 64 hidden states was selected as the final model. A 64 hidden state model was chosen as a trade-off between total marginal log-likelihood on the test dataset, computational time and the number of model parameters. I... |
3.2 Stationary distributions over dihedral angles capture the empirical distribution Figure illustrates the sampled and empirical dihedral angle distributions. There is a good correspondence between dihedral angles sampled under the model (Figure , left) and the empirical distribution of dihedral angles in our training... |
3.3 Estimates of evolutionary time from dihedral angles are consistent with estimates from sequence Figure compares evolutionary times estimated using only pairs of homologous amino acid sequences only versus pairs of homologous dihedral angles only. As desired, the two estimates of evolutionary time for each protein p... |
A paired [MATH] -test gave a [MATH] -value of 0.578, thus failing to reject the null hypothesis that there is no difference between branch lengths estimated using sequence only vs. angles only. This indicates that there is sufficient evolutionary information in the dihedral angles to estimate the evolutionary times and... |
Interestingly, the variance in the sampled evolutionary times is higher when dihedral angles only are used, as compared to sequence only (see Figure 11 ). |
3.4 The relationship between evolutionary time and angular distance is adequately modelled We investigated the relationship between evolutionary time and angular distance between real protein pairs and protein pairs where the dihedral angles of [MATH] [MATH] ) were treated as missing and hence sampled (Figure ). |
As expected, for both real and sampled pairs, angular distance tends to increase as a function evolutionary time. For larger evolutionary times a plateau begins to emerge, which is expected as the maximum possible theoretical angular distance is [MATH] |
When the evolutionary time is exactly zero ( [MATH] ) under our model, the angular distance between sampled dihedral angles is exactly zero (not shown in Figure ), however, this is not expected to be the case for real protein pairs when the two sequences are identical (due to the inherently flexible nature of proteins,... |
For small evolutionary times ( [MATH] ) the curves for the real and sampled protein pairs show a good correspondence, however, for larger evolutionary times the model tends to under-estimate angular distances. This likely reflects the fact that the tpd of the WN diffusion specified is localised around its mean when [MA... |
3.5 Evaluation of the model The conditional independence structure in ( ) enables computationally efficient sampling from the model under different combinations of observed or missing data. For example, ETDBN can be used to sample ( i.e. predict) the dihedral angles of a protein from its corresponding amino acid sequen... |
Predictive accuracy was measured using 38 homologous protein pairs in the test dataset. For every protein pair [MATH] , the dihedral angles of [MATH] in each pair were treated as missing, and these missing dihedral angles were sampled under the model given a particular combination of observation types. The average angu... |
Figure gives an example of predictive accuracy under different combinations of observations types overlaid on a cartoon structure of the protein structure being predicted, whereas Figure provides a representative view of predictive accuracy across 10 different protein pairs in the test dataset for different combination... |
Combination 1 refers to random sampling from the model, implying no data observations were conditioned on besides the respective lengths of proteins [MATH] and [MATH] . The average angular distance between the true and predicted dihedral angles was 1.6. Random sampling acts as a baseline for predictive accuracy. It is ... |
Under combination 2, only the amino acid sequence corresponding to [MATH] is observed. As expected in Figures and there is an increase in predictive accuracy with the addition of the amino acid sequence relative to combination 1. |
Under combination 3, we add in the amino acid sequence of a homologous protein ( [MATH] ). In all ten cases there is an improvement in predictive accuracy. The improvement in predictive accuracy is reasonable, as knowledge of the sequence evolutionary trajectory is expected to encode information about structure evoluti... |
Under combination 4, in addition to the two amino acid sequences we treat the homologous secondary structure as observed. This results in a substantial improvement in predictive accuracy as one would expect. Knowledge of the amino acid sequence and a homologous secondary structure strongly informs regions of the Ramach... |
Under combination 5 (which we consider the canonical combination – the standard homology modelling scenario), we treat both amino acid sequences as observed, as well as the dihedral angles of the homologous protein ( [MATH] ) – in all cases the predictive accuracy improves over combination 4. This is anticipated as the... |
Finally, in combination 6, the same data observations as in combination 5 are used, except the alignment is treated as given a priori (by the HOMSTRAD alignment) rather than as unobserved. The HOMSTRAD alignment is based on a structural and sequence alignment of [MATH] and [MATH] and therefore is expected to encode a h... |
The alignment HMM is valuable accounts for alignment uncertainty in a principled manner, this is particularly useful when an appropriate alignment is unavailable. However, it should be noted that inference scales [MATH] when treating the alignment as unobserved, where [MATH] and [MATH] are the lengths of [MATH] and [MA... |
It should be emphasised that we do not expect ETDBN to compete with structure prediction packages such as Rosetta (Rohl et al. 2004 or homology modelling software such as Arnold et al. 2006 in terms of predictive accuracy. Our current model is a local model of structure evolution – it is not even expected capture const... |
3.6 Evolutionary hidden states reveal a common evolutionary motif One benefit of ETDBN is that the 64 evolutionary hidden states learned during the training phase are interpretable. We give an example of a hidden state encoding jump evolution that was subsequently found to represent an evolutionary motif present in a l... |
Evolutionary hidden state 3 (Figure ) was selected from the 64 hidden states as an example of a hidden state encoding jump evolution and capturing angular shift (a large change in dihedral angle). A notable feature of this hidden state is that the change in dihedral angles between evolutionary regimes [MATH] and [MATH]... |
Having selected hidden state 3, positions in 238 protein pairs were analysed for evidence of the corresponding evolutionary motif. 38 protein pairs in the test dataset and a further 200 from the training dataset were analysed using the criteria described in the Methods section. Using the first criterion, 84 protein sit... |
We give an example of a homologous protein pair illustrating the identified evolutionary motif. Two histidine-containing phosphocarriers, 1pch ( M. capricolum ) and 1poh ( E. coli ), were identified as having the evolutionary motif (Figure ) at homologous site E39/G39. |
Most positions in the homologous pair have very low jump probabilities ( [MATH] ), with the exception of positions N38/N38 and E39/G39, which both have high posterior jump probabilities ( [MATH] ). The exchange between a glutamate (at position 39 in 1poh) and a glycine (at position 39 in 1pch) appears to be responsible... |
Regime [MATH] indicates that a number of amino acids (alanine, aspartic acid, glycine, histidine, lysine, asparagine, proline, glutamine, arginine, serine and theorine) other than glutamate plausibly coincide with the particular dihedral angle conformation specified by regime [MATH] . The involvement of glycine in a ju... |
3.7 Using dihedral angles for alignment A valuable feature of our model is its ability to account for alignment uncertainty by summing over possible pairwise alignments using the TKF92 model as a prior distribution over indel histories, whilst simultaneously taking into account neighbouring dependencies amongst aligned... |
ETDBN and several other alignment methods (namely StatAlign, BAli-Phy, MUSCLE and MAFFT) were used to infer pairwise alignments from simulated and real data under various combinations of data observations, for example: an amino acid sequence pair ( [MATH] ), a secondary structure sequence pair ( [MATH] ), a dihedral an... |
In the first set of benchmarks (Figure 10 A), pairs of proteins were simulated from the ETDBN model conditioned on 38 different pairwise alignments and corresponding evolutionary times. This resulted in a set of 38 simulated pairwise alignments together with corresponding observations, implying that the true underlying... |
More revealing in Figure 10 A was the alignment similarity under ETDBN when using different combinations of simulated data observations. It was found that secondary structure alone ( 10 A.6) performed the worst, which is unsurprising given that only three states were available to align the proteins. The second worst in... |
Following that, the various alignment methods were benchmarked against 38 pairwise alignments consisting of real sequence and structure observations in the test dataset. These pairwise alignments were obtained from the HOMSTRAD alignments. The sequence identity of these pairwise alignments ranged from [MATH] to [MATH] ... |
Using [MATH] alone, ETDBN ( 10 B.6) had substantially lower alignment similarity compared to sequence only, which was expected given that a similar result was obtained for the simulated data ( 10 A.6). However, when including the real sequences ( 10 B.7) the predictive accuracy was once again comparable to sequence onl... |
\noclub When using ( [MATH] ) alone ( 10 B.8), the alignment similarity was found to be somewhat worse than the sequence only cases. Furthermore, when introducing the sequences ( 10 .9) and secondary structures ( 10 B.10) in addition to the dihedral angles, the similarity still remained worse than the sequence only met... |
Upon further investigation, the trend was found to reverse (Figure 10 C) when the precision of predicting homologous sites was calculated (the fraction of sites which were predicted as homologous and were correctly predicted as such). Therefore when only dihedral angle observations are used, ETDBN underpredicts the num... |
When interpreting these results it important to note the HOMSTRAD alignments should not be considered the true underlying alignments and may even be strongly biased (e.g. they may favour the most parsimonious alignments, with the fewest number of indels). In practice, it is extremely difficult to obtain the true underl... |
Concluding remarks The main achievement of this work is a computationally tractable, generative and interpretable probabilistic model of protein sequence and structure evolution on a local scale. |
Previous stochastic models of protein sequence and structure evolution emphasised estimation of evolutionary parameters (Challis and Schmidler, 2012 ; Herman et al. 2014 . ETDBN is somewhat of a departure from these previous models, but is likewise capable of estimating evolutionary parameters. We show that estimates o... |
The generative nature of ETDBN allows us to demonstrate that the underlying empirical distributions over dihedral angles (depicted using Ramachandran plots) are captured and that the model is capable of predicting missing observations, such as dihedral angles, from a variety of different data types. For example, an ami... |
Due to its local nature, ETDBN does not constitute a homology modelling method in itself. Rather, it can be used as a building block, much like fragment libraries model local structure in protein structure prediction methods. ETDBN places the homology modelling problem on a statistical footing, enabling a number of app... |
In addition to multi-level modelling, probabilistic models such as ETDBN allow one to account for and to make statements about uncertainty (e.g. with respect to evolutionary time, alignment, etc.) in a rigorous manner. One immediate use of ETDBN from a structure prediction or homology modelling perspective is as an eff... |
A final key feature of our evolutionary model is its interpretable nature. This interpretability enables the identification of potential evolutionary motifs – common patterns of sequence-structure evolution. We identify one such evolutionary motif in 34 different homologous protein pairs. A major direction for future r... |
Future challenges For reasons of computational tractability the implemented model is pairwise, but it is theoretically possible to generalise it to a phylogeny, such as in Herman et al. 2014 . In practice, for three or more sequences on a phylogeny it is necessary to marginalise out the unobserved ancestral protein sta... |
Although we believe our model provides a substantial improvement over current stochastic models of sequence and structural evolution, there is still scope for improvement. The WN diffusions used to model dihedral angle evolution adequately capture angular drift (small local changes in dihedral angle), but are less capa... |
Software availability Julia code (tested on both Windows and Linux platforms) is available upon request from the first author. Supplement |
A supplement contains further details on the methods employed in the paper and additional results. Acknowledgments The authors acknowledge funding from the University of Copenhagen 2016 Excellence Programme for Interdisciplinary Research (UCPH2016-DSIN). The second author acknowledges support from project MTM2016-76969... |
Supplement to “A generative angular model of protein structure evolution” Michael Golden 1,6 , Eduardo García-Portugués 2,3 , Michael Sørensen |
Kanti V. Mardia , Thomas Hamelryck 2,5 , and Jotun Hein Keywords: Evolution; Protein structure; Probabilistic model; Directional statistics. |
Appendix A Supplementary methods A.1 Statistical alignment: modelling insertions and deletions with neighbouring dependencies Protein sequences can not only undergo point mutation events, but also also insertion and deletion (indel) events. We describe a modified pairwise TKF92 alignment HMM that models both local sequ... |
Whilst it is possible to fix the alignment in advance by pre-aligning the sequences using one of the many available optimisation-based alignment methods (Katoh et al. 2002 ; Edgar, 2004 or using a curated alignment (such as from the HOMSTRAD database), doing so ignores alignment uncertainty. |
An alignment can be thought of as a statement about homology, such that when amino acid positions are aligned in order to indicate homology they are considered to have evolved solely via mutation along the evolutionary trajectory linking them and therefore not arising via an indel. As the evolutionary trajectory of ind... |
For the pairwise case, the TKF92 model can be represented as an HMM using the formulation described in Miklós et al. (2008). This HMM formulation allows one to sum over all possible pairwise alignments in [MATH] time using the HMM forward-backward algorithm, where [MATH] and [MATH] are the respective lengths of the two... |
We implemented a modified version of this HMM implementation (Figure 11 ) such that each emitted pair of characters is drawn from one of [MATH] evolutionary hidden states (Figure 11 ). Additionally, we encode neighbouring dependencies amongst evolutionary hidden states along the alignment, by specifying a probability t... |
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