Evolutionary simulations based on a Z-score model introduce rate variation across sites in a natural way by incorporating structural and functional constraints specific for a protein family 21. The simulation procedure is a Monte Carlo simulation of the amino acid substitution process. The fixation of substitutions is dictated by a simple scoring function, which is derived from the template structure and an alignment of its homologs. The number of substitutions occurring at each site can be recorded during the simulation process and the observed α at a site equals the number of recorded substitutions at that site divided by the average substitution number for all sites. To reduce sampling variance, an average observed α vector is calculated from 100 simulations.

For the alignment consisting of all the leaf node sequences generated by the simulation process, an α_AB vector was calculated according to equation (11) (for details see Methods). An average α_AB vector was derived from 100 simulations. Correlation coefficient between the average α_AB vector and the average observed α vector was high (data not shown). However, we found that for large observed α values, the corresponding α_AB values were smaller. A constant β was introduced to correct this underestimation in equation (11).

Here, α_i is Alignment-Based rate factor at site i. K is the number of sites in a given alignment. C_i is the value assigned to site i (for details see Methods).

We optimized the β value by fitting the average α_AB vector and average observed α vector to y = x line. Alignments for three different protein families (trypsin, carboxypeptidase and pdz domain) gave a good empirical estimation for β of about 1.3. The relation between this corrected average α_AB vector and average observed α vector is shown in Figure 1a for a typical example, the pdz domain (correlation coefficient 0.973).

We also estimated an α_ML vector for each alignment generated from the simulation (for details see Methods). The average α_ML vector shows good correlation with the average observed α vector (Figure 1b) (correlation coefficient 0.945). α_AB or α_ML can be incorporated in likelihood calculation in marginal or joint reconstruction. Table 1 shows that improvement of logarithm likelihood of the alignment is significant when α_AB or α_ML is used.

Rate variation across sites can be modeled by assuming that the rate factors follow a certain type of statistical distribution. Gamma distribution 1327 and its discrete approximations 28 are frequently used for DNA or protein sequences. Rate variation for a protein family reflects different selective pressure at different sites to maintain structure and function. Fewer substitutions are expected to occur in more conserved sites. This hypothesis has prompted us to estimate rate factors (α_AB) based on sequence conservation in an empirical way. The α_AB is compared and calibrated using the observed α as standards. Our method of estimating α_ML is similar to the one proposed by Nielson 18. One problem with site-by-site rate factor estimation is the small sample size at each site, especially with a small alignment. We have used α_AB to eliminate outliers with very large α_ML estimates (for details see Methods).

Evolutionary distances tend to be underestimated when rate homogeneity among sites is assumed 22. This was tested using the simulation with structural and functional constraints. For the arbitrarily selected tree shown in Figure 2, we obtained leaf node sequences in the simulation and estimated an evolutionary distance for each sequence pair by Maximum Likelihood, either incorporating α_AB or setting α equal to 1.0 (equation (16)). Evolutionary distances were severely underestimated (average underestimation: 0.894) without considering rate variation among sites (Figure 3a). Introducing α_AB in the maximum likelihood method gave more accurate distance estimation (Figure 3b), although the distances were still underestimated, especially for small distances (average underestimation: 0.286). We believe that more accurate distances will give more accurate phylogeny reconstruction using "Weighbor" 10. Since a tree is required to estimate α_ML, α_ML is not incorporated in estimating evolutionary distance.

A continuous minimization method by simulated annealing was used to optimize the equilibrium frequency vector π, with the objective function being the logarithm likelihood of the alignment. Our π vector optimization program was tested on four alignments, which were taken from the SH2 and SH3 superfamilies in Pfam database (version 7.3) 29. Two alignments from the SH2 superfamily have 44 and 87 sequences respectively and both alignment lengths are 83 amino acids (including gaps). The other two alignments from SH3 superfamily have 39 and 94 sequences respectively and both alignment lengths are 57 amino acids (including gaps). For each alignment, we ran optimization 3 times starting from different random initial points. The optimized π vectors did not converge to exactly the same point, but they had a high correlation with each other (always > 0.95) and the difference of logarithm likelihood function values was small (less than 0.1%). The logarithm likelihood of the alignment, using optimized π vector, increased slightly, but significantly (Table 2), compared with the logarithm likelihood using the π vector calculated from the alignment.

Optimization of the π vector is time consuming. The running time for reconstruction with or without optimizing π vector is 14,902 seconds and 213 seconds for SH2 alignment (44 sequences), respectively, on a Dell PowerEdge 8450 server (CPU 700MHz, RAM 8G) (Table 2). In our program, the default π vector is calculated from the alignment while the user has the option to optimize the π vector for ancestral sequence reconstruction.

Two different methods for simulations of the evolutionary process were used, as described in Methods, to test the reliability of the reconstruction results. In the first simulation method, starting from a randomly generated root sequence, we simulated the evolutionary process to obtain leaf node sequences based on a tree and a rate matrix. This process was repeated 100 times for a given root sequence R to produce 100 alignments consisting of all leaf node sequences. For each of the 100 alignments, we used the marginal reconstruction method to obtain an amino acid probability vector for each site at the root. To reduce sampling variance, the amino acid probability vector was averaged over the 100 simulation trials. At each site, the amino acid with the highest average probability was chosen as our result of the "reconstructed amino acid" at that site. All "reconstructed amino acids" formed the reconstructed sequences R'. There is no difference between R and R', that is, the accuracy of reconstruction is 100% for the tree shown in Figure 2. For each individual simulation and its reconstruction, we checked the amino acid with the highest probability in the reconstructed probability vector of the root. If it is indeed the "reconstructed amino acid", the prediction for that simulation is correct according to the average reconstructed results. The fraction of individual predictions that are correct according to the average reconstructed results is almost always higher than the average probability of the "reconstructed amino acid", suggesting that the average probability of the "reconstructed amino acid" gives a lower estimation of the reconstruction reliability (Figure 4a).

For the second simulation method, we introduced rate heterogeneity across sites with structural and functional constraints 21. For the same tree, the accuracy of reconstruction was about 90%. Sites with larger substitution rates are expected to have less reliable reconstructions. Figure 4b shows the relationship between the average α_AB and the fraction of individual predictions that are correct according to the "reconstructed amino acid". Sites with incorrect "reconstructed amino acids" all have large α_AB values. These values reflect the difficulty of reconstructing sites with large numbers of substitutions. The probabilities of the "reconstructed amino acids" are all small for sites with incorrect reconstructions (less than 0.15), suggesting that the information content of the reconstruction is low.

The second simulation method was also used to test ANCESCON along with the reconstruction programs from PAML 30, PHYLIP 31 and PAUP* 32. All tree topologies used in reconstruction tests were inferred from real alignments. All original root sequences were taken from PDB database 33. We had three different types of alignment testing sets. The first testing set used the same tree topology but different root sequences to generate 100 alignments (for details see Methods). The second testing set used the same root sequence but different tree topologies. The third testing set randomly selected a root sequence and a tree topology to generate 100 alignments. After 100 alignments were generated, we reconstructed the root sequence for each alignment and found the consensus root sequence for the 100 reconstructed root sequences. Finally, the consensus root sequence was compared with the original root sequence to calculate the reconstruction accuracy, i.e. the fraction of correctly reconstructed sites for the root sequence. In addition, for the third test, the paired t-test was used to calculate the one-tail probability between ANCESCON and other three methods. In order to make different tree topologies comparable, those trees were scaled to make the average distance from root to all leaf nodes (d_a) the same for all trees and equal to the tree of pii1 (a signal transduction protein) (d_a = 4.23). If d_a was too small (e.g. 0.5), the reconstruction accuracy was always close to 1 for all reconstruction methods used. The value d_a = 4.23 was large enough to generate diverse sequences to differentiate 4 different ancestral sequence reconstruction methods.

For ANCESCON we had 3 different parameter settings, which included site-specific rate factors estimated by maximum likelihood method (α_ML), Alignment-Based rate factors (α_AB) and no rate factors (equal rates among sites). Different parameters were also used for the reconstruction programs from PAML and PHYLIP to find their best reconstructions. For PAML, reconstruction was tested with parameter α (rate factor) estimated from alignment and without α. For PHYLIP, 4 different parameter settings were tried, which were combinations of with/without α estimated from alignment by PAML and with/without branch length dwelling in input tree topology. For PAUP*, default settings were used.

Table 3 shows a comparison of the reconstruction accuracy for these 4 methods. The reconstruction accuracy of ANCESCON with α_ML is higher than the other three methods in almost every test. Also the reconstruction accuracy of ANCESCON with α_AB and without α is comparable with PAML and PHYLIP methods and is much better than PAUP*. For the first testing set, the best average accuracy for ANCESCON is about 0.5, while the best average reconstruction accuracies for PAML, PHYLIP and PAUP* are 0.45, 0.39 and 0.32 respectively. Testing set 2 and 3 produce similar results. Using the paired t-test in the third testing set, we show that ANCESCON method with α_ML gives significantly better reconstruction than the other 3 methods. Because the site-specific α_ML is very close to the true mutation rate at a site (Figure 1b), using the site-specific α_ML can improve our ability to reconstruct the amino acids for ancestral sequences correctly. These reconstruction tests suggest that ANSCESCON may be a better tool to reconstruct ancestral sequences compared to PAML, PHYLIP and PAUP* if the given alignment contains more diverse sequences.

Thirty-eight OB (Oligonucleotide/oligosaccharide binding)-fold 34 proteins and ten other alignments (adenylyl kinase, gef, globin, pdz, ph, ptb, ras, sh2, sh3 and subtilase) from the Pfam database (version 7.3) 29 were chosen to perform homology detection tests.

Given an alignment with N sequences, we had four different methods, "BEST", "SECOND BEST", "SHUFFLE" and "RANDOM", to generate another N-1 sequences (for details see Methods). For each combined alignment (2N-1 sequences), PSI-BLAST 35 searches were performed starting from each sequence and seeded with the combined alignment (-B option in the program BLASTPGP, e-value cutoff 0.01), and all found hits were pooled together.

The benchmark experiment was PSI-BLAST seeded with the native alignment (N sequences). For each type of the four combined alignments, we checked hits not found by the native alignments. New hits were verified to be true positives or false positives by running PSI-BLAST or HMMER 36, followed by manual inspections.

Using the 48 native alignments, a total of 13973 hits were found by the benchmark. Compared to the benchmark, the "BEST" method detected 120 new homologs and the other three methods found 69, 74 and 9 new homologs, respectively (Figure 5). Among those new homologs, "BEST", "SECOND BEST", "SHUFFLE" and "RANDOM" methods had 3, 2, 6 and 3 false positives, respectively (Figure 5). Also, "BEST", "SECOND BEST", "SHUFFLE" and "RANDOM" methods missed 61, 1070, 60 and 7811 homologs as compared to the benchmark.

Adding non-native sequences to the native alignment results in a change of sequence profile for PSI-BLAST searches. Random sequences can dilute the position-specific amino acid exchange characteristics of native alignments. This effect should not improve the profile. Indeed, few new homologs are found by the "RANDOM" method. However, sequences generated by shuffling each position of the native alignment have the same conservation properties as the native alignment, and the "SHUFFLE" method detects a total of 74 new homologs. Two effects may account for this finding. First, addition of shuffled sequences to the native alignment can slightly change the estimates of pseudocount frequencies of amino acids and thus the position specific scoring matrix 35. Second, the new version of PSI-BLAST program uses composition-based statistics with e-value estimation related to the composition of the query sequence 37. Each shuffled sequence has its own amino acid composition that is different from the native sequences. This difference can affect the e-values of hits. The "BEST" method detects the most number of new homologs, suggesting that the reconstructed ancestral sequences resemble the native sequences. Ancestral sequences may therefore be more similar to some remote homologs than to the native sequences. The "SECOND BEST" method detects less new homologs than the "BEST" method but more than the "RANDOM" method, suggesting that the second most probable amino acids in reconstruction can still reflect some properties of native sequences. Table 4 shows homology detection results of OB-fold structures using reconstructed ancestral sequences.

Ten well-studied protein families (adenylyl kinase, gef, globin, pdz, ph, ptb, ras, sh2, sh3 and subtilase) from the Pfam database (version 7.3) 29 were selected to test the prediction of functional sites. To define functional sites, we considered residues falling within 5Å of any ligand to be functionally important (i.e. AP5 for adenylyl kinase). As a simple quantification of prediction accuracy, we counted the number of predictions that lie within 5Å from the ligands and consider these sites to be true positives.

Our method intends to identify those sites with high similarity within individual sub-trees and high variation among sub-trees. These sites are likely to contribute to functional specificity. Based on a tree partition and the reconstructions at the cutting nodes (details see Methods), we have developed a measure called specificity score (equation (27)). We expect that both highly variable sites and highly conserved sites tend to score low in our method. Ten top-ranking sites were selected as our predicted functional sites for each family. For comparison, we also implemented a simple conservation (SC) method 25, the evolutionary trace (ET) method 26 and the conservation difference (CD) method 21 on the 10 protein families. The results are shown in Table 5. Here, the results from these three methods tend to include invariant or highly conserved sites while the result from our method scores those sites low. Still, the number of true positives of our method is comparable to other methods for several families. For some protein families, such as gef, pdz and subtilase, our method predicts no fewer functional residues than the other three methods.

Figure 6 shows the mapping of our predictions on the structure for the PDZ domain family. In green color is the ligand and in red color are the functional residues predicted by our method. Six of the predicted residues are within 5Å to the peptide ligand. Nine of the predicted residues are around the ligand binding area. Only one is distant from the ligand (Figure 6).

Another example is the family of adenylyl kinases. Our method identified 3 residues within 5 Å to the ligand while the other 3 methods identified more such residues, most of which are in highly conserved positions such as the catalytic residues. Highly conserved residues, however, are not selected by our method since our measure is designed to emphasize on sites contributing to specificity. Figure 7 shows the N-terminal part of the alignment of adenylyl kinases, with our predictions highlighted in red and orange colors. The evolutionary tree for the alignment is shown in Figure 8. The first cutting layer (for details see Methods) results in two well-separated sub-trees. Functional annotations suggest that they contain enzymes with different substrate preferences: adenylyl kinases and uridylate kinases, respectively. Three residues (27, 54 and 89) from our predictions (red colored in Figure 9) contribute to substrate-binding specificity, as have been noted before in the structural studies of the UMP kinases 38. Figure 9 highlights our predicted functional residues on the adenylyl kinase protein structure. Most of our predictions fall within the specificity pocket.

For all models discussed in this paper, we assume all sites in an alignment evolve independently and according to a homogeneous, stationary and time reversible Markov process. The probability of an amino acid i to be replaced by amino acid j after a time interval t is P_ij(t). The transition probability matrix of 20 amino acids is written as P(t), which can be calculated as

P(t) = exp(Qt)     (2)

Here, Q is the rate matrix. The non-diagonal elements q_ij are the instantaneous rates of change from amino acid i to amino acid j and diagonal elements q_ii are such that each matrix row sums up to 0. Q can be calculated by:

Q = S* diag(π)     (3)

S is the matrix of amino acid exchangeability parameters 39. π_i is the equilibrium frequency for amino acid i. Time reversibility implies that S is a symmetric matrix. In our program, the S matrix is taken from Whelan and Goldman 39 and the default π vector is estimated from the given alignment.

Q can be decomposed into eigenvalues (λ_i) and eigenvectors (u_i).

U = (u₁, ..., u₂₀)     (5)

P_ij(t) can be calculated using the following equation,

The likelihood function 8 for an evolutionary tree T shown in Figure 10 is:

Here, is the equilibrium frequency of the amino acid at the node A. is the transition probability from the amino acid at node A to the amino acid at node B after an evolutionary distance d_AB.

Considering that each site i has a rate factor α_i 1318, we have:

t in equation (6) can be expressed as:

t = α·d     (9)

d is the evolutionary distance and α is rate factor. The following restriction on the vector α holds:

Here, K is the number of sites.

Our program supports two methods to estimate a rate factor for each site: Alignment-Based rate factor α (α_AB) and Maximum Likelihood-estimated rate factor α (α_ML).

The estimation of α_AB is empirical and based on the observation that the substitution rate at a site is correlated with the conservation of the site, which, in turn, is correlated with the average transition probability among the amino acids at that site. Conserved sites are dominated by highly similar amino acids and thus have high average transition probabilities among the amino acids. The algorithm to calculate α_AB is as follows:

1. Set t equal to 1.0 and use equation (6) to calculate a transition probability matrix P for 20 amino acids. Equation, , is used to compute a symmetric matrix P'.

2. Calculate the average transition probability for each site and take the reciprocal: , where is the number of non-gapped amino acid pairs in site i and the denominator is the sum over the transition probabilities between all amino acid pairs (j,k) at a site i.

3. For invariant sites, C_i is set to 0 to make it consistent with the Maximum Likelihood estimation.

4. Equation (11) is used to calculate α_AB, so that equation (10) holds.

If an evolutionary tree is assumed for the alignment, we can estimate the α_ML factors by maximizing the likelihood (equation (8)) for each site:

If some sites are highly variable, the α_ML at those sites can be very large, as has been previously noticed 18. We consider these rate factors to be outliers. For these sites, we have observed that likelihood changes very little over a wide range of the α values. An empirical method is used to reduce the values of α_ML outliers, guided by the α_AB values. a Z-score of the ratio of α_ML to α_AB is calculated for each site except invariant sites:

Here, is the ratio of to for site i; is the number of sites excluding the invariant sites. If Z_i is greater than 3, it is reduced to 3 by decreasing the value of . We repeat this procedure until no Z_i for any site i is greater than 3. After removing the outliers, we scale the values so that equation (10) holds.

Two methods are implemented to estimate the equilibrium frequency vector π, one derived directly from the given alignment (Alignment-Based π or π_AB) and the other estimated by Maximum Likelihood (π_ML). The likelihood for the entire alignment is a function of π with 19 variables. A continuous minimization method by simulated annealing 40 is used to optimize π, with the objective function being the logarithm likelihood of the alignment. The simulated annealing is computationally intensive and is the major reason for the long CPU time given in Table 2.

A Maximum Likelihood approach is used to estimate the evolutionary distances among sequences, either considering rate variation across sites or not. The logarithm likelihood for replacing one protein sequence (A) with another protein sequence (B) after an evolutionary distance d can be written as:

Here, is the equilibrium frequency for the amino acid at site j in sequence A. is the transition probability from amino acid at site j in sequence A to amino acid at site j in sequence B after an evolutionary distance α _j·d. α_j is 1 if all sites are assumed to evolve at the same rate; otherwise the α_AB at site j is used for α_j.

An estimate of the evolutionary distance between two sequences is obtained by maximizing the likelihood function of equation (15):

Equation (16) can be solved by the bisection root-finding method 40.

After the distance matrix is calculated, the "Weighbor" method, i.e. weighted neighbor joining, is used to infer an evolutionary tree 10.

Two methods are implemented to reconstruct ancestral sequences. One is a marginal reconstruction method 4, and the other is a joint reconstruction method 5. Below are their brief descriptions.

We calculate P(A_r|{A_l}T), which is the conditional probability of amino acid A_r at the root, given leaf node amino acid set {A_l} and a tree T. Since time reversibility is assumed, any internal node can serve as a root. Using Bayes' theorem, we have:

Here, P(A_r) is used here instead of P(A_r|T) because the frequency of the root amino acid A_r, i.e. π_r, does not depend on tree T. P({A_l}|A_rT) is the conditional probability of the known amino acids at the leaf nodes, given T and A_r. P({A_l}|T) does not depend on A_r, so it is calculated as a normalization constant for P(A_r|{A_l},T) terms over all 20 possible values of A_r to make the sum equal to 1.

For Figure 10, P({A_l}|A_rT) can be expanded as:

Here, is the transition probability from amino acid at node A to amino acid at node B after an evolutionary distance d_AB. Equation (18) can be calculated using a recursive method suggested by Felsenstein 8.

If rate factors are used in the reconstruction of the root sequence, we have:

Here, α_i could be either α_AB or α_ML at site i. P(A_C,A_D,A_E,A_F | A_A,T)_i is the conditional probability P(A_C,A_D,A_E,A_F | A_A,T) at site i.

The objective of a joint reconstruction method is to find the combination of amino acids for an internal node set {A_i} that maximize the conditional probability of this amino acid combination, given the leaf node amino acid set {A_l} and a tree T, P({A_i}|{A_l},T). Using the Bayes' theorem, we have:

Because P({A_l}|T) is the same for all amino acid combination at internal node set {A_i} this problem becomes finding the maximum of P({A_l}|{A_i},T) *P({A_i}).

The details of a fast algorithm to solve equation (20) can be found in Pupko et al. 5. We also incorporated site-specific rate factors in this algorithm, in a similar way as equation (19)

Due to difficulties with the probabilistic models of gaps, a simplified empirical approach is used to alleviate the problem. We assume that gaps are "supersede" letters. Gaps are considered for each site independently. If a leaf node has a gap instead of an amino acid at a site, this node will be deleted from the tree for this site. After dealing with leaves, we check all internal nodes for children. If an internal node has no children or only one child due to the leaf removal because of gaps, it will be removed from the tree and a gap will be assumed as its reconstructed state.

Two methods of simulating amino acid substitution process were used to test the reliability of reconstruction, rate factors and evolutionary distance estimation. The first simulation method was based on a homogeneous time reversible Markov model. The parameters from Whelan and Goldman 39 were chosen for our model, including the equilibrium frequency vector π and the S matrix. Given the length of a branch from a parent node to one of its child nodes and the amino acid for the parent node, we simulated the substitution process to generate an amino acid for the child node based on the transition probabilities that were calculated using equation (6). For the arbitrarily selected tree shown in Figure 2, we first generated a random sequence of 100 amino acids as the root sequence based on the amino acid frequencies from Whelan and Goldman 39. We then simulated the random substitution process to obtain all leaf node sequences. This simulation was repeated 100 times. The resulting 100 alignments were used to test the reliability of the reconstruction result. In this simulation, each site evolved independently according to the same tree topology and branch lengths, thus there was no rate heterogeneity across sites.

The second simulation method, based on a Z-score model, introduced rate variation across sites by using structural and functional information for a specific protein family 21. We selected three protein families for the Z-score simulations under structural and functional constraints: pdz domain (Protein DataBank (PDB) ID: 1g9o) 41, trypsin (PDB ID: 1sgt) 42 and carboxypeptidase A (PDB ID: 2ctb) 43. Given a rooted tree, the native sequence with known structure was used as the root sequence. Simulations were made along the tree to generate sequences at any internal node or leaf node. If the evolutionary distance from a parent node to a child node was d, the child sequence was obtained after l*d accepted substitutions starting from the parent sequence, where l is protein sequence length. Simulations of the substitution process were repeated 100 times. For each site, the number of accepted substitutions was recorded and averaged over 100 simulations. Rate factors (observed α), representing site mutability, were calculated from these average substitution numbers, such that the average of rate factors is 1 (equation (10)). 100 simulated alignments were used to test the rate factor estimators (α_AB and α_ML), distance calculation methods and ancestral sequence reconstruction.

Our objective is to find sites that are well conserved within each sub-tree, but show high variability between different sub-trees. These sites are likely to contribute to functional specificity 264546.