On the group theoretical background of assigning stepwise mutations onto phylogenies

Recently one step mutation matrices were introduced to model the impact of substitutions on arbitrary branches of a phylogenetic tree on an alignment site. This concept works nicely for the four-state nucleotide alphabet and provides an efficient procedure conjectured to compute the minimal number of substitutions needed to transform one alignment site into another. The present paper delivers a proof of the validity of this algorithm. Moreover, we provide several mathematical insights into the generalization of the OSM matrix to multi-state alphabets. The construction of the OSM matrix is only possible if the matrices representing the substitution types acting on the character states and the identity matrix form a commutative group with respect to matrix multiplication. We illustrate this approach by looking at Abelian groups over twenty states and critically discuss their biological usefulness when investigating amino acids.

called character [5], which is sometimes also called site pattern [2].Given a phylogenetic tree and an alignment that evolved along the tree, Klaere et al. [1] showed, for binary alphabets, how a character changes into another character if a substitution occurs on an arbitrary branch of the tree.The impact of such a substitution is summarized by the so-called One Step Mutation (OSM) matrix.Nguyen et al. [2] extended the concept of the OSM matrix to the four-state nucleotide alphabet while developing a method to evaluate the goodness of fit between models and data in phylogenetic inference.There, the OSM matrix is constructed based on the Kimura three parameter (K3ST) substitution model [6].Nguyen et al. [2] illustrated how one can use Maximum Parsimony (i.e.apply the Fitch algorithm [7]) to compute the minimal number of substitutions required to change one character into another character under the OSM setting.In the present paper, we deliver a proof for this transformation.
In addition, the OSM matrix can be constructed only if the substitution matrices and the identity matrix form a commutative or Abelian group (see, e.g., [8,9]) with respect to matrix multiplication [2].We generalize the construction of the OSM matrix for any alphabet.Moreover, we show that the number of substitutions needed to convert one character into another may change if we use different groups.Finally, we provide a means to find an Abelian group for the twenty-state amino acid alphabet.

Notation and Problem Recapitulation 2.1 Notation
Recall that a rooted binary phylogenetic X-tree is a tree T = (V(T ), E(T )) with leaf set (also called taxon set) X = {1, . . ., n} ⊂ V(T ) with only vertices of degree 1 or 3 (internal vertices), where one of the vertices of degree 1 is defined to be the root, and all edges are directed away from it.In this paper, when there is no ambiguity we often just write "phylogenetic tree" or "tree" when referring to a rooted binary phylogenetic tree.Also, when referring to a tree on a leaf set X with |X| = n, we write n-taxon tree for short.Furthermore, recall that a character f is a function f : X → C for some set C := {c 1 , c 2 , c 3 , . . ., c r } of r character states (r ∈ N).We denote by C n the set of all r n possible characters on C and n taxa.For instance, for the four-state DNA alphabet, C DN A = {A, G, C, T} and the set C n consists of 4 n elements.An extension of f to V(T ) is a map g : V(T ) → C such that g(i) = f (i) for all i in X.For such an extension g of f , we denote by l T (g) the number of edges e = {u, v} in T on which a substitution occurs, i.e.where g(u) = g(v).The parsimony score of f on T , denoted by l T ( f ), is obtained by minimizing l T (g) over all possible extensions g.Given a tree T and a character f on the same taxon set, one can easily calculate the parsimony score of f on T with the famous Fitch algorithm [7].Moreover, when a character state changes along one edge of the tree, we refer to this state change as substitution or mutation.As for our purposes only so-called manifest mutations are relevant, i.e. those mutations that can be observed and are not reversed, we do not distinguish between mutations and substitutions, which is why we use these terms synonymously.

Construction of the OSM matrix
We now introduce the OSM framework in a stepwise fashion.The aim of the OSM approach is to determine the effects a single mutation occurring on a rooted tree T has on a character evolving on that tree.
The first task of this approach is to formalize the term mutation and its effects on a single character state in C. A mutation is an operation σ : C → C which is bijective, i.e. it satisfies the following condition: This guarantees that a mutation affects a character state in a unique fashion.It is well-known that any bijective operation on a finite discrete state set is isomorphic to a permutation (e.g., [10]).Therefore, in the following we consider mutations to be permutations.The next step is to establish which permutations we consider admissible in a model.In other words, we next establish conditions on the set Σ of permutations acting on C.
2. For every pair c i , c j ∈ C there is exactly one operation σ ∈ Σ such that σ(c i ) = c j .This guarantees that every character state change can be observed within a single step and that we do not have any ambiguity.If Σ contains the identity, i.e. the mapping σ 0 such that σ 0 (c i ) = c i for all c i ∈ C, then all other permutations in Σ are fix-point free due to Condition 2. Condition 2 also implies that Σ contains exactly r permutations, where r is the number of character states in C. If Σ had more permutations then for all states c i ∈ C there would be a pair of distinct permutations σ 1 , σ 2 ∈ Σ such that σ 1 (c i ) = σ 2 (c i ), which would lead to ambiguity.Condition 2 also concludes that we exclude GTR [11] from the set of admissible models.However, we explain this more in-depth in Section 3.3.
We add some more useful conditions which give Σ a very convenient structure: 3. For all σ 1 , σ 2 ∈ Σ also the product σ 1 • σ 2 ∈ Σ.In other words, Σ is closed with respect to concatenation of its permutations.

For all
Thus, Σ is commutative, and hence the order in which we assign permutations is irrelevant for the outcome.

5.
There is an element σ 0 ∈ Σ such that for all σ 1 ∈ Σ we have σ 1 • σ 0 = σ 0 • σ 1 = σ 1 .As pointed out above, including the identity guarantees that all other permutations will force a state change, a feature which led to the name "One Step Mutation".6.For every σ 1 ∈ Σ there is a σ 2 ∈ Σ such that σ 1 • σ 2 = σ 0 .The existence of such an inverse element guarantees that every operation can be reversed within a single step, which is quite a useful property.

For all
Associativity is needed to enforce a group structure on Σ.
All of these conditions taken together imply that Σ forms an Abelian group of r permutations.From now on we use the matrix form of permutations for illustration of the operations.A permutation matrix σ over C is represented by an r × r matrix such that σ c i c j = 1 if σ(c i ) = c j , and 0 otherwise.In that case, a concatenation "•" is equivalent to the matrix multiplication "•".
Example 2.1.In genetics, the most commonly used character state set is C DNA = {A, G, C, T}.There are two different Abelian groups for four states, namely the Klein-Four-group Z 2 × Z 2 and the cyclic group Z 4 .The Klein-Four-group is constructed from the cyclic group over two elements, identity (τ 0 ) and flip (τ 0 ).These take the matrix form The Klein-Four-group consists of the four Kronecker products of these two matrices, i.e. s 0 = τ 0 ⊗ τ 0 , s 1 = τ 1 ⊗ τ 0 , s 2 = τ 0 ⊗ τ 1 , and s 3 = τ 1 ⊗ τ 1 .In full, they take the form: This construction coincides with the K3ST model of substitution, where s 1 describes transitions within purines (A, G) and pyrimidines (C, T), s 2 represents transversions within pairs (A, C) and (G, T), and s 3 represents the remaining set of transversions within pairs (A, T) and (C, G).
The cyclic group is given by the permutation set Note that the cyclic group Z 4 has a different interpretation with a different ordering of the nucleotides.E.g., our matrix The cyclic group associated to the latter rotation [12] is linked to the K2ST substitution model [13], where s 2 corresponds to the transition within purines and pyrimidines, and s 1 and s 3 are the (not further distinguished) transversions.
The next step in constructing the OSM matrix is to construct a set Σ T of operations over C n governed by T , and based on the permutation set Σ. To this end, we first define Σ n as a set of operations which work This can also be described by the Kronecker product, i.e. equally This means that there are r n different operators in Another noteworthy consequence of using the Kronecker product is that the elements of Σ n are permutations over C n [14,15], and in fact Σ n satisfies our Conditions 1-7, i.e.Σ n is an Abelian group over In the OSM framework we assume that the permutations acting on a character f ∈ C n are derived from the underlying rooted tree T .In particular, regard the pendant edge to a Taxon 1 ∈ X.A permutation on this edge will only affect the character state of f 1 .Therefore, the edge implies permutations of type This construction works analogously for all pendant edges, with all but one factor being the identity σ 0 , while the one non-identity factor is one of the remaining r − 1 permutations in Σ.A permutation at an interior edge affects the character states of all its descendants, i.e. those taxa whose path to the root passes that edge.E.g., assume Taxa 1 and 2 form a cherry, i.e. their most recent common ancestor has no other descendants, and permutation σ i ∈ Σ, i = 1, . . ., r − 1 is acting on the edge leading to this ancestor.Then, we get the permutation The right hand side equation shows that a single permutation on an internal edge has the same effect as simultaneously applying the same permutation on the pendant edges of all descendant taxa.This also shows that the set Σ X of all permutations on the pendant edges is a generator of Σ n , i.e. the closure of Σ X contains all permutations in Σ n .Since Σ n contains a single permutation to transform character f ∈ C n into g ∈ C n , and since Σ X generates Σ n , there is a shortest chain of permutations in Σ X which transforms f into g.Σ X is also the set of permutations implied by the star tree for X.For every X-tree T we have Σ T ⊇ Σ X , and therefore Σ T is a generator for Σ n , too.An illustration of such a generator set Σ T over the character set C n is the so-called Cayley graph [16], which has as vertices the elements of C n , and two elements f , g ∈ C n are connected if there is a permutation σ ∈ Σ T such that σ( f ) = g.In [1] Cayley graphs have been presented as alternative illustrations of the tree T over a binary state set C.
We are now in a position to recall the definition of the OSM matrix M T for a rooted binary phylogenetic tree T as explained in [1] and [17].For an edge e ∈ E(T ) we denote by p e the relative branch length of e, i.e. its actual length divided by the length of T .Thus, one can view p e as the probability that a mutation is observed at edge e given that a mutation occurred on T .Clearly, ∑ e∈E(T ) p e = 1.Further, denote by α e,i the probability that this mutation on e is of type i ∈ {1, . . ., r − 1} with ∑ r−1 i∈1 α e,i = 1 for all e ∈ E(T ).Then the OSM matrix is the convex sum of the elements in Σ T , where each permutation σ e,i is multiplied by p e α e,i , the probability of hitting the edge e with permutation σ i ∈ Σ.Thus, we obtain: ( M T can be regarded as the weighted exchangeability matrix for all characters given that a substitution occurs somewhere on the tree T .Figure 1(e) depicts the OSM matrix for the tree in Figure 1

The transformation problem
With the construction of Σ T we have generated the tools needed to formally describe the computations in Step 4 of the MISFITS method introduced by Nguyen et al. [2].Given a rooted tree T and two characters f and f d in C n , we want to compute the minimal number of substitutions required on the tree to convert f In our framework this corresponds to finding the smallest number k of permutations σ 1 , . . ., Assume that the character GTAGA is to be converted into character ACCTC.By comparing the two characters positionwise, we need a substitution s 1 on the external branch leading to Taxon 1 to convert G into A at the first position.Similarly, we need a substitution s 1 on the external branch leading to Taxon 2, and a substitution s 2 on every external branch leading to Taxa 3, 4, and 5. Thus, the operation s := (s 1 , s 1 , s 2 , s 2 , s 2 ) transfers the character GTAGA into the character ACCTC.As the operation s := (s 1 , s 1 , s 2 , s 2 , s 2 ) also translates the constant character AAAAA into GGCCC, converting GTAGA into ACCTC is equivalent to evolving the character state A at the root along the tree to obtain the character GGCCC at the leaves.The Fitch algorithm applied to the character GGCCC with the constraint that the character state at the root is A produces a unique most parsimonious solution of two substitutions as depicted by Figure 3(c).

The impact of parsimony on the estimation of substitutions.
In this section, we provide some mathematical insights into the role of Maximum Parsimony in the estimation of the number of substitutions needed to convert a character into another one as explained in Section 2.3.In particular, we deliver a proof for Algorithm 1.
Theorem 3.1.Let T be a rooted binary phylogenetic tree on taxon set X and let f be a character that evolved on T due to some evolutionary model and let f d be another character on X.Then, the number of substitutions to be put on T which change the evolution of f in such a way that f d evolves instead of f can be calculated with Algorithm 1.
Proof.Let f , f d , X, T and Σ be as required for the input of Algorithm 1.Then, the number of substitutions needed to evolve f d on T rather than f depends solely on operation σ.In order to see this, note that σ describes an explicit way to translate f into f d step by step, i.e. for each taxon seperately.The basic idea now is that in order to minimize the number of required substitutions, we need to consider the underlying tree T , as this may allow a single substitution to act on an ancestor of taxa that undergo the same substitution type rather than on each taxon separately.This idea has been described above in Section 2.2, and it coincides precisely with the idea of the parsimony principle.
However, in order to avoid confusion regarding the operation σ as a character on which to apply parsimony, Algorithm 1 instead acts on the constant character.Clearly, in order to evolve the constant character h := c 1 • • • c 1 on a tree with root state c 1 , the corresponding operation would be σ := σ 0 • • • • • σ 0 .If instead of σ we let σ act on h, the resulting character c will differ from h in the same way f d differs from f .Note that two character states in c are identical if and only if the corresponding substitutions in σ are identical, too.Therefore, it is possible to let MP act on c rather than directly on σ.
By the definition of Maximum Parsimony, when applied to c on tree T with given root state c 1 , it calculates the minimum number m of substitutions to explain c on T .This number m is therefore precisely the number of substitions needed to generate c on T rather than h.But as explained before, c and h are by definition related the same way as f and f d .Therefore, m also is the number of substitutions needed to generate f d on T rather than f .This completes the proof.

The impact of different groups
For any alphabet, there might be more than one Abelian group.Different groups might result in different numbers of substitutions required to translate a character into another character.We illustrate this in the following examples.For the four-state nucleotide alphabet there are two Abelian groups, namely the Kleinfour group and the cyclic group (see above).The cyclic group Σ c consists of the identity matrix s 0 and the three substitution types s 1 , s 2 , s 3 depicted by Figure 4(a).Hence, Σ c = {s 0 , s 1 , s 2 , s 3 }.Under Σ c , we note that a substitution type which changes a character state c i to c j does not necessarily change c j to c i .Example 3.2.Assume the rooted five-taxon tree in Figure 4(b) and the character GTAGA at the leaves, which is to be converted into character ACCTC.The tree and the two characters are the same as in Example 2.3.By comparing the two characters positionwise under the group Σ c , we need a substitution s 3 (depicted in blue in Figure 4(a)) on the external branch leading to Taxon 1 to convert G into A at the first position.Analogously, we need a substitution s 1 on the external branches leading to Taxon 2 and to Taxon 4, and a substitution s 3 on the external branches leading to Taxon 3 and to Taxon 5. Thus, the operation s := (s 3 , s 1 , s 3 , s 1 , s 3 ) transfers the character GTAGA into the character ACCTC.As the operation s also translates the constant character AAAAA into CGCGC, converting GTAGA into ACCTC is equivalent to evolving the character state A at the root along the tree to obtain the character CGCGC at the leaves.The Fitch algorithm applied to the character CGCGC with the constraint that the character state at the root is A produces a unique most parsimonious solution of three substitutions as depicted by Figure 4(c).
Thus, under the Σ c group we need one substitution more than under the Σ K3ST group.
Note that variation of the minimum number of substitutions needed to translate a character into another one between different groups is not surprising: As different substitution types are needed to translate one pattern into the other one, depending solely on the underlying group, one group might need the same substitution type for some neighboring branches in the tree and another group different ones.Informally speaking, this would imply that in the first case, the substitution could be "pulled up" by the Fitch algorithm to happen on an ancestral branch, whereas in the second case this would not be possible.

The link between substitution models and permutation matrices
In Examples 2.1 and 2.2 we have shown that the K3ST substitution model can be included into our framework.This section aims at discussing alternative models and how to identify their use (or lack thereof) for our approach.The set Σ T contains a set of permutations which act on the characters in C n .
Most substitution models assume the independence of the different branches of a tree to compute the joint probability of the characters in C n .Therefore, they use the probabilities for substitutions among the character states in C along the edges of the tree T .We now establish a probabilistic link between Σ T and C n .This link is provided by Birkhoff's theorem: Theorem 3.3 (Birkhoff's theorem, e.g., [18], Theorem 8.7.1).A matrix M is doubly stochastic, i.e., each column and each row of M sum to 1, if and only if for some N < ∞ there are permutation matrices σ 1 , . . ., σ N and positive Therefore, the weighted sum of the permutation matrices in Σ T yields a doubly stochastic matrix M T as introduced in Section 2.2.M T also describes a random walk on C n governed by T where the single step in C n is illustrated by the associated Cayley graph.Its stationary distribution is uniform, i.e. when we throw sufficiently many mutations on T then we expect to see each pattern with probability 1/r n .Another, even more useful consequence of Birkhoff's theorem is the fact that it tells us which substitution models are suited for the OSM approach.If the transition matrix associated with the model is doubly stochastic, then we find a set of permutations which give rise to the model.

Let us see how this influences the symmetric form of the general time reversible model (GTR). It has the transition matrix
Assigning permutation matrices to the respective parameters yields the set Σ GTR with elements s 0 (identity) and The weighted sum of the non-identity elements yields is to K3ST.However, Σ GTR does not satisfy Condition 2, because it contains more than four elements.
Therefore, it creates ambiguity since for each nucleotide there are three permutations which do not change the nucleotide.It is also not commutative (Condition 4) which means the order in which we assign the permutations matters.And it is not closed under matrix multiplication (Condition 3), which means that a concatenation of permutations in Σ GTR might lead to a new permutation not in Σ GTR , i.e. we would encounter a new mutation type.All of this shows why the overall applicability of GTR to the OSM approach is rather limited.More complex models like Tamura-Nei [19] do not even permit the decomposition of its transition matrix into the convex sum of permutation matrices.However, including the concept of partial permutation matrices [14] can address this problem.While this approach is interesting for future work, it is beyond the scope of this paper.

Application to other biologically interesting sets
As stated in Section 2.2, the OSM model only requires an underlying Abelian group.Thus, the OSM setting is applicable not only to binary data or four-state (DNA or RNA) data, but also to doublet, codon, and amino acid characters.
In particular, there are four Abelian groups for the twenty-state amino acid alphabet, namely Z 2 × Z 2 × Z 5 , Z 4 × Z 5 , Z 2 × Z 10 , and the cyclic group Z 20 (see e.g., [9] for a complete list of all groups with up to 35 elements).Their construction is analogous to the construction of the Klein-Four group in Example 2.1.
For example, the elements of Z 4 × Z 5 are Kronecker products of one of the four permutations in the cyclic group Z 4 with one of the five permutations of the cyclic group Z 5 .
Figure 5 depicts the 20 substitution types, i.e. the 20 operations including the identity acting on the amino acid character states for all four Abelian groups.If we assign probabilities to the substitution types in the matrices, the resulting matrices are doubly stochastic.The matrices show several features of the groups, e.g. that contrary to the Klein-Four group the elements of the group are not self-inverse but instead the effect of a permutation is reversed by a different mutation.Such events are present in some models of nucleotide evolution, like the strand symmetric model [20], and relatively common in amino acid models where the transition matrix is generated by, e.g., counting mutation types in amino acid alignments (see, e.g., [21] for an overview).It might be interesting to see whether any of these can be fitted.The illustrations in Figure 5 also suggest some ordering of the amino acids to fit the model.For instance, Z 2 × Z 2 × Z 5 and Z 4 × Z 5 seem to partition the sets into four groups with five elements each.

Conclusions
In this paper, we provide the necessary mathematical background for the OSM setting which was introduced and used previously [2,17], but had not been analyzed mathematically for more than two character states.Moreover, the present paper also delivers new insight concerning the requirements for the OSM model to work: In fact, we were able to show that mathematically, it is sufficient to have an underlying Abelian group -which shows a generalization of the OSM concept that was believed to be impossible previously [2].Therefore, we show that OSM is applicable to any number of states.
However, note that the original intuition of the authors in [2] was biologically motivated: The authors supposed that the group not only has to be Abelian, but also symmetric in the sense that each operation can be undone by being applied a second time.Thinking about the DNA, for instance, this works: For example, the transition from A to G can be reverted by another substitution of the same type, namely a transition from G to A. This symmetry criterion is fulfilled by the Klein-Four group, but not by the cyclic group on four states.Unfortunately, for 20 states there is no Abelian group fulfilling this criterion, which is why the demonstrated generalization to 20 states does not provide a nice symmetry (r.f. Figure 5).Therefore, it remains unclear at this stage if there are biologically motivated settings for which our twenty-state generalization is directly applicable.In order to convert the character GTAGA into ACCTC under Σ K3ST , we need to introduce the operation s := (s 1 , s 1 , s 2 , s 2 , s 2 ).As the operation s also translates the constant character AAAAA to GGCCC, converting GTAGA into ACCTC is equivalent to evolving the character state A at the root along the tree to obtain the character GGCCC at the leaves.The Fitch algorithm applied to the latter produces a unique most parsimonious solution of two substitutions as depicted by (c).In order to convert the character GTAGA into ACCTC using this group, we need to introduce the operation s := (s 3 , s 1 , s 3 , s 1 , s 3 ).As the operation s also transforms the constant character AAAAA to CGCGC, converting GTAGA into ACCTC is equivalent to evolving the character state A at the root along the tree such that the character CGCGC is attained at the leaves.The Fitch algorithm applied to the latter produces a unique most parsimonious solution of three substitutions as depicted by (c).In each matrix, the 20 different colors ranging from light yellow to dark red can be regarded to represent 20 substitution types, i.e. 20 operations including the identity acting on the amino acid character states or the corresponding probabilities of these substitution types.In the latter case, the matrices are all doubly stochastic.

Figure 1 (
b), (c), and (d) display the permutation matrices for a transition on branch e 1 (s e 1 ,1

1 . 1 .ITERATION 1 : 2 :ITERATION 3 :Example 2 . 3 .
The number k has multiple equivalent interpretations.It is also the length of the shortest path between f and f d in the Cayley graph for Σ T , where this path corresponds exactly to the chain σ 1 • • • • • σ k since each edge in the Cayley graph corresponds to an operation in Σ T .k is also the smallest matrix power such that M j T = 0 for j < k and M k T > 0, because a positive entry in M k T means that there is a concatenation of k permutations connecting the associated characters.Nguyen et al.[2] presented an efficient procedure to compute the minimal number of substitutions as summarized in Algorithm 1, and we prove its correctness in Section 3.Algorithm INPUT: rooted binary phylogenetic tree T on leaf set X, characters f and f d on X, group Σ. Align characters f and f d and find the corresponding substitution type σ i which translates f j into f d j for all positions j = 1, . . ., |X|.Let σ ∈ Σ n be the resulting operation.ITERATION Let h := c 1 . . .c 1 be the constant character on X with some c 1 ∈ C on all positions.Apply σ to h and call the derived character c.Calculate m := l T (c).OUTPUT: Minimum number m of substitutions needed to evolve f d instead of f on T .Figure3demonstrates how Algorithm 1 works under the K3ST model, i.e. when the group is Σ = Σ K3ST (Figure3(a)).Consider the rooted five-taxon tree in Figure3(b) and the character GTAGA at the leaves.

Figure 1 -
Figure 1-Construction of the OSM matrix Figure 2-The Cayley graph for the two-taxon tree from Figure 1(a).

Figure 3 -
Figure 3-Computing the minimal number of substitutions to translate a character into another one.

Figure 4 -
Figure 4-Converting one character into another character using the cyclic group.

Figure 5 -
Figure 5-Matrices illustrate the four Abelian groups for the twenty-state amino acid alphabet.

Figure 2 :
Figure 2: The vertices depict the characters in C 2 DN A .Two vertices are connected by an edge if there is a permutation in Σ T transforming one of the associated characters into the other.(a) depicts the Cayley graph for the Klein Four group Z 2 × Z 2 , and (b) depicts the Cayley graph for the cyclic group Z 4 .

Figure 3 :
Figure3: (a) depicts the Klein-four group Σ K3ST , which consists of the identity s 0 and the three substitution types s 1 , s 2 , s 3 from the K3ST model.(b) In order to convert the character GTAGA into ACCTC under Σ K3ST , we need to introduce the operation s := (s 1 , s 1 , s 2 , s 2 , s 2 ).As the operation s also translates the constant character AAAAA to GGCCC, converting GTAGA into ACCTC is equivalent to evolving the character state A at the root along the tree to obtain the character GGCCC at the leaves.The Fitch algorithm applied to the latter produces a unique most parsimonious solution of two substitutions as depicted by (c).

Figure 4 :
Figure4: (a) depicts the cyclic group Σ c , which consists of the identity s 0 ≡ s 0 and the three substitution types s 1 , s 2 , s 3 for nucleotide character states.(b) In order to convert the character GTAGA into ACCTC using this group, we need to introduce the operation s := (s 3 , s 1 , s 3 , s 1 , s 3 ).As the operation s also transforms the constant character AAAAA to CGCGC, converting GTAGA into ACCTC is equivalent to evolving the character state A at the root along the tree such that the character CGCGC is attained at the leaves.The Fitch algorithm applied to the latter produces a unique most parsimonious solution of three substitutions as depicted by (c).

Figure 5 :
Figure 5: (a) the Z 2 × Z 2 × Z 5 group, (b) Z 4 × Z 5 , (c) Z 2 × Z 10 , and (d) Z 20 .In each matrix, the 20 different colors ranging from light yellow to dark red can be regarded to represent 20 substitution types, i.e. 20 operations including the identity acting on the amino acid character states or the corresponding probabilities of these substitution types.In the latter case, the matrices are all doubly stochastic.
(a).Here, colors indicate relative branch lengths p e , and patterns denote permutation types α i .E.g., a blue square with horizontal lines indicates the product p e 2 α e 2 ,1 , i.e. the probability of observing a Transition s 1 on Edge e 2 .