Phylosymmetric algebras: mathematical properties of a new tool in phylogenetics

Julia A. Shore; Michael Hendriksen

REVIEW 3 major objections 5 minor 15 references

Phylosymmetric algebras: mathematical properties of a new tool in phylogenetics

T0 review · 3 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read Tree-built rate matrices form a commutative matrix algebra when internal rates are distinct.

desk verdict Interesting and potentially fixable, but the central closure proof relies on a false identity and the abstract overclaims; needs major revision before it can support its conclusions. read the letter →

arxiv 2009.11599 v1 pith:3KI6JHCR submitted 2020-09-24 q-bio.PE math.CO

Michael Hendriksen , Julia A. Shore This is my paper · ORCID

classification q-bio.PEmath.CO MSC 92D1505C5015A30

keywords phylosymmetricalgebrasratematricesrootedphylogenetictreesmostrecentcommonancestormatrixtree-inducedgraphsetsLaplacianaminoacidsubstitutionmodels

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces a recipe for building rate matrices from a rooted phylogenetic tree: assign a real rate to each internal node, and let the rate between two leaves be the rate of their most recent common ancestor. It aims to show that the resulting set of matrices is closed under matrix multiplication, so that products of rate matrices stay in the set and the corresponding transition matrices can be described without exponentiating every matrix. The main proved result is that for a binary tree with distinct internal rates the set forms a commutative matrix algebra, called a phylosymmetric algebra. The paper then characterizes exactly when identifying two rates preserves this closure: the two nodes must be parent and child, or siblings with equally many leaf descendants. A motivating application is parameterised amino acid substitution models with few parameters that still carry biological meaning.

What carries the argument

The central object is the phylosymmetric algebra $Q_T$, the real span of the mrca rate matrices $Q_\alpha = -L(G_\alpha)$, where $G_\alpha$ is the $\alpha$-mrca graph and $L$ is its Laplacian. These graphs form a tree-induced graph set, and this representation is what carries the argument: Theorem 3 and Theorem 4 convert matrix multiplication into counts of two-step walks, so the three relative positions of two tree vertices give three product formulas that always land back in the span of the basis matrices.

What would settle it

Take the two-leaf cherry tree with internal rate $\alpha$. Its rate matrix is $\begin{pmatrix}-1 & 1 \\ 1 & -1\end{pmatrix}$, whose square is $-2Q_\alpha$, while the decomposition displayed in Theorem 5 evaluates to $-Q_\alpha$ for this tree. Computing which value the formula returns settles whether the proof's key step is sound.

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Extended reading notes

Core claim

For a binary phylogenetic tree $T$ with a distinct rate on every internal vertex, the mrca construction produces a set $Q_T$ that is a commutative matrix algebra: every product $Q_\alpha Q_\beta$ is again a linear combination of the mrca rate matrices. The proof shows these matrices are negative Laplacians of graphs in a tree-induced graph set, and uses walk-counting on those graphs to obtain product formulas for comparable vertices, incomparable vertices, and the square of one matrix. Because every matrix and every product is symmetric, commutativity follows. When two internal rates are set equal, the merged set is an algebra exactly when the two vertices are parent and child, or siblings whose subtrees contain the same number of leaves.

Load-bearing premise

The proof rests on a stated formula for the square of a rate matrix as a linear combination of the tree's rate matrices; if that formula is incorrect for even one tree shape, the closure theorem is not established by the argument given.

Editorial extensions

If this is right

For any binary tree with unique rates, computing the product of two mrca rate matrices never leaves the set, so the algebra can be described by a finite multiplication table.
Whenever a rate set is a matrix algebra, the corresponding transition matrices are available as $\{I+Q : \det(I+Q) \neq 0\}$, so a phylogenetic model can avoid repeated matrix exponentials.
The algebra survives merging two rates exactly for parent-child pairs and equal-size sibling pairs, giving a catalogue of parameter constraints that are provably safe to impose.
Because K2P arises as a phylosymmetric algebra, the construction links a standard DNA model to tree-induced rate structure and suggests the same route for amino acid models with 3 to 19 parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct check of the squared-matrix decomposition on a two-leaf cherry is the minimal test a reader should run before building applications on the proof; it decides whether the displayed formula in the proof is sound.
If the closure proof is extended to non-binary trees or to multiple simultaneous repeated rates, the construction would produce a much larger family of low-parameter models whose constraints are read directly off the tree.
The characterisation question left open, which abstract matrix algebras arise from trees, would allow model builders to start from a desired algebra and recover the tree that generates it.
The abstract's wording that 'any rate matrix set generated this way' is closed is broader than the proved binary, distinct-rate theorem; reconciling the two is the first thing an application-minded reader would want resolved.

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Formalized claims in Lean

Claim #1: For a binary phylogenetic tree $T$ with a distinct rate on every internal vertex, the mrca construction produces a set $Q_T$ that is a commutative matrix algebra: every product $Q_\alpha Q_\beta$ is again a linear combination of the mrca rate matrices. The proof shows these matrices are negative Laplacians of graphs in a tree-induced graph set, and uses walk-counting on those graphs to obtain prod

/-- @claim 1 For a binary phylogenetic tree $T$ with a distinct rate on every internal vertex, the mrca construction produces a set $Q_T$ that is a commutative matrix algebra: every product $Q_\alpha Q_\beta$ is again a linear combination of the mrca rate matrices. The proof shows these matrices are negative Laplacians of graphs in a tree-induced graph set, and uses walk-counting on those graphs to obtain prod -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know: this paper defines a nice way to build rate matrices from a rooted tree and claims that the resulting set is closed under matrix multiplication. The abstract overstates it (any tree), and the proof for the binary case relies on a false identity in Theorem 5. As written, the central theorem is not proven; the math may still be salvageable, but not in this version.

The construction is genuinely new, at least in this packaging. The mrca-based rate matrices appeared in Shore et al. (2020) as an empirical tool, but the algebraic closure property and the repeated-rates characterization are new. The TIGS graph encoding is a clean way to think about the matrices as Laplacians of complete multipartite graphs, and Theorems 6 and 7 (products for comparable and incomparable vertices) look correct. The entrywise formula in Theorem 5 is also correct. If the square term could be properly decomposed, the binary closure theorem would likely follow.

The soft spot is the second display of Theorem 5, the "equivalently" decomposition. For a two-leaf cherry, Qα^2 = -2Qα, but the formula gives a different matrix. That identity is what Theorem 8 cites for closure under squaring, and Theorem 11 reuses it. The non-binary case is worse: the abstract says "any" tree, but a root with a leaf and two cherries already gives a square outside the span, so the closure claim is false in general. The binary case might be true with a corrected identity—the stress-test note sketches one—but the paper doesn't contain it.

This is a real, load-bearing error, not a minor typo. That said, the paper is worth engaging with: the framework is elegant, the problem is relevant to model building, and the fix looks within reach. I would send it to peer review, but only with the expectation of major revision and a recheck of the core proofs. As is, I wouldn't cite it.

Referee Report

3 major / 5 minor

Summary. The paper introduces a construction of rate-matrix sets from rooted phylogenetic trees: each internal vertex is assigned a rate, and the off-diagonal entry Q_xy is 1 exactly when the most recent common ancestor of x and y is the vertex carrying that rate, with diagonal entries chosen to give zero row sums. The authors call the resulting matrix sets phylosymmetric algebras and claim that, when all rates are distinct, they are closed under matrix multiplication, so the transition matrices can be obtained algebraically without exponentiating each rate matrix. They also analyze what happens when two rates are set equal, giving a characterization of when the resulting set remains a matrix algebra. The presentation is built around a graph-theoretic reformulation using tree-induced graph sets (TIGS) and Laplacians of the associated mrca graphs. The paper is motivated by potential applications to low-parameter amino acid substitution models.

Significance. If the main theorems were correct, the paper would be a useful contribution to phylogenetic model building: it gives an explicit, biologically interpretable way to generate rate-matrix families with few free parameters, and it connects the construction to classical graph theory through the TIGS reformulation. The treatment is self-contained, definitions are explicit, and the K2P model is correctly identified as a phylosymmetric algebra. However, the central closure theorem is not proven as written because a displayed algebraic identity in Theorem 5 is false and is used in the proofs of Theorems 8, 9, and 11. The error appears repairable by replacing that identity with a corrected decomposition, so I regard this as a major-revision situation rather than a rejection, but the current manuscript does not establish its main claim.

major comments (3)

[Theorem 5 (Section 4)] The second display in Theorem 5 is false. For a cherry with two leaves, Qα = [[-1,1],[1,-1]], so Qα² = [[2,-2],[-2,2]] = -2Qα. Evaluating the displayed right-hand side directly for this tree, the children of u are two leaves with |T^β|=1, and Σ_{γ∈ω(Vu)}Qγ = Qα because leaf rate matrices vanish, so the right-hand side is (1-2)Qα + 2Qα = Qα, not Qα². The first (entrywise) display is correct, so the error is in the claimed algebraic decomposition. This is load-bearing: Theorem 8 cites Theorems 5-7 for closure under products, and the square Qα² is exactly the case that requires this identity; Theorem 9 and Theorem 11 (Case 3) reuse the same decomposition. A corrected identity that appears to hold for arbitrary rooted trees is Qα² = -|T^α|Qα + Σ_{β∈ω(Cu)}(|T^α|-|T^β|)J_β, where J_β = Σ_{γ∈ω(Vβ)}Qγ; the proofs of Theorems 8, 9, and 11 need to be rewritten around this corrected identity.
[Abstract and Theorem 8] The abstract claims that 'any rate matrix set generated this way is closed under matrix multiplication', but Theorem 8 is stated only for binary phylogenetic trees, even though the paper says at the start that trees are permitted to be non-binary. With the corrected identity proposed above, the proof appears to go through for all rooted trees, so the authors should either prove the general statement and remove the word 'binary' from Theorem 8, or restrict the abstract and the corresponding claims in Sections 1 and 6. As written, the theorem is narrower than the announced result.
[Theorem 11 (Section 5)] The proof of the 'only if' direction of Theorem 11 is not reliable. Case 3 uses the false second display of Theorem 5 to expand Qγ² and read off coefficients for Qα and Qβ; with the corrected square identity the expansion is different, so that argument must be redone. In addition, Case 1 asserts n1 ≠ n2 without any derivation, and the assertion is false in general: in a tree where u is the root with seven leaf descendants, one child subtree of size five containing w and another child subtree of size two, and w has child subtrees of sizes three and two with v in the size-three subtree, we get n1 = 7-5 = 2 and n2 = 5-3 = 2. The characterization may still be true, but the present proof does not establish it.

minor comments (5)

[Remark 1] The stated diagonal entry of Ju, namely -m, is inconsistent with the definition of Qα and with Example 1: for a vertex with m leaf descendants the diagonal entries of Ju should be -(m-1), since each leaf has exactly m-1 other leaves in its subtree. This remark is also part of the faulty justification of the 'equivalence' in Theorem 5.
[Theorem 6, proof] In the paragraph beginning 'We now consider AαDβ', the expression (DαAβ)ij should be (AαDβ)ij.
[Section 5] The references 'by Lemma 5' and 'by Lemmas 10 and 9' appear to be misnumbered; they should refer to Theorem 5 (or Theorem 6) and to Theorems 9 and 10, respectively.
[Section 3] There is a typo in 'Recall the folloring standard graph-theoretic definitions'; 'folloring' should be 'following'.
[Theorem 11, Case 1] The symbols n1 and n2 are used without being defined; the sentence '∵ Theorem 6 where ni ∈ N' should be replaced with explicit definitions of the two coefficients.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the closure proofs are self-contained derivations; prior work appears only as motivation.

full rationale

The paper's central claim, that the set of mrca rate matrices from a rooted tree is closed under matrix multiplication, is derived internally: QT is defined directly from a tree, and Theorems 5, 6, and 7 prove, respectively, the square of one rate matrix, products of comparable rate matrices, and products of incomparable rate matrices, all as linear combinations of the same basis. Theorem 8 cites these lemmas, and Theorem 11 applies them, with no fitted parameters, no hidden use of the target result inside the derivation, and no definitional identification of output with input. Self-citations to Shore (2015) and Shore et al. (2020) are motivational and contextual; the algebraic claims in Sections 4 and 5 do not depend on them. The possible arithmetical failure of the second display in Theorem 5 is a correctness defect, not circularity: even if the displayed identity is false, it is not a restatement of the theorem being proved, nor was it constructed from the target conclusion, so it does not make the derivation circular. The paper is therefore assessed as containing no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

No data fitting or empirical constants appear. The main load-bearing support is the decomposition in Theorem 5, which is incorrect, plus the binary-tree restriction that is understated in the abstract. The TIGS and phylosymmetric algebra are definitions, not empirically evidenced entities.

assumptions (3)

domain assumption The tree is binary when closure is asserted
Definition 1 permits non-binary trees and the abstract claims closure for any such set, but Theorem 8 proves only binary trees. Non-binary counterexamples exist, so this restriction is load-bearing and understated.
domain assumption Leaf vertices carry zero rate matrices and diagonal pairs are excluded from the mrca partition Cα
Needed for the diagonal sums in Remark 1 and in the examples. Not stated in the definition of Cα, and this is likely the source of the off-by-one error in the formula for J_u.
standard math Standard walk-counting theorems for adjacency matrices
Theorems 3 and 4 are imported from classical graph theory and are used to compute products of Laplacians. These are standard and not a source of circularity.

invented entities (2)

TIGS (tree-induced graph set)
purpose: Graph-theoretic encoding of mrca partitions to prove closure under multiplication
Introduced in Definition 4. Its bijection with hierarchies is asserted in Theorem 1, but no external empirical evidence is relevant because it is a mathematical construction.
Phylosymmetric algebra
purpose: Newly named matrix algebra generated by a rooted tree with rate assignments
Defined in Section 4. Its defining property, closure under multiplication, is the central claim of the paper rather than an independently evidenced fact.

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Cite this review

Pith. "Pith review of Phylosymmetric algebras: mathematical properties of a new tool in phylogenetics." pith.science (2026). https://pith.science/paper/3KI6JHCR

@misc{pith2026200911599,
  author       = {Pith},
  title        = {Pith review of: Phylosymmetric algebras: mathematical properties of a new tool in phylogenetics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KI6JHCR}},
  note         = {Machine review of arXiv:2009.11599}
}

read the original abstract

In phylogenetics it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction of computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most recent common ancestor of those two states. We investigate the properties of these matrix sets from both a linear algebra and a graph theory perspective and show that any rate matrix set generated this way is closed under matrix multiplication. The consequences of setting two rates assigned to internal tree nodes to be equal are then considered. This methodology could be used to develop parameterised models of amino acid substitution which have a small number of parameters but convey biological meaning.

Figures

Figures reproduced from arXiv: 2009.11599 by the authors.

**Figure 1.** A rooted tree on taxa X = {1, 2, 3, 4, 5}, with all non-leaf vertices labelled by their rates. Example 1 We end this section by computing the rate matrix set associated with the tree in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** An example of a TIGS. Additionally, these graphs are the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** A rooted tree on states of DNA with taxa [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

Journal of Molecular Evolution 42(4):459--468

Adachi J, Hasegawa M (1996) Model of amino acid substitution in proteins encoded by mitochondrial DNA . Journal of Molecular Evolution 42(4):459--468

work page 1996

[2] [2]

Springer Science & Business Media

Brouwer AE, Haemers WH (2011) Spectra of graphs. Springer Science & Business Media

work page 2011

[3] [3]

Journal of Mathematical Biology 70(4):855--891

Fern \'a ndez-S \'a nchez J, Sumner JG, Jarvis PD, Woodhams MD (2015) Lie M arkov models with purine/pyrimidine symmetry. Journal of Mathematical Biology 70(4):855--891

work page 2015

[4] [4]

Journal of molecular evolution 16(2):111--120

Kimura M (1980) A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. Journal of molecular evolution 16(2):111--120

work page 1980

[5] [5]

Molecular biology and evolution 25(7):1307--1320

Le SQ, Gascuel O (2008) An improved general amino acid replacement matrix. Molecular biology and evolution 25(7):1307--1320

work page 2008

[6] [6]

Leon SJ (2010) Linear Algebra with Applications, Eighth Edition. Pearson

work page 2010

[7] [7]

Molecular Biology and Evolution 34(1):45--50

Mello B, Tao Q, Tamura K, Kumar S (2016) Fast and accurate estimates of divergence times from big data. Molecular Biology and Evolution 34(1):45--50

work page 2016

[8] [8]

SIAM review 20(4):801--836

Moler C, Van Loan C (1978) Nineteen dubious ways to compute the exponential of a matrix. SIAM review 20(4):801--836

work page 1978