Phylogenetic Distance Models

Correcting observed sequence differences into evolutionary distances

Bioinformatics

Algorithms

Phylogenetics

Evolution

How substitution models turn an alignment’s observed mismatches into evolutionary distance estimates for phylogenetic inference.

Author

Jiahong Sun

Published

August 25, 2026

Modified

August 25, 2026

1 Topic at a Glance

One-sentence summary: A phylogenetic distance model estimates the expected number of substitutions separating two aligned sequences, including changes hidden by repeated mutation at the same site.

Item	Description
Biological question	How evolutionarily separated are two homologous sequences?
Computational problem	Convert an observed fraction of aligned differences into a comparable distance despite unobserved substitutions
Method type	Stochastic substitution model and pairwise estimation
Input	Homologous multiple-sequence alignment, with a stated treatment of gaps and ambiguous bases
Output	Pairwise distance matrix, often used to infer a tree
Prerequisites	Needleman–Wunsch global alignment, Smith–Waterman local alignment, neighbor-joining trees

2 Why mismatch fraction is not an evolutionary distance

For two aligned sequences, the p-distance is the fraction $p$ of comparable columns with different symbols. It is a useful descriptive quantity, but it is not generally the expected number of substitutions per site. A site can mutate from A to G and later from G to A; the observed bases then match even though two changes occurred. Different substitutions can also occur successively at one site, a phenomenon called multiple hits.

As sequences diverge, observed mismatch fraction saturates while actual evolutionary change continues. A distance model supplies a probabilistic relationship between the unobserved substitution process and the alignment. Its estimate is meaningful only relative to the model, alignment, and filtering choices used to obtain it.

homologous alignment
       -> count comparable matching and differing columns
       -> fit / apply a substitution model
       -> corrected pairwise distances
       -> distance-based tree or exploratory analysis

3 A simple correction: Jukes–Cantor

The Jukes–Cantor (JC69) model assumes four nucleotides have equal equilibrium frequencies and that every one-base change has the same instantaneous rate. Under those restrictive assumptions, the corrected distance is

\[ d_{\mathrm{JC69}}=-\frac{3}{4}\ln\left(1-\frac{4p}{3}\right). \]

Here $p$ is the observed mismatch fraction and $d_{\mathrm{JC69}}$ is the estimated expected substitutions per site. For small $p$, $d$ is close to $p$. As $p$ approaches $0.75$, the logarithm becomes unstable: the alignment no longer contains enough information under this model to estimate a finite distance reliably.

3.1 Toy example

If 8 of 100 comparable nucleotide columns differ, then $p=0.08$ and

\[ d_{\mathrm{JC69}}=-\frac{3}{4}\ln(1-4\times0.08/3)\approx0.085. \]

The correction is modest because a second mutation at the same site is unlikely at this divergence. If 60 of 100 columns differ, the same calculation gives about $1.21$ substitutions per site, far larger than $0.60$; uncertainty and model misspecification are also much more consequential there.

4 Models add biological realism

JC69 is often too simple for real DNA. More flexible models distinguish the changes and base compositions that data support.

Model	Main extra feature	Typical use and caution
JC69	Equal base frequencies and equal rates	Pedagogy or closely related sequences; rarely a sufficient final model
K80 / K2P	Separate transition and transversion rates	Useful when transition bias matters; still assumes equal base frequencies
F81	Unequal equilibrium base frequencies	Addresses composition, but not transition bias
HKY85	Unequal frequencies plus transition/transversion bias	Common interpretable DNA model
GTR	A distinct reversible rate for each substitution pair	Flexible baseline; needs enough information and is often combined with rate heterogeneity

Sites do not necessarily evolve at one rate. A gamma-distributed rate model ($+\Gamma$), a proportion of invariant sites, or an explicit codon/protein model can better match a dataset. These additions are models of an average process; they do not repair poor homology, recombination, contamination, or a mixture of unrelated evolutionary histories.

5 Estimating and using distances

For a fixed model, software can estimate a pairwise distance by maximizing the probability of the two sequences separated by a branch length. This uses all observed substitution categories, rather than only mismatch fraction. From $n$ taxa, the result is an $n\times n$ symmetric matrix; diagonal entries are zero.

Distance methods such as neighbor-joining trees use this matrix directly. The matrix is a lossy summary of the alignment: it is fast and useful, but it does not retain site-pattern information that likelihood- and Bayesian-tree methods use. When comparing distances, retain the alignment coordinates, model, gap policy, and any column filtering because changing them changes the estimate.

5.1 Assumptions and diagnostics

Issue	Why it matters	Practical response
Homologous alignment	Misaligned columns are treated as substitutions	Inspect alignment quality; align coding regions by codon where appropriate
One history across sites	Recombination can make one tree and its distances misleading	Screen or partition recombinant/heterogeneous regions
Adequate model	Composition bias and rate variation distort branch lengths	Compare plausible models and report the selected one
Moderate divergence	Saturation erases historical signal	Examine distances and use richer data/markers rather than overinterpreting large corrections
Independent sampling	Duplicate or contaminated sequences can dominate apparent similarity	Curate taxon identity, metadata, and sequence quality

Gaps and ambiguous bases require an explicit policy. Pairwise deletion uses all usable columns for each pair but makes pairs based on different site sets; complete deletion uses one common set but may discard much information. Neither is universally right, and both should be reported.

6 Connections and method choice

Distance models begin after an alignment has defined which residues are compared. They provide the matrix that neighbor-joining trees converts into an unrooted tree. Tree topology alone is not a biological conclusion: bootstrap or other resampling assessments, alternative alignments, and model sensitivity should accompany it.

Goal	Good starting point	Important limitation
Quick visualization of many related sequences	Model-corrected distances plus neighbor joining	Pairwise compression loses site-pattern information
Very close sequences	p-distance or simple correction	Still verify alignment and coverage comparability
Deeply divergent nucleotide sequences	Composition- and rate-aware model; consider amino acids/codons	Saturation and alignment uncertainty can remain limiting
Gene with recombination	Partition or use a recombination-aware analysis	One distance matrix cannot represent several local histories

7 Take-home Messages

Observed differences undercount substitutions when sites experience repeated mutation.
A distance is a model-based estimate, not a property of a sequence pair alone.
Transition bias, base composition, and among-site rate variation motivate models beyond JC69.
Distance matrices are fast, useful inputs to tree algorithms, but they discard alignment-level information and depend on sound homology and model choices.

8 References

Jukes, T. H., & Cantor, C. R. 1969. Evolution of protein molecules. In Mammalian Protein Metabolism.
Kimura, M. 1980. A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. Journal of Molecular Evolution 16:111–120.
Hasegawa, M., Kishino, H., & Yano, T. 1985. Dating of the human-ape splitting by a molecular clock of mitochondrial DNA. Journal of Molecular Evolution 22:160–174.
IQ-TREE documentation: substitution models: model definitions and selection guidance.

--- title: "Phylogenetic Distance Models" subtitle: "Correcting observed sequence differences into evolutionary distances" description: "How substitution models turn an alignment's observed mismatches into evolutionary distance estimates for phylogenetic inference." author: "Jiahong Sun" date: "2026-08-25" date-modified: "2026-08-25" categories: [Bioinformatics, Algorithms, Phylogenetics, Evolution] toc: true number-sections: true code-fold: show draft: false --- # Topic at a Glance > **One-sentence summary:** A phylogenetic distance model estimates the expected number of substitutions separating two aligned sequences, including changes hidden by repeated mutation at the same site. | Item | Description | |---|---| | **Biological question** | How evolutionarily separated are two homologous sequences? | | **Computational problem** | Convert an observed fraction of aligned differences into a comparable distance despite unobserved substitutions | | **Method type** | Stochastic substitution model and pairwise estimation | | **Input** | Homologous multiple-sequence alignment, with a stated treatment of gaps and ambiguous bases | | **Output** | Pairwise distance matrix, often used to infer a tree | | **Prerequisites** | [Needleman--Wunsch global alignment](needleman-wunsch-global-alignment.qmd), [Smith--Waterman local alignment](smith-waterman-local-alignment.qmd), [neighbor-joining trees](neighbor-joining-trees.qmd) | # Why mismatch fraction is not an evolutionary distance For two aligned sequences, the *p-distance* is the fraction $p$ of comparable columns with different symbols. It is a useful descriptive quantity, but it is not generally the expected number of substitutions per site. A site can mutate from A to G and later from G to A; the observed bases then match even though two changes occurred. Different substitutions can also occur successively at one site, a phenomenon called **multiple hits**. As sequences diverge, observed mismatch fraction saturates while actual evolutionary change continues. A distance model supplies a probabilistic relationship between the unobserved substitution process and the alignment. Its estimate is meaningful only relative to the model, alignment, and filtering choices used to obtain it. ```text homologous alignment -> count comparable matching and differing columns -> fit / apply a substitution model -> corrected pairwise distances -> distance-based tree or exploratory analysis ``` # A simple correction: Jukes--Cantor The Jukes--Cantor (JC69) model assumes four nucleotides have equal equilibrium frequencies and that every one-base change has the same instantaneous rate. Under those restrictive assumptions, the corrected distance is $$ d_{\mathrm{JC69}}=-\frac{3}{4}\ln\left(1-\frac{4p}{3}\right). $$ Here $p$ is the observed mismatch fraction and $d_{\mathrm{JC69}}$ is the estimated expected substitutions per site. For small $p$, $d$ is close to $p$. As $p$ approaches $0.75$, the logarithm becomes unstable: the alignment no longer contains enough information under this model to estimate a finite distance reliably. ## Toy example If 8 of 100 comparable nucleotide columns differ, then $p=0.08$ and $$ d_{\mathrm{JC69}}=-\frac{3}{4}\ln(1-4\times0.08/3)\approx0.085. $$ The correction is modest because a second mutation at the same site is unlikely at this divergence. If 60 of 100 columns differ, the same calculation gives about $1.21$ substitutions per site, far larger than $0.60$; uncertainty and model misspecification are also much more consequential there. # Models add biological realism JC69 is often too simple for real DNA. More flexible models distinguish the changes and base compositions that data support. | Model | Main extra feature | Typical use and caution | |---|---|---| | JC69 | Equal base frequencies and equal rates | Pedagogy or closely related sequences; rarely a sufficient final model | | K80 / K2P | Separate transition and transversion rates | Useful when transition bias matters; still assumes equal base frequencies | | F81 | Unequal equilibrium base frequencies | Addresses composition, but not transition bias | | HKY85 | Unequal frequencies plus transition/transversion bias | Common interpretable DNA model | | GTR | A distinct reversible rate for each substitution pair | Flexible baseline; needs enough information and is often combined with rate heterogeneity | Sites do not necessarily evolve at one rate. A gamma-distributed rate model ($+\Gamma$), a proportion of invariant sites, or an explicit codon/protein model can better match a dataset. These additions are models of an average process; they do not repair poor homology, recombination, contamination, or a mixture of unrelated evolutionary histories. # Estimating and using distances For a fixed model, software can estimate a pairwise distance by maximizing the probability of the two sequences separated by a branch length. This uses all observed substitution categories, rather than only mismatch fraction. From $n$ taxa, the result is an $n\times n$ symmetric matrix; diagonal entries are zero. Distance methods such as [neighbor-joining trees](neighbor-joining-trees.qmd) use this matrix directly. The matrix is a lossy summary of the alignment: it is fast and useful, but it does not retain site-pattern information that likelihood- and Bayesian-tree methods use. When comparing distances, retain the alignment coordinates, model, gap policy, and any column filtering because changing them changes the estimate. ## Assumptions and diagnostics | Issue | Why it matters | Practical response | |---|---|---| | Homologous alignment | Misaligned columns are treated as substitutions | Inspect alignment quality; align coding regions by codon where appropriate | | One history across sites | Recombination can make one tree and its distances misleading | Screen or partition recombinant/heterogeneous regions | | Adequate model | Composition bias and rate variation distort branch lengths | Compare plausible models and report the selected one | | Moderate divergence | Saturation erases historical signal | Examine distances and use richer data/markers rather than overinterpreting large corrections | | Independent sampling | Duplicate or contaminated sequences can dominate apparent similarity | Curate taxon identity, metadata, and sequence quality | Gaps and ambiguous bases require an explicit policy. Pairwise deletion uses all usable columns for each pair but makes pairs based on different site sets; complete deletion uses one common set but may discard much information. Neither is universally right, and both should be reported. # Connections and method choice Distance models begin after an alignment has defined which residues are compared. They provide the matrix that [neighbor-joining trees](neighbor-joining-trees.qmd) converts into an unrooted tree. Tree topology alone is not a biological conclusion: bootstrap or other resampling assessments, alternative alignments, and model sensitivity should accompany it. | Goal | Good starting point | Important limitation | |---|---|---| | Quick visualization of many related sequences | Model-corrected distances plus neighbor joining | Pairwise compression loses site-pattern information | | Very close sequences | p-distance or simple correction | Still verify alignment and coverage comparability | | Deeply divergent nucleotide sequences | Composition- and rate-aware model; consider amino acids/codons | Saturation and alignment uncertainty can remain limiting | | Gene with recombination | Partition or use a recombination-aware analysis | One distance matrix cannot represent several local histories | # Take-home Messages 1. Observed differences undercount substitutions when sites experience repeated mutation. 2. A distance is a model-based estimate, not a property of a sequence pair alone. 3. Transition bias, base composition, and among-site rate variation motivate models beyond JC69. 4. Distance matrices are fast, useful inputs to tree algorithms, but they discard alignment-level information and depend on sound homology and model choices. # References - Jukes, T. H., & Cantor, C. R. 1969. [Evolution of protein molecules](https://doi.org/10.1016/B978-1-4832-3211-9.50009-7). In *Mammalian Protein Metabolism*. - Kimura, M. 1980. [A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences](https://doi.org/10.1007/BF01731581). *Journal of Molecular Evolution* 16:111--120. - Hasegawa, M., Kishino, H., & Yano, T. 1985. [Dating of the human-ape splitting by a molecular clock of mitochondrial DNA](https://doi.org/10.1016/0040-5809(85)90020-3). *Journal of Molecular Evolution* 22:160--174. - [IQ-TREE documentation: substitution models](https://iqtree.github.io/doc/Substitution-Models): model definitions and selection guidance.