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Consensus-level substitution rates are distinct from the virion-level rate

This paper distinguishes between the virion-level substitution rate (VLSR), which counts mutational events along lineages, and consensus-level substitution rates (CLSRs), which measure changes in consensus sequences, arguing that these distinct estimands are not interchangeable and that the specific consensus-generation rule used must be routinely reported to ensure informed methodological choices in viral evolutionary analysis.

Original authors: David J Pascall

Published 2026-06-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: David J Pascall

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to measure how fast a virus is evolving. For a long time, scientists have been using a single ruler to do this, but this paper argues that there are actually two different rulers measuring two completely different things. If you use the wrong ruler, your measurement might be mathematically correct for one thing but completely wrong for the other.

Here is the breakdown of the two "rulers" (or rates) described in the paper, using simple analogies.

1. The Two Different Ways to Look at a Virus

To understand the difference, we need to look at a virus population in two ways:

The "Individual" View (Virion-Level): Imagine a massive stadium filled with millions of unique virus particles. Each one is a distinct individual with its own specific genetic code. Some have a tiny mutation here, some there.
The "Headline" View (Consensus-Level): Imagine a news reporter standing in that stadium. Instead of reporting on every single person, the reporter looks at every spot in the stadium and asks, "What is the most common thing happening here?" They write down a single "Headline" (the consensus sequence) that represents the majority. If 51% of the viruses have an "A" at a specific spot, the headline says "A." If 51% have a "G," the headline says "G."

2. The Two Different Rates

The paper says scientists often confuse the speed of change for these two views.

Rate A: The Virion-Level Substitution Rate (VLSR)

What it measures: How fast individual virus particles are mutating as they travel through time.
The Analogy: Think of a family tree. Every time a parent virus splits into two children, there is a chance a typo (mutation) happens in the child's code. This rate counts every single typo that happens along every single branch of the family tree.
How it works: It assumes a standard "tree" structure where one ancestor splits into two, then four, and so on. It counts the steps on the branches.

Rate B: The Consensus-Level Substitution Rate (CLSR)

What it measures: How fast the "Headline" (the majority report) changes over time.
The Analogy: Imagine a crowded room where people are voting on a color.
- If 51% vote Red and 49% vote Blue, the "Headline" is Red.
- If the crowd shifts slightly so 52% vote Blue and 48% vote Red, the "Headline" instantly flips to Blue.
- Crucial Point: The individual people in the room didn't necessarily change their votes (no new mutations happened); the balance of the crowd just shifted enough to flip the majority.
How it works: This rate counts how many times the "Headline" flips from one letter to another. It doesn't care about the individual trees; it cares about the shifting balance of the crowd.

3. Why They Are Not Interchangeable

The paper makes a critical point: These two rates are not the same, and they don't even move in the same direction.

Scenario 1: The Silent Flip
Imagine a virus population where the "Red" and "Blue" voters are constantly switching back and forth, but no one is actually newly mutating (no new genetic errors).
- Virus Rate (VLSR): Zero. No new mutations occurred.
- Headline Rate (CLSR): High. The headline flips from Red to Blue to Red constantly because the crowd balance is wobbling.
- Result: The headline is changing fast, but the viruses aren't mutating.
Scenario 2: The Hidden Mutation
Imagine a new mutation appears in a tiny group of viruses (say, 1 in 10,000). It exists, but it's too rare to change the "Headline."
- Virus Rate (VLSR): High. A mutation happened!
- Headline Rate (CLSR): Zero. The headline didn't change because the mutation was too small to be noticed by the majority rule.
- Result: The viruses are mutating, but the headline stays the same.

4. The "Bottleneck" Problem

The paper also highlights a weird quirk that happens when a virus jumps from one person to another (like from Patient A to Patient B).

The Tree View: Usually, evolution is a smooth tree.
The Headline View: When Patient A infects Patient B, only a few viruses make the jump. This is a "bottleneck."
- In Patient A, the majority might be "Red."
- In Patient B, by pure luck, the few viruses that jumped happen to be mostly "Blue."
- The Result: The "Headline" for Patient B instantly jumps from Red to Blue. In the math of the "Headline Rate," this looks like a massive, sudden jump in evolution, even though it was just a random sampling accident.

5. What This Means for Scientists

The author argues that scientists need to be very careful about which "ruler" they are using:

If you want to know how fast mutations happen in individual viruses (VLSR): You need data on individual virus particles (or a very specific type of consensus data) and you must use standard "family tree" math.
If you want to know how fast the "Headline" changes (CLSR): You need to use different math. Standard tree math assumes a smooth branching process, but "Headline" data can have sudden jumps (like when a new patient gets infected). Using standard tree math on "Headline" data can give you a distorted picture.

The Bottom Line:
The paper concludes that "Consensus-Level Substitution Rates" are a valid thing to measure, but they are not the same as the standard mutation rates we usually talk about. Because the "Headline" depends on a specific rule (e.g., "pick the letter that is over 50%"), scientists must always report exactly what rule they used. If they don't, we won't know if they are measuring the speed of the individuals or the speed of the majority vote.

Technical Summary: Consensus-level vs. Virion-level Substitution Rates

Problem Statement
The paper addresses a critical ambiguity in evolutionary epidemiology regarding the definition and estimation of viral substitution rates. While estimating these rates is central to understanding phenomena such as within-host evolution and the emergence of variants (e.g., in SARS-CoV-2 and HIV), the literature often conflates two distinct classes of evolutionary rate estimands. Current methods oscillate between approaches based on population-level allele frequencies and those based on individual-level molecular evolution along lineages, yet this distinction is frequently implicit or overlooked in phylogenetic analyses. The author argues that this conflation leads to methodological mismatches where standard phylogenetic tools are applied to data types for which they are theoretically unsuited.

Methodological Distinction and Definitions
The paper establishes a rigorous definitional separation between two classes of estimands:

Virion-Level Substitution Rate (VLSR): This is a molecular quantity representing the expected number of mutational events along a lineage per site per unit time. It corresponds to the standard estimand in phylogenetic models, assuming a binary tree structure where tips represent individual virions.
Consensus-Level Substitution Rate (CLSR): This is a population-summary quantity representing the expected number of times an allele crosses a specific consensus-calling threshold per site per unit time. CLSRs are indexed by a specific consensus-generation rule ( $\phi$ ), typically the selection of the site-wise nucleotide at a frequency $>0.5$ . Unlike VLSR, CLSRs do not inherently follow a binary tree structure; they represent a progression of summary statistics of a time-varying allele frequency distribution.

Theoretical Analysis of Differences
The author demonstrates that VLSR and CLSR are mathematically distinct and not interchangeable through several theoretical arguments:

Topology: VLSR analysis assumes a binary tree where tips are related by an underlying genealogy. In contrast, CLSR analysis involves a "chain of descent" of summary statistics. In a well-mixed population, the consensus sequence at time $t_1$ is the unique "ancestor" of the consensus at $t_2$ , precluding the branching structure required for standard phylogenetic models.
Transmission Dynamics: The paper highlights that during transmission (e.g., from host A to host B), the consensus sequence can exhibit large, instantaneous jumps due to the bottlenecking process and the shift in allele frequencies in the founding population. Standard continuous-time Markov chain (CTMC) sequence evolution models, which assume changes accrue along branches of positive length, cannot naturally accommodate these simultaneous, multi-site jumps at population formation.
Divergence in Zero-Mutation Scenarios: The author presents an illustrative case of a two-type population with fluctuating selection but no mutation. In this scenario, the VLSR is zero (no mutational events occur), while the CLSR is non-zero (the consensus sequence changes as allele frequencies oscillate across the calling threshold). Conversely, mutations that emerge and are lost without crossing the threshold increase VLSR but not CLSR. Thus, neither rate strictly dominates the other.

Methodological Implications for Estimation
The paper categorizes appropriate data and modeling frameworks for each estimand:

Estimating VLSR: Requires virion-level sequences or consensus sequences under the assumption that every consensus sequence is instantiated by a virion in the population. Standard phylogenetic machinery is appropriate here.
Estimating CLSR: Requires consensus sequences and non-standard machinery. Because the relational structure of consensus sequences is tree-like but not binary-tree-like (due to population merging and mode loss), sampled ancestor models (e.g., Gavryushkina et al., 2014) are identified as appropriate genealogical models.
Model Limitations: The author notes that even with appropriate genealogical models, standard sequence evolution components remain misspecified due to the potential discontinuity at population formation. However, the paper suggests this may cause only a small bias in practice, as the founder population's mode is likely close to the source population's mode, though this remains an empirical question.

Key Contributions and Claims

Conceptual Clarification: The primary contribution is the explicit separation of VLSR and CLSR as distinct estimands, clarifying that they measure different aspects of the evolutionary process (molecular events vs. population summary changes).
Reporting Standards: The paper argues that because a CLSR is defined by its consensus-generation rule ( $\phi$ ), reporting this rule should be a routine, non-optional requirement in phylogenetic analysis.
Methodological Guidance: The text provides a framework for analysts to choose the correct data type and modeling approach based on their specific research goal, warning that using standard tools on consensus data (or vice versa) without adjustment may yield estimates that are neither accurate VLSRs nor accurate CLSRs.

Significance
The paper serves as a reflection intended to guide analysts in making more informed methodological choices when working with virus sequence sets. It does not propose new experimental data or specific software implementations but rather highlights a definitional gap in current practice. By distinguishing these rates, the author aims to prevent the confusion that arises when population-level summary statistics are treated as if they were individual-level molecular lineages, thereby improving the rigor of evolutionary epidemiological inference.