💻 bioinformatics

Generative continuous time model reveals epistatic signatures in protein evolution

This paper introduces a continuous-time protein evolution model parameterized by a generative Potts model to simulate epistatic effects, revealing that while epistasis slows overall evolution and causes systematic underestimation of evolutionary distances, it does not alter average site-specific rates due to context-dependent rate heterogeneity.

Original authors: Pagnani, A., Barrat-Charlaix, P.

Published 2026-07-10

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CC BY 4.0

Original authors: Pagnani, A., Barrat-Charlaix, P.

Original paper licensed under CC BY 4.0 (https://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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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 protein evolution as a massive, chaotic game of "telephone" played by a billion people, where every time someone whispers a change to the next person, the whole group's reaction depends on who is standing next to them. For decades, scientists have tried to predict how these proteins change over time using a very simple rulebook: they assumed that every single letter in the protein's genetic code changes on its own, completely ignoring its neighbors. It's like assuming a word in a sentence changes meaning regardless of the other words around it.

But this paper suggests that rulebook is missing a huge piece of the puzzle. The authors built a new, super-complex simulator that treats protein evolution like a continuous, real-time dance rather than a series of frozen snapshots. They used a "Potts model," which is basically a mathematical map of how every amino acid (the building blocks of proteins) interacts with every other one. Think of it as a giant social network where every protein part has a best friend, a rival, and a few people it just tolerates. If you change one part, the whole network shudders.

The Big Surprise: The "Slow Motion" Effect
When the authors ran their simulation, they found something weird. Even though this complex, "social" model of evolution is full of these interactions (called epistasis), the average speed at which proteins change looks almost exactly the same as the old, simple models. It's as if, on average, the protein is walking at the same pace whether it's walking alone or in a crowded crowd.

However, the experience of that walk is totally different. In the simple model, everyone walks at a steady, predictable pace. In the authors' complex simulation, the pace is a rollercoaster. Depending on the specific "context" (the company the protein is keeping at that moment), some parts of the protein freeze completely, while others sprint. The authors found that this "context-dependence" creates two types of protein positions:

The "Social Butterflies": These spots change wildly depending on who their neighbors are. In one context, they might be frozen; in another, they might be changing rapidly.
The "Lone Wolves": These spots barely care who is around them and evolve at a steady, predictable rate.

The "Time Travel" Glitch
Here is where it gets tricky for scientists trying to figure out history. The paper shows that if you use the old, simple models to guess how much time has passed between two proteins, you will almost always be wrong. Specifically, you will underestimate the time.

Imagine you are trying to guess how long a car trip took by looking at how much gas was used. If the car is driving in heavy traffic (the complex, epistatic world), it might use the same amount of gas as a car on an open highway (the simple world), but it will take much longer to get there because it's stuck in stop-and-go traffic. The authors' simulations show that because of these "traffic jams" caused by interactions between amino acids, proteins take longer to accumulate the same number of changes. If you ignore the traffic, you'll think the trip was short, when it was actually long.

How Sure Are We?
The authors didn't just guess this; they ran thousands of computer simulations using real protein families (like the "response regulator" and "beta-lactamase"). They started with 1,000 real natural sequences and watched them evolve for a simulated time of 20 units. They found that while the simple models work okay for the "Lone Wolf" spots, they completely fail to predict the behavior of the "Social Butterfly" spots.

The paper explicitly rules out the idea that these interactions are just a minor detail that can be ignored. They argue that while the average speed looks the same, the mechanism is fundamentally different. They also note that their results are based on simulations using a specific algorithm (the Gillespie algorithm) and that the "energy" of their model is a mix of functional constraints and statistical patterns, so they can't claim with 100% certainty that every single interaction they see is purely about biological function, though it likely plays a major role.

The Takeaway
The main finding is that epistasis (the "social" nature of proteins) acts like a brake on evolution, slowing things down in a way that simple models can't see. This leads to a systematic error where scientists think two proteins are more closely related in time than they actually are. The authors suggest that to get the true history of life, we need to stop treating protein parts as independent islands and start seeing them as a crowded, interconnected city where the neighbors always matter.

Technical Summary: Generative Continuous Time Model Reveals Epistatic Signatures in Protein Evolution

Problem Statement
Protein evolution is fundamentally shaped by epistasis, where the fitness effect of a mutation depends on the sequence context. Standard phylogenetic methods typically assume that sites evolve independently, a simplification that neglects these ubiquitous interactions. While modern generative models, such as Potts models, successfully capture epistatic effects and can generate functional sequences, they have historically relied on discrete-time Monte Carlo sampling. This discrete approach creates a barrier to integration with standard evolutionary biology frameworks, which are based on continuous-time Markov chains (CTMCs) and utilize branch lengths measured in expected substitutions per site. Consequently, there is a lack of tools that can simulate realistic, epistasis-driven protein evolution while remaining directly comparable to traditional phylogenetic models.

Methodology
The authors introduce a continuous-time sequence evolution model parameterized by a generative Potts model. The core methodology involves:

Model Definition: The equilibrium distribution $P_{eq}(a)$ is defined by a Potts model with fields ( $h$ ) and couplings ( $J$ ) inferred from multiple sequence alignments using the bmDCA algorithm. This distribution captures both single-site frequencies and pairwise correlations (epistasis).
Continuous-Time Dynamics: To bridge the gap with phylogenetics, the authors formulate the evolution as a CTMC. They employ the Gillespie algorithm to simulate trajectories. The transition rate matrix $Q$ is constructed to satisfy detailed balance with respect to $P_{eq}$ , using Glauber dynamics (though other dynamics like Metropolis or mutation-selection were tested and found to yield similar qualitative results).
Rate Scaling: A scaling factor $\Omega$ is applied to the time parameter $t$ such that the average rate of substitutions per position is one. This ensures that the time unit corresponds directly to the branch lengths used in standard phylogenetic inference.
Comparative Framework: The model is applied to specific protein families (e.g., PF00072, response regulator) and compared against two baselines: a profile model (which retains site-specific frequencies but assumes independence) and standard substitution models (e.g., LG+G4).

Key Contributions and Results
The study provides several quantitative insights into how epistasis alters evolutionary dynamics:

Generative Accuracy: Trajectories simulated under the continuous-time Potts model produce sequences that are statistically indistinguishable from natural sequences in terms of single-site frequencies, pairwise correlations, and principal component projections. In contrast, profile models and standard substitution models fail to maintain these higher-order statistics over long evolutionary times, often generating sequences with high statistical energy (indicating potential non-functionality).
Epistasis Slows Evolution: The presence of epistasis significantly slows down the rate of divergence. While the average substitution rate per site is similar between the Potts and profile models, the Potts model requires a longer evolutionary time to reach a specific Hamming distance from the ancestral sequence. This is attributed to a more restricted set of accessible mutations and a higher frequency of reverse mutations in the epistatic landscape.
Context-Dependent Rate Heterogeneity: A central finding is that while the average site-specific evolutionary rate $\langle R_i \rangle$ is nearly identical between epistatic and non-epistatic models, the context-dependent rate $R_i^C(a_{\setminus i})$ varies drastically.
- Some positions are "frozen" (rate near zero) in specific contexts but highly evolvable in others.
- Other positions evolve independently of the context.
- The authors developed a ranking method to identify sites based on the variance of their context-dependent rates. They found that highly context-dependent sites often cluster structurally (e.g., at binding interfaces), whereas context-independent sites show no such structural clustering.
Systematic Bias in Evolutionary Distance Inference: When evolutionary distances are inferred using standard site-independent models (like profile models) on data generated by the epistatic Potts model, there is a systematic underestimation of the true evolutionary time.
- This bias is not uniform; it is significantly stronger for sequences starting in "slow" contexts (low-energy, highly constrained sequences).
- The bias is driven primarily by the subset of sites that are strongly context-dependent. When inference is restricted to context-independent sites, the bias largely disappears.
- In four-taxon tree experiments, this leads to the underestimation of branch lengths and tree diameters, particularly for trees rooted in slow-evolving contexts.

Significance
The paper claims to bridge the gap between modern generative models of proteins and classical phylogenetics. By formulating epistatic evolution in continuous time, the authors provide a tool that can simulate realistic evolutionary trajectories compatible with standard phylogenetic units. The work demonstrates that neglecting epistasis does not merely affect the variance of evolutionary rates but introduces a systematic bias in the inference of evolutionary timescales. Specifically, it reveals that standard methods underestimate the time required for sequences to diverge when epistasis is present, particularly for highly constrained sequences. This suggests that current phylogenetic reconstructions may be systematically compressing branch lengths and distorting the perceived timing of evolutionary radiations, especially in clades subject to strong functional constraints. The authors position this work as a necessary step toward more realistic models for molecular evolution and a foundation for correcting biases in phylogenetic inference.

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