# Evolutionary Advantage of Diversity-Generating Retroelements in Switching Environments

Léo Régnier, Paul Rochette, Raphaël Laurenceau, David Bikard, Simona

Cocco, Rémi Monasson

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Léo Régnier, Paul Rochette, Raphaël Laurenceau, David Bikard, Simona Cocco, et al. Evolutionary Advantage of Diversity-Generating Retroelements in Switching Environments. 2026. hal-05517354

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# Evolutionary Advantage of Diversity-Generating Retroelements in Switching Environments

Léo Régnier,¹ Paul Rochette,² Raphaël Laurenceau,² David Bikard,² Simona Cocco,¹ and Rémi Monasson¹

¹Laboratory of Physics of the Ecole Normale Supérieure,
PSL & CNRS UMR8023, Sorbonne Université, Paris, France

²Synthetic Biology, Institut Pasteur, Université Paris Cité, Paris, France.

Diversity-Generating Retroelements (DGRs) create rapid, targeted variation within specific genetic regions in phages and bacteria. They operate through stochastic retro-transcription of a template region (TR) into a variable region (VR), which typically encodes ligand-binding proteins. Despite their prevalence, the evolutionary conditions that maintain such hypermutating systems remain unclear. Here we introduce a two-timescale framework separating fast VR diversification from slow TR evolution, allowing the dynamics of DGR-controlled loci to be analytically understood from the TR design point of view. We quantity the fitness gain provided by the diversification mechanism of DGR in the presence of environmental switching with respect to standard mutagenesis. Our framework accounts for observed patterns of DGR activity in human-gut *Bacteroides* and clarifies when constitutive DGR activation is evolutionarily favored.

**Introduction.** Diversity-Generating Retroelements (DGRs), first identified in the *Bordetella* phage [1], represent an important class of hypermutation systems that enable rapid adaptation in phage and microbial populations. Recent work has clarified their structural organization [2, 3], documented their diversity across microbial genomes [4–6], elucidated their biological roles [6, 7], and highlighted their potential for engineering applications [8, 9]. DGRs introduce mutations by copying a template region (TR) into a target variable region (VR) through an error-prone reverse transcription process [10–12] (Fig. 1(a)). The mechanism is inherently asymmetric. While the TR slowly evolves through standard mutagenesis, the VR is quickly and repeatedly diversified and overwritten, with mutations concentrated at positions corresponding to a specific TR nucleotide (typically, Adenine). This localized hypermutating system can generate extensive phenotypic diversity at loci involved in host recognition, cell–cell interactions, or environmental sensing [5–7].

Recent studies on the human gut microbiome [5, 13, 14] illustrate the benefits of high DGR mutation rates [15]. Infants are born with very few DGRs in their microbiome, but by one year, they reach adult-like levels. Remarkably, ~72% of the 388 parental DGRs were observed to switch their VR during this period. Some *Bacteroides* species (notably *B. ovatus* and *B. finegoldii*) maintain high DGR activity: 13–40% of VRs diverge within 14 days. This suggests that diversification operates on a dedicated timescale far faster than standard mutation.

While environmentally regulated DGR activation might explain infant microbiome diversification and context-dependent activity in many species [5, 14, 16], it does not address a more fundamental question: under what conditions is a DGR-based hypermutating system actually advantageous and thus maintained rather than lost? Because DGR activity depends on a functional reverse transcriptase, mutations that impair this enzyme

could in principle silence diversification, highlighting a potential evolutionary fragility of such systems. The long-term selective benefit of such constitutively active diversification is not immediately obvious and requires explicit mechanistic modeling. Although hypermutation has been studied in other biological contexts [17–19], the evolutionary forces governing the dynamics in systems with the distinctive architecture of DGRs remain largely unexplored.

<figure><img src="image_2.png" /><figcaption>FIG. 1. Illustration of the DGR system. (a) The main components of a DGR are shown. The template region (TR, blue) is copied into the variable region (VR, orange) at rate $v$, with mutations introduced specifically at adenine (A) positions. The TR mutates at rate $\mu$. The fitness of an individual—defined by its (TR, VR) pair—depends solely on its VR sequence. (b) Sketch of a fluctuating environment. Every interval $\tau$, the environment switches to favor a different target (VR) sequence. The population frequency $f(t)$ of the fittest sequence (with fitness $S_e$) then adapts to these shifts.</figcaption></figure>

**DGR model and evolutionary dynamics.** Hereafter, a DGR is defined by the nucleotidic sequences of its **VR** and **TR**, of equal length *L* (Fig. 1(a)). In the following, we use bold symbols when refering to the precise sequences of nucleotides. Under standard point mutation process, each nucleotide in the **VR** and **TR** may
mutate with rate (probability per unit of time) $\mu$. In addition, the **VR** varies due to the DGR replacement and hypermutation mechanism, which happens at rate $\nu \gg \mu$. In practice, we assume that a new **VR** replaces the current **VR** after being randomly drawn from the **TR**. In this random process, nucleotides C, G, T in the **TR** are copied as such in the **VR** at the same positions, while A's are equally likely to mutate into C, G, T or to remain unchanged [20].

As proteins are expressed from the **VR**, the fitness (growth rate) $S(\mathbf{VR}, t)$ of the full DGR depends exclusively on the nucleotide sequence of the **VR**. For simplicity, we assume that $S$ is additive across sites (no epistasis) [21, 22], and that fitness contributions depend solely on the nucleotide identity at each position rather than the encoded amino acids. The fitness at time $t$ is given by

$$S(\mathbf{VR}, t) = \sum_{i=1}^{L} s_{i, \mathbf{VR}_i}(t), \quad (1)$$

where $s_{i,q}$ denotes the contribution of a nucleotide $q = A, T, C, G$ at position $i$ along the **VR** sequence. To model environmental changes, we assume below that the $s$ factors vary with time $t$ on a characteristic switching time $\tau$, as represented in Fig. 1(b).

A key insight is that selection acts on the DGR on two distinct timescales:

*   **Fast (at the VR level):** **VR** sequences diversify through repeated overwriting by the TR at rate $\nu$ and compete under environmental fluctuations. The pattern of Adenines (A) on the **TR** determines which **VR** genotypes are accessible.

*   **Slow (at the TR level):** the **TR** sequence mutates at the much lower per-base rate $\mu$. **TR** variants are indirectly selected through the **VR**'s they generate: the mean fitness of the **VR** population shapes the long-term evolutionary potential of the **TR**.

This separation of timescales is strongly supported by experimental observations (see below) and allows for an analytical treatment of the DGR dynamics. In the following, we first analyze **VR selection**, showing how the effective growth rate depends on the DGR replacement rate $\nu$ relative to environmental switching $\tau$. Second, we address **TR selection**, explaining how selection acts on the TR sequence itself, and in particular, we derive the conditions under which the diversifying nucleotide (Adenine) is evolutionarily stable. In this framework, loss of DGR activity is modeled exclusively through selection acting on the TR sequence; we do not consider deactivation arising from mutations in the reverse transcriptase. Lastly, we explore the effects of the TR length on the selection dynamics, identifying the regimes in which DGR-mediated diversification is most advantageous. This framework explains experimental

observations and identifies conditions under which DGRs are maintained, regulated, or lost in natural systems.

**Experimental background.** The authors of [14] studied five Bacteroides DGR, and found two species that maintained a high level of DGR activity both *in vitro* and *in vivo* in the absence of selective pressure (as evidenced by sustained activity in monoconolized mice). After $t=14$ days, a fraction $\phi$ of VR ranging from 13 to 40% had diverged from the initial ones.

Within our modeling framework, this fraction is simply given by $\phi = 1 - e^{-\nu t}$ due to the lack of selection ($s$ independent of the nucleotide content), see details in Supplementary Materials (SM) [23] Sec. 1. Matching this formula with the experimentally observed fraction, we infer the diversification rates for these two Bacteroides to be between $\nu = 10^{-2} \text{ day}^{-1}$ and $\nu = 4 \times 10^{-2} \text{ day}^{-1}$, see Table I.

<table><caption>TABLE I. Approximate rates relevant for DGR dynamics in *Bacteroides*.</caption><thead><tr><th>Quantity</th><th>Notation</th><th>Range (day<sup>-1</sup>)</th></tr></thead><tbody><tr><td>DGR diversification rate</td><td>ν</td><td>10<sup>-2</sup> - 4 × 10<sup>-2</sup></td></tr><tr><td>Growth rate</td><td>S<sub>e</sub></td><td>1 - 10</td></tr><tr><td>Growth rate per variable site</td><td>s</td><td>10<sup>-1</sup> - 10</td></tr><tr><td>Spontaneous mutation rate<br>per base</td><td>μ</td><td>10<sup>-9</sup> - 10<sup>-8</sup></td></tr><tr><td>Mutation rate for full VR</td><td>μL</td><td>10<sup>-7</sup> - 10<sup>-5</sup></td></tr></tbody></table>

As a point of comparison, the actual doubling time of Bacteroides is around 2 hours in optimal conditions and up to a day [24, 25] (in situations where selection plays a role), resulting in a growth rate $S_e$ between 1 and $10 \text{ day}^{-1}$. Because the spontaneous mutation rate per generation per base is about $10^{-9}$, the rate $\mu$ of spontaneous mutation per base is of the order of $10^{-9} - 10^{-8} \text{ day}^{-1}$ [25, 26]. Consequently, the rate $\mu L$ of spontaneous mutation apparition (without DGR) in the region where selection occurs (the VR encoded part, approximately 100-1000 bases) ranges between $10^{-7}$ and $10^{-5} \text{ day}^{-1}$. The effective number of sites $L$ should be restricted to codons containing adenines within the hypermutation interval, not the entire sequence, as the phenotype is defined by the protein product encoded by the VR. This effective number $L$ of variable sites can be estimated from the data of Ref. [5] to be between 1-8 (interval which contains 95% of the data, see SM Sec. 2). Consequently, there is a huge time scale separation (at least 3 orders of magnitude) between the estimated DGR rate, the replication rate and the rate at which spontaneous mutations appear in a sequence: $\mu L \ll \nu \le S_e$. This is a strong indication that DGR mutation and selection work on a time scale different from the classical selection mechanism.
### Fast dynamics of the VR population at fixed TR.

On time scales < 1/(μL), the TR is effectively fixed, while the VR evolves under selection and is periodically overwritten via DGR-mediated diversification at rate ν. Because the model assumes independent sites, we adopt a per-base perspective: we first focus on a single site (L = 1), and then generalize the results to L > 1.

Nucleotides A, C, G, T are labeled as 1, 2, 3, 4 (Q = 4 possible states). For the sake of simplicity, we consider a cyclic environmental dynamics of period Qτ. Here τ is the switching time. For t ∈ [q't, (q' + 1)τ] with q' = 1, ..., Q, the preferred nucleotide is q*(t) = q' and confers a fitness advantage S(q',t) = s > 0 (as L = 1) with respect to the other Q − 1 ones, i.e. s₁,q(t) = sδq,q*(t) in Eq. (1). This model corresponds to a classical framework of phenotypic switching [27–29].

The population sizes n_q(t) of VR carrying nucleotides q at time t obey the Wright-Fisher coupled equations:

$$ \frac{dn_q(t)}{dt} = (s \delta_{q,q^*(t)} - \nu) n_q(t) + \frac{\nu}{Q} \sum_{p=1}^{Q} n_p(t). \quad (2) $$

Standard mutagenesis (no DGR) is described by the same equations with ν replaced with μ. This set of linear differential equations can be solved as explained in SM Sec. 3. Formally speaking, after each cycle of duration Qτ, the Q-dimensional vector of population sizes, n = {n_q}, is multiplied by a (Q×Q)-dimensional matrix M, whose entries encode the transition rates appearing in Eq. (2), see details in SM. Due to the periodicity of the time evolution, the effective growth rate is given by S₁ = ln Λ/(Qτ), where Λ is the largest eigenvalue of M. This growth rate is plotted in Fig. 2(a) as a function of the mutational rate and is in excellent agreement with the results of numerical simulations for finite population size.

As observed from Fig. 2(a), the growth rate S₁ is maximal when the rate ν is roughly equal to 1/τ [27, 28], and largely exceeds its asymptotic value s/Q at both large and small ν. In each of these limits, a fixed (small ν) or a random (large ν) nucleotide is best for a fraction 1/Q of the time, explaining this expression. Notice that, for large τ, S₁ decays slowly as the mutation rate ν is lowered below its optimal value ~ 1/τ: the system has time to adapt to the slowly fluctuating environment even though mutations become rarer.

The above results remain qualitatively correct for L > 1 variable sites, i.e. for a TR with L Adenines. We assume that two successive environments select for sequences with different states at every position. Without DGR, spontaneous mutations happen at rate μ independently on each site, and the effective fitness is just LS₁(μ). With the DGR mechanism, all VR bases are simultaneously redrawn at each mutational event, introducing correlations between the sites. The effective fit-

<figure><img src="image_3.png" /><figcaption>FIG. 2. VR selection. (a) Effective growth rate as a function of the switching rate ν for L = 1. (b) Same as (a) for length L = 10. Takeover times for the DGR system estimated from simulations (more than 99.5% of the population) (c) and from Eq. (4) (d) vs. diversification rate ν for L = 1. Parameter values: Q = 4, s = 1, population size = 10⁸, μ = 10⁻⁵.</figcaption></figure>

ness with diversification rate ν reads

$$ S_L(\nu) \approx \max \left( \frac{sL}{Q}, sL - \nu + \frac{\nu}{Q^L} + \frac{\ln\left(\frac{2\nu}{sLQ^L}\right)}{\tau} \right), \quad (3) $$

see SM Sec. 3 for details about the derivation. Results are shown in Fig. 2(b), and are qualitatively similar to the L = 1 case.

Our analysis unveils the condition under which a population of organisms equipped with DGR ultimately grows much faster than one lacking it (see also phase diagrams in SM Sec. 3). In particular, when ν ~ 1/τ, there is a consistent advantage to the DGR mechanism regardless of the value of S_L. However, if S_L is too small, the takeover time of the DGR-equipped population,

$$ t_{\text{takeover}} \approx \frac{1}{S_L(\nu) - L S_1(\mu)} \quad (4) $$

may become prohibitively long. Two conditions must be met to avoid large t_{takeover} values, as illustrated in Fig. 2(c) and (d) for L = 1. First, ν should not be larger than s, as replacement by DGR-generated variants would occur so frequently that the system would gain little advantage. Second, τ must not be exceedingly larger than 1/s, as low spontaneous mutation rate μ would be sufficient to track very slow environmental changes, as mentioned above. These conditions for
the success of DGR appear to be met in experimental situations [14], where we infer $\nu \lesssim S_e$. This condition stays true considering $L$ below 8 as the effective number of variable sites and $s = S_e/L$. (Table I), ensuring both sufficient positive selection and high growth rate.

**Slow dynamics of the TR.** We now allow the nucleotides in the TR to mutate at a rate $\mu$. A detailed treatment of the Wright-Fisher equations describing the evolution of the population of (TR, VR) pairs in a fluctuating environment is provided in SM Sec. 4. We hereafter summarize the main results and provide intuition that supports them.

Informally speaking, a TR sequence can be evolutionarily fit in two distinct ways. First, it may carry many A's that allow the VR to diversify and adapt to any environment. However, continual diversification of the VR at a rate $\nu$ leads to the frequent loss of adaptation in stable contexts. Second, the TR may carry non-Adenine nucleotides that are well-adapted to the current environment; their copies in the VR then provide a fitness advantage and are far less mutable (rate $\mu \ll \nu$). Analysis of the evolutionary dynamics of a population of sequences of length $L=1$ allows us to assess how these two factors may concur to push the TR to eventually loose its Adenines [30]. In particular, we find that the overwhelming majority of TR sequences either carry a A, or do not carry a A but are adapted to the current environment. The fraction of the latter is approximately given by

$$f_{\text{not A}} = \frac{\tilde{\mu}}{\tilde{\nu}^2 \tau} (e^{\tilde{\nu}\tau} - 1 - \tilde{\nu}\tau), \quad (5)$$

where $\tilde{\mu} = Q\mu$ and $\tilde{\nu} = (1 - \frac{1}{Q})\nu$. We deduce from the previous formula the existence of three regimes depending on the value of $\tilde{\nu}$ and $\tau$: $f_{\text{not A}} \sim \tilde{\mu} \exp(\tilde{\nu}\tau)/(\tilde{\nu}^2\tau)$ for $\nu \gg 1/\tau$, $\tilde{\mu}/\tilde{\nu}$ for $\nu \sim 1/\tau$, $\tilde{\mu}\tau$ for $\tilde{\nu} \ll 1/\tau$. We compare this approximation with simulations in Fig. 3(a), (b), and (c), confirming its validity. Imposing $f_{\text{not A}} \ll 1$ gives the conditions that allow the DGR mechanism to be favorable with respect to standard spontaneous mutational processes. If the switching time $\tau$ of the environment is tightly regulated, that is, if the changes in the environment are periodic, the DGR mechanism gives a fitness advantage provided that

$$\tilde{\mu} \ll \tilde{\nu} \sim \frac{1}{\tau}. \qquad (6)$$

However, the condition above is not sufficient in the case of an environment with a wide distribution of switching times. Adenines in the TR become strongly disfavored if $\tau$ may significantly exceed $1/\tilde{\nu}$, see Eq. (5). For example, consider the case of $\tau$ algebraically distributed, with typical time-scale $\tau_0$ and exponent $\beta$, i.e. with cumulative probability $P(\tau \ge t) = (\tau_0/t)^\beta$. The adenine is lost with a probability of the order of $(\tilde{\nu}\tau_0)^\beta$, condition which corresponds to $f_{\text{not A}} \sim 1$.

This theoretical scaling is confirmed in Fig. 3(e) by simulations estimating the probability $P_{A \text{ lost}}$ that, by the end of the environmental bout of random duration $\tau$, the population of TR carrying A no longer dominates (fraction smaller than $1/2$). This example shows that strongly (algebraically) variable environmental duration can be disastrous for the maintenance of the DGR mechanism, despite the fact that the typical timescale $\tau_0$ of these environmental duration still obey $\tilde{\nu}\tau_0 \approx 1$.

<figure><img src="image_4.png" /><figcaption>FIG. 3. TR selection. (a) to (c). Fraction $f_{\text{not A}}$ of sequences in the TR population carrying non-adenine nucleotides. (a) $\nu\tau \gg 1$ ($\tau = 200$, $\mu = 10^{-7}$, $10^{-6}$, and $10^{-5}$); (b) $\nu\tau = 1$ ($\nu = 1/\tau = 10^{-2}$, $5 \cdot 10^{-2}$ and $10^{-1}$); (c) $\nu\tau \ll 1$ ($\mu = 10^{-6}$, $10^{-5}$ and $10^{-4}$ and $\tau = 5$). Increasing values of $\nu$ or $\mu$ are shown in blue, orange and green successively. Dashed lines correspond to the expressions of Eq. (5) in the respective regimes. (d) Probability $P_{A \text{ lost}}$ that the fraction of TR with an A (initially, equal to 1) is lower than $1/Q = 1/4$ as a function of $\nu\tau_0$ where $\tau_0$ is the typical duration of an environment. Dashed line corresponds to $(\nu\tau_0)^\beta$. We take $\mu = 10^{-4}$, $\beta = 1/2$ and $\nu = 10^{-3}$, $10^{-2}$ and $10^{-1}$. (e) Phase diagram in the $(L, \tau)$ plane. Dashed lines show the theoretical prediction for $\tau_c$ in Eq. (7) Blue points are determined by solving the dynamics for infinite population size. Parameters are $\mu = 10^{-6}$ and $\nu = 10^{-2}$.</figcaption></figure>

The above analysis can be extended to $L > 1$. To do so, we examine the conditions that lead to the loss of sequences with all $L$ sites occupied by adenines. As seen previously, this maximally-diversifying TR competes with the best adapted sequence in the current environment, whose population grows as $(\frac{\mu L}{Q\nu})^L e^{\nu t}$: the prefactor corresponds to the $L$ spontaneous mutation events
needed to reach the best adapted **TR**, and the exponen-
tial term the gain in losing the DGR (SM Sec. 4). In
addition, random sequences invade the population, with
a fraction growing as $e^{\mu L t}$. As a consequence, the all-$L$
DGR is outcompeted when the time $\tau$ exceeds

$$
\tau_c(L) = \min \left[ \frac{L}{\nu} \ln \left( \frac{\nu Q}{\mu L} \right), \frac{1}{\mu L} \right] . \quad (7)
$$

see Fig. 3(e). The phase diagram in the (L, τ) plane
shows the existence of three regimes, corresponding to
the three types of dominant TR sequences. The DGR
phase is best stable over time for the optimal length

$$
L^* \sim \sqrt{\nu/\mu}, \qquad (8)
$$

up to a logarithmic factor. This result stresses that
having extended variable regions in the **TR** can be
beneficial for the resilience of the DGR system, but
that too long **TR** are not robust against spontaneous
mutations. This is consistent with the relatively low
number *L* (up to 10²) of variable sites found in [5, 14]
and displayed in SM Sec. 2. We stress that our analysis
focuses on the potential loss of DGR due to the TR
dynamics only. Other mechanisms, such as reverse
transcriptase inactivation, are left for future studies.

**Conclusion.** Our analysis unveils to what extent and conditions the DGR system offers fitness advantages over standard spontaneous mutation processes in frequently changing environments. If the environmental context is stable or does not vary quickly enough, the DGR becomes progressively inoperant through the loss of the Adenines in the **TR** after some critical time we have estimated. Our study therefore emphasizes the need for being able to tune the activation and desactivation of the DGR mechanism [14, 31], despite its inherent cost *e.g.* due to the sensing of the environment fluctuations. Organisms can enable/disable the DGR apparatus in multiple ways: some might down-regulate diversification when it is unnecessary [5, 14], others may effectively silence the system through mutations that impair the reverse transcriptase [11, 14, 32], while engineered systems can bypass the VR entirely and reintegrate cDNA directly into the TR [8]. An additional relevant context include coevolving DGR populations [33] in which the environmental switching time $\tau$ is replaced by the inverse DGR switching rate $1/\nu$ of the other population. We could also consider scenarios in which finite population size modulates the benefits of hypermutation [18]. All of these possibilities constitute promising directions for future modeling efforts.

Future analysis should take into account the existence
of mutational biases inherent to DGRs and their conse-
quences for protein-level selection. Because selection acts
on amino acids, the position of adenines within codons
can substantially affect the number of accessible amino-
acid variants (for example, AAC codons can diversify into

15 amino acids, whereas having an adenine only at the
third position yields negligible variation). These posi-
tional effects, together with biases in DGR A-to-N sub-
stitutions [8, 9, 14], shape the landscape of possible re-
ceptor or ligand variants. Exploring how diversification
proceeds under these alternative architectures and muta-
tional regimes would be interesting.

**Funding informations.** This work was supported by the French National Research Agency (ANR) under Grant No. ANR-23-CE45-0034.

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[32] S. S. Naorem, J. Han, S. Wang, W. R. Lee, X. Heng, J. F. Miller, and H. Guo, Proceedings of the National Academy of Sciences of the United States of America **114**, E10187 (2017).

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