Dependent-Site Models in Bio Evolution

• Dependent-site models are probabilistic frameworks that integrate context effects, epistasis, and structural constraints to better model biological sequence evolution.
• They employ continuous-time Markov chains, factor graphs, and Monte Carlo methods to evaluate likelihoods and infer complex evolutionary parameters.
• These models enhance phylogenetic inference by reducing search domains, improving sequence alignment, and accurately capturing compensatory changes missed by independent models.

Dependent-site models of biological sequence evolution constitute a class of probabilistic frameworks in which the substitution process at each site explicitly depends on its current state and the states of neighboring sites. Unlike traditional independent-site models, which treat mutations at different sequence positions as conditionally independent given a phylogeny and evolutionary parameters, dependent-site models accommodate context effects, epistasis, co-evolution, and structural constraints. These models are increasingly significant in evolutionary biology, phylogenetics, bioinformatics, and protein modeling, as they better capture the multivariate dependency structure imposed by biochemical, structural, and functional imperatives in macromolecules.

1. Mathematical Formulations of Dependent-site Models

Dependent-site models are most generally formalized as continuous-time Markov chains (CTMCs) on the sequence state space $\mathscr{A}^n$ (where $\mathscr{A}$ is the alphabet, e.g., $\{\mathrm{A},\mathrm{C},\mathrm{G},\mathrm{T}\}$ for DNA). At time $t$, the sequence is $x=(x_1,\dots,x_n)\in\mathscr A^n$. The key defining feature is that the instantaneous rate of change at position $i$ from base $x_i$ to $b\ne x_i$ depends not only on $x_i$ but on a "context" $\tilde x_i$ comprising $k$ neighboring sites:

$\widetilde Q_{x,x'} = \begin{cases} \,_i(b;\tilde x_i) & \text{if $x'$differs from$x$only at site$i$with$x'_i=b\ne x_i$}, \ -\sum_{b\ne x_i}\!_i(b;\tilde x_i) & \text{if $x'=x$}, \ 0 & \text{otherwise}. \end{cases}$

Here, $_i(b;\tilde x_i)$ is the context-dependent instantaneous rate, and $\tilde x_i$ is the tuple specifying the states of $x_i$’s context (e.g., adjacent nucleotides or amino acids). This formulation, generalized across various instantiations, recovers independent-site models at $k=0$, and includes pairwise-contact models, Potts models, and factor-graph approaches when more complex dependencies are required (Mathews et al., 25 Jul 2025, Bordner et al., 2013, Budzynski et al., 2022).

Structural constraints, such as those arising from protein folding, can be encoded as pairwise factors $C(a_i, a_j)$ for neighboring residues $i, j$ in the three-dimensional protein contact graph, yielding a factor-graph joint distribution over the sequence with both site-specific and pairwise terms:

$$p(a_1,\dots,a_N) = \frac{1}{Z} \prod_{i=1}^N \phi_i(a_i)\; \prod_{(i,j)\in E} \psi_{ij}(a_i,a_j),$$

where $E$ is the set of contacting residue pairs, and $Z$ is a normalization constant (Bordner et al., 2013).

2. Likelihood Evaluation and Statistical Inference

Computing likelihoods under dependent-site models is notably challenging due to the exponential size ($a^n$) of the state space. For endpoint-conditioned processes (fixed $x(0)=x$, $x(T)=y$), the transition probability is represented as a sum/integral over possible mutation paths $\mathcal{P}$:

$$p_{(T,\widetilde Q)}(y \mid x) = \int_{\mathscr P}\!\widetilde P_{(T,\widetilde Q)}\left(y,\mathcal P \mid x\right)\, \nu(d\mathcal P),$$

where each path consists of a sequence of single-site mutations subject to the context-dependent rates, and $\widetilde P$ is the path density including waiting times and rate multipliers (Mathews et al., 11 Nov 2025, Mathews et al., 15 Aug 2025).

Approximate inference strategies include:

• Uniformization/Poisson-jump chains: The generator $\widetilde Q$ can be "uniformized" at total rate $\lambda \ge \max_x \Lambda(x)$, where $\Lambda(x)$ is the sum of transition rates out of $x$. Transition probabilities can then be written as weighted sums over the number of mutations ("jumps") using the jump-chain $P$ and Poisson arrival process:

$$\mathcal L(T \mid x, y) = e^{-\lambda T}\sum_{m\ge r} \frac{(\lambda T)^m}{m!}\left(P^m\right)_{x, y}$$

with $r = d_H(x, y)$ the Hamming distance (Mathews et al., 25 Jul 2025).

• Factor graphs and Belief Propagation: For protein evolutionary models incorporating structure, factor-graph representations allow the application of sum-product algorithms (BP/TEP) to compute approximate marginal distributions and likelihoods (Bordner et al., 2013).
• Importance Sampling and Sequential Monte Carlo (SMC): Efficient randomized methods for likelihood estimation under dependent-site models use ISM (independent-site model) endpoint-conditioned path generators as proposals and reweight by the context-dependent likelihood ratios. In the SMC approach, a tempered sequence of models interpolating between the ISM and full dependent-site model enables robust marginal likelihood estimation, with explicit finite-sample error bounds (Mathews et al., 15 Aug 2025, Mathews et al., 11 Nov 2025).

3. Theoretical Properties: Posterior Concentration, Mixing, and Complexity

Recent work has established rigorous bounds for inference under dependent-site models:

• Posterior Concentration on Divergence Time (T): For two observed sequences $x,y$ with Hamming distance $r$, the posterior for $T$ (divergence time) concentrates below $O(\hat p \log n)$, where $\hat p = r/n$ and provided $r\log n/n = o(1)$ for general dependent-site models. In constant-rate independent-site models (JC69-like), the bound sharpens to $T < c_1 \hat p$ with exponentially vanishing mass above this threshold (Mathews et al., 25 Jul 2025).
• Likelihood Estimation Complexity: Importance sampling estimators for context-dependent models exhibit sample complexity exponential in $r$ (number of observed mutations), but polynomial in $n$ (sequence length) in the regime $r \ll n$. In the SMC context, tempering reduces the exponential dependence to the largest contiguous island $r_\star$ of mutations, yielding a fully polynomial randomized approximation scheme (FPRAS) for cases with localized mutational clusters (Mathews et al., 15 Aug 2025, Mathews et al., 11 Nov 2025).
• MCMC Mixing Time: Blockwise component-wise Metropolis proposals targeting the endpoint-conditioned path space of dependent-site models mix in time $O(\exp(c\,r_\star))$ when initialized with a warm start, with $r_\star$ the largest cluster of sites sharing context (Mathews et al., 11 Nov 2025).

4. Epistasis, Structural Constraints, and Biological Consequences

Epistatic interactions are central to dependent-site evolutionary models. Protein evolution studies demonstrate that fitness effects of substitutions are highly contingent on background genotype and entrenched by subsequent mutations. Pairwise and higher-order epistasis, modeled quantitatively by evaluating changes in log-fitness across substitution histories, is ubiquitous even under purifying selection, and classical independent-site models (GY94, PAML) miss strong history-dependent constraints (Shah et al., 2014).

Empirical studies confirm the importance of direct site-site correlations for contact prediction and functional annotation in proteins. Mechanistic codon models with partial correlation scoring of substitution events achieve contact prediction performance comparable to maximum-entropy Potts models, and explicitly capture concurrent and compensatory substitution effects (Miyazawa, 2012).

Structural constraints in factor-graph models (encoding side-chain contacts and solvent accessibility) further enhance the realism and fit of evolutionary models. Likelihood comparisons across thousands of protein families show that models incorporating pairwise-structural factors decisively outperform independent-site rate-matrix models (Bordner et al., 2013).

5. Parameter Estimation, Model Selection, and Computational Strategies

Parameter estimation in dependent-site models leverages maximum pseudolikelihood approaches, direct coupling analysis (DCA), data-squashing MCMC for mixture models, and regularization schemes for factor functions. In factor-graph models, pseudolikelihood maximization over conditional distributions given context neighbors yields consistent and scalable parameter estimates (Bordner et al., 2013). In infinite mixture models, nonparametric Bayesian methods (Dirichlet process, hierarchical DP, infinite HMM) simultaneously infer the number and assignment of site categories, transition matrices, and evolutionary parameters, with high-dimensional MCMC sampling and parallelized likelihood evaluation (Gill et al., 8 Dec 2024).

Empirical studies reveal that infinite hidden Markov mixture models yield superior marginal likelihoods and tree reconstructions compared to both finite mixtures and independent-site DPs, particularly in cases with spatially correlated heterogeneity in sequence alignments (Gill et al., 8 Dec 2024).

6. Practical Applications and Implications for Phylogenetic Inference

Dependent-site models have significant consequences for practical phylogenetics and sequence analysis:

• Reducing Search Domains: Posterior contraction results allow substantial narrowing of divergence time search intervals, accelerating MCMC and optimization in tree inference under explicit site-dependence (Mathews et al., 25 Jul 2025).
• Alignment and Homology Detection: Message-passing algorithms derived from small-coupling expansions of Potts models improve alignment quality in regions dominated by co-evolutionary constraint, especially when standard profile HMMs fail (Budzynski et al., 2022).
• Likelihood Screening and Proposal Limiting: Inference algorithms benefit by limiting computational effort to plausible $T$ values or evolutionary trajectories, avoiding unnecessary likelihood calculations for implausible hypotheses (Mathews et al., 25 Jul 2025, Mathews et al., 11 Nov 2025).
• Efficient Marginal Likelihood Estimation: Randomized schemes (IS, SMC) tailored to the combinatorial structure in context-dependent models enable tractable marginal likelihood estimation and Bayes factor computation for long sequences and complex models (Mathews et al., 15 Aug 2025, Mathews et al., 11 Nov 2025).

7. Extensions, Open Questions, and Future Directions

Dependent-site models continue to evolve with advances in statistical approximation, nonparametric inference, integration of biophysical constraints, and computational scaling strategies. The development of higher-order interaction terms, integration with structural annotations beyond contacts and accessibility (e.g., secondary structure, conformational dynamics), and model selection in infinite-mixture frameworks are prominent directions. Theoretical work on the limits of approximation methods, posterior consistency, and the impact of epistasis on evolutionary dynamics and inference remains active, with rigorous complexity and mixing analyses now informing both applied and theoretical research.

The consensus is that modeling explicit site dependency, whether due to context effects, epistasis, or structure, provides measurable improvements in accuracy and interpretability of evolutionary inference, and prompts continued refinement of methodologies to overcome the inherent computational challenges.