Directed evolution and antibody engineering lack a measurable, predictive index of evolvability. Schulz et al. (2021) showed that the response to selection of antibody libraries in quantitative phage-display experiments is captured by two parameters, μ\mu and σ\sigma, of a log-normal fit to the library's distribution of enrichment values, and proposed the spread σ\sigma as a measure of the "selective potential" of a library: libraries built on germline antibody scaffolds have larger σ\sigma than libraries built on affinity-maturated scaffolds. This characterization, however, concerns selection without mutation. The authors explicitly identify the next step as extending the framework to quantify evolvability proper, i.e., the response to successive cycles of selection and mutation, which would test whether σ\sigma is a general quantitative index of evolvability for biomolecular libraries. No prior work extends the (μ,σ)(\mu, \sigma) framework to iterated selection and mutation. The problem is therefore open.

Background

Evolvability - the capacity of a biological system to generate heritable, adaptive phenotypic variation - has been discussed theoretically since the 1990s (Wagner and Altenberg 1996), and in quantitative genetics it is often identified with the amount of heritable variation available for directional selection (Houle 1992). For biomolecules, however, theory has not translated into an experimental assay: as Schulz et al. (2021) put it, a major limitation is "the absence of a measurable index of evolvability quantifying evolutionary responses to compare to biophysical or structural quantities". Antibodies are a favorable testing ground because they are amenable to controlled in vitro evolution. An antibody's sequence splits into a genome-encoded scaffold and highly diverse, non-genome-encoded sites concentrated in the complementarity determining regions (CDRs, the hypervariable loops that mainly contact the target); during the immune response, somatic mutations accumulate in a process called affinity maturation, which raises binding affinity but is not transmitted across generations. In a quantitative phage-display experiment, a library of antibody variants is displayed on phages (viruses that infect bacteria) and selected for binding to a target; the enrichment of a sequence, the ratio of its frequency after versus before one selection round, is measured by high-throughput sequencing. The precursor study (Boyer et al. 2016) showed that selections from pools of randomized proteins exhibit a reproducible statistical structure, with the top enrichments following the tail of a log-normal distribution. Schulz et al. (2021) developed this into a two-parameter description: the distribution P(s)P(s) of enrichment values within a library is well fitted by a log-normal distribution with parameters μ\mu and σ\sigma, where σ\sigma (the logarithmic standard deviation) captures intra-library differences in response to selection and μ\mu describes inter-library differences. Log-normality is empirically robust, stable under iteration of selection, and justified by an additive model of interaction energies. Comparing three libraries of single-domain antibodies (compact antibody fragments consisting of a single monomeric binding domain) that differ only in scaffold (one germline-based, two affinity-maturated) against four targets, they found that libraries on germline scaffolds have larger σ\sigma in every case; they thus proposed σ\sigma as a measure of the selective potential of a library, apparently selected for over the long-term evolution of germline antibody genes. In mixed-population dynamics, the library with the largest μ\mu dominates short-term selection while the one with the largest σ\sigma dominates long-term. Crucially, all of this characterizes the response to selection alone. The authors state that extending the work to quantify evolvability - the response to successive cycles of selection and mutations - is the required next step, alongside the design question of whether the index can be increased to build better libraries.

Problem Statement

Does the two-parameter log-normal description of selection responses extend to evolvability, i.e., to successive cycles of selection and mutation? Concretely: formulate how the distribution of enrichment values of a biomolecular library - and its log-normal parameters (μ,σ)(\mu, \sigma), the mean and standard deviation of the library's log-enrichment distribution - evolve when mutation steps are interleaved with selection rounds, and determine whether the spread σ\sigma measured from selection-only data predicts the library's evolvability - its capacity to produce improved variants over iterated mutation and selection - across scaffolds with different degrees of affinity maturation and across targets. The two parts are jointly posed rather than independent sub-goals: the formulation of how (μ,σ)(\mu, \sigma) evolve under interleaved mutation and selection, and, within it, the determination of whether σ\sigma predicts the library's evolvability. The resolution criterion is whether σ\sigma provides a general, measurable quantitative index of evolvability for biomolecular libraries, suitable for comparison with biophysical and structural determinants and for guiding the design of libraries with higher evolutionary potential.

References

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