The context-dependence of mutations: a linkage of formalisms

Defining the extent of epistasis - the non-independence of the effects of mutations - is essential for understanding the relationship of genotype, phenotype, and fitness in biological systems. The applications cover many areas of biological research, including biochemistry, genomics, protein and systems engineering, medicine, and evolutionary biology. However, the quantitative definitions of epistasis vary among fields, and its analysis beyond just pairwise effects remains obscure in general. Here, we show that different definitions of epistasis are versions of a single mathematical formalism - the weighted Walsh-Hadamard transform. We discuss that one of the definitions, the backgound-averaged epistasis, is the most informative when the goal is to uncover the general epistatic structure of a biological system, a description that can be rather different from the local epistatic structure of specific model systems. Key issues are the choice of effective ensembles for averaging and to practically contend with the vast combinatorial complexity of mutations. In this regard, we discuss possible approaches for optimally learning the epistatic structure of biological systems.Comment: 6 pages, 3 figures, supplementary informatio

Optimality and evolution of transcriptionally regulated gene expression

<p>Abstract</p> <p>Background</p> <p>How transcriptionally regulated gene expression evolves under natural selection is an open question. The cost and benefit of gene expression are the driving factors. While the former can be determined by gratuitous induction, the latter is difficult to measure directly.</p> <p>Results</p> <p>We addressed this problem by decoupling the regulatory and metabolic function of the <it>Escherichia coli lac </it>system, using an inducer that cannot be metabolized and a carbon source that does not induce. Growth rate measurements directly identified the induced expression level that maximizes the metabolism benefits minus the protein production costs, without relying on models. Using these results, we established a controlled mismatch between sensing and metabolism, resulting in sub-optimal transcriptional regulation with the potential to improve by evolution. Next, we tested the evolutionary response by serial transfer. Constant environments showed cells evolving to the predicted expression optimum. Phenotypes with decreased expression emerged several hundred generations later than phenotypes with increased expression, indicating a higher genetic accessibility of the latter. Environments alternating between low and high expression demands resulted in overall rather than differential changes in expression, which is explained by the concave shape of the cross-environmental tradeoff curve that limits the selective advantage of altering the regulatory response.</p> <p>Conclusions</p> <p>This work indicates that the decoupling of regulatory and metabolic functions allows one to directly measure the costs and benefits that underlie the natural selection of gene regulation. Regulated gene expression is shown to evolve within several hundreds of generations to optima that are predicted by these costs and benefits. The results provide a step towards a quantitative understanding of the adaptive origins of regulatory systems.</p

Predicting evolution using regulatory architecture

The limits of evolution have long fascinated biologists. However, the causes of evolutionary constraint have remained elusive due to a poor mechanistic understanding of studied phenotypes. Recently, a range of innovative approaches have leveraged mechanistic information on regulatory networks and cellular biology. These methods combine systems biology models with population and single-cell quantification and with new genetic tools, and they have been applied to a range of complex cellular functions and engineered networks. In this article, we review these developments, which are revealing the mechanistic causes of epistasis at different levels of biological organization¤mdash¤in molecular recognition, within a single regulatory network, and between different networks¤mdash¤providing first indications of predictable features of evolutionary constraint

Typical Divergence Pathway: Network Changes, Fitness, and Sequence

<div><p>(A) Evolving interaction network, where line thickness denotes binding strength between repressor monomer and operator-half. Dotted lines denote negligible repression. Yellow crosses indicate repressor and operator mutations, which are positioned at the top and bottom of the interaction lines respectively.</p><p>(B) Fitness trajectory (black) and corresponding repression of each repressor on its operator (red and blue). Fitness is normalized to the maximum value (~1 ×10¹⁰).</p><p>(C) Sequences for each round. Mutated positions are colored white.</p></div

Divergence Success Ratio and Path Length Distributions

<div><p>(A) Fraction of starting sequences (numbering 132 in total) that successfully diverge, as a function of the number of networks carried to the next round <i>(L).</i> Dashed line, idem, but with the additional requirement of continued tight binding (<i>F</i> ≥ 100) for both repressors.</p><p>(B) Distribution of path lengths until divergence. Red color map, optimal co-divergence pathways. Blue color map, pathways with the additional requirement of <i>F</i> ≥ 100 for both repressors. Note that a vertical summation of the color maps yields the lines in (A).</p></div

Definitions of genotype, phenotype, and effects of mutations.

<p>Representation of (A) single mutant, (B) double mutant, and (C) triple mutant experiments. Phenotypes are denoted by <i>y</i>_<i>g</i>, where <i>g</i> is the underlying genotype. <i>g</i> = {<i>g</i>_<i>N</i>,…,<i>g</i>₁} with <i>g</i>_<i>i</i> ∈{0,1}; “0” or “1” indicates the state of the mutable site (e.g., amino acid position). The effect of a single, double, and triple mutation is given by the red arrows. Pairwise (or second-order) epistasis is defined as the differential effect of a mutation depending on the background in which it occurs; for example, in (B) it is the degree to which the effect of one mutation (e.g., <i>y</i>₁₀−<i>y</i>₀₀) deviates in the background of the second mutation (<i>y</i>₁₁−<i>y</i>₀₁). Thus, the expression for second-order epistasis is (<i>y</i>₁₁−<i>y</i>₁₀)−(<i>y</i>₀₁−<i>y</i>₀₀). The third order and higher cases are considered in the main text.</p