# Selection for Protein Stability Enriches for Epistatic Interactions

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### Simulations

## 3. Results

#### 3.1. Epistasis Is Essential for Proper Folding of Evolved Sequences

#### 3.2. Enrichment for Epistasis Observed under the Pairwise Model, but Not the Independent Model

#### 3.3. Bivariate Normal Approximation for the Joint Distribution of Additive and Epistatic Contributions to the Free Energy of Folding Captures Impact of Sequence Entropy

## 4. Discussion

^{−7}or a z-score of roughly −5. Eukaryotic TFs have even smaller information content and therefore smaller absolute value z-scores. In contrast, the z-scores of the spontaneously folding sequences observed in our simulations are on the order of −20, which we would expect to result in a roughly 4-fold larger contribution of epistasis to binding energy at stationarity than for a bacterial transcription factor binding site. Such extreme z-scores are not even possible in short DNA elements, e.g., the most extreme z-score possible in a TF binding site of length 20 is only −7. Thus, the essentiality of epistatic interactions observed here is likely possible only because protein sequence space is very large compared to the space of TF binding sites.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Model for the Free Energy of Folding

#### Appendix A.2. Properties of the ${\beta}_{\left\{k\right\},\alpha}$ and ${\beta}_{\{{k}_{1},{k}_{2}\},{\alpha}_{1}{\alpha}_{2}}$

## Appendix B

#### Appendix B.1. Expected Variance of Epistatic Energy of Distance Classes under the Random Field Model

## Appendix C

#### Appendix C.1. Bivariate Normal Approximation

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**Figure 1.**Free energy of folding, stability effects of mutations, and contribution of additive effects to folding stability under the pairwise epistatic effects model: (

**a**) Distribution of free energy of folding for evolved sequences; (

**b**) distribution of stability effects of random mutations, i.e., distribution of $\Delta \Delta G$ values for a random single mutant generated from a random evolved sequence, where $\Delta \Delta G$ is the change in the $\Delta G$ of folding due to the mutation; (

**c**) distribution of stability effects of fixed mutations, i.e., distribution of $\Delta \Delta G$ values corresponding to two distinct neighboring sequences along the simulated trajectory; (

**d**) stability effects of double mutations versus the sum of the stability effects of the two single mutations. 500 random double mutants are shown, ${R}^{2}=0.9997$; (

**e**) effects of single mutations that fixed along the trajectory in two evolved backgrounds that differ by 50% sequence divergence, ${R}^{2}=0.87$; (

**f**) the stability effect of a random mutation on $\Delta G$ is highly correlated with the stability effect of the mutation on the additive contribution $\Delta {G}_{\mathrm{add}}$, ${R}^{2}=0.89$; (

**g**) free energy of folding versus additive contribution to free energy of folding for evolved sequences. The additive contribution to folding is not a good indicator of the total free energy of folding (${R}^{2}=0.06$) and observed sequences cannot fold spontaneously based on the additive contribution alone. The dashed gray curve is derived from our bivariate normal approximation and is predicted to contain 95% of the evolved sequences. Simulations conducted under the pairwise epistasis model with ${\mu}_{\mathrm{add}}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{kcal}/\mathrm{mol}$, ${\sigma}_{\mathrm{add}}^{2}=1\phantom{\rule{0.277778em}{0ex}}{\mathrm{kcal}}^{2}/{\mathrm{mol}}^{2}$, and ${\sigma}_{\mathrm{epi}}^{2}=0.0003\phantom{\rule{0.277778em}{0ex}}{\mathrm{kcal}}^{2}/{\mathrm{mol}}^{2}$.

**Figure 2.**Free energy of folding, stability effects of mutations, and contribution of additive effects to folding stability under the independent epistatic effects model: (

**a**) Distribution of free energy of folding for evolved sequences; (

**b**) distribution of stability effects of random mutations, i.e., distribution of $\Delta \Delta G$ values for a random single mutant generated from a random evolved sequence; (

**c**) distribution of stability effects of fixed mutations, i.e., distribution of $\Delta \Delta G$ values corresponding to two distinct neighboring sequences along the simulated trajectory; (

**d**) stability effects of double mutations versus the sum of the stability effects of the two single mutations. 500 random double mutants are shown, ${R}^{2}=0.993$; (

**e**) effects of single mutations that fixed along the trajectory in two evolved backgrounds that differ by 50% sequence divergence, ${R}^{2}=0.98$; (

**f**) the stability effect of a random mutation on $\Delta G$ is highly correlated with the stability effect of the mutation on the additive contribution $\Delta {G}_{\mathrm{add}}$, ${R}^{2}=0.997$; (

**g**) the additive contribution to folding is a good indicator of the total free energy of folding (${R}^{2}=0.99$) and 95% of observed sequences can fold spontaneously based on the additive contribution alone. The dashed gray curve is derived from our bivariate normal approximation and is predicted to contain 95% of the evolved sequences. Simulations conducted under the independent epistatic effects model with ${\mu}_{\mathrm{add}}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{kcal}/\mathrm{mol}$, ${\sigma}_{\mathrm{add}}^{2}=1\phantom{\rule{0.277778em}{0ex}}{\mathrm{kcal}}^{2}/{\mathrm{mol}}^{2}$, and ${\sigma}_{\mathrm{HOC}}^{2}=0.01\phantom{\rule{0.277778em}{0ex}}{\mathrm{kcal}}^{2}/{\mathrm{mol}}^{2}$.

**Figure 3.**Expected epistatic variance as a function of distance from the focal sequence for amino acid sequences of length $l=400$. Results for the pairwise epistasis model shown in black, results for the independent epistasis model shown in gray. All variances are normalized relative to the expected variance at Hamming distance $d=2$ which is set to 1. Notice that the epistatic variance at large distances is much larger than the epistatic variance at distance $d=2$ for the pairwise epistasis model but not for the independent epistasis model.

**Figure 4.**Illustration of the main mechanism behind the essentiality of epistatic interactions for spontaneous folding: (

**a**) density of random sequences with given additive free energy $\mathrm{P}\left(\right)open="("\; close=")">\Delta {\mathrm{G}}_{\mathrm{add}}=x$; (

**b**) fraction of sequences that fold given additive free energy $\mathrm{P}(\Delta \mathrm{G}<0|\Delta {\mathrm{G}}_{\mathrm{add}}=x)=\mathrm{CDF}\left(\mathcal{N}(0,{\sigma}_{2}^{2})\right)(-x)$; (

**c**) density of random sequences that fold and have the given additive free energy $\mathrm{P}\left(\right)open="("\; close=")">\Delta \mathrm{G}0\phantom{\rule{0.166667em}{0ex}}\cap \phantom{\rule{0.166667em}{0ex}}\Delta {\mathrm{G}}_{\mathrm{add}}=x$. Parameters are identical to those in Figure 1.

**Figure 5.**Joint distribution of $\Delta G$ of folding and the additive contribution to $\Delta G$ of folding for the independent epistasis model with ${\sigma}_{\mathrm{HOC}}^{2}$ chosen so that the bivariate normal approximation matches the bivariate normal approximation shown in Figure 1g. Simulations conducted under the independent epistasis model with ${\mu}_{\mathrm{add}}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{kcal}/\mathrm{mol}$, ${\sigma}_{\mathrm{add}}^{2}=1\phantom{\rule{0.277778em}{0ex}}{\mathrm{kcal}}^{2}/{\mathrm{mol}}^{2}$, ${\sigma}_{\mathrm{HOC}}^{2}=21.6\phantom{\rule{0.277778em}{0ex}}{\mathrm{kcal}}^{2}/{\mathrm{mol}}^{2}$. A dashed curve shows the area predicted to include 95% of sequences at stationarity under the bivariate normal approximation; the dashed vertical line at $\Delta {G}_{\mathrm{add}}=18.8$ reflects a value, derived using a crude percolation theory argument, for the right-most edge of the region where the bivariate normal approximation is expected to be valid.

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**MDPI and ACS Style**

Posfai, A.; Zhou, J.; Plotkin, J.B.; Kinney, J.B.; McCandlish, D.M.
Selection for Protein Stability Enriches for Epistatic Interactions. *Genes* **2018**, *9*, 423.
https://doi.org/10.3390/genes9090423

**AMA Style**

Posfai A, Zhou J, Plotkin JB, Kinney JB, McCandlish DM.
Selection for Protein Stability Enriches for Epistatic Interactions. *Genes*. 2018; 9(9):423.
https://doi.org/10.3390/genes9090423

**Chicago/Turabian Style**

Posfai, Anna, Juannan Zhou, Joshua B. Plotkin, Justin B. Kinney, and David M. McCandlish.
2018. "Selection for Protein Stability Enriches for Epistatic Interactions" *Genes* 9, no. 9: 423.
https://doi.org/10.3390/genes9090423