# Predicting the Effect of Single and Multiple Mutations on Protein Structural Stability

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.1.1. Experimental Mutagenesis

#### 1.1.2. Computational Approaches

#### 1.1.3. Combinatorial, Rigidity Based Methods

#### 1.1.4. Machine Learning Based Approaches

#### 1.1.5. Model Ensembling

_{2}O.ai was the only tool found by the authors which supports a stacked ensembles [35]. However, it was not used in this work, as H

_{2}O’s software stack was not compatible with our existing research pipeline.

#### 1.2. Motivations and Contributions

## 2. Results

#### 2.1. Voting

#### 2.1.1. Two Model Voting

#### 2.1.2. Three Model Voting

_{combined-wa}model consistently improved performance over every RD metric.

#### 2.2. Feature Ablation Study

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Data Preparation

#### 4.1.1. In Silico Mutants

#### 4.1.2. Rigidity Distance Scores

#### 4.1.3. Feature Extraction

- Solvent accessible surface area (SASA): 2 real-valued features (4 in double mutations) indicating how exposed to the surface a residue is (both absolute and percentage).
- Secondary Structure: 4 binary features (8 for double mutations) indicating whether each mutation is part of a sheet, coil, turn or helix.
- Temperature and pH at which the experiment for calculating ddG was performed.
- Rigidity distances (RD): one of lm, sm1, sm2, sm3, sm4, and sm5 (see above and [37]).
- Rigid Cluster Fraction: 48 features giving the fraction of atoms in the WT and MUT that belong to rigid clusters of size 2, 3, ... , 20, 21–30, 31–50, 51–100, 101–1000, and 1001+, respectively.
- Residue type: 8 categorical features (16 in double mutants) indicating whether the mut1wt, mut1target (in double mutants also mut2wt, and mut2target) are Charged (D, E, K, R), Polar (N, Q, S, T), Aromatic (F, H, W, Y), or Hydrophobic (A, C, G, I, L, M, P, V).

#### 4.1.4. Data Split

#### 4.2. Machine Learning Methods

#### 4.2.1. SVR

#### 4.2.2. RF

#### 4.2.3. DNN

#### 4.3. Voting Schemes

- VS
_{uwa}: An unweighted average of all models’ predictions for a given mutation. For m models each with an output ${h}_{i}\left(x\right)\in \mathbb{R}$, where $i=1,2,\cdots ,m$, our voting prediction is:$${\mathrm{VS}}_{\mathrm{uwa}}\left(x\right)=\frac{1}{m}\sum _{i=1}^{m}{h}_{i}\left(x\right)$$ - VS
_{c-wa}: A weighted average of all models’ predictions for a given mutation, adjusting each model’s prediction based on the strength of its Pearson Correlation Coefficient, R, relative to the model with the best R. Again assume we have a set of models ${h}_{i}\left(x\right)$ for $i=1,2,\cdots ,m$ but let ${h}_{*}\left(x\right)$ denote the output from the best performing model, ${c}_{i}$ denote the R for model i and let ${c}_{*}$ denote R for the best performing model, then our voting prediction is:$${\mathrm{VS}}_{\mathrm{c}-\mathrm{wa}}\left(x\right)=\frac{1}{m}\sum _{i=1}^{m}\left(\right)open="("\; close=")">{h}_{i}\left(x\right)+\left(\right)open="("\; close=")">{h}_{*}\left(x\right)-{h}_{i}\left(x\right)$$ - VS
_{rmse-wa}: A weighted average of all models’ predictions for a given mutation analogous to**VS**, except using RMSE instead of R. In this case, ${h}_{*}\left(x\right)$ is the prediction of the best model (according to RMSE), ${r}_{i}$ is the RMSE for model i and ${r}_{*}$ is the RMSE of the best model. Then our voting prediction is:_{c-wa}$${\mathrm{VS}}_{\mathrm{rmse}-\mathrm{wa}}\left(x\right)=\frac{1}{m}\sum _{i=1}^{m}\left(\right)open="("\; close=")">{h}_{i}\left(x\right)+\left(\right)open="("\; close=")">{h}_{*}\left(x\right)-{h}_{i}\left(x\right)$$ - VS
_{combined-wa}: A weighted average of all models’ predictions for a given mutation incorporating both the R and RMSE performance. Letting ${\gamma}_{i}={c}_{i}/{r}_{i}$ and ${\gamma}_{*}$ denote the best (max) ${\gamma}_{i}$, our prediction is:$${\mathrm{VS}}_{\mathrm{combined}-\mathrm{wa}}\left(x\right)=\frac{1}{m}\sum _{i=1}^{m}\left(\right)open="("\; close=")">{h}_{i}\left(x\right)+\left(\right)open="("\; close=")">{h}_{*}\left(x\right)-{h}_{i}\left(x\right)$$

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

DNN | Deep Neural Network |

SVR | Support Vector Regression |

RF | Random Forest |

LRC | Largest Rigid Cluster |

RD | Rigidity Distance |

WT | Wild Type |

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Sample Availability: Samples of the compounds are not available from the authors. |

**Figure 1.**Cartoon (

**left**) and Rigidity analysis (

**right**) of PDB file 1hvr. Atoms in different rigid clusters are colored by cluster membership. The largest rigid cluster (red-brown) spans both halves of the protein.

**Figure 2.**Test set predicted versus actual $\Delta \Delta G$ for single (

**a**) and multiple (

**b**) mutation data.

**Figure 3.**Test set predicted vs. actual $\Delta \Delta G$, R = 0.73, for best RF model using sm4 rigidity distance metric, for the combined single and double mutations dataset

**Figure 4.**Pearson Correlation (R) values for voting schemes using SVR and DNN predictions for a single mutation using various rigidity distances.

**Figure 5.**Pearson Correlation (R) values for machine learning models and voting schemes with information from SVR, DNN, and RF predictions for a single mutation using various rigidity distances.

**Figure 6.**Sigmoid functions for scaling RD metric values. The green sigmoid acts much like a step function and rigid clusters made up of 10 or fewer atoms are weighted by a factor of 0. Using the violet sigmoid, atoms in a cluster size up to 200 atoms would be assigned a near 0 weight, atoms in clusters of size 200–300 would be weighed by 0.1–0.8, and atoms in clusters of 300+ atoms would be weighted by 0.8 or more. Reproduced from [37].

**Table 1.**Test set results for regression models for single and double mutants, as well as the union of the two (combined). RD = Rigidity Distance. The best results are shown in bold font.

Single Mutants | Double Mutants | Combined | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

RD | Measure | SVR | RF | DNN | SVR | RF | DNN | SVR | RF | DNN |

lm | RMSE | 1.53 | 1.34 | 1.60 | 1.61 | 1.41 | 1.74 | 1.54 | 1.39 | 1.71 |

R | 0.60 | 0.71 | 0.58 | 0.76 | 0.79 | 0.66 | 0.65 | 0.72 | 0.52 | |

sm1 | RMSE | 1.52 | 1.35 | 1.60 | 1.60 | 1.37 | 1.64 | 1.54 | 1.39 | 1.80 |

R | 0.60 | 0.71 | 0.57 | 0.76 | 0.81 | 0.71 | 0.65 | 0.72 | 0.46 | |

sm2 | RMSE | 1.53 | 1.35 | 1.71 | 1.61 | 1.36 | 1.90 | 1.54 | 1.40 | 1.87 |

R | 0.60 | 0.71 | 0.55 | 0.76 | 0.81 | 0.60 | 0.65 | 0.72 | 0.46 | |

sm3 | RMSE | 1.52 | 1.35 | 1.60 | 1.60 | 1.38 | 1.93 | 1.54 | 1.39 | 1.81 |

R | 0.60 | 0.71 | 0.58 | 0.76 | 0.80 | 0.60 | 0.65 | 0.72 | 0.44 | |

sm4 | RMSE | 1.52 | 1.34 | 1.57 | 1.56 | 1.38 | 1.83 | 1.54 | 1.38 | 1.77 |

R | 0.60 | 0.71 | 0.57 | 0.77 | 0.80 | 0.64 | 0.66 | 0.73 | 0.55 | |

sm5 | RMSE | 1.53 | 1.35 | 1.70 | 1.60 | 1.35 | 1.89 | 1.54 | 1.39 | 1.74 |

R | 0.60 | 0.71 | 0.52 | 0.76 | 0.81 | 0.52 | 0.65 | 0.72 | 0.51 | |

Avg. | RMSE | 1.53 | 1.35 | 1.63 | 1.60 | 1.38 | 1.82 | 1.54 | 1.39 | 1.78 |

R | 0.60 | 0.71 | 0.56 | 0.76 | 0.80 | 0.62 | 0.65 | 0.72 | 0.49 |

**Table 2.**Comparing training and test set results for regression models, averaged over the six RD metrics.

Accuracy | Measure | SVR | RF | DNN |
---|---|---|---|---|

Single | Avg. Train RMSE | 1.08 | 0.47 | 1.04 |

Avg. Test RMSE | 1.53 | 1.35 | 1.67 | |

Avg. Train R | 0.79 | 0.97 | 0.79 | |

Avg. Test R | 0.60 | 0.71 | 0.56 | |

Double | Avg. Train RMSE | 1.08 | 0.70 | 1.20 |

Avg. Test RMSE | 1.60 | 1.38 | 1.82 | |

Avg. Train R | 0.83 | 0.93 | 0.69 | |

Avg. Test R | 0.76 | 0.80 | 0.62 | |

Combined | Avg. Train RMSE | 1.11 | 0.50 | 1.33 |

Avg. Test RMSE | 1.52 | 1.39 | 1.78 | |

Avg. Train R | 0.79 | 0.96 | 0.62 | |

Avg. Test R | 0.65 | 0.72 | 0.49 |

**Table 3.**Test set results for voting schemes for single mutants using SVR and DNN predictions. The voting schemes that showed the best improvement are highlighted in bold fonts.

RD | Measure | SVR | DNN-AVG | VS_{uwa} | VS_{rmse-wa} | VS_{c-wa} | VS_{combined-wa} |
---|---|---|---|---|---|---|---|

lm | RMSE | 1.53 | 1.46 | 1.43 | 1.43 | 1.43 | 1.43 |

R | 0.60 | 0.64 | 0.66 | 0.66 | 0.66 | 0.65 | |

sm1 | RMSE | 1.52 | 1.55 | 1.46 | 1.46 | 1.46 | 1.46 |

R | 0.60 | 0.59 | 0.64 | 0.64 | 0.64 | 0.64 | |

sm2 | RMSE | 1.53 | 1.57 | 1.45 | 1.45 | 1.45 | 1.45 |

R | 0.60 | 0.60 | 0.64 | 0.65 | 0.65 | 0.65 | |

sm3 | RMSE | 1.52 | 1.57 | 1.46 | 1.46 | 1.46 | 1.46 |

R | 0.60 | 0.59 | 0.64 | 0.64 | 0.64 | 0.64 | |

sm4 | RMSE | 1.52 | 1.56 | 1.48 | 1.48 | 1.48 | 1.48 |

R | 0.60 | 0.57 | 0.62 | 0.63 | 0.63 | 0.63 | |

sm5 | RMSE | 1.53 | 1.63 | 1.50 | 1.49 | 1.49 | 1.49 |

R | 0.60 | 0.55 | 0.61 | 0.62 | 0.62 | 0.62 |

**Table 4.**Test set results for voting schemes for single mutants using SVR, DNN and RF predictions. The best results are highlighted in bold fonts.

RD | Measure | DNN-AVG | SVR | RF | VS_{uwa} | VS_{rmse-wa} | VS_{c-wa} | VS_{combined-wa} |
---|---|---|---|---|---|---|---|---|

lm | RMSE | 1.46 | 1.53 | 1.34 | 1.37 | 1.35 | 1.35 | 1.33 |

R | 0.64 | 0.60 | 0.71 | 0.70 | 0.71 | 0.71 | 0.72 | |

sm1 | RMSE | 1.55 | 1.52 | 1.35 | 1.39 | 1.37 | 1.37 | 1.34 |

R | 0.59 | 0.60 | 0.71 | 0.69 | 0.70 | 0.70 | 0.71 | |

sm2 | RMSE | 1.57 | 1.53 | 1.35 | 1.37 | 1.35 | 1.35 | 1.33 |

R | 0.60 | 0.60 | 0.71 | 0.69 | 0.71 | 0.71 | 0.72 | |

sm3 | RMSE | 1.56 | 1.52 | 1.35 | 1.38 | 1.36 | 1.36 | 1.34 |

R | 0.59 | 0.60 | 0.71 | 0.69 | 0.70 | 0.71 | 0.72 | |

sm4 | RMSE | 1.56 | 1.52 | 1.34 | 1.40 | 1.37 | 1.38 | 1.34 |

R | 0.57 | 0.60 | 0.71 | 0.68 | 0.70 | 0.70 | 0.72 | |

sm5 | RMSE | 1.63 | 1.53 | 1.35 | 1.41 | 1.37 | 1.37 | 1.35 |

R | 0.55 | 0.60 | 0.71 | 0.67 | 0.70 | 0.70 | 0.71 |

Accuracy Measure | Feature 1 | Feature 2 | Feature 3 | No Ablation | |
---|---|---|---|---|---|

lm | SASA | temp | type | ||

RMSE | 1.48 | 1.39 | 1.38 | 1.34 | |

R | 0.63 | 0.68 | 0.69 | 0.71 | |

sm1 | SASA | temp | type | ||

RMSE | 1.50 | 1.39 | 1.38 | 1.35 | |

R | 0.62 | 0.66 | 0.69 | 0.71 | |

sm2 | SASA | temp | type | ||

RMSE | 1.50 | 1.39 | 1.38 | 1.35 | |

R | 0.62 | 0.68 | 0.69 | 0.71 | |

sm3 | SASA | temp | type | ||

RMSE | 1.49 | 1.38 | 1.37 | 1.36 | |

R | 0.62 | 0.68 | 0.69 | 0.70 | |

sm4 | SASA | temp | type | ||

RMSE | 1.49 | 1.38 | 1.38 | 1.35 | |

R | 0.63 | 0.69 | 0.69 | 0.71 | |

sm5 | SASA | temp | type | ||

RMSE | 1.50 | 1.40 | 1.38 | 1.36 | |

R | 0.62 | 0.68 | 0.69 | 0.70 |

Accuracy Measure | Feature 1 | Feature 2 | Feature 3 | No Ablation | |
---|---|---|---|---|---|

lm | mut2SASA | mutClusterFrac16 | wtClusterFrac1001 | ||

RMSE | 1.47 | 1.41 | 1.40 | 1.39 | |

R | 0.77 | 0.79 | 0.79 | 0.80 | |

sm1 | mut2SASA | wtClusterFrac11 | |||

RMSE | 1.46 | 1.39 | 1.39 | ||

R | 0.77 | 0.80 | 0.80 | ||

sm2 | mut2SASA | wtClusterFrac11 | mut1target_type | ||

RMSE | 1.46 | 1.39 | 1.38 | 1.37 | |

R | 0.77 | 0.80 | 0.80 | 0.81 | |

sm3 | mut2SASA | ph | mutClusterFrac11 | ||

RMSE | 1.46 | 1.40 | 1.38 | 1.37 | |

R | 0.77 | 0.80 | 0.80 | 0.80 | |

sm4 | mut2SASA | wtClusterFrac18 | wtClusterFrac101 | ||

RMSE | 1.45 | 1.40 | 1.40 | 1.37 | |

R | 0.78 | 0.79 | 0.80 | 0.80 | |

sm5 | mut2SASA | wtClusterFrac11 | ph | ||

RMSE | 1.47 | 1.39 | 1.39 | 1.36 | |

R | 0.77 | 0.80 | 0.80 | 0.81 |

Accuracy Measure | Feature 1 | Feature 2 | Feature 3 | No Ablation | |
---|---|---|---|---|---|

lm | mut1SASA | temp | mut1type | ||

RMSE | 1.47 | 1.43 | 1.42 | 1.40 | |

R | 0.68 | 0.70 | 0.70 | 0.72 | |

sm1 | mut1SASA | temp | mut1type | ||

RMSE | 1.48 | 1.42 | 1.41 | 1.39 | |

R | 0.68 | 0.71 | 0.71 | 0.72 | |

sm2 | mut1SASA | temp | mut1type | ||

RMSE | 1.48 | 1.42 | 1.42 | 1.39 | |

R | 0.68 | 0.70 | 0.71 | 0.72 | |

sm3 | mut1SASA | mut1type | temp | ||

RMSE | 1.48 | 1.42 | 1.42 | 1.39 | |

R | 0.68 | 0.70 | 0.71 | 0.72 | |

sm4 | mut1SASA | temp | mut1type | ||

RMSE | 1.47 | 1.42 | 1.41 | 1.39 | |

R | 0.69 | 0.71 | 0.71 | 0.72 | |

sm5 | mut1SASA | temp | mut1type | ||

RMSE | 1.48 | 1.43 | 1.41 | 1.40 | |

R | 0.68 | 0.70 | 0.71 | 0.72 |

Dataset | Training | Development | Test | Total |
---|---|---|---|---|

Single | 1488 | 331 | 320 | 2139 |

Double | 147 | 60 | 107 | 314 |

Combined | 1635 | 391 | 427 | 2453 |

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Dehghanpoor, R.; Ricks, E.; Hursh, K.; Gunderson, S.; Farhoodi, R.; Haspel, N.; Hutchinson, B.; Jagodzinski, F.
Predicting the Effect of Single and Multiple Mutations on Protein Structural Stability. *Molecules* **2018**, *23*, 251.
https://doi.org/10.3390/molecules23020251

**AMA Style**

Dehghanpoor R, Ricks E, Hursh K, Gunderson S, Farhoodi R, Haspel N, Hutchinson B, Jagodzinski F.
Predicting the Effect of Single and Multiple Mutations on Protein Structural Stability. *Molecules*. 2018; 23(2):251.
https://doi.org/10.3390/molecules23020251

**Chicago/Turabian Style**

Dehghanpoor, Ramin, Evan Ricks, Katie Hursh, Sarah Gunderson, Roshanak Farhoodi, Nurit Haspel, Brian Hutchinson, and Filip Jagodzinski.
2018. "Predicting the Effect of Single and Multiple Mutations on Protein Structural Stability" *Molecules* 23, no. 2: 251.
https://doi.org/10.3390/molecules23020251