# Multi-Objective Genetic Algorithm for Cluster Analysis of Single-Cell Transcriptomes

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

**Multi-objective clustering:**Given d-dimensional transcriptomes, we aim to find k cluster prototypes (cluster centers) by optimizing two objective functions (Equation (1)) as follows.

**Chromosome encoding:**In MOGA, clustering solutions are encoded by the real-valued chromosomes of size $l=k\times d$, where k is the number of clusters and d is the number of the dimensions of the dataset. Here, d denotes the number of Uniform Manifold Approximation and Projection for Dimension Reduction (UMAP) components, derived during preprocessing of the datasets, and it is set to 2. Thus, the first d values of a chromosome represent the prototype of the first cluster and the next d values represent the prototype of the second cluster, and so on. At the beginning of the evolution, a population of chromosomes is created. Each chromosome is initialized with random values, bounded by the range of values of each dimension of the dataset.

**Evolution:**After the initialization of the population, fitness values of the individual chromosomes are computed, namely ${f}_{1}\left(x\right)$ and ${f}_{2}\left(x\right)$, and evolution begins. Parent chromosomes are selected using a tournament selection [37], with the tournament size equal to 20% of the population. Specifically, 20% of chromosomes are randomly chosen from the population pool, and the ones with the best fitness values are selected for breeding. Chromosomes with better fitness values have a greater probability of being selected in the tournament. The tournament selection is repeated to create a population of size P, the same size as the initial population.

**Non-dominance solution sorting:**After the genetic operations of crossover and mutation, the parent population is merged with the offspring population, and NDSS is performed to select chromosomes for the next iteration of the algorithm. NDSS is based on the notion of dominance between solutions. Specifically, one solution dominates another if and only if all objective values of that solution are no worse than the objective values of another solution, and at least one objective value of the solution is better than the other one [38,39]. NDSS may select solutions that are very close to each other. To diversify the population, crowding thresholds are applied to select solutions that are evenly distributed across the entire Pareto front [38,39].

**Selection of the final solution:**In the last step of MOGA, the final clustering solution is selected from the Pareto front. For each Pareto-optimal solution, Davies Bouldin Index (DBI) is computed [40]. DBI measures clustering quality by computing the mean ratio between the intra-cluster and inter-cluster distances over all of the clusters (Equation (2)). DBI values range from 0 to 1, where lower values imply a better clustering quality.

**Hyperparameter tuning:**Population size, number of generations, crowding values, individual mutation rate, mutation probability, and crossover probability are determined by a grid search. Notably, to ensure a fair comparison with baseline methods, including a single-objective GA (SOGA), we tune the parameters for the second objective only, and do not tune the parameters for the first objective (Equation (1)).

**Evaluation of cluster validity:**Two validations of clustering solutions are performed, namely internal and external validation. In internal validation, ground truths are not known, and the quality of clustering solutions is measured using a widely-accepted internal validity metric, the Silhouette Coefficient (Sil) [42]. Sil is the mean silhouette width of all cells (Equation (3)), where $a\left({x}_{i}\right)$ refers to the mean distance between cell ${x}_{i}$ from the other cells in the same cluster, and $b\left({x}_{i}\right)$ refers to the minimum of the mean distances of ${x}_{i}$ from all cells in any other cluster. Sil ranges from −1 to 1, and a higher Sil implies better clustering, with a clear separation and good cohesiveness of clusters. Notably, singletons could exist in clustering solutions, where a single cluster only contains one data instance. Sil handles singletons by setting ${S}_{i}$ equal to 0, where $b\left({x}_{i}\right)=a\left({x}_{i}\right)$.

**Metamorphic evaluation of clustering stability:**The stability of clustering results is validated using metamorphic perturbations of transcriptomes. The purpose of this validation is to assure that small perturbations of the input data do not change cluster memberships when the perturbed transcriptomes are reclustered [47,48]. Using the scrnabench package (version 1.0), six metamorphic perturbations of the experimental transcriptomes are generated. They include permutation of the order of cells (MR1), modification of counts of a single gene (MR2), duplication of a transcriptome of a single cell (MR3), permutation of the order of genes (MR4), addition of a pseudo-gene with zero-variance expression counts (MR5), and negation of gene counts (MR6). Metamorphic datasets are reclustered and cluster validity metrics are computed and compared with the metrics of the original clusterings.

**State-of-the-art and baseline methods:**MOGA is compared with three state-of-the-art and two baseline methods. To demonstrate the value of multi-objective formulation, MOGA is compared with SOGA, which optimizes only the second objective function (Equation (1)). Therefore, NDSS and DBI-based selection are not performed, and the evolution of SOGA consists of the initialization, fitness evaluation, tournament selection, crossover, and mutation. The final solution of SOGA is encoded by a chromosome with the best fitness value in the last iteration of the algorithm. Notably, all hyperparameters of MOGA are set to the same values as the hyperparameters of the SOGA. Both MOGA and SOGA are implemented using the DEAP package (version 1.3.1) [49].

**Datasets:**Two types of single-cell transcriptomic datasets are used, namely experimental and synthetic. These datasets vary in size, sparsity, dimensionality, quality, and the availability of known cluster memberships.

**Data preprocessing:**The same standard preprocessing workflow is used to prepare the experimental and synthetic datasets [18]. Specifically, preprocessing comprises five steps: filtering, highly variable gene selection, transformation, scaling, and dimensionality reduction. In filtering, cells with fewer than 200 expressed genes and genes expressed in fewer than three cells are removed. Additionally, cells with mitochondrial content greater than 10% are filtered out. Moreover, outlier cells and genes are removed. The mRNA counts and gene counts are bounded by ${10}^{mean\left(lo{g}_{1}{}_{0}\left(x\right)\right)\pm 2\times std\left(lo{g}_{1}{}_{0}\left(x\right)\right)}$ to ensure that each cell has meaningful expression data, where x is the total mRNA count or the total gene count per cell.

**Estimation of compute time:**Run-time data are collected during the experimentation, including the datasets’ size, number of clusters, combinations of HPC resources, such as the number of CPUs, number of tasks per node, and the number of CPUs per task. These data are used to train a Random forest regressor [56], an ensemble tree-based algorithm for supervised learning. The accuracy of run-time estimates is validated by training a predictor on the simulated datasets and testing it on the reference datasets. Mean absolute error (MSE) between the estimated and actual values is used to evaluate the accuracy of the predictions.

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ARI | Adjusted Rand Index |

DBI | Davies Bouldin Index |

GA | Genetic Algorithm |

ML | Machine Learning |

MOGA | Multi-Objective Genetic Algorithm |

MOO | Multi-Objective |

MSE | Mean Square Error |

NDSS | Non-dominance Solution Sorting |

NMI | Normalized Mutual Information |

Sil | Silhouette Coefficient |

SOGA | Single Objective Genetic Algorithm |

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**Figure 1.**

**Cluster analysis of single-cell transcriptomes**. Shown are the steps of cluster analysis of high-dimensional single-cell transcriptomes, including data preprocessing and dimensionality reduction.

**Figure 2.**

**Cluster analysis as a multi-objective optimization problem.**A solution to a cluster analysis problem is a set of k cluster prototypes ${c}_{1},\dots ,{c}_{k}$. Solution space of two objective functions and a corresponding objective function space are shown. Objective function ${f}_{1}\left(x\right)$ maximizes inter-cluster distances, and ${f}_{2}\left(x\right)$ minimizes intra-cluster distances. Pareto front encompasses optimal solutions that are not dominated by any other feasible solutions.

**Figure 3.**

**Architecture of the multi-objective Genetic Algorithm.**Shown are main steps of the proposed GA. Initial population of chromosomes is randomly created and inputted to the GA optimizer. After Pareto-optimal solutions are found, the best solution is selected using a predefined criterion.

**Figure 4.**

**Hyperparameter tuning of population size, number of iterations, individual mutation rate, mutation probability, and crossover probability for objective ${f}_{2}\left(x\right)$.**Shown are (

**A**) a line plot of fitness values obtained by varying population size and number of iterations and (

**B**) a three-dimensional heatmap of fitness values obtained by varying mutation probability, crossover probability, and individual mutation rate. A small fitness value is preferred.

**Figure 5.**

**Internal validation and comparison of MOGA, SOGA, KMeans, PhenoGraph, Seurat, and Scanpy.**Shown are the box plots of (

**A**) Sil of 48 scRNA-seq reference datasets with six algorithms and (

**B**) Sil scores of MOGA-based clustering by sequencing technology. Experiments are repeated 30 times, and the best Sil values of each dataset are shown. Cell line A: breast cancer; Cell line B: normal B lymphocytes.

**Figure 6.**

**Cluster stability in metamorphic testing.**Shown are the box plots of the distributions of the Sil of (

**A**) MOGA, (

**B**) SOGA, (

**C**) Kmeans, (

**D**) PhenoGraph, (

**E**) Seurat, and (

**F**) Scanpy in original clustering as well as in metamorphic tests (MR1 to MR6). Experiments are repeated 30 times, and the best Sil of each dataset are shown.

**Figure 7.**

**Internal and external validation of MOGA on synthetic datasets.**Shown are the bar graphs of NMI, ARI, and Sil of (

**A**) 60 synthetic datasets with different number of cells and the same number of clusters and (

**B**) 60 synthetic datasets with different number of clusters and the same number of cells. Experiments are repeated 30 times, and the best metrics were retained.

**Figure 8.**

**Running time analysis of MOGA and SOGA.**Shown are time comparisons of MOGA and SOGA (

**A**) with 48 scRNA-seq reference datasets and (

**B**) 60 synthetic datasets. Experiments are repeated 30 times, and the average computational time is retained.

**Table 1.**

**Internal validity of MOGA, SOGA, KMeans, PhenoGraph, Seurat, and Scanpy.**Shown are Silhouette scores of clustering of 12 reference transcriptomes, where MOGA outperformed other methods.

Dataset | MOGA | SOGA | KMeans | PhenoGraph | Seurat | Scanpy |
---|---|---|---|---|---|---|

C1_FDA_HT_A_featureCounts | 0.60 | 0.55 | 0.55 | 0.51 | 0.51 | 0.50 |

C1_FDA_HT_A_kallisto | 0.59 | 0.48 | 0.47 | 0.43 | 0.45 | 0.43 |

C1_FDA_HT_A_rsem | 0.61 | 0.55 | 0.55 | 0.47 | 0.47 | 0.45 |

C1_LLU_A_featureCounts | 0.68 | 0.54 | 0.5 | 0.67 | 0.68 | 0.68 |

C1_LLU_A_kallisto | 0.69 | 0.54 | 0.49 | 0.68 | 0.70 | 0.69 |

C1_LLU_A_rsem | 0.80 | 0.62 | 0.57 | 0.79 | 0.80 | 0.80 |

ICELL8_PE_A_featureCounts | 0.68 | 0.41 | 0.41 | 0.39 | 0.45 | 0.36 |

ICELL8_PE_A_kallisto | 0.83 | 0.58 | 0.41 | 0.41 | 0.44 | 0.40 |

ICELL8_PE_A_rsem | 0.77 | 0.42 | 0.42 | 0.40 | 0.44 | 0.40 |

ICELL8_SE_A_featureCounts | 0.81 | 0.57 | 0.41 | 0.41 | 0.43 | 0.39 |

ICELL8_SE_A_rsem | 0.55 | 0.43 | 0.44 | 0.37 | 0.42 | 0.37 |

ICELL8_SE_A_kallisto | 0.71 | 0.53 | 0.42 | 0.41 | 0.44 | 0.43 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, K.; Grayson, J.M.; Khuri, N. Multi-Objective Genetic Algorithm for Cluster Analysis of Single-Cell Transcriptomes. *J. Pers. Med.* **2023**, *13*, 183.
https://doi.org/10.3390/jpm13020183

**AMA Style**

Zhao K, Grayson JM, Khuri N. Multi-Objective Genetic Algorithm for Cluster Analysis of Single-Cell Transcriptomes. *Journal of Personalized Medicine*. 2023; 13(2):183.
https://doi.org/10.3390/jpm13020183

**Chicago/Turabian Style**

Zhao, Konghao, Jason M. Grayson, and Natalia Khuri. 2023. "Multi-Objective Genetic Algorithm for Cluster Analysis of Single-Cell Transcriptomes" *Journal of Personalized Medicine* 13, no. 2: 183.
https://doi.org/10.3390/jpm13020183