# The Reasonable Effectiveness of Randomness in Scalable and Integrative Gene Regulatory Network Inference and Beyond

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## Abstract

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## 1. Introduction

“… So there are very complex different ways that genes are regulated. I kind of look at it as playing music: You have chords on a guitar, or you play with a right and a left hand on the piano. It depends what strings you push down and what strings you strum, or what keys are up and what keys are down, that determine what the profile of the gene expression will be or the sound that you hear.”David M. Bodine [1]

## 2. The Molecular Biology of Gene Regulation and the Technologies Used to Study It

## 3. A Primer on Randomness and Randomized Algorithms

#### 3.1. A Mathematical Theory of Randomness

#### The Theory of Computation and Kolmogorov’s Definition of Randomness

#### 3.2. Randomized Algorithms

## 4. On the History and Current Applications of Randomness and Randomized Methods in Computational Biology and Gene Regulation Inference

#### 4.1. Markov Chain Monte Carlo Based Sampling for Gene Regulatory Network Structure Selection

#### Integration of Prior Knowledge on Network Structures

#### 4.2. Random Forest Regression Based Gene Regulatory Network Inference from Transcriptomic Data

#### Extension to Prior Knowledge Integration

#### 4.3. Random Walks for Gene Module Detection in Biological Networks

Algorithm 1 Markov Clustering |

Input: Graph adjacency matrix A, Expansion parameter e, inflation parameter i Output: Transition matrix M $A\leftarrow A+I$ // add self-loops to graph adjacency matrix A $M\leftarrow A{D}^{-1}$ // initialize canonical transition matrix M while M has not converged do$M\leftarrow {M}^{e}$ // Expansion for each $x\in V$ dofor each $y\in V$ do${M}_{xy}\leftarrow \frac{{M}_{xy}^{i}}{{\sum}_{y\in V}{M}_{xy}^{i}}$ // Inflation end forend forend while |

#### Regularized Markov Clustering and Random Walks with Restart

#### 4.4. Randomized Matrix Factorizations and Low Rank Approximations with Applications to High-Dimensional Gene Clustering and Gene Regulatory Network Inference

#### 4.4.1. Randomized Singular Value Decomposition

Algorithm 2 Randomized Singular Value Decomposition. |

Input: Data matrix $X\in {\mathbb{R}}^{n\times m}$, target rank k Output: Left and right singular vector matrices U and ${V}^{T}$, singular value matrix $\mathsf{\Sigma}$ $Q=rp(X,k)$ // compute approximate basis $Q\in {\mathbb{R}}^{n\times k}$, e.g., via Algorithm 3 ${X}^{\prime}={Q}^{T}X$ // project to low-dimensional space ${X}^{\prime}=\widehat{U}\mathsf{\Sigma}{V}^{T}$ // compute SVD of ${X}^{\prime}$ $U=Q\widehat{U}$ // recover left singular values $X\approx U\mathsf{\Sigma}{V}^{T}$ // compute SVD of X |

#### 4.4.2. Random Projection

Algorithm 3 Orthonormal basis estimation via Gaussian Random Projection. |

Input: Data matrix $X\in {\mathbb{R}}^{n\times m}$, target rank k Output: Approximate basis $Q\in {\mathbb{R}}^{n\times k}$ $\Omega =rnorm(m,k)$ // generate Gaussian random matrix $Y=X\Omega $ // generate sketch $Q=qr\left(Y\right)$ // form orthonormal basis Q, e.g., using QR factorization |

#### 4.4.3. Randomized Principal Component Analysis

Algorithm 4 Randomized Principal Component Analysis (via SVD). |

Input: Data matrix $X\in {\mathbb{R}}^{n\times m}$ (centered, scaled), target rank $k<min(m,n)$ Output: Eigenvector matrix ${V}^{T}$, eigenvalue matrix $\Lambda $, k principal component matrix ${P}_{k}$ $U,\mathsf{\Sigma},{V}^{T}=rsvd\left(X\right)$ // compute randomized SVD of X (Algorithm 2) $\Lambda ={\mathsf{\Sigma}}^{2}/(n-1)$ // recover eigenvalues ${P}_{k}={U}_{k}{\mathsf{\Sigma}}_{k}$ // compute k principal components |

## 5. A Note on Deep Learning Based Methods for Computational Biology

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Transcriptional regulation involves a variety of key players, including the polymerase complex, transcription factors and co-factors, activators, chromatin regulators. Transcription factors proteins thereby bind sequence motifs in promoter regions of a target gene, or distal enhancers and enable the recruitment of the transcription machinery, which then initiates a gene’s transcription (Figure adapted from [25]).

**Figure 2.**MCMC algorithms implement Markov chains to explore the geometry of the posterior density. Prior beliefs are expressed as a probability distribution over parameters $\theta $, e.g., $\theta =({\theta}_{1},{\theta}_{2})$, which are updated when data is collected via the likelihood function to give a posterior distribution over $\theta $ (

**left**, figure adapted from [108]). The acceptance-rejection feature of the algorithm allows for targeting the posterior as its stationary distribution, if the resulting Markov chain is irreducible (

**right**, figure adapted from [110]).

**Figure 3.**The generalized version of random forest regression-based gene regulatory network inference. Main rationale is that for each gene g, a number of decision trees are grown over different bootstrapped samples of the complete gene expression dataset, thereby using all remaining genes as putative regulators, trying to best explain the target gene expression profile. The resulting importance scores are subsequently combined into an edge matrix, which defines the final, weighted gene regulatory network. Given the inherent directionality of predicted links, edges not only are weighted, but also not likely to be symmetric (Figure adapted from [144]).

**Figure 4.**Randomness may be used to derive a low-rank approximation of the original data matrix. Then, deterministic approaches are applied on this lower dimensional matrix to compute an approximate matrix decomposition. Finally, a near-optimal set of (high-dimensional) factors may be reconstructed. (Figure adapted from [181]).

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Banf, M.; Hartwig, T.
The Reasonable Effectiveness of Randomness in Scalable and Integrative Gene Regulatory Network Inference and Beyond. *Computation* **2021**, *9*, 146.
https://doi.org/10.3390/computation9120146

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Banf M, Hartwig T.
The Reasonable Effectiveness of Randomness in Scalable and Integrative Gene Regulatory Network Inference and Beyond. *Computation*. 2021; 9(12):146.
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2021. "The Reasonable Effectiveness of Randomness in Scalable and Integrative Gene Regulatory Network Inference and Beyond" *Computation* 9, no. 12: 146.
https://doi.org/10.3390/computation9120146