Identifying transition states of chemical kinetic systems using network embedding techniques

Paula Mercurio; Di Liu; Paula Mercurio; Di Liu

doi:10.3934/mbe.2021046

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 1: 868-887. doi: 10.3934/mbe.2021046

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Identifying transition states of chemical kinetic systems using network embedding techniques

Paula Mercurio ^,,
Di Liu

Department of Mathematics, Michigan State University, East Lansing, MI, USA

Received: 12 October 2020 Accepted: 22 December 2020 Published: 25 December 2020

Many chemical and biochemical systems can be intuitively modeled using networks. Due to the size and complexity of many biochemical networks, we require tools for efficient network analysis. Of particular interest are techniques that embed network vertices into vector spaces while preserving important properties of the original graph. In this article, we {introduce a new method for generating low-dimensional node embeddings for directed graphs, using random walk sampling methods for feature learning on networks. Additionally, we demonstrate the usefulness of this method for identifying transition states of stochastic chemical reacting systems.} Network representations of chemical systems are typically given by weighted directed graphs, and are often complex and high dimensional. In order to deal with networks representing these chemical systems, therefore, we modified objective functions adopted in existing random walk based network embedding methods to handle directed graphs and neighbors of different degrees. Through optimization via gradient ascent, we embed the weighted graph vertices into a low-dimensional vector space $ {\mathbb{R}}^d $ while preserving the neighborhood of each node. These embeddings may then be used to detect relationships between nodes and study the structure of the original network. We then demonstrate the effectiveness of our method on dimension reduction through several examples regarding identification of transition states of chemical reactions, especially for entropic systems.
- networks,
- network embedding,
- feature learning,
- transition states,
- random walks
Citation: Paula Mercurio, Di Liu. Identifying transition states of chemical kinetic systems using network embedding techniques[J]. Mathematical Biosciences and Engineering, 2021, 18(1): 868-887. doi: 10.3934/mbe.2021046

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Abstract

Many chemical and biochemical systems can be intuitively modeled using networks. Due to the size and complexity of many biochemical networks, we require tools for efficient network analysis. Of particular interest are techniques that embed network vertices into vector spaces while preserving important properties of the original graph. In this article, we {introduce a new method for generating low-dimensional node embeddings for directed graphs, using random walk sampling methods for feature learning on networks. Additionally, we demonstrate the usefulness of this method for identifying transition states of stochastic chemical reacting systems.} Network representations of chemical systems are typically given by weighted directed graphs, and are often complex and high dimensional. In order to deal with networks representing these chemical systems, therefore, we modified objective functions adopted in existing random walk based network embedding methods to handle directed graphs and neighbors of different degrees. Through optimization via gradient ascent, we embed the weighted graph vertices into a low-dimensional vector space $ {\mathbb{R}}^d $ while preserving the neighborhood of each node. These embeddings may then be used to detect relationships between nodes and study the structure of the original network. We then demonstrate the effectiveness of our method on dimension reduction through several examples regarding identification of transition states of chemical reactions, especially for entropic systems.

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