Introduction

The high mortality of many diseases prohibits human longevity, and therapies need to be designed to suppress disease progression and aid organisms to recover from abnormal states1. However, the cost of launching new drugs is expensive and increasing, due to the long-term safety procedures in clinical trials2 caused by drug overdose toxicity and off-target side effects3,4. The unknown drug targets in individuals may also cause problems. For example, the drug Torcetrapib has been designed for cardiovascular disease5, but it may cause severe side effects of hypertension6. Analyses of omics, including genomics, proteomics, metabolomics and transcriptomics, contribute to the studies of disease mechanisms and drug responses. While a single omics layer focuses on a specific aspect with less complexity but limited information7,8. For instance, using only the conventional marker or the haplotype association cannot reveal the combined effects of Single Nucleotide Polymorphisms (SNPs), which potentially induces stroke9. The systemic view on dynamic gene regulation shows that genes work as part of complex networks instead of acting alone to perform cellular processes10. The integrative multi-layer omics data, such as transcriptional factors, genes and their expression products, provides a comprehensive map of metabolism and molecular regulation when analyzing and predicting based on complex cellular networks11,12,13,14,15. This leads to the prediction of potential molecular interactions through latent information of omics data. A general figure of omics data interactions is shown in Fig. 1.

Fig. 1: A systemic view of disease.
figure 1

Interactions among genomic, proteomic and transcriptomic levels reveal the regulatory process within organisms. Drug molecules intervene in this process by binding with specific target ligands. These omics data and chemical molecules are required to be analyzed simultaneously to study the entire disease mechanisms and drug reaction.

Full size image

In pharmacology, drug molecules act by binding to specific proteins, thereby changing their biochemical and biophysical activities16. Traditional treatment design based on physical parameters and external modalities17,18 or simple ligand-protein interactions4 are not sufficient for meeting clinical drug safety criteria or specifying variability among individuals. Modeling of the integrated clinical data and multi-layer molecular interactions makes the drug responses predictable3,19.

With multi-layer omics data, a single disease can be studied across different clinical modalities simultaneously (i.e., the horizontal direction in Fig. 2), and different diseases can be explored from a single modality (i.e., the vertical direction in Fig. 2). Systems approach makes chemical molecules and biomolecules more likely to be linked to phenotypes for analyzing diseases and drugs and identifying their potential connections. Modeling of disease pathways and drug responses through different layers of regulation contributes to drug repurposing and drug combination based on known molecular interactions. This review classifies the models for interactions among gene, protein and drug molecules into two main classes: static network and dynamic modeling. Both frameworks integrate biological information. The modeling is served for studying disease mechanisms and drug responses. The two main tasks include (1) deriving potential molecular interactions from disease mechanism and drug response, and (2) designing drug dosages.

Fig. 2: Analyses of disease and drug effect through single and multiple layers of omics data.
figure 2

Systemic view enables scientists to establish disease models from a higher hierarchic level by multi-layer data integration. Through vertical analysis within single layers, the disease-related molecules are identified by abnormal gene expression values. Through horizontal analysis with different layers, the interactive information are used to track diseases or drug effects throughout the entire biological process.

Full size image

Network structure in systems biology

A network structure visualizes a wide range of components such as genes or proteins and their interconnections. Network-based modeling can be established for systematic analysis based on omics data from various scales20, which expands the use of bioinformatics beyond its original meaning by mining structural motifs for novel interaction prediction21. This agrees with the ideas that the networks with hierarchical bio-information consist of the metabolic, signal transduction, and gene regulation pathways all contribute to the analyses of interactions between protein inhibitors22. Diseases with overlapping network modules show significant co-expression patterns, symptom similarity and comorbidity23, whereas diseases residing in separated network neighborhoods are phenotypically distinct24.

A basic network is made up of nodes and edges. For molecular interactions, nodes can be genes25, proteins26, and drugs. Node annotations can be connective properties, including binding affinities27, interactive directions28, and the importance and confidence of the connections29. Edges link the nodes, and edge annotations can be functional interactions between nodes, including protein physical interactions, gene regulatory relations30, mechanism of activation and inhibition31,32, and disease associations33. Besides, network complexities reflect in its size. Large networks with high complexity can be iteratively divided into measurable subunits to reduce the complexity of analysis34, and each subnetwork can be a set of functionally grouped nodes28. The patterns in the known annotations can be used to predict new annotation35, and structural patterns can be obtained by network motif. Motif encodes regulatory behaviors and decreases internal cell noise28, and it also helps identify drug molecules with common reactions, discover unknown drug responses, and predict potential therapeutics36. The regulatory motifs in the gene regulatory network contribute to modeling the cell fate dynamics in the immune system37. The pathway motifs within gene regulatory networks help interpret genetic and epigenetic variation38. The topological motifs in the interaction network of drugs and targets help select target protein candidates for drug synergy39.

Static network of diseases and drugs

A static network models the statically functional interactions from omics data. Network structure provides topological properties from the presented interactions. It integrates intra- and extra-cellular information for identifying the modules’ functional response by multiple network alignment. The overlapped multi-omics data integration is informative for reveal new molecular interactions12,15.

The purpose of constructing a static network is to predict the potential interactions among drug molecules and target proteins through the shared components, as they can be the intermediaries to convey information to different network layers4,40. For example, the diseases can be associated based on the shared genetic associations, the gene-disease interactions, and the disease mechanism23,26,30, such that disease connections can be built through the shared genes for drug repurposing. In a host-pathogen interaction network, the shared enzymes and regulatory components connect the metabolic reactions for predicting drugs for fungal infection7. Compared to a multiplex network, which only contains the same type of nodes and integrates the subnetworks from different layers, a heterogeneous network has the capability to include different types of nodes and edges. The multi-layer connections for the same nodes will result in a multiplex-heterogeneous network. More details about the different network structures can be found in reference41. Additionally, the new interacting pairs may account for variability in disease progression or drug response among individuals. To conclude, the shared components across layers may reveal new findings through multi-layer omics data modeled by a heterogeneous (or multiplex-heterogeneous) network structure.

The absent interactions don’t guarantee the negative interacting relations, since the available binding profiles are limited. One obstacle preventing expanding the database is that the clinical experiments are costly. To avoid the expensive clinic experiments, based on the static network models, machine learning-based methods are used to predict possible interactions using known interaction data.

Interactions from omics data

Due to the expression relations, genes and proteins are always analyzed together for genetic analysis of complex diseases. For genome-wide association, proteins and gene interactions can identify densely connected modules in the human protein interactome9. The protein-protein interaction (PPI) networks encode the information of proteins (nodes) and their interactions (edges) into the network structure. PPI networks help predict the potential disease-related proteins, based on the assumption that shared components in disease-related PPI networks may cause similar disease phenotypes7,33,42,43. For example, PPIs can be used with gene co-expression networks to assess the host-pathogen response for clinical treatment of Covid infections44. To be specific, the HCoV-host interactome was used to predict SARS-CoV-2 pathogenesis and provide a theoretical host-pathogen interaction model for HCoV infections.

The aim of modeling static molecular interactions is to use the interaction profiles to find out the potential interacting pairs. The modeling starts from identifying disease-related regulators using omics data. Consider that the RNA-sequencing data on a disease-related microarray is available. The disease-related genes can be selected from the differentially expressed genes (DEGs) based on the moderated t-statistics analyses and empirical Bayes using Limma in R45. The genes with large variations in expression data can be chosen based on fold-change and p-value, and a PPI network can be mapped. Limma focuses on the statistical meaning of the gene expression level, and its performance is affected by the number of samples.

For gene co-expression analyses based on microarray data, Pearson Correlation Coefficient (PCC) is frequently used. For example, a gene co-expression network for the Z. mays and A. flavus genes can be mapped directly from pairwise PCC masked by a customized cutoff46. WGCNA47 constructs an approximately scale-free network for detecting functional gene clusters based on PCC of gene co-expressions, under the assumption that proteins work together to perform metabolic functions. The disease-related hub genes/proteins with high connectivity are selected from the clusters. However, R2 value and connectivity of the identified gene network are sensitive to gene quantity, and different parameter settings (i.e., soft threshold) will result in different co-expression modules. In the frequent gene co-expression network48, gene pairs with high PCC, which are collected from different cancer and normal microarray dataset, are selected to build subnetworks of tightly co-expressed gene clusters using an iterative greedy algorithm “Quasi-Clique Merger”. The edges in subnetworks are weighted by the frequency of these genes, and similar subnetworks are merged into larger networks that are identified for specific diseases. The researchers noted that compared to differential expression analysis, where normal samples are necessary for comparison, this approach integrates multiple microarray datasets that even make the use of data without normal samples, which makes the constructed network more informative. However, the size of the datasets has to be large enough to ensure a high level of significance for PCC. A decision tree-based method Randomforest GENIE349 can be used to infer gene co-expression network by solving p (i.e., the number of genes) regression subproblems of identifying gene expression patterns and then grouping the genes. It can fast detect gene networks from large gene datasets that have multifactorial expression data. However, this method assumes knowing the transcription factors in the gene dataset of the experimentally confirmed gene interactions. Note that PCC assumes the gene expressions are linearly correlated, which may not be true for biology systems48. In a gene co-expression network for identifying cross-species interactions, mutual information and Z-scores of gene pairs are calculated using Context Likelihood of Relatedness algorithm50, which are used to infer edges in the network. As described by the authors, this algorithm can cope with nonlinear changes of gene expression, and it shows higher accuracy compared to PCC. However, PCC is still needed to discriminate the (positive or negative) directions of correlations of gene pairs. See Table 1 for comparison.

Table 1 Methods for constructing gene co-expression networks.
Full size table

Eventually, the target proteins are obtained based on the gene clusters in the gene co-expression networks. Drugs that potentially intervene in disease progression can then be predicted based on these proteins. The question that remains is how to detect the disease mechanism relevant to small gene expression variations, since small changes in some genes may have more essential contributions to the overall process. Note that gene expression level study excludes the non-transcriptional interactions48, which can be a potential limitation for predicting molecular interactions.

Drugs and targets interaction

Similarly, a static network can model the interactions among drugs and targets. Drug-target interaction (DTI) networks have been applied to study the prediction of drug response51, interaction profiles of new drug-target pairs16,26,27,52, and side effects of unknown drug combination22,53,54. A DTI-based target inhibition model has been proposed to identify the disease-specific target set with possible drug-target combinations by mapping drug inhibition profiles with the use of protein candidates, which were relevant to cancer survival with known drug binding profiles52. Notably, the drug target proteins in a DTI network may have larger degrees (i.e., more interactive molecules) than those proteins in a PPI network16.

Nodes in DTI networks include drugs, drug targets, and off-target proteins. For predicting drug side effects, the off-targets can be linked based on drug clinical relevance54. For drug combinations, different drugs may have interactions that induce side effects in patients or reduce the drug efficacy55,56. This requires us to identify potential drug-drug interactions (DDI), which can be explored by expanding DDIs through the shared targets of the drugs. The enriched DDIs identify potential targets and find new therapeutic uses (that the drugs do not initially aim at) and combination with surprising efficacy4. Figure 3 delineates a general view of the abridged drug and target interactions. Besides, the text mining-based predictions of molecular interactions rely heavily on the existing reports and references57, and insufficient clues would result in inaccurate predictions. Some of the frequently used databases for gene and protein interaction analyses are listed in Table 2.

Fig. 3: Interactions among drugs and target proteins offer chances for drug combination, co-administration, and repurposing.
figure 3

Drug molecules may bind with off-target proteins that induce side effects (labeled by red arrows), which should be avoided. Though, the therapeutic drug molecules should be reserved, as they function as desired to cure diseases (labeled in green arrows). Interactions between drugs provide an opportunity for enhanced therapeutic performance through drug co-administration and combination (labeled by cyan arrow). Drug similarity conveyed by drug-target interactions provides a chance for drug repurposing (labeled by orange arrows). Drug targets may be expanded to similar proteins using protein-protein interactions for drug repurposing (labeled in purple arrows). Drug molecules in this figure include epinephrine (for “Drug 1”) and benzene (for “Drug 2”) as examples, using the icons from Biorender.

Full size image
Table 2 Databases for gene and protein interaction analyses.
Full size table

An example of constructing a static network

Here is an example of constructing a heterogeneous network that integrates interactomics data from different subnetworks. The task is to predict potential drugs for a disease using RNA-sequencing data. Data can be obtained from online databases such as Gene Expression Omnibus database58. DEGs can be identified through statistical analyses (e.g., empirical Bayes using “Limma” in R45) on these gene expression data. The resulting genes are mapped into the signaling pathways database (e.g., KEGG database59) such that the highly perturbed disease-related pathways are selected. Target proteins are selected from the pathways, since we aim at intervening in the disease’s progress by blocking the relevant pathways using drugs. Interactions of target proteins are parsed from database (e.g., STRING database60) for constructing PPI networks and embedding the internal connecting information. DTI data is parsed from the drug database (e.g., DrugBank database61) by inputting the name of target proteins. Drugs that bind with at least one target protein are obtained to construct the DTI network. The DDI subnetwork is constructed by expanding the drugs from DTIs to their interactive partners, using interaction data from the drug database. And the internal connecting information among the expanded drug set is embedded into the subnetwork. The molecules from the three types of subnetworks are then integrated using a heterogeneous network, whose nodes are the drugs and target proteins, and edges are the interactions among proteins and targets. More recent works have focused on graph-based representation of molecular interaction network G(V, E), where V is the set of molecules and E is the set of molecular interactions62,63,64. And information is propagated from nodes to nodes through graph edges. The overview of the delineated process is visualized in Fig. 4.

Fig. 4: Statistic analyses on gene expression Values from RNA-sequencing data identify DEGs.
figure 4

The signaling pathways that contain the highest ratio of DEGs are regarded as disease-related. All or part of proteins in these pathways are selected to form the target protein set. By parsing PPI and DTI data from databases, the PPI network and the DTI network are constructed. The drug molecules in the DTI network are also used to parse and construct the DDI network. Finally, the DDI and the PPI are connected by the DTI, and thus the heterogeneous network is constructed. The task of predicting the potential drugs for a given disease is now transferred to the prediction of the interactions between the proteins and the drug molecules. Abbreviation: DEGs Differentially Expressed Genes, PPI Protein-Protein Interaction, DDI Drug-Drug Interaction, DTI Drug-Target Interaction.

Full size image

Limitation of static network modeling

The dynamic metabolic behaviors in patients result in changing expressions of genes. The interactomics in static modeling may become invalid, since molecule expression levels deviate a lot from the points that the model is built based on, thus leading to the failure of static modeling. This makes the dynamic modeling in Section V necessary. The major purpose of dynamic modeling is to map the regulatory relations among molecules, such that drugs can be used to intervene in disease progression by binding with target proteins, which drive the expression levels of genes to the normal range. Once gene expressions are corrected, the predictive static models will be valid again.

Analysis of static modeling

Although the models of diseases or drugs have been studied for decades, the actual biosystems are far more complicated than complete modeling65, which indicates the potential of exploring more comprehensively models. This section reviews recent techniques that have been used to predict more information based on static models.

Importance quantification

The importance of interactions in the network requires measuring and ranking for reducing network complexity and generating the weights, such that the simplified networks include only the most disease-related molecules65, which improves the efficacy of the learning process. The topological descriptors, such as degree, betweenness, and closeness, are frequently used to quantify the node importance in the network and embed spacial information into the modeling16,66,67. Node degree D, which is the number of connected edges30, measures the connectivity of nodes. In molecular interaction networks, the highly connected nodes (i.e., the hub nodes) usually provide more biological insights, compare to nodes with low degrees45,47. However, when identify disease-specific regulators, the importance will be penalized on hubs, since they don’t have much information66. The unconnected nodes will be discarded, since there is no path available to convey messages. Betweenness that describes the centrality of the given nodes is measured by the shortest path68, and it has more change when removing intermodular hubs compared to the intramodular19. Importance quantification can also be done based on statistics. For example, Z-score (i.e., the harmonic mean of precision and recall) measures the variability of the observations, and it has been used to quantify the importance of shortest paths between drug targets and the cardiovascular disease-related proteins53. The eigenvalues can be used to quantify the importance of data projection basis. In WGCNA47, the significance of gene co-expression clusters is quantified using the “eigen-genes” to reserve the most important genes and modules when modeling.

Similarity analysis

Similarity characterizes how elements are similar to each other in a static network. The assumption is that similar nodes have similar interaction profiles. For example, similar chemical structures of drugs show similar therapeutic effects for diseases33,40,51. Different types of similarities can be used based on the type of molecules. Protein similarity can be obtained based on protein sequence using the Smith-Waterman alignment algorithm27,69. Similarities between small molecules drugs are often calculated based on Jaccard coefficient of chemical structure notated by Simplified Molecular Input Line Entry System (SMILES)70. Similarity between cell lines can be obtained using similarity of gene expression profiles51. Other types of similarities can be the drug phenotypic side-effect similarity71, pathological similarities33, and so on.

Learning-based methods for clustering and classification

The evolution in learning-based networks has shown its ability to efficiently learn from massive datasets4. The learning-based methods can be supervised (with labels for training data), unsupervised (without labels) or semi-supervised (with partially labeled data).

A supervised learning approach enables the model to predict unknown parts (links) of the network based on the known interacting molecules72,73,74. Support Vector Machine (SVM), as a supervised learning model for classification, can refine topological information from network structure. SVM has been used for predicting DTIs based on DDIs, drug chemical structures, and side-effect information75. And the result shows that AUC values reach 0.76 in predicting the interaction between 261 drugs and 2,140 proteins. The multi-class SVM has been used to predict the therapeutic class of FDA-approved compounds using drug similarities, and it shows 78% classification accuracy of level 2 ATC codes among 410 drugs76. Kernels in SVM measure the features between gene pairs to train the classifier77, and the classifier makes the binary prediction for the interactions between the existing molecules and the incoming components72. Random Forest (RF), as an ensemble learning algorithm, can be used to performance classification based on decision trees78, such as the prediction of the contact probability between protein coevolved residues79. RF is robust to noise and it is capable to handle small sample size80. However, RF is less interpretable, and the computational load of RF will increase exponentially as the size of data increases81. The convolutional neural network (CNN), which uses the convolutional kernels and refines molecular features from arbitrary network frames of different sizes and shapes82, has been applied to refine DDI features using text mining on the biomedical information56. In the study, the prediction performance reached a F-score of 70% when evaluating about 900 drug documents. While a single-layer neural network may not have good predicting performance, a deep neural network can be deployed and obtain better results. Deep learning (DL) methods, which are built by multiple layers of neural networks, are gaining more and more attention because of the structural flexibility83 and their capability of extracting molecular patterns by mining latent information from the network structures82,84,85,86. For example, a deep CNN learning architecture87 shows its high concordance index (large value is better) in predicting drug-target binding affinity. While using more layers of neural networks results in more parametric settings, which could be potentially time-consuming. One efficient method to search these network parameters can be Particle swarm optimization88. A scheme of a convolutional neural network is shown in Fig. 5.

Fig. 5: A convolutional neural network.
figure 5

This neural network is comprised of an input layer, convolutional layers (that extract features from hyperplanes of input data by projection/convolution), pooling layers (that reduce the spatial size and mitigate the locational sensitivity), flatten layer (that flattens the features and feeds them into the artificial neural network), fully connected layer (that learns nonlinear function of the extracted features) and an output layer. The input is the raw features such as molecules sequence patterns, gene regulation annotations and patterns, molecular interaction network motifs, molecular structures and structural associations, drug chemical structures, drug side-effect reports, and so on. The output can be the predicted classification (e.g., molecular binding profiles) and regression (e.g., quantified molecular binding affinities) when obtaining new samples.

Full size image

However, supervised learning heavily relies on the size of labeled samples during the training. Consider nd drugs and nt targets that include nd ⋅ nt drug-target pairs, then the number of available DTIs na < < nd ⋅ nt. Moreover, when predicting DDIs base on drug side effect, the positive samples (i.e., drug pairs with known interactions) can be obtained from database, but the negative samples (i.e., drug pairs with with clinically validated safe co-prescriptions) are almost unavailable89. The lack of labeled data will decrease the predicting performance when using supervised learning methods. Though data preprocessing partly copes with the missing data problem, it causes certain information loss90.

Unsupervised learning aims at clustering data based on features, without the use of data labels. Examples are given as follows. When identifying relevance of disease phenotype and treatment response between patients, the number of patient clusters is unknown, and a hierarchical clustering (HC) algorithm with multiple linkage methods has been used for clustering patients based on genome-wise similarity and variability91. The benchmark test on 191 Multiple Sclerosis patient samples reaches Rand index moer than 0.85, and it also shows the capability of reducing feature dimension of Single Nucleotide Polymorphisms of 191 patients from more than 25,000 to about 1500. Advantages of HC includes that it is visualizable and user can customize the granularity by cutting the clustering at the desired level92. As a widely used unsupervised learning method, the autoencoder (AE) learns and refines the low-dimensional features from data to reconstruct the input, so that a minimalistic description can be derived for differentiability among samples84. This encoding-decoding frame can be used to denoise a single-cell RNA-sequencing model93, transform molecules directly into a numerical representation86, and compress computational dimensionality4 when detecting the cell type-specific clusters in an ensemble clustering way94. The stacked AE has been used to extract highly representative features from drug molecular structure and protein sequences, which help identify the potential DTIs95. Instead of capturing the deterministic latent features, the Variational Autoencoder (VAE) aims at capturing the distribution of the latent variables, with the help of variational inference96,97. Though VAE is capable of generating artificial samples for sparsely labeled molecules data, its heavy computational load due to the optimization of hyperparameters may limit the usage92. Note that AE and VAE are not classifiers.

Transductive learning learns to predict labels of unlabeled data in the existing network by training on entire dataset, while inductive learning learns a classifier that make prediction on testing data out of the current network. We focus on transductive learning, because the static network structure is deterministic for predicting potential interactions among existing nodes.

Semi-supervised learning (SSL) combines supervised and unsupervised learning. SSL is widely used for predicting molecular interactions using a small number of known interactions and many unknown interactions. Label propagation is a classical transductive SSL method that utilizes a small labeled dataset and predicts the label of the unlabeled data iteratively98. This method has been used to predict DDIs based on side effect similarity among 569 drugs with 52,416 DDI pairs89. The result showed that when the size of the training dataset is small, instead of the training data, the output of the proposed predictive model mainly relies on the geometric structure of the entire dataset. Also, the Area Under the Precision-Recall Curve value ranged from 0.650 to 0.729 for different ratios of testing and training dataset, which indicates the predictive model is stable. The autoencoder-based semi-supervised learning has been applied for predicting DTI99, DDI100 and PPI101. The unsupervised AEs/VAEs with a supervised deep neural network form a semi-supervised neural network, which shows higher AUC (over 0.8) compared to other learning methods99,100,101. Though, SSL cannot work without proper assumptions, as it will loss generalization from a finite training set to other test cases102,103. Besides, SSL is not always superior to supervised learning. An example is that a supervised logistic regression outperformed a semi-supervised label propagation when evaluating the prediction accuracy of gene functions, relevant disease and gene traits of on the full network connectivity, since the former algorithm efficiently extracts local network patterns, while the latter focuses on network topology104. The comparison of learning-based methods for predicting molecular interactions can be found in Table 3.

Table 3 Learning-based Methods for Predicting Molecular Interactions.
Full size table

Graph-based learning for predicting molecular interactions

Graph modeling is gaining increasing attention, since it encodes the structural and spatial information of data into models. The basic graph components include the sets of nodes, edges and an adjacency matrix that stores node connection information. A heterogeneous network of molecular interactions can be easily represented by a graph model. Graph Embedding (GE) encodes the spatial and topological information of graphs into low-dimensional feature vectors using a parametric function105,106. GE in transductive learning is deterministic because of a fixed graph. While for inductive learning, GE is generated from graph input features105.

Several graph learning methods have been proposed. Similar to CNN, graph convolution extracts features from graphs. Graph Convolutional Network (GCN), as a spatial convolution approach and the first-order approximation of Chebyshev polynomials of the graph spectral filter for semi-supervised classification tasks, aggregates the weighted node features from neighborhoods to the current node being visited using locally convolutional computation107. However, GCN has a relatively shallow structure. In a deep GCN, the oversmoothing property gives similar embedding to nodes from different labels/classes, resulting in mislabeling/misclassification issues108. The spectral graph convolution approaches partition/diffuse graph in Fourier domain using Eigen-decomposition of the graph Laplacian109, which could be computationally expensive when dealing with large graphs, and the graph convolution relies on specific graph structures that are not generalizable to other graphs with different structures110. Graph Attention network (GAT) utilizes the self-attention mechanism to obtain the normalized attention score (i.e., the relative importance) of each node from its neighborhood110, instead of the average aggregation functions in GCN. The attention mechanism solves the oversmoothing problem108. Node level attention is computed in parallel which is time-efficient. And it alleviates the effect brought by the lack of knowledge of the entire graph structure. However, GAT can’t tell the differences between local and global structures well because the aggregators lack of cardinality preservation mechanism111. Graph Autoencoder (GAE) and Variational GAE (VGAE) finds the (distribution of) latent variables of embeddings (which can be encoded by GCN), using a (variational) inference model and a Generative Model (GM), in low-dimensional space to recover the adjacency matrix of a graph in an unsupervised manner112. Loss functions in GAE and VGAE can be the reconstruction loss of the adjacency matrix (i.e., graph connectivity) and the variational lower bound, respectively112. The reduced dimensionality of data speeds up the training/backpropagation processes. However, the autoencoder-based methods focus on capturing most of the information from dataset based on a lossy reconstruction, which may not be relevant to the problem, and a relatively large dataset is needed to train an autoencoder. The inadequate bioinformatic data can be augmented by learning strategies. Graph Generative Adversarial Nets (GraphGAN)113 composed of a GM and a Discriminative Model (DM) can be used for this task. GM in both of GraphGAN and VGAE captures the distribution of the graph connectivity. However, rather than predicting the interacting pairs of nodes in each training epoch in VGAE, the GM in GraphGAN generates fake samples to deceive the discriminiator, such that DM learns to discriminate the true samples (that come from training data) from the generated samples. A Nash equilibrium that balances between two models is desired for convergence. Some limitations of GAN, including the unstable gradient updates and the vanished gradients of generator114, may potentially cause problems when being applied to graphs. See Table 4 for comparison of these graph learning methods.

Table 4 Graph learning methods.
Full size table

Additionally, the negative interacting pairs are rarely available in bioinformatics data, which results in an unbalanced dataset. The negative sampling115 can be applied that randomly assigns negative labels to the unlabeled data during the training process. This step aims at training the classifier to tell the difference between positive samples and the pseudo-negative samples drawn from training data. Lastly, a powerful package “PyTorch Geometric”116 (python), which contains multiple recent graph learning algorithms and uses fast tensor operation in GPU, can be used for implementation.

Compare to a binary classification problem (e.g., predict if the interaction exists or not), the multi-class classification problem is more attractive since more than two classes are involved. Examples of multi-class prediction can be found in predicting the interaction type of protein pairs117, the gene phenotype118, etc. The loss functions are designed based on class or labels. Each sample in a multi-class problem is assigned to one of the classes, and the labels for multi-label problems are usually obtained using categorical one-hot encoding119, which encodes the label as a vector with binary values. The entropy-based loss will be different for multi-class and multi-label classification. The Softmax function is often used to output probabilities of classes for the multi-class problems120, and Sigmoid function is used for multi-label problems.

The integrative static modeling conveys information through networks, which potentially results in a more comprehensive disease model by prediction. Similarities of nodes and edges among the nodes are calculated, and the quantified information is embedded in nodes and edges. Then the learnable models find representative features to predict potential molecular interactions. The benefits of static modeling-based prediction for disease treatment include: (1) the predicted disease-related regulators can be the new drug targets, which potentially improves treatment efficacy by finding new drugs for the same diseases; (2) the known drug target proteins redirect the drug molecules to bind with similar proteins through PPIs for drug re-purposing; (3) proteins/genes identified from the predictive model may explain variability among patients for specific disease phenotype or specific drug responses, which contributes to precision medicine; (4) the predicted off-target activities help avoid side effects; (5) the potential drug interactions contribute to designing novel therapies by drug combination, since the combinations may either reduce or enhance drug therapeutic efficacy. Though hypotheses and predictions can be made in a static network, the clinical use of potential drugs, drug combinations, or drug repurposing is highly concerned with patients’ safety, which means the accuracy and generalization of current predictive algorithms could be problematic. Besides, the commonly acknowledged problem in analyzing interactions between biological and chemical molecules based on database is data scarcity20, and how to use limited data to reliably produce more data for learning and obtain highly accurate predictive models remains an open problem.

Dynamic modeling of regulation in organisms

The term dynamic disease refers to that disease pathogenesis is mainly caused by the appearance of new dynamic behaviors of organism, independently of the underlying pathogenesis121. The dynamics of diseases and drug reactions are informative for treatment design. For example, the viral reservoir dynamics are significant to understand the natural properties of ongoing viral progression with treatment122, which help researchers find the cures for diseases. A kinetic model predicts disease outcome by modeling the time-course behaviors of patients’ interactome19,25, and model performance can be assessed by its ability to generate testable predictions123.

Gene regulatory network

In organisms, the changing gene expression results in phenotypic changes. Genes interact with each other through RNA and protein expression products, thereby governing the rates at which genes in the network are transcribed into messenger RNA121. The causal information in gene expression data, e.g., key drivers of complex traits in phenotype-related gene interaction network13, can be identified by variations in DNA and gene expression-related traits25,37. Gene regulatory networks (GRN) are thus used to detect potential mechanisms based on dynamic behaviors of epigenomic activity between normal and disease state10. For applications, GRNs have been used to analyze infectious diseases by detecting gene regulations related to the infectious and viral mechanism15. A combination of gene regulatory components including myeloid and lymphoid has been studied to identify cell fate specification11. Analyzing GRN is regarded as the reverse engineering for investigating gene regulatory relations by going backward with the observed gene expression28. The causal regulatory relations of genes are desired to be found from genome-wide expression data. A diagram of GRNs is shown in Fig. 6a. A GRN model contains genes and regulators as the nodes, and directional edges as the regulatory relations between the nodes31. To model and reconstruct GRN, the time-course microarray data of gene expression product (e.g., gene expression values, chromatin expression profiles) are required25,28,124, and the quantitative regulation can be obtained from computational models of GRN.

Fig. 6: Schematic diagrams in dynamic modeling.
figure 6

a Gene regulatory network adapted from the work25. Gene 1, 2, and 3 are coding genes. Gene 1 regulates its own expression and those of Gene 2. The protein produced by Gene 1 regulates Gene 3 expression through a signaling factor/protein (that is produced from the protein expressed by Gene 2.) Drugs can intervene in the regulation by binding with proteins that change the gene expressions. b Diagram of signaling transduction. Signals are received and enter the nucleus to change gene expression. Proteins are synthesized to regulate phenotypic behaviors of cells or tissue. Errors (e.g., dysregulation) in signaling pathways (e.g., dysregulation) may cause the ceasing of cell apoptotic that results in unlimited growth and division.

Full size image

Signaling pathway and transcription

Signal transduction pathway contains regulators between molecules in an organism. It attributes to changes in both gene expression and gene connectivity29. Errors in signal transduction lead to altered development and incorrect behavioral decisions in organisms, whose dysfunction may result in uncontrolled cell growth or tumorigenesis28,125,126. At the protein level, signaling pathways are comprised of protein interactions covering the biological functions in living cells, which captures the inter- and intracellular regulatory mechanisms of gene transcription and protein synthesis28,31. The modeling of dynamic signaling pathways measures disease progression. For example, pathway analyses contribute essentially to the systematic profiling of the transcriptome in heart failures29. The fungal signaling pathways model the regulatory behaviors in fungal pathogen infections7. Notably, the pathway-wide association can even extract valuable information from background noise and the context-specific logic of GRN38.

Transcription Factors (TFs) are the keys in modeling gene regulatory relation. For example, TF regulates the development of innate and adaptive cells of the immune system11. The dynamic transcriptional and translational subnetworks have been used to model the trigger mechanism of the innate response regulated by intercellular and intracellular heterogeneity15. A diagram of signal transduction is shown in Fig. 6b.

Analysis of dynamic modeling

Mathematical modeling for dynamic regulation

The dynamics of omics data reflects organism’s response to the changes of interior milieu or environmental factor. Variability in metabolisms and phenotypes can be huge even if most of the corresponding genes are the same127. Dynamic modeling exerts mathematical tools that quantify the rate of state change in gene regulation in different conditions and time sequences25. More logic models and kinetic models using Hill function or piecewise linear differential equations for quantifying the dynamic behaviors of gene network have been reviewed in the references128. This section reviews the computational modeling of diseases using Differential Equations (DEs), which can be used for drug administration by control theory.

DEs capture how the system reacts to the variations caused by disease or drugs. From a systemic view, DEs integrate information from multi-layer omics data into a unified form34. DEs model the dynamic behaviors of organisms, such as transcription31, gene regulation25, metabolite concentrations129, reaction rate34, and factors in signal transduction pathway127. DEs have also been used to quantify signal flow in pathways and explores the effect of oncogenic mutations on dynamics of ligands123. For disease modeling and treatment, DEs have been used to capture the dynamics of HIV viral infection130, drug response of an irradiation-induced cellular senescence131, and cancer cell population132. The components in DEs can be the metabolite concentration in nonlinear biochemical models129, the signal transduction molecules in dynamic cell compartment models34, and transcription factors and regulatory site31. A two-step approach, including the estimation of cell expression velocity using finite difference, and the estimation of a numerical square matrix that depicts the gene regulatory influence using sparse regression, can be used to computationally model a GRN from time-course single-cell RNA-sequencing data133. The accuracy of the approximated model depends on the user-defined sampling rate in finite difference. A robust dynamic model design makes it less sensitive to biological noise and disturbances, allowing us to track abnormal variations21,134. An example of the robust design of a physical system can be found in 134. Modeling noise is usually assumed to be Gaussian. Non-Gaussian stochastic noise can be modeled by solving the Fokker-Planck equation135.

Parameter estimation in dynamic models

The main task of parameter estimation is to find the best-fit parameters to characterize physical processes and reproduce experiments136,137,138. To develop data-driven models, one can optimize the fit of a collection of parameters iteratively to a given dataset with random disturbance137,139, which explores the parameter space extensively and limits the number of non-convergent solutions131. The least-square estimation (LSE) that minimizes quadratic errors between the predicted model values and experimental data129,140,141), and the maximum likelihood estimation142, are widely used for this task. A Kalman Filter (KF) can recursively generate the maximum likelihood estimates for a linear dynamic system from a series of noisy measurements143,144. It handles the approximate modeling of high-dimensional noisy data with small sample sizes145. The Extended KF deals with nonlinear models in biology146. Note that the Gaussian noise is assumed in KF, and Particle Filters (PFs) are more appropriate when dealing with non-Gaussian processes. By randomly drawing samples from numerical simulation, the Monte Carlo method estimates parameter values122 and quantifies their uncertainties147. PFs in sequential Monte Carlo method obtain weighted samples from the non-Gaussian posterior probability of the state in nonlinear systems148. Study shows that a PF with an orthogonal basis (used for approximating the posterior by an orthogonal series expansion) outperformed Extended KF when estimating parameters of a Wiener anesthesia delivery model146. The process of parameter estimation is shown in Fig. 7a.

Fig. 7: Dynamic modeling and analyses.
figure 7

a Scheme of parameter estimation. After data acquisition, parameters are fitted in models by minimizing the difference between experimental data and model output. Sensitivity analysis, uncertainty quantification and identifiability analysis help assess the performance and robustness of the fit. b Loop of Model Predictive Control. Based on the output of model prediction, this control strategy updates control input (e.g., drug administration) to make the system dynamics track reference trajectory (e.g., desired tumor cell decrement) during each time interval. Its essence is to handle the constrained optimization problem (Constraints can be maximal drug doses and minimal normal cell populations).

Full size image

Sensitivity analyses assess how sensitive the models’ outputs are to the fitted models’ parameters changes. Sensitivity can then quantify model uncertainty through finite differencing or variational equations139. Identifiability checks the reliability of the estimates and assesses how well the model explains experimental data144,149. The global optima for parameter estimation are always desired, leading the multiple approaches to converge to the same solution ultimately129. Time of searching parameters is upper bounded to prevent an endless search139. Calling a single data cluster recursively may cause the absence of global parameters34 and get stuck in the misleading local optimum that should be avoided139 due to the generality. Precision medicine requires the patient-specific parameters for individual variability, resulting in a more complicated model. Note that a robust model design134 is desired after adding new parameters.

Drug administration with control theory

Drug dosage is designed for patients’ recovery by control algorithms. Though the dynamics of drug concentration in plasma and drug response are usually nonlinear in real world, a linearized model is usually used to approximate the nonlinear behaviors of the original system due to the reduced complexity150,151,152,153. Several methods can be applied for the linear approximation. For example, when the pharmacokinetic data is available, drug model can be fitted using regression methods4. Linear models can be approximated from a nonlinear model by truncating Taylor expansion at the first-order term. Or, the linear descriptors can be obtained using Koopman operator, which projects the infinite-dimensional time-evolving observables (e.g., time-series data of drug concentration and cell population) into finite-dimensional states, using dynamic mode decomposition154. The Koopman method is driven by data, which can be generated from the nonlinear model155. Then the linear controllers for drug dosages can be designed based on a linear model.

The Proportional, Integral and Derivative (PID) control is a classic control algorithm that takes the error e(t) between closed-loop feedback signal y(t) and the setpoint r(t) (i.e., e(t) = r(t) − y(t)) as the controller input to calculate input needed, such that the model states can be driven to the setpoints156. PID controller has been used to design the drug administration for the chemotherapy treatment in a cancer model18 and the anesthesia in the neuromuscular blockade models146,151. The clinical evaluation and simulation results show that drug concentration levels can reach and be maintained at certain levels in acceptable time horizon. A PID controller copes with the uncertainty in the system’s dynamics caused by interpatient variability157 and time variations151. The concise scheme of PID158 makes it flexible for functional expansion, such as an I-PD controller constructed by cascading18. However, the performance and the robustness of PID controllers depend heavily on tuning146,159, and one of the frequently used tuning approaches is the Ziegler-Nichols method160.

Besides maintaining the drug concentration at certain levels, drug dosage should also balance the therapeutic performance and the toxicity/side effects. Optimal control law minimizes control errors (e.g., drive the systems along trajectories) and control efforts (e.g., less energy consumption)161 subject to system dynamics. The linear quadratic regulator (LQR), as one of the optimal control strategies, can be used for deriving an optimal control sequence (e.g., a sequence of drug infusion rates) when dealing with a linear model in the form \(\dot{x}(t)=A(t)x(t)+B(t)u(t)\), with quadratic objective functions in form of l2-norm ∣∣ ⋅ ∣∣2) from time t0 to tf with the initial state x0, as shown in Eqn. (1)162.

$$J=\frac{1}{2}| | x({t}_{f})| {| }_{{S}_{f}}^{2}+\frac{1}{2}\int\nolimits_{{t}_{0}}^{{t}_{f}}(| | x(t)| {| }_{Q(t)}^{2}+| | u(t)| {| }_{R(t)}^{2}),{{{\rm{subject}}}}\,{{{\rm{to}}}}\,\,x({t}_{0})={x}_{0}$$
(1)

where Sf is the terminal weighting matrix. Q(t) and R(t) are user-defined positive semi-definite and positive definite weighting matrix, respectively. Large Q(t) results in aggressive drug doses, and large R(t) leads to medical conservatism. The optimal feedback control law u(t) = − K(t)x(t) is derived by solving the algebraic Riccati equation and Hamiltonian. Compared to PID control, optimal control balances the drug toxicity and dosage. For chemotherapy treatment, the control problem can be to minimize the kinetic energies of all the cancerous cells with low drug toxicity18,163,164,165 by searching the optimal sequence of drug dosages. In a HIV model, the goal can be to maximize the benefit based on levels of healthy CD4+ T cells and immune response cells by reducing the systemic cost of anti-HIV drugs130, and the cost function consists of beneficiary T Cell population and systemic costs of therapy166. For multi-drug treatment, the ratio of co-administration between different drugs is also considered164,167. More examples of optimal control in cancer treatment can be found in the book168. The control law calculated by minimizing the cost can be piecewise constants or linear in the finite time horizon169.

Drug concentration in patients is delicate. The dangerous therapeutic window of drug concentration in plasma determines whether the level is tolerable and the drug is effective for patients17,170. Thus, state and input constraints are necessary to a drug delivery system. Usually, constraints include the upper boundary of toxicity level and the certain therapeutic window for drug concentration and disease progression18,167. The constrained optimization problems require more complicated control algorithms. In a multi-objective genetic algorithm, the Pareto optimal sets have been used to search for lower values of the objectives simultaneously18. Nonlinear optimization can be solved by Bock’s direct multiple shooting method with a numerical solution on a fixed control discretization grid167. The steepest descent method can be used to search numerical solutions iteratively to minimize or maximize the cost function164, and the numerical solutions can be derived with Miser3/Matlab132. Model Predictive Control (MPC) handles the constraints in optimization problems and the mismatch between nominal and actual processes169, compared to optimal control. MPC solves a finite horizon open-loop optimal control problem to obtain control actions with predicted states from models. The local asymptotic stability of the control law is guaranteed when time horizon is sufficiently long169. The loop of model predictive control is shown in Fig. 7b. Besides, controllers can also “learn” from complex situations through iterative learning-based strategies to obtain optimal parameterized control signals171. The equilibrium in a chemotherapy model refer to the elimination or the stop of cancer cell proliferation, so that no more treatment will be needed164. While using MPC, the feasibility and stability should be carefully considered. Compared to optimal control, MPC handles the constrained optimization problems, which makes it more suitable for the drug administration design subject to patients’ physical constraints.

The advanced control has good performance on drug dose adjustment. When eliminating cancer cell population, it has been found that giving bursts of high-dose abiraterone reduces tumor burden more than 10 times, compared to giving a constant dose163. For robustness, when model parameter error is 25%, MPC reaches 98% success rate (among 100 simulations) of stabilizing HIV infection in 2 years165. When designing the dosage of remdesivir for SARS-CoV-2, an optimal control sequence has been obtained by solving a constrained optimization problem, and simulation shows that the proposed control scheme reduces the treatment horizon from 10 days to 5 days and it also reduces more than 50% of the drug dose, compared to the recommended treatment regimes from FDA and WHO172. See Table 5 for comparing these methods used for drug dosage design. By using the advanced control algorithms, fewer drugs can be used to obtain the same or better treatment efficacy with less toxicity. There are some other control strategies that use feedback control laws based on nonlinear models. For example, a positive semi-definite Lyapunov function, whose gradient requires to be negative semi-definite, can be designed to calculate the controlled vaccination rate that asymptotically stabilizes Covid infection138.

Table 5 Control algorithms for drug administration.
Full size table

Optimal drug dosage designed by control algorithms is used to efficiently intervene in disease progression, based on dynamic models with parameters estimated from sample data. Drug administration is transformed into an optimization problem with or without model constraints. The design of personalized treatments requires the patient-specific parameters for individual variability, which makes control scheme more complicated. Notably, the impulsivity of the model should be considered during the design process, and discrete drug therapy should be considered to allow the normal cells to rebuild in clinical treatment166.

Conclusion and outlook

Launching new drugs could be costly compared to using existing drugs for new therapeutic performance, and drug predictions based on computational systems biology have shown its potential in precision medicine, drug combinations, and repurposing. A static network composed of molecular interactions is able to predict the potential interaction pairs based on known omics data by conveying information through nodes and edges. The new participants in the map of molecular interactions help obtain a more comprehensive view of disease progression and drug response. This results in using drug molecules with better therapeutic performance while avoiding off-target effects. Also, the potential patient-specific regulators can be identified to explain individual variability for personalized treatment. However, the predictive models may fail due to in vivo changing behaviors, which makes dynamic modeling necessary. Dynamic modeling aims at building math models to predict disease progression and drug response. Model parameters are estimated from clinic data. The potential participants identified from static modeling can be the new elements in dynamic models. By applying optimal drug dosages designed by control algorithms, disease progression can be intervened efficiently, which also indicates the predictive model in static modeling will be valid again. The combination of static and dynamic modeling makes it a powerful tool for disease analysis and therapy design. SSL outperforms other learning methods for making static modeling predictive, while the underlying assumptions may not hold in real cases, which makes the model loss generality. The learning algorithms that have an accurate prediction of other testing data are always desired. For drug administration, the simple control algorithms cannot meet the complicated design objectives (e.g., control with constraints), while a complicated control algorithm may not be time-efficient, though it aims at more control objectives. A control algorithm that handles multi-objectives and computes drug dosage needed efficiently is desired.

The modeling of DTIs by expanding drugs and targets based on DDIs and PPIs offer opportunities for (1) finding new targets for the same drug, (2) exploring new drugs for the same disease, and (3) minimizing off-target side effects for safe therapies. Though side effects should be avoided for safety issues, it does not mean all drugs that cause side effects should be abandoned. Treatments like chemotherapy harm normal cells, so we shall choose the drug agents that specifically target on cancer cells, rather than healthy cells. Combined with drug administration for changing conditions of patients using control theory, the treatment with better therapeutic performance and lower impairment can be realized simultaneously.