Structure-based control of complex networks with nonlinear dynamics

Pancreatic cancer mutationscape: revealing the link between modular restructuring and intervention efficacy amidst common mutations

There is increasing evidence that biological systems are modular in both structure and function. Complex biological signaling networks such as gene regulatory networks (GRNs) are proving to be composed of subcategories that are interconnected and hierarchically ranked. These networks contain highly dynamic processes that ultimately dictate cellular function over time, as well as influence phenotypic fate transitions. In this work, we use a stochastic multicellular signaling network of pancreatic cancer (PC) to show that the variance in topological rankings of the most phenotypically influential modules implies a strong relationship between structure and function. We further show that induction of mutations alters the modular structure, which analogously influences the aggression and controllability of the disease in silico. We finally present evidence that the impact and location of mutations with respect to PC modular structure directly corresponds to the efficacy of single agent treatments in silico, because topologically deep mutations require deep targets for control.

Cell reprogramming design by transfer learning of functional transcriptional networks

Recent developments in synthetic biology, next-generation sequencing, and machine learning provide an unprecedented opportunity to rationally design new disease treatments based on measured responses to gene perturbations and drugs to reprogram cells. The main challenges to seizing this opportunity are the incomplete knowledge of the cellular network and the combinatorial explosion of possible interventions, both of which are insurmountable by experiments. To address these challenges, we develop a transfer learning approach to control cell behavior that is pre-trained on transcriptomic data associated with human cell fates, thereby generating a model of the network dynamics that can be transferred to specific reprogramming goals. The approach combines transcriptional responses to gene perturbations to minimize the difference between a given pair of initial and target transcriptional states. We demonstrate our approach's versatility by applying it to a microarray dataset comprising >9,000 microarrays across 54 cell types and 227 unique perturbations, and an RNASeq dataset consisting of >10,000 sequencing runs across 36 cell types and 138 perturbations. Our approach reproduces known reprogramming protocols with an AUROC of 0.91 while innovating over existing methods by pre-training an adaptable model that can be tailored to specific reprogramming transitions. We show that the number of gene perturbations required to steer from one fate to another increases with decreasing developmental relatedness and that fewer genes are needed to progress along developmental paths than to regress. These findings establish a proof-of-concept for our approach to computationally design control strategies and provide insights into how gene regulatory networks govern phenotype.

Cell reprogramming design by transfer learning of functional transcriptional networks

Recent developments in synthetic biology, next-generation sequencing, and machine learning provide an unprecedented opportunity to rationally design new disease treatments based on measured responses to gene perturbations and drugs to reprogram cell behavior. The main challenges to seizing this opportunity are the incomplete knowledge of the cellular network and the combinatorial explosion of possible interventions, both of which are insurmountable by experiments. To address these challenges, we develop a transfer learning approach to control cell behavior that is pre-trained on transcriptomic data associated with human cell fates to generate a model of the functional network dynamics that can be transferred to specific reprogramming goals. The approach additively combines transcriptional responses to gene perturbations (single-gene knockdowns and overexpressions) to minimize the transcriptional difference between a given pair of initial and target states. We demonstrate the flexibility of our approach by applying it to a microarray dataset comprising over 9,000 microarrays across 54 cell types and 227 unique perturbations, and an RNASeq dataset consisting of over 10,000 sequencing runs across 36 cell types and 138 perturbations. Our approach reproduces known reprogramming protocols with an average AUROC of 0.91 while innovating over existing methods by pre-training an adaptable model that can be tailored to specific reprogramming transitions. We show that the number of gene perturbations required to steer from one fate to another increases as the developmental relatedness decreases. We also show that fewer genes are needed to progress along developmental paths than to regress. Together, these findings establish a proof-of-concept for our approach to computationally design control strategies and demonstrate their ability to provide insights into the dynamics of gene regulatory networks.

Modular Control of Biological Networks

The concept of control is central to understanding and applications of biological network models. Some of their key structural features relate to control functions, through gene regulation, signaling, or metabolic mechanisms, and computational models need to encode these. Applications of models often focus on model-based control, such as in biomedicine or metabolic engineering. This paper presents an approach to model-based control that exploits two common features of biological networks, namely their modular structure and canalizing features of their regulatory mechanisms. The paper focuses on intracellular regulatory networks, represented by Boolean network models. A main result of this paper is that control strategies can be identified by focusing on one module at a time. This paper also presents a criterion based on canalizing features of the regulatory rules to identify modules that do not contribute to network control and can be excluded. For even moderately sized networks, finding global control inputs is computationally very challenging. The modular approach presented here leads to a highly efficient approach to solving this problem. This approach is applied to a published Boolean network model of blood cancer large granular lymphocyte (T-LGL) leukemia to identify a minimal control set that achieves a desired control objective.

Macroscopic resting-state brain dynamics are best described by linear models

It is typically assumed that large networks of neurons exhibit a large repertoire of nonlinear behaviours. Here we challenge this assumption by leveraging mathematical models derived from measurements of local field potentials via intracranial electroencephalography and of whole-brain blood-oxygen-level-dependent brain activity via functional magnetic resonance imaging. We used state-of-the-art linear and nonlinear families of models to describe spontaneous resting-state activity of 700 participants in the Human Connectome Project and 122 participants in the Restoring Active Memory project. We found that linear autoregressive models provide the best fit across both data types and three performance metrics: predictive power, computational complexity and the extent of the residual dynamics unexplained by the model. To explain this observation, we show that microscopic nonlinear dynamics can be counteracted or masked by four factors associated with macroscopic dynamics: averaging over space and over time, which are inherent to aggregated macroscopic brain activity, and observation noise and limited data samples, which stem from technological limitations. We therefore argue that easier-to-interpret linear models can faithfully describe macroscopic brain dynamics during resting-state conditions.

Deciphering driver regulators of cell fate decisions from single-cell transcriptomics data with CEFCON

Single-cell technologies enable the dynamic analyses of cell fate mapping. However, capturing the gene regulatory relationships and identifying the driver factors that control cell fate decisions are still challenging. We present CEFCON, a network-based framework that first uses a graph neural network with attention mechanism to infer a cell-lineage-specific gene regulatory network (GRN) from single-cell RNA-sequencing data, and then models cell fate dynamics through network control theory to identify driver regulators and the associated gene modules, revealing their critical biological processes related to cell states. Extensive benchmarking tests consistently demonstrated the superiority of CEFCON in GRN construction, driver regulator identification, and gene module identification over baseline methods. When applied to the mouse hematopoietic stem cell differentiation data, CEFCON successfully identified driver regulators for three developmental lineages, which offered useful insights into their differentiation from a network control perspective. Overall, CEFCON provides a valuable tool for studying the underlying mechanisms of cell fate decisions from single-cell RNA-seq data.

Nonlinear control designs and their application to cancer differentiation therapy

We designed three new controllers: a sigmoid-based controller, a polynomial dynamic inversion-based controller, and a proportional-integral-derivative (PID) impulsive controller for cancer differentiation therapy. We compared these three controllers to existing control strategies to show the improvement in performance and compare their robustness. The sigmoid-based controller adds a sigmoid term associated with the error of the controlled state and a selected observed state. The sigmoid term is multiplied by a control gain, thereby decreasing the control effort for state transition. The polynomial dynamic inversion-based controller adds a cubic error term in the error dynamic aiming to achieve a shorter convergence time to the desired value of the controlled state. The PID impulsive controller considers the accumulated controlled state error and the rate of change of the controlled state error, thereby forcing the controlled state to converge to the desired value and alleviating the damping effect in the steady state. For the considered cancer network, the 3 new cancer control strategies exhibit superior and robust performance. The PID impulsive controller has a significant improvement in robustness compared to the impulsive controller and has greater potential for cancer differentiation therapy.

Dynamical methods for target control of biological networks

Estimating the influence that individual nodes have on one another in a Boolean network is essential to predict and control the system's dynamical behaviour, for example, detecting key therapeutic targets to control pathways in models of biological signalling and regulation. Exact estimation is generally not possible due to the fact that the number of configurations that must be considered grows exponentially with the system size. However, approximate, scalable methods exist in the literature. These methods can be divided into two main classes: (i) graph-theoretic methods that rely on representations of Boolean dynamics into static graphs and (ii) mean-field approaches that describe average trajectories of the system but neglect dynamical correlations. Here, we compare systematically the performance of these state-of-the-art methods on a large collection of real-world gene regulatory networks. We find comparable performance across methods. All methods underestimate the ground truth, with mean-field approaches having a better recall but a worse precision than graph-theoretic methods. Computationally speaking, graph-theoretic methods are faster than mean-field ones in sparse networks, but are slower in dense networks. The preference of which method to use, therefore, depends on a network's connectivity and the relative importance of recall versus precision for the specific application at hand.

Developing the genotype-to-phenotype relationship in evolutionary theory: A primer of developmental features

For decades, there have been repeated calls for more integration across evolutionary and developmental biology. However, critiques in the literature and recent funding initiatives suggest this integration remains incomplete. We suggest one way forward is to consider how we elaborate the most basic concept of development, the relationship between genotype and phenotype, in traditional models of evolutionary processes. For some questions, when more complex features of development are accounted for, predictions of evolutionary processes shift. We present a primer on concepts of development to clarify confusion in the literature and fuel new questions and approaches. The basic features of development involve expanding a base model of genotype-to-phenotype to include the genome, space, and time. A layer of complexity is added by incorporating developmental systems, including signal-response systems and networks of interactions. The developmental emergence of function, which captures developmental feedbacks and phenotypic performance, offers further model elaborations that explicitly link fitness with developmental systems. Finally, developmental features such as plasticity and developmental niche construction conceptualize the link between a developing phenotype and the external environment, allowing for a fuller inclusion of ecology in evolutionary models. Incorporating aspects of developmental complexity into evolutionary models also accommodates a more pluralistic focus on the causal importance of developmental systems, individual organisms, or agents in generating evolutionary patterns. Thus, by laying out existing concepts of development, and considering how they are used across different fields, we can gain clarity in existing debates around the extended evolutionary synthesis and pursue new directions in evolutionary developmental biology. Finally, we consider how nesting developmental features in traditional models of evolution can highlight areas of evolutionary biology that need more theoretical attention.

Identifying cancer driver genes in individual tumours

Cancer is a heterogeneous disease with a strong genetic component making it suitable for precision medicine approaches aimed at identifying the underlying molecular drivers within a tumour. Large scale population-level cancer sequencing consortia have identified many actionable mutations common across both cancer types and sub-types, resulting in an increasing number of successful precision medicine programs. Nonetheless, such approaches fail to consider the effects of mutations unique to an individual patient and may miss rare driver mutations, necessitating personalised approaches to driver-gene prioritisation. One approach is to quantify the functional importance of individual mutations in a single tumour based on how they affect the expression of genes in a gene interaction network (GIN). These GIN-based approaches can be broadly divided into those that utilise an existing reference GIN and those that construct de novo patient-specific GINs. These single-tumour approaches have several limitations that likely influence their results, such as use of reference cohort data, network choice, and approaches to mathematical approximation, and more research is required to evaluate the in vitro and in vivo applicability of their predictions. This review examines the current state of the art methods that identify driver genes in single tumours with a focus on GIN-based driver prioritisation.

Modularity of biological systems: a link between structure and function

This paper addresses two topics in systems biology, the hypothesis that biological systems are modular and the problem of relating structure and function of biological systems. The focus here is on gene regulatory networks, represented by Boolean network models, a commonly used tool. Most of the research on gene regulatory network modularity has focused on network structure, typically represented through either directed or undirected graphs. But since gene regulation is a highly dynamic process as it determines the function of cells over time, it is natural to consider functional modularity as well. One of the main results is that the structural decomposition of a network into modules induces an analogous decomposition of the dynamic structure, exhibiting a strong relationship between network structure and function. An extensive simulation study provides evidence for the hypothesis that modularity might have evolved to increase phenotypic complexity while maintaining maximal dynamic robustness to external perturbations.

Model-free tracking control of complex dynamical trajectories with machine learning

Nonlinear tracking control enabling a dynamical system to track a desired trajectory is fundamental to robotics, serving a wide range of civil and defense applications. In control engineering, designing tracking control requires complete knowledge of the system model and equations. We develop a model-free, machine-learning framework to control a two-arm robotic manipulator using only partially observed states, where the controller is realized by reservoir computing. Stochastic input is exploited for training, which consists of the observed partial state vector as the first and its immediate future as the second component so that the neural machine regards the latter as the future state of the former. In the testing (deployment) phase, the immediate-future component is replaced by the desired observational vector from the reference trajectory. We demonstrate the effectiveness of the control framework using a variety of periodic and chaotic signals, and establish its robustness against measurement noise, disturbances, and uncertainties.

Modularity of biological systems: a link between structure and function

This paper addresses two topics in systems biology, the hypothesis that biological systems are modular and the problem of relating structure and function of biological systems. The focus here is on gene regulatory networks, represented by Boolean network models, a commonly used tool. Most of the research on gene regulatory network modularity has focused on network structure, typically represented through either directed or undirected graphs. But since gene regulation is a highly dynamic process as it determines the function of cells over time, it is natural to consider functional modularity as well. One of the main results is that the structural decomposition of a network into modules induces an analogous decomposition of the dynamic structure, exhibiting a strong relationship between network structure and function. An extensive simulation study provides evidence for the hypothesis that modularity might have evolved to increase phenotypic complexity while maintaining maximal dynamic robustness to external perturbations.

Phenotype Control techniques for Boolean gene regulatory networks

Modeling cell signal transduction pathways via Boolean networks (BNs) has become an established method for analyzing intracellular communications over the last few decades. What's more, BNs provide a course-grained approach, not only to understanding molecular communications, but also for targeting pathway components that alter the long-term outcomes of the system. This has come to be known as phenotype control theory. In this review we study the interplay of various approaches for controlling gene regulatory networks such as: algebraic methods, control kernel, feedback vertex set, and stable motifs. The study will also include comparative discussion between the methods, using an established cancer model of T-Cell Large Granular Lymphocyte Leukemia. Further, we explore possible options for making the control search more efficient using reduction and modularity. Finally, we will include challenges presented such as the complexity and the availability of software for implementing each of these control techniques.

Using network control theory to study the dynamics of the structural connectome

Network control theory (NCT) is a simple and powerful tool for studying how network topology informs and constrains dynamics. Compared to other structure-function coupling approaches, the strength of NCT lies in its capacity to predict the patterns of external control signals that may alter dynamics in a desired way. We have extensively developed and validated the application of NCT to the human structural connectome. Through these efforts, we have studied (i) how different aspects of connectome topology affect neural dynamics, (ii) whether NCT outputs cohere with empirical data on brain function and stimulation, and (iii) how NCT outputs vary across development and correlate with behavior and mental health symptoms. In this protocol, we introduce a framework for applying NCT to structural connectomes following two main pathways. Our primary pathway focuses on computing the control energy associated with transitioning between specific neural activity states. Our second pathway focuses on computing average controllability, which indexes nodes' general capacity to control dynamics. We also provide recommendations for comparing NCT outputs against null network models. Finally, we support this protocol with a Python-based software package called network control theory for python (nctpy).

Inferring gene regulatory networks using transcriptional profiles as dynamical attractors

Genetic regulatory networks (GRNs) regulate the flow of genetic information from the genome to expressed messenger RNAs (mRNAs) and thus are critical to controlling the phenotypic characteristics of cells. Numerous methods exist for profiling mRNA transcript levels and identifying protein-DNA binding interactions at the genome-wide scale. These enable researchers to determine the structure and output of transcriptional regulatory networks, but uncovering the complete structure and regulatory logic of GRNs remains a challenge. The field of GRN inference aims to meet this challenge using computational modeling to derive the structure and logic of GRNs from experimental data and to encode this knowledge in Boolean networks, Bayesian networks, ordinary differential equation (ODE) models, or other modeling frameworks. However, most existing models do not incorporate dynamic transcriptional data since it has historically been less widely available in comparison to "static" transcriptional data. We report the development of an evolutionary algorithm-based ODE modeling approach (named EA) that integrates kinetic transcription data and the theory of attractor matching to infer GRN architecture and regulatory logic. Our method outperformed six leading GRN inference methods, none of which incorporate kinetic transcriptional data, in predicting regulatory connections among TFs when applied to a small-scale engineered synthetic GRN in Saccharomyces cerevisiae. Moreover, we demonstrate the potential of our method to predict unknown transcriptional profiles that would be produced upon genetic perturbation of the GRN governing a two-state cellular phenotypic switch in Candida albicans. We established an iterative refinement strategy to facilitate candidate selection for experimentation; the experimental results in turn provide validation or improvement for the model. In this way, our GRN inference approach can expedite the development of a sophisticated mathematical model that can accurately describe the structure and dynamics of the in vivo GRN.

Reproducibility and FAIR principles: the case of a segment polarity network model

The issue of reproducibility of computational models and the related FAIR principles (findable, accessible, interoperable, and reusable) are examined in a specific test case. I analyze a computational model of the segment polarity network in Drosophila embryos published in 2000. Despite the high number of citations to this publication, 23 years later the model is barely accessible, and consequently not interoperable. Following the text of the original publication allowed successfully encoding the model for the open source software COPASI. Subsequently saving the model in the SBML format allowed it to be reused in other open source software packages. Submission of this SBML encoding of the model to the BioModels database enables its findability and accessibility. This demonstrates how the FAIR principles can be successfully enabled by using open source software, widely adopted standards, and public repositories, facilitating reproducibility and reuse of computational cell biology models that will outlive the specific software used.

Evolutionary Method of Heterogeneous Combat Network Based on Link Prediction

Currently, research on the evolution of heterogeneous combat networks (HCNs) mainly focuses on the modeling process, with little attention paid to the impact of changes in network topology on operational capabilities. Link prediction can provide a fair and unified comparison standard for network evolution mechanisms. This paper uses link prediction methods to study the evolution of HCNs. Firstly, according to the characteristics of HCNs, a link prediction index based on frequent subgraphs (LPFS) is proposed. LPFS have been demonstrated on a real combat network to be superior to 26 baseline methods. The main driving force of research on evolution is to improve the operational capabilities of combat networks. Adding the same number of nodes and edges, 100 iterative experiments demonstrate that the evolutionary method (HCNE) proposed in this paper outperforms random evolution and preferential evolution in improving the operational capabilities of combat networks. Furthermore, the new network generated after evolution is more consistent with the characteristics of a real network.

Controlling complex dynamical systems based on the structure of the networks

Progress of molecular biology resulted in the accumulation of information on biomolecular interactions, which are complex enough to be termed as networks. Dynamical behavior generated by complex network systems is considered to be the origin of the biological functions. One of the largest missions in modern life science is to obtain logical understanding for the dynamics of complex systems based on experimentally identified networks. However, a network does not provide sufficient information to specify dynamics explicitly, i.e. it lacks information of mathematical formulae of functions or parameter values. One has to develop mathematical models under assumptions of functions and parameter values to know the detail of dynamics of network systems. In this review, on the other hand, we introduce our own mathematical theory to understand the behavior of biological systems from the information of regulatory networks alone. Using the theory, important aspects of dynamical properties can be extracted from networks. Namely, key factors for observing/controlling the whole dynamical system are determined from network structure alone. We also show an application of the theory to a real biological system, a gene regulatory network for cell-fate specification in ascidian. We demonstrate that the system was completely controllable by experimental manipulations of the key factors identified by the theory from the information of network alone. This review article is an extended version of the Japanese article, Controlling Cell-Fate Specification System Based on a Mathematical Theory of Network Dynamics, published in SEIBUTSU BUTSURI Vol. 60, p. 349-351 (2020).

Phenotype control techniques for Boolean gene regulatory networks

Modeling cell signal transduction pathways via Boolean networks (BNs) has become an established method for analyzing intracellular communications over the last few decades. What’s more, BNs provide a course-grained approach, not only to understanding molecular communications, but also for targeting pathway components that alter the long-term outcomes of the system. This has come to be known as phenotype control theory . In this review we study the interplay of various approaches for controlling gene regulatory networks such as: algebraic methods, control kernel, feedback vertex set, and stable motifs. The study will also include comparative discussion between the methods, using an established cancer model of T-Cell Large Granular Lymphocyte (T-LGL) Leukemia. Further, we explore possible options for making the control search more efficient using reduction and modularity. Finally, we will include challenges presented such as the complexity and the availability of software for implementing each of these control techniques.

SWEET: a single-sample network inference method for deciphering individual features in disease

Recently, extracting inherent biological system information (e.g. cellular networks) from genome-wide expression profiles for developing personalized diagnostic and therapeutic strategies has become increasingly important. However, accurately constructing single-sample networks (SINs) to capture individual characteristics and heterogeneity in disease remains challenging. Here, we propose a sample-specific-weighted correlation network (SWEET) method to model SINs by integrating the genome-wide sample-to-sample correlation (i.e. sample weights) with the differential network between perturbed and aggregate networks. For a group of samples, the genome-wide sample weights can be assessed without prior knowledge of intrinsic subpopulations to address the network edge number bias caused by sample size differences. Compared with the state-of-the-art SIN inference methods, the SWEET SINs in 16 cancers more likely fit the scale-free property, display higher overlap with the human interactomes and perform better in identifying three types of cancer-related genes. Moreover, integrating SWEET SINs with a network proximity measure facilitates characterizing individual features and therapy in diseases, such as somatic mutation, mut-driver and essential genes. Biological experiments further validated two candidate repurposable drugs, albendazole for head and neck squamous cell carcinoma (HNSCC) and lung adenocarcinoma (LUAD) and encorafenib for HNSCC. By applying SWEET, we also identified two possible LUAD subtypes that exhibit distinct clinical features and molecular mechanisms. Overall, the SWEET method complements current SIN inference and analysis methods and presents a view of biological systems at the network level to offer numerous clues for further investigation and clinical translation in network medicine and precision medicine.

Dynamic thresholding search for the feedback vertex set problem

Given a directed graph G = (V, E), a feedback vertex set is a vertex subset C whose removal makes the graph G acyclic. The feedback vertex set problem is to find the subset C* whose cardinality is the minimum. As a general model, this problem has a variety of applications. However, the problem is known to be NP-hard, and thus computationally challenging. To solve this difficult problem, this article develops an iterated dynamic thresholding search algorithm, which features a combination of local optimization, dynamic thresholding search, and perturbation. Computational experiments on 101 benchmark graphs from various sources demonstrate the advantage of the algorithm compared with the state-of-the-art algorithms, by reporting record-breaking best solutions for 24 graphs, equally best results for 75 graphs, and worse best results for only two graphs. We also study how the key components of the algorithm affect its performance of the algorithm.

Dauer fate in a Caenorhabditis elegans Boolean network model

Cellular fates are determined by genes interacting across large, complex biological networks. A critical question is how to identify causal relationships spanning distinct signaling pathways and underlying organismal phenotypes. Here, we address this question by constructing a Boolean model of a well-studied developmental network and analyzing information flows through the system. Depending on environmental signals Caenorhabditis elegans develop normally to sexual maturity or enter a reproductively delayed, developmentally quiescent 'dauer' state, progressing to maturity when the environment changes. The developmental network that starts with environmental signal and ends in the dauer/no dauer fate involves genes across 4 signaling pathways including cyclic GMP, Insulin/IGF-1, TGF-β and steroid hormone synthesis. We identified three stable motifs leading to normal development, each composed of genes interacting across the Insulin/IGF-1, TGF-β and steroid hormone synthesis pathways. Three genes known to influence dauer fate, daf-2, daf-7 and hsf-1, acted as driver nodes in the system. Using causal logic analysis, we identified a five gene cyclic subgraph integrating the information flow from environmental signal to dauer fate. Perturbation analysis showed that a multifactorial insulin profile determined the stable motifs the system entered and interacted with daf-12 as the switchpoint driving the dauer/no dauer fate. Our results show that complex organismal systems can be distilled into abstract representations that permit full characterization of the causal relationships driving developmental fates. Analyzing organismal systems from this perspective of logic and function has important implications for studies examining the evolution and conservation of signaling pathways.

Global stabilizing control of large-scale biomolecular regulatory networks

MOTIVATION

Cellular behavior is determined by complex non-linear interactions between numerous intracellular molecules that are often represented by Boolean network models. To achieve a desired cellular behavior with minimal intervention, we need to identify optimal control targets that can drive heterogeneous cellular states to the desired phenotypic cellular state with minimal node intervention. Previous attempts to realize such global stabilization were based solely on either network structure information or simple linear dynamics. Other attempts based on non-linear dynamics are not scalable.

RESULTS

Here, we investigate the underlying relationship between structurally identified control targets and optimal global stabilizing control targets based on non-linear dynamics. We discovered that optimal global stabilizing control targets can be identified by analyzing the dynamics between structurally identified control targets. Utilizing these findings, we developed a scalable global stabilizing control framework using both structural and dynamic information. Our framework narrows down the search space based on strongly connected components and feedback vertex sets then identifies global stabilizing control targets based on the canalization of Boolean network dynamics. We find that the proposed global stabilizing control is superior with respect to the number of control target nodes, scalability, and computational complexity.

AVAILABILITY AND IMPLEMENTATION

We provide a GitHub repository that contains the DCGS framework written in Python as well as biological random Boolean network datasets (https://github.com/sugyun/DCGS).

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Mindful attention promotes control of brain network dynamics for self-regulation and discontinues the past from the present

Mindful attention is characterized by acknowledging the present experience as a transient mental event. Early stages of mindfulness practice may require greater neural effort for later efficiency. Early effort may self-regulate behavior and focalize the present, but this understanding lacks a computational explanation. Here we used network control theory as a model of how external control inputs-operationalizing effort-distribute changes in neural activity evoked during mindful attention across the white matter network. We hypothesized that individuals with greater network controllability, thereby efficiently distributing control inputs, effectively self-regulate behavior. We further hypothesized that brain regions that utilize greater control input exhibit shorter intrinsic timescales of neural activity. Shorter timescales characterize quickly discontinuing past processing to focalize the present. We tested these hypotheses in a randomized controlled study that primed participants to either mindfully respond or naturally react to alcohol cues during fMRI and administered text reminders and measurements of alcohol consumption during 4 wk postscan. We found that participants with greater network controllability moderated alcohol consumption. Mindful regulation of alcohol cues, compared to one's own natural reactions, reduced craving, but craving did not differ from the baseline group. Mindful regulation of alcohol cues, compared to the natural reactions of the baseline group, involved more-effortful control of neural dynamics across cognitive control and attention subnetworks. This effort persisted in the natural reactions of the mindful group compared to the baseline group. More-effortful neural states had shorter timescales than less effortful states, offering an explanation for how mindful attention promotes being present.

Target Control of Asynchronous Boolean Networks

We study the target control of asynchronous Boolean networks, to identify interventions that can drive the dynamics of a given Boolean network from any initial state to the desired target attractor. Based on the application time, the control can be realised with three types of perturbations, including instantaneous, temporary and permanent perturbations. We develop efficient methods to compute the target control for a given target attractor with these three types of perturbations. We compare our methods with the stable motif-based control method on a variety of real-life biological networks to evaluate their performance. We show that our methods scale well for large Boolean networks and they are able to identify a rich set of solutions with a small number of perturbations.

Learning in Transcriptional Network Models: Computational Discovery of Pathway-Level Memory and Effective Interventions

Trainability, in any substrate, refers to the ability to change future behavior based on past experiences. An understanding of such capacity within biological cells and tissues would enable a particularly powerful set of methods for prediction and control of their behavior through specific patterns of stimuli. This top-down mode of control (as an alternative to bottom-up modification of hardware) has been extensively exploited by computer science and the behavioral sciences; in biology however, it is usually reserved for organism-level behavior in animals with brains, such as training animals towards a desired response. Exciting work in the field of basal cognition has begun to reveal degrees and forms of unconventional memory in non-neural tissues and even in subcellular biochemical dynamics. Here, we characterize biological gene regulatory circuit models and protein pathways and find them capable of several different kinds of memory. We extend prior results on learning in binary transcriptional networks to continuous models and identify specific interventions (regimes of stimulation, as opposed to network rewiring) that abolish undesirable network behavior such as drug pharmacoresistance and drug sensitization. We also explore the stability of created memories by assessing their long-term behavior and find that most memories do not decay over long time periods. Additionally, we find that the memory properties are quite robust to noise; surprisingly, in many cases noise actually increases memory potential. We examine various network properties associated with these behaviors and find that no one network property is indicative of memory. Random networks do not show similar memory behavior as models of biological processes, indicating that generic network dynamics are not solely responsible for trainability. Rational control of dynamic pathway function using stimuli derived from computational models opens the door to empirical studies of proto-cognitive capacities in unconventional embodiments and suggests numerous possible applications in biomedicine, where behavior shaping of pathway responses stand as a potential alternative to gene therapy.

What Makes a Functional Gene Regulatory Network? A Circuit Motif Analysis

One of the key questions in systems biology is to understand the roles of gene regulatory circuits in determining cellular states and their functions. In previous studies, some researchers have inferred large gene networks from genome wide genomics/transcriptomics data using the top-down approach, while others have modeled core gene circuits of small sizes using the bottom-up approach. Despite many existing systems biology studies, there is still no general rule on what sizes of gene networks and what types of circuit motifs a system would need to achieve robust biological functions. Here, we adopt a gene circuit motif analysis to discover four-node circuits responsible for multiplicity (rich in dynamical behavior), flexibility (versatile to alter gene expression), or both. We identify the most reoccurring two-node circuit motifs and the co-occurring motif pairs. Furthermore, we investigate the contributing factors of multiplicity and flexibility for large gene networks of different types and sizes. We find that gene networks of intermediate sizes tend to have combined high levels of multiplicity and flexibility. Our study will contribute to a better understanding of the dynamical mechanisms of gene regulatory circuits and provide insights into rational designs of robust gene circuits in synthetic and systems biology.

Uncovering the biological basis of control energy: Structural and metabolic correlates of energy inefficiency in temporal lobe epilepsy

Network control theory is increasingly used to profile the brain's energy landscape via simulations of neural dynamics. This approach estimates the control energy required to simulate the activation of brain circuits based on structural connectome measured using diffusion magnetic resonance imaging, thereby quantifying those circuits' energetic efficiency. The biological basis of control energy, however, remains unknown, hampering its further application. To fill this gap, investigating temporal lobe epilepsy as a lesion model, we show that patients require higher control energy to activate the limbic network than healthy volunteers, especially ipsilateral to the seizure focus. The energetic imbalance between ipsilateral and contralateral temporolimbic regions is tracked by asymmetric patterns of glucose metabolism measured using positron emission tomography, which, in turn, may be selectively explained by asymmetric gray matter loss as evidenced in the hippocampus. Our investigation provides the first theoretical framework unifying gray matter integrity, metabolism, and energetic generation of neural dynamics.

Probabilistic edge weights fine-tune Boolean network dynamics

Biological systems are noisy by nature. This aspect is reflected in our experimental measurements and should be reflected in the models we build to better understand these systems. Noise can be especially consequential when trying to interpret specific regulatory interactions, i.e. regulatory network edges. In this paper, we propose a method to explicitly encode edge-noise in Boolean dynamical systems by probabilistic edge-weight (PEW) operators. PEW operators have two important features: first, they introduce a form of edge-weight into Boolean models through the noise, second, the noise is dependent on the dynamical state of the system, which enables more biologically meaningful modeling choices. Moreover, we offer a simple-to-use implementation in the already well-established BooleanNet framework. In two application cases, we show how the introduction of just a few PEW operators in Boolean models can fine-tune the emergent dynamics and increase the accuracy of qualitative predictions. This includes fine-tuning interactions which cause non-biological behaviors when switching between asynchronous and synchronous update schemes in dynamical simulations. Moreover, PEW operators also open the way to encode more exotic cellular dynamics, such as cellular learning, and to implementing edge-weights for regulatory networks inferred from omics data.

Leveraging network structure in nonlinear control

Over the last twenty years, dynamic modeling of biomolecular networks has exploded in popularity. Many of the classical tools for understanding dynamical systems are unwieldy in the highly nonlinear, poorly constrained, high-dimensional systems that often arise from these modeling efforts. Understanding complex biological systems is greatly facilitated by purpose-built methods that leverage common features of such models, such as local monotonicity, interaction graph sparsity, and sigmoidal kinetics. Here, we review methods for controlling the systems of ordinary differential equations used to model biomolecular networks. We focus on methods that make use of the structure of the network of interactions to help inform, which variables to target for control, and highlight the computational and experimental advantages of such approaches. We also discuss the importance of nonperturbative methods in biomedical and experimental molecular biology applications, where finely tuned interventions can be difficult to implement. It is well known that feedback loops, and positive feedback loops in particular, play a major determining role in the dynamics of biomolecular networks. In many of the methods we cover here, control over system trajectories is realized by overriding the behavior of key feedback loops.

Uncovering potential interventions for pancreatic cancer patients via mathematical modeling

Pancreatic Ductal Adenocarcinoma (PDAC) is widely known for its poor prognosis because it is often diagnosed when the cancer is in a later stage. We built a Boolean model to analyze the microenvironment of pancreatic cancer in order to better understand the interplay between pancreatic cancer, stellate cells, and their signaling cytokines. Specifically, we have used our model to study the impact of inducing four common mutations: KRAS, TP53, SMAD4, and CDKN2A. After implementing the various mutation combinations, we used our stochastic simulator to derive aggressiveness scores based on simulated attractor probabilities and long-term trajectory approximations. These aggression scores were then corroborated with clinical data. Moreover, we found sets of control targets that are effective among common mutations. These control sets contain nodes within both the pancreatic cancer cell and the pancreatic stellate cell, including PIP3, RAF, PIK3 and BAX in pancreatic cancer cell as well as ERK and PIK3 in the pancreatic stellate cell. Many of these nodes were found to be differentially expressed among pancreatic cancer patients in the TCGA database. Furthermore, literature suggests that many of these nodes can be targeted by drugs currently in circulation. The results herein help provide a proof of concept in the path towards personalized medicine through a means of mathematical systems biology. All data and code used for running simulations, statistical analysis, and plotting is available on a GitHub repository athttps://github.com/drplaugher/PCC_Mutations.

The expanding horizons of network neuroscience: From description to prediction and control

The field of network neuroscience has emerged as a natural framework for the study of the brain and has been increasingly applied across divergent problems in neuroscience. From a disciplinary perspective, network neuroscience originally emerged as a formal integration of graph theory (from mathematics) and neuroscience (from biology). This early integration afforded marked utility in describing the interconnected nature of neural units, both structurally and functionally, and underscored the relevance of that interconnection for cognition and behavior. But since its inception, the field has not remained static in its methodological composition. Instead, it has grown to use increasingly advanced graph-theoretic tools and to bring in several other disciplinary perspectives-including machine learning and systems engineering-that have proven complementary. In doing so, the problem space amenable to the discipline has expanded markedly. In this review, we discuss three distinct flavors of investigation in state-of-the-art network neuroscience: (i) descriptive network neuroscience, (ii) predictive network neuroscience, and (iii) a perturbative network neuroscience that draws on recent advances in network control theory. In considering each area, we provide a brief summary of the approaches, discuss the nature of the insights obtained, and highlight future directions.

Describing and Controlling Multivariate Nonlinear Dynamics: A Boolean Network Approach

We introduce a discrete-time dynamical system method, the Boolean network method, that may be useful for modeling, studying, and controlling nonlinear dynamics in multivariate systems, particularly when binary time-series are available. We introduce the method in three steps: inference of the temporal relations as Boolean functions, extraction of attractors and assignment of desirability based on domain knowledge, and design of network control to direct a psychological system toward a desired attractor. To demonstrate how the Boolean network can describe and prescribe control for emotion regulation dynamics, we applied this method to data from a study of how children use bidding to an adult and/or distraction to regulate their anger during a frustrating task (N = 120, T = 480 seconds). Network control strategies were designed to move the child into attractors where anger is OFF. The sample shows heterogeneous emotion regulation dynamics across children in 22 distinct Boolean networks, and heterogeneous control strategies regarding which behavior to perturb and how to perturb it. The Boolean network method provides a novel method to describe nonlinear dynamics in multivariate psychological systems and is a method with potential to eventually inform the design of interventions that can guide those systems toward desired goals.

Prevalence and scalable control of localized networks

The ability to control network dynamics is essential for ensuring desirable functionality of many technological, biological, and social systems. Such systems often consist of a large number of network elements, and controlling large-scale networks remains challenging because the computation and communication requirements increase prohibitively fast with network size. Here, we introduce a notion of network locality that can be exploited to make the control of networks scalable, even when the dynamics are nonlinear. We show that network locality is captured by an information metric and is almost universally observed across real and model networks. In localized networks, the optimal control actions and system responses are both shown to be necessarily concentrated in small neighborhoods induced by the information metric. This allows us to develop localized algorithms for determining network controllability and optimizing the placement of driver nodes. This also allows us to develop a localized algorithm for designing local feedback controllers that approach the performance of the corresponding best global controllers, while incurring a computational cost orders-of-magnitude lower. We validate the locality, performance, and efficiency of the algorithms in Kuramoto oscillator networks, as well as three large empirical networks: synchronization dynamics in the Eastern US power grid, epidemic spreading mediated by the global air-transportation network, and Alzheimer's disease dynamics in a human brain network. Taken together, our results establish that large networks can be controlled with computation and communication costs comparable to those for small networks.

Multi-modal optimization to identify personalized biomarkers for disease prediction of individual patients with cancer

Finding personalized biomarkers for disease prediction of patients with cancer remains a massive challenge in precision medicine. Most methods focus on one subnetwork or module as a network biomarker; however, this ignores the early warning capabilities of other modules with different configurations of biomarkers (i.e. multi-modal personalized biomarkers). Identifying such modules would not only predict disease but also provide effective therapeutic drug target information for individual patients. To solve this problem, we developed a novel model (denoted multi-modal personalized dynamic network biomarkers (MMPDNB)) based on a multi-modal optimization mechanism and personalized dynamic network biomarker (PDNB) theory, which can provide multiple modules of personalized biomarkers and unveil their multi-modal properties. Using the genomics data of patients with breast or lung cancer from The Cancer Genome Atlas database, we validated the effectiveness of the MMPDNB model. The experimental results showed that compared with other advanced methods, MMPDNB can more effectively predict the critical state with the highest early warning signal score during cancer development. Furthermore, MMPDNB more significantly identified PDNBs containing driver and biomarker genes specific to cancer tissues. More importantly, we validated the biological significance of multi-modal PDNBs, which could provide effective drug targets of individual patients as well as markers for predicting early warning signals of the critical disease state. In conclusion, multi-modal optimization is an effective method to identify PDNBs and offers a new perspective for understanding tumor heterogeneity in cancer precision medicine.

Conserved Control Path in Multilayer Networks

The determination of directed control paths in complex networks is important because control paths indicate the structure of the propagation of control signals through edges. A challenging problem is to identify them in complex networked systems characterized by different types of interactions that form multilayer networks. In this study, we describe a graph pattern called the conserved control path, which allows us to model a common control structure among different types of relations. We present a practical conserved control path detection method (CoPath), which is based on a maximum-weighted matching, to determine the paths that play the most consistent roles in controlling signal transmission in multilayer networks. As a pragmatic application, we demonstrate that the control paths detected in a multilayered pan-cancer network are statistically more consistent. Additionally, they lead to the effective identification of drug targets, thereby demonstrating their power in predicting key pathways that influence multiple cancers.

NETISCE: a network-based tool for cell fate reprogramming

The search for effective therapeutic targets in fields like regenerative medicine and cancer research has generated interest in cell fate reprogramming. This cellular reprogramming paradigm can drive cells to a desired target state from any initial state. However, methods for identifying reprogramming targets remain limited for biological systems that lack large sets of experimental data or a dynamical characterization. We present NETISCE, a novel computational tool for identifying cell fate reprogramming targets in static networks. In combination with machine learning algorithms, NETISCE estimates the attractor landscape and predicts reprogramming targets using signal flow analysis and feedback vertex set control, respectively. Through validations in studies of cell fate reprogramming from developmental, stem cell, and cancer biology, we show that NETISCE can predict previously identified cell fate reprogramming targets and identify potentially novel combinations of targets. NETISCE extends cell fate reprogramming studies to larger-scale biological networks without the need for full model parameterization and can be implemented by experimental and computational biologists to identify parts of a biological system relevant to the desired reprogramming task.

Influence maximization in Boolean networks

The optimization problem aiming at the identification of minimal sets of nodes able to drive the dynamics of Boolean networks toward desired long-term behaviors is central for some applications, as for example the detection of key therapeutic targets to control pathways in models of biological signaling and regulatory networks. Here, we develop a method to solve such an optimization problem taking inspiration from the well-studied problem of influence maximization for spreading processes in social networks. We validate the method on small gene regulatory networks whose dynamical landscapes are known by means of brute-force analysis. We then systematically study a large collection of gene regulatory networks. We find that for about 65% of the analyzed networks, the minimal driver sets contain less than 20% of their nodes.

Structure-based approach to identifying small sets of driver nodes in biological networks

In network control theory, driving all the nodes in the Feedback Vertex Set (FVS) by node-state override forces the network into one of its attractors (long-term dynamic behaviors). The FVS is often composed of more nodes than can be realistically manipulated in a system; for example, only up to three nodes can be controlled in intracellular networks, while their FVS may contain more than 10 nodes. Thus, we developed an approach to rank subsets of the FVS on Boolean models of intracellular networks using topological, dynamics-independent measures. We investigated the use of seven topological prediction measures sorted into three categories-centrality measures, propagation measures, and cycle-based measures. Using each measure, every subset was ranked and then evaluated against two dynamics-based metrics that measure the ability of interventions to drive the system toward or away from its attractors: To Control and Away Control. After examining an array of biological networks, we found that the FVS subsets that ranked in the top according to the propagation metrics can most effectively control the network. This result was independently corroborated on a second array of different Boolean models of biological networks. Consequently, overriding the entire FVS is not required to drive a biological network to one of its attractors, and this method provides a way to reliably identify effective FVS subsets without the knowledge of the network dynamics.

Network Control Models With Personalized Genomics Data for Understanding Tumor Heterogeneity in Cancer

Due to rapid development of high-throughput sequencing and biotechnology, it has brought new opportunities and challenges in developing efficient computational methods for exploring personalized genomics data of cancer patients. Because of the high-dimension and small sample size characteristics of these personalized genomics data, it is difficult for excavating effective information by using traditional statistical methods. In the past few years, network control methods have been proposed to solve networked system with high-dimension and small sample size. Researchers have made progress in the design and optimization of network control principles. However, there are few studies comprehensively surveying network control methods to analyze the biomolecular network data of individual patients. To address this problem, here we comprehensively surveyed complex network control methods on personalized omics data for understanding tumor heterogeneity in precision medicine of individual patients with cancer.

Discovering sparse control strategies in neural activity

Biological circuits such as neural or gene regulation networks use internal states to map sensory input to an adaptive repertoire of behavior. Characterizing this mapping is a major challenge for systems biology. Though experiments that probe internal states are developing rapidly, organismal complexity presents a fundamental obstacle given the many possible ways internal states could map to behavior. Using C. elegans as an example, we propose a protocol for systematic perturbation of neural states that limits experimental complexity and could eventually help characterize collective aspects of the neural-behavioral map. We consider experimentally motivated small perturbations—ones that are most likely to preserve natural dynamics and are closer to internal control mechanisms—to neural states and their impact on collective neural activity. Then, we connect such perturbations to the local information geometry of collective statistics, which can be fully characterized using pairwise perturbations. Applying the protocol to a minimal model of C. elegans neural activity, we find that collective neural statistics are most sensitive to a few principal perturbative modes. Dominant eigenvalues decay initially as a power law, unveiling a hierarchy that arises from variation in individual neural activity and pairwise interactions. Highest-ranking modes tend to be dominated by a few, “pivotal” neurons that account for most of the system’s sensitivity, suggesting a sparse mechanism of collective control.

Topology Identification of Time-Scales Complex Networks

This paper studies a topology identification problem of complex networks with dynamics on different time scales. Using the adaptive synchronization method, some criteria for a successful estimation are obtained. In particular, by regulating the original network to synchronize with an auxiliary chaotic network, this work further explores a way to avoid the precondition of linear independence. When the adaptive controller fails to achieve the outer synchronization, an impulsive control method is used. In the end, we conclude with three numerical simulations. The results obtained in this paper generalize continuous, discrete with arbitrary time step size and mixed cases.

Identification of dynamic driver sets controlling phenotypical landscapes

Controlling phenotypical landscapes is of vital interest to modern biology. This task becomes highly demanding because cellular decisions involve complex networks engaging in crosstalk interactions. Previous work on control theory indicates that small sets of compounds can control single phenotypes. However, a dynamic approach is missing to determine the drivers of the whole network dynamics. By analyzing 35 biologically motivated Boolean networks, we developed a method to identify small sets of compounds sufficient to decide on the entire phenotypical landscape. These compounds do not strictly prefer highly related compounds and show a smaller impact on the stability of the attractor landscape. The dynamic driver sets include many intervention targets and cellular reprogramming drivers in human networks. Finally, by using a new comprehensive model of colorectal cancer, we provide a complete workflow on how to implement our approach to shift from in silico to in vitro guided experiments.