Critical dynamics in genetic regulatory networks: examples from four kingdoms

Reviewing mathematical models of sperm signaling networks

Dave Garbers' work significantly contributed to our understanding of sperm's regulated motility, capacitation, and the acrosome reaction. These key sperm functions involve complex multistep signaling pathways engaging numerous finely orchestrated elements. Despite significant progress, many parameters and interactions among these elements remain elusive. Mathematical modeling emerges as a potent tool to study sperm physiology, providing a framework to integrate experimental results and capture functional dynamics considering biochemical, biophysical, and cellular elements. Depending on research objectives, different modeling strategies, broadly categorized into continuous and discrete approaches, reveal valuable insights into cell function. These models allow the exploration of hypotheses regarding molecules, conditions, and pathways, whenever they become challenging to evaluate experimentally. This review presents an overview of current theoretical and experimental efforts to understand sperm motility regulation, capacitation, and the acrosome reaction. We discuss the strengths and weaknesses of different modeling strategies and highlight key findings and unresolved questions. Notable discoveries include the importance of specific ion channels, the role of intracellular molecular heterogeneity in capacitation and the acrosome reaction, and the impact of pH changes on acrosomal exocytosis. Ultimately, this review underscores the crucial importance of mathematical frameworks in advancing our understanding of sperm physiology and guiding future experimental investigations.

Canalization reduces the nonlinearity of regulation in biological networks

Biological networks, such as gene regulatory networks, possess desirable properties. They are more robust and controllable than random networks. This motivates the search for structural and dynamical features that evolution has incorporated into biological networks. A recent meta-analysis of published, expert-curated Boolean biological network models has revealed several such features, often referred to as design principles. Among others, the biological networks are enriched for certain recurring network motifs, the dynamic update rules are more redundant, more biased, and more canalizing than expected, and the dynamics of biological networks are better approximable by linear and lower-order approximations than those of comparable random networks. Since most of these features are interrelated, it is paramount to disentangle cause and effect, that is, to understand which features evolution actively selects for, and thus truly constitute evolutionary design principles. Here, we compare published Boolean biological network models with different ensembles of null models and show that the abundance of canalization in biological networks can almost completely explain their recently postulated high approximability. Moreover, an analysis of random N-K Kauffman models reveals a strong dependence of approximability on the dynamical robustness of a network.

Sensory-Motor Loop Adaptation in Boolean Network Robots

Recent technological advances have made it possible to produce tiny robots equipped with simple sensors and effectors. Micro-robots are particularly suitable for scenarios such as exploration of hostile environments, and emergency intervention, e.g., in areas subject to earthquakes or fires. A crucial desirable feature of such a robot is the capability of adapting to the specific environment in which it has to operate. Given the limited computational capabilities of a micro-robot, this property cannot be achieved by complicated software but it rather should come from the flexibility of simple control mechanisms, such as the sensory-motor loop. In this work, we explore the possibility of equipping simple robots controlled by Boolean networks with the capability of modulating their sensory-motor loop such that their behavior adapts to the incumbent environmental conditions. This study builds upon the cybernetic concept of homeostasis, which is the property of maintaining essential parameters inside vital ranges, and analyzes the performance of adaptive mechanisms intervening in the sensory-motor loop. In particular, we focus on the possibility of maneuvering the robot's effectors such that both their connections to network nodes and environmental features can be adapted. As the actions the robot takes have a feedback effect to its sensors mediated by the environment, this mechanism makes it possible to tune the sensory-motor loop, which, in turn, determines the robot's behavior. We study this general setting in simulation and assess to what extent this mechanism can sustain the homeostasis of the robot. Our results show that controllers made of random Boolean networks in critical and chaotic regimes can be tuned such that their homeostasis in different environments is kept. This outcome is a step towards the design and deployment of controllers for micro-robots able to adapt to different environments.

A meta-analysis of Boolean network models reveals design principles of gene regulatory networks

Gene regulatory networks (GRNs) play a central role in cellular decision-making. Understanding their structure and how it impacts their dynamics constitutes thus a fundamental biological question. GRNs are frequently modeled as Boolean networks, which are intuitive, simple to describe, and can yield qualitative results even when data are sparse. We assembled the largest repository of expert-curated Boolean GRN models. A meta-analysis of this diverse set of models reveals several design principles. GRNs exhibit more canalization, redundancy, and stable dynamics than expected. Moreover, they are enriched for certain recurring network motifs. This raises the important question why evolution favors these design mechanisms.

Models of Cell Processes are Far from the Edge of Chaos

Complex living systems are thought to exist at the "edge of chaos" separating the ordered dynamics of robust function from the disordered dynamics of rapid environmental adaptation. Here, a deeper inspection of 72 experimentally supported discrete dynamical models of cell processes reveals previously unobserved order on long time scales, suggesting greater rigidity in these systems than was previously conjectured. We find that propagation of internal perturbations is transient in most cases, and that even when large perturbation cascades persist, their phenotypic effects are often minimal. Moreover, we find evidence that stochasticity and desynchronization can lead to increased recovery from regulatory perturbation cascades. Our analysis relies on new measures that quantify the tendency of perturbations to spread through a discrete dynamical system. Computing these measures was not feasible using current methodology; thus, we developed a multipurpose CUDA-based simulation tool, which we have made available as the open-source Python library cubewalkers. Based on novel measures and simulations, our results suggest that-contrary to current theory-cell processes are ordered and far from the edge of chaos.

Potential long consequences from internal and external ecology: loss of gut microbiota antifragility in children from an industrialized population compared with an indigenous rural lifestyle

Human health is strongly mediated by the gut microbiota ecosystem, which, in turn, depends not only on its state but also on its dynamics and how it responds to perturbations. Healthy microbiota ecosystems tend to be in criticality and antifragile dynamics corresponding to a maximum complexity configuration, which may be assessed with information and network theory analysis. Under this complex system perspective, we used a new analysis of published data to show that a children's population with an industrialized urban lifestyle from Mexico City exhibits informational and network characteristics similar to parasitized children from a rural indigenous population in the remote mountainous region of Guerrero, México. We propose then, that in this critical age for gut microbiota maturation, the industrialized urban lifestyle could be thought of as an external perturbation to the gut microbiota ecosystem, and we show that it produces a similar loss in criticality/antifragility as the one observed by internal perturbation due to parasitosis by the helminth A. lumbricoides. Finally, several general complexity-based guidelines to prevent or restore gut ecosystem antifragility are discussed.

Effective Connectivity and Bias Entropy Improve Prediction of Dynamical Regime in Automata Networks

Biomolecular network dynamics are thought to operate near the critical boundary between ordered and disordered regimes, where large perturbations to a small set of elements neither die out nor spread on average. A biomolecular automaton (e.g., gene, protein) typically has high regulatory redundancy, where small subsets of regulators determine activation via collective canalization. Previous work has shown that effective connectivity, a measure of collective canalization, leads to improved dynamical regime prediction for homogeneous automata networks. We expand this by (i) studying random Boolean networks (RBNs) with heterogeneous in-degree distributions, (ii) considering additional experimentally validated automata network models of biomolecular processes, and (iii) considering new measures of heterogeneity in automata network logic. We found that effective connectivity improves dynamical regime prediction in the models considered; in RBNs, combining effective connectivity with bias entropy further improves the prediction. Our work yields a new understanding of criticality in biomolecular networks that accounts for collective canalization, redundancy, and heterogeneity in the connectivity and logic of their automata models. The strong link we demonstrate between criticality and regulatory redundancy provides a means to modulate the dynamical regime of biochemical networks.

Characterizing the Impact of Communication on Cellular and Collective Behavior Using a Three-Dimensional Multiscale Cellular Model

Communication between cells enables the coordination that drives structural and functional complexity in biological systems. Both single and multicellular organisms have evolved diverse communication systems for a range of purposes, including synchronization of behavior, division of labor, and spatial organization. Synthetic systems are also increasingly being engineered to utilize cell-cell communication. While research has elucidated the form and function of cell-cell communication in many biological systems, our knowledge is still limited by the confounding effects of other biological phenomena at play and the bias of the evolutionary background. In this work, our goal is to push forward the context-free understanding of what impact cell-cell communication can have on cellular and population behavior to more fully understand the extent to which cell-cell communication systems can be utilized, modified, and engineered. We use an in silico model of 3D multiscale cellular populations, with dynamic intracellular networks interacting via diffusible signals. We focus on two key communication parameters: the effective interaction distance at which cells are able to interact and the receptor activation threshold. We found that cell-cell communication can be divided into six different forms along the parameter axes, three asocial and three social. We also show that cellular behavior, tissue composition, and tissue diversity are all highly sensitive to both the general form and specific parameters of communication even when the cellular network has not been biased towards that behavior.

Unsupervised learning of aging principles from longitudinal data

Age is the leading risk factor for prevalent diseases and death. However, the relation between age-related physiological changes and lifespan is poorly understood. We combined analytical and machine learning tools to describe the aging process in large sets of longitudinal measurements. Assuming that aging results from a dynamic instability of the organism state, we designed a deep artificial neural network, including auto-encoder and auto-regression (AR) components. The AR model tied the dynamics of physiological state with the stochastic evolution of a single variable, the "dynamic frailty indicator" (dFI). In a subset of blood tests from the Mouse Phenome Database, dFI increased exponentially and predicted the remaining lifespan. The observation of the limiting dFI was consistent with the late-life mortality deceleration. dFI changed along with hallmarks of aging, including frailty index, molecular markers of inflammation, senescent cell accumulation, and responded to life-shortening (high-fat diet) and life-extending (rapamycin) treatments.

Relative importance of composition structures and biologically meaningful logics in bipartite Boolean models of gene regulation

Boolean networks have been widely used to model gene networks. However, such models are coarse-grained to an extent that they abstract away molecular specificities of gene regulation. Alternatively, bipartite Boolean network models of gene regulation explicitly distinguish genes from transcription factors (TFs). In such bipartite models, multiple TFs may simultaneously contribute to gene regulation by forming heteromeric complexes, thus giving rise to composition structures. Since bipartite Boolean models are relatively recent, an empirical investigation of their biological plausibility is lacking. Here, we estimate the prevalence of composition structures arising through heteromeric complexes. Moreover, we present an additional mechanism where composition structures may arise as a result of multiple TFs binding to cis-regulatory regions and provide empirical support for this mechanism. Next, we compare the restriction in BFs imposed by composition structures and by biologically meaningful properties. We find that though composition structures can severely restrict the number of Boolean functions (BFs) driving a gene, the two types of minimally complex BFs, namely nested canalyzing functions (NCFs) and read-once functions (RoFs), are comparatively more restrictive. Finally, we find that composition structures are highly enriched in real networks, but this enrichment most likely comes from NCFs and RoFs.

On the Criticality of Adaptive Boolean Network Robots

Systems poised at a dynamical critical regime, between order and disorder, have been shown capable of exhibiting complex dynamics that balance robustness to external perturbations and rich repertoires of responses to inputs. This property has been exploited in artificial network classifiers, and preliminary results have also been attained in the context of robots controlled by Boolean networks. In this work, we investigate the role of dynamical criticality in robots undergoing online adaptation, i.e., robots that adapt some of their internal parameters to improve a performance metric over time during their activity. We study the behavior of robots controlled by random Boolean networks, which are either adapted in their coupling with robot sensors and actuators or in their structure or both. We observe that robots controlled by critical random Boolean networks have higher average and maximum performance than that of robots controlled by ordered and disordered nets. Notably, in general, adaptation by change of couplings produces robots with slightly higher performance than those adapted by changing their structure. Moreover, we observe that when adapted in their structure, ordered networks tend to move to the critical dynamical regime. These results provide further support to the conjecture that critical regimes favor adaptation and indicate the advantage of calibrating robot control systems at dynamical critical states.

Network Physiology of Exercise: Beyond Molecular and Omics Perspectives

Molecular Exercise Physiology and Omics approaches represent an important step toward synthesis and integration, the original essence of Physiology. Despite the significant progress they have introduced in Exercise Physiology (EP), some of their theoretical and methodological assumptions are still limiting the understanding of the complexity of sport-related phenomena. Based on general principles of biological evolution and supported by complex network science, this paper aims to contrast theoretical and methodological aspects of molecular and network-based approaches to EP. After explaining the main EP challenges and why sport-related phenomena cannot be understood if reduced to the molecular level, the paper proposes some methodological research advances related to the type of studied variables and measures, the data acquisition techniques, the type of data analysis and the assumed relations among physiological levels. Inspired by Network Physiology, Network Physiology of Exercise provides a new paradigm and formalism to quantify cross-communication among diverse systems across levels and time scales to improve our understanding of exercise-related phenomena and opens new horizons for exercise testing in health and disease.

Attractor-Specific and Common Expression Values in Random Boolean Network Models (with a Preliminary Look at Single-Cell Data)

Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), which have also been widely studied as abstract models of complex systems and have been used to simulate different phenomena. We define the "common sea" (CS) as the set of nodes that take the same value in all the attractors of a given network realization, and the "specific part" (SP) as the set of all the other nodes, and we study their properties in different ensembles, generated with different parameter values. Both the CS and of the SP can be composed of one or more weakly connected components, which are emergent intermediate-level structures. We show that the study of these sets provides very important information about the behavior of the model. The distribution of distances between attractors is also examined. Moreover, we show how the notion of a "common sea" of genes can be used to analyze data from single-cell experiments.

When to Be Critical? Performance and Evolvability in Different Regimes of Neural Ising Agents

It has long been hypothesized that operating close to the critical state is beneficial for natural and artificial evolutionary systems. We put this hypothesis to test in a system of evolving foraging agents controlled by neural networks that can adapt the agents' dynamical regime throughout evolution. Surprisingly, we find that all populations that discover solutions evolve to be subcritical. By a resilience analysis, we find that there are still benefits of starting the evolution in the critical regime. Namely, initially critical agents maintain their fitness level under environmental changes (for example, in the lifespan) and degrade gracefully when their genome is perturbed. At the same time, initially subcritical agents, even when evolved to the same fitness, are often inadequate to withstand the changes in the lifespan and degrade catastrophically with genetic perturbations. Furthermore, we find the optimal distance to criticality depends on the task complexity. To test it we introduce a hard task and a simple task: For the hard task, agents evolve closer to criticality, whereas more subcritical solutions are found for the simple task. We verify that our results are independent of the selected evolutionary mechanisms by testing them on two principally different approaches: a genetic algorithm and an evolutionary strategy. In summary, our study suggests that although optimal behaviour in the simple task is obtained in a subcritical regime, initializing near criticality is important to be efficient at finding optimal solutions for new tasks of unknown complexity.

Effective connectivity determines the critical dynamics of biochemical networks

Living systems comprise interacting biochemical components in very large networks. Given their high connectivity, biochemical dynamics are surprisingly not chaotic but quite robust to perturbations-a feature C.H. Waddington named canalization. Because organisms are also flexible enough to evolve, they arguably operate in a critical dynamical regime between order and chaos. The established theory of criticality is based on networks of interacting automata where Boolean truth values model presence/absence of biochemical molecules. The dynamical regime is predicted using network connectivity and node bias (to be on/off) as tuning parameters. Revising this to account for canalization leads to a significant improvement in dynamical regime prediction. The revision is based on effective connectivity, a measure of dynamical redundancy that buffers automata response to some inputs. In both random and experimentally validated systems biology networks, reducing effective connectivity makes living systems operate in stable or critical regimes even though the structure of their biochemical interaction networks predicts them to be chaotic. This suggests that dynamical redundancy may be naturally selected to maintain living systems near critical dynamics, providing both robustness and evolvability. By identifying how dynamics propagates preferably via effective pathways, our approach helps to identify precise ways to design and control network models of biochemical regulation and signalling.

Mathematical model reveals that heterogeneity in the number of ion transporters regulates the fraction of mouse sperm capacitation

Capacitation is a complex maturation process mammalian sperm must undergo in the female genital tract to be able to fertilize an egg. This process involves, amongst others, physiological changes in flagellar beating pattern, membrane potential, intracellular ion concentrations and protein phosphorylation. Typically, in a capacitation medium, only a fraction of sperm achieve this state. The cause for this heterogeneous response is still not well understood and remains an open question. Here, one of our principal results is to develop a discrete regulatory network, with mostly deterministic dynamics in conjunction with some stochastic elements, for the main biochemical and biophysical processes involved in the early events of capacitation. The model criterion for capacitation requires the convergence of specific levels of a select set of nodes. Besides reproducing several experimental results and providing some insight on the network interrelations, the main contribution of the model is the suggestion that the degree of variability in the total amount and individual number of ion transporters among spermatozoa regulates the fraction of capacitated spermatozoa. This conclusion is consistent with recently reported experimental results. Based on this mathematical analysis, experimental clues are proposed for the control of capacitation levels. Furthermore, cooperative and interference traits that become apparent in the modelling among some components also call for future theoretical and experimental studies.

The Impact of Self-Loops on Boolean Networks Attractor Landscape and Implications for Cell Differentiation Modelling

Boolean networks are a notable model of gene regulatory networks and, particularly, prominent theories discuss how they can capture cellular differentiation processes. One frequent motif in gene regulatory networks, especially in those circuits involved in cell differentiation, is autoregulation. In spite of this, the impact of autoregulation on Boolean network attractor landscape has not yet been extensively discussed in literature. In this paper we propose to model autoregulation as self-loops, and analyse how the number of attractors and their robustness may change once they are introduced in a well-known and widely used Boolean networks model, namely random Boolean networks. Results show that self-loops provide an evolutionary advantage in dynamic mechanisms of cells, by increasing both number and maximal robustness of attractors. These results provide evidence to the hypothesis that autoregulation is a straightforward functional component to consolidate cell dynamics, mainly in differentiation processes.

Engineering self-organized criticality in living cells

Complex dynamical fluctuations, from intracellular noise, brain dynamics or computer traffic display bursting dynamics consistent with a critical state between order and disorder. Living close to the critical point has adaptive advantages and it has been conjectured that evolution could select these critical states. Is this the case of living cells? A system can poise itself close to the critical point by means of the so-called self-organized criticality (SOC). In this paper we present an engineered gene network displaying SOC behaviour. This is achieved by exploiting the saturation of the proteolytic degradation machinery in E. coli cells by means of a negative feedback loop that reduces congestion. Our critical motif is built from a two-gene circuit, where SOC can be successfully implemented. The potential implications for both cellular dynamics and behaviour are discussed.

Parity and time reversal elucidate both decision-making in empirical models and attractor scaling in critical Boolean networks

We present new applications of parity inversion and time reversal to the emergence of complex behavior from simple dynamical rules in stochastic discrete models. Our parity-based encoding of causal relationships and time-reversal construction efficiently reveal discrete analogs of stable and unstable manifolds. We demonstrate their predictive power by studying decision-making in systems biology and statistical physics models. These applications underpin a novel attractor identification algorithm implemented for Boolean networks under stochastic dynamics. Its speed enables resolving a long-standing open question of how attractor count in critical random Boolean networks scales with network size and whether the scaling matches biological observations. Via 80-fold improvement in probed network size (N = 16,384), we find the unexpectedly low scaling exponent of 0.12 ± 0.05, approximately one-tenth the analytical upper bound. We demonstrate a general principle: A system's relationship to its time reversal and state-space inversion constrains its repertoire of emergent behaviors.

Transcriptional and Microenvironmental Landscape of Macrophage Transition in Cancer: A Boolean Analysis

The balance between pro- and anti-inflammatory immune system responses is crucial to face and counteract complex diseases such as cancer. Macrophages are an essential population that contributes to this balance in collusion with the local tumor microenvironment. Cancer cells evade the attack of macrophages by liberating cytokines and enhancing the transition to the M2 phenotype with pro-tumoral functions. Despite this pernicious effect on immune systems, the M1 phenotype still exists in the environment and can eliminate tumor cells by liberating cytokines that recruit and activate the cytotoxic actions of TH1 effector cells. Here, we used a Boolean modeling approach to understand how the tumor microenvironment shapes macrophage behavior to enhance pro-tumoral functions. Our network reconstruction integrates experimental data and public information that let us study the polarization from monocytes to M1, M2a, M2b, M2c, and M2d subphenotypes. To analyze the dynamics of our model, we modeled macrophage polarization in different conditions and perturbations. Notably, our study identified new hybrid cell populations, undescribed before. Based on the in vivo macrophage behavior, we explained the hybrid macrophages’ role in the tumor microenvironment. The in silico model allowed us to postulate transcriptional factors that maintain the balance between macrophages with anti- and pro-tumoral functions. In our pursuit to maintain the balance of macrophage phenotypes to eliminate malignant tumor cells, we emulated a theoretical genetically modified macrophage by modifying the activation of NFκB and a loss of function in HIF1-α and discussed their phenotype implications. Overall, our theoretical approach is as a guide to design new experiments for unraveling the principles of the dual host-protective or -harmful antagonistic roles of transitional macrophages in tumor immunoediting and cancer cell fate decisions.

Reconciling Non-Genetic Plasticity with Somatic Evolution in Cancer

Post-treatment progression of tumors is commonly explained by somatic Darwinian evolution (i.e., selection of cells carrying genetic mutations that create more aggressive cell traits). But cancer genome and transcriptome analyses now paint a picture far more complex, prompting us to see beyond the Darwinian scheme: non-genetic cell phenotype plasticity explained by alternative stable gene expression states ('attractors'), may also produce aggressive phenotypes that can be selected for, without mutations. Worse, treatment may even induce cell state transitions into more malignant attractors. We review recent evidence for non-genetic mechanisms of progression, explain the theoretical foundation of attractor transitions behind treatment-induced increase of aggressiveness, and provide a framework for unifying genetic and non-genetic dynamics in tumor progression.

Guiding the Self-Organization of Cyber-Physical Systems

Self-organization offers a promising approach for designing adaptive systems. Given the inherent complexity of most cyber-physical systems, adaptivity is desired, as predictability is limited. Here I summarize different concepts and approaches that can facilitate self-organization in cyber-physical systems, and thus be exploited for design. Then I mention real-world examples of systems where self-organization has managed to provide solutions that outperform classical approaches, in particular related to urban mobility. Finally, I identify when a centralized, distributed, or self-organizing control is more appropriate.

Random Parametric Perturbations of Gene Regulatory Circuit Uncover State Transitions in Cell Cycle

Many biological processes involve precise cellular state transitions controlled by complex gene regulation. Here, we use budding yeast cell cycle as a model system and explore how a gene regulatory circuit encodes essential information of state transitions. We present a generalized random circuit perturbation method for circuits containing heterogeneous regulation types and its usage to analyze both steady and oscillatory states from an ensemble of circuit models with random kinetic parameters. The stable steady states form robust clusters with a circular structure that are associated with cell cycle phases. This circular structure in the clusters is consistent with single-cell RNA sequencing data. The oscillatory states specify the irreversible state transitions along cell cycle progression. Furthermore, we identify possible mechanisms to understand the irreversible state transitions from the steady states. We expect this approach to be robust and generally applicable to unbiasedly predict dynamical transitions of a gene regulatory circuit.

Statistical physics of liquid brains

Liquid neural networks (or 'liquid brains') are a widespread class of cognitive living networks characterized by a common feature: the agents (ants or immune cells, for example) move in space. Thus, no fixed, long-term agent-agent connections are maintained, in contrast with standard neural systems. How is this class of systems capable of displaying cognitive abilities, from learning to decision-making? In this paper, the collective dynamics, memory and learning properties of liquid brains is explored under the perspective of statistical physics. Using a comparative approach, we review the generic properties of three large classes of systems, namely: standard neural networks (solid brains), ant colonies and the immune system. It is shown that, despite their intrinsic physical differences, these systems share key properties with standard neural systems in terms of formal descriptions, but strongly depart in other ways. On one hand, the attractors found in liquid brains are not always based on connection weights but instead on population abundances. However, some liquid systems use fluctuations in ways similar to those found in cortical networks, suggesting a relevant role for criticality as a way of rapidly reacting to external signals. This article is part of the theme issue 'Liquid brains, solid brains: How distributed cognitive architectures process information'.

The Emergence of Integrated Information, Complexity, and 'Consciousness' at Criticality

Integrated Information Theory (IIT) posits that integrated information ( Φ ) represents the quantity of a conscious experience. Here, the generalized Ising model was used to calculate Φ as a function of temperature in toy models of fully connected neural networks. A Monte-Carlo simulation was run on 159 normalized, random, positively weighted networks analogous to small five-node excitatory neural network motifs. Integrated information generated by this sample of small Ising models was measured across model parameter spaces. It was observed that integrated information, as an order parameter, underwent a phase transition at the critical point in the model. This critical point was demarcated by the peak of the generalized susceptibility (or variance in configuration due to temperature) of integrated information. At this critical point, integrated information was maximally receptive and responsive to perturbations of its own states. The results of this study provide evidence that Φ can capture integrated information in an empirical dataset, and display critical behavior acting as an order parameter from the generalized Ising model.

Disturbance in human gut microbiota networks by parasites and its implications in the incidence of depression

If you think you are in control of your behavior, think again. Evidence suggests that behavioral modifications, as development and persistence of depression, maybe the consequence of a complex network of communication between macro and micro-organisms capable of modifying the physiological axis of the host. Some parasites cause significant nutritional deficiencies for the host and impair the effectiveness of cognitive processes such as memory, teaching or non-verbal intelligence. Bacterial communities mediate the establishment of parasites and vice versa but this complexity approach remains little explored. We study the gut microbiota-parasite interactions using novel techniques of network analysis using data of individuals from two indigenous communities in Guerrero, Mexico. Our results suggest that Ascaris lumbricoides induce a gut microbiota perturbation affecting its network properties and also subnetworks of key species related to depression, translating in a loss of emergence. Studying these network properties changes is particularly important because recent research has shown that human health is characterized by a dynamic trade-off between emergence and self-organization, called criticality. Emergence allows the systems to generate novel information meanwhile self-organization is related to the system's order and structure. In this way, the loss of emergence means a depart from criticality and ultimately loss of health.

Evolution Towards Criticality in Ising Neural Agents

Criticality is thought to be crucial for complex systems to adapt at the boundary between regimes with different dynamics, where the system may transition from one phase to another. Numerous systems, from sandpiles to gene regulatory networks to swarms to human brains, seem to work towards preserving a precarious balance right at their critical point. Understanding criticality therefore seems strongly related to a broad, fundamental theory for the physics of life as it could be, which still lacks a clear description of how life can arise and maintain itself in complex systems. In order to investigate this crucial question, we model populations of Ising agents competing for resources in a simple 2D environment subject to an evolutionary algorithm. We then compare its evolutionary dynamics under different experimental conditions. We demonstrate the utility that arises at a critical state and contrast it with the behaviors and dynamics that arise far from criticality. The results show compelling evidence that not only is a critical state remarkable in its ability to adapt and find solutions to the environment, but the evolving parameters in the agents tend to flow towards criticality if starting from a supercritical regime. We present simulations showing that a system in a supercritical state will tend to self-organize towards criticality, in contrast to a subcritical state, which remains subcritical though it is still capable of adapting and increasing its fitness.

A universal transcriptomic signature of age reveals the temporal scaling of Caenorhabditis elegans aging trajectories

We collected 60 age-dependent transcriptomes for C. elegans strains including four exceptionally long-lived mutants (mean adult lifespan extended 2.2- to 9.4-fold) and three examples of lifespan-increasing RNAi treatments. Principal Component Analysis (PCA) reveals aging as a transcriptomic drift along a single direction, consistent across the vastly diverse biological conditions and coinciding with the first principal component, a hallmark of the criticality of the underlying gene regulatory network. We therefore expected that the organism's aging state could be characterized by a single number closely related to vitality deficit or biological age. The "aging trajectory", i.e. the dependence of the biological age on chronological age, is then a universal stochastic function modulated by the network stiffness; a macroscopic parameter reflecting the network topology and associated with the rate of aging. To corroborate this view, we used publicly available datasets to define a transcriptomic biomarker of age and observed that the rescaling of age by lifespan simultaneously brings together aging trajectories of transcription and survival curves. In accordance with the theoretical prediction, the limiting mortality value at the plateau agrees closely with the mortality rate doubling exponent estimated at the cross-over age near the average lifespan. Finally, we used the transcriptomic signature of age to identify possible life-extending drug compounds and successfully tested a handful of the top-ranking molecules in C. elegans survival assays and achieved up to a +30% extension of mean lifespan.

Functionality of Disorder in Muscle Mechanics

A salient feature of skeletal muscles is their ability to take up an applied slack in a microsecond timescale. Behind this fast adaptation is a collective folding in a bundle of elastically interacting bistable elements. Since this interaction has a long-range character, the behavior of the system in force and length controlled ensembles is different; in particular, it can have two distinct order-disorder-type critical points. We show that the account of the disregistry between myosin and actin filaments places the elementary force-producing units of skeletal muscles close to both such critical points. The ensuing "double criticality" contributes to the system's ability to perform robustly and suggests that the disregistry is functional.

Ensembles, dynamics, and cell types: Revisiting the statistical mechanics perspective on cellular regulation

Genetic regulatory networks control ontogeny. For fifty years Boolean networks have served as models of such systems, ranging from ensembles of random Boolean networks as models for generic properties of gene regulation to working dynamical models of a growing number of sub-networks of real cells. At the same time, their statistical mechanics has been thoroughly studied. Here we recapitulate their original motivation in the context of current theoretical and empirical research. We discuss ensembles of random Boolean networks whose dynamical attractors model cell types. A sub-ensemble is the critical ensemble. There is now strong evidence that genetic regulatory networks are dynamically critical, and that evolution is exploring the critical sub-ensemble. The generic properties of this sub-ensemble predict essential features of cell differentiation. In particular, the number of attractors in such networks scales as the DNA content raised to the 0.63 power. Data on the number of cell types as a function of the DNA content per cell shows a scaling relationship of 0.88. Thus, the theory correctly predicts a power law relationship between the number of cell types and the DNA contents per cell, and a comparable slope. We discuss these new scaling values and show prospects for new research lines for Boolean networks as a base model for systems biology.

Modeling the Role of the Microbiome in Evolution

There is undeniable evidence showing that bacteria have strongly influenced the evolution and biological functions of multicellular organisms. It has been hypothesized that many host-microbial interactions have emerged so as to increase the adaptive fitness of the holobiont (the host plus its microbiota). Although this association has been corroborated for many specific cases, general mechanisms explaining the role of the microbiota in the evolution of the host are yet to be understood. Here we present an evolutionary model in which a network representing the host adapts in order to perform a predefined function. During its adaptation, the host network (HN) can interact with other networks representing its microbiota. We show that this interaction greatly accelerates and improves the adaptability of the HN without decreasing the adaptation of the microbial networks. Furthermore, the adaptation of the HN to perform several functions is possible only when it interacts with many different bacterial networks in a specialized way (each bacterial network participating in the adaptation of one function). Disrupting these interactions often leads to non-adaptive states, reminiscent of dysbiosis, where none of the networks the holobiont consists of can perform their respective functions. By considering the holobiont as a unit of selection and focusing on the adaptation of the host to predefined but arbitrary functions, our model predicts the need for specialized diversity in the microbiota. This structural and dynamical complexity in the holobiont facilitates its adaptation, whereas a homogeneous (non-specialized) microbiota is inconsequential or even detrimental to the holobiont's evolution. To our knowledge, this is the first model in which symbiotic interactions, diversity, specialization and dysbiosis in an ecosystem emerge as a result of coevolution. It also helps us understand the emergence of complex organisms, as they adapt more easily to perform multiple tasks than non-complex ones.

Flexibility of Boolean Network Reservoir Computers in Approximating Arbitrary Recursive and Non-Recursive Binary Filters

Reservoir computers (RCs) are biology-inspired computational frameworks for signal processing that are typically implemented using recurrent neural networks. Recent work has shown that Boolean networks (BN) can also be used as reservoirs. We analyze the performance of BN RCs, measuring their flexibility and identifying the factors that determine the effective approximation of Boolean functions applied in a sliding-window fashion over a binary signal, both non-recursively and recursively. We train and test BN RCs of different sizes, signal connectivity, and in-degree to approximate three-bit, five-bit, and three-bit recursive binary functions, respectively. We analyze how BN RC parameters and function average sensitivity, which is a measure of function smoothness, affect approximation accuracy as well as the spread of accuracies for a single reservoir. We found that approximation accuracy and reservoir flexibility are highly dependent on RC parameters. Overall, our results indicate that not all reservoirs are equally flexible, and RC instantiation and training can be more efficient if this is taken into account. The optimum range of RC parameters opens up an angle of exploration for understanding how biological systems might be tuned to balance system restraints with processing capacity.

Criticality Distinguishes the Ensemble of Biological Regulatory Networks

The hypothesis that many living systems should exhibit near-critical behavior is well motivated theoretically, and an increasing number of cases have been demonstrated empirically. However, a systematic analysis across biological networks, which would enable identification of the network properties that drive criticality, has not yet been realized. Here, we provide a first comprehensive survey of criticality across a diverse sample of biological networks, leveraging a publicly available database of 67 Boolean models of regulatory circuits. We find all 67 networks to be near critical. By comparing to ensembles of random networks with similar topological and logical properties, we show that criticality in biological networks is not predictable solely from macroscale properties such as mean degree ⟨K⟩ and mean bias in the logic functions ⟨p⟩, as previously emphasized in theories of random Boolean networks. Instead, the ensemble of real biological circuits is jointly constrained by the local causal structure and logic of each node. In this way, biological regulatory networks are more distinguished from random networks by their criticality than by other macroscale network properties such as degree distribution, edge density, or fraction of activating conditions.

Dynamics of the Gene Regulatory Network of HIV-1 and the Role of Viral Non-coding RNAs on Latency Reversion

The use of latency reversing agents (LRAs) is currently a promising approach to eliminate latent reservoirs of HIV-1. However, this strategy has not been successful in vivo. It has been proposed that cellular post-transcriptional mechanisms are implicated in the underperformance of LRAs, but it is not clear whether proviral regulatory elements like viral non-coding RNAs (vncRNAs) are also implicated. In order to visualize the complexity of the HIV-1 gene expression, we used experimental data to construct a gene regulatory network (GRN) of latent proviruses in resting CD4+ T cells. We then analyzed the dynamics of this GRN using Boolean and continuous mathematical models. Our simulations predict that vncRNAs are able to counteract the activity of LRAs, which may explain the failure of these compounds to reactivate latent reservoirs of HIV-1. Moreover, our results also predict that using inhibitors of histone methyltransferases, such as chaetocin, together with releasers of the positive transcription elongation factor (P-TEFb), like JQ1, may increase proviral reactivation despite self-repressive effects of vncRNAs.

Emergence of diversity in homogeneous coupled Boolean networks

The origin of multicellularity in metazoa is one of the fundamental questions of evolutionary biology. We have modeled the generic behaviors of gene regulatory networks in isogenic cells as stochastic nonlinear dynamical systems-coupled Boolean networks with perturbation. Model simulations under a variety of dynamical regimes suggest that the central characteristic of multicellularity, permanent spatial differentiation (diversification), indeed can arise. Additionally, we observe that diversification is more likely to occur near the critical regime of Lyapunov stability.

How Criticality of Gene Regulatory Networks Affects the Resulting Morphogenesis under Genetic Perturbations

Whereas the relationship between criticality of gene regulatory networks (GRNs) and dynamics of GRNs at a single-cell level has been vigorously studied, the relationship between the criticality of GRNs and system properties at a higher level has not been fully explored. Here we aim at revealing a potential role of criticality of GRNs in morphogenesis, which is hard to uncover through the single-cell-level studies, especially from an evolutionary viewpoint. Our model simulated the growth of a cell population from a single seed cell. All the cells were assumed to have identical intracellular GRNs. We induced genetic perturbations to the GRN of the seed cell by adding, deleting, or switching a regulatory link between a pair of genes. From numerical simulations, we found that the criticality of GRNs facilitated the formation of nontrivial morphologies when the GRNs were critical in the presence of the evolutionary perturbations. Moreover, the criticality of GRNs produced topologically homogeneous cell clusters by adjusting the spatial arrangements of cells, which led to the formation of nontrivial morphogenetic patterns. Our findings correspond to an epigenetic viewpoint that heterogeneous and complex features emerge from homogeneous and less complex components through the interactions among them. Thus, our results imply that highly structured tissues or organs in morphogenesis of multicellular organisms might stem from the criticality of GRNs.

Perspectives on theory at the interface of physics and biology

Theoretical physics is the search for simple and universal mathematical descriptions of the natural world. In contrast, much of modern biology is an exploration of the complexity and diversity of life. For many, this contrast is prima facie evidence that theory, in the sense that physicists use the word, is impossible in a biological context. For others, this contrast serves to highlight a grand challenge. I am an optimist, and believe (along with many colleagues) that the time is ripe for the emergence of a more unified theoretical physics of biological systems, building on successes in thinking about particular phenomena. In this essay I try to explain the reasons for my optimism, through a combination of historical and modern examples.

The CD4+ T cell regulatory network mediates inflammatory responses during acute hyperinsulinemia: a simulation study

Background

Obesity is frequently linked to insulin resistance, high insulin levels, chronic inflammation, and alterations in the behaviour of CD4+ T cells. Despite the biomedical importance of this condition, the system-level mechanisms that alter CD4+ T cell differentiation and plasticity are not well understood.

Results

We model how hyperinsulinemia alters the dynamics of the CD4+ T regulatory network, and this, in turn, modulates cell differentiation and plasticity. Different polarizing microenvironments are simulated under basal and high levels of insulin to assess impacts on cell-fate attainment and robustness in response to transient perturbations. In the presence of high levels of insulin Th1 and Th17 become more stable to transient perturbations, and their basin sizes are augmented, Tr1 cells become less stable or disappear, while TGFβ producing cells remain unaltered. Hence, the model provides a dynamic system-level framework and explanation to further understand the documented and apparently paradoxical role of TGFβ in both inflammation and regulation of immune responses, as well as the emergence of the adipose Treg phenotype. Furthermore, our simulations provide new predictions on the impact of the microenvironment in the coexistence of the different cell types, suggesting that in pro-Th1, pro-Th2 and pro-Th17 environments effector and regulatory cells can coexist, but that high levels of insulin severely diminish regulatory cells, especially in a pro-Th17 environment.

Conclusions

This work provides a first step towards a system-level formal and dynamic framework to integrate further experimental data in the study of complex inflammatory diseases.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-017-0436-y) contains supplementary material, which is available to authorized users.

A dynamic genetic-hormonal regulatory network model explains multiple cellular behaviors of the root apical meristem of Arabidopsis thaliana

The study of the concerted action of hormones and transcription factors is fundamental to understand cell differentiation and pattern formation during organ development. The root apical meristem of Arabidopsis thaliana is a useful model to address this. It has a stem cell niche near its tip conformed of a quiescent organizer and stem or initial cells around it, then a proliferation domain followed by a transition domain, where cells diminish division rate before transiting to the elongation zone; here, cells grow anisotropically prior to their final differentiation towards the plant base. A minimal model of the gene regulatory network that underlies cell-fate specification and patterning at the root stem cell niche was proposed before. In this study, we update and couple such network with both the auxin and cytokinin hormone signaling pathways to address how they collectively give rise to attractors that correspond to the genetic and hormonal activity profiles that are characteristic of different cell types along A. thaliana root apical meristem. We used a Boolean model of the genetic-hormonal regulatory network to integrate known and predicted regulatory interactions into alternative models. Our analyses show that, after adding some putative missing interactions, the model includes the necessary and sufficient components and regulatory interactions to recover attractors characteristic of the root cell types, including the auxin and cytokinin activity profiles that correlate with different cellular behaviors along the root apical meristem. Furthermore, the model predicts the existence of activity configurations that could correspond to the transition domain. The model also provides a possible explanation for apparently paradoxical cellular behaviors in the root meristem. For example, how auxin may induce and at the same time inhibit WOX5 expression. According to the model proposed here the hormonal regulation of WOX5 might depend on the cell type. Our results illustrate how non-linear multi-stable qualitative network models can aid at understanding how transcriptional regulators and hormonal signaling pathways are dynamically coupled and may underlie both the acquisition of cell fate and the emergence of hormonal activity profiles that arise during complex organ development.

Adaptive Local Information Transfer in Random Boolean Networks

Living systems such as gene regulatory networks and neuronal networks have been supposed to work close to dynamical criticality, where their information-processing ability is optimal at the whole-system level. We investigate how this global information-processing optimality is related to the local information transfer at each individual-unit level. In particular, we introduce an internal adjustment process of the local information transfer and examine whether the former can emerge from the latter. We propose an adaptive random Boolean network model in which each unit rewires its incoming arcs from other units to balance stability of its information processing based on the measurement of the local information transfer pattern. First, we show numerically that random Boolean networks can self-organize toward near dynamical criticality in our model. Second, the proposed model is analyzed by a mean-field theory. We recognize that the rewiring rule has a bootstrapping feature. The stationary indegree distribution is calculated semi-analytically and is shown to be close to dynamical criticality in a broad range of model parameter values.

Integer programming-based method for observability of singleton attractors in Boolean networks

Boolean network (BN) is a popular mathematical model for revealing the behaviour of a genetic regulatory network. Furthermore, observability, an important network feature, plays a significant role in understanding the underlying network. Several studies have been done on analysis of observability of BNs and complex networks. However, the observability of attractor cycles, which can serve as biomarker detection, has not yet been addressed in the literature. This is an important, interesting and challenging problem that deserves a detailed study. In this study, a novel problem was first proposed on attractor observability in BNs. Identification of the minimum set of consecutive nodes can be used to discriminate different attractors. Furthermore, it can serve as a biomarker for different disease types (represented as different attractor cycles). Then a novel integer programming method was developed to identify the desired set of nodes. The proposed approach is demonstrated and verified by numerical examples. The computational results further illustrates that the proposed model is effective and efficient.

Estimating Propensity Parameters Using Google PageRank and Genetic Algorithms

Stochastic Boolean networks, or more generally, stochastic discrete networks, are an important class of computational models for molecular interaction networks. The stochasticity stems from the updating schedule. Standard updating schedules include the synchronous update, where all the nodes are updated at the same time, and the asynchronous update where a random node is updated at each time step. The former produces a deterministic dynamics while the latter a stochastic dynamics. A more general stochastic setting considers propensity parameters for updating each node. Stochastic Discrete Dynamical Systems (SDDS) are a modeling framework that considers two propensity parameters for updating each node and uses one when the update has a positive impact on the variable, that is, when the update causes the variable to increase its value, and uses the other when the update has a negative impact, that is, when the update causes it to decrease its value. This framework offers additional features for simulations but also adds a complexity in parameter estimation of the propensities. This paper presents a method for estimating the propensity parameters for SDDS. The method is based on adding noise to the system using the Google PageRank approach to make the system ergodic and thus guaranteeing the existence of a stationary distribution. Then with the use of a genetic algorithm, the propensity parameters are estimated. Approximation techniques that make the search algorithms efficient are also presented and Matlab/Octave code to test the algorithms are available at http://www.ms.uky.edu/~dmu228/GeneticAlg/Code.html.

Arnica montana Stimulates Extracellular Matrix Gene Expression in a Macrophage Cell Line Differentiated to Wound-Healing Phenotype

Arnica montana (Arnica m.) is used for its purported anti-inflammatory and tissue healing actions after trauma, bruises, or tissue injuries, but its cellular and molecular mechanisms are largely unknown. This work tested Arnica m. effects on gene expression using an in vitro model of macrophages polarized towards a "wound-healing" phenotype. The monocyte-macrophage human THP-1 cell line was cultured and differentiated with phorbol-myristate acetate and Interleukin-4, then exposed for 24h to Arnica m. centesimal (c) dilutions 2c, 3c, 5c, 9c, 15c or Control. Total RNA was isolated and cDNA libraries were sequenced with a NextSeq500 sequencer. Genes with significantly positive (up-regulated) or negative (down-regulated) fold changes were defined as differentially expressed genes (DEGs). A total of 20 DEGs were identified in Arnica m. 2c treated cells. Of these, 7 genes were up-regulated and 13 were down-regulated. The most significantly up-regulated function concerned 4 genes with a conserved site of epidermal growth factor-like region (p<0.001) and three genes of proteinaceous extracellular matrix, including heparin sulphate proteoglycan 2 (HSPG2), fibrillin 2 (FBN2), and fibronectin (FN1) (p<0.01). Protein assay confirmed a statistically significant increase of fibronectin production (p<0.05). The down-regulated transcripts derived from mitochondrial genes coding for some components of electron transport chain. The same groups of genes were also regulated by increasing dilutions of Arnica m. (3c, 5c, 9c, 15c), although with a lower effect size. We further tested the healing potential of Arnica m. 2c in a scratch model of wound closure based on the motility of bone marrow-derived macrophages and found evidence of an accelerating effect on cell migration in this system. The results of this work, taken together, provide new insights into the action of Arnica m. in tissue healing and repair, and identify extracellular matrix regulation by macrophages as a therapeutic target.

Intrinsic noise and deviations from criticality in Boolean gene-regulatory networks

Gene regulatory networks can be successfully modeled as Boolean networks. A much discussed hypothesis says that such model networks reproduce empirical findings the best if they are tuned to operate at criticality, i.e. at the borderline between their ordered and disordered phases. Critical networks have been argued to lead to a number of functional advantages such as maximal dynamical range, maximal sensitivity to environmental changes, as well as to an excellent tradeoff between stability and flexibility. Here, we study the effect of noise within the context of Boolean networks trained to learn complex tasks under supervision. We verify that quasi-critical networks are the ones learning in the fastest possible way -even for asynchronous updating rules- and that the larger the task complexity the smaller the distance to criticality. On the other hand, when additional sources of intrinsic noise in the network states and/or in its wiring pattern are introduced, the optimally performing networks become clearly subcritical. These results suggest that in order to compensate for inherent stochasticity, regulatory and other type of biological networks might become subcritical rather than being critical, all the most if the task to be performed has limited complexity.

Identification of control targets in Boolean molecular network models via computational algebra

BACKGROUND

Many problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type considered is that of Boolean networks. The potential control targets can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system.

RESULTS

This paper presents a method for the identification of potential intervention targets in Boolean molecular network models using algebraic techniques. The approach exploits an algebraic representation of Boolean networks to encode the control candidates in the network wiring diagram as the solutions of a system of polynomials equations, and then uses computational algebra techniques to find such controllers. The control methods in this paper are validated through the identification of combinatorial interventions in the signaling pathways of previously reported control targets in two well studied systems, a p53-mdm2 network and a blood T cell lymphocyte granular leukemia survival signaling network. Supplementary data is available online and our code in Macaulay2 and Matlab are available via http://www.ms.uky.edu/~dmu228/ControlAlg .

CONCLUSIONS

This paper presents a novel method for the identification of intervention targets in Boolean network models. The results in this paper show that the proposed methods are useful and efficient for moderately large networks.

Sport science integration: An evolutionary synthesis

The aim of the paper is to point out one way of integrating the supposedly incommensurate disciplines investigated in sports science. General, common principles can be found among apparently unrelated disciplines when the focus is put on the dynamics of sports-related phenomena. Dynamical systems approaches that have recently changed research in biological and social sciences among others, offer key concepts to create a common pluricontextual language in sport science. This common language, far from being homogenising, offers key synthesis between diverse fields, respecting and enabling the theoretical and experimental pluralism. It forms a softly integrated sports science characterised by a basic dynamic explanatory backbone as well as context-dependent theoretical flexibility. After defining the dynamic integration in living systems, unable to be captured by structural static approaches, we show the commonalities between the diversity of processes existing on different levels and time scales in biological and social entities. We justify our interpretation by drawing on some recent scientific contributions that use the same general principles and concepts, and diverse methods and techniques of data analysis, to study different types of phenomena in diverse disciplines. We show how the introduction of the dynamic framework in sport science has started to blur the boundaries between physiology, biomechanics, psychology, phenomenology and sociology. The advantages and difficulties of sport science integration and its consequences in research are also discussed.