Bifurcation analysis of bistable and oscillatory dynamics in biological networks using the root‐locus method

Most of the biological systems including gene regulatory networks can be described well by ordinary differential equation models with rational non‐linearities. These models are derived either based on the reaction kinetics or by curve fitting to experimental data. This study demonstrates the applicability of the root‐locus‐based bifurcation analysis method for studying the complex dynamics of such models. The effectiveness of the bifurcation analysis in determining the exact parameter regions in each of which the system shows a certain dynamical behaviour, such as bistability, oscillation, and asymptotically equilibrium dynamics is shown by considering two mostly studied gene regulatory networks, namely Gardner''s genetic toggle switch and p53 gene network possessing two‐phase (mono‐stable/oscillation) dynamics.Inspec keywords: oscillations, curve fitting, differential equations, bifurcation, genetics, nonlinear dynamical systemsOther keywords: nonlinearities, reaction kinetics, root‐locus‐based bifurcation analysis method, complex dynamics, exact parameter regions, dynamical behaviour, equilibrium dynamics, studied gene regulatory networks, p53 gene network, bistable dynamics, oscillatory dynamics, biological networks, root‐locus method, biological systems, ordinary differential equation models     

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.     

Functional tunability of biological circuits from additional toggle switches

In many complex regulatory networks with interlinked feedback loops, the simple core circuits are sufficient to achieve the specific biological functions of the whole networks, naturally raising a question: what is the role of the additional feedback loops. By investigating the effect of an additional toggle switch on the auto‐activation circuit responsible for competent switch in Bacillus subtilits and on the activator–repressor circuit responsible for cell cycle in Xenopus embryonic, the authors show that the additional toggle switch can elaborate the dynamical behaviour of both circuits. Specifically, the additional toggle switch in B. subtilits does not significantly affect the saturation level of the competent state but can tune the activation threshold (i.e. the minimal stimulus required to switch the system from the non‐competent state to the competent state). For the activator–repressor circuit in X. embryonic cell cycle, the additional toggle switch can tune the oscillation frequency but does not change the oscillation amplitude. The proposed detailed results not only provide guidelines to the engineering of synthetic genetic circuits, but also imply a significant fact that additional toggle switches in a complex network are not really redundant but play a role of tuning network functions.Inspec keywords: biochemistry, cellular biophysics, microorganismsOther keywords: functional tunability, biological circuits, toggle switches, complex regulatory networks, interlinked feedback loops, Bacillus subtilits, autoactivation circuit, activator‐repressor circuit, Xenopus embryonic     

New scaling relation for information transfer in biological networks

We quantify characteristics of the informational architecture of two representative biological networks: the Boolean network model for the cell-cycle regulatory network of the fission yeast Schizosaccharomyces pombe (Davidich et al. 2008 PLoS ONE 3, e1672 (doi:10.1371/journal.pone.0001672)) and that of the budding yeast Saccharomyces cerevisiae (Li et al. 2004 Proc. Natl Acad. Sci. USA 101, 4781–4786 (doi:10.1073/pnas.0305937101)). We compare our results for these biological networks with the same analysis performed on ensembles of two different types of random networks: Erdös–Rényi and scale-free. We show that both biological networks share features in common that are not shared by either random network ensemble. In particular, the biological networks in our study process more information than the random networks on average. Both biological networks also exhibit a scaling relation in information transferred between nodes that distinguishes them from random, where the biological networks stand out as distinct even when compared with random networks that share important topological properties, such as degree distribution, with the biological network. We show that the most biologically distinct regime of this scaling relation is associated with a subset of control nodes that regulate the dynamics and function of each respective biological network. Information processing in biological networks is therefore interpreted as an emergent property of topology (causal structure) and dynamics (function). Our results demonstrate quantitatively how the informational architecture of biologically evolved networks can distinguish them from other classes of network architecture that do not share the same informational properties.     

The evolvability of programmable hardware

In biological systems, individual phenotypes are typically adopted by multiple genotypes. Examples include protein structure phenotypes, where each structure can be adopted by a myriad individual amino acid sequence genotypes. These genotypes form vast connected ‘neutral networks’ in genotype space. The size of such neutral networks endows biological systems not only with robustness to genetic change, but also with the ability to evolve a vast number of novel phenotypes that occur near any one neutral network. Whether technological systems can be designed to have similar properties is poorly understood. Here we ask this question for a class of programmable electronic circuits that compute digital logic functions. The functional flexibility of such circuits is important in many applications, including applications of evolutionary principles to circuit design. The functions they compute are at the heart of all digital computation. We explore a vast space of 10⁴⁵ logic circuits (‘genotypes’) and 10¹⁹ logic functions (‘phenotypes’). We demonstrate that circuits that compute the same logic function are connected in large neutral networks that span circuit space. Their robustness or fault-tolerance varies very widely. The vicinity of each neutral network contains circuits with a broad range of novel functions. Two circuits computing different functions can usually be converted into one another via few changes in their architecture. These observations show that properties important for the evolvability of biological systems exist in a commercially important class of electronic circuitry. They also point to generic ways to generate fault-tolerant, adaptable and evolvable electronic circuitry.     

A comparative analysis of synthetic genetic oscillators

Synthetic biology is a rapidly expanding discipline at the interface between engineering and biology. Much research in this area has focused on gene regulatory networks that function as biological switches and oscillators. Here we review the state of the art in the design and construction of oscillators, comparing the features of each of the main networks published to date, the models used for in silico design and validation and, where available, relevant experimental data. Trends are apparent in the ways that network topology constrains oscillator characteristics and dynamics. Also, noise and time delay within the network can both have constructive and destructive roles in generating oscillations, and stochastic coherence is commonplace. This review can be used to inform future work to design and implement new types of synthetic oscillators or to incorporate existing oscillators into new designs.     

Modelling co-translational dimerization for programmable nonlinearity in synthetic biology

Nonlinearity plays a fundamental role in the performance of both natural and synthetic biological networks. Key functional motifs in living microbial systems, such as the emergence of bistability or oscillations, rely on nonlinear molecular dynamics. Despite its core importance, the rational design of nonlinearity remains an unmet challenge. This is largely due to a lack of mathematical modelling that accounts for the mechanistic basis of nonlinearity. We introduce a model for gene regulatory circuits that explicitly simulates protein dimerization—a well-known source of nonlinear dynamics. Specifically, our approach focuses on modelling co-translational dimerization: the formation of protein dimers during—and not after—translation. This is in contrast to the prevailing assumption that dimer generation is only viable between freely diffusing monomers (i.e. post-translational dimerization). We provide a method for fine-tuning nonlinearity on demand by balancing the impact of co- versus post-translational dimerization. Furthermore, we suggest design rules, such as protein length or physical separation between genes, that may be used to adjust dimerization dynamics in vivo. The design, build and test of genetic circuits with on-demand nonlinear dynamics will greatly improve the programmability of synthetic biological systems.     

Phase transitions and assortativity in models of gene regulatory networks evolved under different selection processes

We study a simplified model of gene regulatory network evolution in which links (regulatory interactions) are added via various selection rules that are based on the structural and dynamical features of the network nodes (genes). Similar to well-studied models of ‘explosive’ percolation, in our approach, links are selectively added so as to delay the transition to large-scale damage propagation, i.e. to make the network robust to small perturbations of gene states. We find that when selection depends only on structure, evolved networks are resistant to widespread damage propagation, even without knowledge of individual gene propensities for becoming ‘damaged’. We also observe that networks evolved to avoid damage propagation tend towards disassortativity (i.e. directed links preferentially connect high degree ‘source’ genes to low degree ‘target’ genes and vice versa). We compare our simulations to reconstructed gene regulatory networks for several different species, with genes and links added over evolutionary time, and we find a similar bias towards disassortativity in the reconstructed networks.     

Dynamics on higher-order networks: a review

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.     

Bow-tie architecture of gene regulatory networks in species of varying complexity

The gene regulatory network (GRN) architecture plays a key role in explaining the biological differences between species. We aim to understand species differences in terms of some universally present dynamical properties of their gene regulatory systems. A network architectural feature associated with controlling system-level dynamical properties is the bow-tie, identified by a strongly connected subnetwork, the core layer, between two sets of nodes, the in and the out layers. Though a bow-tie architecture has been observed in many networks, its existence has not been extensively investigated in GRNs of species of widely varying biological complexity. We analyse publicly available GRNs of several well-studied species from prokaryotes to unicellular eukaryotes to multicellular organisms. In their GRNs, we find the existence of a bow-tie architecture with a distinct largest strongly connected core layer. We show that the bow-tie architecture is a characteristic feature of GRNs. We observe an increasing trend in the relative core size with species complexity. Using studied relationships of the core size with dynamical properties like robustness and fragility, flexibility, criticality, controllability and evolvability, we hypothesize how these regulatory system properties have emerged differently with biological complexity, based on the observed differences of the GRN bow-tie architectures.     

Deconstructing the core dynamics from a complex time-lagged regulatory biological circuit

Complex regulatory dynamics is ubiquitous in molecular networks composed of genes and proteins. Recent progress in computational biology and its application to molecular data generate a growing number of complex networks. Yet, it has been difficult to understand the governing principles of these networks beyond graphical analysis or extensive numerical simulations. Here the authors exploit several simplifying biological circumstances which thereby enable to directly detect the underlying dynamical regularities driving periodic oscillations in a dynamical nonlinear computational model of a protein?protein network. System analysis is performed using the cell cycle, a mathematically well-described complex regulatory circuit driven by external signals. By introducing an explicit time delay and using a `tearing-and-zooming? approach the authors reduce the system to a piecewise linear system with two variables that capture the dynamics of this complex network. A key step in the analysis is the identification of functional subsystems by identifying the relations between statevariables within the model. These functional subsystems are referred to as dynamical modules operating as sensitive switches in the original complex model. By using reduced mathematical representations of the subsystems the authors derive explicit conditions on how the cell cycle dynamics depends on system parameters, and can, for the first time, analyse and prove global conditions for system stability. The approach which includes utilising biological simplifying conditions, identification of dynamical modules and mathematical reduction of the model complexity may be applicable to other well-characterised biological regulatory circuits.     

Structure and formation of ant transportation networks

Many biological systems use extensive networks for the transport of resources and information. Ants are no exception. How do biological systems achieve efficient transportation networks in the absence of centralized control and without global knowledge of the environment? Here, we address this question by studying the formation and properties of inter-nest transportation networks in the Argentine ant (Linepithema humile). We find that the formation of inter-nest networks depends on the number of ants involved in the construction process. When the number of ants is sufficient and networks do form, they tend to have short total length but a low level of robustness. These networks are topologically similar to either minimum spanning trees or Steiner networks. The process of network formation involves an initial construction of multiple links followed by a pruning process that reduces the number of trails. Our study thus illuminates the conditions under and the process by which minimal biological transport networks can be constructed.     

Effect of static local distortions vs. dynamic motions on the stability and band gaps of cubic oxide and halide perovskites

Ternary ABX₃ perovskites made of corner-sharing BX₆ octahedra have long featured prominently in solid-state chemistry and condensed matter physics. Still, the joint understanding of their two main subgroups—halides and oxides—has not been fully developed. Indeed, unlike the case in simpler compounds having a single, robust repeated motif (“monomorphous”), certain cubic perovskites can manifest a non-thermal (= intrinsic) distribution of local motifs (“polymorphous networks”). Such static deformations can include positional degrees of freedom (e.g., atomic displacements and octahedral tilting) or magnetic moment degrees of freedom in paramagnets. Unlike thermal motion, such static distortions do not time-average to zero, being an expression of the intrinsic symmetry breaking preference of the chemical bonding. The present study compares electronic structure features of oxide and halide perovskites starting from the static polymorphous distribution of motifs described by Density Functional Theory (DFT) minimization of the internal energy, continuing to finite temperature thermal disorder modeled via finite temperature DFT molecular dynamics. We find that (i) different oxide vs. halide ABX₃ compounds adopt different energy-lowering symmetry-breaking modes. The calculated pair distribution function (PDF) of SrTiO₃ from the first-principles agrees with recently measured PDF. (ii) In both oxides and halides, such static distortions lead to band gap blueshifts with respect to undistorted cubic Pm-3m structure. (iii) For oxide perovskites, high-temperature molecular dynamics simulations initiated from the statically distorted polymorphous structures reveal that the thermally-induced distortions can lead to a band gap redshift. (iv) In contrast, for cubic halide perovskite CsPbI₃, both the intrinsic distortions and the thermal distortions contribute in tandem to band gap blueshift, the former, intrinsic effect being dominant. (v) In the oxide SrTiO₃ and CaTiO₃ (but not in halide) perovskites, octahedral tilting leads to the emergence of a distinct Γ–Γ direct band gap component as a secondary valley minimum to the well-known indirect R–Γ gap. Understanding such intrinsic vs. thermal effects on oxide vs. halide perovskites holds the potential for designing target electronic properties.     

Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics

Linearized catalytic reaction equations (modelling, for example, the dynamics of genetic regulatory networks), under the constraint that expression levels, i.e. molecular concentrations of nucleic material, are positive, exhibit non-trivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems, an inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity, which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow us to understand the fundamental dynamical properties of complex biological reaction networks. We analyse the Lyapunov spectrum, determine the probability of finding stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network, and study how the frequency distributions of oscillatory modes of such a system depend on the average connectivity.     

Spectrum of genetic diversity and networks of clonal organisms.

Clonal reproduction characterizes a wide range of species including clonal plants in terrestrial and aquatic ecosystems, and clonal microbes such as bacteria and parasitic protozoa, with a key role in human health and ecosystem processes. Clonal organisms present a particular challenge in population genetics because, in addition to the possible existence of replicates of the same genotype in a given sample, some of the hypotheses and concepts underlying classical population genetics models are irreconcilable with clonality. The genetic structure and diversity of clonal populations were examined using a combination of new tools to analyse microsatellite data in the marine angiosperm Posidonia oceanica. These tools were based on examination of the frequency distribution of the genetic distance among ramets, termed the spectrum of genetic diversity (GDS), and of networks built on the basis of pairwise genetic distances among genets. Clonal growth and outcrossing are apparently dominant processes, whereas selfing and somatic mutations appear to be marginal, and the contribution of immigration seems to play a small role in adding genetic diversity to populations. The properties and topology of networks based on genetic distances showed a 'small-world' topology, characterized by a high degree of connectivity among nodes, and a substantial amount of substructure, revealing organization in subfamilies of closely related individuals. The combination of GDS and network tools proposed here helped in dissecting the influence of various evolutionary processes in shaping the intra-population genetic structure of the clonal organism investigated; these therefore represent promising analytical tools in population genetics.     

汽轮机故障诊断的遗传神经网络法

阐明了遗传算法与神经网络结合的必要必邮多层感知器神经网络作为遗传搜索的问题表示方式的思想。设计了遗传神经网络故障诊断模型,用这种新方法解决了汽轮机多故障诊断问题。新方法可以简化神经网络的结构并逃逸局部极小。诊断结果与实际相符,从而验证了该方法的有效性和实用性。     

Unsupervised learning of control signals and their encodings in Caenorhabditis elegans whole-brain recordings

A major goal of computational neuroscience is to understand the relationship between synapse-level structure and network-level functionality. Caenorhabditis elegans is a model organism to probe this relationship due to the historic availability of the synaptic structure (connectome) and recent advances in whole brain calcium imaging techniques. Recent work has applied the concept of network controllability to neuronal networks, discovering some neurons that are able to drive the network to a certain state. However, previous work uses a linear model of the network dynamics, and it is unclear if the real neuronal network conforms to this assumption. Here, we propose a method to build a global, low-dimensional model of the dynamics, whereby an underlying global linear dynamical system is actuated by temporally sparse control signals. A key novelty of this method is discovering candidate control signals that the network uses to control itself. We analyse these control signals in two ways, showing they are interpretable and biologically plausible. First, these control signals are associated with transitions between behaviours, which were previously annotated via expert-generated features. Second, these signals can be predicted both from neurons previously implicated in behavioural transitions but also additional neurons previously unassociated with these behaviours. The proposed mathematical framework is generic and can be generalized to other neurosensory systems, potentially revealing transitions and their encodings in a completely unsupervised way.