The effect of negative feedback loops on the dynamics of boolean networks

Feedback loops play an important role in determining the dynamics of biological networks. To study the role of negative feedback loops, this article introduces the notion of distance-to-positive-feedback which, in essence, captures the number of independent negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks, it is shown that distance-to-positive-feedback has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same number of nodes and same connectivity.     

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.     

Robustness in phenotypic plasticity and heterogeneity patterns enabled by EMT networks

Epithelial-mesenchymal plasticity (EMP) is a key arm of cancer metastasis and is observed across many contexts. Cells undergoing EMP can reversibly switch between three classes of phenotypes: epithelial (E), mesenchymal (M), and hybrid E/M. While a large number of multistable regulatory networks have been identified to be driving EMP in various contexts, the exact mechanisms and design principles that enable robustness in driving EMP across contexts are not yet fully understood. Here, we investigated dynamic and structural robustness in EMP networks with regard to phenotypic heterogeneity and plasticity. We use two different approaches to simulate these networks: a computationally inexpensive, parameter-independent continuous state space Boolean model, and an ODE-based parameter-agnostic framework (RACIPE), both of which yielded similar phenotypic distributions. While the latter approach is useful for measurements of plasticity, the former model enabled us to extensively investigate robustness in phenotypic heterogeneity. Using perturbations to network topology and by varying network parameters, we show that multistable EMP networks are structurally and dynamically more robust compared with their randomized counterparts, thereby highlighting their topological hallmarks. These features of robustness are governed by a balance of positive and negative feedback loops embedded in these networks. Using a combination of the number of negative and positive feedback loops weighted by their lengths, we identified a metric that can explain the structural and dynamical robustness of these networks. This metric enabled us to compare networks across multiple sizes, and the network principles thus obtained can be used to identify fragilities in large networks without simulating their dynamics. Our analysis highlights a network topology-based approach to quantify robustness in the phenotypic heterogeneity and plasticity emergent from EMP networks.     

Boolean dynamics of biological networks with multiple coupled feedback loops

Boolean networks have been frequently used to study the dynamics of biological networks. In particular, there have been various studies showing that the network connectivity and the update rule of logical functions affect the dynamics of Boolean networks. There has been, however, relatively little attention paid to the dynamical role of a feedback loop, which is a circular chain of interactions between Boolean variables. We note that such feedback loops are ubiquitously found in various biological systems as multiple coupled structures and they are often the primary cause of complex dynamics. In this article, we investigate the relationship between the multiple coupled feedback loops and the dynamics of Boolean networks. We show that networks have a larger proportion of basins corresponding to fixed-point attractors as they have more coupled positive feedback loops, and a larger proportion of basins for limit-cycle attractors as they have more coupled negative feedback loops.     

Coherent coupling of feedback loops: a design principle of cell signaling networks

MOTIVATION: It is widely accepted that cell signaling networks have been evolved to be robust against perturbations. To investigate the topological characteristics resulting in such robustness, we have examined large-scale signaling networks and found that a number of feedback loops are present mostly in coupled structures. In particular, the coupling was made in a coherent way implying that same types of feedback loops are interlinked together. RESULTS: We have investigated the role of such coherently coupled feedback loops through extensive Boolean network simulations and found that a high proportion of coherent couplings can enhance the robustness of a network against its state perturbations. Moreover, we found that the robustness achieved by coherently coupled feedback loops can be kept evolutionarily stable. All these results imply that the coherent coupling of feedback loops might be a design principle of cell signaling networks devised to achieve the robustness.     

Robust,tunable genetic memory from protein sequestration combined with positive feedback

Natural regulatory networks contain many interacting components that allow for fine-tuning of switching and memory properties. Building simple bistable switches, synthetic biologists have learned the design principles of complex natural regulatory networks. However, most switches constructed so far are so simple (e.g. comprising two regulators) that they are functional only within a limited parameter range. Here, we report the construction of robust, tunable bistable switches in Escherichia coli using three heterologous protein regulators (ExsADC) that are sequestered into an inactive complex through a partner swapping mechanism. On the basis of mathematical modeling, we accurately predict and experimentally verify that the hysteretic region can be fine-tuned by controlling the interactions of the ExsADC regulatory cascade using the third member ExsC as a tuning knob. Additionally, we confirm that a dual-positive feedback switch can markedly increase the hysteretic region, compared to its single-positive feedback counterpart. The dual-positive feedback switch displays bistability over a 10⁶-fold range of inducer concentrations, to our knowledge, the largest range reported so far. This work demonstrates the successful interlocking of sequestration-based ultrasensitivity and positive feedback, a design principle that can be applied to the construction of robust, tunable, and predictable genetic programs to achieve increasingly sophisticated biological behaviors.     

Flexible and robust networks

We consider networks with two types of nodes. The v-nodes, called centers, are hyperconnected and interact with one another via many u-nodes, called satellites. This centralized architecture, widespread in gene networks, possesses two fundamental properties. Namely, this organization creates feedback loops that are capable of generating practically any prescribed patterning dynamics, chaotic or periodic, or having a number of equilibrium states. Moreover, this organization is robust with respect to random perturbations of the system.     

Bidirectional Long Short-Term Memory Networks for predicting the subcellular localization of eukaryotic proteins

An algorithm called bidirectional long short-term memory networks (BLSTM) for processing sequential data is introduced. This supervised learning method trains a special recurrent neural network to use very long-range symmetric sequence context using a combination of nonlinear processing elements and linear feedback loops for storing long-range context. The algorithm is applied to the sequence-based prediction of protein localization and predicts 93.3 percent novel nonplant proteins and 88.4 percent novel plant proteins correctly, which is an improvement over feedforward and standard recurrent networks solving the same problem. The BLSTM system is available as a Web service at http://stepc.stepc.gr/-synaptic/blstm.html.     

Recurrent fractal neural networks: a strategy for the exchange of local and global information processing in the brain

The regulation of biological networks relies significantly on convergent feedback signaling loops that render a global output locally accessible. Ideally, the recurrent connectivity within these systems is self-organized by a time-dependent phase-locking mechanism. This study analyzes recurrent fractal neural networks (RFNNs), which utilize a self-similar or fractal branching structure of dendrites and downstream networks for phase-locking of reciprocal feedback loops: output from outer branch nodes of the network tree enters inner branch nodes of the dendritic tree in single neurons. This structural organization enables RFNNs to amplify re-entrant input by over-the-threshold signal summation from feedback loops with equivalent signal traveling times. The columnar organization of pyramidal neurons in the neocortical layers V and III is discussed as the structural substrate for this network architecture. RFNNs self-organize spike trains and render the entire neural network output accessible to the dendritic tree of each neuron within this network. As the result of a contraction mapping operation, the local dendritic input pattern contains a downscaled version of the network output coding structure. RFNNs perform robust, fractal data compression, thus coping with a limited number of feedback loops for signal transport in convergent neural networks. This property is discussed as a significant step toward the solution of a fundamental problem in neuroscience: how is neuronal computation in separate neurons and remote brain areas unified as an instance of experience in consciousness? RFNNs are promising candidates for engaging neural networks into a coherent activity and provide a strategy for the exchange of global and local information processing in the human brain, thereby ensuring the completeness of a transformation from neuronal computation into conscious experience.     

Computational neurogenetic modelling: a pathway to new discoveries in genetic neuroscience

The paper presents a methodology for using computational neurogenetic modelling (CNGM) to bring new original insights into how genes influence the dynamics of brain neural networks. CNGM is a novel computational approach to brain neural network modelling that integrates dynamic gene networks with artificial neural network model (ANN). Interaction of genes in neurons affects the dynamics of the whole ANN model through neuronal parameters, which are no longer constant but change as a function of gene expression. Through optimization of interactions within the internal gene regulatory network (GRN), initial gene/protein expression values and ANN parameters, particular target states of the neural network behaviour can be achieved, and statistics about gene interactions can be extracted. In such a way, we have obtained an abstract GRN that contains predictions about particular gene interactions in neurons for subunit genes of AMPA, GABAA and NMDA neuro-receptors. The extent of sequence conservation for 20 subunit proteins of all these receptors was analysed using standard bioinformatics multiple alignment procedures. We have observed abundance of conserved residues but the most interesting observation has been the consistent conservation of phenylalanine (F at position 269) and leucine (L at position 353) in all 20 proteins with no mutations. We hypothesise that these regions can be the basis for mutual interactions. Existing knowledge on evolutionary linkage of their protein families and analysis at molecular level indicate that the expression of these individual subunits should be coordinated, which provides the biological justification for our optimized GRN.     

Quantitative analysis of robustness and fragility in biological networks based on feedback dynamics

Motivation: It has been widely reported that biological networksare robust against perturbations such as mutations. On the contrary,it has also been known that biological networks are often fragileagainst unexpected mutations. There is a growing interest inthese intriguing observations and the underlying design principlethat causes such robust but fragile characteristics of biologicalnetworks. For relatively small networks, a feedback loop hasbeen considered as an important motif for realizing the robustness.It is still, however, not clear how a number of coupled feedbackloops actually affect the robustness of large complex biologicalnetworks. In particular, the relationship between fragilityand feedback loops has not yet been investigated till now. Results: Through extensive computational experiments, we foundthat networks with a larger number of positive feedback loopsand a smaller number of negative feedback loops are likely tobe more robust against perturbations. Moreover, we found thatthe nodes of a robust network subject to perturbations are mostlyinvolved with a smaller number of feedback loops compared withthe other nodes not usually subject to perturbations. This topologicalcharacteristic eventually makes the robust network fragile againstunexpected mutations at the nodes not previously exposed toperturbations. Contact: ckh{at}kaist.ac.kr Supplementary information: Supplementary data are availableat Bioinformatics online. Associate Editor: Thomas Lengauer     

Parkinsonian tremor and simplification in network dynamics

We explore the behavior of richly connected inhibitory neural networks under parameter changes that correspond to weakening of synaptic efficacies between network units, and show that transitions from irregular to periodic dynamics are common in such systems. The weakening of these connections leads to a reduction in the number of units that effectively drive the dynamics and thus to simpler behavior. We hypothesize that the multiple interconnecting loops of the brain’s motor circuitry, which involve many inhibitory connections, exhibit such transitions. Normal physiological tremor is irregular while other forms of tremor show more regular oscillations. Tremor in Parkinson’s disease, for example, stems from weakened synaptic efficacies of dopaminergic neurons in the nigro-striatal pathway, as in our general model. The multiplicity of structures involved in the production of symptoms in Parkinson’s disease and the reversibility of symptoms by pharmacological and surgical manipulation of connection parameters suggest that such a neural network model is appropriate. Furthermore, fixed points that can occur in the network models are suggestive of akinesia in Parkinson’s disease. This model is consistent with the view that normal physiological systems can be regulated by robust and richly connected feedback networks with complex dynamics, and that loss of complexity in the feedback structure due to disease leads to more orderly behavior.     

生物分子网络弹性研究进展

弹性是生物分子网络重要且基础的属性之一,一方面弹性赋予生物分子网络抵抗内部噪声与环境干扰并维持其自身基本功能的能力,另一方面,弹性为网络状态的恢复制造了阻力。生物分子网络弹性研究试图回答如下3个问题：a. 生物分子网络弹性的产生机理是什么？b. 弹性影响下生物分子网络的状态如何发生转移？c. 如何预测生物网络状态转换临界点,以防止系统向不理想的状态演化？因此,研究生物分子网络弹性有助于理解生物系统内部运作机理,同时对诸如疾病发生临界点预测、生物系统状态逆转等临床应用具有重要的指导意义。鉴于此,本文主要针对以上生物分子网络弹性领域的3个热点研究问题,在研究方法和生物学应用上进行了系统地综述,并对未来生物分子网络弹性的研究方向进行了展望。     

Transient dynamics of genetic regulatory networks

We present an approximation scheme for deriving reaction rate equations of genetic regulatory networks. This scheme predicts the timescales of transient dynamics of such networks more accurately than does standard quasi-steady state analysis by introducing prefactors to the ODEs that govern the dynamics of the protein concentrations. These prefactors render the ODE systems slower than their quasi-steady state approximation counterparts. We introduce the method by examining a positive feedback gene regulatory network, and show how the transient dynamics of this network are more accurately modeled when the prefactor is included. Next, we examine the repressilator, a genetic oscillator, and show that the period, amplitude, and bifurcation diagram defining the onset of the oscillations are better estimated by the prefactor method. Finally, we examine the consequences of the method to the dynamics of reduced models of the phage lambda switch, and show that the switching times between the two states is slowed by the presence of the prefactor that arises from protein multimerization and DNA binding.