Autonomous Boolean modelling of developmental gene regulatory networks

A comprehensive survey of developmental programs reveals a dearth of tree-like lineage graphs and ubiquitous regeneration

Background

Multicellular organisms are characterized by a wide diversity of forms and complexity despite a restricted set of key molecules and mechanisms at the base of organismal development. Development combines three basic processes—asymmetric cell division, signaling, and gene regulation—in a multitude of ways to create this overwhelming diversity of multicellular life forms. Here, we use a generative model to test the limits to which such processes can be combined to generate multiple differentiation paths during development, and attempt to chart the diversity of multicellular organisms generated.

Results

We sample millions of biologically feasible developmental schemes, allowing us to comment on the statistical properties of cell differentiation trajectories they produce. We characterize model-generated “organisms” using the graph topology of their cell type lineage maps. Remarkably, tree-type lineage differentiation maps are the rarest in our data. Additionally, a majority of the “organisms” generated by our model appear to be endowed with the ability to regenerate using pluripotent cells.

Conclusions

Our results indicate that, in contrast to common views, cell type lineage graphs are unlikely to be tree-like. Instead, they are more likely to be directed acyclic graphs, with multiple lineages converging on the same terminal cell type. Furthermore, the high incidence of pluripotent cells in model-generated organisms stands in line with the long-standing hypothesis that whole body regeneration is an epiphenomenon of development. We discuss experimentally testable predictions of our model and some ways to adapt the generative framework to test additional hypotheses about general features of development.

Supplementary Information

The online version contains supplementary material available at (10.1186/s12915-021-01013-4).

Optical Boolean chaos

Boolean chaos is widely used in physical systems for its digital-like behavior and complex dynamics. However, electronic logic devices limit the bandwidth of Boolean chaos and its development. Based on an autonomous optical Boolean network, a method of generating optical Boolean chaos with 14 GHz bandwidth is proposed, exploring the physical mechanism of the chaos generated by the system and analyzing the influences of external parameters on the dynamic characteristics of the system. The output status is mainly affected by the detection optical power, carrier recovery time of the semiconductor optical amplifier, and difference between the two self-feedback time delays.

General method to find the attractors of discrete dynamic models of biological systems

Analyzing the long-term behaviors (attractors) of dynamic models of biological networks can provide valuable insight. We propose a general method that can find the attractors of multilevel discrete dynamical systems by extending a method that finds the attractors of a Boolean network model. The previous method is based on finding stable motifs, subgraphs whose nodes' states can stabilize on their own. We extend the framework from binary states to any finite discrete levels by creating a virtual node for each level of a multilevel node, and describing each virtual node with a quasi-Boolean function. We then create an expanded representation of the multilevel network, find multilevel stable motifs and oscillating motifs, and identify attractors by successive network reduction. In this way, we find both fixed point attractors and complex attractors. We implemented an algorithm, which we test and validate on representative synthetic networks and on published multilevel models of biological networks. Despite its primary motivation to analyze biological networks, our motif-based method is general and can be applied to any finite discrete dynamical system.

Computer modeling in developmental biology: growing today, essential tomorrow

D'Arcy Thompson was a true pioneer, applying mathematical concepts and analyses to the question of morphogenesis over 100 years ago. The centenary of his famous book, On Growth and Form, is therefore a great occasion on which to review the types of computer modeling now being pursued to understand the development of organs and organisms. Here, I present some of the latest modeling projects in the field, covering a wide range of developmental biology concepts, from molecular patterning to tissue morphogenesis. Rather than classifying them according to scientific question, or scale of problem, I focus instead on the different ways that modeling contributes to the scientific process and discuss the likely future of modeling in developmental biology.

LoTo: a graphlet based method for the comparison of local topology between gene regulatory networks

One of the main challenges of the post-genomic era is the understanding of how gene expression is controlled. Changes in gene expression lay behind diverse biological phenomena such as development, disease and the adaptation to different environmental conditions. Despite the availability of well-established methods to identify these changes, tools to discern how gene regulation is orchestrated are still required. The regulation of gene expression is usually depicted as a Gene Regulatory Network (GRN) where changes in the network structure (i.e., network topology) represent adjustments of gene regulation. Like other networks, GRNs are composed of basic building blocks; small induced subgraphs called graphlets. Here we present LoTo, a novel method that using Graphlet Based Metrics (GBMs) identifies topological variations between different states of a GRN. Under our approach, different states of a GRN are analyzed to determine the types of graphlet formed by all triplets of nodes in the network. Subsequently, graphlets occurring in a state of the network are compared to those formed by the same three nodes in another version of the network. Once the comparisons are performed, LoTo applies metrics from binary classification problems calculated on the existence and absence of graphlets to assess the topological similarity between both network states. Experiments performed on randomized networks demonstrate that GBMs are more sensitive to topological variation than the same metrics calculated on single edges. Additional comparisons with other common metrics demonstrate that our GBMs are capable to identify nodes whose local topology changes between different states of the network. Notably, due to the explicit use of graphlets, LoTo captures topological variations that are disregarded by other approaches. LoTo is freely available as an online web server at http://dlab.cl/loto.

Transient dynamics and their control in time-delay autonomous Boolean ring networks

Biochemical systems with switch-like interactions, such as gene regulatory networks, are well modeled by autonomous Boolean networks. Specifically, the topology and logic of gene interactions can be described by systems of continuous piecewise-linear differential equations, enabling analytical predictions of the dynamics of specific networks. However, most models do not account for time delays along links associated with spatial transport, mRNA transcription, and translation. To address this issue, we have developed an experimental test bed to realize a time-delay autonomous Boolean network with three inhibitory nodes, known as a repressilator, and use it to study the dynamics that arise as time delays along the links vary. We observe various nearly periodic oscillatory transient patterns with extremely long lifetime, which emerge in small network motifs due to the delay, and which are distinct from the eventual asymptotically stable periodic attractors. For repeated experiments with a given network, we find that stochastic processes give rise to a broad distribution of transient times with an exponential tail. In some cases, the transients are so long that it is doubtful the attractors will ever be approached in a biological system that has a finite lifetime. To counteract the long transients, we show experimentally that small, occasional perturbations applied to the time delays can force the trajectories to rapidly approach the attractors.

Estimation of delays in generalized asynchronous Boolean networks

A new generalized asynchronous Boolean network (GABN) model has been proposed in this paper. This continuous-time discrete-state model captures the biological reality of cellular dynamics without compromising the computational efficiency of the Boolean framework. The GABN synthesis procedure is based on the prior knowledge of the logical structure of the regulatory network, and the experimental transcriptional parameters. The novelty of the proposed methodology lies in considering different delays associated with the activation and deactivation of a particular protein (especially the transcription factors). A few illustrative examples of some well-studied network motifs have been provided to explore the scope of using the GABN model for larger networks. The GABN model of the p53-signaling pathway in response to γ-irradiation has also been simulated in the current paper to provide an indirect validation of the proposed schema.

Graphlet Based Metrics for the Comparison of Gene Regulatory Networks

Understanding the control of gene expression remains one of the main challenges in the post-genomic era. Accordingly, a plethora of methods exists to identify variations in gene expression levels. These variations underlay almost all relevant biological phenomena, including disease and adaptation to environmental conditions. However, computational tools to identify how regulation changes are scarce. Regulation of gene expression is usually depicted in the form of a gene regulatory network (GRN). Structural changes in a GRN over time and conditions represent variations in the regulation of gene expression. Like other biological networks, GRNs are composed of basic building blocks called graphlets. As a consequence, two new metrics based on graphlets are proposed in this work: REConstruction Rate (REC) and REC Graphlet Degree (RGD). REC determines the rate of graphlet similarity between different states of a network and RGD identifies the subset of nodes with the highest topological variation. In other words, RGD discerns how th GRN was rewired. REC and RGD were used to compare the local structure of nodes in condition-specific GRNs obtained from gene expression data of Escherichia coli, forming biofilms and cultured in suspension. According to our results, most of the network local structure remains unaltered in the two compared conditions. Nevertheless, changes reported by RGD necessarily imply that a different cohort of regulators (i.e. transcription factors (TFs)) appear on the scene, shedding light on how the regulation of gene expression occurs when E. coli transits from suspension to biofilm. Consequently, we propose that both metrics REC and RGD should be adopted as a quantitative approach to conduct differential analyses of GRNs. A tool that implements both metrics is available as an on-line web server (http://dlab.cl/loto).

Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions

The biomolecules inside or near cells form a complex interacting system. Cellular phenotypes and behaviors arise from the totality of interactions among the components of this system. A fruitful way of modeling interacting biomolecular systems is by network-based dynamic models that characterize each component by a state variable, and describe the change in the state variables due to the interactions in the system. Dynamic models can capture the stable state patterns of this interacting system and can connect them to different cell fates or behaviors. A Boolean or logic model characterizes each biomolecule by a binary state variable that relates the abundance of that molecule to a threshold abundance necessary for downstream processes. The regulation of this state variable is described in a parameter free manner, making Boolean modeling a practical choice for systems whose kinetic parameters have not been determined. Boolean models integrate the body of knowledge regarding the components and interactions of biomolecular systems, and capture the system's dynamic repertoire, for example the existence of multiple cell fates. These models were used for a variety of systems and led to important insights and predictions. Boolean models serve as an efficient exploratory model, a guide for follow-up experiments, and as a foundation for more quantitative models.

Regulatory logic and pattern formation in the early sea urchin embryo

We model the endomesoderm tissue specification process in the vegetal half of the early sea urchin embryo using Boolean models with continuous-time updating to represent the regulatory network that controls gene expression. Our models assume that the network interaction rules remain constant over time and the dynamics plays out on a predetermined program of cell divisions. An exhaustive search of two-node models, in which each node may represent a module of several genes in the real regulatory network, yields a unique network architecture that can accomplish the pattern formation task at hand--the formation of three latitudinal tissue bands from an initial state with only two distinct cell types. Analysis of an eight-gene model constructed from available experimental data reveals that it has a modular structure equivalent to the successful two-node case. Our results support the hypothesis that the gene regulatory network provides sufficient instructions for producing the correct pattern of tissue specification at this stage of development (between the fourth and tenth cleavages in the urchin embryo).

Piecewise linear and Boolean models of chemical reaction networks

Models of biochemical networks are frequently complex and high-dimensional. Reduction methods that preserve important dynamical properties are therefore essential for their study. Interactions in biochemical networks are frequently modeled using Hill functions ([Formula: see text]). Reduced ODEs and Boolean approximations of such model networks have been studied extensively when the exponent [Formula: see text] is large. However, while the case of small constant [Formula: see text] appears in practice, it is not well understood. We provide a mathematical analysis of this limit and show that a reduction to a set of piecewise linear ODEs and Boolean networks can be mathematically justified. The piecewise linear systems have closed-form solutions that closely track those of the fully nonlinear model. The simpler, Boolean network can be used to study the qualitative behavior of the original system. We justify the reduction using geometric singular perturbation theory and compact convergence, and illustrate the results in network models of a toggle switch and an oscillator.

Boolean modelling reveals new regulatory connections between transcription factors orchestrating the development of the ventral spinal cord

We have assembled a network of cell-fate determining transcription factors that play a key role in the specification of the ventral neuronal subtypes of the spinal cord on the basis of published transcriptional interactions. Asynchronous Boolean modelling of the network was used to compare simulation results with reported experimental observations. Such comparison highlighted the need to include additional regulatory connections in order to obtain the fixed point attractors of the model associated with the five known progenitor cell types located in the ventral spinal cord. The revised gene regulatory network reproduced previously observed cell state switches between progenitor cells observed in knock-out animal models or in experiments where the transcription factors were overexpressed. Furthermore the network predicted the inhibition of Irx3 by Nkx2.2 and this prediction was tested experimentally. Our results provide evidence for the existence of an as yet undescribed inhibitory connection which could potentially have significance beyond the ventral spinal cord. The work presented in this paper demonstrates the strength of Boolean modelling for identifying gene regulatory networks.

Regenerative orthopaedics: in vitro, in vivo...in silico

In silico, defined in analogy to in vitro and in vivo as those studies that are performed on a computer, is an essential step in problem-solving and product development in classical engineering fields. The use of in silico models is now slowly easing its way into medicine. In silico models are already used in orthopaedics for the planning of complicated surgeries, personalised implant design and the analysis of gait measurements. However, these in silico models often lack the simulation of the response of the biological system over time. In silico models focusing on the response of the biological systems are in full development. This review starts with an introduction into in silico models of orthopaedic processes. Special attention is paid to the classification of models according to their spatiotemporal scale (gene/protein to population) and the information they were built on (data vs hypotheses). Subsequently, the review focuses on the in silico models used in regenerative orthopaedics research. Contributions of in silico models to an enhanced understanding and optimisation of four key elements-cells, carriers, culture and clinics-are illustrated. Finally, a number of challenges are identified, related to the computational aspects but also to the integration of in silico tools into clinical practice.

Stabilization of perturbed Boolean network attractors through compensatory interactions

Background

Understanding and ameliorating the effects of network damage are of significant interest, due in part to the variety of applications in which network damage is relevant. For example, the effects of genetic mutations can cascade through within-cell signaling and regulatory networks and alter the behavior of cells, possibly leading to a wide variety of diseases. The typical approach to mitigating network perturbations is to consider the compensatory activation or deactivation of system components. Here, we propose a complementary approach wherein interactions are instead modified to alter key regulatory functions and prevent the network damage from triggering a deregulatory cascade.

Results

We implement this approach in a Boolean dynamic framework, which has been shown to effectively model the behavior of biological regulatory and signaling networks. We show that the method can stabilize any single state (e.g., fixed point attractors or time-averaged representations of multi-state attractors) to be an attractor of the repaired network. We show that the approach is minimalistic in that few modifications are required to provide stability to a chosen attractor and specific in that interventions do not have undesired effects on the attractor. We apply the approach to random Boolean networks, and further show that the method can in some cases successfully repair synchronous limit cycles. We also apply the methodology to case studies from drought-induced signaling in plants and T-LGL leukemia and find that it is successful in both stabilizing desired behavior and in eliminating undesired outcomes. Code is made freely available through the software package BooleanNet.

Conclusions

The methodology introduced in this report offers a complementary way to manipulating node expression levels. A comprehensive approach to evaluating network manipulation should take an "all of the above" perspective; we anticipate that theoretical studies of interaction modification, coupled with empirical advances, will ultimately provide researchers with greater flexibility in influencing system behavior.

Causal structure of oscillations in gene regulatory networks: Boolean analysis of ordinary differential equation attractors

A common approach to the modeling of gene regulatory networks is to represent activating or repressing interactions using ordinary differential equations for target gene concentrations that include Hill function dependences on regulator gene concentrations. An alternative formulation represents the same interactions using Boolean logic with time delays associated with each network link. We consider the attractors that emerge from the two types of models in the case of a simple but nontrivial network: a figure-8 network with one positive and one negative feedback loop. We show that the different modeling approaches give rise to the same qualitative set of attractors with the exception of a possible fixed point in the ordinary differential equation model in which concentrations sit at intermediate values. The properties of the attractors are most easily understood from the Boolean perspective, suggesting that time-delay Boolean modeling is a useful tool for understanding the logic of regulatory networks.