Accuracy of parameter estimation for auto-regulatory transcriptional feedback loops from noisy data

Models of Gene Regulatory Networks (GRNs) capture the dynamics of the regulatory processes that occur within the cell as a means to understand the variability observed in gene expression between different conditions. Arguably the simplest mathematical construct used for modeling is the Boolean network, which dictates a set of logical rules for transition between states described as Boolean vectors. Due to the complexity of gene regulation and the limitations of experimental technologies, in most cases knowledge about regulatory interactions and Boolean states is partial. In addition, the logical rules themselves are not known a-priori. Our goal in this work is to create an algorithm that finds the network that fits the data optimally, and identify the network states that correspond to the noise-free data. We present a novel methodology for integrating experimental data and performing a search for the optimal consistent structure via optimization of a linear objective function under a set of linear constraints. In addition, we extend our methodology into a heuristic that alleviates the computational complexity of the problem for datasets that are generated by single-cell RNA-Sequencing(scRNA-Seq). We demonstrate the effectiveness of these tools using a public scRNA-Seq dataset and the GRN that is associated with it. Our methodology wm enable researchers to obtain a better understanding of the dynamics of gene regulatory networks and their biological role.

BackgroundThe emergent dynamics of complex gene regulatory networks govern various cellular processes. However, understanding these dynamics is challenging due to the difficulty of parameterizing the computational models for these networks, especially as the network size increases. Here, we introduce a simulation library, Gene Regulatory Interaction Network Simulator (GRiNS), to address these challenges.ResultsGRiNS integrates popular parameter-agnostic simulation frameworks, RACIPE and Boolean Ising formalism, into a single Python library capable of leveraging GPU acceleration for efficient and scalable simulations. GRiNS extends the ordinary differential equations (ODE) based RACIPE framework with a more modular design, allowing users to choose parameters, initial conditions, and time-series outputs for greater customisability and accuracy in simulations. For large networks, where ODE-based simulation formalisms do not scale well, GRiNS implements Boolean Ising formalism, providing a simplified, coarse-grained alternative, significantly reducing the computational cost while capturing key dynamical behaviours of large regulatory networks.ConclusionGRiNS enables parameter-agnostic modeling of gene regulatory networks to study their dynamic and steady-state behaviors in a scalable and efficient manner. The documentation and installation instructions for GRiNS can be found at https://moltenecdysone09.github.io/GRiNS/.

Inferring transcriptional gene regulatory networks from transcriptomic datasets is a key challenge of systems biology, with potential impacts ranging from medicine to agronomy. There are several techniques used presently to experimentally assay transcription factors to target relationships, defining important information about real gene regulatory networks connections. These techniques include classical ChIP-seq, yeast one-hybrid, or more recently, DAP-seq or target technologies. These techniques are usually used to validate algorithm predictions. Here, we developed a reverse engineering approach based on mathematical and computer simulation to evaluate the impact that this prior knowledge on gene regulatory networks may have on training machine learning algorithms. First, we developed a gene regulatory networks-simulating engine called FRANK (Fast Randomizing Algorithm for Network Knowledge) that is able to simulate large gene regulatory networks (containing 104 genes) with characteristics of gene regulatory networks observed in vivo. FRANK also generates stable or oscillatory gene expression directly produced by the simulated gene regulatory networks. The development of FRANK leads to important general conclusions concerning the design of large and stable gene regulatory networks harboring scale free properties (built ex nihilo). In combination with supervised (accepting prior knowledge) support vector machine algorithm we (i) address biologically oriented questions concerning our capacity to accurately reconstruct gene regulatory networks and in particular we demonstrate that prior-knowledge structure is crucial for accurate learning, and (ii) draw conclusions to inform experimental design to performed learning able to solve gene regulatory networks in the future. By demonstrating that our predictions concerning the influence of the prior-knowledge structure on support vector machine learning capacity holds true on real data (Escherichia coli K14 network reconstruction using network and transcriptomic data), we show that the formalism used to build FRANK can to some extent be a reasonable model for gene regulatory networks in real cells.

Genes encode proteins, some of which in turn regulate other genes. Such interactions make up a gene regulatory network. The understanding and unraveling of gene regulatory networks have been proven very useful in disease diagnosis and genomic drug design. Due to the complexity of gene regulatory networks, the completely understanding of their dynamics is difficult to achieve only through biological experiments without any computational aids. As a consequence, computational models for gene regulatory networks are indispensable. Recently a wide variety of different computational models have been proposed for inferring gene regulatory networks. This paper surveys some of computational models for inferring large gene regulatory networks, in particular, Boolean network model, differential/ difference equation models, and state-space models. Some advantages and disadvantages of these models are commented on. Some criteria for validating the inferred gene regulatory networks are also discussed from the bioinformatics perspective. Finally, several directions of the future work for modeling gene regulatory networks are proposed.

Perturbation avalanches and criticality in gene regulatory networks

Background and Objectives: Gene regulatory networks model the interactions among the genes and provide a decision rule describing activation and repression of each gene via various proteins. In order to be able to capture the complex gene interactions efficiently, it is imperative to develop algorithms that model the nonlinear interactions among the genes. This work considers the problem of inferring gene regulatory networks using time series data. A nonlinear model is assumed for the gene expression profiles, whereas the microarray data follows a linear Gaussian model. Methods: A particle filter based approach is proposed to estimate the gene expression profiles and the parameters are estimated online using the Kalman filter. In order to capture the inherent sparsity of gene networks, a least squares shrinkage selection operator (LASSO) based regression and model selection algorithm is proposed. Results: The performance of the aforementioned algorithm is rigorously evaluated for synthetic as well as real biological data sets arising from Drosophila melanogaster time series gene expression profiles. The results are contrasted with those reported in the literature. The performance of the proposed algorithm is compared with the extended Kalman filter (EKF) algorithm using mean square error (MSE) as the fidelity criterion. The proposed algorithm is observed to outperform the EKF in the scenarios considered. Conclusions: This work considered the problem of modeling and learning of gene regulatory networks using a nonlinear dynamical model. This represents a quite general modeling set-up. The gene network is modeled using a state space approach, and particle filtering is used for state estimation. The parameters regulating the interaction among genes are supplied by an online Kalman filter. Since the parameter vector is sparse, LASSO identifies the subset of these parameters pertaining to the relevant system coefficients. Extensive performance evaluations demonstrate that the proposed particle filter based approach outperforms EKF in terms of MSE.

BackgroundA key problem in systems biology is estimating dynamical models of gene regulatory networks. Traditionally, this has been done using regression or other ad-hoc methods when the model is linear. More detailed, realistic modeling studies usually employ nonlinear dynamical models, which lead to computationally difficult parameter estimation problems. Functional data analysis methods, however, offer a means to simplify fitting by transforming the problem from one of matching modeled and observed dynamics to one of matching modeled and observed time derivatives–a regression problem, albeit a nonlinear one.ResultsWe formulate a functional data analysis approach for estimating the parameters of nonlinear dynamical models and evaluate this approach on data from two real systems, the gap gene system of Drosophila melanogaster and the synthetic IRMA network, which was created expressly as a test case for genetic network inference. We also evaluate the approach on simulated data sets generated by the GeneNetWeaver program, the basis for the annual DREAM reverse engineering challenge. We assess the accuracy with which the correct regulatory relationships within the networks are extracted, and consider alternative methods of regularization for the purpose of overfitting avoidance. We also show that the computational efficiency of the functional data analysis approach, and the decomposability of the resulting regression problem, allow us to explicitly enumerate and evaluate all possible regulator combinations for every gene. This gives deeper insight into the the relevance of different regulators or regulator combinations, and lets one check for alternative regulatory explanations.ConclusionsFunctional data analysis is a powerful approach for estimating detailed nonlinear models of gene expression dynamics, allowing efficient and accurate estimation of regulatory architecture.

Development and comparison in uncertainty assessment based Bayesian modularization method in hydrological modeling

BackgroundGenetic canalization reflects the capacity of an organism’s phenotype to remain unchanged in spite of mutations. As selection on genetic canalization is weak and indirect, whether or not genetic canalization can reasonably evolve in complex genetic architectures is still an open question. In this paper, we use a quantitative model of gene regulatory network to describe the conditions in which substantial canalization is expected to emerge in a stable environment.ResultsThrough an individual-based simulation framework, we confirmed that most parameters associated with the network topology (complexity and size of the network) have less influence than mutational parameters (rate and size of mutations) on the evolution of genetic canalization. We also established that selecting for extreme phenotypic optima (nil or full gene expression) leads to much higher canalization levels than selecting for intermediate expression levels. Overall, constrained networks evolve less canalization than networks in which some genes could evolve freely (i.e. without direct stabilizing selection pressure on gene expression).ConclusionsTaken together, these results lead us to propose a two-fold mechanism involved in the evolution of genetic canalization in gene regulatory networks: the shrinkage of mutational target (useless genes are virtually removed from the network) and redundancy in gene regulation (so that some regulatory factors can be lost without affecting gene expression).Electronic supplementary materialThe online version of this article (doi:10.1186/s12862-016-0801-2) contains supplementary material, which is available to authorized users.

Interaction among the gene products contributes to the regulation of gene expression.

The aim of this paper is two-fold. First, we propose a new computational method to investigate the particularities of evolution. Second, we apply this method to a model of gene regulatory networks (GRNs) and explore the evolution of mutational robustness and bistability. Living systems have developed their functions through evolutionary processes. To understand the particularities of this process theoretically, evolutionary simulation (ES) alone is insufficient because the outcomes of ES depend on evolutionary pathways. We need a reference system for comparison. An appropriate reference system for this purpose is an ensemble of the randomly sampled genotypes. However, generating high-fitness genotypes by simple random sampling is difficult because such genotypes are rare. In this study, we used the multicanonical Monte Carlo method developed in statistical physics to construct a reference ensemble of GRNs and compared it with the outcomes of ES. We obtained the following results. First, mutational robustness was significantly higher in ES than in the reference ensemble at the same fitness level. Second, the emergence of a new phenotype, bistability, was delayed in evolution. Third, the bistable group of GRNs contains many mutationally fragile GRNs compared with those in the non-bistable group. This suggests that the delayed emergence of bistability is a consequence of the mutation-selection mechanism.

Gene regulatory networks (GRN) have been studied by computational scientists and biologists over 20 years to gain a fine map of gene functions. With large-scale genomic and epigenetic data generated under diverse cells, tissues, and diseases, the integrative analysis of multi-omics data plays a key role in identifying casual genes in human disease development. Bayesian inference (or integration) has been successfully applied to inferring GRNs. Learning a posterior distribution than making a single-value prediction of model parameter makes Bayesian inference a more robust approach to identify GRN from noisy biomedical observations. Moreover, given multi-omics data as input and a large number of model parameters to estimate, the automatic preference of Bayesian inference for simple models that sufficiently explain data without unnecessary complexity ensures fast convergence to reliable results. In this chapter, we introduced GRN modeling using hierarchical Bayesian network and then used Gibbs sampling to identify network variables. We applied this model to breast cancer data and identified genes relevant to breast cancer recurrence. In the end, we discussed the potential of Bayesian inference as well as Bayesian deep learning for large-scale and complex GRN inference.

Quantification of protein levels in absolute units will allow testing quantitative models of gene regulatory networks. Better quantitative models have application in fundamental plant biology but also in synthetic biology. Models in absolute units can, for example, help guide the design of promoter modifications using CRISPR genome editing technologies, such as base editing. Absolute protein quantification can also be applied to assessing the quality and biological meaning of affinity-constant measurements of transcription factors for DNA- or Protein-Protein interactions. Some absolute quantification methods using targeted proteomics have been described for mammalian cells and plant cells. However, gathering data using such methods is expensive, making it prohibitive for experimental set ups in which dynamic data is required (e.g.: inferring model parameters from time series in chronobiology applications). To overcome these shortcomings, we developed a Nano Luciferase based toolkit and methodology for performing absolute quantifications using recombinant NanoLUC and tagged lines. The steps presented in this chapter are extensively exemplified in the Doctoral thesis “A Mathematical Model in Absolute Units For the Arabidopsis Circadian Clock” (Urquiza Garcia 2018) and preprint (Urquiza Garcia et al. 2024; to be published in Molecular Systems Biology, 2025). Here we described steps to follow and tips for using NanoLUC for absolute quantification of a protein of interest. Introduction In systems and synthetic biology, quantitative data is important for model calibration and testing. In plant biology the largest source of semi-quantitative transcript data generated by RT-qPCR. Which has provided valuable data for quantitative model building and testing. This can be exemplified by many mathematical models of gene regulatory networks (circadian clock, flowering). In contrast, the generation of quantitative protein data has lagged significantly behind. However, having quantitative abundance data of proteins will aid in model refinement by providing the material for challenging model dynamics and parameters for plant systems (Andrew J. Millar, 2016; Andrew J. Millar et al., 2019). The gathering of reliable, dynamic and quantitative protein data remains very challenging. Quantitative Western blots can be used to extract information in absolute units. However, quantitative Western blots can only be performed if the reagents are available (native antibodies, epitope tagged stable lines) and, although possible, there are practical limits with regards to the dynamic resolution that can be achieved with this technique. This leaves vast room for improvement in terms of sensitivity, reproducibility and ease in experimental design. Mass-spectrometry (MS) has been applied for absolute quantification of low abundance protein-like transcription factors in mammalian systems (Simicevic et al., 2013). Further work providing better control on the generation of multiplexed absolute quantification has been reported for animal clock proteins using MS-based Quantification by isotope-labeled cell-free products (MS-QBiC) (Narumi et al., 2016), in which multiplexed production of internal standards for quantification is achieved through Cell-free synthesis using the PURE system (Shimizu et al., 2001). The advantage of this approach is that it does not require the generation of time consuming stable transgenic lines. However, this approach still requires access to expensive equipment and highly trained personnel. So extended time series over different experimental conditions that would be informative for model building and parameter inference can become prohibitively expensive. Therefore a methodology that provides quantitative data of enough quality for easy to access equipment and minimal training would be of high value. Reporter based approaches are important tools in the elucidation of molecular mechanisms. In chronobiology and photobiology the use of Firefly luciferase (FLUC) has served as an invaluable tool. Data generated with this automatable reporter assay boosted the elucidation of key molecular mechanisms governing the emergence of circadian and photobiological responses in plants (A. J. Millar et al., 1995; Ulm et al., 2004). The instability of firefly provides a suitable reporter for tracking changes in transcriptional activity in a relatively straightforward and inexpensive way. However, the instability of FLUC has its drawbacks when studying molecular events that present half-lives significantly larger than the FLUC half-life in vivo. Therefore the use of a more stable reporter was desirable, for which the half-life is mainly determined by the process it is reporting. In particular, we noticed that the NanoLUC technology offers several advantages for tracking protein dynamics in plants (Hall et al., 2012; Urquiza-García & Millar, 2019). Here we describe in detail the process for performing absolute protein quantification, which is publicly available in the Doctoral thesis “A Mathematical Model in Absolute Units For the Arabidopsis Circadian Clock” (Urquiza García, 2018).

Boolean Kalman filter and smoother under model uncertainty

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.