Fluctuation-induced corrections to the Hill function: implications for gene regulatory network dynamics

Stem cell regeneration is crucial for development and maintaining tissue homeostasis in self-renewing tissues. The dynamics of gene regulatory networks (GRNs) play a vital role in regulating stem cell renewal and differentiation. However, integrating the quantitative dynamics of GRNs at the single-cell level with population-level stem cell regeneration poses significant challenges. This study presents a computational framework that links GRN dynamics to stem cell regeneration through an inheritance function. This function captures epigenetic state transitions during cell division in heterogeneous stem cell populations. Our model derives this function using a hybrid approach that integrates cross-cell-cycle GRN dynamics, effectively connecting cellular-level GRN structures with population-level regeneration processes. By incorporating GRN structure directly into stem cell regeneration dynamics, this framework simulates cross-cell-cycle gene regulation using individual-cell-based models. The scheme is adaptable to various GRNs, providing insights into the relationship between gene regulatory dynamics and stem cell regeneration. Additionally, we propose a future perspective that integrates single-cell ribonucleic acid sequencing data, GRN analysis, and cell regeneration dynamics using AI-driven tools to enhance the precision of regenerative studies.

Gene Networks with Transcriptional Bursting Recapitulate Rare Transient Coordinated High Expression States in Cancer.

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/.

In this chapter, passive molecular communication (PMC) is introduced for the cases in which the receiver nanomachine (RN) has receptors on its surface and receives molecules by means of the ligand–receptor binding phenomenon. The deterministic and probabilistic models of the ligand–receptor binding are first introduced. Then, PMC in gene regulatory networks is discussed and a unified model incorporating the diffusion and degradation of molecules and ligand–receptor binding is introduced. Accuracies of the concentration and gradient sensing with ligand–receptor binding are also investigated. Finally, the communication theories and techniques are given for PMC with ligand–receptor binding.

How can cells shape and utilize dynamic gene regulation to enable complex cellular behaviors? I study this question in natural and synthetic contexts. The first project studies how a natural genetic network can imbue cells with a sense of ‘time’. It has long been known that environmental signals induce diverse cellular differentiation programs. In certain systems, cells defer differentiation for extended time periods after the signal appears, proliferating through multiple rounds of cell division before committing to a new fate. How can cells set a deferral time much longer than the cell cycle? Here we study Bacillus subtilis cells that respond to sudden nutrient limitation with multiple rounds of growth and division before differentiating into spores. A well characterized genetic circuit controls the concentration and phosphorylation of the master regulator Spo0A, which rises to a critical concentration to initiate sporulation. However, it remains unclear how this circuit enables cells to defer sporulation for multiple cell cycles. Using quantitative time-lapse fluorescence microscopy of Spo0A dynamics in individual cells, we observed pulses of Spo0A phosphorylation at a characteristic cell cycle phase. Pulse amplitudes grew systematically and cell-autonomously over multiple cell cycles leading up to sporulation. This pulse growth required a key positive feedback loop involving the sporulation kinases, without which the deferral of sporulation became ultrasensitive to kinase expression. Thus, deferral is controlled by a pulsed positive feedback loop in which kinase expression is activated by pulses of Spo0A phosphorylation. This pulsed positive feedback architecture provides a more robust mechanism for setting deferral times than constitutive kinase expression. Finally, using mathematical modeling, we show how pulsing and time delays together enable ‘polyphasic’ positive feedback, in which different parts of a feedback loop are active at different times. Polyphasic feedback can enable more accurate tuning of long deferral times. Together, these results suggest that Bacillus subtilis uses a pulsed positive feedback loop to implement a timer that operates over time scales much longer than a cell cycle. The second project proposes a method to rapidly generate and test complex genetic network dynamics in living cells. Existing microorganisms have evolved genetic circuitry to meet diverse challenges and maximize their survival and fitness. These challenges arise from external environmental pressures, or internal evolved constraints. Furthermore, these challenges may be either static or dynamic in nature. While existing circuits have likely evolved to be ‘good enough’ to respond to historical challenges, it remains unclear if they can be improved upon, and whether they respond well to novel situations. Synthetic biology seeks to engineer organisms with complex novel phenotypes, both to harness these novel organisms for a function and to understand their underlying biology. Dynamic gene expression strategies may be necessary to successfully generate these phenotypes. Unfortunately, generating novel dynamic gene expression patterns with conventional genetic engineering remains a challenge. Here I propose and describe progress towards a computerized feedback control setup to enable the programming and rapid testing of dynamic gene regulatory patterns in living cells. Small sets of genes will be regulated optogenetically based on programmed control laws, and past and present cellular state. This setup will enable us to explore the functions and limits of engineered dynamic gene regulation, while hopefully, in the process, providing lessons about the underlying biology.

A stochastic genetic toggle switch model that consists of two identical, mutually repressive genes is built using the Gillespie algorithm with time delays as an example of a simple stochastic gene regulatory network. The stochastic kinetics of this model is investigated, and it is found that the delays for the protein productions can highly weaken the global fluctuations for the expressions of the two genes, making the two mutually repressive genes coexist for a long time. Starting from this model, we propose a practical modeling strategy for more complex gene regulatory networks. Unlike previous applications of the Gillespie algorithm to simulate specific genetic networks dynamics, this modeling strategy is proposed for an ensemble approach to study the dynamical properties of these networks. The model allows any combination of gene expression products, forming complex multimers, and each one of the multimers is assigned to a randomly chosen gene promoter site as an activator or inhibitor. In addition, each gene, although it has only one promoter site, can have multiple regulatory sites and distinct rates of translation and transcription. Also, different genes have different time delays for transcription and translation and all reaction constant rates are initially randomly chosen from a range of values. Therefore, the general strategy here proposed may be used to simulate real genetic networks.

Within systems biology there is an increasing interest in the stochastic behavior of genetic and biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous time Markov chain (CTMC). In this paper we consider the stochastic properties of a toggle switch, involving a protein compound (E2Fs and Myc) and a miRNA cluster (miR-17-92), known to control the eukaryotic cell cycle and possibly involved in oncogenesis, recently proposed in the literature within a deterministic framework. Due to the inherent stochasticity of biochemical processes and the small number of molecules involved, the stochastic approach should be more correct in describing the real system: we study the agreement between the two approaches by exploring the system parameter space. We address the problem by proposing a simplified version of the model that allows analytical treatment, and by performing numerical simulations for the full model. We observed optimal agreement between the stochastic and the deterministic description of the circuit in a large range of parameters, but some substantial differences arise in at least two cases: (1) when the deterministic system is in the proximity of a transition from a monostable to a bistable configuration, and (2) when bistability (in the deterministic system) is "masked" in the stochastic system by the distribution tails. The approach provides interesting estimates of the optimal number of molecules involved in the toggle switch. Our discussion of the points of strengths, potentiality and weakness of the chemical master equation in systems biology and the differences with respect to deterministic modeling are leveraged in order to provide useful advice for both the bioinformatician and the theoretical scientist.

Hill functions follow from the equilibrium state of the reaction in which n ligands simultaneously bind a single receptor. This result if often employed to interpret the Hill coefficient as the number of ligand binding sites in all kinds of reaction schemes. Here, we study the equilibrium states of the reactions in which n ligand bind a receptor sequentially, both non-cooperatively and in a cooperative fashion. The main outcomes of such analysis are that: n is not a good estimate, but only an upper bound, for the Hill coefficient; while the Hill coefficient depends quite strongly on the cooperativity level among ligands. We finally use these results to discuss the feasibility and constrains of using Hill functions to model the regulatory functions in mathematical models of gene regulatory networks.

How gene regulatory networks (GRNs) encode gene expression dynamics and how GRNs evolve are not well understood, although these problems have been studied extensively. We created a digital twin that accurately reproduces expression dynamics of 13 genes that initiate expression in 32-cell ascidian embryos. We first showed that gene expression patterns can be manipulated according to predictions by this digital model. Next, to simulate GRN rewiring, we changed regulatory functions that represented their regulatory mechanisms in the digital twin, and found that in 55 of 100 cases, removal of a single regulator from a conjunctive clause of Boolean functions did not theoretically alter qualitative expression patterns of these genes. In other words, we found that more than half the regulators gave theoretically redundant temporal or spatial information to target genes. We experimentally substantiated that the expression pattern of Nodal was maintained without one of these factors, Zfpm, by changing the upstream regulatory sequence of Nodal. Such robust buffers of regulatory mechanisms may provide a basis of enabling developmental system drift, or rewiring of GRNs without changing expression patterns of downstream genes, during evolution.

Gene expression and differentiation were shown to be stochastic processes. However, cells in a tissue can coordinate their behavior, including gene expression and differentiation pathways choice. A tissue of coupled cells is modeled as a two-dimensional regular square lattice of identical cells, each a three-dimensional compartment with a gene regulatory network (GRN) and a toggle switch (TS). The dynamics is driven by a delayed stochastic simulation algorithm, nearest neighbor cells are coupled by normally distributed time delayed reactions allowing interchange of proteins, and gene expression is a multiple time delayed reaction. It is defined the coupling strength (C), to characterize the lattice structure as a function of the rate constants of the reactions coupling nearest neighbor cells. Conditions are investigated for the emergence of synchronization and stable differentiation of cells within a tissue. From the time series of the cells GRNs, the tissue dynamical stability (S) is computed from the average toggling period of each GRN. The synchronization of cells' proteins expression levels is measured by their time series entropy (H). It is shown that the tissue goes through various dynamical regimes as C is increased, by measuring H and S . For null C, the cells GRNs toggle asynchronously. For weak C, cells synchronize by regions of space. Increasing C, the tissue becomes homogeneously synchronous. As C is further increased, S goes through a phase transition, from synchronized toggling to stable, where all cells produce one and the same protein. Finally, increasing C even further, a new stable state emerges where both genes of all cells are expressed and bistability is lost. This state, resembling an infinitely long transient, is an emergent behavior not observable in a single TS. The results provide an explanation of how cells with bistable GRNs, inherently stochastic, can synchronize or uniformly differentiate into stable states, by interacting with nearest neighbors.

A new phenotypic variant may appear first in organisms through plasticity, that is, as a response to an environmental signal or other nongenetic perturbation. If such trait is beneficial, selection may increase the frequency of alleles that enable and facilitate its development. Thus, genes may take control of such traits, decreasing dependence on nongenetic disturbances, in a process called genetic assimilation. Despite an increasing amount of empirical studies supporting genetic assimilation, its significance is still controversial. Whether genetic assimilation is widespread depends, to a great extent, on how easily mutation and recombination reduce the trait's dependence on nongenetic perturbations. Previous research suggests that this is the case for mutations. Here we use simulations of gene regulatory network dynamics to address this issue with respect to recombination. We find that recombinant offspring of parents that produce a new phenotype through plasticity are more likely to produce the same phenotype without requiring any perturbation. They are also prone to preserve the ability to produce that phenotype after genetic and nongenetic perturbations. Our work also suggests that ancestral plasticity can play an important role for setting the course that evolution takes. In sum, our results indicate that the manner in which phenotypic variation maps unto genetic variation facilitates evolution through genetic assimilation in gene regulatory networks. Thus, we contend that the importance of this evolutionary mechanism should not be easily neglected.

We investigate the effects of intrinsic noise on Turing pattern formation near the onset of bifurcation from the homogeneous state to Turing pattern in the reaction-diffusion Brusselator. By performing stochastic simulations of the master equation and using Gillespie's algorithm, we check the spatiotemporal behaviour influenced by internal noises. We demonstrate that the patterns of occurrence frequency for the reaction and diffusion processes are also spatially ordered and temporally stable. Turing patterns are found to be robust against intrinsic fluctuations. Stochastic simulations also reveal that under the influence of intrinsic noises, the onset of Turing instability is advanced in comparison to that predicted deterministically.

Modeling stochasticity in gene regulatory networks is an important and complex problem in molecular systems biology. To elucidate intrinsic noise, several modeling strategies such as the Gillespie algorithm have been used successfully. This article contributes an approach as an alternative to these classical settings. Within the discrete paradigm, where genes, proteins, and other molecular components of gene regulatory networks are modeled as discrete variables and are assigned as logical rules describing their regulation through interactions with other components. Stochasticity is modeled at the biological function level under the assumption that even if the expression levels of the input nodes of an update rule guarantee activation or degradation there is a probability that the process will not occur due to stochastic effects. This approach allows a finer analysis of discrete models and provides a natural setup for cell population simulations to study cell-to-cell variability. We applied our methods to two of the most studied regulatory networks, the outcome of lambda phage infection of bacteria and the p53-mdm2 complex.

The patterning of many developing tissues is orchestrated by gradients of morphogens through a variety of elaborate regulatory interactions. Such interactions are thought to make gradients robust, that is, resistant to changes induced by genetic or environmental perturbations; but just how this might be done is a major unanswered question. Recently extensive numerical simulations suggest that robustness of signaling gradients cannot be attained by negative feedback (of the Hill's function type) on signaling receptors but can be achieved through binding with nonsignaling receptors (or nonreceptors for short) such as heparan sulfate proteoglycans with the resulting complexes degrading after endocytosis. These were followed by a number of analytical and numerical studies in support of the aforementioned observations. However, evidence of feedback regulating signaling gradients has been reported in literature. The present paper undertakes a different approach to the role of feedback in robust signaling gradients. The overall goal of the project is to investigate the effectiveness of feedback mechanisms on ligand synthesis, receptor synthesis, nonreceptor synthesis, and other regulatory processes in the morphogen gradient system. As a first step, we embark herein a proof-of-concept examination of a new spatially uniform feedback process that is distinctly different from the conventional spatially nonuniform Hill function approach.

Cells in an organism interact with each other and with the environment through a complex set of signals, which triggers responses and activates cellular regulation mechanisms. Models obtained by computational mathematics for cellular signaling dynamics are used to understand factors and causes of deregulation of internal biological processes, which is a relevant knowledge in a disease such as cancer. Gene regulatory networks describe gene interactions and how these relationships control cellular processes such as growth and cell division, which relate this disease to the regulatory network. Despite its simplicity, Boolean networks may accurately model some biological phenomena, such as gene regulatory network dynamics. Indeed, several reports in the literature show that they are accurate enough to build models of regulatory networks of cell lines related to breast cancer. In this chapter, we present a methodology for building cellular regulatory networks based on the Boolean paradigm, which uses entropy as a criterion for selecting genes that are included in the network. The main objective is to understand dynamical behaviors related to situations that cause breast cancer and tumor lineages and to suggest experimentations to verify the outcome of interventions in networks, in order to support the identification of new therapeutic targets.