Modeling gene expression in time and space

Enlightening the blind spot of the Michaelis-Menten rate law: The role of relaxation dynamics in molecular complex formation

The century-long Michaelis-Menten rate law and its modifications in the modeling of biochemical rate processes stand on the assumption that the concentration of the complex of interacting molecules, at each moment, rapidly approaches an equilibrium (quasi-steady state) compared to the pace of molecular concentration changes. Yet, in the case of actively time-varying molecular concentrations with transient or oscillatory dynamics, the deviation of the complex profile from the quasi-steady state becomes relevant. A recent theoretical approach, known as the effective time-delay scheme (ETS), suggests that the delay from the relaxation time of molecular complex formation contributes to the substantial breakdown of the quasi-steady state assumption. Here, we systematically expand this ETS and inquire into the comprehensive roles of relaxation dynamics in complex formation. Through the modeling of rhythmic protein-protein and protein-DNA interactions and the mammalian circadian clock, our analysis reveals the effect of the relaxation dynamics beyond the time delay, which extends to the dampening of changes in the complex concentration with a reduction in the oscillation amplitude compared to the quasi-steady state. Interestingly, the combined effect of the time delay and amplitude reduction shapes both qualitative and quantitative oscillatory patterns such as the emergence and variability of the mammalian circadian rhythms. These findings highlight the downside of the routine assumption of quasi-steady states and enhance the mechanistic understanding of rich time-varying biomolecular processes.

Beyond homogeneity: Assessing the validity of the Michaelis-Menten rate law in spatially heterogeneous environments

The Michaelis-Menten (MM) rate law has been a fundamental tool in describing enzyme-catalyzed reactions for over a century. When substrates and enzymes are homogeneously distributed, the validity of the MM rate law can be easily assessed based on relative concentrations: the substrate is in large excess over the enzyme-substrate complex. However, the applicability of this conventional criterion remains unclear when species exhibit spatial heterogeneity, a prevailing scenario in biological systems. Here, we explore the MM rate law's applicability under spatial heterogeneity by using partial differential equations. In this study, molecules diffuse very slowly, allowing them to locally reach quasi-steady states. We find that the conventional criterion for the validity of the MM rate law cannot be readily extended to heterogeneous environments solely through spatial averages of molecular concentrations. That is, even when the conventional criterion for the spatial averages is satisfied, the MM rate law fails to capture the enzyme catalytic rate under spatial heterogeneity. In contrast, a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA), is accurate. Specifically, the tQSSA-based modified form, but not the original MM rate law, accurately predicts the drug clearance via cytochrome P450 enzymes and the ultrasensitive phosphorylation in heterogeneous environments. Our findings shed light on how to simplify spatiotemporal models for enzyme-catalyzed reactions in the right context, ensuring accurate conclusions and avoiding misinterpretations in in silico simulations.

Generalized Michaelis-Menten rate law with time-varying molecular concentrations

The Michaelis-Menten (MM) rate law has been the dominant paradigm of modeling biochemical rate processes for over a century with applications in biochemistry, biophysics, cell biology, systems biology, and chemical engineering. The MM rate law and its remedied form stand on the assumption that the concentration of the complex of interacting molecules, at each moment, approaches an equilibrium (quasi-steady state) much faster than the molecular concentrations change. Yet, this assumption is not always justified. Here, we relax this quasi-steady state requirement and propose the generalized MM rate law for the interactions of molecules with active concentration changes over time. Our approach for time-varying molecular concentrations, termed the effective time-delay scheme (ETS), is based on rigorously estimated time-delay effects in molecular complex formation. With particularly marked improvements in protein-protein and protein-DNA interaction modeling, the ETS provides an analytical framework to interpret and predict rich transient or rhythmic dynamics (such as autogenously-regulated cellular adaptation and circadian protein turnover), which goes beyond the quasi-steady state assumption.

Single-cell Bayesian deconvolution

Individual cells exhibit substantial heterogeneity in protein abundance and activity, which is frequently reflected in broad distributions of fluorescently labeled reporters. Since all cellular components are intrinsically fluorescent to some extent, the observed distributions contain background noise that masks the natural heterogeneity of cellular populations. This limits our ability to characterize cell-fate decision processes that are key for development, immune response, tissue homeostasis, and many other biological functions. It is therefore important to separate the contributions from signal and noise in single-cell measurements. Addressing this issue rigorously requires deconvolving the noise distribution from the signal, but approaches in that direction are still limited. Here, we present a non-parametric Bayesian formalism that performs such a deconvolution efficiently on multidimensional measurements, providing unbiased estimates of the resulting confidence intervals. We use this approach to study the expression of the mesodermal transcription factor Brachyury in mouse embryonic stem cells undergoing differentiation.

Model of neural induction in the ascidian embryo

How cell specification can be controlled in a reproducible manner is a fundamental question in developmental biology. In ascidians, a group of invertebrate chordates, geometry plays a key role in achieving this control. Here, we use mathematical modeling to demonstrate that geometry dictates the neural-epidermal cell fate choice in the 32-cell stage ascidian embryo by a two-step process involving first the modulation of ERK signaling and second, the expression of the neural marker gene, Otx. The model describes signal transduction by the ERK pathway that is stimulated by FGF and attenuated by ephrin, and ERK-mediated control of Otx gene expression, which involves both an activator and a repressor of ETS-family transcription factors. Considering the measured area of cell surface contacts with FGF- or ephrin-expressing cells as inputs, the solutions of the model reproduce the experimental observations about ERK activation and Otx expression in the different cells under normal and perturbed conditions. Sensitivity analyses and computations of Hill coefficients allow us to quantify the robustness of the specification mechanism controlled by cell surface area and to identify the respective role played by each signaling input. Simulations also predict in which conditions the dual control of gene expression by an activator and a repressor that are both under the control of ERK can induce a robust ON/OFF control of neural fate induction.

Data-driven dynamical modelling of a pathogen-infected plant gene regulatory network: A comparative analysis

Recent advances in synthetic biology have enabled the design of genetic feedback control circuits that could be implemented to build resilient plants against pathogen attacks. To facilitate the proper design of these genetic feedback control circuits, an accurate model that is able to capture the vital dynamical behaviour of the pathogen-infected plant is required. In this study, using a data-driven modelling approach, we develop and compare four dynamical models (i.e. linear, Michaelis-Menten with Hill coefficient (Hill Function), standard S-System and extended S-System) of a pathogen-infected plant gene regulatory network (GRN). These models are then assessed across several criteria, i.e. ease of identifying the type of gene regulation, the predictive capability, Akaike Information Criterion (AIC) and the robustness to parameter uncertainty to determine its viability of balancing between biological complexity and accuracy when modelling the pathogen-infected plant GRN. Using our defined ranking score, we obtain the following insights to the modelling of GRN. Our analyses show that despite commonly used and provide biological relevance, the Hill Function model ranks the lowest while the extended S-System model ranks highest in the overall comparison. Interestingly, the performance of the linear model is more consistent throughout the comparison, making it the preferred model for this pathogen-infected plant GRN when considering data-driven modelling approach.

Misuse of the Michaelis-Menten rate law for protein interaction networks and its remedy

For over a century, the Michaelis-Menten (MM) rate law has been used to describe the rates of enzyme-catalyzed reactions and gene expression. Despite the ubiquity of the MM rate law, it accurately captures the dynamics of underlying biochemical reactions only so long as it is applied under the right condition, namely, that the substrate is in large excess over the enzyme-substrate complex. Unfortunately, in circumstances where its validity condition is not satisfied, especially so in protein interaction networks, the MM rate law has frequently been misused. In this review, we illustrate how inappropriate use of the MM rate law distorts the dynamics of the system, provides mistaken estimates of parameter values, and makes false predictions of dynamical features such as ultrasensitivity, bistability, and oscillations. We describe how these problems can be resolved with a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA). Furthermore, we show that the tQSSA can be used for accurate stochastic simulations at a lower computational cost than using the full set of mass-action rate laws. This review describes how to use quasi-steady state approximations in the right context, to prevent drawing erroneous conclusions from in silico simulations.

PhysiBoSS: a multi-scale agent-based modelling framework integrating physical dimension and cell signalling

MOTIVATION

Due to the complexity and heterogeneity of multicellular biological systems, mathematical models that take into account cell signalling, cell population behaviour and the extracellular environment are particularly helpful. We present PhysiBoSS, an open source software which combines intracellular signalling using Boolean modelling (MaBoSS) and multicellular behaviour using agent-based modelling (PhysiCell).

RESULTS

PhysiBoSS provides a flexible and computationally efficient framework to explore the effect of environmental and genetic alterations of individual cells at the population level, bridging the critical gap from single-cell genotype to single-cell phenotype and emergent multicellular behaviour. PhysiBoSS thus becomes very useful when studying heterogeneous population response to treatment, mutation effects, different modes of invasion or isomorphic morphogenesis events. To concretely illustrate a potential use of PhysiBoSS, we studied heterogeneous cell fate decisions in response to TNF treatment. We explored the effect of different treatments and the behaviour of several resistant mutants. We highlighted the importance of spatial information on the population dynamics by considering the effect of competition for resources like oxygen.

AVAILABILITY AND IMPLEMENTATION

PhysiBoSS is freely available on GitHub (https://github.com/sysbio-curie/PhysiBoSS), with a Docker image (https://hub.docker.com/r/gletort/physiboss/). It is distributed as open source under the BSD 3-clause license.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Differential transcriptional regulation by alternatively designed mechanisms: A mathematical modeling approach

Cells maintain cellular homeostasis employing different regulatory mechanisms to respond external stimuli. We study two groups of signal-dependent transcriptional regulatory mechanisms. In the first group, we assume that repressor and activator proteins compete for binding to the same regulatory site on DNA (competitive mechanisms). In the second group, they can bind to different regulatory regions in a noncompetitive fashion (noncompetitive mechanisms). For both competitive and noncompetitive mechanisms, we studied the gene expression dynamics by increasing the repressor or decreasing the activator abundance (inhibition mechanisms), or by decreasing the repressor or increasing the activator abundance (activation mechanisms). We employed delay differential equation models. Our simulation results show that the competitive and noncompetitive inhibition mechanisms exhibit comparable repression effectiveness. However, response time is fastest in the noncompetitive inhibition mechanism due to increased repressor abundance, and slowest in the competitive inhibition mechanism by increased repressor level. The competitive and noncompetitive inhibition mechanisms through decreased activator abundance show comparable and moderate response times, while the competitive and noncompetitive activation mechanisms by increased activator protein level display more effective and faster response. Our study exemplifies the importance of mathematical modeling and computer simulation in the analysis of gene expression dynamics.

Computational prediction of gene regulatory networks in plant growth and development

Plants integrate a wide range of cellular, developmental, and environmental signals to regulate complex patterns of gene expression. Recent advances in genomic technologies enable differential gene expression analysis at a systems level, allowing for improved inference of the network of regulatory interactions between genes. These gene regulatory networks, or GRNs, are used to visualize the causal regulatory relationships between regulators and their downstream target genes. Accordingly, these GRNs can represent spatial, temporal, and/or environmental regulations and can identify functional genes. This review summarizes recent computational approaches applied to different types of gene expression data to infer GRNs in the context of plant growth and development. Three stages of GRN inference are described: first, data collection and analysis based on the dataset type; second, network inference application based on data availability and proposed hypotheses; and third, validation based on in silico, in vivo, and in planta methods. In addition, this review relates data collection strategies to biological questions, organizes inference algorithms based on statistical methods and data types, discusses experimental design considerations, and provides guidelines for GRN inference with an emphasis on the benefits of integrative approaches, especially when a priori information is limited. Finally, this review concludes that computational frameworks integrating large-scale heterogeneous datasets are needed for a more accurate (e.g. fewer false interactions), detailed (e.g. discrimination between direct versus indirect interactions), and comprehensive (e.g. genetic regulation under various conditions and spatial locations) inference of GRNs.

A multiscale model of early cell lineage specification including cell division

Embryonic development is a self-organised process during which cells divide, interact, change fate according to a complex gene regulatory network and organise themselves in a three-dimensional space. Here, we model this complex dynamic phenomenon in the context of the acquisition of epiblast and primitive endoderm identities within the inner cell mass of the preimplantation embryo in the mouse. The multiscale model describes cell division and interactions between cells, as well as biochemical reactions inside each individual cell and in the extracellular matrix. The computational results first confirm that the previously proposed mechanism by which extra-cellular signalling allows cells to select the appropriate fate in a tristable regulatory network is robust when considering a realistic framework involving cell division and three-dimensional interactions. The simulations recapitulate a variety of in vivo observations on wild-type and mutant embryos and suggest that the gene regulatory network confers differential plasticity to the different cell fates. A detailed analysis of the specification process emphasizes that developmental transitions and the salt-and-pepper patterning of epiblast and primitive endoderm cells from a homogenous population of inner cell mass cells arise from the interplay between the internal gene regulatory network and extracellular signalling by Fgf4. Importantly, noise is necessary to create some initial heterogeneity in the specification process. The simulations suggest that initial cell-to-cell differences originating from slight inhomogeneities in extracellular Fgf4 signalling, in possible combination with slightly different concentrations of the key transcription factors between daughter cells, are able to break the original symmetry and are amplified in a flexible and self-regulated manner until the blastocyst stage.

Elements of biological oscillations in time and space

Oscillations in time and space are ubiquitous in nature and play critical roles in dynamic cellular processes. Although the molecular mechanisms underlying the generation of the dynamics are diverse, several distinct regulatory elements have been recognized as being critical in producing and modulating oscillatory dynamics. These include negative and positive feedback, time delay, nonlinearity in regulation, and random fluctuations ('noise'). Here we discuss the specific roles of these five elements in promoting or attenuating oscillatory dynamics, by drawing on insights from quantitative analyses of natural or synthetic biological networks.

Methods for Modeling Brassinosteroid-Mediated Signaling in Plant Development

Mathematical modeling of biological processes is a useful tool to draw conclusions that are contained in the data, but not directly reachable, as well as to make predictions and select the most efficient follow-up experiments. Here we outline a method to model systems of a few proteins that interact transcriptionally and/or posttranscriptionally, by representing the system as Ordinary Differential Equations and to study the model dynamics and stationary states. We exemplify this method by focusing on the regulation by the brassinosteroid (BR) signaling component BRASSINOSTEROID INSENSITIVE1 ETHYL METHYL SULFONATE SUPPRESSOR1 (BES1) of BRAVO, a quiescence-regulating transcription factor expressed in the quiescent cells of Arabidopsis thaliana roots. The method to extract the stationary states and the dynamics is provided as a Mathematica code and requires basic knowledge of the Mathematica software to be executed.

Kinetic aspects of enzyme-mediated repair of DNA single-strand breaks

In cells and bacteria, DNA can be damaged in different ways. The efficient damage repair, mediated by various enzymes, is crucial for their survival. Most frequently, the damage is reduced to single-strand breaks. In human cells, according to the experiments, the repair of such breaks can mechanistically be divided into four steps including (i) the break detection, (ii) processing of damaged ends, (iii) gap filling, and (iv) ligation of unbound ends of the broken strand. The first and second steps run in parallel while the third and fourth steps are sequential. The author proposes a kinetic model describing these steps. It allows one to understand the likely dependence of the number of breaks in different states on enzyme concentrations. The dependence of these concentrations on the rate of the formation of breaks can be understood as well. In addition, the likely role of unzipping and zipping of the fragments of broken ends of the strand in the ligation step has been scrutinized taking the specifics of binding of DNA stands into account.

Evolving gene regulatory networks into cellular networks guiding adaptive behavior: an outline how single cells could have evolved into a centralized neurosensory system

Understanding the evolution of the neurosensory system of man, able to reflect on its own origin, is one of the major goals of comparative neurobiology. Details of the origin of neurosensory cells, their aggregation into central nervous systems and associated sensory organs and their localized patterning leading to remarkably different cell types aggregated into variably sized parts of the central nervous system have begun to emerge. Insights at the cellular and molecular level have begun to shed some light on the evolution of neurosensory cells, partially covered in this review. Molecular evidence suggests that high mobility group (HMG) proteins of pre-metazoans evolved into the definitive Sox [SRY (sex determining region Y)-box] genes used for neurosensory precursor specification in metazoans. Likewise, pre-metazoan basic helix-loop-helix (bHLH) genes evolved in metazoans into the group A bHLH genes dedicated to neurosensory differentiation in bilaterians. Available evidence suggests that the Sox and bHLH genes evolved a cross-regulatory network able to synchronize expansion of precursor populations and their subsequent differentiation into novel parts of the brain or sensory organs. Molecular evidence suggests metazoans evolved patterning gene networks early, which were not dedicated to neuronal development. Only later in evolution were these patterning gene networks tied into the increasing complexity of diffusible factors, many of which were already present in pre-metazoans, to drive local patterning events. It appears that the evolving molecular basis of neurosensory cell development may have led, in interaction with differentially expressed patterning genes, to local network modifications guiding unique specializations of neurosensory cells into sensory organs and various areas of the central nervous system.

Suppression and enhancement of transcriptional noise by DNA looping

DNA looping has been observed to enhance and suppress transcriptional noise but it is uncertain which of these two opposite effects is to be expected for given conditions. Here, we derive analytical expressions for the main quantifiers of transcriptional noise in terms of the molecular parameters and elucidate the role of DNA looping. Our results rationalize paradoxical experimental observations and provide the first quantitative explanation of landmark individual-cell measurements at the single molecule level on the classical lac operon genetic system [Choi, L. Cai, K. Frieda, and X. S. Xie, Science 322, 442 (2008)].

Hormonal crosstalk for root development: a combined experimental and modeling perspective

Plants are sessile organisms and therefore they must adapt their growth and architecture to a changing environment. Understanding how hormones and genes interact to coordinate plant growth in a changing environment is a major challenge in developmental biology. Although a localized auxin concentration maximum in the root tip is important for root development, auxin concentration cannot change independently of multiple interacting hormones and genes. In this review, we discuss the experimental evidence showing that the POLARIS peptide of Arabidopsis plays an important role in hormonal crosstalk and root growth, and review the crosstalk between auxin and other hormones for root growth with and without osmotic stress. Moreover, we discuss that experimental evidence showing that, in root development, hormones and the associated regulatory and target genes form a network, in which relevant genes regulate hormone activities and hormones regulate gene expression. We further discuss how it is increasingly evident that mathematical modeling is a valuable tool for studying hormonal crosstalk. Therefore, a combined experimental and modeling study on hormonal crosstalk is important for elucidating the complexity of root development.

microRNA as a potential vector for the propagation of robustness in protein expression and oscillatory dynamics within a ceRNA network

microRNAs (miRNAs) are small noncoding RNAs that are important post-transcriptional regulators of gene expression. miRNAs can induce thresholds in protein synthesis. Such thresholds in protein output can be also achieved by oligomerization of transcription factors (TF) for the control of gene expression. First, we propose a minimal model for protein expression regulated by miRNA and by oligomerization of TF. We show that miRNA and oligomerization of TF generate a buffer, which increases the robustness of protein output towards molecular noise as well as towards random variation of kinetics parameters. Next, we extend the model by considering that the same miRNA can bind to multiple messenger RNAs, which accounts for the dynamics of a minimal competing endogenous RNAs (ceRNAs) network. The model shows that, through common miRNA regulation, TF can control the expression of all proteins formed by the ceRNA network, even if it drives the expression of only one gene in the network. The model further suggests that the threshold in protein synthesis mediated by the oligomerization of TF can be propagated to the other genes, which can increase the robustness of the expression of all genes in such ceRNA network. Furthermore, we show that a miRNA could increase the time delay of a "Goodwin-like" oscillator model, which may favor the occurrence of oscillations of large amplitude. This result predicts important roles of miRNAs in the control of the molecular mechanisms leading to the emergence of biological rhythms. Moreover, a model for the latter oscillator embedded in a ceRNA network indicates that the oscillatory behavior can be propagated, via the shared miRNA, to all proteins formed by such ceRNA network. Thus, by means of computational models, we show that miRNAs could act as vectors allowing the propagation of robustness in protein synthesis as well as oscillatory behaviors within ceRNA networks.

Functional roles of pulsing in genetic circuits

A fundamental problem in biology is to understand how genetic circuits implement core cellular functions. Time-lapse microscopy techniques are beginning to provide a direct view of circuit dynamics in individual living cells. Unexpectedly, we are discovering that key transcription and regulatory factors pulse on and off repeatedly, and often stochastically, even when cells are maintained in constant conditions. This type of spontaneous dynamic behavior is pervasive, appearing in diverse cell types from microbes to mammalian cells. Here, we review recent work showing how pulsing is generated and controlled by underlying regulatory circuits and how it provides critical capabilities to cells in stress response, signaling, and development. A major theme is the ability of pulsing to enable time-based regulation analogous to strategies used in engineered systems. Thus, pulsatile dynamics is emerging as a central, and still largely unexplored, layer of temporal organization in the cell.

A general model for toxin-antitoxin module dynamics can explain persister cell formation in E. coli

Toxin-Antitoxin modules are small operons involved in stress response and persister cell formation that encode a “toxin” and its corresponding neutralizing “antitoxin”. Regulation of these modules involves a complex mechanism known as conditional cooperativity, which is supposed to prevent unwanted toxin activation. Here we develop mathematical models for their regulation, based on published molecular and structural data, and parameterized using experimental data for F-plasmid ccdAB, bacteriophage P1 phd/doc and E. coli relBE. We show that the level of free toxin in the cell is mainly controlled through toxin sequestration in toxin-antitoxin complexes of various stoichiometry rather than by gene regulation. If the toxin translation rate exceeds twice the antitoxin translation rate, toxins accumulate in all cells. Conditional cooperativity and increasing the number of binding sites on the operator serves to reduce the metabolic burden of the cell by reducing the total amounts of proteins produced. Combining conditional cooperativity and bridging of antitoxins by toxins when bound to their operator sites allows creation of persister cells through rare, extreme stochastic spikes in the free toxin level. The amplitude of these spikes determines the duration of the persister state. Finally, increases in the antitoxin degradation rate and decreases in the bacterial growth rate cause a rise in the amount of persisters during nutritional stress.

Bacterial persistence plays an important role in many chronic infections. Persisters are subpopulations of bacteria which are tolerant to biological stresses such as antibiotics because they are in a dormant, non-dividing state. Toxin-antitoxin (TA) modules play a pivotal role in persister generation and bacterial stress response. These small genetic loci, ubiquitous in bacterial genomes and plasmids, code for a toxin that slows down or halts bacterial metabolism and a corresponding antitoxin that regulates this activity. In order to further unravel the intricate autoregulation of TA modules and their role in persister cell formation, we built stochastic models describing the transcriptional regulation including conditional cooperativity. This is a complex mechanism in which the molar ratio between both proteins determines whether the toxin will behave as a co-repressor or as a de-repressor for the antitoxin. We found that the necessary protein production and therefore the energetic cost decreases with increased binding site number. Finally, these models allow us to simulate the formation of persister cells through rare, stochastic increases in the free toxin level. We believe that our analysis provides a fresh view and contributes to our understanding of TA regulation and how it may be related to the emergence of persisters.