Noise-based switches and amplifiers for gene expression

Nonlocal models in biology and life sciences: Sources, developments, and applications

Mathematical modeling is one of the fundamental techniques for understanding biophysical mechanisms in developmental biology. It helps researchers to analyze complex physiological processes and connect like a bridge between theoretical and experimental observations. Various groups of mathematical models have been studied to analyze these processes, and the nonlocal models are one of them. Nonlocality is important in realistic mathematical models of physical and biological systems when local models fail to capture the essential dynamics and interactions that occur over a range of distances (e.g., cell-cell, cell-tissue adhesions, neural networks, the spread of diseases, intra-specific competition, nanobeams, etc.). This review illustrates different nonlocal mathematical models applied to biology and life sciences. The major focus has been given to sources, developments, and applications of such models. Among other things, a systematic discussion has been provided for the conditions of pattern formations in biological systems of population dynamics. Special attention has also been given to nonlocal interactions on networks, network coupling and integration, including brain dynamics models that provide an important tool to understand neurodegenerative diseases better. In addition, we have discussed nonlocal modeling approaches for cancer stem cells and tumor cells that are widely applied in the cell migration processes, growth, and avascular tumors in any organ. Furthermore, the discussed nonlocal continuum models can go sufficiently smaller scales, including nanotechnology, where classical local models often fail to capture the complexities of nanoscale interactions, applied to build biosensors to sense biomaterial and its concentration. Piezoelectric and other smart materials are among them, and these devices are becoming increasingly important in the digital and physical world that is intrinsically interconnected with biological systems. Additionally, we have reviewed a nonlocal theory of peridynamics, which deals with continuous and discrete media and applies to model the relationship between fracture and healing in cortical bone, tissue growth and shrinkage, and other areas increasingly important in biomedical and bioengineering applications. Finally, we provided a comprehensive summary of emerging trends and highlighted future directions in this rapidly expanding field.

Phase separation provides a mechanism to drive phenotype switching

Phenotypic switching plays a crucial role in cell fate determination across various organisms. Recent experimental findings highlight the significance of protein compartmentalization via liquid-liquid phase separation in influencing such decisions. However, the precise mechanism through which phase separation regulates phenotypic switching remains elusive. To investigate this, we established a mathematical model that couples a phase separation process and a gene expression process with feedback. We used the chemical master equation theory and mean-field approximation to study the effects of phase separation on the gene expression products. We found that phase separation can cause bistability and bimodality. Furthermore, phase separation can control the bistable properties of the system, such as bifurcation points and bistable ranges. On the other hand, in stochastic dynamics, the droplet phase exhibits double peaks within a more extensive phase separation threshold range than the dilute phase, indicating the pivotal role of the droplet phase in cell fate decisions. These findings propose an alternative mechanism that influences cell fate decisions through the phase separation process. As phase separation is increasingly discovered in gene regulatory networks, related modeling research can help build biomolecular systems with desired properties and offer insights into explaining cell fate decisions.

Dynamical Modeling and Qualitative Analysis of a Delayed Model for CD8 T Cells in Response to Viral Antigens

Although the immune effector CD8 T cells play a crucial role in clearance of viruses, the mechanisms underlying the dynamics of how CD8 T cells respond to viral infection remain largely unexplored. Here, we develop a delayed model that incorporates CD8 T cells and infected cells to investigate the functional role of CD8 T cells in persistent virus infection. Bifurcation analysis reveals that the model has four steady states that can finely divide the progressions of viral infection into four states, and endows the model with bistability that has ability to achieve the switch from one state to another. Furthermore, analytical and numerical methods find that the time delay resulting from incubation period of virus can induce a stable low-infection steady state to be oscillatory, coexisting with a stable high-infection steady state in phase space. In particular, a novel mechanism to achieve the switch between two stable steady states, time-delay-based switch, is proposed, where the initial conditions and other parameters of the model remain unchanged. Moreover, our model predicts that, for a certain range of initial antigen load: 1) under a longer incubation period, the lower the initial antigen load, the easier the virus infection will evolve into severe state; while the higher the initial antigen load, the easier it is for the virus infection to be effectively controlled and 2) only when the incubation period is small, the lower the initial antigen load, the easier it is to effectively control the infection progression. Our results are consistent with multiple experimental observations, which may facilitate the understanding of the dynamical and physiological mechanisms of CD8 T cells in response to viral infections.

Effects of microRNA-mediated negative feedback on gene expression noise

MicroRNAs (miRNAs) are small noncoding RNAs that regulate gene expression post-transcriptionally in eukaryotes by binding with target mRNAs and preventing translation. miRNA-mediated feedback motifs are ubiquitous in various genetic networks that control cellular decision making. A key question is how such a feedback mechanism may affect gene expression noise. To answer this, we have developed a mathematical model to study the effects of a miRNA-dependent negative-feedback loop on mean expression and noise in target mRNAs. Combining analytics and simulations, we show the existence of an expression threshold demarcating repressed and expressed regimes in agreement with earlier studies. The steady-state mRNA distributions are bimodal near the threshold, where copy numbers of mRNAs and miRNAs exhibit enhanced anticorrelated fluctuations. Moreover, variation of negative-feedback strength shifts the threshold locations and modulates the noise profiles. Notably, the miRNA-mRNA binding affinity and feedback strength collectively shape the bimodality. We also compare our model with a direct auto-repression motif, where a gene produces its own repressor. Auto-repression fails to produce bimodal mRNA distributions as found in miRNA-based indirect repression, suggesting the crucial role of miRNAs in creating phenotypic diversity. Together, we demonstrate how miRNA-dependent negative feedback modifies the expression threshold and leads to a broader parameter regime of bimodality compared to the no-feedback case.

Out-of-equilibrium gene expression fluctuations in the presence of extrinsic noise

Cell-to-cell variability in protein concentrations is strongly affected by extrinsic noise, especially for highly expressed genes. Extrinsic noise can be due to fluctuations of several possible cellular factors connected to cell physiology and to the level of key enzymes in the expression process. However, how to identify the predominant sources of extrinsic noise in a biological system is still an open question. This work considers a general stochastic model of gene expression with extrinsic noise represented as fluctuations of the different model rates, and focuses on the out-of-equilibrium expression dynamics. Combining analytical calculations with stochastic simulations, we characterize how extrinsic noise shapes the protein variability during gene activation or inactivation, depending on the prevailing source of extrinsic variability, on its intensity and timescale. In particular, we show that qualitatively different noise profiles can be identified depending on which are the fluctuating parameters. This indicates an experimentally accessible way to pinpoint the dominant sources of extrinsic noise using time-coarse experiments.

Systematic analysis and optimization of early warning signals for critical transitions using distribution data

Abrupt shifts between alternative regimes occur in complex systems, from cell regulation to brain functions to ecosystems. Several model-free early warning signals (EWS) have been proposed to detect impending transitions, but failure or poor performance in some systems have called for better investigation of their generic applicability. Notably, there are still ongoing debates whether such signals can be successfully extracted from data in particular from biological experiments. In this work, we systematically investigate properties and performance of dynamical EWS in different deteriorating conditions, and we propose an optimized combination to trigger warnings as early as possible, eventually verified on experimental data from microbiological populations. Our results explain discrepancies observed in the literature between warning signs extracted from simulated models and from real data, provide guidance for EWS selection based on desired systems and suggest an optimized composite indicator to alert for impending critical transitions using distribution data.

Applications of synthetic biology in medical and pharmaceutical fields

Synthetic biology aims to design or assemble existing bioparts or bio-components for useful bioproperties. During the past decades, progresses have been made to build delicate biocircuits, standardized biological building blocks and to develop various genomic/metabolic engineering tools and approaches. Medical and pharmaceutical demands have also pushed the development of synthetic biology, including integration of heterologous pathways into designer cells to efficiently produce medical agents, enhanced yields of natural products in cell growth media to equal or higher than that of the extracts from plants or fungi, constructions of novel genetic circuits for tumor targeting, controllable releases of therapeutic agents in response to specific biomarkers to fight diseases such as diabetes and cancers. Besides, new strategies are developed to treat complex immune diseases, infectious diseases and metabolic disorders that are hard to cure via traditional approaches. In general, synthetic biology brings new capabilities to medical and pharmaceutical researches. This review summarizes the timeline of synthetic biology developments, the past and present of synthetic biology for microbial productions of pharmaceutics, engineered cells equipped with synthetic DNA circuits for diagnosis and therapies, live and auto-assemblied biomaterials for medical treatments, cell-free synthetic biology in medical and pharmaceutical fields, and DNA engineering approaches with potentials for biomedical applications.

Buffering variability in cell regulation motifs close to criticality

Bistable biological regulatory systems need to cope with stochastic noise to fine tune their function close to bifurcation points. Here, we study stability properties of this regime in generic systems to demonstrate that cooperative interactions buffer system variability, hampering noise-induced regime shifts. Our analysis also shows that, in the considered cooperativity range, impending regime shifts can be generically detected by statistical early warning signals from distributional data. Our generic framework, based on minimal models, can be used to extract robustness and variability properties of more complex models and empirical data close to criticality.

Inferring structural and dynamical properties of gene networks from data with deep learning

The reconstruction of gene regulatory networks (GRNs) from data is vital in systems biology. Although different approaches have been proposed to infer causality from data, some challenges remain, such as how to accurately infer the direction and type of interactions, how to deal with complex network involving multiple feedbacks, as well as how to infer causality between variables from real-world data, especially single cell data. Here, we tackle these problems by deep neural networks (DNNs). The underlying regulatory network for different systems (gene regulations, ecology, diseases, development) can be successfully reconstructed from trained DNN models. We show that DNN is superior to existing approaches including Boolean network, Random Forest and partial cross mapping for network inference. Further, by interrogating the ensemble DNN model trained from single cell data from dynamical system perspective, we are able to unravel complex cell fate dynamics during preimplantation development. We also propose a data-driven approach to quantify the energy landscape for gene regulatory systems, by combining DNN with the partial self-consistent mean field approximation (PSCA) approach. We anticipate the proposed method can be applied to other fields to decipher the underlying dynamical mechanisms of systems from data.

Using single-cell models to predict the functionality of synthetic circuits at the population scale

SignificanceAt the single-cell level, biochemical processes are inherently stochastic. For many natural systems, the resulting cell-to-cell variability is exploited by microbial populations. In synthetic biology, however, the interplay of cell-to-cell variability and population processes such as selection or growth often leads to circuits not functioning as predicted by simple models. Here we show how multiscale stochastic kinetic models that simultaneously track single-cell and population processes can be obtained based on an augmentation of the chemical master equation. These models enable us to quantitatively predict complex population dynamics of a yeast optogenetic differentiation system from a specification of the circuit's components and to demonstrate how cell-to-cell variability can be exploited to purposefully create unintuitive circuit functionality.

Volumetric compression develops noise-driven single-cell heterogeneity

Recent studies have revealed that extensive heterogeneity of biological systems arises through various routes ranging from intracellular chromosome segregation to spatiotemporally varying biochemical stimulations. However, the contribution of physical microenvironments to single-cell heterogeneity remains largely unexplored. Here, we show that a homogeneous population of non-small-cell lung carcinoma develops into heterogeneous subpopulations upon application of a homogeneous physical compression, as shown by single-cell transcriptome profiling. The generated subpopulations stochastically gain the signature genes associated with epithelial-mesenchymal transition (EMT; VIM, CDH1, EPCAM, ZEB1, and ZEB2) and cancer stem cells (MKI67, BIRC5, and KLF4), respectively. Trajectory analysis revealed two bifurcated paths as cells evolving upon the physical compression, along each path the corresponding signature genes (epithelial or mesenchymal) gradually increase. Furthermore, we show that compression increases gene expression noise, which interplays with regulatory network architecture and thus generates differential cell-fate outcomes. The experimental observations of both single-cell sequencing and single-molecule fluorescent in situ hybridization agrees well with our computational modeling of regulatory network in the EMT process. These results demonstrate a paradigm of how mechanical stimulations impact cell-fate determination by altering transcription dynamics; moreover, we show a distinct path that the ecology and evolution of cancer interplay with their physical microenvironments from the view of mechanobiology and systems biology, with insight into the origin of single-cell heterogeneity.

Different Patterns of mRNA Nuclear Retention during Meiotic Prophase in Larch Microsporocytes

Recent studies show a crucial role of post-transcriptional processes in the regulation of gene expression. Our research has shown that mRNA retention in the nucleus plays a significant role in such regulation. We studied larch microsporocytes during meiotic prophase, characterized by pulsatile transcriptional activity. After each pulse, the transcriptional activity is silenced, but the transcripts synthesized at this time are not exported immediately to the cytoplasm but are retained in the cell nucleus and especially in Cajal bodies, where non-fully-spliced transcripts with retained introns are accumulated. Analysis of the transcriptome of these cells and detailed analysis of the nuclear retention and transport dynamics of several mRNAs revealed two main patterns of nuclear accumulation and transport. The majority of studied transcripts followed the first one, consisting of a more extended retention period and slow release to the cytoplasm. We have shown this in detail for the pre-mRNA and mRNA encoding RNA pol II subunit 10. In this pre-mRNA, a second (retained) intron is posttranscriptionally spliced at a precisely defined time. Fully mature mRNA is then released into the cytoplasm, where the RNA pol II complexes are produced. These proteins are necessary for transcription in the next pulse to occur.mRNAs encoding translation factors and SERRATE followed the second pattern, in which the retention period was shorter and transcripts were rapidly transferred to the cytoplasm. The presence of such a mechanism in various cell types from a diverse range of organisms suggests that it is an evolutionarily conserved mechanism of gene regulation.

A mathematical modelling framework for the regulation of intra-cellular OCT4 in human pluripotent stem cells

Human pluripotent stem cells (hPSCs) have the potential to differentiate into all cell types, a property known as pluripotency. A deeper understanding of how pluripotency is regulated is required to assist in controlling pluripotency and differentiation trajectories experimentally. Mathematical modelling provides a non-invasive tool through which to explore, characterise and replicate the regulation of pluripotency and the consequences on cell fate. Here we use experimental data of the expression of the pluripotency transcription factor OCT4 in a growing hPSC colony to develop and evaluate mathematical models for temporal pluripotency regulation. We consider fractional Brownian motion and the stochastic logistic equation and explore the effects of both additive and multiplicative noise. We illustrate the use of time-dependent carrying capacities and the introduction of Allee effects to the stochastic logistic equation to describe cell differentiation. We conclude both methods adequately capture the decline in OCT4 upon differentiation, but the Allee effect model has the advantage of allowing differentiation to occur stochastically in a sub-set of cells. This mathematical framework for describing intra-cellular OCT4 regulation can be extended to other transcription factors and developed into predictive models.

Topologies of synthetic gene circuit for optimal fold change activation

Computations widely exist in biological systems for functional regulations. Recently, incoherent feedforward loop and integral feedback controller have been implemented into Escherichia coli to achieve a robust adaptation. Here, we demonstrate that an indirect coherent feedforward loop and mutual inhibition designs can experimentally improve the fold change of promoters, by reducing the basal level while keeping the maximum activity high. We applied both designs to six different promoters in E. coli, starting with synthetic inducible promoters as a proof-of-principle. Then, we examined native promoters that are either functionally specific or systemically involved in complex pathways such as oxidative stress and SOS response. Both designs include a cascade having a repressor and a construct of either transcriptional interference or antisense transcription. In all six promoters, an improvement of up to ten times in the fold change activation was observed. Theoretically, our unitless models show that when regulation strength matches promoter basal level, an optimal fold change can be achieved. We expect that this methodology can be applied in various biological systems for biotechnology and therapeutic applications.

Reconstructing an epigenetic landscape using a genetic pulling approach

Cells use genetic switches to shift between alternate stable gene expression states, e.g., to adapt to new environments or to follow a developmental pathway. Conceptually, these stable phenotypes can be considered as attractive states on an epigenetic landscape with phenotypic changes being transitions between states. Measuring these transitions is challenging because they are both very rare in the absence of appropriate signals and very fast. As such, it has proved difficult to experimentally map the epigenetic landscapes that are widely believed to underly developmental networks. Here, we introduce a nonequilibrium perturbation method to help reconstruct a regulatory network's epigenetic landscape. We derive the mathematical theory needed and then use the method on simulated data to reconstruct the landscapes. Our results show that with a relatively small number of perturbation experiments it is possible to recover an accurate representation of the true epigenetic landscape. We propose that our theory provides a general method by which epigenetic landscapes can be studied. Finally, our theory suggests that the total perturbation impulse required to induce a switch between metastable states is a fundamental quantity in developmental dynamics.

A Dimension Reduction Approach for Energy Landscape: Identifying Intermediate States in Metabolism-EMT Network

Dimension reduction is a challenging problem in complex dynamical systems. Here, a dimension reduction approach of landscape (DRL) for complex dynamical systems is proposed, by mapping a high-dimensional system on a low-dimensional energy landscape. The DRL approach is applied to three biological networks, which validates that new reduced dimensions preserve the major information of stability and transition of original high-dimensional systems. The consistency of barrier heights calculated from the low-dimensional landscape and transition actions calculated from the high-dimensional system further shows that the landscape after dimension reduction can quantify the global stability of the system. The epithelial-mesenchymal transition (EMT) and abnormal metabolism are two hallmarks of cancer. With the DRL approach, a quadrastable landscape for metabolism-EMT network is identified, including epithelial (E), abnormal metabolic (A), hybrid E/M (H), and mesenchymal (M) cell states. The quantified energy landscape and kinetic transition paths suggest that for the EMT process, the cells at E state need to first change their metabolism, then enter the M state. The work proposes a general framework for the dimension reduction of a stochastic dynamical system, and advances the mechanistic understanding of the underlying relationship between EMT and cellular metabolism.

Quantifying the Stability of Coupled Genetic and Epigenetic Switches With Variational Methods

The Waddington landscape provides an intuitive metaphor to view development as a ball rolling down the hill, with distinct phenotypes as basins and differentiation pathways as valleys. Since, at a molecular level, cell differentiation arises from interactions among the genes, a mathematical definition for the Waddington landscape can, in principle, be obtained by studying the gene regulatory networks. For eukaryotes, gene regulation is inextricably and intimately linked to histone modifications. However, the impact of such modifications on both landscape topography and stability of attractor states is not fully understood. In this work, we introduced a minimal kinetic model for gene regulation that combines the impact of both histone modifications and transcription factors. We further developed an approximation scheme based on variational principles to solve the corresponding master equation in a second quantized framework. By analyzing the steady-state solutions at various parameter regimes, we found that histone modification kinetics can significantly alter the behavior of a genetic network, resulting in qualitative changes in gene expression profiles. The emerging epigenetic landscape captures the delicate interplay between transcription factors and histone modifications in driving cell-fate decisions.

A redox-based electrogenetic CRISPR system to connect with and control biological information networks

Electronic information can be transmitted to cells directly from microelectronics via electrode-activated redox mediators. These transmissions are decoded by redox-responsive promoters which enable user-specified control over biological function. Here, we build on this redox communication modality by establishing an electronic eCRISPR conduit of information exchange. This system acts as a biological signal processor, amplifying signal reception and filtering biological noise. We electronically amplify bacterial quorum sensing (QS) signaling by activating LasI, the autoinducer-1 synthase. Similarly, we filter out unintended noise by inhibiting the native SoxRS-mediated oxidative stress response regulon. We then construct an eCRISPR based redox conduit in both E. coli and Salmonella enterica. Finally, we display eCRISPR based information processing that allows transmission of spatiotemporal redox commands which are then decoded by gelatin-encapsulated E. coli. We anticipate that redox communication channels will enable biohybrid microelectronic devices that could transform our abilities to electronically interpret and control biological function.

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Autoregulatory feedback loops are one of the most common network motifs. A wide variety of stochastic models have been constructed to understand how the fluctuations in protein numbers in these loops are influenced by the kinetic parameters of the main biochemical steps. These models differ according to 1) which subcellular processes are explicitly modeled, 2) the modeling methodology employed (discrete, continuous, or hybrid), and 3) whether they can be analytically solved for the steady-state distribution of protein numbers. We discuss the assumptions and properties of the main models in the literature, summarize our current understanding of the relationship between them, and highlight some of the insights gained through modeling.

New Method to Global Exponential Stability Analysis for Switched Genetic Regulatory Networks With Mixed Delays

In this paper, the sufficient conditions for the global exponential stability of the switched genetic regulatory networks with mixed time delays are obtained. The proposed method does not need the construction of Lyapunov-Krasovskii functional, but is directly proceeded by the definition of global exponential stability. The derived sufficient conditions can easily be verified by checking the eigenvalues of a constant matrix or solving several simple linear matrix inequalities. Finally, two numerical examples are presented to illustrate that the obtained global exponential stability criteria are available.

Transitions in the cell-fate induction induced by colored noise associated with the inductive stimulus

The cell-fate induction based on the saddle-node bifurcation is undoubtedly a very important concept in developmental biology, which provides a possible mechanism to explain the intrinsic irreversibility in the developmental process. In this paper, the effect of a colored noise, which is associated with the inductive stimulus, on the saddle-node landscape of cell-fate induction is investigated, especially, the effect of the change of correlation time of colored noise on cell-fate induction. The main results show clearly that the change of correlation time of colored noise could induce the transitions of the system. This implies that the colored noise associated with inductive stimulus may have a profound effect on the saddle-node bifurcation landscape of cell-fate induction. This will also help us to understand more deeply the role of cell-fate induction in developmental biology.

Improvement of the memory function of a mutual repression network in a stochastic environment by negative autoregulation

BACKGROUND

Cellular memory is a ubiquitous function of biological systems. By generating a sustained response to a transient inductive stimulus, often due to bistability, memory is central to the robust control of many important biological processes. However, our understanding of the origins of cellular memory remains incomplete. Stochastic fluctuations that are inherent to most biological systems have been shown to hamper memory function. Yet, how stochasticity changes the behavior of genetic circuits is generally not clear from a deterministic analysis of the network alone. Here, we apply deterministic rate equations, stochastic simulations, and theoretical analyses of Fokker-Planck equations to investigate how intrinsic noise affects the memory function in a mutual repression network.

RESULTS

We find that the addition of negative autoregulation improves the persistence of memory in a small gene regulatory network by reducing stochastic fluctuations. Our theoretical analyses reveal that this improved memory function stems from an increased stability of the steady states of the system. Moreover, we show how the tuning of critical network parameters can further enhance memory.

CONCLUSIONS

Our work illuminates the power of stochastic and theoretical approaches to understanding biological circuits, and the importance of considering stochasticity when designing synthetic circuits with memory function.

STOCHASTIC DYNAMICS OF CELL LINEAGE IN TISSUE HOMEOSTASIS

During epithelium tissue maintenance, lineages of cells differentiate and proliferate in a coordinated way to provide the desirable size and spatial organization of different types of cells. While mathematical models through deterministic description have been used to dissect role of feedback regulations on tissue layer size and stratification, how the stochastic effects influence tissue maintenance remains largely unknown. Here we present a stochastic continuum model for cell lineages to investigate how both layer thickness and layer stratification are affected by noise. We find that the cell-intrinsic noise often causes reduction and oscillation of layer size whereas the cell-extrinsic noise increases the thickness, and sometimes, leads to uncontrollable growth of the tissue layer. The layer stratification usually deteriorates as the noise level increases in the cell lineage systems. Interestingly, the morphogen noise, which mixes both cell-intrinsic noise and cell-extrinsic noise, can lead to larger size of layer with little impact on the layer stratification. By investigating different combinations of the three types of noise, we find the layer thickness variability is reduced when cell-extrinsic noise level is high or morphogen noise level is low. Interestingly, there exists a tradeoff between low thickness variability and strong layer stratification due to competition among the three types of noise, suggesting robust layer homeostasis requires balanced levels of different types of noise in the cell lineage systems.

Time delay and cross-correlated Gaussian noises-induced stochastic stability and regime shift between steady states for an insect outbreak system

In this paper, we focus on investigating the stochastic stability and the regime transition between the endangered state and the boom state for a time-delayed insect growth system driven by correlated external and internal noises. By use of the Fokker–Planck equation, the method of small time delay approximation and the fast descent method, we explore in detail the joint action of noise terms and time delay on the mean reproduction and depression time for the insect population. Our investigations indicate that the pseudo-resonance phenomenon of the mean first-passage time (MFPT) occurs because of the impact of different noises and time delay. Through the numerical calculation, it is discovered that multiplicative noise can speed up the shift of the insect population from the boom state to the endangered one, while the noise correlation and time delay can propel the insect system to evolve from the endangered state to the boom state and improve the biological stability. In addition, the impact of the additive noise on the stability of the biological system depends on the positive and negative situation of the noise correlation. On the other hand, during the process of suppressing the insect explosion, it is beneficial to the pest control to amplify the association noise strength and weaken the intensities of the multiplicative, additive noises and time delay. However, during the process of eliminating the pests, it can produce nice effect on the disinsection to increase time delay, the intensities of multiplicative and additive noises and weaken the strength of noise correlation.

Stability Analysis of Genetic Regulatory Networks With Switching Parameters and Time Delays

This paper is concerned with the exponential stability analysis of genetic regulatory networks (GRNs) with switching parameters and time delays. In this paper, a new integral inequality and an improved reciprocally convex combination inequality are considered. By using the average dwell time approach together with a novel Lyapunov-Krasovskii functional, we derived some conditions to ensure the switched GRNs with switching parameters and time delays are exponentially stable. Finally, we give two numerical examples to clarify that our derived results are effective.

Bottom-up approaches in synthetic biology and biomaterials for tissue engineering applications

Synthetic biologists use engineering principles to design and construct genetic circuits for programming cells with novel functions. A bottom-up approach is commonly used to design and construct genetic circuits by piecing together functional modules that are capable of reprogramming cells with novel behavior. While genetic circuits control cell operations through the tight regulation of gene expression, a diverse array of environmental factors within the extracellular space also has a significant impact on cell behavior. This extracellular space offers an addition route for synthetic biologists to apply their engineering principles to program cell-responsive modules within the extracellular space using biomaterials. In this review, we discuss how taking a bottom-up approach to build genetic circuits using DNA modules can be applied to biomaterials for controlling cell behavior from the extracellular milieu. We suggest that, by collectively controlling intrinsic and extrinsic signals in synthetic biology and biomaterials, tissue engineering outcomes can be improved.

Landscape reveals critical network structures for sharpening gene expression boundaries

Background

Spatial pattern formation is a critical issue in developmental biology. Gene expression boundary sharpening has been observed from both experiments and modeling simulations. However, the mechanism to determine the sharpness of the boundary is not fully elucidated.

Results

We investigated the boundary sharpening resulted by three biological motifs, interacting with morphogens, and uncovered their probabilistic landscapes. The landscape view, along with calculated average switching time between attractors, provides a natural explanation for the boundary sharpening behavior relying on the noise induced gene state switchings. To possess boundary sharpening potential, a gene network needs to generate an asymmetric bistable state, i.e. one of the two stable states is less stable than the other. We found that the mutual repressed self-activation model displays more robust boundary sharpening ability against parameter perturbation, compared to the mutual repression or the self-activation model. This is supported by the results of switching time calculated from the landscape, which indicate that the mutual repressed self-activation model has shortest switching time, among three models. Additionally, introducing cross gradients of morphogens provides a more stable mechanism for the boundary sharpening of gene expression, due to a two-way switching mechanism.

Conclusions

Our results reveal the underlying principle for the gene expression boundary sharpening, and pave the way for the mechanistic understanding of cell fate decisions in the pattern formation processes of development.

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0595-5) contains supplementary material, which is available to authorized users.

Time-delay-induced dynamical behaviors for an ecological vegetation growth system driven by cross-correlated multiplicative and additive noises

In this paper, the modified potential function, the stationary probability distribution function (SPDF), the mean growth time and the mean degeneration time for a vegetation growth system with time delay are investigated, where the vegetation system is assumed to be disturbed by cross-correlated multiplicative and additive noises. The results reveal some fact that the multiplicative and additive noises can both reduce the stability and speed up the decline of the vegetation system, while the strength of the noise correlation and time delay can both enhance the stability of the vegetation and slow down the depression process of the ecological system. On the other hand, with regard to the impacts of noises and time delay on the mean development and degeneration processes of the ecological system, it is discovered that 1) in the development process of the vegetation population, the increase of the noise correlation strength and time delay will restrain the regime shift from the barren state to the boom one, while the increase of the additive noise can lead to the fast regime shift from the barren state to the boom one. 2) Conversely, in the depression process of the ecological system, the increase of the strength of the correlation noise and time delay will prevent the regime shift from the boom state to the barren one. Comparatively, the increase of the additive and multiplicative noises can accelerate the regime shift from the boom state to the barren state.

Threshold response and bistability in gene regulation by small noncoding RNA

In this paper, we study through mathematical modelling the combined effect of transcriptional and translational regulation by proteins and small noncoding RNAs (sRNA) in a genetic feedback motif that has an important role in the survival of E. coli under stress associated with oxygen and energy availability. We show that subtle changes in this motif can bring in drastically different effects on the gene expression. In particular, we show that a threshold response in the gene expression changes to a bistable response as the regulation on sRNA synthesis or degradation is altered. These results are obtained under deterministic conditions. Next, we study how the gene expression is altered by additive and multiplicative noise which might arise due to probabilistic occurrences of different biochemical events. Using the Fokker-Planck formulation, we obtain steady-state probability distributions for sRNA concentration for the network motifs displaying bistability. The probability distributions are found to be bimodal with two peaks at low and high concentrations of sRNAs. We further study the variations in the probability distributions under different values of noise strength and correlations. The results presented here might be of interest for designing synthetic network for artificial control.

Likelihood for transcriptions in a genetic regulatory system under asymmetric stable Lévy noise

This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case (β = 0). While in the multiplicative noise case, non-Gaussianity index value α = 1 is a separating point or line for both the MFET and the FEP. We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in "regulating" the likelihood of gene transcription by tuning asymmetric stable Lévy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.

Underlying Principles of Natural Selection in Network Evolution: Systems Biology Approach

Systems biology is a rapidly expanding field that integrates diverse areas of science such as physics, engineering, computer science, mathematics, and biology toward the goal of elucidating the underlying principles of hierarchical metabolic and regulatory systems in the cell, and ultimately leading to predictive understanding of cellular response to perturbations. Because post-genomics research is taking place throughout the tree of life, comparative approaches offer a way for combining data from many organisms to shed light on the evolution and function of biological networks from the gene to the organismal level. Therefore, systems biology can build on decades of theoretical work in evolutionary biology, and at the same time evolutionary biology can use the systems biology approach to go in new uncharted directions. In this study, we present a review of how the post-genomics era is adopting comparative approaches and dynamic system methods to understand the underlying design principles of network evolution and to shape the nascent field of evolutionary systems biology. Finally, the application of evolutionary systems biology to robust biological network designs is also discussed from the synthetic biology perspective.

Entropy production selects nonequilibrium states in multistable systems

Far-from-equilibrium thermodynamics underpins the emergence of life, but how has been a long-outstanding puzzle. Best candidate theories based on the maximum entropy production principle could not be unequivocally proven, in part due to complicated physics, unintuitive stochastic thermodynamics, and the existence of alternative theories such as the minimum entropy production principle. Here, we use a simple, analytically solvable, one-dimensional bistable chemical system to demonstrate the validity of the maximum entropy production principle. To generalize to multistable stochastic system, we use the stochastic least-action principle to derive the entropy production and its role in the stability of nonequilibrium steady states. This shows that in a multistable system, all else being equal, the steady state with the highest entropy production is favored, with a number of implications for the evolution of biological, physical, and geological systems.

Core signalling motif displaying multistability through multi-state enzymes

Bistability, and more generally multistability, is a key system dynamics feature enabling decision-making and memory in cells. Deciphering the molecular determinants of multistability is thus crucial for a better understanding of cellular pathways and their (re)engineering in synthetic biology. Here, we show that a key motif found predominantly in eukaryotic signalling systems, namely a futile signalling cycle, can display bistability when featuring a two-state kinase. We provide necessary and sufficient mathematical conditions on the kinetic parameters of this motif that guarantee the existence of multiple steady states. These conditions foster the intuition that bistability arises as a consequence of competition between the two states of the kinase. Extending from this result, we find that increasing the number of kinase states linearly translates into an increase in the number of steady states in the system. These findings reveal, to our knowledge, a new mechanism for the generation of bistability and multistability in cellular signalling systems. Further the futile cycle featuring a two-state kinase is among the smallest bistable signalling motifs. We show that multi-state kinases and the described competition-based motif are part of several natural signalling systems and thereby could enable them to implement complex information processing through multistability. These results indicate that multi-state kinases in signalling systems are readily exploited by natural evolution and could equally be used by synthetic approaches for the generation of multistable information processing systems at the cellular level.

Regime shifts driven by dynamic correlations in gene expression noise

Gene expression is a noisy process that leads to regime shifts between alternative steady states among individual living cells, inducing phenotypic variability. The effects of white noise on the regime shift in bistable systems have been well characterized, however little is known about such effects of colored noise (noise with nonzero correlation time). Here, we show that noise correlation time, by considering a genetic circuit of autoactivation, can have a significant effect on the regime shift between distinct phenotypic states in gene expression. We demonstrate this theoretically, using stochastic potential, stationary probability density function, and first-passage time based on the Fokker-Planck description, where the Ornstein-Uhlenbeck process is used to model colored noise. We find that an increase in noise correlation time in the degradation rate can induce a regime shift from a low to a high protein concentration state and enhance the bistable regime, while an increase in noise correlation time in the basal rate retains the bimodal distribution. We then show how cross-correlated colored noises in basal and degradation rates can induce regime shifts from a low to a high protein concentration state, but reduce the bistable regime. We also validate these results through direct numerical simulations of the stochastic differential equation. In gene expression understanding the causes of regime shift to a harmful phenotype could improve early therapeutic intervention in complex human diseases.

Mathematical comparison of memory functions between mutual activation and repression networks in a stochastic environment

Biological memory is a ubiquitous function that can generate a sustained response to a transient inductive stimulus. To better understand this function, we must consider the mechanisms by which different structures of genetic networks achieve memory. Here, we investigated two competitive gene regulatory network models: the regulated mutual activation network (MAN) and the regulated mutual repression network (MRN). Stochasticity deteriorated the persistence of memory of both the MAN and the MRN. Mathematical comparison by simulation and theoretical analysis identified functional differences in the stochastic memory between the competitive models: specifically, the MAN provided much more robust, persistent memory than the MRN. The stochastic persistent memory pattern of the MAN can be adjusted by changing the binding strength of the activators, whereas the MRN required highly cooperative and strong binding repressors for robust memory.

Information-based measures for logical stochastic resonance in a synthetic gene network under Lévy flight superdiffusion

We investigate the logical information transmission of a synthetic gene network under Lévy flight superdiffusion by an information-based methodology. We first present the stochastic synthetic gene network model driven by a square wave signal under Lévy noise caused by Lévy flight superdiffusion. Then, to quantify the potential of logical information transmission and logical stochastic resonance, we theoretically obtain an information-based methodology of the symbol error rate, the noise entropy, and the mutual information of the logical information transmission. Consequently, based on the complementary "on" and "off" states shown in the logical information transmission for the repressive proteins, we numerically calculate the symbol error rate for logic gates, which demonstrate that the synthetic gene network under Lévy noise can achieve some logic gates as well as logical stochastic resonance. Furthermore, we calculate the noise entropy and the mutual information between the square wave signal and the logical information transmission, which reveal and quantify the potential of logical information transmission and logical stochastic resonance. In addition, we analyze the synchronization degree of the mutual information for the accomplished logical stochastic resonance of two repressive proteins of the synthetic gene network by synchronization variances, which shows that those mutual information changes almost synchronously.

Delayed bet-hedging resilience strategies under environmental fluctuations

Many biological populations, such as bacterial colonies, have developed through evolution a protection mechanism, called bet hedging, to increase their probability of survival under stressful environmental fluctuation. In this context, the concept of preadaptation refers to a common type of bet-hedging protection strategy in which a relatively small number of individuals in a population stochastically switch their phenotypes to a dormant metabolic state in which they increase their probability of survival against potential environmental shocks. Hence, if an environmental shock took place at some point in time, preadapted organisms would be better adapted to survive and proliferate once the shock is over. In many biological populations, the mechanisms of preadaptation and proliferation present delays whose influence in the fitness of the population are not well understood. In this paper, we propose a rigorous mathematical framework to analyze the role of delays in both preadaptation and proliferation mechanisms in the survival of biological populations, with an emphasis on bacterial colonies. Our theoretical framework allows us to analytically quantify the average growth rate of a bet-hedging bacterial colony with stochastically delayed reactions with arbitrary precision. We verify the accuracy of the proposed method by numerical simulations and conclude that the growth rate of a bet-hedging population shows a nontrivial dependency on their preadaptation and proliferation delays. Contrary to the current belief, our results show that faster reactions do not, in general, increase the overall fitness of a biological population.

Reconstruction of noise-driven nonlinear networks from node outputs by using high-order correlations

Many practical systems can be described by dynamic networks, for which modern technique can measure their outputs, and accumulate extremely rich data. Nevertheless, the network structures producing these data are often deeply hidden in the data. The problem of inferring network structures by analyzing the available data, turns to be of great significance. On one hand, networks are often driven by various unknown facts, such as noises. On the other hand, network structures of practical systems are commonly nonlinear, and different nonlinearities can provide rich dynamic features and meaningful functions of realistic networks. Although many works have considered each fact in studying network reconstructions, much less papers have been found to systematically treat both difficulties together. Here we propose to use high-order correlation computations (HOCC) to treat nonlinear dynamics; use two-time correlations to decorrelate effects of network dynamics and noise driving; and use suitable basis and correlator vectors to unifiedly infer all dynamic nonlinearities, topological interaction links and noise statistical structures. All the above theoretical frameworks are constructed in a closed form and numerical simulations fully verify the validity of theoretical predictions.

Noise Decomposition Principle in a Coherent Feed-Forward Transcriptional Regulatory Loop

Coherent feed-forward loops exist extensively in realistic biological regulatory systems, and are common signaling motifs. Here, we study the characteristics and the propagation mechanism of the output noise in a coherent feed-forward transcriptional regulatory loop that can be divided into a main road and branch. Using the linear noise approximation, we derive analytical formulae for the total noise of the full loop, the noise of the branch, and the noise of the main road, which are verified by the Gillespie algorithm. Importantly, we find that (i) compared with the branch motif or the main road motif, the full motif can effectively attenuate the output noise level; (ii) there is a transition point of system state such that the noise of the main road is dominated when the underlying system is below this point, whereas the noise of the branch is dominated when the system is beyond the point. The entire analysis reveals the mechanism of how the noise is generated and propagated in a simple yet representative signaling module.

Stochasticity in the Genotype-Phenotype Map: Implications for the Robustness and Persistence of Bet-Hedging

Nongenetic variation in phenotypes, or bet-hedging, has been observed as a driver of drug resistance in both bacterial infections and cancers. Here, we study how bet-hedging emerges in genotype-phenotype (GP) mapping through a simple interaction model: a molecular switch. We use simple chemical reaction networks to implement stochastic switches that map gene products to phenotypes, and investigate the impact of structurally distinct mappings on the evolution of phenotypic heterogeneity. Bet-hedging naturally emerges within this model, and is robust to evolutionary loss through mutations to both the expression of individual genes, and to the network itself. This robustness explains an apparent paradox of bet-hedging-why does it persist in environments where natural selection necessarily acts to remove it? The structure of the underlying molecular mechanism, itself subject to selection, can slow the evolutionary loss of bet-hedging to ensure a survival mechanism against environmental catastrophes even when they are rare. Critically, these properties, taken together, have profound implications for the use of treatment-holidays to combat bet-hedging-driven resistant disease, as the efficacy of breaks from treatment will ultimately be determined by the structure of the GP mapping.

Bursting noise in gene expression dynamics: linking microscopic and mesoscopic models

The dynamics of short-lived mRNA results in bursts of protein production in gene regulatory networks. We investigate the propagation of bursting noise between different levels of mathematical modelling and demonstrate that conventional approaches based on diffusion approximations can fail to capture bursting noise. An alternative coarse-grained model, the so-called piecewise deterministic Markov process (PDMP), is seen to outperform the diffusion approximation in biologically relevant parameter regimes. We provide a systematic embedding of the PDMP model into the landscape of existing approaches, and we present analytical methods to calculate its stationary distribution and switching frequencies.

Lévy-noise-induced transport in a rough triple-well potential

Rough energy landscape and noisy environment are two common features in many subjects, such as protein folding. Due to the wide findings of bursting or spiking phenomenon in biology science, small diffusions mixing large jumps are adopted to model the noisy environment that can be properly described by Lévy noise. We combine the Lévy noise with the rough energy landscape, modeled by a potential function superimposed by a fast oscillating function, and study the transport of a particle in a rough triple-well potential excited by Lévy noise, rather than only small perturbations. The probabilities of a particle staying in the middle well are considered under different amplitudes of roughness to find out how roughness affects the steady-state probability density function. Variations in the mean first passage time from the middle well to the right well have been investigated with respect to Lévy parameters and amplitudes of the roughness. In addition, we have examined the influences of roughness on the splitting probabilities of the first escape from the middle well. We uncover that the roughness can enhance significantly the first escape of a particle from the middle well, especially for different skewness parameters, but weak differences are found for stability index and noise intensity on the probabilities a particle staying in the middle well and splitting probability to the right.

Patterns of patterns of synchronization: Noise induced attractor switching in rings of coupled nonlinear oscillators

Following the long-lived qualitative-dynamics tradition of explaining behavior in complex systems via the architecture of their attractors and basins, we investigate the patterns of switching between distinct trajectories in a network of synchronized oscillators. Our system, consisting of nonlinear amplitude-phase oscillators arranged in a ring topology with reactive nearest-neighbor coupling, is simple and connects directly to experimental realizations. We seek to understand how the multiple stable synchronized states connect to each other in state space by applying Gaussian white noise to each of the oscillators' phases. To do this, we first analytically identify a set of locally stable limit cycles at any given coupling strength. For each of these attracting states, we analyze the effect of weak noise via the covariance matrix of deviations around those attractors. We then explore the noise-induced attractor switching behavior via numerical investigations. For a ring of three oscillators, we find that an attractor-switching event is always accompanied by the crossing of two adjacent oscillators' phases. For larger numbers of oscillators, we find that the distribution of times required to stochastically leave a given state falls off exponentially, and we build an attractor switching network out of the destination states as a coarse-grained description of the high-dimensional attractor-basin architecture.