Noise contributions in an inducible genetic switch: a whole-cell simulation study

Switching off: The phenotypic transition to the uninduced state of the lactose uptake pathway

The lactose uptake pathway of E. coli is a paradigmatic example of multistability in gene regulatory circuits. In the induced state of the lac pathway, the genes comprising the lac operon are transcribed, leading to the production of proteins that import and metabolize lactose. In the uninduced state, a stable repressor-DNA loop frequently blocks the transcription of the lac genes. Transitions from one phenotypic state to the other are driven by fluctuations, which arise from the random timing of the binding of ligands and proteins. This stochasticity affects transcription and translation, and ultimately molecular copy numbers. Our aim is to understand the transition from the induced to the uninduced state of the lac operon. We use a detailed computational model to show that repressor-operator binding and unbinding, fluctuations in the total number of repressors, and inducer-repressor binding and unbinding all play a role in this transition. Based on the timescales on which these processes operate, we construct a minimal model of the transition to the uninduced state and compare the results with simulations and experimental observations. The induced state turns out to be very stable, with a transition rate to the uninduced state lower than 2×10^-9 per minute. In contrast to the transition to the induced state, the transition to the uninduced state is well described in terms of a 2D diffusive system crossing a barrier, with the diffusion rates emerging from a model of repressor unbinding.

Heterogeneity is not always a source of noise: Stochastic gene expression in regulatory heterozygote

Zygosity of diploid genome (i.e., degree to which two parental alleles of a gene have varied genetic sequences) adds another dimension to stochastic gene expression. The allelic imbalance in chromatin accessibility or divergence in regulatory sequences leads to fitness effects but the quantitative aspects thereof are largely left unexplored. We investigate diploid gene expression systems with homozygous (the same) and heterozygous (varied) combination of alleles in cis-regulatory sequences, not in structural gene loci, and characterize the zygosity-associated stochastic fluctuations in protein abundance. An emerging feat of heterozygosity is its counterintuitive capacity for genetic noise control. Especially when the noise is dominantly contributed to by the fluctuations in duty cycle ("reliability") rather than in process speed ("productivity") of gene expression machinery, its interallelic discrepancy acts to reduce the gene expression noise. These findings offer a novel insight into the rich repertoire of balancing selection enriched in the regulatory elements of immune response genes.

Polymer brush-induced depletion interactions and clustering of membrane proteins

We investigate the effect of mobile polymer brushes on proteins embedded in biological membranes by employing both Asakura-Oosawa type of theoretical model and coarse-grained molecular dynamics simulations. The brush polymer-induced depletion attraction between proteins changes non-monotonically with the size of brush. The depletion interaction, which is determined by the ratio of the protein size to the grafting distance between brush polymers, increases linearly with the brush size as long as the polymer brush height is shorter than the protein size. When the brush height exceeds the protein size, however, the depletion attraction among proteins is slightly reduced. We also explore the possibility of the brush polymer-induced assembly of a large protein cluster, which can be related to one of many molecular mechanisms underlying recent experimental observations of integrin nanocluster formation and signaling.

Generating Chromosome Geometries in a Minimal Cell From Cryo-Electron Tomograms and Chromosome Conformation Capture Maps

JCVI-syn3A is a genetically minimal bacterial cell, consisting of 493 genes and only a single 543 kbp circular chromosome. Syn3A’s genome and physical size are approximately one-tenth those of the model bacterial organism Escherichia coli’s, and the corresponding reduction in complexity and scale provides a unique opportunity for whole-cell modeling. Previous work established genome-scale gene essentiality and proteomics data along with its essential metabolic network and a kinetic model of genetic information processing. In addition to that information, whole-cell, spatially-resolved kinetic models require cellular architecture, including spatial distributions of ribosomes and the circular chromosome’s configuration. We reconstruct cellular architectures of Syn3A cells at the single-cell level directly from cryo-electron tomograms, including the ribosome distributions. We present a method of generating self-avoiding circular chromosome configurations in a lattice model with a resolution of 11.8 bp per monomer on a 4 nm cubic lattice. Realizations of the chromosome configurations are constrained by the ribosomes and geometry reconstructed from the tomograms and include DNA loops suggested by experimental chromosome conformation capture (3C) maps. Using ensembles of simulated chromosome configurations we predict chromosome contact maps for Syn3A cells at resolutions of 250 bp and greater and compare them to the experimental maps. Additionally, the spatial distributions of ribosomes and the DNA-crowding resulting from the individual chromosome configurations can be used to identify macromolecular structures formed from ribosomes and DNA, such as polysomes and expressomes.

A forecast for large-scale, predictive biology: Lessons from meteorology

Quantitative systems biology, in which predictive mathematical models are constructed to guide the design of experiments and predict experimental outcomes, is at an exciting transition point, where the foundational scientific principles are becoming established, but the impact is not yet global. The next steps necessary for mathematical modeling to transform biological research and applications, in the same way it has already transformed other fields, is not completely clear. The purpose of this perspective is to forecast possible answers to this question-what needs to happen next-by drawing on the experience gained in another field, specifically meteorology. We review here a number of lessons learned in weather prediction that are directly relevant to biological systems modeling, and that we believe can enable the same kinds of global impact in our field as atmospheric modeling makes today.

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.

Simultaneous cross-evaluation of heterogeneous E. coli datasets via mechanistic simulation

The extensive heterogeneity of biological data poses challenges to analysis and interpretation. Construction of a large-scale mechanistic model of Escherichia coli enabled us to integrate and cross-evaluate a massive, heterogeneous dataset based on measurements reported by various groups over decades. We identified inconsistencies with functional consequences across the data, including that the total output of the ribosomes and RNA polymerases described by data are not sufficient for a cell to reproduce measured doubling times, that measured metabolic parameters are neither fully compatible with each other nor with overall growth, and that essential proteins are absent during the cell cycle-and the cell is robust to this absence. Finally, considering these data as a whole leads to successful predictions of new experimental outcomes, in this case protein half-lives.

An in-silico human cell model reveals the influence of spatial organization on RNA splicing

Spatial organization is a characteristic of all cells, achieved in eukaryotic cells by utilizing both membrane-bound and membrane-less organelles. One of the key processes in eukaryotes is RNA splicing, which readies mRNA for translation. This complex and highly dynamical chemical process involves assembly and disassembly of many molecules in multiple cellular compartments and their transport among compartments. Our goal is to model the effect of spatial organization of membrane-less organelles (specifically nuclear speckles) and of organelle heterogeneity on splicing particle biogenesis in mammalian cells. Based on multiple sources of complementary experimental data, we constructed a spatial model of a HeLa cell to capture intracellular crowding effects. We then developed chemical reaction networks to describe the formation of RNA splicing machinery complexes and splicing processes within nuclear speckles (specific type of non-membrane-bound organelles). We incorporated these networks into our spatially-resolved human cell model and performed stochastic simulations for up to 15 minutes of biological time, the longest thus far for a eukaryotic cell. We find that an increase (decrease) in the number of nuclear pore complexes increases (decreases) the number of assembled splicing particles; and that compartmentalization is critical for the yield of correctly-assembled particles. We also show that a slight increase of splicing particle localization into nuclear speckles leads to a disproportionate enhancement of mRNA splicing and a reduction in the noise of generated mRNA. Our model also predicts that the distance between genes and speckles has a considerable effect on the mRNA production rate, with genes located closer to speckles producing mRNA at higher levels, emphasizing the importance of genome organization around speckles. The HeLa cell model, including organelles and sub-compartments, provides a flexible foundation to study other cellular processes that are strongly modulated by spatiotemporal heterogeneity.

Automatic error control during forward flux sampling of rare events in master equation models

Enhanced sampling methods, such as forward flux sampling (FFS), have great capacity for accelerating stochastic simulations of nonequilibrium biochemical systems involving rare events. However, the description of the tradeoffs between simulation efficiency and error in FFS remains incomplete. We present a novel and mathematically rigorous analysis of the errors in FFS that, for the first time, covers the contribution of every phase of the simulation. We derive a closed form expression for the optimally efficient count of samples to take in each FFS phase in terms of a fixed constraint on sampling error. We introduce a new method, forward flux pilot sampling (FFPilot), that is designed to take full advantage of our optimizing equation without prior information or assumptions about the phase weights and costs along the transition path. In simulations of both single and multidimensional gene regulatory networks, FFPilot is able to completely control sampling error. We then discuss how memory effects can introduce additional error when relaxation along the transition path is slow. This extra error can be traced to correlations between the FFS phases and can be controlled by monitoring the covariance between them. Finally, we show that, in sets of simulations with matched error, FFPilot is on the order of tens-to-hundreds of times faster than direct sampling and noticeably more efficient than previous FFS methods.

On Protein Folding in Crowded Conditions

The interior of a cell is a highly packed environment that can be occupied up to 40% by different macromolecules. Such crowded media influence different biochemical processes like protein folding, enzymatic activity, and gene regulation. In this work, we use simulations to study protein stability under the presence of crowding agents that interact with the protein by excluded volume interactions. In general, the presence of crowding agents in the solution enhances the stability of the protein's native state. However, we find that the effects of excluded volume depend not only on crowding occupancy but also the crowders' geometry and size. Specifically, we find that polymeric crowders have stronger influence than spherical crowders and that this effect increases with polymer length, while it decreases with increasing size of spherical crowders. These opposing size effects are explained by the interplay of decreasing excluded volume and demixing, which together determine the change in the entropy of the crowders upon folding of the protein.

Model reduction methods for population dynamics with fast-switching environments: Reduced master equations, stochastic differential equations, and applications

We study stochastic population dynamics coupled to fast external environments and combine expansions in the inverse switching rate of the environment and a Kramers-Moyal expansion in the inverse size of the population. This leads to a series of approximation schemes, capturing both intrinsic and environmental noise. These methods provide a means of efficient simulation and we show how they can be used to obtain analytical results for the fluctuations of population dynamics in switching environments. We place the approximations in relation to existing work on piecewise-deterministic and piecewise-diffusive Markov processes. Finally, we demonstrate the accuracy and efficiency of these model-reduction methods in different research fields, including systems in biology and a model of crack propagation.

ReaDDy 2: Fast and flexible software framework for interacting-particle reaction dynamics

Interacting-particle reaction dynamics (iPRD) combines the simulation of dynamical trajectories of interacting particles as in molecular dynamics (MD) simulations with reaction kinetics, in which particles appear, disappear, or change their type and interactions based on a set of reaction rules. This combination facilitates the simulation of reaction kinetics in crowded environments, involving complex molecular geometries such as polymers, and employing complex reaction mechanisms such as breaking and fusion of polymers. iPRD simulations are ideal to simulate the detailed spatiotemporal reaction mechanism in complex and dense environments, such as in signalling processes at cellular membranes, or in nano- to microscale chemical reactors. Here we introduce the iPRD software ReaDDy 2, which provides a Python interface in which the simulation environment, particle interactions and reaction rules can be conveniently defined and the simulation can be run, stored and analyzed. A C++ interface is available to enable deeper and more flexible interactions with the framework. The main computational work of ReaDDy 2 is done in hardware-specific simulation kernels. While the version introduced here provides single- and multi-threading CPU kernels, the architecture is ready to implement GPU and multi-node kernels. We demonstrate the efficiency and validity of ReaDDy 2 using several benchmark examples. ReaDDy 2 is available at the https://readdy.github.io/ website.

Phenotypic instability in fungi

Fungi are prone to phenotypic instability, that is, the vegetative phase of these organisms, be they yeasts or molds, undergoes frequent switching between two or more behaviors, often with different morphologies, but also sometime having different physiologies without any obvious morphological outcome. In the context of industrial utilization of fungi, this can have a negative impact on the maintenance of strains and/or on their productivity. Instabilities have been shown to result from various mechanisms, either genetic or epigenetic. This chapter will review different types of instabilities and discuss some lesser-known ones, mostly in filamentous fungi, while it will direct readers to additional literature in the case of well-known phenomena such as the amyloid prions or fungal senescence. It will present in depth the "white/opaque" switch of Candida albicans and the "crippled growth" degeneration of the model fungus Podospora anserina. These are two of the most thoroughly studied epigenetic phenotypic switches. I will also discuss the "sectors" presented by many filamentous ascomycetes, for which a prion-based model exists but is not demonstrated. Finally, I will also describe intriguing examples of phenotypic instability for which an explanation has yet to be provided.

Elucidating effects of reaction rates on dynamics of the lac circuit in Escherichia coli

Gene expression is regulated by a complex transcriptional network. It is of interest to quantify uncertainty of not knowing accurately reaction rates of underlying biochemical reactions, and to understand how they affect gene expression. Assuming a kinetic model of the lac circuit in Escherichia coli, regardless of how many reactions are involved in transcription regulation, transcription rate is shown to be the most important parameter affecting steady state production of mRNA and protein in the cell. In particular, doubling the transcription rate approximately doubles the number of mRNA synthesized at steady state for any rates of transcription inhibition and activation. On the other hand, increasing the rate of transcription inhibition by 10% reduces the average steady state count of mRNA by about 7%, whereas changes in the rate of transcription activation appear to have no such effect. Furthermore, for wide range of reaction rates in the kinetic model of the lac genetic switch considered, protein production was observed to always reach a maximum before the degradation reduces its count to zero, and this maximum was found to be always at least 27 protein molecules. Such value appears to be a fundamental structural property of genetic circuits making it very robust against changes in the internal and external conditions.

Determining the Transcription Rates Yielding Steady-State Production of mRNA in the Lac Genetic Switch of Escherichia coli

To elucidate the regulatory dynamics of the gene expression activation and inactivation, an in silico biochemical model of the lac circuit in Escherichia coli was used to evaluate the transcription rates that yield the steady-state mRNA production in active and inactive states of the lac circuit. This result can be used in synthetic biology applications to understand the limits of the genetic synthesis. Since most genetic networks involve many interconnected components with positive and negative feedback control, intuitive understanding of their dynamics is often difficult to obtain. Although the kinetic model of the lac circuit considered involves only a single positive feedback, the developed computational framework can be used to evaluate supported ranges of other reaction rates in genetic circuits with more complex regulatory networks. More specifically, the inducible lac gene switch in E. coli is regulated by unbinding and binding of the inducer-repressor complexes to or from the DNA operator to switch the gene expression on and off. The dependency of mRNA production at steady state on different transcription rates and the repressor complexes has been studied by computer simulations in the Lattice Microbe software. Provided that the lac circuit is in active state, the transcription rate is independent of the inducer-repressor complexes present in the cell. In inactive state, the transcription rate is dependent on the specific inducer-repressor complex bound to the operator that inactivates the gene expression. We found that the repressor complex with the largest affinity to the operator yields the smallest range of the feasible transcription rates to yield the steady state while the lac circuit is in inactive state. In contrast, the steady state in active state can be obtained for any value of the transcription rate.

Simulating biological processes: stochastic physics from whole cells to colonies

The last few decades have revealed the living cell to be a crowded spatially heterogeneous space teeming with biomolecules whose concentrations and activities are governed by intrinsically random forces. It is from this randomness, however, that a vast array of precisely timed and intricately coordinated biological functions emerge that give rise to the complex forms and behaviors we see in the biosphere around us. This seemingly paradoxical nature of life has drawn the interest of an increasing number of physicists, and recent years have seen stochastic modeling grow into a major subdiscipline within biological physics. Here we review some of the major advances that have shaped our understanding of stochasticity in biology. We begin with some historical context, outlining a string of important experimental results that motivated the development of stochastic modeling. We then embark upon a fairly rigorous treatment of the simulation methods that are currently available for the treatment of stochastic biological models, with an eye toward comparing and contrasting their realms of applicability, and the care that must be taken when parameterizing them. Following that, we describe how stochasticity impacts several key biological functions, including transcription, translation, ribosome biogenesis, chromosome replication, and metabolism, before considering how the functions may be coupled into a comprehensive model of a 'minimal cell'. Finally, we close with our expectation for the future of the field, focusing on how mesoscopic stochastic methods may be augmented with atomic-scale molecular modeling approaches in order to understand life across a range of length and time scales.

A Digitally Programmable Cytomorphic Chip for Simulation of Arbitrary Biochemical Reaction Networks

Prior work has shown that compact analog circuits can faithfully represent and model fundamental biomolecular circuits via efficient log-domain cytomorphic transistor equivalents. Such circuits have emphasized basis functions that are dominant in genetic transcription and translation networks and deoxyribonucleic acid (DNA)-protein binding. Here, we report a system featuring digitally programmable 0.35 μm BiCMOS analog cytomorphic chips that enable arbitrary biochemical reaction networks to be exactly represented thus enabling compact and easy composition of protein networks as well. Since all biomolecular networks can be represented as chemical reaction networks, our protein networks also include the former genetic network circuits as a special case. The cytomorphic analog protein circuits use one fundamental association-dissociation-degradation building-block circuit that can be configured digitally to exactly represent any zeroth-, first-, and second-order reaction including loading, dynamics, nonlinearity, and interactions with other building-block circuits. To address a divergence issue caused by random variations in chip fabrication processes, we propose a unique way of performing computation based on total variables and conservation laws, which we instantiate at both the circuit and network levels. Thus, scalable systems that operate with finite error over infinite time can be built. We show how the building-block circuits can be composed to form various network topologies, such as cascade, fan-out, fan-in, loop, dimerization, or arbitrary networks using total variables. We demonstrate results from a system that combines interacting cytomorphic chips to simulate a cancer pathway and a glycolysis pathway. Both simulations are consistent with conventional software simulations. Our highly parallel digitally programmable analog cytomorphic systems can lead to a useful design, analysis, and simulation tool for studying arbitrary large-scale biological networks in systems and synthetic biology.

Emerging whole-cell modeling principles and methods

Whole-cell computational models aim to predict cellular phenotypes from genotype by representing the entire genome, the structure and concentration of each molecular species, each molecular interaction, and the extracellular environment. Whole-cell models have great potential to transform bioscience, bioengineering, and medicine. However, numerous challenges remain to achieve whole-cell models. Nevertheless, researchers are beginning to leverage recent progress in measurement technology, bioinformatics, data sharing, rule-based modeling, and multi-algorithmic simulation to build the first whole-cell models. We anticipate that ongoing efforts to develop scalable whole-cell modeling tools will enable dramatically more comprehensive and more accurate models, including models of human cells.

Challenges of Integrating Stochastic Dynamics and Cryo-Electron Tomograms in Whole-Cell Simulations

Cryo-electron tomography (cryo-ET) has rapidly emerged as a powerful tool to investigate the internal, three-dimensional spatial organization of the cell. In parallel, the GPU-based technology to perform spatially resolved stochastic simulations of whole cells has arisen, allowing the simulation of complex biochemical networks over cell cycle time scales using data taken from -omics, single molecule experiments, and in vitro kinetics. By using real cell geometry derived from cryo-ET data, we have the opportunity to imbue these highly detailed structural data-frozen in time-with realistic biochemical dynamics and investigate how cell structure affects the behavior of the embedded chemical reaction network. Here we present two examples to illustrate the challenges and techniques involved in integrating structural data into stochastic simulations. First, a tomographic reconstruction of Saccharomyces cerevisiae is used to construct the geometry of an entire cell through which a simple stochastic model of an inducible genetic switch is studied. Second, a tomogram of the nuclear periphery in a HeLa cell is converted directly to the simulation geometry through which we study the effects of cellular substructure on the stochastic dynamics of gene repression. These simple chemical models allow us to illustrate how to build whole-cell simulations using cryo-ET derived geometry and the challenges involved in such a process.

Effects of cell cycle noise on excitable gene circuits

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a three-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

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.

Ribosome biogenesis in replicating cells: Integration of experiment and theory

Ribosomes-the primary macromolecular machines responsible for translating the genetic code into proteins-are complexes of precisely folded RNA and proteins. The ways in which their production and assembly are managed by the living cell is of deep biological importance. Here we extend a recent spatially resolved whole-cell model of ribosome biogenesis in a fixed volume [Earnest et al., Biophys J 2015, 109, 1117-1135] to include the effects of growth, DNA replication, and cell division. All biological processes are described in terms of reaction-diffusion master equations and solved stochastically using the Lattice Microbes simulation software. In order to determine the replication parameters, we construct and analyze a series of Escherichia coli strains with fluorescently labeled genes distributed evenly throughout their chromosomes. By measuring these cells' lengths and number of gene copies at the single-cell level, we could fit a statistical model of the initiation and duration of chromosome replication. We found that for our slow-growing (120 min doubling time) E. coli cells, replication was initiated 42 min into the cell cycle and completed after an additional 42 min. While simulations of the biogenesis model produce the correct ribosome and mRNA counts over the cell cycle, the kinetic parameters for transcription and degradation are lower than anticipated from a recent analytical time dependent model of in vivo mRNA production. Describing expression in terms of a simple chemical master equation, we show that the discrepancies are due to the lack of nonribosomal genes in the extended biogenesis model which effects the competition of mRNA for ribosome binding, and suggest corrections to parameters to be used in the whole-cell model when modeling expression of the entire transcriptome. © 2016 Wiley Periodicals, Inc. Biopolymers 105: 735-751, 2016.

Predictive biophysical modeling and understanding of the dynamics of mRNA translation and its evolution

mRNA translation is the fundamental process of decoding the information encoded in mRNA molecules by the ribosome for the synthesis of proteins. The centrality of this process in various biomedical disciplines such as cell biology, evolution and biotechnology, encouraged the development of dozens of mathematical and computational models of translation in recent years. These models aimed at capturing various biophysical aspects of the process. The objective of this review is to survey these models, focusing on those based and/or validated on real large-scale genomic data. We consider aspects such as the complexity of the models, the biophysical aspects they regard and the predictions they may provide. Furthermore, we survey the central systems biology discoveries reported on their basis. This review demonstrates the fundamental advantages of employing computational biophysical translation models in general, and discusses the relative advantages of the different approaches and the challenges in the field.

A biophysical model of supercoiling dependent transcription predicts a structural aspect to gene regulation

Background

Transcription in Escherichia coli generates positive supercoiling in the DNA, which is relieved by the enzymatic activity of gyrase. Recently published experimental evidence suggests that transcription initiation and elongation are inhibited by the buildup of positive supercoiling. It has therefore been proposed that intermittent binding of gyrase plays a role in transcriptional bursting. Considering that transcription is one of the most fundamental cellular processes, it is desirable to be able to account for the buildup and release of positive supercoiling in models of transcription.

Results

Here we present a detailed biophysical model of gene expression that incorporates the effects of supercoiling due to transcription. By directly linking the amount of positive supercoiling to the rate of transcription, the model predicts that highly transcribed genes’ mRNA distributions should substantially deviate from Poisson distributions, with enhanced density at low mRNA copy numbers. Additionally, the model predicts a high degree of correlation between expression levels of genes inside the same supercoiling domain.

Conclusions

Our model, incorporating the supercoiling state of the gene, makes specific predictions that differ from previous models of gene expression. Genes in the same supercoiling domain influence the expression level of neighboring genes. Such structurally dependent regulation predicts correlations between genes in the same supercoiling domain. The topology of the chromosome therefore creates a higher level of gene regulation, which has broad implications for understanding the evolution and organization of bacterial genomes.

Electronic supplementary material

The online version of this article (doi:10.1186/s13628-016-0027-0) contains supplementary material, which is available to authorized users.

Effects of DNA replication on mRNA noise

There are several sources of fluctuations in gene expression. Here we study the effects of time-dependent DNA replication, itself a tightly controlled process, on noise in mRNA levels. Stochastic simulations of constitutive and regulated gene expression are used to analyze the time-averaged mean and variation in each case. The simulations demonstrate that to capture mRNA distributions correctly, chromosome replication must be realistically modeled. Slow relaxation of mRNA from the low copy number steady state before gene replication to the high steady state after replication is set by the transcript's half-life and contributes significantly to the shape of the mRNA distribution. Consequently both the intrinsic kinetics and the gene location play an important role in accounting for the mRNA average and variance. Exact analytic expressions for moments of the mRNA distributions that depend on the DNA copy number, gene location, cell doubling time, and the rates of transcription and degradation are derived for the case of constitutive expression and subsequently extended to provide approximate corrections for regulated expression and RNA polymerase variability. Comparisons of the simulated models and analytical expressions to experimentally measured mRNA distributions show that they better capture the physics of the system than previous theories.

Sources of Variability in a Synthetic Gene Oscillator

Synthetic gene oscillators are small, engineered genetic circuits that produce periodic variations in target protein expression. Like other gene circuits, synthetic gene oscillators are noisy and exhibit fluctuations in amplitude and period. Understanding the origins of such variability is key to building predictive models that can guide the rational design of synthetic circuits. Here, we developed a method for determining the impact of different sources of noise in genetic oscillators by measuring the variability in oscillation amplitude and correlations between sister cells. We first used a combination of microfluidic devices and time-lapse fluorescence microscopy to track oscillations in cell lineages across many generations. We found that oscillation amplitude exhibited high cell-to-cell variability, while sister cells remained strongly correlated for many minutes after cell division. To understand how such variability arises, we constructed a computational model that identified the impact of various noise sources across the lineage of an initial cell. When each source of noise was appropriately tuned the model reproduced the experimentally observed amplitude variability and correlations, and accurately predicted outcomes under novel experimental conditions. Our combination of computational modeling and time-lapse data analysis provides a general way to examine the sources of variability in dynamic gene circuits.

Dynamics of simple gene-network motifs subject to extrinsic fluctuations

Cellular processes do not follow deterministic rules; even in identical environments genetically identical cells can make random choices leading to different phenotypes. This randomness originates from fluctuations present in the biomolecular interaction networks. Most previous work has been focused on the intrinsic noise (IN) of these networks. Yet, especially for high-copy-number biomolecules, extrinsic or environmental noise (EN) has been experimentally shown to dominate the variation. Here, we develop an analytical formalism that allows for calculation of the effect of EN on gene-expression motifs. We introduce a method for modeling bounded EN as an auxiliary species in the master equation. The method is fully generic and is not limited to systems with small EN magnitudes. We focus our study on motifs that can be viewed as the building blocks of genetic switches: a nonregulated gene, a self-inhibiting gene, and a self-promoting gene. The role of the EN properties (magnitude, correlation time, and distribution) on the statistics of interest are systematically investigated, and the effect of fluctuations in different reaction rates is compared. Due to its analytical nature, our formalism can be used to quantify the effect of EN on the dynamics of biochemical networks and can also be used to improve the interpretation of data from single-cell gene-expression experiments.

Why Build Whole-Cell Models?

Our ability to build computational models that account for all known gene functions in a cell has increased dramatically. But why build whole-cell models, and how can they best be used? In this forum, we enumerate several areas in which whole-cell modeling can significantly impact research and technology.

Toward a Whole-Cell Model of Ribosome Biogenesis: Kinetic Modeling of SSU Assembly

Central to all life is the assembly of the ribosome: a coordinated process involving the hierarchical association of ribosomal proteins to the RNAs forming the small and large ribosomal subunits. The process is further complicated by effects arising from the intracellular heterogeneous environment and the location of ribosomal operons within the cell. We provide a simplified model of ribosome biogenesis in slow-growing Escherichia coli. Kinetic models of in vitro small-subunit reconstitution at the level of individual protein/ribosomal RNA interactions are developed for two temperature regimes. The model at low temperatures predicts the existence of a novel 5'→3'→central assembly pathway, which we investigate further using molecular dynamics. The high-temperature assembly network is incorporated into a model of in vivo ribosome biogenesis in slow-growing E. coli. The model, described in terms of reaction-diffusion master equations, contains 1336 reactions and 251 species that dynamically couple transcription and translation to ribosome assembly. We use the Lattice Microbes software package to simulate the stochastic production of mRNA, proteins, and ribosome intermediates over a full cell cycle of 120 min. The whole-cell model captures the correct growth rate of ribosomes, predicts the localization of early assembly intermediates to the nucleoid region, and reproduces the known assembly timescales for the small subunit with no modifications made to the embedded in vitro assembly network.

Unexpected Swelling of Stiff DNA in a Polydisperse Crowded Environment

We investigate the conformations of DNA-like stiff chains, characterized by contour length (L) and persistence length (lp), in a variety of crowded environments containing monodisperse soft spherical (SS) and spherocylindrical (SC) particles, a mixture of SS and SC, and a milieu mimicking the composition of proteins in the Escherichia coli cytoplasm. The stiff chain, whose size modestly increases in SS crowders up to ϕ ≈ 0.1, is considerably more compact at low volume fractions (ϕ ≤ 0.2) in monodisperse SC particles than in a medium containing SS particles. A 1:1 mixture of SS and SC crowders induces greater chain compaction than the pure SS or SC crowders at the same ϕ, with the effect being highly nonadditive. We also discover a counterintuitive result that the polydisperse crowding environment, mimicking the composition of a cell lysate, swells the DNA-like polymer, which is in stark contrast to the size reduction of flexible polymers in the same milieu. Trapping of the stiff chain in a fluctuating tube-like environment created by large-sized crowders explains the dramatic increase in size and persistence length of the stiff chain. In the polydisperse medium, mimicking the cellular environment, the size of the DNA (or related RNA) is determined by L/lp. At low L/lp, the size of the polymer is unaffected, whereas there is a dramatic swelling at an intermediate value of L/lp. We use these results to provide insights into recent experiments on crowding effects on RNA and also make testable predictions.

The principles of whole-cell modeling

Whole-cell models which comprehensively predict how phenotypes emerge from genotype promise to enable rational bioengineering and precision medicine. Here, we outline the key principles of whole-cell modeling which have emerged from our work developing bacterial whole-cell models: single-cellularity; functional, genetic, molecular, and temporal completeness; biophysical realism including temporal dynamics and stochastic variation; species-specificity; and model integration and reproducibility. We also outline the whole-cell model construction process, highlighting existing resources. Numerous challenges remain to achieving fully complete models including developing new experimental tools to more completely characterize cells and developing a strong theoretical understanding of hybrid mathematics. Solving these challenges requires collaboration among computational and experimental biologists, biophysicists, biochemists, applied mathematicians, computer scientists, and software engineers.

Iterative experiment design guides the characterization of a light-inducible gene expression circuit

Systems biology rests on the idea that biological complexity can be better unraveled through the interplay of modeling and experimentation. However, the success of this approach depends critically on the informativeness of the chosen experiments, which is usually unknown a priori. Here, we propose a systematic scheme based on iterations of optimal experiment design, flow cytometry experiments, and Bayesian parameter inference to guide the discovery process in the case of stochastic biochemical reaction networks. To illustrate the benefit of our methodology, we apply it to the characterization of an engineered light-inducible gene expression circuit in yeast and compare the performance of the resulting model with models identified from nonoptimal experiments. In particular, we compare the parameter posterior distributions and the precision to which the outcome of future experiments can be predicted. Moreover, we illustrate how the identified stochastic model can be used to determine light induction patterns that make either the average amount of protein or the variability in a population of cells follow a desired profile. Our results show that optimal experiment design allows one to derive models that are accurate enough to precisely predict and regulate the protein expression in heterogeneous cell populations over extended periods of time.

A Complete Pathway Model for Lipid A Biosynthesis in Escherichia coli

Lipid A is a highly conserved component of lipopolysaccharide (LPS), itself a major component of the outer membrane of Gram-negative bacteria. Lipid A is essential to cells and elicits a strong immune response from humans and other animals. We developed a quantitative model of the nine enzyme-catalyzed steps of Escherichia coli lipid A biosynthesis, drawing parameters from the experimental literature. This model accounts for biosynthesis regulation, which occurs through regulated degradation of the LpxC and WaaA (also called KdtA) enzymes. The LpxC degradation signal appears to arise from the lipid A disaccharide concentration, which we deduced from prior results, model results, and new LpxK overexpression results. The model agrees reasonably well with many experimental findings, including the lipid A production rate, the behaviors of mutants with defective LpxA enzymes, correlations between LpxC half-lives and cell generation times, and the effects of LpxK overexpression on LpxC concentrations. Its predictions also differ from some experimental results, which suggest modifications to the current understanding of the lipid A pathway, such as the possibility that LpxD can replace LpxA and that there may be metabolic channeling between LpxH and LpxB. The model shows that WaaA regulation may serve to regulate the lipid A production rate when the 3-deoxy-D-manno-oct-2-ulosonic acid (KDO) concentration is low and/or to control the number of KDO residues that get attached to lipid A. Computation of flux control coefficients showed that LpxC is the rate-limiting enzyme if pathway regulation is ignored, but that LpxK is the rate-limiting enzyme if pathway regulation is present, as it is in real cells. Control also shifts to other enzymes if the pathway substrate concentrations are not in excess. Based on these results, we suggest that LpxK may be a much better drug target than LpxC, which has been pursued most often.

Space-induced bifurcation in repression-based transcriptional circuits

BACKGROUND

Albeit the molecular mechanisms of gene expression are well documented, our understanding of their dynamics is much less advanced. Recent experimental evidence has revealed that gene expression might be accurately organized in space, with several molecular actors localized to specific positions in the cell. However, the influence of this spatial localization on the dynamics of gene expression is unclear. This issue is also central in synthetic biology, where one usually considers the spatial localization in the cell of the genes of the inserted synthetic construct as irrelevant for its temporal dynamics.

RESULTS

Here, we assessed the influence of the spatial distribution of the genes on the dynamics of 3-gene transcriptional ring networks regulated by repression, i.e. repressilator circuits, using individual-based modelling to simulate their dynamics in two and three space dimensions. Our simulations suggest that variations of spatial parameters - namely the degree of demixing of the positions of the gene or the spatial range of the mRNA and proteins (i.e. the typical distance they travel before degradation) - have dramatic effects by switching the dynamical regime from spontaneous oscillations to a stationary state where each species fluctuates around a constant value. By analogy with the bifurcations arising from the variation of kinetic parameters, we referred to those transitions as space-induced bifurcations.

CONCLUSIONS

Taken together, our results strongly support the idea that the spatial organization of the molecular actors of transcriptional networks is crucial for the dynamics of gene expression and suggest that the spatial localization of the synthetic genes in the cell could be used as an additional toggle to control the dynamics of the inserted construct in synthetic biology experiments.

Diagnostics for stochastic genome-scale modeling via model slicing and debugging

Modeling of biological behavior has evolved from simple gene expression plots represented by mathematical equations to genome-scale systems biology networks. However, due to obstacles in complexity and scalability of creating genome-scale models, several biological modelers have turned to programming or scripting languages and away from modeling fundamentals. In doing so, they have traded the ability to have exchangeable, standardized model representation formats, while those that remain true to standardized model representation are faced with challenges in model complexity and analysis. We have developed a model diagnostic methodology inspired by program slicing and debugging and demonstrate the effectiveness of the methodology on a genome-scale metabolic network model published in the BioModels database. The computer-aided identification revealed specific points of interest such as reversibility of reactions, initialization of species amounts, and parameter estimation that improved a candidate cell's adenosine triphosphate production. We then compared the advantages of our methodology over other modeling techniques such as model checking and model reduction. A software application that implements the methodology is available at http://gel.ym.edu.tw/gcs/.

What makes the lac-pathway switch: identifying the fluctuations that trigger phenotype switching in gene regulatory systems

Multistable gene regulatory systems sustain different levels of gene expression under identical external conditions. Such multistability is used to encode phenotypic states in processes including nutrient uptake and persistence in bacteria, fate selection in viral infection, cell-cycle control and development. Stochastic switching between different phenotypes can occur as the result of random fluctuations in molecular copy numbers of mRNA and proteins arising in transcription, translation, transport and binding. However, which component of a pathway triggers such a transition is generally not known. By linking single-cell experiments on the lactose-uptake pathway in E. coli to molecular simulations, we devise a general method to pinpoint the particular fluctuation driving phenotype switching and apply this method to the transition between the uninduced and induced states of the lac-genes. We find that the transition to the induced state is not caused only by the single event of lac-repressor unbinding, but depends crucially on the time period over which the repressor remains unbound from the lac-operon. We confirm this notion in strains with a high expression level of the lac-repressor (leading to shorter periods over which the lac-operon remains unbound), which show a reduced switching rate. Our techniques apply to multistable gene regulatory systems in general and allow to identify the molecular mechanisms behind stochastic transitions in gene regulatory circuits.

Automated single cell microbioreactor for monitoring intracellular dynamics and cell growth in free solution

We report an automated microfluidic-based platform for single cell analysis that allows for cell culture in free solution with the ability to control the cell growth environment. Using this approach, cells are confined by the sole action of gentle fluid flow, thereby enabling non-perturbative analysis of cell growth away from solid boundaries. In addition, the single cell microbioreactor allows for precise and time-dependent control over cell culture media, with the combined ability to observe the dynamics of non-adherent cells over long time scales. As a proof-of-principle demonstration, we used the platform to observe dynamic cell growth, gene expression, and intracellular diffusion of repressor proteins while precisely tuning the cell growth environment. Overall, this microfluidic approach enables the direct observation of cellular dynamics with exquisite control over environmental conditions, which will be useful for quantifying the behaviour of single cells in well-defined media.

Whole Cell Modeling: From Single Cells to Colonies

A great deal of research over the last several years has focused on how the inherent randomness in movements and reactivity of biomolecules can give rise to unexpected large-scale differences in the behavior of otherwise identical cells. Our own research has approached this problem from two vantage points - a microscopic kinetic view of the individual molecules (nucleic acids, proteins, etc.) diffusing and interacting in a crowded cellular environment; and a broader systems-level view of how enzyme variability can give rise to well-defined metabolic phenotypes. The former led to the development of the Lattice Microbes software - a GPU-accelerated stochastic simulator for reaction-diffusion processes in models of whole cells; the latter to the development of a method we call population flux balance analysis (FBA). The first part of this article reviews the Lattice Microbes methodology, and two recent technical advances that extend the capabilities of Lattice Microbes to enable simulations of larger organisms and colonies. The second part of this article focuses on our recent population FBA study of Escherichia coli, which predicted variability in the usage of different metabolic pathways resulting from heterogeneity in protein expression. Finally, we discuss exciting early work using a new hybrid methodology that integrates FBA with spatially resolved kinetic simulations to study how cells compete and cooperate within dense colonies and consortia.

An integrative, multi-scale, genome-wide model reveals the phenotypic landscape of Escherichia coli

Given the vast behavioral repertoire and biological complexity of even the simplest organisms, accurately predicting phenotypes in novel environments and unveiling their biological organization is a challenging endeavor. Here, we present an integrative modeling methodology that unifies under a common framework the various biological processes and their interactions across multiple layers. We trained this methodology on an extensive normalized compendium for the gram-negative bacterium Escherichia coli, which incorporates gene expression data for genetic and environmental perturbations, transcriptional regulation, signal transduction, and metabolic pathways, as well as growth measurements. Comparison with measured growth and high-throughput data demonstrates the enhanced ability of the integrative model to predict phenotypic outcomes in various environmental and genetic conditions, even in cases where their underlying functions are under-represented in the training set. This work paves the way toward integrative techniques that extract knowledge from a variety of biological data to achieve more than the sum of their parts in the context of prediction, analysis, and redesign of biological systems.

Simulation of reaction diffusion processes over biologically relevant size and time scales using multi-GPU workstations

Simulation of in vivo cellular processes with the reaction-diffusion master equation (RDME) is a computationally expensive task. Our previous software enabled simulation of inhomogeneous biochemical systems for small bacteria over long time scales using the MPD-RDME method on a single GPU. Simulations of larger eukaryotic systems exceed the on-board memory capacity of individual GPUs, and long time simulations of modest-sized cells such as yeast are impractical on a single GPU. We present a new multi-GPU parallel implementation of the MPD-RDME method based on a spatial decomposition approach that supports dynamic load balancing for workstations containing GPUs of varying performance and memory capacity. We take advantage of high-performance features of CUDA for peer-to-peer GPU memory transfers and evaluate the performance of our algorithms on state-of-the-art GPU devices. We present parallel e ciency and performance results for simulations using multiple GPUs as system size, particle counts, and number of reactions grow. We also demonstrate multi-GPU performance in simulations of the Min protein system in E. coli. Moreover, our multi-GPU decomposition and load balancing approach can be generalized to other lattice-based problems.

Towards a computational model of a methane producing archaeum

Progress towards a complete model of the methanogenic archaeum Methanosarcina acetivorans is reported. We characterized size distribution of the cells using differential interference contrast microscopy, finding them to be ellipsoidal with mean length and width of 2.9 μm and 2.3 μm, respectively, when grown on methanol and 30% smaller when grown on acetate. We used the single molecule pull down (SiMPull) technique to measure average copy number of the Mcr complex and ribosomes. A kinetic model for the methanogenesis pathways based on biochemical studies and recent metabolic reconstructions for several related methanogens is presented. In this model, 26 reactions in the methanogenesis pathways are coupled to a cell mass production reaction that updates enzyme concentrations. RNA expression data (RNA-seq) measured for cell cultures grown on acetate and methanol is used to estimate relative protein production per mole of ATP consumed. The model captures the experimentally observed methane production rates for cells growing on methanol and is most sensitive to the number of methyl-coenzyme-M reductase (Mcr) and methyl-tetrahydromethanopterin:coenzyme-M methyltransferase (Mtr) proteins. A draft transcriptional regulation network based on known interactions is proposed which we intend to integrate with the kinetic model to allow dynamic regulation.

The future of whole-cell modeling

Integrated whole-cell modeling is poised to make a dramatic impact on molecular and systems biology, bioengineering, and medicine--once certain obstacles are overcome. From our group's experience building a whole-cell model of Mycoplasma genitalium, we identified several significant challenges to building models of more complex cells. Here we review and discuss these challenges in seven areas: first, experimental interrogation; second, data curation; third, model building and integration; fourth, accelerated computation; fifth, analysis and visualization; sixth, model validation; and seventh, collaboration and community development. Surmounting these challenges will require the cooperation of an interdisciplinary group of researchers to create increasingly sophisticated whole-cell models and make data, models, and simulations more accessible to the wider community.

Cellular and molecular structure as a unifying framework for whole-cell modeling

Whole-cell modeling has the potential to play a major role in revolutionizing our understanding of cellular biology over the next few decades. A computational model of the entire cell would allow cellular biologists to integrate data from many disparate sources in a single consistent framework. Such a comprehensive model would be useful both for hypothesis testing and in the discovery of new behaviors that emerge from complex biological networks. Cellular and molecular structure can and should be a key organizing principle in a whole-cell model, connecting models across time and length scales in a multiscale approach. Here I present a summary of recent research centered around using molecular and cellular structure to model the behavior of cells.

FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW

A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

Comparing protein folding in vitro and in vivo: foldability meets the fitness challenge

In this review, we compare and contrast current knowledge about in vitro and in vivo protein folding. Major advances in understanding fundamental principles underlying protein folding in optimized in vitro conditions have yielded detailed physicochemical principles of folding landscapes for small, single domain proteins. In addition, there has been increased research focusing on the key features of protein folding in the cell that differentiate it from in vitro folding, such as co-translational folding, chaperone-facilitated folding, and folding in crowded conditions with many weak interactions. Yet these two research areas have not been bridged effectively in research carried out to date. This review points to gaps between the two that are ripe for future research. Moreover, we emphasize the biological selection pressures that impact protein folding in vivo and how fitness drives the evolution of protein sequences in ways that may place foldability in tension with other requirements on a given protein. We suggest that viewing the physicochemical process of protein folding through the lens of evolution will unveil new insights and pose novel challenges about in-cell folding landscapes.

Parameter discovery in stochastic biological models using simulated annealing and statistical model checking

Stochastic models are increasingly used to study the behaviour of biochemical systems. While the structure of such models is often readily available from first principles, unknown quantitative features of the model are incorporated into the model as parameters. Algorithmic discovery of parameter values from experimentally observed facts remains a challenge for the computational systems biology community. We present a new parameter discovery algorithm that uses simulated annealing, sequential hypothesis testing, and statistical model checking to learn the parameters in a stochastic model. We apply our technique to a model of glucose and insulin metabolism used for in-silico validation of artificial pancreata and demonstrate its effectiveness by developing parallel CUDA-based implementation for parameter synthesis in this model.