Distance matters: the impact of gene proximity in bacterial gene regulation

Probing the modulation in facilitated diffusion guided by DNA-protein interactions in target search processes

Many fundamental biophysical processes involving gene regulation and gene editing rely, at the molecular level, on an intricate methodology of searching and locating the precise target base pair sequence on the genome by specific binding proteins. A unique mechanism, known as 'facilitated diffusion', which is a combination of 1D sliding along with 3D movement, is considered to be the key step for such events. This also explains the relatively much shorter timescale of the target searching process, compared to other diffusion-controlled biophysical processes. In this work, we aim to probe the modulation of target search dynamics of a protein moiety by estimating the rate of the target search process, and the statistics of the search rounds and timescales accomplished by the 1D and 3D motions, based on first passage time (FPT) calculations. This is studied with its characteristics getting influenced by various given conditions such as, when the DNA is rigid or flexible, and when the target is placed at different locations on the DNA. The current theoretical framework includes a Brownian dynamics simulation setup adopting a straightforward coarse-grained model for a diffusing protein on DNA. Moreover, this theoretical analysis provides insights into the complex target search dynamics by highlighting the significance of the chain dynamics in the mechanistic details of the facilitated diffusion process.

Fast Rare Events in Exit Times Distributions of Jump Processes

Rare events in the first-passage distributions of jump processes are capable of triggering anomalous reactions or series of events. Estimating their probability is particularly important when the jump probabilities have broad-tailed distributions, and rare events are therefore not so rare. We formulate a general approach for estimating the contribution of fast rare events to the exit probabilities in the presence of fat-tailed distributions. Using this approach, we study three jump processes that are used to model a wide class of phenomena ranging from biology to transport in disordered systems, ecology, and finance: discrete time random walks, Lévy walks, and the Lévy-Lorentz gas. We determine the exact form of the scaling function for the probability distribution of fast rare events, in which the jump process exits from an interval in a very short time at a large distance opposite to the starting point. In particular, we show that events occurring on timescales orders of magnitude smaller than the typical timescale of the process can make a significant contribution to the exit probability. Our results are confirmed by extensive numerical simulations.

Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking

A panoply of new tools for tracking single particles and molecules has led to an explosion of experimental data, leading to novel insights into physical properties of living matter governing cellular development and function, health and disease. In this Perspective, we present tools to investigate the dynamics and mechanics of living systems from the molecular to cellular scale via single-particle techniques. In particular, we focus on methods to measure, interpret, and analyse complex data sets that are associated with forces, materials properties, transport, and emergent organisation phenomena within biological and soft-matter systems. Current approaches, challenges, and existing solutions in the associated fields are outlined in order to support the growing community of researchers at the interface of physics and the life sciences. Each section focuses not only on the general physical principles and the potential for understanding living matter, but also on details of practical data extraction and analysis, discussing limitations, interpretation, and comparison across different experimental realisations and theoretical frameworks. Particularly relevant results are introduced as examples. While this Perspective describes living matter from a physical perspective, highlighting experimental and theoretical physics techniques relevant for such systems, it is also meant to serve as a solid starting point for researchers in the life sciences interested in the implementation of biophysical methods.

Theoretical insights into the full description of DNA target search by subdiffusing proteins

DNA binding proteins (DBPs) diffuse in the cytoplasm to recognise and bind with their respective target sites on the DNA to initiate several biologically important processes. The first passage time distributions (FPTDs) of DBPs are useful in quantifying the timescales of the most-probable search paths in addition to the mean value of the distribution which, strikingly, are decades of order apart in time. However, extremely crowded in vivo conditions or the viscoelasticity of the cellular medium among other factors causes biomolecules to exhibit anomalous diffusion which is usually overlooked in most theoretical studies. We have obtained approximate analytical expressions of a general FPTD and the two characteristic timescales that are valid for any single subdiffusing protein searching for its target in vivo. Our results can be applied to single-particle tracking experiments of target search.

First-passage times in conical varying-width channels biased by a transverse gravitational force: Comparison of analytical and numerical results

We study the crossing time statistic of diffusing point particles between the two ends of expanding and narrowing two-dimensional conical channels under a transverse external gravitational field. The theoretical expression for the mean first-passage time for such a system is derived under the assumption that the axial diffusion in a two-dimensional channel of smoothly varying geometry can be approximately described as a one-dimensional diffusion in an entropic potential with position-dependent effective diffusivity in terms of the modified Fick-Jacobs equation. We analyze the channel crossing dynamics in terms of the mean first-passage time, combining our analytical results with extensive two-dimensional Brownian dynamics simulations, allowing us to find the range of applicability of the one-dimensional approximation. We find that the effective particle diffusivity decreases with increasing amplitude of the external potential. Remarkably, the mean first-passage time for crossing the channel is shown to assume a minimum at finite values of the potential amplitude.

The Spatial Organization of Bacterial Transcriptional Regulatory Networks

The transcriptional regulatory network (TRN) is the central pivot of a prokaryotic organism to receive, process and respond to internal and external environmental information. However, little is known about its spatial organization so far. In recent years, chromatin interaction data of bacteria such as Escherichia coli and Bacillus subtilis have been published, making it possible to study the spatial organization of bacterial transcriptional regulatory networks. By combining TRNs and chromatin interaction data of E. coli and B. subtilis, we explored the spatial organization characteristics of bacterial TRNs in many aspects such as regulation directions (positive and negative), central nodes (hubs, bottlenecks), hierarchical levels (top, middle, bottom) and network motifs (feed-forward loops and single input modules) of the TRNs and found that the bacterial TRNs have a variety of stable spatial organization features under different physiological conditions that may be closely related with biological functions. Our findings provided new insights into the connection between transcriptional regulation and the spatial organization of chromosome in bacteria and might serve as a factual foundation for trying spatial-distance-based gene circuit design in synthetic biology.

Influence of Nonspecific Interactions between Proteins and In Vivo Cytoplasmic Crowders in Facilitated Diffusion of Proteins: Theoretical Insights

The binding of proteins to their respective specific sites on the DNA through facilitated diffusion serves as the initial step of various important biological processes. While this search process has been thoroughly investigated via in vitro studies, the cellular environment is complex and may interfere with the protein's search dynamics. The cytosol is heavily crowded, which can potentially modify the search by nonspecifically interacting with the protein that has been mostly overlooked. In this work, we probe the target search dynamics in the presence of explicit crowding agents that have an affinity toward the protein. We theoretically investigate the role of such protein-crowder associations in the target search process using a discrete-state stochastic framework that allows for the analytical description of dynamic properties. It is found that stronger nonspecific associations between the crowder and proteins can accelerate the facilitated diffusion of proteins in comparison with a purely inert, rather weakly interacting cellular environment. This effect depends on how strong these associations are, the spatial positions of the target with respect to the crowders, and the size of the crowded region. Our theoretical results are also tested with Monte Carlo computer simulations. Our predictions are in qualitative agreement with existing experimental observations and computational studies.

The economy of chromosomal distances in bacterial gene regulation

In the transcriptional regulatory network (TRN) of a bacterium, the nodes are genes and a directed edge represents the action of a transcription factor (TF), encoded by the source gene, on the target gene. It is a condensed representation of a large number of biological observations and facts. Nonrandom features of the network are structural evidence of requirements for a reliable systemic function. For the bacterium Escherichia coli we here investigate the (Euclidean) distances covered by the edges in the TRN when its nodes are embedded in the real space of the circular chromosome. Our work is motivated by 'wiring economy' research in Computational Neuroscience and starts from two contradictory hypotheses: (1) TFs are predominantly employed for long-distance regulation, while local regulation is exerted by chromosomal structure, locally coordinated by the action of structural proteins. Hence long distances should often occur. (2) A large distance between the regulator gene and its target requires a higher expression level of the regulator gene due to longer reaching times and ensuing increased degradation (proteolysis) of the TF and hence will be evolutionarily reduced. Our analysis supports the latter hypothesis.

Trading bits in the readout from a genetic network

In the regulation of gene expression, information of relevance to the organism is represented by the concentrations of transcription factor molecules. To extract this information the cell must effectively "measure" these concentrations, but there are physical limits to the precision of these measurements. We use the gap gene network in the early fly embryo as an example of the tradeoff between the precision of concentration measurements and the transmission of relevant information. For thresholded measurements we find that lower thresholds are more important, and fine tuning is not required for near-optimal information transmission. We then consider general sensors, constrained only by a limit on their information capacity, and find that thresholded sensors can approach true information theoretic optima. The information theoretic approach allows us to identify the optimal sensor for the entire gap gene network and to argue that the physical limitations of sensing necessitate the observed multiplicity of enhancer elements, with sensitivities to combinations rather than single transcription factors.

Stochastic resetting by a random amplitude

Stochastic resetting, a diffusive process whose amplitude is reset to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. Here we generalize the resetting step by introducing a random resetting amplitude such that the diffusing particle may be only partially reset towards the trajectory origin or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) and population dynamics or financial markets, as well as generic search processes.

Stochastic Kinetic Treatment of Protein Aggregation and the Effects of Macromolecular Crowding

Investigation of protein self-assembly processes is important for understanding the growth processes of functional proteins as well as disease-causing amyloids. Inside cells, intrinsic molecular fluctuations are so high that they cast doubt on the validity of the deterministic rate-equation approach. Furthermore, the protein environments inside cells are often crowded with other macromolecules, with volume fractions of the crowders as high as 40%. We have developed a stochastic kinetic framework using Gillespie's algorithm for general systems undergoing particle self-assembly, including particularly protein aggregation at the cellular level. The effects of macromolecular crowding are investigated using models built on scaled-particle and transition-state theories. The stochastic kinetic method can be formulated to provide information on the dominating aggregation mechanisms in a method called reaction frequency (or propensity) analysis. This method reveals that the change of scaling laws related to the lag time can be directly related to the change in the frequencies of reaction mechanisms. Further examination of the time evolution of the fibril mass and length quantities unveils that maximal fluctuations occur in the periods of rapid fibril growth and the fluctuations of both quantities can be sensitive functions of rate constants. The presence of crowders often amplifies the roles of primary and secondary nucleation and causes shifting in the relative importance of elongation, shrinking, fragmentation, and coagulation of linear aggregates. We also show a dual effect of changing volume on the halftime of aggregation for ApoC2 which is reduced in the presence of crowders. A comparison of the results of stochastic simulations with those of rate equations gives us information on the convergence relation between them and how the roles of reaction mechanisms change as the system volume is varied.

Quasi-robust control of biochemical reaction networks via stochastic morphing

One of the main objectives of synthetic biology is the development of molecular controllers that can manipulate the dynamics of a given biochemical network that is at most partially known. When integrated into smaller compartments, such as living or synthetic cells, controllers have to be calibrated to factor in the intrinsic noise. In this context, biochemical controllers put forward in the literature have focused on manipulating the mean (first moment) and reducing the variance (second moment) of the target molecular species. However, many critical biochemical processes are realized via higher-order moments, particularly the number and configuration of the probability distribution modes (maxima). To bridge the gap, we put forward the stochastic morpher controller that can, under suitable timescale separations, morph the probability distribution of the target molecular species into a predefined form. The morphing can be performed at a lower-resolution, allowing one to achieve desired multi-modality/multi-stability, and at a higher-resolution, allowing one to achieve arbitrary probability distributions. Properties of the controller, such as robustness and convergence, are rigorously established, and demonstrated on various examples. Also proposed is a blueprint for an experimental implementation of stochastic morpher.

Discrete-state stochastic kinetic models for target DNA search by proteins: Theory and experimental applications

To perform their functions, transcription factors and DNA-repair/modifying enzymes randomly search DNA in order to locate their specific targets on DNA. Discrete-state stochastic kinetic models have been developed to explain how the efficiency of the search process is influenced by the molecular properties of proteins and DNA as well as by other factors such as molecular crowding. These theoretical models not only offer explanations on the relation of microscopic processes to macroscopic behavior of proteins, but also facilitate the analysis and interpretation of experimental data. In this review article, we provide an overview on discrete-state stochastic kinetic models and explain how these models can be applied to experimental investigations using stopped-flow, single-molecule, nuclear magnetic resonance (NMR), and other biophysical and biochemical methods.

Modelling experimentally measured of ciprofloxacin antibiotic diffusion in Pseudomonas aeruginosa biofilm formed in artificial sputum medium

We study the experimentally measured ciprofloxacin antibiotic diffusion through a gel-like artificial sputum medium (ASM) mimicking physiological conditions typical for a cystic fibrosis layer, in which regions occupied by Pseudomonas aeruginosa bacteria are present. To quantify the antibiotic diffusion dynamics we employ a phenomenological model using a subdiffusion-absorption equation with a fractional time derivative. This effective equation describes molecular diffusion in a medium structured akin Thompson's plumpudding model; here the 'pudding' background represents the ASM and the 'plums' represent the bacterial biofilm. The pudding is a subdiffusion barrier for antibiotic molecules that can affect bacteria found in plums. For the experimental study we use an interferometric method to determine the time evolution of the amount of antibiotic that has diffused through the biofilm. The theoretical model shows that this function is qualitatively different depending on whether or not absorption of the antibiotic in the biofilm occurs. We show that the process can be divided into three successive stages: (1) only antibiotic subdiffusion with constant biofilm parameters, (2) subdiffusion and absorption of antibiotic molecules with variable biofilm transport parameters, (3) subdiffusion and absorption in the medium but the biofilm parameters are constant again. Stage 2 is interpreted as the appearance of an intensive defence build-up of bacteria against the action of the antibiotic, and in the stage 3 it is likely that the bacteria have been inactivated. Times at which stages change are determined from the experimentally obtained temporal evolution of the amount of antibiotic that has diffused through the ASM with bacteria. Our analysis shows good agreement between experimental and theoretical results and is consistent with the biologically expected biofilm response. We show that an experimental method to study the temporal evolution of the amount of a substance that has diffused through a biofilm is useful in studying the processes occurring in a biofilm. We also show that the complicated biological process of antibiotic diffusion in a biofilm can be described by a fractional subdiffusion-absorption equation with subdiffusion and absorption parameters that change over time.

Lévy noise-driven escape from arctangent potential wells

The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, Lévy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Lévy noise-driven escape from an almost rectangular, arctangent potential well restricted by two absorbing boundaries, mostly under the action of the Cauchy noise. We unveil analogies of the observed transient dynamics to the general properties of stationary states of Lévy processes in single-well potentials. The first-escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness, and height of the arctangent potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point.

Network Rewiring: Physiological Consequences of Reciprocally Exchanging the Physical Locations and Growth-Phase-Dependent Expression Patterns of the Salmonella fis and dps Genes

We assessed the impact on Salmonella physiology of reciprocally translocating the genes encoding the Fis and Dps nucleoid-associated proteins (NAPs) and of inverting their growth-phase production patterns such that Fis was produced in stationary phase (like Dps) and Dps was produced in exponential phase (like Fis). Changes to peak binding of Fis were detected by ChIP-seq on the chromosome, as were widespread impacts on the transcriptome, especially when Fis production mimicked Dps production. Virulence gene expression and the expression of a virulence phenotype were altered. Overall, these radical changes to NAP gene expression were well tolerated, revealing the robust and well-buffered nature of global gene regulation networks in the bacterium.

The Fis nucleoid-associated protein controls the expression of a large and diverse regulon of genes in Gram-negative bacteria. Fis production is normally maximal in bacteria during the early exponential phase of batch culture growth, becoming almost undetectable by the onset of stationary phase. We tested the effect on the Fis regulatory network in Salmonella of moving the complete fis gene from its usual location near the origin of chromosomal replication to the position normally occupied by the dps gene in the right macrodomain of the chromosome, and vice versa, creating the gene exchange (GX) strain. In a parallel experiment, we tested the effect of rewiring the Fis regulatory network by placing the fis open reading frame under the control of the stationary-phase-activated dps promoter at the dps genetic location within the right macrodomain, and vice versa, creating the open reading frame exchange (OX) strain. Chromatin immunoprecipitation sequencing (ChIP-seq) was used to measure global Fis protein binding levels and to determine gene expression patterns. Strain GX showed few changes compared with the wild type, although we did detect increased Fis binding at Ter, accompanied by reduced binding at Ori. Strain OX displayed a more pronounced version of this distorted Fis protein-binding pattern together with numerous alterations in the expression of genes in the Fis regulon. OX, but not GX, had a reduced ability to infect cultured mammalian cells. These findings illustrate the inherent robustness of the Fis regulatory network with respect to the effects of rewiring based on gene repositioning alone and emphasize the importance of fis expression signals in phenotypic determination.

Elucidating the Influence of Chromosomal Architecture on Transcriptional Regulation in Prokaryotes - Observing Strong Local Effects of Nucleoid Structure on Gene Regulation

Both intrinsic and extrinsic mechanisms regulating bacterial expression have been elucidated and described, however, such studies have mainly focused on local effects on the two-dimensional structure of the prokaryote genome while long-range as well as spatial interactions influencing gene expression are still only poorly understood. In this paper, we investigate the association between co-expression and distance between genes, using RNA-seq data at multiple growth phases in order to illuminate whether such conserved patterns are an indication of a gene regulatory mechanism relevant for prokaryotic cell proliferation, adaption, and evolution. We observe recurrent sinusoidal patterns in correlation of pairwise expression as function of genomic distance and rule out that these are caused by transcription-induced supercoiling gradients, gene clustering in operons, or association with regulatory transcription factors (TFs). By comparing spatial proximity for pairs of genomic bins with their correlation of pairwise expression, we further observe a high co-expression proportional with the spatial proximity. Based on these observations, we propose that the observed patterns are related to nucleoid structure as a product of transcriptional spilling, where genes actively influence transcription of spatially proximal genes through increases within shared local pools of RNA polymerases (RNAP), and actively spilling transcription onto neighboring genes.

Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.

CRISPR-Cas9 Genome Interrogation: A Facilitated Subdiffusive Target Search Strategy

The functional application of RNA-guided CRISPR-associated Cas9 protein, a bacterial immune system-based protein complex, via which in vivo, highly specific, and well-regulated, gene-editing processes are being monitored at an unprecedented level, has led to remarkable progress in genetic engineering and technology. The complicated in vivo process of genome interrogation followed by gene editing by the Cas9 complex was recently reported by Knight et al. (Science, 2015, 350, 823-826) using an elegant single-particle tracking method, aided by the two-photon fluorescence correlation spectroscopic technique. In contrast to the usually observed fast target-searching and protein-binding events in biophysical systems, an interesting slow genome-interrogation process by the RNA-guided CRISPR-Cas9 system through a crowded chromatin environment of a mammalian cell has been revealed in Knight et al.'s study. Motivated by this experiment, in this paper, we provide a generalized theoretical framework to capture this particular target-searching mechanism of the CRISPR-Cas9 protein complex. We show that an analysis on the basis of 3D subdiffusion under a cylindrical volume, created by several nonspecific off-target interactions from the DNA strands, can capture the essential details of the process. Moreover, on the basis of this model, we quantify the dynamics of this process and estimate the survival probability, first passage time, and the intensity correlation function of the tagged proteins of the experiment. The results from our theoretical predictions are found to be consistent with the experimental observations, and hence, seem to provide a plausible microscopic picture of the process.

Physical properties of phospholipids and integral proteins and their biofunctional roles in pulmonary surfactant from molecular dynamics simulation

This work deals with a quantitative investigation of the physical properties of pulmonary surfactant near melting temperature. To this end, we make use of molecular dynamics simulations, using the MARTINI coarse-grained model, for determining the physical properties of the system, such as the potential energy, the specific heat, the microstructure, the diffusion laws, and the elastic properties of the surfactant. The microstructure is studied by computation of the radial-distribution-function upon varying the distance between constituents (lipid molecules or proteins). The diffusion phenomenon is investigated by determination of the mean-squared-displacement and the time dependent velocity-autocorrelation-function for various values of temperature. We show that the dynamics of lipids and proteins exhibit a subdiffusion regime (slow movement) due to the cage effect within pulmonary surfactant. From the obtained mean-squared-displacement, we get the values of the self-diffusion-coefficients and the anomalous exponents at different temperatures close to the melting temperature. For the mathematical description of the cage effect, we make use of the scale relations in terms of the waiting time probability distribution. The last study is concerned with determination of the dependence of the lateral stress upon the strain of pulmonary surfactant, which is found to be linear, and from which we deduce the lateral-elastic-modulus.

The effect of obstacles in multi-site protein target search with DNA looping

Many fundamental biological processes are regulated by protein-DNA complexes called synaptosomes, which possess multiple interaction sites. Despite the critical importance of synaptosomes, the mechanisms of their formation are not well understood. Because of the multisite nature of participating proteins, it is widely believed that their search for specific sites on DNA involves the formation and breaking of DNA loops and sliding in the looped configurations. In reality, DNA in live cells is densely covered by other biological molecules that might interfere with the formation of synaptosomes. In this work, we developed a theoretical approach to evaluate the role of obstacles in the target search of multisite proteins when the formation of DNA loops and the sliding in looped configurations are possible. Our theoretical method is based on analysis of a discrete-state stochastic model that uses a master equations approach and extensive computer simulations. It is found that the obstacle slows down the search dynamics in the system when DNA loops are long-lived, but the effect is minimal for short-lived DNA loops. In addition, the relative positions of the target and the obstacle strongly influence the target search kinetics. Furthermore, the presence of the obstacle might increase the noise in the system. These observations are discussed using physical-chemical arguments. Our theoretical approach clarifies the molecular mechanisms of formation of protein-DNA complexes with multiple interactions sites.

Structural Basis of Enhanced Facilitated Diffusion of DNA-Binding Protein in Crowded Cellular Milieu

Although the fast association between DNA-binding proteins (DBPs) and DNA is explained by a facilitated diffusion mechanism, in which DBPs adopt a weighted combination of three-dimensional diffusion and one-dimensional (1D) sliding and hopping modes of transportation, the role of cellular environment that contains many nonspecifically interacting proteins and other biomolecules is mostly overlooked. By performing large-scale computational simulations with an appropriately tuned model of protein and DNA in the presence of nonspecifically interacting bulk and DNA-bound crowders (genomic crowders), we demonstrate the structural basis of the enhanced facilitated diffusion of DBPs inside a crowded cellular milieu through, to our knowledge, novel 1D scanning mechanisms. In this one-dimensional scanning mode, the protein can float along the DNA under the influence of nonspecific interactions of bulk crowder molecules. The search mode is distinctly different compared to usual 1D sliding and hopping dynamics in which protein diffusion is regulated by the DNA electrostatics. In contrast, the presence of genomic crowders expedites the target search process by transporting the protein over DNA segments through the formation of a transient protein-crowder bridged complex. By analyzing the ruggedness of the associated potential energy landscape, we underpin the molecular origin of the kinetic advantages of these search modes and show that they successfully explain the experimentally observed acceleration of facilitated diffusion of DBPs by molecular crowding agents and crowder-concentration-dependent enzymatic activity of transcription factors. Our findings provide crucial insights into gene regulation kinetics inside the crowded cellular milieu.

Guided nuclear exploration increases CTCF target search efficiency

The enormous size of mammalian genomes means that for a DNA-binding protein, the number of non-specific, off-target sites vastly exceeds the number of specific, cognate sites. How mammalian DNA-binding proteins overcome this challenge to efficiently locate their target sites is not known. Here through live-cell single-molecule tracking, we show that CCCTC-binding factor, CTCF, is repeatedly trapped in small zones that likely correspond to CTCF clusters, in a manner that is largely dependent on an internal RNA-binding region (RBR_i). We develop a new theoretical model, Anisotropic Diffusion through transient Trapping in Zones (ADTZ), to explain CTCF dynamics. Functionally, transient RBR_i-mediated trapping increases the efficiency of CTCF target search by ~2.5 fold. Overall, our results suggest a “guided” mechanism where CTCF clusters concentrate diffusing CTCF proteins near cognate binding sites, thus increasing the local ON-rate. We suggest that local guiding may allow DNA-binding proteins to more efficiently locate their target sites.

Survival and confinement under quenched disorder

We study the survival and confinement of random walkers under quenched disorder characterized by spatially-varying waiting times and decay rates. Spatial heterogeneity and segregation lead to a dynamic coupling between transport and reaction, resulting in history-dependent dynamics exhibiting long survivals and confinement. The survival probability decays as a power law, in contrast to the classical exponential law for decay at a homogeneous rate. The mean squared displacement shows dimension-dependent subdiffusive growth followed by localization, with stronger confinement in higher dimensions.

Target search on DNA by interacting molecules: First-passage approach

Gene regulation is one of the most important fundamental biological processes in living cells. It involves multiple protein molecules that locate specific sites on DNA and assemble gene initiation or gene repression multimolecular complexes. While the protein search dynamics for DNA targets has been intensively investigated, the role of intermolecular interactions during the genetic activation or repression remains not well quantified. Here, we present a simple one-dimensional model of target search for two interacting molecules that can reversibly form a dimer molecular complex, which also participates in the search process. In addition, the proteins have finite residence times on specific target sites, and the gene is activated or repressed when both proteins are simultaneously present at the target. The model is analyzed using first-passage analytical calculations and Monte Carlo computer simulations. It is shown that the search dynamics exhibit a complex behavior depending on the strength of intermolecular interactions and on the target residence times. We also found that the search time shows a nonmonotonic behavior as a function of the dissociation rate for the molecular complex. Physical-chemical arguments to explain these observations are presented. Our theoretical approach highlights the importance of molecular interactions in the complex process of gene activation/repression by multiple transcription factor proteins.

A Model for the Spatiotemporal Design of Gene Regulatory Circuits †

Mathematical modeling assists the design of synthetic regulatory networks by providing a detailed mechanistic understanding of biological systems. Models that can predict the performance of a design are fundamental for synthetic biology since they minimize iterations along the design-build-test lifecycle. Such predictability depends crucially on what assumptions (i.e., biological simplifications) the model considers. Here, we challenge a common assumption when it comes to the modeling of bacterial-based gene regulation: considering negligible the effects of intracellular physical space. It is commonly assumed that molecules, such as transcription factors (TF), are homogeneously distributed inside a cell, so there is no need to model their diffusion. We describe a mathematical model that accounts for molecular diffusion and show how simulations of network performance are decisively affected by the distance between its components. Specifically, the model focuses on the search by a TF for its target promoter. The combination of local searches, via one-dimensional sliding along the chromosome, and global searches, via three-dimensional diffusion through the cytoplasm, determine TF-promoter interplay. Previous experimental results with engineered bacteria in which the distance between TF source and target was minimized or enlarged were successfully reproduced by the spatially resolved model we introduce here. This suggests that the spatial specification of the circuit alone can be exploited as a design parameter in synthetic biology to select programmable output levels.

Burst statistics in an early biofilm quorum sensing model: the role of spatial colony-growth heterogeneity

Quorum-sensing bacteria in a growing colony of cells send out signalling molecules (so-called “autoinducers”) and themselves sense the autoinducer concentration in their vicinity. Once—due to increased local cell density inside a “cluster” of the growing colony—the concentration of autoinducers exceeds a threshold value, cells in this clusters get “induced” into a communal, multi-cell biofilm-forming mode in a cluster-wide burst event. We analyse quantitatively the influence of spatial disorder, the local heterogeneity of the spatial distribution of cells in the colony, and additional physical parameters such as the autoinducer signal range on the induction dynamics of the cell colony. Spatial inhomogeneity with higher local cell concentrations in clusters leads to earlier but more localised induction events, while homogeneous distributions lead to comparatively delayed but more concerted induction of the cell colony, and, thus, a behaviour close to the mean-field dynamics. We quantify the induction dynamics with quantifiers such as the time series of induction events and burst sizes, the grouping into induction families, and the mean autoinducer concentration levels. Consequences for different scenarios of biofilm growth are discussed, providing possible cues for biofilm control in both health care and biotechnology.

Energetic funnel facilitates facilitated diffusion

Transcription factors (TFs) are able to associate to their binding sites on DNA faster than the physical limit posed by diffusion. Such high association rates can be achieved by alternating between three-dimensional diffusion and one-dimensional sliding along the DNA chain, a mechanism-dubbed facilitated diffusion. By studying a collection of TF binding sites of Escherichia coli from the RegulonDB database and of Bacillus subtilis from DBTBS, we reveal a funnel in the binding energy landscape around the target sequences. We show that such a funnel is linked to the presence of gradients of AT in the base composition of the DNA region around the binding sites. An extensive computational study of the stochastic sliding process along the energetic landscapes obtained from the database shows that the funnel can significantly enhance the probability of TFs to find their target sequences when sliding in their proximity. We demonstrate that this enhancement leads to a speed-up of the association process.

Quantifying the two-state facilitated diffusion model of protein-DNA interactions

The current report extends the facilitated diffusion model to account for conflict between the search and recognition binding modes adopted by DNA-binding proteins (DBPs) as they search DNA and subsequently recognize and bind to their specific binding site. The speed of the search dynamics is governed by the energetic ruggedness of the protein-DNA landscape, whereas the rate for the recognition process is mostly dictated by the free energy barrier for the transition between the DBP's search and recognition binding modes. We show that these two modes are negatively coupled, such that fast 1D sliding and rapid target site recognition probabilities are unlikely to coexist. Thus, a tradeoff occurs between optimizing the timescales for finding and binding the target site. We find that these two kinetic properties can be balanced to produce a fast timescale for the total target search and recognition process by optimizing frustration. Quantification of the facilitated diffusion model by including a frustration term enables it to explain several experimental observations concerning search and recognition speeds. The extended model captures experimental estimate of the energetic ruggedness of the protein-DNA landscape and predicts how various molecular properties of protein-DNA binding affect recognition kinetics. Particularly, point mutations may change the frustration and so affect protein association with DNA, thus providing a means to modulate protein-DNA affinity by manipulating the protein's association or dissociation reactions.

Optimizing the dynamics of protein expression

Heterologously expressed genes require adaptation to the host organism to ensure adequate levels of protein synthesis, which is typically approached by replacing codons by the target organism’s preferred codons. In view of frequently encountered suboptimal outcomes we introduce the codon-specific elongation model (COSEM) as an alternative concept. COSEM simulates ribosome dynamics during mRNA translation and informs about protein synthesis rates per mRNA in an organism- and context-dependent way. Protein synthesis rates from COSEM are integrated with further relevant covariates such as translation accuracy into a protein expression score that we use for codon optimization. The scoring algorithm further enables fine-tuning of protein expression including deoptimization and is implemented in the software OCTOPOS. The protein expression score produces competitive predictions on proteomic data from prokaryotic, eukaryotic, and human expression systems. In addition, we optimized and tested heterologous expression of manA and ova genes in Salmonella enterica serovar Typhimurium. Superiority over standard methodology was demonstrated by a threefold increase in protein yield compared to wildtype and commercially optimized sequences.

Disparity in anomalous diffusion of proteins searching for their target DNA sites in a crowded medium is controlled by the size, shape and mobility of macromolecular crowders

Using extensive computer simulations, we analyzed the role of physical properties of molecular crowding agents such as size, shape and mobility in the target search dynamics of DNA binding proteins. Our main result is that the sub-diffusive dynamics of a protein inside a crowded medium strongly depends on the crowder properties and also on the protein's mode of diffusion. For instance, while scanning the DNA one-dimensionally, the protein dynamics does not vary with the change in crowder properties. Conversely, the diffusion exponent varies non-monotonically during 3D diffusion and is maximally affected when the crowders match the protein physically. The investigation shows that the effect stems from the ruggedness of the associated potential energy landscape, which is regulated by the protein-crowder and DNA-crowder interactions. Our findings have broad significance in understanding the target search dynamics of proteins on DNA in crowded cellular milieu and selecting appropriate crowding agents when designing in vitro experiments.

Gene Expression Is Not Random: Scaling, Long-Range Cross-Dependence, and Fractal Characteristics of Gene Regulatory Networks

Gene expression is a vital process through which cells react to the environment and express functional behavior. Understanding the dynamics of gene expression could prove crucial in unraveling the physical complexities involved in this process. Specifically, understanding the coherent complex structure of transcriptional dynamics is the goal of numerous computational studies aiming to study and finally control cellular processes. Here, we report the scaling properties of gene expression time series in Escherichia coli and Saccharomyces cerevisiae. Unlike previous studies, which report the fractal and long-range dependency of DNA structure, we investigate the individual gene expression dynamics as well as the cross-dependency between them in the context of gene regulatory network. Our results demonstrate that the gene expression time series display fractal and long-range dependence characteristics. In addition, the dynamics between genes and linked transcription factors in gene regulatory networks are also fractal and long-range cross-correlated. The cross-correlation exponents in gene regulatory networks are not unique. The distribution of the cross-correlation exponents of gene regulatory networks for several types of cells can be interpreted as a measure of the complexity of their functional behavior.

Impact of Chromosomal Architecture on the Function and Evolution of Bacterial Genomes

The bacterial nucleoid is highly condensed and forms compartment-like structures within the cell. Much attention has been devoted to investigating the dynamic topology and organization of the nucleoid. In contrast, the specific nucleoid organization, and the relationship between nucleoid structure and function is often neglected with regard to importance for adaption to changing environments and horizontal gene acquisition. In this review, we focus on the structure-function relationship in the bacterial nucleoid. We provide an overview of the fundamental properties that shape the chromosome as a structured yet dynamic macromolecule. These fundamental properties are then considered in the context of the living cell, with focus on how the informational flow affects the nucleoid structure, which in turn impacts on the genetic output. Subsequently, the dynamic living nucleoid will be discussed in the context of evolution. We will address how the acquisition of foreign DNA impacts nucleoid structure, and conversely, how nucleoid structure constrains the successful and sustainable chromosomal integration of novel DNA. Finally, we will discuss current challenges and directions of research in understanding the role of chromosomal architecture in bacterial survival and adaptation.

Towards a full quantitative description of single-molecule reaction kinetics in biological cells

The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying first-passage events, in particular, inherent cell-to-cell fluctuations. We here present a robust explicit approach for obtaining the distribution of FPTs to a small partially reactive target in cylindrical-annulus domains, which represent typical bacterial and neuronal cell shapes. We investigate various asymptotic behaviours of this FPT distribution and show that it is typically very broad in many biological situations, thus, the mean FPT can differ from the most probable FPT by orders of magnitude. The most probable FPT is shown to strongly depend only on the starting position within the geometry and to be almost independent of the target size and reactivity. These findings demonstrate the dramatic relevance of knowing the full distribution of FPTs and thus open new perspectives for a more reliable description of many intracellular processes initiated by the arrival of one or few biomolecules to a small, spatially localised region inside the cell.

Acceleration of bursty multiprotein target search kinetics on DNA by colocalisation

Proteins are capable of locating specific targets on DNA by employing a facilitated diffusion process with intermittent 1D and 3D search steps. We here uncover the implications of colocalisation of protein production and DNA binding sites via computer simulations.

Kinetics of lipid-nanoparticle-mediated intracellular mRNA delivery and function

mRNA delivery into cells forms the basis for one of the new and promising ways to treat various diseases. Among suitable carriers, lipid nanoparticles (LNPs) with a size of about 100 nm are now often employed. Despite high current interest in this area, the understanding of the basic details of LNP-mediated mRNA delivery and function is limited. To clarify the kinetics of mRNA release from LNPs, the author uses three generic models implying (i) exponential, (ii) diffusion-controlled, and (iii) detachment-controlled kinetic regimes, respectively. Despite the distinct differences in these kinetics, the associated transient kinetics of mRNA translation to the corresponding protein and its degradation are shown to be not too sensitive to the details of the mRNA delivery by LNPs (or other nanocarriers). In addition, the author illustrates how this protein may temporarily influence the expression of one gene or a few equivalent genes. The analysis includes positive or negative regulation of the gene transcription via the attachment of the protein without or with positive or negative feedback in the gene expression. Stable, bistable, and oscillatory schemes have been scrutinized in this context.

Tracking Low-Copy Transcription Factors in Living Bacteria: The Case of the lac Repressor

Transcription factors control the expression of genes by binding to specific sites in DNA and repressing or activating transcription in response to stimuli. The lac repressor (LacI) is a well characterized transcription factor that regulates the ability of bacterial cells to uptake and metabolize lactose. Here, we study the intracellular mobility and spatial distribution of LacI in live bacteria using photoactivated localization microscopy combined with single-particle tracking. Since we track single LacI molecules in live cells by stochastically photoactivating and observing fluorescent proteins individually, there are no limitations on the copy number of the protein under study; as a result, we were able to study the behavior of LacI in bacterial strains containing the natural copy numbers (∼40 monomers), as well as in strains with much higher copy numbers due to LacI overexpression. Our results allowed us to determine the relative abundance of specific, near-specific, and non-specific DNA binding modes of LacI in vivo, showing that all these modes are operational inside living cells. Further, we examined the spatial distribution of LacI in live cells, confirming its specific binding to lac operator regions on the chromosome; we also showed that mobile LacI molecules explore the bacterial nucleoid in a way similar to exploration by other DNA-binding proteins. Our work also provides an example of applying tracking photoactivated localization microscopy to studies of low-copy-number proteins in living bacteria.

Diffusive escape through a narrow opening: new insights into a classic problem

We study the mean first exit time (T_ε) of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size ε.

How nonspecifically DNA-binding proteins search for the target in crowded environments

We investigate how a tracer particle searches a target located in DNA modeled by a stiff chain in crowded environments using theoretical analysis and Langevin dynamics simulations. First, we show that the three-dimensional (3D) diffusion coefficient of the tracer only depends on the density of crowders ϕ, while its one-dimensional (1D) diffusion coefficient is affected by not only ϕ but also the nonspecific binding energy ε. With increasing ϕ and ε, no obvious change in the average 3D diffusion time is observed, while the average 1D sliding time apparently increases. We propose theoretically that the 1D sliding of the tracer along the chain could be well captured by the Kramers' law of escaping rather than the Arrhenius law, which is verified directly by the simulations. Finally, the average search time increases monotonously with an increase in ϕ while it has a minimum as a function of ε, which could be understood from the different behaviors of the average number of search rounds with the increasing ϕ or ε. These results provide a deeper understanding of the role of facilitated diffusion in target search of proteins on DNA in vivo.

Dichotomous noise models of gene switches

Molecular noise in gene regulatory networks has two intrinsic components, one part being due to fluctuations caused by the birth and death of protein or mRNA molecules which are often present in small numbers and the other part arising from gene state switching, a single molecule event. Stochastic dynamics of gene regulatory circuits appears to be largely responsible for bifurcations into a set of multi-attractor states that encode different cell phenotypes. The interplay of dichotomous single molecule gene noise with the nonlinear architecture of genetic networks generates rich and complex phenomena. In this paper, we elaborate on an approximate framework that leads to simple hybrid multi-scale schemes well suited for the quantitative exploration of the steady state properties of large-scale cellular genetic circuits. Through a path sum based analysis of trajectory statistics, we elucidate the connection of these hybrid schemes to the underlying master equation and provide a rigorous justification for using dichotomous noise based models to study genetic networks. Numerical simulations of circuit models reveal that the contribution of the genetic noise of single molecule origin to the total noise is significant for a wide range of kinetic regimes.

Intrinsic phenotypic stability of a bi-stable auto regulatory gene

Even under homogenous conditions clonal cells can assume different distinct states for generations to follow, also known as epigenetic inheritance. Such long periods of different phenotypic states can be formed due to the existence of more than one stable state in the molecule concentration, where the different states are explored through molecular fluctuations. By formulating a single reaction variable representing the birth and death of molecules, including transcription, translation and decay, we calculate the escape time from the phenotypic states attained from autocatalytic synthesis through a Fokker- Planck formulation and integration of an effective pseudo-potential. We calculate the stability of the phenotypic states both for cooperative binding feedback and dimer binding feedback, resulting in non-linear decay.

First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time

The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes one step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.