AgentCell: a digital single-cell assay for bacterial chemotaxis

Developing stochastic models for spatial inference: bacterial chemotaxis

BACKGROUND

Biological systems are inherently inhomogeneous and spatial effects play a significant role in processes such as pattern formation. At the cellular level proteins are often localised either through static attachment or via a dynamic equilibrium. As well as spatial heterogeneity many cellular processes exhibit stochastic fluctuations and so to make inferences about the location of molecules there is a need for spatial stochastic models. A test case for spatial models has been bacterial chemotaxis which has been studied extensively as a model of signal transduction.

RESULTS

By creating specific models of a cellular system that incorporate the spatial distributions of molecules we have shown how the fit between simulated and experimental data can be used to make inferences about localisation, in the case of bacterial chemotaxis. This method allows the robust comparison of different spatial models through alternative model parameterisations.

CONCLUSIONS

By using detailed statistical analysis we can reliably infer the parameters for the spatial models, and also to evaluate alternative models. The statistical methods employed in this case are particularly powerful as they reduce the need for a large number of simulation replicates. The technique is also particularly useful when only limited molecular level data is available or where molecular data is not quantitative.

An agent-based model of signal transduction in bacterial chemotaxis

We report the application of agent-based modeling to examine the signal transduction network and receptor arrays for chemotaxis in Escherichia coli, which are responsible for regulating swimming behavior in response to environmental stimuli. Agent-based modeling is a stochastic and bottom-up approach, where individual components of the modeled system are explicitly represented, and bulk properties emerge from their movement and interactions. We present the Chemoscape model: a collection of agents representing both fixed membrane-embedded and mobile cytoplasmic proteins, each governed by a set of rules representing knowledge or hypotheses about their function. When the agents were placed in a simulated cellular space and then allowed to move and interact stochastically, the model exhibited many properties similar to the biological system including adaptation, high signal gain, and wide dynamic range. We found the agent based modeling approach to be both powerful and intuitive for testing hypotheses about biological properties such as self-assembly, the non-linear dynamics that occur through cooperative protein interactions, and non-uniform distributions of proteins in the cell. We applied the model to explore the role of receptor type, geometry and cooperativity in the signal gain and dynamic range of the chemotactic response to environmental stimuli. The model provided substantial qualitative evidence that the dynamic range of chemotactic response can be traced to both the heterogeneity of receptor types present, and the modulation of their cooperativity by their methylation state.

Quantitative modeling of Escherichia coli chemotactic motion in environments varying in space and time

Escherichia coli chemotactic motion in spatiotemporally varying environments is studied by using a computational model based on a coarse-grained description of the intracellular signaling pathway dynamics. We find that the cell's chemotaxis drift velocity v(d) is a constant in an exponential attractant concentration gradient [L] proportional, variantexp(Gx). v(d) depends linearly on the exponential gradient G before it saturates when G is larger than a critical value G(C). We find that G(C) is determined by the intracellular adaptation rate k(R) with a simple scaling law: G(C) infinity k(1/2)(R). The linear dependence of v(d) on G = d(ln[L])/dx directly demonstrates E. coli's ability in sensing the derivative of the logarithmic attractant concentration. The existence of the limiting gradient G(C) and its scaling with k(R) are explained by the underlying intracellular adaptation dynamics and the flagellar motor response characteristics. For individual cells, we find that the overall average run length in an exponential gradient is longer than that in a homogeneous environment, which is caused by the constant kinase activity shift (decrease). The forward runs (up the gradient) are longer than the backward runs, as expected; and depending on the exact gradient, the (shorter) backward runs can be comparable to runs in a spatially homogeneous environment, consistent with previous experiments. In (spatial) ligand gradients that also vary in time, the chemotaxis motion is damped as the frequency omega of the time-varying spatial gradient becomes faster than a critical value omega(c), which is controlled by the cell's chemotaxis adaptation rate k(R). Finally, our model, with no adjustable parameters, agrees quantitatively with the classical capillary assay experiments where the attractant concentration changes both in space and time. Our model can thus be used to study E. coli chemotaxis behavior in arbitrary spatiotemporally varying environments. Further experiments are suggested to test some of the model predictions.

Mathematical modeling and experimental validation of chemotaxis under controlled gradients of methyl-aspartate in Escherichia coli

Escherichia coli has evolved an intracellular pathway to regulate its motion termed as chemotaxis so as to move towards a favorable environment such as regions with higher concentration of nutrients. Chemotaxis is a response to temporal and spatial variation of extracellular ligand concentration and randomness in motion induced by collisions with solvent molecules. Previous studies have reported average drift velocities for a given gradient and do not measure drift velocities as a function of time and space. To address this issue, a novel experimental technique was developed to quantify the motion of E. coli cells to varying concentrations and gradients of methyl-aspartate so as to capture the spatial and temporal variation of the drift velocity. A two-state receptor model accounting for the intracellular signaling pathway predicted the experimentally observed increase in drift velocity with gradient and the subsequent adaptation. Our study revealed that the rotational diffusivity induced by the extracellular environment is crucial in determining the drift velocity of E. coli. The model predictions matched with experimental observations only when the response of the intracellular pathway was highly ultra-sensitive to overcome the extracellular randomness. The parametric sensitivity of the pathway indicated that the dissociation constant for the binding of the ligand and the rate constants of the methylation/demethylation of the receptor are key to predict the performance of the chemotactic behavior. The study also indicates a possible role of oxygen in the chemotaxis response and that the response to a ligand may have to account for effects of oxygen.

Reverse engineering of bacterial chemotaxis pathway via frequency domain analysis

Chemotaxis is defined as a behavior involving organisms sensing attractants or repellents and leading towards or away from them. Therefore, it is possible to reengineer chemotaxis network to control the movement of bacteria to our advantage. Understanding the design principles of chemotaxis pathway is a prerequisite and an important topic in synthetic biology. Here, we provide guidelines for chemotaxis pathway design by employing control theory and reverse engineering concept on pathway dynamic design. We first analyzed the mathematical models for two most important kinds of E. coli chemotaxis pathway—adaptive and non-adaptive pathways, and concluded that the control units of the pathway de facto function as a band-pass filter and a low-pass filter, respectively, by abstracting the frequency response properties of the pathways. The advantage of the band-pass filter is established, and we demonstrate how to tune the three key parameters of it—A (max amplification), ω₁ (down cut-off frequency) and ω₂ (up cut-off frequency) to optimize the chemotactic effect. Finally, we hypothesized a similar but simpler version of the dynamic pathway model based on the principles discovered and show that it leads to similar properties with native E. coli chemotactic behaviors. Our study provides an example of simulating and designing biological dynamics in silico and indicates how to make use of the native pathway's features in this process. Furthermore, the characteristics we discovered and tested through reverse engineering may help to understand the design principles of the pathway and promote the design of artificial chemotaxis pathways.

Biocharts: a visual formalism for complex biological systems

We address one of the central issues in devising languages, methods and tools for the modelling and analysis of complex biological systems, that of linking high-level (e.g. intercellular) information with lower-level (e.g. intracellular) information. Adequate ways of dealing with this issue are crucial for understanding biological networks and pathways, which typically contain huge amounts of data that continue to grow as our knowledge and understanding of a system increases. Trying to comprehend such data using the standard methods currently in use is often virtually impossible. We propose a two-tier compound visual language, which we call Biocharts, that is geared towards building fully executable models of biological systems. One of the main goals of our approach is to enable biologists to actively participate in the computational modelling effort, in a natural way. The high-level part of our language is a version of statecharts, which have been shown to be extremely successful in software and systems engineering. The statecharts can be combined with any appropriately well-defined language (preferably a diagrammatic one) for specifying the low-level dynamics of the pathways and networks. We illustrate the language and our general modelling approach using the well-studied process of bacterial chemotaxis.

Chemotaxis: how bacteria use memory

Bacterial chemotaxis represents one of the simplest and best studied examples of unicellular behavior. Chemotaxis allows swimming bacterial cells to follow chemical gradients in the environment by performing temporal comparisons of ligand concentrations. The process of chemotaxis in the model bacteriumEscherichia colihas been studied in great molecular detail over the past 40 years, using a large range of experimental tools to investigate physiology, genetics and biochemistry of the system. The abundance of quantitative experimental data enabled detailed computational modeling of the pathway and theoretical analyses of such properties as robustness and signal amplification. Because of the temporal mode of gradient sensing in bacterial chemotaxis, molecular memory is an essential component of the chemotaxis pathway. Recent studies suggest that the memory time scale has been evolutionary optimized to perform optimal comparisons of stimuli while swimming in the gradient. Moreover, noise in the adaptation system, which results from variations of the adaptation rate both over time and among cells, might be beneficial for the overall chemotactic performance of the population.

E. coli superdiffusion and chemotaxis-search strategy, precision, and motility

Escherichia coli motion is characterized by a sequence of consecutive tumble-and-swim events. In the absence of chemical gradients, the length of individual swims is commonly believed to be distributed exponentially. However, recently there has been experimental indication that the swim-length distribution has the form of a power-law, suggesting that bacteria might perform superdiffusive Lévy-walk motion. In E. coli, the power-law behavior can be induced through stochastic fluctuations in the level of CheR, one of the key enzymes in the chemotaxis signal transmission pathway. We use a mathematical model of the chemotaxis signaling pathway to study the influence of these fluctuations on the E. coli behavior in the absence and presence of chemical gradients. We find that the population with fluctuating CheR performs Lévy-walks in the absence of chemoattractants, and therefore might have an advantage in environments where nutrients are sparse. The more efficient search strategy in sparse environments is accompanied by a generally larger motility, also in the presence of chemoattractants. The tradeoff of this strategy is a reduced precision in sensing and following gradients, as well as a slower adaptation to absolute chemoattractant levels.

Logarithmic sensing in Escherichia coli bacterial chemotaxis

We studied the response of swimming Escherichia coli (E. coli) bacteria in a comprehensive set of well-controlled chemical concentration gradients using a newly developed microfluidic device and cell tracking imaging technique. In parallel, we carried out a multi-scale theoretical modeling of bacterial chemotaxis taking into account the relevant internal signaling pathway dynamics, and predicted bacterial chemotactic responses at the cellular level. By measuring the E. coli cell density profiles across the microfluidic channel at various spatial gradients of ligand concentration grad[L] and the average ligand concentration [L] near the peak chemotactic response region, we demonstrated unambiguously in both experiments and model simulation that the mean chemotactic drift velocity of E. coli cells increased monotonically with grad [L]/[L] or approximately grad(log[L])--that is E. coli cells sense the spatial gradient of the logarithmic ligand concentration. The exact range of the log-sensing regime was determined. The agreements between the experiments and the multi-scale model simulation verify the validity of the theoretical model, and revealed that the key microscopic mechanism for logarithmic sensing in bacterial chemotaxis is the adaptation kinetics, in contrast to explanations based directly on ligand occupancy.

Dependence of bacterial chemotaxis on gradient shape and adaptation rate

Simulation of cellular behavior on multiple scales requires models that are sufficiently detailed to capture central intracellular processes but at the same time enable the simulation of entire cell populations in a computationally cheap way. In this paper we present RapidCell, a hybrid model of chemotactic Escherichia coli that combines the Monod-Wyman-Changeux signal processing by mixed chemoreceptor clusters, the adaptation dynamics described by ordinary differential equations, and a detailed model of cell tumbling. Our model dramatically reduces computational costs and allows the highly efficient simulation of E. coli chemotaxis. We use the model to investigate chemotaxis in different gradients, and suggest a new, constant-activity type of gradient to systematically study chemotactic behavior of virtual bacteria. Using the unique properties of this gradient, we show that optimal chemotaxis is observed in a narrow range of CheA kinase activity, where concentration of the response regulator CheY-P falls into the operating range of flagellar motors. Our simulations also confirm that the CheB phosphorylation feedback improves chemotactic efficiency by shifting the average CheY-P concentration to fit the motor operating range. Our results suggest that in liquid media the variability in adaptation times among cells may be evolutionary favorable to ensure coexistence of subpopulations that will be optimally tactic in different gradients. However, in a porous medium (agar) such variability appears to be less important, because agar structure poses mainly negative selection against subpopulations with low levels of adaptation enzymes. RapidCell is available from the authors upon request.

A hybrid agent-based approach for modeling microbiological systems

Models for systems biology commonly adopt Differential Equations or Agent-Based modeling approaches for simulating the processes as a whole. Models based on differential equations presuppose phenomenological intracellular behavioral mechanisms, while models based on Multi-Agent approach often use directly translated, and quantitatively less precise if-then logical rule constructs. We propose an extendible systems model based on a hybrid agent-based approach where biological cells are modeled as individuals (agents) while molecules are represented by quantities. This hybridization in entity representation entails a combined modeling strategy with agent-based behavioral rules and differential equations, thereby balancing the requirements of extendible model granularity with computational tractability. We demonstrate the efficacy of this approach with models of chemotaxis involving an assay of 10(3) cells and 1.2x10(6) molecules. The model produces cell migration patterns that are comparable to laboratory observations.

A computational model of chemotaxis-based cell aggregation

We present a computational model that successfully captures the cell behaviors that play important roles in 2-D cell aggregation. A virtual cell in our model is designed as an independent, discrete unit with a set of parameters and actions. Each cell is defined by its location, size, rates of chemoattractant emission and response, age, life cycle stage, proliferation rate and number of attached cells. All cells are capable of emitting and sensing a chemoattractant chemical, moving, attaching to other cells, dividing, aging and dying. We validated and fine-tuned our in silico model by comparing simulated 24-h aggregation experiments with data derived from in vitro experiments using PC12 pheochromocytoma cells. Quantitative comparisons of the aggregate size distributions from the two experiments are produced using the Earth Mover's Distance (EMD) metric. We compared the two size distributions produced after 24 h of in vitro cell aggregation and the corresponding computer simulated process. Iteratively modifying the model's parameter values and measuring the difference between the in silico and in vitro results allow us to determine the optimal values that produce simulated aggregation outcomes closely matched to the PC12 experiments. Simulation results demonstrate the ability of the model to recreate large-scale aggregation behaviors seen in live cell experiments.

Relationship between cellular response and behavioral variability in bacterial chemotaxis

Over the last decades, bacterial chemotaxis in Escherichia coli has emerged as a canonical system for the study of signal transduction. A remarkable feature of this system is the coexistence of a robust adaptive behavior observed at the population level with a large fluctuating behavior in single cells [Korobkova E, Emonet T, Vilar JMG, Shimizu TS, Cluzel P (2004) Nature 428:574-578]. Using a unified stochastic model, we demonstrate that this coexistence is not fortuitous but a direct consequence of the architecture of this adaptive system. The methylation and demethylation cycles that regulate the activity of receptor-kinase complexes are ultrasensitive because they operate outside the region of first-order kinetics. As a result, the receptor-kinase that governs cellular behavior exhibits a sigmoidal activation curve. We propose that the steepness of this kinase activation curve simultaneously controls the behavioral variability in nonstimulated individual bacteria and the duration of the adaptive response to small stimuli. We predict that the fluctuating behavior and the chemotactic response of individual cells both peak within the transition region of this sigmoidal curve. Large-scale simulations of digital bacteria suggest that the chemotaxis network is tuned to simultaneously maximize both the random spread of cells in the absence of nutrients and the cellular response to gradients of attractant. This study highlights a fundamental relation from which the behavioral variability of nonstimulated cells is used to infer the timing of the cellular response to small stimuli.

Individual-based modeling of phytoplankton: Evaluating approaches for applying the cell quota model

Present phytoplankton models typically use a population-level (lumped) modeling (PLM) approach that assumes average properties of a population within a control volume. For modern biogeochemical models that formulate growth as a nonlinear function of the internal nutrient (e.g. Droop kinetics), this averaging assumption can introduce a significant error. Individual-based (agent-based) modeling (IBM) does not make the assumption of average properties and therefore constitutes a promising alternative for biogeochemical modeling. This paper explores the hypothesis that the cell quota (Droop) model, which predicts the population-average specific growth or cell division rate, based on the population-average nutrient cell quota, can be applied to individual algal cells and produce the same population-level results. Three models that translate the growth rate calculated using the cell quota model into discrete cell division events are evaluated, including a stochastic model based on the probability of cell division, a deterministic model based on the maturation velocity and fraction of the cell cycle completed (maturity fraction), and a deterministic model based on biomass (carbon) growth and cell size. The division models are integrated into an IBM framework (iAlgae), which combines a lumped system representation of a nutrient with an individual representation of algae. The IBM models are evaluated against a conventional PLM (because that is the traditional approach) and data from a number of steady and unsteady continuous (chemostat) and batch culture laboratory experiments. The stochastic IBM model fails the steady chemostat culture test, because it produces excessive numerical randomness. The deterministic cell cycle IBM model fails the batch culture test, because it has an abrupt drop in cell quota at division, which allows the cell quota to fall below the subsistence quota. The deterministic cell size IBM model reproduces the data and PLM results for all experiments and the model parameters (e.g. maximum specific growth rate, subsistence quota) are the same as those for the PLM. In addition, the model-predicted cell age, size (carbon) and volume distributions are consistent with those derived analytically and compare well to observations. The paper discusses and illustrates scenarios where intra-population variability in natural systems leads to differences between the IBM and PLM models.

Accounting for intrapopulation variability in biogeochemical models using agent-based methods

Present biogeochemical models typically use a lumped-system (population-level) modeling (LSM) approach that assumes average properties of a population within a control volume. For modern models that formulate phytoplankton growth as a nonlinear function of the internal nutrient (e.g., Droop kinetics), this averaging assumption can introduce a significant error. Agent-based (individual-based) modeling (ABM) is an alternative approach that does not make the assumption of average properties. This paper presents a new agent-based phytoplankton model called iAlgae. The model is contrasted to a conventional lumped-system model, constructed based on identical underlying sub-models of nutrient uptake (including luxury uptake) and growth (cell quota, Droop model). The two models are validated against laboratory data and applied to a realistic scenario, consisting of a point source nutrient discharge into a river. For the realistic scenario, the ABM-predicted phytoplankton bloom is significantly lower than the LSM-predicted one, which is due to the intrapopulation distribution in cell quotas (due to different life histories of individuals) and nonlinearity of the growth rate model. In the ABM, a fraction of the population accumulates nutrients in excess of their immediate growth requirement (luxury uptake), leaving less for the remainder. Because the model is nonlinear, this results in a suboptimal (from a population perspective) utilization of nutrient and a lower population-level growth rate, compared to the case of no intrapopulation variability assumed by the LSM model. In general, the ABM and LSM approaches can produce significantly different results when incompletely mixed conditions lead to intrapopulation variability in cell properties (i.e., cell quota) and the model equations are nonlinear.

Computational systems biology in drug discovery and development: methods and applications

Computational systems biology is an emerging field in biological simulation that attempts to model or simulate intra- and intercellular events using data gathered from genomic, proteomic or metabolomic experiments. The need to model complex temporal and spatiotemporal processes at many different scales has led to the emergence of numerous techniques, including systems of differential equations, Petri nets, cellular automata simulators, agent-based models and pi calculus. This review provides a brief summary and an assessment of most of these approaches. It also provides examples of how these methods are being used to facilitate drug discovery and development.

Hidden stochastic nature of a single bacterial motor

The rotary flagellar motor of Escherichia coli bacterium switches stochastically between the clockwise (CW) and counterclockwise (CCW) direction. We found that the CW and CCW intervals could be described by a gamma distribution, suggesting the existence of hidden Markov steps preceding each motor switch. Power spectra of time series of switching events exhibited a peaking frequency instead of the Lorentzian profile expected from standard kinetic two-state models. Our analysis indicates that the number of hidden steps may be a key dynamical parameter underlying the switching process in a single bacterial motor as well as in large cooperative molecular systems.