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1.  Bistable Forespore Engulfment in Bacillus subtilis by a Zipper Mechanism in Absence of the Cell Wall 
PLoS Computational Biology  2014;10(10):e1003912.
To survive starvation, the bacterium Bacillus subtilis forms durable spores. The initial step of sporulation is asymmetric cell division, leading to a large mother-cell and a small forespore compartment. After division is completed and the dividing septum is thinned, the mother cell engulfs the forespore in a slow process based on cell-wall degradation and synthesis. However, recently a new cell-wall independent mechanism was shown to significantly contribute, which can even lead to fast engulfment in 60 of the cases when the cell wall is completely removed. In this backup mechanism, strong ligand-receptor binding between mother-cell protein SpoIIIAH and forespore-protein SpoIIQ leads to zipper-like engulfment, but quantitative understanding is missing. In our work, we combined fluorescence image analysis and stochastic Langevin simulations of the fluctuating membrane to investigate the origin of fast bistable engulfment in absence of the cell wall. Our cell morphologies compare favorably with experimental time-lapse microscopy, with engulfment sensitive to the number of SpoIIQ-SpoIIIAH bonds in a threshold-like manner. By systematic exploration of model parameters, we predict regions of osmotic pressure and membrane-surface tension that produce successful engulfment. Indeed, decreasing the medium osmolarity in experiments prevents engulfment in line with our predictions. Forespore engulfment may thus not only be an ideal model system to study decision-making in single cells, but its biophysical principles are likely applicable to engulfment in other cell types, e.g. during phagocytosis in eukaryotes.
Author Summary
When the bacterium B. subtilis runs out of food, it undergoes a fundamental development process by which it forms durable spores. Sporulation is initiated by asymmetric cell division after which the larger mother cell engulfs the smaller forespore, followed by spore maturation and release. This survival strategy is so robust that engulfment even proceeds when cells are deprived of their protective cell wall. Under these severe perturbations, 60 of the mother cells still engulf their forespores in only 10 of the normal engulfment time, while the remaining 40 of mother cells withdraw from engulfment. This all-or-none outcome of engulfment suggests decision-making, which was recently also identified in other types of engulfment, e.g. during phagocytosis when immune cells engulf and destroy pathogens. Here, we developed a biophysical model to explain fast bistable forespore engulfment in absence of the cell wall and energy sources. Our discovered principles may prove very general, thus predicting key ingredients of successful engulfment across all kingdoms of life.
PMCID: PMC4214620  PMID: 25356555
2.  Predicting Chemical Environments of Bacteria from Receptor Signaling 
PLoS Computational Biology  2014;10(10):e1003870.
Sensory systems have evolved to respond to input stimuli of certain statistical properties, and to reliably transmit this information through biochemical pathways. Hence, for an experimentally well-characterized sensory system, one ought to be able to extract valuable information about the statistics of the stimuli. Based on dose-response curves from in vivo fluorescence resonance energy transfer (FRET) experiments of the bacterial chemotaxis sensory system, we predict the chemical gradients chemotactic Escherichia coli cells typically encounter in their natural environment. To predict average gradients cells experience, we revaluate the phenomenological Weber's law and its generalizations to the Weber-Fechner law and fold-change detection. To obtain full distributions of gradients we use information theory and simulations, considering limitations of information transmission from both cell-external and internal noise. We identify broad distributions of exponential gradients, which lead to log-normal stimuli and maximal drift velocity. Our results thus provide a first step towards deciphering the chemical nature of complex, experimentally inaccessible cellular microenvironments, such as the human intestine.
Author Summary
Outside the laboratory, bacteria live in complex microenvironments characterized by competition for space and available nutrients. Although often inaccessible by experiments, understanding the spatio-temporal dynamics of bacterial microenvironments is biomedically important. For instance, the chemical environment that symbiotic Escherichia coli encounter in the human gut relates to health of the gastrointestinal tract, gut metabolism, immune response, and tissue homeostasis. Other complex microenvironments include soil and biofilms. Assuming that bacterial sensory systems have evolved to optimally sense typical gradients, we treat signaling data, the signaling pathway with its architecture and reaction rates, and computer simulations of swimming bacteria in different gradients as “prior knowledge” to “reverse engineer” E. coli's habitat. Our identified gradients are exponentially shaped with wide-ranging rate values. These microenvironments most likely stem from local fluctuating nutrient sources and degradation by competing species, in which bacteria have evolved to swim with optimal performance.
PMCID: PMC4207464  PMID: 25340783
3.  Memory improves precision of cell sensing in fluctuating environments 
Scientific Reports  2014;4:5688.
Biological cells are often found to sense their chemical environment near the single-molecule detection limit. Surprisingly, this precision is higher than simple estimates of the fundamental physical limit, hinting towards active sensing strategies. In this work, we analyse the effect of cell memory, e.g. from slow biochemical processes, on the precision of sensing by cell-surface receptors. We derive analytical formulas, which show that memory significantly improves sensing in weakly fluctuating environments. However, surprisingly when memory is adjusted dynamically, the precision is always improved, even in strongly fluctuating environments. In support of this prediction we quantify the directional biases in chemotactic Dictyostelium discoideum cells in a flow chamber with alternating chemical gradients. The strong similarities between cell sensing and control engineering suggest universal problem-solving strategies of living matter.
PMCID: PMC4097367  PMID: 25023459
4.  Unraveling Adaptation in Eukaryotic Pathways: Lessons from Protocells 
PLoS Computational Biology  2013;9(10):e1003300.
Eukaryotic adaptation pathways operate within wide-ranging environmental conditions without stimulus saturation. Despite numerous differences in the adaptation mechanisms employed by bacteria and eukaryotes, all require energy consumption. Here, we present two minimal models showing that expenditure of energy by the cell is not essential for adaptation. Both models share important features with large eukaryotic cells: they employ small diffusible molecules and involve receptor subunits resembling highly conserved G-protein cascades. Analyzing the drawbacks of these models helps us understand the benefits of energy consumption, in terms of adjustability of response and adaptation times as well as separation of cell-external sensing and cell-internal signaling. Our work thus sheds new light on the evolution of adaptation mechanisms in complex systems.
Author Summary
Adaptation is a common feature in sensory systems, well familiar to us from light and dark adaptation of our visual system. Biological cells, ranging from bacteria to complex eukaryotes, including single-cell organisms and human sensory receptors, adopt different strategies to fulfill this property. However, all of them require substantial amounts of energy to adapt. Here, we compare the different biological strategies and design two minimal models which allow adaptation without requiring energy consumption. Schemes similar to the ones we proposed in our minimal models could have been adopted by ancient protocells, that have evolved into the pathways we now know and study. Analyzing our models can thus help elucidate the advantages brought to the cells by consumption of energy, including the bypassing of hard-wired cell parameters such as diffusion constants with increased control over time scales.
PMCID: PMC3812047  PMID: 24204235
5.  Distinct cell shapes determine accurate chemotaxis 
Scientific Reports  2013;3:2606.
The behaviour of an organism often reflects a strategy for coping with its environment. Such behaviour in higher organisms can often be reduced to a few stereotyped modes of movement due to physiological limitations, but finding such modes in amoeboid cells is more difficult as they lack these constraints. Here, we examine cell shape and movement in starved Dictyostelium amoebae during migration toward a chemoattractant in a microfluidic chamber. We show that the incredible variety in amoeboid shape across a population can be reduced to a few modes of variation. Interestingly, cells use distinct modes depending on the applied chemical gradient, with specific cell shapes associated with shallow, difficult-to-sense gradients. Modelling and drug treatment reveals that these behaviours are intrinsically linked with accurate sensing at the physical limit. Since similar behaviours are observed in a diverse range of cell types, we propose that cell shape and behaviour are conserved traits.
PMCID: PMC3764443  PMID: 24008441
6.  Intracellular chemical gradients: morphing principle in bacteria 
BMC Biophysics  2012;5:18.
Advances in computational biology allow systematic investigations to ascertain whether internal chemical gradients can be maintained in bacteria – an open question at the resolution limit of fluorescence microscopy. While it was previously believed that the small bacterial cell size and fast diffusion in the cytoplasm effectively remove any such gradient, a new computational study published in BMC Biophysics supports the emerging view that gradients can exist. The study arose from the recent observation that phosphorylated CtrA forms a gradient prior to cell division in Caulobacter crescentus, a bacterium known for its complicated cell cycle. Tropini et al. (2012) postulate that such gradients can provide an internal chemical compass, directing protein localization, cell division and cell development. More specifically, they describe biochemical and physical constraints on the formation of such gradients and explore a number of existing bacterial cell morphologies. These chemical gradients may limit in vitro analyses, and may ensure timing control and robustness to fluctuations during critical stages in cell development.
PMCID: PMC3443414  PMID: 22954369
7.  Noise characteristics of the Escherichia coli rotary motor 
BMC Systems Biology  2011;5:151.
The chemotaxis pathway in the bacterium Escherichia coli allows cells to detect changes in external ligand concentration (e.g. nutrients). The pathway regulates the flagellated rotary motors and hence the cells' swimming behaviour, steering them towards more favourable environments. While the molecular components are well characterised, the motor behaviour measured by tethered cell experiments has been difficult to interpret.
We study the effects of sensing and signalling noise on the motor behaviour. Specifically, we consider fluctuations stemming from ligand concentration, receptor switching between their signalling states, adaptation, modification of proteins by phosphorylation, and motor switching between its two rotational states. We develop a model which includes all signalling steps in the pathway, and discuss a simplified version, which captures the essential features of the full model. We find that the noise characteristics of the motor contain signatures from all these processes, albeit with varying magnitudes.
Our analysis allows us to address how cell-to-cell variation affects motor behaviour and the question of optimal pathway design. A similar comprehensive analysis can be applied to other two-component signalling pathways.
PMCID: PMC3224245  PMID: 21951560
8.  The zipper mechanism in phagocytosis: energetic requirements and variability in phagocytic cup shape 
BMC Systems Biology  2010;4:149.
Phagocytosis is the fundamental cellular process by which eukaryotic cells bind and engulf particles by their cell membrane. Particle engulfment involves particle recognition by cell-surface receptors, signaling and remodeling of the actin cytoskeleton to guide the membrane around the particle in a zipper-like fashion. Despite the signaling complexity, phagocytosis also depends strongly on biophysical parameters, such as particle shape, and the need for actin-driven force generation remains poorly understood.
Here, we propose a novel, three-dimensional and stochastic biophysical model of phagocytosis, and study the engulfment of particles of various sizes and shapes, including spiral and rod-shaped particles reminiscent of bacteria. Highly curved shapes are not taken up, in line with recent experimental results. Furthermore, we surprisingly find that even without actin-driven force generation, engulfment proceeds in a large regime of parameter values, albeit more slowly and with highly variable phagocytic cups. We experimentally confirm these predictions using fibroblasts, transfected with immunoreceptor FcγRIIa for engulfment of immunoglobulin G-opsonized particles. Specifically, we compare the wild-type receptor with a mutant receptor, unable to signal to the actin cytoskeleton. Based on the reconstruction of phagocytic cups from imaging data, we indeed show that cells are able to engulf small particles even without support from biological actin-driven processes.
This suggests that biochemical pathways render the evolutionary ancient process of phagocytic highly robust, allowing cells to engulf even very large particles. The particle-shape dependence of phagocytosis makes a systematic investigation of host-pathogen interactions and an efficient design of a vehicle for drug delivery possible.
PMCID: PMC2991294  PMID: 21059234
9.  Chemotactic Response and Adaptation Dynamics in Escherichia coli 
PLoS Computational Biology  2010;6(5):e1000784.
Adaptation of the chemotaxis sensory pathway of the bacterium Escherichia coli is integral for detecting chemicals over a wide range of background concentrations, ultimately allowing cells to swim towards sources of attractant and away from repellents. Its biochemical mechanism based on methylation and demethylation of chemoreceptors has long been known. Despite the importance of adaptation for cell memory and behavior, the dynamics of adaptation are difficult to reconcile with current models of precise adaptation. Here, we follow time courses of signaling in response to concentration step changes of attractant using in vivo fluorescence resonance energy transfer measurements. Specifically, we use a condensed representation of adaptation time courses for efficient evaluation of different adaptation models. To quantitatively explain the data, we finally develop a dynamic model for signaling and adaptation based on the attractant flow in the experiment, signaling by cooperative receptor complexes, and multiple layers of feedback regulation for adaptation. We experimentally confirm the predicted effects of changing the enzyme-expression level and bypassing the negative feedback for demethylation. Our data analysis suggests significant imprecision in adaptation for large additions. Furthermore, our model predicts highly regulated, ultrafast adaptation in response to removal of attractant, which may be useful for fast reorientation of the cell and noise reduction in adaptation.
Author Summary
Bacterial chemotaxis is a paradigm for sensory systems, and thus has attracted immense interest from biologists and modelers alike. Using this pathway, cells can sense chemical molecules in their environment, and bias their movement towards nutrients and away from toxins. To avoid over- or understimulation of the signaling pathway, receptors adapt to current external conditions by covalent receptor modification, ultimately allowing cells to chemotax over a wide range of background concentrations. While the robustness and precision in adaptation was previously explained, we quantify the dynamics of adaptation, important for cell memory and behavior, as well as noise filtering in the pathway. Specifically, we study the intracellular signaling response and subsequent adaptation to concentration step changes in attractant chemicals. We combine measurements of signaling in living cells with a dynamic model for strongly coupled receptors, even including the effects of concentration flow in the experiment. Using a novel way of summarizing time-dependent data, we derive a new adaptation model, predicting additional layers of feedback regulation. As a consequence, adaptation to sudden exposure of unfavorable conditions is very fast, which may be useful for a quick reorientation and escape of the cell.
PMCID: PMC2873904  PMID: 20502674
10.  Biophysical Mechanism for Ras-Nanocluster Formation and Signaling in Plasma Membrane 
PLoS ONE  2009;4(7):e6148.
Ras GTPases are lipid-anchored G proteins, which play a fundamental role in cell signaling processes. Electron micrographs of immunogold-labeled Ras have shown that membrane-bound Ras molecules segregate into nanocluster domains. Several models have been developed in attempts to obtain quantitative descriptions of nanocluster formation, but all have relied on assumptions such as a constant, expression-level independent ratio of Ras in clusters to Ras monomers (cluster/monomer ratio). However, this assumption is inconsistent with the law of mass action. Here, we present a biophysical model of Ras clustering based on short-range attraction and long-range repulsion between Ras molecules in the membrane. To test this model, we performed Monte Carlo simulations and compared statistical clustering properties with experimental data. We find that we can recover the experimentally-observed clustering across a range of Ras expression levels, without assuming a constant cluster/monomer ratio or the existence of lipid rafts. In addition, our model makes predictions about the signaling properties of Ras nanoclusters in support of the idea that Ras nanoclusters act as an analog-digital-analog converter for high fidelity signaling.
PMCID: PMC2704371  PMID: 19587789
11.  Variable sizes of Escherichia coli chemoreceptor signaling teams 
Like many sensory receptors, bacterial chemotaxis receptors form clusters. In bacteria, large-scale clusters are subdivided into signaling teams that act as ‘antennas' allowing detection of ligands with remarkable sensitivity. The range of sensitivity is greatly extended by adaptation of receptors to changes in concentrations through covalent modification. However, surprisingly little is known about the sizes of receptor signaling teams. Here, we combine measurements of the signaling response, obtained from in vivo fluorescence resonance energy transfer, with the statistical method of principal component analysis, to quantify the size of signaling teams within the framework of the previously successful Monod–Wyman–Changeux model. We find that size of signaling teams increases 2- to 3-fold with receptor modification, indicating an additional, previously unrecognized level of adaptation of the chemotaxis network. This variation of signaling-team size shows that receptor cooperativity is dynamic and likely optimized for sensing noisy ligand concentrations.
PMCID: PMC2538909  PMID: 18682701
adaptation; chemotaxis; FRET; principal component analysis; receptor clusters
12.  Chemotaxis in Escherichia coli: A Molecular Model for Robust Precise Adaptation 
The chemotaxis system in the bacterium Escherichia coli is remarkably sensitive to small relative changes in the concentrations of multiple chemical signals over a broad range of ambient concentrations. Interactions among receptors are crucial to this sensitivity as is precise adaptation, the return of chemoreceptor activity to prestimulus levels in a constant chemoeffector environment. Precise adaptation relies on methylation and demethylation of chemoreceptors by the enzymes CheR and CheB, respectively. Experiments indicate that when transiently bound to one receptor, these enzymes act on small assistance neighborhoods (AN) of five to seven receptor homodimers. In this paper, we model a strongly coupled complex of receptors including dynamic CheR and CheB acting on ANs. The model yields sensitive response and precise adaptation over several orders of magnitude of attractant concentrations and accounts for different responses to aspartate and serine. Within the model, we explore how the precision of adaptation is limited by small AN size as well as by CheR and CheB kinetics (including dwell times, saturation, and kinetic differences among modification sites) and how these kinetics contribute to noise in complex activity. The robustness of our dynamic model for precise adaptation is demonstrated by randomly varying biochemical parameters.
Author Summary
Bacteria swim in relatively straight lines and change directions through tumbling. In the process of chemotaxis, a network of receptors and other proteins controls the tumbling frequency to direct an otherwise random walk toward nutrients and away from repellents. Receptor clustering and adaptation to persistent stimuli through covalent modification allow chemotaxis to be sensitive over a large range of ambient concentrations. The individual components of the chemotaxis network are well characterized, and signaling measurements by fluorescence microscopy quantify the network's response, making the system well suited for modeling and analysis. In this paper, we expand upon a previous model based on experiments indicating that the covalent modifications required for adaptation occur through the action of enzymes on groups of neighboring receptors, referred to as assistance neighborhoods. Simulations show that our proposed molecular model of a strongly coupled complex of receptors produces accurate responses to different stimuli and is robust to parameter variation. Within this model, the correct adaptation response is limited by small assistance-neighborhood size as well as enzyme kinetics. We also explore how these kinetics contribute to noise in the chemotactic response.
PMCID: PMC2174977  PMID: 18179279
13.  Chemotaxis Receptor Complexes: From Signaling to Assembly 
PLoS Computational Biology  2007;3(7):e150.
Complexes of chemoreceptors in the bacterial cytoplasmic membrane allow for the sensing of ligands with remarkable sensitivity. Despite the excellent characterization of the chemotaxis signaling network, very little is known about what controls receptor complex size. Here we use in vitro signaling data to model the distribution of complex sizes. In particular, we model Tar receptors in membranes as an ensemble of different sized oligomer complexes, i.e., receptor dimers, dimers of dimers, and trimers of dimers, where the relative free energies, including receptor modification, ligand binding, and interaction with the kinase CheA determine the size distribution. Our model compares favorably with a variety of signaling data, including dose-response curves of receptor activity and the dependence of activity on receptor density in the membrane. We propose that the kinetics of complex assembly can be measured in vitro from the temporal response to a perturbation of the complex free energies, e.g., by addition of ligand.
Author Summary
Chemotaxis allows bacteria to sense and swim toward nutrients and away from toxins. The remarkable sensing properties of the chemotaxis network, such as high sensitivity to small changes in the chemical environment, are thought to originate from receptor complexes in the membrane, which act as antennas to magnify weak signals. To adapt to persistent stimulation, receptors are covalently modified. While the individual protein components of the chemotaxis network are well characterized, making the system well suited for quantitative and computational analysis, direct experimental visualization of receptors and receptor complexes is difficult within the current limits of fluorescence and electron microscopy. To address questions such as how large are complexes and why do they assemble, we analyze in vitro signaling data using a previously developed model of signaling by receptor complexes. Based on the data, we propose a statistical physics model for the distribution of complex sizes in the membrane. Within this model, complex size depends on the receptor free energy with contributions from receptor modification level, ligand binding, receptor–receptor coupling, and binding to accessory proteins. Our model results compare favorably with a variety of different signaling data, and suggest new experiments to measure the kinetics of assembly of receptor complexes.
PMCID: PMC1933480  PMID: 17676982
14.  Kinetic Analysis of the Assembly of the Outer Membrane Protein LamB in Escherichia coli Mutants Each Lacking a Secretion or Targeting Factor in a Different Cellular Compartment▿  
Journal of Bacteriology  2006;189(2):446-454.
Outer membrane β-barrel proteins in gram-negative bacteria, such as Escherichia coli, must be translocated from their site of synthesis in the cytoplasm to the periplasm and finally delivered to the outer membrane. At least a dozen proteins located in the cytoplasm, the periplasm, and both the inner and outer membranes are required to catalyze this complex assembly process. At normal growth temperatures and conditions the transport and assembly processes are so fast that assembly intermediates cannot be detected. Using cells grown at a low temperature to slow the assembly process and pulse-chase analysis with immunodetection methods, we followed newly synthesized LamB molecules during their transit through the cell envelope. The quality and reproducibility of the data allowed us to calculate rate constants for three different subassembly reactions. This kinetic analysis revealed that secB and secD mutants exhibit nearly identical defects in precursor translocation from the cytoplasm. However, subsequent subassembly reaction rates provided no clear evidence for an additional role for SecD in LamB assembly. Moreover, we found that surA mutants are qualitatively indistinguishable from yfgL mutants, suggesting that the products of both of these genes share a common function in the assembly process, most likely the delivery of LamB to the YaeT assembly complex in the outer membrane.
PMCID: PMC1797403  PMID: 17071751

Results 1-14 (14)