81 results found
Endres R, Cavanagh H, Mosbach A, et al., 6424, Physics-informed deep learning characterizes morphodynamics of Asian soybean rust disease, Nature Communications, Vol: 12, Pages: 1-8, ISSN: 2041-1723
Medicines and agricultural biocides are often discovered using large phenotypic screens across hundreds of compounds, where visible effects of whole organisms are compared to gauge efficacy and possible modes of action. However, such analysis is often limited to human-defined and static features. Here, we introduce a novel framework that can characterize shape changes (morphodynamics) for cell-drug interactions directly from images, and use it to interpret perturbed development of Phakopsora pachyrhizi, the Asian soybean rust crop pathogen. We describe population development over a 2D space of shapes (morphospace) using two models with condition-dependent parameters: a top-down Fokker-Planck model of diffusive development over Waddington-type landscapes, and a bottom-up model of tip growth. We discover a variety of landscapes, describing phenotype transitions during growth, and identify possible perturbations in the tip growth machinery that cause this variation. This demonstrates a widely-applicable integration of unsupervised learning and biophysical modeling.
Cavanagh H, Kempe D, Mazalo JK, et al., 2022, T cell morphodynamics reveal periodic shape oscillations in three-dimensional migration, JOURNAL OF THE ROYAL SOCIETY INTERFACE, Vol: 19, ISSN: 1742-5689
Cook J, Pawar S, Endres R, 2021, Thermodynamic constraints on the assembly and diversity of microbial ecosystems are different near to and far from equilibrium, PLOS COMPUTATIONAL BIOLOGY, Vol: 17, ISSN: 1553-734X
Cook J, Pawar S, Endres RG, 2021, Thermodynamic constraints on the assembly and diversity of microbial ecosystems are different near to and far from equilibrium
<jats:title>Abstract</jats:title><jats:p>Non-equilibrium thermodynamics has long been an area of substantial interest to ecologists because most fundamental biological processes, such as protein synthesis and respiration, are inherently energy-consuming. However, most of this interest has focused on developing coarse ecosystem-level maximisation principles, providing little insight into underlying mechanisms that lead to such emergent constraints. Microbial communities are a natural system to decipher this mechanistic basis because their interactions in the form of substrate consumption, metabolite production, and cross-feeding can be described explicitly in thermodynamic terms. Previous work has considered how thermodynamic constraints impact competition between pairs of species, but restrained from analysing how this manifests in complex dynamical systems. To address this gap, we develop a thermodynamic microbial community model with fully reversible reaction kinetics, which allows direct consideration of free-energy dissipation. This also allows species to interact via products rather than just substrates, increasing the dynamical complexity, and allowing a more nuanced classification of interaction types to emerge. Using this model, we find that community diversity increases with substrate lability, because greater free-energy availability allows for faster generation of niches. Thus, more niches are generated in the time frame of community establishment, leading to higher final species diversity. We also find that allowing species to make use of near-to-equilibrium reactions increases diversity in a low free-energy regime. In such a regime, two new thermodynamic interaction types that we identify here reach comparable strengths to the conventional (competition and facilitation) types, emphasising the key role that thermodynamics plays in community dynamics. Our results suggest that accounting for realistic thermodynamic constraints is vital fo
Ding SS, Muhle LS, Brown A, et al., 2020, Comparison of solitary and collective foraging strategies of Caenorhabditis elegans in patchy food distributions, Philosophical Transactions of the Royal Society B: Biological Sciences, Vol: 375, ISSN: 0962-8436
Collective foraging has been shown to benefit organisms in environments where food is patchily distributed, but whether this is true in the case where organisms do not rely on long range communications to coordinate their collective behaviour has been understudied. To address this question, we use the tractable laboratory model organism Caenorhabditis elegans, where a social strain (npr-1 mutant) and a solitary strain (N2) are available for directcomparison of foraging strategies. We first developed an on-lattice minimal model for comparing collective and solitary foraging strategies, finding that social agents benefit from feeding faster and more efficiently simply due to group formation. Our laboratory foraging experiments with npr-1 and N2 worm populations, however, show an advantage for solitary N2 in all food distribution environments that we tested. We incorporated additional strain43 specific behavioural parameters of npr-1 and N2 worms into our model and computationally identified N2’s higher feeding rate to be the key factor underlying its advantage, without which it is possible to recapitulate the advantage of collective foraging in patchy environments. Our work highlights the theoretical advantage of collective foraging due to group formation alone without long-range interactions, and the valuable role of modelling to guide experiments.
Bodor DL, Ponisch W, Endres RG, et al., 2020, Of Cell Shapes and Motion: The Physical Basis of Animal Cell Migration, DEVELOPMENTAL CELL, Vol: 52, Pages: 550-562, ISSN: 1534-5807
Cook J, Endres R, 2020, Thermodynamics of switching in multistable non-equilibrium systems, Journal of Chemical Physics, Vol: 152, ISSN: 0021-9606
Multistable non-equilibrium systems are abundant outcomes of nonlinear dynamics with feedback, but still relatively little is known about what determines the stability of the steady states and their switching rates in terms of entropy and entropy production. Here, we will link fluctuation theorems for the entropy production along trajectories with the action obtainable from the Freidlin–Wentzell theorem to elucidate the thermodynamics of switching between states in the large volume limit of multistable systems. We find that the entropy production at steady state plays no role, but the entropy production during switching is key. Steady-state entropy and diffusive noise strength can be neglected in this limit. The relevance to biological, ecological, and climate models is apparent.
Ni B, Colin R, Link H, et al., 2020, Growth-rate dependent resource investment in bacterial motile behavior quantitatively follows potential benefit of chemotaxis, PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, Vol: 117, Pages: 595-601, ISSN: 0027-8424
Endres R, Micali G, 2019, Maximal information transmission is compatible with ultrasensitive biological pathways, Scientific Reports, Vol: 9, ISSN: 2045-2322
Cells are often considered input-output devices that maximize the transmission of informationby converting extracellular stimuli (input) via signaling pathways (communication channel) to cellbehavior (output). However, in biological systems outputs might feed back into inputs due to cellmotility, and the biological channel can change by mutations during evolution. Here, we showthat the conventional channel capacity obtained by optimizing the input distribution for a fixedchannel may not reflect the global optimum. In a new approach we analytically identify both inputdistributions and input-output curves that optimally transmit information, given constraints fromnoise and the dynamic range of the channel. We find a universal optimal input distribution onlydepending on the input noise, and we generalize our formalism to multiple outputs (or inputs).Applying our formalism to Escherichia coli chemotaxis, we find that its pathway is compatible withoptimal information transmission despite the ultrasensitive rotary motors.
Yolland L, Burki M, Marcotti S, et al., 2019, Persistent and polarized global actin flow is essential for directionality during cell migration, Nature Cell Biology, Vol: 21, Pages: 1370-1381, ISSN: 1465-7392
Cell migration is hypothesized to involve a cycle of behaviours beginning with leading edge extension. However, recent evidence suggests that the leading edge may be dispensable for migration, raising the question of what actually controls cell directionality. Here, we exploit the embryonic migration of Drosophila macrophages to bridge the different temporal scales of the behaviours controlling motility. This approach reveals that edge fluctuations during random motility are not persistent and are weakly correlated with motion. In contrast, flow of the actin network behind the leading edge is highly persistent. Quantification of actin flow structure during migration reveals a stable organization and asymmetry in the cell-wide flowfield that strongly correlates with cell directionality. This organization is regulated by a gradient of actin network compression and destruction, which is controlled by myosin contraction and cofilin-mediated disassembly. It is this stable actin-flow polarity, which integrates rapid fluctuations of the leading edge, that controls inherent cellular persistence.
Ding SS, Muhle L, Brown A, et al., 2019, Comparison of solitary and collective foraging strategies of Caenorhabditis elegansin patchy food distributions, Publisher: bioRxiv
Abstract The benefits of social behaviour in insects and vertebrates are well-documented in terms of mating success and predator avoidance. Social foraging has also been shown to benefit organisms in environments where food is patchily distributed, but whether this is true in the case where organisms do not rely on long-range communications to coordinate their social behaviour has been understudied. To address this question, we use the tractable laboratory model organism Caenorhabditis elegans , where a social strain ( npr-1 mutant) and a solitary strain (N2) are available for direct comparison of foraging strategies. We first develop an on-lattice minimal model for comparing social and solitary feeding strategies, finding that social agents benefit from feeding faster and more efficiently simply due to group formation. To compare these simulation results with real experimental data, we modify our minimal model to incorporate the specific feeding behaviours of the npr-1 and N2 strains. Surprisingly, the resultant strain-specific model predicts that the solitary strain performs better than the social one in all food distribution environments that we tested, which we confirm with lab experiments. Additional computational experiments identify the N2 strain’s higher feeding rate to be the key factor underlying its advantage over npr-1 worms. Our work highlights the difficulties in addressing questions of optimal behaviour, and the valuable role of modelling as a guiding principle.
Tweedy L, Witzel P, Heinrich D, et al., 2019, Screening by changes in stereotypical behavior during cell motility, Scientific Reports, Vol: 9, ISSN: 2045-2322
Stereotyped behaviors are series of postures that show very little variability between repeats. They have been used to classify the dynamics of individuals, groups and species without reference to the lower-level mechanisms that drive them. Stereotypes are easily identified in animals due to strong constraints on the number, shape, and relative positions of anatomical features, such as limbs, that may be used as landmarks for posture identification. In contrast, the identification of stereotypes in single cells poses a significant challenge as the cell lacks these landmark features, and finding constraints on cell shape is a non-trivial task. Here, we use the maximum caliber variational method to build a minimal model of cell behavior during migration. Without reference to biochemical details, we are able to make behavioral predictions over timescales of minutes using only changes in cell shape over timescales of seconds. We use drug treatment and genetics to demonstrate that maximum caliber descriptors can discriminate between healthy and aberrant migration, thereby showing potential applications for maximum caliber methods in automated disease screening, for example in the identification of behaviors associated with cancer metastasis.
Ding SS, Schumacher L, Javer A, et al., 2019, Shared behavioral mechanisms underlie C. elegans aggregation and swarming, eLife, Vol: 8, ISSN: 2050-084X
In complex biological systems, simple individual-level behavioral rules can give rise to emergent group-level behavior. While collective behavior has been well studied in cells and larger organisms, the mesoscopic scale is less understood, as it is unclear which sensory inputs and physical processes matter a priori. Here, we investigate collective feeding in the roundworm C. elegans at this intermediate scale, using quantitative phenotyping and agent-based modeling to identify behavioral rules underlying both aggregation and swarming—a dynamic phenotype only observed at longer timescales. Using fluorescence multi-worm tracking, we quantify aggregation in terms of individual dynamics and population-level statistics. Then we use agent-based simulations and approximate Bayesian inference to identify three key behavioral rules for aggregation: cluster-edge reversals, a density-dependent switch between crawling speeds, and taxis towards neighboring worms. Our simulations suggest that swarming is simply driven by local food depletion but otherwise employs the same behavioral mechanisms as the initial aggregation.
Lopez-Garrido J, Ojkic N, Khanna K, et al., 2018, Chromosome translocation inflates bacillus forespores and impacts cellular morphology, Cell, Vol: 172, Pages: 758-770.e14, ISSN: 0092-8674
The means by which the physicochemical properties of different cellular components together determine bacterial cell shape remain poorly understood. Here, we investigate a programmed cell-shape change during Bacillus subtilis sporulation, when a rod-shaped vegetative cell is transformed to an ovoid spore. Asymmetric cell division generates a bigger mother cell and a smaller, hemispherical forespore. The septum traps the forespore chromosome, which is translocated to the forespore by SpoIIIE. Simultaneously, forespore size increases as it is reshaped into an ovoid. Using genetics, timelapse microscopy, cryo-electron tomography, and mathematical modeling, we demonstrate that forespore growth relies on membrane synthesis and SpoIIIE-mediated chromosome translocation, but not on peptidoglycan or protein synthesis. Our data suggest that the hydrated nucleoid swells and inflates the forespore, displacing ribosomes to the cell periphery, stretching septal peptidoglycan, and reshaping the forespore. Our results illustrate how simple biophysical interactions between core cellular components contribute to cellular morphology.
Endres RG, 2017, Entropy production selects nonequilibrium states in multistable systems, Scientific Reports, Vol: 7, ISSN: 2045-2322
Far-from-equilibrium thermodynamics underpins the emergence of life, but how has been a long-outstanding puzzle. Best candidate theories based on the maximum entropy production principle could not be unequivocally proven, in part due to complicated physics, unintuitive stochastic thermodynamics, and the existence of alternative theories such as the minimum entropy production principle. Here, we use a simple, analytically solvable, one-dimensional bistable chemical system to demonstrate the validity of the maximum entropy production principle. To generalize to multistable stochastic system, we use the stochastic least-action principle to derive the entropy production and its role in the stability of nonequilibrium steady states. This shows that in a multistable system, all else being equal, the steady state with the highest entropy production is favored, with a number of implications for the evolution of biological, physical, and geological systems.
Chemotaxis of the bacterium Escherichia coli is well understood in shallow chemical gradients, but its swimming behavior remains difficult to interpret in steep gradients. By focusing on single-cell trajectories from simulations, we investigated the dependence of the chemotactic drift velocity on attractant concentration in an exponential gradient. Whereas maxima of the average drift velocity can be interpreted within analytical linear-response theory of chemotaxis in shallow gradients, limits in drift due to steep gradients and finite number of receptor-methylation sites for adaptation go beyond perturbation theory. For instance, we found a surprising pinning of the cells to the concentration in the gradient at which cells run out of methylation sites. To validate the positions of maximal drift, we recorded single-cell trajectories in carefully designed chemical gradients using microfluidics.
Richards DM, Endres RG, 2017, How cells engulf: a review of theoretical approaches to phagocytosis., Reports on Progress in Physics, Vol: 80, ISSN: 0034-4885
Phagocytosis is a fascinating process whereby a cell surrounds and engulfs particles such as bacteria and dead cells. This is crucial both for single-cell organisms (as a way of acquiring nutrients) and as part of the immune system (to destroy foreign invaders). This whole process is hugely complex and involves multiple coordinated events such as membrane remodelling, receptor motion, cytoskeleton reorganisation and intracellular signalling. Because of this, phagocytosis is an excellent system for theoretical study, benefiting from biophysical approaches combined with mathematical modelling. Here, we review these theoretical approaches and discuss the recent mathematical and computational models, including models based on receptors, models focusing on the forces involved, and models employing energetic considerations. Along the way, we highlight a beautiful connection to the physics of phase transitions, consider the role of stochasticity, and examine links between phagocytosis and other types of endocytosis. We cover the recently discovered multistage nature of phagocytosis, showing that the size of the phagocytic cup grows in distinct stages, with an initial slow stage followed by a much quicker second stage starting around half engulfment. We also address the issue of target shape dependence, which is relevant to both pathogen infection and drug delivery, covering both one-dimensional and two-dimensional results. Throughout, we pay particular attention to recent experimental techniques that continue to inform the theoretical studies and provide a means to test model predictions. Finally, we discuss population models, connections to other biological processes, and how physics and modelling will continue to play a key role in future work in this area.
Yap L, Endres RG, 2017, A model of cell-wall dynamics during sporulation in Bacillus subtilis, Soft Matter, Vol: 13, Pages: 8089-8095, ISSN: 1744-683X
To survive starvation, Bacillus subtilis forms durable spores. After asymmetric cell division, the septum grows around the forespore in a process called engulfment, but the mechanism of force generation is unknown. Here, we derived a novel biophysical model for the dynamics of cell-wall remodeling during engulfment based on a balancing of dissipative, active, and mechanical forces. By plotting phase diagrams, we predict that sporulation is promoted by a line tension from the attachment of the septum to the outer cell wall, as well as by an imbalance in turgor pressures in the mother-cell and forespore compartments. We also predict that significant mother-cell growth hinders engulfment. Hence, relatively simple physical principles may guide this complex biological process.
Schumacher LJ, Ding S, Brown AEX, et al., 2017, Collective feeding in C. elegans, 19th IUPAB Congress / 11th EBSA Congress, Publisher: SPRINGER, Pages: S74-S74, ISSN: 0175-7571
Schumacher LJ, Ding S, Brown AEX, et al., 2017, Collective feeding in C-elegans, 19th IUPAB Congress / 11th EBSA Congress, Publisher: Springer (part of Springer Nature), Pages: S74-S74, ISSN: 0175-7571
Rotrattanadumrong R, Endres RG, 2017, Emergence of cooperativity in a model biofilm, Journal of Physics D: Applied Physics, Vol: 50, ISSN: 0022-3727
Evolution to multicellularity from an aggregate of cells involves altruistic cooperation between individual cells, which is in conflict with Darwinian evolution. How cooperation arises and how a cell community resolves such conflicts remains unclear. In this study, we investigated the spontaneous emergence of cell differentiation and the subsequent division of labour in evolving cellular metabolic networks. In spatially extended cell aggregates, our findings reveal that resource limitation can lead to the formation of subpopulations and cooperation of cells, and hence multicellular communities. A specific example of our model can explain the recently observed oscillatory growth in Bacillus subtilis biofilms.
De Palo G, Yi D, Endres RG, 2017, A critical-like collective state leads to long-range cell communication in Dictyostelium discoideum aggregation, PLOS Biology, Vol: 15, ISSN: 1544-9173
The transition from single-cell to multicellular behavior is important in early development but rarely studied. The starvation-induced aggregation of the social amoeba Dictyostelium discoideum into a multicellular slug is known to result from single-cell chemotaxis towards emitted pulses of cyclic adenosine monophosphate (cAMP). However, how exactly do transient, short-range chemical gradients lead to coherent collective movement at a macroscopic scale? Here, we developed a multiscale model verified by quantitative microscopy to describe behaviors ranging widely from chemotaxis and excitability of individual cells to aggregation of thousands of cells. To better understand the mechanism of long-range cell—cell communication and hence aggregation, we analyzed cell—cell correlations, showing evidence of self-organization at the onset of aggregation (as opposed to following a leader cell). Surprisingly, cell collectives, despite their finite size, show features of criticality known from phase transitions in physical systems. By comparing wild-type and mutant cells with impaired aggregation, we found the longest cell—cell communication distance in wild-type cells, suggesting that criticality provides an adaptive advantage and optimally sized aggregates for the dispersal of spores.
Ojkic N, Lopez-Garrido J, Pogliano K, et al., 2016, Cell-wall remodeling drives engulfment during Bacillus subtiliss porulation, eLife, Vol: 5, ISSN: 2050-084X
When starved, the Gram-positive bacterium Bacillus subtilis forms durable spores forsurvival. Sporulation initiates with an asymmetric cell division, creating a large mother cell and asmall forespore. Subsequently, the mother cell membrane engulfs the forespore in a phagocytosislikeprocess. However, the force generation mechanism for forward membrane movement remainsunknown. Here, we show that membrane migration is driven by cell wall remodeling at the leadingedge of the engulfing membrane, with peptidoglycan synthesis and degradation mediated bypenicillin binding proteins in the forespore and a cell wall degradation protein complex in themother cell. We propose a simple model for engulfment in which the junction between the septumand the lateral cell wall moves around the forespore by a mechanism resembling the ‘templatemodel’. Hence, we establish a biophysical mechanism for the creation of a force for engulfmentbased on the coordination between cell wall synthesis and degradation.
Richards D, Endres RG, 2016, Target-shape dependence in a simple model of receptor-mediated endocytosis and phagocytosis, Proceedings of the National Academy of Sciences of the United States of America, Vol: 113, Pages: 6113-6118, ISSN: 1091-6490
Along with other forms of internalisation, phagocytosis and receptormediatedendocytosis are vitally important for many cell types, rangingfrom single-cell organisms to immune cells. It is known experimentallythat engulfment in both cases depends critically on particleshape and orientation. However, most previous theoretical workhas focused only on spherical particles and hence disregards the widerangingparticle shapes occurring in nature, such as those of bacteria.Here, by implementing a simple model in one- and two-dimensions, wecompare and contrast receptor-mediated endocytosis and phagocytosisfor a range of biologically-relevant shapes, including spheres, ellipsoids,capped-cylinders and hourglasses. We find a whole range of different engulfmentbehaviours with some ellipsoids engulfing quicker than spheres,and that phagocytosis is able to engulf a greater range of target shapesthan other types of endocytosis. Further, the two-dimensional modelcan explain why some non-spherical particles engulf quickest (not at all)when presented to the membrane tip-first (lying flat). Our work revealshow some bacteria may avoid being internalised simply by their shape,and suggests shapes for optimal drug delivery.
Micali G, Endres RG, 2015, Bacterial chemotaxis: information processing, thermodynamics, and behavior., Current Opinion in Microbiology, Vol: 30, Pages: 8-15, ISSN: 1879-0364
Escherichia coli has long been used as a model organism due to the extensive experimental characterization of its pathways and molecular components. Take chemotaxis as an example, which allows bacteria to sense and swim in response to chemicals, such as nutrients and toxins. Many of the pathway's remarkable sensing and signaling properties are now concisely summarized in terms of design (or engineering) principles. More recently, new approaches from information theory and stochastic thermodynamics have begun to address how pathways process environmental stimuli and what the limiting factors are. However, to fully capitalize on these theoretical advances, a closer connection with single-cell experiments will be required.
Eismann S, Endres RG, 2015, Protein connectivity in chemotaxis receptor complexes, PLOS Computational Biology, Vol: 11, ISSN: 1553-734X
The chemotaxis sensory system allows bacteria such as Escherichia coli to swim towards nutrients and away from repellents. The underlying pathway is remarkably sensitive in detecting chemical gradients over a wide range of ambient concentrations. Interactions among receptors, which are predominantly clustered at the cell poles, are crucial to this sensitivity. Although it has been suggested that the kinase CheA and the adapter protein CheW are integral for receptor connectivity, the exact coupling mechanism remains unclear. Here, we present a statistical-mechanics approach to model the receptor linkage mechanism itself, building on nanodisc and electron cryotomography experiments. Specifically, we investigate how the sensing behavior of mixed receptor clusters is affected by variations in the expression levels of CheA and CheW at a constant receptor density in the membrane. Our model compares favorably with dose-response curves from in vivo Förster resonance energy transfer (FRET) measurements, demonstrating that the receptor-methylation level has only minor effects on receptor cooperativity. Importantly, our model provides an explanation for the non-intuitive conclusion that the receptor cooperativity decreases with increasing levels of CheA, a core signaling protein associated with the receptors, whereas the receptor cooperativity increases with increasing levels of CheW, a key adapter protein. Finally, we propose an evolutionary advantage as explanation for the recently suggested CheW-only linker structures.
Aquino G, Wingreen NS, Endres RG, 2015, Know the Single-Receptor Sensing Limit? Think Again, Journal of Statistical Physics, Vol: 162, Pages: 1353-1364, ISSN: 0022-4715
How cells reliably infer information about their environment is a fundamentallyimportant question. While sensing and signaling generally start with cell-surface receptors,the degree of accuracy with which a cell can measure external ligand concentration with eventhe simplest device—a single receptor—is surprisingly hard to pin down. Recent studies provideconflicting results for the fundamental physical limits. Comparison is made difficult asdifferent studies either suggest different readout mechanisms of the ligand-receptor occupancy,or differ on how ligand diffusion is implemented. Here we critically analyse thesestudies and present a unifying perspective on the limits of sensing, with wide-ranging biologicalimplications.
Micali G, Aquino G, Richards DM, et al., 2015, Accurate encoding and decoding by single cells: amplitude versus frequency modulation., Plos Computational Biology, Vol: 11, Pages: e1004222-e1004222, ISSN: 1553-7358
Cells sense external concentrations and, via biochemical signaling, respond by regulating the expression of target proteins. Both in signaling networks and gene regulation there are two main mechanisms by which the concentration can be encoded internally: amplitude modulation (AM), where the absolute concentration of an internal signaling molecule encodes the stimulus, and frequency modulation (FM), where the period between successive bursts represents the stimulus. Although both mechanisms have been observed in biological systems, the question of when it is beneficial for cells to use either AM or FM is largely unanswered. Here, we first consider a simple model for a single receptor (or ion channel), which can either signal continuously whenever a ligand is bound, or produce a burst in signaling molecule upon receptor binding. We find that bursty signaling is more accurate than continuous signaling only for sufficiently fast dynamics. This suggests that modulation based on bursts may be more common in signaling networks than in gene regulation. We then extend our model to multiple receptors, where continuous and bursty signaling are equivalent to AM and FM respectively, finding that AM is always more accurate. This implies that the reason some cells use FM is related to factors other than accuracy, such as the ability to coordinate expression of multiple genes or to implement threshold crossing mechanisms.
Endres RG, 2015, Bistability: Requirements on Cell-Volume, Protein Diffusion, and Thermodynamics, PLOS ONE, Vol: 10, ISSN: 1932-6203
Fan S, Endres RG, 2014, A minimal model for metabolism-dependent chemotaxis in Rhodobacter sphaeroides, INTERFACE FOCUS, Vol: 4, ISSN: 2042-8898
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