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1.
EMBO J ; 42(14): e112168, 2023 07 17.
Artículo en Inglés | MEDLINE | ID: mdl-37260169

RESUMEN

All bacterial cells must expand their envelopes during growth. The main load-bearing and shape-determining component of the bacterial envelope is the peptidoglycan cell wall. Bacterial envelope growth and shape changes are often thought to be controlled through enzymatic cell wall insertion. We investigated the role of cell wall insertion for cell shape changes during cell elongation in Gram-negative bacteria. We found that both global and local rates of envelope growth of Escherichia coli remain nearly unperturbed upon arrest of cell wall insertion-up to the point of sudden cell lysis. Specifically, cells continue to expand their surface areas in proportion to biomass growth rate, even if the rate of mass growth changes. Other Gram-negative bacteria behave similarly. Furthermore, cells plastically change cell shape in response to differential mechanical forces. Overall, we conclude that cell wall-cleaving enzymes can control envelope growth independently of synthesis. Accordingly, the strong overexpression of an endopeptidase leads to transiently accelerated bacterial cell elongation. Our study demonstrates that biomass growth and envelope forces can guide cell envelope expansion through mechanisms that are independent of cell wall insertion.


Asunto(s)
Pared Celular , Escherichia coli , Pared Celular/metabolismo , Membrana Celular/metabolismo , Escherichia coli/metabolismo , Ciclo Celular , Bacterias Gramnegativas/metabolismo , Peptidoglicano/metabolismo
2.
EMBO J ; 39(5): e102246, 2020 03 02.
Artículo en Inglés | MEDLINE | ID: mdl-32009249

RESUMEN

The peptidoglycan (PG) sacculus provides bacteria with the mechanical strength to maintain cell shape and resist osmotic stress. Enlargement of the mesh-like sacculus requires the combined activity of peptidoglycan synthases and hydrolases. In Escherichia coli, the activity of two PG synthases is driven by lipoproteins anchored in the outer membrane (OM). However, the regulation of PG hydrolases is less well understood, with only regulators for PG amidases having been described. Here, we identify the OM lipoprotein NlpI as a general adaptor protein for PG hydrolases. NlpI binds to different classes of hydrolases and can specifically form complexes with various PG endopeptidases. In addition, NlpI seems to contribute both to PG elongation and division biosynthetic complexes based on its localization and genetic interactions. Consistent with such a role, we reconstitute PG multi-enzyme complexes containing NlpI, the PG synthesis regulator LpoA, its cognate bifunctional synthase, PBP1A, and different endopeptidases. Our results indicate that peptidoglycan regulators and adaptors are part of PG biosynthetic multi-enzyme complexes, regulating and potentially coordinating the spatiotemporal action of PG synthases and hydrolases.


Asunto(s)
Proteínas de Escherichia coli/metabolismo , Escherichia coli/enzimología , Lipoproteínas/metabolismo , Complejos Multienzimáticos , N-Acetil Muramoil-L-Alanina Amidasa/metabolismo , Pared Celular/enzimología , Endopeptidasas/genética , Endopeptidasas/metabolismo , Escherichia coli/genética , Proteínas de Escherichia coli/genética , Lipoproteínas/genética , N-Acetil Muramoil-L-Alanina Amidasa/genética , Peptidoglicano/metabolismo
3.
Proc Natl Acad Sci U S A ; 118(32)2021 08 10.
Artículo en Inglés | MEDLINE | ID: mdl-34341116

RESUMEN

During growth, cells must expand their cell volumes in coordination with biomass to control the level of cytoplasmic macromolecular crowding. Dry-mass density, the average ratio of dry mass to volume, is roughly constant between different nutrient conditions in bacteria, but it remains unknown whether cells maintain dry-mass density constant at the single-cell level and during nonsteady conditions. Furthermore, the regulation of dry-mass density is fundamentally not understood in any organism. Using quantitative phase microscopy and an advanced image-analysis pipeline, we measured absolute single-cell mass and shape of the model organisms Escherichia coli and Caulobacter crescentus with improved precision and accuracy. We found that cells control dry-mass density indirectly by expanding their surface, rather than volume, in direct proportion to biomass growth-according to an empirical surface growth law. At the same time, cell width is controlled independently. Therefore, cellular dry-mass density varies systematically with cell shape, both during the cell cycle or after nutrient shifts, while the surface-to-mass ratio remains nearly constant on the generation time scale. Transient deviations from constancy during nutrient shifts can be reconciled with turgor-pressure variations and the resulting elastic changes in surface area. Finally, we find that plastic changes of cell width after nutrient shifts are likely driven by turgor variations, demonstrating an important regulatory role of mechanical forces for width regulation. In conclusion, turgor-dependent cell width and a slowly varying surface-to-mass coupling constant are the independent variables that determine dry-mass density.


Asunto(s)
Escherichia coli/química , Escherichia coli/citología , Microscopía de Contraste de Fase/métodos , Bacterias/química , Bacterias/citología , Bacterias/crecimiento & desarrollo , Escherichia coli/efectos de los fármacos , Escherichia coli/crecimiento & desarrollo , Procesamiento de Imagen Asistido por Computador , Modelos Biológicos , Ósmosis , Análisis de la Célula Individual , Imagen de Lapso de Tiempo
4.
Mol Syst Biol ; 14(3): e7899, 2018 03 08.
Artículo en Inglés | MEDLINE | ID: mdl-29519933

RESUMEN

Over the past few years, tools that make use of the Cas9 nuclease have led to many breakthroughs, including in the control of gene expression. The catalytically dead variant of Cas9 known as dCas9 can be guided by small RNAs to block transcription of target genes, in a strategy also known as CRISPRi. Here, we reveal that the level of complementarity between the guide RNA and the target controls the rate at which RNA polymerase "kicks out" dCas9 from the target and completes transcription. We use this mechanism to precisely and robustly reduce gene expression by defined relative amounts. Alternatively, tuning repression by changing dCas9 concentration is noisy and promoter-strength dependent. We demonstrate broad applicability of this method to the study of genetic regulation and cellular physiology. First, we characterize feedback strength of a model auto-repressor. Second, we study the impact of amount variations of cell-wall synthesizing enzymes on cell morphology. Finally, we multiplex the system to obtain any combination of fractional repression of two genes.


Asunto(s)
Bacterias/genética , Genes Bacterianos , ARN Guía de Kinetoplastida/genética , Sistemas CRISPR-Cas , Regulación Bacteriana de la Expresión Génica , Técnicas de Silenciamiento del Gen , Regiones Promotoras Genéticas , Activación Transcripcional
5.
Proc Natl Acad Sci U S A ; 113(14): E2066-72, 2016 Apr 05.
Artículo en Inglés | MEDLINE | ID: mdl-26933214

RESUMEN

Many bacterial species colonize surfaces and form dense 3D structures, known as biofilms, which are highly tolerant to antibiotics and constitute one of the major forms of bacterial biomass on Earth. Bacterial biofilms display remarkable changes during their development from initial attachment to maturity, yet the cellular architecture that gives rise to collective biofilm morphology during growth is largely unknown. Here, we use high-resolution optical microscopy to image all individual cells in Vibrio cholerae biofilms at different stages of development, including colonies that range in size from 2 to 4,500 cells. From these data, we extracted the precise 3D cellular arrangements, cell shapes, sizes, and global morphological features during biofilm growth on submerged glass substrates under flow. We discovered several critical transitions of the internal and external biofilm architectures that separate the major phases of V. cholerae biofilm growth. Optical imaging of biofilms with single-cell resolution provides a new window into biofilm formation that will prove invaluable to understanding the mechanics underlying biofilm development.


Asunto(s)
Biopelículas/crecimiento & desarrollo , Análisis de la Célula Individual , Vibrio cholerae/fisiología , Vibrio cholerae/crecimiento & desarrollo
6.
Bioessays ; 34(5): 427-36, 2012 May.
Artículo en Inglés | MEDLINE | ID: mdl-22415868

RESUMEN

Fluorescence microscopy is the primary tool for studying complex processes inside individual living cells. Technical advances in both molecular biology and microscopy have made it possible to image cells from many genetic and environmental backgrounds. These images contain a vast amount of information, which is often hidden behind various sources of noise, convoluted with other information and stochastic in nature. Accessing the desired biological information therefore requires new tools of computational image analysis and modeling. Here, we review some of the recent advances in computational analysis of images obtained from fluorescence microscopy, focusing on bacterial systems. We emphasize techniques that are readily available to molecular and cell biologists but also point out examples where problem-specific image analyses are necessary. Thus, image analysis is not only a toolkit to be applied to new images but also an integral part of the design and implementation of a microscopy experiment.


Asunto(s)
Bacterias/ultraestructura , Rastreo Celular/métodos , Procesamiento de Imagen Asistido por Computador/métodos , Microscopía Fluorescente/métodos , Bacterias/crecimiento & desarrollo , Bacterias/metabolismo , Proteínas Bacterianas/metabolismo , Proteínas Bacterianas/ultraestructura , Movimiento Celular , Difusión , Sustancias Macromoleculares/metabolismo , Sustancias Macromoleculares/ultraestructura
7.
Proc Natl Acad Sci U S A ; 108(38): 15822-7, 2011 Sep 20.
Artículo en Inglés | MEDLINE | ID: mdl-21903929

RESUMEN

Bacterial cells possess multiple cytoskeletal proteins involved in a wide range of cellular processes. These cytoskeletal proteins are dynamic, but the driving forces and cellular functions of these dynamics remain poorly understood. Eukaryotic cytoskeletal dynamics are often driven by motor proteins, but in bacteria no motors that drive cytoskeletal motion have been identified to date. Here, we quantitatively study the dynamics of the Escherichia coli actin homolog MreB, which is essential for the maintenance of rod-like cell shape in bacteria. We find that MreB rotates around the long axis of the cell in a persistent manner. Whereas previous studies have suggested that MreB dynamics are driven by its own polymerization, we show that MreB rotation does not depend on its own polymerization but rather requires the assembly of the peptidoglycan cell wall. The cell-wall synthesis machinery thus either constitutes a novel type of extracellular motor that exerts force on cytoplasmic MreB, or is indirectly required for an as-yet-unidentified motor. Biophysical simulations suggest that one function of MreB rotation is to ensure a uniform distribution of new peptidoglycan insertion sites, a necessary condition to maintain rod shape during growth. These findings both broaden the view of cytoskeletal motors and deepen our understanding of the physical basis of bacterial morphogenesis.


Asunto(s)
Actinas/metabolismo , Pared Celular/metabolismo , Proteínas de Escherichia coli/metabolismo , Escherichia coli/metabolismo , Actinas/genética , Amdinocilina/farmacología , Antibacterianos/farmacología , Simulación por Computador , Depsipéptidos/farmacología , Relación Dosis-Respuesta a Droga , Proteínas de Escherichia coli/genética , Glicosilación , Cinética , Proteínas Luminiscentes/genética , Proteínas Luminiscentes/metabolismo , Microscopía Fluorescente , Modelos Biológicos , Péptidos/metabolismo , Peptidoglicano/metabolismo , Rotación , Vancomicina/farmacología
8.
Proc Natl Acad Sci U S A ; 107(32): 14194-8, 2010 Aug 10.
Artículo en Inglés | MEDLINE | ID: mdl-20660743

RESUMEN

Despite its fundamental nature, bacterial chromosome segregation remains poorly understood. Viewing segregation as a single process caused multiple proposed mechanisms to appear in conflict and failed to explain how asymmetrically dividing bacteria break symmetry to move only one of their chromosomes. Here, we demonstrate that the ParA ATPase extends from one cell pole and pulls the chromosome by retracting upon association with the ParB DNA-binding protein. Surprisingly, ParA disruption has a specific effect on chromosome segregation that only perturbs the latter stages of this process. Using quantitative high-resolution imaging, we demonstrate that this specificity results from the multistep nature of chromosome translocation. We propose that Caulobacter chromosome segregation follows an ordered pathway of events with distinct functions and mechanisms. Initiation releases polar tethering of the origin of replication, distinction spatially differentiates the two chromosomes, and commitment irreversibly translocates the distal centromeric locus. Thus, much as eukaryotic mitosis involves a sequence of distinct subprocesses, Caulobacter cells also segregate their chromosomes through an orchestrated series of steps. We discuss how the multistep view of bacterial chromosome segregation can help to explain and reconcile outstanding puzzles and frame future investigation.


Asunto(s)
Caulobacter/genética , Segregación Cromosómica , Cromosomas Bacterianos , Adenosina Trifosfatasas/metabolismo , Adenosina Trifosfatasas/fisiología , Proteínas Bacterianas , Proteínas de Unión al ADN/metabolismo , Proteínas de Unión al ADN/fisiología
9.
Curr Opin Microbiol ; 74: 102326, 2023 08.
Artículo en Inglés | MEDLINE | ID: mdl-37279609

RESUMEN

For bacteria to increase in size, they need to enzymatically expand their cell envelopes, and more concretely their peptidoglycan cell wall. A major task of growth is to increase intracellular space for the accumulation of macromolecules, notably proteins, RNA, and DNA. Here, we review recent progress in our understanding of how cells coordinate envelope growth with biomass growth, focusing on elongation of rod-like bacteria. We first describe the recent discovery that surface area, but not cell volume, increases in proportion to mass growth. We then discuss how this relation could possibly be implemented mechanistically, reviewing the role of envelope insertion for envelope growth. Since cell-wall expansion requires the well-controlled activity of autolysins, we finally review recent progress in our understanding of autolysin regulation.


Asunto(s)
Proteínas Bacterianas , N-Acetil Muramoil-L-Alanina Amidasa , N-Acetil Muramoil-L-Alanina Amidasa/genética , N-Acetil Muramoil-L-Alanina Amidasa/metabolismo , Proteínas Bacterianas/genética , Proteínas Bacterianas/metabolismo , Pared Celular/metabolismo , Membrana Celular/metabolismo , Ciclo Celular , Peptidoglicano/metabolismo
10.
J Cell Biol ; 222(5)2023 05 01.
Artículo en Inglés | MEDLINE | ID: mdl-36880553

RESUMEN

Single-particle tracking microscopy is a powerful technique to investigate how proteins dynamically interact with their environment in live cells. However, the analysis of tracks is confounded by noisy molecule localization, short tracks, and rapid transitions between different motion states, notably between immobile and diffusive states. Here, we propose a probabilistic method termed ExTrack that uses the full spatio-temporal information of tracks to extract global model parameters, to calculate state probabilities at every time point, to reveal distributions of state durations, and to refine the positions of bound molecules. ExTrack works for a wide range of diffusion coefficients and transition rates, even if experimental data deviate from model assumptions. We demonstrate its capacity by applying it to slowly diffusing and rapidly transitioning bacterial envelope proteins. ExTrack greatly increases the regime of computationally analyzable noisy single-particle tracks. The ExTrack package is available in ImageJ and Python.


Asunto(s)
Proteínas Bacterianas , Microscopía , Difusión , Cinética
11.
PNAS Nexus ; 1(4): pgac134, 2022 Sep.
Artículo en Inglés | MEDLINE | ID: mdl-36082236

RESUMEN

All cells must increase their volumes in response to biomass growth to maintain intracellular mass density within physiologically permissive bounds. Here, we investigate the regulation of volume growth in the Gram-positive bacterium Bacillus subtilis. To increase volume, bacteria enzymatically expand their cell envelopes and insert new envelope material. First, we demonstrate that cell-volume growth is determined indirectly, by expanding their envelopes in proportion to mass growth, similarly to the Gram-negative Escherichia coli, despite their fundamentally different envelope structures. Next, we studied, which pathways might be responsible for robust surface-to-mass coupling: We found that both peptidoglycan synthesis and membrane synthesis are required for proper surface-to-mass coupling. However, surprisingly, neither pathway is solely rate-limiting, contrary to wide-spread belief, since envelope growth continues at a reduced rate upon complete inhibition of either process. To arrest cell-envelope growth completely, the simultaneous inhibition of both envelope-synthesis processes is required. Thus, we suggest that multiple envelope-synthesis pathways collectively confer an important aspect of volume regulation, the coordination between surface growth, and biomass growth.

12.
Elife ; 102021 10 06.
Artículo en Inglés | MEDLINE | ID: mdl-34612203

RESUMEN

Cells must control the cell cycle to ensure that key processes are brought to completion. In Escherichia coli, it is controversial whether cell division is tied to chromosome replication or to a replication-independent inter-division process. A recent model suggests instead that both processes may limit cell division with comparable odds in single cells. Here, we tested this possibility experimentally by monitoring single-cell division and replication over multiple generations at slow growth. We then perturbed cell width, causing an increase of the time between replication termination and division. As a consequence, replication became decreasingly limiting for cell division, while correlations between birth and division and between subsequent replication-initiation events were maintained. Our experiments support the hypothesis that both chromosome replication and a replication-independent inter-division process can limit cell division: the two processes have balanced contributions in non-perturbed cells, while our width perturbations increase the odds of the replication-independent process being limiting.


Asunto(s)
División Celular , Replicación del ADN , Escherichia coli/citología , Escherichia coli/genética , Ciclo Celular , Cromosomas Bacterianos , ADN Bacteriano
14.
Bio Protoc ; 10(19): e3780, 2020 Oct 05.
Artículo en Inglés | MEDLINE | ID: mdl-33659436

RESUMEN

Bacteria are surrounded by a protective peptidoglycan cell wall. Provided that this structure and the enzymes involved are the preferred target for our most successful antibiotics, determining its structural and chemical complexity is of the highest interest. Traditionally, high-performance liquid chromatography (HPLC) analyses have been performed, but these methods are very time consuming in terms of sample preparation and chromatographic separation. Here we describe an optimized method for preparation of Gram-negative bacteria peptidoglycan and its subsequent analysis by ultra-performance liquid chromatography (UPLC). The use of UPLC in peptidoglycan analyses provides a dramatic reduction of the sample volume and hands-on time required and, furthermore, permits in-line mass spectrometry (MS) of the UPLC resolved muropeptides, thus facilitating their identification. This method improves our capability to perform high throughput analysis to better understand the cell-wall biology.

15.
Elife ; 92020 02 20.
Artículo en Inglés | MEDLINE | ID: mdl-32077853

RESUMEN

Bacterial shape is physically determined by the peptidoglycan cell wall. The cell-wall-synthesis machinery responsible for rod shape in Escherichia coli is the processive 'Rod complex'. Previously, cytoplasmic MreB filaments were thought to govern formation and localization of Rod complexes based on local cell-envelope curvature. Using single-particle tracking of the transpeptidase and Rod-complex component PBP2, we found that PBP2 binds to a substrate different from MreB. Depletion and localization experiments of other putative Rod-complex components provide evidence that none of those provide the sole rate-limiting substrate for PBP2 binding. Consistently, we found only weak correlations between MreB and envelope curvature in the cylindrical part of cells. Residual correlations do not require curvature-based Rod-complex initiation but can be attributed to persistent rotational motion. We therefore speculate that the local cell-wall architecture provides the cue for Rod-complex initiation, either through direct binding by PBP2 or through an unknown intermediate.


Asunto(s)
Pared Celular/metabolismo , Proteínas de Escherichia coli/metabolismo , Escherichia coli/metabolismo , Proteínas de Unión a las Penicilinas/metabolismo , Fracciones Subcelulares/metabolismo , Proteínas de Escherichia coli/biosíntesis , Proteínas de Unión a las Penicilinas/biosíntesis
16.
Sci Signal ; 13(632)2020 05 19.
Artículo en Inglés | MEDLINE | ID: mdl-32430292

RESUMEN

In nature, bacteria form biofilms-differentiated multicellular communities attached to surfaces. Within these generally sessile biofilms, a subset of cells continues to express motility genes. We found that this subpopulation enabled Bacillus subtilis biofilms to expand on high-friction surfaces. The extracellular matrix (ECM) protein TasA was required for the expression of flagellar genes. In addition to its structural role as an adhesive fiber for cell attachment, TasA acted as a developmental signal stimulating a subset of biofilm cells to revert to a motile phenotype. Transcriptomic analysis revealed that TasA stimulated the expression of a specific subset of genes whose products promote motility and repress ECM production. Spontaneous suppressor mutations that restored motility in the absence of TasA revealed that activation of the biofilm-motility switch by the two-component system CssR/CssS antagonized the TasA-mediated reversion to motility in biofilm cells. Our results suggest that although mostly sessile, biofilms retain a degree of motility by actively maintaining a motile subpopulation.


Asunto(s)
Bacillus subtilis/fisiología , Proteínas Bacterianas/metabolismo , Biopelículas/crecimiento & desarrollo , Proteínas de la Matriz Extracelular/metabolismo
17.
Elife ; 92020 01 06.
Artículo en Inglés | MEDLINE | ID: mdl-31904338

RESUMEN

Cell shape and cell-envelope integrity of bacteria are determined by the peptidoglycan cell wall. In rod-shaped Escherichia coli, two conserved sets of machinery are essential for cell-wall insertion in the cylindrical part of the cell: the Rod complex and the class-A penicillin-binding proteins (aPBPs). While the Rod complex governs rod-like cell shape, aPBP function is less well understood. aPBPs were previously hypothesized to either work in concert with the Rod complex or to independently repair cell-wall defects. First, we demonstrate through modulation of enzyme levels that aPBPs do not contribute to rod-like cell shape but are required for mechanical stability, supporting their independent activity. By combining measurements of cell-wall stiffness, cell-wall insertion, and PBP1b motion at the single-molecule level, we then present evidence that PBP1b, the major aPBP, contributes to cell-wall integrity by repairing cell wall defects.


Asunto(s)
Pared Celular/fisiología , Proteínas de Escherichia coli/genética , Escherichia coli/fisiología , Proteínas de Unión a las Penicilinas/genética , Proteínas de Escherichia coli/metabolismo , Proteínas de Unión a las Penicilinas/metabolismo
18.
Phys Rev E Stat Nonlin Soft Matter Phys ; 79(5 Pt 1): 051404, 2009 May.
Artículo en Inglés | MEDLINE | ID: mdl-19518453

RESUMEN

The phase-field-crystal model is by now widely used in order to predict crystal nucleation and growth. For colloidal solidification with completely overdamped individual particle motion, we show that the phase-field-crystal dynamics can be derived from the microscopic Smoluchowski equation via dynamical density-functional theory. The different underlying approximations are discussed. In particular, a variant of the phase-field-crystal model is proposed which involves less approximations than the standard phase-field-crystal model. We finally test the validity of these phase-field-crystal models against dynamical density-functional theory. In particular, the velocities of a linear crystal front from the undercooled melt are compared as a function of the undercooling for a two-dimensional colloidal suspension of parallel dipoles. Good agreement is only obtained by a drastic scaling of the free energies in the phase-field-crystal model in order to match the bulk freezing transition point.

19.
Nat Microbiol ; 4(8): 1294-1305, 2019 08.
Artículo en Inglés | MEDLINE | ID: mdl-31086310

RESUMEN

Rod-shaped bacteria grow by adding material into their cell wall via the action of two spatially distinct enzymatic systems: the Rod complex moves around the cell circumference, whereas class A penicillin-binding proteins (aPBPs) do not. To understand how the combined action of these two systems defines bacterial dimensions, we examined how each affects the growth and width of Bacillus subtilis as well as the mechanical anisotropy and orientation of material within their sacculi. Rod width is not determined by MreB, rather it depends on the balance between the systems: the Rod complex reduces diameter, whereas aPBPs increase it. Increased Rod-complex activity correlates with an increased density of directional MreB filaments and a greater fraction of directional PBP2a enzymes. This increased circumferential synthesis increases the relative quantity of oriented material within the sacculi, making them more resistant to stretching across their width, thereby reinforcing rod shape. Together, these experiments explain how the combined action of the two main cell wall synthetic systems builds and maintains rods of different widths. Escherichia coli Rod mutants also show the same correlation between width and directional MreB filament density, suggesting this model may be generalizable to bacteria that elongate via the Rod complex.


Asunto(s)
Bacillus subtilis/citología , Bacillus subtilis/metabolismo , Pared Celular/metabolismo , Bacillus subtilis/crecimiento & desarrollo , Proteínas Bacterianas/metabolismo , Escherichia coli/metabolismo , Proteínas de Escherichia coli/metabolismo , Proteínas de Unión a las Penicilinas/metabolismo
20.
Phys Rev E Stat Nonlin Soft Matter Phys ; 78(2 Pt 1): 020101, 2008 Aug.
Artículo en Inglés | MEDLINE | ID: mdl-18850771

RESUMEN

Self-propelled particles move along circles rather than along a straight line when their driving force does not coincide with their propagation direction. Examples include confined bacteria and spermatozoa, catalytically driven nanorods, active, anisotropic colloidal particles and vibrated granulates. Using a non-Hamiltonian rate theory and computer simulations, we study the motion of a Brownian "circle swimmer" in a confining channel. A sliding mode close to the wall leads to a huge acceleration as compared to the bulk motion, which can further be enhanced by an optimal effective torque-to-force ratio.

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