View source for Run-and-tumble motion

{{Short description|Type of bacterial motion}}
{{microbial and microbot movement|taxa}}

'''Run-and-tumble motion''' is a movement pattern exhibited by certain [[bacteria]] and other microscopic agents. It consists of an alternating sequence of "runs" and "tumbles": during a run, the agent propels itself in a fixed (or slowly varying) direction, and during a tumble, it remains stationary while it reorients itself in preparation for the next run.{{sfn | Berg | 2004}}

The tumbling is erratic or "random" in the sense of a [[stochastic process]]—that is, the new direction is sampled from a [[probability density function]], which may depend on the organism's local environment (e.g., chemical gradients). The duration of a run is usually random in the same sense. An example is wild-type ''[[E. coli]]'' in a dilute aqueous medium, for which the run duration is [[exponentially distributed]] with a mean of about 1 second.{{sfn | Berg | 2004}}

Run-and-tumble motion forms the basis of certain mathematical models of [[self-propelled particles]], in which case the particles themselves may be called '''run-and-tumble particles'''.{{sfn | Cates | Tailleur | 2015}}

==Description==
Many bacteria swim, propelled by rotation of the flagella outside the cell body. In contrast to [[protist flagella]], bacterial flagella are [[Wiktionary:rotor|rotors]] and—irrespective of species and type of flagellation—they have only two modes of operation: clockwise  or counterclockwise rotation. Bacterial swimming is used in [[#Bacterial taxis: Directed motion|bacterial taxis]] (mediated by specific receptors and [[signal transduction]] pathways) for the bacterium to move in a directed manner along gradients and reach more favorable conditions for life.{{sfn | Sowa | Berry | 2008 | p=}}{{sfn | Krell | Lacal | Muñoz-Martínez | Reyes-Darias | 2011 | p=}} The direction of flagellar rotation is controlled by the type of molecules detected by the receptors on the surface of the cell: in the presence of an attractant gradient, the rate of smooth swimming increases, while the presence of a repellent gradient increases the rate of tumbling.{{sfn | Berg | 2004}}{{sfn | Bastos-Arrieta | Revilla-Guarinos | Uspal | Simmchen | 2018 | p=}}

==Biological examples==
Run-and-tumble motion is found in many [[peritrichous]] bacteria, including ''E. coli'', ''[[Salmonella typhimurium]]'', and ''[[Bacillus subtilis]]''.{{sfn | Guasto | Rusconi | Stocker | 2012}} It has also been observed in the [[alga]] ''[[Chlamydomonas reinhardtii]]''{{sfn | Polin | Tuval | Drescher | Gollub | 2009}} and the [[cyanobacterium]] ''[[Synechocystis]]''.<ref name=Mehdizadeh2022 />

===Directed motility (taxis)===
Genetically diverse groups of microorganisms rely upon [[directed motility]] (taxis), such as [[chemotaxis]] or [[phototaxis]], to optimally navigate through complex environments or [[Colonisation (biology)|colonise]] host tissues.<ref name=Cooper2022 /><ref name=Mehdizadeh2022 />  In the model organisms ''[[Escherichia coli]]'' and ''[[Salmonella]]'', bacteria swim in a random pattern produced by alternating counterclockwise (CCW) and clockwise (CW) [[Flagellum|flagellar rotation]]. [[Chemoreceptor]]s detect attractants or repellents and stimulate responses through a [[signalling cascade]] that controls the direction of the flagellar motor. This can result in a chemotaxis, where attractant gradients extend the length of time flagellar motors rotate CCW, resulting in more smooth swimming in a favourable direction, while repellents cause an increase of CW rotations, resulting in more tumbling and changes in direction.<ref name=Cooper2022>{{cite journal | last1=Cooper | first1=Kendal G. | last2=Chong | first2=Audrey | last3=Kari | first3=Laszlo | last4=Jeffrey | first4=Brendan | last5=Starr | first5=Tregei | last6=Martens | first6=Craig | last7=McClurg | first7=Molly | last8=Posada | first8=Victoria R. | last9=Laughlin | first9=Richard C. | last10=Whitfield-Cargile | first10=Canaan | last11=Garry Adams | first11=L. | last12=Bryan | first12=Laura K. | last13=Little | first13=Sara V. | last14=Krath | first14=Mary | last15=Lawhon | first15=Sara D. | last16=Steele-Mortimer | first16=Olivia | title=Regulatory protein HilD stimulates Salmonella Typhimurium invasiveness by promoting smooth swimming via the methyl-accepting chemotaxis protein McpC | journal=Nature Communications | publisher=Springer Science and Business Media LLC | volume=12 | issue=1 | date=2021-01-13 | page=348 | issn=2041-1723 | doi=10.1038/s41467-020-20558-6| pmid=33441540 | pmc=7806825 }} [[File:CC-BY icon.svg|50px]] Modified material was copied from this source, which is available under a [https://creativecommons.org/licenses/by/4.0/ Creative Commons Attribution 4.0 International License].</ref> The cyanobacterium ''[[Synechocystis]]'' uses run-and-tumbling in a manner which can result in phototaxis.<ref name=Mehdizadeh2022 />

===''Escherichia coli''===
An archetype of bacterial swimming is represented by the well-studied model organism ''[[Escherichia coli]]''.{{sfn | Bastos-Arrieta | Revilla-Guarinos | Uspal | Simmchen | 2018 | p=}} With its [[peritrichous]] flagellation, ''E. coli'' performs a [[run-and-tumble]] swimming pattern, as shown in the diagrams below. Counterclockwise rotation of the [[flagellar motor]]s leads to [[Flagellar bundle|flagellar bundle formation]] that pushes the cell in a forward run, parallel to the long axis of the cell. Clockwise rotation disassembles the bundle and the cell rotates randomly (tumbling). After the tumbling event, straight swimming is recovered in a new direction.{{sfn | Berg | 2004}} That is, counterclockwise rotation results in steady motion and clockwise rotation in tumbling; counterclockwise rotation in a given direction is maintained longer in the presence of molecules of interest (like sugars or aminoacids).{{sfn | Berg | 2004}}{{sfn | Bastos-Arrieta | Revilla-Guarinos | Uspal | Simmchen | 2018 | p=}}

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 | image1    = Tumbling and running modes associated with Escherichia coli and Salmonella typhimurium.png
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 | caption1  = {{center| ''[[E. coli]]'' have [[flagella]] which project in all directions. Clockwise (CW) rotation of flagellar motors results in random re-orientation for the bacterium, but counter-clockwise (CCW) rotation produces approximate straight-line motion.<ref>{{cite journal | last1=Egbert | first1=Matthew D. | last2=Barandiaran | first2=Xabier E. | last3=Di Paolo | first3=Ezequiel A. | title=A Minimal Model of Metabolism-Based Chemotaxis | journal=PLOS Computational Biology | volume=6 | issue=12 | date=2010-12-02 | issn=1553-7358 | doi=10.1371/journal.pcbi.1001004 | page=e1001004| pmid=21170312 | pmc=3000427 | bibcode=2010PLSCB...6E1004E | doi-access=free }}</ref>}}
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[[File:Escherichia coli 2.jpg|thumb|190px|right|Diagram of ''E. coli'', with [[flagella]] projecting in all directions]]
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In a uniform medium, run-and-tumble trajectories appear as a sequence of nearly straight segments interspersed by erratic reorientation events, during which the bacterium remains stationary. The straight segments correspond to the runs, and the reorientation events correspond to the tumbles. Because they exist at low [[Reynolds number]], bacteria starting at rest quickly reach a fixed [[terminal velocity]], so the runs can be approximated as constant velocity motion. The deviation of real-world runs from straight lines is usually attributed to [[rotational diffusion]], which causes small fluctuations in the orientation over the course of a run.

In contrast with the more gradual effect of rotational diffusion, the change in orientation (turn angle) during a tumble is large; for an isolated ''E. Coli'' in a uniform aqueous medium, the mean turn angle is about 70 degrees, with a relatively broad distribution. In more complex environments, the tumbling distribution and run duration may depend on the agent's local environment, which allows for goal-oriented navigation ([[taxis]]).{{sfn | Wadhams | Armitage | 2004 }}{{sfn | Jensen | 2015 }} For example, a tumbling distribution that depends on a chemical gradient can guide bacteria toward a food source or away from a repellant, a behavior referred to as [[chemotaxis]].{{sfn | Wadhwa | Berg | 2021}} Tumbles are typically faster than runs: tumbling events of ''E. Coli'' last about 0.1 seconds, compared to ~ 1 second for a run.

===''Synechocystis''===
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 | caption1  = Cross section of a ''[[Synechocystis]]'' cell with [[Pilus|pili]]. These cells lack flagella, but achieve motility using retractile [[type IV pili]]{{hsp}}<ref name=Jakob2017>{{cite journal | last1=Jakob | first1=Annik | last2=Schuergers | first2=Nils | last3=Wilde | first3=Annegret | title=Phototaxis Assays of Synechocystis sp. PCC 6803 at Macroscopic and Microscopic Scales | journal=Bio-Protocol | volume=7 | issue=11 | year=2017 | pages=e2328 | issn=2331-8325 | doi=10.21769/bioprotoc.2328| pmid=34541089 | pmc=8410341 }}</ref> 
 | image2    = Run-and-tumble motion of a Synechocystis cell.webp
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 | caption2  = Run-and-tumble motion of a ''[[Synechocystis]]'' cyanobacterium. During run the cell moves quickly from one point to another, while during tumble it remains constrained in a given area and tends to change directions.<ref name=Mehdizadeh2022>{{cite journal | last1=Mehdizadeh Allaf | first1=Malihe | last2=Peerhossaini | first2=Hassan | title=Cyanobacteria: Model Microorganisms and Beyond | journal=Microorganisms | volume=10 | issue=4 | date=2022-03-24 | issn=2076-2607 | doi=10.3390/microorganisms10040696 | page=696| pmid=35456747 | pmc=9025173 | doi-access=free }} [[File:CC-BY icon.svg|50px]] Modified material was copied from this source, which is available under a [https://creativecommons.org/licenses/by/4.0/ Creative Commons Attribution 4.0 International License].</ref>
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Another example is ''[[Synechocystis]]'', a genus of [[cyanobacterium]]. Cyanobacterium do not have flagella. Nonetheless, ''Synechocystis'' species can move in cell suspensions and on moist surfaces and by using retractile [[type IV pili]], displaying an intermittent two phase motion; a high-motility run and a low-motility tumble ''(see diagram)''.<ref name=Jakob2017 /><ref name="Vourc'h2020">{{cite journal | last1=Vourc'h | first1=Thomas | last2=Léopoldès | first2=Julien | last3=Peerhossaini | first3=Hassan | title=Light Control of the Diffusion Coefficient of Active Fluids | journal=Journal of Fluids Engineering | volume=142 | issue=3 | date=2020-02-03 | issn=0098-2202 | doi=10.1115/1.4045951| arxiv=2003.02207 | s2cid=211987677 }}</ref> The two phases can be modified under various external stressors. Increasing the [[Photometry (optics)#Photometric quantities|light intensity]], uniformly over the space, increases the probability of ''Synechocystis'' being in the run state randomly in all directions. This feature, however, vanishes after a typical characteristic time of about one hour, when the initial probability is recovered. These results were well described by a mathematical model based on a linear response theory proposed by Vourc’h et al.<ref name="Vourc'h2020" /><ref name=Mehdizadeh2022 />

''Synechocystis'' cells can also undergo biased motility under directional illumination. Under directional [[radiant flux|light flux]], ''Synehcocystis'' cells perform phototactic motility and head toward the light source (in positive [[phototaxis]]). Vourc’h et al. (2020) showed that this biased motility stems from the averaged displacements during run periods, which is no longer random (as it was in the uniform illumination).<ref name="Vourc'h2020" /> They showed the bias is the result of the number of runs, which is greater toward the light source, and not of longer runs in this direction.<ref name="Vourc'h2020" /> Brought together, these results suggest distinct pathways for the recognition of light intensity and light direction in this prokaryotic microorganism. This effect can be used in the active control of bacterial flows.<ref name=Mehdizadeh2022 />

It has also been observed that very strong local illumination inactivates the motility apparatus.<ref>{{cite journal | last1=Schuergers | first1=Nils | last2=Lenn | first2=Tchern | last3=Kampmann | first3=Ronald | last4=Meissner | first4=Markus V | last5=Esteves | first5=Tiago | last6=Temerinac-Ott | first6=Maja | last7=Korvink | first7=Jan G | last8=Lowe | first8=Alan R | last9=Mullineaux | first9=Conrad W | last10=Wilde | first10=Annegret | title=Cyanobacteria use micro-optics to sense light direction | journal=eLife | volume=5 | date=2016-02-09 | issn=2050-084X | doi=10.7554/elife.12620| pmid=26858197 | pmc=4758948 | doi-access=free }}</ref> Increasing the light intensity of more than ~475 μmol m<sup>−2</sup> s<sup>−1</sup> reverses the direction of ''Synechocystis'' cells to move away from the high levels of radiation source.<ref>{{cite journal | last1=Ng | first1=Wing-On | last2=Grossman | first2=Arthur R. | last3=Bhaya | first3=Devaki | title=Multiple Light Inputs Control Phototaxis in Synechocystis sp. Strain PCC6803 | journal=Journal of Bacteriology | volume=185 | issue=5 | year=2003 | issn=0021-9193 | doi=10.1128/jb.185.5.1599-1607.2003 | pages=1599–1607| pmid=12591877 | pmc=148062 }}</ref><ref name=Moon2012>{{cite journal | last1=Moon | first1=Yoon-Jung | last2=Kim | first2=Seung | last3=Chung | first3=Young-Ho | title=Sensing and Responding to UV-A in Cyanobacteria | journal=International Journal of Molecular Sciences | volume=13 | issue=12 | date=2012-12-03 | issn=1422-0067 | doi=10.3390/ijms131216303 | pages=16303–16332| pmid=23208372 | pmc=3546692 | doi-access=free }}</ref> Moreover, ''Synechocystis'' cells show a negative phototaxis behavior under [[ultraviolet]] radiation as an effective escape mechanism to avoid [[DNA damage|damage to DNA]] and other cellular components of ''Synechocystis''.<ref>Chau, R.M.W., Bhaya, D. and Huang, K.C., 2017. ''Emergent phototactic responses of cyanobacteria under complex light regimes''. '''mBio 8''': e02330-16.</ref><ref name=Moon2012 /><ref>{{cite journal | last1=Choi | first1=Jong-Soon | last2=Chung | first2=Young-Ho | last3=Moon | first3=Yoon-Jung | last4=Kim | first4=Changhoon | last5=Watanabe | first5=Masakatsu | last6=Song | first6=Pill-Soon | last7=Joe | first7=Cheol-O | last8=Bogorad | first8=Lawrence | last9=Park | first9=Young Mok | title=Photomovement of the Gliding Cyanobacterium Synechocystis sp. PCC 6803 | journal=Photochemistry and Photobiology | volume=70 | issue=1 | year=1999 | issn=0031-8655 | doi=10.1111/j.1751-1097.1999.tb01954.x | pages=95–102| pmid=10420848 | s2cid=25364218 }}</ref><ref name=Mehdizadeh2022 /> Contrary to the run phase that can extend from a fraction of a second to several minutes, the tumble lasts only a fraction of a second. The tumbling phase is a clockwise rotation that allows the cell to change the motility direction of the next run.<ref name="Parkinson2009">{{cite journal | last=Parkinson | first=John S. | title=Signal transduction schemes of bacteria | journal=Cell | volume=73 | issue=5 | year=1993 | issn=0092-8674 | doi=10.1016/0092-8674(93)90267-t | pages=857–871| pmid=8098993 | s2cid=205020855 }}</ref><ref>{{cite journal | last1=Copeland | first1=Matthew F. | last2=Weibel | first2=Douglas B. | title=Bacterial swarming: a model system for studying dynamic self-assembly | journal=Soft Matter | volume=5 | issue=6 | year=2009 | pages=1174–1187 | issn=1744-683X | doi=10.1039/b812146j | pmid=23926448 | pmc=3733279 | bibcode=2009SMat....5.1174C }}</ref><ref name=Mehdizadeh2022 />

[[Chemotaxis]] is another scheme that allows an organism to move toward or away from gradients of nutrients or other chemical stimuli. Detecting by [[transmembrane]] [[chemoreceptor]]s{{hsp}}<ref>{{cite journal | last=Mattick | first=John S. | title=Type IV Pili and Twitching Motility | journal=Annual Review of Microbiology | volume=56 | issue=1 | year=2002 | issn=0066-4227 | doi=10.1146/annurev.micro.56.012302.160938 | pages=289–314| pmid=12142488 }}</ref><ref>{{cite journal |last1=Webre |first1=D.J. |last2=Wolanin |first2=P.M |last3=Stock |first3=J.B. |year=2003 |title=Bacterial chemotaxis |url=https://www.cell.com/current-biology/pdf/S0960-9822(02)01424-0.pdf |journal=Current Biology |volume=13 |issue=2 |pages=R47–R49|doi=10.1016/S0960-9822(02)01424-0 |pmid=12546801 |s2cid=3203488 }}</ref><ref>{{cite journal | last1=Sourjik | first1=Victor | last2=Wingreen | first2=Ned S | title=Responding to chemical gradients: bacterial chemotaxis | journal=Current Opinion in Cell Biology | volume=24 | issue=2 | year=2012 | issn=0955-0674 | doi=10.1016/j.ceb.2011.11.008 | pages=262–268| pmid=22169400 | pmc=3320702 }}</ref> the microorganism performs a three-dimensional [[random walk]] is observed in a homogenous environment, and the direction of each run is identified after a tumble.<ref name="Parkinson2009" /><ref name=Mehdizadeh2022 />

==Mathematical modeling==
Theoretically and computationally, run-and-tumble motion can be modeled as a stochastic process. One of the simplest models is based on the following assumptions:{{sfn | Solon | Cates | Tailleur | 2015}}<ref>{{Cite journal |last1=Santra |last2=Basu |last3=Sabhapandit |date=2020-06-15 |title=Run-and-tumble particles in two dimensions: Marginal position distributions |url=https://link.aps.org/doi/10.1103/PhysRevE.101.062120 |journal=Physical Review E |volume=101 |issue=6 |pages=062120 |doi=10.1103/PhysRevE.101.062120|pmid=32688530 |arxiv=2004.07562 |bibcode=2020PhRvE.101f2120S |s2cid=219636018 }}</ref>

* Runs are straight and performed at constant velocity {{math|''v''<sub>0</sub>}} (initial speed-up is instantaneous)
* Tumbling events are [[uncorrelated]] and occur at average rate {{math|α}}, i.e., the number of tumbling events in a given time interval has a [[Poisson distribution]]. This implies that the run durations are [[exponentially distributed]] with mean {{math|α<sup>−1</sup>}}.
* Tumble duration is negligible
* Interactions with other agents are negligible (dilute limit)

With a few other simplifying assumptions, an [[integro-differential equation]] can be derived for the probability density function {{math|''f'' ('''r''', '''ŝ''', ''t'')}}, where {{math|'''r'''}} is the particle position and {{math|'''ŝ'''}} is the unit vector in the direction of its orientation. In {{mvar|d}}-dimensions, this equation is
:<math>
\frac{\partial f(\mathbf{r},\hat{\mathbf{s}}, t)}{\partial t} + v_0 \, \hat{\mathbf{s}} \cdot \nabla f(\mathbf{r},\hat{\mathbf{s}}, t) =  \xi^{-1} \nabla \cdot (\nabla V(\mathbf{r}) f(\mathbf{r},\hat{\mathbf{s}}, t)) -\alpha f(\mathbf{r},\hat{\mathbf{s}}, t) + \frac{\alpha}{\Omega_d} \int g(\hat{\mathbf{s}} - \hat{\mathbf{s}}')  f(\mathbf{r},\hat{\mathbf{s}}', t) d\hat{\mathbf{s}}'
</math>
where {{math|Ω<sub>''d''</sub> {{=}} 2π<sup>''d''/2</sup>/Γ(''d''/2)}} is the {{math|''d''}}-dimensional [[solid angle]], {{math|''V''('''r''')}} is an external potential, {{math|''ξ''}} is the friction, and the function {{math|''g'' ('''ŝ''' - '''ŝ'''{{'}})}} is a [[Cross section (physics)|scattering cross section]] describing transitions from orientation {{math|'''ŝ'''{{'}}}} to {{math|'''ŝ'''}}. For complete reorientation, {{math|''g'' {{=}} 1}}. The integral is taken over all possible unit vectors, i.e., the {{mvar|d}}-dimensional [[unit sphere]].

In free space (far from boundaries), the [[mean squared displacement]] {{math|⟨'''r'''(''t'')<sup>2</sup>⟩}} generically scales as {{math|⟨'''r'''(''t'')<sup>2</sup>⟩ ~ ''t''<sup>2</sup>}} for small {{math|''t''}} and {{math|⟨'''r'''(''t'')<sup>2</sup>⟩ ~ ''t''}} for large {{math|''t''}}. In two dimensions, the mean squared displacement corresponding to initial condition {{math|''f'' ('''r''', '''ŝ''', 0) {{=}} δ(''r'')/(2π)}} is
:<math>
\langle \mathbf{r}^2 \rangle = \frac{2 v_0^2}{\alpha^2(1 - \sigma_1)} \left[\alpha \, t + \frac{e^{-\alpha(1-\sigma_1)t} - 1}{1 - \sigma_1} \right]
</math>
where 
:<math>\sigma_1 = \int_{0}^{2\pi} g(\hat{\mathbf{s}}) (\cos \theta) d \theta</math> 
with {{math|'''ŝ'''}} parametrized as {{math|'''ŝ''' {{=}} (cos ''θ'', sin ''θ'')}}.{{sfn | Villa-Torrealba | Chávez-Raby | de Castro | Soto | 2020}}

In real-world systems, more complex models may be required. In such cases, specialized analysis methods have been developed to infer model parameters from experimental trajectory data.{{sfn | Rosser | Fletcher | Wilkinson | de Beyer | 2013}}{{sfn | Seyrich | Alirezaeizanjani | Beta | Stark | 2018}}<ref>{{Cite journal |last1=Santra |last2=Basu |last3=Sabhapandit |date=2021-07-13 |title=Active Brownian motion with directional reversals |url=https://link.aps.org/doi/10.1103/PhysRevE.104.L012601 |journal=Physical Review E |volume=104 |issue=1 |pages=L012601 |doi=10.1103/PhysRevE.104.L012601|pmid=34412243 |arxiv=2101.11327 |bibcode=2021PhRvE.104a2601S |s2cid=231718971 }}</ref>

The mathematical abstraction of run-and-tumble motion also appears outside of biology—for example, in idealized models of [[radiative transfer]] and [[neutron transport]].{{sfn | Chandrasekhar | 1960}}{{sfn | Case | Zweifel | 1963}}{{sfn | Paasschens | 1997}}

==See also==
{{Columns-list|colwidth=30em|
*{{annotated link|Animal navigation}}
*{{annotated link|Bacterial motility}}
*{{annotated link|Aquatic locomotion}}
*{{annotated link|Quorum sensing}}
*{{annotated link|Microswimmer}}
*{{annotated link|Active matter}}
*{{annotated link|Squirmer}}
*{{annotated link|Active Brownian particle}}
*{{annotated link|Telegraph process}}
}}

==Notes==
{{reflist|2}}

==Sources==
{{refbegin|2|indent=yes}}
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* {{cite book | last=Berg | first=Howard | title=E. coli in motion | publisher=Springer | publication-place=New York | year=2004 | isbn=978-0-387-21638-6 | oclc=56124142}}
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* {{cite journal | last1=Cates | first1=Michael E. | last2=Tailleur | first2=Julien | title=Motility-Induced Phase Separation | journal=[[Annual Review of Condensed Matter Physics]] | volume=6 | issue=1 | date=2015-03-01 | issn=1947-5454 | doi=10.1146/annurev-conmatphys-031214-014710 | pages=219–244| arxiv=1406.3533 | bibcode=2015ARCMP...6..219C | s2cid=15672131 }}
* {{cite book | first = Subrahmanyan |last=Chandrasekhar | year = 1960 | title = Radiative Transfer | publisher = [[Dover Publications]] | pages = | isbn = 978-0-486-31845-5 | oclc = 1084019191 | url = https://books.google.com/books?id=1YHCAgAAQBAJ&pg=PP1}}
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{{refend}}

[[Category:Bacteriology]]