Long-range nematic order & anomalous fluctuations in collective - - PowerPoint PPT Presentation
Daiki Nishiguchi 1 Department of Physics, The University of Tokyo PhD student (D3) at Sano Lab. Ken H. Nagai (JAIST) Hugues Chat (CEA-Saclay, CSRC-Beijing) Masaki Sano (The University of Tokyo) Big Waves of Theoretical Sciences in Okinawa @
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Department of Physics, The University of Tokyo PhD student (D3) at Sano Lab. Ken H. Nagai (JAIST) Hugues Chaté (CEA-Saclay, CSRC-Beijing) Masaki Sano (The University of Tokyo) Big Waves of Theoretical Sciences in Okinawa @ OIST
(arXiv:1604.04247)
・Consume some kind of energy at the particle level (intrinsically nonequilibrium) ・Direction of motion is determined not solely by external force, but by internal degree of freedom.
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■Examples of “Collective motion”
school of jack tuna
(www.youtube.com/watch?v=D6HdoIsLMFg)
bird flock
(www.youtube.com/watch?v=_tEFRAI9WSE)
polarity・shape・cell cycle... ■Characteristics of “self-propelled particles”
3 Vicsek, et al. Phys. Rev. Lett. 75, 6 (1995)
■Our motivation
↑average angle of particles inside radius R
: white noise uniformly distributed on density ↗
noise ↘
■Vicsek model
・Many studies on Vicsek-like models regarding collective behavior. ・However, no corresponding experiments yet. →We want to realize such experiments, if possible, with biological systems!
transition (1st order)
3 Vicsek, et al. Phys. Rev. Lett. 75, 6 (1995)
■Our motivation
↑average angle of particles inside radius R
: white noise uniformly distributed on density ↗
noise ↘
■Vicsek model
・Many studies on Vicsek-like models regarding collective behavior. ・However, no corresponding experiments yet. →We want to realize such experiments, if possible, with biological systems!
transition (1st order)
3 Vicsek, et al. Phys. Rev. Lett. 75, 6 (1995)
■Our motivation
↑average angle of particles inside radius R
: white noise uniformly distributed on density ↗
noise ↘
■Vicsek model
・Many studies on Vicsek-like models regarding collective behavior. ・However, no corresponding experiments yet. →We want to realize such experiments, if possible, with biological systems!
transition (1st order)
3 Vicsek, et al. Phys. Rev. Lett. 75, 6 (1995)
■Our motivation
↑average angle of particles inside radius R
: white noise uniformly distributed on density ↗
noise ↘
■Vicsek model
・Many studies on Vicsek-like models regarding collective behavior. ・However, no corresponding experiments yet. →We want to realize such experiments, if possible, with biological systems!
transition (1st order)
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■What is number fluctuation?
：# of observed particles at time t.
：mean value ：fluctuation (standard deviation) in equilibrium or random systems (Central limit theorem)
■What about in Vicsek model?
Chaté, et al. Eur. Phys. J. B 64, 451 (2008), Toner and Tu Phys. Rev. E 58, 4 (1998)
・Exponent larger than 0.5 in the homogeneous ordered state. →giant number fluctuation (GNF) ・Due to numerics and continuum theory:
Normal fluctuations
・Note: GNF is associated with Nambu- Goldstone modes in the ordered state.
Giant fluctuations
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Vicsek'model ac-ve'nema-cs self1propelled'rods mo-lity polar apolar polar interac-on polar nema-c nema-c GNF'(numerics) ~'0.8 ~'0.8 ~'0.75 GNF'(con-nuum) 0.8'(exact) 1'(linear'theory) controversial
True'LRO Quasi1LRO True1LRO
■Active nematics
e.g.：shaken rods Narayan, et al. Science, 317, 6 (2007).
cf：Vicsek model
Ngo, et al.
■Self-propelled rods nematic ordered state→
Ginelli, et al.
e.g.: Bacteria
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■Vicsek model (Toner-Tu theory) ■Active nematics (Ramaswamy, et al)
・From symmetry arguments, hydrodynamic equations can be written: ・Dynamical renomalization group method can derive the exponent in 2D. ・Exponents of GNF is only known for linear theory. ・Some theoreticians believes that phenomenologies on active nematics and self-propelled rods should be the same.
Toner & Tu, PRL, 75, 23 (1995). Ramaswamy, et al. EPL, 62(2), 196, (2003). Mishra, et al., J. Stat. Mech., (2010) P02003. Vicsek'model ac-ve'nema-cs self1propelled'rods mo-lity polar apolar polar interac-on polar nema-c nema-c GNF'(numerics) ~'0.8 ~'0.8 ~'0.75 GNF'(con-nuum) 0.8'(exact) 1'(linear'theory) controversial
True'LRO Quasi1LRO True1LRO ↑Symmetric Traceless part
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Bacillus subtilis Zhang, Beér, Florin & Swinney (PNAS 2010) shaken rods Narayan, Ramaswamy & Menon (Science 2007)
←
These GNF is trivial because of their obvious structures.
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motility assay shaken polar disks
Schaller, et al. Nature, 467, 73 (2008) Schaller and Bausch, PNAS, 110, 4488 (2013) Deseigne, et al. Phys. Rev. Lett. 05,098001 (2010)
: variance
motility assay shaken polar disks
・Many experiments report GNF, although it’s not their main claims. ・However, these GNFs came from effects of boundary or clustering. →Looking at fluctuations not in homogeneous ordered states.
Bacillus subtilis shaken rods not ordered clustering not ordered boundary effects clustering banding clustering boundary effects Zhang, et al. PNAS, 107, 31, 13626 (2010).
Schaller and Bausch, PNAS, 110, 4488 (2013) Deseigne, et al.
105, 098001 (2010)
Narayan, et al. Science, 317, 6 (2007).
→Realize experimental systems that exhibit homogeneous ordered states!
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・As simple as possible to compare with numerical models. (avoid complicated interactions) ・Far from boundaries, a large number of particles to see statistics. →µm-scale particles are preferable. →microswimmers (e.g. self-propelled colloids or bacteria) (→Demonstrations with biological systems is important!) →Use bacteria. ・Flow fields of bacteria (pusher-type swimmer) destabilize ordered states. →Sandwich them between two walls to suppress flow. (The thinner, the more collisions. 2D is easier to observe.) ・To realize an ordered state, stronger alignment is required. →use higher-aspect-ratio particles ■Design principles for homogeneous ordered phases
Janus colloids
DN & M. Sano,
92, 052309 (2015)
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10μm real speed
20-100 μm
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・Add an antibiotic that inhibits cell division to the suspension of E.coli (strain: RP437 or RP4979). ・The length of E.coli can be controlled by changing the incubation time. real speed
usual E.coli（2-3 µm） filamentous cells（30-40 µm） antibiotic
20 µm
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・As simple as possible to compare with numerical models. (avoid complicated interactions) ・Far from boundaries, a large number of particles to see statistics. →µm-scale particles are preferable. →microswimmers (e.g. self-propelled colloids or bacteria) (→Demonstrations with biological systems is important!) →First introduce bacterial experiments!! ・Flow fields of bacteria (pusher-type swimmer) destabilize ordered states. →Sandwich them between two walls to suppress flow. (The thinner, the more collisions. 2D is easier to observe.) ・To realize an ordered state, stronger alignment is required. →use higher-aspect-ratio particles ■Experimental Systems for realizing homogeneous ordered states
Janus colloids
Nishiguchi & Sano,
92, 052309 (2015)
17 Concentrated suspension Coverslip PDMS seal Objective lens excitation light
O2
・Oxygen is required for high motility. ・PDMS(polymeric organo silicon) is transparent to oxygen. ・Make patterns on PDMS so that excessive suspension escapes into halls. ・Gap between glass and PDMS: smaller than 2µm
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・Used strain: RP4979(ΔcheY, YFP), chemotactic mutant →no tumbling, swim smoothly, exclude artifacts.
some cells swim together after collisions→
50μm
x40 objective lens captured@10fps, real speed wildtype with tumbling mutants without tumbling →motility is polar!
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・Each particle has polarity. ・Collisions lead to global nematic order. →self-propelled rods model ! real speed captured@10fps
Vicsek'model ac-ve'nema-cs self1propelled'rods mo-lity polar apolar polar interac-on polar nema-c nema-c exponent'of'GNF 0.8 ~'0.8 ~'0.75 100 µm
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Ordered state in large area! →measure GNF!! x3 fast x10 objective captured@10fps
100μm
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x3 fast x10 objective captured @10fps
100μm
Disordered state is random. Number fluctuation obeys (central limit theorem)
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■Experimental Result
Ginelli, et al. Phys. Rev. Lett. 104, 184502 (2010)
・n: number of pixels in a box that bacteria exist (1 bacterium ~ 100 pixels). ・We obtained giant number fluctuations over 3-4 decades.
→fitting: 0.632±0.014
■Self-propelled rods model
(small due to exclusive volume?)
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disordered →fitting: 0.511±0.012
Exponent ~ 0.75
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■Do they have long-range order？ ■How to distinguish
・Plot nematic order parameter against calculation box size and see how they decay. →quasi long-range order: algebraic decay (with small exponent) toward 0 true long-range order: converge to a positive value
LRO:'Long1Range'Order
Vicsek'model ac-ve'nema-cs self1propelled'rods mo-lity polar apolar polar interac-on polar nema-c nema-c GNF'(numerics) ~'0.8 ~'0.8 ~'0.75 GNF'(con-nuum) 0.8'(exact) 1'(linear'theory) controversial
True'LRO Quasi1LRO True1LRO
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evaluate
■How to distinguish
・Plot nematic order parameter against calculation box size and see how they decay. →quasi long-range order: algebraic decay (with small exponent) toward 0 true long-range order: converge to a positive value
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evaluate
■How to distinguish
・Plot nematic order parameter against calculation box size and see how they decay. →quasi long-range order: algebraic decay (with small exponent) toward 0 true long-range order: converge to a positive value
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evaluate
■How to distinguish
・Plot nematic order parameter against calculation box size and see how they decay. →quasi long-range order: algebraic decay (with small exponent) toward 0 true long-range order: converge to a positive value
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■Evaluating nematic order ・Direct calculation of S requires detecting all the cells (difficult). ・Evaluate by ”structure tensor” method (find max/min of intensity gradient). ：image data in the box e.g.: Coherency: →regard this as nematic order S →take average of Coherency
Rezakhaniha, et al.
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■Nematic Order vs Box Size ■Self-propelled rods model
Ginelli, et al.
104, 184502 (2010)
・Slower decay than algebraic decay. ・Converge to a positive value. →true long-range order! (consistent with self-propelled rods model) ・Obtained GNF should remain in large system size limit.
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due to spatial inhomogeneity
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■Correlation function
10 100 800 0.03 0.10 1.00
Correlation should reflect the properties of Nambu-Goldstone mode. →What is the theoretical dependence of Corr(r)? (on-going) power-law? local director↑ ↑global(mean) director fluctua-on'of'director:
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Janus'par-cles filamentous'bacteria ■Classification of Vicsek-like models (Poster'by'J.'Iwasawa) (This'Talk'&'my'poster)
Vicsek'model ac-ve'nema-cs self1propelled'rods mo-lity polar apolar polar interac-on polar nema-c nema-c GNF'(numerics) ~'0.8 ~'0.8 ~'0.75 GNF'(con-nuum) 0.8'(exact) 1'(linear'theory) controversial
True'LRO Quasi1LRO True1LRO Janus colloids
DN & M. Sano,
92, 052309 (2015)
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・We have realized an ordered state of collective motion with a biological system using filamentous cells of E. coli. ・The obtained ordered state has true long-range order. ・“GNF in the sense of numerics/theories”(not from clustering etc.), which had not been experimentally observed, is observed over 3-4 decades. ・All the results are consistent with self-propelled rods model (numerics).
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・See the phase transition. ・Other properties suggested by numerics & theory (superdiffusion, dispersion, etc.) ・See how aspect ratio affects collective dynamics.
・Transformation of E.coli (fluorescent protein): Yusuke Maeda (Kyushu Univ.)
Vicsek'model ac-ve'nema-cs self1propelled'rods mo-lity polar apolar polar interac-on polar nema-c nema-c GNF'(numerics) ~'0.8 ~'0.8 ~'0.75 GNF'(con-nuum) 0.8'(exact) 1'(linear'theory) controversial
True'LRO Quasi1LRO True1LRO
Preprint available!! → arXiv:1604.04247