facilities needed for cloud applications that require advanced business logic
and coordination evolution20 . One example is stateful function-as-a-service,
which demands ACID transactions, global state consolidation, and debugging and
auditing capabilities katsifodimos2019operational ; akhter2019stateful . These
requirements are similar to those of transactional stream processing (TSP).
However, it’s unclear if TSP systems like S-Store S-Store  can fully satisfy
these requirements. Further exploration is needed, including supporting
debugging stream_debug  and isolation 254432  in stateful stream processing as
microservices.

###  6.2 Novel Hardware Platforms

Modern hardware advancements have made servers with hundreds of cores and
several terabytes of main memory available. Such advancements have driven
researchers to rethink TSP systems and put emerging hardware platforms to good
use survey_hwstream . Next, we take a closer look at multi-/many-core
architectures, non-volatile storage, and trusted computing platforms.

Multi-/Many-core Architectures. Supporting shared mutable states in TSP
systems can create bottlenecks due to concurrent state accesses. TStream
tstream  is a recent example that effectively utilizes multicore CPUs to
improve concurrent shared state access performance through dual-mode
scheduling and a dynamic transaction restructuring mechanism. However, current
TSP systems still face scalability challenges with complex workloads and input
dependencies. Further research is needed to enhance TSP systems for complex
workloads, emerging multi-/many-core architectures with high-bandwidth memory,
and multi-node settings while maintaining correctness guarantees Tatbul2018 .

Non-Volatile Storage. Non-Volatile Memory (NVM) is an emerging technology
offering byte-addressability and low latency of DRAM along with persistence
and density of block-based storage media, but with limited cell endurance and
read-write latency asymmetry. Fernando et al.NVM  explored efficient
approaches for analytical workloads on NVM, potentially laying the foundation
for future TSP systemsTSM_NVM . NVMe-based SSDs can deliver high performance
in terms of latency and peak bandwidth. Lee et al. SSD_Stream  investigated
performance limitations of SPEs managing application states on SSDs, showing
query-aware optimization can improve stateful stream processing on SSDs. Their
work is valuable for TSP systems with strict ACID and streaming property
requirements, but more research is needed.

Trusted Computing Platforms. The need for low latency and local processing of
sensitive IoT data calls for edge stream processing. However, edge devices are
vulnerable to attacks due to limited power and computing capacity, posing
severe security threats to sensitive data. A potential solution streamboxtz
is trusted computing platforms (TCPs), which protect data and code within
isolated, encrypted memory areas. Bringing TSP to TCPs is nontrivial and
requires further research, particularly in handling limited memory for
transactional stateful stream processing a15060183 . Additionally, scaling
systems to multiple TCPs in a distributed environment presents challenges due
to each computing node’s computational limits.

##  7 Conclusion

††margin:

In this survey, we provided a comprehensive overview of Transactional Stream
Processing (TSP), and addressed key concepts, techniques, and challenges to be
overcome, in order to ensure reliable and consistent data stream processing.
We introduced terms, definitions, and a conceptual framework for TSP systems
and presented a taxonomy that offers a structured understanding of various
approaches and models for integrating transactional properties with streaming
requirements. We also discussed several notable TSP systems, each showcasing
unique features and design choices that were made, to cater to different
application requirements. These systems offer insight and inform designers
about the alternative choices they will need to make when designing and
implementing a novel TSP system. We highlighted various TSP applications and
use cases, such as stream processing optimization, concurrent stateful
processing, and stream and DBMS integration. Finally, we explored some open
challenges and suggest future directions for TSP research and development,
including novel applications and hardware platforms. This survey serves as a
valuable resource for researchers and practitioners. It aims to inspire others
to pursue work in this field and develop efficient, reliable, and scalable TSP
systems for diverse application domains.

##  8 Declarations

The authors have no financial or proprietary interests in any material
discussed in this article.

###### Acknowledgements.

This work is supported by the National Research Foundation, Singapore and
Infocomm Media Development Authority under its Future Communications Research
& Development Programme (FCP-SUTD-RG-2021-005), the SUTD Start-up Research
Grant (SRT3IS21164), the DFG Priority Program (MA4662-5), the German Federal
Ministry of Education and Research (BMBF) under grants 01IS18025A (BBDC -
Berlin Big Data Center) and 01IS18037A (BIFOLD - Berlin Institute for the
Foundations of Learning and Data). Shuhao Zhang’work is partially done while
working as a Postdoc at TU Berlin.

## References

  * (1) Rocksdb. http://rocksdb.org/
  * (2) Coral8, inc, http://www.coral8.com/ (2008)
  * (3) Data Artisans Streaming Ledger Serializable ACID Transactions on Streaming Data, https://www.data-artisans.com/blog/serializable-acid-transactions-on-streaming-data (2018)
  * (4) Abadi, D.J., Ahmad, Y., Balazinska, M., Cetintemel, U., Cherniack, M., Hwang, J.H., Lindner, W., Maskey, A., Rasin, A., Ryvkina, E., et al.: The design of the borealis stream processing engine.  In: CIDR’05, vol. 5, pp. 277–289 (2005)
  * (5) Abadi, D.J., Carney, D., Çetintemel, U., Cherniack, M., Convey, C., Lee, S., Stonebraker, M., Tatbul, N., Zdonik, S.: Aurora: A new model and architecture for data stream management.  The VLDB Journal 12(2), 120–139 (2003).  DOI 10.1007/s00778-003-0095-z.  URL http://dx.doi.org/10.1007/s00778-003-0095-z
  * (6) Affetti, L., Margara, A., Cugola, G.: Flowdb: Integrating stream processing and consistent state management.  In: Proceedings of the 11th ACM International Conference on Distributed and Event-based Systems, Debs ’17, pp. 134–145. Acm, New York, NY, USA (2017).  DOI 10.1145/3093742.3093929.  URL http://doi.acm.org/10.1145/3093742.3093929
  * (7) Affetti, L., Margara, A., Cugola, G.: Tspoon: Transactions on a stream processor.  Journal of Parallel and Distributed Computing 140, 65–79 (2020).  DOI https://doi.org/10.1016/j.jpdc.2020.03.003.  URL http://www.sciencedirect.com/science/article/pii/S0743731518305082
  * (8) Akhter, A., Fragkoulis, M., Katsifodimos, A.: Stateful functions as a service in action.  Proceedings of the VLDB Endowment 12(12), 1890–1893 (2019)
  * (9) Akidau, T., Balikov, A., Bekiroğlu, K., Chernyak, S., Haberman, J., Lax, R., McVeety, S., Mills, D., Nordstrom, P., Whittle, S.: Millwheel: Fault-tolerant stream processing at internet scale.  Proc. VLDB Endow. 6(11), 1033–1044 (2013-08).  DOI 10.14778/2536222.2536229
  * (10) Arasu, A., Babu, S., Widom, J.: The cql continuous query language: Semantic foundations and query execution.  The VLDB Journal 15(2), 121–142 (2006).  DOI 10.1007/s00778-004-0147-z.  URL http://dx.doi.org/10.1007/s00778-004-0147-z
  * (11) Armbrust, M., Das, T., Sun, L., Yavuz, B., Zhu, S., Murthy, M., Torres, J., van Hovell, H., Ionescu, A., Łuszczak, A., Świtakowski, M., Szafrański, M., Li, X., Ueshin, T., Mokhtar, M., Boncz, P., Ghodsi, A., Paranjpye, S., Senster, P., Xin, R., Zaharia, M.: Delta lake: High-performance acid table storage over cloud object stores.  Proc. VLDB Endow. 13(12), 3411–3424 (2020).  DOI 10.14778/3415478.3415560.  URL https://doi.org/10.14778/3415478.3415560
  * (12) Ayad, A.M., Naughton, J.F.: Static optimization of conjunctive queries with sliding windows over infinite streams.  In: Proceedings of the 2004 ACM SIGMOD International Conference on Management of Data, SIGMOD ’04, p. 419–430. Association for Computing Machinery, New York, NY, USA (2004).  DOI 10.1145/1007568.1007616.  URL https://doi.org/10.1145/1007568.1007616
  * (13) Babcock, B., Babu, S., Datar, M., Motwani, R., Widom, J.: Models and issues in data stream systems.  In: Proceedings of the Twenty-First ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS ’02, p. 1–16. Association for Computing Machinery, New York, NY, USA (2002).  DOI 10.1145/543613.543615.  URL https://doi.org/10.1145/543613.543615
  * (14) Bailis, P., Ghodsi, A.: Eventual consistency today: Limitations, extensions, and beyond: How can applications be built on eventually consistent infrastructure given no guarantee of safety?  Queue 11(3), 20–32 (2013).  DOI 10.1145/2460276.2462076.  URL https://doi.org/10.1145/2460276.2462076
  * (15) Bernstein, P., Newcomer, E.: Principles of Transaction Processing: For the Systems Professional.  Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (1997)
  * (16) Bernstein, P.A., Goodman, N.: Concurrency control in distributed database systems.  ACM Comput. Surv. 1981 13(2), 185–221 (1981).  DOI 10.1145/356842.356846.  URL http://doi.acm.org/10.1145/356842.356846
  * (17) Botan, I., Alonso, G., Fischer, P.M., Kossmann, D., Tatbul, N.: Flexible and scalable storage management for data-intensive stream processing.  In: Proceedings of the 12th International Conference on Extending Database Technology: Advances in Database Technology, EDBT ’09, p. 934–945. Association for Computing Machinery, New York, NY, USA (2009).  DOI 10.1145/1516360.1516467.  URL https://doi.org/10.1145/1516360.1516467
  * (18) Botan, I., Cho, Y., Derakhshan, R., Dindar, N., Haas, L., Kim, K., Lee, C., Mundada, G., Shan, M.C., Tatbul, N., Yan, Y., Yun, B., Zhang, J.: Design and implementation of the maxstream federated stream processing architecture (2009).  DOI 10.1007/978-3-642-14559-9˙2
  * (19) Botan, I., Fischer, P.M., Kossmann, D., Tatbul, N.: Transactional stream processing.  In: Proceedings of the 15th International Conference on Extending Database Technology, Edbt ’12, pp. 204–215. Acm, New York, NY, USA (2012).  DOI 10.1145/2247596.2247622.  URL http://doi.acm.org/10.1145/2247596.2247622
  * (20) Braun, L., Etter, T., Gasparis, G., Kaufmann, M., Kossmann, D., Widmer, D., Avitzur, A., Iliopoulos, A., Levy, E., Liang, N.: Analytics in motion: High performance event-processing and real-time analytics in the same database.  In: Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data, SIGMOD ’15, p. 251–264. Association for Computing Machinery, New York, NY, USA (2015).  DOI 10.1145/2723372.2742783.  URL https://doi.org/10.1145/2723372.2742783
  * (21) Brito, A., Fetzer, C., Sturzrehm, H., Felber, P.: Speculative out-of-order event processing with software transaction memory.  In: R. Baldoni (ed.) Proceedings of the Second International Conference on Distributed Event-Based Systems, DEBS 2008, Rome, Italy, July 1-4, 2008, _ACM International Conference Proceeding Series_ , vol. 332, pp. 265–275. ACM (2008).  DOI 10.1145/1385989.1386023.  URL https://doi.org/10.1145/1385989.1386023
  * (22) Carbone, P., Ewen, S., Fóra, G., Haridi, S., Richter, S., Tzoumas, K.: State management in apache flink: Consistent stateful distributed stream processing.  Proc. VLDB Endow. 10(12), 1718–1729 (2017-08).  DOI 10.14778/3137765.3137777.  URL https://doi.org/10.14778/3137765.3137777
  * (23) Carbone, P., Fragkoulis, M., Kalavri, V., Katsifodimos, A.: Beyond analytics: The evolution of stream processing systems.  In: Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data, SIGMOD ’20, p. 2651–2658. Association for Computing Machinery, New York, NY, USA (2020).  DOI 10.1145/3318464.3383131.  URL https://doi.org/10.1145/3318464.3383131
  * (24) Carbone, P., Katsifodimos, A., Ewen, S., Markl, V., Haridi, S., Tzoumas, K.: Apache flink: Stream and batch processing in a single engine.  Bulletin of the IEEE Computer Society Technical Committee on Data Engineering 36(4) (2015)
  * (25) Cetintemel, U., Du, J., Kraska, T., Madden, S., Maier, D., Meehan, J., Pavlo, A., Stonebraker, M., Sutherland, E., Tatbul, N., Tufte, K., Wang, H., Zdonik, S.: S-store: A streaming newsql system for big velocity applications.  Proc. VLDB Endow. 7(13), 1633–1636 (2014).  DOI 10.14778/2733004.2733048.  URL http://dx.doi.org/10.14778/2733004.2733048
  * (26) Chandrasekaran, S., Franklin, M.: Remembrance of streams past: Overload-sensitive management of archived streams.  In: VLDB (2004)
  * (27) Chang, F., Dean, J., Ghemawat, S., Hsieh, W.C., Wallach, D.A., Burrows, M., Chandra, T., Fikes, A., Gruber, R.E.: Bigtable: A distributed storage system for structured data.  In: 7th USENIX Symposium on Operating Systems Design and Implementation (OSDI), pp. 205–218 (2006)
  * (28) Chang, F., Dean, J., Ghemawat, S., Hsieh, W.C., Wallach, D.A., Burrows, M., Chandra, T., Fikes, A., Gruber, R.E.: Bigtable: A distributed storage system for structured data.  ACM Transactions on Computer Systems (TOCS) 26(2), 1–26 (2008)
  * (29) Chen, H., Migliavacca, M.: Streamdb: A unified data management system for service-based cloud application.  In: 2018 IEEE International Conference on Services Computing (SCC), pp. 169–176. IEEE (2018)
  * (30) Chen, Q., Hsu, M.: Experience in extending query engine for continuous analytics.  In: T. Bach Pedersen, M.K. Mohania, A.M. Tjoa (eds.) Data Warehousing and Knowledge Discovery, pp. 190–202. Springer Berlin Heidelberg, Berlin, Heidelberg (2010)
  * (31) Chen, Q., Hsu, M.: Query engine grid for executing sql streaming process.  In: International Conference on Data Management in Grid and P2P Systems, pp. 95–107. Springer (2011)
  * (32) Chen, Q., Hsu, M., Zeller, H.: Experience in continuous analytics as a service (caaas).  In: Proceedings of the 14th International Conference on Extending Database Technology, EDBT/ICDT ’11, p. 509–514. Association for Computing Machinery, New York, NY, USA (2011).  DOI 10.1145/1951365.1951426.  URL https://doi.org/10.1145/1951365.1951426
  * (33) Conway, N.: Cisc 499*: Transactions and data stream processing.  Apr 6, 28 (2008)
  * (34) Corbett, J.C., Dean, J., Epstein, M., Fikes, A., Frost, C., Furman, J.J., Ghemawat, S., Gubarev, A., Heiser, C., Hochschild, P., et al.: Spanner: Google’s globally distributed database.  ACM Transactions on Computer Systems (TOCS) 31(3), 1–22 (2013)
  * (35) Franklin, M., Krishnamurthy, S., Conway, N., Li, A., Russakovsky, A., Thombre, N.: Continuous analytics: Rethinking query processing in a network-effect world.  In: CIDR (2009)
  * (36) Garcia-Molina, H., Salem, K.: Sagas.  SIGMOD Rec. 16(3), 249–259 (1987).  DOI 10.1145/38714.38742.  URL https://doi.org/10.1145/38714.38742
  * (37) Golab, L., Bijay, K.G., Özsu, M.T.: On concurrency control in sliding window queries over data streams.  In: Y. Ioannidis, M.H. Scholl, J.W. Schmidt, F. Matthes, M. Hatzopoulos, K. Boehm, A. Kemper, T. Grust, C. Boehm (eds.) Advances in Database Technology - EDBT 2006, pp. 608–626. Springer Berlin Heidelberg, Berlin, Heidelberg (2006)
  * (38) Golab, L., Özsu, M.T.: Update-pattern-aware modeling and processing of continuous queries.  In: Proceedings of the 2005 ACM SIGMOD International Conference on Management of Data, SIGMOD ’05, p. 658–669. Association for Computing Machinery, New York, NY, USA (2005).  DOI 10.1145/1066157.1066232.  URL https://doi.org/10.1145/1066157.1066232
  * (39) Götze, P., Sattler, K.: Snapshot isolation for transactional stream processing.  In: EDBT (2019)
  * (40) Group, S., et al.: Stream: The stanford stream data manager.  Tech. rep., Stanford InfoLab (2003)
  * (41) Gürgen, L., Roncancio, C., Labbé, C., Olive, V.: Transactional issues in sensor data management.  In: Proceedings of the 3rd Workshop on Data Management for Sensor Networks: In Conjunction with VLDB 2006, DMSN ’06, p. 27–32. Association for Computing Machinery, New York, NY, USA (2006).  DOI 10.1145/1315903.1315910.  URL https://doi.org/10.1145/1315903.1315910
  * (42) Ishikawa, Y., Sugiura, K., Takao, D.: Fault tolerant data stream processing in cooperation with oltp engine.  In: A. Mondal, H. Gupta, J. Srivastava, P.K. Reddy, D. Somayajulu (eds.) Big Data Analytics, pp. 3–14. Springer International Publishing, Cham (2018)
  * (43) Katsifodimos, A., Fragkoulis, M.: Operational stream processing: Towards scalable and consistent event-driven applications. (2019)
  * (44) Krishnamurthy, S., Franklin, M.J., Davis, J., Farina, D., Golovko, P., Li, A., Thombre, N.: Continuous analytics over discontinuous streams.  In: Proceedings of the 2010 ACM SIGMOD International Conference on Management of Data, SIGMOD ’10, p. 1081–1092. Association for Computing Machinery, New York, NY, USA (2010).  DOI 10.1145/1807167.1807290.  URL https://doi.org/10.1145/1807167.1807290
  * (45) Kumar, A., Wang, Z., Ni, S., Li, C.: Amber: A debuggable dataflow system based on the actor model.  Proc. VLDB Endow. 13(5), 740–753 (2020).  DOI 10.14778/3377369.3377381.  URL https://doi.org/10.14778/3377369.3377381
  * (46) Lee, G., Eo, J., Seo, J., Um, T., Chun, B.G.: High-performance stateful stream processing on solid-state drives.  In: Proceedings of the 9th Asia-Pacific Workshop on Systems, APSys ’18, pp. 9:1–9:7. Acm, New York, NY, USA (2018).  DOI 10.1145/3265723.3265739.  URL http://doi.acm.org/10.1145/3265723.3265739
  * (47) Liarou, E., Goncalves, R., Idreos, S.: Exploiting the power of relational databases for efficient stream processing.  In: Proceedings of the 12th International Conference on Extending Database Technology: Advances in Database Technology, pp. 323–334 (2009)
  * (48) Liarou, E., Kersten, M.: Datacell: Building a data stream engine on top of a relational database kernel.  In: VLDB PhD Workshop (2009)
  * (49) Madden, S., Franklin, M.J.: Fjording the stream: An architecture for queries over streaming sensor data.  In: Proceedings 18th International Conference on Data Engineering, pp. 555–566. IEEE (2002)
  * (50) Mao, Y., Zhao, J., Zhang, S., Liu, H., Markl, V.: Morphstream: Adaptive scheduling for scalable transactional stream processing on multicores.  In: Proceedings of the 2023 International Conference on Management of Data (SIGMOD), SIGMOD ’23. Association for Computing Machinery, New York, NY, USA (2023)
  * (51) Marz, N.: Trident API Overview: github.com/nathanmarz/storm/wiki/trident-apioverview.  github.com/nathanmarz/storm/wiki/Trident-APIOverview
  * (52) Meehan, J., Aslantas, C., Zdonik, S., Tatbul, N., Du, J.: Data ingestion for the connected world.  In: CIDR (2017)
  * (53) Meehan, J., Tatbul, N., Zdonik, S., Aslantas, C., Cetintemel, U., Du, J., Kraska, T., Madden, S., Maier, D., Pavlo, A., Stonebraker, M., Tufte, K., Wang, H.: S-store: Streaming meets transaction processing.  Proc. VLDB Endow. 8(13), 2134–2145 (2015).  DOI 10.14778/2831360.2831367.  URL https://doi.org/10.14778/2831360.2831367
  * (54) Meftah, S., Zhang, S., Veeravalli, B., Aung, K.M.M.: Revisiting the design of parallel stream joins on trusted execution environments.  Algorithms 15(6) (2022).  DOI 10.3390/a15060183.  URL https://www.mdpi.com/1999-4893/15/6/183
  * (55) Motwani, R., Widom, J., Arasu, A., Babcock, B., Babu, S., Datar, M., Manku, G., Olston, C., Rosenstein, J., Varma, R.: Query processing, resource management, and approximation ina data stream management system.  In: CIDR 2003. Stanford InfoLab (2002)
  * (56) Ooi, B.C., Tan, K.L., Tung, A., Chen, G., Shou, M.Z., Xiao, X., Zhang, M.: Sense the physical, walkthrough the virtual, manage the metaverse: A data-centric perspective.  arXiv preprint arXiv:2206.10326 (2022)
  * (57) Oyamada, M., Kawashima, H., Kitagawa, H.: Efficient invocation of transaction sequences triggered by data streams.  In: 2011 International Conference on P2P, Parallel, Grid, Cloud and Internet Computing, pp. 332–337. IEEE (2011)
  * (58) Oyamada, M., Kawashima, H., Kitagawa, H.: Continuous query processing with concurrency control: Reading updatable resources consistently.  In: Proceedings of the 28th Annual ACM Symposium on Applied Computing, SAC ’13, p. 788–794. Association for Computing Machinery, New York, NY, USA (2013).  DOI 10.1145/2480362.2480514.  URL https://doi.org/10.1145/2480362.2480514
  * (59) Oyamada, M., Kawashima, H., Kitagawa, H.: Data stream processing with concurrency control.  SIGAPP Appl. Comput. Rev. 13(2), 54–65 (2013).  DOI 10.1145/2505420.2505425.  URL https://doi.org/10.1145/2505420.2505425
  * (60) Park, H., Zhai, S., Lu, L., Lin, F.X.: Streambox-tz: Secure stream analytics at the edge with trustzone.  In: Proceedings of the 2019 USENIX Conference on Usenix Annual Technical Conference, USENIX ATC ’19, p. 537–554. USENIX Association, USA (2019)
  * (61) Philipp, G., Stephan, B., Kai-Uwe, S.: An nvm-aware storage layout for analytical workloads.  In: 2018 IEEE 34th International Conference on Data Engineering Workshops (ICDEW), pp. 110–115 (2018).  DOI 10.1109/icdew.2018.00025
  * (62) Poess, M., Rabl, T., Jacobsen, H.A., Caufield, B.: Tpc-di: The first industry benchmark for data integration.  Proc. VLDB Endow. 7(13), 1367–1378 (2014).  DOI 10.14778/2733004.2733009.  URL https://doi.org/10.14778/2733004.2733009
  * (63) Ray, M., Lei, C., Rundensteiner, E.A.: Scalable pattern sharing on event streams*.  In: Proceedings of the 2016 International Conference on Management of Data, SIGMOD ’16, p. 495–510. Association for Computing Machinery, New York, NY, USA (2016).  DOI 10.1145/2882903.2882947.  URL https://doi.org/10.1145/2882903.2882947
  * (64) Sahin, O.C., Karagoz, P., Tatbul, N.: Streaming event detection in microblogs: Balancing accuracy and performance.  In: M. Bakaev, F. Frasincar, I.Y. Ko (eds.) Web Engineering, pp. 123–138. Springer International Publishing, Cham (2019)
  * (65) Sattler, K.U.: Transactional stream processing on non-volatile memory (2019).  URL https://www.tu-ilmenau.de/dbis/research/active-projects/transactional-stream-processing/
  * (66) Shahvarani, A., Jacobsen, H.A.: Parallel index-based stream join on a multicore cpu.  In: Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data, SIGMOD ’20, p. 2523–2537. Association for Computing Machinery, New York, NY, USA (2020).  DOI 10.1145/3318464.3380576.  URL https://doi.org/10.1145/3318464.3380576
  * (67) Shaikh, S.A., Chao, D., Nishimura, K., Kitagawa, H.: Incremental continuous query processing over streams and relations with isolation guarantees.  In: Proceedings, Part I, 27th International Conference on Database and Expert Systems Applications - Volume 9827, DEXA 2016, p. 321–335. Springer-Verlag, Berlin, Heidelberg (2016).  DOI 10.1007/978-3-319-44403-1˙20.  URL https://doi.org/10.1007/978-3-319-44403-1%5F20
  * (68) Shaikh, S.A., Kitagawa, H.: Streamingcube: Seamless integration of stream processing and olap analysis.  IEEE Access 8, 104,632–104,649 (2020).  DOI 10.1109/ACCESS.2020.2999572
  * (69) Shillaker, S., Pietzuch, P.: Faasm: Lightweight isolation for efficient stateful serverless computing.  In: 2020 USENIX Annual Technical Conference (USENIX ATC 20), pp. 419–433. USENIX Association (2020).  URL https://www.usenix.org/conference/atc20/presentation/shillaker
  * (70) Silvestre, P.F., Fragkoulis, M., Spinellis, D., Katsifodimos, A.: Clonos: Consistent Causal Recovery for Highly-Available Streaming Dataflows, p. 1637–1650.  Association for Computing Machinery, New York, NY, USA (2021).  URL https://doi.org/10.1145/3448016.3457320
  * (71) Stonebraker, M., Çetintemel, U., Zdonik, S.: The 8 requirements of real-time stream processing.  SIGMOD Rec. 34(4), 42–47 (2005).  DOI 10.1145/1107499.1107504.  URL http://doi.acm.org/10.1145/1107499.1107504
  * (72) Stonebraker, M., Madden, S., Abadi, D.J., Harizopoulos, S., Hachem, N., Helland, P.: The end of an architectural era: (it’s time for a complete rewrite).  In: Proc VLDB Endow. 2007
  * (73) Tatbul, N.: Streaming data integration: Challenges and opportunities.  In: 2010 IEEE 26th International Conference on Data Engineering Workshops (ICDEW 2010), pp. 155–158 (2010).  DOI 10.1109/ICDEW.2010.5452751
  * (74) Tatbul, N.: Transactional Stream Processing, pp. 4205–4211.  Springer New York, New York, NY (2018).  DOI 10.1007/978-1-4614-8265-9˙80704.  URL https://doi.org/10.1007/978-1-4614-8265-9%5F80704
  * (75) To, Q.C., Soto, J., Markl, V.: A survey of state management in big data processing systems.  The VLDB Journal 27(6), 847–872 (2018).  DOI 10.1007/s00778-018-0514-9.  URL https://doi.org/10.1007/s00778-018-0514-9
  * (76) Verheijde, J., Karakoidas, V., Fragkoulis, M., Katsifodimos, A.: S-query: Opening the black box of internal stream processor state.  In: 2022 IEEE 38th International Conference on Data Engineering (ICDE), pp. 1314–1327. IEEE (2022)
  * (77) Vidyasankar, K.: Transactional properties of compositions of internet of things services.  pp. 1–6 (2015).  DOI 10.1109/ISC2.2015.7366218
  * (78) Vidyasankar, K.: A transaction model for executions of compositions of internet of things services.  Procedia Computer Science 83, 195–202 (2016).  DOI 10.1016/j.procs.2016.04.116
  * (79) Vidyasankar, K.: Transactional composition of executions in stream processing.  In: 2016 27th International Workshop on Database and Expert Systems Applications (DEXA), pp. 114–118 (2016).  DOI 10.1109/DEXA.2016.037
  * (80) Vossen, G.: ACID Properties, pp. 1–3.  Springer New York, New York, NY (2016).  DOI 10.1007/978-1-4899-7993-3˙831-2.  URL https://doi.org/10.1007/978-1-4899-7993-3%5F831-2
  * (81) Wang, D., Rundensteiner, E.A., Ellison III, R.T.: Active complex event processing over event streams.  Proc. VLDB Endow. 4(10), 634–645 (2011).  DOI 10.14778/2021017.2021021.  URL http://dx.doi.org/10.14778/2021017.2021021
  * (82) Winter, C., Schmidt, T., Neumann, T., Kemper, A.: Meet me halfway: Split maintenance of continuous views.  Proceedings of the VLDB Endowment 13(11)
  * (83) Wu, Y., Arulraj, J., Lin, J., Xian, R., Pavlo, A.: An empirical evaluation of in-memory multi-version concurrency control.  Proc. VLDB Endow. 10(7), 781–792 (2017-03).  DOI 10.14778/3067421.3067427
  * (84) Zeuch, S., Chaudhary, A., Monte, B.D., Gavriilidis, H., Giouroukis, D., Grulich, P.M., Breß, S., Traub, J., Markl, V.: The nebulastream platform for data and application management in the internet of things.  In: CIDR 2020, 10th Conference on Innovative Data Systems Research, Amsterdam, The Netherlands, January 12-15, 2020, Online Proceedings. www.cidrdb.org (2020).  URL http://cidrdb.org/cidr2020/papers/p7-zeuch-cidr20.pdf
  * (85) Zhang, S., Wu, Y., Zhang, F., He, B.: Towards concurrent stateful stream processing on multicore processors.  In: 2020 IEEE 36th International Conference on Data Engineering (ICDE), pp. 1537–1548 (2020).  DOI 10.1109/ICDE48307.2020.00136
  * (86) Zhang, S., Zhang, F., Wu, Y., He, B., Johns, P.: Hardware-conscious stream processing: A survey.  SIGMOD Rec. 48(4), 18–29 (2020).  DOI 10.1145/3385658.3385662.  URL https://doi.org/10.1145/3385658.3385662


# Directed collective motion of bacteria under channel confinement

Hugo Wioland1111Present address: Institut Jacques Monod, Centre Nationale pour
la Recherche Scientifique (CNRS), UMR 7592, Université Paris Diderot, Sorbonne
Paris Cité, F-75205 Paris, France, Enkeleida Lushi2 and Raymond E. Goldstein1
1 Department of Applied Mathematics and Theoretical Physics, Centre for
Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge
CB3 0WA, UK
2 School of Engineering, Brown University, 182 Hope Street, Providence, Rhode
Island 02912, USA

(March 3, 2016)

###### Abstract

Dense suspensions of swimming bacteria are known to exhibit collective
behaviour arising from the interplay of steric and hydrodynamic interactions.
Unconfined suspensions exhibit transient, recurring vortices and jets, whereas
those confined in circular domains may exhibit order in the form of a spiral
vortex. Here we show that confinement into a long and narrow macroscopic
‘racetrack’ geometry stabilises bacterial motion to form a steady
unidirectional circulation. This motion is reproduced in simulations of
discrete swimmers that reveal the crucial role that bacteria-driven fluid
flows play in the dynamics. In particular, cells close to the channel wall
produce strong flows which advect cells in the bulk against their swimming
direction. We examine in detail the transition from a disordered state to
persistent directed motion as a function of the channel width, and show that
the width at the crossover point is comparable to the typical correlation
length of swirls seen in the unbounded system. Our results shed light on the
mechanisms driving the collective behaviour of bacteria and other active
matter systems, and stress the importance of the ubiquitous boundaries found
in natural habitats.

##  I Introduction

Spreading and survival of populations of bacteria often depend on their
ability to behave collectively: cells aggregate into swarms to seek and
migrate towards nutrient-rich regions BenJacob2004 ; Ariel2013 , organise into
biofilms resistant to antibiotics Mah2001 ; Wilking2013 , respond to
starvation by building fruiting bodies Shimkets1999 ; Branda2001  or opt for
cannibalism Gonzales2011 . In such organisations, the surrounding environment
often plays a major role, through its chemical composition Mah2001 ; Ariel2013
or geometrical constraints Wilking2013 ; Grant2014 . A complex and fascinating
issue is how the various chemical or mechanical interactions between the
microorganisms and their environment can guide the intricate dynamics of
populations.

Theoretical approaches to this question may utilize methods from the emerging
field of ‘active matter’, which has taken motivation from collective behaviour
in suspensions of flagellated bacteria (e.g., Escherichia coli and Bacillus
subtilis) Dombrowski2004 ; Cisneros2011 ; Sokolov2012 ; Aranson2013 ;
Gachelin2013 ; Wioland2013 ; Lushi2014 ; Wioland2016  as well as those
composed of molecular motors and the filaments along which they move Nedelec ;
Schaller . Using discrete discrete , continuum SaintillanShelley , and
phenomenological models, this growing field of research has studied self-
organisation of populations of interacting motile organisms or other kinds of
active and driven objects, often giving rise to striking collective
behaviours. Recent studies have predicted that physical confinement can have a
strong impact on the spatio-temporal organization, and indeed may result in
unidirectional flows Voiturez2005 ; Fielding2011 ; Woodhouse2012 ; Ravnik2013
; Neef2014 ; Tsang2016 . Experimental realisations of confined active matter
however remain relatively rare Wioland2013 ; Lushi2014 ; Wioland2016 ;
Bricard2013 ; Kumar2014 ; Vladescu2014 ; Bricard2015 .

The interactions between swimming cells in bacterial suspensions have two main
components: direct steric repulsion and long-range hydrodynamic forces created
by the action of multiple flagella. These interactions lead to complex
collective motion: while direct repulsion between rod-like bacteria produces
local alignment akin to nematic liquid crystal ordering, hydrodynamic forces
can advect and reorient nearby bacteria Ramaswamy ; PedleyKessler ;
Drescher2011  and power macroscopic turbulent patterns such as jets and swirls
much faster than an individual bacterium Dombrowski2004 ; Cisneros2011 ;
Sokolov2012 . Yet, these collective structures are not permanent, instead they
are recurrent and transient.

Previous experiments have been successful in controlling the bacterial
migration through gradients of chemoattractants BenJacob2004 ; Ariel2013  or
by varying external flow and environment properties Rusconi2014 . An
alternative that we consider here takes advantage of interfaces, as can be
commonly encountered in bacterial habitats like soil. Recent studies have
explained how a single microorganism interacts with surfaces Drescher2011 ;
Kantsler2013 , yet little has been done to experimentally study the collective
dynamics of motile bacteria and other types of active matter under confinement
Wioland2013 ; Lushi2014 ; Wioland2016 ; Bricard2013 ; Bricard2015 . Different
interfaces (fluid, solid) or confinement topologies (e.g. planar, linear,
circular) could lead to various macroscopic organisations.

Here we describe an experimental setup able to stabilise a dense suspension of
Bacillus subtilis into a persistent stream through physical confinement alone.
As shown in figure 1, bacteria are introduced into an array of a thin periodic
millimetre-long racetracks, allowing for quantification of many realizations
of the resultant flow patterns. Our chief finding is that there exists a clear
transition from a ‘turbulent’ state to stable circulation for channel widths
$\lesssim 70\,\mu$m, or a scale comparable to correlation length of cell
orientations, or equivalently the diameter of the characteristic swirls seen
in effectively unbounded suspensions Dombrowski2004 ; Cisneros2011 ;
Dunkel2013 . Numerical studies of a discrete model of swimmers which
incorporates both steric and hydrodynamic interactions between the cells
reveal that the suspension motion is dominated by the bacteria-driven fluid
stream. In particular, cells close to the bounding walls create a strong fluid
flow which advects the cells in the bulk of the channel against their swimming
direction in a manner similar to that seen in circular drops of suspension
Wioland2013 ; Lushi2014 . Analysis of the suspension kinetics sheds light on
the mechanism that drives bacterial collective behaviour. This organisation
shows strong connections with previous work on bacterial pumping in
microfluidic chambers Kim2008  and cell swimming or alignment against a shear
fluid flow Rusconi2014 ; Hill2007 ; Kaya2012 ; Marcos2012 ; Kantsler2014 .

Figure 1:  Experimental setup. (a) Layout of the entire microfluidic circuit.
Bacteria are injected through the inlet, which is sealed prior to recording.
(b) Detail of one racetrack in the PDMS chamber device. Arrow indicates one of
two equivalent directions for spontaneous circulation. (c) close-up of the
circulation. Arrows: instantaneous bacterial flow field as measured by PIV.

##  II Materials and Methods

###  II.1 Experiments

Dense suspension of motile B. subtilis (wild-type, strain 168) were prepared
from an overnight culture, diluted $200$-fold into fresh Terrific Broth (TB,
Sigma) and incubated at $35^{\circ}$C for $\approx 5$h. When the bacteria
reached a high motility fraction ( $>90\%$ motile cells), $10\,$ml of the
suspension was concentrated by centrifugation ($1500\,$g, $10\,$min), and the
pellet was used in the experiments without further dilution. We estimate the
cell volume fraction to be $\sim 20\%$.

Polydimethylsiloxane (PDMS) microchambers were fabricated with racetracks
($20\,\mu$m high, $20-130\,\mu$m wide and more than a $1$ mm long, figure
1(a)), linked by thin inlets ($\approx 10\,\mu$m wide). Bacteria were
introduced into the chamber with a syringe and the main inlets were closed
prior to recording to avoid any unwanted motion (visual inspection verified
the absence of any net flow between racetracks).

Each preparation was allowed to reach a steady state over two minutes and
videos lasting $5$ seconds were acquired with a high-speed camera (Fastcam,
Photron, $125$ frames/sec) under bright field illumination, using a $63\times$
oil-immersion objective (Zeiss). The velocity field of the bacteria was
determined by Particle Image Velocimetry (using a customised version of mPIV
mpiv ) with no time averaging (figure 1(b)). This method yields the properties
of the bacterial motion, not the associated fluid flow.

###  II.2 Measuring the bacterial orientation

Bright field images yield only the local velocity field of the bacteria and
their rough alignment Wioland2013 , not their head-tail orientation. To
determine the relationship between swimming and motion directions we labelled
the flagella and membrane of a subset of the population with two different
fluorophores, as described previously Lushi2014 . The mutant strain
amyE::hag(T209C) DS1919 3610 Guttenplan2013  was grown in the same conditions
as wild-type bacteria and labelled following the protocol of Guttenplan et al.
Guttenplan2013 . $1\,$ml of the suspension was centrifuged ($1000\,$g,
$2\,$min) and resuspended in $50\,\mu$L of Phosphate Buffered Saline (PBS)
containing $5\,\mu$g/mL Alexa Fluor 488 $\textrm{C}_{5}$ Maleimide (Molecular
Probes) and incubated at room temperature for $5\,$minutes. This fluorophore
reacts with the cysteine added to the flagella protein flagellin (gene hag).
Bacteria were then washed in $1\,$mL PBS and resuspended in PBS containing
$5\,\mu$g/mL FM 4-64 (Molecular Probes). This fluorophore incorporates itself
into lipid bilayers and labels cell membranes. The excess of fluorophore was
removed in a final wash and the bacteria were resuspended into $50\,\mu$l PBS.
A fraction of the labelled mutants was then gently mixed with unlabelled wild-
type bacteria. Cells cannot swim in PBS, which lacks a carbon source, but
recover their initial motility once transferred into Terrific Broth.

Images were acquired on a spinning disc confocal microscope (Zeiss Axio
Observer Z1, camera Photometrics Evolve 512 Delta) at $6$ fps. Both
fluorophores were excited with a $488\,nm$ laser and the emission was filtered
with a GFP filter (barrier filter $500-550\,nm$, Zeiss) for Alexa Fluor 488
$\textrm{C}_{5}$ Maleimide and DsRed filter (barrier filter $570-640\,nm$,
Zeiss) for FM 4-64. We took a sequence of three images: membrane (false
coloured red), flagella (false coloured green) and again membrane (false
coloured blue, figure 6) and measured the bacterial motion from the membrane
displacement between the first and last image. We then deduced the orientation
of the cell (its swimming direction) from the bundling of the flagella (at the
rear of the cell) and from the average position of the membrane relative to
this bundle.

###  II.3 Modelling and Simulations

We used numerical simulations, adapted from a recent method Lushi2014 , to
understand the self-organisation of the suspension. Bacteria are represented
as motile ellipses, subject to steric and hydrodynamic interactions and
confined into a periodic channel. Each swimmer generates a dipolar
“pusher”-flow field and is affected by the fluid flow disturbances created by
the other swimmers.

We use a 2D domain with periodic boundaries of length $L$ and width $2w$ where
$y=0,w$ are the channel walls. Each swimmer is modelled as an ellipse of
length $\ell=1$, width $\ell/4$, described by its centre of mass
$\mathbf{X}_{i}$ and orientation $\mathbf{P}_{i}$. $\mathbf{X}_{i}$ and
$\mathbf{P}_{i}$ are randomly initiated and follow the dynamics Lushi2014 ;
Lushi2013 :

$\displaystyle\partial_{t}\mathbf{X}_{i}$
$\displaystyle=U_{0}\mathbf{P}_{i}+\mathbf{v}+\Xi^{-1}_{i}\sum_{j\neq
i}\mathbf{F}^{e}_{ij}~{},$ (1) $\displaystyle\partial_{t}\mathbf{P}_{i}$
$\displaystyle=(\mathbf{I}-\mathbf{P}_{i}\mathbf{P}_{i}^{T})(\gamma\mathbf{E}+\mathbf{W})\mathbf{P}_{i}+k\sum_{j\neq
i}(\mathbf{T}^{e}_{ij}\times\mathbf{P}_{i}).$ (2)

Equation (1) describes self-propulsion with constant speed (chosen as
$U_{0}=1$) along the cell direction $\mathbf{P}_{i}$, advection by the fluid
velocity $\mathbf{v}$ interpolated at the swimmer position, and pairwise
steric repulsion with force $\mathbf{F}^{e}_{ij}$ between swimmers. Here,
$\Xi=m_{||}\mathbf{P}_{i}\mathbf{P}_{i}^{T}+m_{\perp}(\mathbf{I}-\mathbf{P}_{i}\mathbf{P}_{i}^{T})$
with the mobility parameters $m_{\perp}$=$2m_{||}\approx 2$. The first term of
equation (2) is Jeffery’s equation and describes rotation of the particle by
the fluid flow $\mathbf{v}$ with
$2\mathbf{E}=\nabla\mathbf{v}+\nabla\mathbf{v}^{T}$,
$2\mathbf{W}=\nabla\mathbf{v}-\nabla\mathbf{v}^{T}$; $\gamma\approx 0.9$ for
ellipses with aspect ratio 4. The last term of equation (2) describes swimmer
rotation due to torques from steric interactions with neighbours with
$k\approx 5$. The purely repulsive steric forces $\mathbf{F}^{e}_{ij}$ and
torques $\mathbf{T}^{e}_{ij}$ between swimmers are obtained using the method
described by Constanzo et al.Constanzo2012 . Each swimmer is discretized into
$n_{b}=4$ beads that interact with other swimmers through a soft capped
Lennard-Jones potential Lushi2014 . This potential allows some overlaps but
does not over-restrict the time-stepping. Noise terms are not included in the
dynamics.

Figure 2:  Examples of the fluid flow generated by one swimmer inside a
channel (a) at the boundary and (b) in the bulk.

To approximate the boundary condition at $y=0,w$, we use a system of images.
For each swimmer $i$ in the channel $(x,y)\in([0,L],[0,w])$, a mirror swimmer
is placed in $(x,y)\in([0,L],[0,-w])$ with centre of mass
$\mathbf{X}_{i}-2(\mathbf{X}_{i}\cdot\mathbf{e}_{y})\mathbf{e}_{y}$ and
orientation
$\mathbf{P}_{i}-2(\mathbf{P}_{i}\cdot\mathbf{e}_{y})\mathbf{e}_{y}$. We
calculate the active stress tensor
$\mathbf{S}^{a}_{i}$=$\sigma\mathbf{P}_{i}\mathbf{P}_{i}^{T}$ that results
from locomotion for both the swimmers and their images. The non-dimensional
stresslet strength is set to $\sigma=-1$ for a slender pusher swimmer with
length $\ell=1$ and speed $U_{0}=1$ Lushi2013 . The fluid velocity
$\mathbf{v}$ is obtained by solving with Fourier series the (non-dimensional)
2D Stokes Equations with the extra active stresses

$\displaystyle-\nabla^{2}\mathbf{v}+\nabla
q=\nabla\cdot\sum_{i}\mathbf{S}^{a}_{i}\delta(\mathbf{x}-\mathbf{X}_{i}),\quad\nabla\cdot\mathbf{v}=0$
(3)

in the entire (real and mirror) domain $(x,y)\in([0,L],[-w,w])$ which is
assumed bi-periodic. Using the mirror domain in the $y$-direction for solving
equation 3 is a computationally efficient way to approximate the effects of
the PDMS interface as a no-stress boundary condition (figure 2). Note that to
guarantee a no-slip boundary condition for the fluid flow, other methods, e.g.
discrete  are more appropriate, but require higher computational power for
many swimmer simulations.

The free parameters in the simulations are the swimmer length $\ell:=1$, its
shape parameter, its swimming speed $U_{0}:=1$, the stresslet strength
$\sigma:=-1$ which determines the magnitude of the generated fluid flows, as
well as the channel dimensions. We note that we did not match these with the
experimental values, hence the comparisons between the results are qualitative
and not quantitative.

Figure 3:  Normalised net flow $\Phi$ for a variety of channel widths in both
experiments (circles) and simulations (crosses). Experimental points are
averaged over at least five movies, recorded on at least two different days,
simulations are averaged over a time $5U_{0}/\ell$. $\infty$: an unconfined
quasi-2D chamber.

##  III Results

###  III.1 Confinement stabilises a bacterial stream

We inject a dense suspension of swimming B. subtilis into $20\,\mu$m high
racetracks. When confinement is strong enough, the turbulent collective motion
is stabilised and forms a persistent stream (figure 1). This circulation is
stable for tens of minutes until bacteria stop swimming due to oxygen
depletion.

We quantify the overall motion by measuring the normalised net flow:

$\Phi=\left|\frac{\sum\mathbf{u}\cdot\mathbf{e}_{x}}{\sum||\mathbf{u}||}\right|,$
(4)

where $\mathbf{u}$ is the bacterial flow as measured by PIV, $\mathbf{e}_{x}$
the unit vector along the channel main direction (figure 1(b)) and summing
over all PIV sub-windows inside the channel and over $\approx 70$ frames of a
$5$ seconds long movie. We observed CW and CCW circulation with equal
probability and therefore study the absolute value of $\Phi$. $\Phi=1$
indicates a purely longitudinal flow (e.g. $\mathbf{u}=\|u\|\mathbf{e}_{x}$)
and $\Phi=0$ indicates a globally stationary suspension.

Figure 3 shows a clear dependence of the net flow $\Phi$ on the channel width
$w$: while the suspension exhibits a strong circulation ($\Phi>0.7$) for thin
chambers, it quickly transitions into a stationary turbulent state
($\Phi<0.2$) around $w^{*}_{\textrm{exp}}=70\,\mu$m.

We compared these results with simulations in a 2D periodic channel domain, to
mimic the straight section of the racetracks (figure 1(b)). Ellipsoid
swimmers, subject to both steric and hydrodynamic interactions, self-organise
into a stream similar to the observed experimental motion, as seen in figure
5(d). We computed the net flow $\Phi$, this time summing over bulk swimmers
only. We found the same qualitative behaviour with strong circulation only for
thin channels $w<w^{*}_{\textrm{sim}}=17\ell$ (where $\ell$ is the swimmer
length).

Figure 3 shows that the transition from a stable stream to an overall
stationary state can be matched between experiments and simulation
($w^{*}_{\textrm{exp}}=w^{*}_{\textrm{sim}}$) by choosing $\ell=4\,\mu$m. Note
that other outcomes, e.g. flow strength, are not necessarily matched.

Figure 4:  Bacterial flow pattern and profiles depending on the channels
width. (a) Bacterial circulation measured by PIV for channels width
$35,45,55,75,90\,\mu$m. (b) Averaged bacterial flow profile $F(y)$ in
experiments. Measurements (dots) are fitted as the sum (solid line) of two
beta distribution functions, $f^{(1)}_{\mathrm{beta}}$ and
$f^{(2)}_{\mathrm{beta}}$ (dotted lines). (c) Averaged bacterial flow profile
$F(y)$ in simulations. The distance is expressed in terms of swimmer length
$\ell$ and the fluid flow in terms of the single cell swimming speed $U_{0}$.
Sketches of the swimmers at the edges indicate their orientations (green
arrow) and the direction of the generated fluid flow (purple arrow). (d)
Distance $y_{m}$ between the PDMS interface and the maximum of the first beta
distribution function, as indicated in (b) by a thin black line. Each point is
an average taken over at least five movies, recorded on at least two different
days. Error bars: standard error.

###  III.2 Bacterial flow patterns under confinement

In unconfined environments, dense bacterial suspensions exhibit typical
patterns of jets and swirls Cisneros2011 ; Sokolov2012 ; Aranson2013 ;
Gachelin2013 . PDMS interfaces in racetracks change this pattern in order to
allow for the net circulation. We now study the pattern stabilisation, through
the averaged flow profile, depending on the channel width. If the flow were
driven by a pressure gradient, a passive Newtonian fluid would produce a
parabolic or pump flow Brody1996 . Yet bacteria collectively generate
irregular flow patterns.

We measure the profile or cross section of the flow patterns, averaged over
time and space:

$F(y)=\left<\mathbf{u}(x,y,t)\cdot\mathbf{e}_{x}\right>_{(x,t)},$ (5)

where the direction of $\mathbf{e}_{x}$ is chosen such that $F(y)$ is on
average positive. Figure 4(a) shows different behaviours depending on the
channel width $w$.

For the thinnest channels ($w\approx 35\,\mu$m) the flow is directed parallel
to the channel length and $F$ takes a quasi-parabolic profile. $F$ then
flattens for $w\approx 45\,\mu$m and splits into two peaks for $w\approx
55\,\mu$m. At such width, small swirls start to emerge but are not strong
enough to alter the circulation. When the width reaches the critical value
$w>w^{*}_{\textrm{exp}}=70\,\mu$m, the suspension is able to form full
vortices. $F$ takes a sinusoidal form such that bacteria on each side move in
opposite directions and most vortices circulate in the same direction (CCW in
figure 4 (a)). For wider racetrack ($w\approx 100\,\mu$m) the suspension
recovers its quasi-turbulent state and $F$ exhibits several oscillations. In
this last case, if we averaged over a longer period of time, the resulting
profile should come to zero.

For channels between $50$ and $70\,\mu$m in width, we observe that the profile
$F$ displays two maximum values, at a distance $y_{m}$ from the PDMS interface
(figure 4(b)). In order to measure this length for different widths, we fit
$F$ with a combination of two symmetric beta distribution functions:

$\displaystyle F_{fit}(y)$ $\displaystyle=f^{(1)}_{beta}+f^{(2)}_{beta}$
$\displaystyle=A_{1}(\frac{y}{w})(1-\frac{y}{w})^{\beta-1}+A_{2}(1-\frac{y}{w})(\frac{y}{w})^{\beta-1},$
(6)

where $A_{1}$ and $A_{2}$ are the amplitude of each function and $\beta$ is a
parameters setting the shape of the distribution. Each beta function is the
symmetric of the other except for the amplitude:
$f^{(1)}_{beta}(y)=\frac{A_{1}}{A_{2}}f^{(2)}_{beta}(w-y)$. The beta
distribution functions are merely used for simplicity and have no physical
connection to the bacterial suspensions.

We found that $F_{fit}$ correctly describes the flow profiles until $w\approx
75\,\mu$m (figure 4(a)). For $w\approx 35\,\mu$m, the two beta functions are
almost identically and mostly overlap. They then come apart to form a double-
peaked function ($w\approx 55\,\mu$m) which then becomes anti-correlated
($A_{1}A_{2}<0$) for $w\approx 75\,\mu$m. For the largest racetrack, $F_{fit}$
only captures the first and last peaks but not the bulk behaviour.

We then measure the distance $y_{m}$ at which $f^{(1)}_{beta}$ reaches its
maximum (represented as a thin line on figure 4(b)). Figure 4(d) shows that
$y_{m}$ does not depend on the racetrack width: the flow profile is simply the
sum of two beta functions, of width-independent shape, and that move apart as
the channels widen, suggesting that the circulation is driven by bacteria
close to the wall.

Simulations show a similar behaviour, with parabolic ($w=6\ell$), flattened
($w=9\ell$), double-peaked ($w=15\ell$) and sinusoidal profiles ($w=19\ell$,
figure 4(b)). The only significant difference lies in the bacteria motion at
the boundary: simulations reveal that these swimmers move against the bulk
circulation: $F_{\mathrm{simu}}(0)\cdot F_{\mathrm{simu}}(3\ell)<0$. We also
observed experimentally that bacteria at interfaces swim against the bulk
circulation, but PIV measurements could not catch this feature.

Both the swimmer motion at the boundary and the flow profile fitting suggest
that bacteria close to the wall play an important role in the suspension
ordering.

###  III.3 Boundaries and fluid flow drive the bacterial circulation

Figure 5:  Snapshots from simulations of micro-swimmers in a 2D channel. (a,
b) Dilute and dense suspensions of motile ellipses without hydrodynamical
interactions ($\sigma=0$). (c, d) Dilute and dense suspensions with
hydrodynamical interactions ($\sigma=-1$). Collectively-generated fluid flow
fields are shown superimposed (the magnitude in plot (d) has been halved). In
both cases the fluid flow is not purely longitudinal and a wavy-like pattern
is discernible.

We turn to simulations to understand how boundaries affect the bacteria
organisation and in particular to find the causes of the bulk circulation as
observed in experiments. We consider different scenarios: dilute or dense
suspensions, with or without hydrodynamic interactions. The examples in figure
5 have a channel width of $w=6\ell$.

We first neglect hydrodynamics and consider swimmers interacting between
themselves and with the boundary through direct steric repulsions only. In
both dilute and dense cases (figure 5(a),(b)), swimmers aggregate at the
boundary to form packs or hedgehog-like clusters, as previously observed
Constanzo2012 ; Wensink2008 . Due to their elongated shape, swimmers at the
boundary exhibit a local nematic alignment, with heads facing the interface,
and slowly move alongside it. Cells in the bulk jam in clusters with little
organisation discernible.

Figure 6:  Fluorescently labelled bacteria indicate orientation and motion
direction of cells at the PDMS wall and in bulk. Mutant B. subtilis (strain
amyE::hag(T204C) DS1919 3610 Guttenplan2013 ) were tagged at the body and
flagella, respectively with FM 4-64 (false coloured red and blue) and Alexa
488 $\textrm{C}_{5}$ maleimide (false coloured green)

We next include fluid flows generated by bacteria. In dilute suspensions, as
shown in figure 5(c), swimmers aggregate at the boundary and form packed
layers similar to the steric-only case. However they generate a strong fluid
stream, against their swimming direction. If the channel width is sufficiently
small, the fluid reorients the opposite layer such that swimmers on each
channel side move in the same direction. This packing and organisation is not
observed experimentally at low densities, most probably because 2D simulations
do not allow for overlap whereas bacteria in $20\,\mu$m deep racetracks can
easily swim and roll-over over one-another.

When the density is further increased (figure 5(d)), the same mechanism is at
work: the two boundary-bound layers of swimmers drive a strong fluid flow,
reorienting and advecting the cells that now fill up the centre of the
channel. Bulk swimmers have a biased orientation against this flow but do not
swim fast enough to overcome the advection. They are thus transported against
the boundary layers giving rise to a macroscopic bulk circulation similar to
the experimental observations (figure 1(b)).

The simulations also clarify the profile shapes (figure 4). Sketches of the
swimmers, drawn at the channel walls in figure 4(c), indicate their swimming
orientation and the direction of the fluid flow they push back. This flow
advects over a distance $\sim y_{m}$ the cells in the bulk, whose overall
collective motion can be fitted with the double beta function profiles.

Experimentally, bacteria can flow over $60\,\mu$m/s, significantly faster than
the single cell swimming speed of $10\,\mu$m/s. This high velocity suggests
that the bulk bacterial motion is indeed dominated by fluid advection.

To verify the organisation predicted by simulations, we track fluorescently
labelled bacteria (figure 6). We deduced the cell orientation from the
relative position of the body and flagella and the overall motion from the
body displacement. We found the same biased orientation against the flow in
the bulk of thin channels ($w<w^{*}=70\,\mu$m): out of the $24$ labelled
bacteria tracked, $4$ were oriented along the $\mathbf{e}_{y}$ direction, $4$
were swimming with the flow and $16$ were swimming against the flow (oriented
against their overall motion direction). Cells at the surface had a forward
motion, against the bulk circulation, confirming the organisation found in
simulations.

###  III.4 Spatial and temporal variations in bacterial flow patterns

Our current model - edge bacteria driving the bulk flow through hydrodynamic
interactions - explains the overall circulation. As described earlier,
bacteria in thin racetracks also form partial and travelling swirls,
reminiscent of the unconfined turbulent state. We study this flow pattern by
computing both the correlation lengths and the motion of swirls.

Figure 7:  Bacterial flow correlation depending on the channel width $w$. (a)
measurement of the correlation lengths depending on the racetrack width $w$.
Both oscillation period and amplitude decay length scale reach a plateau state
at large width (dashed lines). Points are averaged over at least five movies,
recorded on at least two different days. Error bars: standard error. $\infty$:
unconfined quasi-2D chambers. (b) Example of a correlation function $C(s)$
experimentally measured and fitted.

We first compute the spatial correlation from which we extract two
characteristic lengths: the oscillation period $L_{c}$ and the amplitude decay
length $L_{e}$. To do so, we take advantage of the lateral confinement and
modify the classical two-point velocity correlation function to focus on the
variation of the orthogonal flow $u_{y}$ along the channel length
$\mathbf{e}_{x}$:

$C(s)=\frac{\sum u_{y}(x,y,t)\cdot u_{y}(x+s,y,t)}{\sum||u_{y}(x,y,t)||^{2}},$
(7)

where $u_{y}=\mathbf{u}\cdot\mathbf{e}_{y}$ and taking the sum over PIV sub-
windows in the bulk of a channel of a full $5$ second long movie. $C$ exhibits
decaying oscillations (figure 7(b)), a behaviour that was observed in previous
work Cisneros2007  but, to our knowledge, has not been fully analysed.

Two main approaches have been used to compute the correlation length: either
fitting with a decaying exponential Sokolov2012 ; Gachelin2013  or taking the
distance at which $C$ reaches zero or its first minimum Wensink2008 . Instead
we fit $C$ with a function that describes the decay and the oscillations:

$C_{\textrm{fit}}=\left[Ae^{-s/L_{e}}+(1-A)\right]\cdot\cos(2\pi s/L_{c}),$
(8)

where the first term indicates the amplitude decaying over the length $L_{e}$
and the cosine term highlights the oscillation period $L_{c}$ (swirl size).

Figure 7(a) shows that both $L_{e}$ and $L_{c}$ increase with the racetrack
width and reach a plateau around $w\approx 85\,\mu$m. This size is comparable
with $w^{*}=70\,\mu$m when streaming ceases and the suspension is able to form
full vortices. Moreover values at the plateau, $L^{*}_{e}\approx 70\,\mu$m and
$L^{*}_{c}\approx 90\,\mu$m, are comparable in large racetracks and unconfined
2D chambers (noted $\infty$ on the graphs).

Even though we also do observe a wave pattern in simulations, the correlation
function mostly reflects the domain periodicity.

The bacterial flow patterns then are not static: while the bacteria stream in
the channel, the partial swirls they generate travel as well. To quantify this
effect we measure the flow pattern motion by the spatial correlation between
two time points:

$C_{\mathrm{swirl}}(s)=\frac{\sum u_{y}(x,y,t)\cdot u_{y}(x+s,y,t+\Delta
t)}{\sum||u_{y}(x,y,t)||^{2}},$ (9)

taking the sum over all PIV sub-windows inside the channel and over a full $5$
second long movie. $C_{\mathrm{swirl}}$ takes a maximum value at $s^{*}$ which
gives a wave speed: $U_{\mathrm{wave}}=s^{*}/\Delta t$. We compare this speed
to the averaged circulation speed $<u_{x}>_{x,y,t}$, and find that the wave
travels slightly faster than the bacteria:
Median$\left(\frac{U_{\mathrm{wave}}}{<u_{x}>}\right)=1.2$ (standard error:
$0.1$). Yet, a more detailed analysis is required to fully understand the
dynamics between cell motion and wave propagation in this particular
confinement.

Figure 8:  Illustration of the bacterial organization in racetracks of varying
width.

##  IV Discussion

We have shown that confining a dense bacterial suspension into a thin periodic
racetrack leads to the spontaneous formation of a stable circulation along it.
A similar setup has been previously used to study two different active matter
systems: marching locusts Buhl2006  and rolling colloidal particles
Bricard2013 . As for bacteria, collective colloid motion is driven by
hydrodynamic interactions, but can result in the formation of travelling waves
and density shocks while the bacterial suspension always appears spatially
homogeneous.

To understand how the suspension self-organises in racetracks, we have
measured the net flow, flow correlation and profile depending on the channel
width. We reproduced these results with simulations, revealing the fluid flows
generated by the bacteria. Combining these different approaches, we draw a
picture in Figure 8 of how bacterial motion, hydrodynamic interactions and
confinement drive the macroscopic circulation.

Independently of the channel size, bacteria at the PDMS surface move along a
given direction while propelling the fluid in the opposite direction, over a
distance $y_{m}\approx 14\,\mu$m. This fluid flow, much faster than the single
cell’s swimming speed, reorients and advects cells in the suspension. As a
result bulk bacteria are transported against their swimming orientation in a
backward motion. Different behaviours then arise for varying channel widths.
For thin channels ($w<w^{*}=70\,\mu$m) the fluid flow on one side reaches the
opposite interface and reorients the bacteria such that all cells at the
different surfaces move in the opposite direction to the bulk circulation, and
give rise to the parabolic, flat and double-peaked profiles (figure 8). Inside
the channel, confinement restricts the turbulent bacterial motion such that
bacteria form partial swirls that mostly travel with the bacterial stream,
without affecting the overall circulation. For intermediate channel widths
($70<w<90\,\mu$m) the suspension forms full vortices, showing no net
circulation ($\Phi<0.2$). Yet the coordination between opposite sides is not
lost but now appears to be anti-correlated (as shown in the sinusoidal flow
profiles, figure 8). In particular, vortices in the channels have a preferred
rotation direction, either clockwise or counterclockwise, depending on the
orientation of the bacteria at the interface. For even larger channels,
$w>90\,\mu$m, the suspension recovers its unconfined turbulent state and no
correlation is observed between opposite PDMS walls. However it still remains
unclear how the critical length-scale of $w^{*}=70\,\mu$m of the vortical
macroscopic structures in the collective motion of this bacterium is selected.

Some continuum models of active matter in confinement have previously
predicted unidirectional flow or net circulation Voiturez2005 ; Fielding2011 ;
Woodhouse2012 ; Ravnik2013 ; Neef2014 . Ravnik et al. Ravnik2013  found that
the bacterial flow in a pipe has a weak ($\sim 1\%$) component along the $y$
and $z$ directions, but with vortex patterns quite different from the wave-
like stream we observe here. Fielding et al.Fielding2011  considered 2D
channels of varying width but observed circulation for channels wider than the
suspension vortex size, in contradiction with our net flow measurements
(figure 3). Neef and Kruse Neef2014  consider active polar fluids in annular
domains and see surprisingly similar patterns to ours such as unidirectional
streaming, moving or stationary vortices for increasing channel widths in the
extensile particle case. Many of the differences between these theoretical
predictions may arise from the difficulty of setting realistic boundary
conditions and on including both the swimmer-generated fluid flow and
bacterial motion in such continuum models.

Despite many previous simulation studies on micro-swimmers in channels or
between plates, this specific collective behaviour we report here has not been
observed before in experiments or simulations. Menzel Menzel2013  performed
simulations of self-propelled particles using a Vicsek model with aligning
interactions, no hydrodynamics, and observed collective migration along the
channel in the form of particle clusters and lanes. Costanzo et
al.Constanzo2012  simulated elongated swimmers in a periodic channel,
interacting directly and through dipolar-generated fluid flows, but did not
observe any collective behaviour and net circulation when external flow was
not imposed. One possible explanation is that they computed the hydrodynamic
interactions without taking into account boundaries. Here we include mirror
images to approximate the PDMS interface which enhances the bulk fluid stream.
These small differences between these studies highlight the key roles played
not only by the surfaces and the cell-driven flows arising from them but also
of the elongated particle shapes that are needed for the swimmers orienting
and ordering alongside each-other at the wall.

Yet, the behaviour of bacteria in racetracks is reminiscent of some previous
experiments and can be understood as a combination of insights deriving from
them. Swimming bacteria glued to a surface can coordinate their orientations
and create a net flow far from the wall. Darnton et al.Darnton2004  used this
effect to propel small objects while Kim et al.Kim2008  used it to turn a
microfluidic channel into a bacteria-powered pump. This phenomenon is
qualitatively similar to what we observed in the racetracks, except here the
bacteria are free and swimming along the surface while creating fluid flow in
the bulk. Other studies have shown that the motion of swimmers in micro-
channels is affected by an external shear flow Rusconi2014 ; Hill2007 ;
Kaya2012 ; Marcos2012 ; Kantsler2014 . In particular, swimmers there are
biased to swim against the flow as we also observed in simulations and
confirmed with fluorescence labelling. Finally, the self-organisation in
racetracks is comparable to what was observed when bacteria were confined in
flattened drops: the bacteria at the interface move in opposite direction to
the bulk, which itself is advected by the fluid Wioland2013 ; Lushi2014 . In
both experiments (drops and racetracks), we found that the net circulation
breaks down around $70\,\mu$m, the typical size of swirls in unconfined
chambers Dunkel2013  and also the critical diameter of circular drops below
which bacterial motion stabilizes into one vortex Wioland2013 .

Microfluidic channels have been used previously to study the behaviour of
micro-swimmers, notably to sort bacteria E. coli by length Hulme2008  or
direct spermatozoa Denissenko2012 ; Guidobaldi2014  by making use of
ratcheting channels. The curves of the micro-channels in those cases are
designed to guide the swimmers in one specified direction. Here we used
straight walls, such that the spontaneously emerging stream can occur in
either direction with equal probability. As an extension of our work, one
could design a circular channel with ratcheted surfaces to direct the
suspension stream in a chosen left or right direction.

Dense bacterial suspensions have been extensively studied without confinement
and yet their self-organisation presents new challenges and surprises. The
experiments presented here not only give insight into the effect of
confinement but also help in understanding the general behaviour of such
suspensions. In particular, we have shown that the correlation function could
be analysed as two parts: an oscillating and a decaying term. To our
knowledge, other studies have described only one of the two, fitting the
vortex size or the persistence length. Here confinement into a racetrack
stabilises the suspension to form a more regular pattern, emphasising this
structure. With the added insight of our work, some previous results could be
reinterpreted: for example, Gachelin et al.Gachelin2013  have shown the
persistence length of an E. coli suspension to increase with the density,
which translates to a slower decay of the correlation function $C(s)$ at
higher densities. However the first zero of the $C(s)$ occurred, over all
their measurements, at the constant value $s^{\ast}\approx 55\,\mu$m,
indicating that the swirl size does not depend on the concentration but is an
intrinsic length scale associated with the swimming bacteria.

Finally, this confinement setup opens new avenues for the study of collective
behaviours in more complex natural or artificial environments, that could
include networks of channels, various solid or fluid interfaces, and chemical
attractants. Notably, Wilking et al. showed B. subtilis biofilms to form
channels, on average $90\,\mu$m in diameter Wilking2013 , comparable with our
critical width $w^{\ast}\approx 70\,\mu$m. Although the primary role of these
channels is to transport liquid and nutrients, they could also direct the
migration of single or groups of swimming bacteria. A better understanding of
the interactions and how they drive phenomena in these systems can guide us
into better control of collective bacterial behaviour and possible use in
technological applications.

## Acknowledgments

This work was supported in part by the European Research Council Advanced
Investigator Grant 247333 (H.W. and R.E.G.) and the National Science
Foundation grant CBET-1544196 (E.L).

## References

## References

  * (1) Ben-Jacob E, Becker I, Shapira Y and Levine H 2004 Trends Microbiol. 12 366-372
  * (2) Ariel G, Shklarsh A, Kalisman O, Ingham C and Ben-Jacob E 2013 New J. Phys. 15 125019-125036
  * (3) Mah T F C and O’Toole G A 2001 Trends Microbiol. 9 34-39
  * (4) Wilking J N, Zaburdaev V, De Volder M, Losick R, Brenner M P and Weitz D A 2013 Proc. Natl. Acad. Sci. U.S.A. 110 848-852
  * (5) Shimkets L J 1999 Annu. Rev. Microbiol. 53 525-549
  * (6) Branda S S, González-Pastor J E, Ben-Yehuda S, Losick R and Kolter R 2001 Proc. Natl. Acad. Sci. U.S.A. 98 11621-11626
  * (7) González-Pastor J E 2011 FEMS Microbiol. Rev. 35 415-424
  * (8) Grant M A, Waclaw B, Allen R J and Cicuta P. 2014 J. R. Soc. Interface 11 20140400
  * (9) Dombrowski C, Cisneros L, Chatkaew S, Goldstein R E and Kessler J O 2004 Phys. Rev. Lett. 93 098103
  * (10) Cisneros L H, Kessler J O, Ganguly S and Goldstein R E 2011 Phys. Rev. E 83 061907
  * (11) Sokolov A and Aranson I 2012 Phys. Rev. Lett. 109 248109
  * (12) Aranson I 2013 Physics 6 61
  * (13) Gachelin J, Rousselet A, Lindner A and Clement E 2014 New J. Phys. 16 025003-025011
  * (14) Wioland H, Woodhouse F G, Dunkel J, Kessler J O and Goldstein R E 2013 Phys. Rev. Lett. 110 268102-268106
  * (15) Lushi E, Wioland H and Goldstein R E 2014 Proc. Natl. Acad. Sci. U.S.A. 111 9733-9738
  * (16) Wioland H, Woodhouse F G, Dunkel J and Goldstein R E 2016 Nat. Phys.
  * (17) Nédélec F, Surrey T, Maggs A and Leibler S 1997 Nature 389 305
  * (18) Schaller V, Weber C, Semmrich C, Frey E and Bausch A R 2010 Nature 467 73-77
  * (19) Underhill P T, Hernandez-Ortiz J P and Graham M D 2008 Phys. Rev. Lett. 100 248101
  * (20) Saintillan D and Shelley M J 2008 Phys. Rev. Lett. 100 178103
  * (21) Voituriez R, Joanny J F and Prost J 2005 Europhys. Lett. 70 404
  * (22) Fielding S M, Marenduzzo D and Cates M E 2011 Phys Rev E 83, 041910-041923
  * (23) Woodhouse F G and Goldstein R E 2012 Phys. Rev. Lett. 109 168105
  * (24) Ravnik M and Yeomans J M 2013 Phys. Rev. Lett. 110 026001-026005
  * (25) Neef M and Kruse K 2014 Phys. Rev. E 90 052703
  * (26) Tsang A C H and Kanso E 2016 Phys. Rev. Lett. 116 048101
  * (27) Bricard A, Caussin J B, Desreumaux N, Dauchot O and Bartolo D 2013 Nature 503 95-98
  * (28) Kumar N, Soni H, Ramaswamy S and Sood AK 2014 Nat. Comm. 5 4688
  * (29) Vladescu I D, EMarsden E J, Schwarz-Linek J, Martinez V A, Arlt J, Morozov A N, Marenduzzo D, Cates M E and Poon W C K 2014 Phys. Rev. Lett. 113, 268101
  * (30) Bricard A, Caussin J-B, Das D, Savoie C, Chikkadi V, Shitara K, Chepizhko O, Peruani F, Saintillan D and Bartolo D 2015 Nat. Comm. 6 7470
  * (31) Ramaswamy S 2010 Annu. Rev. Condens. Matter Phys. 1 323
  * (32) Pedley T J and Kessler J O 1992 Annu. Rev. Fluid Mech. 24 313
  * (33) Drescher K, Dunkel J, Cisneros LH, Ganguly S and Goldstein R E 2011 Proc. Natl. Acad. Sci. U.S.A. 108 10940-10954
  * (34) Rusconi R, Guasto J S and Stocker R 2014 Nat. Phys. 10 212-217
  * (35) Kantsler V, Dunkel J, Polin M and Goldstein R E 2013 Proc. Natl. Acad. Sci. U.S.A. 110 1187-1192
  * (36) Dunkel J, Heidenreich S, Drescher K, Wensink H H, Bär M and Goldstein R E 2013 Phys. Rev. Lett. 110 228102
  * (37) Kim M J and Breuer K S 2008 Small 4 111-118
  * (38) Hill J, Kalkanci O, McMurry J L and Koser H 2007 Phys. Rev. Lett. 98 068101-068104
  * (39) Kaya T and Koser H 2012 Biophys. J. 102 1514-1523
  * (40) Marcos, Fu H C, Powers T R and Stocker R 2012 Proc. Natl. Acad. Sci. U.S.A. 109 4780-4785
  * (41) Kantsler V, Dunkel J, Blayney M and Goldstein R E 2014 eLife 3 e02403
  * (42) Mori N and Chang K-A 2003 Introduction to MPIV. User reference manual, 14p Available from: http://www.oceanwave.jp/softwares/mpiv
  * (43) Guttenplan S B, Shaw S and Kearns D B 2013 Mol. Microbiol. 87 211-229
  * (44) Lushi E and Peskin C S 2013 Comput. Struct. 122 239-248
  * (45) Costanzo A, Di Leonardo R, Ruocco G and Angelani L 2012 J. Phys.: Condens. Matter 24 065101-065108
  * (46) Brody J P, Yager P, Goldstein R E and Austin R H 1996 Biophys. J. 71 3430-3441
  * (47) Wensink H H and Lowen H 2008 Phys. Rev. E 78 031409
  * (48) Cisneros L H, Cortez R, Dombrowski C, Goldstein R E and Kessler J O 2007 Exp. Fluids 43 737-753
  * (49) Buhl J, Sumpter D J, Couzin I D, Hale J J, Despland E, Miller E R and Simpson S J 2006 Science 312 1402-1406
  * (50) Menzel A 2013 J. Phys.: Condens. Matter 25 505103
  * (51) Darnton N, Turner L, Breuer K and Berg H C 2004 Biophys. J. 86 1863-1870
  * (52) Hulme S E, DiLuzio W R, Shevkoplyas S S, Turner L, Mayer M, Berg H C and Whitesides G M 2008 Lab Chip 8 1888-1895
  * (53) Denissenko P, Kantsler V, Smith D J and Kirkman-Brown J 2012 Proc. Natl. Acad. Sci. U.S.A. 109 8007-8010
  * (54) Guidobaldi A, Jeyaram Y, Berdakin I, Moshchalkov V V, Condat C A, Marconi V I, Giojalas L and Silhanek A V 2014 Phys. Rev. E 89 032720