Majority Consensus Thresholds in Competitive Lotka-Volterra Populations
Pages 76 - 86
Abstract
One of the key challenges in synthetic biology is devising robust signaling primitives for engineered microbial consortia. In such systems, a fundamental signal amplification problem is the majority consensus problem: given a system with two input species with initial difference of Δ in population sizes, what is the probability that the system reaches a state in which only the initial majority species is present?
In this work, we consider a discrete and stochastic version of competitive Lotka-Volterra dynamics, a standard model of microbial community dynamics. We identify new threshold properties for majority consensus under different types of interference competition:
• We show that under so-called self-destructive interference competition between the two input species, majority consensus can be reached with high probability if the initial difference satisfies Δ ∈ Ω (log2 n), where n is the initial population size. This gives an exponential improvement compared to the previously known bound of [EQUATION] by Cho et al. [Distributed Computing, 2021] given for a special case of the competitive Lotka-Volterra model. In contrast, we show that an initial gap of [EQUATION] is necessary.
• On the other hand, we prove that under non-self-destructive interference competition, an initial gap of [EQUATION] is necessary to succeed with high probability and that a [EQUATION] gap is sufficient.
This shows a strong qualitative gap between the performance of self-destructive and non-self-destructive interference competition. Moreover, we show that if in addition the populations exhibit interference competition between the individuals of the same species, then majority consensus cannot always be solved with high probability, no matter what the difference in the initial population counts.
References
[1]
Dan Alistarh, James Aspnes, David Eisenstat, Rati Gelashvili, and Ronald L. Rivest. 2017. Time-space trade-offs in population protocols. In Proc. 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 2560--2579.
[2]
Dan Alistarh, James Aspnes, and Rati Gelashvili. 2018. Space-optimal majority in population protocols. In Proc. 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA).
[3]
Dan Alistarh and Rati Gelashvili. 2018. Recent algorithmic advances in population protocols. ACM SIGACT News (2018).
[4]
Dan Alistarh, Rati Gelashvili, and Milan Vojnović. 2015. Fast and exact majority in population protocols. In Proc. 2015 ACM Symposium on Principles of Distributed Computing (PODC). 47--56.
[5]
Razan N. Alnahhas, Mehdi Sadeghpour, Ye Chen, Alexis A. Frey, William Ott, Krešimir Josić, and Matthew R. Bennett. 2020. Majority sensing in synthetic microbial consortia. Nature Communications 11, 1 (2020), 1--10.
[6]
Victoria Andaur, Janna Burman, Matthias Függer, Manish Kushwaha, Bilal Manssouri, Thomas Nowak, and Joel Rybicki. 2021. Reaching Agreement in Competitive Microbial Systems. arXiv:2103.07450 [cs.DC]
[7]
Dana Angluin, James Aspnes, Zoë Diamadi, Michael J. Fischer, and René Peralta. 2006. Computation in networks of passively mobile finite-state sensors. Distributed Computing 18, 4 (2006), 235--253.
[8]
Dana Angluin, James Aspnes, and David Eisenstat. 2008. A simple population protocol for fast robust approximate majority. Distributed Computing 21, 2 (2008), 87--102.
[9]
Gregor Bankhamer, Petra Berenbrink, Felix Biermeier, Robert Elsässer, Hamed Hosseinpour, Dominik Kaaser, and Peter Kling. 2022. Population protocols for exact plurality consensus. In Proc. 2022 ACM Symposium on Principles of Distributed Computing (PODC).
[10]
Yaakov Benenson. 2012. Biomolecular computing systems: principles, progress and potential. Nature Reviews Genetics 13, 7 (2012), 455--468.
[11]
Petra Berenbrink, George Giakkoupis, and Peter Kling. 2020. Optimal time and space leader election in population protocols. In Proc. 52nd ACM Symposium on Theory of Computing (STOC).
[12]
Philip Bittihn, M. Omar Din, Lev S. Tsimring, and Jeff Hasty. 2018. Rational engineering of synthetic microbial systems: from single cells to consortia. Current Opinion in Microbiology 45 (2018), 92--99.
[13]
Andrew J Black and Alan J McKane. 2012. Stochastic formulation of ecological models and their applications. Trends in ecology & evolution 27, 6 (2012), 337--345.
[14]
Pierre Brémaud. 1999. Markov Chains. Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, Heidelberg.
[15]
Katie Brenner, Lingchong You, and Frances H. Arnold. 2008. Engineering micro-bial consortia: a new frontier in synthetic biology. Trends in Biotechnology 26, 9 (2008), 483--489.
[16]
D. Ewen Cameron, Caleb J. Bashor, and James J. Collins. 2014. A brief history of synthetic biology. Nature Reviews Microbiology 12, 5 (2014), 381--390.
[17]
Luca Cardelli and Attila Csikász-Nagy. 2012. The cell cycle switch computes approximate majority. Scientific reports 2, 1 (2012), 656.
[18]
Bernard Chazelle. 2012. Natural algorithms and influence systems. Commun. ACM 55, 12 (2012), 101--110.
[19]
Da-Jung Cho, Matthias Függer, Corbin Hopper, Manish Kushwaha, Thomas Nowak, and Quentin Soubeyran. 2021. Distributed computation with continual population growth. Distributed Computing (2021), 1--23.
[20]
Anne Condon, Monir Hajiaghayi, David Kirkpatrick, and Ján Maňuch. 2020. Approximate majority analyses using tri-molecular chemical reaction networks. Natural Computing 19, 1 (2020), 249--270.
[21]
Colin Cooper, Robert Elsässer, and Tomasz Radzik. 2014. The power of two choices in distributed voting. In International Colloquium on Automata, Languages, and Programming. Springer, 435--446.
[22]
Jurek Czyzowicz, Leszek Gąsieniec, Adrian Kosowski, Evangelos Kranakis, Paul G. Spirakis, and Przemysław Uznański. 2022. On convergence and threshold properties of discrete Lotka-Volterra population protocols. J. Comput. System Sci. 130 (2022), 1--25.
[23]
Tal Danino, Octavio Mondragón-Palomino, Lev Tsimring, and Jeff Hasty. 2010. A synchronized quorum of genetic clocks. Nature 463, 7279 (2010), 326--330.
[24]
Sandra Dedrick, Vaishnavi Warrier, Katherine P Lemon, and Babak Momeni. 2023. When does a Lotka-Volterra model represent microbial interactions? Insights from in vitro nasal bacterial communities. Msystems (2023), e00757--22.
[25]
Domitilla Del Vecchio, Yili Qian, Richard M. Murray, and Eduardo D. Sontag. 2018. Future systems and control research in synthetic biology. Annual Reviews in Control 45 (2018), 5--17.
[26]
Alexander Dobrinevski and Erwin Frey. 2012. Extinction in neutrally stable stochastic Lotka-Volterra models. Physical Review E 85, 5 (2012), 051903.
[27]
David Doty. 2012. Theory of algorithmic self-assembly. Commun. ACM (2012).
[28]
David Doty, Mahsa Eftekhari, Leszek Gąsieniec, Eric Severson, Przemyslaw Uznański, and Grzegorz Stachowiak. 2022. A time and space optimal stable population protocol solving exact majority. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 1044--1055.
[29]
Moez Draief and Milan Vojnović. 2012. Convergence speed of binary interval consensus. SIAM Journal on Control and Optimization 50, 3 (2012), 1087--1109.
[30]
Michael B. Elowitz and Stanislas Leibler. 2000. A synthetic oscillatory network of transcriptional regulators. Nature 403, 6767 (2000), 335--338.
[31]
Ofer Feinerman, Bernhard Haeupler, and Amos Korman. 2017. Breathe before speaking: efficient information dissemination despite noisy, limited and anonymous communication. Distributed Computing 30, 5 (2017), 339--355.
[32]
Ofer Feinerman and Amos Korman. 2013. Theoretical distributed computing meets biology: A review. In Proc. 9th International Conference on Distributed Computing and Internet Technology (ICDIT).
[33]
Jasmin Fisher, David Harel, and Thomas A. Henzinger. 2011. Biology as reactivity. Commun. ACM 54, 10 (2011), 72--82.
[34]
Matthias Függer, Thomas Nowak, and Joel Rybicki. 2024. Majority consensus thresholds in competitive Lotka-Volterra populations. arXiv:2405.03568 [cs.DC]
[35]
Timothy S. Gardner, Charles R. Cantor, and James J. Collins. 2000. Construction of a genetic toggle switch in Escherichia coli. Nature 403, 6767 (2000), 339--342.
[36]
Stefan Geritz and Éva Kisdi. 2012. Mathematical ecology: why mechanistic models? Journal of mathematical biology 65, 6-7 (2012), 1411.
[37]
Daniel T. Gillespie. 1977. Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81, 25 (1977), 2340--2361.
[38]
Didier Gonze, Katharine Z. Coyte, Leo Lahti, and Karoline Faust. 2018. Microbial communities as dynamical systems. Current Opinion in Microbiology 44 (2018), 41--49.
[39]
Elisa T. Granato, Thomas A. Meiller-Legrand, and Kevin R. Foster. 2019. The evolution and ecology of bacterial warfare. Current Biology 29, 11 (2019), R521--R537.
[40]
Nicolas E. Grandel, Kiara Reyes Gamas, and Matthew R. Bennett. 2021. Control of synthetic microbial consortia in time, space, and composition. Trends in Microbiology (2021).
[41]
Lewis Grozinger, Martyn Amos, Thomas E. Gorochowski, Pablo Carbonell, Diego A. Oyarzún, Ruud Stoof, Harold Fellermann, Paolo Zuliani, Huseyin Tas, and Angel Goñi-Moreno. 2019. Pathways to cellular supremacy in biocomputing. Nature Communications (2019).
[42]
Jiliang Hu, Daniel R Amor, Matthieu Barbier, Guy Bunin, and Jeff Gore. 2022. Emergent phases of ecological diversity and dynamics mapped in microcosms. Science 378, 6615 (2022), 85--89.
[43]
Weini Huang, Christoph Hauert, and Arne Traulsen. 2015. Stochastic game dynamics under demographic fluctuations. Proceedings of the National Academy of Sciences 112, 29 (2015), 9064--9069.
[44]
Peter Jagers. 2010. A plea for stochastic population dynamics. Journal of Mathematical Biology 60 (2010), 761--764.
[45]
Behzad D. Karkaria, Neythen J. Treloar, Chris P. Barnes, and Alex J. H. Fedorec. 2020. From microbial communities to distributed computing systems. Frontiers in Bioengineering and Biotechnology (2020).
[46]
Russell Lande, Steinar Engen, and Bernt-Erik Saether. 2003. Stochastic population dynamics in ecology and conservation. Oxford University Press, USA.
[47]
Anselm Levskaya, Aaron A. Chevalier, Jeffrey J. Tabor, Zachary Booth Simpson, Laura A. Lavery, Matthew Levy, Eric A. Davidson, Alexander Scouras, Andrew D. Ellington, Edward M. Marcotte, and Christopher A. Voigt. 2005. Engineering Escherichia coli to see light. Nature 438, 7067 (2005), 441--442.
[48]
Shuyao Li, Jing Xiao, Tianzheng Sun, Fangjian Yu, Kaihang Zhang, Yuantao Feng, Chenchao Xu, Baojun Wang, and Lei Cheng. 2022. Synthetic microbial consortia with programmable ecological interactions. Methods in Ecology and Evolution 13, 7 (2022), 1608--1621.
[49]
Javier Macía, Francesc Posas, and Ricard V. Solé. 2012. Distributed computation: the new wave of synthetic biology devices. Trends in Biotechnology 30, 6 (2012), 342 -- 349.
[50]
Junwen Mao, Andrew E Blanchard, and Ting Lu. 2015. Slow and steady wins the race: a bacterial exploitative competition strategy in fluctuating environments. ACS Synthetic Biology 4, 3 (2015), 240--248.
[51]
John P Marken and Richard M Murray. 2023. Addressable and adaptable intercellular communication via DNA messaging. Nature Communications 14, 1 (2023), 2358.
[52]
Alan J McKane and Timothy J Newman. 2004. Stochastic models in population biology and their deterministic analogs. Physical Review E 70, 4 (2004), 041902.
[53]
George B. Mertzios, Sotiris E. Nikoletseas, Christoforos Raptopoulos, and Paul G. Spirakis. 2014. Determining Majority in Networks with Local Interactions and Very Small Local Memory. In Proc. 41st International Colloquium on Automata, Languages, and Programming (ICALP). 871--882.
[54]
Saket Navlakha and Ziv Bar-Joseph. 2014. Distributed information processing in biological and computational systems. Commun. ACM (2014).
[55]
Sabrina Rashid, Gadi Taubenfeld, and Ziv Bar-Joseph. 2021. The epigenetic consensus problem. In International Colloquium on Structural Information and Communication Complexity. Springer, 146--163.
[56]
Sergi Regot, Javier Macia, Núria Conde, Kentaro Furukawa, Jimmy Kjellén, Tom Peeters, Stefan Hohmann, Eulãlia De Nadal, Francesc Posas, and Ricard Solé. 2011. Distributed biological computation with multicellular engineered networks. Nature (2011).
[57]
Tobias Reichenbach, Mauro Mobilia, and Erwin Frey. 2006. Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model. Physical Review E 74, 5 (2006), 051907.
[58]
Bruno Sericola. 2013. Birth-and-Death Processes. Markov Chains: Theory, Algorithms and Applications (2013), 191--233.
[59]
Alvin Tamsir, Jeffrey J Tabor, and Christopher A Voigt. 2011. Robust multicellular computing using genetically encoded NOR gates and chemical 'wires'. Nature 469, 7329 (2011), 212--215.
[60]
Darren J Wilkinson. 2018. Stochastic Modelling for Systems Biology. CRC press.
[61]
Jizhong Zhou and Daliang Ning. 2017. Stochastic community assembly: does it matter in microbial ecology? Microbiology and Molecular Biology Reviews 81, 4 (2017), 10--1128.
Index Terms
- Majority Consensus Thresholds in Competitive Lotka-Volterra Populations
Recommendations
Dynamics of two-element populations in the space of population states
An analysis of population dynamics in the space of population states is presented. The simplest case of the phenotypic evolution-a population consisting of two individuals with one real-valued trait, evolving under proportional selection and mutation ...
Reduction to a Single Closed Equation for 2-by-2 Reaction-Diffusion Systems of Lotka--Volterra Type
We consider general models of coupled reaction-diffusion systems for interacting variants of a species. When the total population becomes large with intensive competition, we prove that the frequency (i.e., proportion) of a variant can be approached by ...
Evolutionary and population dynamics of 3 parents differential evolution (3PDE) using self-adaptive tuning methodologies
Differential Evolution is known for its simplicity and effectiveness as an evolutionary optimizer. In recent years, many researchers have focused on the exploration of Differential Evolution (DE). The objective of this paper is to show the evolutionary ...
Comments
Information & Contributors
Information
Published In
June 2024
570 pages
ISBN:9798400706684
DOI:10.1145/3662158
- Chair:
- Ran Gelles,
- Proceedings Chair:
- Dennis Olivetti,
- Program Chair:
- Petr Kuznetsov
This work is licensed under a Creative Commons Attribution International 4.0 License.
Sponsors
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 17 June 2024
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- Agence Nationale de la Recherche
Conference
Acceptance Rates
Overall Acceptance Rate 740 of 2,477 submissions, 30%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 51Total Downloads
- Downloads (Last 12 months)51
- Downloads (Last 6 weeks)31
Reflects downloads up to 30 Aug 2024
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in