1
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Royzenblat SK, Freddolino L. Spatio-temporal organization of the E. coli chromosome from base to cellular length scales. EcoSal Plus 2024:eesp00012022. [PMID: 38864557 DOI: 10.1128/ecosalplus.esp-0001-2022] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2023] [Accepted: 04/17/2024] [Indexed: 06/13/2024]
Abstract
Escherichia coli has been a vital model organism for studying chromosomal structure, thanks, in part, to its small and circular genome (4.6 million base pairs) and well-characterized biochemical pathways. Over the last several decades, we have made considerable progress in understanding the intricacies of the structure and subsequent function of the E. coli nucleoid. At the smallest scale, DNA, with no physical constraints, takes on a shape reminiscent of a randomly twisted cable, forming mostly random coils but partly affected by its stiffness. This ball-of-spaghetti-like shape forms a structure several times too large to fit into the cell. Once the physiological constraints of the cell are added, the DNA takes on overtwisted (negatively supercoiled) structures, which are shaped by an intricate interplay of many proteins carrying out essential biological processes. At shorter length scales (up to about 1 kb), nucleoid-associated proteins organize and condense the chromosome by inducing loops, bends, and forming bridges. Zooming out further and including cellular processes, topological domains are formed, which are flanked by supercoiling barriers. At the megabase-scale both large, highly self-interacting regions (macrodomains) and strong contacts between distant but co-regulated genes have been observed. At the largest scale, the nucleoid forms a helical ellipsoid. In this review, we will explore the history and recent advances that pave the way for a better understanding of E. coli chromosome organization and structure, discussing the cellular processes that drive changes in DNA shape, and what contributes to compaction and formation of dynamic structures, and in turn how bacterial chromatin affects key processes such as transcription and replication.
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Affiliation(s)
- Sonya K Royzenblat
- Cellular and Molecular Biology Program, University of Michigan Medical School, Ann Arbor, Michigan, USA
| | - Lydia Freddolino
- Cellular and Molecular Biology Program, University of Michigan Medical School, Ann Arbor, Michigan, USA
- Department of Biological Chemistry, University of Michigan Medical School, Ann Arbor, Michigan, USA
- Department of Computational Medicine & Bioinformatics, University of Michigan Medical School, Ann Arbor, Michigan, USA
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2
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Nieto C, Vargas-García CA, Pedraza JM, Singh A. Mechanisms of cell size regulation in slow-growing Escherichia coli cells: discriminating models beyond the adder. NPJ Syst Biol Appl 2024; 10:61. [PMID: 38811603 PMCID: PMC11137094 DOI: 10.1038/s41540-024-00383-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2023] [Accepted: 05/09/2024] [Indexed: 05/31/2024] Open
Abstract
Under ideal conditions, Escherichia coli cells divide after adding a fixed cell size, a strategy known as the adder. This concept applies to various microbes and is often explained as the division that occurs after a certain number of stages, associated with the accumulation of precursor proteins at a rate proportional to cell size. However, under poor media conditions, E. coli cells exhibit a different size regulation. They are smaller and follow a sizer-like division strategy where the added size is inversely proportional to the size at birth. We explore three potential causes for this deviation: degradation of the precursor protein and two models where the propensity for accumulation depends on the cell size: a nonlinear accumulation rate, and accumulation starting at a threshold size termed the commitment size. These models fit the mean trends but predict different distributions given the birth size. To quantify the precision of the models to explain the data, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. We found that none of the models alone can consistently explain the data. However, the degradation model better explains the division strategy when cells are larger, whereas size-related models (power-law and commitment size) account for smaller cells. Our methodology proposes a data-based method in which different mechanisms can be tested systematically.
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Affiliation(s)
- César Nieto
- Department of Physics, Universidad de los Andes, Bogotá, Colombia
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, 19716, USA
| | | | | | - Abhyudai Singh
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, 19716, USA.
- Department of Electrical and Computer Engineering, Biomedical Engineering, Mathematical Sciences, Center of Bioinformatic and Computational Biology, University of Delaware, Newark, DE, 19716, USA.
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3
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ElGamel M, Mugler A. Effects of Molecular Noise on Cell Size Control. PHYSICAL REVIEW LETTERS 2024; 132:098403. [PMID: 38489620 DOI: 10.1103/physrevlett.132.098403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2023] [Accepted: 02/12/2024] [Indexed: 03/17/2024]
Abstract
Cells employ control strategies to maintain a stable size. Dividing at a target size (the "sizer" strategy) is thought to produce the tightest size distribution. However, this result follows from phenomenological models that ignore the molecular mechanisms required to implement the strategy. Here we investigate a simple mechanistic model for exponentially growing cells whose division is triggered at a molecular abundance threshold. We find that size noise inherits the molecular noise and is consequently minimized not by the sizer but by the "adder" strategy, where a cell divides after adding a target amount to its birth size. We derive a lower bound on size noise that agrees with publicly available data from six microfluidic studies on Escherichia coli bacteria.
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Affiliation(s)
- Motasem ElGamel
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - Andrew Mugler
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
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4
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Govers SK, Campos M, Tyagi B, Laloux G, Jacobs-Wagner C. Apparent simplicity and emergent robustness in the control of the Escherichia coli cell cycle. Cell Syst 2024; 15:19-36.e5. [PMID: 38157847 DOI: 10.1016/j.cels.2023.12.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2023] [Revised: 06/15/2023] [Accepted: 12/01/2023] [Indexed: 01/03/2024]
Abstract
To examine how bacteria achieve robust cell proliferation across diverse conditions, we developed a method that quantifies 77 cell morphological, cell cycle, and growth phenotypes of a fluorescently labeled Escherichia coli strain and >800 gene deletion derivatives under multiple nutrient conditions. This approach revealed extensive phenotypic plasticity and deviating mutant phenotypes were often nutrient dependent. From this broad phenotypic landscape emerged simple and robust unifying rules (laws) that connect DNA replication initiation, nucleoid segregation, FtsZ ring formation, and cell constriction to specific aspects of cell size (volume, length, or added length) at the population level. Furthermore, completion of cell division followed the initiation of cell constriction after a constant time delay across strains and nutrient conditions, identifying cell constriction as a key control point for cell size determination. Our work provides a population-level description of the governing principles by which E. coli integrates cell cycle processes and growth rate with cell size to achieve its robust proliferative capability. A record of this paper's transparent peer review process is included in the supplemental information.
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Affiliation(s)
- Sander K Govers
- Howard Hughes Medical Institute, Stanford University, Stanford, CA 94305, USA; de Duve Institute, UCLouvain, Brussels, Belgium; Department of Biology, KU Leuven, Leuven, Belgium
| | - Manuel Campos
- Centre de Biologie Intégrative de Toulouse, Laboratoire de Microbiologie et Génétique Moléculaires, Université de Toulouse, Toulouse, France
| | - Bhavyaa Tyagi
- Howard Hughes Medical Institute, Stanford University, Stanford, CA 94305, USA; Department of Biology, Stanford University, Stanford, CA 94305, USA
| | | | - Christine Jacobs-Wagner
- Howard Hughes Medical Institute, Stanford University, Stanford, CA 94305, USA; Department of Biology, Stanford University, Stanford, CA 94305, USA; Sarafan Chemistry, Engineering Medicine for Human Health Institute, Stanford University, Stanford, CA 94305, USA; Department of Microbiology and Immunology, Stanford School of Medicine, Stanford, CA 94305, USA.
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5
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Niault T, Czarnecki J, Lambérioux M, Mazel D, Val ME. Cell cycle-coordinated maintenance of the Vibrio bipartite genome. EcoSal Plus 2023; 11:eesp00082022. [PMID: 38277776 PMCID: PMC10729929 DOI: 10.1128/ecosalplus.esp-0008-2022] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2024]
Abstract
To preserve the integrity of their genome, bacteria rely on several genome maintenance mechanisms that are co-ordinated with the cell cycle. All members of the Vibrio family have a bipartite genome consisting of a primary chromosome (Chr1) homologous to the single chromosome of other bacteria such as Escherichia coli and a secondary chromosome (Chr2) acquired by a common ancestor as a plasmid. In this review, we present our current understanding of genome maintenance in Vibrio cholerae, which is the best-studied model for bacteria with multi-partite genomes. After a brief overview on the diversity of Vibrio genomic architecture, we describe the specific, common, and co-ordinated mechanisms that control the replication and segregation of the two chromosomes of V. cholerae. Particular attention is given to the unique checkpoint mechanism that synchronizes Chr1 and Chr2 replication.
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Affiliation(s)
- Théophile Niault
- Bacterial Genome Plasticity Unit, CNRS UMR3525, Institut Pasteur, Université Paris Cité, Paris, France
- Collège Doctoral, Sorbonne Université, Paris, France
| | - Jakub Czarnecki
- Bacterial Genome Plasticity Unit, CNRS UMR3525, Institut Pasteur, Université Paris Cité, Paris, France
| | - Morgan Lambérioux
- Bacterial Genome Plasticity Unit, CNRS UMR3525, Institut Pasteur, Université Paris Cité, Paris, France
- Collège Doctoral, Sorbonne Université, Paris, France
| | - Didier Mazel
- Bacterial Genome Plasticity Unit, CNRS UMR3525, Institut Pasteur, Université Paris Cité, Paris, France
| | - Marie-Eve Val
- Bacterial Genome Plasticity Unit, CNRS UMR3525, Institut Pasteur, Université Paris Cité, Paris, France
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6
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Nieto C, Vargas-García C, Pedraza JM, Singh A. Mechanisms of Cell Size Regulation in Slow-Growing Escherichia coli Cells: Discriminating Models Beyond the Adder. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.09.11.557238. [PMID: 37745550 PMCID: PMC10515837 DOI: 10.1101/2023.09.11.557238] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/26/2023]
Abstract
Under ideal conditions, Escherichia coli cells divide after adding a fixed cell size, a strategy known as the adder. This concept applies to various microbes and is often explained as the division that occurs after a certain number of stages, associated with the accumulation of precursor proteins at a rate proportional to cell size. However, under poor media conditions, E. coli cells exhibit a different size regulation. They are smaller and follow a sizer-like division strategy where the added size is inversely proportional to the size at birth. We explore three potential causes for this deviation: precursor protein degradation, nonlinear accumulation rate, and a threshold size termed the commitment size. These models fit mean trends but predict different distributions given the birth size. To validate these models, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. the degradation model could explain the division strategy for media where cells are larger, while the commitment size model could account for smaller cells. The power-law model, finally, better fits the data at intermediate regimes.
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Affiliation(s)
- César Nieto
- Department of Physics, Universidad de los Andes, Bogotá, Colombia
- Department of Electrical and Computing Engineering, University of Delaware. Newark, DE 19716, USA
| | - César Vargas-García
- AGROSAVIA Corporación Colombiana de Investigación Agropecuaria. Mosquera. Colombia
| | | | - Abhyudai Singh
- Department of Electrical and Computing Engineering, University of Delaware. Newark, DE 19716, USA
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7
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Kohiyama M, Herrick J, Norris V. Open Questions about the Roles of DnaA, Related Proteins, and Hyperstructure Dynamics in the Cell Cycle. Life (Basel) 2023; 13:1890. [PMID: 37763294 PMCID: PMC10532879 DOI: 10.3390/life13091890] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2023] [Revised: 08/29/2023] [Accepted: 09/06/2023] [Indexed: 09/29/2023] Open
Abstract
The DnaA protein has long been considered to play the key role in the initiation of chromosome replication in modern bacteria. Many questions about this role, however, remain unanswered. Here, we raise these questions within a framework based on the dynamics of hyperstructures, alias large assemblies of molecules and macromolecules that perform a function. In these dynamics, hyperstructures can (1) emit and receive signals or (2) fuse and separate from one another. We ask whether the DnaA-based initiation hyperstructure acts as a logic gate receiving information from the membrane, the chromosome, and metabolism to trigger replication; we try to phrase some of these questions in terms of DNA supercoiling, strand opening, glycolytic enzymes, SeqA, ribonucleotide reductase, the macromolecular synthesis operon, post-translational modifications, and metabolic pools. Finally, we ask whether, underpinning the regulation of the cell cycle, there is a physico-chemical clock inherited from the first protocells, and whether this clock emits a single signal that triggers both chromosome replication and cell division.
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Affiliation(s)
- Masamichi Kohiyama
- Institut Jacques Monod, Université Paris Cité, CNRS, 75013 Paris, France;
| | - John Herrick
- Independent Researcher, 3 rue des Jeûneurs, 75002 Paris, France;
| | - Vic Norris
- CBSA UR 4312, University of Rouen Normandy, University of Caen Normandy, Normandy University, 76000 Rouen, France
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8
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ElGamel M, Vashistha H, Salman H, Mugler A. Multigenerational memory in bacterial size control. Phys Rev E 2023; 108:L032401. [PMID: 37849186 DOI: 10.1103/physreve.108.l032401] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2022] [Accepted: 07/25/2023] [Indexed: 10/19/2023]
Abstract
Cells maintain a stable size as they grow and divide. Inspired by the available experimental data, most proposed models for size homeostasis assume size-control mechanisms that act on a timescale of one generation. Such mechanisms lead to short-lived autocorrelations in size fluctuations that decay within less than two generations. However, recent evidence from comparing sister lineages suggests that correlations in size fluctuations can persist for many generations. Here we develop a minimal model that explains these seemingly contradictory results. Our model proposes that different environments result in different control parameters, leading to distinct inheritance patterns. Multigenerational memory is revealed in constant environments but obscured when averaging over many different environments. Inferring the parameters of our model from Escherichia coli size data in microfluidic experiments, we recapitulate the observed statistics. Our paper elucidates the impact of the environment on cell homeostasis and growth and division dynamics.
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Affiliation(s)
- Motasem ElGamel
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania, 15260, USA
| | - Harsh Vashistha
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania, 15260, USA
| | - Hanna Salman
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania, 15260, USA
| | - Andrew Mugler
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania, 15260, USA
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9
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Fu H, Xiao F, Jun S. Bacterial Replication Initiation as Precision Control by Protein Counting. PRX LIFE 2023; 1:013011. [PMID: 38550259 PMCID: PMC10977104 DOI: 10.1103/prxlife.1.013011] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 04/09/2024]
Abstract
Balanced biosynthesis is the hallmark of bacterial cell physiology, where the concentrations of stable proteins remain steady. However, this poses a conceptual challenge to modeling the cell-cycle and cell-size controls in bacteria, as prevailing concentration-based eukaryote models are not directly applicable. In this study, we revisit and significantly extend the initiator-titration model, proposed 30 years ago, and we explain how bacteria precisely and robustly control replication initiation based on the mechanism of protein copy-number sensing. Using a mean-field approach, we first derive an analytical expression of the cell size at initiation based on three biological mechanistic control parameters for an extended initiator-titration model. We also study the stability of our model analytically and show that initiation can become unstable in multifork replication conditions. Using simulations, we further show that the presence of the conversion between active and inactive initiator protein forms significantly represses initiation instability. Importantly, the two-step Poisson process set by the initiator titration step results in significantly improved initiation synchrony with C V ~ 1 / N scaling rather than the standard 1 / N scaling in the Poisson process, where N is the total number of initiators required for initiation. Our results answer two long-standing questions in replication initiation: (i) Why do bacteria produce almost two orders of magnitude more DnaA, the master initiator proteins, than required for initiation? (ii) Why does DnaA exist in active (DnaA-ATP) and inactive (DnaA-ADP) forms if only the active form is competent for initiation? The mechanism presented in this work provides a satisfying general solution to how the cell can achieve precision control without sensing protein concentrations, with broad implications from evolution to the design of synthetic cells.
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Affiliation(s)
- Haochen Fu
- Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
| | - Fangzhou Xiao
- Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
| | - Suckjoon Jun
- Department of Physics and Department of Molecular Biology, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA
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10
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Fu H, Xiao F, Jun S. Replication initiation in bacteria: precision control based on protein counting. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.05.26.542547. [PMID: 37292844 PMCID: PMC10246017 DOI: 10.1101/2023.05.26.542547] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Balanced biosynthesis is the hallmark of bacterial cell physiology, where the concentrations of stable proteins remain steady. However, this poses a conceptual challenge to modeling the cell-cycle and cell-size controls in bacteria, as prevailing concentration-based eukaryote models are not directly applicable. In this study, we revisit and significantly extend the initiator-titration model, proposed thirty years ago, and explain how bacteria precisely and robustly control replication initiation based on the mechanism of protein copy-number sensing. Using a mean-field approach, we first derive an analytical expression of the cell size at initiation based on three biological mechanistic control parameters for an extended initiator-titration model. We also study the stability of our model analytically and show that initiation can become unstable in multifork replication conditions. Using simulations, we further show that the presence of the conversion between active and inactive initiator protein forms significantly represses initiation instability. Importantly, the two-step Poisson process set by the initiator titration step results in significantly improved initiation synchrony with C V ~ 1 / N scaling rather than the standard 1 / N scaling in the Poisson process, where N is the total number of initiators required for initiation. Our results answer two long-standing questions in replication initiation: (1) Why do bacteria produce almost two orders of magnitude more DnaA, the master initiator proteins, than required for initiation? (2) Why does DnaA exist in active (DnaA-ATP) and inactive (DnaA-ADP) forms if only the active form is competent for initiation? The mechanism presented in this work provides a satisfying general solution to how the cell can achieve precision control without sensing protein concentrations, with broad implications from evolution to the design of synthetic cells.
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Affiliation(s)
- Haochen Fu
- Department of Physics, University of California San Diego, 9500 Gilman Dr, La Jolla, CA 92093
| | - Fangzhou Xiao
- Department of Physics, University of California San Diego, 9500 Gilman Dr, La Jolla, CA 92093
| | - Suckjoon Jun
- Department of Physics and Department of Molecular Biology, University of California San Diego, 9500 Gilman Dr, La Jolla, CA 92093
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11
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Leonard AC. Recollections of a Helmstetter Disciple. Life (Basel) 2023; 13:life13051114. [PMID: 37240759 DOI: 10.3390/life13051114] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2023] [Revised: 04/25/2023] [Accepted: 04/28/2023] [Indexed: 05/28/2023] Open
Abstract
Nearly fifty years ago, it became possible to construct E. coli minichromosomes using recombinant DNA technology. These very small replicons, comprising the unique replication origin of the chromosome oriC coupled to a drug resistance marker, provided new opportunities to study the regulation of bacterial chromosome replication, were key to obtaining the nucleotide sequence information encoded into oriC and were essential for the development of a ground-breaking in vitro replication system. However, true authenticity of the minichromosome model system required that they replicate during the cell cycle with chromosome-like timing specificity. I was fortunate enough to have the opportunity to construct E. coli minichromosomes in the laboratory of Charles Helmstetter and, for the first time, measure minichromosome cell cycle regulation. In this review, I discuss the evolution of this project along with some additional studies from that time related to the DNA topology and segregation properties of minichromosomes. Despite the significant passage of time, it is clear that large gaps in our understanding of oriC regulation still remain. I discuss some specific topics that continue to be worthy of further study.
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Affiliation(s)
- Alan C Leonard
- Department of Biological Sciences, Florida Institute of Technology, 150 W. University Blvd., Melbourne, FL 32952, USA
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12
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Kar P, Tiruvadi-Krishnan S, Männik J, Männik J, Amir A. Using conditional independence tests to elucidate causal links in cell cycle regulation in Escherichia coli. Proc Natl Acad Sci U S A 2023; 120:e2214796120. [PMID: 36897981 PMCID: PMC10089181 DOI: 10.1073/pnas.2214796120] [Citation(s) in RCA: 7] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2022] [Accepted: 01/09/2023] [Indexed: 03/12/2023] Open
Abstract
How cells regulate their cell cycles is a central question for cell biology. Models of cell size homeostasis have been proposed for bacteria, archaea, yeast, plant, and mammalian cells. New experiments bring forth high volumes of data suitable for testing existing models of cell size regulation and proposing new mechanisms. In this paper, we use conditional independence tests in conjunction with data of cell size at key cell cycle events (birth, initiation of DNA replication, and constriction) in the model bacterium Escherichia coli to select between the competing cell cycle models. We find that in all growth conditions that we study, the division event is controlled by the onset of constriction at midcell. In slow growth, we corroborate a model where replication-related processes control the onset of constriction at midcell. In faster growth, we find that the onset of constriction is affected by additional cues beyond DNA replication. Finally, we also find evidence for the presence of additional cues triggering initiations of DNA replication apart from the conventional notion where the mother cells solely determine the initiation event in the daughter cells via an adder per origin model. The use of conditional independence tests is a different approach in the context of understanding cell cycle regulation and it can be used in future studies to further explore the causal links between cell events.
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Affiliation(s)
- Prathitha Kar
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA02134
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA02138
| | | | - Jaana Männik
- Department of Physics and Astronomy, University of Tennessee, Knoxville, TN37996
| | - Jaan Männik
- Department of Physics and Astronomy, University of Tennessee, Knoxville, TN37996
| | - Ariel Amir
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA02134
- Department of Complex Systems, Weizmann Institute of Science, Rehovot7610001, Israel
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