1
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Kirby D. Reply to Xiao and Galstyan: Reliable ligand discrimination involves many trade-offs. Proc Natl Acad Sci U S A 2024; 121:e2411465121. [PMID: 39052852 DOI: 10.1073/pnas.2411465121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 07/27/2024] Open
Affiliation(s)
- Duncan Kirby
- Department of Physics, University of Toronto, Toronto, ON, M5S 1A7, Canada
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2
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Yin R, Wang Q, Barkai E. Instability in the quantum restart problem. Phys Rev E 2024; 109:064150. [PMID: 39020895 DOI: 10.1103/physreve.109.064150] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2024] [Accepted: 06/06/2024] [Indexed: 07/20/2024]
Abstract
Repeatedly monitored quantum walks with a rate 1/τ yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random-walk process. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus τ, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time is controlled not only by the restart time, as in classical problems, but also by the sampling time τ. We provide numerical evidence that this global minimum occurs for the τ minimizing the mean hitting time, given restarts taking place after each measurement. Last, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time τ.
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3
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Barkai E, Flaquer-Galmés R, Méndez V. Ergodic properties of Brownian motion under stochastic resetting. Phys Rev E 2023; 108:064102. [PMID: 38243500 DOI: 10.1103/physreve.108.064102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2023] [Accepted: 09/26/2023] [Indexed: 01/21/2024]
Abstract
We study the ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin- or fat-tailed distributions the normalized or non-normalized invariant density of this process. The former case corresponds to known results in the resetting literature and the latter to infinite ergodic theory. Two types of ergodic transitions are found in this system. The first is when the mean waiting time between resets diverges, when standard ergodic theory switches to infinite ergodic theory. The second is when the mean of the square root of time between resets diverges and the properties of the invariant density are drastically modified. We then find a fractional integral equation describing the density of particles. This finite time tool is particularly useful close to the ergodic transition where convergence to asymptotic limits is logarithmically slow. Our study implies rich ergodic behaviors for this nonequilibrium process which should hold far beyond the case of Brownian motion analyzed here.
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Affiliation(s)
- E Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
| | - R Flaquer-Galmés
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
| | - V Méndez
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
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4
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Yin R, Barkai E. Restart Expedites Quantum Walk Hitting Times. PHYSICAL REVIEW LETTERS 2023; 130:050802. [PMID: 36800468 DOI: 10.1103/physrevlett.130.050802] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2022] [Accepted: 12/22/2022] [Indexed: 06/18/2023]
Abstract
Classical first-passage times under restart are used in a wide variety of models, yet the quantum version of the problem still misses key concepts. We study the quantum hitting time with restart using a monitored quantum walk. The restart strategy eliminates the problem of dark states, i.e., cases where the particle evades detection, while maintaining the ballistic propagation which is important for a fast search. We find profound effects of quantum oscillations on the restart problem, namely, a type of instability of the mean detection time, and optimal restart times that form staircases, with sudden drops as the rate of sampling is modified. In the absence of restart and in the Zeno limit, the detection of the walker is not possible, and we examine how restart overcomes this well-known problem, showing that the optimal restart time becomes insensitive to the sampling period.
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Affiliation(s)
- R Yin
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - E Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
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5
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Diffusion control in biochemical specificity. Biophys J 2022; 121:1541-1548. [PMID: 35278424 PMCID: PMC9072584 DOI: 10.1016/j.bpj.2022.03.005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2021] [Revised: 09/28/2021] [Accepted: 03/04/2022] [Indexed: 11/20/2022] Open
Abstract
Biochemical specificity is critical in enzyme function, evolution, and engineering. Here we employ an established kinetic model to dissect the effects of reactant geometry and diffusion on product formation speed and accuracy in the presence of cognate (correct) and near-cognate (incorrect) substrates. Using this steady-state model for spherical geometries, we find that, for distinct kinetic regimes, the speed and accuracy of the reactions are optimized on different regions of the geometric landscape. From this model we deduce that accuracy can be strongly dependent on reactant geometric properties even for chemically limited reactions. Notably, substrates with a specific geometry and reactivity can be discriminated by the enzyme with higher efficacy than others through purely diffusive effects. For similar cognate and near-cognate substrate geometries (as is the case for polymerases or the ribosome), we observe that speed and accuracy are maximized in opposing regions of the geometric landscape. We also show that, in relevant environments, diffusive effects on accuracy can be substantial even far from extreme kinetic conditions. Finally, we find how reactant chemical discrimination and diffusion can be related to simultaneously optimize steady-state flux and accuracy. These results highlight how diffusion and geometry can be employed to enhance reaction speed and discrimination, and similarly how they impose fundamental restraints on these quantities.
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6
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Bel G, Zilman A, Kolomeisky AB. Different time scales in dynamic systems with multiple outcomes. J Chem Phys 2020; 153:054107. [PMID: 32770919 DOI: 10.1063/5.0018558] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Stochastic biochemical and transport processes have various final outcomes, and they can be viewed as dynamic systems with multiple exits. Many current theoretical studies, however, typically consider only a single time scale for each specific outcome, effectively corresponding to a single-exit process and assuming the independence of each exit process. However, the presence of other exits influences the statistical properties and dynamics measured at any specific exit. Here, we present theoretical arguments to explicitly show the existence of different time scales, such as mean exit times and inverse exit fluxes, for dynamic processes with multiple exits. This implies that the statistics of any specific exit dynamics cannot be considered without taking into account the presence of other exits. Several illustrative examples are described in detail using analytical calculations, mean-field estimates, and kinetic Monte Carlo computer simulations. The underlying microscopic mechanisms for the existence of different time scales are discussed. The results are relevant for understanding the mechanisms of various biological, chemical, and industrial processes, including transport through channels and pores.
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Affiliation(s)
- G Bel
- Department of Solar Energy and Environmental Physics, BIDR, and Department of Physics, Ben-Gurion University of the Negev, Sede Boqer Campus 8499000, Israel
| | - A Zilman
- Department of Physics and Institute for Biomaterials and Bioengineering, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
| | - A B Kolomeisky
- Department of Chemistry and Center for Theoretical Biological Physics, Department of Chemical and Biomolecular Engineering, Department of Physics and Astronomy, Rice University, Houston, Texas 77005-1892, USA
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7
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Schnoerr D, Cseke B, Grima R, Sanguinetti G. Efficient Low-Order Approximation of First-Passage Time Distributions. PHYSICAL REVIEW LETTERS 2017; 119:210601. [PMID: 29219406 DOI: 10.1103/physrevlett.119.210601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2017] [Indexed: 06/07/2023]
Abstract
We consider the problem of computing first-passage time distributions for reaction processes modeled by master equations. We show that this generally intractable class of problems is equivalent to a sequential Bayesian inference problem for an auxiliary observation process. The solution can be approximated efficiently by solving a closed set of coupled ordinary differential equations (for the low-order moments of the process) whose size scales with the number of species. We apply it to an epidemic model and a trimerization process and show good agreement with stochastic simulations.
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Affiliation(s)
- David Schnoerr
- School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, United Kingdom
| | - Botond Cseke
- Microsoft Research, Cambridge CB1 2FB, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, United Kingdom
| | - Guido Sanguinetti
- School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, United Kingdom
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8
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Michel D, Boutin B, Ruelle P. The accuracy of biochemical interactions is ensured by endothermic stepwise kinetics. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2016; 121:35-44. [PMID: 26867859 DOI: 10.1016/j.pbiomolbio.2016.02.001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2016] [Accepted: 02/01/2016] [Indexed: 11/18/2022]
Abstract
The discerning behavior of living systems relies on accurate interactions selected from the lot of molecular collisions occurring in the cell. To ensure the reliability of interactions, binding partners are classically envisioned as finely preadapted molecules, selected on the basis of their affinity in one-step associations. But the counterselection of inappropriate interactions can in fact be much more efficiently obtained through difficult multi-step adjustment, whose final high energy state is locked by a fluctuation ratchet. The progressive addition of molecular bonds during stereo-adjustment can be modeled as a predominantly backward random walk whose first arrival is frozen by a micro-irreversible transition. A new criterion of ligand specificity is presented, that is based on the ratio rejection/incorporation. In addition to its role in the selectivity of interactions, this generic recipe can underlie other important biological phenomena such as the regular synthesis at low level of supramolecular complexes, monostable kinetic bimodality, substrate concentration thresholds or the preparation of rapidly depolymerizable structures with stored energy, like microtubules.
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Affiliation(s)
- Denis Michel
- Universite de Rennes1, IRSET, Campus Santé de Villejean, 35000 Rennes, France.
| | - Benjamin Boutin
- Universite de Rennes1, Institut de Recherche Mathématiques de Rennes (IRMAR), Campus de Beaulieu Bat. 22/23, 35042, Rennes, France
| | - Philippe Ruelle
- Université catholique de Louvain - UCL, Institut de Recherche en Mathématique et Physique, IRMP, Chemin du Cyclotron, 2, B-1348, Louvain-la-Neuve, Belgium
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9
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Rotbart T, Reuveni S, Urbakh M. Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:060101. [PMID: 26764608 DOI: 10.1103/physreve.92.060101] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2015] [Indexed: 05/27/2023]
Abstract
We study the effect of restart, and retry, on the mean completion time of a generic process. The need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of Michaelis and Menten. Stopping a process in its midst-only to start it all over again-may prolong, leave unchanged, or even shorten the time taken for its completion. Here we are interested in the optimal restart problem, i.e., in finding a restart rate which brings the mean completion time of a process to a minimum. We derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest. We then continue to discover regimes at which solutions to the problem take on universal, details independent forms which further give rise to optimal scaling laws. The formalism we develop, and the results obtained, can be utilized when optimizing stochastic search processes and randomized computer algorithms. An immediate connection with kinetic proofreading is also noted and discussed.
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Affiliation(s)
- Tal Rotbart
- School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel
| | - Shlomi Reuveni
- Department of Systems Biology, Harvard Medical School, 200 Longwood Avenue, Boston, Massachusetts 02115, USA
| | - Michael Urbakh
- School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel
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10
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Kumar A, Maity H, Dua A. Parallel versus Off-Pathway Michaelis–Menten Mechanism for Single-Enzyme Kinetics of a Fluctuating Enzyme. J Phys Chem B 2015; 119:8490-500. [DOI: 10.1021/acs.jpcb.5b03752] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Ashutosh Kumar
- Department of Chemistry, Indian Institute of Technology, Madras, Chennai 600036, India
| | - Hiranmay Maity
- Department of Chemistry, Indian Institute of Technology, Madras, Chennai 600036, India
| | - Arti Dua
- Department of Chemistry, Indian Institute of Technology, Madras, Chennai 600036, India
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11
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Lang AH, Fisher CK, Mora T, Mehta P. Thermodynamics of statistical inference by cells. PHYSICAL REVIEW LETTERS 2014; 113:148103. [PMID: 25325665 DOI: 10.1103/physrevlett.113.148103] [Citation(s) in RCA: 45] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2014] [Indexed: 06/04/2023]
Abstract
The deep connection between thermodynamics, computation, and information is now well established both theoretically and experimentally. Here, we extend these ideas to show that thermodynamics also places fundamental constraints on statistical estimation and learning. To do so, we investigate the constraints placed by (nonequilibrium) thermodynamics on the ability of biochemical signaling networks to estimate the concentration of an external signal. We show that accuracy is limited by energy consumption, suggesting that there are fundamental thermodynamic constraints on statistical inference.
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Affiliation(s)
- Alex H Lang
- Physics Department, Boston University, Boston, Massachusetts 02215, USA
| | - Charles K Fisher
- Physics Department, Boston University, Boston, Massachusetts 02215, USA
| | - Thierry Mora
- Laboratoire de physique statistique, CNRS, UPMC and École normale supérieure, 75005 Paris, France
| | - Pankaj Mehta
- Physics Department, Boston University, Boston, Massachusetts 02215, USA
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12
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Cheng X, Merchan L, Tchernookov M, Nemenman I. A large number of receptors may reduce cellular response time variation. Phys Biol 2013; 10:035008. [PMID: 23735700 DOI: 10.1088/1478-3975/10/3/035008] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
Cells often have tens of thousands of receptors, even though only a few activated receptors can trigger full cellular responses. Reasons for the overabundance of receptors remain unclear. We suggest that, under certain conditions, the large number of receptors can result in a competition among receptors to be the first to activate the cell. The competition decreases the variability of the time to cellular activation, and hence results in a more synchronous activation of cells. We argue that, in simple models, this variability reduction does not necessarily interfere with the receptor specificity to ligands achieved by the kinetic proofreading mechanism. Thus cells can be activated accurately in time and specifically to certain signals. We predict the minimum number of receptors needed to reduce the coefficient of variation for the time to activation following binding of a specific ligand. Furthermore, we predict the maximum number of receptors so that the kinetic proofreading mechanism still can improve the specificity of the activation. These predictions fall in line with experimentally reported receptor numbers for multiple systems.
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Affiliation(s)
- Xiang Cheng
- Department of Physics, Emory University, Atlanta, GA 30322, USA.
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13
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Saha S, Sinha A, Dua A. Single-molecule enzyme kinetics in the presence of inhibitors. J Chem Phys 2012; 137:045102. [DOI: 10.1063/1.4737634] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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14
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Munsky B, Nemenman I, Bel G. Erratum: “Specificity and completion time distributions of biochemical processes” [J. Chem. Phys. 131, 235103 (2009)]. J Chem Phys 2012. [DOI: 10.1063/1.4722990] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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15
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Structure–function relations are subtle in genetic regulatory networks. Math Biosci 2011; 231:61-8. [DOI: 10.1016/j.mbs.2011.02.003] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2010] [Revised: 01/10/2011] [Accepted: 02/09/2011] [Indexed: 02/01/2023]
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16
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Rinott R, Jaimovich A, Friedman N. Exploring transcription regulation through cell-to-cell variability. Proc Natl Acad Sci U S A 2011; 108:6329-34. [PMID: 21444810 PMCID: PMC3076844 DOI: 10.1073/pnas.1013148108] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The regulation of cellular protein levels is a complex process involving many regulatory mechanisms, each introducing stochastic events, leading to variability of protein levels between isogenic cells. Previous studies have shown that perturbing genes involved in transcription regulation affects the amount of cell-to-cell variability in protein levels, but to date there has been no systematic characterization of variability in expression as a phenotype. In this research, we use single-cell expression levels of two fluorescent reporters driven by two different promoters under a wide range of genetic perturbations in Saccharomyces cerevisiae, to identify proteins that affect variability in the expression of these reporters. We introduce computational methodology to determine the variability caused by each perturbation and distinguish between global variability, which affects both reporters in a coordinated manner (e.g., due to cell size variability), and local variability, which affects the individual reporters independently (e.g., due to stochastic events in transcription initiation). Classifying genes by their variability phenotype (the effect of their deletion on reporter variability) identifies functionally coherent groups, which broadly correlate with the different stages of transcriptional regulation. Specifically, we find that most processes whose perturbation affects global variability are related to protein synthesis, protein transport, and cell morphology, whereas most processes whose perturbations affect local variability are related to DNA maintenance, chromatin regulation, and RNA synthesis. Moreover, we demonstrate that the variability phenotypes of different protein complexes provide insights into their cellular functions. Our results establish the utility of variability phenotype for dissecting the regulatory mechanisms involved in gene expression.
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Affiliation(s)
- Ruty Rinott
- School of Computer Science and Engineering and
- Institute of Life Sciences, Hebrew University, Jerusalem 91904, Israel; and
| | - Ariel Jaimovich
- School of Computer Science and Engineering and
- Department of Molecular Genetics and Biotechnology, Faculty of Medicine, Hebrew University, Jerusalem 91120, Israel
| | - Nir Friedman
- School of Computer Science and Engineering and
- Institute of Life Sciences, Hebrew University, Jerusalem 91904, Israel; and
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