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Nieto C, Täuber S, Blöbaum L, Vahdat Z, Grünberger A, Singh A. Coupling Cell Size Regulation and Proliferation Dynamics of C. glutamicum Reveals Cell Division Based on Surface Area. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.12.26.573217. [PMID: 38234762 PMCID: PMC10793411 DOI: 10.1101/2023.12.26.573217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/19/2024]
Abstract
Single cells actively coordinate growth and division to regulate their size, yet how this size homeostasis at the single-cell level propagates over multiple generations to impact clonal expansion remains fundamentally unexplored. Classical timer models for cell proliferation (where the duration of the cell cycle is an independent variable) predict that the stochastic variation in colony size will increase monotonically over time. In stark contrast, implementing size control according to adder strategy (where on average a fixed size added from cell birth to division) leads to colony size variations that eventually decay to zero. While these results assume a fixed size of the colony-initiating progenitor cell, further analysis reveals that the magnitude of the intercolony variation in population number is sensitive to heterogeneity in the initial cell size. We validate these predictions by tracking the growth of isogenic microcolonies of Corynebacterium glutamicum in microfluidic chambers. Approximating their cell shape to a capsule, we observe that the degree of random variability in cell size is different depending on whether the cell size is quantified as per length, surface area, or volume, but size control remains an adder regardless of these size metrics. A comparison of the observed variability in the colony population with the predictions suggests that proliferation matches better with a cell division based on the cell surface. In summary, our integrated mathematical-experimental approach bridges the paradigms of single-cell size regulation and clonal expansion at the population levels. This innovative approach provides elucidation of the mechanisms of size homeostasis from the stochastic dynamics of colony size for rod-shaped microbes.
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Affiliation(s)
- César Nieto
- Department of Electrical and Computing Engineering, University of Delaware. Newark, DE 19716, USA
| | - Sarah Täuber
- CeBiTec, Bielefeld University. Bielefeld, Germany
- Multiscale Bioengineering, Technical Faculty, Bielefeld University. Bielefeld, Germany
| | - Luisa Blöbaum
- CeBiTec, Bielefeld University. Bielefeld, Germany
- Multiscale Bioengineering, Technical Faculty, Bielefeld University. Bielefeld, Germany
| | - Zahra Vahdat
- Department of Electrical and Computing Engineering, University of Delaware. Newark, DE 19716, USA
| | - Alexander Grünberger
- CeBiTec, Bielefeld University. Bielefeld, Germany
- Multiscale Bioengineering, Technical Faculty, Bielefeld University. Bielefeld, Germany
- Institute of Process Engineering in Life Sciences: Microsystems in Bioprocess Engineering, Karlsruhe Institute of Technology. Karlsruhe, Germany
| | - Abhyudai Singh
- Department of Electrical and Computing Engineering, University of Delaware. Newark, DE 19716, USA
- Center for Bioinformatics and Computational Biology, University of Delaware, Newark, DE 19716 USA
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2
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Jones EW, Derrick J, Nisbet RM, Ludington WB, Sivak DA. First-passage-time statistics of growing microbial populations carry an imprint of initial conditions. Sci Rep 2023; 13:21340. [PMID: 38049502 PMCID: PMC10696051 DOI: 10.1038/s41598-023-48726-w] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2023] [Accepted: 11/29/2023] [Indexed: 12/06/2023] Open
Abstract
In exponential population growth, variability in the timing of individual division events and environmental factors (including stochastic inoculation) compound to produce variable growth trajectories. In several stochastic models of exponential growth we show power-law relationships that relate variability in the time required to reach a threshold population size to growth rate and inoculum size. Population-growth experiments in E. coli and S. aureus with inoculum sizes ranging between 1 and 100 are consistent with these relationships. We quantify how noise accumulates over time, finding that it encodes-and can be used to deduce-information about the early growth rate of a population.
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Affiliation(s)
- Eric W Jones
- Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada.
| | - Joshua Derrick
- Department of Biological Sciences and Engineering, Carnegie Institution for Science, Baltimore, MD, 21218, USA
| | - Roger M Nisbet
- Department of Ecology, Evolution, and Marine Biology, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA
| | - William B Ludington
- Department of Biological Sciences and Engineering, Carnegie Institution for Science, Baltimore, MD, 21218, USA
- Department of Biology, Johns Hopkins University, Baltimore, MD, 21218, USA
| | - David A Sivak
- Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada
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3
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Jafarpour F, Levien E, Amir A. Evolutionary dynamics in non-Markovian models of microbial populations. Phys Rev E 2023; 108:034402. [PMID: 37849168 DOI: 10.1103/physreve.108.034402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2023] [Accepted: 06/07/2023] [Indexed: 10/19/2023]
Abstract
In the past decade, great strides have been made to quantify the dynamics of single-cell growth and division in microbes. In order to make sense of the evolutionary history of these organisms, we must understand how features of single-cell growth and division influence evolutionary dynamics. This requires us to connect processes on the single-cell scale to population dynamics. Here, we consider a model of microbial growth in finite populations which explicitly incorporates the single-cell dynamics. We study the behavior of a mutant population in such a model and ask: can the evolutionary dynamics be coarse-grained so that the forces of natural selection and genetic drift can be expressed in terms of the long-term fitness? We show that it is in fact not possible, as there is no way to define a single fitness parameter (or reproductive rate) that defines the fate of an organism even in a constant environment. This is due to fluctuations in the population averaged division rate. As a result, various details of the single-cell dynamics affect the fate of a new mutant independently from how they affect the long-term growth rate of the mutant population. In particular, we show that in the case of neutral mutations, variability in generation times increases the rate of genetic drift, and in the case of beneficial mutations, variability decreases its fixation probability. Furthermore, we explain the source of the persistent division rate fluctuations and provide analytic solutions for the fixation probability as a multispecies generalization of the Euler-Lotka equation.
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Affiliation(s)
- Farshid Jafarpour
- Institute for Theoretical Physics, Utrecht University, 3584 CC Utrecht, The Netherlands
| | - Ethan Levien
- Mathematics Department, Dartmouth College, Hanover, New Hampshire 03755, USA
| | - Ariel Amir
- Department of Complex Systems, Faculty of Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel
- John A. Paulson, School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
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4
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Jędrak J, Rubin M, Ochab-Marcinek A. Generalization of Powell's results to population out of steady state. Phys Rev E 2023; 108:024405. [PMID: 37723697 DOI: 10.1103/physreve.108.024405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2022] [Accepted: 07/27/2023] [Indexed: 09/20/2023]
Abstract
Since the seminal work of Powell, the relationships between the population growth rate, the probability distributions of generation time, and the distribution of cell age have been known for the bacterial population in a steady state of exponential growth. Here we generalize these relationships to include an unsteady (transient) state for both the batch culture and the mother machine experiment. In particular, we derive a time-dependent Euler-Lotka equation (relating the generation-time distributions to the population growth rate) and a generalization of the inequality between the mean generation time and the population doubling time. To do this, we use a model proposed by Lebowitz and Rubinow, in which each cell is described by its age and generation time. We show that our results remain valid for a class of more complex models that use other state variables in addition to cell age and generation time, as long as the integration of these additional variables reduces the model to Lebowitz-Rubinow form. As an application of this formalism, we calculate the fitness landscapes for phenotypic traits (cell age, generation time) in a population that is not growing exponentially. We clarify that the known fitness landscape formula for the cell age as a phenotypic trait is an approximation to the exact time-dependent formula.
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Affiliation(s)
- Jakub Jędrak
- Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warsaw, Poland
| | - Marcin Rubin
- Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warsaw, Poland
| | - Anna Ochab-Marcinek
- Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warsaw, Poland
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5
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Gavagnin E, Vittadello ST, Gunasingh G, Haass NK, Simpson MJ, Rogers T, Yates CA. Synchronized oscillations in growing cell populations are explained by demographic noise. Biophys J 2021; 120:1314-1322. [PMID: 33617836 DOI: 10.1016/j.bpj.2021.02.017] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2020] [Revised: 12/19/2020] [Accepted: 02/08/2021] [Indexed: 01/14/2023] Open
Abstract
Understanding synchrony in growing populations is important for applications as diverse as epidemiology and cancer treatment. Recent experiments employing fluorescent reporters in melanoma cell lines have uncovered growing subpopulations exhibiting sustained oscillations, with nearby cells appearing to synchronize their cycles. In this study, we demonstrate that the behavior observed is consistent with long-lasting transient phenomenon initiated and amplified by the finite-sample effects and demographic noise. We present a novel mathematical analysis of a multistage model of cell growth, which accurately reproduces the synchronized oscillations. As part of the analysis, we elucidate the transient and asymptotic phases of the dynamics and derive an analytical formula to quantify the effect of demographic noise in the appearance of the oscillations. The implications of these findings are broad, such as providing insight into experimental protocols that are used to study the growth of asynchronous populations and, in particular, those investigations relating to anticancer drug discovery.
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Affiliation(s)
- Enrico Gavagnin
- School of Biological Sciences, University of Bristol, Bristol, United Kingdom.
| | - Sean T Vittadello
- School of BioSciences, University of Melbourne, Melbourne, Victoria, Australia
| | - Gency Gunasingh
- The University of Queensland, The University of Queensland Diamantina Institute, Brisbane, Queensland, Australia
| | - Nikolas K Haass
- The University of Queensland, The University of Queensland Diamantina Institute, Brisbane, Queensland, Australia
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia
| | - Tim Rogers
- Department of Mathematical Sciences, University of Bath, Bath, United Kingdom
| | - Christian A Yates
- Department of Mathematical Sciences, University of Bath, Bath, United Kingdom
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Nozoe T, Kussell E. Cell Cycle Heritability and Localization Phase Transition in Growing Populations. PHYSICAL REVIEW LETTERS 2020; 125:268103. [PMID: 33449732 PMCID: PMC8528515 DOI: 10.1103/physrevlett.125.268103] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2020] [Accepted: 10/27/2020] [Indexed: 06/12/2023]
Abstract
The cell cycle duration is a variable cellular phenotype that underlies long-term population growth and age structures. By analyzing the stationary solutions of a branching process with heritable cell division times, we demonstrate the existence of a phase transition, which can be continuous or first order, by which a nonzero fraction of the population becomes localized at a minimal division time. Just below the transition, we demonstrate the coexistence of localized and delocalized age-structure phases and the power law decay of correlation functions. Above it, we observe the self-synchronization of cell cycles, collective divisions, and the slow "aging" of population growth rates.
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Affiliation(s)
- Takashi Nozoe
- Department of Biology, New York University, 12 Waverly Place, New York, New York 10003, USA
| | - Edo Kussell
- Department of Biology, New York University, 12 Waverly Place, New York, New York 10003, USA
- Department of Physics, New York University, 726 Broadway, New York, New York 10003, USA
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7
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Banerjee M, Volpert V. Stochastic intracellular regulation can remove oscillations in a model of tissue growth. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2020; 37:551-568. [PMID: 32735317 DOI: 10.1093/imammb/dqaa009] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2019] [Revised: 03/28/2020] [Accepted: 06/19/2020] [Indexed: 01/06/2023]
Abstract
The work is devoted to the analysis of cell population dynamics where cells make a choice between differentiation and apoptosis. This choice is based on the values of intracellular proteins whose concentrations are described by a system of ordinary differential equations with bistable dynamics. Intracellular regulation and cell fate are controlled by the extracellular regulation through the number of differentiated cells. It is shown that the total cell number necessarily oscillates if the initial condition in the intracellular regulation is fixed. These oscillations can be suppressed if the initial condition is a random variable with a sufficiently large variation. Thus, the result of the work suggests a possible answer to the question about the role of stochasticity in the intracellular regulation.
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Affiliation(s)
- M Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India
| | - V Volpert
- Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France.,INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France.,Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation
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8
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Jędrak J, Ochab-Marcinek A. Contributions to the 'noise floor' in gene expression in a population of dividing cells. Sci Rep 2020; 10:13533. [PMID: 32782314 PMCID: PMC7419568 DOI: 10.1038/s41598-020-69217-2] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2019] [Accepted: 05/26/2020] [Indexed: 11/14/2022] Open
Abstract
Experiments with cells reveal the existence of a lower bound for protein noise, the noise floor, in highly expressed genes. Its origins are still debated. We propose a minimal model of gene expression in a proliferating bacterial cell population. The model predicts the existence of a noise floor and it semi-quantitatively reproduces the curved shape of the experimental noise vs. mean protein concentration plots. When the cell volume increases in a different manner than does the mean protein copy number, the noise floor level is determined by the cell population’s age structure and by the dependence of the mean protein concentration on cell age. Additionally, the noise floor level may depend on a biological limit for the mean number of bursts in the cell cycle. In that case, the noise floor level depends on the burst size distribution width but it is insensitive to the mean burst size. Our model quantifies the contributions of each of these mechanisms to gene expression noise.
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Affiliation(s)
- Jakub Jędrak
- Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224, Warsaw, Poland.
| | - Anna Ochab-Marcinek
- Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224, Warsaw, Poland
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9
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Levien E, GrandPre T, Amir A. Large Deviation Principle Linking Lineage Statistics to Fitness in Microbial Populations. PHYSICAL REVIEW LETTERS 2020; 125:048102. [PMID: 32794821 DOI: 10.1103/physrevlett.125.048102] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2020] [Accepted: 06/18/2020] [Indexed: 06/11/2023]
Abstract
In exponentially proliferating populations of microbes, the population doubles at a rate less than the average doubling time of a single-cell due to variability at the single-cell level. It is known that the distribution of generation times obtained from a single lineage is, in general, insufficient to determine a population's growth rate. Is there an explicit relationship between observables obtained from a single lineage and the population growth rate? We show that a population's growth rate can be represented in terms of averages over isolated lineages. This lineage representation is related to a large deviation principle that is a generic feature of exponentially proliferating populations. Due to the large deviation structure of growing populations, the number of lineages needed to obtain an accurate estimate of the growth rate depends exponentially on the duration of the lineages, leading to a nonmonotonic convergence of the estimate, which we verify in both synthetic and experimental data sets.
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Affiliation(s)
- Ethan Levien
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, Harvard 02138, USA
| | - Trevor GrandPre
- Department of Physics, University of California, Berkeley, California, Berkeley 94720, USA
| | - Ariel Amir
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, Harvard 02138, USA
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10
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Levien E, Kondev J, Amir A. The interplay of phenotypic variability and fitness in finite microbial populations. J R Soc Interface 2020; 17:20190827. [PMID: 32396808 DOI: 10.1098/rsif.2019.0827] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
In isogenic microbial populations, phenotypic variability is generated by a combination of stochastic mechanisms, such as gene expression, and deterministic factors, such as asymmetric segregation of cell volume. Here we address the question: how does phenotypic variability of a microbial population affect its fitness? While this question has previously been studied for exponentially growing populations, the situation when the population size is kept fixed has received much less attention, despite its relevance to many natural scenarios. We show that the outcome of competition between multiple microbial species can be determined from the distribution of phenotypes in the culture using a generalization of the well-known Euler-Lotka equation, which relates the steady-state distribution of phenotypes to the population growth rate. We derive a generalization of the Euler-Lotka equation for finite cultures, which relates the distribution of phenotypes among cells in the culture to the exponential growth rate. Our analysis reveals that in order to predict fitness from phenotypes, it is important to understand how distributions of phenotypes obtained from different subsets of the genealogical history of a population are related. To this end, we derive a mapping between the various ways of sampling phenotypes in a finite population and show how to obtain the equivalent distributions from an exponentially growing culture. Finally, we use this mapping to show that species with higher growth rates in exponential growth conditions will have a competitive advantage in the finite culture.
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Affiliation(s)
- Ethan Levien
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.,Department of Physics, Brandeis University, Waltham, MA 02453, USA
| | - Jane Kondev
- Department of Physics, Brandeis University, Waltham, MA 02453, USA
| | - Ariel Amir
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
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11
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Hubbard JB, Halter M, Sarkar S, Plant AL. The role of fluctuations in determining cellular network thermodynamics. PLoS One 2020; 15:e0230076. [PMID: 32160263 PMCID: PMC7065797 DOI: 10.1371/journal.pone.0230076] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2019] [Accepted: 02/20/2020] [Indexed: 12/28/2022] Open
Abstract
The steady state distributions of phenotypic responses within an isogenic population of cells result from both deterministic and stochastic characteristics of biochemical networks. A biochemical network can be characterized by a multidimensional potential landscape based on the distribution of responses and a diffusion matrix of the correlated dynamic fluctuations between N-numbers of intracellular network variables. In this work, we develop a thermodynamic description of biological networks at the level of microscopic interactions between network variables. The Boltzmann H-function defines the rate of free energy dissipation of a network system and provides a framework for determining the heat associated with the nonequilibrium steady state and its network components. The magnitudes of the landscape gradients and the dynamic correlated fluctuations of network variables are experimentally accessible. We describe the use of Fokker-Planck dynamics to calculate housekeeping heat from the experimental data by a method that we refer to as Thermo-FP. The method provides insight into the composition of the network and the relative thermodynamic contributions from network components. We surmise that these thermodynamic quantities allow determination of the relative importance of network components to overall network control. We conjecture that there is an upper limit to the rate of dissipative heat produced by a biological system that is associated with system size or modularity, and we show that the dissipative heat has a lower bound.
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Affiliation(s)
- Joseph B. Hubbard
- Biosystems and Biomaterials Division, National Institute of Standards and Technology, Gaithersburg, MD, United States of America
| | - Michael Halter
- Biosystems and Biomaterials Division, National Institute of Standards and Technology, Gaithersburg, MD, United States of America
| | - Swarnavo Sarkar
- Biosystems and Biomaterials Division, National Institute of Standards and Technology, Gaithersburg, MD, United States of America
| | - Anne L. Plant
- Biosystems and Biomaterials Division, National Institute of Standards and Technology, Gaithersburg, MD, United States of America
- * E-mail:
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