October  2020, 25(10): 3931-3961. doi: 10.3934/dcdsb.2020078

Estimating the division rate from indirect measurements of single cells

1. 

Sorbonne Université, Inria, Université Paris-Diderot, CNRS, Laboratoire Jacques-Louis Lions, 75005 Paris, France

2. 

Laboratoire de Mathematiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

3. 

Laboratoire Jean Perrin, Sorbonne Université, 75005 Paris, France

* Corresponding author

Received  June 2019 Revised  December 2019 Published  January 2020

Fund Project: The first author has been supported by the ERC Starting Grant SKIPPERAD (number 306321). The second author was on leave ("délégation") at the french National Research Centre for Science (CNRS) during the finalization of this work

Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? This article adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.

Citation: Marie Doumic, Adélaïde Olivier, Lydia Robert. Estimating the division rate from indirect measurements of single cells. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3931-3961. doi: 10.3934/dcdsb.2020078
References:
[1]

H. T. BanksK. L. SuttonW. C. ThompsonG. BocharovD. RooseT. Schenkel and A. Meyerhans, Estimation of cell proliferation dynamics using CFSE data, Bulletin of mathematical biology, 73 (2011), 116-150.  doi: 10.1007/s11538-010-9524-5.  Google Scholar

[2]

H. T. Banks and W. C. Thompson, A division-dependent compartmental model with cyton and intracellular label dynamics, Int. J. Pure Appl. Math, 77 (2012), 119-147.   Google Scholar

[3]

J. Baumeister and A. Leitão, Topics in Inverse Problems, 25th edition, Publicações Matemáticas do IMPA. [IMPA Mathematical Publication], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2005.  Google Scholar

[4]

D. Belomestny and A. Goldenshluger, Nonparametric density estimation from observations with multiplicative measurement errors, preprint, arXiv: 1709.00629. Google Scholar

[5]

E. Bernard, M. Doumic and P. Gabriel, Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, Kinetic and Related Models, 12 (2019), 551–571, https://hal.archives-ouvertes.fr/hal-01363549. doi: 10.3934/krm.2019022.  Google Scholar

[6]

T. Bourgeron, M. Doumic and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel, Inverse Problems, 30 (2014), 025007, 28pp, http://stacks.iop.org/0266-5611/30/i=2/a=025007. doi: 10.1088/0266-5611/30/2/025007.  Google Scholar

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[8]

C. Cadart, S. Monnier, J. Grilli, P. J. S{á}ez, N. Srivastava, R. Attia, E. Terriac, B. Baum, M. Cosentino-Lagomarsino and M. Piel, Size control in mammalian cells involves modulation of both growth rate and cell cycle duration, Nature Communications, 9 (2018), 3275. doi: 10.1038/s41467-018-05393-0.  Google Scholar

[9]

M. CamposI. V. SurovtsevS. KatoA. PaintdakhiB. BeltranS. E. Ebmeier and C. Jacobs-Wagner, A constant size extension drives bacterial cell size homeostasis, Cell, 159 (2014), 1433-1446.  doi: 10.1016/j.cell.2014.11.022.  Google Scholar

[10]

M. DeforetD. Van Ditmarsch and J. B. Xavier, Cell-size homeostasis and the incremental rule in a bacterial pathogen, Biophysical journal, 109 (2015), 521-528.  doi: 10.1016/j.bpj.2015.07.002.  Google Scholar

[11]

M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2 (2007), 121-152.  doi: 10.1051/mmnp:2007006.  Google Scholar

[12]

M. Doumic, M. Escobedo and M. Tournus, Estimating the division rate and kernel in the fragmentation equation, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 35 (2018), 1847–1884, https://hal.archives-ouvertes.fr/hal-01501811. doi: 10.1016/j.anihpc.2018.03.004.  Google Scholar

[13]

M. DoumicM. HoffmannN. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21 (2015), 1760-1799.  doi: 10.3150/14-BEJ623.  Google Scholar

[14]

M. DoumicM. HoffmannP. Reynaud and V. Rivoirard, Nonparametric estimation of the division rate of a size-structured population, SIAM J. on Numer. Anal., 50 (2012), 925-950.  doi: 10.1137/110828344.  Google Scholar

[15]

M. Doumic, B. Perthame and J. Zubelli, Numerical solution of an inverse problem in size-structured population dynamics, Inverse Problems, 25 (2009), 045008, 25pp. doi: 10.1088/0266-5611/25/4/045008.  Google Scholar

[16]

Y.-J. EunP.-Y. HoM. KimS. LaRussaL. RobertL. D. RennerA. SchmidE. Garner and A. Amir, Archaeal cells share common size control with bacteria despite noisier growth and division, Nature Microbiology, 3 (2018), 148-154.  doi: 10.1038/s41564-017-0082-6.  Google Scholar

[17]

J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems, The Annals of Statistics, 19 (1991), 1257-1272.  doi: 10.1214/aos/1176348248.  Google Scholar

[18]

A. Fievet, A. Ducret, T. Mignot, O. Valette, L. Robert, R. Pardoux, A. R. Dolla and C. Aubert, Single-cell analysis of growth and cell division of the anaerobe desulfovibrio vulgaris hildenborough, Frontiers in Microbiology, 6 (2015), 1378. doi: 10.3389/fmicb.2015.01378.  Google Scholar

[19]

P. Gabriel and H. Martin, Steady distribution of the incremental model for bacteria proliferation, Netw. Heterog. Media, 14 (2019), 149–171, https://hal.archives-ouvertes.fr/hal-01742140. doi: 10.3934/nhm.2019008.  Google Scholar

[20]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, in Mathematics Applied to Science (eds. J. Goldstein, S. Rosencrans and G. Sod), Academic Press, 1988, 79–105, https://www.sciencedirect.com/science/article/pii/B9780122895104500124.  Google Scholar

[21]

A. J. HallG. C. Wake and P. W. Gandar, Steady size distributions for cells in one-dimensional plant tissues, Journal of Mathematical Biology, 30 (1991), 101-123.  doi: 10.1007/BF00160330.  Google Scholar

[22]

V. H. Hoang, T. M. Pham Ngoc, V. Rivoirard and V. C. Tran, Nonparametric estimation of the fragmentation kernel based on a PDE stationary distribution approximation, 2017, https://hal.archives-ouvertes.fr/hal-01623403, Working paper or preprint. Google Scholar

[23]

M. Hoffmann and A. Olivier, Nonparametric estimation of the division rate of an age dependent branching process, Stochastic Processes and their Applications, 126 (2016), 1433–1471, https://hal.archives-ouvertes.fr/hal-01254203. doi: 10.1016/j.spa.2015.11.009.  Google Scholar

[24]

J. Johannes, Deconvolution with unknown error distribution, The Annals of Statistics, 37 (2009), 2301-2323.  doi: 10.1214/08-AOS652.  Google Scholar

[25]

C. LacourP. Massart and V. Rivoirard, Estimator selection: A new method with applications to kernel density estimation, Sankhya A, 79 (2017), 298-335.  doi: 10.1007/s13171-017-0107-5.  Google Scholar

[26]

J. Monod, The growth of bacterial cultures, Selected Papers in Molecular Biology by Jacques Monod, (1978), 139–162. doi: 10.1016/B978-0-12-460482-7.50020-8.  Google Scholar

[27]

M. Nussbaum and S. Pereverzev, The degrees of ill-posedness in stochastic and deterministic noise models, Preprint WIAS 509. Google Scholar

[28]

B. Perthame and J. Zubelli, On the inverse problem for a size-structured population model, Inverse Problems, 23 (2007), 1037-1052.  doi: 10.1088/0266-5611/23/3/012.  Google Scholar

[29]

L. Robert, M. Hoffmann, N. Krell, S. Aymerich, J. Robert and M. Doumic, Division in Escherichia coli is triggered by a size-sensing rather than a timing mechanism, BMC Biology, 12 (2014), 17pp, https://bmcbiol.biomedcentral.com/articles/10.1186/1741-7007-12-17. doi: 10.1186/1741-7007-12-17.  Google Scholar

[30]

I. SoiferL. Robert and A. Amir, Single-cell analysis of growth in budding yeast and bacteria reveals a common size regulation strategy, Current Biology, 26 (2016), 356-361.  doi: 10.1016/j.cub.2015.11.067.  Google Scholar

[31]

E. Stewart, R. Madden, G. Paul and F. Taddei, Aging and death in an organism that reproduces by morphologically symmetric division, PLoS Biol., 3 (2005), e45. doi: 10.1371/journal.pbio.0030045.  Google Scholar

[32]

S. Taheri-Araghi, S. Bradde, J. T. Sauls, N. S. Hill, P. A. Levin, J. Paulsson, M. Vergassola and S. Jun, Cell-size control and homeostasis in bacteria, Current Biology, 11679. doi: 10.1016/j.cub.2014.12.009.  Google Scholar

[33]

A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer {S}eries in {S}tatistics, Springer, 2009. doi: 10.1007/b13794.  Google Scholar

[34]

S. Varet, C. Lacour, P. Massart and V. Rivoirard, Numerical performance of penalized comparison to overfitting for multivariate kernel density estimation, arXiv preprint, arXiv: 1902.01075. Google Scholar

show all references

References:
[1]

H. T. BanksK. L. SuttonW. C. ThompsonG. BocharovD. RooseT. Schenkel and A. Meyerhans, Estimation of cell proliferation dynamics using CFSE data, Bulletin of mathematical biology, 73 (2011), 116-150.  doi: 10.1007/s11538-010-9524-5.  Google Scholar

[2]

H. T. Banks and W. C. Thompson, A division-dependent compartmental model with cyton and intracellular label dynamics, Int. J. Pure Appl. Math, 77 (2012), 119-147.   Google Scholar

[3]

J. Baumeister and A. Leitão, Topics in Inverse Problems, 25th edition, Publicações Matemáticas do IMPA. [IMPA Mathematical Publication], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2005.  Google Scholar

[4]

D. Belomestny and A. Goldenshluger, Nonparametric density estimation from observations with multiplicative measurement errors, preprint, arXiv: 1709.00629. Google Scholar

[5]

E. Bernard, M. Doumic and P. Gabriel, Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, Kinetic and Related Models, 12 (2019), 551–571, https://hal.archives-ouvertes.fr/hal-01363549. doi: 10.3934/krm.2019022.  Google Scholar

[6]

T. Bourgeron, M. Doumic and M. Escobedo, Estimating the division rate of the growth-fragmentation equation with a self-similar kernel, Inverse Problems, 30 (2014), 025007, 28pp, http://stacks.iop.org/0266-5611/30/i=2/a=025007. doi: 10.1088/0266-5611/30/2/025007.  Google Scholar

[7]

F. B. BrikciJ. Clairambault and B. Perthame, Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle, Mathematical and Computer Modelling, 47 (2008), 699-713.  doi: 10.1016/j.mcm.2007.06.008.  Google Scholar

[8]

C. Cadart, S. Monnier, J. Grilli, P. J. S{á}ez, N. Srivastava, R. Attia, E. Terriac, B. Baum, M. Cosentino-Lagomarsino and M. Piel, Size control in mammalian cells involves modulation of both growth rate and cell cycle duration, Nature Communications, 9 (2018), 3275. doi: 10.1038/s41467-018-05393-0.  Google Scholar

[9]

M. CamposI. V. SurovtsevS. KatoA. PaintdakhiB. BeltranS. E. Ebmeier and C. Jacobs-Wagner, A constant size extension drives bacterial cell size homeostasis, Cell, 159 (2014), 1433-1446.  doi: 10.1016/j.cell.2014.11.022.  Google Scholar

[10]

M. DeforetD. Van Ditmarsch and J. B. Xavier, Cell-size homeostasis and the incremental rule in a bacterial pathogen, Biophysical journal, 109 (2015), 521-528.  doi: 10.1016/j.bpj.2015.07.002.  Google Scholar

[11]

M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2 (2007), 121-152.  doi: 10.1051/mmnp:2007006.  Google Scholar

[12]

M. Doumic, M. Escobedo and M. Tournus, Estimating the division rate and kernel in the fragmentation equation, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 35 (2018), 1847–1884, https://hal.archives-ouvertes.fr/hal-01501811. doi: 10.1016/j.anihpc.2018.03.004.  Google Scholar

[13]

M. DoumicM. HoffmannN. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21 (2015), 1760-1799.  doi: 10.3150/14-BEJ623.  Google Scholar

[14]

M. DoumicM. HoffmannP. Reynaud and V. Rivoirard, Nonparametric estimation of the division rate of a size-structured population, SIAM J. on Numer. Anal., 50 (2012), 925-950.  doi: 10.1137/110828344.  Google Scholar

[15]

M. Doumic, B. Perthame and J. Zubelli, Numerical solution of an inverse problem in size-structured population dynamics, Inverse Problems, 25 (2009), 045008, 25pp. doi: 10.1088/0266-5611/25/4/045008.  Google Scholar

[16]

Y.-J. EunP.-Y. HoM. KimS. LaRussaL. RobertL. D. RennerA. SchmidE. Garner and A. Amir, Archaeal cells share common size control with bacteria despite noisier growth and division, Nature Microbiology, 3 (2018), 148-154.  doi: 10.1038/s41564-017-0082-6.  Google Scholar

[17]

J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems, The Annals of Statistics, 19 (1991), 1257-1272.  doi: 10.1214/aos/1176348248.  Google Scholar

[18]

A. Fievet, A. Ducret, T. Mignot, O. Valette, L. Robert, R. Pardoux, A. R. Dolla and C. Aubert, Single-cell analysis of growth and cell division of the anaerobe desulfovibrio vulgaris hildenborough, Frontiers in Microbiology, 6 (2015), 1378. doi: 10.3389/fmicb.2015.01378.  Google Scholar

[19]

P. Gabriel and H. Martin, Steady distribution of the incremental model for bacteria proliferation, Netw. Heterog. Media, 14 (2019), 149–171, https://hal.archives-ouvertes.fr/hal-01742140. doi: 10.3934/nhm.2019008.  Google Scholar

[20]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, in Mathematics Applied to Science (eds. J. Goldstein, S. Rosencrans and G. Sod), Academic Press, 1988, 79–105, https://www.sciencedirect.com/science/article/pii/B9780122895104500124.  Google Scholar

[21]

A. J. HallG. C. Wake and P. W. Gandar, Steady size distributions for cells in one-dimensional plant tissues, Journal of Mathematical Biology, 30 (1991), 101-123.  doi: 10.1007/BF00160330.  Google Scholar

[22]

V. H. Hoang, T. M. Pham Ngoc, V. Rivoirard and V. C. Tran, Nonparametric estimation of the fragmentation kernel based on a PDE stationary distribution approximation, 2017, https://hal.archives-ouvertes.fr/hal-01623403, Working paper or preprint. Google Scholar

[23]

M. Hoffmann and A. Olivier, Nonparametric estimation of the division rate of an age dependent branching process, Stochastic Processes and their Applications, 126 (2016), 1433–1471, https://hal.archives-ouvertes.fr/hal-01254203. doi: 10.1016/j.spa.2015.11.009.  Google Scholar

[24]

J. Johannes, Deconvolution with unknown error distribution, The Annals of Statistics, 37 (2009), 2301-2323.  doi: 10.1214/08-AOS652.  Google Scholar

[25]

C. LacourP. Massart and V. Rivoirard, Estimator selection: A new method with applications to kernel density estimation, Sankhya A, 79 (2017), 298-335.  doi: 10.1007/s13171-017-0107-5.  Google Scholar

[26]

J. Monod, The growth of bacterial cultures, Selected Papers in Molecular Biology by Jacques Monod, (1978), 139–162. doi: 10.1016/B978-0-12-460482-7.50020-8.  Google Scholar

[27]

M. Nussbaum and S. Pereverzev, The degrees of ill-posedness in stochastic and deterministic noise models, Preprint WIAS 509. Google Scholar

[28]

B. Perthame and J. Zubelli, On the inverse problem for a size-structured population model, Inverse Problems, 23 (2007), 1037-1052.  doi: 10.1088/0266-5611/23/3/012.  Google Scholar

[29]

L. Robert, M. Hoffmann, N. Krell, S. Aymerich, J. Robert and M. Doumic, Division in Escherichia coli is triggered by a size-sensing rather than a timing mechanism, BMC Biology, 12 (2014), 17pp, https://bmcbiol.biomedcentral.com/articles/10.1186/1741-7007-12-17. doi: 10.1186/1741-7007-12-17.  Google Scholar

[30]

I. SoiferL. Robert and A. Amir, Single-cell analysis of growth in budding yeast and bacteria reveals a common size regulation strategy, Current Biology, 26 (2016), 356-361.  doi: 10.1016/j.cub.2015.11.067.  Google Scholar

[31]

E. Stewart, R. Madden, G. Paul and F. Taddei, Aging and death in an organism that reproduces by morphologically symmetric division, PLoS Biol., 3 (2005), e45. doi: 10.1371/journal.pbio.0030045.  Google Scholar

[32]

S. Taheri-Araghi, S. Bradde, J. T. Sauls, N. S. Hill, P. A. Levin, J. Paulsson, M. Vergassola and S. Jun, Cell-size control and homeostasis in bacteria, Current Biology, 11679. doi: 10.1016/j.cub.2014.12.009.  Google Scholar

[33]

A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer {S}eries in {S}tatistics, Springer, 2009. doi: 10.1007/b13794.  Google Scholar

[34]

S. Varet, C. Lacour, P. Massart and V. Rivoirard, Numerical performance of penalized comparison to overfitting for multivariate kernel density estimation, arXiv preprint, arXiv: 1902.01075. Google Scholar

Figure 1.  Protocol 1 – Reconstruction of $ B $ when both $ U_{B,x} $ and $ \mathcal{L}_B $ are (almost) exactly known. The oracle choice for $ h $ gives us the value 1/4.75
Figure 2.  Results of Protocols 1 and 2. $ x $ stands for size, $ \xi $ for frequency and $ a $ for increment of size. Estimation of the division rate $ B(a) = a^2 $ in function of the increment of size $ a $ (Subfigure 2g), and of all the intermediate functions necessary (Subfigures 2a to 2b)
Figure 3.  Protocol 2 – Reconstruction of $ B $ when $ U_{B,x} $ is (almost) exactly known but not $ \mathcal{L}_B $. The oracle choice for $ h $ gives us the value 1/5
Figure 4.  Protocol 3 – Reconstruction of $ B $ when $ U_{B,x} $ is reconstructed from $ X_1,\ldots,X_n $ i.i.d. $ \sim $ $ U_{B,x} $ but $ \mathcal{L}_B $ is (almost) exactly known. The oracle choice for $ h_3 $ gives us values that range between $ 1/3.25 $ for $ n = 500 $ and $ 1/4.75 $ for $ n = 50\; 000 $. We set $ \varpi_n = 1/n $
Figure 5.  Results of Protocol 3 for $ n = 2000 $ and $ M = 100 $ Monte Carlo samples. ($ x $ stands for size, $ \xi $ for frequency and $ a $ for increment of size). Estimation of the division rate $ B(a) = a^2 $ in function of the increment of size $ a $ (Subfigure 5e), and of intermediate functions (Subfigures 5a to 5d). In beige, the zone where 95% of the 100 Monte Carlo samples lie
Figure 6.  Results of Protocols 3 (Left) and 4 (Right), – Estimation of the division rate $ B(a) = a^2 $ in function of the increment of size $ a $ for different $ n $ (from up to bottom, 500, 5 000, 10 000) and $ M = 100 $ Monte Carlo samples (the beige zone representing the zone of 95% of the 100 reconstructions). We see that the result improves for $ x\leq\approx 2 $ when $ n $ increases, but remains very poor for larger $ x $, the difference between the blue curve (Protocol 1 or 2) and the beige zone showing the influence of the sampling noise. However, Protocol 4 does not significantly worsen the results of Protocol 3
Figure 7.  Results of Protocols 3 and 4 – Reduction of the mean error over $ M = 100 $ samples (in log-scale) in function of the sample size (from $ n = 500 $ to $ n = 50\; 000 $). Empirical errors are computed over the following regular grids: (a)-(e) $ [0;6] $, $ \Delta x = \tfrac{6}{500} $; (f)-(h) $ [-10;10] $, $ \Delta \xi = 0.05 $; (i)-(j) $ [0;2.25] $, $ \Delta a = \tfrac{1}{\sqrt{n}} $; (k) $ [0;2] $, $ \Delta a = \tfrac{1}{\sqrt{n}} $
Figure 8.  Protocol 4 – Reconstruction of $ B $ when both $ U_{B,x} $ and $ \mathcal{L}_B $ are reconstructed from $ X_1,\ldots,X_n $ i.i.d. $ \sim $ $ U_{B,x} $. The parameter $ h_1 $ is automatically chosen by the kernel smoothing function ${\text ksdensity}$; $ h_2 $ is deduced from $ h_1 $. The oracle choice for $ h_3 $ gives us values that range between $ 1/3.25 $ for $ n = 500 $ and $ 1/4.5 $ for $ n = 50\; 000 $. We set $ \varpi_n = 1/n $
Figure 9.  Results of Protocol 4 for $ n = 2000 $ and $ M = 100 $ Monte Carlo samples, with an inital division rate $ B(a) = a^2 $. From (a) to (j): successive steps of the Protocol 4. We see that the main errors come from the estimation of the Fourier transform (Subfigures 9g and 9h)
Figure 10.  Testing the procedure on experimental data

Initial step: estimation of the size distribution.

Figure 12.  Speed of convergence of each step of Protocol 4 in a log-log scale
Figure 11.  Testing the procedure on experimental data

Final step: estimation of the increment-structured division rate.

Table 1.  Errors of Protocols 1 and 2 for the intermediate steps
Reconstruction of $ \mathcal L_B $ $ \mathcal G_B $ $ \mathcal G^*_B $
Numerical [0;6] [0;6] [-50;50]
sampling $ \Delta x = \tfrac{6}{500} $ $ \Delta x = \tfrac{6}{500} $ $ \Delta \xi = 0.05 $
Protocol 1 - - -
Protocol 2 0.0478 0.0417 0.0417
Reconstruction of $ \mathcal L_B $ $ \mathcal G_B $ $ \mathcal G^*_B $
Numerical [0;6] [0;6] [-50;50]
sampling $ \Delta x = \tfrac{6}{500} $ $ \Delta x = \tfrac{6}{500} $ $ \Delta \xi = 0.05 $
Protocol 1 - - -
Protocol 2 0.0478 0.0417 0.0417
Table 2.  Errors of Protocols 1 and 2 for $ B $ in function of the numerical sampling
Reconstruction of $ B $ $ B $
Numerical [0;2] [0;2.5 ]
sampling $ \Delta a = 0.01 $ $ \Delta a = 0.01 $
Protocol 1 0.0730 0.2065
Protocol 2 0.0849 0.1321
Reconstruction of $ B $ $ B $
Numerical [0;2] [0;2.5 ]
sampling $ \Delta a = 0.01 $ $ \Delta a = 0.01 $
Protocol 1 0.0730 0.2065
Protocol 2 0.0849 0.1321
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