American Institute of Mathematical Sciences

Advanced
October  2019, 24(10): 5539-5552. doi: 10.3934/dcdsb.2019070

Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay

Pavol Bokes

Department of Applied Mathematics and Statistics, Comenius University, Bratislava 842 48, Slovakia

Received  July 2018 Revised  November 2018 Published  April 2019

Fund Project: The author is supported by the Slovak Research and Development Agency under the contract No. APVV-14-0378 and also by the VEGA grant 1/0347/18

Synthesis of individual molecules in the expression of genes often occurs in bursts of multiple copies. Gene regulatory feedback can affect the frequency with which these bursts occur or their size. Whereas frequency regulation has traditionally received more attention, we focus specifically on the regulation of burst size. It turns out that there are (at least) two alternative formulations of feedback in burst size. In the first, newly produced molecules immediately partake in feedback, even within the same burst. In the second, there is no within-burst regulation due to what we call infinitesimal delay. We describe both alternatives using a minimalistic Markovian drift-jump framework combining discrete and continuous dynamics. We derive detailed analytic results and efficient simulation algorithms for positive non-cooperative autoregulation (whether infinitesimally delayed or not). We show that at steady state both alternatives lead to a gamma distribution of protein level. The steady-state distribution becomes available only after a transcritical bifurcation point is passed. Interestingly, the onset of the bifurcation is postponed by the inclusion of infinitesimal delay.

Keywords: Stochastic gene expression, bursting, positive feedback, MichaelisɃMenten kinetics, hill function, delay, transcritical bifurcation, hybrid model, piecewise deterministic Markov process, partial integro-differential equation, Volterra integral equation.
Mathematics Subject Classification: Primary: 92C40; Secondary: 60J75, 45D05.
Citation: Pavol Bokes. Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5539-5552. doi: 10.3934/dcdsb.2019070
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, D.C., 1972. Google Scholar

[2]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Garland Science New York, 2002. Google Scholar

[3]

U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman & Hall/CRC, 2007.  Google Scholar

[4]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation, PLoS Comput. Biol., 2 (2006), e117. Google Scholar

[5]

P. Bokes and A. Singh, Gene expression noise is affected differentially by feedback in burst frequency and burst size, J. Math. Biol., 74 (2017), 1483-1509.  doi: 10.1007/s00285-016-1059-4.  Google Scholar

[6]

P. Bokes, Y. T. Lin and A. Singh, High cooperativity in negative feedback can amplify noisy gene expression, B. Math. Biol., 80 (2018), 1871-1899.  doi: 10.1007/s11538-018-0438-y.  Google Scholar

[7]

P. Bokes, J. R. King, A. T. A. Wood and M. Loose, Transcriptional bursting diversifies the behaviour of a toggle switch: hybrid simulation of stochastic gene expression, B. Math. Biol., 75 (2013), 351-371.  doi: 10.1007/s11538-013-9811-z.  Google Scholar

[8]

L. Cai, N. Friedman and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362.   Google Scholar

[9]

A. Crudu, A. Debussche, A. Muller and O. Radulescu, others, Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann. Appl. Probab., 22 (2012), 1822Ƀ1859. doi: 10.1214/11-AAP814.  Google Scholar

[10]

P. Czuppon and P. Pfaffelhuber, Limits of noise for autoregulated gene expression, J. Math. Biol., 77 (2018), 1153-1191.  doi: 10.1007/s00285-018-1248-4.  Google Scholar

[11]

R. D. Dar, B. S. Razooky, A. Singh, T. V. Trimeloni, J. M. McCollum, C. D. Cox, M. L. Simpson and L. S. Weinberger, Transcriptional burst frequency and burst size are equally modulated across the human genome, P. Natl. Acad. Sci. USA, 109 (2012), 17454-17459.   Google Scholar

[12]

R. Dessalles, V. Fromion and P. Robert, A stochastic analysis of autoregulation of gene expression, J. Math. Biol., 75 (2017), 1253-1283.  doi: 10.1007/s00285-017-1116-7.  Google Scholar

[13]

M. B. Elowitz, A. J. Levine, E. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.   Google Scholar

[14]

N. Friedman, L. Cai and X. S. Xie, Linking stochastic dynamics to population distribution: An analytical framework of gene expression, Phys. Rev. Lett., 97 (2006), 168302. Google Scholar

[15]

M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2004.  Google Scholar

[16]

G. C. P. Innocentini, M. Forger, O. Radulescu and F. Antoneli, Protein synthesis driven by dynamical stochastic transcription, B. Math. Biol., 78 (2016), 110-131.  doi: 10.1007/s11538-015-0131-3.  Google Scholar

[17]

G. C. P. Innocentini, S. Guiziou, J. Bonnet and O. Radulescu, Analytic framework for a stochastic binary biological switch, Phys. Rev. E, 94 (2016), 062413. Google Scholar

[18]

S. Intep and D. J. Higham, Zero, one and two-switch models of gene regulation, Discrete Cont. Dyn-B., 14 (2010), 495-513.  doi: 10.3934/dcdsb.2010.14.495.  Google Scholar

[19]

J. Jedrak and A. Ochab-Marcinek, Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound poisson process, Phys. Rev. E, 94 (2016), 032401. Google Scholar

[20]

C. Jia, H. Qian, M. Chen and M. Q. Zhang, Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks, J. Chem. Phys., 148 (2018), 095102. Google Scholar

[21]

N. Kumar, T. Platini and R. V. Kulkarni, Exact distributions for stochastic gene expression models with bursting and feedback, Phys. Rev. Lett., 113 (2014), 268105. Google Scholar

[22]

D. R. Larson, R. H. Singer and D. Zenklusen, A single molecule view of gene expression, Trends Cell Biol., 19 (2009), 630-637.   Google Scholar

[23]

G. Lin, J. Yu, Z. Zhou, Q. Sun and F. Jiao, Fluctuations of mrna distributions in multiple pathway activated transcription, Discrete Cont. Dyn-B., 2018. Google Scholar

[24]

Y. T. Lin and N. E. Buchler, Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic markov processes, J. Roy. Soc. Interface, 15 (2018), 20170804. Google Scholar

[25]

Y. T. Lin, P. G. Hufton, E. J. Lee and D. A. Potoyan, A stochastic and dynamical view of pluripotency in mouse embryonic stem cells, PloS Comput. Biol., 14 (2018), e1006000. Google Scholar

[26]

Y. T. Lin and C. R. Doering, Gene expression dynamics with stochastic bursts: Construction and exact results for a coarse-grained model, Phys. Rev. E, 93 (2016), 022409, 10pp. doi: 10.1103/physreve.93.022409.  Google Scholar

[27]

M. Masujima, Applied Mathematical Methods in Theoretical Physics, John Wiley & Sons, 2009. doi: 10.1002/9783527627745.  Google Scholar

[28]

N. A. M. Monk, Oscillatory expression of hes1, p53, and nf-$ \kappa $b driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413.   Google Scholar

[29]

M. Pájaro, A. A. Alonso, I. Otero-Muras and C. Vázquez, Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting, J. Theor. Biol., 421 (2017), 51-70.  doi: 10.1016/j.jtbi.2017.03.017.  Google Scholar

[30]

J. Peccoud and B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.   Google Scholar

[31]

A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. Google Scholar

[32]

J. Ren, F. Jiao, Q. Sun, M. Tang and J. Yu, The dynamics of gene transcription in random environments, Discrete Cont. Dyn-B., 23 (2018), 3167-3194.  doi: 10.3934/dcdsb.2018224.  Google Scholar

[33]

R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, Springer, 2017. doi: 10.1007/978-3-319-61295-9.  Google Scholar

[34]

M. A. Schikora-Tamarit, C. Toscano-Ochoa, J. D. Espinos, L. Espinar and L. B. Carey, A synthetic gene circuit for measuring autoregulatory feedback control, Integr. Biol., 8 (2016), 546-555.   Google Scholar

[35]

F. Veerman, C. Marr and N. Popović, Time-dependent propagators for stochastic models of gene expression: an analytical method, J. Math. Biol., 77 (2018), 261-312.  doi: 10.1007/s00285-017-1196-4.  Google Scholar

[36]

H. Wang, P. Liu, Q. Li and T. Zhou, Entangled signal pathways can both control expression stability and induce stochastic focusing, FEBS Lett., 592 (2018), 1135-1149.   Google Scholar

[37]

S. Winkelmann and C. Schütte, Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems, J. Chem. Phys., 147 (2017), 114115. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, D.C., 1972. Google Scholar

[2]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Garland Science New York, 2002. Google Scholar

[3]

U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman & Hall/CRC, 2007.  Google Scholar

[4]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation, PLoS Comput. Biol., 2 (2006), e117. Google Scholar

[5]

P. Bokes and A. Singh, Gene expression noise is affected differentially by feedback in burst frequency and burst size, J. Math. Biol., 74 (2017), 1483-1509.  doi: 10.1007/s00285-016-1059-4.  Google Scholar

[6]

P. Bokes, Y. T. Lin and A. Singh, High cooperativity in negative feedback can amplify noisy gene expression, B. Math. Biol., 80 (2018), 1871-1899.  doi: 10.1007/s11538-018-0438-y.  Google Scholar

[7]

P. Bokes, J. R. King, A. T. A. Wood and M. Loose, Transcriptional bursting diversifies the behaviour of a toggle switch: hybrid simulation of stochastic gene expression, B. Math. Biol., 75 (2013), 351-371.  doi: 10.1007/s11538-013-9811-z.  Google Scholar

[8]

L. Cai, N. Friedman and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362.   Google Scholar

[9]

A. Crudu, A. Debussche, A. Muller and O. Radulescu, others, Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann. Appl. Probab., 22 (2012), 1822Ƀ1859. doi: 10.1214/11-AAP814.  Google Scholar

[10]

P. Czuppon and P. Pfaffelhuber, Limits of noise for autoregulated gene expression, J. Math. Biol., 77 (2018), 1153-1191.  doi: 10.1007/s00285-018-1248-4.  Google Scholar

[11]

R. D. Dar, B. S. Razooky, A. Singh, T. V. Trimeloni, J. M. McCollum, C. D. Cox, M. L. Simpson and L. S. Weinberger, Transcriptional burst frequency and burst size are equally modulated across the human genome, P. Natl. Acad. Sci. USA, 109 (2012), 17454-17459.   Google Scholar

[12]

R. Dessalles, V. Fromion and P. Robert, A stochastic analysis of autoregulation of gene expression, J. Math. Biol., 75 (2017), 1253-1283.  doi: 10.1007/s00285-017-1116-7.  Google Scholar

[13]

M. B. Elowitz, A. J. Levine, E. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.   Google Scholar

[14]

N. Friedman, L. Cai and X. S. Xie, Linking stochastic dynamics to population distribution: An analytical framework of gene expression, Phys. Rev. Lett., 97 (2006), 168302. Google Scholar

[15]

M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2004.  Google Scholar

[16]

G. C. P. Innocentini, M. Forger, O. Radulescu and F. Antoneli, Protein synthesis driven by dynamical stochastic transcription, B. Math. Biol., 78 (2016), 110-131.  doi: 10.1007/s11538-015-0131-3.  Google Scholar

[17]

G. C. P. Innocentini, S. Guiziou, J. Bonnet and O. Radulescu, Analytic framework for a stochastic binary biological switch, Phys. Rev. E, 94 (2016), 062413. Google Scholar

[18]

S. Intep and D. J. Higham, Zero, one and two-switch models of gene regulation, Discrete Cont. Dyn-B., 14 (2010), 495-513.  doi: 10.3934/dcdsb.2010.14.495.  Google Scholar

[19]

J. Jedrak and A. Ochab-Marcinek, Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound poisson process, Phys. Rev. E, 94 (2016), 032401. Google Scholar

[20]

C. Jia, H. Qian, M. Chen and M. Q. Zhang, Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks, J. Chem. Phys., 148 (2018), 095102. Google Scholar

[21]

N. Kumar, T. Platini and R. V. Kulkarni, Exact distributions for stochastic gene expression models with bursting and feedback, Phys. Rev. Lett., 113 (2014), 268105. Google Scholar

[22]

D. R. Larson, R. H. Singer and D. Zenklusen, A single molecule view of gene expression, Trends Cell Biol., 19 (2009), 630-637.   Google Scholar

[23]

G. Lin, J. Yu, Z. Zhou, Q. Sun and F. Jiao, Fluctuations of mrna distributions in multiple pathway activated transcription, Discrete Cont. Dyn-B., 2018. Google Scholar

[24]

Y. T. Lin and N. E. Buchler, Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic markov processes, J. Roy. Soc. Interface, 15 (2018), 20170804. Google Scholar

[25]

Y. T. Lin, P. G. Hufton, E. J. Lee and D. A. Potoyan, A stochastic and dynamical view of pluripotency in mouse embryonic stem cells, PloS Comput. Biol., 14 (2018), e1006000. Google Scholar

[26]

Y. T. Lin and C. R. Doering, Gene expression dynamics with stochastic bursts: Construction and exact results for a coarse-grained model, Phys. Rev. E, 93 (2016), 022409, 10pp. doi: 10.1103/physreve.93.022409.  Google Scholar

[27]

M. Masujima, Applied Mathematical Methods in Theoretical Physics, John Wiley & Sons, 2009. doi: 10.1002/9783527627745.  Google Scholar

[28]

N. A. M. Monk, Oscillatory expression of hes1, p53, and nf-$ \kappa $b driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413.   Google Scholar

[29]

M. Pájaro, A. A. Alonso, I. Otero-Muras and C. Vázquez, Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting, J. Theor. Biol., 421 (2017), 51-70.  doi: 10.1016/j.jtbi.2017.03.017.  Google Scholar

[30]

J. Peccoud and B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.   Google Scholar

[31]

A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. Google Scholar

[32]

J. Ren, F. Jiao, Q. Sun, M. Tang and J. Yu, The dynamics of gene transcription in random environments, Discrete Cont. Dyn-B., 23 (2018), 3167-3194.  doi: 10.3934/dcdsb.2018224.  Google Scholar

[33]

R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, Springer, 2017. doi: 10.1007/978-3-319-61295-9.  Google Scholar

[34]

M. A. Schikora-Tamarit, C. Toscano-Ochoa, J. D. Espinos, L. Espinar and L. B. Carey, A synthetic gene circuit for measuring autoregulatory feedback control, Integr. Biol., 8 (2016), 546-555.   Google Scholar

[35]

F. Veerman, C. Marr and N. Popović, Time-dependent propagators for stochastic models of gene expression: an analytical method, J. Math. Biol., 77 (2018), 261-312.  doi: 10.1007/s00285-017-1196-4.  Google Scholar

[36]

H. Wang, P. Liu, Q. Li and T. Zhou, Entangled signal pathways can both control expression stability and induce stochastic focusing, FEBS Lett., 592 (2018), 1135-1149.   Google Scholar

[37]

S. Winkelmann and C. Schütte, Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems, J. Chem. Phys., 147 (2017), 114115. Google Scholar

Figure 1.  A: Random telegraph model for gene expression. Protein concentration increases if gene is On and decreases if gene is Off. Short On periods accompanied by fast production lead to burst-like gene expression. B: The burst dynamics can be deciphered by looking on the fast timescale (the scaling factor is set to $ \varepsilon = 0.05 $ here). C: Protein dynamics in the reduced drift-jump model. Between bursts (vertical dotted lines) the protein level decays exponentially (solid lines; one such period is demarcated by the braces). D: Positive feedback response function of the Michaelis-Menten type (11) as function of protein level $ x $ for selected values of the threshold level $ \kappa $. The dotted black lines ending in coloured markers indicate that the response is half-maximal at $ x = \kappa $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Transcritical bifurcation of steady-state protein mean in the undelayed and infinitesimally delayed models. Analytical results (13) and (23) (solid lines) are cross-validated with kinetic Monte Carlo estimates (34) (discrete markers). Each individual marker gives the average value based on one hundred independent estimates (34)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[2]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907
[3]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885
[4]

Jagadeesh R. Sonnad, Chetan T. Goudar. Solution of the Michaelis-Menten equation using the decomposition method. Mathematical Biosciences & Engineering, 2009, 6 (1) : 173-188. doi: 10.3934/mbe.2009.6.173
[5]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217
[6]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537
[7]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129
[8]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
[9]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[10]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[11]

Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015
[12]

István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134
[13]

Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826
[14]

Karl Peter Hadeler. Michaelis-Menten kinetics, the operator-repressor system, and least squares approaches. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1541-1560. doi: 10.3934/mbe.2013.10.1541
[15]

Jean-Michel Roquejoffre, Juan-Luis Vázquez. Ignition and propagation in an integro-differential model for spherical flames. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 379-387. doi: 10.3934/dcdsb.2002.2.379
[16]

Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417
[17]

Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591
[18]

Urszula Ledzewicz, Helen Moore. Optimal control applied to a generalized Michaelis-Menten model of CML therapy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 331-346. doi: 10.3934/dcdsb.2018022
[19]

Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261
[20]

Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (34)
  • HTML views (282)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences