Research article Special Issues

Modeling stochastic gene expression: From Markov to non-Markov models

  • Gene expression is an inherently noisy process due to low copy numbers of mRNA or protein. The involved reaction events may happen in a Markov fashion but also in a non-Markov manner, depending on waiting-time distributions for the occurrence of reaction events. In recent years, many mechanistic models of stochastic gene expression have been developed to forecast fluctuations in mRNA or protein levels. Here we discus commonalities between these models as well as their extensions from Markov to non-Markov models, focusing on the contributions of noisy sources to the protein level. We derive a useful formula for the protein noise quantified by the ratio of the variance over the squared mean. This formula, expressed in terms of the frequencies of the probabilistic events, can be used in the fast evaluation of fluctuations in the protein abundance. Although the detail of the formula may vary from gene to gene, it highlights sources of the protein noise, which can be decomposed into two parts: spontaneous fluctuations resulting from the birth and death of the protein and forced fluctuations originated from switching between the promoter states.

    Citation: Zhenquan Zhang, Junhao Liang, Zihao Wang, Jiajun Zhang, Tianshou Zhou. Modeling stochastic gene expression: From Markov to non-Markov models[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5304-5325. doi: 10.3934/mbe.2020287

    Related Papers:

    [1] Qiqi Deng, Aimin Chen, Huahai Qiu, Tianshou Zhou . Analysis of a non-Markov transcription model with nuclear RNA export and RNA nuclear retention. Mathematical Biosciences and Engineering, 2022, 19(8): 8426-8451. doi: 10.3934/mbe.2022392
    [2] Dawid Czapla, Sander C. Hille, Katarzyna Horbacz, Hanna Wojewódka-Ściążko . Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process. Mathematical Biosciences and Engineering, 2020, 17(2): 1059-1073. doi: 10.3934/mbe.2020056
    [3] ZongWang, Qimin Zhang, Xining Li . Markovian switching for near-optimal control of a stochastic SIV epidemic model. Mathematical Biosciences and Engineering, 2019, 16(3): 1348-1375. doi: 10.3934/mbe.2019066
    [4] H.Thomas Banks, Shuhua Hu . Nonlinear stochastic Markov processes and modeling uncertainty in populations. Mathematical Biosciences and Engineering, 2012, 9(1): 1-25. doi: 10.3934/mbe.2012.9.1
    [5] Linard Hoessly, Carsten Wiuf . Fast reactions with non-interacting species in stochastic reaction networks. Mathematical Biosciences and Engineering, 2022, 19(3): 2720-2749. doi: 10.3934/mbe.2022124
    [6] Kseniia Kravchuk, Alexander Vidybida . Non-Markovian spiking statistics of a neuron with delayed feedback in presence of refractoriness. Mathematical Biosciences and Engineering, 2014, 11(1): 81-104. doi: 10.3934/mbe.2014.11.81
    [7] Jiangtao Dai, Ge Guo . A leader-following consensus of multi-agent systems with actuator saturation and semi-Markov switching topologies. Mathematical Biosciences and Engineering, 2024, 21(4): 4908-4926. doi: 10.3934/mbe.2024217
    [8] Linda J. S. Allen, Vrushali A. Bokil . Stochastic models for competing species with a shared pathogen. Mathematical Biosciences and Engineering, 2012, 9(3): 461-485. doi: 10.3934/mbe.2012.9.461
    [9] Yan Wang, Tingting Zhao, Jun Liu . Viral dynamics of an HIV stochastic model with cell-to-cell infection, CTL immune response and distributed delays. Mathematical Biosciences and Engineering, 2019, 16(6): 7126-7154. doi: 10.3934/mbe.2019358
    [10] Nikolai Leonenko, Enrica Pirozzi . The time-changed stochastic approach and fractionally integrated processes to model the actin-myosin interaction and dwell times. Mathematical Biosciences and Engineering, 2025, 22(4): 1019-1054. doi: 10.3934/mbe.2025037
  • Gene expression is an inherently noisy process due to low copy numbers of mRNA or protein. The involved reaction events may happen in a Markov fashion but also in a non-Markov manner, depending on waiting-time distributions for the occurrence of reaction events. In recent years, many mechanistic models of stochastic gene expression have been developed to forecast fluctuations in mRNA or protein levels. Here we discus commonalities between these models as well as their extensions from Markov to non-Markov models, focusing on the contributions of noisy sources to the protein level. We derive a useful formula for the protein noise quantified by the ratio of the variance over the squared mean. This formula, expressed in terms of the frequencies of the probabilistic events, can be used in the fast evaluation of fluctuations in the protein abundance. Although the detail of the formula may vary from gene to gene, it highlights sources of the protein noise, which can be decomposed into two parts: spontaneous fluctuations resulting from the birth and death of the protein and forced fluctuations originated from switching between the promoter states.


    1. Introduction

    Hafnia alvei is a Gram-negative facultatively anaerobic bacillus that belongs to the family Enterobacteriaceae. It is known to be among the Enterobacteriaceaespecies most commonly isolated from vacuum-packed chilled meat samples, raw milk, raw fish and other foods [1,2,3]. H. alvei is also an opportunistic human pathogen involved in various types of nosocomical infections [4,5,6,7]. Though H. alvei isolation from human clinical specimens remains uncommon, development of drug resistance by this species is emerging and it is likely that this organism will gain increasing importance in the future [8]. Moreover, these bacteria have been considered as opportunistic pathogens in several animal species including mammals, fish, insects and birds [4,9,10]. Although some virulence traits have been studied in H. alvei, little is known about the factors that contribute to their pathogenesis within a host, including biofilm formation [11,12,13]. Biofilms can be defined as a structured community of bacterial cells enclosed in a self-produced polymeric matrix and adherent to an inert or living surface. Growth in biofilm enables bacterial populations to survive better in hospital environments, food-processing environments and during host infections, increasing the probability of causing infections. In addition, some antiseptics and disinfectants extensively used in hospitals are ineffective against pathogens growing as biofilms attached to surfaces [14]. Furthermore, biofilm formation has been connected to infections associated with indwelling medical devices, such as central venous or urinary catheters [15]. We recently used the next generation sequencing (NGS) technologies to obtain the complete genome sequence of H. alvei HUMV-5920 isolated from the urine sample from a human patient [16]. The goal of the present study was to analyze the capacity of an H. alvei HUMV-5920 human isolate to form biofilms. Our results confirmed the presence of genes already described for the modulation of multicellular behavior in Salmonella Typhimurium and other Enterobacteriaceae species, but also revealed new aspects of the biofilm formation process. This should contribute to a better understanding of the correlation between adherence capabilities and the pathogenicity of this bacterium.


    2. Materials and Methods


    2.1. Bacterial strains

    The strain used in this study (HUMV-5920) was isolated from the urine sample from a woman at the Hospital Universitario Marqués de Valdecilla in Santander, Spain [16]. The strain was routinely cultured in Luria-Bertani (LB) agar or broth and frozen at -80 °C with 20% glycerol. Strains Salmonella Enteritidis 3934 with rdar, bdar, pdar and saw morphotypes (kindly provided by Dr. Toledo-Arana) were used as positive controls for the different phenotypes.


    2.2. Growth conditions and morphotype identification

    Hafnia alvei was directly streaked on Congo Red (CR) plates, which are LB plates without salt supplemented with CR (40 µg/ml) and Coomassie brilliant blue (20 µg/ml). Colony morphology was assessed on CR plates after growth at 37 °C for 24 h and at 28 °C for 48 h according to the basic morphotypes detected in S. Enteritidis 3934: rdar (expresses curli fimbriae and cellulose), pdar (expresses cellulose), bdar (expresses curli fimbriae) and saw (no expression of curli fimbriae nor cellulose) [17,18]. Indication for cellulose production was obtained when fluorescent colonies were observed under a 366-nm UV light source after growth on calcofluor (fluorescent brightener) plates (LB agar plates with 50 µM calcofluor).


    2.3. Biofilm formation

    Biofilm formation was estimated in 48-well Ubottom polystyrene microtiter plates (Nunc, Thermo Fisher Scientific) by the method of O’Toole and Kolter with some modifications [19]. H. alvei was grown in LB medium for 24 h at 37 °C with shaking (175 rpm), and a 1:1,000 dilution was prepared in PBS (OD620, 0.01). Twenty-five microliters were placed in each well containing 500 µl of culture medium. The microplates were incubated under static conditions for 48 h at 28 °C or 37 °C. Wells containing biofilms were rinsed three times with distilled water (1 ml/well), and stained with 1ml/well of crystal violet (CV) (0.7% [wt/vol] solution; Sigma-Aldrich) for 12 min. Excess stain was removed by three washes with distilled water. Crystal violet was extracted by an ethanol-acetone solution (80:20, vol/vol), and the plates were incubated at RT in an orbital shaker for 1 min at 400 rpm (Thermomixer comfort; Eppendorf) to release the dye into the solution. Then, a sample of 100 µl from each well was transferred to another 96-well flat-bottom plate, and the amount of dye (proportional to the density of adherent cells) was determined at 620 nm using a microplate reader (Multiskan FC; Thermo Fisher). In each experiment, results were corrected for background staining by subtracting the value for crystal violet bound to uninoculated controls. The biofilm assay was performed three times, with duplicates in each assay. To confirm the role of cellulose in the biofilm formation process of the wild-type strain H. alvei HUMV-5920, the biofilm-forming assay was carried out in the presence of 0.1% cellulase (Sigma) as described elsewhere [20]. Biofilm formation was studied also using Tryptic Soy Broth (TSB) and Brain Heart Infusion Broth (BHIB) as culture media.


    2.4. Confocal Laser Scanning Microscopy (CLSM)

    Bacteria were grown in 4-well μ-chamber uncoated slides (Ibidi, Martinsried) without shaking, in presence or absence of 0.1% cellulase (Sigma). The slides were incubated at 37 °C for 48 h. After 48 h, planktonic cells were removed by rinsing with saline (0.85% NaCl) and bacterial viability within biofilms in chambers with or without cellulose was determined by adding 200 ml of the BacLight LIVE/DEAD bacterial viability kit (Molecular Probes Inc.) per well for 25 min. For LIVE/DEAD a 488/561 nm excitation, 500-550/570-620 nm emission filters were used respectively. A series of optical sections were obtained with a Nikon A1R confocal scanning laser microscope. Images were captured at random with a ×20 Plan Apo 0.75 NA objective. Reconstructions of confocal sections were assembled using the NIS-Elements 3.2 software. Z-stacks of confocal images were rendered into 3D mode using the ImageJ software.


    2.5. Sequence analysis of cellulose biosynthesis genes

    Genome-wide comparative and interspecies clustering analyses of protein coding sequences (CDSs) related to cellulose production were performed using a new version of the Conserved Domain Database (CDD) Version 3.15 and UGENE Version 1.24.


    3. Results

    Hafnia alvei strain HUMV-5920 was able to form biofilms in microtiter plates and glass. The amount of biofilm formed in LB or in TSB medium was higher than that of cells grown in BHIB (Figure 1). Interestingly, 28 °C was the most favorable temperature to form biofilms in all culture media under static conditions in comparison with 37 °C.

    Figure 1. Biofilm formation in H. alvei HUMV-5920 in conventional media and polystyrene plates. Biofilm formation on polystyrene surface after 48 h was assessed by crystal violet staining. Each bar indicates the mean values with standard deviations. H. alvei was cultivated in LB (black bars), BHIB (grey bars) or TSB (white bars) at both temperatures.

    Calcofluor binding assays indicate putative production of cellulose in this strain (Figure 2).

    Figure 2. Calcofluor staining of morphotype in H. alvei 5920. Strain HUMV-5920 fluoresced on calcofluor agar plates (left) in comparison with other H. alvei strains from different origin. Morphotype was compared with the phenotype of different mutants of S. Enteritidis 3934 (right). A, B, C and D corresponding to S. Enteritidis 3934 (pdar morphotype), S. Enteritidis 3934 (rdar morphotype), S. Enteritidis 3934 (bdar morphotype) and S. Enteritidis 3934 (saw morphotype) respectively.

    Since calcofluor binding is not absolutely specific for cellulose (a 1,4-β-glucan), we confirmed cellulose production by an enzymatic assay. The presence of cellulase in the media during the biofilm formation process in H. alvei led to a total absence of a visible biofilm in LB medium without salts (Figure 3).

    Figure 3. Cellulase treatment. Example of a CV solubilisation after biofilm formation by H. alvei HUMV-5920 and Salmonella strains during growth on LB without salts, in presence (+), or absence (-) of cellulase. C, controls (uninoculated wells).

    Similar results were obtained after staining of Hafnia alvei and Salmonella biofilms and processed by CLSM. Our findings also showed that bacterial attachment and biofilm formation can be greatly prevented by cellulase treatment in H. alvei as well as in the rdar morphotype of S. Enteritidis (expresses curli fimbriae and cellulose), whereas pdar (expresses only cellulose) and bdar (expresses only curli fimbriae) were less affected. The saw morphotype (no expression of curli fimbriae nor cellulose) was not significantly affected. Representative CLSM images are shown in Figure 4.

    Figure 4. Confocal Laser Scanning Microscopic images of strains (H. alvei and S. Enteritidis 3934 (pdar, rdar, bdar, and saw morphotypes) after growth on uncoated 4-well chamber slides in presence or absence of cellulase and stained with the LIVE/DEAD viability kit. Live cells are stained green with Syto 9 dye and dead cells are stained red with propidium iodide. Original magnification ×200.

    Genome-wide comparative and interspecies clustering analyses of CDSs related to cellulose production were performed with E. coli and Salmonella enterica, yielding interesting information. The basic structural genomic organization of the cellulose biosynthesis system present on both reference species (operons bcsABZC and bcsEFG) is also present in the H. alvei HUMV-5920 chromosome (Figure 5).

    Figure 5. Comparison of the genomic loci bearing conserved genes related to cellulose production and their background. The displayed operons are from (top to down): Escherichia coli O157:H7 (accession no: NC_002695.1); Salmonella enterica subsp. enterica serovar Typhimurium str. LT2 (accession no NCBI: NC_003197.1) and Hafnia alvei HUMV-5920 (accession no: CP015379.1). Products of the core genes of various cellulose synthase operons (protein name and functional annotation) were: BcsA, Cellulose synthase catalytic subunit A; BcsB, Cellulose synthase subunit B (periplasmic); BcsC, Cellulose synthase subunit C, spans periplasm and outer membrane; BcsE, Cellulose synthase cytoplasmic subunit E, binds c-di-GMP; BcsF, Membrane-anchored subunit (1 TM segment); BcsG, Contains 4 TM segments and a periplasmic AlkP domain; BcsR, Likely regulatory subunit; BcsZ, Endo-β-1,4-glucanase (cellulase), periplasmic.

    4. Conclusion

    The ability of opportunistic pathogens to form biofilms is of significant clinical interest, since biofilm formation influences the efficacy of antimicrobial therapy and the outcome of an infection. Moreover, biofilm formation may contribute to the establishment and long-term survival of bacterial pathogens in the hospital environment or in food processing environments [21]. A number of previous studies have shown that the nutrient content of the growth medium influences biofilm development in different organisms. More specifically, environmental factors such as glucose and temperature affect biofilm development in Enterobacteriaceae and other bacteria [22,23]. In this line, we have shown that environmental factors such as temperature affect biofilm development in H. alvei [12,13]. In agreement with our previous reports, low temperature induces high biofilm formation in strain HUMV-5920.

    A four-gene bcsABCD operon involved in cellulose biosynthesis was initially identified in Acetobacter xylinus. An early analysis of the cellulose synthase operons in E. coli and S. enterica indicated the presence of the same bcsA, bcsB, and bcsC genes and two additional genes, yhjQ and bcsZ conforming a new variant, the bcsABZC operon. This locus also contains a divergent operon, bcsEFG, which is also required for cellulose production [24]. All the four proteins in the bcsABZC operon were required for maximal cellulose production in vivo with BcsC involved in exporting the glucan molecules and packing them at the cell surface [24], and all these genes are present in the H. alvei HUMV-5920 chromosome [16]. The production of an extracellular matrix is recognized as a key element in determining the mature biofilm architecture [25,26]. Cellulose production prepares bacteria for surface colonization because it has a high capacity of water retention, mediates cell-cell interactions and cell adherence [27].

    To date, only two other H. alvei genomes have been fully sequenced: strain FB1, from a kind of fish food in Asia [28] and strain FDAARGOS-158 (GenBank sequence CP014031.1) from stool. Our genomic analysis also suggests a high frequency of putative adhesins and toxins produced by Hafnia isolates and also several genes implicated in antimicrobial resistance such as metallo-and beta-lactamases, multidrug export proteins and efflux pumps, and penicillin-binding proteins (PBPs) [16]. The genotypic and phenotypic characteristics of Hafnia alvei strains, in particular our clinical isolate, could reflect the organism’s ability to sense and adapt to changes in its environment. Moreover, the genomics of this strain will accelerate research on H. alvei in numerous domains and will provide new insights into the genetic mechanisms responsible for the virulence in this species. These data and recent preoccupant findings on antimicrobial resistance in Hafnia strains [8] claim for further efforts to examine the precise significance of Hafnia strains in the clinical settings.


    Acknowledgements

    J.R.-V. holds a Miguel Servet II contract from the Instituto de Salud Carlos III, Spain. M.L.-D. holds a contract from the Instituto de Investigación Sanitaria Valdecilla IDIVAL and Universidad de Cantabria (no. PREVAL16/05). S.R.-S. holds a contract from the Instituto de Investigación Sanitaria Valdecilla IDIVAL.


    Conflict of Interest

    All authors declare no conflicts of interest in this paper.




    [1] A. Sanchez, S. Choubey, J. Kondev, Stochastic models of transcription: From single molecules to single cells, Methods, 62 (2013), 13-25.
    [2] J. Zhang, T. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.
    [3] T. Zhou, J. Zhang, Analytical results for a multistate gene model, SIAM J. Appl. Math., 72 (2012), 789-818.
    [4] L. Cai, N. Friedman, X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362.
    [5] T. Liu, J. Zhang, T. Zhou, Effect of interaction between chromatin loops on cell-to-cell variability in gene expression, PLoS Comput. Biol., 12 (2016), e1004917.
    [6] D. R. Rigney, W. C. Schieve, Stochastic model of linear, continuous protein-synthesis in bacterial populations, J. Theor. Biol., 69 (1977), 761-766.
    [7] O. G. Berg, A model for the statistical fluctuations of protein numbers in a microbial population, J. Theor. Biol., 71 (1978), 587-603.
    [8] P. K. Tapaswi, R. K. Roychoudhury, T. Prasad, A stochastic model of gene activation and RNA synthesis during embryogenesis, Sankhyā Indian J. Statist. Ser. B, 49 (1987), 51-67.
    [9] J. Peccoud, B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.
    [10] D. R. Rigney, Stochastic model of constitutive protein levels in growing and dividing bacterial cells, J. Theor. Biol., 76 (4), 453-480.
    [11] D. R. Rigney, Stochastic models of cellular variability. In Kinetic Logic A Boolean Approach to the Analysis of Complex Regulatory Systems, Springer, Berlin, Heidelberg, 1979.
    [12] T. B. Kepler, T. C. Elston, Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations, Biophys. J., 81 (2001), 3116-3136.
    [13] M. Thattai, A. Van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. U.S.A., 98 (2001), 8614-8619.
    [14] P. S. Swain, M. B. Elowitz, E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Natl. Acad. Sci. U.S.A., 99 (2002), 12795-12800.
    [15] M. Sasai, P. G. Wolynes, Stochastic gene expression as a many-body problem, Proc. Natl. Acad. Sci. U.S.A., 100 (2003), 2374-2379.
    [16] T. Jia, R. V. Kulkarni, Intrinsic noise in stochastic models of gene expression with molecular memory and bursting, Phys. Rev. Lett., 106 (2011), 058102.
    [17] Z. Cao, R. Grima, Linear mapping approximation of gene regulatory networks with stochastic dynamics, Nat. Commun., 9 (2018), 1-15.
    [18] C. V. Harper, B. Finkenstädt, D. J. Woodcock, S. Friedrichsen, S. Semprini, L. Ashall, et al., Dynamic analysis of stochastic transcription cycles, PLoS Biol., 9 (2011), e1000607.
    [19] M. R. Green, Eukaryotic transcription activation: Right on target, Mol. Cell, 18 (2005), 399-402.
    [20] J. Paulsson, Models of stochastic gene expression, Phys. Life Rev., 2 (2005), 157-175.
    [21] G. Hornung, R. Bar-Ziv, D. Rosin, N. Tokuriki, D. S. Tawfik, M. Oren, et al., Noise-mean relationship in mutated promoters, Genom. Res., 22 (2012), 2409-2417.
    [22] Q. Li, G. Barkess, H. Qian, Chromatin looping and the probability of transcription, Trend. Genet., 22 (2006), 197-202.
    [23] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer, Berlin, Heidelberg, 2004.
    [24] J. D. Jordan, E. M. Landau, R. Iyengar, Signaling networks: The origins of cellular multitasking, Cell, 103 (2000), 193-200.
    [25] L. Bintu, J. Yong, Y. E. Antebi, K. McCue, Y. Kazuki, N. Uno, et al., Dynamics of epigenetic regulation at the single-cell level, Science, 351 (2016), 720-724.
    [26] C. W. Gardiner, Stochastic Methods: a handbook for the natural and social sciences, Springer, New York, 2009.
    [27] N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 2007.
    [28] E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, vol 796, John Wiley & Sons, New York, 2008.
    [29] H. Andersson, T. Britton, Stochastic epidemic models and their statistical analysis, vol. 151, Springer Science & Business Media, 2012.
    [30] M. Salathé, M. Kazandjieva, J. W. Lee, P. Levis, M. W. Feldman, J. H. Jones, A high-resolution human contact network for infectious disease transmission, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), 22020-22025.
    [31] A. Corral, Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes, Phys. Rev. Lett., 92 (2004), 108501.
    [32] P. S. Stumpf, R. C. Smith, M. Lenz, A. Schuppert, F. J. Müller, A. Babtie, et al., Stem cell differentiation as a non-Markov stochastic process, Cell Syst., 5 (2017), 268-282.
    [33] D. M. Suter, N. Molina, D. Gatfield, K. Schneider, U. Schibler, F. Naef, Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474.
    [34] T. Guérin, O. Bénichou, R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem., 4 (2012), 568-573.
    [35] A. L. Barabasi, The origin of bursts and heavy tails in human dynamics, Nature, 435 (2005), 207-211.
    [36] J. M. Pedraza, J. Paulsson, Effects of molecular memory and bursting on fluctuations in gene expression, Science, 319 (2008), 339-343.
    [37] G. Srinivasan, D. M. Tartakovsky, B. A. Robinson, A. B. Aceves, Quantification of uncertainty in geochemical reactions, Water Resour. Res., 43 (2007), W12415.
    [38] S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, J. Klafter, First-passage times in complex scale-invariant media, Nature, 450 (2007), 77-80.
    [39] G. Guigas, M. Weiss, Sampling the cell with anomalous diffusion—the discovery of slowness, Biophys. J., 94 (2008), 90-94.
    [40] Y. Meroz, I. M. Sokolov, J. Klafter, Distribution of first-passage times to specific targets on compactly explored fractal structures, Phys. Rev. E, 83 (2011), 020104.
    [41] M. Dentz, A. Russian, P. Gouze, Self-averaging and ergodicity of subdiffusion in quenched random media, Phys. Rev. E, 93 (2016), 010101.
    [42] A. A. Ovchinnikov, Y. B. Zeldovich, Role of density fluctuations in bimolecular reaction kinetics, Chem. Phys., 28 (1978), 215-218.
    [43] M. Dobrzyński, F. J. Bruggeman, Elongation dynamics shape bursty transcription and translation, Proc. Natl. Acad. Sci. U.S.A., 106 (2009), 2583-2588.
    [44] D. R. Larson, D. Zenklusen, B. Wu, J. A. Chao, R. H. Singer, Real-time observation of transcription initiation and elongation on an endogenous yeast gene, Science, 332 (2011), 475-478.
    [45] S. Yunger, L. Rosenfeld, Y. Garini, Y. Shav-Tal, Single-allele analysis of transcription kinetics in living mammalian cells, Nat. Methods, 7 (2010), 631-633.
    [46] I. Golding, J. Paulsson, S. M. Zawilski, E. C. Cox, Real-time kinetics of gene activity in individual bacteria, Cell, 123 (2005), 1025-1036.
    [47] T. Muramoto, D. Cannon, M. Gierliński, A. Corrigan, G. J. Barton, J. R. Chubb, Live imaging of nascent RNA dynamics reveals distinct types of transcriptional pulse regulation, Proc. Natl. Acad. Sci. U.S.A., 109 (2012), 7350-7355.
    [48] A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas, S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309.
    [49] D. G. Spiller, C. D. Wood, D. A. Rand, M. R. White, Measurement of single-cell dynamics, Nature, 465 (2010), 736-745.
    [50] A. Eldar, M. B. Elowitz, Functional roles for noise in genetic circuits, Nature, 467 (2010), 167-173.
    [51] B. Zoller, D. Nicolas, N. Molina, F. Naef, Structure of silent transcription intervals and noise characteristics of mammalian genes, Mol. Syst. Biol., 11 (2015), 823.
    [52] T. R. Sokolowski, T. Erdmann, P. R. Ten Wolde, Mutual repression enhances the steepness and precision of gene expression boundaries, PLoS Comput. Biol., 8 (2012), e1002654.
    [53] J. Paulsson, Summing up the noise in gene networks, Nature, 427 (2001), 415-418.
    [54] D. R. Larson, What do expression dynamics tell us about the mechanism of transcription?, Curr. Opin. Gen. Dev., 21 (2011), 591-599.
    [55] V. Shahrezaei, P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. U.S.A., 105 (2008), 17256-17261.
    [56] N. Kumar, A. Singh, R. V. Kulkarni, Transcriptional bursting in gene expression: analytical results for general stochastic models, PLoS Comput. Biol., 11 (2015), e1004292.
    [57] Z. Wang, Z. Zhang, T. Zhou, Exact distributions for stochastic models of gene expression with arbitrary regulation, Sci. China Math., 63 (2020), 485-500.
    [58] P. Liu, Z. Yuan, L. Huang, T. Zhou, Roles of factorial noise in inducing bimodal gene expression, Phys. Rev. E, 91 (2015), 062706.
    [59] J. Zhang, Q. Nie, T. Zhou, A moment-convergence method for stochastic analysis of biochemical reaction networks, J. Chem. Phys., 144 (2016), 194109.
    [60] A. B. O. Daalhuis, Confluent hypergeometric functions, NIST Handb. Math. Funct., 2010.
    [61] T. Aquino, M. Dentz, Chemical continuous time random walks, Phys. Rev. Lett., 119 (2017), 230601.
    [62] N. Masuda, M. A. Porter, R. Lambiotte, Random walks and diffusion on networks, Phys. Rep., 716 (2017), 1-58.
    [63] R. Kutner, J. Masoliver, The continuous time random walk, still trendy: fifty-year history, state of art and outlook, Eur. Phys. J. B, 90 (2017), 50.
    [64] L. Liu, B. R. K. Kashyap, J. G. C. Templeton, On the GIX/G/∞ system, J. Appl. Prob., 27 (1990), 671-683.
    [65] A. R. Stinchcombe, C. S. Peskin, D. Tranchina, Population density approach for discrete mRNA distributions in generalized switching models for stochastic gene expression, Phys. Rev. E, 85 (2012), 061919.
    [66] N. Masuda, L. E. Rocha, A Gillespie algorithm for non-Markovian stochastic processes, SIAM Rev., 60 (2018), 95-115.
    [67] C. Deneke, R. Lipowsky, A. Valleriani, Complex degradation processes lead to non-exponential decay patterns and age-dependent decay rates of messenger RNA, PloS One, 8 (2013), e55442.
    [68] B. C. Arnold, Majorization: Here, there and everywhere, Statist. Sci., 22 (2007), 407-413.
    [69] A. David, S. Larry, The least variable phase type distribution is Erlang, Stochastic Models, 3 (1987), 467-473.
    [70] J. Zhang, T. Zhou, Markovian approaches to modeling intracellular reaction processes with molecular memory, Proc. Natl. Acad. Sci. U.S.A., 116 (2019), 23542-23550.
    [71] H. Qiu, B. Zhang, T. Zhou, Analytical results for a generalized model of bursty gene expression with molecular memory, Phys. Rev. E, 100 (2019), 012128.
    [72] A. Coulon, C. C. Chow, R. H. Singer, D. R. Larson, Eukaryotic transcriptional dynamics: From single molecules to cell populations, Nat. Rev. Genet., 14 (2013), 572-584.
    [73] W. J. Blake, M. Kærn, C. R. Cantor, J. J. Collins, Noise in eukaryotic gene expression, Nature, 422 (2003), 633-637.
    [74] J. M. Raser, E. K. O'Shea, Control of stochasticity in eukaryotic gene expression, Science, 304 (2004), 1811-1814.
    [75] N. Friedman, L. Cai, X. S. Xie, Linking stochastic dynamics to population distribution: an analytical framework of gene expression, Phys. Rev. Lett., 97 (2006), 168302.
    [76] A. M. Kringstein, F. M. Rossi, A. Hofmann, H. M. Blau, Graded transcriptional response to different concentrations of a single transactivator, Proc. Natl. Acad. Sci. U.S.A., 95 (1998), 13670-13675.
    [77] J. Stewart-Ornstein, C. Nelson, J. DeRisi, J. S. Weissman, H. El-Samad, Msn2 coordinates a stoichiometric gene expression program, Curr. Biol., 23 (2013), 2336-2345.
    [78] J. Paulsson, M. Ehrenberg, Noise in a minimal regulatory network: plasmid copy number control, Quart. Rev. Biophys., 34 (2001), 1-59.
    [79] M. B. Elowitz, A. J. Levine, E. D. Siggia, P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.
  • This article has been cited by:

    1. Washington Luiz Caneschi, Angélica Bianchini Sanchez, Érica Barbosa Felestrino, Camila Gracyelle de Carvalho Lemes, Isabella Ferreira Cordeiro, Natasha Peixoto Fonseca, Morghana Marina Villa, Izadora Tabuso Vieira, Lauro Ângelo Gonçalves Moraes, Renata de Almeida Barbosa Assis, Flávio Fonseca do Carmo, Luciana Hiromi Yoshino Kamino, Robson Soares Silva, Jesus Aparecido Ferro, Maria Inês Tiraboschi Ferro, Rafael Marini Ferreira, Vera Lúcia Santos, Ubiana de Cássia Mourão Silva, Nalvo Franco Almeida, Alessandro de Mello Varani, Camila Carrião Machado Garcia, João Carlos Setubal, Leandro Marcio Moreira, Serratia liquefaciens FG3 isolated from a metallophyte plant sheds light on the evolution and mechanisms of adaptive traits in extreme environments, 2019, 9, 2045-2322, 10.1038/s41598-019-54601-4
    2. Annika Cimdins, Roger Simm, 2017, Chapter 18, 978-1-4939-7239-5, 225, 10.1007/978-1-4939-7240-1_18
    3. José Ramos-Vivas, Olga Tapia, María Elexpuru-Zabaleta, Kilian Tutusaus Pifarre, Yasmany Armas Diaz, Maurizio Battino, Francesca Giampieri, The Molecular Weaponry Produced by the Bacterium Hafnia alvei in Foods, 2022, 27, 1420-3049, 5585, 10.3390/molecules27175585
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5896) PDF downloads(260) Cited by(4)

Article outline

Figures and Tables

Figures(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog