zbMATH — the first resource for mathematics

Mean field analysis of a spatial stochastic model of a gene regulatory network. (English) Zbl 1350.92020
Summary: A gene regulatory network may be defined as a collection of DNA segments which interact with each other indirectly through their RNA and protein products. Such a network is said to contain a negative feedback loop if its products inhibit gene transcription, and a positive feedback loop if a gene product promotes its own production. Negative feedback loops can create oscillations in mRNA and protein levels while positive feedback loops are primarily responsible for signal amplification. It is often the case in real biological systems that both negative and positive feedback loops operate in parameter regimes that result in low copy numbers of gene products. In this paper we investigate the spatio-temporal dynamics of a single feedback loop in a eukaryotic cell. We first develop a simplified spatial stochastic model of a canonical feedback system (either positive or negative). Using a Gillespie’s algorithm, we compute sample trajectories and analyse their corresponding statistics. We then derive a system of equations that describe the spatio-temporal evolution of the stochastic means. Subsequently, we examine the spatially homogeneous case and compare the results of numerical simulations with the spatially explicit case. Finally, using a combination of steady-state analysis and data clustering techniques, we explore model behaviour across a subregion of the parameter space that is difficult to access experimentally and compare the parameter landscape of our spatio-temporal and spatially-homogeneous models.

92C40 Biochemistry, molecular biology
92C42 Systems biology, networks
93B52 Feedback control
Full Text: DOI
[1] Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P (2008) Molecular biology of the cell. Garland Science, 5th edn. Taylor and Francis Group Ltd, Oxford · Zbl 1405.92085
[2] Barik, D; Baumann, WT; Paul, MR; Novak, B; Tyson, JJ, A model of yeast cell-cycle regulation based on multisite phosphorylation, Mol Syst Biol, 6, 405, (2010)
[3] Barik, D; Paul, MR; Baumann, WT; Cao, Y; Tyson, JJ, Stochastic simulation of enzyme-catalyzed reactions with disparate timescales, Biophys J, 95, 3563-3574, (2008)
[4] Barrio, M; Burrage, K; Leier, A; Tian, T, Oscillatory regulation of hes1: discrete stochastic delay modelling and simulation, PLoS ONE, 2, e117, (2006)
[5] Boyer, L; Lee, T; Cole, M; Johnstone, S; Levine, S; Zucker, J; Guenther, M; Kumar, R; Murray, H; Jenner, R; Gifford, D; Melton, D; Jaenisch, R; Young, R, Core transcriptional regulatory circuitry in human embryonic stem cells, Cell, 122, 947-956, (2005)
[6] Buratti, E; Baralle, FE, New aspects of autoregulation mechanisms in RNA binding proteins and their connection with human disease, FEBS J, 278, 3530-3538, (2011)
[7] Busenberg, S; Mahaffy, JM, Interaction of spatial diffusion and delays in models of genetic control by repression, J Math Biol, 22, 313-333, (1985) · Zbl 0593.92010
[8] Cabal, GG; Genovesio, A; Rodriguez-Navarro, S; Zimmer, C; Gadal, O; Lesne, A; Buc, H; Feuerbach-Fournier, F; Olivo-Marin, J; Hurt, EC; Nehrbass, U, SAGA interacting factors confine sub-diffusion of transcribed genes to the nuclear envelope, Nature, 441, 770-773, (2006)
[9] Caravagna, G; Mauri, G; d’Onofrio, A, The interplay of intrinsic and extrinsic bounded noises in bimolecular networks, PLoS ONE, 8, e51174, (2013)
[10] Ciliberto, A; Novak, B; Tyson, JJ, Steady states and oscillations in the p53/mdm2 network, Cell Cycle, 4, 488-493, (2005)
[11] Franciscis, S; d’Onofrio, A, Cellular polarization: interaction between extrinsic noises and the wave-pinning mechanism, Phys Rev E, 88, 032709, (2013)
[12] Eichenberger, P; Fujita, M; Jensen, S; Conlon, E; Rudner, D; Wang, S; Ferguson, C; Haga, K; Sato, T; Liu, J; Losick, R, The program of gene transcription for a single differentiating cell type during sporulation in bacillus subtilis, PLoS Biol., 2, e328, (2004)
[13] Erban R, Chapman J, Maini P (2007) A practical guide to stochastic simulations of reaction-diffusion processes. arXiv preprint. arXiv:0704.1908
[14] Fall CP, Marland ES, Wagner JM, Tyson JJ (2002) Computational cell biology, 5th edn. Springer, New York · Zbl 1010.92019
[15] Fusco, D; Accornero, N; Lavoie, B; Shenoy, SM; Blanchard, J; Singer, RH; Bertrand, E, Single mrna molecules demonstrate probabilistic movement in living Mammalian cells, Curr Biol, 13, 161-167, (2003)
[16] Gamba, A; Candia, A; Talia, SD; Coniglio, A; Bussolino, F; Serini, G, Diffusion-limited phase separation in eukarytoic chemotaxis, Proc Natl Acad Sci USA, 47, 16927-16932, (2005)
[17] Geva-Zatorsky, N; Dekel, E; Batchelor, E; Lahav, G; Alon, U, Fourier analysis and systems identification of the p53 feedback loop, Proc Natl Acad Sci USA, 107, 13550-13555, (2010)
[18] Goodwin, BC, Oscillatory behavior in enzymatic control processes, Adv Enzyme Regul, 3, 425-428, (1965)
[19] Griffith, JS, Mathematics of cellular control processes. I. negative feedback to one gene, J Theor Biol, 20, 202-208, (1968)
[20] Griffith, JS, Mathematics of cellular control processes. II. positive feedback to one gene, J Theor Biol, 20, 209-216, (1968)
[21] Harris, S; Levine, A, The p53 pathway: positive and negative feedback loops, Oncogene, 24, 2899-2908, (2005)
[22] Hirata, H; Yoshiura, S; Ohtsuka, T; Bessho, Y; Harada, T; Yoshikawa, K; Kageyama, R, Oscillatory expression of the bhlh factor hes1 regulated by a negative feedback loop, Science, 298, 840-843, (2002)
[23] Ingolia, NT; Murray, AW, Positive-feedback loops as a flexible biological module, Curr Biol, 17, 668-677, (2007)
[24] Jensen, MH; Sneppen, J; Tiana, G, Sustained oscillations and time delays in gene expression of protein hes1, FEBS Lett, 541, 176-177, (2003)
[25] Kagemyama, R; Ohtsuka, T; Kobayashi, T, The hes1 gene family: repressors and oscillators that orchestrate embryogenesis, Development, 134, 1243-1251, (2007)
[26] Kar, S; Baumann, WT; Paul, MR; Tyson, JJ, Exploring the roles of noise in the eukaryotic cell cycle, Proc Natl Acad Sci USA, 106, 6471-6476, (2009)
[27] Keller, AD, Specifying epigenetic states with autoregulatory transcription factors, J Theor Biol, 170, 175-181, (1994)
[28] Keller, AD, Model genetic circuits encoding autoregulatory transcription factors, J Theor Biol, 172, 169-185, (1995)
[29] Kepler, TB; Elston, TC, Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations, Biophys J, 81, 3116-3136, (2001)
[30] Klonis, N; Rug, M; Harper, I; Wickham, M; Cowman, A; Tilley, L, Fluorescence photobleaching analysis for the study of cellular dynamics, Eur Biophys J, 31, 36-51, (2002)
[31] Kobayashi T, Kageyama R (2010) Hes1 regulates embryonic stem cell differentiation by suppressing notch signaling. Genes Cells 15:689-698. doi:10.1111/j.1365-2443.2010.01413.x. ISSN 1356-9597
[32] Kobayashi T, Kageyama R (2011) Hes1 oscillations contribute to heterogeneous differentiation responses in embryonic stem cells. Genes 2:219-228. doi:10.3390/genes2010219. ISSN 2073-4425
[33] Kobayashi, T; Mizuno, H; Imayoshi, I; Furusawa, C; Shirahige, K; Kageyama, R, The cyclic gene hes1 contributes to diverse differentiation responses of embryonic stem cells, Genes Dev, 23, 1870-1875, (2009)
[34] Jacob Kogan (2007) Introduction to clustering large and high-dimensional data. Cambridge University Press, New York · Zbl 1183.62106
[35] Lahav, G; Rosenfeld, N; Sigal, A; Geva-Zatorsky, N; Levine, AJ; Elowitz, MB; Alon, U, Dynamics of the p53-mdm2 feedback loop in individual cells, Nature Genet, 36, 147-150, (2004)
[36] Lee, T; Rinaldi, N; Robert, F; Odom, D; Bar-Joseph, Z; Gerber, G; Hannett, N; Harbison, C; Thompson, C; Simon, I; Zeitlinger, J; Jennings, E; Murray, H; Gordon, B; Ren, B; Wyrick, J; Tagne, J; Volkert, T; Fraenkel, E; Gifford, D; Young, R, Transcriptional regulatory networks in saccharomyces cerevisiae, Science, 298, 799-804, (2002)
[37] Liao, TW, Clustering of time series data-a survey, Pattern Recogn, 38, 1857-1874, (2005) · Zbl 1077.68803
[38] Lipniacki, T; Paszek, P; Marciniak-Czochra, A; Brasier, AR; Kimmel, M, Transcriptional stochasticity in gene expression, J Theor Biol, 21, 348-367, (2006)
[39] Liu, S; Matzavinos, A; Sethuraman, S, Random walk distances in data clustering and applications, Adv Data Anal Classif, 7, 83-108, (2013) · Zbl 1261.62059
[40] Mahaffy, JM, Genetic control models with diffusion and delays, Math Biosci, 90, 519-533, (1988) · Zbl 0684.92012
[41] Mahaffy, JM; Pao, CV, Models of genetic control by repression with time delays and spatial effects, J Math Biol, 20, 39-57, (1984) · Zbl 0577.92010
[42] Masamizu, Y; Ohtsuka, T; Takashima, Y; Nagahara, H; Takenaka, Y; Yoshikawa, K; Okamura, H; Kageyama, R, Real-time imaging of the somite segmentation clock: revelation of unstable oscillators in the individual presomitic mesoderm cells, Proc Natl Acad Sci USA, 103, 1313-1318, (2006)
[43] Matsuda, T; Miyawaki, A; Nagai, T, Direct measurement of protein dynamics inside cells using a rationally designed photoconvertible protein, Nature Meth, 5, 339-345, (2008)
[44] Milo, R; Shen-Orr, S; Itzkovitz, S; Kashtan, N; Chklovskii, D; Alon, U, Network motifs: simple building blocks of complex networks, Science, 298, 824-827, (2002)
[45] Monk, NAM, Oscillatory expression of hes1, p53, and NF-\(κ \)B driven by transcriptional time delays, Curr Biol, 13, 1409-1413, (2003)
[46] Nelson, DE; Ihekwaba, AEC; Elliott, M; Johnson, JR; Gibney, CA; Foreman, BE; Nelson, G; See, V; Horton, CA; Spiller, DG; Edwards, SW; McDowell, HP; Unitt, JF; Sullivan, E; Grimley, R; Benson, N; Broomhead, D; Kell, DB; White, MRH, Oscillations in NF-\(κ \)B signaling control the dynamics of gene expression, Science, 306, 704-708, (2004)
[47] Nguyen, LK; Kulasiri, D, On the functional diversity of dynamical behaviour in genetic and metabolic feedback systems, BMC Syst Biol, 3, 51, (2009)
[48] Pirim, H; Ekşioğlu, B; Perkins, A; Yüceer, Ç, Clustering of high throughput gene expression data, Comput Oper Res, 39, 3046-3061, (2012) · Zbl 1349.62554
[49] Saddic, L; Huvermann, B; Bezhani, S; Su, Y; Winter, C; Kwon, C; Collum, R; Wagner, D, The LEAFY target LMI1 is a meristem identity regulator and acts together with LEAFY to regulate expression of CAULIFLOWER, Development, 133, 1673-1682, (2006)
[50] Seksek, O; Biwersi, J; Verkman, AS, Translational diffusion of macromolecule-sized solutes in cytoplasm and nucleus, J Cell Biol, 138, 131-142, (1997)
[51] Shahrezaei, V; Ollivier, JF; Swain, PS, Colored extrinsic fluctuations and stochatic gene expression, Mol Syst Biol, 4, 1-9, (2008)
[52] Shahrezaei, V; Swain, PS, The stochastic nature of biochemical networks, Curr Opin Biotechnol, 19, 369-374, (2008)
[53] Shimojo H, Maeda Y, Ohtsuka T, Kageyama R (2013) Dynamic notch signaling in neural progenitor cells. Springer, Japan
[54] Smolen, P; Baxter, DA; Byrne, JH, Effects of macromolecular transport and stochastic fluctuations on the dynamics of genetic regulatory systems, Am J Physiol, 277, c777-c790, (1999)
[55] Smolen, P; Baxter, DA; Byrne, JH, Modeling clarifies the role of delays and feedback in Circadian oscillators, Soc Neurosci Abstr, 25, 867, (1999)
[56] Smolen, P; Baxter, DA; Byrne, JH, Modeling transcriptional control in gene networks - methods, recent results and future directions, Bull Math Biol, 62, 247-292, (2000) · Zbl 1323.92088
[57] Snoussi, EH; Thomas, R, Logical identification of all steady states: the concept of feedback loop characteristic states, Bull Math Biol, 55, 973-991, (1993) · Zbl 0784.92002
[58] Sturrock, M; Terry, AJ; Xirodimas, DP; Thompson, AM; Chaplain, MAJ, Spatio-temporal modelling of the hes1 and p53-mdm2 intracellular signalling pathways, J Theor Biol, 273, 15-31, (2011) · Zbl 1405.92085
[59] Sturrock, M; Hellander, A; Aldakheel, S; Petzold, L; Chaplain, MAJ, The role of dimerisation and nuclear transport in the hes1 gene regulatory network, Bull Math Biol, 76, 766-798, (2014) · Zbl 1297.92032
[60] Sturrock, M; Terry, AJ; Xirodimas, DP; Thompson, AM; Chaplain, MAJ, Influence of the nuclear membrane, active transport and cell shape on the hes1 and p53-mdm2 pathways: insights from spatio-temporal modelling, Bull Math Biol, 74, 1531-1579, (2012) · Zbl 1312.92021
[61] Tafvizi, A; Mirny, LA; Oijen, AMV, Dancing on DNA: kinetic aspects of search processes on DNA, Chem Phys Chem, 12, 1481-1489, (2011)
[62] Terry, AJ; Sturrock, M; Dale, JK; Maroto, M; Chaplain, MAJ, A spatio-temporal model of notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics, PLoS ONE, 6, e16980, (2011)
[63] Thomas, R, The role of feedback circuits: positive feedback circuits are a necessary condition for positive real eigenvalues of the Jacobian matrix, Ber Busenges Phys Chem, 98, 1148-1151, (1994)
[64] Thomas, R; Thieffry, D; Kauffman, M, Dynamical behaviour of biological regulatory networks-I. biological role of feedback loops and practical use of the concept of the loop-characteristic state, Bull Math Biol, 57, 247-276, (1995) · Zbl 0821.92009
[65] Tiana, G; Jensen, MH; Sneppen, K, Time delay as a key to apoptosis induction in the p53 network, Eur Phys J B, 29, 135-140, (2002)
[66] Tibshirani, R; Walther, G; Hastie, T, Estimating the number of clusters in a data set via the gap statistic, J R Stat Soc, 63, 411-423, (2001) · Zbl 0979.62046
[67] Tyson, JJ; Othmer, HG, The dynamics of feedback control circuits in biochemical pathways, Prog Theor Biol, 5, 1-62, (1978)
[68] Wachsmuth, M; Waldeck, W; Langowski, J, Anomalous diffusion of fluorescent probes inside living cell nuclei investigated by spatially-resolved fluorescence correlation spectroscopy, J Mol Biol, 298, 677-689, (2000)
[69] Walther, GR; Marée, AFM; Edelstein-Keshet, L; Grieneisen, VA, Deterministic versus stochastic cell polarisation through wave-pinning, Bull Math Biol, 74, 2570-2599, (2012) · Zbl 1362.92018
[70] Wang J, Lefranc M, Thommen Q (2012) Stochastic oscillations induced by intrinsic fluctuations in a self-repressing gene: a deterministic approach
[71] Weiner, OD; Neilsen, PO; Prestwich, GD; Kirschner, MW; Cantley, LC; Bourne, HR, A ptdinsp3- and rho gtpase-mediated positive feedback loop regulates neutrophil polarity, Nat Cell Biol, 4, 509-513, (2002)
[72] Weiss, M; Hashimoto, H; Nilsson, T, Anomalous protein diffusion is a measure for cytoplasmic crowding in living cells, Biophys J, 87, 3518-3524, (2004)
[73] Wilkinson, D, Stochastic modelling for quantitative description of heterogeneous biological systems, Nat Rev Genet, 10, 122-133, (2009)
[74] Xiong, W; Ferrell, JE, A positive-feedback-based bistable ‘memory module’ that governs a cell fate decision, Nature, 426, 460-465, (2003)
[75] Zeiser, S; Muller, J; Liebscher, V, Modeling the hes1 oscillator, J Comp Biol, 14, 984-1000, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.