Chapter 5 The Burst Production Model of Single Gene Expression
5.3 Numerical Validation of the Langevin form in burst production model
We notice that since the Langevin form in Eq. (5.32) “directly” describe the protein level in single gene expression model, and the description of mRNA level is effectively included in the mean burst size β . It means that Eq. (5.32) bypass the direct description of mRNA and avoid the problems in rare event case. Owing to this reason, Eq. (5.32) has a more robust simulating ability in rare event cases, and it successfully describes the original two-step model in single gene expression. The following figures show the simulation result of the single gene expression model using parameter set in Table 4-1.
Figure 5-7 A sample trajectory for Langevin simulation in burst model. The
parameters used in simulation are listed in Table 4-1 of section4.1. The time step used in simulation is 150 sec.
0 0.5 1 1.5 2
x 104 0
20 40 60 80 100 120 140 160
Sample trajectories (protein)
time(sec)
Protein particle numbers
burst-Langevin (dt = 150 sec) Deterministic O.D.E.
Figure 5-8 Steady state distribution of protein in different simulation approaches.
Shown are distributions from 10,000 trajectories simulated with the parameters listed in Table 4-1 of section 4.1. The time step used in both Langevin forms is 150 sec.
The distributions in Figure 5-8 compare different simulating approaches and it shows a better approximation of Langevin in burst model, compared to the Langevin in two step which suffers from rare events problem in mRNA discussed in Chapter 4.
The scanning of the accuracy of approximation for different transcription rate and different dt is shown in the next figure (Figure 5-9). It shows different error levels of approximation under two different Langevin form. Left column: Langevin form in two-step model. Right column: Langevin form in burst model. The figure is generated by scanning the combination of (km , dt) and compare the result to analytical solution of the master equation. The rest of parameters are the same as Table 4-1. It shows that burst-Langevin not only provides a better approximation, but also stay more stable when transcription rate km is changing. It means that for simulating gene regulation, the
0 100 200 300 400 500
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Steady state distribution of protein (dt=150 sec)
protein particle number
Occurrence
protein (Gillespie) mean = 98.9499 Fano = 20.1247 ---protein (burst-Gillespie) mean = 99.644 Fano = 20.1569 ---protein (Langevin) mean = 208.1909 Fano = 12.4628 ---protein (burst-Langevin) mean = 101.4331 Fano = 19.4846
Figur wn are error gevin (Eq. ( Burst-Lange
le gene expr rence betwe een the stati ults at stead
km (M/sec) Error in mean
4 6 8
km (M/sec) ror in Fano fac
4 6 8
n of differen upper panels ction 3.3.2 o
.32) in secti del with the
istic values dy state (Eq
n
nt error leve s) and in Fa or Eq. (4.9)
ion 5.2.2, pa parameters
of proteins q. (4.8) in se ano factor (l
in section 4 anels to the s listed in Ta
from 10,00 ection 4.1).
km (M/sec) Error in mean
4 6 8
km (M/sec) rror in Fano fa
4 6 8
fferent Lang lower panel 4.2, panels t right) in sim able 4-1. Th ls) for tradit to the left) a mulating a he error are ies as comp
2
REFERENCE
[1] A. Raj and A. van Oudenaarden, Nature, Nurture, or Chance:
Stochastic Gene Expression and Its Consequences, Cell, 135: 216-226, 2008.
[2] M. B. Elowitz, A. J. Levine, E. D. Siggia and P. S. Swain, Stochastic Gene Expression in a Single Cell, Science, 297: 1183-1186, 2002.
[3] A. Raj, S. A. Rifkin, E. Andersen, and A. van Oudenaarden, Variability in gene expression underlies incomplete penetrance, Nature, 463: 913–918, 2010
[4] T. Kalmar, C. Lim, P. Hayward, S. M. Descalzo, J. Nichols, J. G. Ojalvo, A. M.
Arias, Regulated Fluctuations in Nanog Expression Mediate Cell Fate Decisions in Embryonic Stem Cells, Public Library of Science Biology, 7(7): e1000149, 2009.
[5] G. Chalancon, C. N. Ravarani, S. Balaji, A. Martinez-Arias, L. Aravind, R. Jothi and M. M. Babu, Interplay between gene expression noise and regulatory network, Trends in Genetics, 28(5): 221-232, 2012.
[6] S. Hooshangi, S. Thiberge, and R. Weiss, Ultrasensitivity and noise propagation in a synthetic transcriptional cascade, Proceedings of the National Academy of Sciences, 102(10): 3581-3586, 2005.
[7] M. Thattai and A. van Oudenaarden, Attenuation of Noise in Ultrasensitive Signaling Cascades, Biophysical Journal, 82: 2943-2950, 2002.
[8] J. M. Pedraza and A. van Oudenaarden, Noise Propagation in Gene Networks, Science, 307: 1965-1969, 2005.
[9] A. Eldar and M. B. Elowitz, Functional roles for noise in genetic circuits, Nature, 467:167–173, 2010.
[10] R. Losick and C. Desplan, Stochasticity and Cell Fate, Science, 320: 65, 2008.
[11] C. J. Davidson and M. G. Surette, Individuality in Bacteria, Annual Review of Genetics, 42: 253–268, 2008.
[12] Z. Hensel, H. Feng, B. Han, C. Hatem, J. Wang and J. Xiao, Stochastic expression dynamics of a transcription factor revealed by single-molecule noise analysis, Nature Structural and Molecular Biology 19: 797-802, 2012.
[13] E. B. Jacob and D. Schultz, Bacteria determine fate by playing dice with controlled odds, Proceedings of the National Academy of Sciences, 107(30):
13197-13198, 2010.
[14] U. Alon, Network motifs: theory and experimental approaches, Nature Reviews Genetics, 8: 450-461, 2007.
[15] N. G. van Kampen, Stochastic Processes in Physics and Chemistry 3rd edition, Amsterdam, The Netherlands, 2007.
[16] N. Maheshri and E. K. O'Shea, Living with Noisy Genes: How Cells Function Reliably with Inherent Variability in Gene Expression, Annual Review of Biophysics and Biomolecular Structure, 36: 413–434, 2007.
[17] D. T. Gillespie, A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions, Journal of Computational Physics, 2:403-434, 1976.
[18] D. T. Gillespie, Exact Stochastic Simulation of Coupled Chemical Reactions, The Journal of Physical Chemistry, 81(25): 2340, 1977.
[19] D. T. Gillespie, The chemical Langevin equation, Journal of Chemical Physics, 113: 297, 2000.
[20] D. A. McQuarrie, Stochastic Approach to Chemical Kinetics, Journal of Applied
[21] D. T. Gillespie, A rigorous derivation of the chemical master equation, Physica A, 188: 404-425, 1992.
[22] D. J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, Society for Industrial and Applied Mathematics Review, 43(3): 525–546, 2001.
[23] D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, Journal of Chemical Physics, 115: 1716, 2001.
[24] D. T. Gillespie, Perspective: Stochastic algorithms for chemical kinetics, Journal of Chemical Physics, 138: 170901, 2013.
[25] M. Thattai and A. van Oudenaarden, Intrinsic noise in gene regulatory networks, Proceedings of the National Academy of Sciences, 98(15): 8614-8619, 2001.
[26] N. Friedman, L. Cai, and X. S. Xie, Linking Stochastic Dynamics to Population Distribution: An Analytical Framework of Gene Expression, Physical review letters (PRL), 97: 168302, 2006.
[27] J. Paulsson, Summing up the noise in gene networks, Nature 427: 415-418, 2004.
[28] E. M. Ozbudak, M. Thattai, I. Kurtser, A. D. Grossman and A. van Oudenaarden, Regulation of noise in the expression of a single gene, Nature Genetics, 31: 69 – 73, 2002.
[29] I. Golding, J. Paulsson, S. M. Zawilski, and E. C. Cox, Real-Time Kinetics of Gene Activity in Individual Bacteria, Cell, 123(6): 1025-1236, 2005.
[30] L. Cai, N. Friedman and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440: 358-362, 2006.
[31] J. Yu, J. Xiao, X.J. Ren, K. Q. Lao, X. S. Xie, Probing Gene Expression in Live Cells, One Protein Molecule at a time, Science 311: 1600-1603, 2006.
Barkai, Noise in protein expression scales with natural protein abundance, Nature Genetics, 38: 636, 2006.
[33] J. R. Newman, S. Ghaemmaghami, J. Ihmels, D. K. Breslow, M. Noble, J. L.
DeRisi and J. S. Weissman, Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise, Nature, 441: 840, 2006.
[34] J. M. Pedraza and J. Paulsson, Effects of Molecular Memory and Bursting on Fluctuations in Gene Expression, Nature, 455: 485–490, 2008.
[35] A. Maaref and R. Annavajjala, The Gamma Variate with Random Shape Parameter and Some Applications, Institute of Electrical and Electronics Engineers Communications Letters, 14(12): 1146-1148, 2010.