Parameter asymmetry and time-scale separation in core genetic commitment circuits

doi:10.1007/s40484-015-0042-1

Quant. Biol.

2015, Vol. 3

Issue (1) : 19-45 https://doi.org/10.1007/s40484-015-0042-1

RESEARCH ARTICLE

Parameter asymmetry and time-scale separation in core genetic commitment circuits

Hongguang Xi¹,Marc Turcotte^1,^2,^*(

)

1. Department of Mathematics, University of Texas at Arlington, Arlington, TX 76109, USA
2. Biological Sciences, University of Texas at Dallas, Richardson, TX 75080, USA

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Abstract

Theory allows studying why Evolution might select core genetic commitment circuit topologies over alternatives. The nonlinear dynamics of the underlying gene regulation together with the unescapable subtle interplay of intrinsic biochemical noise impact the range of possible evolutionary choices. The question of why certain genetic regulation circuits might present robustness to phenotype-delivery breaking over others, is therefore of high interest. Here, the behavior of systematically more complex commitment circuits is studied, in the presence of intrinsic noise, with a focus on two aspects relevant to biology: parameter asymmetry and time-scale separation. We show that phenotype delivery is broken in simple two- and three-gene circuits. In the two-gene circuit, we show how stochastic potential wells of different depths break commitment. In the three-gene circuit, we show that the onset of oscillations breaks the commitment phenotype in a systematic way. Finally, we also show that higher dimensional circuits (four-gene and five-gene circuits) may be intrinsically more robust.

Keywords systems biology theoretical biology gene regulation nonlinear dynamics stochasticity

Corresponding Author(s): Marc Turcotte

Issue Date: 06 May 2015

Cite this article:

Marc Turcotte,Hongguang Xi. Parameter asymmetry and time-scale separation in core genetic commitment circuits[J]. Quant. Biol., 2015, 3(1): 19-45.

URL:

https://academic.hep.com.cn/qb/EN/10.1007/s40484-015-0042-1
https://academic.hep.com.cn/qb/EN/Y2015/V3/I1/19

Fig.1 (A) Canonical two-gene mutually repressive, self-promoting differentiation circuit. The notation DS1 refers to the deterministic version of the circuit. CircuitThree refers to the stochastic implementation. (B) Canonical three-gene differentiation circuit generalized from the corresponding two-gene version shown on panel A. The mutual regulation is negative. The auto-regulation is positive. Parameter asymmetry provider bcommon = b13= b32= b21 (blue boxes). (C) Canonical four-gene differentiation circuit generalized from the corresponding three-gene version shown on panel B. The mutual regulation is negative. The auto-regulation is positive. Parameter asymmetry provider bcommon = b13= b32= b21= b14= b43= b42 (blue boxes). (D) Canonical five-gene differentiation circuit generalized from the corresponding four-gene version shown on panel C. The mutual regulation is negative. The auto-regulation is positive. Parameter asymmetry provider bcommon = b13= b32= b21= b14= b43= b42= b35= b52= b51= b45 (blue boxes).

Fig.2 A and B show the bifurcation diagram of DS1, 2D. Blue and black are for bmod=1. Red and green are for bmod=0.9. Stable steady states are shown in solid line. Unstable steady states are shown in dashed line. (A) Displays X₁ vs. the repression parameter “b”. (B) Displays X₂ vs. “b”. The effect of a 10% difference in parameters on the bifurcation is discussed in the text. The equations and parameters of the calculation are in Supplementary Table 1. C and D show the bifurcation diagram of DS1, 4D. Blue and black are for bmod=1. Red and green for bmod=0.9. Stable steady states are shown in solid line. Unstable steady states are shown in dashed line. (C) Displays X₁ vs. the repression parameter “b”. (D) Displays X₂ vs. “b”. Negative concentration solutions paired with positive ones have no biological meaning and can be ignored, as explained in the text. C shows the equations and the parameter values used in the study. The equations and parameters of the calculation are in Supplementary Table 2.

Fig.3 DS1 phase plane showing vector field (arrows omitted). Green and red lines show X₁ and X₂ nullclines. The small circles of matching color on the nullclines are “4D nullclines” are explained in the text. The fixed points of the dynamics are indicated by blue circles and are numbered 1, 2 and 3 (stable node, saddle node and stable node, respectively). The cyan line is the location of the 2D separatrix as computed by time-reversed integration originating from the saddle (middle fixed point). The solid magenta lines are 2D integrations. The dashed magenta lines are 4D integrations. 2D and 4D trajectories originate at the same locations, S1 and S2, in the X₁X₂ plane. However, for S1, the end point of the trajectory differs: the 2D trajectory goes to fixed point #1 while the 4D trajectory goes to fixed point #3. In the case of S2, both 2D and 4D trajectories terminate at fixed point #3. The parameters of the simulation, fixed point locations, eigenvalues and stability assignment are in Supplementary Table 3.

Fig.4 Average residency over 100 statistically independent stochastic simulations. The axes are linear. All simulations set bmod=0.9 and Ω = 100. The color indicates the base 10 logarithm of the residency as shown on the scales to the right. All simulations are started near the upper fixed point. (A) Run75.1 through 75.100 (configuration #5): Low Time-Scale Separation. The 2D and 4D separatrix locations differ significantly. The upper fixed point stochastic potential well is much deeper than that of the lower fixed point. (B) Run76.1 through 76.100 (configuration #5.2): High Time-Scale Separation. The 2D and 4D separatrix overlap. The fixed point stochastic potential wells are of much more similar depths.

Fig.5 Computation of the location of the 2D (thin red) and 4D (thin blue) separatrix in DS1. (A) Show computations performed with the asymmetry parameter bmod = 0.9. (B) Show computations done with bmod=1.0. Both A and B computations have the mutual suppression parameter b=0.4. At this value of suppression strength (b=0.4), in the presence of asymmetry in the suppression (bmod=0.9), the 4D separatrix differs markedly from the 2D separatrix (A). As “b” is increased, the agreement is increased (C). (C and D) Similar to A and B except the mutual suppression strength “b” is much higher (x10); parameter b=4.0. At this value of suppression strength (b=4.0), even in the presence of asymmetry in the suppression (bmod=0.9), the 2D and 4D separatrix agree over a wide (but not complete) range. As the strength of mutual suppression “b” is increased, 2D and 4D agreement is increased (compare A and C). When there is no asymmetry in suppression (bmod=1), the 2D and 4D separatrix always agree (compare C and D). All computations were performed on a 50x50 grid using a numerical search algorithm as explained in Methods.

Fig.6 (A) Phase diagram of DS2 dynamics for configuration #5. Here, bcommon=1. This configuration does not enforce full time-scale separation. The three null-surfaces are shown in cyan, yellow and blue respectively. The fixed points numbered 1, 2, 3 and 4 are located at their common intersections (stable node, saddle node, stable node and stable node, respectively). A stochastic track showing dwelling in the saddle area, eventually reaching fixed-point #3 is shown. (B) Phase diagram with null-surfaces removed and general separatrix surface added. Numerical fixed point locators are shown as “clouds” of black dots used to compute the accurate location of fixed points (shown by red crosses), as explained in the text. The generalized separatrix surface (three planes of red dots) is a symmetrical three-leaved surface centered on the saddle (fixed point #2). The same stochastic track is shown, traveling from the saddle area to stable fixed-point #3. The fixed point locations, eigenvalues and stability assignments are given in Supplementary Table 4.

Fig.7 (Aand B) Bifurcation diagrams of the 6D DS2 circuit in parameter set 12.1 at reference time-scale separation. Computation performed with MATCONT. “H” stands for “Hopf”, “(H)” for “neutral saddle”, “LP” for “limit point” bifurcations; “u” for unstable; “s” for stable. The leftmost unstable limit cycle is entirely in the negative “bcommon” range. The stable limit cycle (l. c.) born at the second Hopf on the left ends in homoclinic orbit/saddle collisions (α,β). Minima and maxima of l. c. are shown in red. On the right, a large l. c. exists throughout the dynamics, ending in homoclinic orbit saddle collisions (γ,δ). (B) Blowup of central dynamical region (dashed rectangle on A). The homoclinic orbit/saddle collision points are shown by α,β,γ,δ. (C) Bifurcation diagram of the 3D DS2 circuit. Computation performed with Oscill8 using configuration #12.1 (infinite time-scale separation). There are two oscillation regimes (one on the left, and one on the right) interspaced by a non-oscillatory regime in the middle. Oscillations begin on the left at a Hopf bifurcation and on the right, at another Hopf bifurcation. “H” stands for Hopf. The saddle homoclinic orbit collisions that demark the limits of the left (right) oscillatory regime are shown by α,β (γ,δ). SN stands for “saddle node collision”, “s” for stable, “u” for unstable.

Fig.8 (A) The 3D DS2 circuit behavior in parameter set 12.1 (bcommon=0.85). Three stochastic tracks with different initial seeds originating at the same location near fixed point “2” alternatively reach any of the three stable fixed points of the dynamics (#1, #3 and #4). However, 3D and 6D deterministic tracks both reach fixed point #1. (B) Details of the trajectories. Due to finite noise in the system, the stochastic tracks wander considerably. The “cloud” of black dots numerically aproximate the location of fixed point “2”. The 6D and 3D deterministic tracks are close because there is high time-scale separation intrinsic to the 6D. They do not overlap perfectly because the time-scale separation is not infinite.

Fig.9 (A) DS2 circuit in configuration #15 for which bcommon=1.2 but otherwise similar to #12.1, hence with high time-scale separation. There are seven fixed points #1 to #7 (stable node, stable node, stable node, saddle node, saddle node, saddle node, stable spiral). (B) Details showing a 3D deterministic track originating near fixed point #7, but not exactly at the fixed point (X3 is displaced slightly: start= [0.62373,0.62373,0.61805]). The track reaches a surrounding stable limit cycle. Similar tracks started anywhere in the basin of attraction of the limit cycle behave similarly. The basin of attraction is the triangular region defined by saddle #4, #5 and #6. However, any track started anywhere directly along the eigenvector #1 direction, centered on fixed point #7, falls into fixed point #7 (data omitted). (C) Diagram explaining the stability of fixed point #7. Tracks started on the first eigenvector direction fall into fixed point #7. But tracks started elsewhere spiral outward to the surrounding stable limit cycle. These results are also valid in 6D. More details in text. The fixed point locations, eigenvalues and stabilities are in Supplementary Table 5. For fixed point #7, the eigenvector elements are also listed.

Fig.10 (A) DS2 circuit in configuration #18 for which bcommon=1.6, but otherwise similar to #12.1, hence with high time-scale separation. Elongated clouds of locator points along the bi-null-surface intersections locate 7 fixed points. (B) The surface in red is the generalized separatrix computed on a 30x30x30 grid. (C) 3D and 6D tracks. In one case, both 3D and 6D tracks reach stable fixed point #5. In the other, the 6D track falls into the stable orbit surrounding (6D-unstable) fixed point #4 while the 3D track falls into the (3D-stable) fixed point #4. (D) Same as C, but showing only the 6D deterministic tracks, for clarity. The 6D limit cycle is clearly visible. In 3D however, at bcommon=1.6, there is no limit cycle.

Fig.11 DS3 phase volume at asymmetry bcommon=1.0 (fully symmetric). (A) X₁, X₂, X₃ phase volume. (B) X₁, X₂, X₄ phase volume. The colored clouds of dots on both panels locate the 3-variable null-surfaces (yellow: 123, cyan: 124, green: 134 and magenta: 234). The black dots locate the global (1, 2, 3, 4) fixed points of the 4D dynamics. At this value of asymmetry (bcommon=1), there are five stable fixed points (four peripheral #1, #3, #5, and #7) and one central (#9) dynamically separated by four intervening saddles (#2, #4, #6 and #8). The fixed point locations eigenvalues, eigenvectors and stability assignments are in Supplementary Table 6.

Fig.12 The locations of all fixed points in the DS2 6D bifurcation analyses performed over widely different time-scale separations (reference, x2, x10 and x1000) are invariant. All branches of the steady-state dynamics overlap. However, the rotational dynamics differs due to the changing location of the Hopf bifurcations (shown by black *). Time-scale separation affects the rotational dynamics on the right of the central region, more than on the left. However, the homoclinic orbit/saddle collision points (α, β, γ and δ) locations are invariant.

Fig.13 DS2 in the reference time-scale separation, A (bcommon=.7250, configuration #24.2) and B (bcommon=1.2, configuration #24) show out of 10 stochastically independent tracks all starting at the same location in phase space denoted by X on B (hidden on A), only those 6 tracks that fluctuate out of the stable limit cycle. The other four tracks remain on the limit cycle. At asymmetry bcommon=.7250, stable fixed point #3 is selected, at bcommon=1.2, stable fixed point #1 is selected.

Fig.14 Panel A shows the DS2 phase space at bcommon=0.725 where one large limit cycle co-exists with the three stable fixed points labeled 1, 2 and 3. Here, one single stochastic track (shown in green) imparted with 10x previous noise level now visits all three fixed points in a row, without end, and without getting permanently trapped in any. This occurs because the dynamics is driven by the presence of the inner limit cycle (3D in magenta, 6D in blue). The commitment phenotype is broken. The initial conditions of the 3D trajectory are: X₁=X₂=X₃=0.1. The initial conditions of the 6D and of the stochastic trajectories are: X₁=X₂=X₃=0.1; mRNA1=mRNA2=mRNA3=.01 In contrast, Panel B shows the phase space at bcommon=0.9 where the dynamics is absent of any limit cycles. Here, ten independent stochastic tracks are shown in blue. Each track gets permanently trapped into one single fixed point, and never escapes. This is due to the absence of any limit cycle; there is no driving effect. The commitment phenotype is preserved. In both Panel A and B, the initial conditions of all tracks is the same.

Fig.15 DS2 phase space at parameter asymmetry bcommon=1.6, in reference time-scale separation. The time evolution of one single stochastic track is shown in different colors to indicate the progression and is split into two panels for clarity. The initial condition is shown by the red dot. (A) In blue, 59% of the total simulation time is first spent accessing and then hopping in and out of the “quasi stable” central state (maintained by the limit cycle), and stable fixed point #3. (B) In green, a further 35% of the simulation time is then spent first in the basin of attraction of stable fixed point #3, then hopping over to that of stable fixed point #2. In magenta, the remaining 6% of the simulation is finally spend in the basins of attraction of stable fixed point #2, then hopping over to that of #1, and finally back to #3. The clockwise hopping order and direction 3→2→1→3 is imposed by the influence of the limit cycle direction of rotation (shown in orange). At this value of parameter asymmetry and reference time-scale separation, there is no 3D limit cycle.

Fig.16 (A) Bifurcation analysis of the DS3 8D system in reference time-scale separation. The X₁, X₂, X₃ and X₄ (black, red, green and blue) fixed point locations, and stability (thick: stable, thin: unstable), are shown versus asymmetry parameter “bcommon”. The four mRNAs are omitted for clarity. Large amplitude stable oscillatory behavior develops at the Hopf bifurcation (HB12 at bcommon~6.24); the upper and lower extend of the limit cycle are shown. However, there is no overlap of rotation with the stable/unstable (node and saddle) dynamical regime of fixed points located at lower asymmetry. Furthermore, in infinite time-scale separation, there is no limit cycle (data omitted for brevity). (B) The dynamical regime at lower asymmetry; the various saddle/node bifurcations are shown in details; the vertical lines indicate their locations. Stability is shown by stick lines, un-stability by thin lines. The system of coupled ordinary differential equations describing the DS3 8D dynamics and the parameters are in Supplementary Table 7.

Fig.17 Bifurcation diagram of 10D DS4 in reference time-scale separation. (A) The fixed points of X₁, X₂, X₃, X₄ and X₅ (values are in black, red, green, blue, magenta respectively and stability, thick: stable, thin: unstable) are shown as a function of the asymmetry parameter “bcommon”. The mRNAs are not shown for clarity. The dynamics develops a large amplitude limit cycle at bcommon~8.39, where a Hopf bifurcation occurs. The limit cycle does not overlap with the dynamical regime at low asymmetry, where saddle/node bifurcations dominate the dynamics. (B) Details of the dynamics at lower asymmetry. The red dotted line is at bcommon=1.81, in the center of the fold. There only one fixed point exists. The green line is at bcommon=1.76, just below the saddle/node bifurcations. There, the dynamics presents two stable fixed points separated by a saddle. This feature results in extreme sensitivity of the commitment phenotype to asymmetry, in a limited range of asymmetry.

Tab.1 DS1 Parameter configuration details.

Tab.2 DS2 Parameter configuration details.

1	Waddington, C. H. (1957) The Strategy of the Genes. London: Routledge
2	Ferrell, J. E. Jr. (2012) Bistability, bifurcations, and Waddington’s epigenetic landscape. Curr. Biol., 22, R458–R466 https://doi.org/10.1016/j.cub.2012.03.045. pmid: 22677291
3	Strogatz, S. H. (1994) Nonlinear Dynamics and Chaos. Cambridge: Perseus Books Publishing
4	Jaeger, J., Monk, N. (2014) Bioattractors: Dynamical systems theory and the evolution of regulatory processes. J. Physiol., 592, 2267–2281
5	?a?atay, T., Turcotte, M., Elowitz, M. B., Garcia-Ojalvo, J. and Süel, G. M. (2009) Architecture-dependent noise discriminates functionally analogous differentiation circuits. Cell, 139, 512–522 https://doi.org/10.1016/j.cell.2009.07.046. pmid: 19853288
6	Elowitz, M. B., Levine, A. J., Siggia, E. D. and Swain, P. S. (2002) Stochastic gene expression in a single cell. Science, 297, 1183–1186 https://doi.org/10.1126/science.1070919 pmid: 12183631
7	Süel, G. M., Garcia-Ojalvo, J., Liberman, L. M. and Elowitz, M. B. (2006) An excitable gene regulatory circuit induces transient cellular differentiation. Nature, 440, 545–550 https://doi.org/10.1038/nature04588 pmid: 16554821
8	Süel, G. M., Kulkarni, R. P., Dworkin, J., Garcia-Ojalvo, J. and Elowitz, M. B. (2007) Tunability and noise dependence in differentiation dynamics. Science, 315, 1716–1719 https://doi.org/10.1126/science.1137455. pmid: 17379809
9	Thattai, M. and van Oudenaarden, A. (2004) Stochastic gene expression in fluctuating environments. Genetics, 167, 523–530 https://doi.org/10.1534/genetics.167.1.523. pmid: 15166174
10	Turcotte, M., Garcia-Ojalvo, J. and Süel, G. M. (2008) A genetic timer through noise-induced stabilization of an unstable state. Proc. Natl. Acad. Sci. USA, 105, 15732–15737 https://doi.org/10.1073/pnas.0806349105. pmid: 18836072
11	Xi, H., Duan, L. and Turcotte, M. (2013) Point-cycle bistability and stochasticity in a regulatory circuit for Bacillus subtilis competence. Math. Biosci., 244, 135–147 https://doi.org/10.1016/j.mbs.2013.05.002. pmid: 23693123
12	Xi, H., Yang, Z. and Turcotte, M. (2013) Subtle interplay of stochasticity and deterministic dynamics pervades an evolutionary plausible genetic circuit for Bacillus subtilis competence. Math. Biosci., 246, 148–163 https://doi.org/10.1016/j.mbs.2013.08.007. pmid: 24012503
13	Li, C., Wang, E. and Wang, J. (2011) Landscape and flux decomposition for exploring global natures of non-equilibrium dynamical systems under intrinsic statistical fluctuations. Chem. Phys. Lett., 505, 75–80 https://doi.org/10.1016/j.cplett.2011.02.020
14	Li, C., Wang, E. and Wang, J. (2011) Landscape, flux, correlation, resonance, coherence, stability, and key network wirings of stochastic circadian oscillation. Biophys. J., 101, 1335–1344 https://doi.org/10.1016/j.bpj.2011.08.012. pmid: 21943414
15	Li, C., Wang, E. and Wang, J. (2012) Landscape topography determines global stability and robustness of a metabolic network. ACS Synth Biol, 1, 229–239 https://doi.org/10.1021/sb300020f. pmid: 23651205
16	Li, C. and Wang, J. (2013) Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation. J. R. Soc. Interface, 10, 20130787 https://doi.org/10.1098/rsif.2013.0787. pmid: 24132204
17	Li, C. and Wang, J. (2014) Landscape and flux reveal a new global view and physical quantification of mammalian cell cycle. Proc. Natl. Acad. Sci. USA, 111, 14130–14135 https://doi.org/10.1073/pnas.1408628111. pmid: 25228772
18	Li, C. and Wang, J. (2014) Quantifying the underlying landscape and paths of cancer. J. R. Soc. Interface, 11, 20140774 https://doi.org/10.1098/rsif.2014.0774. pmid: 25232051
19	Wang, J., Zhang, K., Xu, L. and Wang, E. (2011) Quantifying the Waddington landscape and biological paths for development and differentiation. Proc. Natl. Acad. Sci. USA, 108, 8257–8262 https://doi.org/10.1073/pnas.1017017108. pmid: 21536909
20	Wu, W. and Wang, J. (2013) Landscape framework and global stability for stochastic reaction diffusion and general spatially extended systems with intrinsic fluctuations. J. Phys. Chem. B, 117, 12908–12934 https://doi.org/10.1021/jp402064y. pmid: 23865936
21	Wu, W. and Wang, J. (2013) Potential and flux field landscape theory. I. Global stability and dynamics of spatially dependent non-equilibrium systems. J. Chem. Phys., 139, 121920 https://doi.org/10.1063/1.4816376. pmid: 24089732
22	Xu, L., Zhang, F., Zhang, K., Wang, E. and Wang, J. (2014) The potential and flux landscape theory of ecology. PLoS One, 9, e86746 https://doi.org/10.1371/journal.pone.0086746. pmid: 24497975
23	Zhang F., Xu L., Zhang K., Wang E., Wang J., (2012) The potential and flux landscape theory of evolution. J. Chem. Phys., 137, 065102
24	Beard, D. A. D., Babson, E., Curtis, E. and Qian, H. (2004) Thermodynamic constraints for biochemical networks. J. Theor. Biol., 228, 327–333 https://doi.org/10.1016/j.jtbi.2004.01.008. pmid: 15135031
25	Beard, D. A. and Qian H. (2008) Chemical Biophysics, Cambridge: Cambridge University Press
26	Qian, H. and Cooper, J. A. (2008) Temporal cooperativity and sensitivity amplification in biological signal transduction. Biochemistry, 47, 2211–2220 https://doi.org/10.1021/bi702125s. pmid: 18193898
27	Qian, H. (2007) Phosphorylation energy hypothesis: open chemical systems and their biological functions. Annu. Rev. Phys. Chem., 58, 113–142 https://doi.org/10.1146/annurev.physchem.58.032806.104550. pmid: 17059360
28	Qian, H. and Beard, D. A. (2005) Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium. Biophys. Chem., 114, 213–220 https://doi.org/10.1016/j.bpc.2004.12.001. pmid: 15829355
29	Qian, H., Beard, D. A. and Liang, S. D. (2003) Stoichiometric network theory for nonequilibrium biochemical systems. Eur. J. Biochem., 270, 415–421 https://doi.org/10.1046/j.1432-1033.2003.03357.x. pmid: 12542691
30	Ma, W., Trusina, A., El-Samad, H., Lim, W. A. and Tang, C. (2009) Defining network topologies that can achieve biochemical adaptation. Cell, 138, 760–773 https://doi.org/10.1016/j.cell.2009.06.013. pmid: 19703401
31	Zhang, J., Yuan, Z., Li, H. X. and Zhou, T. (2010) Architecture-dependent robustness and bistability in a class of genetic circuits. Biophys. J., 99, 1034–1042 https://doi.org/10.1016/j.bpj.2010.05.036. pmid: 20712986
32	Snoussi, E. H. (1998) Necessary Conditions for Multistationarity and Stable Periodicity. J. Biol. Syst., 06, 3–9 https://doi.org/10.1142/S0218339098000042
33	Gardner, T. S. and Faith, J. J. (2005) Reverse-engineering transcription control networks. Phys. Life Rev., 2, 65–88 https://doi.org/10.1016/j.plrev.2005.01.001. pmid: 20416858
34	Chickarmane, V., Troein, C., Nuber, U. A., Sauro, H. M. and Peterson, C. (2006) Transcriptional dynamics of the embryonic stem cell switch. PLoS Comput. Biol., 2, e123 https://doi.org/10.1371/journal.pcbi.0020123. pmid: 16978048
35	Gillespie, D. T. (1976) A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. J. Comput. Phys., 22, 403–434 https://doi.org/10.1016/0021-9991(76)90041-3
36	Gillespie, D. T. (1977) Exact Stochastic Simulation of Coupled Chemical Reactions. J. Phys. Chem., 81, 2340–2361 https://doi.org/10.1021/j100540a008
37	Gillespie Markov Processes, D. T. An Introduction for Physical Scientists, Academic Press, 1991
38	Gillespie, D. T. (2007) Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem., 58, 35–55 https://doi.org/10.1146/annurev.physchem.58.032806.104637. pmid: 17037977

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