Time-scale separation and stochasticity conspire to impact phenotypic dynamics in the canonical and inverted <i>Bacillus subtilis</i> core genetic regulation circuits

doi:10.1007/s40484-018-0151-8

Quant. Biol.

2019, Vol. 7

Issue (1) : 54-68 https://doi.org/10.1007/s40484-018-0151-8

RESEARCH ARTICLE

Time-scale separation and stochasticity conspire to impact phenotypic dynamics in the canonical and inverted Bacillus subtilis core genetic regulation circuits

Lijie Hao¹, Zhuoqin Yang¹, Marc Turcotte²(

)

¹. School of Mathematics and Systems Science and LMIB, Beihang University, Beijing 100191, China
². University of Texas at Dallas, Richardson, TX 75080, USA

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Abstract

Background: In this work, we study two seemingly unrelated aspects of core genetic nonlinear dynamical control of the competence phenotype in Bacillus subtilis, a common Gram-positive bacterium living in the soil.

Methods: We focus on hitherto unchartered aspects of the dynamics by exploring the effect of time-scale separation between transcription and translation and, as well, the effect of intrinsic molecular stochasticity. We consider these aspects of regulatory control as two possible evolutionary handles.

Results: Hence, using theory and computations, we study how the onset of oscillations breaks the excitability-based competence phenotype in two topologically close evolutionary-competing circuits: the canonical “wild-type” regulation circuit selected by Evolution and the corresponding indirect-feedback inverted circuit that failed to be selected by Evolution, as was shown elsewhere, due to dynamical reasons.

Conclusions: Relying on in-silico perturbation of the living state, we show that the canonical core genetic regulation of excitability-based competence is more robust against switching to phenotype-breaking oscillations than the inverted feedback organism. We show how this is due to time-scale separation and stochasticity.

Keywords Bacillus subtilis competence gene regulation deterministic dynamics stochastic dynamics

Corresponding Author(s): Marc Turcotte

Online First Date: 22 October 2018 Issue Date: 22 March 2019

Cite this article:

Lijie Hao,Zhuoqin Yang,Marc Turcotte. Time-scale separation and stochasticity conspire to impact phenotypic dynamics in the canonical and inverted Bacillus subtilis core genetic regulation circuits[J]. Quant. Biol., 2019, 7(1): 54-68.

URL:

https://academic.hep.com.cn/qb/EN/10.1007/s40484-018-0151-8
https://academic.hep.com.cn/qb/EN/Y2019/V7/I1/54

Fig.1 Core genetic regulation circuits.(A) Depiction of the competence phenotype core genetic regulation circuit of the canonical (native) Bacillus subtilis bacterium. (B) Depiction of a circuit with inverted indirect regulation. The term “inverted” refers to the inverted order of indirect feedback regulation compared to the canonical circuit. In the canonical circuit (panel (A)), the indirect feedback regulation order is suppression followed by activation whereas in the inverted circuit (panel (B)), the indirect feedback regulation order is opposite: activation followed by repression. ComK is the master regulatory gene of the competence phenotype; when protein ComK exceeds a certain threshold, the organism becomes competent to accept exogenous DNA into its own genome. In both the canonical and inverted circuits, ComK directly auto-activates itself. The organism controlled by the circuit depicted in panel (A) is wild-type. The organism controlled by the circuit depicted in panel (B), as far as we know, does not occur in Nature but it has been re-engineered in the laboratory [²]. More details are in the text and references.

Fig.2 Canonical Bacillus subtilis 2D circuit dynamics.（A?C) Phase portraits with several sample tracks started at locations indicated by the black dots. The ComK nullcline is shown in red. The ComS nullcline is shown in green. Nullcline intersections indicated by the labels “1”, “2” and “3” are the fixed points of the dynamics. #1 is a stable spiral, #2 is a saddle node and #3 is an unstable spiral. In panel (A), k₁=0.0333. In panel (B), k₁=0.0405. In panels (A?C), the separatrix is shown by the cyan curve. Tracks (in magenta) started on the left of the separatrix on panels (A) and (B) fall into the stable fixed point #1. In panel (A), tracks (in brown) started on the right side of the separatrix undergo an excitable trajectory about fixed point #3 and fall into fixed point #1. In panel (B) however, similar tracks (in brown) also started on the right side of the separatrix, do not undergo an excitable trajectory. They instead fall into a limit cycle. Axes on panels (A) and (B) are in log base 10. Panel (C) is a detailed view of panel (B) in the region surrounding the saddle node #2. Axes in panel (C) are linear. (D) Shows the corresponding bifurcation diagram of the dynamics of fixed point #3 vs. k₁. ComS is shown in orange. ComK is shown in black. Solid (dotted) orange and black lines indicate a stable (unstable) fixed point. A limit cycle clearly exists between the Hopf on the right (indicated by “H” at k₁~.06) and its appearance/disappearance point occurring on the left of the diagram (where the maximum amplitude purple and green curves and the minimum amplitude red and blue curves, all begin/end abruptly). At k₁=.0333, there is no limit cycle (as in panel (A)). But at k₁=.0405, the dynamics already exhibits a wide amplitude limit cycle, as in panel (B) and panel (C).

Fig.3 Deterministic and stochastic phase portraits of the canonical Bacillus subtilis circuit.(A)Deterministic 2D phase portrait at k₁=0.05, well in the presence of the limit cycle (see Figure 1C). The ComK and ComS nullclines are the curves in red and green, respectively. Their intersections (“1”, “2” and “3”) are fixed points of the dynamics. Point #1 is a stable spiral, point #2 is a saddle node, and point # 3 is unstable spiral. The magenta curve is a 2D computation of a trajectory started on the right side of the separatrix (shown in cyan), at the location indicated by the black dot. In this 2D simulation, by construction as explained in the text, the mRNA sub-manifold is at rest; time-scale separation is infinite. The limit cycle is obvious. The units in panel (A) are dimensionless. (B) Shows the corresponding stochastic simulation of the same system but, as explained in the text, by construction, the mRNA dynamics is not constrained to be at rest. In this particular case, time-scale separation is finite. Unlike panel (A), the units in panel (B) are dimensioned but for ease of comparison, the locations of the 2D fixed points (from panel (A)) are shown in panel (B) by three white circles. The color on the plot is a measure of the probability density for the stochastic system to occupy a certain state. The color bar on the right indicates the log base 10 value of the density corresponding to each color in the range. It is clear that the dynamics of the stochastic system, at this level of time-scale separation does not exhibit a limit cycle.

Fig.4 Dynamical behavior of 6D system vs. slowing down of translation (SDT) factors. As explained in the text, in order to facilitate dynamical exploration, a 6D deterministic system was derived from the stochastic discrete event system. This cognate 6D system corresponds to the thermodynamic limit of the stochastic system: it is the infinite number of molecule limit of the stochastic system. The 6D system displays the evolution of the mean of the stochastic system in the limit of infinite number of molecules, therefore with zero fluctuations. Panels (A) and (B) are bifurcation diagrams of the 6D system showing the ComK fixed point locations vs. two slowing down factors on the translation manifolds: SDT_ComK acting on the ComK mRNA sub-manifold and SDT_ComS acting on the ComS sub-manifold. In panel (A), SDT_ComS = 0.9, and in panel (B), SDT_ComK = 0.14. Panels(C) and (D) are the equivalent bifurcation diagrams for ComS. All bifurcation diagrams exhibit a Hopf bifurcation indicated by “H” and delineate the dynamical regions where rotation exists (low and high amplitudes of the concomitant limit cycle are shown in green). These four diagrams make it emphatically clear that the choice of SDT_ComS = 0.9 and SDT_ComK = 0.14 are slowing down translation factors on the dynamics of the 6D system that result in healthy limit cycle dynamics located in mid-range of amplitude, away from the Hopf and away from the appearance/disappearance points of oscillations (end points of green curves on right side of the diagram). The diagrams show stable fixed points in solid red or black, and unstable fixed points in dashed red or black. The fixed point with the Hopf bifurcation and associated rotation are denoted by #3 in Figures 2 and 3.

Fig.5 Effect of increasing time-scale separation by slowing down translation (SDT) and effect of reducing fluctuations.(A) Shows the phase portrait of the canonical Bacillus subtilis circuit with k₁ = 0.05. The nullclines and the fixed points of the dynamics are the same as in Figure 3A. Here, a 2D computation of a trajectory initiated at the location of the black dot is shown in magenta. The 2D system exhibits built-in infinite time-scale separation: the mRNA sub-manifold is at rest. The 2D magenta track clearly falls into a limit cycle. Shown in cyan is the corresponding 6D computation (also initiated at the same location as the 2D) for which the slowdown factors SDT_ComS = 0.9 and SDT_ComK = 0.14 are enforced. The values for the slowdown factors are determined using bifurcation analysis (see Figure 4). As explained in the text, the 6D system is the infinite number of molecule limit of the corresponding stochastic system, shown in panel (B). In the 6D system, while the mRNA sub-manifold is not at rest by construction, the presence of the slowdown factors on the translation dynamics enforces a large time-scale separation that brings the behavior close to that of the 2D system. Therefore, the 6D endowed with translation slowdown cyan track also falls into a limit cycle. Note that a 6D track without translation slowdown (SDT_ComS = 1 and SDT_ComK = 1) is excitatory (see Supplementary Figure S1). As stated above, panel (B) shows the corresponding stochastic simulation with the same translation slowdowns as in panel (A) but with a canonical level of noise. It is clear that, in this noise regime, there is no limit cycle. Panel (C) however shows the stochastic simulation with the translation slowdown factors but for which both the number of molecules and the simulation volume are multiplied by a factor of 10, thus leaving the concentrations intact yet reducing the intrinsic fluctuations by roughly 10^1/2. As expected, the limit cycle appears. Thus both time-scale separation and biochemical noise conspire to hide oscillatory dynamics. Panel (A is in dimensionless unit and panels (B) and (C) are in dimensioned units. For convenience, in panels (B) and (C), the locations of the 2D system dynamical fixed points from panel (A) are indicated by black circles.

Fig.6 2D Dynamics of the inverted Bacillus subtilis circuit. The ComK nullcline is shown in red and the MecAnullcline is shown in green. The nullcline intersections labeled “1”, “2” and “3” are stable spiral, saddle, and unstable spiral fixed points of the dynamics, respectively. The location of the separatrix is indicated by the cyan curve. In panel (A), k_m = 0.1474 and in panel (B) k_m =0.1475. In panels (A) and (B) a 2D system track is started at the same location indicated by a black dot. In panel (A) the system is the excitatory regime so the track winds around fixed point #3 and falls back into fixed point #1.In panel (B) however, the system is in oscillatory mode so the track falls into a limit cycle. Panels (A) and (B) have dimensionless units. Panel (C) is a bifurcation analysis of the 2D dynamics showing in orange and red the location of fixed point #3 for MecA and ComK, respectively. Solid lines indicate this fixed point is stable; dashed lines indicate the fixed point is unstable. There is a Hopf bifurcation located at k_m ~0 .1559. To the left of the Hopf develops a limit cycle whose maximum/minimum amplitudes are denoted in purple/blue and green/red lines for MecA and ComK, respectively. Towards the left of the diagram is the appearance/disappearance point of the limit cycle located between k_m = 0.1474 and k_m = 0.1475. The dynamical behavior observed in panels (A) and (B) is therefore corroborated by the bifurcation analysis. At k_m = 0.1474 no limit cycle is expected while at k_m =0.1475 one observes a large amplitude limit cycle. The 2D system has built-in infinite time-scale separation; its mRNA sub-manifold is at rest.

Fig.7 Stochastic dynamics of the inverted Bacillus subtilis circuit. (A–C) Show probability densities as a function of the k_m. In panel (A), k_m =0.1, in panel (B, k_m=0.15 and in panel (C) k_m= 0.2. The units on these plots are dimensioned so the effective K_m = k_m× sf where the scaling factor “sf” takes care of dimensions. In panel (A) the system is in excitatory regime, in panel (B) the system is in oscillatory regime (notice the presence of limit cycle behavior) and in panel (C) the system is bi-stable as the upper fixed point’s stability has now shifted to stable spiral. These three dynamical regimes are roughly predicted by the bifurcation analysis shown in Figure 6C). However, the correspondence is only expected to be approximate because whereas in the 2D system the time-scale separation is infinite, in the canonical system shown here, it is finite. The colorbars on the right of each plot indicate the log base 10 of the density.

Fig.8 Appearance/disappearance of the limit cycle in the stochastic dynamics of the inverted Bacillus subtilis circuit. (A?C) show probability densities as a function of the k_m. In panel (A), k_m = 0.125, in panel (B), k_m=0.1474 and in panel (C) k_m=0.1475. The units on these plots are dimensioned so the effective k_m = k_m× sf where the scaling factor “sf” takes care of dimensions. In panel (A), as expected from the 2D bifurcation analysis (Figure 6C), the dynamics does not exhibit a limit cycle: it is purely excitatory. In panel (B, just before the appearance/disappearance point of the limit cycle, and in panel (C) just following it, both diagrams exhibit similar limit cycle behavior. Whereas in panel (B, according to the 2D bifurcation analysis, there should not be any limit cycle behavior, there clearly is, as much as there is in panel (C) where oscillations are expected and observed. Panel (D) offers a stochastic simulation also for k_m=0.1474 but including a translation slowdown factor SDT=0.5. Comparing panel (B) to panel (D), it is clear that the inclusion, in panel (D), of significant time-scale separation in the system makes the limit cycle behavior disappear. Hence, the stochastic system with translation slowdown has acquired increased similarity with the 2D infinite time-scale simulation.

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[1]	Hao Tian, Ying Yang, Sirui Liu, Hui Quan, Yi Qin Gao. Toward an understanding of the relation between gene regulation and 3D genome organization[J]. Quant. Biol., 2020, 8(4): 295-311.
[2]	Marc Turcotte. Delineating the respective impacts of stochastic curl- and grad-forces in a family of idealized core genetic commitment circuits[J]. Quant. Biol., 2016, 4(2): 69-83.
[3]	Hongguang Xi, Marc Turcotte. Parameter asymmetry and time-scale separation in core genetic commitment circuits[J]. Quant. Biol., 2015, 3(1): 19-45.
[4]	Arwen Meister, Chao Du, Ye Henry Li, Wing Hung Wong. Modeling stochastic noise in gene regulatory systems[J]. Quant. Biol., 2014, 2(1): 1-29.

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