Adiabatic elimination of the fast variables <i>q</i>_<i>r</i> and <i>q</i>_<i>a</i>.

<p>Due to the fast dynamics that HetR and NtcA exhibit, we can approach the treatment of the system by adopting a point of view that follows the slower variables <i>q</i>_<i>s</i> and <i>q</i>_<i>n</i>. From this viewpoint, the time-evolution of the pair (<i>q</i>_<i>s</i>(<i>t</i>), <i>q</i>_<i>n</i>(<i>t</i>)) is considered by assuming that <i>q</i>_<i>r</i> and <i>q</i>_<i>a</i> instantaneously relax to an equilibrium, which corresponds to a sink (<math><mrow><msubsup><mi>q</mi><mi>r</mi><mo>*</mo></msubsup></mrow></math>, <math><mrow><msubsup><mi>q</mi><mi>a</mi><mo>*</mo></msubsup></mrow></math>) for the fixed pair (<i>q</i>_<i>s</i>(<i>t</i>), <i>q</i>_<i>n</i>(<i>t</i>)). Depending on the region of the (<i>q</i>_<i>s</i>, <i>q</i>_<i>n</i>)-plane, there are three fixed points (two sinks corresponding to the highest and the lowest concentrations respectively and a saddle in the middle) or one (a sink) for <i>q</i>_<i>r</i> and <i>q</i>_<i>a</i> (I and II). There are two one-sink regions that are separated from the two-sink region by saddle-node bifurcations (A-F). Sinks and saddles are represented by filled and unfilled circles respectively and arrows indicate the flow of the dynamics. We can then imagine the dynamics of <i>q</i>_<i>s</i> and <i>q</i>_<i>n</i> as evolving either in the bottom or in the top branch of I. In the two-sink region, both branches are plausible and the history of the dynamics determine the solution (hysteresis effect): a dynamics in a branch will continue in it until experiencing a bifurcation in the (<i>q</i>_<i>r</i>, <i>q</i>_<i>a</i>) plane (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004129#pcbi.1004129.g005" target="_blank">Fig. 5</a> for examples).</p

Heterocyst pattern.

<p>Time-evolution of the pattern of heterocysts (A) and of the probability distribution of the distance between consecutive heterocysts (B). Green and blue curves represent the concentration profiles of HetR and PatS (NtcA and cN are not presented since their behavior along the filament is comparable to that of HetR and PatS, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004129#pcbi.1004129.g006" target="_blank">Fig. 6</a> to see the similarities). Small perturbations along the filament of vegetative cells (initially in the steady state B of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004129#pcbi.1004129.g005" target="_blank">Fig. 5</a>) are amplified due to diffusion processes in a demonstration of Turing’s theory [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004129#pcbi.1004129.ref051" target="_blank">51</a>]. New heterocysts appear in regions that are not dominated by the action of other heterocysts. Finally, the competition between nearby differentiating cells ceases the differentiation of some of them, as observed in B: consecutive heterocysts, which are created by strong perturbations, finally disapear due to the aforementioned competition. The final pattern presents localized levels of HetR (heterocysts) and a diffusive-like behavior of PatS, as expected.</p

Time evolution of molecular concentrations.

<p>Time evolution of the main components of the differentiation in heterocysts (green) and vegetative cells (blue). Averages along the filament are also presented (black). Heterocysts, due to the early diffusion, evolve toward steady states of the type D of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004129#pcbi.1004129.g005" target="_blank">Fig. 5</a> characterized by high levels of HetR and NtcA while vegetative cells present very low concentrations of them (A and B). The levels of PatS and cN in vegetative cells depend on their distance to close heterocysts: C and D show the concentrations of PatS and cN in a heterocyst and in its first two neighbouring vegetative cells, which clearly highlight the effect of diffusion along the filament.</p

GS/GOGAT cycle.

<p>2-OG and cN indirectly interact through the GS/GOGAT cycle. Glutamine is transformed into glutamate by means of 2-OG through the 2-OG amidotransferase (GOGAT) while cN converts glutamate into glutamine through the glutamine synthetase (GS). The importance of the cycle in heterocyst differentiation is twofold. From one side, it constitutes the early one-cell sensor to nitrogen starvation: the absence of cN breaks the cycle down and 2-OG starts to accumulate, whose action leads to the cascade of processes that provoke the differentiation (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004129#pcbi.1004129.g001" target="_blank">Fig. 1</a>). Additionally, later during the differentiation, it processes the cN created by the heterocysts decreasing the levels of 2-OG. The latter is crucial for the formation of the heterocyst pattern (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004129#pcbi.1004129.g003" target="_blank">Fig. 3</a>).</p

Main components and interactions involved in the reaction to combined nitrogen deprivation in cyanobacteria.

<p>Rectangular boxes represent genes (<i>ntcA, hetR</i> and <i>patS</i>) while rounded boxes and circles represent transcription factors (NtcA, HetR and PatS) and smaller molecules (2-OG and cN) respectively. Normal-tipped and flat-tipped arrows stand for up-regulating and down-regulating processes respectively. Dashed lines stand for indirect or imperfectly understood interactions. The accumulation of 2-OG enhances the DNA-binding activity of NtcA, which in turn up-regulates the transcription of <i>ntcA</i> and <i>hetR</i>. HetR activates <i>ntcA</i> and <i>hetR</i> (composing the central NtcA-HetR autoregulatory loop), the inhibitor <i>patS</i> and other genes that lead to nitrogen fixation and the morphological changes involved in heterocyst differentiation. 2-OG and cN levels are linked through the GS/GOGAT cycle (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004129#pcbi.1004129.g002" target="_blank">Fig. 2</a>).</p

States of a cyanobacterium when subjected to different conditions of nitrogen and diffusion.

<p>When the cell is provided of cN (<i>l</i>_<i>n</i> = 0.03), there is only one stable fixed point (A) in the bottom branch, which corresponds to a state in which the production of both HetR and PatS is minimum (vegetative state). When subjected to nitrogen deprivation (<i>l</i>_<i>n</i> = 0), there are two stable fixed points (B and C) each one in a different branch. The first point (B) is a vegetative state in which there exists an equilibrium between a small production of HetR, PatS and cN. The same kind of equilibrium is present in the second fixed point (C) but in this case the production of all TFs and cN is high (heterocyst steady state). When the cell is exposed to nitrogen stress its trajectory evolves from A to the steady state B and thus it remains vegetative. Assuming some diffusion of cN and PatS from the cell (<i>l</i>_<i>s</i> = −0.2 and <i>l</i>_<i>n</i> = −0.002), the only stable state (D) corresponds to a heterocyst state with high levels of production of HetR, cN and PatS, being the latter transported to the surroundings of the cell.</p