Correction: Systems analysis of the CO_{2} concentrating mechanism in cyanobacteria
Main text
Mangan NM, Brenner MP. 2014. Systems analysis of the CO_{2} concentrating mechanism in cyanobacteria. eLife 3:e02043. doi: 10.7554/eLife.02043
Published 29 April 2014
We would like to issue the following corrections to our research article entitled, ‘Systems analysis of the CO_{2} concentrating mechanism in cyanobacteria.’ The code used to generate the plots in the paper, mistakenly used 6.022 × 10^{22} instead of 6.022 × 10^{23} for Avogadro's number. This correction changes the quantitative numbers in the original paper but none of the conclusions are affected. Here are the corrected tables, figures, and text that changed. We thank Avi Flamholz for catching this error.
The article has been corrected accordingly. The original version published on 29 April 2014 is provided as Supplementary file 1 (file held on figshare under doi: 10.6084/m9.figshare.1352030).
Table 2. Enzymatic rates
We recalculated the enzymatic rates for carbonic anhydrase and RuBisCO. The values for V_{max} and K_{1/2} listed in the previous version of Table 2 used the incorrect 6.022 × 10^{22} value for Avogadro's number. All plots in the original paper were for those values. All new plots are for the values listed in the updated table, using Avogadro's number 6.022 × 10^{23}. The V_{max} values are all about an order of magnitude lower than previously calculated.
Enzyme reaction  active sites  k_{cat} $\left[\frac{1}{s}\right]$  V_{max} in ‘cell’ $\left[\frac{\mu M}{s}\right]$  V_{max} in carboxysome $\left[\frac{\mu M}{s}\right]$  K_{1/2} $\left[\mu M\right]$ 

carbonic anhydrase hydration  80  8 × 10^{4}  8.8 × 10^{3}  1.5 × 10^{7}  3.2 × 10^{3} 
carbonic anhydrase dehydration  80  4.6 × 10^{4}  1.5 × 10^{4}  8.8 × 10^{6}  9.3 × 10^{3} 
RuBisCO carboxylation  2160  26  178  1.8 × 10^{5}  270 
Figure 2. Phase space for ${\text{HCO}}_{3}^{}$ transport and carboxysome permeability
The main effect of correcting Avagadro's number is that the peak defining optimal carboxysome permeability is lower and broader. The optimal carboxhysome permeability for a target carboxysomal CO_{2} concentration is found by looking at the leftmost value on a line of constant concentration on the permeability vs ${\text{HCO}}_{3}^{}$ transport plot. This point represents the carboxysome permeability where the least amount of ${\text{HCO}}_{3}^{}$ transport is needed to achieve a given carboxysomal CO_{2} concentration. For 99% carboxylation (cyan curve below) this value is now around k_{c} = 10^{−3} cm/s (previously k_{c} = 6 × 10^{−3} cm/s). Carbonic anhydrase also saturates at a lower ${\text{HCO}}_{3}^{}$ transport value.
Figure 2—figure supplement 1. Effect of decreased diffusion in the carboxysome
With the correction, decreasing the diffusion constant in the carboxysome has much less of an effect on the phase space plot. For optimal or larger carboxysome permeability, there is very little effect (k_{c} ≥ 10^{−3} cm/s).
Figure 2—figure supplement 2. Effect of CO_{2} scavenging or facilitated uptake
The correction reduces the effect of facilitated uptake. At the optimal permeability it has nearly no effect (difference between dashed and solid lines near k_{c} = 10^{−3} cm/s).
Figure 3. CO_{2} concentration in the carboxysome as function of ${\text{HCO}}_{3}^{}$ transport: comparison of numeric and analytic solutions
We plot Figure 3 assuming the new optimal carboxysome permeability of k_{c} = 10^{−3} cm/s (previously k_{c} = 6 × 10^{−3} cm/s). There is very little change from the previous Figure 3, with the updated carboxysome permeability. Had we plotted at the same k_{c} = 6 × 10^{−3} cm/s, the CO_{2} would have been lower, because the system would not be at the optimal permeability.
Figure 3—figure supplement 1. No effect of localizing carbonic anhydrase to the shell of the carboxysome
The correction did not at all effect the conclusion that the organization of carbonic anhydrase within the carboxysome has no effect on the resulting concentrations. Consistent with updated Figure 2—figure supplement 1, the effect of decreasing the diffusion in carboxysome is much less prominent than in the original manuscript.
Figure 4. CO_{2} concentration in the carboxysome with varying carboxysome permeability
The correction does not effect the conclusion that an optimal carboxysome permeability exists, or what causes it, but it changes its absolute value. The correction has two main effects: (1) it decreases the optimal carboxysome permeability (seen as a shift in the peak) by a factor of 6, and (2) for carboxysome permeabilities higher than optimal (k_{c} > 0.1 cm/s) the CO_{2} concentration is an order of magnitude lower. Therefore the optimal carboxysome permeability peak is at a lower k_{c} value and more prominent.
Table 3. Fate of carbon brought into the cell for j_{c} = 0.6 cm/s and k_{c} = 1 × 10^{−3} cm/s
We updated the flux to the value at which cytosolic ${\text{HCO}}_{3}^{}$ is 30 mM and carboxysome permeability is optimal. Previously they were j_{c} = 0.7 cm/s and k_{c} = 6 × 10^{−3} cm/s.
Table 4. Fate of carbon brought into the cell for j_{c} = 0.06 cm/s and k_{c} = 1 × 10^{−3} cm/s
Discussion of fluxes and resulting 2phosphoglycolate, and 3phosphoglycerate production
The discussion of the number of transporters needed for this magnitude of ${\text{HCO}}_{3}^{}$ influx increases by an order of magnitude from ${10}^{3}\text{\hspace{0.17em}}\frac{\text{transporters}}{\mu {\text{m}}^{2}}$ to ${10}^{4}\text{\hspace{0.17em}}\frac{\text{transporters}}{\mu {\text{m}}^{2}}$. While the change in uptake is less than a factor of two different, the conversion from picomoles/s to molecules/s for a cell used the incorrect value of Avogadro's number. So this is now about an order of magnitude larger than the number of ATP synthase complexes on the thylakoid membrane of spinach.
The numbers for net flux of ${\text{HCO}}_{3}$ change slightly. For j_{c} = 0.6 cm/s the net flux is $6\times {10}^{6}\frac{\text{pmol}}{\text{cell\hspace{0.17em}s}}$ compared to the previous value of ${10}^{5}\frac{\text{pmol}}{\text{cell\hspace{0.17em}s}}$ and for j_{c} = 0.06 cm/s the net flux is $9\times {10}^{7}\text{\hspace{0.17em}}\frac{\text{pmol}}{\text{cell\hspace{0.17em}s}}$ compared with the previous value of $1\times {10}^{6}\text{\hspace{0.17em}}\frac{\text{pmol}}{\text{cell\hspace{0.17em}s}}$. So the conclusion that measured net fluxes of ${10}^{6}\frac{\text{pmol}}{\text{cell\hspace{0.17em}s}}$ compares best with the lower ${\text{HCO}}_{3}^{}$ flux still holds.
Since Avagadro's number effects the rate of carboxylation and oxygenation, the recalculated values for 2phosphoglycolate and 3phosphoglycerate production are lower. This exacerbates the large amount of energy that appears to go into CO_{2} concentration compared to the actual fixation. Now around 0.03% of concentrated inorganic carbon is fixed into 3phosphoglycerate. Over 99.9% is lost to leakage either in the form of CO_{2} or ${\text{HCO}}_{3}^{}$ (previously these values were calculated to be 99%). The higher ${\text{HCO}}_{3}^{}$ flux assumption (Table 3) predicts that only 217 2phosphoglycolate are produced a second, and the lower ${\text{HCO}}_{3}^{}$ flux assumption (Table 4) predicts that 1.4 × 10^{3} 2phosphoglycolate are produced a second. So the higher flux allows the production of many fewer 2phosphoglycolate than previously estimated. Recalculating the time to fix the necessary carbon to replicate a cell results in 7–21 hr for the higher flux rate (Table 3) and 11–35 hr for the lower flux rate (Table 4). Both are consistant with the division times of cyanobacteria, but the higher flux rate is a better match.
In conclusion, the correction further highlights the puzzle of how much energy cyanobacteria must use ${\text{HCO}}_{3}^{}$ to achieve a level of CO_{2} which creates adequate fixation rates. Either cyanobacteria use a much larger amount of energy to achieve this internal concentration than to fix each CO_{2}, the permeability of the cell must be much lower for ${\text{HCO}}_{3}^{}$ than assumed here, or there is another mechanism not covered in this model.
Figure 5. Flux of ${\text{HCO}}_{3}^{}$ from varying sources as the proportion of CO_{2} to ${\text{HCO}}_{3}^{}$ outside the cell changes
The correction decreases the effect of CO_{2} scavenging at high permeabilities (k_{c} = 1 cm/s and 10^{−2} cm/s). It now appears to be negligible compared to ${\text{HOC}}_{3}^{}$ uptake and CO_{2} uptake for all permeabilities. Only when CO_{2} concentration is a very small percentage of the external carbon, does the scavenging contribute more than the CO_{2} uptake, and both are far less than ${\text{HCO}}_{3}^{}$ uptake.
Figure 6. CO_{2} concentration with various organizations of enzymes in the cell
With the corrected Avagadro's number the CO_{2} concentration is significantly lower in all but the optimal permeability case, making the error rate drastically higher. The effect of optimal permeability on CO_{2} concentration is, therefore, much larger.
Corrections to text
Throughout the text the default ${\text{HCO}}_{3}^{}$ flux has been changed from ${j}_{c}=0.7\frac{\text{cm}}{\text{s}}$ to ${j}_{c}=0.6\frac{\text{cm}}{\text{s}}$ and optimal carboxysome permeability from ${k}_{c}=6\times {10}^{3}\frac{\text{cm}}{\text{s}}$ to ${k}_{c}={10}^{3}\frac{\text{cm}}{\text{s}}$.
In the discussion of fluxes and concentrations we previously had written:
‘Our simulated cell has a flux of 2 × 10^{7} molecules/s. Assuming the rate of transport per transporter of ${10}^{3}\frac{\text{molecules}}{\text{s}}$ and our cell’s surface area this requires about ${10}^{3}\frac{\text{transporters}}{\mu {\text{m}}^{2}}$. This is actually not that far off from the number of ATP synthase complexes on the thylakoid membrane in spinach, $700\frac{\text{complexes}}{\mu {\text{m}}^{2}}$ (Miller and Staehelin, 1979), although it is still quite high.’
This is now: ‘Our simulated cell has a flux of 2 × 10^{8}molecules/s. Assuming the rate of transport per transporter of ${10}^{3}\frac{\text{molecules}}{\text{s}}$ and our cell’s surface area this requires about ${10}^{4}\frac{\text{transporters}}{\mu {\text{m}}^{2}}$. This is about an order of magnitude higher than the number of ATP synthase complexes on the thylakoid membrane in spinach, $700\frac{\text{complexes}}{\mu {\text{m}}^{2}}$ (Miller and Staehelin, 1979).’
In the discussion of fluxes and concentrations we previously had written:
‘At the higher flux rate (Table 3) this means that a cell could replicate every 1–2 hr, so faster than cyanobacteria replicate. The lower flux rate (Table 4) would produce fix enough CO_{2} for the cell to replicate every 8 to 21 hr, which is similar to the division times of cyanobacteria.’
This is now: ‘At the higher flux rate (Table 3) this means that a cell could replicate every 7–21 hr and the lower flux rate (Table 4) allows replication every 11 to 35 hr. Both are consistent with the division times of cyanobacteria.’
In the caption of Figure 5 we had written:
‘When the carboxysome permeability is larger than optimal, k_{c} = 1 cm/s, scavenging can contribute more than facilitated uptake at low external CO_{2} concentrations. However, when the carboxysome permeability at or below our geometric bound, k_{c} <= 0.02 cm/s, scavenging is negligibly small.’
This is now: ‘Scavenging is negligibly small for all values of k_{c} shown.’
In the text of the Benefit of CO_{2} to ${\text{HCO}}_{3}^{}$ conversion: facilitated uptake or scavenging of CO_{2} we had written: ‘Scavenging only contributes significantly to total incoming ${\text{HCO}}_{3}^{\mathrm{}}$ when the carboxysome permeability is higher than optimal, Figure 5, and does not contribute significantly below our calculated upper bound of ${k}_{c}<=0.02cm/s$. In these ranges for carboxysome permeability, there is very little CO2 leaking out of the carboxysome into the cytosol, so there is very little CO2 to scavenge, Figure 5.’
This is now: ‘Scavenging is negligibly small for all values of k_{c} shown. There is very little CO_{2} in the cytosol, so there is very little CO_{2} to scavenge, Figure 5.’
There were also a few typos we have now corrected:
In the Reaction diffusion model section we had: ‘RuBisCO also requires ribulose1,5bisphosphate, the substrate which CO_{2} reacts with to produce 3phosphoglycolate.’ This has been corrected to ‘… CO_{2} reacts with to produce 3phosphoglycerate.’
in Table 1: ‘${k}_{m}^{H}$ permeability of cell membrane to CO_{2}’ has been corrected to: ‘${k}_{m}^{H}$ permeability of cell membrane to ${\text{HCO}}_{3}^{}$’.
Table 2 header said k_{cut} which has been corrected to k_{cat}.
In the Varying ${\text{HCO}}_{3}^{}$ transport saturates enzyme section we had:
The chemical equilibrium is ${K}_{eq}=H/C=\left({K}_{ba}{V}_{ca}\right)\left({K}_{ca}{V}_{ba}\right)\approx 5$, for pH around 7 (DeVoe and Kistiakowsky, 1961), so that ${\text{HCO}}_{3}^{}$ > CO_{2} in the carboxysome.
The equation is now: ${K}_{eq}=H/C=\left(KbaVca\right)/\left(KcaVba\right)\approx 5$.
In Figure 3—figure supplement 1 the units for D_{c} on both the plot and caption are cm/s, they should be cm^{2}/s.
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 Version of Record published: March 26, 2015 (version 1)
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© 2015, Mangan and Brenner
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