Systems analysis of the CO2 concentrating mechanism in cyanobacteria

Cyanobacteria are photosynthetic bacteria with a unique CO2 concentrating mechanism (CCM), enhancing carbon fixation. Understanding the CCM requires a systems level perspective of how molecular components work together to enhance CO2 fixation. We present a mathematical model of the cyanobacterial CCM, giving the parameter regime (expression levels, catalytic rates, permeability of carboxysome shell) for efficient carbon fixation. Efficiency requires saturating the RuBisCO reaction, staying below saturation for carbonic anhydrase, and avoiding wasteful oxygenation reactions. We find selectivity at the carboxysome shell is not necessary; there is an optimal non-specific carboxysome shell permeability. We compare the efficacy of facilitated CO2 uptake, CO2 scavenging, and HCO3− transport with varying external pH. At the optimal carboxysome permeability, contributions from CO2 scavenging at the cell membrane are small. We examine the cumulative benefits of CCM spatial organization strategies: enzyme co-localization and compartmentalization. DOI: http://dx.doi.org/10.7554/eLife.02043.001


Introduction
Intracellular compartments are critical for directing and protecting biochemical reactions. One of the simplest and most striking known examples of compartmentalization are the carboxysomes (Cannon et al., 2001;Yeates et al., 2008) of cyanobacteria and other autotrophic proteobacteria (Savage et al., 2010;Rosgaard et al., 2012). These small, 100-200 nm compartments separate the principal reaction of the Calvin cycle, the RuBisCO catalyzed fixation of carbon dioxide (CO 2 ) into 3-phosphoglycerate, from the rest of the cell (Cannon et al., 1991). CO 2 and oxygen (O 2 ) competitively bind as substrates of RuBisCO, and the reaction with O 2 produces phosphoglycolate, a waste product which must be recycled by the cell (Jordan & Ogren, 1981;Tcherkez et al., 2006;Savir et al., 2010). To maximize carboxylation and minimize oxygenation, the carboxysome is believed to act as a diffusion barrier to CO 2 (Reinhold et al., 1989;Dou et al., 2008). There is much interest in the design and function of such compartments and whether they can be used to enhance carbon fixation in other organisms such as plants or to improve reaction rates in other metabolic systems (Ducat & Silver, 2012;Agapakis et al., 2012;Papapostolou & Howorka, 2009;Frank et al., 2013). Increased efficiency of biochemical reactions will lead to better yield in bioengineered bacterial systems, creating new possibilities for production of high-value products such as biofuels. Enhancing carbon fixation in plants or other organisms could lead to increased carbon sequestration, or crop yield.
The concentrating mechanism in cyanobacteria relies on the interaction of a number of well characterized components, as shown in Figure 1, transferring inorganic carbon from outside the cell into cytosol and carboxysomes (Allen, 1984;Badger & Price, 2003;Kaplan & Reinhold, 1999;Price et al., 2007). Due to this mechanism, inorganic carbon concentration is elevated well above 200-300 μM, the CO 2 concentration required for saturating the RuBisCO. Additionally a high CO 2 concentration increases the ratio of CO 2 to O 2 so that carboxylation dominates over oxygenation. Concentrations of 20-40 mM inorganic carbon, up to 4000-fold higher than external levels, have been energy from the sun to convert carbon dioxide into sugars and other useful compounds. This process-called photosynthesis-releases oxygen as a by-product. Cyanobacteria were crucial in making the atmosphere of the early Earth habitable for other organisms, and they created the vast carbon-rich deposits that now supply us with fossil fuels. Modern cyanobacteria continue to sustain life on Earth by providing oxygen and food for other organisms, and researchers are trying to bioengineer cyanobacteria to produce alternative, cleaner, fuels.
Understanding how cyanobacteria can be as efficient as possible at harnessing sunlight to 'fix' carbon dioxide into carbon-rich molecules is an important step in this endeavor. Carbon dioxide can readily pass through cell membranes, so instead cyanobacteria accumulate molecules of bicarbonate inside their cells. This molecule is then converted back into carbon dioxide by an enzyme found in specials compartments within cells called carboxysomes. The enzyme that fixes the carbon is also found in the carboxysomes. However, several important details in this process are not fully understood.
Here, Mangan and Brenner further extend a mathematical model of the mechanism that cyanobacteria use to concentrate carbon dioxide in order to explore the factors that optimize carbon fixation by these microorganisms. Carbon fixation is deemed efficient when there is more carbon dioxide in the carboxysome than the carbon-fixing enzyme can immediately use (which also avoids wasteful side-reactions that use oxygen instead of carbon dioxide). However, there should not be too much bicarbonate, otherwise the enzyme that converts it to carbon dioxide is overwhelmed and cannot take advantage of the extra bicarbonate.
Mangan and Brenner's model based the rates that carbon dioxide and bicarbonate could move in and out of the cell, and the rates that the two enzymes work, on previously published experiments. The model varied the location of the enzymes (either free in the cell or inside a carboxysome), and the rate at which carbon dioxide and bicarbonate could diffuse in and out of the carboxysome (the carboxysome's permeability). Mangan and Brenner found that containing the enzymes within a carboxysome increased the concentration of carbon dioxide inside the cell by an order of magnitude. The model also revealed the optimal permeability for the carboxysome outer-shell that would maximize carbon fixation.
In addition to being of interest to researchers working on biofuels, if the model can be adapted to work for plant photosynthesis, it may help efforts to boost crop production to feed the world's growing population.
The goal of this study is to further develop a mathematical model of the CCM (Reinhold et al., 1989;Fridlyand et al., 1996;Reinhold et al., 1991) that uses recent experimental progress on the CCM to untangle the relative roles of the different molecular components, and predict the region of parameter space where efficient carbon fixation occurs. We are considering conditions where CO 2 is limiting (15μM external inorganic carbon) and, for the moment, ignore other biological pressures. In this context, efficient carbon fixation requires two conditions: First, the CO 2 concentration must be high enough that RuBisCO is saturated, and the competitive reaction with O 2 is negligible. We emphasize that for the oxygenation reaction to be negligible the CO 2 concentration should be higher than needed to merely saturate RuBisCO. Secondly, the carbonic anhydrase within the carboxysome must be unsaturated, so that extra energy isn't wasted transporting unused -3 HCO into the cell. Examination of the system performance with varying expression levels of -3 HCO transporters, carboxysome permeability, and conversion from CO 2 to -3 HCO , reveals a parameter window where these conditions are simultaneously satisfied. We comment on the relation of this window to measured carbon pools, carbon fixation rates, and -3 HCO transporters. We find that the -3 HCO concentration in the cytosol is constant across the cell, set by the -3 HCO transport and leakage rates, and depends very little on the carboxysome permeability. The carboxysome permeability does, however, set how the CO 2 is partitioned between the carboxysome and cytosol. At optimal carboxysome permeability, -3 HCO diffusion into the carboxysome is fast enough to supply inorganic carbon for fixation, but the rate of CO 2 leakage out of the carboxysome is low. We explore the fluxes from CO 2 facilitated uptake and scavenging with varying ratios of external CO 2 and -3 HCO . Finally we discuss the proportion the carbon concentration comes from different methods of spatial organization such as co-localization, encapsulation, and spatial location of carboxysomes. Concentration of carbonic anhydrase increases the maximum rate of reaction for carbonic anhydrase per volume, causing carbonic anhydrase to saturate at a  HCO transport into the cell is indicated (in light blue), as well as active conversion from CO 2 to -3 HCO , sometimes called 'facilitated uptake' or 'scavenging', at membranes (in orange). Both CO 2 and -3 HCO can leak in and out of the cell, with CO 2 leaking out much more readily. Both species passively diffuse across the carboxysome shell. Carbonic anhydrase (red) and RuBisCO (blue) are contained in the carboxysomes and facilitate reactions as shown. DOI: 10.7554/eLife.02043.003 higher level of -3 HCO , and achieve an order of magnitude higher local CO 2 concentrations. Encapsulation of the reactions in an optimally permeable carboxysome shell achieves another order of magnitude of CO 2 concentration.

Reaction diffusion model
We present our mathematical model, which captures all aspects of the CCM as described above. This model is an expansion of previously developed models (Reinhold et al., 1989;Fridlyand et al., 1996;Reinhold et al., 1991). Our three dimensional model of the CCM solves for both the -3 HCO and CO 2 concentration throughout a spherical cell. We solve this model numerically and analytically at steady state for three different spatial organizations of carbonic anhydrase and RuBisCO in the cell (See Figure 6): enzymes distributed evenly throughout the cell, enzymes localized to the center of the cell but not encapsulated (as they would be on a scaffold), enzymes encapsulated in a carboxysome. We compare the effects of these scenarios in the discussion section, and for now consider a spherical cell of radius R b = 0.5 μm with a single spherical carboxysome of radius R c = 50 nm containing RuBisCO and carbonic anhydrase. Numerical computations are carried out with finite difference methods in MATLAB. The details of analytic solutions are given in the Supplementary file 1.
We include the effects of diffusion, active transport and leakage through the cell membrane, and reactions with carbonic anhydrase and RuBisCO. In the carboxysome (r < R c ), the equations governing the -3 HCO and CO 2 , here H and C respectively, are where here D is the diffusion constant, and R CA is the carbonic anhydrase reaction, and R Rub is the RuBisCO reaction. The carbonic anhydrase reaction follows reversible Michaelis-Menten kinetics (Kaplan & Reinhold, 1999;Price et al., 2007), where V ca and V ba are hydration and dehydration rates, proportional to the local carbonic anhydrase concentration. K ca and K ba are the concentration at which hydration and dehydration are half maximum. The RuBisCO reaction follows Michaelis-Menten kinetics with competitive binding with O 2 , Here V max is the maximum rate of carbon fixation and K m is the apparent half maximum concentration value, which has been modified to include competitive binding with O 2 , O. K i is the dissociation constant of O 2 with the RuBisCO and ′ m K is the half maximum concentration with no O 2 present. RuBisCO also requires ribulose-1,5-bisphosphate, the substrate which CO 2 reacts with to produce 3-phosphoglycolate. Under CO 2 limiting conditions it has been shown that there is sufficient ribulose-1,5-bisphosphate to saturate all RuBisCO active sites, and the reaction rates are independent of ribulose-1,5-bisphosphate concentrations (Mayo et al., 1989;Whitehead et al., 2014).
In the cytosol there is no carbonic anhydrase or RuBisCO activity, so R CA = 0 and = 0 Rub R , and there is only diffusion of CO 2 and -3 HCO . We do not include the natural, but slow, interconversion of CO 2 and -3 HCO in the cytosol. This assumption is a good one given that the -3 HCO concentration is known to be held out of equilibrium in the cell (Volokita et al., 1984;Price & Badger, 1989). In agreement with this experimental observation, we find that all the other processes effecting the concentration of -3 HCO in the cytosol happen much faster than the natural interconversion.
Boundary conditions prescribe the inorganic carbon fluxes into the cell and the diffusion across the carboxysome boundary. We treat the inorganic carbon fluxes at cell and thylakoid membranes together. At this cell boundary, there is passive leakage of both CO 2 and -3 HCO : the velocity of CO 2 across the cell membrane, k C m is about 1000-fold higher than that of -3 HCO , k H m , due to the lower permeability of the membrane to charged molecules. To account for active import of HCO transporters. These transporters include BCT1 (encoded by cpm), which is thought to be powered by ATP; and BicA and SbtA which are thought to be symporters between -3 HCO and Na + , driven by the highly controlled electrochemical gradient for Na + , Research article 2004Omata et al., 1999). Additionally, there are two complexes NDH-1 3 and NDH-1 4 responsible for converting CO 2 to -3 HCO . This conversion is thought to either decrease CO 2 , creating a gradient across the membranes and 'facilitating uptake' of CO 2 , or 'scavenge' CO 2 which has escaped from the carboxysome. These are localized to the thylakoid and possibly the plasma membrane. They have been linked to the photosynthetic linear and cyclic electron transport chain Maeda et al., 2002;Shibata et al., 2001). It has been proposed that electron transport drives the formation of local alkaline pockets where CO 2 more rapidly converts to -3 HCO . We more simply describe the conversion with a maximal reaction rate α, and concentration of half maximal activity of K α . Combining these effects, the boundary condition setting diffusive flux of -3 HCO and CO 2 at the cell membrane is where the subscript cytosol and out indicate we are taking the concentration immediately inside and outside the cell boundary respectively. The diffusion constant times partial derivatives with respect to the radial coordinate, r, define the diffusive flux at the membrane.
The carboxysome shell is composed of proteins with a radius R c ≈ 50 nm. As of yet, there have been no direct measurements of the carboxysome permeability to small molecules. Using the carboxysome geometry, we can calculate an upper bound for the diffusive velocity across the carboxysome shell, which is directly related to the carboxysome permeability. Crystal structures Cheng et al., 2008;yeates et al., 2007) show the surface has approximately N pores = 4800 small pores with radius 0.35 pore r ≈ nm, and thickness l = 1.8 nm. If k c is the characteristic velocity that small molecules pass through the shell, these numbers imply the upper bound for diffusive transport . This calculation is done by taking the probability that a molecule will encounter a pore on the carboxysome shell ×pore surface area carboxysome surface area and multipling it by the speed a small molecule will diffuse through the length of the pore (D/l). Since it does not take into account any charge effects, which would add an additional energy barrier, it is an upper bound. Although there has been much speculation that the positively charged pores might enhance diffusion of negatively charged -3 HCO Dou et al., 2008;Cheng et al., 2008), here we explore the simplest assumption, that both -3 HCO and CO 2 have the same permeability. Namely, the boundary conditions at the carboxysome shell are We will vary k c (henceforth called carboxysome permeability, although it is a velocity) within our model and see that there is a range of k c where the CCM is effective even with k c identical for CO 2 and -3 HCO .

Analysis of model: Finding functional parameter space
Now that we have defined our model, we wish to find the range of parameters where efficient carbon fixation occurs. In what follows, we fix the enzymatic rates, cell membrane permeability, and diffusion constant as reported in the literature (Jordan & Ogren, 1981;Missner et al., 2008;Gutknecht et al., 1977;Heinhorst et al., 2006) (see Table 1 and Table 2). Note that full analytic solutions are available in Supplementary file 1 sections S3 and S4, so the effect of varying other parameters can be analyzed. We consider the efficacy of the CCM as a function of j c , the flux of -3 HCO into the cell, k c , the carboxysome permeability, and the parameters (α, K α ) governing the CO 2 facilitated uptake mechanism. Both α and j c can be regulated by the organism and vary depending on environmental conditions, whereas the carboxysome permeability, k c , is the parameter with the largest uncertainty and debate (Cannon et al., 2001;Yeates et al., 2008;Cheng et al., 2008).
For any given pair of k c and j c , we ask whether the CO 2 concentrating mechanism is effective, using the criteria of saturating RuBisCO, reducing oxidation reactions, and not increasing the -3 HCO concentration beyond carbonic anhydrase saturation. Our central result is presented in Figure 2, which shows the range of k c and j c where these conditions are met, assuming that there is no facilitated uptake, α = 0. The blue shaded region shows where RuBisCO is unsaturated, and the red shaded region shows where carbonic anhydrase is saturated. There is a crescent shaped region between these regions, where the CCM is effective according to our criteria. In the white region oxygenation reactions happen at a rate of greater than 1%. In the green shaded region oxygenation reactions occur at a rate of less than 1%. Within the white and green regions the CO 2 concentration in the carboxysome varies greatly.  (Fridlyand et al., 1996) c m k permeability of cell membrane to CO 2 0.3 cm s (Missner et al., 2008;Gutknecht et al., 1977) H m k permeability of cell membrane to -3 HCO -4 3×10 cm s (Missner et al., 2008;Gutknecht et al., 1977) R c radius of carboxysome 5×10 −6 cm (Cheng et al., 2008;Schmid et al., 2006) R b radius of bacteria 5 × 10 −5 cm (Savage et al., 2010) (Woodger et al., 2005;Sultemeyer et al., 1995;Heinhorst et al., 2006) Enzyme reaction active sites the entire cell or only carboxysome. V ba (V max for carbonic anhydrase dehydration) is estimated by assuming K eq = 5 and using parameters found in (Heinhorst et al., 2006). V ca is V max for carbonic anhydrase hydration. Similarly, K ba , and K ca are K 1/2 for dehydration and hydration respectively. DOI: 10.7554/eLife.02043.006 The lines dividing the regions in Figure 2 are lines of constant carboxysomal CO 2 concentration in j c and k c parameter space. The dark blue line is where CO 2 = K m , the CO 2 concentration for half-maximum RuBisCO reactions. The light blue line indicates parameter values resulting in the CO 2 concentration (C 99% ) where rate of oxygenation reactions is 1% for O 2 concentration of 260 μM. Varying carboxysome permeability, k c values, require more or less HCO 3 transport, j c , to achieve the same carboxysomal CO 2 concentration.
We can calculate an amplification factor for the C 99% carboxysomal CO 2 concentration as Cout + Hout = 133. Any combination of j c and k c which produce C = C 99% , make 133 times more CO 2 available in the carboxysome than there is total inorganic carbon outside the cell. Generally, increasing -3 HCO transport, below the carbonic anhydrase saturation point results in higher CO 2 concentration in the carboxysome.

HCO transport saturates enzymes
The basic physics of the phase diagram Figure 2 follows from examining how CO 2 and -3 HCO in the carboxysome change as j c is varied. Figure 3 shows the response to varying j c , with -3 = 10 c cm k s (the optimal value in Figure 4).
When j c is low, the ratio of CO 2 and -3 HCO is constant, set by the chemical equilibrium at a given pH. In this case the rate of the carbonic anhydrase reaction is much faster than diffusion within the carboxysome, so that V ba KcaH = VcaK ba C . Unlike the uncatalyzed interconversion of CO 2 and -3 HCO in the cytosol, carbonic anhydrase brings the concentrations in the carboxysome to equilibrium very quickly. The chemical equilibrium is for pH around 7 [Heinhorst et al., 2006;DeVoe & Kistiakowsky, 1961)], so that -3 HCO > CO 2 in the carboxysome. Increased pH would increase K eq and the proportion of -3 HCO , while decreased pH would decrease K eq and the proportion of -3 HCO . Such variations do not substantially effect the subsequent discussion and mechanisms, although they will change the absolute values of CO 2 concentration in the carboxysome.
The K m dashed line in Figure 3 shows the CO 2 level above which RuBisCO reaction is saturated: this gives the RuBisCO saturated (blue) boundary in Figure 2. We have similarly marked the concentration C 99% where there is a 1% oxygen reaction error rate with a dash-doted line.
At higher levels, the CO 2 concentration no longer increases with increasing j c , because the carbonic anhydrase is saturated. The saturated regime occurs in Figure 3 when > carboxysome ba H K , so that increasing carboxysome H (controlled directly by j c ) no longer increases the rate of production of carboxysome C . This transition from unsaturated to saturated carbonic anhydrase defines the line for the carbonic anhydrase saturated region in Figure 2. All other parameters, such as reaction rates are held fixed and the value can be found in the Table 1 and Table 2.

Carboxysome permeability has optimal value
For each line of constant concentration in Figure 2 there is an optimal permeability value, where the least -3 HCO transport is required to achieve the same CO 2 concentration. The optimal permeability value shifts downward with increasing CO 2 concentration (compare light and dark blue curves).
For C 99% the optimal permeability is 3 = 10 c cm k s -, below the calculated upper bound: < 0.02 c cm k s obtained above from the carboxysome structure. To further understand the effect of permeability, we examine the CO 2 concentration in the carboxysome for varying carboxysome permeabilities and a fixed -3 HCO transport rate in Figure 4. Figure 4A, shows that there is a broad range of k c where the CCM has maximal efficacy. Figure 4 shows the distribution of inorganic carbon throughout the cell when the permeability is low (B), optimal (C), and high (D). At high permeability, the CO 2 produced in the carboxysome rapidly leaks out of the carboxysome, and the CO 2 concentration in the cytosol, shown in Figure 4D, is relatively high. When the carboxysome permeability decreases to near the optimal value, Figure 4C, the carboxysome traps CO 2 , and the cytosolic levels are lower, decreasing leakage out of the cell. This transition occurs when diffusion across the cell (and carboxysome) takes a shorter time than diffusion through the carboxysome shell; or the CO 2 in the carboxysome is effectively partitioned from the CO 2 in the cell.
If the carboxysome permeability is below optimal, diffusion of -3 HCO into the carboxysome cannot keep up with consumption from RuBisCO, Figure 4B. The existence of an optima requires RuBisCO consumption to be low enough that there is a k c where the cytosol and carboxysome are partitioned, but -3 HCO diffusion in can keep up. When such an optima exists, the carboxysome permeability can improve the CO 2 concentration in the carboxysome without any special selectivity between -3 HCO and CO 2 . The location and concentrating power of the optimal regime, is dependent on the size of the cell and the membrane permeabilities to CO 2 and -3 HCO .

Discussion
Are the fluxes and concentrations reasonable?
While we have solved our model to describe a vast parameter space it is instructive to compare the fluxes and concentrations we find within our optimal parameter space (the green region in Figure 2) to actual numbers. At low external inorganic carbon conditions, internal inorganic carbon pools due to CCM activity are regularly measured as high as C i = 30 mM. The inorganic carbon is predominantly in the form of -3 HCO , and measurements do not distinguish between the cytosol and carboxysome (Kaplan & Reinhold, 1999;Woodger et al., 2005;Sultemeyer et al., 1995;Price et al., 1998Price et al., , 2008. In our model, we find that the cytosolic  Figure 2). From Figure 4 we can also see that the cytosolic -3 HCO concentration is the dominate form of inorganic carbon in the -3 HCO transport is varied, and all other system parameters are held constant. The CO 2 concentration above which RuBisCO is saturated is K m (grey dashed line). The CO 2 concentration where the oxygen reaction error rate will be 1% is C 99% (grey dash-dotted line). The transition between carbonic anyhdrase being unsatruated and saturated happens where the two analytic solutions meet (where the dashed and solid red lines meet). Below a critical value of transport, ≈ -3 10 c cm j s the level of transport is lower than the HCO leaking out, CO 2 leaking out, CO 2 fixation or carboxylation, and O 2 fixation or oxygenation (Table 3). HCO fluxes (transport -leakage) are measured 5 pmol 10 mgChl s , with CO 2 net flux being slightly lower but the same order of magnitude (Whitehead et al., 2014;Badger et al., 1994). High external inorganic carbon conditions produce slightly higher net -3

For cells grown under low inorganic carbon conditions net
HCO rates (Tchernov et al., 1997). Assuming chlorophyll per cell volume of around -11 mgChl 10 cell for cells of our size we can convert this into a flux of -6 10 pmol (cell s) (Whitehead et al., 2014;Keren et al., 2004Keren et al., , 2002. The net -3 HCO flux for our model cell is HCO transport value one order of magnitude smaller, we will get net fluxes of the same order of magnitude as the measurements at the cost of slightly lower carboxylation rates and higher oxygenation rates ( Table 4). This would also mean a lower internal . This is about an order of magnitude higher than the number of ATP synthase complexes on the thylakoid membrane in spinach, 2 complexes 700 m μ (Miller and Staehelin, 1979).
According to our calculation only around 1 % of the carbon transported into the cell is fixed into 3-phosophoglycerate. Even in this highly CO 2 concentrating regime, 5 × 10 4 2-phosophoglycolate produced per second. Cyanobacteria have been shown to have multiple pathways for recycling 2-phosophoglycolate (Hackenberg et al., 2009). Our system fixes CO 2 at a rate of 0.14 pg/hour.   Given the volume of our cell, and the fact that between 115-300 fg/μm 3 of carbon are needed to produce a new cyanobacterial cell (Mahlmann et al., 2008) we need between 0.1 and 0.3 picograms of carbon per cell. 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 11-35 hr. Both are consistent with the division times of cyanobacteria.
It is possible that since the scavenging mechanism is associated with the electron transport chain of the light reactions of photosynthesis scavenging can be ramped up more easily when there is excess light energy. If this were the case, a comparison of c cm j s =1 and =1 cm k s α α is deceiving and K α α could be much larger. Indeed it has been suggested that the cell uses this mechanism as a way to dissipate excess light energy (Tchernov et al., 1997(Tchernov et al., , 2003.

Cellular organization
The most striking aspect of the CCM is the way that spatial organization is used to increase the efficacy of the reactions. Figure 6 compares the effect of different enzymatic reaction organizations. Concentrating carbonic anhydrase and RuBisCO to a small region in the center of the cell, on a scaffold for example, leads to an order of magnitude increase in the concentration of CO 2 . Localizing the carbonic anhydrase to a small volume concentrates it, increasing the maximum reaction rate per volume, V ca and V ba . A larger V ba increases the -3 HCO concentration at which carbonic anhydrase is saturated allowing the mechanism to take advantage of a larger -3 HCO flux, j c . A small increase can be gained from encapsulating the enzymes in a permeable carboxysome shell and another order of magnitude is gained at the optimal permeability. At optimal carboxysome permeability, the CO 2 is effectively partitioned into the carboxysome and conversion can act only as facilitated uptake as shown in Figure 5.
Another advantage of localizing the enzymes in a small region at the center of the cell is separating carbonic anhydrase from the α (CO 2 to -3 HCO ) conversion mechanism, preventing a futile cycle. The futile cycle is most detrimental when the enzymes are distributed through out the cytosol, and increases the oxygenation error rate (data not shown). Concentrating the enzymes away from the cell and thylakoid membranes, where conversion happens, removes this effect. On a scaffold the oxygenation rate is almost exactly the same with and without the α conversion mechanism. This is consistent with the HCO transport level such that the -3 HCO concentration in the cytosol is 30 mM. O 2 concentration is 260 μM. The oxygenation error rates, as a percent of total RuBisCO reactions are indicated on the concentration bars. The cellular organizations investigated are RuBisCO and carbonic anhydrase distributed throughout the entire cytosol, co-localizing RuBisCO and carbonic anhydrase on a scaffold at the center of the cell without a carboxysome shell, RuBisCO and carbonic anhydrase encapsulated in a carboxysome with high permeability at the center of the cell, and RuBisCO and carbonic anhydrase encapsulated in a carboxysome with optimal permeability at the center of the cell. DOI: 10.7554/eLife.02043.004 previously shown detrimental effect of having active carbonic anhydrase free within the cytosol (Price & Badger, 1989). It would be impossible to keep the cytosol completely free from carbonic anhydrase enzyme, so there must be a way of activating it within the carboxysome only. Carbonic anhydrase is inactivated under reducing conditions (Peña et al., 2010). Recently it was shown that carboxysomes oxidize after assembly, providing a way to keep carbonic anhydrase inactive until fully enclosed in a carboxysome (Chen et al., 2013).

Effects of pH
Cyanobacteria must regulate pH as almost all biochemical reactions are pH sensitive. We do not attempt to model this regulation or potential pH variation within the cell, however pH may be included implicitly in a couple ways. We have already explored the effect of varying external pH, and the effects of pH on carbonic anhydrase. Cytosolic pH would have little direct effect on the CO 2 and -3 HCO levels since the non-enzymatic interconversion is very slow as previously discussed. The effect of internal pH could also be explored by varying the reaction rate of RuBisCO, which is pH sensitive. Varying the reaction rate of RuBisCO greatly could change the range where a non-specific carboxysome permeability can increase the concentration of CO 2 in the carboxysome. It would be unexpected that the RuBisCO rate be much faster than we assume, as we have assumed a rate on the high end. A lower RuBisCO rate would increase the range of effective carboxysome permeabilities. As previously mentioned the CO 2 facilitated uptake mechanism functions by creating local alkaline pockets. Diffusion of hydrogen ions across the cell would be very fast, so such pockets would require a massive reduction from the light reactions to maintain local alkalinity. Whether such pH gradients are possible, is certainly a subject of future interest. The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Author contributions NMM, preformed calculations, Conception and design, Analysis and interpretation of data, Drafting or revising the article; MPB, oversaw and mentored research, Conception and design, Drafting or revising the article

Additional files
Supplementary file • Supplementary file 1. Mathematical derivation appendix. Mathematical derivations of analytic solutions for a spherical cell with reactions organized in a variety of ways. We present analytic solutions for the concentration of CO 2 and HCO 3 − a carboxysome located at the center of the cell. We derive analytic solutions assuming a number of different cases for the enzymatic rates in the carboxysome: RuBisCO reaction rate negligible with carbonic anhydrase saturated and unsaturated, RuBisCO reaction rate non-negligible with carbonic anhydrase unsaturated. Additionally we derive analytic solutions for the enzymatic reactions throughout the cell and localized to a scaffold without a carboxysome shell.