Effects of microcompartmentation on flux distribution and metabolic pools in Chlamydomonas reinhardtii chloroplasts
Authors: Mark Stitt
Stefan Geimer
Tabea Mettler-Altmann
Luke Cm Mackinder
Liliya Yaneva-Roder
Martin C Jonikas
Michael Schroda
Anika Küken
Zoran Nikoloski
Melanie Höhne
Frederik Sommer
Journal: eLife (2018)
Discipline: BIOLOGY & BIOCHEMISTRY
Keywords: Chlamydomonas reinhardtii; biochemistry; carbon fixation; chemical biology; chlamydomonas reinhardtii
Affiliations: Max Planck Institute of Molecular Plant Physiology, Potsdam-Golm, Germany., Bioinformatics Group, Institute of Biochemistry and Biology, University of Potsdam, Potsdam, Germany., Department of Plant Biology, Carnegie Institution for Science, Stanford, United States., Institute of Cell Biology, University of Bayreuth, Bayreuth, Germany., Cluster of Excellence on Plant Sciences, Heinrich-Heine University, Düsseldorf, Germany., Institute of Plant Biochemistry, Heinrich-Heine University, Düsseldorf, Germany.

Abstract:  Cells and organelles are not homogeneous but include microcompartments that alter the spatiotemporal characteristics of cellular processes. The effects of microcompartmentation on metabolic pathways are however difficult to study experimentally. The pyrenoid is a microcompartment that is essential for a carbon concentrating mechanism (CCM) that improves the photosynthetic performance of eukaryotic algae. Using Chlamydomonas reinhardtii, we obtained experimental data on photosynthesis, metabolites, and proteins in CCM-induced and CCM-suppressed cells. We then employed a computational strategy to estimate how fluxes through the Calvin-Benson cycle are compartmented between the pyrenoid and the stroma. Our model predicts that ribulose-1,5-bisphosphate (RuBP), the substrate of Rubisco, and 3-phosphoglycerate (3PGA), its product, diffuse in and out of the pyrenoid, respectively, with higher fluxes in CCM-induced cells. It also indicates that there is no major diffusional barrier to metabolic flux between the pyrenoid and stroma. Our computational approach represents a stepping stone to understanding microcompartmentalized CCM in other organisms. 

Retracted: No

PubPeer comments:
 # Abstract Recently, the carbon concentrating mechanism (CCM) in _Chlamydomonas reinhardtii_ was investigated by studying metabolite concentrations and the distribution of Calvin-Benson cycle enzymes between pyrenoid and stroma under two different conditions. High-light adapted cells display significantly different metabolite levels and enzyme distributions compared to low-light adapted cells. A mathematical model simulating the flux distributions was constructed to predict microcompartment-specific metabolite concentrations in free and enzyme-bound state, rate constants of enzymes, and rates of metabolite transport between pyrenoid and stroma. From the theoretical model, it was concluded that RuBisCO in the pyrenoid is more active for low-light adapted (or CCM-induced) cells, and that the substrate of RuBisCO, RuBP, is transported along the concentration gradient from stroma to pyrenoid, while its product, 3PGA, is transported from pyrenoid to stroma, without a concentration gradient. Here, we express three main concerns regarding the theoretical study. Firstly, the main conclusions derived from the model are direct consequences from the data and the way the model is constructed. In fact, they do not require any model to be supported. Secondly, the model is contradicting fundamental laws of physics. The solutions of the model may violate energy conservation, and thus cannot be used to derive any conclusions about the real system. Thirdly, the source code introduces constraints not discussed in the main paper. The respective lines of code can be used to perfectly tune the model behaviour to reflect experimental values. Thus, the model is fitted to experimental values rather than predicting them, as claimed in the paper. # Introduction In the paper by [Ku00fcken _et al._] , the authors present a combined experimental and theoretical study to elucidate fluxes through the Calvin-Benson cycle through microcompartments. In particular, the pyrenoid of _Chlamydomonas reinhardtii_, a structure responsible for a carbon concentrating mechanism, is considered, and by separating this structure experimentally from the rest of the stroma, local enzyme concentrations could be determined. In the following, the modelling approach presented in this paper is critically reviewed. The review will begin with explaining and questioning the general approach and the ideas behind it, and discussing the main conclusions drawn from the model. Next, it will be demonstrated that the constructed models contradict fundamental physical laws, and consequences will be discussed. Finally, a detailed analysis of the source code is presented and it will be demonstrated how hard-coded code segments critically tune the results, leading to the illusion that the models almost perfectly reproduce experimental values, while in fact they are essentially fitted. # General approach and conclusions A mathematical model of the carbon concentrating mechanism (CCM) is constructed as follows. First, a stoichiometric model, i.e. a model that does not contain any kinetic parameters, but only the reaction s stoichiometries, of the Calvin cycle is constructed. Two copies of this model are connected. The copies reflect the reactions occurring in the stroma and the pyrenoid, respectively. The connection is described by assumed transport processes between stroma and pyrenoid. Subsequently only those reactions in the pyrenoid are considered (all others removed from the model), for which there exists experimental evidence that the respective enzyme is expressed in the pyrenoid. The main model version presented in the paper is that, for which only a single reaction still occurs in the pyrenoid (depicted schematically in Figure 6A). The only reaction left in the pyrenoid is the carbon fixation catalysed by RuBisCO, transforming a molecule of RuBP and one molecule of CO$_2$ into two molecules of 3PGA. Already here it is evident and trivial that, if the model carries any stationary flux through the RuBisCO-catalysed reaction in the pyrenoid, the substrate (RuBP) must be transported into the pyrenoid and the product (3PGA) out of the pyrenoid. This observation is presented as a major conclusion from the model. It is already stressed in the abstract that  ribulose-1,5-bisphosphate (RuBP), the substrate of Rubisco, and 3-phosphoglycerate (3PGA), its product, diffuse in and out of the pyrenoid, respectively, with higher fluxes in CCM-induced cells. Also the fact that higher fluxes are observed in CCM-induced cells is a direct consequence of the model assumptions. In the presented model, only the enzyme RuBisCO has activity in the pyrenoid, and the ratio of fluxes between pyrenoid and stroma RuBisCO is constrained to the experimentally determined ratio of the enzyme abundance in pyrenoid and stroma (see Equation 1). Therefore, in LC (CCM-induced) cells the flux of RuBisCO in the pyrenoid must be larger than HC (non CCM-induced) cells, simply because in the first case 61.8% of enzyme are found in the pyrenoid, while in the latter only 21.8% (see page 4). Because the total carbon fixation rate is fixed (to 398 $mu$M s - see Equation 1), the rate of flux through RuBisCO in the stroma is also fixed (to a higher value in LC compared to HC cells). Since this is the only reaction in the pyrenoid, a steady state entails that the substrate RuBP must be imported from the stroma with the same rate, and the product 3PGA must be exported with twice that rate (RuBisCO releases two molecules of 3PGA). Thus, these observations are direct consequences of the model assumptions and constraints. No further model activities are required to derive these conclusions. For the further calculations, the authors follow a strategy similar to the Ensemble Modelling approach proposed by [Tran _et al._] , without however citing the paper. This approach basically creates an ensemble of models that can all reproduce some, possibly experimentally determined, stationary state. It works in two steps. First, from a stoichiometric model, many flux distributions are sampled that fulfil the stationarity condition $mathbf{N}cdot v=0$. This is always possible, because the problem is underdetermined and has a solution space of a high dimension. The dimension reads $r-n+c$, if $r$ denotes the number of reactions in the model, $n$ the number of metabolites, and $c$ the number of linearly independent conserved quantities (see e.g. Heinrich Schuster, _The Regulation of Cellular Systems_, Springer, 1996, __ISBN 978-0412032615__). Then, for each of the sampled flux distributions parameters for a _kinetic_ model are identified, so that the kinetic model exhibits exactly the chosen flux distribution. The original idea of this ensemble modelling approach is to create an ensemble of models that are in agreement with experimental data, despite a lack of knowledge of enzymatic parameters. The ensemble of models can for example be analysed statistically, and most likely scenarios regarding the distribution of parameters values may be inferred. In order to define the kinetic model, all enzymatic reactions are replaced by elementary steps, which are all assumed to follow mass-action kinetics. This was also proposed in [Tran _et al._] . In the present paper by [Ku00fcken _et al._] , two important details are different, which lead to thermodynamic inconsistencies. These will be discussed in Section __Violation of Thermodynamic laws__. Usually, in the ensemble modelling approach, emsembles of models are created that are constrained as much as possible by experimental data. These emsembles are then carefully investigated, often with statistical methods, to learn something about the system s behaviour or model parameters. Interpreting results from ensemble modelling is often challenging. In [Ku00fcken _et al._] , the authors chose a different path and decide to constrain the model only with a part of the available experimental information and to predict the other part. Specifically,  [f]or reactions with known enzymatic turnover obtained from BRENDA (Chang et al., 2015) (Supplementary file 3), [the authors] integrated this information by restricting the respective parameter boundaries. (page 18). The general approach how to determine kinetic model parameters is given by Equation 2. Here, intermediate concentrations $c$ and rate constants $k$ are the free parameters, constrained by some boundaries $c_mathrm{min max}$ and $k_mathrm{min max}$, respectively, which are not further specified except by Supplementary file 3 (see above). The second line of Equation 2 ensures that all rate expressions yield the correct flux (as sampled in the first step). Under these very generic constraints, those solutions are identified in which the kinetic parameters for enzymes active in both compartments are most similar in both compartments. In Section __Fit and Predictions__ it will be demonstrated that these constraints are very general and do not allow to make model predictions. In fact, several lines of code are used to tune the model behaviour to create the illusion of actual model predictions. # Violation of Thermodynamic laws The model is constructed by splitting enzyme-catalysed reactions into elementary steps, which are assumed to follow mass-action kinetics. For a reversible reaction $Cleftrightarrow D$, two reversible and two irreversible reactions are introduced (Section  Model construction ). If $E$ denotes the enzyme, these are $C+Eleftrightarrow CE$, $CEto D+E$, $D+Eleftrightarrow DE$ and $DEto C+E$. This entails that in equilibrium there is a constant cyclic flow $C+Eto CEto D+Eto DEto C+E$, which violates the metabolic analogon of Kirchhoff s loop law in electrodynamics (see [Schellenberger _et al._](http: dx.doi.org 10.1016 j.bpj.2010.12.3707)). It is important to mention that in the original introduction of the ensemble modelling approach by [Tran _et al._] , splitting of reversible reactions is performed correctly into a series of reversible elementary reactions. ![file] Interestingly, if one ignores the violation of Kirchhoff s loop law and uses the incorrect dissection of reversible reactions according to the scheme above to derive an enzymatic rate law, one arrives at $v = E^Tcdotfrac{frac{k_1k_3}{k_2+k_3}C+frac{k_4k_6}{k_5+k_6}D}{1+frac{k_1}{k_2+k_3}C+frac{k_4}{k_5+k_6}D},$ where $E^T=E+[CE]+[DE]$ is the total enzyme concentration. Now this equation has the same form as the standard reversible Michaelis-Menten kinetics. However, if in equilibrium, the reactions in the scheme will all carry a positive flux, meaning that also in equilibrium there is a constant energy dissipation. Moreover, the rate expression above clearly demonstrates that the six parameters $k_1ldots k_6$ are not entirely free but constrained in analogy to the famous Haldane relationship, and it must hold that $frac{k_1k_3}{k_4k_6}frac{(k_5+k_6)}{(k_2+k_3)}=K_mathrm{eq}$, where $K_mathrm{eq}$ is the equilibrium constant. Thus, in any approach to determine parameter values that fit some flux distribution, only 5 out of the 6 parameters can be freely chosen. Now it may be argued that the precise values of equilibrium constants are only approximately known. However, if an ensemble of models is created, which all differ in the equilibrium constants, then these models reflect different sets of physical laws and are hardly comparable. Moreover, and even more problematic, not ensuring Haldane s relationship means that it is not guaranteed that Hess s law is fulfilled. Hess s law states that the Gibbs free energy difference only depends on the substrates and products, but not on the precise chemical reaction path used to convert substrates into products. Thus, if there exist two routes converting a the same set of substrates into the same set of products, and it is not guaranteed that for all reactions along the route Haldane s relationship is fulfilled, the routes may exhibit different free energy gradients. As a consequence, energy dissipating and generating cycles emerge. Thus, the models do not conserve energy and thus violate the first law of thermodynamics. Interestingly, the observation by the authors that some reactions seemingly proceed with a positive $Delta G$ ( The positive $Delta G$ for reactions RPE and RPI may point to a substrate channeling - see page 9) is exactly a manifestation of this missing constraint: metabolite concentrations are simulated with a different equilibrium constant than that, which is used to calculate the Gibbs free energy of reaction. Of course, this observation does not point to substrate channelling, but simply indicates errors in model construction. In summary, the model is thermodynamically inconsistent and may well violate fundamental laws of physics. All model simulations are therefore irrelevant for any realistic scenario and conclusions can not be drawn on a sound basis. It is interesting to note that [Tran _et al._] , who originally proposed the ensemble modelling approach, were well aware of the constraints of parameters through Haldane s relationship. In their approach to identify parameters that reproduce a certain flux distribution, they ensure that every reaction proceeds in a direction that corresponds to the respective sign of $Delta G$ (i.e. forward for negative $Delta G$, reverse for positive $Delta G$). However, thermodynamic constraints are only enforced on a reaction-by-reaction basis, and also in the original description of the ensemble modelling approach thermodynamically infeasible loops violating Hess s law are not excluded. # Fit and Predictions In this chapter, we take a closer look at the source code and analyse in particular how Figure 5 in [Ku00fcken _et al._] was generated. Figure 5 displays  predicted and measured concentrations (see page 9) of Calvin cycle intermediates. The stunning agreement of simulated and experimental value appears surprising considering the algorithm according to which the models were produced. Moreover, the variation of the simulated values among 1000 models is so small is cannot even be seen in Figure 5. It is also puzzling how 1000 models, generated from randomly sampled flux distributions, produce almost identical results, unless they are fitted to these values. The models, from which the simulation results used to generate Figure 5 were obtained, have been generated as described above. To recapitulate, an ensemble of models is constructed by first sampling flux distributions fulfilling the steady-state condition $mathbf{N}cdot v=0$. Then, for each of the (5000) sampled flux distributions, kinetic parameters $k$ and concentrations $c$ are determined by an optimisation procedure described by Equation 2. Here, the free variables are the $c$ and $k$, which are constrained such that the rate equations yield the respective fluxes $v$. Because mass-action kinetics are assumed for all reactions, these constraints can be formulated as linear equations in the logarithmic variables (the second line of Equation 2). Apart from the information that kinetic constants corresponding to known catalytic rates have been constrained to reflect experimental values, only generic constraints for the variables $c$ and $k$ are given ($c_mathrm{min max}$, $k_mathrm{min max}$). Equation 2 represents an underdetermined problem, because the number of free variables is larger than the number of equations. The optimisation criterion states that the differences between kinetic parameters of the same enzymes in the pyrenoid and in the stroma are minimal. Then, from these 5000 parameterised models, which all exhibit one of the sampled steady-state fluxes, 1000 are selected, which display the best agreement with experimental data. For these, the simulated concentrations are compared to the experimentally determined concentrations in Figure 5. Because the problem to parameterise the models is underdetermined, there must be a high degree of freedom in both sets of variables, $c$ and $k$. It is therefore surprising that exactly the concentrations of Calvin cycle intermediates are so close to the experimental values and display such a low variability. A closer inspection of the source code reveals how the agreement is established. The MATLAB code works as follows: In `main.m`, flux sampling is performed (or, alternatively, the flux distributions are loaded from a provided data file). This routine calls `getParameterValues.m`, where the optimisation problem defined in Equation 2 is solved for every flux distribution, and the $chi^2$-values to determine the goodness of fit are calculated. In the first part of the file, the optimisation problem is defined as a linear program, which is solved by calling the Cobra function `optimizeCbModel` in line 199. A piece of code which drew my attention is found in lines 163-181: ``` % bounds on concentrations x_lb = log(ones(size(alpha,2),1)*0.01); x_ub = log(ones(size(alpha,2),1)*1e5); for i=1:size(M,1) x_ub(M{i}) = log(str2double(ExpDataConcentrations{idx_c,i})); if i==2 x_lb(M{i}) = log(str2double(ExpDataConcentrations{idx_c,i})*0.5); end if strcmp(cond, lowCO2 ) if i==4 || i==5 || i==6 || i==7 || i==8 || i==9 || i==10 || i==11 || i==12 || i==13 || i==14 x_lb(M{i}) = log(str2double(ExpDataConcentrations{idx_c,i})*0.99); end else if i==4 || i==5 || i==7 || i==8 || i==10 || i==11 || i==12 || i==13 || i==14 x_lb(M{i}) = log(str2double(ExpDataConcentrations{idx_c,i})*0.99); elseif i==6 x_lb(M{i}) = log(str2double(ExpDataConcentrations{idx_c,i})*0.95); elseif i==9 x_lb(M{i}) = log(str2double(ExpDataConcentrations{idx_c,i})*0.9); end end end ``` Here, constraints are defined for the free intermediate concentrations. The variables `x_lb` and `x_ub` define lower and upper bounds and are used in the call to the optimisation routine `optimizeCbModel` a few lines of code later. Remarkably, many of the metabolite concentrations are constrained between 99% and 100% of the experimentally determined values. These constraints have not been mentioned anywhere in the manuscript. Rather, on page 9 it is explicitly stated that  [g]iven a steady-state flux distribution sampled from $C(N)$ the linear system could be solved using a constraint-based optimization approach (see Materials and methods) to obtain predictions of metabolite concentrations and rate constants under the constraint that the rate constants between stroma and pyrenoid were close to each other. Moreover, later on page 9 the values presented in Figure 5 are explicitly classified as  predicted and measured concentrations . The only difference between the concentrations depicted in Figure 5 and the constrained values is that in Figure 5 the sum of free and bound metabolites are depicted. However, under any realistic physiological condition, in which enzyme concentrations are orders of magnitude lower than the respective substrate concentrations, these narrow constrains must have a direct effect on the values plotted in Figure 5. To study how significant the constraints influence the simulation results, we executed the source code, first to reproduce Figure 5, then to repeat the analysis without lines 163-181. ![file] In order to execute the code, some modifications were necessary to skip commands depending on files that were not provided. Specifically, we changed the path to the COBRA toolbox, we used the provided pre-calculated sampled flux distributions instead of calculating them de novo, and we commented out all lines that depend on reading the file `Escher_met_list.csv` in line 297 (which is exectued after production of Figure 5 in lines 126-148 of `main.m`), because this file was not included in the provided repository. In `getParameterValues.m` we commented out the call to `MyCobraFVA`, because this routine was also not provided. The results of the flux variability analysis are irrelevant for the production of Figure 5, in fact the returned values (in variables `mi_l_00`, etc.) are used nowhere else in `main.m`. All implemented changes were therefore not influencing the reproduction of Figure 5. The resulting figure is shown above, and represents an exact analogon apart from stylistic features to Figure 5 in the article. In a second simulation, we commented out lines 163--181 in `getParameterValues.m` and repeated the analysis. The resulting figure is displayed below. This is how Figure 5 of the manuscript would look like if the calculations were performed as described. Now, metabolite concentrations are not fitted, and therefore the simulated values can be considered as model predictions. It can clearly be seen that predicted and measured values deviate significantly, in some cases by a factor of $10^5$. ![file] Only the values of ribulose-1,5 bisphosphate (RuBP) still agree with the measured values. This, however, can be explained by the fact that the rate of carbon fixation is fixed (to 398$mu$M s, see  Model parameterization on page 17), and the $k_mathrm{cat}$ of RuBisCO is constrained to be close to the literature value of 5.8s$^{-1}$. Because carbon fixation rate is identical to the rate of RuBisCO, and thus the rate of consumption of RuBP, fixing the flux and the rate constant directly determines the concentration of the enzyme-substrate complex serving as substrate for the catalytic step. Considering that the authors found  that the vast majority of the RuBP was bound , it is not surprising that total RuBP values agree with measured quantities. The model is _designed_ to produce correct RuBP values, it does not _predict_ it. # Conclusions The motivation to develop a mathematical model of any system should always be to increase our understanding of the system under investigation. As a simplified representation of the real system, a model provides a framework to understand the essential features that are responsible for the emergent system behaviour. The models presented by [Ku00fcken _et al._] do not increase understanding nor do they generate insight. The model construction process reveals many beginner mistakes. The dissection of reversible reactions into elementary steps, for example, illustrates that most elementary thermodynamic considerations have not been considered. This could easily have been avoided by studying more carefully the original publication by [Tran _et al._] (which the authors do not cite), in which the ensemble modelling approach was introduced. The observation that reactions proceed against their energy gradient is yet another indication that fundamental knowledge regarding dynamic systems and thermodynamics are lacking. The modelling strategy is not adequate to gain deeper understanding. In fact, the models can easily be adjusted to show any kind of desired bahaviour by chosing appropriate constraints and hard-coding these in lines 163-181. Therefore, the model predictions are worthless, because with the model, as it is presented, any result can easily be generated. The apparent conclusions that the substrate of RuBisCO must diffuse in, and its substrate out, to facilitate a reaction in the pyrenoid are trivial and do not require any mathematical analysis. The models have not been constructed for any scientific purpose other than creating some nice plots, which conjure up the illusion of predictive power. In fact, the models are simply tuned to show the desired behaviour. Of course, this could have been achieved far more directly, but the apparent model complexity makes it hard to see through the approach. Because of the apparent complexity of the approach and the poorly documented code, which does not adhere to any professional programming standards, it costs a significant amount of time to a) understand the author s approach, and b) to identify and specify the errors, mistakes, and manipulations. In summary, the modelling efforts are not insightful and miss the point. They create the illusion of predictive power, while in fact the models are essentially fitted by hard-coding constraints in the source code. ,