A few methods for dealing with this uncertainty have been suggested previously [8-12]. In this context, we have previously analyzed the existing model of T. brucei glycolysis taking into account uncertainty about the enzyme parameter values but keeping the topology of the model fixed . In addition, we incorporate a second tier of uncertainty into the model with regard to its topology (i.e. considering which reactions occur, and in which subcellular compartment). Topological uncertainty is regularly dealt with in the context of genome-scale metabolic reconstructions  but is usually disregarded in dynamic model analysis. For this purpose, we constructed alternative model versions with different topologies. We then simulated each model version using a large range of plausible parameter sets, which were sampled from distributions reflecting our degree of uncertainty about each parameter .
Exploring the effect of a permeable glycosome
In the original model published in 1997 and in all of its subsequent iterations [1, 3, 7, 14-17], the glycosomes have been considered to be impermeable (Fig. 1). However, the recently discovered pores  would allow free diffusion of the smaller metabolites between the cytosolic and the glycosomal lumen. This has been considered to be incompatible with the survival of the parasite as a result of the accumulation of sugar phosphates observed when the glycosomal enzymes are mislocalized to the cytosol and therefore exposed to a high ATP/ADP ratio. This phenomenon, referred to as a ‘turbo explosion’, is assumed to result from the loss of specific regulatory feedbacks for critical enzymes [3, 4]. Consequently, after the discovery of unspecific pores, mechanisms that would nonetheless permit all metabolites to be retained inside the glycosomes have been hypothesized, although no evidence for any of them has been found . Here, we explore the consequences of increasing the permeability of glycosomes in a series of alternative computational models and reveal an important role for residual cytosolic activities of several glycolytic enzymes whose function had previously remained unclear.
Figure 1. Aerobic glycolysis in bloodstream form T. brucei. Metabolites: Glc-6-P, glucose 6-phosphate; Fru-6-P, fructose 6-phos-phate; Fru-1,6-BP, fructose 1,6-bisphosphate; DHAP, dihydroxyacetone phosphate; GA-3-P, glyceraldehyde 3-phosphate; Gly-3-P, glycerol 3-phosphate; 1,3-BPGA, 1,3-bisphosphoglycerate; 3-PGA, 3-phosphoglycerate; 2-PGA, 2-phosphoglycerate; PEP, phosphoenolpyruvate. Reactions: 1, transport of glucose across the cytosolic membrane; 2, transport of glucose across the glycosomal membrane; 3, hexokinase; 4, phosphoglucose isomerase; 5, phosphofructokinase; 6, aldolase; 7,TPI; 8, GAPDH; 9, PGK; 10, transport of 3-PGA across the glycosomal membrane; 11, PGAM; 12, enolase; 13, pyruvate kinase; 14, transport of pyruvate across the cytosolic membrane; 15, glycerol 3-phosphate dehydrogenase; 16, glycerol kinase; 17, DHAP-Gly-3-P antiporter; 18, glycerol-3-phosphate oxidation; 19, ATP utilization; 20, adenylate kinase; 21, glycosomal glycerol transporter; 22, cytosolic glycerol transporter.
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To represent our uncertainty about the permeability of the glycosome, we constructed different versions of the model in which different groups of metabolites can freely diffuse across the glycosomal membrane. We defined the different groups of metabolites based on their molecular weight (Fig. 2). Based on these groups and on the last published version of the model  (which we updated by explicitly including glycerol transporters; for details, see Materials and methods and Doc. S1), we built seven different model versions, each permitting an increasingly large set of metabolites to diffuse freely.
Figure 2. Definition of the different types of models based on the size of metabolites that can be transported across the glycosomal membrane. Models 1a and 1b differ by the nature of the transport of glycerol 3-phosphate and dihydroxyacetone phosphate. In model 1a, their transport is linked via an antiporter, as introduced in 1999 , whereas, in model 1b, they are transported independently, as was the case in the original model of 1997 .
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Among these seven models, we included two versions of the original impermeable model: model 1a, in which the transport of glycerol 3-phosphate and dihydroxyacetone phosphate is linked via an antiporter in the glycosomal membrane, as introduced in 1999 , and model 1b, in which the transport of glycerol 3-phosphate and dihydroxyacetone phosphate is independent, as was the case in the first published version of the model in 1997 . The glycerol 3-phosphate/dihydroxyacetone phosphate antiporter was introduced to lower the effect of the inhibition of the glucose consumption flux by glycerol, thus allowing a better agreement of the model with experimental results . Therefore, we expect model 1a (with antiporter) to perform better than model 1b (without). Experimental evidence for the antiporter is lacking. However, the requirement for such a transporter disappears if glycerol 3-phosphate and dihydroxyacetone phosphate are able to diffuse using the newly identified permeability pores. Therefore, it is important to evaluate the effect on model behaviour of the replacement of the specific antiporter by diffusion pores.
Because exposing the glycosomal enzymes to the ATP/ADP ratio of the cytosol leads to a ‘turbo explosion’ when cells are exposed to high concentrations of external glucose , we know that at least model 6 (which allows ATP and ADP to diffuse freely) should show this lethal increase in sugar phosphate concentrations when the external glucose concentration is high. We simulated all model versions at both 5 and 50 mm of glucose in the medium and in both aerobic and anaerobic-like conditions. In our model, the anaerobic case was simulated by including an inhibitor (salicylhydroxamic acid; SHAM) of the trypanosomal glycerophosphate oxidase system (i.e. the only oxygen-using reaction in the model that covers two reactions in vivo: glycerol 3-phosphate and trypanosome alternative oxidase) to allow our simulation process to remain as close to the experimental setting as possible (Doc. S1).
Under aerobic conditions (Fig. 3A,B), most models lead to similar predictions for the glucose consumption flux. Only model 4 (in which glycosomes are permeable to all metabolites other than AMP, ADP, ATP, NAD and NADH) and model 5 (similar to model 4, although glycosomes are also permeable to AMP) show lower fluxes for some parameter sets. It has been shown that the parasite dies when the glucose consumption flux is inhibited by more than 50% ; therefore, models 4 and 5 are not compatible with the experimental data for some parameter sets. Under anaerobic conditions, however, all models show lower glucose consumption fluxes than under aerobic conditions, except for model 6 (in which glycosomes are permeable to ADP and ATP). Models 1b, 2, 3, 4 and 5 (which are models with increasingly permeable glycosomes) show significantly lower fluxes than model 1a (the impermeable model with a glycerol 3-phosphate/dihydroxyacetone phosphate antiporter). Hence, the absence of the antiporter is sufficient to render the performance of the impermeable model as poor as the semi-permeable models (models 2 to 5). Model 6 is the only one to show a similar distribution of glucose consumption fluxes under all conditions (Fig. 3), which is consistent with experimental data [15, 20]. However, model 6 shows a dramatic increase in the intracellular sugar phosphate concentrations, especially at higher glucose concentrations, corresponding to the expected ‘turbo explosion’ phenomenon, which would be lethal in vivo (i.e. see the example of fructose 1,6-bisphosphate concentration in Fig. 4); hence, model 6 is not a realistic description of intact trypanosomes, as expected.
Figure 3. Glucose consumption flux in models with increasing glycosome permeability. (A, B) Aerobic conditions (−SHAM). (C, D) Anaerobic-like conditions (+SHAM). (A, C) 5 mm of external glucose. (B, D) 50 mm of external glucose. The glucose consumption flux is lower under anaerobic conditions than under aerobic conditions for all models, and more so for the permeable models, which is in contradiction with the experimental results.
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Figure 4. Fructose 1,6-biphosphate concentration in models with increasing glycosome permeability. (A, B) Aerobic conditions (−SHAM). (C, D) Anaerobic-like conditions (+SHAM). (A, C) 5 mm of external glucose. (B, D) 50 mm of external glucose. A vertical grey line indicates the mean concentration of fructose 1,6-bisphosphate as measured by Visser and Opperdoes . All models maintain realistic concentrations except model 6, which accumulated fructose 1,6-bisphosphate under all conditions but, more importantly, at 50 mm of external glucose.
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From these results, the model in which the glycosome is impermeable with a glycerol 3-phosphate/di-hydroxyacetone phosphate antiporter (model 1a), allows the best match between the experimental results and simulations. However, the discovery of nonspecific metabolite pores in the glycosomal membrane and the absence of experimental evidence for the antiporter means that a dilemma remains. Moreover, model 1a is inconsistent with several other facts. Most importantly, all parameter sets result in a lower glucose consumption flux under anaerobic conditions than under aerobic conditions, which is contrary to experimental observations [15, 20]. Model 1a did not show any decrease in glucose consumption flux under anaerobic conditions in previous versions of the model. Indeed, in earlier versions, the glycerol concentration was fixed to zero inside the glycosome, which means that the transport of glycerol outside was considered infinitely fast and the glycerol concentration outside as infinitely low. Both of these assumptions are unrealistic. When introducing the glycerol transporter, even considering that the transport can favour glycerol export over import and setting its Vmax up to more than twice as fast as the measured Vmax, some inhibition of glucose consumption already appears with a very small concentration of glycerol outside the cell (Fig. S1). Even when the glycerol transporter Vmax is increased to much higher values (up to 1000 nmol·min−1·mg−1 cell protein, which is ten times the rate of the glucose transporter), glucose consumption is still predicted to be lower under anaerobic conditions (Fig. S2).
In addition, model 1a neglects the need for glucose 6-phosphate and fructose 6-phosphate in the cytosol. Indeed, previous versions of the model also assumed that the glycosomal enzymes have no activity in the cytosol. However, several glycolytic metabolites are required in the cytosol as input to other pathways; for example, glucose 6-phosphate for the cytosolic branch of the pentose phosphate pathway  or fructose 6-phosphate for the synthesis of fructose 2,6-bisphosphate . These metabolites either come from the glycosome or are synthesized directly in the cytosol, implying the presence of some glycolytic enzymes in the cytosol. Moreover, it has been known for many years that specific isoforms are present for key enzymes including glyceraldehyde 3-phosphate dehydrogenase (GAPDH). The glycosomal enzymes are imported fully folded and functional [23, 24]. Therefore, even when specific cytosolic isoforms have not been identified, the activity of the freshly synthesized enzymes, before their entry into the glycosomes, will be found in the cytosol.
Exploring the effect of specific or residual activity of the glycosomal enzymes in the cytosol
Uncertainty pertaining to the topology of the model means that some fraction of activity of the glycosomal enzymes in the cytosol should be included. Data about the percentage of activity in the cytosol are very limited except for GAPDH, which is present in the cytosol as the product of a different gene than that encoding the glycosomal version (Doc. S1). Although measurements of these fractions have been attempted [2, 25], the fragility of glycosomes and the properties of some enzymes (such as the tendency of hexokinase to stick to membranes)  have precluded any reliable quantification. We assumed, however, that the percentage of cytosolic activity of these enzymes is probably small, and certainly more than 50% of each of these enzymes is in the glycosomes. Therefore, percentages were sampled from a log uniform distribution between 0.01% and 50% (using a log uniform distribution allows each order of magnitude to be sampled in similar proportions, and also ensures that the unlikely very high values are only rarely sampled; 80% of the values will be < 5%; for details, see Doc. S1).
Figure 5A,B shows that most models with an explicit fraction of cytosolic activity of glycolytic enzymes have similar distributions of glucose consumption flux under aerobic conditions, with only a small proportion of some models with some sampled parameter sets showing a drop in flux that differs from experimental observations. However, under anaerobic conditions (Fig. 5C,D), models 3, 4 and 5 (which have glycosomes permeable to metabolites smaller than, respectively, fructose 6-phosphate, fructose 1,6-bisphosphate and AMP) allow higher glucose consumption fluxes at both 5 and 50 mm of glucose than when the cytosolic activities of the glycolytic enzymes were not considered (Fig. 3). Therefore, introducing a fraction of cytosolic activity of the glycolytic enzymes allows models with semi-permeable glycosomes to simulate the experimental data (i.e. fluxes and metabolite concentrations) much better than when the cytosolic fraction is ignored. We went on to test whether these higher fluxes are always associated with lethal accumulation of sugar phosphates at 50 mm of glucose under aerobic conditions. Figure 6B shows that, under both aerobic and anaerobic conditions, most parameter sets do indeed lead to an increased fructose 1,6-bisphosphate concentration compared to the same model without any glycolytic activity in the cytosol (Fig. 4). As expected, model 6 still shows a dramatic increase in fructose 1,6-bisphosphate. This shows that adding the cytosolic activity of the glycolytic enzymes can allow the glucose consumption flux to be higher anaerobically, although some of these parameter sets lead to an increase of the sugar phosphate concentrations (sometimes by up to several hundred millimolars). To better simulate experimental observation, a model should allow both higher fluxes and avoid these accumulations of sugar phosphate. Therefore, to be able to determine which model is the most realistic, every model and every parameter set needs to be evaluated by comparing its simulated results with all of the experimental data available (metabolite concentrations and fluxes).
Figure 5. Glucose consumption flux in models with increasing glycosome permeability and cytosolic activities of the glycosomal enzymes. The distributions shown include all sampled fractions of cytocolic activities of the glycosomal enzymes for each model topology. (A, B) Aerobic conditions (−SHAM). (C, D) Anaerobic-like conditions (+SHAM). (A, C) 5 mm of external glucose. (B, D) 50 mm of external glucose. More models allow a similar flux in both aerobic and anaerobic conditions when a fraction of cytosolic activity of the glycosomal enzymes is included in the model (compare with Fig. 3).
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Figure 6. Fructose 1,6-bisphosphate concentration in models with increasing glycosome permeability and cytosolic activities of the glycosomal enzymes. The distributions shown include all sampled fractions of cytocolic activities of the glycosomal enzymes for each model topology. (A, B) Aerobic conditions (−SHAM). (C, D) Anaerobic-like conditions (+SHAM). (A, C) 5 mm of external glucose. (B, D) 50 mm of external glucose. A vertical grey line indicates the mean whole-cell concentration of fructose 1,6-bisphosphate as measured by Visser and Opperdoes . When a fraction of cytosolic activity of the glycosomal enzymes is included in the models, all model topologies have parameter sets that lead to the accumulation of high concentrations of fructose 1,6-bisphosphate. Hence, if glycosomal enzymes have a fraction of activity in the cytosol, it is probably regulated.
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However, matching experimental data of this complexity to model predictions offers great challenges because some aspects of the metabolic phenotype can match experimental results, whereas, simultaneously, others do not. To quantify the match between the models and the experimentally measured concentrations and fluxes, we computed log-likelihoods for each model and for each parameter set. These log-likelihoods are computed by comparing the simulated values of the whole-cell metabolite concentrations and fluxes to distributions based on a wide range of experimental data. These distributions represent our (uncertain) knowledge of the intracellular metabolite concentrations, the glucose consumption flux, and the glycerol production flux relative to the pyruvate production flux under both aerobic and anaerobic conditions, at 50 mm of external glucose (which is the concentration of glucose used in the experiments measuring the intracellular metabolite concentrations; for details, see Materials and methods). The higher the log-likelihood (closer to zero), the better the match is between model prediction and experimental data.
The distributions of the log-likelihoods for each model topology are shown in Fig. 7. As expected, model 6 never performs as well as the best models. Indeed, because ATP and ADP can diffuse freely in this model, the model is less sensitive to the proportion of activity of the glycosomal enzymes in the cytosol. The models that allow the highest log likelihood (Fig. 7) for the largest number of parameter sets are models 3 and 4 (the models with glycosomes permeable to small metabolites up to the size of fructose 6-phosphate and fructose 1,6-bisphosphate, respectively). This shows that, based on our current knowledge, permeable models can yield matches to the experimental data that are equally good as the original model that depends on an impermeable membrane.
Figure 7. Log-likelihood of models with increasing glycosome permeability and cytosolic activities of the glycosomal enzymes. The distributions shown include all sampled fractions of cytocolic activities of the glycosomal enzymes for each model topology. Inserts on the right show the fraction of parameter sets leading to a log-likelihood higher than −100 (top) higher than −30 (bottom) for each model topology. The best parameter sets produces the largest log-likelihoods (closer to 0); hence, the models producing the closest match to experimental data are models 1a (impermeable with antiporter), 3 (permeable up to fructose 6-phosphate) and 4 (permeable up to fructose 1,6-bisphosphate).
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Identification of the best parameter sets for each model topology
It is striking that, no matter which topology we preferred, there is always a large fraction of perfectly plausible parameter sets that lead to highly unlikely model predictions. We aim to exploit this observation to update our beliefs about the most likely parameter values. Using the log-likelihoods, we tested whether the top performing 1% of parameter sets for each model have significantly different parameter values compared to the other parameter sets. The results are shown in Table 1 (selected Vmax only) and Table S1 (for two examples, see Fig. S3; for the distribution of the proportion of activities in the top 1% of parameter sets, see Fig. S4). Indeed, the best matches to experimental data tend to occur for particular choices of parameters, and some of these are similar for all model topologies. First, the phosphoglycerate mutase (PGAM) Vmax is usually higher in the best 1% of parameter sets for all models. This is expected because previous analyses showed that decreasing this parameter leads to an accumulation of 3-phosphoglycerate . The top 1% of parameter sets for all models also tend to have glucose transporter activity close to the minimal plausible value, allowing a flux of apprximately 100 nmol·min−1·mg−1 cell protein (Fig. S3A). The effect of this reduction in transporter activity is to lower the concentration of free glucose in the cytosol, thus limiting the concentration of glucose 6-phosphate that would accumulate as a result of cytosolic hexokinase. This limited activity of the transporter is also in agreement with the finding that the glucose transporter has a significant control over the glucose consumption flux . The fraction of parameter sets that allows the models to reach steady-state is close to 100% for most model topologies (Fig. S5).
Table 1. Fold-change of the median maximal activities significantly different in the best parameter sets (highest 1% log-likelihood) compared to the other sampled values. Columns represents the different model topologies. For all results, see Table S1. The fold-changes are computed as the median value of the best sets over the median value of all sets. When the fold-change is smaller then 1 (the parameter values are smaller in the best sets compared to all sets) then −1/fold-change is displayed. The value is missing when the difference between the best parameter values and all parameter values is not statistically significant (Mann–Whitney U-test)
|ALDc||4.5||9.0||14.4||9.4|| || ||−2.0|
|TPIc||1.9||−1.6|| ||−1.5||−1.4|| ||−2.8|
Also common to all models, the glycerol transporter Vmax is usually set to higher values in the best parameter sets, so that glycerol is rapidly exported. This relates to the best models being those whose flux under anaerobic conditions is higher, and hence where the inhibition of the flux by intracellular glycerol is the lowest. Under anaerobic conditions, for 1 mol of glucose consumed, 1 mol of glycerol and 1 mol of pyruvate are produced. This exact stoichiometric match is a result of the fact that, for each glucose consumed, two ATP molecules are used in the glycosomes (by hexokinase and phosphofructokinase); this ATP needs to be recovered by phosphoglycerate kinase (pyruvate branch) and glycerol kinase (glycerol branch). However, because the glycerol kinase equilibrium strongly favours ATP consumption using glycerol as its substrate, even a little accumulated glycerol inhibits ATP production. Indeed, it has been shown that the addition of glycerol under anaerobic conditions kills the parasites, whereas, without the addition of external glycerol, the glucose consumption flux under aerobic and anaerobic conditions is similar [27-29]. This inhibitory effect can be reduced by the increase of glycerol export, at the same time as increasing the fraction of glycerol kinase in the glycosome. This also explains that the glycerol kinase Vmax is higher in the glycosome of the top 1% of parameter sets of each model.
A particularly unexpected observation is the fact that models 1 to 3 favour having a higher proportion of aldolase activity in the cytosol in the best 1% of parameter sets: there appears to be no intuitive reason why the level of aldolase activity would lead to reduced accumulation of sugar phosphates (including fructose 1,6-bisphosphate). The fraction of aldolase in the cytosol appears to be important, as long as fructose 1,6-bisphosphate cannot be transported across the glycosomal membrane. The fraction of hexokinase in the cytosol of the top 1% of parameter sets is also different depending on the model topology. In the best parameter sets of model 1a, 1b and 6, it is relatively small (the median value is approximately 4 nmol·min−1·mg−1 cell protein for model 1a; 9 nmol·min−1·mg−1 cell protein for models 1b and 6; Table S1), whereas, in models 2, 3 and 5, it is higher (the median value is 40 nmol·min−1·mg−1 cell protein for model 2; 53 nmol·min−1·mg−1 cell protein for model 3; and 190 nmol·min−1·mg−1 cell protein for model 5; model 4 shows no significant difference in cytosolic hexokinase activity between the top 1% of parameter sets and the other parameters sets: the median value of all parameter sets is approximately 13 nmol·min−1·mg−1 cell protein). However, the highest cytosolic hexokinase Vmax values never lead to a good match with the experimental data (Fig. S3B).
Differences are also observed for phosphoglycerate kinase, which is significantly higher in the cytosol of the best parameter sets of model 1a and 3 to 5 but lower in models 2 and 6. Additionally, GAPDH activity in the cytosol is also significantly higher in the top 1% of parameter sets of models 1a, 3 and 4. Therefore, the best model topologies (models 1a, 3 and 4) appear to favour a higher flux in the cytosolic branch of the lower part of glycolysis (producing 3-phosphoglycerate from glyceraldehyde 3-phosphate). Such a topology could allow these models to compensate for the loss of bound phosphates in the glycosome as a result of the the leakiness of the glycosomal membrane (models 3 and 4). This may appear to be inconsistent with experiments previously showing  that expressing the cytosolic version of phosphoglycerate kinase in the bloodstream form parasite causes cell death, even for a low level of expression. However, in those experiments, no difference in metabolite concentration or flux related to glycolysis could be measured in the overexpressing lines, and a general toxicity effect could not be excluded .
Many other parameters are distributed differently, depending on the model, and could potentially be used to design discriminatory experiments to decide in favour of one or the other topology, should the necessary measurement technology become available. For example, the triosephosphate isomerase (TPI) Vmax in the cytosol is significantly different for the best 1% of all models except models 2 and 5. Higher cytosolic TPI Vmax values are favoured for model 1a (the impermeable model with antiporter), whereas lower values are favoured for models 1b (no antiporter) to 6. This suggests that the free diffusion of dihydroxyacetone phosphate and glycerol 3-phosphate makes the models sensitive to the presence of TPI in the cytosol. It has been shown that overexpressing this enzyme in the cytosol kills the parasite. However, overexpression was important and, as is the case for phosphoglycerate kinase (PGK), the cause of death for cells with overexpression of TPI was not established, and a toxic effect of the overexpression per se (in contrast to simply mis-targeting) could not be excluded. Establishing how sensitive trypanosomes are to the expression of TPI (and other glycosomal enzymes) in the cytosol, as well as what the associated metabolic changes are, would comprise valuable information for helping to establish which of our models is the closest mechanistically correct representation of the biological system.