We propose a hierarchical modelling approach to construct models for disease states at the whole-body level. Such models can simulate effects of drug-induced inhibition of reaction steps on the whole-body physiology. We illustrate the approach for glucose metabolism in malaria patients, by merging two detailed kinetic models for glucose metabolism in the parasite Plasmodium falciparum and the human red blood cell with a coarse-grained model for whole-body glucose metabolism. In addition we use a genome-scale metabolic model for the parasite to predict amino acid production profiles by the malaria parasite that can be used as a complex biomarker.

Multi-scale hierarchical modelling of disease states

Diseases manifest as phenotypic changes in whole-body physiology that are experienced as illness and can be caused by external factors (e.g. infectious diseases caused by bacteria) or by internal dysfunction (e.g. type II diabetes, cancer). Medical treatment of a disease can be in the form of medication, such as a pharmaceutical drug that affects specific reactions in target cells. In the case of a so-called ‘magic bullet’ drug, displaying complete specificity and complete inhibition of an essential reaction in a parasite or diseased cell, the outcome of the drug effect is direct and simple to predict and no detailed analysis is required. However, most drugs affect more than one reaction step, do not lead to complete inhibition and have side effects [1]. The response to such a drug is more complicated and a quantitative analysis of the combined effects on the whole-body physiology is necessary to evaluate the treatment. Analysis of whole-body responses to partial inhibition of one or more reaction steps is challenging; typically, mathematical models at the whole-body level are not fine-grained and do not include individual chemical reaction steps. We propose a hierarchical approach where parts of a system are resolved with sufficient detail to analyse drug effects at the individual reaction step, whereas for other parts of the system coarse-grained models are used. This approach is illustrated with a multi-scale hierarchical model for glucose metabolism in malaria patients (Figure 1).

Hierarchical multi-scale model for malaria patients

Figure 1
Hierarchical multi-scale model for malaria patients

The whole-body model consists of several modules at the organ level, each described with input-output functions. The red blood cell compartment is modelled at the cellular and detailed metabolic level, including P. falciparum metabolism.

Figure 1
Hierarchical multi-scale model for malaria patients

The whole-body model consists of several modules at the organ level, each described with input-output functions. The red blood cell compartment is modelled at the cellular and detailed metabolic level, including P. falciparum metabolism.

From diagnosis to drug

Medical treatment usually starts with a diagnosis and classification of a disease. Typically these are initially made on the basis of simple phenotypic descriptions. Relating these symptoms to a mechanistic interpretation of the disease is not a trivial task and will involve a systems based approach [2], especially when multiple molecular aetiologies with patient-dependent contributions are involved. Biomarkers play an important role in such a diagnosis and are likely to be found at the metabolomics level [3]. A mechanistic, systems level interpretation of a disease could point to multiple drug targets, moving away from the single drug single target paradigm [4]. Such systems approaches to pharmacology [5] and toxicology [6] could lead to more effective drug development strategies [7].

Genome-scale network analysis of metabolism

Metabolomics and metabolic modelling are important tools in following and predicting disease progress and understanding drug efficacy and mode of action [3,8]. In the last decade, enormous progress has been made in the genome-scale analysis of metabolic networks for a large number of species ranging from bacteria (e.g. E. coli [9]), to eukaryotes (e.g. S. cerevisiae [10]) to humans [11,12] and includes important human pathogens (e.g. Haemophilus influenzae [13], Mycobacterium tuberculosis [14] and Plasmodium falciparum [1518]). These networks are very large, up to several thousands of reactions and the analyses are restricted to topological and constraint based modelling techniques, such as flux balance analysis [19,20]. Such models have been very useful for calculating metabolic phenotypes [21], such as the prediction of changes in metabolite biomarkers for inborn errors of metabolism [12] or, for instance, in analysing medium composition requirements for bacterial growth [22]. The models are typically analysed for steady state conditions and optimization criteria (e.g. growth rate) are used to minimize the solution space. Choosing suitable constraints on exchange reactions, biomass composition and maintenance reactions can have important effects on model predictions and should be done carefully.

Why study metabolism in malaria patients?

Malaria is a dreaded disease that is widespread across tropical and sub-tropical regions and responsible for the death of between 500000 and 1000000 people per year, mostly children in sub-Saharan countries. One might not immediately think of malaria as a metabolic disease; the classic symptom of 48-h cyclical fever attacks and diagnosis via blood smears has no relation to metabolism. However, the key-diagnostics for poor chances of survival, lactic acidosis and hypoglycaemia [23] are clearly linked to metabolism. In addition, in malaria patients, blood concentrations of glycerol [24] and alanine are increased and arginine concentration is decreased [25], indicating more general metabolic changes [26].

To what extent can these metabolic changes be related to metabolic activity of the parasite? Plasmodium cannot synthesize its own amino acids and is dependent on the host's haemoglobin for protein biosynthesis and on the host's glucose for its free energy production. As such, the metabolic activity of the parasite has a direct effect on the host, but, in addition, the parasites cause indirect damage by lysis of red blood cells and sequestration of parasitized red blood cells in the vasculature leading to reduced blood perfusion [27].

Although the pathophysiology used to be attributed to two main syndromes, cerebral malaria and severe anaemia malaria, it has become clear that severe malaria is complicated and involves several syndromes [2729]. Ultimately, the goal is to delineate the individual contributions of these syndromes to the pathophysiology of malaria. Such an analysis would point at the best points of intervention to relieve the burden of the disease. In an attempt to estimate the direct contribution of Plasmodium activity we set out to analyse its amino acid and carbohydrate metabolism.

Plasmodium biomass production from haemoglobin

The genome of P. falciparum was sequenced in 2002 and several genome-scale metabolic maps have been reconstructed [30]. Plasmodium is severely limited in its biosynthetic reactions and is largely dependent on the host's supply of amino acids for protein synthesis, for which the parasite degrades almost all haemoglobin in the red blood cell during its 48-h growth cycle. The specific condition of Plasmodium growing in the red blood cell and using the available haemoglobin for protein synthesis, leads to an elegant set of constraints that can be used in a genome-scale analysis. We used an existing genome-scale model [18] with a set rate of haemoglobin consumption (assuming a 75% consumption of total haemoglobin in 48-h [31]) and optimized for biomass production, only allowing uptake of the amino acids isoleucine and arginine. Isoleucine is not present in haemoglobin and must be taken up from the blood. P. falciparum is known to convert arginine to ornithine [32], leading to hypoargininaemia [25]. Under these conditions, a specific growth rate of 0.049 h−1 was calculated for P. falciparum, which is close to the expected value of 0.058 h−1 (calculated on the basis of 16 merozoites formed in 48-h). A glucose consumption rate of 1.6 mmol·h−1·gDW−1 was obtained which is somewhat lower than the experimentally measured value of 2.1 mmol·h−1·gDW−1 [33].

The complete set of reactions for the genome-scale network is shown in Figure 2(A), where the red lines indicate the fluxes through the reactions. In Figure 2(B), a subset of reactions involved in amino acid metabolism is shown. These reactions fall in three classes: (1) degradation of haemoglobin, (2) the synthesis of biomass and (3) export and inter-conversion of amino acids. The export fluxes for the amino acids are indicated in Figure 2(C).

P. falciparum genome-scale network analysis

Figure 2
P. falciparum genome-scale network analysis

The steady state solution space for the genome-scale metabolic network for P. falciparum [18] with a set influx rate of 0.83 μmol haemoglobin·h−1·gDW−1 was optimized for biomass formation and ornithine production. Fluxes are indicated in red on the complete network structure in (A). A subset of reactions for amino acid metabolism, indicating the haemoglobin degradation, biomass formation and the amino acid inter-conversion and export are indicated in (B). The export fluxes of the different amino acids are indicated in (C).

Figure 2
P. falciparum genome-scale network analysis

The steady state solution space for the genome-scale metabolic network for P. falciparum [18] with a set influx rate of 0.83 μmol haemoglobin·h−1·gDW−1 was optimized for biomass formation and ornithine production. Fluxes are indicated in red on the complete network structure in (A). A subset of reactions for amino acid metabolism, indicating the haemoglobin degradation, biomass formation and the amino acid inter-conversion and export are indicated in (B). The export fluxes of the different amino acids are indicated in (C).

The relative amino acid production rates compare well with rates observed in P. falciparum culture [32]. Note that the arginine to ornithine conversion was part of the objective and is therefore not a validation for the model. One cannot immediately compare these amino acid product formation rates to changes in blood amino acid concentrations in malaria patients since these concentrations are also dependent on the consumption rates in the body. However, the high capacity of the network to consume arginine and the high alanine production rates are in good agreement with the observed hypoargininaemia and high alanine blood concentrations in malaria patients. Interestingly, a high alanine blood concentration in malaria patients is usually attributed to a reduced alanine to glucose conversion in the liver [26], but our network analysis shows that a high alanine production by Plasmodium could contribute to this symptom. For an accurate prediction of blood concentration changes of amino acids a full body implementation of a model is required. However, high production rates (e.g. alanine) or consumption rates (e.g. arginine) due to Plasmodium activity can point to potential metabolic biomarkers for malaria progression. In addition, one can simulate the effect of a drug by setting a constraint on a metabolic flux in the network. If such an inhibition is complete then the network analysis can be accurate and the effect is dependent on whether the inhibited step is essential or not. If the inhibition is not complete, which is the likely scenario, it is better to analyse the effect in a kinetic model.

Kinetic modelling of Plasmodium glucose metabolism in malaria patients

Currently no detailed kinetic models exist for genome-scale networks, mostly due to limited kinetic information. Kinetic models do exist for smaller systems, such as central carbon metabolism and in a more coarse-grained form for organ and organism level metabolism. To analyse the effect of increased glycolytic activity of Plasmodium infected red blood cells in malaria patients, we merged three existing kinetic models: a detailed kinetic model for glycolysis of P. falciparum [34], a detailed kinetic model for central carbon metabolism of the red blood cell [3537] and a coarse grained kinetic model for whole-body glucose metabolism [38]. The models were obtained from the JWS Online [39] and Biomodels [40] model repositories, corrected for units and shared-variable-names inconsistencies and integrated. No adaptations were made to the P. falciparum and red blood cell model and for the whole-body model only the fixed metabolites alanine and non-esterified fatty acids were changed. A detailed description of the merged model will be published elsewhere (K. Green, D.C. Palm, F. du Toit and D.D. van Niekerk and Snoep, manuscript in preparation).

Figure 3(A) shows a schema for the combined model with the compartmentalized whole-body model and the added Plasmodium infected red blood model. A simulation of the effect of increased parasitaemia on blood glucose concentration is given in Figure 3(B), together with patient data (and rat model data) obtained from the literature. The patient data show a lot of scatter, which is indicative of large intermittent variance (no longitudinal data for a patient followed over time was available). Most papers make reference to hypoglycaemia in severe malaria patients, but then do not report both parasitaemia and blood glucose levels. The model prediction does simulate the reference state and the few patient data with hypoglycaemia, for which data were available, quite well. Similarly, the lactate data for malaria patients shows a lot of scatter (Figure 3C) and much more consistent data for the longitudinal rat study was obtained. The model prediction of lactate is low for the reference state but seems to follow the trend of lactate increase (and the rat data) reasonably well. In Figure 3(D), the results of an inhibition of the glucose transporter (to 50% of non-inhibited lactate flux) in P. falciparum are simulated.

Modelling glucose and lactate metabolism in malaria patients

Figure 3
Modelling glucose and lactate metabolism in malaria patients

(A) kinetic model for whole-body glucose metabolism [38] was merged with two detailed kinetic models for glucose metabolism in P. falciparum [34] and the red blood cell [3537] (A). The effect of increasing levels of parasitaemia on steady state blood glucose and lactate concentrations was analysed and shown in (B and C) respectively, together with concentrations measured in malaria patients (black symbols, calculated from [4152]) and rat data (red symbols, calculated from [53]). The shaded boxes indicate the severe malaria (> 5% parasitaemia) and hypoglycaemia (<2 mM glucose, B) and lactic acidosis (>5 mM lactate, C) areas. (D) The effect of inhibition of the glucose transport step (starting at t=50 min, resulting in 50% reduction in glycolytic flux in the parasite) on blood glucose and lactate concentrations in a malaria patient (5% parasitaemia) was simulated.

Figure 3
Modelling glucose and lactate metabolism in malaria patients

(A) kinetic model for whole-body glucose metabolism [38] was merged with two detailed kinetic models for glucose metabolism in P. falciparum [34] and the red blood cell [3537] (A). The effect of increasing levels of parasitaemia on steady state blood glucose and lactate concentrations was analysed and shown in (B and C) respectively, together with concentrations measured in malaria patients (black symbols, calculated from [4152]) and rat data (red symbols, calculated from [53]). The shaded boxes indicate the severe malaria (> 5% parasitaemia) and hypoglycaemia (<2 mM glucose, B) and lactic acidosis (>5 mM lactate, C) areas. (D) The effect of inhibition of the glucose transport step (starting at t=50 min, resulting in 50% reduction in glycolytic flux in the parasite) on blood glucose and lactate concentrations in a malaria patient (5% parasitaemia) was simulated.

Discussion and conclusion

To understand the pathophysiology of complex diseases, whole-body mathematical models can be strong tools to integrate and analyse the numerous effects that lead to the disease state. Specifically, when personal parameters can be added to such a model, they can be instrumental in choosing a correct treatment. Currently only very few molecularly informed models exist for the whole-body level that are detailed enough to be useful in medical applications (e.g. http://www.entelos.com). Specifically for the simulation of pharmacological drug effects on the disease state there is a big challenge in terms of modelling at the correct level of detail. To simulate the drug effect at the reaction step, a high level of detail is needed at the drug target level, which cannot be sustained up to the whole-body level.

In the present paper, we illustrated how a hierarchical model, with a high level of detail at the drug target site and more coarse-grained for the whole-body level, can be used to simulate the effect of a pharmaceutical drug on blood glucose concentration. Clearly, the model is still in a very rudimentary stage; we only simulate the direct metabolic effect of parasite activity and have ignored any effects on blood perfusion or reduced red blood cell contents due to cell lysis or any of the large number of secondary effects caused by the malaria parasites. In addition, we simulated the drug effect by simply assuming a constant inhibitor concentration in the blood, a much more realistic simulation would have to include a full PK/PD (pharmacokinetics/ pharmacodynamics) model to evaluate the drug efficacy [5,54].

The aim to mechanistically simulate drug effects (inhibiting a specific target reaction) at the whole-body level (physiological disease state) is very ambitious. However, we feel the time is right for this. Some large-scale projects have existed for quite some time and produced detailed kinetic models at the organ level that can be extended to the whole-body level (e.g. the physiome project, http://physiomeproject.org [55]; the virtual liver, http://www.virtual-liver.de [56]). In addition, a much stronger adherence to modelling standards, such as description in standard formats [systems biology markup language (SBML) and CellML)] and storage in curated model databases (JWS Online, Biomodels and CellML), makes model reuse much easier. For our initial model construction, we merged three existing models and this enabled us to make some preliminary simulations at different hierarchical levels. Of course, one cannot simple merge all existing models; they must be compatible and constructed for similar physiological conditions [57]. After merging of existing models, one can start a number of iterative cycles to improve the integral model and adapt it to specific disease states.

With the present concept paper, we hope to have illustrated the approach we follow towards whole-body modelling of blood glucose and lactate metabolism in malaria patients. Clearly, much work is still needed and especially at the whole-body level the model needs to be validated more thoroughly. For this, we will first work in a rat model system for which it is much easier to obtain longitudinal data. Although the specific model for malaria patients is still very preliminary, the suggested approach of a hierarchical model structure with a high level of detail at the drug target level and more coarse-grained models at the whole-body physiological level, is generic and could be applied to other complex metabolic diseases such as type II diabetes.

We thank Gunnar Cedersund and Brett Olivier, who were involved in the initial stages of the construction of the dynamic and structural model respectively and Barbara Bakker for discussing potential modelling strategies at the whole-body disease state.

Funding

This work was supported by the National Research Foundation, South Africa [grant numbers SARCHI 82813 (to J.L.S.) and TTK14051967526 (to D.D.v.N.)].

Metabolic Pathways Analysis 2015: Held at Bom Jesus, Braga, Portugal., 8–12 June 2015.

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