In recent years, with important advances in molecular biology, experimental and measurement technologies, it has become possible to generate the quantitative data that are needed for building mathematical models of complex biochemical processes. Cartoon-like diagrams of biological pathways can be turned into dynamical models, allowing simulation and analysis to gain an insight into the underlying control mechanisms and the behaviour of the overall system. This kind of system-level understanding has not been reachable from the study of the components of pathways in isolation. However, mathematical modelling does not only integrate the available knowledge about a certain system with newly generated experimental results. During the process of modelling, questions need to be addressed that lead to an increased quantitative understanding of the system. Models can be used to optimize experimental approaches and protocols and to test different hypotheses about the underlying biological mechanisms. Finally, a validated mathematical model can be used to perform in silico experiments that might be hard or impossible to do in the laboratory. In this chapter we present a case study of a systematic modelling approach applied to the thiamine uptake system of the yeast Saccharomyces cerevisiae. This example is part of our broader effort to model the whole of thiamine metabolism in yeast, which involves several additional processes such as thiamine utilization, biosynthesis and gene regulation. Our main goal is to describe how systematic modelling has improved the knowledge about the system under study.
Thiamine metabolism in yeast is a complex dynamic system involving thiamine uptake, utilization, biosynthesis and gene regulation. This metabolic network is interesting from several points of view: it represents a model for microbial nutrient sensing and a regulatory network with well-defined and characterized components; it consists of a relatively small number of proteins with rather specific functions restricted to this system; and it involves genes which are under the most stringent transcriptional control known in eukaryotic cells. These features make this network an attractive system to be studied using a systematic modelling approach. More specifically, it was interesting to investigate whether by using model simulations we could explain, or at least propose, certain underlying reaction mechanisms of key enzymes of the system such as those of Thi80 and Thi7 where traditional in vitro biochemical assays often fail. In fact, it turned out that the development of the mathematical model for thiamine uptake, which we discuss in the present chapter, led to significant new biological insights. In the following section we will ‘set the scene’ for this chapter by giving a short overview of thiamine metabolism in yeast and the modelling approach taken.
Yeast thiamine-related metabolism
Thiamine, also known as vitamin B1, is a water-soluble vitamin crucial for carbon metabolism. In addition to its functions in central metabolism, it is also necessary for the metabolism of some amino acids. Thiamine, in the form of ThDP (thiamine diphosphate), is involved as a cofactor in numerous enzymes such as pyruvate decarboxylase (fermentation), pyruvate dehydrogenase and 2-oxo-glutarate dehydrogenase [both from the TCA (tricarboxylic acid) cycle] and transketolase (from the pentose phosphate pathway). The significance of ThDP is particularly evident for metabolism of pyruvate, which cannot be further processed without ThDP. Yeast and other organisms (such as bacteria or plants, but not animals) can produce thiamine de novo by diverse biochemical pathways [1–4].
Since ThDP is a crucial molecule in cellular metabolism, its constant availability is essential for the cell. Thiamine biosynthesis, which serves as an internal source of ThDP, is a particularly demanding process which requires stringent regulation to avoid unnecessary waste of cellular resources, and yeast obviously prefers to take up thiamine from the environment (Figure 1). For this reason, a tightly controlled system for thiamine uptake and biosynthesis has evolved in yeast. This is particularly evident when analysing the expression profile of the THI genes, which belong to the most strongly expressed genes in the cell under thiamine-limiting conditions, whereas expression is virtually undetectable in the presence of external thiamine. Once thiamine becomes limited in the yeast environment, a sequence of regulatory processes is activated. An early response includes the transcriptional activation of genes whose products are involved in thiamine uptake (A. Ericsson, D. Mojzita, H. Schmidt and S. Hohmann, unpublished work). The expression of Thi7 and Pho3 is strongly up-regulated leading to more efficient uptake of thiamine and its phosphoesters from the surroundings. During this process, which inevitably leads to a complete consumption of the external thiamine followed by a depletion of the internal thiamine pool, expression of genes encoding enzymes involved in de novo thiamine biosynthesis are strongly up-regulated (Figure 1) [5,6]. The thiamine biosynthetic pathway consists of four processes: (i) parallel synthesis of two ring structures, HMP (4-amino-5-hydroxymethyl-2-methylpyrimidine) and HET [4-methyl-5-(2-hydroxyethyl)thiazole]; (ii) their concurrent condensation to ThMP (thiamine monophosphate) [2,5–7]; (iii) its dephosphorylation by an unknown phosphatase; and (iv) pyrophosphorylation by Thi80 kinase (Figure 1).
With the purpose of constructing a mathematical model of this system we split the system into four modules as depicted in Figure 1. In the present chapter, we focus our modelling efforts on the thiamine transport and phosphorylation module, or in short the thiamine uptake module. Experimental settings to be considered for modelling are thiamine availability and depletion in the extracellular medium, capturing thiamine uptake and the switch from uptake to de novo synthesis of ThMP.
Systematic modelling approach
Mathematical modelling of biochemical networks is a highly interdisciplinary task that can be subdivided into several steps (Figure 2) that need to be iterated until a validated predictive model is obtained. It is important to point out that a prerequisite for modelling is the presence of questions to be answered about the system or process under investigation. It is these questions that define the scope of the model and the operating conditions of the system that are to be considered. Modelling starts by setting up a static network diagram of the system, named here the ‘model network structure’, based on previously available knowledge and pathway databases (e.g. KEGG PATHWAY Database; http://www.genome.jp/kegg/pathway.html). Such a model network structure defines the components of the system and the interactions between these components that are judged to be important for the system’ behaviour of interest. Often, not enough functional knowledge is available to draw a very detailed network diagram but it usually is possible to depict different distinct modules and the main features of the system (Figure 1). By taking the boundaries of these modules into account they can be modelled independently, prior to integrating them into the final overall model. This approach has the advantage that the modelling complexity is considerably reduced. The next step is to turn the static model, i.e. the network structure, into a dynamic model by choosing reasonable mathematical rate expressions for the interactions between the components. These expressions can be determined based on the type of the interaction (e.g. transport, enzymatic reaction, transcription, diffusion), the properties of the interaction (e.g. number of participating species and modifiers, saturation) and the directionality. Once this is done, the parameters appearing in the rate expressions need to be estimated based on quantitative time-series data. This is done by comparing simulations of experiments with measured time-series data and optimizing the parameter values in order to obtain the best fit between model and reality. Finally, the obtained model needs to be validated. This means that the model’ predictive capabilities are tested by comparing simulations to experimental data that have not been used in the model building process. An important paradigm in mathematical modelling is that if a model is able to satisfactorily describe the measured data used for modelling, and is able to make experimentally validated predictions, then this indicates that the mechanistic assumptions about the system might be correct and that the model might be accurate in some of its other (non-validated) predictions as well.
It should be pointed out that modelling is not a sequential process, but that the different steps have to be iterated many times in order to arrive at a validated model (Figure 2). Analysis and simulation of intermediate models is used in order to set up hypotheses to be tested by experiments and to optimize experimental settings for the generation of data, required for the estimation of parameters. Depending on the properties of the system, different mathematical frameworks might be chosen by the modeller. The discussion of these is out of the scope of the present chapter. In the following section we will go through the different modelling steps based on thiamine uptake as an example system. We employ ordinary differential equations as the mathematical framework.
Model network structure
In this section, the main components of the system and their interactions will be presented as a background for the construction of a model network.
Thiamine uptake is mediated by an active mechanism, performed by the high-affinity and high-capacity thiamine membrane transporter Thi7 [8,9]. THI7 is expressed under all conditions and its expression is strongly up-regulated upon intracellular ThDP depletion [5,6]. This regulation is a link between the thiamine sensing, gene regulation and uptake modules in the overall system (Figure 1). For the uptake system, we model this regulation simplistically by a single reaction producing Thi7 which is inhibited by intracellular ThDP (Figure 3).
Thi7 is responsible for the bulk of thiamine transport; however, two other isoforms have been identified that are capable of thiamine transport as well. These are Thi71 and Thi72, which are expressed to a much lesser extent and contribute only to a minor fraction of thiamine uptake [5,8] and are therefore neglected in our model.
The periplasmic acid phosphatase Pho3 is also important for the uptake of thiamine . Pho3 is highly active towards extracellular thiamine phosphoesters converting them into free thiamine, which can be readily internalized by Thi7 [5,11]. Interestingly, a pho3Δ strain is able to accumulate all three forms of thiamine (thiamine, ThMP and ThDP) from the surrounding medium (A. Ericsson, D. Mojzita, H. Schmidt and S. Hohmann, unpublished work). Also HMP, a thiamine biosynthetic precursor, can be taken up by Thi7 , indicating a more promiscuous specificity of Thi7-mediated thiamine uptake. Owing to the high activity of Pho3 (de-phosphorylation is so quick that it can almost be assumed to be instantaneous) we did not model the regulation of Pho3.
Once in the cell, free thiamine becomes rapidly phosphorylated to ThDP by Thi80 (thiamine pyrophosphokinase). Thi80 is a cytoplasmic constitutively expressed kinase, which is indispensable for intracellular ThDP production and hence essential. Two mutants of Thi80 with considerably lower thiamine pyrophosphokinase activity have been isolated (thi80-1 and thi80-2). These mutants confer constitutively high thiamine transport and Pho3 activity indicating the crucial role of intracellular ThDP in the regulation of transcription [6,13].
Other components in the system are two unspecific phosphatases PX and PY (Figure 3), which catalyse two dephosphorylation steps from ThDP to ThMP and from ThMP to thiamine respectively. The PX phosphatase catalyses an essential step downstream in the thiamine biosynthetic pathway, where ThMP is the end product. The PY phosphatase was introduced to the system’ network in order to explain the experimentally observed behaviour of slow in vivo ThDP dephosphorylation. There are no indications in the literature of thiamine-related regulation of these phosphatases so we have modelled them as constitutively expressed and active.
In order to obtain a functional model of the thiamine uptake, we also had to include reactions and/or processes which connect this module with the other modules. Apart from the Thi7 production mentioned above, thiamine biosynthesis, with ThMP as the end product, was also included as a single reaction inhibited by intracellular ThDP (Figure 3).
Putting all of this together we obtained the final model network structure of the thiamine uptake system (Figure 3).
Mathematical rate expressions
The static model, obtained in the previous step, is turned into a dynamic model by the choice of reasonable mathematical expressions for the reaction rates. In the present chapter we limit the discussion to two example reactions, namely the thiamine uptake reaction performed by Thi7 and the pyrophosphorylation of thiamine performed by Thi80.
The Thi7 reaction
At the initial stage, our experimental findings, together with some previous work , indicated that Thi7 is inhibited when a certain amount of thiamine is accumulated in the cell; however, the model was not able to explain this phenomenon. Further experiments, such as the assessment of the ratios between the volumes of cells and media and the analysis of extracellular thiamine depletion, showed that Thi7 is not being inhibited in this manner. The only regulation which occurs is at the THI7 expression level. A classical Michaelis–Menten kinetics was not suitable for fitting the model to the experimental data, whereas using Hill-type kinetics proved to be optimal for the description of the observed behaviour of Thi7. The reaction rate expression for thiamine (Th) transport by Thi7 is thus given by eqn (1): where ThE determines the extracellular thiamine concentration, VThi7 the maximum reaction rate, KThi7 the concentration of the substrate for which the rate is half the maximum rate and hThi7 is the Hill coefficient. For ThMP and ThDP transport the same expression is used, however with different parameters.
The Thi80 reaction
As indicated above, formation of ThDP in the cell is catalysed by Thi80. The kinase forms a dimer with two thiamine-binding reaction sites [15,16]. This observation implies Hill-type kinetics of the Thi80 reaction, which has been employed in our initial model. Our experiments indicated that the Thi80 reaction is inhibited shortly after accumulation of ThDP in the cells. Since the inhibition correlates with the accumulation of ThDP, we have used ThDP as the inhibitor in our model. We have extended the original Hill-type kinetics for this kind of inhibition (allosteric) and employed it in our model.
The equation for irreversible allosteric inhibition type kinetics (Monod–Wyman–Changeux) is given by eqn (2): where Th and ThDP determine the intracellular thiamine and thiamine diphosphate concentrations, VThi80 the maximum reaction rate, Ks,Thi80 and Ki,Thi80, the concentration of the substrate (Th) and inhibitor (ThDP) respectively, for which the rate is half the maximum rate, LThi80 is the ratio between the concentrations of the enzyme in the weak and strong binding states, and nThi80 is the Hill coefficient.
Once a dynamic model is available, it can be simulated by numeric integration for arbitrary values of the model parameters. Sensitivity analysis techniques are used to find the parameters to which the model predictions are most sensitive and that thus should be considered in the estimation process. Parameter estimation techniques are used in order to identify the parameter values that lead to the best agreement between simulated data and experimental measurement data. Posterior identifiability techniques are employed in order to detect correlations between the estimated parameters, leading to a reduction of the search space and to better confidence intervals for the estimated parameters. For each experiment the model needs to be adapted to the particular experimental settings and to be simulated to return time-series data that can be directly compared with the measurements. In order to quantify the agreement between model and measurements we employed a cost function based on the sum of squared errors. The total cost has been determined as the sum of the costs, determined for the individual experiments. The optimal parameters have then been found by minimization of the total cost function.
As a starting point in estimating the parameters, previously published values, e.g. apparent Km for thiamine in the case of Thi7 , were used. We made sure that these values will not be considerably diverged by the estimation process. The measurement data employed in this case study for the estimation of parameters have been primarily time-traces of intracellular thiamine and its derivates, obtained by HPLC.
In this section we briefly describe selected experimental layouts used for obtaining quantitative data. These data are crucial for estimation of the parameter values in the model. Description of specific experimental techniques is outside the scope of this case study.
Sets of time course experiments were performed in order to determine the parameter values. To examine the properties of Pho3, Thi7 and Thi80, we followed extracellular thiamine dephosphorylation, thiamine uptake, its concurrent phosphorylation in the cell and eventually its gradual consumption/depletion leading to activation of de novo biosynthesis. Moreover, we investigated the uptake of ThMP and ThDP in the pho3 Δ strain. To estimate the phosphatase activities of PX and PY, we followed in vivo dephosphorylation of ThMP and ThDP accumulated inside the cell. We also followed levels of selected proteins in these conditions.
The results of the above modelling procedure have been obtained in an iterative manner. A comparison between simulations of the final model and the experimental measurement data are shown in Figure 4. The first five plots correspond to experiments that have been used for parameter estimation. The last plot shows an example of an experiment that has not been used during modelling but instead for the validation of the model.
A further validation has been performed for the Thi80 reaction. The model suggested that the reaction has to be irreversible, which was previously not known. To test this assumption, in vitro experiments were performed, where purified Thi80 was incubated with ThDP. No detectable degradation of ThDP was found after 2 h of the reaction, which would suggest irreversibility.
As suggested by the model, Thi7 is not directly inhibited by ThDP accumulation in the cell. To confirm this, a sequential addition of thiamine and/or ThDP was carried out. After the first addition of thiamine, a high accumulation of thiamine and ThDP was observed in the cell. This was followed by a second addition of thiamine and further accumulation of ThDP in the cell. Thi7 does not seem to be affected by previously accumulated ThDP and thus is not inhibited by ThDP.
In the present chapter we have applied a systematic modelling approach to the thiamine uptake system in the yeast S. cerevisiae. Apart from capturing the previously available knowledge and the reproduction of the experimental data, the modelling effort has resulted in significant new insights into this system. It provided the basis for new experiments, which probably would not have been performed otherwise and served as a testing environment for the system. The now verified hypothesis concerning the allosteric inhibition of Thi80 by ThDP is also a result of the model prediction.
Although, the above procedure has been presented in a sequential way, the modelling process has been based on several rounds of iterations between experiment and simulation and between the different steps (Figure 2).
It should be noted that even though this system is relatively simple, involving only seven species, there are thirty parameters to be estimated. With respect to the limited amount of measurement data available for the parameter estimation, the model is certainly over-parameterized. This means that there is an infinite number of sets of parameter values that produce an equally good agreement with the data . The treatment of this important issue of wide significance in systems biology is out of the scope of the present chapter.
For handling measurement uncertainties during parameter estimation, it is very valuable to know the variance of each estimated data point as obtained by repeating the same experiment a number of times. However, during this modelling project this has not been possible. The qualitative and quantitative fit of our model is, nevertheless, reasonably close to the measured data for a variety of different experiments (not all data are shown in Figure 4). Certainly, additional quantitative measurements of the components involved in the uptake system, such as determination of the absolute number of all proteins, metabolites etc. in different conditions will constrain the model and improve its predictive power.
As indicated above, the model is a starting point for our wider effort of modelling the entire thiamine-related metabolism. Since ThDP and thiamine-dependent enzymes play an important role in central metabolism, such a model could serve as a valuable complement to already existing models for glycolysis and other essential metabolic pathways. In addition, it is expected to become the first quantitative description of the interplay between enzyme cofactor demand and control of uptake and biosynthesis.
• Principles in systematic modelling.
• Thiamine metabolism, in particular the thiamine uptake system in yeast S. cerevisiae.
• Step-by-step construction of a dynamical model for the thiamine uptake system.
- © The Authors Journal compilation © 2008 Biochemical Society