Abstract
Metabolic modeling has been particularly efficient to understand the conditions affecting the metabolism of an organism. But so far, metabolic models have mainly considered static situations, assuming balanced growth. Some organisms are always far from equilibrium, and metabolic modeling must account for their dynamics. This leads to high-dimensional models in which metabolic fluxes are no more constant but vary depending on the intracellular concentrations. Such metabolic models must be reduced and simplified so that they can be calibrated and analyzed. Reducing these models of large dimension down to a model of smaller dimension is very challenging, specially, when dealing with nonlinear metabolic rates. Here, we propose a rigorous approach to reduce metabolic models using quasi-steady-state reduction based on Tikhonov’s theorem, with a characterized and bounded reduction error. We assume that the metabolic network can be represented with Michaelis-Menten enzymatic reactions that evolve at different time scales. In this simplest approach, some metabolites can accumulate. We consider the case with a continuous varying input in the model, such as light for microalgae, so that the system is never at a steady state. Furthermore, our analysis proves that metabolites in the slow part of the metabolic system reach higher concentrations (by one order of magnitude) than metabolites in the fast part under some flux conditions. A simple example illustrates our approach and the resulting accuracy of the reduction method.
1. Introduction
Metabolic models have considerably helped in understanding the metabolism of an organism and enhancing its production capability. These models are based on simplified metabolic networks and generally include several hundreds of reactions associated to many metabolic compounds. For example, metabolic models to better understand the production of triacylglycerols and carbohydrates from microalgae (both compounds can then be turned into biofuel) [1] use between 56 and 2190 reactions and between 46 and 1862 metabolites, depending on authors and studies. In order to manage the large dimension of these models, some simplifying assumptions are generally necessary.
The most classical hypothesis is balanced growth, that is, global steady-state assumption (SSA). This means that the derivatives, with respect to time, of all variables are put to zero. For instance, flux balance analysis (FBA) [2] or macroscopic bioreaction models (MBM) [3] are based on linear algebra to solve the equation , where is the stoichiometric matrix and is the vector of intracellular reaction rates.
Yet, metabolisms of microalgae and cyanobacteria are directly related to solar light providing the energy for incorporating CO2 through Calvin cycle. Periodic fluctuation of light induces unstationarity and permanent accumulation and reuse of metabolites (specially lipids and carbohydrates). Therefore, such metabolisms are never at a steady state, and the classical approaches based on balanced growth hypothesis cannot be used to describe their metabolisms.
Here, we propose a rigorous mathematical approach to reduce the dimension of a dynamical metabolic system, in order to analyze its behavior and calibrate it. The reduction that we propose allows to characterize the approximation error, and it is appropriate to model-based control strategies. The idea is to keep some dynamical components of the model that are necessary specially when dealing with microalgae and cyanobacteria.
A first attempt in this direction was carried out with the Dynamic Reduction of Unbalanced Metabolism (DRUM) method [4]. DRUM considers subnetworks in quasi-steady state (QSS), which are interconnected by metabolites that can accumulate. Then, elementary flux modes (EFM) are computed in each subnetwork to reduce them using quasi-steady-state assumption (QSSA). As a result, the dynamics of accumulative metabolites form a reduced system of ordinary differential equations (ODE). It provided sound results, specially to describe accumulation of lipids and carbohydrates in microalgae. However, as almost all the methods, it also relies on a series of assumptions whose mathematical bases have not been rigorously established [5]. Beyond QSSA which is not rigorously defined from a mathematical viewpoint, these approaches also neglect intracellular dilution due to growth.
Models of metabolic networks are nonlinear and high-dimensional systems, which make their dynamical behavior difficult to determine and calibrate. The main objective of our work is to provide mathematical foundations for the reduction of metabolic networks down to low-dimensional dynamical models.
Here, we study a class of metabolic models of dimension , where the enzymatic reaction rates are represented by Michaelis-Menten reactions. This class of models is the simplest nonlinear one to get accumulation of some intermediate compounds. The objective is to reduce this model accounting for a permanently fluctuating input and rigorously including dilution of the metabolic compounds due to the growth rate. The system is not closed and never reaches a steady state. At the end, we can express a slow dynamical system of small dimension and a fast system as a function of the variables of the slow system. The error in this reduction is characterized and bounded.
In Section 2, we introduce the class of models we consider, which is composed of two (general) subnetworks of fast reactions connected by metabolites with slow dynamics. In Section 3, we develop a mathematical model for these metabolic systems.
In Section 4, with proper mathematical hypotheses, after a change of variables for the metabolites with fast dynamics, the system becomes a slow-fast system. The conditions for applying Tikhonov’s theorem for singularly perturbed systems are verified and we end up with a reduced dynamical model and a bound of the approximation error.
In Section 5, we prove that metabolites in QSS have a concentration one order of magnitude lower than slow metabolites. Additionally, in Section 6, we propose an identification algorithm to estimate the parameters of the reduced system from available data.
Finally, we apply our method to a toy metabolic model in Section 7. This simple model is forced by a periodic input and includes standard bricks in metabolic networks: combination of reversible and nonreversible reactions, with chains and cycles.
2. Network of Enzymatic Reactions
In this section, we present the class of metabolic networks studied all over the paper, which are illustrated in Figure 1. These networks are composed of two subnetworks of fast reactions, which are interconnected by several metabolites with slow rates of consumption. The subnetworks have an arbitrary finite number of metabolites and reactions between them.
These subnetworks are not assumed to have a specific topology. Therefore, they represent a generic case of metabolic networks. The only condition on them is that their metabolites are consumed by fast reactions.
The class of systems addressed in this paper can be considered as a simplification or one part of a larger network. However, the results presented through this paper can be extended, allowing the study of more complex systems on the bases of this approach.
2.1. Summary of the Methodology for Reducing Slow-Fast Dynamical Metabolic Models
We consider a general class of metabolic models allowing internal accumulation, represented with the network in Figure 1. In order to rigorously reduce this large dimensional model, our objective is to take benefit of the two time scales and finally rewrite it in the canonical form of singularly perturbed systems. Then, the theorem of Tikhonov [6] can be applied, and a reduced system is derived with an accurate bound of the error.
The first challenge is to find the appropriate change of variable for the metabolites with fast dynamics, to end up with a slow-fast system: where is the vector of metabolites with slow dynamics, is the vector of metabolites with fast dynamics, and is a very small parameter. Actually, results from a rescaling of fast dynamic metabolites in the following model:
When the system is under this general form, we prove some conditions necessary to apply Tikhonov’s theorem [6], and finally we obtain a quasi-steady-state reduction of system (1): where is a root of the equation
If is a solution for (3), the quasi-steady-state approximation to the solution of (1) satisfies after an initial fast transient. In other words, the error of the quasi-steady-state approximation has order of magnitude , which is supposed to be a small positive number. In the manuscript, we show that the reduced system differs from existing approaches, mainly because we do not neglect the metabolite dilution due to cell growth.
The mathematical validity of the quasi-steady-state reduction (QSSR) for the class of systems considered in this paper (Figure 1) is showed from Section 3 to Section 4.
As a new striking result, this approach allows to prove that the concentration of the metabolites in quasi-steady state is one order of magnitude lower than that of the metabolites with slow dynamics, that is,
The conditions under which this assertion holds are given in Section 5.
3. Considered Class of Networks
In this section, we describe the systems of the network class considered in this work. Then, in Section 4 and Section 5, we deduce a QSSR and prove some conclusions about the magnitude of metabolite concentrations (see Theorem 1) for these systems.
The results obtained in the following sections can be extended to more complex networks. For instance, considering additional slow reactions or more subnetworks of fast reactions connected by metabolites with slow dynamics.
3.1. Notations
Consider the network of enzymatic reactions depicted in Figures 1 and 2, where an arrow from to represents an enzymatic reaction catalyzed by , with substrate , product , and product formation rate or . Then, every enzymatic reaction can be described with the Michaelis-Menten model (see Appendix A).
However, it is necessary justify the quasi-steady-state approximation for the Michaelis-Menten model. For this purpose, many solutions have been presented. For example, this holds if the initial substrate concentration is sufficiently large compared with the initial enzyme concentration [7] or if the product formation rate is small enough [8].
We suppose that among the product formation rates, there are two scales of magnitude. Reactions with large rate are within two subnetworks, which are interconnected by the metabolites , , and . We suppose that the metabolites connecting the subnetworks are consumed by reactions with low rates.
In this context, we say that a reaction is fast if its rate is large, while a reaction is slow if its rate is low. Moreover, we assume the rates of fast reactions sufficiently larger than those of the slow reactions. Then, we denote fast reaction rates by and slow reaction rates by where is a small positive number.
Additionally, a continuously varying nonnegative input (e.g., the CO2 uptake in a microalgae submitted to light/dark cycles) and a growth rate , which acts as a dilution factor, are taken into account for the models.
3.2. Dynamical Model
According to the standard quasi-steady-state reduction for Michaelis-Menten enzymatic reactions described in Appendix A, we write the ODE system for the model in Figure 1 as where for and for every , .
The variable describes the th metabolite cell concentration; is a nonnegative continuous function; is a small positive number; , , and are nonnegative parameters; and is the growth rate. When there is no reaction with substrate and product , we define and also for every .
Note. In our model, we can include first-order (linear) reactions. In this case, instead of writing as for enzymatic reactions, we have to write respectively, in the algebraic equation (9). For the sake of simplicity, in this paper we only consider the more general case with Michaelis-Menten reactions.
In line with the QSSR of Michaelis-Menten system, we recall that (or for the fast reactions) and are parameters related to the enzyme reaction with substrate and product . Indeed, is the initial enzyme concentration, (or ) is the product formation rate, and is the specific Michaelis-Menten constant defined as (see Appendix A).
An important preliminary property that the dynamical system (9) has to obey is that the concentration has to remain nonnegative over the time if the initial conditions are nonnegative. In our model, this depends on the input . This is stated in the following property:
Property 1. If the initial condition is nonnegative for every and for every , then system (9) is positively invariant in .
Proof. To verify this, we show that system (9) is positively invariant in if is nonnegative over any interval .
Recall that all is supposed to be positive and every parameter, , , and , is nonnegative. Then, we have for any if for every , . Therefore, system (9) is positively invariant in .
3.3. Parameter Order of Magnitude
With our notations, to represent different time scales in the reactions, we fix a small positive number highlighting the difference between the parameter scale orders. We suppose that the parameters are of standard range, that is, where denotes the Big O or Landau symbol. For the definition and some properties of , we refer to [9].
Also, we suppose that the input has a magnitude not larger than the slow reactions. In other words,
The rate of growth is considered as a parameter smaller than any reaction rate (a standard hypothesis [10]). Here, we assume
4. Quasi-Steady-State Reduction
In this section, we propose a rigorous quasi-steady-state reduction (QSSR) of (9). Its mathematical validity is proved, thanks to the theorem of Tikhonov [6]. In other words, this theorem states that the error of this quasi-steady-state approximation is bounded by the small parameter .
We formally define the QSSR after Tikhonov’s theorem, of the metabolic network in Figure 1 and system (9), as the following system of dimension three: with initial conditions , , and , and for the metabolites in QSS, for every . The definition of the parameter is given later in this section (see Proposition 1 and its proof).
4.1. Slow-Fast System
In order to write system (9) in the canonical form of singularly perturbed systems, we define a change of variable for the fast metabolites by
Let us set the initial conditions for these new variables as and growth rate as
Therefore, after the change of variables (23), (9) can be rewritten as follows:
Since is a very small positive number, the dynamics of are faster than those of . Hence, the equations of form the slow part of system (26), while the equations of constitute its fast part.
The previous (26) is written with further details in the next subsection. The goal is to expose how the quasi-steady-state reduction is obtained and validated using Tikhonov’s theorem.
4.2. Canonical Form of Singularly Perturbed Systems
The slow-fast system (26) is in the class of singularly perturbed systems of the exact form: with initial conditions for every , .
Note. Equations (27)-(28) above are a more detailed expression of (26). Indeed, we obtain system (26) when is substituted for in (27)-(28).
An approximation to the solution of system (27)-(28) can be obtained considering the limit when . Then, the dynamics in (28) are considered as fast, and the QSSA is applied to the metabolites for every , .
Hereafter, we say that (27) is the slow part and (28) the fast part of system (9).
4.3. Hypotheses Necessary for Quasi-Steady State
In the following two subsections, we check the assumptions of Tikhonov’s theorem [6]. First, we demonstrate that the system has a single steady state (which is not straightforward for nonlinear systems). Then, we demonstrate that this steady state is asymptotically stable. Eventually, once all the conditions have been established, in Section 4.5 we present the result of Tikhonov’s theorem.
Consider the following algebraic system of equations, obtained from equating to 0 the terms in square brackets in (28) and substituting :
In order to apply Tikhonov’s theorem [6], we have to prove that (29) has an isolated root for any nonnegative constant values and and that this root is asymptotically stable for the following system:
The purpose of finding a root of (29) is to write the fast variables in terms of the slow variables . In this case, it is possible to find an analytic solution of this algebraic system, because it is a linear equation for the variables . Similarly, the asymptotical stability of this root for system (30) can be verified with the theory of linear systems of ODE.
Proposition 1. Consider and as nonnegative constant values. Then, system (30) has a single equilibrium point which is globally asymptotically stable. Moreover, where are nonnegative coefficients.
Proof. First, notice that system (30) is a linear system for under the hypotheses of Proposition 1. Then, we just have to show that its Jacobian matrix is stable, that is, that all its eigenvalues have negative real part [11].
The Jacobian matrix of (30) is where
But is a strictly column diagonally dominant matrix, because . In other words, for every column of the matrix , the sum of the entries out of the diagonal is strictly less than the absolute value of the entry in the diagonal. Hence, by the theorem of Gershgorin, is a stable matrix [12].
The matrix form of (29) is
Then, the solution of the algebraic system (29) is
Therefore, the variables of the solution can be written as with . Moreover, since is strictly column diagonally dominant, by the theorem of Gershgorin, is a stable matrix [12]. Then, its inverse matrix is nonpositive [13] (i.e., each entry of is nonpositive). Therefore, all entries in are nonnegative. We conclude that coefficients in (32) are nonnegative.
Note. Although Proposition 1 is proved for nonnegative constant values and , we consider in (32) also as functions of . Then, we have the functions in (22), defined for the QSSR.
4.4. Study of the Slowly Varying System
The dynamics of the slow system (metabolites which do accumulate) are obtained by setting in (27) and substituting the fast variables for the expression given by (32):
Then, we obtain the remaining dynamical system (19)–(21), which provides the dynamics to the overall network.
The other variables of the metabolic network, which are the fast variables (indeed, most of the variables are fast) can then be reconstructed after the solution of the algebraic system (29), given in (22). Finally, these fast variables rely on system (19)–(21). This system is also referred as the quasi-steady-state system [6].
Proposition 2. If system (19)–(21) has nonnegative initial conditions, then it has a unique nonnegative solution () defined on the interval .
For the proof of Proposition 2, see Appendix C.
4.5. Tikhonov’s Theorem
Propositions 1 and 2 prove that the class of systems with the form (27)-(28) satisfies the hypothesis of Tikhonov’s theorem [6]. Then, we can apply this theorem to system (26).
The following proposition is a consequence of Tikhonov’s theorem [6]. The proposition states that the approximation given by the QSSR (19)–(22) has an error with order , after a fast initial transient for the fast variables.
Proposition 3 (deduction of Tikhonov’s theorem). If is a nonnegative continuous function over , then
Moreover, there exists such that for every , where are the solutions of the original system (9), are the solutions of (19)–(21), and are the functions defined in (32).
Note. The solution of the boundary layer problem for system (27)-(28) is similar to that of system (30). We include its demonstration in Appendix B.
5. Magnitude of Concentrations throughout the Metabolic Network
In this section, we study the magnitude of metabolite concentrations, depending if they are associated to slow or fast reactions. They are deduced from the reduced system after Tikhonov’s theorem (19)–(22). We now show that the concentration of metabolites in QSS (that do not trap the input flux) is one order of magnitude lower than that of metabolites with slow dynamics. In order to prove this assertion, we define the conditions under which to obtain for every .
5.1. Parameter Orders
We show that all off-diagonal entries of the Jacobian matrix have the same order of magnitude, for both matrices defined in (34)-(35).
Lemma 1. Suppose that the parameters of each Michaelis-Menten enzymatic reaction (see Appendix A) satisfy then
Proof. As a consequence of (16),
Moreover, by the definition of the Michaelis-Menten constant (14) and (45), we have and then
Hence, combining (47) and (48), we have
Actually, all the entries of the matrix have the same order of magnitude, as asserts the following corollary.
Proposition 4. Consider the matrices defined in (34)-(35). All the entries of and have the same order.
Proof. According to Lemma 1, for the sums in the diagonal of the matrices, we have
Moreover, . Then,
For the off-diagonal entries, consider (48). Therefore, all the entries of and have the order .
5.2. A Theorem for Magnitude of Concentrations
In order to prove that a metabolite in QSS does not reach high concentrations, we have to suppose that it is not in a trap for the input flux. The definition of trap was introduced in [14], and we formally adapt it to the class of models considered in this article (see Appendix D.1). Then, we define a flux trap, which is a trap reached by the flux.
Assumption 1. There exists a flux from to in the system of enzymatic reaction (9) (depicted in Figure 1). Moreover, we define as the set of indices such that is in a flux trap, for every , and is not in a flux trap for every.
Notice that the presence of the flux from to implies
Then, flux traps are only possible within the subnetworks in QSS. Also, if there is no flux trap.
The following lemma sets down the order of magnitude of the parameters in (22), for the metabolites which are not in a flux trap. These parameters are used for writing the expression of fast metabolites in the QSSR.
Lemma 2. Suppose the system of enzymatic reaction (9) (Figure 1) under Assumption 1. Consider the parameters of (22), obtained in Section 4. Then, if , it holds
Proof. From the results stated in Appendix D, particularly Theorem 2, we have for ,
Using equality (48), we conclude that, for ,
The next theorem is a powerful conclusion obtained after the QSSR (19)–(22). Theorem 1 states that the concentration of a metabolite in QSS, which is not in a flux trap, is one order of magnitude lower than that of the concentration of a metabolite with slow dynamics. This result holds even if there is a trap or a flux trap in the system.
Theorem 1 (magnitude of concentration theorem). Consider the system of enzymatic reactions (9) (Figure 1). Under Assumption 1, the following inequalities hold:
Proof of Theorem 1. Since the reduction from Tikhonov’s theorem, we have (22), that is,
Then, as stated in Lemma 2, for such that ,
But , because system (19)–(21) is positively invariant and . Hence,
We conclude that
Note. The approach presented in this work can be used to reduce a metabolic network which has flux traps, obtaining an error characterization (as established in Proposition 3) and the conclusion of Theorem 1. But, in agreement with Theorem 1, the magnitude of concentrations of the metabolites in the flux traps cannot be bounded by the concentration of the metabolites in the slow part of the system. This fact can be inferred from the proof of Theorem 1 (see Appendix D.2.)
The presence of a flux trap leads to accumulation, without reuse, of compounds in the metabolic network. However, accumulation of some compounds to large concentrations often results to cell death. For example, the accumulation of lactate has been recognized as one cause of cell death [15, 16].
6. Reduced Model Calibration
Now that we have described the way to synthesize the initial model of the metabolic network into a small dynamical system (for accumulating metabolites) and a set of algebraic equations, we will explain how to calibrate this reduced model from experimental data. Of course, we assume that the initial stoichiometric coefficients are known, but the parameters associated to reaction rates are unknown.
Here, we propose a method to estimate the parameters of the reduced system. In a first stage, we identify the parameters of the reduced dynamical system representing the accumulating metabolites (19)–(21). The identification method is based on the minimization of a cost function, computing the error model with respect to experimental data.
Furthermore, if data of any metabolite in QSS is available, we can also estimate the respective parameters in (22), to write its concentration as a linear combination of the slow metabolites.
6.1. Calibration of the Slow Dynamics
We suppose experimental data of the metabolites in the slow part of the system (27), denoted by where is the solution of the original system (9) and represents an error of measurements. In order to estimate the parameters of the reduced system (19)–(21), we rewrite it as
Let and define a cost function . This cost function has to measure the error between the solution of (62) and the data , for every value in a domain . For example, we can define as
Then, we have to find such that
Note. For obtaining the vector of parameters to calibrate (62), it is not necessary to have data of any metabolite in QSS, with , . Only the data (61) of the metabolites in the slow part, , is used.
6.2. Fast Dynamics Parameters
In some (rare) cases, measurements of some fast metabolites can be available. Generally, these data are only obtained at quasi-steady state after the initial transient and for a subset of the metabolic compounds.
Supposing that we have experimental data of the metabolites in QSS after the initial fast transient, and that we have obtained after calibrating (62), we can estimate the parameters in (22), as a matter of fact, in line with the reduced system.
In (19)–(22) and the calibrated system (62), for the metabolites in QSS, we have where are the parameters to be estimated.
Here, we can explicitly resolve the linear least square problem. The least squares solution that minimizes the difference between the data and the expressions in (66) is the following [17]:
Indeed, we look for values of that minimize the differences
Note. To obtain the parameter , we only need the data (of the corresponding metabolites in QSS, ) and the calibrated system (62) with .
7. Illustrative Example with a Toy Enzymatic Network
In this section, we apply the method developed in this paper to the toy network represented in Figure 3. This toy network accounts for one reversible enzymatic reaction and a cycle of enzymatic reactions. Moreover, the toy network contains two subnetworks in QSS (in blue in Figure 3), which are interconnected by metabolites with slow rates of consumption (in black in Figure 3).
First, we consider the ODE of the toy enzymatic network, as in Section 2. Then, using the time-scale separation hypothesis, we reduce this ODE with the method described in Section 4. Finally, we estimate the parameters of the reduced system as it is suggested in Section 6.
All the parameters in the toy network are supposed to satisfy the conditions established in (16) and Section 5. The periodic and continuous input considered is given by where is a parameter with the same order of magnitude as the slow reactions rates.
7.1. Reduction
We apply to the toy network our reduction scheme, as described in Section 4.
First, to simplify the notation, we define the following parameters:
Then, we obtain the following reduced system for the toy network, and the expressions for the metabolites in QSS,
7.2. Calibration of the Reduced Toy Network
We follow the procedure in Section 6. For simplicity, we suppose that the data are measured at the same time instants (we assume that 48 measurement instants are available) for the slow and the fast parts of the system.
The measurements are the variables (unit g/L) of the original system (9) (for the toy network in Figure 3) plus a white noise: where and for every .
As in Section 6, to estimate the parameters of (71), we use the reduced system (62), with and . The cost function considered is , defined in (63), with and .
The function fminsearch in Scilab (www.scilab.org) was used for minimizing . This function is based on the Nelder-Mead algorithm to compute the unconstrained minimum of a given function. For the simulations in Figure 4, the value obtained is in Table 1 and .
Here, for illustration purposes, we suppose that the metabolites in QSS are also measured; we calculate the parameters to estimate their concentrations as explained in Section 6.2. Then, their concentrations are obtained according to (66).
We computed the numerical solution of the systems describing the dynamics in the toy network of Figure 3. The results are represented in Figures 4 and 5. As expected, the concentrations of the metabolites in QSS are one order of magnitude lower than those of the metabolites in the slow part.
Note that the parameters and are affinity constants in Michaelis-Menten functions, whose sensitivity is low [18]. Here, we have used 48 samples for parameter identification.
It is worth noting that the identification process results in a satisfying agreement between simulations of the calibrated system (62)–(66) and recorded data, as represented in Figures 4 and 5.
8. Discussion
8.1. Time-Scale Hypotheses
Metabolic networks can involve much more than two different time scales. Actually, our method considers the division of these in two groups of reaction rates, the kinetics slower than a certain threshold and the kinetics faster than this threshold. Our approach eventually preserves the dynamics of the slower kinetics (keeping the different time scales), while the fastest dynamics are lumped and approximated.
Also, to better illustrate this important aspect, we have considered several reaction rate orders in the toy network. The reaction rates are divided into slow and fast, and each group of reactions has different scales (see Table 3 and Table 4). The simulation results illustrate Theorem 1 (see Figure 4), and the reduced system accurately represents all different time scales (see Figures 4 and 5).
Finally, note that it would be possible to set up a finer approximation considering several time scales for Tikhonov’s theorem, but at the risk of higher mathematical complexity. Indeed, extended versions of Tikhonov’s theorem exist for several time scales, using powers of [6, 19, 20] or even different epsilons [21]. But computations with this method highly complicate the reduction.
8.2. Comparison with Experimental Data
To the best knowledge of the authors, there are to date no examples of metabolome measured at high frequency, at least for a large number of metabolites to assess the kinetics. In general, only a very limited number of macromolecules (typically proteins, carbohydrates, lipids, chlorophyll, etc.) are recorded, specially for microalgae. However, to show that our findings are in agreement with experimental studies, we considered the results from [4] for an autotrophic microalgae metabolic network.
The authors in [4] fitted the parameters of a metabolic model to the set of available experimental data. We examined the reaction rates which ranged from 102 to 10−1 (h−1·mM·B−1) and compared them with the level of concentrations in the cell. Indeed (see Table 5 and Table 6), the concentration of carbohydrates has a magnitude 102 (mM) times higher than that of the intermediate metabolites (GAP, PEP, and G6P). Moreover, GAP, PEP, and G6P are consumed by reactions with a rate order 101 or 102 (h−1·mM·B−1), while carbohydrates are consumed at a rate of order 100 (h−1·mM·B−1). Additionally, carbohydrates are produced by a single reaction with a rate of order 101 (h−1·mM·B−1), as well as GAP, and G6P, and PEP by reactions of order 102 (h−1·mM·B−1). This evidences that the concentration is related to the rate of consumption, in the way predicted by Theorem 1 in our paper.
Nevertheless, we emphasize that the reduction method proposed in this paper can be used even if only some metabolites with large concentrations have been measured. Indeed, such data will support the calibration of the reduced model, that is, describing the dynamics of the slow metabolites (see Section 6.1).
8.3. Extensions of Results
In order to obtain reduced metabolic systems by a rigorous procedure, many extensions of the results can be obtained. Particularly, considering more reactions between the metabolites with slow dynamics is possible (as long as these reactions are slow), without changing the equations of the fast part.
Hence, modifications in the reactions between slow metabolites do not alter the equations of the metabolites in QSS, and the slow dynamics remain in the reduced system. Moreover, the result obtained in Theorem 1 still holds if the equations of the fast part are not changed.
Furthermore, effects such as inhibition can be considered in the slow part of the system, for example, using the model of Haldane or feedback inhibition in enzyme-catalyzed subnetworks.
In addition, models with more subnetworks of fast reactions, connected by metabolites with slow dynamics, can be reduced and analyzed using the present approach.
9. Conclusions
Quasi-steady-state assumption without verifying mathematical conditions can lead to erroneous conclusion and strongly biased reduced systems [22, 23]. The aim of our work was to define the mathematical foundations of quasi-steady-state reduction for metabolic networks.
We reduced a general class of dynamical metabolic systems using time-scale separation and Tikhonov’s theorem. The considered models include the Michaelis-Menten reaction rates and the possibility for some compounds to accumulate. The reduction leads to a simpler model given by a small system of differential equations: regardless of the initial dimension of the network, we end up with a low-dimensional dynamical system, representing the dynamics of the slow variables. The dilution due to growth plays an important role and must not be neglected. It is worth noting that keeping the growth rate in the equations strongly improves approximation precision and preserves qualitative (stability) features of the original system.
We show that a metabolite in QSS has a concentration one order of magnitude lower than a metabolite in the slow part of the system. This is indirectly a way to validate the hypotheses on the magnitude of the reaction kinetics.
Eventually, the calibration algorithm is very simple. It is remarkable that the reduced model can predict all the fast compounds which have been measured, regardless of the other compounds whose concentrations cannot be recorded.
This approach covers a large class of metabolic enzymatic networks. But more work remains to be done to treat further metabolic systems. For example, networks with more reactions between fast and slow metabolites can be studied in detail. Moreover, to obtain models that rigorously describe several hierarchies in metabolic networks, systems with more than two time scales can be analyzed on the basis of the present paper.
Appendix
A. Michaelis-Menten Reaction
In this paper, we present a metabolic network which contains enzymatic reactions. Therefore, we present the Michaelis-Menten enzymatic reaction to set the notation that we use throughout the text.
The Michaelis-Menten model considers a substrate which reacts with an enzyme to produce a complex . Then, this complex is transformed into a product and the enzyme . This enzymatic reaction is abstracted as follows:
It is necessary to justify the quasi-steady-state approximation for the Michaelis-Menten model. For example, this holds if the initial substrate concentration is sufficiently large compared with the initial enzyme concentration [7] or if the product formation rate is small enough [8]. A widely used quasi-steady-state reduction of system (A.2) is the following [7, 8]: where is the Michaelis-Menten constant.
B. Boundary Layer
A second condition related to the uniform convergence of approximations when has to be verified with the boundary layer of (30) [6]. For this, we define the boundary layer correction and the boundary layer problem: with initial conditions for every , .
Proposition 5. The equilibrium point of system (B.1) is asymptotically stable.
Proof. First, notice that system (B.1) is linear, since (30) is linear. That is an equilibrium point of system (B.1) is a consequence of (37). Moreover, the Jacobian matrix of system (B.1) is the same with (30). Therefore, as in the proof of Proposition 1, we conclude that the origin is asymptotically stable for system (B.1).
On the other hand, the boundary layer correction allows to correct the error of the approximation (22) at the initial fast transition. Indeed, notice that the initial condition in (28) can be different from in (22). But
Moreover, the boundary layer correction vanishes quickly [6] since
C. Solution of the Slow System
Proof of Proposition 2. As in the proof of Proposition 1, we use the fact that is a nonnegative continuous function on and all the parameters in (19)–(21) are nonnegative real numbers. Hence, system (19)–(21) is positively invariant in .
Let us denote the right hand of (19)–(21). Then, we have and that are continuous on . Moreover, is uniformly bounded on .
As a consequence, we can deduce from the global existence and uniqueness theorem [19] that (19)–(21) has a unique solution over .
D. Supplement for the Proof of Theorem 1
D.1. Fluxes, Traps, and Flux Traps
In order to see when the metabolites in QSS do not accumulate, we have to introduce the following definitions.
Definition 1. We define a directed graph related to the network in Figure 1, equivalent to system (9), as follows: the substrates and products , are the nodes of the . Then, if (i.e., if there is a reaction with substrate and product ), there is an edge with initial node and final node . In a similar way, we define the graph associated to a subsystem of (9), with metabolites .
The concept of graph allows the following definitions.
Definition 2 (flux). A flux from to is a directed path which has an initial vertex and a final vertex .
Definition 3 (trap). Consider a graph with a set of vertices and a subset of this . We say that is a trap if (i)for every vertex , there is no flux from to any metabolite of \;(ii)no is the initial vertex of an edge with final vertex .
In this case, we also say that is in a trap for every .
Definition 4 (flux trap). Consider a flux with initial vertex and final vertex in a graph with vertices . We say that the graph has a trap for the flux if there is a subset , such that (i) is a trap (hence, there is no flux from to for every vertex );(ii)there is a flux from to for every vertex .
When it is clear which flux is taken into account, we only say that the graph has a flux trap. We also say that is in a flux trap for every vertex .
If the graph associated to a network has a flux, trap, or flux trap, we also say that the network has a flux, trap, or flux trap, respectively.
D.2. Matrix Analysis
Consider the Jacobian matrix defined in (33). For the sake of simplicity, we denote where if there is no reaction from to . Then, where
Theorem 2. Suppose that the graph associated to (9) satisfies Assumption 1. Consider the expression of the metabolites in QSS (22) and define Then, for every ,
We recall that means the metabolite is not in a flux trap.
Before proving Theorem 2, we demonstrate several propositions. The proof of Theorem 2 is in Appendix D.3. For this, we have to analyze the order of the parameters where and are the cofactors of and , respectively.
Lemma 3. Consider a singular matrix of dimension and . Suppose when , for every entry of . Then,
Proof. Define as the function
Since when for every entry of , is infinitely differentiable at 0. Then, considering its Taylor series around zero, it follows
But and when , for every , as a consequence of the hypothesis on the orders of entries. We conclude that
Lemma 4. Suppose that is a column diagonally dominant matrix of size , such that . If every off-diagonal entry of is nonnegative, then all the cofactors of have the same sign equal to (−1)n−1 and sgn(det(M)) = (−1)n.
Proof. Since is nonsingular and column diagonally dominant, by the theorem of Gershgorin, is a positive stable matrix [12]. Then, its inverse matrix is nonnegative [13] (i.e., each entry of (−M)−1 is nonnegative). But where is the transpose matrix of cofactors of [24]. Then, which implies that all the cofactors , with the minor of obtained from removing the th row and the th column [24], have the same sign. Moreover, since all the principal minors of are positive [13], then . We conclude that and that is negative if is odd and positive if is even.
Proposition 6. Let where . Consider the directed graph associated to as a graph with nodes and an edge with origin and final if . Suppose that has no traps and that . Then,
Proof. Notice that an output from the th metabolite is equivalent to . Here, without loss of generality, we begin by supposing that the th metabolite has an output. Then .
We prove the proposition by induction over . For , consider the matrix of a system with two metabolites and one output. The determinant of is
If , then . We examine in which cases . If , the system has no output, contrary to our hypothesis. On the other hand, implies that is in a trap (see Figure 6). We conclude that . The case in dimension with more than one output is verified immediately.
We make the following induction hypothesis: consider a graph of metabolites with no traps and one output at least. If is the matrix of size associated to , then .
Now we prove the case of a network with metabolites. We take into account that all the cofactors of have the same sign, as claimed by Lemma 4. It holds where are cofactors of [24].
Suppose that and for every . Then, is isolated and the rest of the metabolites form a trap. Hence, or for some , and we can apply the hypothesis of induction to deduce that
On the other hand, the term in the squared brackets in (C.19) is the determinant of the matrix , where is a matrix of size with zero at every entry, except for the entry which is equal to 1. If for every , then
according to Lemma 3, and the statement of Proposition 6 is proved. In other case, suppose without loss of generality. Hence, if we develop the determinant of by the th column and we substitute in (C.19), we have where are minors of . Moreover, the matrix satisfies the conditions of Lemma 4. Then, all its cofactors have the same sign. Particularly, and then
Once again, the term in the square brackets in (C.22) is equal to . We proceed as for to extract the following term: which has the same sign as . In steps, we arrive to an expression of the determinant where all the terms have the same sign and one term is the determinant of a matrix whose entries by column sum to . That is to say, if we define for every , where we define , then with a principal minor of , and the term in square brackets represents or some , according to Lemma 3. Moreover, for every , as a consequence of Lemma 4. Therefore, we conclude
The goal of the following proposition is to define the order of some minors, as required for the proof of Theorem 2.
Proposition 7. Let us suppose that represents a graph with no traps. Moreover, assume a flux from to . Consider the minor of resulting from removing the first line and the th column: Then,
Proof. The demonstration is by induction over the squared matrix size. For the case of a minor with dimension two, we have since there is a flux from to and no traps. We then suppose the validity of this lemma for a minor of dimension up to (induction hypothesis).
If we develop the determinant by the first column, we verify that the minor resulting from striking the first column and the th row satisfies the hypothesis of this lemma after changes of columns, for . Hence, applying the induction hypothesis to these minors, we obtain that they are quantities equal to , where is the struck row index.
Since there are no traps by hypothesis, the minor obtained after omitting the first line and column and the last line and row of has a column which is strictly diagonally dominant. We can then apply Proposition 6 and conclude that it has the order .
Therefore, we conclude that the determinant is the sum of positive quantities of the order :
For the other minors, we obtain a similar result. Indeed, every minor obtained from striking the first row and the th column can be transformed in a matrix of the form , by changes of rows. Therefore, the following assertion holds.
Corollary 1. When the graph related to has no traps, the minor has the order , for every .
Recall that in Assumption 1, we only take into consideration flux traps.
For this reason, we also analyze the determinant of the matrix associated to a system with traps. For instance, with the matrix defined in (C.17), if has a trap, and its determinant have the order .
In general, we can expect that a graph with a trap has a determinant with order . As a consequence, the matrix is ill-conditioned. This happens because a trap implies a block of zeros in the matrix. Indeed, the th column of the matrix system represents the edges whose origin is the metabolite . Then, if is in a trap, for every with out of the trap.
Proposition 8. Let be a matrix defined as in (C.15). If has a trap, then where and are a square submatrices of , which correspond to metabolites in a trap and to metabolites not in a trap, respectively.
Proof. If there is a trap in , the matrix is reducible [24]. Then, after the same number of interchanges of rows than columns, can be transformed in a square block triangular matrix (keeping the dominant diagonal structure): where and are square submatrices that correspond to the metabolites which are not in a trap and the metabolites which are in a trap, respectively. Since the matrix in (C.35) is square block triangular, its determinant is the product of the determinants of the diagonal blocks [25].
Corollary 2. If has a trap, then
Proof. The square block is equal to a singular matrix minus . Then, by Lemma 3, its determinant has order less or equal to .
If is a cofactor of and has order , then the coefficients. can be affected by a factor of order . However, in the following proposition, we prove that when there is a trap , is also a factor of the cofactor if is not in the trap.
Proposition 9. Let be a matrix defined as in (C.15) and a flux from to . If has traps (not reached by ) or flux traps for , then has the form where is a matrix with no traps, is the square block corresponding to metabolites in traps not reached by , corresponds to metabolites which are in flux traps, and C2 corresponds to metabolites that connect the traps to the rest of the network but which do not have a flux from the input. Then,
Furthermore, its minors satisfy with a minor of and
Note. Notice that the block is different from zero if there is a flux from to the flux trap ().
Proof. Since defined in (C.38) is a square block triangular matrix, its determinant is the product of the determinants of the diagonal blocks [25]. Then,
For , the submatrix obtained from deleting the first row and the th column of is also a square block triangular matrix. Then, its determinant is. where is a minor of and is the matrix without its first row. On the other hand, for , the minor is also the determinant of a square block triangular matrix, that is,
On the other hand, we have the minor as a consequence of the block of zeros below . We conclude
Finally, to analyze the minors of , the block of corresponding to the subgraph with no traps, we refer to Proposition 7 and Corollary 1.
D.3. Proof of Theorem 2
Proof of Theorem 2. Since is a nonsingular matrix, where is the transpose matrix of cofactors of [24] (i.e., ).
We then have according to (37)
Then, by definition of ,
If has no traps (i.e., the subnetwork with metabolites has no traps), then as stated by Proposition 6. Moreover, Corollary 1 implies that the cofactors have the order
On the other hand, if has a trap not reached by the flux or a flux trap , as a consequence of Corollary 1 and Propositions 6 and 9,
We conclude that if , for . The same reasoning applies for and the variables of the second subnetwork .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Claudia López Zazueta was supported by the National Council of Science and Technology of Mexico through the program Conacyt-Secretaría de Energía-Sustentabilidad Energética 2015 (Scholarship no. 427010). The authors acknowledge the support of the Investissements d'Avenir Bio-informatique program for the project RESET (ANR-11-BINF-0005), the ANR project Maximic (ANR-17-CE40-0024-01), and the Inria Project Lab Algae in silico.