Framework IV Programme. We thank Hans Westerhoff (Free University ... Bellgardt, K. H., Hopf, N., Luttman, R., and Deckwer, W. D. (1986). A new approach for ...
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137, available online at http:www.idealibrary.com on
Dynamic Optimal Control of Homeostasis: An Integrative System Approach for Modeling of the Central Nitrogen Metabolism in Saccharomyces cerevisiae N. A. W. van Riel,* , 1 M. L. F. Giuseppin, - and C. T. Verrips* , *Department of Molecular Cell Biology, Institute of Biomembranes, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands; and Unilever Research Vlaardingen, Olivier van Noortlaan 120, 3133 AT Vlaardingen, The Netherlands Received April 16, 1999; accepted October 11, 1999
closely related area of cell science in which there is need for new modeling tools is in functional genomics (a.o., the assignment of function to open reading frames in the sequenced genomes of organisms). Among others, Bailey (1998) and Edwards and Palsson (1998) recognize the need for a mathematical framework to structure those data. Mathematical models are tools to structure knowledge. In fundamental sciences, mathematical models are used for analysis of experimental data, theory development, and to design new experiments. In engineering, the knowledge captured in fundamental models is used to manipulate systems or to create new ones. In the current situation in metabolic engineering and functional genomics, there is still a lack of basic understanding of the systems and a lot of (very reliable) metabolic analysis is needed. This is a complex starting point for an engineering approach. Fundamental science (analysis) and engineering should be combined and the models are tools both to increase our understanding and to be used to optimize organisms and processes. The kind of mathematical model to be developed depends on the goal for which it will be used. For advanced metabolic engineering, a (quantitative) understanding of the microorganism as a self-regulating system is necessary and therefore the applied mathematical modeling tools should have the same characteristics. The modeling tools used so far for metabolic engineering do not provide a general framework to include that kind of complexity in a structured way. The simple kinetic models for intracellular metabolism found in literature are completely predefined. The (nonlinear) parametric expressions are often based on biochemical knowledge on in vitro studied enzymes, such as Michaelis Menten kinetics. 2 It is an essential question if in vitro kinetic submodels can be combined in a straightforward way to build a complex in vivo model. Nevertheless these
The theory of dynamic optimal metabolic control (DOMC), as developed by Giuseppin and Van Riel (Metab . Eng., 2000), is applied to model the central nitrogen metabolism (CNM) in Saccharomyces cerevisiae. The CNM represents a typical system encountered in advanced metabolic engineering. The CNM is the source of the cellular amino acids and proteins, including flavors and potentially valuable biomolecules; therefore, it is also of industrial interest. In the DOMC approach the cell is regarded as an optimally controlled system. Given the metabolic genotype, the cell faces a control problem to maintain an optimal flux distribution in a changing environment. The regulation is based on strategies and balances feedback control of homeostasis and feedforward regulation for adaptation. The DOMC approach is an integrative, holistic approach, not based on mechanistic descriptions and (therefore) not biased by the variation present in biochemical and molecular biological data. It is an effective tool to structure the rapidly increasing amount of data on the function of genes and pathways. The DOMC model is used successfully to predict the responses of pulses of ammonia and glutamine to nitrogen-limited continuous cultures of a wild-type strain and a glutamine synthetase-negative mutant. The simulation results are validated with experimental data. 2000 Academic Press
1. INTRODUCTION Metabolic reprogramming or metabolic engineering of yeast (Bailey, 1991; Stephanopoulos and Vallino, 1991) has many future applications in the production of yeast-derived products and may provide useful basic knowledge on modification of metabolism in higher organisms. So far, no complete set of reliable techniques is available to enable (rational) metabolic pathway engineering. Besides molecular biological tools, especially modeling tools must be developed which enable the prediction of the results of the modification of complex metabolic networks. Another
2 Also arbitrarily defined nonlinear mathematical relations are used for the enzymatic reactions, such as the power-law modeling in biochemical system theory (e.g., Savageau, 1969).
1
To whom correspondence should be addressed. Fax:+ +31 10 460 5383. E-mail: natal-van.rielunilever.com.
49
1096-717600 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
Van Riel, Giuseppin, and Verrips
In the new DOMC approach, the optimal flux distribution is used to predict the dynamic network responses to relatively small and short-term perturbations, without the need for data from dynamic experiments. DOMC is focused on a dynamic optimal control of homeostasis (DOCH) of the metabolic pools. For this, the metabolic strategy applied for the static situation in FBA with optimization must be extended with strategies and constraints maintaining an optimal flux distribution during the dynamic response. In the implementation, the numerical integration steps for computer simulation are combined with optimization of the flux distribution, under control of the metabolic regulation function. The DOMC includes three regulatory concepts, which are present in microorganisms and any (more complex) biological system in general. (The terminology is adopted from control engineering.)
approaches, also combined with system sensitivity analysis (metabolic control analysis, e.g., Fell, 1997; Kell and Westerhoff, 1986), have proved to be useful to gain insight in microorganisms. In some cases, this information was successfully used for metabolic engineering. To model microorganisms as autonomous systems, information flows and control need to be included. Kinetic models based on in vivo determined parameters in combination with regulation (at different levels) are an improvement of the kinetic modeling approach (Rizzi et al., 1997; Van Riel et al., 1998). The central nitrogen metabolism (CNM) of Saccharomyces cerevisiae is a complex part of metabolism, embedded in various other pathways and almost not studied quantitatively in defined continuous cultures. For the CNM in yeast and for metabolic engineering problems in general, the level of in vivo kinetic information needed for kinetic modeling is not available. It is extremely laborious to generate all the necessary (high quality) data in dynamic experiments. Furthermore, most of the parameters are valid for limited working conditions such as media, strains, and cultivation conditions. Nevertheless, the setup of such kinetic models is a useful analysis tool to structure the available knowledge and to identify the open questions. In contrast to the reductionistic kinetic models, the new approach of dynamic optimal metabolic control (DOMC), as developed by Giuseppin and Van Riel (2000), focuses on the functionality of the system as a whole and its subsystems (a holistic approach). The cell is regarded as an adaptive, cybernetic system (cybernetics was originally defined as the science of communication and control; Wiener, 1948) with strategies. The strategies are used in the allocation problem which the cells face: balancing maximization of the growth rate and related synthesis of building blocks given the available nutrients in the environment. In engineering terms, this is a multidimensional nonlinear optimal control problem. Based on the principle of flux balance analysis with optimization (FBA, e.g., Bonarius et al., 1998; Fell and Small, 1986; Savinell and Palsson, 1992), a DOMC model is able to describepredict the dynamic behavior, without the need of detailed mechanistic information. In FBA with optimization the problem of underdetermined mass balances is solved by optimization of the static, steady-state flux distribution, where a metabolic strategy forms the cost function. One paper has been reported (Varma and Palsson, 1994) in which FBA combined with a macroscopic model was used to describe the (slowly) changing extracellular substrate and product concentrations during batch and fedbatch fermentation. Edwards and Palsson (1998) already mentioned that FBA also has the potential to play an important role in dealing with the emerging genome information for metabolic analysis and engineering.
(1) Feedback control is present to maintain homeostasis of the metabolites. Homeostasis is an important concept in biology to explain processes like hormonal balance, maintenance of temperature, etc. From a control engineering point of view, the feedback loop is a tracking controller, which reduces the effect of environmental noise. (2) Biological systems deal with growth, change, and emergence and react to changes in their environment, i.e., they are adaptive. Biological systems include preprogrammed responses (feedforward regulation), which cause adaptation, 3 and are the driving forces behind change. From a system analytical point of view, feedback control enables the organism to maintain itself in its changing environment, and feedforward regulation gives it a chance to adapt to environmental changes. (3) A metabolic system consists of thousands of different components operating in multiple compartments and at time scales between 10 &6 and 10 4 s (for phosphorylation dephosphorylation reactions and biomass growth, respectively). A cell is not a well-stirred reactor; many reactions occur through vectorial processes. Such a system only can build up to a stable, adaptive organism if several hierarchical levels of regulation are present. Therefore, also a hierarchy in the regulation in the DOMC model is included (although a simulation model never can cover a time window of 10 decades). Control of slow processes generally takes place at a higher level than that of fast processes. In nature, the possibilities for control in a cell are constrained. Control of homeostasis of the metabolite pools is dependent on the environment (availability of nutrients, 3 Adaptation is often called ``morphogenesis'' in cybernetic terminology, which should not be confused with the microbiological meaning.
50
Optimal Control of Central Nitrogen Metabolism
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
the ratio of the flux from glutamine and glutamate toward protein synthesis. ! and are the ratios of the part of glutamine and glutamate, respectively, which acts as amino group donor for protein synthesis versus the part of glutamineglutamate which is directly built into proteins. Metabolic flux analysis (MFA) and the stoichiometry used by Giuseppin and Van Riel (2000) result in :=0.22, !=0.25, and =0.77.
presence of toxins, etc.). Constraints for adaptability are set by evolutionary history, i.e., the metabolic genotype. The application field of our work is focused on the modification of the amino acid level in S. cerevisiae and the primary metabolism of which amino acids are derived. To use a molecule as nitrogen source, yeast cells must convert this molecule into glutamate and glutamine (Cooper, 1982; Magasanik, 1992; Ter Schure et al., 1999). The glutamate and glutamine nodes are the center of the nitrogen metabolism (CNM) and the source of the cellular building blocks. From these two amino acids all other nitrogen-containing compounds in the cell can be produced (Magasanik, 1992). The CNM consists of a highly regulated, compartmentalized network which has been studied relatively little. Knowledge and (quantitative) data are scarce and ambiguous and fundamental understanding is lacking. The combination of a complex network and limited knowledge represents a typical system encountered in ``advanced'' metabolic engineering. The same situation will be encountered for most of the pathways of other (micro)organisms that could be of future interest for metabolic engineering. The CNM of S. cerevisiae is a suitable model system for the development of mathematical modeling techniques for metabolic analysis and engineering. A second reason to use the CNM as a model system is that nitrogen metabolism is of industrial interest because many flavors and valuable functional biomolecules, such as enzymes, are derived from amino acids. It becomes more and more clear that modification of the primary metabolic pathways, sourcing the pathways in which the products of interest are formed, often is essential to really use microorganisms as efficient cell factories. In this work the theory of DOMC is applied to model the CNM in yeast. It is shown how DOMC can be used to develop predictive mathematical models with application in metabolic analysis and engineering. This shows the value of this modeling tool and gives more insight in, and understanding of, the theory of DOMC.
2.2. Experimental Setup The quantitative data (high quality) necessary for DOMC are steady-state network fluxes and intracellular metabolite concentrations. The steady-state data for the CNM are taken from Ter Schure et al. (1998) for glutamine-limited continuous cultures (D=0.1 h &1 ). The data of Ter Schure et al. (1998) of ammonia and glutamine pulses to continuous cultures of both 71278b wild-type and a gln1-37 mutant strain are used to validate the DOMC model. 2.3. Mathematical Techniques Dynamic Optimal Metabolic Control In general, a (nonlinear) system can be described by a set of nonlinear differential equations, x* =f (x, u ),
(1)
of the states x and with system inputs u. The biochemical reaction network is described with mass balances for the intracellular metabolites x in [mmol } gX &1 ] and the extracellular components x ex [mmol } liter &1 ] with a stoichiometric matrix related to the (intracellular) reactions E in and a matrix containing the transport stoichiometry E ex . The stoichiometric network combined with mass balances forms the linear biochemical subsystem. Of this subsystem, certain (biological) inputs and outputs are known in steady state (measured). The biological system inputs are the substrate uptake fluxes r ex [mmol } gX &1 } h &1 ]. In the DOMC approach intracellular metabolism is regarded as an (optimally) controlled system (Fig. 2). Metabolic regulation is a separate subsystem and is incorporated in the metabolic control function F. The metabolic regulation is based on a hierarchical combination of metabolic strategies and is a nonlinear subsystem. The exact form of F will be specified later. The dynamic flux distribution in the metabolic network is directly determined by the metabolic control function, i.e., the rates are the outputs of the controller. No a priori knowledge or assumptions about specific dynamic relations between the system nodes are included in the model (i.e., any parametric expression to
2. SYSTEM, MATERIALS, AND METHODS 2.1. The Central Nitrogen Metabolism in S. cerevisiae The central nitrogen metabolism of S. cerevisiae, which is used as an example system to demonstrate the use of the DOMC approach, is described in general by, e.g., Ter Schure et al. (1998) and more specific for modeling by Van Riel et al. (1998). For the model, the actual amino acid and protein synthesis are lumped (Fig. 1). The lumped amino acid and protein synthesis is described with one flux, r prot , and three parameters, :, !, and . (The nomenclature and abbreviations used are listed at the end of the article.) : is 51
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
Van Riel, Giuseppin, and Verrips
FIG. 1. Scheme of the most simplified model structure of the CNM in S. cerevisiae. :KG, :-ketoglutarate; glu, glutamate; gln, glutamine; NH 4 , ammonia; NADPHGDH, NADPH-dependent glutamate dehydrogenase; NADGDH, NAD-dependent glutamate dehydrogenase; GS, glutamine synthetase; GDA, glutamine deaminase; GOGAT, glutamate synthase; NH 2 , the amino groups which are transferred from glutamine and glutamate toward protein synthesis; :; ratio of the flux from glutamine and glutamate toward protein synthesis; ! and , ratios of the part of glutamine and glutamate, respectively, which acts as amino group donor for protein synthesis versus the part of glutamineglutamate which is directly built into proteins.
model node interactions is left out). In principle, the metabolic regulation is allowed to modify all network fluxes: the uptake rates (biological system inputs) and the intracellular reaction rates r in [mmol } gX &1 } h &1 ]. The intracellular rates are additional controlled inputs of the biochemical subsystem. The total controlled system input u(F) of the biochemical reaction network is a combination r and r . The system can be written as of ex in x* =f (x, F)=E in r in (F)&+x +E ex r ex (F).
As indicated in the Introduction, regulation in DOMC models is assumed to consist of feedback control maintaining an optimal flux distribution in the metabolic network after a perturbation and feedforward regulation, stimulating adaptation to changes in the environment. For the DOMC approach, such a combination of conflicting control strategies is required. The balance between the strategies determines the steady state (therefore, DOMC can be called a ``self-tuning'' model). Plausible cell strategies of an organism like yeast can be quite easily deduced from experimental data and should then be translated into mathematical expressions for the metabolic control function. The development of mathematical
(2)
The term &+x describes the dilution of the component pools through growth of the cell.
FIG. 2. Schematic overview of DOMC. The total modelsystem consists of a stoichiometric mass balance model and a metabolic control F regulating the fluxes. The controlled system inputs u of the biochemical reaction network are a combination of r ex and rin . The inputs of the total system are the extracellular concentrations x ex and the steady-state concentrations x 0 . 52
File: 741J 013704 . By:XX . Date:22:12:99 . Time:15:36 LOP8M. V8.B. Page 01:01 Codes: 3809 Signs: 2461 . Length: 57 pic 1 pts, 240 mm
Optimal Control of Central Nitrogen Metabolism
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
strategies will involve some trial and error. Various strategies have been formulated and applied to metabolic problems (Giuseppin and Van Riel, 2000) and strategies can be extended to specific observed behavior. In practice, microorganisms display a combination of strategies depending on the environment and cellular status. Only the dominant strategies are likely to be observed experimentally. The time constants of the processes in the cell put constraints on the controllability of these processes during propagation. Only those metabolic processes that have a time constant in the range of the propagation process can be manipulated and controlled with ease. The constraints, as defined by Giuseppin and Van Riel (2000), for the maximal allowed changes in time for each component pool (i.e., constraints on the absolute value of the state derivatives) are grouped in a vector G, which is constrained to be G 0 during operation of the metabolic regulation. In the implementation, the numerical integration steps applied to the mass balances for computer simulation are combined with an optimization of the flux distribution by the metabolic control. For every step k, an optimal flux distribution r^ k is calculated, based on the cellular control strategies and given the cellular environment x ex . During each simulation interval the rates are kept constant, resulting in a staircase-like pattern for the profiles of the rates vs time. The optimal fluxes r^ k for simulation interval k result from minimization of the metabolic control function Fk , r^ k =arg min Fk , rlb r^ rub
TABLE 1 Computer Algorithm Parameters for the Matlab Routines Simulation parameters Relative error Minimum step size t $ Maximum step size t $ Total simulation time
10 &3 [] 10 &2 [h] 10 &2 [h] 2 [h]
Optimization parameters
QLP
Termination for r (the worst-case precision required of the variables x ) Termination for F (precision required of the objective function F at the solution) Termination for G (a measure of the worst-case constraint violation that is acceptable) Optimization (time) interval t $ Selection algorithm Mutation algorithm Recombination algorithm Generation gap Number of chromosomes
GA
10 &4 [] 10 &4 [] 10 &4 [] 10 &4 [] 10 &4 []
10 &2 [h]
10 &2 [h] Stochastic Real Discrete 1.0 [] 10 []
computer algorithm parameters as used during simulation are given in Table 1. The optimization of the flux distribution for each simulation interval is computer-time-consuming and the choice of a proper, constrained optimization routine is important to speed up the simulation. In the quadratic linear programming (QLP) minimization algorithm, constraints can be explicitly included. The search space of the optimization, which is formed by the metabolic control function F, is irregular (i.e., has a discontinuous gradient). Especially at the moment when the different competitive strategies arrive at the same value (become equally important), optimization algorithms based on the gradient of the search space (such as QLP) converge slowly. The better global performance of optimization techniques such as simulated annealing (SA) and genetic algorithms (GA) is advantageous, especially for relatively low-dimensional optimization problems (such as for the model of the CNM). However, SAs and GAs cannot deal with explicit constraints. Then constraints are included as penalty functions in the cost function. For search algorithms which make use of the gradient of the search landscape, it is important that a penalty function does not cause discontinuities in the landscape, i.e., a smooth function is used:
(3)
with a bounded search space and under constraints G k 0. The hat 7 denotes that the calculated rates are best estimates. (In optimization algorithms F is usually called the cost function.) Overall, the estimated network fluxes for the time interval k are a function of the environment x ex , the steady-state reference x 0 of the network, the applied network strategies constraints included in G. in F, and the Optimization Algorithm The size of the optimization time interval t $ (equal to the integration step size) is an important time constant of the resulting model. It determines the smallest time constant of the model and should be in agreement with the smallest (relevant) system time constant. This is a common notion in system analysis and engineering and was introduced in biochemistry with the modal analysis approach (Palsson and Joshi, 1987). A fixed optimization and integration step of 0.01 h has been used. Hereby the model at least includes dynamic effects slower than 3 min. The values of the
F*= 53
F
{F+10
6
7G
2 i
\G 0 \G >0.
(4)
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
Van Riel, Giuseppin, and Verrips
All model approaches, including optimization, were implemented in MATLAB (versions 4.2c and 5.2, The Mathworks Inc., Natick, MA). For simulation of the models, Gear's method with a fixed step integration was used in MATLAB 4.2c and ode15s in MATLAB 5.2.
TABLE 2 Steady-State Data from Glutamine-Limited Continuous Culture (D =0.1 h &1 ) of 71278b (Ter Schure et al., 1998) Value Glucose feed (glc feed )
3. RESULTS
Glutamine feed (gln feed )
3.1. Flux Analysis
Biomass (X) Yield on Glucose (Y Sc X)
The steady state of the metabolic network is the reference situation for the DOMC. Based on the measured fluxes, the unknown steady-state fluxes in the central nitrogen metabolism are calculated by metabolic flux analysis. The scheme of the CNM as considered here can be found in Fig. 1. The number of intracellular compounds (n) is 4 and the number of reaction rates considered (m) is 5. Based on the steady-state data from glutamine-limited continuous cultures (D=0.1 h &1 ) of 71278b in Table 2, the transport fluxes r glc , r glu , r gln , and r NH4 [mmol } gX &1 } h &1 ] can be calculated: r xi =
D(x i, feed &x i, ex ) , X
Yield on glutamine (Y Sn X ) Ex. ammonia (NH + 4ex. ) Ex. glutamate (glu ex. ) Ex. glutamine (gln ex. ) Ex. glucose (glc ex. ) In. glutamine (gln in. ) In. glutamate (glu in. ) Total in. :-ketoglutarate (:KG in. )
0 1 0 0
0 0 . 1 0
_ &
20 g } liter &1 111.1 mM 3 g } liter &1 20.5 mM 9.2 g } liter &1 0.46 gX } gS &1 0.0828 gX } mmolS &1 3.07 gX } gS &1 0.15 gX } mmolS &1 0,
+
i=1, ..., 4.
(11)
To prevent the summation of the deviation 72 i from becoming too large, an anti-wind-up mechanism for the PI 56
Optimal Control of Central Nitrogen Metabolism
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
TABLE 5 Parameters of DOMC Model of CNM in S. cerevisiae { glu [h &1 ] { gln [h &1 ] { NH4 [h &1 ] { :KG [h &1 ] # glu # gln # NH4 # :KG 2k
0.1472 0.0448 0.1309 0.1003 2 2 2 2 25
= glu [mmol } gX &1 ] = gln [mmol } gX &1 ] = NH4 [mmol } gX &1 ] = :KG [mmol } gX &1 ] ; glu ; gln ; NH4 ; :KG h homeostasis
10 &4 10 &4 10 &4 10 &4 0.1 0.1 0.1 0.1 10 4
P glu P gln P NH4 P :KG * glu * gln * NH4 * :KG h uptake
1 0.07 0.7 0.7 5 5 5 5 10 3
I glu I gln I NH4 I :KG
0.4 0.02 0.2 0.2
where E + is the stoichiometric matrix related to production, E & contains the stoichiometry for consumption reactions, and r ex + and r ex & are the uptake and secretion rates, respectively. The proportional and integration control parameters used can be found in Table 5. Since in the control function the product x* i 2 i w i f (2 i ) is used, with both w i and 2 i nonlinear functions of the state x i , the resulting control is nonlinear. The principle of the control of homeostasis is indicated in Fig. 3. The total control function for homeostasis of the metabolic pools is a summation of the (positive) terms for the individual pools:
controller is used. Instead of the approach of Giuseppin and Van Riel (2000), it is implemented as a moving time window 2 k which only takes into account the deviations at the 2 k previous time intervals. The values of the proportional and integration control parameters P i and I i depend on the pool time constant { i =x 0, i , 0, i . To calculate the time constants, either the summed formation or summed consumption reactions for each pool in steady state, , 0, i , need to be calculated (summed formation is equal to summed consumption in steady state):
4
8 0 =E +r in, 0 +r ex + or 8 0 =&(E & r in, 0 + r ex & ),
F homeostasis = : f homeostasis, i .
(13)
i=1
The objectives related to control of homeostasis and maximization of the substrate uptake (adaptation) are
(12)
FIG. 3. Control of homeostasis and constraints for simulation of interval k to k+1. (Control) When the current derivative is x* i 0. Otherwise, the controller is ``off.'' (Constraints) The maximal change of the pool concentration in a certain time { i (the time constant) is limited. ; i is the minimal level of the steady state of one compound and * i is the maximal level. Factor # i is the maximal deviation from the MFA flux for one compound. 57
File: 741J 013709 . By:XX . Date:12:01:00 . Time:15:48 LOP8M. V8.B. Page 01:01 Codes: 4238 Signs: 2474 . Length: 57 pic 1 pts, 240 mm
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
Van Riel, Giuseppin, and Verrips
58
File: 741J 013710 . By:XX . Date:22:12:99 . Time:15:36 LOP8M. V8.B. Page 01:01 Codes: 666 Signs: 96 . Length: 57 pic 1 pts, 240 mm
Optimal Control of Central Nitrogen Metabolism
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
FIG. 4. (A) Simulation results of the DOMC model (full lines), simulation results of the kinetic model (Van Riel et al., 1998) (dashed lines), and experimental data (Ter Schure et al., 1998) of a 40 mM ammonia pulse to the glutamine-limited continuous culture (D=0.1 h &1 ) of 71278b. The error bars indicate the standard deviations in the data. (B) Representation of the control functions for homeostasis (m) and adaptation (+) after a 40 mM ammonia pulse to the wild type. (C) The intracellular reaction rates and the relevant uptake rates after the ammonia pulse to the wild type culture growing on glutamine. The spikes in the fluxes result from the numerical techniques for optimization of the distribution.
should be higher than the weight for substrate uptake (10 4 vs 10 3, Table 5). When substrate uptake would be dominant in the model, the resulting higher uptake rate (which, besides from the threshold value x ex, 0 , is independent of the substrate concentration in the DOMC model) would also consume the residual limiting substrate, leading to complete depletion.
competitive. A hierarchy is included in the regulation. The total nonlinear cost function Fk for simulation interval k consists of a weighted sum of the relevant terms F i, k for control (Fk = h i F i, k ). It depends on the system under investigation, whose objectives are included and with what weight h i , i.e., what the hierarchy is. To describe a steady state, the weight of control of homeostasis of the metabolic pools 59
File: 741J 013711 . By:XX . Date:22:12:99 . Time:15:37 LOP8M. V8.B. Page 01:01 Codes: 2470 Signs: 1593 . Length: 57 pic 1 pts, 240 mm
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
Van Riel, Giuseppin, and Verrips
Metabolic Constraints
ammonia either. From measurements of the pools of all amino acids after pulses (data to be published) it is clear that the assumption of a constant flux toward protein synthesis and a constant ratio of the amino acid pools cannot be completely justified. To correct the model, to some extent, for this knowledge an extra flux is added which withdraws glutamate, just like in the kinetic model of the CNM (Van Riel et al., 1998). The results of a 40 mM ammonia pulse to the wild-type strain can be found in Fig. 4. The derivatives for the first optimization step of a steady-state simulation are not zero. Since the dilution through growth was ignored in metabolic flux analysis, but included in the dynamic model, the derivatives are equal to &+ } x i, 0 . The initial x* i 0 and I i >0. (The rate of
(Fig. 4C). This is not possible in the gln1-37 mutant and therefore the responses are different. 4. DISCUSSION The NADPH-dependent glutamate dehydrogenase is the major anabolic enzyme in S. cerevisiae. The glutamine synthetaseGOGAT pathway is believed to be of minor importance (Roon et al., 1974; Holmes et al., 1991; Ter Schure et al., 1998), mainly because GOGAT null mutants grow as well as a wild-type strain in shake flasks. Parallel pathways in microorganisms often operate under different physiological conditions or in different compartments, hereby giving the cell a flexible means to control efficiency and growth. The results of Helling (1998) suggest that NADPHGDH and GSGOGAT, the two pathways for glutamate synthesis in E. coli, give the cell the possibility to deal with rapid fluctuations in (certain) metabolites. Functional analysis with dynamic models as presented here and by Van Riel et al. (1998) indicates that GOGAT could have the same physiological role in S. cerevisiae under nitrogen-limited conditions with fast changes in the availability of nitrogen. In the model results, the flux through glutamate synthase (GOGAT) is 0 in steady state, but during the dynamic responses this flux increased significantly (to a maximum of 0.25 mmol } gX &1 } h &1; Fig. 4C). The conclusion of a specific function of GOGAT, not revealed by classical analysis, is supported by Valenzuela et al. (1998). The operation of GOGAT andor glutamine deaminase under dynamic conditions will be experimentally verified by comparing a wild-type strain and a GOGAT negative mutant, using 15N NMR. In the central nitrogen metabolism of S. cerevisiae, the (in vivo) interaction between cytosol and mitochondria is unclear. Mathematical modeling can be a very useful tool to test the feasibility of hypotheses with respect to, for example, the model structure. The CNM of yeast contains different compartments because for growth on ammonia, C5 carbon skeletons for the amino acids are supplied by the TCA cycle through the mitochondrial :-ketoglutarate pool, whereas both the NAD- and NADPHdependent glutamate dehydrogenases of S. cerevisiae are cytosolic (Hollenberg et al., 1970; Perlman and Mahler, 1970). The structure has a large influence on the model results and therefore different options need to be studied and the ``final'' choice needs to be carefully motivated and experimentally verified. In the kinetic model of Van Riel et al. (1998) it was necessary to include hypothetical interactions between the compartments. With the DOMC approach, the model without intracellular compartmentation (as depicted in Fig. 1) and the related assumptions has enough degrees of freedom to allow the metabolic control to
TABLE 6 The Relative Efects on the Area under the (x, t ) Plots for a 10 0 Decrease in the Parameters Parameter
Sensitivity [0]
vlb uptake glutamine vlb r GDA vlb r prot vlb r TCA vub uptake glutamine vub r GDH vub r GDA vub r GOGAT vub r prot { glu { :KG td INH4 I:KG =x * #
177 33 331 181 373 &14 11 53 174 62 &12 159 &13 &13 279 119 32
Note. Sensitivities below \100 are omitted. 64
Optimal Control of Central Nitrogen Metabolism
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
convergence of the QLP optimization algorithm is also parameter independent. For the simulation of the steady state the solution is obtained within eight evaluationsiterations.) As a simple and representative measure for the dynamic response after a substrate pulse, the area under the (x, t) plots of the internal states is calculated for parameter changes \$% of the nominal values % 0. The relative effects for a 10 0 decrease in the parameters are shown in Table 6. Changes smaller than 10 0 (the alteration of the parameter is attenuated) have been omitted from the table. When the actual simulation results are inspected (e.g., Fig. 8), it is clear that (only) the largest sensitivities correspond to qualitative different responses. The small sensitivities result in responses which can hardly be discriminated from the nominal response (results not shown). The relative change in the summed areas under the (x, t) plots is a good (qualitative) measure for the parameter sensitivity. The parameters with a high sensitivity show bifurcation in the range in which they have been modified. It is remarkable that all parameter changes which have been studied and resulted in bifurcation generally yielded the same different response pattern, indicating the model is operating in a different mode. These bifurcations will be analyzed in a future paper, where the model properties of the DOMC approach are compared with those of a kinetic model. The simulation results of the dynamic optimal metabolic control model of the CNM in S. cerevisiae, for different pulses to both a wild-type and a mutant strain, are acceptable for our purpose. The most dominant dynamics during the first 2 h after pulses of good nitrogen sources to nitrogen-limited continuous cultures are the downregulation of the uptake and of most of the enzymes of the CNM. This phenomenon is known as nitrogen catabolic repression (Cooper, 1982). Also in the kinetic model, this metabolic
regulation was most important. This reduces the importance of the correct description of the in vivo enzyme kinetics for engineering purposes. No explicit kinetic formulas have been used and no other a priori knowledge or assumptions about the dynamic relations between the system nodes were included in the DOMC model. The dynamics depend on the optimization solution found for the flux distribution at each interval. This can be a major advantage because the data sets available for metabolic modeling are often limited and inconsistent from experiments not designed for that purpose. Also for the various strains and conditions the responses are different. For the DOMC model only steady-state data (both fluxes and concentrations) have been used, which can be easier and better generated with a high quality than data of in vivo dynamics. The DOMC is a flexible approach in which a certain observation or idea of the experimentator for the biological system can be incorporated and tested as a thought experiment. Besides an advantage, this is a risk of the DOMC approach. When the approach is not carefully used and validated, very unrealistic results can be obtained. However, validation is essential for all models. The different optimization algorithms as discussed in Section 2.3 yielded the same model responses (results not shown). The model with QLP is much faster than the implementation with a GA (which is a ``global'' search algorithm). For the optimal control problem in the CNM a nonlinear, local search algorithm, such as QLP, is sufficient. This indicates that, given the metabolic network and the measured uptake fluxes, the estimated steady state fluxes are globally optimal and are the best reference flux distribution to start dynamic simulations. The (slowly) increasing number of positive results with a cybernetic approach indicates its validity as a tool, especially for engineering purposes. The acceptance and use of flux balance analysis with linear programming is increasing. The cybernetic principle was also successfully applied for dynamic models [initiated by Ramkrishna and co-workers in the mid 1980s, e.g., Kompala et al. (1984)], but this is not yet followed by many others. The cybernetic approach of Varner et al. (1998) is a significant extension of the original framework of Ramkrishna. In 1986 Bellgardt introduced the metabolic regulator concept (Bellgardt et al., 1986), but no use of this approach has been reported. Recently, Van den Berg et al. (1998) also used the principle of optimal allocation between nutrient uptake and growth in a macroscopic model describing growth of a microbial trichome. Most likely it is impossible to really prove that a cybernetic approach of biological systems is correct. However, it is a plausible approach. In nature, microorganisms have evolved to survive under a wide range of external conditions. The various types of microorganisms in ecological
FIG. 8. (A) Nominal response of intracellular glutamate after a 15 mM glutamine pulse to a glutamine-limited continuous culture (D=0.1 h &1 ) of 7 1278b (see Fig. 5). Data and corresponding error bars have been included. (B) Response after a 10 0 decrease in the maximum glutamine uptake (vub uptake glutamine). 65
File: 741J 013717 . By:XX . Date:13:01:00 . Time:08:23 LOP8M. V8.B. Page 01:01 Codes: 6314 Signs: 5321 . Length: 57 pic 1 pts, 240 mm
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
Van Riel, Giuseppin, and Verrips
niches have developed strategies to survive and to compete adequately. They can adapt their metabolism within their genetic capabilities (genotype) with a given response time. It is probably much more efficient to focus on these functional system characteristics to understand and quantitatively model cellular metabolism, instead of a reductionistic approach. Metabolic engineering is usually aimed at forcing the cells to a steady state in a different, more profitable operating point, within the evolutionary boundaries. However, efforts to redirect metabolic flux using recombinant DNA methods often fail. From a cybernetic perspective this is not strange. The specific goal of metabolic engineering is not the optimum outcome from the microorganism's ``point of view.'' Accordingly, the microorganism will resist through its internal control machinery. The control of the microorganism is sufficiently redundant andor robust to maintain the original homeostasis in the same operating point (the same metabolic phenotype results). These robustness and redundancy in metabolism also cause problems in genome projects for gene assignment, because the physiological function often cannot be straightforwardly revealed. On the other hand, the possibilities for control in a cell are clearly constrained. Some knock-out mutations are lethal, i.e., the modified cells are not able to adapt to the new genotype, within the constraints set by evolution. The DOMC framework is a heuristic approach not suited to study metabolism in mechanistic detail. The DOMC models describe the global behavior of an optimally controlled biochemical network. Models such as these should not be considered as definitive descriptions of metabolic networks, but rather as an approach that allows to understand the capabilities of complex systems. The approach is (especially) scientific at the system level, in the sense that it gives fundamental insight in the emergent properties and functionality of complex metabolic networks. The DOMC principle can very well be used in a modular andor hybrid modeling approach in which various modeling techniques are combined. Structurally, the approach can also be extended to subcellular modules such as mitochondria, vacuoles, and peroxisomes. When it is necessary to study a particular (small) part of metabolism in mechanistic detail, the DOMC approach can be used to model (approximate) its intracellular surrounding. This makes it a valuable tool to be combined with kinetic modeling, in which necessary assumptions on the intracellular surrounding usually severely limit the quality of the model (e.g., Van Riel et al., 1998). Apart from modeling the substrate uptake r ex of the CNM as free fluxes determined by the metabolic control, the observed uptake kinetics have also been modeled with a MichaelisMenten relation for the transporters. Then the regulation of the transporters must be
modeled explicitly (according to Van Riel et al., 1998). This hybrid model yielded the same responses as shown with a strictly DOMC approach. This shows that the DOMC model not only yields acceptable responses when compared to experimental data, but also is realistic with respect to a mechanistic (sub)model of the underlying system.
5. CONCLUSIONS Dynamic optimal metabolic control is an integrative system analytical approach yielding holistic models. A gene and its product are modeled in the context of their physiological function. Both for metabolic engineering and functional genomics there is the need for mathematical modeling tools with such characteristics. Our motivation to develop the new approach matches well to the personal commentary of Bailey (1998) on the subject of mathematical modeling in biochemical engineering. The DOMC framework is a flexible approach, suitable to structure knowledge and test related hypotheses based on the rapidly increasing amount of ``semiquantitative'' information from functional genomics and ``global metabolite pool analysis'' techniques (e.g., Tweedale et al., 1998). Bioinformatics with analysis of the hierarchical genetics-to-physiology relationship will lead to the discovery of biological ``rules'' and ``principles'' upon which design of biological systems will rely (Edwards and Palsson, 1998). This kind of information can be extracted with an approach such as DOMC and, in return, it will also improve the quality of the DOMC models. The major advantage of DOMC above FBA with optimization is that it can analyze dynamic responses, which can be very important to reveal the function of certain gene products, such as for example glutamate synthase (GOGAT) in S. cerevisiae.
APPENDIX Abbreviations and Nomenclature CNM DOMC DOCH MFA FBA GA LP NMR QLP SA 66
Central nitrogen metabolism Dynamic optimal metabolic control Dynamic optimal control of homeostasis Metabolic flux analysis Flux balance analysis Genetic algorithm Linear programming Nuclear magnetic resonance Quadratic linear programming Simulated annealing
Optimal Control of Central Nitrogen Metabolism
Model components glc :KG glu gln NADPHGDH NADGDH GS GDA GOGAT NH 2 X
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
Glucose :-Ketoglutarate Glutamate Glutamine The NADPH-dependent glutamate dehydrogenase The NAD-dependent glutamate dehydrogenase Glutamine synthetase Glutamine deaminase Glutamate amide-2-oxoglutarate amino transferaseglutamate synthase Amino group Biomass concentration [g } liter &1 ]
7
h t$ k, N 2k 2i wi = x, i
Subscripts in Intracellular ex Extracellular lb Lower bounds obs Observed ub Upper bounds prot Protein S Substrate C Carbon N Nitrogen 0 Initial (steady state) + Formation reaction & Consumption reaction
Pi Ii {i ,i ;i *i #i
Ratio of the part of glutamate which acts as amino group donor for protein synthesis versus the part of glutamate which is built into proteins Estimated Weights for the controlcost function (determine hierarchy) Simulation and optimization time interval [h] kth and N th simulation and optimization time interval Time window Relative deviation from steady state [mmol } gX &1 ] Normalized weight of the deviation from steady state The smallest concentration which is relevant for the system [mmol } gX &1 ] Proportional control parameter Integration control parameter Time constant of the metabolic pool [h] Summed formation or summed consumption reactions for a pool [mmol } gX &1 } h &1 ] Minimal level of the steady state of a compound Maximal level of the steady state of a compound Maximal deviation from MFA flux for a compound ACKNOWLEDGMENTS
This work was part of EC Cell Factory Project ``From Gene to Product in Yeast: A Quantitative Approach,'' sponsored by the DGXII Framework IV Programme. We thank Hans Westerhoff (Free University of Amsterdam, The Netherlands) for his useful suggestions when preparing the manuscript.
Process Parameters Biomass yield on a substrate S [gX } gS &1 ] Y SX D Dilution rate [h &1 ] + Specific growth rate [h &1 ] + max Maximal specific growth rate [h &1 ]
REFERENCES
Mathematics x State vector of metabolic pool concentrations [mmol } gX &1 ] u System inputs [mmol } gX &1 } h &1 ] r Rate vector [mmol } gX &1 } h &1 ] % Parameter vector E Stoichiometric matrix m Number of reaction rates n Number of intracellular compounds F Costcontrol function f Term of cost function F G Vector with constraints : Ratio of the flux from glutamine and glutamate toward protein synthesis ! Ratio of the part of glutamine which acts as amino group donor for protein synthesis versus the part of glutamine which is built into proteins
Bailey, J. E. (1991). Towards a science of metabolic engineering. Science 252, 16681675. Bailey, J. E. (1998). Mathematical modeling and analysis in biochemical engineering: Past accomplishments and future opportunities. Biotechnol. Prog. 14, 820. Bellgardt, K. H., Hopf, N., Luttman, R., and Deckwer, W. D. (1986). A new approach for development of structured growth models. Proc. Comput. Appl. Ferment. Technol. Modelling Control Biotechnol. Processes, 7992. Bonarius, H. P. J., Timmerarends, B., de Gooijer, C. D., and Tramper, J. (1998). Metabolic-balancing techniques vs. 13C tracer experiments to determine metabolic fluxes in hybridoma cells. Biotechnol. Bioeng. 58, 258262. Cooper, T. G. (1982). Nitrogen metabolism in Saccharomyces cerevisiae. In ``The Molecular Biology of the Yeast Saccharomyces cerevisiae'' (J. N. Strathern et al., Eds.), pp. 3999, Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY. Edwards, J. S., and Palsson, B. O. (1998). How will bioinformatics influence metabolic engineering? Biotechnol. Bioeng. 58, 162169. 67
Metabolic Engineering 2, 4968 (2000) Article ID mben.1999.0137
Van Riel, Giuseppin, and Verrips
Fell, D. A., and Small, J. R. (1986). Fat synthesis in adipose tissue. An examination of stoichiometric constraints. Biochem. J. 238, 781786. Fell, D. (1997). ``Understanding the Control of Metabolism,'' Portland Press, London and Miami. Giuseppin, M. L. F., and Van Riel, N. A. W. (2000). Metabolic modeling of Saccharomyces cerevisiae using the optimal control of homeostasis: A cybernetic model definition. Metab. Eng. 2, 1433. Helling, R. B. (1998). Pathway choice in glutamate synthesis in Escherichia coli. J. Bacteriol. 180, 45714575. Hollenberg, C. P., Riks, W. F., and Borst, P. (1970). The glutamate dehydrogenases of yeast: Extra-mitochondrial enzymes. Biochim. Biophys. Acta 201, 1319. Holmes, A. R., Mcaughton, G. S., More, R. D., and Sepherd, M. G. (1991). Ammonium assimilation by Candida albicans and other yeasts: A 13N isotope study. Can. J. Microbiol. 37, 226232. Kell, D. B., and Westerhoff, H. V. (1986). Metabolic control theory: Its role in microbiology and biotechnology. FEMS Microbiol. Rev. 39, 305320. Kompala, D. S., Ramkrishna, D., and Tsao, G. T. (1984). Cybernetic modeling of microbial growth on multiple substrates. Biotechnol. Bioeng. 26, 12721281. Magasanik, B. (1992). Regulation of nitrogen utilization. In ``The Molecular and Cellular Biology of the Yeast Saccharomyces cerevisiae: Gene Expression'' (E. W. Jones et al., Eds.), pp. 283317, Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY. Palsson, B. O., and Joshi, A. (1987). On the dynamic order of structured Escherichia coli growth models. Biotechnol. Bioeng. 29, 789792. Perlman, P. S., and Mahler, H. R. (1970). Intracellular localization of enzymes in yeast. Arch. Biochem. Biophys. 136, 245259. Rizzi, M., Baltes, M., Theobald, U., and Reuss, M. (1997). In vivo analysis of metabolic dynamics in Saccharomyces cerevisiae: II. Mathematical model. Biotechnol. Bioeng. 55, 592608. Roon, R. J., Even, H. L., and Larimore, F. (1974). Glutamate synthase: properties of the reduced nicotinamide adenine dinucleotidedependent enzyme from Saccharomyces cerevisiae. J. Bacteriol. 118, 8995. Savageau, M. A. (1969). Biochemical systems analysis I. Some mathematical properties of the rate law for the component enzymatic reactions. J. Theor. Biol. 25, 365369. Savinell, J. M., and Palsson, B. O. (1992). Network analysis of intermediary metabolism using linear optimisation. I. Development of mathematical formalism. J. Theor. Biol. 154, 421454.
ter Schure, E. G., Sillje, H. H. W., Vermeulen, E. E., Kalhorn, J., Verkleij, A. J., Boonstra, J., and Verrips, C. T. (1998). Repression of nitrogen catabolic genes by ammonia and glutamine in nitrogen-limited continuous cultures of Saccharomyces cerevisiae. Microbiology 144, 14511462. ter Schure, E. G., van Riel, N. A. W., and Verrips, C. T. (1999). The role of ammonia metabolism for nitrogen catabolite repression in Saccharomyces cerevisiae. FEMS Microbiol. Rev., in press. Soberon, M., and Gonzalez, A. (1987). Physiological role of glutaminase activity in Saccharomyces cerevisiae. J. Gen. Microbiol. 133, 18. Stephanopoulos, G., and Vallino, J. (1991). Network rigidity and metabolic engineering in metabolite overproduction. Science 252, 16751681. Tweeddale, H., Notley-McRobb, L., and Ferenci, T. (1998). Effect of slow growth on metabolism of Escherichia coli, as revealed by global metabolite pool (``Metabolome'') analysis. J. Bacteriol. 180, 51095116. Valenzuela, L., Ballario, P., Aranda, C., Filetici, P., and Gonzalez, A. (1998). Regulation of expression of GLT1, the gene encoding glutamate synthase in Saccharomyces cerevisiae. J. Bacteriol. 180, 35333540. van den Berg, H. A., Kiselev, Y. N., Kooijman, S. A. L. M., and Orlov, M. V. (1998). Optimal allocation between nutrient uptake and growth in a micobial trichome. J. Math. Biol. 37, 2848. van Riel, N. A. W., Giuseppin, M. L. F., ter Schure, E. G., and Verrips, C. T. (1998). A structured, minimal parameter model of the central nitrogen metabolism in Saccharomyces cerevisiae: The prediction of the behaviour of mutants. J. Theor. Biol. 191, 397414. Varma, A., and Palsson, B. O. (1994). Stoichiometric flux balance models quantitatively predict growth and metabolic by-product secretion in wild-type Escherichia coli W3110. Appl. Environ. Microbiol. 60, 37243731. Varner, J., and Ramkrishna, D. (1998). Application of cybernetic models to metabolic engineering: Investigation of storage pathways. Biotechnol. Bioeng. 58, 282291. Wiener, N. (19481961). ``Cybernetics, or Control and Communication in the Animal and the Machine,'' MIT Press, Cambridge. Woldringh, C. L., Huls, P. G., and Vischer, N. O. E. (1993). Volume growth of daughter and parent cells during the cell cycle of Saccharomyces cerevisiae a: as determined by image cytometry. J. Bacteriol. 175, 31743181.
68
