New types of experimental data shape the use of enzyme kinetics for dynamic network modeling

Transcription

1 REVIEW ARTICLE New types of experimental data shape the use of enzyme kinetics for dynamic network modeling Katja Tummler*, Timo Lubitz*, Max Schelker and Edda Klipp Theoretical Biophysics, Humboldt-Universit at zu Berlin, Germany Keywords databases for kinetics; history of rate laws; mathematical modeling; Michaelis Menten kinetics; network related approach; parameter estimation; systems biology; standardization; sensitivity analysis; use of omics data Correspondence E. Klipp, Theoretical Biophysics, Humboldt- Universit at zu Berlin, Invalidenstraße 42, Berlin, Germany Fax: Tel: *These authors contributed equally to this work. (Received 21 June 2013, revised 27 August 2013, accepted 10 September 2013) Since the publication of Leonor Michaelis and Maude Menten s paper on the reaction kinetics of the enzyme invertase in 1913, molecular biology has evolved tremendously. New measurement techniques allow in vivo characterization of the whole genome, proteome or transcriptome of cells, whereas the classical enzyme essay only allows determination of the two Michaelis Menten parameters V and K m. Nevertheless, Michaelis Menten kinetics are still commonly used, not only in the in vitro context of enzyme characterization but also as a rate law for enzymatic reactions in larger biochemical reaction networks. In this review, we give an overview of the historical development of kinetic rate laws originating from Michaelis Menten kinetics over the past 100 years. Furthermore, we briefly summarize the experimental techniques used for the characterization of enzymes, and discuss web resources that systematically store kinetic parameters and related information. Finally, describe the novel opportunities that arise from using these data in dynamic mathematical modeling. In this framework, traditional in vitro approaches may be combined with modern genome-scale measurements to foster thorough understanding of the underlying complex mechanisms. doi: /febs Introduction Molecular biology has advanced tremendously since Michaelis and Menten published their now famous observations on the kinetics of enzyme-catalyzed reactions in 1913 [1]. The complete metabolic and proteomic composition of cells have been revealed, and whole genomes and transcriptomes may be sequenced by high-throughput set-ups, presenting researchers with an enormous amount of high-quality data with high information density [2]. New technologies are being developed and rapidly improved to resolve open biological questions on the functionality and regulation of the whole cell. The characterization of single enzymes appears to be outdated in the age of -omics, requiring a highly complicated experimental procedure to obtain only one or two kinetic constants. However, from the modeler s point of view, this is not the case at all. On the contrary, quantitative modeling of dynamic biological networks requires knowledge about the kinetic properties of the single reactions to be as precise as possible. Dynamic models may directly incorporate experimentally derived kinetic parameters in their rate equations and use them to describe and predict the behavior of a biochemical system under changing conditions [3,4]. The dynamic response of the networks to altered nutrient supplies, endogenous stresses, regulatory gene expression changes or progression of the cell cycle provides deep insights into cellular function, and a thorough understanding of these details has advantages compared to the more extensive but far more static perspective of genome-scale approaches. Nevertheless, systems biological models try to be as comprehensive as possible, and this review describes how the classical enzyme kinetic approach may be complemented and improved by the integration of novel data types. FEBS Journal 281 (2014) ª 2013 FEBS 549

2 New data new enzyme kinetics K. Tummler et al. Pragmatically speaking, Michaelis Menten kinetics is a mathematical model with assumptions and simplifications. The kinetics of the consumption of a substrate are described as a mathematical function of specific parameters that are derived from a general biological mechanism, subject to very specific assumptions: Michaelis and Menten based their equation on the assumption that the substrate and the enzyme substrate complex are in an instantaneous equilibrium (equilibrium approximation) [1], whereas Briggs and Haldane formulated the quasi-steady-state assumption in 1925 [5], whereby the concentration of the complex is thought to be constant over time. Both simplifications yield the same general form of the equation, as illustrated in Fig. 1. The reaction velocity v describes the change in the substrate concentration as a result of its consumption by the enzymatic reaction with respect to time, i.e. ds/dt. In some cases, it is defined as the concentration increase of the product over time, dp/dt, which is an equivalent formulation. The reaction velocity is dependent on the current concentration of the substrate, as well as on the Michaelis Menten parameters V and K m. The mathematical form describes a saturation curve that asymptotically approaches the value V for high values of S. V is thereby a combined parameter, being the product of the catalytic activity of the enzyme k cat and the total enzyme amount E 0. The steepness of the curve in the lower substrate concentration range is defined by the half-saturation constant K m, specifying the substrate concentration at which the enzyme operates at half limiting velocity. Table 1 summarizes the notations used. Outline of this review The Michaelis Menten model is widely accepted in the biological community, and its use has resulted in many valuable scientific results during the past century. However, other models of the same type and often based on the same assumptions frequently face consolidated skepticism. We hope that this review helps to resolve some of these doubts, and to encourage more researchers for the iterative cycle of knowledge acquisition in systems biology. First, we review the roots of the Michaelis Menten kinetics and kinetic pathway modeling. A short summary of the classical experimental techniques of in vitro enzymology, highlighting aspects important to the modeler, is followed by consideration of how the Michaelis Menten parameters were estimated before the age of mathematical computation. From these linearized regression methods, it becomes evident that, for many enzymes, the Michaelis Menten model cannot completely describe the observed behavior. Based on this observation, and the need to describe enzymatic reactions with more than one substrate or product, a number of more complex kinetic models arose. We summarize the most commonly used forms and comment on efforts to derive generally applicable rate laws that may be used for a wider variety of reaction types. These rate laws also account for thermodynamic consistency, an essential feature of biological reactions that is captured in dynamic mathematical models. This concept may be extended to network or even genome-scale models, which put the single characterized enzymes in the context of their biochemical pathway and hence their physiological function. The aim is to describe the functionality and regulation of the biological processes in vivo. We explain how kinetic parameters measured in vitro may be re-scaled to match the physiological situation. As shown schematically in Fig. 1, these models may additionally incorporate data obtained from new experimental techniques that directly probe the in vivo situation, such as Fig. 1. The classical Michaelis Menten equation and the various data types that may be used to estimate the kinetic parameters. The notation used is summarized in Table FEBS Journal 281 (2014) ª 2013 FEBS

3 K. Tummler et al. New data new enzyme kinetics genome-wide metabolomics, proteomics, transcriptomics, and others. We explain how and where this information enters the models, and describe the workflow and the mathematical methods used to consistently estimate kinetic parameters from these mixed data collections. Finally, we highlight the impact that calibrated kinetic models may have for the experimental community. They allow appraisal of how accurate the kinetic information for a specific enzyme is based on all available data for a reaction, which is not possible for parameters derived from in vitro measurements alone. In addition, it is possible to identify poorly characterized parts of the pathway, and even to propose the most efficient experimental strategy to resolve these gaps in our knowledge. In the age of systems biology, Michaelis and Menten s model of enzyme kinetics is taken to the next level by mathematical extensions, thermodynamic constraints, and combination with a whole new realm of high-quality data. As a means to obtain the most comprehensive understanding of biochemical pathways in vivo, this is a promising perspective for modelers and experimentalists alike. Historical summary: isolated enzymes and linearized plots Enzyme kinetic measurements have a long tradition in biochemistry. Numerous enzymes have been characterized to shed light on their catalytic mechanism and to understand how and why biological reactions function so precisely and effectively. Here, we provide a short overview of the experimental techniques and the widely used linearizing estimation methods to illustrate how modern kinetic network modeling arose from these sound experimental foundations during the last century. This is by no means an exhaustive collection, and further detailed reviews [6 8] are highly recommended for additional reading. In vitro enzyme characterization Most enzymes are characterized in vitro. The purified enzyme is mixed with its substrate in a test tube, and the velocity of the reaction occurring is calculated from the initial linear decrease of the substrate concentration. Repeating the experiment with different substrate concentrations yields the classical Michaelis Menten form that asymptotically approaches the limiting reaction speed V. The second parameter of the equation, K m, may also be calculated easily from the obtained curve: it is defined by the substrate concentration at which the velocity reaches half of its limiting value. Because the Michaelis Menten parameters are directly derivable from the experimental data, the vast majority of enzyme characterizations are based on this mathematical formulation. However, if only the resulting parameter values are available from an enzyme characterization, it is impossible to use them in other, more complex or more mechanistic kinetic rate laws that apply a different mathematical formulation. This is only possible if the raw data of the experiment are also provided, as described below. Enzyme kinetic measurements are challenging and difficult to perform. The enzyme needs to be isolated to a very high purity either from the organism of interest or from recombinant bacteria, such as Escherichia coli, expressing the respective gene. During the purification process, there are many steps that potentially influence the activity of the enzyme. Detergents or high salt concentrations may have a denaturing effect, and changing temperatures or altered timing of the experiments may introduce systematic errors to the measurement [7]. The reaction velocity for one given substrate concentration may be measured in a photometer by monitoring the changes in fluorescence intensity or light absorbance caused by clearance of the substrate or the increase in product concentration. If neither substrate nor product is photometrically detectable, coupled enzyme assays may be used. Thereby, the product of the analyzed reaction is consumed by a subsequent enzyme, resulting in a measurable secondary product such as NADH. As only the initial phase of the linear concentration change is used to estimate the reaction speed, the time lag introduced by such coupled assays may lead to an under-estimation of the reaction speed and requires careful consideration. In addition, efficient coupling and thorough calculation of the parameters is only possible if the reaction of interest is the rate-limiting step in a set of coupled reactions, which may be ensured by using a sufficiently high abundance of the coupling enzyme. With the same experimental set-up, the effect of further chemical compounds on the enzyme activity may be tested. Metal ions, such as manganese, copper, iron or magnesium, may be added at various concentrations to investigate their influence on the catalytic efficiency. Similarly, the effect of activators or inhibitors may be tested. These are mostly small chemical compounds, intermediary metabolites or substrates of downstream reactions, which are a priori suspected to alter the activity. Unbiased screens for potential but unknown effectors require a very high experimental effort. To test the regulatory effect of other cellular FEBS Journal 281 (2014) ª 2013 FEBS 551

4 New data new enzyme kinetics K. Tummler et al. components, such as other enzymes, phosphatases or kinases, is difficult in the in vitro context, as they require optimum reaction conditions for both enzymes or at least the modifying enzyme to be known. This is a major drawback, as it excludes all higher-order regulatory processes, such as post-translational modifications, that play an essential role in the biological scenario. Some studies have been performed to assess the importance of specific phosphorylation sites, for example for the trehalase of Saccharomyces cerevisiae [9], but they are highly complex and required subsequent knockout of the single phosphorylation sites. In addition, many data for the dependence of activity on changing ph or temperature ranges have been obtained. In many cases, insights from those measurements are used to define the ph and temperature optima for the enzymatic activity, which are then used in the set-up for further experiments. However, these optimized in vitro conditions [10] do not necessarily correspond to the intracellular situation in vivo. Data obtained from the described experiments may be used to extract values for the Michaelis Menten parameters for the given enzyme, and also to quantify the effect of the investigated inhibitors and activators. From a mathematical point of view, the estimation of the kinetic parameters based on the measured data is a non-linear optimization problem, with the aim of minimizing the deviation of the theoretical description from the data points. This formalism may be applied to the simple form of the Michaelis Menten equation, to more complex descriptions of enzymatic activity, or to models of large networks as described below. Today, very efficient mathematical theories and computational power are available to solve optimization problems, which was not the case in the times of Michaelis and Menten. Nevertheless, the elegant form of their kinetic equation allows the problem to be easily reduced to a linear regression problem that may be solved with a ruler and a pencil by transforming the Table 1. Mathematical notation used in this review. Entity Symbol Fluxes/reaction velocities v i v~ ¼ðv 1 ;...; v r Þ Substrates/products S j S~ ¼ðS 1 ;...; S l ; S lþ1 ;...; S n Þ Kinetic parameters p k p~ ¼ðp 1 ;...; p m Þ Michaelis Menten parameters V or k cat, K m Dissociation constant, K D, K I, K A inhibition/activation constant Total enzyme amount E 0 Stoichiometric matrix N, with n ij as the stoichiometric coefficient of substrate j in reaction i data to a logarithmic or reciprocal scale. Various versions of transformations have been established during the past century and have found wide application. Here, we provide a short overview of the various linearization methods, which may seem outdated in the era of mathematical computation, but still yield a number of valuable insights, especially with regard to assessing the quality of the experimental set-up, as illustrated using a comparative example. Linearized plotting There are several ways of linearized plotting, each with their own benefits and drawbacks. Figure 2 and Tables 2 show linearized plots of literature-derived kinetic data, including the underlying equation for their creation. The direct plot may be created by plotting the reaction rate v against the concentration S, which yields a hyperbolic curve of saturation (Fig. 2, column 1). Nevertheless, this plot may be unsatisfying, as it is difficult to locate the asymptotes correctly and potential deviations from the expected curve are also hard to track. Thus, Michaelis and Menten introduced the Michaelis Menten plot [1], in which the reaction rate is plotted against the logarithm of the substrate concentration. The benefit of using the logarithm of S rather than the actual concentration is that the slope remains close to its limiting value over a wide range and thus may be estimated easily. This plot also allows comparison of different isoenzymes that may be catalyzing the same reaction but have different substrate affinities. However, this plot is not included in Fig. 2, as it is not used frequently and we focus on the more commonly employed plots. Various linearization techniques for the Michaelis Menten equation prove useful for visualization and error handling. A very common linearization is the double-reciprocal plot, also called the Lineweaver Burke plot (Fig. 2, column 2) [11]. For this linearization, 1/v is plotted against 1/S, which yields a straight line with slope K m /V. This plot is used widely, but the handling of the experimental error is problematic. If v is very small, a small error in v leads to huge deviations in 1/v [12]. A means to at least ease this problem is to assign proper weights [11]. A further improvement of the aforementioned plot may be realized by multiplying both sides of the Lineweaver Burke plot by the substrate concentration S. This shows that the plot of S/v against S is also a straight line. It is often called a Woolf plot [13] or Hanes plot [14] (Fig. 2, column 3) and allows a reasonable error handling of S/v. 552 FEBS Journal 281 (2014) ª 2013 FEBS

5 K. Tummler et al. New data new enzyme kinetics Fig. 2. Plots of the various linearization methods described in the text for three enzymes of S. cerevisiae. Colored dots (red for enolase 1, ENO1; blue for enolase 2, ENO2; green for fructose-bisphosphate aldolase 1, FBA1) represent data points [18]; black lines represent the respective regressions. The units for S and v are lm and lms 1, respectively. For the Cornish Bowden plot, black dots represent the intersection point of two measurements projected to the axes, and the gray areas indicate the density distributions of the dots. As mentioned in the text, the estimates for K m and V are the medians of the distributions shown. Values for the parameters estimated using the various methods are shown in Table 2. Table 2. Summary of the regressions shown in Fig. 2 (linearized equations, quantities plotted on the x and y axis and estimation results for V in lms 1 and K m in lm for the three enzymes). Non-linearized fitting Lineweaver Burke Eadie Hofstee Hanes Woolf Cornish Bowden 1 Linearized equation v ¼ 1 v þ Km 1 v s v ¼ v K m v s s v ¼ Km v þ 1 v S v ¼ v þ v s K m x S 1/S v/s S S y v 1/v v S/v v ENO1 V K m ENO2 V K m FBA1 V K m A third linearization of the Michaelis Menten equation is called the Eadie Hofstee plot (Fig. 2, column 4) [15,16], and may be created by multiplying both sides of the equation of the Woolf plot by the factors vv. In this plot, the slope of v against v/s equals K m.an intrinsic feature of this plot is that errors in v influence both coordinates. Hence, systematic deviations of the data become obvious, and allow detection of discrepancies from Michaelis Menten behavior [12]. Finally, the Michaelis Menten equation may be plotted in terms of a direct linear plot. For each observation, the negative value of the applied substrate concentration is marked on the x-axis and linked to the value of the measured reaction speed on the y-axis by a straight line with slope v/s (Fig. 2, column 5). The resulting plot was introduced by Eisenthal and Cornish-Bowden [17] and shows one line for each observation, which discriminates this plot from the former FEBS Journal 281 (2014) ª 2013 FEBS 553

6 New data new enzyme kinetics K. Tummler et al. ones. In a fictitious experimental environment without measurement error, all these lines intersect at one point, defining the unique values of K m and V for all observations. Obviously, in a natural environment, the lines do not intersect at one point. Thus, the experimental error is visualized by how much the intersecting points differ from each other. It may be quantified by taking the median estimate. In addition, confidence regions for the estimated parameters may be estimated based on the width of the area of intersections, i.e. this linearization allows quantification of the accuracy of the obtained kinetic parameters. Complex systems require complex kinetics As stated above, the linearization plots allow scientists to approximate kinetic parameters by hand. However, not only are many reactions more complex than the reactions we have mentioned so far (one substrate, one product), but also computational power has increased immensely in the last decades. Thus, researchers are able and recommended to generate and evaluate more complex kinetic mechanisms with a multitude of substrates and products. In general, the approaches for this topic may be split into three major areas: (a) the attempt to comprehend the complexity of kinetic mechanisms by generating more and more complicated kinetic laws, (b) the contrasting approach of reducing the complex mechanisms by proposing simplifying assumptions, and (c) the use of generalized kinetic rate laws, which facilitate application of kinetic laws to large networks. Here we introduce general aspects of these three approaches, with examples. In analyses of more complex reactions, it is practical to introduce notation to name the reactions according to their order, i.e. in a first-order reaction, the rate is proportional to one concentration, in a second-order reaction, it is proportional to the product of two concentrations or to the square of one concentration, and so on [6]. Complicated mechanistic kinetic laws While Michaelis Menten kinetics give very good approximations for first-order reactions, most reactions in nature comprise more than one substrate and product. With regard to the complexity of kinetic mechanisms, it is understandable to try and explain them by mechanistic models with a growing level of detail. The following examples differ in their complexity and applicability. Cleland notation For differentiation between enzymatic reactions of different orders, it may be useful to use a unified notation. In the Cleland notation [19], the reactions are preceded by the number of involved substrates and products, using the prefixes uni, bi, ter etc. For example, a reaction with two substrates and two products is called a bi bi reaction, while one with two substrates and three products is a bi ter reaction. Moreover, if a bi bi reaction requires its two substrates to bind in a certain order, it is called an ordered or sequential bi bi reaction, otherwise it is a random bi bi reaction. The mechanism may be more complex if products are released before certain substrates are bound. In this case, it is referred to as a ping pong mechanism, which allows further clarification of the binding details: a reaction with three substrates and three products, in which one product is released after the first substrate binds, is called a uni uni bi bi ping pong reaction. The term uni uni relates to the first substrate and product, while the term bi bi relates to the following two substrates to be bound and products to be released. This simple notation facilitates the naming of numerous enzymatic binding mechanisms. Ternary complex mechanism The most common reactions in nature have two substrates and two products (a bi bi reaction in the Cleland notation) [20], and, while the majority of them are oxidation/reduction reactions, group-transfer reactions are very frequent. A group-transfer reaction indicates how a chemical group G is transferred from a donor molecule S 1 to an acceptor molecule S 2 [21]: S 1 G þ S 2 $ S 1 þ S 2 G The so-called ternary complex mechanism is a grouptransfer reaction that contains the enzyme and both substrates in a single complex ES 1 GS 2, with E being the enzyme. The order in which the substrates bind the enzyme was originally assumed to be random [13,14]. However, this is a simplification of the actual process, as it is now known that the conformation of the enzyme as well as any substrate is likely to be changed by binding mechanisms. Thus, binding of a substrate to an enzyme is often only facilitated by binding of another substrate and a corresponding change in conformation of some or all of the involved players [22]. Ping pong mechanism In contrast to the ternary complex mechanism, the aforementioned ping pong mechanism (also termed a 554 FEBS Journal 281 (2014) ª 2013 FEBS

7 K. Tummler et al. New data new enzyme kinetics substituted-enzyme mechanism) postulates that, in an intermediate state, the transfer group is bound to the enzyme in a single EG complex, without being bound to either of the substrates [22]. A ternary complex cannot be generated, probably due to the binding sites of the substrates S 1 and S 2 being the same (or overlapping). In practice, two group transfers are occurring, and hence the term double-displacement reaction was established [23]. At high substrate concentrations, this may lead to a saturation effect: the substrates are bound to the wrong enzyme. The enzymes that do not carry the transfer group may bind to the substrates that do not carry the transfer group either, thus blocking the group transfer. Cooperativity models The concerted or symmetry model of Monod, Wyman and Changeaux (MWC model) [24,25] contrasts the sequential model of subsequent substrate binding and product release. It postulates that a protein consists of subunits, and each of these single subunits may exist in an inactive (tense, T) and an active (relaxed, R) state. Independent of the subunit s state, several molecules of the same substrate may bind to the individual subunits. At a certain time, all subunits of a specific molecule must be in the same conformation. Thus, the number of states that a protein may adopt depends on the number of different molecules. An equilibrium assumption for the unbound states (R0, T0) enables derivation of a rate law. In the limiting case that the inactive form is not present, this rate law approaches the Michaelis Menten rate law. The algebraic analyses, the effects and the properties of the various complexes are a broad field that has been discussed elsewhere [6,26]. A more complicated version of kinetics for enzymes with subunits was suggested by Koshland, Nemety and Filmer (1966), the so-called KNF model [27]. Simplified empirical kinetic laws Simplified kinetic laws are based on empirical considerations. Instead of going into more and more detail of the mechanisms, they allow simple mathematical calculation of the reaction rate v. Hill kinetics The Hill equation [28] is used for sigmoidal, saturated non-michaelis Menten kinetics, and its application in enzyme kinetics is very common. It represents the enzyme cooperativity for the reaction rate v in terms of v ¼ V SH K H D þ SH with H being the Hill coefficient, and K H D resembling the K m value of the Michaelis Menten equation. Nevertheless, it should not be named K m, as this equation does not equal the Michaelis Menten equation. Using the Hill equation, it is possible to quantify the cooperative binding effect and characterize the possibility that the binding affinity of a substrate to an enzyme may be enhanced or decreased by another substrate that is already bound. The Hill coefficient plays a key role in defining the substrate dependency of the kinetics: it is set to H > 1 if the substrate binding affinity is increased by other substrate molecules; if H < 1, the affinity is decreased and negative cooperative binding is assumed. A Hill coefficient of 1 indicates non-cooperative binding, and the Hill equation equals the Michaelis Menten equation. In 1997, Hofmeyr and Cornish-Bowden introduced a reversible form of the Hill equation, which allows incorporation of modifiers [29]. The direction of the rate depends on the thermodynamic state, and, if either the substrate or product concentration are zero, it degenerates to the Hill equation. However, the number and measurability of the parameters in the reversible Hill equation may be obstacles when applying it to bigger networks. In practice, many reactions with high equilibrium constants are treated as irreversible [30,31], which may also lead to problems in modeling pathways realistically. As an example, Cornish-Bowden and Cardenas investigated the shortcomings (i.e. missing feedbacks in a pathway) that arise in a model of glycolysis in Trypanosoma brucei [32]. A quick route through several simple approximative kinetic rate laws The kinetics derived from studies on purified enzymes are non-linear, and contain many parameters that complicate the search for analytical solutions for steady states of metabolic network models. This may make the numerical simulations that are required for thorough analysis of metabolic models unreliable. Thus, there is a need for approximate kinetic rate laws that (a) are non-linear (to account for the nonlinear kinetic properties of enzyme-catalyzed reactions or more specifically the downward concave behavior of rate versus concentration at high metabolite FEBS Journal 281 (2014) ª 2013 FEBS 555

8 New data new enzyme kinetics K. Tummler et al. concentrations), (b) have a small number of kinetic parameters, (c) allow analytical solutions of the metabolic network mass balances (analyzing the mass fluxes of the system by exploiting the laws of mass conservation), (d) have a reaction rate proportional to the enzyme amount, and finally (e) are in agreement with experimental evidence from metabolic networks [33]. In general, the velocity of an enzyme-catalyzed reaction may be formulated as v i ¼ E fðs~; p~þ (1) where E is the enzyme concentration, p~ are the kinetic parameters, and S~ are the metabolite concentrations. The approximation of function f is usually performed around a reference state (here a steady state) that is characterized by the reference-state reaction rates (or fluxes) J~, 0 the metabolite levels S~, 0 and the enzyme concentrations E~ 0. This may be extended by introducing a matrix for scaled concentration elasticities E S,in which an entry e ij holds the derivative of reaction rate v j with respect to metabolite concentration S i. These elasticities are convenience parameters with fixed values at the reference state; they change only if the system state changes [34]. Regarding these deviations from the steady state and using the normalized quantities ^v i ¼ vi J 0;i, ^E i ¼ Ei E 0;i and ^S j ¼ S j S 0;j, Eqn (1) may be reformulated to ^v~ ¼ð^E ~ Þgð ^S ~ ; E s Þ Thus, the general non-linear rate is most easily linearized around the reference state with respect to S and E, which facilitates a general derivation of all known metabolic control analysis equations [35 37] by solving the network mass balances for fluxes and dependent metabolites: ^v~ i ~¼ð^E ~ i~þþe s ð ^S ~ i~þ where i~ is a unit vector. The steady-state network mass balances may thus be solved for fluxes and dependent metabolites [38]. However, this easy linear approximation only produces reliable results for very small concentration changes around the steady state, which cannot be realized experimentally. This may be improved by a log-linear approximation as proposed by Hatzimanikatis and Bailey [39], using the approach y 1 ffi ln y. The obtained set of dynamic mass balances is completely linear in logarithmic quantities; the derived equation reads ^v~ i ~¼ lnð ^E ~ ÞþE s lnð ^S ~ Þ The advantage of this approach is that small changes in the natural logarithmic counterparts of the metabolites still account for large changes in their original measuring unit. In contrast, the enzyme level may still only be changed less than twofold without significantly influencing the accuracy of the predictions [39]. This is a serious limitation of the log-linear approximation, as large changes in enzyme concentration are a common trait of in vivo systems. By applying the large-perturbation approach described by Small and Kacser [40] and generalizing the linear metabolite treatment by introduction of an approximative kinetic rate law [41], the equation is changed to ^v~ ¼ð^E ~ Þ 1 þ E s ð ~ ^S ~ ^i Þ in which the rate remains proportional to the enzyme level. The main benefit of this approach is to facilitate analysis of large perturbations in enzyme level. Conversely, the prediction of metabolite control coefficients appears comparably weak [40]. This suggests that the effect of metabolite concentrations on the reaction rate should be approximated in a non-linear fashion. A possible solution to this problem is the generalized mass action approach [42], where each reaction is described by using power law kinetics that have been successfully applied to a metabolic model in S. cerevisiae [43]. It may be formulated as the generalized mass action approximation: ln^v ~ ¼ lnð ^E ~ ÞþE s lnð ^S ~ Þ Using this equation for the rate determination, the rate is proportional to enzyme level, and the metabolite concentrations are approximated by non-linear power functions. On the downside, a change in metabolite concentration does not cause a change in the elasticities. Furthermore, in the likely case of the network containing branched pathways, all reactions originating from a metabolite must be combined into a single aggregated power law, just like the reactions producing the named metabolite. This approximation based on the power-law approach is called the S-system approach [44]. It produces only small errors for large metabolite concentration changes, but the errors for the aggregated rates may be rather high [33]. In 2002, Visser and Heijnen proposed the lin-log approach [38], introducing independent empirical coefficients for the metabolites. For a reference state 556 FEBS Journal 281 (2014) ª 2013 FEBS

9 K. Tummler et al. New data new enzyme kinetics with flux J 0 and reference elasticities E S, this may be formulated as ^v~ ¼½^E ~ Š i~þ E s lnð ~ ^S Þ with the diagonal matrix ½ ^E ~ Š. The rate and enzyme concentration are assumed to be proportional, and the non-linear logarithmic concentration terms are summed. The lin-log approach accounts for large changes in enzyme activity and metabolite concentration, and furthermore only requires a comparably small number of kinetic parameters. The elasticity parameters may be obtained from steady state and dynamic experiments [38]. Convenience kinetics [45] is a saturable rate law that is derived from a simple random-order enzyme mechanism. It may be considered as a generalized Michaelis Menten kinetics [1] for all possible stoichiometries, and may include allosteric regulations. The kinetic parameters used are thermodynamically independent, so they do not violate the system s thermodynamic constraints [46]. The equilibrium constants are either expressed as Gibbs free energies of formation or a set of independent equilibrium constants. Defining the concentration vector S~ ¼ðS 1 ; S l ; S ðlþ1þ ; ; S n Þ T for l reaction substrates and n l products, the convenience kinetics is formulated as vðsþ ¼E Q l j¼1 k þ Q l ~ cat j¼1 S j k Q n ~ cat j¼lþ1 S j 1 þ Q ~S j þ n j¼lþ1 1 þ ~S j 1 The variable S~ j is a normalized reactant constant, namely ~S j ¼ S j K m ; in analogy to the Michaelis constant K m. The kinetics may be easily extended to serve practical applications. If more than one molecule of the same substance participates in a reaction, one may introduce stoichiometric coefficients a j and b j to the respective substances and reformulate the convenience kinetics to vðsþ ¼E Q l j¼1 k þ Q l cat j¼1 1 þ ~S j þþ ~S a j j S ~ aj j k cat þ Q n j¼lþ1 Q n j¼lþ1 ~ S b j j 1 þ ~S j þþ ~S b j j 1 Generalized rate laws: comparability and mathematical simplicity The use of generalized rate laws has become popular and widespread in kinetic modeling. They serve as a substitute for the actual mechanism, which is in many cases either unknown or tedious to characterize and use due to the lack of kinetic data [8]. Usually, these rate laws underlie a certain set of assumptions or simplifications, which must be kept in mind when evaluating (a) which generalized rate law is suitable for the current need, and (b) how the results of the modeling may be understood. Ideally, the number of kinetic parameters should be as small as possible, and the approximate kinetic rate law should allow analytical solutions of the network s mass balances. Convenience kinetics Furthermore, possible reaction modifiers may be introduced by multiplying this equation with activation or inhibition prefactors that introduce the inhibition and activation constants K I and K A : h A ðd; K A Þ¼ d K A þ d and h Iðd; K I Þ¼ K I K I þ d where d is the modifier concentration. The kinetic parameters used for convenience kinetics may be measured in experiments and correspond to the parameters of Michaelis and Menten. Thus, they may be retrieved from the vast amounts of kinetic data in online databases [47,48]. Missing parameter values may be complemented by coupling with existing parameters. Modular rate laws for enzymatic reactions In 2010, Liebermeister et al. [49] proposed a family of reversible rate laws for reactions with arbitrary stoichiometries and various types of regulation. The parameter sets are thermodynamically safe and guaranteed to lead to consistent chemical equilibrium states. The five proposed rate laws are characterized by the different formulae of their denominators. The rate laws are exemplified for a simple unregulated reaction S 1 þ S 2! 2S 3 Firstly, the common modular rate law may be considered as a general form of a reversible FEBS Journal 281 (2014) ª 2013 FEBS 557

10 New data new enzyme kinetics K. Tummler et al. Michaelis Menten kinetic, applicable to any desired reaction stoichiometry. For the simple unregulated reaction it reads k þ cat v ¼ E S0 1 S0 2 k cat ðs0 3 Þ2 ð1 þ S 0 1 Þð1 þ S0 2 Þþð1þS0 3 Þ2 1 0 where S 0 j ¼ S j=k S j m, with the reaction constant K S j m (mm) for metabolite S j. K m values are dissociation constants comparable to the Michaelis Menten kinetic constant. The terms of the denominator represent the different binding states of the enzyme. The second rate law of the family of modular rate laws is the direct-binding modular rate law, in which the denominator only contains the substrate and product terms of highest order: v ¼ E kþ cat S0 1 S0 2 k þ k cat ðs0 3 Þ2 cat S 1S 2 K 1 þ S 0 1 ¼ E mþ k cat ðs 3Þ 2 K m S0 2 þðs0 3 Þ2 1 þ S1S2 K mþ þ ðs 3Þ 2 K m where K mþ ¼ K S1 m KS2 m and K m ¼ðK S3 m Þ2. At low concentrations, the reaction rate is higher than in the common modular rate law if the same parameters are chosen. Furthermore, for a reaction with simultaneous binding properties, the simultaneous binding modular rate law reads v ¼ E kþ cat S0 1 S0 2 k cat ðs0 3 Þ2 ð1 þ S 0 1 Þð1 þ S0 2 Þð1 þ S0 3 Þ2 and yields a lower reaction rate, which becomes even more apparent at high concentrations. This is realized by additional terms in the denominator. The simple power-law modular rate law sets the denominator to 1, resulting in v ¼ E kþ cat S0 1 S0 2 k cat ðs0 3 Þ2 1 which in this case is just mass-action kinetics v ¼ Eðk þ0 cat S 1 S 2 k 0 cat S 3Þ, and may be useful for a case of exponential growth. Finally, the force-dependent modular rate law serves as an illustration for a more thorough thermodynamic realization: v ¼ E kþ cat S0 1 S0 2 k cat ðs0 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ2 S 0 1 S0 2 ðs0 3 Þ2 This rate law is not supported by a biochemical mechanism and includes the risk of the reaction rate going to infinity (i.e. if the substrate or product concentration is set to zero). However, it may be a valuable alternative to other rate laws, as it may be easily adjusted to stationary flux distributions. Finally, the choice of the kinetic constants has a direct effect on the thermodynamic consistency of the reactions of a model. The thermodynamic laws as well as the Wegscheider condition [50a] and Haldane relationship [50b] must be accounted for to avoid unrealistic or even impossible model behavior. The introduced modular rate laws use thermodynamically safe independent parameter sets to overcome this obstacle [45,46]. Thus, all kinetic constants may be written as linear functions of the independent parameters and automatically ensure correct Wegscheider conditions and Haldane relations [50]. Naturally, this parameterization requires a higher numerical effort, but the results are advantageous. New data, new perspectives The increasing complexity of the mathematical formulation of kinetic rate laws is mirrored and even exceeded by the advance in enzyme-related experimental techniques. In the -omics era, kinetic measurements are only a small part of the extensive amount of data collected from a vast variety of cell types and test scenarios [2]. Genome-wide data are en vogue and are starting to substitute for conventional bottom-up approaches such as single enzyme characterization, which are becoming an add-on to reinforce findings from large-scale screens. High-throughput experiments allow quantitative assessment of the whole set of expressed genes, proteins or metabolites within a cell at a certain time point. However, the datasets produced are of such immense size and information density that intuitive conclusions are almost impossible to draw. A possible solution to overcome these limitations is systems biological network modeling. Computational models may be used to collect all topological information available for a pathway or even for the whole metabolic or gene regulatory network of a cell and effectively link them to various types of -omics data (e.g. [51], compare also to Fig. 3). One of the main benefits is thereby an unbiased view of the in vivo situation, as it is no longer necessary to purify single molecules and transfer them to an optimized in vitro environment. This also holds new promise for kinetic characterization of enzymes and their framing pathways: the network perspective allows characterization of enzymes in their natural environment, connecting specific activity kinetics to the function of a whole pathway. However, the realization of this new scenario is as stunning as it is challenging. Combining various data 558 FEBS Journal 281 (2014) ª 2013 FEBS

11 K. Tummler et al. New data new enzyme kinetics A B C D E F Fig. 3. Illustrative biological example combining various available data sources [122]. FEBS Journal 281 (2014) ª 2013 FEBS 559

12 New data new enzyme kinetics K. Tummler et al. types within one consistent and thermodynamically correct model is a non-trivial task. Here, we provide an overview of how conventional enzyme kinetic data may be linked to new in vivo measurements of enzymes, metabolites or gene expression patterns in order to obtain rigorous understanding of a biochemical pathway. Despite the decreasing recognition and number of enzyme kinetic measurements in the scientific community, we wish to highlight that these are exactly the type of data that form the core of a thorough dynamic model. As soon as the in vitro conditions are well documented and ideally mimic the in vivo situation, the characterized kinetic parameters may be incorporated directly into mathematical models and hence provide highly valuable information for systems biologists. Figure 3 shows an example referring to the enzyme enolase of yeast to demonstrate how various data types may be integrated. Enzyme kinetic data in the light of network modeling Dynamic network models aim to describe reaction systems within the cell. Hence, it is essential to carefully consider the effect that the altered environment of the in vitro test tube may have on the isolated enzyme. Many factors are neglected in the defined, buffered solutions used for in vitro enzyme characterization and are discussed below. However, the definable conditions are also a major advantage of classical kinetic measurements. They allow systematic testing of the catalytic activity over a wide range of substrate and effector concentrations, with the potential to reveal important mechanistic insights. For use in mathematical models of the in vivo situation, the derived parameters may be used but may need to be rescaled according to the intracellular ph, redox state, ion concentrations, and the abundances of substrates, effectors and regulatory compounds (as discussed below). How far are the in vitro conditions from the intracellular situation? Enzyme assays are often optimized to allow maximum activity of the analyzed enzyme. Although beneficial for the in vitro tests, this does not necessarily translate well to the intracellular functionality of the enzyme. Recent studies [52,53] have shown that in a medium resembling the intracellular conditions (see below), enzymatic activities are significantly different from the optimized in vitro set-up. As not all characterized enzymes show the same ph or temperature optimum, it is likely that some enzymes operate under sub-optimal conditions in the cytoplasm, which may be used for regulatory or temporal fine-tuning of the enzymatic activity. The influx into a metabolic pathway, for example, may be regulated by controlling the flux through a rate-limiting bottleneck enzyme that operates very slowly under normal conditions. As mentioned above, the purification process severely affects the stability and functionality of the proteins, but the conditions of the assay itself may also introduce artifacts. The abundance of co-factors and metal ions strongly affects the activity of the enzyme. Their effect has been tested separately in many kinetic studies, which allows extrapolation of the activity to physiological concentrations. Other ions such as calcium, potassium or sodium, which are important for the electrical charge of the enzyme, are usually added in standard concentrations to the medium, and it is therefore difficult to estimate how the activity may change in solutions of physiological ionic strength. The intracellular space has a very high macromolecular content, described by the term macromolecular crowding [54]. No single macromolecule in the cell has a very high concentration, but the total macromolecular concentration may reach up to 400 mgml 1, resulting in 40% of the cytoplasmic volume being physically filled with molecules. A typical in vitro setup of an enzyme assay totals up to 1 10 mgml 1, which may cause serious dilution artifacts [55]. The high proportion of occupied space also results in an excluded volume effect that may significantly affect the reaction rates and equilibrium constants [56]. Another spatial effect occurring in the cytoplasm is substrate channeling [57,58]. Consecutive enzymes of metabolic pathways may form multi-enzyme complexes, within which the product of one enzymatic reaction is directly passed to the active site of the next enzyme, where it serves as the substrate for the following reaction step. In this way, a local concentration increase may be achieved, which again affects the equilibrium constant of the pathway s enzymes. Many catalytic processes are surface-associated, which means that the reaction space is reduced to two dimensions. This scenario may result in diffusion limitations if the diffusion constant of the substrate is smaller than the velocity of the catalysis. In addition, theoretical studies have shown that a lower dimensional reaction space increases the kinetic order of the reaction such that a one-substrate reaction is no longer of the first kinetic order, as assumed for the law of mass action on which the Michaelis Menten equation is based [59,60]. 560 FEBS Journal 281 (2014) ª 2013 FEBS

13 K. Tummler et al. New data new enzyme kinetics An intermediate step between in vitro and in vivo investigation of enzyme catalysis is enzymatic characterization of crude cell extracts. These experiments aim to identify activity changes under various conditions such as different growth and cell-cycle phases, drug treatment, stress responses or nutrient conditions. However, assaying the whole cell content for a specific enzymatic activity combines the effects of changes in activity, enzyme amount and regulatory mechanisms [7]. Crude cell assays have been regarded with skepticism as many unwanted side-effects may occur by keeping all the cell s constituents. Non-specific substrates may compete for binding at the active site, other enzymes in the extract may non-specifically consume the substrate of interest, and free proteases may damage the enzyme under consideration. However, in the light of network modeling, these analyses represent a snapshot of the enzyme activity under the current intracellular conditions and may yield insights into the functionality of the biological process, albeit in a more qualitative manner. Modelers should nevertheless be cautious with the parameters derived from such experiments, as they do not correspond one-to-one to the values found in in vitro assays, where it is possible for results to be normalized to the total amount of the enzyme in question. Standardization efforts In the wake of the rising popularity of computational biology, the need for comparable and standardized enzyme kinetic experiments has become clearly evident. Systems biological models collect information and data from various sources which are not always easily comparable, especially in enzymology. However, the comparability of datasets from different publications may be greatly increased by detailed and complete documentation of the method used [61,62]. As for many other experimental techniques [63], a standard reporting format for enzymological experiments has been developed in a community effort project. The result is summarized in the STRENDA guidelines ( strenda/guidelines). The format proposes annotation categories that allow unique identification of enzymes and substrates, a detailed and reproducible description of the assay used, as well as standardized specifications for the derived kinetic parameters and the fitting method by which they were obtained. For the modeling community, enzyme kinetic measurements with such detailed meta-data are the ideal foundations of sound kinetic models, and allow comprehensive integration of datasets from various studies and laboratories. Ideally, these standardized procedures may then be used to perform experiments that mimic the intracellular situation as closely as possible, as has been performed for the glycolytic enzymes of S. cerevisiae [52]. In this study, the intracellular ion concentrations, ph and charge state were measured for the yeast cell, leading to development of an assay medium that is as close to the measured features as possible. The activities of the glycolytic enzymes under these in vivo-like conditions differed significantly from the values obtained under optimized conditions, and are much more likely to give an accurate picture of the intracellular functionality. A similar study has been performed for central metabolic enzymes in E. coli [53], which highlighted the dramatic pleiotropic influence of the experimental conditions: addition of K + and phosphate to the in vivo-like E. coli medium altered the activity of almost all enzymes examined. Complementing scarce kinetic measurements with genome-scale data For the majority of biochemical pathways, not all enzymes have been experimentally characterized. In order to fill these knowledge gaps, mathematical models may integrate new data types to consistently estimate the missing rate constants. Metabolomics, flux measurements, proteomics and transcriptomics provide an unbiased insight into the in vivo state of the network, and hence allow rigorous adjustment of the results obtained from in vitro studies. In addition, the modeling framework accounts for thermodynamic consistency and excludes the risk of obtaining biologically infeasible network representations. Dynamic models usually describe the time-dependent change of compound concentrations (e.g. Eqn 1 and Figs 1 and 3). Hence, to calibrate unknown parameters, measurements of those concentrations are the first choice. For metabolic systems, highly efficient methods to quantify all internal metabolite concentrations have been developed in the past decades, based on distinguishing specific molecular properties of single molecules. Nuclear magnetic resonance spectroscopy [64,65] exploits the magnetic properties and capillary electrophoresis [66] exploits the charge and mobility differences of molecules, but both methods have recently been outperformed by quantitative mass spectrometry (MS) [67,68]. MS uses the specific mass and ionization properties of molecules to separate them in an electric field. The technique is not limited to small compounds such as metabolites but may also measure proteins, peptides, fatty acids and carbohydrates, and their polymers. Pre-fractionation by liquid or gas FEBS Journal 281 (2014) ª 2013 FEBS 561