TY - JOUR
AU - Rajon, Etienne
AB - Abstract Enzymes speed up reactions that would otherwise be too slow to sustain the metabolism of selfreplicators. Yet, most enzymes seem only moderately efficient, exhibiting kinetic parameters orders of magnitude lower than their expected physically achievable maxima and spanning over surprisingly large ranges of values. Here, we question how these parameters evolve using a mechanistic model where enzyme efficiency is a key component of individual competition for resources. We show that kinetic parameters are under strong directional selection only up to a point, above which enzymes appear to evolve under near-neutrality, thereby confirming the qualitative observation of other modeling approaches. While the existence of a large fitness plateau could potentially explain the extensive variation in enzyme features reported, we show using a population genetics model that such a widespread distribution is an unlikely outcome of evolution on a common landscape, as mutation–selection–drift balance occupy a narrow area even when very moderate biases towards lower efficiency are considered. Instead, differences in the evolutionary context encountered by each enzyme should be involved, such that each evolves on an individual, unique landscape. Our results point to drift and effective population size playing an important role, along with the kinetics of nutrient transporters, the tolerance to high concentrations of intermediate metabolites, and the reversibility of reactions. Enzyme concentration also shapes selection on kinetic parameters, but we show that the joint evolution of concentration and efficiency does not yield extensive variance in evolutionary outcomes when documented costs to protein expression are applied. enzyme evolution, enzyme kinetics, kcat/KM, kcat, facilitated diffusion, Michaelis–Menten Introduction Livingorganisms need to uptake and metabolize nutrients, relying on enzymes to catalyze chemical reactions along metabolic pathways. Accordingly, and despite being intrinsically reversible (Haldane 1930; Klipp and Heinrich 1994), in vivo enzyme-catalyzed reactions are commonly thought of as an irreversible two-step process (Michaelis and Menten 1913; Bar-Even et al. 2011, 2015; Michaelis et al. 2011): E+S⇌krkfES→kcatE+P,(1) where kf and kr are the rates of association and dissociation between enzyme and substrate, and kcat is the turnover number, that is the rate of formation of the product P from ES complexes. The first part of this chemical equation describes the encounters between the enzyme E and the substrate S; the enzyme will be efficient if ES complexes form often and do not dissociate before the substrate has been turned into a product, which is reflected by the constant KM=kr+kcatkf ⁠. The efficiency v of an enzyme—the rate at which it makes a product P from S—depends on these two constants through equation (2): v=kcat.[Etot].[S]KM+[S],(2) under the assumption that the concentration [S] is approximately constant and that of [ES] is at a steady state (Michaelis and Menten 1913; Briggs and Haldane 1925). At first glance, Natural Selection is presumed to optimize enzymatic efficiency by pushing kcat upwards and KM downwards to universal physical limits. Enzyme efficiencies are for instance limited by the diffusion properties of their aqueous environment, which sets an upper bound of approximately 108−1010M−1s−1 for the ratio kcat/KM (Alberty and Hammes 1958; Zhou and Zhong 1982). Nearly optimal enzymes indeed seem to exist, as exemplified by triosephosphate isomerase (TIM) whose ratio is close to this theoretical ceiling (Knowles and Albery 1977). But they are uncommon: most enzymes appear to be only moderately efficient and far off these physical limits—including enzymes immediately flanking TIM in the glycolysis metabolic pathway (Davidi et al. 2018). Indeed, Bar-Even et al. (2011) have analyzed a data set of several hundreds of enzymes and found a wide diversity among enzyme parameters, thus sketching a puzzling pattern that has far more in common with a zoo than it looks like variations around an archetypal form. This wide distribution of enzyme features could partly be explained by differences between enzyme behavior in vivo and in vitro. Such differences are expected, first because diffusion in a test tube is hardly comparable to diffusion in the cytoplasm (Ellis 2001; Rivas et al. 2004; Zhou et al. 2008; Rivas and Minton 2018). As the cytoplasm gets packed, the cell approaches a state where molecules are less mobile, hindering substrate-enzyme encounters (Muramatsu and Minton 1988; Zimmerman and Minton 1993; Blanco et al. 2018). In this regard, KM values are likely underestimated in vitro, and enzymes should perform less efficiently in vivo. Simultaneously, macromolecular crowding can sometimes improve catalytic activity in vivo, making specificity constants kcat/KM higher than their in vitro estimates (Ralston 1990; Ellis 2001; Jiang and Guo 2007; Pozdnyakova and Wittung-Stafshede 2010). Crowding effects are obviously important for our understanding of enzyme evolution but, alone, they are definitely too weak to explain the wide variability across enzymes insofar as their reported estimates typically lie in the range of one order of magnitude (Davidi et al. 2016). Another source of explanation to the observed distribution of enzymes (in)efficiencies is a failure of Evolution to consistently optimize them, possibly due to physical constraints. Indeed, Heckmann et al. (2018) have shown how a variety of kcats may evolve provided that some of them are physically constrained. Besides the diffusion limit already mentioned, constraints on enzyme evolution might include an intrinsic trade-off (Gudelj et al. 2010; Stiffler et al. 2015) that originates from the dependency of both kcat and KM on intermediate energy profiles (Heinrich et al. 1991). Nonetheless, this trade-off is scarcely observable among evolved enzymes—Bar-Even et al. (2011) report a coefficient of determination around 0.09 for the correlation between  log 10(kcat) and  log 10(KM)—suggesting that it can be overcome. Other constraints may exist and be specific of a given reaction (Klipp and Heinrich 1994)—for example, reaction reversibility—potentially explaining a part of the variance in enzyme efficiencies. It remains that estimating constraints on all individual enzymes appears like a daunting task, which could be guided, in part, by the identification of deviations from evolutionary predictions. Following this idea, the premise of our theoretical investigation into the origins of enzyme diversity is that it results mainly from unconstrained evolution, such that the reported differences may be caused by the combined action of selection and genetic drift. It is important to notice that the information we have is partial, as an enzyme’s activity is the joint result of its kinetic constants and cellular concentration, perhaps also contributing to the reported variance in the former. In fact, Davidi et al. (2016)’s method to determine in vivo turn-over rates lends some credence to the idea that increased levels of expression make up for lower kinetic constants (Davidi et al. 2018). It is therefore obvious that an enzyme’s expression needs to be considered as another dimension of its activity, especially since it has been shown that the evolutionary tuning of gene expression can happen very quickly (Dekel and Alon 2005). Concomitantly, an enzyme’s activity can be impacted by protein misfolding, which reduces the effective enzyme concentration (Drummond et al. 2005; Yue et al. 2005; Tokuriki and Tawfik 2009; Echave and Wilke 2017) while also impacting fitness by enhancing protein erroneous interactions (Yang et al. 2012) and the formation of toxic protein aggregates (Bucciantini et al. 2002; Sabate et al. 2010; Geiler-Samerotte et al. 2011). Protein stability is thus under strong purifying selection to avoid the deleterious effects of misfolding (Drummond and Wilke 2008). Accordingly, it has been shown that proteins have evolved to stand beyond a stability threshold (Bloom et al. 2005), although marginally (Taverna and Goldstein 2002). Because mutations are on average destabilizing, this definitely narrows down the spectrum of adaptive mutations (Shoichet et al. 1995; DePristo et al. 2005; Weinreich et al. 2006; Tokuriki et al. 2007, 2008; Lunzer et al. 2010). Nevertheless, several studies have reported the existence of a genotype space where activity can be optimized without compromising stability (Schreiber et al. 1994; van den Burg and Eijsink 2002; Bloom et al. 2004; Knies et al. 2017; Miller 2017). Even when improving function requires the fixation of destabilizing mutations, compensatory mutations can in principle cancel out stability losses arising from active site evolution (DePristo et al. 2005; Tokuriki et al. 2008; Tokuriki and Tawfik 2009; Storz 2018). Adaptive evolution may even be facilitated by preexisting mutational robustness against misfolding (Bloom et al. 2006, 2007). Therefore, although the requirement of a stable, correctly folding protein may sometimes slow down the evolutionary process, it is rather unlikely that stability explains the distribution of enzyme kinetic parameters albeit marginally. Enzyme kinetics evolution has often been considered theoretically through the lens of flux control (Burns et al. 1985; Clark 1991; Fell 1992; Kacser et al. 1995; Yi and Dean 2019). Indeed, the control of the flux in a metabolic pathway is shared between all enzymes, in what is known as the summation theorem (Kacser and Burns 1973; Heinrich and Rapoport 1974). Thence, because the sum of control coefficients must equal 1 within a pathway, if all enzymes have similar kinetic parameters, none of them exerts a strong influence (Dean 1995). But if one enzyme departs from this trend and becomes inefficient, it exerts a strong control at the expense of others (Dykhuizen and Dean 1990). This leads to diminishing-returns epistasis in which the fitness landscape flattens because, as an enzyme becomes more efficient, subsequent mutations have smaller effects (Kacser and Burns 1973; Dykhuizen et al. 1987; Tokuriki et al. 2012), a finding that has since received empirical confirmation (Fell 1992; Dean 1995; Lunzer et al. 2005; Yi and Dean 2019; Chou et al. 2014). Hartl et al. (1985) and Dean et al. (1986) have considered such a fitness landscape under a population genetics framework to conclude that enzymes may quickly reach a fitness plateau and evolve on nearly neutral landscapes (Ohta 1992). Nonetheless, these studies fall short of explaining why inefficient enzymes having stronger control do not evolve higher activities (Yi and Dean 2019). In these models as in most, an enzyme’s efficiency is captured by its activity, generally represented by a composite of kcat, KM and enzyme concentration (Hartl et al. 1985; Clark 1991; Chou et al. 2014; Kaltenbach and Tokuriki 2014), such that the distinct evolutionary dynamics of these parameters, together with an enzyme’s concentration, is ignored. This reduction of an enzyme’s dimensionality goes against the empirical observation that each dimension may have a differential impact on fitness in the context of antibiotic resistance (Walkiewicz et al. 2012; Stiffler et al. 2015; Rodrigues et al. 2016) and that each is thus necessary to predict evolutionary outcomes (Walkiewicz et al. 2012). Perhaps more importantly, Heinrich et al. (1991) and Schuster et al. (2008) have pointed out that these modeling frameworks assume a constant value for either or both concentrations of the first substrate and of the final product (Orth et al. 2010), whereas evolution should instead maximize the amount of final products generated. Klipp and Heinrich (1994) found that the aforementioned concentrations indeed have a major influence on optimal rate constants under certain assumptions. Likewise, nutrient uptake is most often not considered explicitly in existing models of enzyme evolution, while it is obviously critical in the competition for resources (Dykhuizen and Dean 1994). Nutrient uptake occurs when metabolites move inwards across cell membranes; it may rely on membrane permeability only (passive diffusion) or involve channels and carrier proteins, be they transporters or cotransporters (Stein 1986a). Here, we build a model that explicitly includes passive (PD hereafter) or facilitated diffusion (FD) followed by an unbranched metabolic pathway to study how resource availability coupled to transport modulates the evolution of enzymes along the pathway. In ecological scenarios where individuals compete for resources, natural selection should favor genotypes that maximize the net intake of molecules and their transformation, which are linked under both PD and FD. Based on this premise, we confirm that the evolution of enzyme kinetic parameters kf and kcat takes place on cliff-like fitness landscapes where a fitness plateau covers a wide part of the relevant parameter space. Kinetic parameters have codependent but distinct evolutionary dynamics—and thus distinct sensitivities to certain parameters of the model—such that the shape of the plateau can be modulated by changing parameters of the model within realistic ranges. We show that this fitness landscape depends on features of transporters that initiate a metabolic pathway, along with parameters that vary among enzymes within a pathway, like the tolerance to high concentrations of intermediate metabolites or the reversibility of reactions. We further demonstrate, using a simple population genetics model, that the evolutionarily expected features of an enzyme should be predictable, even though enzymes evolve near-neutrally on the fitness plateau. This is because the model includes slightly biased mutations that tend to produce a majority of less efficient enzymes. We thus postulate that the wide variety of enzyme features reported might be explained in a large part by differences in the shape of their fitness landscapes. While testing this hypothesis will require extensive information about individual enzymes, we made a small step in this direction, showing that enzymes involved in metabolic pathways with different types of transporters exhibit differences that our model qualitatively predicts. Results Passive Diffusion is Generally Inadequate to Sustain Cell Metabolism In the version of our model in which intake relies on passive diffusion (PD), the net uptake of a nutrient is a direct outcome of its concentration gradient, and therefore of how efficiently the first enzyme catalyzes its transformation inside the cell. Assuming that fitness is proportional to the flux of product of this reaction, we find that the fitness landscape has a cliff-like shape with fitness increasing steeply as parameters kcat and kf increase (see Supplementary materials, section Text S1). The precise shape will not be commented in detail here, for it is very similar to landscapes obtained under facilitated diffusion (FD, treated in the rest of this manuscript). Importantly, our results indicate that PD can only sustain a small part of the metabolism of most living cells given cell permeabilities reported in the literature (Wood et al. 1968; Milo et al. 2010), suggesting that this process may not be a determining factor in the evolution of enzymes along metabolic pathways. Indeed, even extremely efficient enzymes, harboring values of kcat and kcat/KM close to their physical limits, yield low inward fluxes that approach 10−2 mM.s−1 when considering a spherical cell with a diameter D=1μm ⁠. To get a sense of how low these fluxes are, we calculated the maximum cell size they can theoretically sustain. Considering that basal metabolic demands are approximately proportional to the cell volume and using estimates by Lynch and Marinov (2015) for this relationship, we predicted the maximum size enabled by sugar passive diffusion (see Materials and Methods section). Setting a (conservatively high) medium concentration in glucose [G]=1M yields a theoretical volume ceiling Vest=0.84μm3 ⁠. Nearly all eukaryotes, and most prokaryotes are de facto larger than this threshold (Heim et al. 2017), which might help explain the apparent ubiquity of FD. While this demonstration hinges on numbers for sugar uptake, which may arguably be the task requiring the highest flux, PD may be limiting for many other metabolites (Boer et al. 2010), depending on their permeability and availability in the environment: even for very high amino-acids concentrations that may only be met in multicellular organisms (Schmidt et al. 2016) and assuming the highest observed permeability for such metabolites (Chakrabarti 1994), these levels are orders of magnitude lower than with FD (see Supplementary material—section Text S1 for PD results). General Shape of the Fitness Landscape under Facilitated Diffusion For most metabolites, FD relies on the specific binding of the substrate to transmembrane carrier proteins (transporters hereafter), followed by its translocation to the other side of the membrane (Danielli 1954; Kotyk 1967; Stein 1986b). Our model builds on ter Kuile and Cook (1994)’s approach to model FD, considering the simplifying assumption of symmetric transport. Within this framework, FD operates on the concentration gradient (Bosdriesz et al. 2018) and is susceptible to saturation, represented by constant KT—similar to KM in the Michaelis–Menten equation—and an interaction constant α (see Materials and Methods section for details). We assessed how this saturation phenomenon influences the selection pressure acting on forward enzyme kinetic parameters (kf and kcat ⁠) under various scenarios. In order to depict a fitness landscape representative of an average enzyme, we first consider a situation where transporters induce a moderately low rate VTm and saturate with a relatively high affinity—corresponding to a low KT (fig. 1). In this situation, the inward flux at steady-state (which, as argued in the introduction, can be considered representative of fitness) forms a plateau when the upstream enzyme in the metabolic pathway has high kcat and kf. This low equilibrium flux elasticity coincides with the saturation theory (Wright 1934; Kacser and Burns 1973; Hartl et al. 1985; Dykhuizen et al. 1987; Dean 1995; Yi and Dean 2019), especially with its version incorporating facilitated diffusion (ter Kuile and Cook 1994; Dean 1995). The flux plateau is delineated by parallel isoclines (solid and interrupted lines in fig. 1) oriented in the bottom-right direction of the landscape for intermediate values of kcat and kf, such that decreasing kf by one order of magnitude can be compensated by a similar increase in kcat ⁠. While this mutual dependency holds even for high kf values as long as kcat is not critically low (i.e., when kcat>10−3 ⁠), it stops when kcat≥103 ⁠, where increasing kcat no longer improves fitness. Besides, the influence of kcat and kf is not strictly equivalent, since the increase in flux is more gradual in response to kf. Fig. 1 Open in new tabDownload slide The flux of product following substrate uptake by transporters and conversion by a dedicated enzyme depends on kinetic parameters kf and kcat ⁠. This landscape is based on a moderately low flux at saturation VTm=1μM.s−1 close to those measured for amino acids and nucleosides in E. coli (Zampieri et al. 2019). We also set the transport saturation ratio [Sout]/KT to 10 such that the FD process approaches saturation, and relatively high transporter affinity KT=50μM ⁠, also in line with estimates for nucleosides (Griffith and Jarvis 1996; Xie et al. 2004). Other parameter values include kr=103s−1 and [Etot]=1mM ⁠. The color gradient indicates the absolute and normalized (with a maximum flux of 1) values of equilibrium flux. Fig. 1 Open in new tabDownload slide The flux of product following substrate uptake by transporters and conversion by a dedicated enzyme depends on kinetic parameters kf and kcat ⁠. This landscape is based on a moderately low flux at saturation VTm=1μM.s−1 close to those measured for amino acids and nucleosides in E. coli (Zampieri et al. 2019). We also set the transport saturation ratio [Sout]/KT to 10 such that the FD process approaches saturation, and relatively high transporter affinity KT=50μM ⁠, also in line with estimates for nucleosides (Griffith and Jarvis 1996; Xie et al. 2004). Other parameter values include kr=103s−1 and [Etot]=1mM ⁠. The color gradient indicates the absolute and normalized (with a maximum flux of 1) values of equilibrium flux. Furthermore, and contrary to the textbook picture whereby most biological reactions are not limited by diffusion at all (Bar-Even et al. 2011; Sweetlove and Fernie 2018), increasing an enzyme’s association rate kf—be it through its diffusivity or its binding rate—may still enhance the equilibrium flux when diffusion is substantially faster than catalysis. Properties of Facilitated Diffusion Modulate the Landscape To explore the effect of FD kinetics on the evolution of enzymes in the metabolic pathway, we studied the influence of KT—the affinity of the transporter for the substrate—and VTm—the maximum transport rate—still assuming that the substrate is close to saturation (⁠ [Sout]/KT=10 ⁠). We find that increasing the transport flux VTm exerts a positive selection pressure on kinetic parameters for the upstream enzyme (i.e., for increasing kcat and kf). The plateau is shifted accordingly (see fig. 2A), towards the top-right corner of the landscape, at a distance that corresponds to the magnitude of the change in VTm. Increasing the affinity of the transporter (i.e., decreasing KT), however, selects for higher kf (the isoclines are displaced to the right and the fold change is similar to that of KT) but has no other visible influence on kcat than increasing its codependency with kf, a result that holds regardless of the flux at saturation VTm (notice that we only considered high VTms, larger than in the average case, because these cases are more likely to be under directional selection). Fig. 2 Open in new tabDownload slide Features of a transporter have an impact on the flux landscape for upstream enzymes, as shown by the 0.9 isoclines—above which the relative flux is >90% – that delineate the fitness plateau for each set of parameter. (A) low (⁠ KT=0.1M ⁠) and high (⁠ 10μM ⁠) transporter affinities are considered, in combination with low (⁠ VTm=10−6M ⁠), moderate (⁠ 10−4.5M ⁠) or high maximum flux (⁠ 10−3M ⁠). Increasing KT extends the plateau only towards the left part of the landscape, allowing enzymes with lower kf on the plateau, whereas decreasing VTm extends the plateau in both directions. (B) the shape of the fitness plateau is however little dependent on the saturation of the transporter, for a transporter with moderate flux (⁠ VTm=10−4.5M.s−1 ⁠; the effect is identical for higher VTm, see supplementary fig. S2, Supplementary Material online). Other parameter values: kr=1000/s, [Etot]=1mM and [Senv]=10×KT ⁠. Fig. 2 Open in new tabDownload slide Features of a transporter have an impact on the flux landscape for upstream enzymes, as shown by the 0.9 isoclines—above which the relative flux is >90% – that delineate the fitness plateau for each set of parameter. (A) low (⁠ KT=0.1M ⁠) and high (⁠ 10μM ⁠) transporter affinities are considered, in combination with low (⁠ VTm=10−6M ⁠), moderate (⁠ 10−4.5M ⁠) or high maximum flux (⁠ 10−3M ⁠). Increasing KT extends the plateau only towards the left part of the landscape, allowing enzymes with lower kf on the plateau, whereas decreasing VTm extends the plateau in both directions. (B) the shape of the fitness plateau is however little dependent on the saturation of the transporter, for a transporter with moderate flux (⁠ VTm=10−4.5M.s−1 ⁠; the effect is identical for higher VTm, see supplementary fig. S2, Supplementary Material online). Other parameter values: kr=1000/s, [Etot]=1mM and [Senv]=10×KT ⁠. This specific effect on the affinity of the upstream enzyme is likely due to a competition between the transporter—which can transport the substrate in both directions—and the enzyme, which harvests the substrate at a rate that depends on the dissociation constant KD=kr/kf ⁠. It should be noted that nutrients under lower demands—for example, amino acids—are generally less concentrated in the environment, often coinciding with a higher affinity of their transporter. Therefore, the possible combinations of flux and affinity likely occupy a restricted space of possibilities where flux and affinity are negatively linked (Gudelj et al. 2010; Bosdriesz et al. 2018), which as can be seen in supplementary figure S3A, E, and I results in landscapes that mainly differ by the minimum value of kcat on the plateau. In figure 2A, we have considered ranges of empirical estimates for sugars (high flux with low to moderate affinity) (Stein 1986b; Maier et al. 2002), nucleosides (Griffith and Jarvis 1996) and amino acids (Stein 1986b; Zampieri et al. 2019) (weak to moderate flux with moderate to high affinity), which indeed mainly correspond to these combinations. So far, we have considered transporters saturated by high external substrate concentrations. Relaxing this assumption has little impact on the fitness landscape, except that very low values of kcat (lower than 10−2 in fig. 2B) can only sustain the low influx of transporters far from saturation, but fail to keep up with higher influxes in richer environments. Enzymes Differ among Metabolic Pathways We then superimposed empirical estimates of kinetic parameters over our theoretical fitness landscapes, after substituting parameter kf for its usual empirical counterpart, kcat/KM ⁠. Because kcat/KM=kfkcat/(kr+kcat) ⁠, this approximation only holds when kcat≫kr ⁠. Representing the fitness landscape in this parameter space sets an inaccessible area in the bottomright part of the landscapes where kf would exceed the diffusion limit (gray area on fig. 3). For purposes of inclusiveness, we used kr=102s−1 by default—noting that this limit would be displaced upwards for larger kr (and downwards otherwise). Fig. 3 Open in new tabDownload slide In vitro experimental estimates of kinetic parameters kcat and kcat/KM exhibit different distributions for enzymes involved in different categories of pathways—as identified by Bar-Even et al. (2011)—namely (AFN): amino acids, fatty acids, and nucleotides and (CE): carbohydrates and energy. Corresponding fitness landscapes—differing by transporter features—are superimposed, with the parameter space narrowed down due to the diffusion limit (grey area, set for kr=102s−1 ⁠). The isoclines shown correspond to parameter values typical of sugar transporters (⁠ KT=5mM, VTm=1mM.s−1 ⁠, in red) (Maier et al. 2002) or amino acids transporters (same as in fig. 1, in black). Fig. 3 Open in new tabDownload slide In vitro experimental estimates of kinetic parameters kcat and kcat/KM exhibit different distributions for enzymes involved in different categories of pathways—as identified by Bar-Even et al. (2011)—namely (AFN): amino acids, fatty acids, and nucleotides and (CE): carbohydrates and energy. Corresponding fitness landscapes—differing by transporter features—are superimposed, with the parameter space narrowed down due to the diffusion limit (grey area, set for kr=102s−1 ⁠). The isoclines shown correspond to parameter values typical of sugar transporters (⁠ KT=5mM, VTm=1mM.s−1 ⁠, in red) (Maier et al. 2002) or amino acids transporters (same as in fig. 1, in black). We otherwise used sets of parameters that correspond to typical features of sugar and amino acid/nucleoside transporters to obtain fig. 3. Because we have previously shown that changing the affinity or maximum flux of transporters may move the fitness plateau, our model predicts that enzymes involved in the corresponding pathways (e.g., of sugars and amino acids) should have their own specific distributions. We see that enzymes involved in the central carbohydrate metabolism as categorized by Bar-Even et al. (2011) have on average higher kcat and KM than those metabolizing amino-acids and nucleotides. Our superimposition with the predicted fitness plateaus in figure 3 suggests that there may indeed be an explainable difference between enzymes contributing to carbohydrate processing (in red) and to that of other primary metabolites (in black, e.g., amino acids). We acknowledge that this result implicitly suggests that enzymes within a pathway have evolved on a common fitness landscape, spreading neutrally onto the fitness plateau. This is by no means our interpretation, as this subset of the full data set includes enzymes that differ in many other ways that, as we will see, make each enzyme evolve on its own fitness landscape and thereby potentially explain a large part of this observed variance. Downstream Enzymes also Evolve on Cliff-like Fitness Landscapes One of the factors that make enzymes different along a pathway is their position, such that the fitness landscape in figure 1 may only hold for the most upstream enzyme in a pathway. Indeed, because the flux of the first product in a pathway increases with the substrate gradient across the cell membrane, the upstream enzyme of a given metabolic pathway is selected for efficiency as described above. In contrast, this selection pressure does not apply directly downstream; at steady-state, even inefficient enzymes can in principle process newly formed substrate molecules at an elevated rate, assuming that the concentration of the substrate is allowed to reach any steady-state value. This is an obviously unreasonable assumption, since a part of this standing substrate should be lost by outward diffusion or degradation (Jones et al. 2015; Bosdriesz et al. 2018). The loss of fitness may therefore result from the loss of metabolites in a way that can be modeled by a constant degradation rate ηd (Chou et al. 2014) (assuming that the external environment is infinite, the degradation term can as well represent an efflux). Highly concentrated metabolites may also be involved in widespread nonspecific (Keller et al. 2015) or promiscuous interactions (Khersonsky and Tawfik 2010; Schäuble et al. 2013; Peracchi 2018) that may interfere with other cellular processes; this is well captured by the linear cost as nonspecific interactions should follow Michaelis–Menten kinetics albeit with much lower affinities, hence following an approximately linear relationship up to very high cellular concentrations (see Materials and Methods section for more details). However, for some reactions the accumulation of metabolites may result in the production of toxic compounds (Lilja and Johnson 2017; Niehaus and Hillmann 2020), hence triggering toxicity best modeled as a nonlinear fitness cost (Clark 1991; Wright and Rausher 2010). We first consider a “perfect,” highly concentrated upstream enzyme (⁠ kf=1010M−1s−1, kcat=106s−1, kr=103s−1, [Etot]=10−3M ⁠) and focus on the second enzyme in the pathway, showing that it evolves on a fitness landscape that has a similar shape than described above, still hitting a plateau (fig. 4, with the same parameterization as fig. 1). The degradation rate creates a ceiling for the concentration of the product of the first reaction, such that reducing ηd allows for higher concentrations (see supplementary fig. S4, Supplementary Material online) and makes the flux tolerant to second enzymes with lower kfs, whereas selection on kcat is barely impacted by this parameter. The plateau is therefore extended to the left when high product concentrations are enabled at low ηd (see fig. 4B). The shape of the plateau is little impacted by changes in the efficiency of the first enzyme, especially when it stands on the plateau. These results are almost independent of the transporter initiating the pathway (see supplementary fig. S6, Supplementary Material online for the case of moderate affinity, high flux transporters). Fig. 4 Open in new tabDownload slide Downstream enzymes exhibit similar fitness landscapes as those upstream, with a dependency to degradation parameter ηd. (A) A high degradation rate (⁠ ηd=10−2/s ⁠) results in a fitness plateau for the second enzyme very similar to that of the first enzyme; in the case presented the first enzyme is considered “perfect” in order to draw the fitness landscape of the second enzyme (⁠ kf=1010M−1.s−1, kcat=106s−1, kr=103s−1, [Etot]=1mM ⁠). (B) decreasing the degradation rate allows less efficient enzymes (with lower kcat or kf) to reach the fitness plateau. Considering the first enzyme to be inefficient (⁠ kf=102M−1.s−1, kcat=10−2s−1, kr=103s−1 ⁠) instead of perfect marginally changes the fitness landscape by making organisms tolerant to extremely low kcat ⁠. Other parameter values are identical to fig. 1 (findings are relatively similar for sugar-like transporters, as reported in supplementary fig. S6, Supplementary Material online). Fig. 4 Open in new tabDownload slide Downstream enzymes exhibit similar fitness landscapes as those upstream, with a dependency to degradation parameter ηd. (A) A high degradation rate (⁠ ηd=10−2/s ⁠) results in a fitness plateau for the second enzyme very similar to that of the first enzyme; in the case presented the first enzyme is considered “perfect” in order to draw the fitness landscape of the second enzyme (⁠ kf=1010M−1.s−1, kcat=106s−1, kr=103s−1, [Etot]=1mM ⁠). (B) decreasing the degradation rate allows less efficient enzymes (with lower kcat or kf) to reach the fitness plateau. Considering the first enzyme to be inefficient (⁠ kf=102M−1.s−1, kcat=10−2s−1, kr=103s−1 ⁠) instead of perfect marginally changes the fitness landscape by making organisms tolerant to extremely low kcat ⁠. Other parameter values are identical to fig. 1 (findings are relatively similar for sugar-like transporters, as reported in supplementary fig. S6, Supplementary Material online). The shape of the negative relationship between metabolite concentration and fitness can be important (supplementary figs. S7–S9, Supplementary Material online), as it can make the fitness landscape of an enzyme dependent on the overall flux of the metabolic pathway, and therefore on other enzymes in the pathway. Indeed, low general fluxes (as modeled by an inefficient first enzyme in supplementary figs. S7 and S8, Supplementary Material online) make the metabolite concentration below its toxicity threshold, therefore making organisms tolerant to enzymes with lower kf and kcat ⁠. Taken together, these results show that the precise epistatic relationship between enzymes in a pathway will depend on the exact cost function applied, with a linear cost generating epistasis for kcat only and a nonlinear cost possibly impacting both kf and kcat ⁠. The Reversibility of Reactions also Matters Reversibility is an intrinsic feature of chemical reactions that cannot be directly overcome by Evolution (Haldane 1930; Cornish-Bowden 1979). A highly reversible reaction corresponds to a large intrinsic equilibrium constant Keq=[S]eq/[P]eq (Klipp and Heinrich 1994), and results in higher backward than forward rates in the following chemical equation: E+S⇌krkfES⇌kinhkcatE+P1,(3) where kinh represents the rate at which enzyme and product combine back. Such a (reversible) reaction could in principle influence the selective pressure acting on the following enzyme in the pathway, for both enzymes compete to process the same metabolite P1. We thus quantified how reversibility affects the evolution of an enzyme downstream (supplementary figs. S10 and S11, Supplementary Material online). The equilibrium constant Keq has a similar (nonlinear) impact on the fitness landscape of the second enzyme to that of the degradation rate, with a highly reversible upstream enzyme exerting a selection pressure downstream towards an increase of kinetic parameters (supplementary fig. S10A, Supplementary Material online). Indeed, increasing Keq moves the fitness plateau toward the upper-right corner in the (kf, kcat ⁠) parameter space, hence selecting for more efficient downstream enzymes. The effect appears linear, except for very low values of Keq where metabolite accumulation exerts a dominant role in shaping the fitness landscape (through the degradation rate ηd, set to a low residual value). Therefore, the reversibility of the upstream reaction appears like a critical parameter for the evolution of an enzyme. Evolutionary Dynamics of Enzyme Kinetic Parameters How much variance in evolutionary outcomes these differences in fitness landscapes may explain is contingent on the interplay between selection, mutation, and drift. Small differences in an isocline position should indeed be of little importance if populations perform random walks on the fitness plateau, for instance. To approach how populations evolve on our mathematically derived fitness landscapes, we built a simple population genetics model in which absolute fitness is directly proportional to the flux arising from the first enzyme at steady-state—which itself equals the net inward flux of nutrients. Two different levels of metabolic demands were considered, corresponding to parameter values of amino acids/nucleosides and sugar transporters (supplementary fig. S3A and I, Supplementary Material online). In this instance of the model, only kcat and kf were susceptible to evolve through mutations. Mutational effects on  log 10kcat and  log 10kf were drawn from independent normal distributions with mean b≤0 ⁠, and the absolute value of b setting the intensity of a mutational bias towards less efficient parameter values, as has been widely documented in many contexts (Eyre-Walker and Keightley 2007; Serohijos et al. 2012; Heckmann et al. 2018). The standard deviation of the distribution of mutational effects equals 0.3 such that most mutations explore the neighboring parameter space, seldom changing a parameter by more than one order of magnitude (one  log 10 unit) in compliance with empirical estimates (Carlin et al. 2016). Since the relation between kinetic parameters may be constrained—for example, due to shared properties of the energy profile of a reaction—we tested the influence of negative and positive relationships using bivariate normal distributions, with three different values of ρ (see Materials and Methods section for details). In the absence of mutational bias (b = 0), simulated enzymes spread over the fitness plateau, as expected (supplementary fig. S16A, Supplementary Material online for low flux, supplementary fig. S17A, Supplementary Material online otherwise). The onset of the plateau depends on the strength of drift and hence derive from the effective population size Ne, following the classical expectation that selection becomes inefficient when Ne×s<1 (Wright 1931; Kimura 1968). Introducing a mutational bias that makes enzyme kinetics less efficient on average has a strong effect on both kcat and kf, preventing simulated enzymes from improving far above the drift barrier (fig. 5A for low flux, fig. 5B otherwise). Even weak biases (⁠ b=−0.1 ⁠) lead to enzymes evolving in the vicinity of the isocline where Ne×s≈1 ⁠. Increasing the strength of this bias to 0.2 only slightly decreases the population variance around this expectation. Finally, mutational correlations do not impact much the distribution of evolutionary outcomes (supplementary fig. S18, Supplementary Material online). Fig. 5 Open in new tabDownload slide Population genetic simulations predict that enzymes should reach a predictable set of features when mutation biases towards lower efficiencies are considered (see supplementary fig. S17, Supplementary Material online for the case of an absence of bias). Indeed, the mutation selection drift equilibrium establishes close to an isocline indicative of effective selection that depends on the effective population size Ne. The cases considered here are that of a transporter with a low flux at saturation and high affinity (A; VTm=1μMs−1 and KT=10μM ⁠) and one with a high flux at saturation but low affinity (B; VTm=1mMs−1 and KT=100mM ⁠) with effective population sizes ranging from 102 to 105 (different colors) and two strengths of the mutational bias (the absence of mutational bias was also considered, see Supplementary Material online). Each of 30 independent simulations for each scenario is represented a dot in the “empirical” parameter space (⁠ kcat, kcat/KM ⁠), but only kcat and kf were susceptible to evolve. kr is set to 103s−1 such that the grey part of the parameter space is inaccessible to enzymes that would otherwise exceed the diffusion limit. Fig. 5 Open in new tabDownload slide Population genetic simulations predict that enzymes should reach a predictable set of features when mutation biases towards lower efficiencies are considered (see supplementary fig. S17, Supplementary Material online for the case of an absence of bias). Indeed, the mutation selection drift equilibrium establishes close to an isocline indicative of effective selection that depends on the effective population size Ne. The cases considered here are that of a transporter with a low flux at saturation and high affinity (A; VTm=1μMs−1 and KT=10μM ⁠) and one with a high flux at saturation but low affinity (B; VTm=1mMs−1 and KT=100mM ⁠) with effective population sizes ranging from 102 to 105 (different colors) and two strengths of the mutational bias (the absence of mutational bias was also considered, see Supplementary Material online). Each of 30 independent simulations for each scenario is represented a dot in the “empirical” parameter space (⁠ kcat, kcat/KM ⁠), but only kcat and kf were susceptible to evolve. kr is set to 103s−1 such that the grey part of the parameter space is inaccessible to enzymes that would otherwise exceed the diffusion limit. Our results suggest a strong effect of the effective population size on enzyme evolution, such that species with Ne above 105 (Bobay and Ochman 2018, most unicellular organisms) should express extremely efficient enzymes. This appears to not be the case, as for instance Eukaryotes and Prokaryotes display similar enzymes despite large differences in effective population sizes (Bar-Even et al. 2011). As we will later discuss, this conundrum might resolve when considering the smaller size of organisms forming larger populations, making them more sensitive to noise in gene expression and favoring higher concentrations. Notwithstanding this issue, the prediction of enzymes evolving a predictable set of kinetic parameters strongly suggests that a large part of the broad variance in enzyme features is due to differences in the selective context experienced by each, thereupon requiring further investigation on the dependency of the position of the fitness plateau to parameters of our model. The Joint Evolution of Enzyme Concentrations and Kinetic Parameters Hitherto, we have considered enzymes to be highly concentrated, an assumption that we now relax since it is an important component of the presumed kinetic activity (Koshland 2002). Predictably, increasing the concentration of the first or second enzyme in a pathway releases the selection on their kinetic parameters (Noor et al. 2016), producing larger fitness plateaus as an enzyme concentration increases (see supplementary figs. S12B and S13B, Supplementary Material online for this influence in different contexts). Due to the compensatory effects between concentration and activity, we anticipate that the joint evolutionary dynamics of the concentration and kinetic parameters should yield a negative correlation between them, as reported by Davidi et al. (2016, 2018). Despite their common role on reaction efficiency, enzyme concentration expectedly responds to very different selection pressures than kinetic parameters, as increased gene expression levels come with costs (Wagner 2005; Lang et al. 2009; Scott et al. 2010; Noor et al. 2016; Kafri et al. 2016). Indeed, producing extra proteins requires both energy and matter (Novick and Weiner 1957; Stoebel et al. 2008; Wagner 2005; Lynch and Marinov 2015) and may impede the efficiency of physical processes that rely on an optimal intermediate content (Dong et al. 1995; Dill et al. 2011; Andrews 2020). We designed a new instance of our population genetics model to study the tangled evolution of kinetic constants and enzyme concentration, introducing two of these costs: 1) the cost of producing proteins c, considered to be proportional to concentration (Wagner 2005; Chou et al. 2014; Lynch and Marinov 2015) and 2) the exponential cost of an increase in macromolecular crowding, which hinders diffusion and thus slows down reactions (Dill et al. 2011; Schavemaker et al. 2018; Andrews 2020) (see supplementary fig. S15, Supplementary Material online for the resulting fitness landscapes of enzyme concentration). The two types of costs result in a different shape of the fitness landscape, with the noticeable difference that evolutionarily expected concentration depends on Ne when the cost of production is considered (supplementary fig. S19, Supplementary Material online) but not with crowding effects (supplementary fig. S20, Supplementary Material online). With a combination of the two costs, enzyme concentrations decrease with Ne and production costs, resulting in the evolution of higher kinetic constants (fig. 6). This is because at higher effective sizes, direct costs of protein production are large enough to incur effective selection for lower protein expression. This is no longer the case when Ne decreases, such that the major force driving the optimization of enzyme concentration becomes that opposing macromolecular crowding, which is less sensitive to Ne (as shown in supplementary fig. S19, Supplementary Material online). The balance between these two selective forces, and the dependency to Ne, obviously depend on the relative importance of these costs (supplementary fig. S20, Supplementary Material online), itself depending on many parameters (protein length, molecular weight, etc.) that should only make enzymes marginally different within a given species (when their activity evolves on similar fitness landscapes). Fig. 6 Open in new tabDownload slide Simulations of the joint evolution of enzyme concentration and kinetic parameters, with a 2-fold cost of enzyme overexpression (the direct metabolic cost and the indirect cost of cell packing). The case considered here is that of a transporter with a high flux at saturation and low affinity (⁠ VTm=1mMs−1 and KT=1mM ⁠) under a high mutational bias on kinetic constants (⁠ b=−0.2 ⁠). Two different costs of protein production c are considered along with four effective population sizes ranging from 102 to 105. We ran 30 independent simulations for each scenario, each represented by a dot in the “empirical” parameter space as described in fig. 5. Fig. 6 Open in new tabDownload slide Simulations of the joint evolution of enzyme concentration and kinetic parameters, with a 2-fold cost of enzyme overexpression (the direct metabolic cost and the indirect cost of cell packing). The case considered here is that of a transporter with a high flux at saturation and low affinity (⁠ VTm=1mMs−1 and KT=1mM ⁠) under a high mutational bias on kinetic constants (⁠ b=−0.2 ⁠). Two different costs of protein production c are considered along with four effective population sizes ranging from 102 to 105. We ran 30 independent simulations for each scenario, each represented by a dot in the “empirical” parameter space as described in fig. 5. Discussion Most enzymes have been considered to be only moderately efficient (Bar-Even et al. 2011), if not sloppy (Bar-Even et al. 2015). This claim was put into perspective by Newton et al. (2018) who argued that the link between fitness and enzyme efficiencies is complex and may be partly enzyme dependent, such that all enzymes may not evolve on a common fitness landscape. Through this work, we have developed a model where enzyme efficiencies are mechanistically linked to fitness through the impact of nutrient gradients on the production of metabolites. Our results emphasize that an enzyme’s fitness landscape—and the resulting mutation–selection–drift balance—may indeed be largely context dependent, possibly explaining a large part of the extreme observed variance in enzyme features. At first sight, all enzymes evolve on fitness landscapes that have the same general shape, with a fitness plateau surrounded by a steep slope. While this shape is usual in models of enzyme evolution (Hartl et al. 1985; Kaltenbach and Tokuriki 2014; Yi and Dean 2019), in our model the landscape is drawn in the parameter space formed by the two forward kinetic parameters kcat and kf, instead of a composite “efficiency” whose relevance is questionable (Koshland 2002; Eisenthal et al. 2007). Our model allows to predict the precise position of the fitness plateau in various contexts, showing that model parameters may have a selective impact on kf, kcat ⁠, or both, thereby confirming the relevance of considering their distinct evolutionary dynamics. We have shown that the exact position of the plateau is important through a population genetics model including mutational biases that produce less efficient enzymes at a slightly higher frequency. Despite their small effect, these biases are sufficient to have a significant impact on the evolutionary dynamics occurring on the fitness plateau, preventing enzymes to explore the parameter space far away from an isocline whose precise value can be predicted. Because the mutation–selection–drift balance occupies a narrow part of the landscape, this makes the evolution of an enzyme, in principle, highly predictable. Likewise, we anticipate that differences between enzymes should largely be explained by differences in the shapes of their individual fitness landscapes. Overall, the selective pressure acting on an enzyme results from an interplay between several biochemical factors. We have effectively found that the shape of the fitness landscape is first governed by features of the transporter initiating a pathway, especially the maximum flux they can sustain. Using parameters that correspond to empirical estimates for sugars and amino acids/nucleosides, we have found that enzymes contributing to subsequent metabolic pathways should be different, with those in the “sugars” pathway being selected for faster kinetics. While sharing a common transporter, enzymes along a pathway are also subject to a variety of local chemical contexts, making each evolve on its own unique fitness landscape. This could explain, at least in part, the large within-pathway variance of enzyme kinetic parameters. Physical constraints may for instance act differentially on different enzymes, as exemplified by the intrinsic reversibility of a reaction that fuels the selective pressure towards higher efficiency in downstream enzymes. This may contribute to explain the high efficiency of a few enzymes like TIM (Williamson et al. 1967; Davidi et al. 2018). One way to compensate for low kinetic constants is to enhance the level of expression of an enzyme, as confirmed by our model—concentration indeed has a strong influence on the fitness landscape of kf and kcat ⁠. Nonetheless, concentration and kinetic parameters face very distinct selection regimes: while the latter are both under directional selection, vanishing at high efficiencies, concentration is under stabilizing selection—owing to a combination between its positive impact on the flux and the adverse costs to high expression—as already pinpointed by Chou et al. (2014). Their joint evolution is complex because the position of the concentration optimum depends on an enzyme’s kinetic constants, whose fitness landscape itself depends on concentration. This results in a slightly increased variance in enzyme efficiencies compared to simulations with fixed concentrations, along with a complex relationship with genetic drift, because smaller populations tend to tolerate higher enzyme concentrations and, therefore, evolve less efficient enzymes. It should be noted that our model does not consider another selection pressure on enzyme concentrations that arises from noise in gene expression, as argued by Wang and Zhang (2011). Indeed, low expression results in detrimental noise that should be avoided by pushing enzyme concentrations towards higher values in small organisms like Prokaryotes (see supplementary Text S6, Supplementary Material online for an estimate of this effect). This could result in a different relationship between Ne and enzyme efficiencies than considered here, possibly explaining the confusing observation that species with larger populations (and smaller sizes) do not express markedly more efficient catalysts. Furthermore, an enzyme’s effective concentration can also increase through compartmentalization (Ovádi and Saks 2004; Diekmann and Pereira-Leal 2013; Cornejo et al. 2014) and substrate channeling (Welch and Easterby 1994; Huang et al. 2001; Sweetlove and Fernie 2018), within the limits imposed by noise, and modify the selective pressure acting on kinetic parameters. This illustrates that rather than making precise predictions, our study aims at making the strong claim that selection acting on enzyme features is important for their diversity, possibly largely overcoming the diversity arising from neutral processes. If this is indeed the case, trends in enzyme evolution can be predicted—as it was shown empirically in the context of antibiotic resistance (Walkiewicz et al. 2012)—and further improvements of this model should allow one to predict the expected features of individual enzymes. Such improvements are made easier by the use of a mechanistic framework, where fitness arises as enzymatic efficiency helps ingesting nutrients and win the competition for resources. This framework could even be enriched by other dimensions relevant to the genotype-phenotype-fitness map (Bershtein et al. 2017; Echave 2019; Kinsler et al. 2020). Unfortunately, mechanistic does not mean free of a definition of fitness, as we have here assumed that the latter is proportional to metabolic flux, hence considering each flux in isolation. Fitness instead results from a wide range of metabolic pathways that combine together and should all be competitive to certain degrees. How global epistasis builds up (Weinreich et al. 2013; Otwinowski et al. 2018; Reddy and Desai 2021), and genetic drift acts in this context, is far from obvious (Iwasa et al. 2004; Weinreich and Chao 2005; Weissman et al. 2009). But this should not impact much how enzymes evolve in old, overall efficient pathways, as any impediment in efficiency should have a relatively independent effect on fitness in this context, as captured by our model. Understanding these complex interactions between pathways would nevertheless be crucial to understand how metabolic pathways arose and improved, likely from a highly inefficient state during early steps in the evolution of life on Earth (Kacser and Beeby 1984; Schmidt et al. 2003; Heckmann et al. 2018). Materials and Methods Quantifying the Maximum Size for Cells Using Passive Diffusion If a cell is to be viable, it has, at least, to uptake enough glucose to compensate for basal metabolism—metabolism that allows to maintain the same cell size for nonactively growing cells (Lynch and Marinov 2015)—leading to the following equation: ϕPD=CM ⁠, with ϕPD the uptake through passive diffusion and CM the basal metabolism demand. To calculate the maximum size a cell can reach using only passive diffusion, we relied on the formula CM=0.39V0.88(109ATP/hr) estimated in (Lynch and Marinov 2015). We also assumed the cell to be of spherical shape, both concentrations—inside and outside the cell—to be constant with the cellular concentration staying so low that it can be overlooked, meaning that the uptake resulting from passive diffusion can merely be written as ϕPD=P.[Sout].SAsphereVsphere ⁠, where SAsphere and Vsphere are the surface area and the volume of a sphere, and P represents the cell permeability and was measured to 10−6μm−1 (Wood et al. 1968) for glucose. Finally, we considered that each glucose yields 30 ATP molecules (Rich 2003). Flux Sustained by the First Enzyme When assessing the flux of product made by the first enzyme in a pathway, both (PD) and (FD) result in similar sets of equations; we focus on FD here (see supplementary Text S5 - Mathematical appendix, Supplementary material online for a comparison with PD). FD typically relies on the specific binding of substrate molecules—located outside the cell—by transmembrane carrier proteins, followed by their translocation into the cytoplasm (Danielli 1954; Wilbrandt and Rosenberg 1961; Kotyk 1967; Bosdriesz et al. 2018). This specific process obeys Michaelis–Menten-like kinetics when transport is assumed to be symmetric (Kotyk 1967), which can be modeled through Brigg–Haldane equations (Briggs and Haldane 1925; Haldane 1930; Stein 1986b): d[Sin]dt=VTm.[Sout]−[Sin]KT+([Sout]+[Sin])+α.[Sout][Sin]KT(4) with: {VTm: the maximum rate of a given carrier protein;KT: a constant inversely proportional to the transpor ter efficiency;α : the Kotyk interactive constant capturing the disequilibrium between boundand free transporters.  By construction, α cannot exceed 1 Kotyk (1967) and is close to this upper limit for sugars (e.g., α=0.91 for glucose (Teusink et al. 1998), so we set α = 1 by default in this study, maximizing the effect of interaction). A model including both FD and irreversible substrate conversion by an enzyme therefore corresponds to the following chemical equation: Sout⇌αVTm,KTSin+E⇌krkfES→kcatE+P(5) Using analytical tools (see ter Kuile and Cook (1994) and Bosdriesz et al. (2018), rederived in supplementary —Text S5 Mathematical appendix, Supplementary material online), the flux can be determined through the following set of equations: [ES]*=kf[Sin]*kr+kcat+kf[Sin]*.[Etot](6) v=d[P]dt=kcat[ES]*,(7) where: [Sin]*=−b+b2−4ac2a(8) with: {a=kfkcat[Etot](1+[Sout]KT)+kfVTmb=kfkcat[Etot]([Sout]+KT)+(kcat+kr−kf[Sout])VTmc=−VTm[Sout](kr+kcat)(9) Multiple Enzymes Model In order to study the evolution of downstream enzymes, we considered an unbranched metabolic pathway in which the product formed by the first reaction serves as the substrate for a second reaction whose flux determines fitness. Theoretically, as there is nothing prohibiting increase in product concentrations—for it is not considered reversible at this point—any second enzyme should be able to sustain any metabolic demand. We penalized large increases in cellular concentrations through a degradation process of the product of the first reaction, occurring at rate ηd (× this concentration). The chemical reactions occurring after uptake (Michaelis Menten part of equation 5) are described by the following equations: Sin+E1⇌kr1kf1E1S→kcat1E1+P1(10) P1+E2⇌kr2kf2E2P1→kcat2E2+P2 ↓ηd P1out(11) Such a system may reach a steady-state at which the cellular concentrations of the substrate Sin and of the first product P1 are constant. At this point, the net instantaneous uptake of substrate equals the instantaneous production of P1 which, in turn, equals the sum of the instantaneous amount of degraded P1 and the instantaneous production of P2, according to the principle of mass conservation. It yields the following system of equations: VTm.([Sout]−[Sin])KT+([Sout]+[Sin])+α.[Sout][Sin]KT=Vm1.[Sin]KM1+[Sin](12) Vm1.[Sin]KM1+[Sin]=Vm2.[P1]KM2+[P1]+ηd.[P1],(13) where appear the traditional Michaelis–Menten kinetic parameters for the (ieth) reaction: {Vmi=kcati[Etoti]KMi=kri+kcatikfi . To test the potential influence of toxicity, we defined the absolute fitness as a function of both the flux and a toxicity factor whose influence is represented through a sigmoid function and reflects the nonlinearity nature of toxic effects (Clark 1991; Wright and Rausher 2010): f=ϕ(1−[P][P]+T) In an independent section, we also introduced reversibility through the modification of equation (10), which becomes: Sin+E1⇌kr1kf1E1S⇌kinh1kcat1E1+P1(14) Such a phenomenon is described by the more general form of Haldane equation (Haldane 1930; Cornish-Bowden 1979), which changes the contribution of the first reaction (⁠ Vm1.[Sin]KM1+[Sin] ⁠) in both equations (12) and (13) to: Vm1+.[Sin]KM1++[Sin]+KI[P1]−Vm1−.[P1]KM1−+[P1]+[Sin]/KI with Vm1+ and KM1+ respectively corresponding to the previous Vm1 and KM1 ⁠, while: {Vm1−=kr1[Etot1]KI=kinh1/kf1KM1−=KM1+/KI . To solve these systems, we implemented the Newton method (Atkinson 1989) aiming to find [Sin]* and [P1]* ⁠. We ran the algorithm starting from very low values of concentration (both set to 10−20M ⁠) to determine numerically the equilibrium without facing convergence problems. The final flux can then be determined through the “production” part of equation (13), that is, Vm2.[P1]KM2+[P1] ⁠. Validation of the Method and Computing of the Fitness Landscapes To validate the approach, we compared equilibrium results obtained with Raphson–Newton algorithm to those obtained when simulating the process with Euler explicit schemes for a set of (3x3) kinetic values – kcat and kf - encompassing three orders of magnitude (see supplementary Section Text 5, Supplementary material online for further details). We then drew fitness landscapes after determining the flux—assumed to be to be linearly related to fitness—achieved for enzyme kinetic parameters kcat and kf varying by 10 orders of magnitude, setting kr to 103s−1—within the range found for several enzymes (Klipp and Heinrich 1994; Knowles and Albery 1977)—unless stated otherwise. Except in the section dedicated to the influence of enzyme concentration, we set the enzyme concentration such that [Etot]=1mM ⁠, lying nearby the highest observed values (Albe et al. 1990; Noor et al. 2016). Other parameters are detailed on a case-by-case basis as they may change depending on the goal of each section. To compare with the data and visualize results in the experimenter’s parameter space, we also determined the flux and plotted simulation results using kcat and kcatKM ⁠, also making them vary by 10 orders of magnitude. We divided the parameter space in 100 log-equidistant values (250 for representations with kcat/KM to obtain a cleaner demarcation for the diffusion limit). Population Genetics Model Evolutionary simulations rely on a Wright–Fisher process including the selective effects of mutations displacing enzymes on mathematically derived fitness landscapes. Two fitness landscapes were considered: weak flux, high affinity (supplementary fig. S3A, Supplementary material online) and high flux, low affinity (supplementary fig. S3I, Supplementary material online), both with saturated facilitated diffusion (⁠ [Sout]=10KT ⁠) and the following constant parameters: kr=103s−1 and [Etot]=1mM ⁠. Mutations occur at a rate μ=10−1/Ne following reproduction, with an effect sampled in Gaussian distributions with dispersion (⁠ σ=0.3 ⁠). The mean effect of a mutation is given by parameter b, hence representing the mutation bias—absent with b = 0, low (⁠ b=−0.1 ⁠) or high (⁠ b=−0.2 ⁠). Kinetic parameters were initiated to the inefficient values of kcat=10−3s−1 and kf=102M−1s−1 and kf was limited to values under the diffusion limit— 1010M−1s−1 (kcat was also limited to 106s−1 when b = 0 to avoid physical outliers). To analyses simulation outcomes, we picked out the kinetic and fitness values of the most represented genotype when multiple variants were segregating. Thirty simulations were ran for each set of parameters. Finally, we verified that evolutionary steady-states were reached and considered it was the case when at least the average fitnesses (over all simulations) of the last three time-steps were not significantly different one from another (supplementary figs. S5 and S6, Supplementary material online). We also simulated the case where mutations between parameters are correlated. We tested both positive and negative mutational relationships through a bivariate Gaussian distribution whose correlation coefficient were set to ρ=−0.8, ρ=−0.5, ρ=+0.5 (see supplementary fig. 18, Supplementary material online for the results with a moderate flux). Evolution of Enzyme Concentrations Finally, we simulated the joint evolution between kinetic parameters and enzyme concentration, repeating the previous procedure with concentration as an evolvable quantity and the fitness function including deleterious effects of extra gene expression (see supplementary section Text S5, Supplementary material online for the effect of each cost considered independently one from another). Mutations affected either levels of expression or kinetic constants, with those affecting levels of expression (in log values) being drawn from Gaussian distributions with mean 0 and σ=0.2 to comply with existing estimates (Landry et al. 2007; Metzger et al. 2016; Hodgins-Davis et al. 2019). Because sugars are directly involved in energy metabolism, we computed these simulations for the case of a high flux which can more readily be linked to the cost of expression. As explained above, producing extra proteins is costly, both energetically and because it may increase a cell’s crowding. The cost of protein production is considered to be proportional to the steady-state enzyme concentration, with a slope c. Empirical estimates suggest that c should be in the range [10−4,10−3] (Wagner 2005; Lynch and Marinov 2015), such that producing an extra mM of a protein would impede the whole energy budget by one 10,000th to one 1,000th (we also consider c=10−5 and 10−2 in the SM). Accordingly, we calculate the absolute fitness f=Φ−[Etot]c ⁠, where Φ is the flux generated by the enzyme. The influence of crowding was computed by penalizing kf through an effective kf,act=kf.10−([Etot]+[Mb])/[2Mb] ⁠, where [Mb]=2.5.10−3M represents the basal protein concentration of a viable cell and also constitutes a scaling factor that complies with Andrews (2020) log-linear influence of crowding or glass transition effects described by Dill et al. (2011). Data Availability The models have been implemented using R. The source code for these models and the simulated data are available from https://gitlab.in2p3.fr/florian.labourel/ruemee. All the enzyme data used in this work to compare fitness landscapes and measured values were recovered from (Bar-Even et al. 2011) and so was the classification of reactions with regards to metabolic groups. Thanks to their authors and publisher, data sets are publicly available at https://pubs.acs.org/doi/10.1021/bi2002289. Apart from that, no new empirical data were generated in support of this research. 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This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com © The Author(s) 2021. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution.
TI - Resource Uptake and the Evolution of Moderately Efficient Enzymes
JF - Molecular Biology and Evolution
DO - 10.1093/molbev/msab132
DA - 2021-08-23
UR - https://www.deepdyve.com/lp/oxford-university-press/resource-uptake-and-the-evolution-of-moderately-efficient-enzymes-92AL07Fw3h
SP - 3938
EP - 3952
VL - 38
IS - 9
DP - DeepDyve
ER -