A kinetic model for analysis of physical tunnels in sequentially acting enzymes with direct proximity channeling

Biochemical Engineering Journal 105 (2016) 242–248

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Biochemical Engineering Journal journal homepage: www.elsevier.com/locate/bej

Regular article

A kinetic model for analysis of physical tunnels in sequentially acting enzymes with direct proximity channeling Gang Li a,b , Chong Zhang a,b,∗ , Xin-Hui Xing a,b a b

Key Laboratory of Industrial Biocatalysis, Ministry of Education, China Institute of Biochemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing 10084, China

a r t i c l e

i n f o

Article history: Received 2 June 2015 Received in revised form 21 September 2015 Accepted 23 September 2015 Available online 28 September 2015 Keywords: Proximity channeling Modelling Diffusion-reaction Enzymes Kinetic parameters Mass transfer

a b s t r a c t Direct channeling is a well-known process in which intermediates are funneled between enzyme active sites through a physical tunnel and can be a potential way to enhance the biocatalytic efficiency for cascading bioreactions. However, the exact mechanism of the substrate channeling remains unclear. In this work, we used mathematical models to describe the mass transfer in the physical tunnels and to gain further understanding of direct proximity channeling. Simulation with a diffusion-reaction model showed that the reduction of the diffusion distance of intermediates could not cause proximity channeling. A second kinetic model, which considered the physical tunnel as a small sphere capable of preventing diffusion of the intermediate into the bulk, was then constructed. It was used to show that the maximum channeling degree in branched pathways depends on the strength of the side reactions, suggesting that proximity channeling in a physical tunnel is more suitable for a pathway with strong side reactions. On the other hand, for a linear pathway, proximity channeling is more beneficial when the constituting enzymes have relatively low activities and expression levels. Our kinetic model provides a theoretical basis for engineering proximity channeling between sequentially acting enzymes in microbial cell factories and enzyme engineering. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In cascading enzymatic reactions, substrate channeling is a wellknown phenomenon in which the product of an enzyme passes directly into the active site of the subsequent enzyme without diffusing away [1–3]. This phenomenon normally results in an accelerated reaction rate. As substrate channeling can confine a metabolic intermediate to a small region and reduce its concentration in the bulk, engineering of substrate channeling has potential uses in metabolic engineering, multi-enzyme-mediated biocatalysis, and cell-free biosynthetic systems, by avoiding the toxicity of intermediates, side reactions, allosteric regulation and the escape of the intermediates into the bulk [3–5]. Substrate channeling generally occurs when sequentially acting enzymes are positioned close together, which is called ‘proximity channeling’ [1–3]. Some researchers have suggested that proximity channeling occurs by reducing the diffusion distance of the inter-

∗ Corresponding author at: Institute of Biochemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing 10084, China. E-mail address: [email protected] (C. Zhang). http://dx.doi.org/10.1016/j.bej.2015.09.020 1369-703X/© 2015 Elsevier B.V. All rights reserved.

mediates, therefore enabling the intermediates to be consumed immediately before diffusing away [6–8]. However, proximity channeling is only likely to occur when the two active sites of sequential enzymes are aligned to each other and are as close as 1 nm apart [9]. However, the large size of proteins means that it is sterically quite difficult to obtain such spatial proximity of active sites [10]. Direct substrate channeling, in which enzymes are funneled between enzyme active sites through a physical tunnel, was then proposed, and it has been proved to be a possible mechanism of proximity channeling [1,11,12]. Another possible mechanism is that the sequential enzymes are assembled to form a large agglomerate, which restricts the reaction to a diffusion-limited regime; this is known as agglomerate channeling [1,4]. Several previous attempts to understand substrate channeling have used diffusion-reaction models to explain the transport of intermediates between sequential enzymes. To illustrate the mechanism of agglomerate channeling, Castellana et al. divided the cell cytoplasm into several basins, with each basin containing an enzyme cluster and its surrounding volume. Diffusion-reaction equations and boundary conditions were then used to describe the mass transfer of the substrates and intermediates in the enzyme cluster [1]. Castellana et al. used this model to show that the optimal

G. Li et al. / Biochemical Engineering Journal 105 (2016) 242–248

spacing between clusters, namely that which would maximize the metabolic efficiency of the agglomerate channeling, was 6.5 ␮m. Such a large spacing implied that agglomerate channeling could only be possible in large cells such as human cells. In the case of enzymes that have high catalytic constants, for example, a catalytic efficiency of more than 107 M−1 s−1 , agglomerate channeling could also occur in small cells, and this was confirmed by an engineered CarB–PyrB enzyme cluster in Escherichia coli [1]. However, in terms of direct substrate channeling, although there is evidence supporting the existence of physical tunnels that are favorable for substrate transport [2,13,14], there were few studies to elucidate the transport of intermediates through these tunnels. Bauler et al. used Brownian dynamics to simulate the transport of intermediates between the two active sites of sequential enzymes and found that, in order to maximize the catalytic efficiency, the two active sites should be aligned to each other very closely [9]. Eun et al. combined the concentration field and electrostatic field to study the transfer of the intermediate between two consecutive active sites in a diffusion limited system, and found that attractive electrostatic interactions could confine the intermediate to the vicinity of the enzymes and thus elevate its local concentration [15]. However, there has been no study that used mathematical models to explain the process of mass transfer in physical tunnelbased proximity channeling, and the exact mechanisms still remain unclear. In this study, we consider fusion enzymes as a strategy to obtain a closer distance between two consecutive active sites, with the assumption that the fusion of the enzymes does not significantly affect their activity. We first establish a diffusion-reaction model based on a two-step pathway with a branch point, in order to test the effect of the diffusion distance on proximity channeling. We then present a kinetic model that ignores the diffusion rates and introduces a small volume to represent the physical tunnel and we use this model to elucidate the process of substrate channeling with a physical tunnel. Finally, we apply the kinetic model to gain an understanding of the role of proximity channeling in branched pathways and linear pathways. Our approach provides a clear explanation of how physical tunnel-based proximity channeling occurs, and can potentially be a useful tool for guiding protein engineering with a view to optimizing substrate channeling between sequentially acting enzymes.

2. Model development

243

Fig. 1. The models for proximity channeling. (A) A metabolic pathway with a branch point was chosen as the model system. The geometry for (B) the fusion enzyme and (C) individual enzymes considered in this paper. Two fused active sites are placed in the orange circle to describe their close proximity, and the active site for side reactions is placed in the green region. The symbols ‘S’, ‘M’ and ‘P’ represent ‘substrate’, ‘intermediate’ and ‘product’, respectively, and ‘Ps’ is the final product of the side reactions. E1 and E2 are two enzymes involved in the coupled reaction and Es is the enzyme for the side reaction that competes with E2 for M. R0 is the radius of the region where E2 consumes the intermediate when fused with E1 . R is the radius of the basin. v2 and vs are the reaction rates of E2 and Es, respectively.

For the fusion enzyme:

⎧ 2 ∂C ⎪ ⎪ ⎪ −4␲r D ∂r = N1 ⎪ ⎪ ⎪ ⎪ ∂C ⎪ 1 ∂ 2 ∂c ⎪ ⎪ = D[ 2 (r )] − v2 ⎪ ⎪ r ∂r ∂ t ∂r ⎪ ⎪ ⎪ ⎪ ∂C ∂C ⎪ ⎪ = | | ⎪ ⎪ + ∂ ∂r r→R1− r ⎪ r→R ⎪ 1    ⎪ ⎨ ∂C 1 ∂ ∂C r2

=D

r→0 0 < r < R0 r = R0 R0 < r < R1

r 2 ∂r ∂t ∂r ⎪ ⎪ ⎪ ⎪ ⎪ ∂C ∂C ⎪ ⎪ = r = R1 | | ⎪ + ⎪ ∂ r ∂r r→R2− r→R ⎪ 2 ⎪    ⎪ ⎪ 1 ∂ ⎪ ∂C 2 ∂C ⎪ D r − vs R1 < r < R = ⎪ ⎪ r 2 ∂r ⎪ ∂t ∂r ⎪ ⎪ ⎪ ⎪ ∂C ⎪ ⎩ r=R =0 |

∂r

(1)

r=R

For individual enzymes:

2.1. Diffusion-reaction model Here, we consider a two-step metabolic pathway with a branch point as a model system to study substrate channeling between two sequential enzymes (Fig. 1A). Such a model could be used to describe pathways with side reactions, unstable intermediates and reversible reactions. As shown in Fig. 1B and C, to simplify the system, the cell was divided into many spherical basins [1], where each basin contains an E1 enzyme located at its center and is delimited by a thin boundary layer. The radius of the basin is determined by the concentration of E1 . As the concentration of overexpressed proteins in E. coli is approximately 2000 enzymes per cell [1], equal to 0.5 ␮M, each E1 occupies a volume with a radius of about 100 nm (R in Figs. 1 and 3 A). The second enzyme, E2 , is placed in the center of the basin for the E1 –E2 fusion protein system and in the thin boundary layer of the basin for individual enzymes. Es, which catalyzes the side reaction, is placed in the thin boundary layer of the basin (Fig. 1B and C). The diffusion-reaction equations presented below were used to describe the transfer of the intermediate M [1,16].

⎧ 2 ∂C ⎪ ⎪ ⎪ −4r D ∂r = N1 ⎪ ⎪ ⎪ ⎪ ⎪ ∂C 1 ∂ 2 ∂C ⎪ ⎪ (r )] = D[ 2 ⎪ ⎪ r ∂r ∂r ∂t ⎪ ⎪ ⎨ ∂C

=

|

∂C

|

r→0 0 < r < R0 r = R1

− ∂r r→R2+ ⎪ ⎪  ∂rr→R2  ⎪ ⎪ ⎪ ⎪ ∂C = D 1 ∂ r 2 ∂C − v2 − vs R1 < r < R ⎪ ⎪ ⎪ r 2 ∂r ∂r ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎩ ∂C

∂r

|

=0

(2)

r=R

r=R

In these equations, N1 is the rate of the reaction catalyzed by the first enzyme and D is the diffusion coefficient of the intermediate. In our model, we assume that the substrate of the first enzyme is present in excess, thus N1 remains nearly unchanged with time. v2 and vs are the rates of the reactions catalyzed by E2 and Es , respectively. The basin has spherical symmetry, that is, the concentration of the intermediates depends only on the distance r. The no-flux bound-

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Table 1 The parameters used in the model. Symbol

Description

Value

Unit

NA kcat1 kcat2 kcat3 kcat /Km N1 R R0 D

Avogadro’s constant Catalytic constant of E1 Catalytic constant of E2 Catalytic constant of Es Catalytic efficiency of enzymes The catalytic rate of E1 in diffusion-reaction model. N1 = kcat1 /NA The radius of the basin The radius of the tunnel The diffusion constant of the intermediates The reaction rate of E1 in kinetic model The reaction rate of E2 in kinetic model The reaction rate of Es in kinetic model The rate of diffusion of intermediate from the tunnel to the bulk The rate coefficient of vdiff The concentration of the substrate for E1 The concentration of the final product The concentration of the intermediates in tunnel The concentration of the intermediates in the bulk The volume of the basin The volume of the tunnel

6.022 × 1023 100 100 100 106 1.661 × 10−22 100 5 Variable(10−9 ∼ 10−14 ) Eq. (5) Eq. (6) Eq. (6) Variable, Eq. (4) Variable 10 Dependent variable Dependent variable Dependent variable 4.189 × 10−21 5.236 × 10−25

mol−1 s−1 s−1 s−1 s−1 M−1 mol s−1 nm nm m2 s−1 M s−1 M s−1 M s−1 mol s−1 L s−1 mM mM mM mM m3 m3

v1 v2 vs vdiff kdiff S P Min Mout V V0

ary condition at the edge is given. For the boundaries separating the different regions within the basin (i.e. r = R0 and r = R1 ), the concentration derivatives are continuous. These coupled differential equations were solved numerically by MATLAB. In the calculation, a slightly unbalanced mass, caused by error from the difference method, was found. To correct the mass balance, a scaling factor was introduced to ensure that the output flux was exactly equal to the input flux; this method of correction is commonly used in such numerical calculations [15]. 2.2. Kinetic model for a fusion enzyme with a physical tunnel In the second model, we assume that the active sites of E1 and E2 are located in a connecting tunnel, while the active site of Es is placed in the bulk solution (Fig. 3A). The properties of the tunnel are described by two parameters: size and “leakproofness”. In the model, a small sphere with radius R0 is introduced to represent the physical tunnel; R0 is set as 5 nm [9]. kdiff is introduced to evaluate the potential for intermediate transfer from the tunnel to the bulk.

⎧ dS ⎪ = −v1 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ dM ⎪ ⎨ in = v1 − v2 − vdiff /V0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

dt

dMout = vdiff /V − vs dt

(3)

dP = v2 dt The rate of diffusion of intermediate from the tunnel to the bulk,

vdiff , is given by: vdiff = kdiff (Min − Mout )

(4)

in these equations, Min and Mout are the concentrations of the intermediate in the tunnel and the bulk, respectively. v1 , v2 and vs are the reaction rates of E1 , E2 and Es , while V and V0 are the volumes of the basin and the physical tunnel, respectively. Thus, kdiff /V0 is an intrinsic property of a physical tunnel. 2.3. Model parameters The typical turnover number of an enzyme is approximately 100 s−1 , and the typical kcat /Km is about 105 ∼ 106 s−1 M−1 [17]. Thus, in our model, we assume that kcat1 , kcat2 and kcats have the same value of 100 s−1 and the kcat /Km values for all three enzymes

are 106 s−1 M−1 . In the d iffusion-reaction model, N1 =kcat1 /NA , where NA is Avogadro’s constant. All the parameters used in our model can be found in Table 1. In the kinetic models, v1 is the rate-determining step of the pathway, and is described with the Michaelis–Menten equation:

v1 =

kcat1 [S] (Km1 + [S]) NA V

(5)

In both models, v2 and vs are not the rate-determining steps and are described as follows:

v2 , vs =

kcat 1 [M] × Km NA V

(6)

3. Results and discussion 3.1. The effect of diffusion on proximity channeling We firstly focused on how diffusion distance affects the catalytic rate of the two individual enzymes in sequence. To elucidate this, the basin was divided into several concentric regions: For the fusion enzymes, E2 is linked to E1 and thus can only consume the intermediate in the region near E1 (r < R0 ) (Fig. 1B). For the individual enzymes, the average distance between E1 and E2 is R, thus, in our model E2 is placed in the boundary of the basin and the small region of R1 < r < R was chosen for E2 consuming the intermediates (Fig. 1C). In both cases, the intermediates diffuse freely in the whole basin. Fig. 2 shows the numerical solutions of the model with the diffusion coefficient varied from 10−10 to 10−14 m2 s−1 . Fig. 2A shows the distributions of the intermediate M with radial distance (r). For both fusion enzymes and individual enzymes, M is produced in the center (r = 0) and then diffuses along the radius. As shown in Fig. 2A, in all cases with different diffusion coefficients, the concentration of M in the region of r < 5 nm forms a sharp concentration gradient, while in the region of r > 5 nm, this concentration gradient becomes gentle and the intermediate concentration at different distances is nearly the same. For fusion enzymes, in the region of r < 5 nm, the curves with smaller diffusion coefficients are located at higher positions compared with the curves with larger diffusion coefficients, indicating that the average concentration of M in this region increases with the decrease in the value of the diffusion coefficient. This is most apparent when D ≤ 10−12 m2 s−1 , which gives significant higher intermediate concentrations compared with the situation with D = 10−10 m2 s−1 in this region.

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Fig. 2. The numerical solutions for diffusion-reaction mode. (A) The predicted spatial distribution of the intermediate for the fusion enzyme system (solid line) and the individual-enzyme system (dashed line), with different diffusion coefficients. (B) The progress curve of intermediate formation for the fusion enzyme system (solid line) and the individual-enzyme system (dashed line), with different diffusion coefficients. For (A) and (B), kcat /Km is 106 L mol−1 s−1 and kcat is 100 s−1 . The units of the diffusion coefficient are m2 s−1 .

Fig. 3. The model for proximity channeling with a physical tunnel. (A) The geometry for direct channeling. Two active sites are in the orange circle, which represents the physical tunnel. The side reaction occurs in the yellow bulk region. (B) The accumulation curves for the final product with different values of kdiff /V0 . The inset shows the transient time for different values of kdiff /V0 . (C) The accumulation curves for the intermediate for different values of kdiff /V0 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The global catalytic efficiency of the system is reflected by the rate of accumulation of the final product (Fig. 2B). When D > 10−12 m2 s−1 , the global catalytic efficiencies of fusion enzymes and individual enzymes are almost the same; while when D ≤ 10−12 m2 s−1 , the global catalytic efficiency of the fusion enzymes is larger than that of the individual enzymes. Particularly, when D reaches 10−14 m2 s−1 , there is an approximately twofold improvement in global catalytic efficiency. These results indicate that the reactions of sequential enzymes follow a diffusion-limited regime only when D is less than 10−12 m2 s−1 , which would result in better catalytic efficiency when the enzymes are fused compared with individual enzymes. Usually, the diffusion coefficient (D) for biomolecules in water is around 10−9 m2 s−1 at room temperature. However, compared with the value in water, the diffusion coefficient in the cytoplasm is reduced three- to fourfold for small molecules and 10–1000fold for large molecules, compared with D in water, due to the higher viscosity [18–20]. As a result, the diffusion coefficient in the cytoplasm is approximately 10−10 m2 s−1 for small molecules and 10−10 −10−12 m2 s−1 for large molecules. This implies that the catalytic reactions in the cytoplasm generally follow a reaction-limited regime [18,21]; therefore, the reduction of the diffusion distance would not lead to proximity channeling in these cases. In conclusion, simply locating two sequential enzymes close to one another is not sufficient to affect substrate channeling. This conclusion is

contrary to the suggestion of previous researchers that the reduced diffusion distance causes the proximity channeling [6–8].

3.2. The kinetic model for direct channeling with a physical tunnel As simply reducing the diffusion distance does not give efficient proximity channeling, we speculated that there might exist a physical tunnel between two active sites that limits the diffusion of the intermediates into the bulk. To test our hypothesis, we proposed a kinetic model that ignores diffusion. In this kinetic model, a small sphere with a physical barrier in the center of the basin is used to represent the physical tunnel, with the reactions catalyzed by E1 and E2 only occurring in the tunnel (Fig. 3A). The property of mass transfer in the physical tunnel is described by a kinetic parameter kdiff , which represents the “leakproofness” of the tunnel. Fig. 3B shows the accumulation curve of the final product with the change of kdiff /V0 . It shows that the product accumulation rate for the fusion enzyme increases as kdiff /V0 decreases; when kdiff /V0 approaches 0 s−1 , the fusion enzyme gives the maximum product accumulation rate, which is nearly 2-fold higher than that of the system of individual enzymes. When kdiff /V0 approaches 104 s−1 , the product accumulation rate of the fusion enzyme is decreased and approaches that of the individual enzymes; these results indicate that only when kdiff /V0 is less than 104 s−1 will the global catalytic efficiency of the fused enzymes be better than that of

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Fig. 4. The effect of proximity channeling on the branched pathway. (A) The effect of the side reactions and kdiff /V0 on the channeling degree . (B) The effect of kdiff /V0 on /m, in which m is defined as m = (k2 + ks )/k2 . (C) Fitting of the model to the experimental data reported for the GPD1–GPP2 fusion enzymes. Table 2 The global diffusion coefficient in physical tunnel based proximity channeling. kdiff /V0 (s−1 )

Global diffusion coefficient (m2 s−1 )

0 102 103 104

0 1.67 × 10−14 1.67 × 10−13 1.67 × 10−12

the system of individual enzymes. When kdiff /V0 = 0, there exists a perfect tunnel in the fused enzymes, in the sense that the intermediate cannot diffuse out of the tunnel, and the maximum reaction efficiency is reached. The ratio kdiff /V0 can be transformed into the diffusion coefficient by: C kdiff S CV0 = D V0 L

(7)

where C = Min − Mout , which represents the difference in the concentrations of the intermediate in the tunnel and the bulk, and L is the length of the diffusion path. S is the surface area of the tunnel. Thus, D=

kdiff LR0 × V0 3

(8)

where L = R, and D represents the global diffusion coefficient of the fusion enzymes. As shown in Table 2, kdiff/ V0 = 104 s−1 corresponds to a global diffusion coefficient of 10−12 m2 s−1 . As discussed in the diffusion-reaction model, when D ≤ 10−12 m2 s−1 , the global catalytic efficiency of the fusion enzymes is increased compared with individual enzymes, this being identical to the prediction of the kinetic model. This analysis shows that the kinetic model proposed in this study is feasible for explaining the physical tunnels in fusion enzymes. The concentration of the intermediate in the bulk also decreases with the decrease in kdiff /V0 (Fig. 3C). When kdiff/ V0 = 0 s−1 , the concentration of the intermediate in the bulk is 0 mM. This is reasonable, as the formation of a physical tunnel prevents the intermediate from diffusing to the bulk. This phenomenon is supported by many experimental results [22,23]. For example, tryptophan synthase has been proven by structural analysis to have a physical tunnel [14], in which only a trace amount of intermediate (indole) can be detected in the bulk in a single turnover experiment [23]. Transient time (), namely the time required for accumulation of intermediates to concentrations necessary to establish the steady state, is a key phenomenological parameter for coupled enzymatic reactions [24,25]. As shown in Fig. 3B, the transient time of the channeling enzymes decreases from 0.8 s to 0 s with the decrease of kdiff /V0 , while the transient time of individual enzymes is significantly larger ( = 1.3 s). These results suggest that the formation of

the physical tunnels can significantly reduce the transient time of the coupled reactions. The physical tunnel considered in our model is represented with a spherical geometry in order to make the model formulation easier. The volume of the tunnel (V0 ) is necessary for the kinetic model that corresponds to Eq. (3); in practice, this volume could be obtained from the crystal structure of the protein. Moreover, the mass transfer properties (kdiff ) depend on the structure of the tunnel: amongst other factors, they will be affected by the shape of the tunnel and the amino acid composition of the tunnel. Thus, in future studies, information about the three-dimensional structure should be considered in our model. This will give us a deeper understanding of proximity channeling and further guide our design of proximity channeling through metabolic engineering approaches. 3.3. The effect of proximity channeling on a branched pathway Branched pathways exist widely in nature and are important for metabolic modeling and engineering [26–28]. The kinetic model was used to study the effects of the side reactions, through varik . The ations of the value of ks /k2 , in which k2 = Kcat2 and ks = kKcats ms m2 channeling degree () is used to evaluate the effect of proximity channeling on the efficiency of catalysis in producing the desired end product [29,30]. Here,  is defined as the ratio of the global rate, at steady state, of the system when the enzymes are in close proximity with channeling (vwithc ), to that of the individual enzymes without channeling (vwithoutc ): =

vwith c | vwithout c steady

(9)

As shown in Fig. 4A,  decreases as ks /k2 decreases. Also, when ks /k2 approaches 0,  approaches 1. These results indicate that  is proportional to the strength of the side reaction: when the side reaction is very weak and makes the pathway essentially linear, the introduction of substrate channeling would have little effect on the whole catalytic efficiency at steady state compared with that for systems comprised of the individual enzymes. We then defined a parameter m, equal to (k2 + ks)/ k2, to represent the relative strength of the side reaction. As shown in Fig. 4B, /m increases with the reduction of kdiff /V0 , and when kdiff /V0 approaches 0, /m approaches 1, indicating that the value of the channeling degree  approaches the value of m. It is important to note that the whole analysis that is undertaken in this work assumes that E1 catalyzes the rate-determining step and that the reactions catalyzed by E2 and Es follow first-order kinetics with respect to the concentration of the intermediate (see Eq. (6)). Thereby, in the individual enzyme systems: k2 [M]steady v2 k2 | = = vs steady ks [M]steady ks

(10)

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247

Fig. 5. The effect of proximity channeling on the linear pathway. (A) Fitting of the model to the experimental data reported for the PutA proteins. (B) The numerical solutions for the models with low activity enzymes, for which kca t is 0.1 s−1 and kcat /Km is 103 m2 s−1 . The enzyme concentration is 0.2 ␮M. (C) The distribution of the transient time for different enzyme concentrations and activities.

therefore, k2 /ks also represents the ratio of the catalytic rates of E2 and Es at steady state. Thus, for a reaction system with proximity channeling, the maximum channeling degree for the fusion enzyme would be: v2 + vs m = |steady (11)

v2

We then applied our model to a system comprised of GPD1 and GPP2, which are two key enzymes in the glycerol pathway of Saccharomyces cerevisiae. GPD1 is a reversible Glycerol–3-P dehydrogenase (G3P), which makes this pathway branched, and GPP2 converts the intermediate G3P into the final product glycerol: GPDI

GPP2

Dihydroxyacetone − P ↔ Glycerol − 3 − P → Glycerol + P Salles et al. obtained a fused GPD1–GPP2 and found that the system with the fusion protein had a seven-fold reduced transient time, a five-fold reduced concentration of intermediate G3P in the bulk and a twofold enhanced rate of accumulation of the final product glycerol, compared with the individual-enzyme system, indicating the existence of proximity channeling in the fused GPD1–GPP2 [31]. The product accumulation curve predicted by our model fits the experimental data very well (Fig. 4C, the details of the simulation can be found in the Supplementary materials). The value of kdiff /V0 for GPD1–GPP2 was estimated as 10 s−1 , indicating that a nearly perfect physical tunnel exists in GPD1–GPP2. For this system, the maximum channeling degree of GPD1–GPP2 is, m =

v2 + vs 0.0058 + 0.0054 | = = 1.93 v2 steady 0.0058

which agrees well with the twofold enhanced rate of accumulation of glycerol that was observed experimentally. Our results suggest that proximity channeling is more beneficial in branched pathways that have strong side reactions. 3.4. The effect of proximity channeling on a linear pathway Proline utilization A (PutA) proteins were taken as an example to study proximity channeling in linear pathways [2]. They are bifunctional peripheral membrane flavoenzymes that catalyze the oxidation of l-proline to l-glutamate by two sequential active sites, PRODH and P5CDH: PRODH

P5CDH

l − proline−−−−−−→P5C ↔ GSA−−−−−−→Glutamate ±H2 O

Singh et al. showed that there was a 75-Å-long tunnel between the active sites [2]. The experimental results showed that PutA could accumulate the final product more efficiently than a nonchanneling PutA [2,32].

We used our kinetic model to study this system. The calculated product accumulation curve fitted the experimental data well for both channeling and non-channeling PutA (Fig. 5A, the details of the simulation can be found in the Supplementary materials). Our kinetic model suggests that the value of kdiff /V0 is 0 s−1 for the channeling system. This indicates that there is a perfect tunnel in PutA. As a linear pathway without any side reaction, the channeling degree of PutA is 1 (in other words, the rate of accumulation of the final product is the same for both channeling and non-channeling PutA). However, the catalytic efficiency in the channeling system is significantly higher than that in non-channeling system before steady state, thereby leading to an improved final product titer. As shown in Fig. 5B, the transient time of the non-channeling system is 0.69 h and there is no perceptible transient time in the channeling system; even after 4 h, the channeling system still has a 1.2-fold higher titer of the final product. In a system with a long transient time, for example, of several hours, the introduction of a physical tunnel by protein engineering could improve the global catalytic efficiency in a linear pathway. The transient time was estimated by the following equation [25]: ␶=

[M]

v

(12)

where [M] is the concentration of the intermediate at steady state, and v is the reaction rate of the upstream enzyme. Fig. 5C shows that when kcat /Km is less than 104 L mol−1 s−1 and the enzyme concentration is less than 1 ␮M, the transient time would be 1 h or longer. According to published data, there are many enzymes with kcat /Km less than 104 L mol−1 s−1 [17] and the enzyme concentration in the cytoplasm is around 1 ␮M, ranging from 0.01 to 100 ␮M [33]. Thus, a transient time of more than 1 h would generally occur in metabolic pathways, and the existence of proximity channeling would therefore lead to an improved titer of the final product (Fig. 5B). These results suggest that, for enzymes in linear pathways, the introduction of proximity channeling by protein engineering would be more promising for those situations in which the constituting enzymes have relatively low activities and expression levels. Previous workers have suggested that proximity channeling is caused by the reduction of the diffusion distance of the intermediates and that, as a consequence, simply locating two sequential enzymes close to one another would be a direct way to engineer proximity channeling [6–8]. However, our results show that diffusion distance has little effect on proximity channeling and that simply locating two sequential enzymes close together is not sufficient. Rather, an additional physical tunnel connecting the two consecutive active sites is necessary for proximity channeling. For the engineering of proximity channeling to enhance the catalytic rate of coupled reactions, our results show that the strength of

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side reactions, the concentrations and the activities of the enzymes must be taken into consideration, however, these factors are usually ignored by researchers. 4. Conclusions A diffusion-reaction model showed that simply reducing the diffusion distance does not improve the global catalytic efficiency of a coupled reaction, rather, a physical tunnel between two consecutive active sites of enzymes is necessary if direct proximity channeling is to give improved efficiency. A kinetic model using the parameter kdiff /V0 to describe the mass transfer properties of the physical tunnel showed that kdiff /V0 is of crucial importance: a physical tunnel will only give a significant improvement in global catalytic efficiency when kdiff /V0 is less than 104 s−1 . Also, the channeling degree reaches a maximum when kdiff /V0 is 0 s−1 . For branched pathways, since the maximum channeling degree depends on the strength of the side reactions, the activity of side reactions must be taken into consideration when attempts are made to introduce proximity channeling where it does not exist in nature. For linear pathways, proximity channeling would be more beneficial for systems in which the constituting enzymes have relatively low activities and expression levels, such that the coupled reaction has a long transient time. Conflict of interest The authors have no conflicts of interest. Acknowledgements This work was supported by the National Natural Science Foundation of China (NSFC 21376137) and Tsinghua University Initiative Scientific Research Program (2013Z02-1). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.bej.2015.09.020. References [1] M. Castellana, M.Z. Wilson, Y. Xu, P. Joshi, I.M. Cristea, J.D. Rabinowitz, et al., Enzyme clustering accelerates processing of intermediates through metabolic channeling, Nat. Biotechnol. 32 (2014) 1–10. [2] H. Singh, B.W. Arentson, D.F. Becker, J.J. Tanner, Structures of the PutA peripheral membrane flavoenzyme reveal a dynamic substrate-channeling tunnel and the quinone-binding site, Proc. Natl. Acad. Sci. U. S. A. 111 (2014) 3389–3394. [3] Y.-H.P. Zhang, Substrate channeling and enzyme complexes for biotechnological applications, Biotechnol. Adv. 29 (2011) 715–725. [4] A.H. Chen, P.A. Silver, Designing biological compartmentalization, Trends Cell Biol. 22 (2012) 662–670. [5] H. Lee, W.C. DeLoache, J.E. Dueber, Spatial organization of enzymes for metabolic engineering, Metab. Eng. 14 (2012) 242–251. [6] R.J. Conrado, J.D. Varner, M.P. DeLisa, Engineering the spatial organization of metabolic enzymes: mimicking nature’s synergy, Curr. Opin. Biotechnol. 19 (2008) 492–499. [7] J. Shi, L. Zhang, Z. Jiang, Facile construction of multicompartment multienzyme system through layer-by-layer self-assembly and biomimetic mineralization, ACS Appl. Mater. Interfaces 3 (2011) 881–889.

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