Enzyme Cascade Reactions - ACS Publications - American Chemical

Subscriber access provided by UNIV TEXAS SW MEDICAL CENTER

Article

A queueing theory-based perspective of the kinetics of “channeled” enzyme cascade reactions Stanislav Tsitkov, Theo Pesenti, Henri Palacci, Jose Blanchet, and Henry Hess ACS Catal., Just Accepted Manuscript • DOI: 10.1021/acscatal.8b02760 • Publication Date (Web): 03 Oct 2018 Downloaded from http://pubs.acs.org on October 9, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

A Queueing Theory-Based Perspective of the Kinetics of “Channeled” Enzyme Cascade Reactions Stanislav Tsitkov1‡, Theo Pesenti1,2‡, Henri Palacci1, Jose Blanchet3, Henry Hess1*

1. Department of Biomedical Engineering, Columbia University, New York, NY 10027, USA

2. École Supérieure de Physique et de Chimie Industrielles (ESPCI), 75231 Paris Cedex 05, France

3. Management Science and Engineering, Stanford University, Palo Alto, CA 94305, USA

ABSTRACT: Queueing approaches can capture the stochastic dynamics of chemical reactions and provide a more accurate picture of the reaction kinetics than coupled 1

ACS Paragon Plus Environment

ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 72

differential equations in situations where the number of molecules is small. A striking example of such a situation is an enzyme cascade with substrate channeling, where reaction intermediates are directly passed from one enzyme to the next via tunnels or surface paths with limited capacity. In order to better understand the contribution of the stochastic dynamics to the observed enhancement in cascade throughput as a result from substrate channeling, we compare the results of a model using differential equations to describe concentration changes with a queueing model. The continuum model and the queueing model yield identical results, except when the maximum rate of reaction of the enzymes are similar. In two enzyme cascades, the queueing model predicts at most a 50% smaller throughput than the continuum model even if the waiting room size (the maximum number of molecules that can fit in the tunnel or surface path between enzymes) is limited to only one molecule and the enzymes are perfectly matched in their kinetic rates. In longer cascades the discrepancy increases, reaching a five-fold difference for a 10 enzyme cascade. In line with theoretical results from queueing theory, stochastic effects are found to always reduce cascade throughput, which means they 2


Page 3 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

cannot contribute to the experimentally observed enhancement in throughput due to channeling.

KEYWORDS: enzyme cascade, queueing theory, Gillespie algorithm, Michaelis-Menten model, stochasticity

Introduction Cascade reactions catalyzed by enzymes are reminiscent of factory processes where parts are transformed and combined as they travel from one workstation to the next.1-3 The promise of nanotechnology is to enable the construction of "molecular assembly lines", where a molecule produced by one enzyme is directed straight to the next enzyme, which processes it in order of arrival.4-7 Biological systems have already evolved structures enabling such sequential transformations of individual molecules.8 Specifically, these structures enable "substrate channeling" either via tunnels connecting active sites,

3


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 72

or via surface paths which attract intermediate substrate molecules and prevent them from leaving the enzyme complex as they travel from one active site to the next.9 Since the sequential transformation of substrates within an enzyme cascade is similar to the processing of parts at different work stations in a factory, it is reasonable to expect that natural and synthetic enzyme complexes with substrate channeling would exhibit complexities in the organization of the workflow reminiscent of a factory, where the description and optimization of the production processes has given rise to the engineering discipline of logistics.10,

11

Traditionally, chemical reactions are described by coupled

differential equations, whose evolution represents the time-varying concentrations of the different molecular species. It is likely that these differential equations are only imperfect approximations of the discrete transformation processes occurring in a "molecular assembly line."12 Therefore, the goal of this paper is to explore the impact of stochastic effects on an Nstep reaction cascade catalyzed by N enzymes immobilized on a scaffold using a queueing model.13 Queueing models have been recently utilized to model a variety of 4


Page 5 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

biological systems, including multi-site enzymes 14, gene expression and regulation 15, 16, the spread of invasive species

17,

and physiological insulin levels.18 The traditional

queueing model describes the dynamics of customers being served in a waiting room.19 In our analysis, the molecule is the customer, the enzyme is the server, and the waiting room is the inter-enzyme space of limited capacity where the molecule is held while it is waiting to be processed by the next enzyme (Figure 1a). Examples of this interenzyme space can be the tunnel found within the naturally occurring tryptophan synthase enzyme complex 20 (Figure 1b), or the DNA-origami compartment created to house the synthetic glucose-oxidase and horseradish-peroxidase cascade 21 (Figure 1c). Other examples of naturally occurring molecular tunnels connecting enzyme active sites include those used in the formation of carbamoyl phosphate22, in the activation of glutamine phosphoribosylpyrophsphate amidotransferase23, in the formation of asparagine by asparagine synthetase B24, and more.20, 25-28

5


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 72

The discrete character of the queueing model captures the stochastic effects and predicts significant deviations in throughput from the predictions of a system of differential equations under specific circumstances.

Figure 1. Examples of enzyme cascades exhibiting confinement of intermediate substrates. (A) Equivalence between an enzyme cascade and a queueing model. (B)

6


Page 7 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

Substrate channeling through a tunnel connecting two active sites in tryptophan synthase.29 Copyright 2012 Elsevier. Adapted with permission from reference 29. All rights reserved. (C) Glucose oxidase and horseradish peroxidase encased in an artificial DNA scaffold.

Copyright 2016 Nature Publishing Group.

Adapted with permission from

reference 21. All rights reserved.

The study of enzyme cascades on scaffolds30, and in particular on artificial scaffolds assembled from DNA31 has received exponentially growing interest in the past decade.32 Indeed, the increased throughput of the cascade after the attachment of the enzymes to the scaffold has been interpreted as evidence of channeling due to enzyme proximity. A prominent example of the hundreds of publications is the study by Fu et al.,5 where the rate of final product formation of a two-enzyme cascade on a DNA scaffold was enhanced several-fold relative to the same enzymes diffusing freely in solution. While the finding that the positioning of two coupled enzymes on a DNA scaffold with an interenzyme spacing on the order of 10 nm results in an increased final product formation has been 7


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 72

reproduced by many researchers, the theoretical interpretation of these experimental results has been frequently unconvincing. Our previous work has aimed to provide some theoretical insight using basic analytical models of enzyme kinetics as well as numerical solutions of partial differential equations describing the reaction-diffusion process.33 Our central finding was that the rapid free diffusion of intermediate substrate molecules rules out any simple proximity effect. Instead, enhancement must originate from confinement of the intermediate substrates (true "channeling"), competition by other reactions involving the intermediate substrates, aggregation of large numbers of enzyme pairs, and the effect of the scaffold on the activity of the individual enzymes.34-37 Irrespective of the mechanisms at work in artificial systems, biological systems achieve channeling via confinement of intermediate substrates in tunnels and on pathways as described above, and these examples motivate our analysis. Since some of these substrate-scaffold interactions are weak enough to permit mobility38, they also permit the escape of intermediate substrate molecules. The rate of intermediate substrate loss to the environment is likely dependent on the system geometry and interactions, as well as 8


Page 9 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

the number of intermediate substrate molecules present. We therefore aim to describe both complete and partial confinement of the intermediate substrates.

Methods 1. Deterministic description Before describing our complete discrete stochastic model, we will begin by modeling an N-enzyme cascade with scaffolds and partial confinement in a deterministic setting. Each enzyme catalyzes the following reaction: 𝑘𝑓

S+E

𝑘𝑟

ES

𝑘𝑐𝑎𝑡

P + E#(1)

Where S is the substrate, E the (free) enzyme, ES the substrate-enzyme complex, P the product, kf the rate constant associated to the forward reaction, kr the rate constant associated to the reverse reaction and kcat the enzyme catalytic rate constant. The classical description of enzymatic reactions kinetics is the Michaelis-Menten (MM) model.39 Assuming the quasi steady-state approximation (QSSA), the output rate of one enzyme rMM is given by40 9


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

𝑟𝑀𝑀 =

𝑘𝑐𝑎𝑡[𝑆] 𝐾𝑀 + [𝑆]

𝐾𝑀 =

Page 10 of 72

⋅ [𝐸]𝑇#(2)

𝑘𝑟 + 𝑘𝑐𝑎𝑡 #(3) 𝑘𝑓

where [𝑆] is the substrate concentration, [𝐸]𝑇 = [𝐸] + [𝐸𝑆] the total enzyme concentration and KM the Michaelis constant. The validity of this approximation for our system will be discussed in the Discussion. For N-enzyme cascades, we model each enzyme output rate using the MM model. Since there is only one enzyme molecule at each step of the cascade, the production rate of Ei is given by 𝑟𝑀𝑀,𝑖 =

𝑘𝑐𝑎𝑡,𝑖[𝑆𝑖 ― 1] 𝐾𝑀,𝑖 + [𝑆𝑖 ― 1]

1

⋅ 𝑁𝐴𝑉#(4)

Where NA is Avogadro's number and V the volume accessible to the confined intermediate substrates. The net production of intermediate substrate Si is given by the difference between its production by Ei and its consumption by Ei+1. Our first assumption in the modelling of the net substrate production is that there is a small accessible volume between enzymes 10


Page 11 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

which places a limit on the number of intermediate substrates which can be waiting to be processed by the next enzyme; from now on, we will refer to this inter-enzyme space as the “waiting room.” An example of the waiting room is the tunnel found in tryptophan synthase (Figure 1b), which can accommodate up to four indole molecules.20 If a waiting room is full, the preceding enzyme will release the substrate to the bulk. We will also assume that enzymes in the cascade can only recruit substrate molecules from their waiting rooms and not from the bulk solution. This assumption will be justified in the Discussion. Finally, we model the possibility of escape of a substrate molecule from the waiting room with a first order rate constant koff; following queueing theory terminology, we will refer to this loss rate constant as the impatience:

Si

𝑘𝑜𝑓𝑓

ø#(5)

Therefore, for each substrate Si - except S0 -, we would like to express the net production

𝑑[𝑆𝑖(𝑡)] 𝑑𝑡

as:

=

{

𝑘𝑐𝑎𝑡,𝑖[𝑆𝑖 ― 1(𝑡)]

𝑘𝑐𝑎𝑡,𝑖 + 1[𝑆𝑖(𝑡)]

𝑛𝑊𝑅,𝑖

𝑘𝑐𝑎𝑡,𝑖 + 1[𝑆𝑖(𝑡)]

𝑛𝑊𝑅,𝑖#(6)

[ ] [ ] 𝐾𝑀,𝑖 + [𝑆𝑖 ― 1(𝑡)] 𝐸𝑖 𝑇 ― 𝐾𝑀,𝑖 + 1 + [𝑆𝑖(𝑡)] 𝐸𝑖 + 1 𝑇 ― 𝑘𝑜𝑓𝑓[𝑆𝑖(𝑡)], [𝑆𝑖(𝑡)] < 𝑁𝐴𝑉 ― 𝐾𝑀,𝑖 + 1 + [𝑆𝑖(𝑡)][𝐸𝑖 + 1]𝑇 ― 𝑘𝑜𝑓𝑓[𝑆𝑖(𝑡)], [𝑆𝑖(𝑡)] ≥

𝑁 𝐴𝑉

11


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 72

where nWR,i is the capacity (in number of molecules) of the waiting room. However, due to the discontinuity in [𝑆𝑖], the solution to this differential equation does not exist. Instead, we model the discontinuity with a logistic term, 𝐿([𝑆𝑖(𝑡)]) = (1 + 𝑒𝛾( 𝑆𝑖(𝑡) [

high steepness parameter, 𝛾 = 10, and [𝑆𝑚𝑎𝑥] =

𝑑[𝑆𝑖(𝑡)] 𝑑𝑡

=

𝑘𝑐𝑎𝑡,𝑖[𝑆𝑖 ― 1(𝑡)] 𝐾𝑀,𝑖 + [𝑆𝑖 ― 1(𝑡)]

[𝐸𝑖]𝑇 ⋅ 𝐿([𝑆𝑖(𝑡)]) ―

𝑛𝑊𝑅,𝑖

𝑁 𝐴𝑉 .

] ― [𝑆𝑚𝑎𝑥]) ―1

)

with a

Then, the set of equations turns into:

𝑘𝑐𝑎𝑡,𝑖 + 1[𝑆𝑖(𝑡)] 𝐾𝑀,𝑖 + 1 + [𝑆𝑖(𝑡)]

[𝐸𝑖 + 1]𝑇 ― 𝑘𝑜𝑓𝑓[𝑆𝑖(𝑡)] #(7)

We further assume that there is no waiting room for E0, so it produces S1 at a constant velocity41, 42 𝑣0 until the waiting room is filled. Hence, we have the following simplification for S1 net production:

𝑑[𝑆1(𝑡)] 𝑑𝑡

= 𝑣1 ⋅ 𝐿([𝑆1(𝑡)]) ―

𝑘𝑐𝑎𝑡,2[𝑆1(𝑡)] 𝐾𝑀,2 + [𝑆1(𝑡)]

[𝐸2]𝑇 ― 𝑘𝑜𝑓𝑓[𝑆1(𝑡)]#(8)

For simplicity, 𝑣0 will be approximated by a tunable enzymatic catalytic rate constant kcat,1 divided by the waiting room volume V. 12


Page 13 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

With [𝑃] denoting the final product concentration, the output rate of the enzymatic cascade, 𝑟, is given by

𝑟=

𝑘𝑐𝑎𝑡,𝑁[𝑆𝑁 ― 1(𝑡)] 𝑑[𝑃(𝑡)] [𝐸 ] #(9) = 𝑑𝑡 𝐾𝑀,𝑁 + [𝑆𝑁 ― 1(𝑡)] 𝑁 𝑇

Note that no loss rate is considered for the final product P since it is not confined. The system is solved using either the ode45 or ode23s solver of the MATLAB® software by setting all the concentrations to 0 at t=0.

2. Stochastic Description A stochastic process is defined by the transition probabilities between states. These transition probabilities can be written down in the form of the master equation, which describes the time dependent evolution of the distribution of the stochastic process. These transition probabilities also form the basis of processes examined in queueing theory.19 The stochastic evolution of a set of chemical reactions can be described by the chemical master equation.43,

44

We solve the chemical master equation using the 13


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 72

stochastic simulation algorithm (SSA) also known as the Gillespie algorithm.45 The main difference with the deterministic approach is that chemical reactions are not described by a rate constant but by their probability of occurring between time t and t+dt. This probability is characterized by a likelihood, or propensity function. The definition of these functions is similar to the definition of an elementary chemical reaction rate. Here, we highlight the main features of the discrete stochastic approach (see SI for details). We model an enzymatic cascade as a succession of N servers, each with a service rate equal to the catalytic rate of the enzyme (Figure 2). The first enzyme, E1, consumes its substrate S0 to produce the substrate S1 for the following enzyme E2. It is followed by N2 enzymes which transform Si into Si+1. The last enzyme EN produces the final product P from SN-1. For each enzyme, we assume Poisson arrival times, exponential service times and that the enzyme can only be in a complex with one substrate molecule at a time. In order to consider the same set of reactions as in the deterministic description (Eq. (7)), each waiting room is modelled with a finite capacity as shown in Figure 2. When a

14


Page 15 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

substrate molecule is to be produced, the occupation of the corresponding waiting room is checked. If the i+1-th waiting room is full, enzyme Ei will release its product into the bulk instead of into the i+1-th waiting room. For the reverse reaction, the substrate molecule Si released from the ESi complex will be considered lost into the bulk if the i-th waiting room is full.

Figure 2. Schematic representation of the stochastic model. (A) The enzyme cascade is modeled as a series of N servers with service rates equal to the stochastic enzyme catalytic rates. In between enzymes, there is a tunnel or surface path which stores molecules; we refer to this space as the waiting room. Molecules may exit the waiting room either by forming a complex with the next enzyme, or by being lost to the 15


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 72

environment (impatience). (B) Model of the waiting room i between Ei-1 and Ei. cf, cr, ccat,i-1, and coff are stochastic rate constants associated to the simulated chemical reactions (Eq. (1)) used in the computation of the propensity function. Note that no other simplifications are assumed in the stochastic model.

3. Simulations To determine the rate of product throughput for each condition, the stochastic simulation was independently run 100 times over a time scale of 100s (unless the cascade consisted of more than two enzymes, in which case runs were performed over a time period of 1000s to ensure steady state). The calculated throughputs from each run were averaged to attain the final, presented result. The standard error of the mean is provided with each data point. The kinetic parameters in the simulation were chosen to represent the widely studied glucose oxidase-horseradish peroxidase (GOx-HRP) cascade in which hydrogen

16


Page 17 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

peroxide is the intermediate produced by GOx, and consumed by HRP. The rate constant at which hydrogen peroxide is produced by GOx is varied between 0 and 180 s-1, and the catalytic rate constant of HRP is set to 30 s-1.34 The volume accessible to the intermediate substrate is assumed to be 10 nm3 for all waiting rooms, based on typical tunnel dimensions in enzymes.20 The KM of HRP is set to 2.5 μM.34 This KM, as the KM of almost every enzyme, is below the concentration created by a single intermediate substrate molecule in the waiting room (170 mM). Since the MM model generally assumes the rapid establishment of equilibrium between the enzyme, the substrate and the enzyme substrate complex, we set the substrate unbinding rate constant kr to be 100-fold higher than the kcat and calculated the substrate binding rate constant kf from kcat and KM using Eq. (3). The computations of stochastic rate constants from deterministic rate constants are performed according to Sanft et al.46 All parameters are listed in Table 1.

17


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 72

Table 1. Values of Parameters Used in Stochastic Simulations.

Parameter

Symbol

Number enzymes

Value

of in

the N

2-10

cascade Waiting

room

capacity Interenzyme volume Michaelis constant (enzyme E2)

nWR

1-10 (or +∞)

V

10 nm3

KM,2

2.5 μM or 250 mM

Deterministic rate kcat,1

catalytic

0 to 180 s-1

constant (E1) Deterministic enzyme

catalytic kcat,2

30 s-1

rate constant (E2) Deterministic reverse

reaction kr = 100 kcat,2

3000 s-1

rate constant

18


Page 19 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

Deterministic forward

reaction 𝑘𝑓 =

𝑘𝑟 + 𝑘𝑐𝑎𝑡,2

rate constant

𝐾𝑀,2

12.12 to 12.12*105 mM-1.s-1

Stochastic catalytic

rate ccat,1 = kcat,1

0-180 s-1

rate ccat,2 = kcat,2

30 s-1

constant (E1) Stochastic catalytic constant (E2) Stochastic reverse reaction

rate cr = kr

3000 s-1

constant Stochastic forward reaction constant

rate

𝑐𝑓 =

𝑘𝑓 𝑁 𝐴𝑉

2013 to 2013*105 s-1

19


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 72

Results 1. Cascades without impatience The most basic system is a two-enzyme cascade with no impatience in the waiting room and a KM of the second enzyme far below the concentration corresponding to just one intermediate molecule in the accessible volume of the waiting room (170 mM for a volume of 10 nm3). Under these conditions, any intermediate substrate molecule entering the waiting room is accepted by the second enzyme without delay if the second enzyme is not occupied. We computed the output rate of the cascade with both deterministic and stochastic models. In the deterministic model, the overall output rate is equal to the smaller of the production rate of the first enzyme (kcat,1) or the catalytic rate constant of the second enzyme (kcat,2). The stochastic model shows deviations from the deterministic model only when the two enzymes have similar catalytic rate constants and when the waiting room capacity is small (Figure 3). For a waiting room with infinite capacity, both models agree within the error of the stochastic simulation.

20


Page 21 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

Figure 3. Deterministic model solutions and simulations of a two-enzyme cascade with a

KM = 2.5 μM (cf = 2013*105 s-1) and a finite waiting room in between enzymes. The solid line represents the solutions of the MM ODEs system (Eq. (7)) and a maximum concentration in the intermediate volume representing 1, 2, 4, 10 and ∞ molecules (solutions overlap). Symbols represent the results of the stochastic simulations for a waiting room capacity of 1 (orange triangles), 2 (blue squares), 4 (green triangles), 10 (red circles) and ∞ molecules (black diamonds). Error bars for standard error are smaller than the marker.

21


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 72

Figure 4. Deterministic model solutions and simulations of a two-enzyme cascade with a

KM = 250 mM (cf = 2013 s-1) and a finite waiting room in between enzymes. Solid lines represent the solutions of the MM ODEs system (Eq. (7)) and a maximum concentration in the intermediate volume representing 1 (orange), 2 (blue), 4 (green), 10 (red) and ∞ (black) molecules. Symbols represent the results of the stochastic simulations for a waiting room capacity of 1 (orange triangles), 2 (blue squares), 4 (green triangles), 10 (red circles) and ∞ molecules (black diamonds). Error bars for standard error are smaller than the marker.

22


Page 23 of 72 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

The system increases slightly in complexity if the KM of the second enzyme is chosen so high (e.g. 250 mM), that it requires a significant population (an average of 1.5 molecules for 250 mM) in the waiting room (Figure 4). In this case, also the results of the deterministic model depend on the waiting room capacity if the first enzyme is producing faster than the second enzyme, because the second enzyme cannot be saturated without filling the waiting room and triggering the overflow condition (molecules produced by the first enzyme are returned to the bulk). The maximal output rate for the downstream enzyme is achieved when the number of intermediate molecules available is equal to the waiting room capacity; then, the maximal output rate, based on the MM model, 𝑟𝑚𝑎𝑥 𝑀𝑀,2, for

kcat,2

Enzyme Cascade Reactions - ACS Publications - American Chemical

Recommend Documents