Computational Analysis of the Kinetic Requirements for Coupled Reaction Systems

Abstract

The art of designing coupling systems to drive reactions for endergonic synthesis is a subject of great interest in the scientific community, but it still presents major challenges. The aim of this kinetic study was to run simulations in COPASI 4.39 to test the behavior of hypothetical models for a system that couples two independent reactions, one exergonic and the other endergonic. In our computational study, we unraveled the qualitative and quantitative conditions that allow and benefit coupling, considering all possible reaction pathways within the network. Optimal conditions were reached by assigning favorable directionalities and low activation energies to six reaction steps within a network that featured twenty reaction steps. Moreover, different models were designed and tested in order to investigate the availability of coupling with different reaction steps.

Keywords: computational model; endergonic synthesis; kinetics; simulation.

Figure 1

Representation of a possible coupling system featuring the different states of the molecular system that allows the coupling. In its starting state (upper left), the system can only bind the reagent of the endergonic reaction (blue sphere); then it can bind the substrate of the exergonic reaction (green square). Together, these binding reactions drive the system’s conformational change (yellow to orange). In this state, the system is able to complete the endergonic reaction (sphere turning from blue to purple) and allow the release of that product. Afterward, the system completes the exergonic reaction (green square to red square) through product release. This system therefore features coupling in which the endergonic reaction is completed before the exergonic one, provided that the reagents of both reactions are binding at the same time before release of the first product.

Figure 2

The general reaction network that was designed to study coupling between the exergonic reaction and the endergonic one. Each reaction is labelled with its own reaction name (e.g., R1). A and X are the reagents of the overall exergonic reaction, and B and Y are its products. C and W are the reagent and product of the overall endergonic reaction, and D and Z are its products. B and D are the products of interest whereas Y and Z are treated as waste. M acts as the complex coupling the reactions. All possible reaction paths for an overall A-to-B (exergonic) and C-to-D (endergonic) conversion are featured. For the sake of readability, only one direction of each reaction is labelled with reagents binding to M and/or products being released from the complex (e.g., +A). It is implied that the reaction in the opposite direction has opposite signs for said species.

Figure 3

Reaction network of Model-0 and Model-0-B. Each reaction is labelled with its own reaction name (e.g., R1). A and X are the reagents of the overall exergonic reaction, and B and Y are its products. C and W are the reagent and product of the overall endergonic reaction, and D and Z are its products. B and D are the products of interest whereas Y and Z are treated as waste. M acts as the complex coupling the reactions together. All possible reaction paths for an overall A-to-B (exergonic) and C-to-D (endergonic) conversion are featured. For the sake of readability, only one direction of each reaction is labelled with reagents binding to M and/or products being released from the complex (e.g., +A). It is implied that the reaction in the opposite direction has opposite signs for said species.

Figure 4

Plot of concentration vs. time of Model-0. A is converted to B (exergonic) and C is converted to D (endergonic), thanks to coupling. There is great overlap between the concentrations of the two products and of the two reagents, respectively. The products’ final yield is over 70%, which makes this model efficient, but its timescale makes it extremely slow.

Figure 5

Plot of concentration vs. time of Model-0-B. A is converted to B (exergonic) and C is converted to D (endergonic), thanks to coupling. Model-0-B exclusively contains the R2 + R6 + R8 + R10 + R11 + R3 pathway, whereas Model-0 also includes other pathway possibilities featured in Figure 3.

Figure 6

Reaction network of Model-1. The pathway highlighted in green (R2 + R6 + R8 + R10 + R11 + R3) is called Track 1 and it represents the following sequence in the direction of product synthesis: C + M → CM, CM + A → AMC, AMC + W → AMD + Z, AMD → AM + D, AM + X → BM + Y, BM → B + M. Each reaction is labelled with its own reaction name (e.g., R1). A and X are the reagents of the overall exergonic reaction, B and Y are its products. C and W are the reagent and product of the overall endergonic reaction, D and Z are its products. B and D are the products of interest whereas Y and Z are treated as waste. M acts as the complex coupling the reactions together. All possible reaction paths for an overall A-to-B (exergonic) and C-to-D (endergonic) conversion are featured. For the sake of readability, only one direction of each reaction is labelled with reagents binding to M and/or products being released from the complex (e.g., +A). It is implied that the reaction in the opposite direction has opposite signs for said species.

Figure 7

Plot of concentration vs. time of Model-1. A is converted to B (exergonic) and C is converted to D (endergonic), thanks to coupling. There is great overlap between the concentrations of the two products and of the two reagents, respectively. The final concentration of the products is 100%, a complete conversion.

Figure 8

Reaction network with the same directionalities as the ones in Figure 6. The R2 + R6 + R8 + R18 + R20 + R3 pathway is highlighted in purple and is called Track 2. Track 2 represents the following sequence in the direction of product synthesis: C + M → CM, CM + A → AMC, AMC + W → AMD + Z, AMD + X → BMD + Y, BMD → BM + D, BM → B + M.

Figure 9

Plot of concentration vs. time of Model-2. A is converted to B (exergonic) and C is converted to D (endergonic), thanks to coupling. This plot shows that the performance of Model-2 is on par with Model-1. Model-2 relies on Track 2, a different pathway compared to Model-1, whose performance depends on Track 1.

Figure 10

Reaction network with the same directionalities as the ones in Figure 6. The pathway highlighted in green (R2 + R6 + R8 + R10 + R11 + R3) is called Track 1. The pathway highlighted in pink (R1 + R5 + R7 + R19 + R20 + R3) is called Track 4. Track 1 and Track 4 share the final reaction step: R3. When only one of the two tracks has low activation barriers, coupling is observable (Track 4 shows coupling, but at a slower timescale than Track 1). However, if Track 1 and Track 4 both have low activation barriers, then coupling is lost because their combination allows the exergonic reaction to complete independently via R1 + R11 + R3.

Figure 11

(a) Plot obtained by keeping all activation barriers high, except for Track 1 (R2 + R6 + R8 + R10 + R11 + R3) and Track 4 (R1 + R5 + R7 + R19 + R20 + R3), whose reaction steps have low activation barriers. (b) Plot obtained by keeping all activation barriers high, except for Track 4, whose reaction steps have low activation barriers. Overlap is observable. Coupling is therefore computationally possible with just Track 4, but it is very slow.

Figure 12

Reaction network with the same directionalities as the ones in Figure 6. The pathway highlighted in green (R2 + R6 + R8 + R10 + R11 + R3) is called Track 1. The pathway highlighted in brown (R2 + R6 + R7 + R19 + R20 + R3) is called Track 7. Track 1 and Track 7 share the three reaction steps R2, R6 and R3. When Track 1 and Track 7 both have low activation barriers, coupling is observable, but Track 7 severely slows down the process. Track 7 alone shows coupling, but at a slower timescale than Track 1.

Figure 13

Reaction network of Model-3. The pathway highlighted in orange (R1 + R5 + R7 + R19 + R17 + R4) is called Track 3. In this model, Track 3 features all directionalities in favor of product synthesis.

Figure 14

Plot of concentration vs. time of Model-3. A is converted to B (exergonic) and C is converted to D (endergonic), thanks to coupling.

Figure 15

(a) Schematic representation of the Janusian enzyme, called M. A and X are the reagents of the exergonic reaction and B and Y are its products. C and W are the reagents of the endergonic reaction and D and Z are its products. M changes state when the other species are binding, allowing coupling to occur, resulting in the release of both B and D. The various states of M are omitted here because this scheme would otherwise become as intricate as the network in Figure 2. (b) Schematic representation of an information ratchet mechanism, like those studied by Leigh and collaborators [20], featuring A-to-B as the fuel-to-waste reaction. The ratchet changes states (M-1, M-2, M-3 and M-4) thanks to the following directional path: M-1 is converted to M-2, then it becomes M-3, which then relaxes to M-4.