Introduction
We developed a mathematical model to understand the relationship between the enzymes in our genetic constructs and their immediate microenvironment a priori. Using the MATLAB numerical solver ode23 and kinetic information from in vitro experiments found in literature (Table 1 and 2), we modeled the bioluminescence reaction network as an isolated system under steady state conditions and acquired preliminary light emission results for our different genetic constructs.
Chemical Reaction Network
The microbial Lux system can be interpreted as a two-component module, the aldehyde synthesis pathway and the light production pathway, coupled by an aldehyde and fatty acid.
Figure 1. The reactions in the aldehyde synthesis pathway are tightly coupled by the LuxCDE genes given the 1:1 ratio amongst substrate and product15.
In the aldehyde synthesis pathway (Fig. 1), the transferase (LuxD) first cleaves acyl-ACP and forms a fatty acid (RCOOH), via hydrolysis, which is subsequently activated by the synthetase subunit (LuxE) and reversibly transferred to the reductase subunit (LuxC) before being reduced with NADPH to an aldehyde (RCHO)5,8,18,19,20. The fatty acid activation process in the synthetase subunit occurs at very low levels in the absence of reductase15. However, when the synthetase and reductase subunits are co-expressed with luciferase, the system proceeds to emit light in the absence of transferase16; albeit at lower levels.
Figure 2. The light production pathway is controlled by an enzyme complex luciferase (LuxA+LuxB, we used a LuxA+B fusion protein for enhanced luminescence23) and its immediate microenvironment (molecular oxygen, aldehyde, and reduced flavin concentrations).
In effort to parallel our in vivo experiments, the light production network (Fig. 2) was extended to incorporate alternate reactions, such as aldehyde inhibition and the autoxidation of reduced flavin.
In the absence of RCHO, the reduced enzyme-oxygen adduct complex (intermediate II) dissociates and shifts the light producing pathway to the dark pathway4,7,12,11,17. Under conditions in which the aldehyde concentration is high and the reduced flavin concentration insufficient, the aldehyde binds to luciferase prior to reduced flavin or to luciferase-bound hyrdroperoxyflavin and inhibits the system1,2. Further light emission is limited by the autoxidation of reduced flavin and the slow dissociation of the aldehyde from the aldehyde-luciferase complex10 in addition to the release of fatty acid from peroxyhemiacetal (LuxAB-FMNH2O2-RCHO)11,13. The presence of functional luciferase, which governs light production, is therefore determined controlled by the presence of specific cofactors.
Modeling
To study the possible behavior of our genetic constructs, we constructed a model following the work of Stekel6. We reduced the complexity of our system to only enzymatic interactions via reactant(s) and product(s) exchange using the following assumptions:
- The reversible reactions in the Lux pathway proceed at a negligible rate such that the forward reaction constant far exceeds the reverse reaction constant.
- In low to zero concentrations of aldehyde, the LuxAB-FMNH2-O2 complex can proceed towards the dark pathway, in which flavin and hydrogen peroxide are produced. However, we have two sources of aldehyde in our experiment: (1) the endogenous supply from yeast via acyl-CoA and (2) the recycled fatty acid in the Lux pathway. With these two sources, it is unlikely that the concentration of aldehyde in our system will be close enough to zero so that the dark pathway will proceed therefore we did not incorporate the dark pathway. Furthermore, the incorporation of the frp gene will provide sufficient amounts of reduced flavin therefore we neglected aldehyde inhibition.
- Intermediate species were removed using quasi-steady state assumptions.
- Cellular heterogeneity is ignored.
- Reactants are equally distributed throughout the cell. By assuming spatial homogeneity, the rate of each equation is independent of position in space. As a result, we explicitly refer to the reaction rate in the volume instead of having to specify different rates at different positions.
- By the continuum hypothesis, we describe changes in molecular abundance over time as a function of concentration.
- The system in study is isolated and under steady state conditions. Therefore, environmental co-factor concentrations, such as [O2] , [NADPH], and [H2O], are fixed.
Use of these assumptions resulted in the following chemical reaction network and corresponding reactions.
Figure 3. LuxD increases the production of aldehyde via fatty acid synthesis while the frp enzyme (f) provides a sufficient amount of reduced flavin enabling light production without inhibition. Furthermore, co-factors, such as ATP and H2O, were not included in the reaction network since their concentrations are assumed to be fixed.
Figure 4. Given the cooperative and structural22 nature of the synthetase and reductase subunit, the individual reactions (fatty acid activation, transfer and reduction), were reduced to one equation.
In effort to capture the catalytic activity of an individual enzyme or complex, we used the modified steady state velocity equations (unpublished) provided by D. Stekel and M. Iqbal21.
Figure 5. The steady state velocity equations were developed using the Alberty Rate Law.
Differential equations describing the flux of individual substrates throughout the reduced reaction (Fig. 3) were then derived using the previous velocity equations (Fig. 5) and simulated using custom MATLAB analysis code (luxAB.m, luxABfrp_CDE.m, luxABfrp_CDECE.m, luxABfrp_CDED.m, and stoichiometry_analysis.m, GitHub link at end of documentation).
Figure 6. The distribution of substrates throughout the reduced reaction network can be characterized as the difference in reaction rates.
Due to the equal molar expression of enzyme provided by our genetic strategy, we can assume that the overexpression of a certain enzyme would approximately double its corresponding reaction rate. Since the subunits cannot perform effectively when isolated, we expected that the overexpression of LuxC and LuxE would produce the same emission, at steady state, as our control (LuxCDE). Therefore, we decided to simulate the overexpression of the LuxEC complex in attempt to show a change in light emission.
Results
Figure 7. Change in substrate and product concentration and light yield for the different genetic constructs. (A) Change in aldehyde concentration. (B) Change in fatty acid concentration. (C) Change in light production at steady state levels of aldehyde (A). (D) Summary plot of light production at steady state. Note: LuxCDE is also representative of the LuxCDEC and LuxCDEE constructs.
The molar overproduction of LuxEC lead to a higher steady state production of aldehyde, which was expected since the aldehyde-producing enzyme complex concentration was increase thereby increasing its maximum yield. Interestingly, however, the change in RCOOH concentration for both the control and the overproduced LuxEC construct was the same. LuxD overproduction, lead to a faster production RCOOH, as expected; however, the lowered RCHO steady state concentration was unforeseen. The results seem to suggest that high LuxD concentrations saturate the LuxEC complex, which effectively reduces the production of RCHO and therefore light emission (Fig. 7C). However, in Gupta et al., the expression of LuxD in yeast lead to an approximate 33-fold increase in emission16. Further experimentation would be required to confirm such results.
Future Directions
We expect to improve our model’s predictive power by deriving a steady-state equation for the complex light reaction mechanism (Fig. 2) without barring intermediate interactions using the King-Altman method. As a preliminary effort, we constructed a model for the light production pathway to study light emission under different conditions in addition to studying the sensitivity of certain parameters that have been suggested in literature to control light production. The following system of equations were simulated using our custom MATLAB analysis code (light_rxn.m and light_rxn_sim.m, GitHub link at end of documentation)
Figure 8. The Law of Mass Action was implemented to model the rate of individual chemical processes as the instantaneous change in reactant and product concentration. Remaining equations are in script file light_rxn.m.
We studied light emission while varying substrate concentration (RCHO and FMNH2) at fixed O2 (550 µM).
Figure 9. Induction curves for different aldehyde concentration and steady state results
The steady state results for aldehyde indicate that it has an immediate effect on the system (Fig. 9).
Figure 10. Induction curves for reduced flavin indicate a gradual influence on light production relative to aldehyde (Fig. 9).
Note that under saturating conditions the limiting factor becomes the luciferase (LuxAB) since both RCHO and FMNH2 converge to a similar light production state.
As suggested by Li et. al3 the dissociation of FMN controls the availability of functional luciferase and therefore light production. To test the concept, we simulated our system while varying the rate constant in control of FMN release (k19).
Figure 11. Results show that the release of FMN can in fact control light emission.
Another source for a potential discrepancy in light emission from the different genetic constructs would be differing enzyme degradation rates. Further experimentation by the UC San Diego team would help confirm such a hypothesis and further improve our model.
Enzyme Table
| Enzyme | Parameter | Value |
|---|---|---|
| lux AB | VmaxluxAB | 71.58 |
| r22 | 0.62 | |
| k41 | 0.22 | |
| k42 | 81.5 | |
| k43 | 72.2 | |
| lux EC | VmaxluxEC | 198.93 |
| r44 | 0.04 | |
| k61 | 90.9 | |
| k62 | 95.3 | |
| k63 | 24.35 | |
| k64 | 76.5 | |
| frp | Vmaxfrp | 51.8 |
| r12 | 1 | |
| k31 | 0.72 | |
| k32 | 49.5 | |
| lux D | VmaxluxD | 45.8 |
| r33 | arbitrary | |
| k51 | 0.37 | |
| k52 | arbitrary |
Parameter Table
| Rate Constant | Value | Reference |
|---|---|---|
| k1 | ||
| k2 | ||
| k3 | ||
| k4 | ||
| k5 | ||
| k6 | ||
| k7 | 21.2 s-1 | (14) |
| k8 | 10 s-1 | (9) |
| k9 | 6.0*105 M-1s-1 | (12) |
| k10 | 4.6 s-1 | (12) |
| k11 | 2.4*106 M-1s-1 | (12) |
| k12 | 0.1 s-1 | (12) |
| k13 | 1.2*105 M-1s-1 | (12) |
| k14 | 0.1 s-1 | (12) |
| k15 | 9.5 s-1 | (12) |
| k16 | 0.5 s-1 | (12) |
| k17 | 3*103 M-1s-1 | (2) |
| k18 | 0.06 s-1 | (2) |
| k19 | 1.9*10-3 s-1 | (3) |
| k20 | 1*105 M-1s-1 | (12) |
| k21 | 40 s-1 | (12) |
Cofactor Table
| Chemical Species | Concentration (µM) | Reference |
|---|---|---|
| NADPH | 560 | (6) |
| ATP | 1310 | (6) |
| O2 | 550 | (27) |
| H2O | 500 | arbitrary |
| H+ | 300 | arbitrary |
Scripts
Link to Enzyme Kinetics Scripts
References
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