Team:UC San Diego/Modeling/Kinetics

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 pathway^4,7,12,11,17. 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 system^1,2. 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-FMNH2O2 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.

To study the relationship between light emission and different conditions, a model for the light production pathway was constructed to analyze single-burst assays in addition to studying the sensitivity of certain parameters that have been suggested to control light emission.

We used mass action approximation to model the rate of individual chemical processes as the instantaneous change in reactant and product concentration. Furthermore, we made the following assumptions to reduce the complexity of our system to only enzymatic interactions:

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, concentrations, such as [O2] , [NADPH], and [H2O], are fixed.

The following system of equations were used to simulate the light production pathway under varying environmental conditions using our custom MATLAB analysis code (light_rxn.m and light_rxn_sim.m, GitHub link at end of documentation)

Induction Curves and Parameter Sensitivity

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FIGURES

Model Reduction

SYSTEM OF EQUATIONS

Velocity Equations

EQUATIONS

Results

Future Directions

Enzyme Table

Enzyme	Parameter	Value
lux AB	Vmax_luxAB	71.58
	r₂₂	0.62
	k₄₁	0.22
	k₄₂	81.5
	k₄₃	72.2
lux EC	Vmax_luxEC	198.93
	r₄₄	0.04
	k₆₁	90.9
	k₆₂	95.3
	k₆₃	24.35
	k₆₄	76.5
frp	Vmax_frp	51.8
	r₁₂	1
	k₃₁	0.72
	k₃₂	49.5
lux D	Vmax_luxD	45.8
	r₃₃	unknown
	k₅₁	0.37
	k₅₂	unknown

Parameter Table

Rate Constant	Value	Reference
k₁	unknown
k₂	unknown
k₃	unknown
k₄	unknown
k₅	unknown
k₆	unknown
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 (uM)	Reference
NADPH	560	(6)
ATP	1310	(6)
O₂	550	(27)
H₂O	500	unknown
H+	300	unknown

Scripts

Link to Enzyme Kinetics Scripts

References

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[2] Abu-Soud HM, Clark AC, Francisco WA, Baldwin TO, Raushel FM. (1993) Kinetic destabilization of the hydroperoxy flavin intermediate by site-directed modification of the reactive thiol in bacterial luciferase. J. Biol. Chem. 268:7699–706.
[3] Z. Li and E. A. Meighen (1994) The turnover of bacterial luciferase is limited by a slow decomposition of the ternary enzyme-product complex of luciferase, FMN, and fatty acid.J. Biol. Chem. 269: 6640-6644.
[4] J W Hastings and C Balny (1975)The oxygenated bacterial luciferase-flavin intermediate. Reaction products via the light and dark pathways.J. Biol. Chem. 250: 7288-7293.
[5] L. Wall, A. Rodriguez, and E. Meighen (1986) Intersubunit transfer of fatty acyl groups during fatty acid reduction.J. Biol. Chem. 261: 15981-15988.
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[9] Meighen, E. A. and Hastings, J. W. (1971) Binding Site Determination from Kinetic Data: Reduced Flavin Mononucleotide Binding to Bacterial Luciferase J. Biol. Chem. 246: 7666-7674.
[10] B. Lei, K. W. Cho, and S. C. Tu. (1994) Mechanism of aldehyde inhibition of Vibrio harveyi luciferase. Identification of two aldehyde sites and relationship between aldehyde and flavin binding. J. Biol. Chem. 269: 5612-5618.
[11] Husam Abu-Soud, Leisha S. Mullins, Thomas O. Baldwin, and Frank M. Raushe. (1992) “Stopped-flow kinetic analysis of the bacterial luciferase reaction”. Biochemistry 31 (15), 3807-3813
[12] Frank M. Raushel, Husman M. Abu-Soud, Leisha S. Mullins, Wilson A. Francisco, Thomas O. Baldwin. “Kinetic and Mechanistic Investigation of the Bacterial Luciferase Reaction”
[13] Kurfurst, M., Ghisla, S. & Hastings, J. W. (1984).”Characterization and postulated structure of the primary emitter in the bacterial luciferase reaction”. Proc. Nail. Acad. Sci. USA 81, 2990-2994.
[14] S. Inouye and H. Nakamura, (1994) Stereospecificity of hydride transfer and substrate specificity for FMN-containing NAD(P)H-flavin oxidoreductase from the luminescent bacterium, Vibrio fischeri ATCC 7744, Biochem. Biophys. Res. Commun., 205, 275–281
[15] Meighen EA & Dunlap PV (1993) Physiological, biochemical and genetic control of bacterial bioluminescence. Adv Microb Physiol 34: 1–67
[16] Gupta RK, Patterson SS, Ripp S, Simpson ML, Sayler GS (2003) Expression of the Photorhabdus luminescens lux genes (luxA, B, C, D, and E) in Saccharomyces cerevisiae. FEMS Yeast Res 4: 305–313
[17 H. Watanabe, T. Nagoshi, H. Inaba (1993).Luminescence of a bacterial luciferase intermediate by reaction with H2O2: the evolutionary origin of luciferase and source of endogenous light emission Biochim. Biophys. Acta, 1141, pp. 297–302
[18] Byers, D.M. and Meighen, E.A. (1985). Acyl-Acyl carrier protein as a source of fatty acid for bacterial bioluminescence. Proc. Natl. Acad. Sci. USA 82, pp. 6085-6089
[19] Soly, R.R. and Meighen, E.A. (1991). Identification of the Acyl Transfer Site of Fatty Acyl-Protein Synthetase from Bioluminescent Bacteria. J. Mol. Biol. 219, 69-77.
[20] Wall, L. , Rodriguez, A. , and Meighen E. (1986) Intersubunit Transfer of Fatty Acyl Groups during Fatty Acid Reduction. J. Biol. Chem. 261: 15981-15988
[21] Iqbal M. , Stekel D. (2015). An extended mathematical model of Lux bioluminescence in bacteria. [unpublished]
[22] Wall, L. and Meighen E.A. (1986). Subunit Structure of the Fatty Acid Reductase Complex from Photobacterium phosphoreum. Biochem. 25, 4315-4321
[23] Cui, Boyu, et al. "Engineering an Enhanced, Thermostable, Monomeric Bacterial Luciferase Gene As a Reporter in Plant Protoplasts." (2014): e107885.