Modeling in synthetic biology is a crucial tool that helps us to get a comprehensive vision of biological systems and the working principles while providing ways to improve the system. In our case, it significantly led to the design of our project while improving the part choices. Our project was based on the enhancement of fermentation by increasing the lifespan of an ethanologenic bacteria; E.coli strain KO11. Thus, we used simple models to simulate the kinetics of fermentation and enzymatic activities which provided insights on the formation of our experiments and the improvement of our project design.

What we have achieved

We constructed our gene circuit with the help of toxicity analysis and estimation of enzymatic reactions by Michaelis and Menten enzyme kinetics.
We used microbial growth and fermentation kinetics to simulate the expected behaviors of our system and the effects of our genes.
We used microbial growth and fermentation kinetics to simulate the expected behaviors of our system and the effects of our genes.
We improved the understanding of our project by demonstrating the pathways and effects of our genes.

Formation of Our Kinetic Models

Cell Growth Kinetic

The cell growth rate of bacteria depends on the number of cells and their specific growth rate. Thus, it can be expressed as:

dX/dt = µ * X

when we arranged the equation and integrated the both sides we get:

X = e ^ µ*t

We decided to use the Monod kinetics to make substrate and cell concentration-dependent equation. It is shown below:

dX/dt = (µmax * X * S) / Ks + S

“Ks” represents the saturation concentration where “µmax” represents the maximum specific growth rate. In the Monod equation, we can observe the lag, exponential and stationary phase. However, because our system also includes inhibitor by-products, the Monod equation turned into a new form where the cell growth also depends on the inhibitor concentration:

dX/dt = (1 - (C / Cm)) * [(µmax * X * S) / (Ks * ( 1 - (C / Cm))]

X = Number of Cell,

C = inhibitor concentration

Cm = Maximum inhibitor concentration cell can live

µmax = Maximum specific growth rate

Ks = Saturation concentration

S = Sugar concentration.

Enzymatic Reaction Kinetics:

Enzymatic reactions were one of the main components of our project where we aimed to catalyze the furfural reduction. The enzymatic reactions depend on the presence of the substrate and the enzyme which are multiplied by forward and reverse reaction rate constants. We can show the reaction by using mass kinetics.

E + S → ES → P + E
d[S]/ dt = -k1 * [E] * [S] + k-1* [ES]
d[E]/dt = -k1[E]*[S] + (k-1 + k2) * [ES] - (k-2 * [E]*[P])
d[P] /dt = k2 * [ES] - k-2 * [E] * [P]

However, like in the Monod model, the formula can be developed by adding different constants like “Km” and “Vmax” to form Michaelis and Menten equations which represent the enzyme reaction rate.

V = (Vmax * [S]) / Km + [S])

V = Reaction Rate

S = Substrate Concentration

Km = Michaelis Menten constant

Vmax = Maximum Reaction Rate

Fermentation Kinetics:

Ethanol production rate depends on the cell and substrate concentration. However, we consider the mutual inhibition of xylose and glucose fermentation along with alcohol inhibition. Thus, the formula developed by Olsson, Hagerdalt & Zacchi (1995) expressed as below:

Rpg =( X* Vmg * S ) / (Kms + S[ 1 + S / Kis + G / Kigs ) * (1 - P / Pm) ^ n

Rps = ( X* Vms * S ) / (Kms + S[ 1 + S / Kis + G / Kisg ) * [(1 - P / Pm) ^ n] * K1

Rp = Rpg * K2 + Rps * K3

G = Glucose concentration (g/L)
K1 = inhibition constant for xylose fermentation
K2 = inhibition constant for nonspecific inhibitors in the condensate in glucose fermentation
K3 = inhibition constant for nonspecific inhibitors in condensate in xylose fermentation
Kms = Monod constant for pentose fermentation (g/L)
Kigs = inhibition constant for pentoses in glucose fermentation (g/L)
Kisg = inhibition constant for glucose in pentose fermentation (g/L)
X = final cell mass concentration
P = product (ethanol) concentration
Rps = ethanol production rate from xylose g/L * h)
Rpg = ethanol production rate from glucose (g/L * h)
Rp = ethanol production rate from both glucose and xylose (g/L *h)
Vmg = maximum ethanol production rate in glucose fermentation(h^-1)
Vms = maximum ethanol production rate in pentose fermentation(h^-1)
S = pentose concentration
Pm = highest amount of ethanol observed

Toxicity Analysis:

Our main issue was to determine what genes to use that would have a positive effect on bacterial fermentation. Thus, we analyzed the toxicity of different inhibitory molecules and substances produced during pretreatment: Furfural, 5-hydroxymethylfurfural (5-HMF) and aliphatic acids; such as formic acid, and levulinic acid.

Figure 1: The results were obtained from Kim et al. (2013). Inhibitor substances were put individually into the medium in 5g/L and relative cell growths were calculated. In the presence of furfural and HMF, relative growth was lower than in the presence of formic acid, and levulinic acid.

Toxicity Analysis Results:

Formic acid and Levulinic acid were shown to inhibit the growth significantly but furans were worse (Kim et al., 2013). Hence, we chose to focus on furans. When we discard the lag phase, it was possible to calculate what percentage the cell growth was inhibited as demonstrated below:

HMF	80% inhibition
Furfural	83% inhibition
Levulinic Acid	10% inhibition
Formic Acid	60% inhibition

Furans (HMF & Furfural):

According to analysis, furans were found to be the most toxic substances when it comes to cell growth (Kim et al., 2013). The most used pretreatment process, dilute acid, gives rise to the formation of furanic aldehydes (Palmqvist and Hahn-Hagerdal, 2000; Larsson et al., 1999; Thomsen et al., 2009; Klinke et al., 2004). Furans are highly reactive, contributing to the birth of reactive oxygen species (ROS) which damage proteins, nucleic acids and cell organelles (Wierckx et al., 2011). Because of the toxicity provided by furans, cell mass and productivity of fermentation decreases (Almeida et al., 2009; Palmqvist and Hahn-Hagerdal, 2000b; Thomsen et al., 2009). Thus, we examined the reaction pathways of furfural and HMF below.

Reaction Pathways:

It was shown that furfural can be reduced to a less toxic form, furfuryl alcohol, by NAD(P)H dependent oxidoreductases which are coded by FucO and YqhD genes (Wierckx et al., 2011). The pathways are shown below:

Furfural Reduction Pathway of YqhD

Enzyme = NADPH Dependent Oxidoreductase:

Furfural + NADPH + Enzyme → NADP+ + Enzyme + Furfuryl Alcohol

HMF + NADPH + Enzyme → NADP+ + Enzyme + 5-Hydroxymethyl-2-furfuryl alcohol

Furfural Reduction Pathway of FucO

Enzyme = NADH dependent Oxidoreductase:

Furfural + NADH + Enzyme → NAD+ + Enzyme + Furfuryl Alcohol

HMF + NADH + Enzyme → NAD+ + Enzyme + 5-Hydroxymethyl-2-furfuryl alcohol

Reaction Kinetics:

The comparison between the reaction of furfural and HMF reductions with different oxidoreductases were done by Michaelis-Menten equation with the data obtained from Miller et al. (2009) and Wang et al. (2011). The results are shown below:

Michaelis & Menten Equation:

V = (Vmax * [S]) / Km + [S])

Results:

The Km of YqhD was approximately 0.9 mM whereas the Km of FucO was approximately 0.4 mM. Km is directly proportional to Vmax and the affinity to the substrate. Though YqhD has higher affinity with the NADPH and furfural, it computes for NADPH biosynthesis which causes cell death (Miller et al., 2009). Thus we decided to use the FucO gene coding for L-1,2-propanediol oxidoreductase that is responsible for the furanic compound degradation. However, there was a need for a strong promoter that would increase the gene expression, because of the low reactivity of L-1,2-propanediol oxidoreductases. Thus, we looked through the existing promoters in iGEM database and examined their strengths that were compared by LMU-Munich 2012 iGEM team.

Decision:

Thanks to the modeling of different promoter strengths, we decided to use the promoter J23100 because of it being the most powerful promoter among the Anderson promoter family.

Furthermore, because furans’ high reactiveness eventually leads to the formation of ROS, the GSH gene producing glutathione synthetase was decided to be utilized in order to decrease the harmful effects and raise tolerance to environmental toxicity.

GSH - Glutathione synthetase

GSH gene codes for Bifunctional gamma-glutamate-cysteine ligase/glutathione synthetase which is responsible for the mass production of the main antioxidant, glutathione. It is one of the most important antioxidant that works in order to reduce ROSs that are produced during metabolic activities such as furfural reduction and fermentation (Pizzorno, 2014; Forman, 2009). It prevents oxidative stress from building up in the cell metabolism which causes cell damage and eventually, death (Pizzorno, 2014; Forman, 2009). The pathway of glutathione synthetase and glutathione are demonstrated below:

Glutathione Synthesis by Glutathione Synthetase

Glycine + y-glutamyl cysteine + Glutathione Synthetases → Glutathione Synthetase Glutathione

Glutathione Oxidation and Reduction Pathways

Glutathione + ROS → Water + Oxidized Glutathione

Oxidized Glutathione + Glutathione Peroxidases + NADPH → NADP+ + Glutathione

Growth and Fermentation

Growth:

The growth rate of our bacteria is important to determine the effects of our genes because we aimed to increase the toxicity tolerance of E.coli KO11 and consequently leading to a better cell growth rate and lifespan.

The average specific growth rate of our four modules were calculated with the help of our wet lab results by using a simple equation as demonstrated below:

µ = (lnX2 − lnX1)/(t2−t1)

X2 is the biomass concentration (g/L) at time t2(h)

X1 is the biomass concentration (g/L) at time t1 (h)

Then, the specific growth rate and the substrate concentration data were used to define the maximum specific growth rate and Monod Constant. In order to obtain the parameters, Monod growth model is linearized by inverting and factoring out umax and we get the Lineweaver-Burk plot (Rorke & Kana, 2017 ). Monod growth model and Lineweaver-Burk plot are shown below:

Monod Cell Growth Model: µ = µmax * S / (Ks + S)

Lineweaver-Burk plot: 1 / µ =(1 / µmax) + (Ks / µmax) * (1 / S)

After defining the parameters, it was important to use a cell growth model with inhibition thus, the below model was used.

dX/dt = k * ( 1 - (C / Cm)) * [(µmax * X * S) / (Ks * ( 1 - (C / Cm)) ]

Fermentation:

Ethanol production rate depends on the cell mass, sugar concentration and the inhibitor substances. In our project, the improvement of ethanol production was decided to be accomplished by increasing the tolerance of bacteria to inhibitors. Thus, the best model that would accommodate all of the variables needed was chosen to demonstrate the ethanol production.

An empirical kinetic model developed by Olsson, Hagerdalt & Zacchi (1995) was used to emphasize the ethanol production rate of normal and improved E.coli KO11. The model that uses the Monod kinetics including substrate and product inhibition, is demonstrated below.

Rpg =( X* Vmg * S ) / (Kms + S[ 1 + S / Kis + G / Kigs ) * (1 - P / Pm) ^ n

Rps = ( X* Vms * S ) / (Kms + S[ 1 + S / Kis + G / Kisg ) * [(1 - P / Pm) ^ n] * K1

There is mutual inhibition of the sugar utilization on fermentation of E.coli KO11 (Olsson, Hagerdalt & Zacchi, 1995). Thus, the corresponding production rates are shown with Kisg and Kigs parameters respectively. Furthermore, because even when the glucose is consumed completely, the xylose utilization is slower, the phenomenon is shown with K1 (Olsson, Hagerdalt & Zacchi, 1995).

The above models describing the ethanol production rate from glucose and xylose respectively are combined to calculate the ethanol production rate from both xylose and glucose demonstrated below:

Rp = Rpg * K2 + Rps * K3

Because the utilized fermentation kinetic model is based on the final cell mass concentration, the final cell mass was calculated by using the following empirical equation:

∆ x = [ (4.30 * M) / (37.6 + M) ] * [ ( 0.77 * X0) / (X0 - 0.11) ]

∆ x = Growth

M = total initial sugar concentration

X0 = initial cell mass concentration

References

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