Team:CLS CLSG UK/Total Breakdown
Credit - John Moeses Bauan@unsplash
Total Breakdown
Constants
E - enzyme
kd - rate constant for unbinding
ka - rate constant for binding
E∙S -enzyme-substrate complex
P - products of cocaine breakdown
Kcat - -turnover number
Vmax - the maximum rate of the reaction
KD - the disociation constant
Total breakdown
From the previous two parts to our model, we had the concentration of enzyme and the assumption that cocaine concentration is constant inside the cell. Calculating the rate at which an enzyme breaks down substrate involves the use of Michaelis-Menten Kinetics. Chemically, this equation can be written as:
The rate of breakdown of cocaine by the enzyme is: where E is the enzyme, S is the associated substrate (in this case cocaine), E·S is the enzyme-substrate complex, and P is the products produced, here are benzoate and ecgonine methyl ester. ka and kd are the association and dissociation constants, respectively, for the first reversible reaction, and kcat is the speed at which E and P are produced in the second reaction.
The maximum rate of breakdown is usually described as the maximum speed of the reaction between the enzyme and substrate, Vmax. This occurs when the enzyme is saturated, i.e. when all the enzyme is in a complex with the cocaine.
Using mass action kinetics, and assuming an equilibrium between the cocaine and the enzyme-substrate complex:
It can also be said that [E]=[E]total-[E∙S]. This gives the equation:
Dividing both the numerator and denominator of the fraction by ka gives:
The value kd/ka is the dissociation constant KD. And from equation 3.1:
From equation 3.2:
This is the well-established Michaelis-Menten equation. The major differences from normal Michaelis-Menten kinetics are that [S] is being assumed constant. This results in dP/dt being constant. We have also assumed, as stated earlier in equation 3.3, that the binding of the enzyme and cocaine is in equilibrium which can be well approximated since the substrate reaches equilibrium on a much faster time-scale than the product is formed. This was the reason for the previous model, an iterative model to calculate the initial enzyme concentration in the E. coli. We also couldn’t find a value for KD so we used a value for a different bacterial enzyme involved in breakdown of a similar substrate. Through our research we discovered that all bacterial enzymes bind to substrates at similar rates and so this was a fair approximation. These values are shown in the table here and putting them into equation 3.8 gives d[P]/dt equal to 2.5×10-11 Ms-1 for CocE and 1.1×10-2Ms-1. This proves that the mutant enzyme breaks down cocaine at a rate an order of 109 faster than the CocE which proves it to be far more effective as a filter.
Integrating both sides of equation 3.7 with respect to t gives:
Where the constant of integration is equal to 0. From here we arrived at graph 4 and graph 5 for the two enzymes.
With this rate we worked out the total number of cells in order to calculate the total rate of breakdown by the entire filter.
Using the images we took of the wells on our ELISA plates, we adapted the ecological technique of quadrating to determine an average density of cells per unit area. We realise that this method to determine the coverage of our bacteria is not a particularly accurate one, but given the maximum magnification we were able to focus on was 100x we were unable to get objective quantitative data from an image processing software, see hardware for more on how we dealt with this problem. In conclusion, we ended up with a value of 740.88 cells/m2.
If we assume the lamella takes up the top meter of the filter. This means the total volume of the lamella is 4.3×24=117.6 m3
The surface area of the lamella has been quoted as 6.3 m2m-3 by Hewi manufacturer. Which gives a total surface area of 740.88 m2 for our lamella.
Using our calculated rate of breakdown for CocE, 7.6μgm-3s-1 which, using our volume of the cell from the first model is , 4.9×10-25g cell-1 s-1 and the value for the number of cells per meter squared which we calculated in the lab, we can estimated there are:
The total rate of breakdown= 4.9×10-25×6.59×109×740.88 = 2.3 pgs-1
For the mutant enzyme the equation is exactly the same except the rate of breakdown is now 2.2×10-16 g cell-1 s-1.
This gives a total rate of breakdown = 1.1 mgs-1
The total flow rate of the cocaine can be calculated
Using this data, we know that the cross-sectional area of the tank = 4.9m × 4.3m = 21.07m2
Given a flight speed of 0.9m/min, total flow = 18.96m3min-1, which = 0.316m3s-1
0.316m3s-1 = 316 Ls-1
At a concentration of 1000ng L-1, the flow rate of cocaine can be calculated as 1000ng L-1 × 316 Ls-1= 316000ng s-1 = 0.000316 g s-1
0.000316g s-1 = 316000000 pg s-1 which is larger than the breakdown rate calculated for the CocE enzyme (2.3 pg s-1)
0.000316g s-1 = 0.316mg s-1 which is smaller than the breakdown rate calculated for the mutant enzyme (1.1mg s-1)
From this we can conclude that according to our model, while the wildtype CocE enzyme would not be able to break down all of the cocaine the mutant form would be able to do so.