Calculating total DCM degraded

1) Obtaining theoretical growth curve

To start this calculation, we needed to know how many bacteria we could reasonably expect to have in our system. To do this, we used reasonable bead dimensions and reasonable bead numbers to calculate the volume of bacteria infused agarose that we were likely to have in the system. We then used the assumption that the bacteria would grow to an optimum density of 10^7 bacteria per ml of agarose (REFERENCE) and combined these to give us an approximation of how to scale the growth curve:

A = Gompertz vertical scaling constant

N = number of beads

V = volume of each bead in ml

ρ = number of bacteria per ml

Our theoretical growth curves were based on Gompertz functions for the reasons explained when you follow this link: (what are Gompertz functions?). An example output growth curve of the model is shown here.

The scaling of the growth rate of the Gompertz function comes directly from growth curves of the DM4 bacteria that we obtained in the lab. See more about our work with growth curves here.

2) Calculating the volume of DCM that these bacteria could degrade

Our next task was to model the rate of DCM degradation by the average bacteria. Using Michaelis-Menten kinetics[1], this was predicted to be:

d[Ndcm]/dt = rate of DCM molecule degradation (s-1)

Kcat = dcmA turnover rate (= 0.6 s-1 for DM4)

[DCM] = DCM concentration (= 0.02M for our system)

[DcmA] = Number of DcmA molecules per cell (87576) Where did this number come from?

Km = Michaelis constant ( = 9 x 10^-6 M for this reaction)

Through the use of diffusion-limiting beads, [DCM] is kept constant at 0.02M. This is significantly larger than our Michaelis constant so this equation can be simplified by using the following assumptions:

Multiplying this by our population function, the total rate of DCM molecule degradation is given as:

Turning this into a more recognisable value (a volume) gives the total rate of DCM degradation as:

Where:

When all of these calculations were modelled in Matlab with the input conditions shown above, the total volume of DCM that we would predict the native bacteria DM4 to degrade in 24 hours is shown by the red line below. This is before you account for the possible toxicity of the pH drop. This is taken into account in the sections below.

Reference:

  1. Michaelis L. and Menten M.L. Kinetik der Invertinwirkung Biochem. Z. 1913; 49:333–369 English translation Accessed 6 April 2007

Calculating the pH change

The degradation of DCM by DcmA produces hydrochloric acid (HCl) according to the reaction below: 

There is a resulting pH change as a result of this. Because we are dealing with an organic system which cannot tolerate excessive acidity (pH<6), we must track the anticipated HCl production and resulting pH change. The following relationships were used:

NHCL = molecules of HCl

NDCM = molecules of DCM

V1(t) = aqueous layer volume (ml)

V2(t) = DCM layer volume (ml)

y(t) = bacteria population

MRDCM = DCM molar mass

NA = Avogadro’s constant (mol-1)

ρDCM = DCM density (kg/dm3)

[HCl] = HCl concentration (M)

These equations were then simulated using a series of linked functions on MatLab and the results are displayed below:

Summary:

As you can see from the above graph, the native bacteria (Methylobacterium extorquens DM4) will not be able to degrade a large volume of DCM. This bacteria will not be suitable for our purposes as the solution to chlorinated waste disposal. There are several reasons for this, these include:

The degradation of DCM is a stress response for the DM4 bacteria.

The DM4 bacteria grow very slowly (time to grow up a culture is approximately 2 weeks) and are not very robust to external conditions at all.

The DM4 bacteria are very vulnerable to the toxic intermediates of this reaction.

DM4 are not very well known characterised or known about.

However, using synthetic biology, we can dramatically increase the amount of chlorinated solvents that certain bacteria can degrade. This is because:

We will remove the stress response dependence for the degradation of chlorinated solvents. Also, due to this being a reaction driven by the removal of product, we will be able to significantly increase the rate of degradation.

The bacteria that we’re using are E-coli and pseudomonas strains. These bacteria are very fast growing (relatively) and are much more robust to changes in the external conditions.

These strains are less vulnerable to our toxic intermediates.

This model proves the power of computer modelling and shows the importance of using synthetic biology to solve global problems.

Gompertz Functions

We used a variation of a sigmoid function called a Gompertz function to model the theoretical growth of our bead-encapsulated bacterial population. These functions are well established[1] as a method of predicting population growth in a confined space, as birth rates first increase and then slow as resource limits are reached. As this is how our bacteria will be growing (when confined in the beads), we took this information and assumed that the bacteria’s population over time will follow one of these functions (when scaled correctly).

Gompertz functions are of the form:

y(t) = population size as a function of time

A = maximum sustainable population

B = shift on time axis

C = growth rate

Using this theoretical form, we could then calibrate the values of our variables through comparison with actual growth curve data from wet lab experiments. This was an important step because it would then allow us to calculate the total theoretical degradation rate of DCM that our kit could support.

Varying each of the three constants allows us to fit our Gompertz function to the actual growth data. The effect of varying each constant is shown below:

Reference:

Zwietering, M. H.; Jongenburger, I.; Rombout, F. M.; van 't Riet, K. (1990), "Modeling of the Bacterial Growth Curve", Applied and Environmental Microbiology 56 (6): 1875–1881

Using buffers to reduce the pH change of our system

As one of the products of our reaction is hydrochloric acid, we have been able to calculate the pH change of the system. However, as pH change is bad for the bacteria, we have investigated the effect of using pH buffers in the aqueous part of our system.

The pH change of our system in the presence of the buffer HEPES is described by:

Upon solving the equation in Matlab, it was clear that only a relatively low concentration (0.05 M) of buffer was needed to significantly reduce the pH change of the solution:

The other method of reducing the overall pH change is adding much more water to the system at the start. This is the easiest and cheapest method, indeed it could be used for single use DCM disposal kits. However it is impractical in large scale applications due to the huge advantages of using buffer. On the other hand, because the buffer itself is toxic in high enough concentrations, a compromise between the amount of water added and buffer concentration had to be reached.

Calculating the pH change

We then used our model to predict the effect on the system if you simply increased the amount of water in the aqueous layer. It does show how much water is necessary to prevent the pH change being too large and helps show how useful adding a pH buffer to the aqueous solution would be.

The graph here is for non specific inputs and is for demonstration purposes only. It shows well how the model responds to changing the input values.

Buffers?

The apparent uni-molecular rate constant kcat is also called turnover number and denotes the maximum number of enzymatic reactions catalysed per second.

We used our model to predict the effect on the response of the system if the kcat value of the bacteria was varied. Changing the kcat of our system by a significant amount is unrealistic in the length of our project. However, in future work, the kcat could be altered gradually.

In the graph shown here, the total volume degraded doesn’t change. This is because the amount of HCl that the system requires to reach a toxic pH level is constant. This is because we are not varying the volume of the aqueous layer. To increase the total amount of DCM degraded, you simply need to add more water or a pH buffer to the system. However, increasing the kcat value dramatically increases the rate of the degradation. This hints towards a valid future area of research.

The graph here is for non specific inputs and is for demonstration purposes only. It shows well how the model responds to changing the input values.

Increasing the kcat of the degradation would be a very valuable improvement to the system. Above you can see how we have modelled specifically how it does improve the system.

It can be shown by simple adjustment of the parameters that this model could be adapted to simulate the processing of some other reactions by different bacteria with different Kcat values. This modelling technique is particularly powerful because if you know certain parameters about the system, you can simulate the amount of product that could be produced.

More broadly, the potential benefit of months of synthetic biology research could be analysed in a few hours with this model. On top of this, even if the relevant parameters weren’t known exactly, finding these parameters is much easier (CHECK?) than completing the biological system.

To demonstrate what we mean by this, here are some other processes with different kcat values[1]:

As you can see, some of these values are many orders of magnitude higher. Using processes with these values would improve the amount of product produced by the same number of orders of magnitude and therefore could be highly beneficial to a system. Future work could definitely involve modelling these reactions and investigating the potential benefits before the wet lab work begins.

Reference:

1.  Mathews, C.K.; van Holde, K.E.; Ahern, K.G. (10 Dec 1999). Biochemistry (3 ed.). Prentice Hall. ISBN 978-0-8053-3066-3.

Calculating the pH change