Modelling

**Overview **

Mathematical modelling is
fundamental to synthetic biology, a tool that allows for deeper understanding
of biological systems, acting as a link between the conception and the physical
realisation of a biological circuit. Being able stimulate and understand our
system behaviour before actual implementation saves both time and resources.

Through our modelling we tried to gain insight into our system so that we could improve it and make it realistically achievable.

We tried to describe the
whole system with a mathematical system of linear ODEs which could
characterizes the expression and secretion of all enzymes, the associated
substrate – enzyme kinetics.

In order to realize the long term goal of developing a detergent biodegradation device for household and commercial use we tried to implement continuous culture modelling on our bioreactor design in order to estimate yearly cost of detergent biodegradation.

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**Single Cell Modelling**

Single Cell Modelling allows to model our gene regulatory network (GRN) and the extracellular secretion of our enzyme alkyl sulfatase (SdsA1).

This model helped us gain
insight into our system in order to understand dependence of rate of secretion
of alkyl sulfatase in media under varying promoter strengths and secretion
efficiencies due to the different secretion extracellular secretion tags PelB
and OmpT.

Our model is based on overexpression of alkyl sulfatase under constitutive promoters from Anderson promoter collection in iGEM registry. The concentration of our enzyme SdsA1 is then predicted using the rates of transcription, translation and degradation (of both mRNA and protein) that are known in literature.

Since under a constitutive gene expression is unregulated, it is always on and its strength could be modelled through the transcription rate constant k1.

Using the law of mass action

1. Transcription rate k1 is estimated from literature.

2. Translation rate k2 is estimated from the literature

3. mRNA degradation (d1) and Protein degradation rate (d2) are known for Ecoli through literature.

Since SdsA1 (alkyl sulfatase) a extracellular enzyme, it was essential to understand effect of secretion efficiencies of our enzyme from Ecoli cells, in order to determine concentration of SdsA1 in the media, which would be needed to model our enzyme substrate kinetics in order to understand SDS(Sodium Dodecyl Sulfate ) degradation.

We tried to model our protein secretion using a empirical secretion law used by iGEM Stuttgart 2017 team.

r_{secretion} :
secretion rate

[enzyme] : enzyme concentration

s : secretion efficiency

t : time

Here the secretion efficiency is a value between zero and one.

Since reliable data on expression of SdsA1 and its extracellular expression in Ecoli was not available we concluded that our model could only provide a qualitative understanding various factors on these.

Fig 1.Simbiology implementation of our Model

Fig.2 Effect of secretion efficiency on enzyme production Fig.3 Effect of promoter strength on enzyme production

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**Enzyme Kinetics**

We use the simple Michaelis-Menten formula to describe our enzymes’ kinetics.

Here, *V*_{max} represents
the maximum velocity achieved by the system, at maximum (saturating) substrate
concentrations. *K _{M}*
(the Michaelis constant; sometimes represented as

*K*instead) is the substrate concentration at which the reaction velocity is 50% of the

_{S}*V*

_{max}. [

*S*] is the concentration of the substrate

*S*.

Our enzyme (E) being SDS, substrate S being SdsA1 and P being our final product 1-Dodecanol.

Here k1 is rate of forward and k-1 being rate of backword reactions and k2 being rate of product formation.

In terms of specific Michaelis-Menten reaction, these constants are quoted in the literature as:

We assumed an average SDS concentration of 5-10 mg/L in domestic wastewater discharges.

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**Continuous Culture
Modelling**

** **

Fig4. Our SDS biodegradation chemostat illustration.

In order to understand if our project could be implemented in a real world we decided to check its economic sustainability by trying to estimate the yearly cost of operation of our bioreactor.

To do so we implement a model based on previous model developed by iGEM 2017 Manchester team who were trying to estimate cost of chemostat operation for cleaning Phosphate in wastewater.

The growth of bacteria in its exponential phase can be represented in the following exponential growth equation:

where:

*x*
is the *bacteria concentration* (dry weight mass/unit volume) at time

* μ*
is the

*specific growth rate*

*is the*td

*doubling time*(time required for the concentration of organism to double)

Monod showed that there is a relationship between the specific growth rate and the concentration of a limiting growth substrate that can be represented in this equation:

where:

* s*
the

*concentration of a limiting growth substrate*

μ

*max*

*maximum growth rate*(growth rate when organism is placed in excess nutrients without any limiting factors)

*K**s**
*is the *saturation constant* –
the value of * s*
when:

Bacterial growth and utilization of substrate is depicted by the Monod by the equation:

where *Y* is known as
the *yield constant*.

In the chemostat fresh
growth medium is added into the vessel at a *steady flow-rate* (* F*)
and culture liquid exits at the same rate and the growth medium is uniformly
dispersed. The rate of nutrient is exchange is given by the

*dilution rate*(

*):*D

Assuming every organism
will have an equal probability of leaving the vessel within a given time. The *wash-out
rate* (rate in which organism initially present in the vessel will be washed
out) can be expressed as:

where * x*
is the concentration of organisms in the vessel

**1. Changes in
concentration of organism**

In a continuous culture, combining growth (1) and washout rate (5) we have the net rate of increase is therefore:

** **

**2. Changes in substrate
concentration**

Assuming substrate enters
the vessel at a concentration * Sin*,
consumed by the bacterial cell in the vessel and then exits the vessel at
concentration

*. The net rate of change is therefore:*Sout

When * dx/dt*
and

*is 0, the system is said to be in a ‘steady state’ because the concentration of organism and substrate within the continuous culture is kept constant. The values of steady state*ds/dt

*and*x

*, designated as*s

x~ and s~

* Sin*
control the steady state within the chemostat. Since we have been also using
E.coli for SDS degradation we use values constants of (growth constant

*μ*

*max*

Ks

*same as used by team iGEM Manchester 2017.*Y

We use values of these constants as referenced here by iGEM Manchester 2017.

Finally output is modelled through the equation:

So there two design
parameters in our bioreactor design model required for consideration.

Cost Estimation

In order to achieve economically viable bioreactor we need to use a cheap easily available source of growth medium like molasses.

-Molasses cost $0.07/kg

-The density of molasses is roughly 1.4 kg/L

-Therefore, 1 L of molasses will cost $0.07 x 1.4 = $0.098

1-Dodecanol manufactured by conventional methods usually is contaminated by long carbon chain compounds, hence are associated with expensive purification costs. The low downstream processing costs of our final product 1-dodecanol obtained from bacterial degradation of SDS may help us cut down our costs, hence this could lead to development of a economically viable product.

A rough estimate of price of operation can be given by based on the substrate consumption and enzyme production rate

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*References:*

1.Towards the Identification of Type II Secretion Signals in a Nonacylated Variant of Pullulanase from Klebsiella oxytoca (2005), Olivera Francetić and Anthony P. Pugsley.

2.http://www.bg.ic.ac.uk/research/g.stan/2010_Course_MiB_article.pdf, Accessed 04/08/2017.

3.Commercial Laundry Water CharacterisationJ. K. Braga*, M. B. A. Varesche

Department of Hydraulics and Sanitation, Engineering School of São Carlos, São Paulo University,

São Carlos, Brazil

4.Guy-Bart Stan. Modelling in Biology. Lecture notes, 2017.

5.https://math.la.asu.edu/~halsmith/bacteriagrow.pdf