Team:Judd UK/Pages/Model

Modelling the series of reactions relating Iron (II) ion concentration in saliva to amilCP concentrations.

Abstract

This year, our team created a mathematical model to represent the series of reactions from Iron(II) ion concentrations in saliva to amilCP concentrations for our team’s system. We first had to decide whether we should model our final cell-free system (in vitro), or our in vivo lab work. An in vivo model would be advantageous, as it would allow us to compare the lab work we did to to a model. However, we decided an in vitro model would be more appropriate. This is because the in vivo lab work was just to test whether our construct worked, not to find a relationship between iron (II) ions and amilCP. We therefore felt a model to represent the relationship in an in vitro environment would better represent the final product our project aimed to create.

It should be noted that in a cell-free system, we would have to provide the FUR (Ferric Uptake Regulator) protein- in an in vivo system, the bacteria provides the FUR. Although it means we can easily increase the FUR concentration, FUR is very expensive (£130/5μg) [11].

With this in mind, these are the aims of our model:

(1) Prove that our construct would or wouldn’t work an in in vitro environment in a cell-free system.

If the system does work, we then aim to:

(1a) Find an appropriate FUR concentration for an effective efficient test, but while minimising its cost.

(1b) Find an appropriate time to take a measurement.

(1c) Show a relationship between initial Iron (II) ion concentration and amilCP concentration on a graph.

Model formation

We used mass action kinetics to model our reactions as a function of the velocity of the reaction. We first needed to establish what our reactions were. They can be seen in figure 1.

Dimerisation of the FUR protein.

(2) Transcription of LacI that is inhibited by the activated FUR.

(3) Translation of LacI protein.

(4) Transcription of amiCP that is inhibited by the LacI protein.

(5) Translation of amilCP protein.

If we let initial FUR concentration be a constant, then more iron leads to more activated FUR, less LacI, so more amilCP. In other words, because we have two inhibitors, a lower iron concentration in saliva leads to a less intense blue, while a high iron concentration leads to a more intense blue.

We can break down our model into 9 different reactions to consider separately, then put them together once they are all defined.

For the transcription of LacI and amilCP, the transcription factors (FURactivated and LacI respectively) are not constant. We therefore cannot describe their rate’s of formation as a constant- we must consider the concentration of the transcription factors (TF). To do this, we use the hill equation, Where θ is the fraction of transcription factors that are bound to the promoter, [TF] is the concentration of the transcription factor, and Kd is the dissociation constant of the transcription factor to the promoter.

h is the hill coefficient (we will define it has “h” from now onwards). This can be approximated to the number of TF that binds to one promotor.

Chandran et al. discussed in their review “Mathematical modelling and synthetic biology” [1] that the rate of transcription, also known as the velocity (v) of a reaction, will be proportional to this fraction(1) if the TF is an activator, v = kTX(θ), (2) if the TF is a repressor, v = kTX(1-θ), where KTX is the maximum transcription rate of the mRNA.

Because both of our transcription reactions are regulated by a repressor TF, both our velocities will be equal to kTX(1-θ).

The hill equation is actually used to describe the fraction of a macromolecule saturated by ligand as a function of the ligand concentration. In this way, we can also describe our FURactivated reaction by using the hill equation, where [TF] becomes the iron (II) ions and KTX becomes initial FUR concentrations (FURt)

We can now consider each reaction. We will use v as an arbitrary variable, simply there to denote a rate of reaction. The r0x notation will be used later in the programming to remove clutter from the code

(0) Dimerisation of the FUR protein

For the FURactivated reaction, we make a quasi-steady state assumption. This is used when a part of a system reacts much quicker than the other parts, and thus reaches completion on a much smaller time scale. While FURactivated reaction will reach equilibrium in a the scale of seconds and minutes, translation and transcription takes hours, so instead of considering the velocity of the reaction, we just assume it happens instantly.

(1) Transcription of LacI mRNA (r01)

(2) Degradation of LacI mRNA (r02)

Note that this rate will be negative. This is because it will be decreasing the concentration of the the mRNA. We can now put them together, to form an ordinary differential equation (ODE) for the rate of mRNA production.

(3) Translation of LacI protein (r03)

The translation reactions are fairly simple, as they are just affected by the concentration of the mRNA of the corresponding protein and can be modelled as follows:

(4) Degradation of LacI protein (r04)

These two together form the ODE for the rate of LacI production:

(5) Transcription of amilCP mRNA (r05)

(6) Degradation of amilCP mRNA (r06)

These come together to form an ODE for the rate of amilCP mRNA production:

(7) Translation of amilCP protein (r07)

(8) Degradation of amilCP protein (r08)

These two form the ODE for amilCP concentration:

Altogether, we have 5 reaction equations:

Defining our constants

Now that we have our equations, we must now define our constants. This step is very important, so a lot of effort was put into ensuring we had accurate values based on the literature we could find online. These are the values we found, along with their source (note that if multiple values or a range of values were found, we indicated the value we used in a bracket {x} after the values): The rate of degradation (half-life of 60 min or faster) and the total amount of protein undergoing degradation (2 to 7%) was the same during growth and during various kinds of starvation.

For the transcription and translation rates, we used the table from Eyal Karzbrun et al [7], as it describes the rate in a cell-free system using RNA Polymerase. Transcription is described in terms of rNTP/s (1.0±0.5 -we used 1.5, as it was closer to other literature values [10]). We could then work out kTX by dividing the number of rNTPs in each mRNA by 1.5s ((1128/1.5) = 752s for LacI, (669/1.5) = 446s for amilCP) to get the time take to transcribe one mRNA, then finding it’s reciprocal ((1/752)~1.33e-3/s for LacI, (1/446) = 2.24e-3/s) to find the transcription rate.

We then did the same process for translation. Eyal Karzbrun et al [7] also describes the rate of protein synthesis in terms of a rate of amino acid production. It describes the rate as >4, but from other literature [10], their maximum is 21, so we calculated a range of values and estimated that the rate would be somewhere in the middle. LacI is 360 amino acids long [13], so 0.014≤ kTL ≤ 0.058. We used 0.04 as our value. AmilCP is 223 amino acids long [14], so 0.022≤ kTL ≤ 0.094. We used 0.06 as our value.

Solving the equations

There are fundamentally two ways to solve a series of ODEs: analytically and by using software. We first attempted to solve analytically. This involves integrating the equations so that you get a variable as a function of time. We actually have 4 variables: FURt, Fe2+. time and amilCP. The first 3 are all independent and amilCP is dependent. Trying to solve this with 4 variables is near impossible, so we let FURt and Fe2+ be constant and let time and amilCP be our two variables. In this way, after integrating fully, we would get amilCP concentration as a function of time. An attempt was made to do this. However, it proved to be unsolvable. The final equation was a second order differential of amilCP as a function of time and amilCP mRNA and even Wolfram Alpha couldn’t solve it. This being the case, we then opted to try using software, namely MatLab.

Note that the k_tx and k_tl values are 0 up until a specified time. This is because in Eyal Karzbrun et al [7], they observed a delay in mRNA and protein synthesis.

“Under TX/ degredation, mRNA appeared after a delay of τ0 = 15 ± 5 … We attribute the delay to the time required for TX of a single mRNA” [Fig. 2(c)]

“Assuming TX and TL are concurrent [20], we attribute the additional delay τf ≤ 5 min to the protein folding time, which is similar to previous measurements [21,22]” [Fig. 2(d)]

(a) Information flow from DNA (D) to mRNA (m) to protein (p) is carried by synthesis enzymes RNA polymerase Rp and ribosome R. Degradation by RNAse Xm and protease Xp.

(b) Exponential degradation of 200 nM mRNA with lifetime m ¼ 12 min .

(c) mRNA dynamics in the presence of 30 nM DNA exhibiting a delay and exponential accumulation with a time scale m.

(d) Kinetics of protein synthesis at various DNA concentrations.

(e) Normalized protein synthesis rate p_ syn=p_ max fitted to an exponential rise with a time scale m.

(f) Protein maximal accumulation rate as a function of DNA fitted with a MM curve using Rp ¼ 29 nM and KTX ¼ 1:5 nM.

These delays are not accounted for in our equations, so instead, we let k_tx and k_tl equal 0 up until the delay has ended- we estimated these to be 15 minutes for k_tx and 17 minutes for k_tl.

Because it is a cell-free system, transcription is 10 to 100 times slower than in vivo [7], which could be an issue, as the protein may reach steady state at a much lower concentration than we need. However, it should also be noted that because it is cell-free, we do not have to account for dilution of the proteins. Since degradation of proteins is so slow [23, 24], we can essentially ignore the degradation rates of the proteins. In our code, we set d_LacI and d_CP to 0, but we could just as well have removed r04 and r08 from the ODEs.

Unfortunately, when we ran the code, we noticed that the model had very little sensitivity to changes in FURt and iron (II) ion concentrations, as can be seen below: