Receptor and Surface protein model

The aim of this model is to determine the concentration of Schistosoma elastase or TEV protease that should be added to bacteria to trigger the response. It is also attempted to model how long does it take for protease or elastase to cleave enough peptides.

Cleavage of protein is an enzymatic reaction, which can be written as:

• S + E <--> E-S --> P + E
• Substrate (S) = Protein
• Enzyme (E) = TEV (Protease)
• Product (P) = Peptide

This can be modelled in a very similar way to the 1-step amplification model, however, all the constants and initial concentrations will be different.

[TEV](t=0) - initial concentration of TEV will be arbitrary for us to choose. However, ultimately we would need to measure the concentration of elastase the schistosoma releases.

Threshold concentration of peptide (20/08/2010)

The optimal peptide concentration required to activate ComD is 10 ng/ml (see this paper). This is the chosen threshold value for ComD activation. However, the minimum concntration of peptide to give 1st detectable activation is 0.5ng/ml. We want to know how long it takes until the threshold will be reached.

• The mass of a peptide is 2.24kDa = 3.7184*10^-21g.
• The number of molecules in a ml is 10ng/3.7184*10^-21g = 2.6893*10^12. In a litre, the number of molecules is 2.6893*10^15.
• Dividing this value by Avogadro's constant gives the threshold concentration of c_th=4.4658*10^-9 mol/L.
• The threshold for minimal activation of receptor is 2.2329*10^-10 mol/L.

Protein production in B.sub (23/08/2010)

• The paper mentions that each cell displays 2.4*10^5 peptides.
• 2.4*10^5 molecules = 2.4*10^5/6.02*10^23 mol = 0.398671*10^-18 mol
• Volume of B.sub: 2.79*10^-15 dm^3
• Concentration = [mol/L]
• c = 1.4289*10^-4 mol*dm^(-3)
• This is the concentration of protein that will be produced in a single cell of B.sub.

Hence, we can deduce the concentration that the protein expression will tend to (c = 1.4289*10^-4 mol/dm^3 = c_final). Therefore, we can model the protein production by transcription and translation and adjust the production constant so the concentration value will converge to c_final.

Using a similar model to the simple production of Dioxygenase for the Output Amplification Model (Model preA), we obtain the following graph:

Production of protein by transcription and translation

The degradation rate was kept constant, and the production rate was changed according to the final concentration.

Control volume 1 - initial choice (23/08/2010)

All enzymatic reactions that we have modelled so far were happening in the fixed volume (whithin bacterial cell). This case is different as the molecules are not confined by the bacterial membrane and can diffuse freely out of the cell.

The control volume: In order to avoid trying to account for transient diffusion, we decided to determine combination of imaginary and real boundaries for the ssytem. The inner boundary is determined the bacterial cell (proteins after being displayed and cleaved cannot diffuse back into bacterium). The outer boundary is more time scale dependent. We have decided that after mass cleaving of display-proteins by TEV many of them will bind to receptors quite quickly (eg.8 seconds). Our volume is determined by the distance that AIPs could travel outwards by diffusion within that short time. In that way, we are sure that the concentration of AIPs outside our control volume within given time is approximately 0.

This approach is not very accurate and can lead us to false negative conclusions (as in reality there will be concentration gradient with highest conentration at the call wall, not evenly spread concentration).

Control volume (volume of B.sub. to be excluded. x indicates the distance travelled by AIPs from the surface in time t)

Difussion distance was calculated using Fick's 1st Law: x=sqrt(2*D*t), where: x - diffusion distance, D - diffusion constant, t - time of diffusion

D_average = 10.7*10^(-11) m^2/s - for a protein in agarose gel for pH=5.6 according to paper

t = 8s (for now chosen arbitrarily by us - we hope this is long enough time for AIP to bind to ComD)

This gives: x = 4.14*10^(-5) m

The control volume can be calculated by adding 2x to the length and the diamter of the cell. This gives a control volume (CV) = 4.81*10^(-7) m^3

Production of protein by transcription and translation in control volume

Protein production in Control Volume (23/08/2010)

The previously determined constants of protein production in B.sub to obtain the concentration of proteins per cell indicated by paper is not valid in Control Volume. It has to be adjusted (multiplied) by the following factor:

factor= V_bacillus/V_control_volume = 5.7974*10^(-6) (for particular numbers presented above)

It is enough to adjust the production rate by the 'factor'. The resulting concentration will follow.

Control volume final choice (23/08/2010)

We have realised that initial choice of control volume was not reflecting the reality well. It was treating single bacterium as if it was by itself in the medium. However, in reality bacteria live in colonies very close to each other. They are closer to each other that the diffusion distance (1.9596*10^(-5) m) derived above even if placed in water solution.

Using CFU to estimate the spacing between cells (24/08/2010)

CFU stands for Colony-forming unit. It is a measure of bacterial numbers. For liquids, CFU is measured per ml. Since we already have data of CFU/ml from the Imperial iGEM 2008, this is an easy way to estimate the number of cells in a given volume using spectrometer at 600nm wavelength. The graph below is taken from the Imperial iGEM 2008 Wiki page.

CFU/ml vs. Optical Density at 600nm (OD600)

The graph shows values of CFU/ml for different optical densities. The range of CFU/ml is therefore between 0.5*10^8 - 5*10^8.

In this calculation, we will assume that only one cell will grow and become one colony (i.e. no more than one cell will form no more than one colony). Therefore, the maximum number of cells in 1ml of solution is 5*10^8. Taking the volume of 1 ml = 10^-3 dm^3 and dividing by the (maximum) number of cells in 1ml gives the average control volume (CV) around each cell: 2*10^(-12) dm^3/cell. As this volume is 3 magnitudes bigger than the cell itself, its shape is arbitrary. For simplicity we choose it to be cubic. Taking the 3rd root of this value gives the length of one side of the control volume. Side length of CV = y = 1.26*10^(-4) dm = 1.26*10^(-5) m.

Choice of Control Volume allows simplifications (24/08/2010)

• Firstly, let us assume that on average the cells will be placed in the centre of the cube. This gives that protein after cleavage will have distance y/2 (assume the bacterium is dimensionless for a moment) to travel to get out of control volume. This is calculated to happen within 0.18s (diffusion coefficient for protein in water). Even if the bacterium was not in the centre, the protein will travel from one end of the cube to the other in less than a second (~0.74s). Hence, it will take between 0.18 and 0.74s for concentration of AIPs around the cell to be uniform. Noting that those time values are really small, we can approximate our model to be having uniform concentration across the volume. In that way we are understimating the value of AIP concentration right next to the cell's surface. Hence, we are overestimating the time required for the AIP concentration to reach the threshold level. If our model will happen not to be able to reach the threshold we will consider this approximation as one of the reasons.
• We can neglect the diffusive fluxes across the CV border (see figure below). Assuming that adjacent cells are producing the peptide at the same rate and the concentration of TEV is the same around them, then the fluxes should be of the same value giving net 0. Hence, we can neglect diffusion and have our model limited to 1 bacterium!
Figure showing 2 cells with their control volumes. It shows the reciprocal fluxes in the opposite directions. The situation is true for all 3 dimensions. In our model we concentrate on 1 cell that is surrounded by others

Conditions as a result of assumptions

Most our assumptions in choice of control volume were possible due to careful choice of cell density = 5*10^8 CFU/ml. If density is changed too much form the chosen value, it happens that simplifications stop holding. However, that does not mean that our system cannot function for lower cell densities. It just means that our model would not be very good for predicting situations with cell densities very different from 5*10^8 CFU/ml.

It was decided that the model should not be used for cell densities lower 10^7 CFU/ml. Below that value AIP takes more than a 1 second to diffuse accross the half of control volume (assuming that cell is inside control volume). We agreed that below 10^7 CFU/ml the approximation about uniform concentration throughout the control volume could be wrong and that concentration gradients could become significant. If our model, would be applied to that situation it would overestimate the time of receptor activation.

It is not possible to increase cell density by more than 10^9 CFU/ml, because of the cell size. They would not fit inside.

Matlab Simulation (24/08/2010)

Graphs showing the simulation using [TEV]0 = 4*10^-4 mol/dm^3. The graph on the right hand-side below shows that the AIP threshold (red line) is reached after 22 s.

Sensitivity of our model (24/08/2010)

• Changing initial concentration of TEV

Whether the threshold concentration of AIP is reached is highly dependent on the initial concentration of TEV. The smallest initial concentration of TEV, [TEV]0, for which the threshold is reached is 6.0*10^-6 mol/dm^3. On the graoh below, it can be seen that the optimal [TEV]0 is a concentration higher than 10^-4 mol/dm^3, which corresponds to a threshold being reached within 1.5 minutes.

Graph showing when threshold AIP concentration is reached (for different initial TEV concentrations). Notice log-log scale.
• Changing the production rate

1 order of magnitude change in production rate results in at least 50s (50 seconds is the smallest step for 1 order change - for others is way beigger) delay of the AIP concentration riching the threshold concentration.

• Changing production rate

It influences a lot the time duration of the AIP concentration being above the threshold level. The higher it is the shorter the receptor is activated (at extreme values, AIP concentration never get to the threshold). However, it has not much influence on how fast the threshold is being reached.

• Changing control volume

Our model is extremely sensitive to that factor. 1 order of magnitude change in CV results in several orders of magnitude change in AIP concentration. Hence, special care should be taken in determination of that value. If model is to be compared with the experimental results, the CFU/ml has to be the same as used in model. Otherwise, the CV has to be readjusted.

Risk of False positives (31/08/2010)

It was pointed out that we should assess a risk of false positives. We are concerned in particular about the display protein not binding into the cell wall, but freeing themselves into extra-cellular environment.

In order to be able to assess the risk of false positives:

• We need to do further research into the affinity of AIP with attached linker and transmembrane protein for the receptor as compared to the affinity of the AIP itself for the receptor.

Have a look at this paper and look for affinity comparison.

• We need to get to know how proteins are being transported from intracellular to transmembrane space. Understanding of that concept could potentially give us an idea of what could go wrong.

Constants for Modelling Protein Display

Type of constant Derivation of value
TEV Enzyme dynamics Enzymatic Reaction:

E + S <-> ES -> E + P

Derivation of these values is made in Variables for Amplification Module Section

• k1 = rate constant for E + S -> ES = 10^8 M^-1 s^-1
• k2 = rate constant for E + S <- ES = 10^3 s^-1
• kcat = rate constant for ES -> E + P = 0.16 +/- 0.01 s^-1

We are assuming the same cleaving rates of TEV as on other substrates. However, we are planning to measure them to gain more confidence in the model.

(common for all)

Assumption: To be approximated by cell division (dilution of media) as none of the proteins are involved in any active degradation pathways

Derived in Variables for Amplification Module Section: k_deg= 0.000289 s^(-1)

For all proteins that are outside of cells or the timescale that is short enough to neglect cell division effect: k_deg=0

Control volume Control volume seems to be the weakest point of this model. We have tried to rationalise it as much as we could. However, error seems to be unavoidable. It is important to realise that the Control Volume needs to be adjusted if different than 5*10^8 CFU/ml concentration of bacteria is used.
Production rate of surface protein It was found that each cell displays 2.4*10^5 peptides (Reference for protein production (Page 4, Paragraph 1)).

Hence, we adjusted our simple production of display protein model to converge to that value. As production rate was the constant that we didn't have a clue about, that value was manipulated.

The resulting 4.13e-8 mol*dm^-3*s^-1 seemed to be of the probable order of magnitude, so we kept it. Ideally, we would like to get this value measured as it is resulting from really vague estimate.

Diffusion coefficient of protein

We have found to references which quote very similar values for very different media.

For protein in agarose gel: D_average = 1.07*10^(-10) m^2/s - for a protein in agarose gel for pH=5.6 according to paper

In the final model the following was used:

For protein in water: D=10^(-10) m^2/s reference

Experiments requested

1. Kinetic ocnstants (kcat, Km) of TEV acting on display peptide.
2. Production rate of peptide.
3. Total average number of proteins on the cell wall
4. Number of free floating proteins without cleaving by TEV(or the ratio with the above)