Latest revision as of 18:29, 16 September 2015

Micelle model

Introduction

The product of second to last enzyme of our pathway, butyraldehyde, is toxic to the cell. Because of that and about 15 naturally occurring butyraldehyde-consuming enzymes in the cell, it is essential for the propane production that butyraldehyde goes swiftly to the enzyme we want it to go to, ADO. As the solution to this our team wanted to put CAR and ADO close together in a micelle so butyraldehyde would go to ADO with a higher probability than to any other competing enzyme in the cell.

We have made a model of effectiveness of having enzymes close together, but our team also wanted to know if the micelle structure was possible in the first place. We know that it is possible to form the micelle without any proteins at the end and with green fluorecent protein (GFP), but could CAR and ADO be part of this kind of structure?

Geometrical approach

Approach I

**Figure 2:**
2d simplification of micelle

Micelle structure

The micelle is formed by amphiphilic proteins that have both hydrophilic and hydrophobic parts. At the end of the hydrophilic part there is a short (8 aa) oligopeptide linker that attaches CAR or ADO to the amphiphilic part.

**Figure 3:**
Image from structure
prediction software

As the bilayer structures formed by amphiphilic proteins have been reported to be 10 nm thick, we can deduce that the amphiphilic proteins are 5 nm long, 2.5 nm for both hydrophilic and hydrophobic parts. The linker consists of eight amino acids (GSPTGAST), and for each amino acid, the maximum lenght is 0.38 nm. From this we can calculate that at most the length of one linker is 2.8 nm. If the linker would form an α-helical structure, then the length for one amino acid would be about 0.15 nm so one 8 amino acid linker would be 1.2 nm long. However, we can estimate that the linkers are rather straight, since running the structure in peptide structure prediction software doesn't yield strong folding or helical structure. Thus we predict our linker lenght to be 2.8 nm. CAR uses two subsequent linkers whereas ADO uses one.

One problem we are facing here is that we need some sort of approximations for the enzymes’ radii. Since we don’t know the exact three-dimensional structure of the proteins, we approximated the enzymes as perfect spheres. If we assume that our enzymes have a density that is common to enzymes, our task becomes easier. The average density of proteins is 1.37 g/ml. Because we want to calculate the volume, and in effect, their radius, we invert this value, thus getting 0.73 ml/g.

Using some clever calculation, we get that the relationship between mass and volume for proteins is \[V(nm^3)=\frac{0.73\tfrac{cm^3}{g}10^{21}\tfrac{nm^3}{cm^3}}{6.023\cdot 10^{23} \tfrac{Da}{g}} M(Da).\] If we know the volume of a sphere, we can calculate the radius by \[R_{min(nm)} = \left( \frac{3V}{4\pi} \right)^{1/3}\ = \left( \frac{3\cdot \frac{0.73\tfrac{cm^3}{g}10^{21}\tfrac{nm^3}{cm^3}}{6.023\cdot 10^{23} \tfrac{Da}{g}}}{4\pi} \right)^{1/3} \cdot \left( M(Da) \right) ^{1/3} = 0.066 \cdot (M(Da))^{1/3} \].

The mass of CAR is 127 797 DA and the mass of ADO is 27 569.15 Da. Now the radius would be 3,5 nm for CAR and 2 nm for ADO. The mass of GFP is 26 890 Da, which makes its radius roughly 2 nm.

Calculations for ADO and CAR

We can estimate how many amphiphilic proteins we can theoretically fit in one micelle by calculating how large solid angles they take with attached enzymes. The easiest way to estimate the solid angles is to think the amphiphilic proteins linked with enzymes as cones. We can calculate the solid angle \( \Omega \) for these by \[ \Omega = 2\pi \left( 1-\cos(\theta) \right), \] where \( \theta \) is half of the apex angle. So for CAR we get \[ \Omega_{cone\text{-}CAR} = 2\pi \left( 1-\cos\left( \arctan\left(\frac{3.5}{14.1}\right)\right) \right) \approx 0.185 \text{ sr} \] and for ADO \[ \Omega_{cone\text{-}ADO} = 2\pi \left( 1-\cos\left( \arctan\left(\frac{2}{9.8}\right)\right) \right) \approx 0.127 \text{ sr}.\]

This means that there would be at most 40 of both ADO and CAR, totalling 80 fusion proteins in one micelle by this method of calculation.

However, when approximating enzymes with spheres it is not possible for them to fill the whole surface of the micelle; there will always be gaps. This is why it might be better to approximate the solid angle these complexes of amphiphilic proteins and enzymes by using pyramids instead of cones.

The solid angle \( \Omega\) for this kind of structure can be calculated by \[\Omega = 4\arcsin\left( \sin\left(\theta\right) ^2 \right),\] where \( \theta\) is again half of the apex angle. This yields us \[\Omega_{pyramid\text{-}CAR} = 4\arcsin\left( \sin\left(\arctan\left( \frac{3.5}{14.1} \right) \right) ^2 \right) \approx 0.232 \text{ sr}\] and \[\Omega_{pyramid\text{-}ADO} = 4\arcsin\left( \sin\left(\arctan\left( \frac{2}{9.8} \right) \right) ^2 \right) \approx 0.16 \text{ sr}.\]

By this method of calculation we could get at most 32 of both fusion proteins in one micelle, yielding a total size of 64.

Above isn’t a perfect arrangement of these fusion proteins either, but the problem is too difficult on a spherical surface with two different sizes of enzymes. The real maximum value if we think the problem this way is somewhere between the ones obtained, so somewhere between 64 and 80 fusion proteins in a micelle.

The previous calculations have not taken into account that CAR and ADO might overlap because CAR has two linkers when ADO has just one. We don’t know the ideal structure of the overlapping, but we can estimate it by the structure shown below.

**Figure 6:**
Estimation of the overlapping micelle structure.

We can think that this structure consists of a single cone whose centre is the centre of CAR fusion protein and the side goes along ADO fusion protein. We can further take some of the empty areas into account by thinking this as pyramid instead of cone. This yields us \[\Omega_{CAR\&ADO} = 4\arcsin\left( \sin\left( \arccos\left( \frac{14.1^2+9.8^2-5.5^2}{2\cdot14.1\cdot 9.8} \right) \right) ^2 \right) \approx 0.3336 \text{ sr}.\]

This means that about 37 of these pyramid stuctures fit in one micelle, meaning 37 CAR enzymes per micelle. For ADO we can approximate that there are about twice as many of them than CAR fusion proteins (this is justified in infinite field so we approximate with it here), so the amount of ADO would be 74 and the whole amount of fusion proteins in this micelle 111. Since there is probably an even more efficient way of packing these proteins in one micelle, the real upper bound might be even larger.

Calculations for GFP

For comparison we calculated how many green fluorescent proteins could fit into a micelle. Since the GFP is same size as ADO, we can use values from previous calculations. With cone-approximation we get \[\frac{4\pi}{ \Omega_{cone\text{-}ADO}} \approx 98\] of these fusion proteins in one micelle, and with pyramid-approximation \[\frac{4\pi}{ \Omega_{pyramid\text{-}ADO}} \approx 78\] fusion proteins. The real value is thus probably somewhere between them.