Geometrical approach

Approach I

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 protein, a linker that attaches CAR or ADO to the amphiphilic part.

Amphiphilic proteins are 5 nm long, 2.5 nm for both hydrophilic and hydrophobic parts [1]. The linker consists of eight amino acids, and for each amino acid, the maximum lenght is 0.38nm. 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 0.15nm so one linker would be 1,2 nm long [2]. However, we can estimate that the linkers are 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 2.8nm. 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. Based on [2], 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.

Fix the pictures above (length of amphiphilic part is 5, not 10!)

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 \[ \Omega_{cone\text{-}ADO} = 2\pi \left( 1-\cos\left( \arctan\left(\frac{2}{9.8}\right)\right) \right) \approx 0.127 \text{ sr}.\]

--picture of this cone-like structure? is it needed or can this be understood without it?--

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.

--picture of pyramid structure? is it needed or can this be understood without it?--

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.

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.

Approach II

Micelle structure

In the first approach we didn't take into account anything else from amphiphilic proteins than the lengths. However we can calculate the width of the protein a bit similarly as we calculated the sizes of the enzymes.

We know that the length of the amphiphilic proteins is 5 nm. If we assume their density to be that of other proteins and assume that they are roughly cylinder-shaped, we can calculate the radius for the cylinder.

If the volume of the amphiphilic protein is V, then we get it simply form the equation \[ V = \pi \cdot r^2 \cdot h \] , where r is the radius of the cylinder and h is the length of the cylinder. By solving the equation for r, we get \[ r = \sqrt{\frac{V}{h \cdot \pi}} \]

Now that we approximate the amphiphilic proteins as a cylinder there will be empty space in the middle of the micelle. insert pic of this general shape somewhere here; the middle should be as close to a perfect circle as possible. This shouldn't be a problem since in real world there are many non-polar molecules in the cell that are ready to take that space

Now that we think that there is empty space in the middle of the micelle we must calculate the radius of it. This can be calculated with similar triangles from ADO fusion protein. See the figure X below.

Here be pic of our situation, explaining the lengths of things.

Calculations for ADO and CAR

We can again approximate the micelle structure with cones and pyramids representing the fusion proteins and as well with the overlapping structure. With cones we get \[ \Omega_{cone\text{-}CAR} = 2 \pi \left( 1-\cos \left( \arctan \left( \tfrac{3.5}{21.34} \right)\right)\right) \approx 0.0828 \text{ sr} \] \[ \Omega_{cone\text{-}ADO} = 2 \pi \left( 1-\cos \left( \arctan \left( \tfrac{2}{17.04} \right)\right)\right) \approx 0.0428 \text{ sr}\] and with these values we could fit \( (4\pi)/(0.0828+0.0428)\approx 100\) of both fusion proteins in one micelle, yielding a total size of 200.

With pyramid structure we get solid angles \[\Omega_{pyramid\text{-}CAR} = 4\arcsin\left( \sin\left(\arctan\left( \frac{3.5}{21.34} \right) \right) ^2 \right) \approx 0.1048 \text{ sr}\] \[\Omega_{pyramid\text{-}ADO} = 4\arcsin\left( \sin\left(\arctan\left( \frac{2}{17.04} \right) \right) ^2 \right) \approx 0.0544 \text{ sr}\] and thus we could fit \( (4\pi)/(0.1048+0.0544)\approx 78\) of both CAR and ADO in one micelle, 156 in total.

With overlapping structure explained before, we had to check whether we would need to make any changes into this approach. We calculated the angle \(\gamma\) from figure Y, the same structure than in previous overlapping situation, no amphiphilic protein sizes matter and compared it to \(\gamma\) from figure Z, what the angle should be that the two amphiphilic proteins could fit side-by-side.. The latter is bigger with \(\gamma = 2 \tan(\tfrac{0.85}{7.24}) \approx 0.236\). This yields us \[\Omega_{CAR\&ADO} = 4\arcsin\left( \sin\left( 2 \tan \left(\tfrac{0.85}{7.24} \right) \right) ^2 \right) \approx 0.2186 \text{ sr}\] and thus about \( (4\pi)/0.2186\approx 57\) CAR fusion proteins and about 114 ADO fusion proteins making the whole micelle size 171.

Calculations for GFP

Since ADO and GFP are about the same size we can again use the values calculated in previous section for ADO.With cone-approximation we get \[\frac{4\pi}{ \Omega_{cone\text{-}ADO}} \approx 293\] of these fusion proteins in one micelle, and with pyramid-approximation \[\frac{4\pi}{ \Omega_{pyramid\text{-}ADO}} \approx 230\] fusion proteins. The real value is again probably somewhere between them.

Discussion

The goal of this geometrical model was to understand if it was possible to form micelles which have CAR or ADO at the end of the amphiphilic proteins. The main thing was to prove that the proteins aren’t too big to have an impact on micelle formation, and since we already knew that this arrangement works with green fluorescent protein it was only natural to compare these two. Because based on our calculations the green fluorescent protein micelles have upper bound of building blocks somewhere between 78 and 98 (approach I) and the micelles with CAR and ADO somewhere between 64 and 111 (approach I) we can say that geometrically it is possible for CAR and ADO to be part of a micelle structure according to our first approach.

However, the second approach is not so optimistic. When considering also the amphiphilic protein sizes, we get that sizes of micelles with CAR and ADO are somewhere between 156 and 200 when only with GFP the size would be somewhere between 230 and 293. This doesn't outright prove that our micelles can't be constructed, but as a conclusion we can say that it is not certain geometrically.

It needs to be noted that we didn’t even aim to be accurate in the assembly of ADO and CAR on a spherical surface. The best possible formation is very hard to find in this situation and there wasn’t any need to be that accurate in our calculations. Furthermore, since we don't know what shapes the enzymes or amphiphilic proteins are, we had to estimate.

Even though our model seems to prove that the formation of these micelles is possible, there are lots of things we couldn’t take into account that might have effects on micelle formation and make it impossible. We didn't consider any forces that might form between our proteins, thus rendering micelles impossible. It might well be that even though this is geometrically possible in reality micelles can not form. However, finding out the forces between these enzymes might prove to be extremely difficult, which is why we did not pursue that avenue of thought.