Geometrical approach

Micelle structure

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

Amphiphilic proteins are 10 nm long, 5 nm for both hydrophilic and hydrophobic parts. (Here where we got amphiphilic proteins sizes.) The linker (here link for more info about this. Structure and such, does lab have that somewhere?) 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 then one linker would be 1,2 nm long. (we need some source for the Å-lengths) 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 this paper, 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 \[V(nm^3)=\frac{0.73\tfrac{nm^3}{g}10^{21}\tfrac{nm^3}{cm^3}}{6.023\cdot 10^-3 \tfrac{nm^3}{Da}} M(Da).\] We can calculate the radius by \[r_{min} = \left( \frac{3V}{4\pi} \right)^{1/3}\] if we know the volume of a sphere, so we get that \( R_{min}=0.066\cdot M(Da) \).

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 big 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 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 pyramides 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.

Above isn’t a perfect arrangement of these fusion proteins either, but the problem is too hard in ball 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 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.

Discussion

The goal of this modeling approach 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 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 and the micelles with CAR and ADO somewhere between 64 and 111 we can say that geometrically it is possible for CAR and ADO to be part of a micelle structure.

There are some assumptions of the model that might have some effect on its accuracy. We have assumed that the amphiphilic protein could be approximated by just its length, and it has no width that could have any effect on our calculations. It is also to be noted that we didn’t even aim to be accurate in assembly of ADO and CAR in ball 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. Further, since we don't know what shapes the enzymes are, we have estimated them as spheres.

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 we feel that taking these things into account is not worth the effort especially when mere geometrical model could predict things so well.