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Used Parameters
Before we could generate results with the model, we had to define some parameter, like concentrations and Kd-values. The Kd-values we found in the paper of den Hamer[1] and appeared to be 66 μM for the Inducer to the scaffold, 1 μM for the Binding Partner with the Center Point and 0.25 μM for the Binding Partner to the Scaffold. The Kd-value could then be used to calculate the association or dissociation rate, as long as you know the other. For the association rate, we used 105 M-1s-1 as found in the paper of Schlosshauer [2] and is the average diffusion limited association rate they calculated for different proteins. In reality, our constructs probably are more diffusion limited than they calculated, as our construct are very large proteins and therefore will experience much more resistance.
Results Description
In Figure 1 you can see that we use an excess of the Inducers and that the amount of inducers decreases over time but this decrease stagnates after some time. This is because all the pockets of the scaffold construct are already filled with the inducer. Data supporting the inducer excess is also shown in the figure, as the amount of empty scaffold sites decreases at the same rate as the inducer. Furthermore, we see that the induced scaffold sites are very high in the beginning. Our initial state is defines as that the induced amount of scaffold sites starts at 0, this is supported if you look closely to the data, but it increases so fast, that it is not visible with a logarithmic time scale. Another thing that stands out in the graph, is that the green graph (the one of the Binding Sites for the CenterPoint) is not visible in the figure, this is because these values are about the same as the ones of the CenterPoint Sites. They decrease at the same rate, because they are both used for the formation of a bond to create a complex and have about the same initial amount of molecules. Another thing that supports the graph is that the sum of the Induced Scaffold Sites and the Empty Scaffold sites is comparable with the dark blue graph (Binding Sites for Scaffold). The decrease of the dark blue graph means that there are bonds formed between the Scaffold and the Binding Partner.
In Figure 2 we can see how many complexes are formed. With the programs provided by NFsim, we could extract the total amount of molecules and the average complex size. They also provided us with a method to determine the amount of single molecules (they call it complexes with size one) and we used that to determine the amount of formed complexes and the average complex size, where the single molecules are not taken into account. Just as we mentioned as comment for the first figure, the initial values of Complexes and Average Complex size >1 start at zero, but is not visible in the figure due to the logarithmic scale.
Simulation for Potsdam
The team from Potsdam has a project with some similarities with our own project. This leaded to collaboration on different fields of the iGEM competition, including modelling. As there was not much known about their system, a model would help with getting more insights and a better understanding. The main problem was that they did not know much about the aggregation principle, so we had to make some assumptions and test different approximations, and see how well they fit with the wet-lab results.
Approximations
The first approximation was the concentration of their protein in the yeast cells. They found a rough estimation of a range, which was 0.1 - 1.4 μM. We therefore set the initial concentration to be 0.5 μM, which lies somewhere in between. For the association rate, we took the same as we used for our own simulation, namely 105 M-1s-1 as found in the paper of Schlosshauer [2].
The known information of the system includes the molecular weight of the protein, being 60.4 kDa and the droplet size, with an average diameter of 1.19 μm. The weight of a single molecule can be used to approximate the volume of a single molecule with Protein Volume = 1.21 Molecular Weight [3]. This leads us to a volume of 73.084 Å3 per molecule. As 10 Å=1 nm this means that 73.084 A3 = 73,1 · 10-3 nm3 per molecule.
Like the formula for the volume of the protein, there is a formula for the volume of spheres, which is more generally known, and defined as Volume Sphere = 4/3 · π · r3 . Using this formula, we got a volume of 0.88 μm3 = 0.88 · 109 nm3. When dividing the volumes, you would think that this means that there are about 1,2 · 1010 proteins in one droplet. However, this is not quite the case, as you have to take into account that sphere packing has a lower density and that there are different ways of packing, and we do not know which is applicable yet. If we assume there is a surrounding of 4 molecules, as in a tetrahedral lattice, there are only 4,1 · 109 molecules that should fit in the droplets.
Analysis of Results
Because simulating that many molecules takes a lot of computer memory, which we did not have, we rescaled the system. A yeast cell has a volume of 42 μm3 [4] and we used a simulation box with the size of 10 μm 3. If we rescale the measured droplet size, which is 2% of the yeast volume, we expect to have cluster that also have a size of 2% of the initial amount of molecules. This would indicate that the simulated complexed would have an average size of somewhere around 60 molecules, as the initial amount of molecules was 3010. In Table 1 you can see what the results of the simulation for different valencies are, and that the average complex size of 60 lies between a valency of 3 and of 4.
Table 1: Influence of valency on average complex size
|
Average Complex Size |
Total Amount of Complexes |
| Valency 2
| 2.7 |
1117 |
| Valency 3 |
18.0 |
167 |
| Valency 4 |
250.8 |
12 |
| Valency 5 |
1505 |
2 |
The team of Potsdam used disordered regions that can interact with itself, the valency is not specified and indeed can differ for each protein.
[1] A. den Hamer, L.J.M. Lemmens, M.A.D. Nijenhuis, C. Ottman, M.Merkx, T.F.A. de Greef and Luc Brunsveld, Small-Molecule-Induced and Cooperative Enzyme Assembly on a 14-3-3 Scaffold. ChemBioChem, vol 18, pp 331-335, 2017
[2] M. Schlosshauer, D. Baker, Realistic protein-protein association rates from a simple diffusional model neglecting long-range interactions, free energy barriers, and landscape ruggedness. Protein Science vol 13, pp 1660-1669, 2004
[3] Y. Harpaz, M. Gerstein and C. Chothia, Volume Changes on Protein Folding. Structure, vol 2, pp 641-649, 1994
[4] R. Milo and R. Phillips, Cell Biology by the Numbers. ISBN: 9780815345374, 2015
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