Team:GreatBay SCIE/Optimum Aptamer Concentration
The aim for this modeling is to utilize the least amount of aptamer to reach the highest binding affinity.
As a single liposome molecule can bind with numerous aptamers, we built up a mathematic model to estimate the best ratio between aptamer and liposome. Which we can use the least amount of aptamer to reach the highest binding affinity.
We use pymol(A mathematical modelling software) to measure the length between the epitopes on Her2 and cell surface which we estimate it as 5nm.
(The yellow line shows the distance between the sequence of epitope and the sequence on the cell membrane, unit: Å)
Thus two adjust aptamer on the liposome surface linear distance should be 10nm. In this distance, no matter which part of the liposome collides with HER2, there is at least one aptamer can bind with HER2.
By using the law of cosin, we can obtain the arc distance of two adjust aptamer. The radius of the liposome r is 150nm
$$\frac{r^2+r^2-(2d)^2}{2r^2}=\cos\theta$$ $$\theta=\cos^{-1}(1-\frac{2d^2}{r^2})$$ $$S=\theta\times r$$
As the arc distance compared to the liposome radius is extremely small, hence we can assume that the sphere is built up by numerous hexagons. Each aptamer is on the peak of the hexagon and its center.
(assume that the hexagon separated on the sphere averagely）
Using the area of the sphere surface divided by the surface area of the equilateral triangle, we can obtain how many triangles are presented on its surface. Then times 3 as each triangle contain three vertex, then divided by 6 as each vertex act as center as well. Therefore we can obtain the amount of the optimum aptamer number. Hence the ratio can be estimated.
$$(\frac{4\pi r^2}{\frac{\sqrt{3}}{4}S^2}\times\frac{1}{2}):1=n:1$$
(n is the concentration of the aptamer, S is the arc of two adjacent aptamers. r is the radius of the liposome. d is the length between an aptamer and cell membrane)
After we conjugated the aptamer and the drug we then designed an experiment based on ELISA to ensure our model is valid.
By using the data from liposome group we can know that the radius of liposome r=158. Therefore we can calculate that the optimal ratio between this size of liposome and aptamer is about 1:3600
In order to make sure the aptamer can allow these number of binding, we also generally calculate the maximum number of aptamer can bind with. BY using the data of the surface area of phospholipid is approximately 0.502nm, assume the surface of phospholipid is an oval
$$S=\pi\times a\times b$$
a=8Å b=2Å
The ratio between phospholipid and PEG molecule is 10:1, thus there is approximately 56810 individual PEG molecules on the surface of liposome
$$N_{PEG}=\frac{4\pi r^2}{11S}$$
The width of aptamer is 17.6Å, hence there is 8059 individual aptamer can bind with liposome.
$$N_{aptamer}=\frac{4\pi r^2}{4\pi 1.76^2}$$
Therefore the liposome can approximately tolerance the number of 8059 individual aptamer.
The result is shown on the graph
Thus we conduct the experiment by modify the protocol of ELISA, which we change the independent variable from concentration gradient to time gradient and get the result.
In the result we can see the intensity reach the maximum when incubation time is 60mins but that can be an anomalous data, as the intensity is a bit high in the whole group of 60mins.
The overall result shows our modeling could be right as when ratio is 1:3.6 and 1:4.5, the intensity is almost the same when incubation time is the same. However, the characteristic of non-specific bind doesn't illustrate in the result.
click here to obtain more information on dissociation constant.