The field of synthetic biology relies heavily on mathematical models that help simulate and predict the process and consequence of biological systems. We developed two models, the radial domino model and the blooming model to describe the potential perturbation mechanism and interaction modes between protein particles and liquid crystal molecules. The protein density threshold can be calculated by this model.
Purpose
An important part of our liquid crystal experiment is to determine an optimal antigen concentration, so we creatively proposed two models to calculate the β2-M density microscopically. The trouble is that we do not know the range of antigen molecules that can perturb liquid crystal molecules. Therefore, we consulted the relevant parameters, proposed two models, calculated the range of the single antigen molecule to perturb the liquid crystal, and finally determined the threshold of the antigen concentration, which was consistent with our experimental results.
Parameters
 |
|
 |
|
|
Fig.1 The chemical structure (A) and the simulated structure (B) of
5CB |
|
|
|
|
Fig.2 The simulated structure of Nanoantibody |
Fig.3 The simulated structure of β2-MG |
Hypothesis
|
|
Fig.4 Two possible mechanisms for β2-M to disturb liquid crystal |
The process of protein perturbation of liquid crystal molecules is particularly complicated, so we can only guess the disturbance mechanism on our own. Inspired by dominoes and the blossoming of flowers, we proposed that there might be two mechanisms of disruption. We called them RADIAL DOMINO MODEL and BLOOMING MODEL
|
|
Fig.5 RADIAL DOMINO MODEL |
Simplify the β2-M and 5CB as ball and stick. The distance between each two sticks is equal. Due to the interference of the ball, the first stick next to the ball will be inclined at an angle. The first stick will affect the next stick until the last stick is aligned vertically.
|
|
Fig.6 BLOOMING MODEL |
This model is actually an extension of the previous model. Simplify the 5CB into one side of the polygons. As the molecular shells increases, the angle between the two sides of the polygon will tend to be 90°.
Calculation
Obtain the spacing of 5CB molecules
The forces between two molecules are covalent bonds, ionic bonds, metal bonds, hydrogen bonds and Van der Waals force. Because no other forces are present, the main force is Van der Waals force. The point at which the force becomes repulsive rather than attractive as two molecules near one another is called the van der Waals contact distance. Van der Waals force includes electrostatic force, dispersion force and induction force.
|
|
Fig.7 van der Waals potential energy |
We can use a simplified formula: the Lennard-Jones potential[1].
$$E=\frac{A}{r^{12}}-\frac{B}{r^6}$$
where \(A=4\epsilon\sigma^{12}\), \(B=4\epsilon\sigma^6\).
However, there is no sigma Lennard-Jones parameters (Angstrom) epsilon Lennard-Jones parameters (kcal/mol) about 5CB, so we calculate the van der Waals contact distance(\(r_0\)) by statistical mechanics[2].
In statistical mechanics, the radial distribution function \(g(r)\) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.
|
|
Fig.8 The sketch map of radial distribution function |
The radial distribution function:
$$\rho g(r)4\pi r^2=dN$$
where \(\rho\) is the density of 5CB, \(N\) is the total number of 5CB molecules. Integrate the above formula:
$$\int_0^\infty \rho g(r)4\pi r^2 \mathrm{d}r=\int_0^N\mathrm{d}N=N$$
$$g(r)=\frac{\mathrm{d}N}{\rho 4\pi r^2}$$
Molecular dynamics calculation:
$$g(r)=\frac{1}{\rho 4\pi r^2}\frac{\sum_{i=1}^r\sum_{j=1}^N \Delta N(r\to r+\delta r)}{N\times T}$$
Where \(N\) is the number of molecules and \(T\) is the calculated time (steps), \(\Delta N\) is the number of molecules between \(r\) and \(r+\delta r\).
The radial distribution function can be computed via computer simulation methods. Because the peak of the radial distribution function appears at 0.5nm, so the spacing of 5CB molecules is around 0.5nm.
Calculate the range of β2-M that can perturb 5CB
We assume 5CB molecule as a ellipse (\(a\)=1nm, \(b\)=0.1nm), we got the tilt angles of three molecules by Matlab.
|
|
Fig.9 The tilt angles are 43°, 64°, 90° respectively. |
|
|
\(\therefore\)Maximum radius of β2-M that can perturb 5CB (Fig. 5):
\(R_1=[2\times\sin (64^\circ )+1]\mathrm{nm}=\)2.8nm
Verification
In the liquid crystal experiment, the droplets dropped on the slide have a radius of about 2 mm.
$$\therefore S=\pi R^2=4\pi\times 10^{12}\mathrm{nm^2}$$
The molecular weight of β2-M is 11800, the volume of the droplet is 10μL.
The area disturbed by each protein molecule is:
$$r=2.8\mathrm{nm}$$
$$S_0=\pi r^2=7.84\pi\mathrm{nm^2}$$
When the picture is just full colorful, the number of molecules contained in the droplet is:
$$N=\frac{S}{S_0}=5.1\times 10^{11}$$
$$m=\frac{11800\times 5.1\times 10^{11}}{6.02\times 10^{23}}=10^{-8}\mathrm{g}$$
\(\therefore\)Maximum concentration is:
\(c=\frac{m}{V}=\)1000ng/mL
That is to say, when the concentration reaches 1000ng/mL, the image starts to be completely bright. This is in good agreement with our experimental results. The following are the results of the liquid crystal experiment.
|
|
|
A. 500ng/ml(β2-M) |
B. 600ng/ml(β2-M) |
|
|
|
C. 700ng/ml(β2-M) |
D. 1000ng/ml(β2-M) |
Photographs of different antigen concentrations(photographed through a polarizing microscope) |
|
[1] Jones J E. On the determination of molecular fields. From the equation of state of a gas[J]. Proc. R. Soc. Lond. A, 1924, 106(738): 463-477.
[2] Haynes W M. CRC handbook of chemistry and physics[M]. CRC press, 2014.
