# Team:NTNU Trondheim/Collaborations

Collaborations

### I. Brixells modeling (University of Warwick)

Team Warwick has shown interest in our offer to help in modeling and quantitative analysis that we have posted on the iGEM Matchmaker. Team Warwick is aiming to provide precision control over spatial arrangement of cells by designing a tool that enables drawing and building with them.

Zinc finger proteins are intracellular molecules which recognize and bind unique dsDNA sequences. We have engineered these proteins to be expressed on the surface of an E. coli cell, such that dsDNA can be used as mortar to cement cells together. Team Warwick plans to demonstrate this principle by assembling fluorescent cells onto a 2D surface and producing microscopic images, with the ultimate goal being to build complex 3D structures comprised of different cell types. This level of control over cellular localisation is useful in multiple fields including research into cell-cell interactions in microbial communities, multicellularity, and the construction of 3D cell structures in tissue engineering.

The collaboration focused on two elements:

1. Calculating the probability of formation, including the probability of E. Coli bonding correctly.

2. Calculating the effect of arm length on structure changes increases. This has been motivated by the observation that the longer the arms the higher the likelihood that the e.coli cells can attach because they are less closely packed and such wouldn't get in each others way.

There are many methods to look at this kind of problems, but the most realistic may require huge computational resources. So it is necessary to find an approach that is experimentally relevant, computationally tractable, and reflecting the biophysics underlying Brixells.

We, Team NTNU Trondheim have offered ideas, modeling frameworks, software simulation, and data analysis to answer these questions, based on information theory and thermodynamics. For Problem 1, we have suggested an approach in terms of binding affinity and specificity where the information entropy is used as a measure of specificity. For Problem 2, we have suggested a thermodynamics approach using the Poisson-Boltzmann theory where the zinc fingers and E. Coli are approximated by a set of beads (one large bead for the E. Coli, small beads for each zinc finger, and a rod of beads for the DNA arm. Team Warwick has been enthusiastic about this approach since it is a stochastic method that they have not considered, and they have provided that us with data related to the geometry of E. Coli-arm formation.

#### I.1) Probability of bond formation

The probability of bond formation $P$ is equal for all different types of fingers $i$ can be expressed as the product of the probability of formation for each zinc finger ${P}_{i}$:

$P=\prod _{i}{P}_{i}$

This is a clear measure of how strong the structure is.

The probability of formation for a zinc finger $i$ is expressed as a function of the specificity.

${P}_{i}=\frac{{S}_{i}}{W\left({S}_{i}\right)}$

where $W$ is the product log function and ${S}_{i}$ is the specificity which is the information theoretical Shannon "mutual information", and is expressed as follows:

${S}_{i}=\sum _{j}\frac{{K}_{i,j}}{\sum _{j}{K}_{i,j}}\mathrm{ln}\left(\frac{{K}_{i,j}}{⟨{K}_{i,j}⟩}\right)$

That calculates the specificity of one zinc finger, i.e. how well a zinc finger binds specifically to an arm when it is competing with other zinc fingers. ${K}_{i,j}$ is the binding affinity of zinc finger $i$ specifically to arm $j$.

The binding affinity is related to the binding probability ${P}_{i,j}^{\mathit{bound}}$ according to the following equation:

${P}_{i,j}^{\mathit{bound}}=\frac{\left[{F}_{i}\right]}{\left[{F}_{i}\right]+{K}_{i,j}}$

where ${A}_{i}$ is the the arm $i$ and ${F}_{j}$ is the the finger $j$, which are governed by the following ligand-binding reaction:

${A}_{i}+{F}_{j}⇔{A}_{i}{F}_{j}$

#### I.2) Effect of arm length

In order to account for the arm length. It is important to express the binding affinity to the geometry of the system consisting of one arm and one zinc finger. We dropp the indices $i$ and $j$ in the following for simplicity.

${K}_{i,j}={e}^{\frac{\mathrm{-\Delta }{G}_{}}{\mathit{RT}}}$

We approximate the bacteria, the zinc fingers, and the DNA arm by a set of beads. The bacteria becomes one large sphere, the zinc finger one small sphere, and the DNA arm becomes a rod consisting of a cascade of beads.

Using these assumptions, the free energy can be calculated from geometrical and electrochemical considerations. In other words, knowing the positions, radii, and charge magnitudes on the beads, the binding free energy of the system is directly calculated as follows.

$\mathrm{\Delta }G={G}^{\mathit{bound}}-{G}^{\mathit{unbound}}$

where ${G}^{\mathit{bound}}$ is the bound free energy expressed as:

${G}^{\mathit{bound}}=\sum _{\begin{array}{c}m,n=1\\ n\ne m\end{array}}^{M+N}\left({S}_{m,n}+{V}_{m,n}+{E}_{m,n}\right)$

and ${G}^{\mathit{unbound}}$ is the unbound free energy expressed as:

${G}^{\mathit{unbound}}=\sum _{\begin{array}{c}m,n=1\\ n\ne m\end{array}}^{N}\left({S}_{m,n}+{V}_{m,n}+{E}_{m,n}\right)+\sum _{\begin{array}{c}m,n=N+1\\ n\ne m\end{array}}^{M+N}\left({S}_{m,n}+{V}_{m,n}+{E}_{m,n}\right)$

where $m$ and $n$ are bead indices, $M$ and $N$ are the number of beads in each subsystem.

$Rm,n$ is the distance between two beads. ${\mathrm{\rho }}_{m}$ is the radius of a bead $m$, and ${\mathrm{q}}_{m}$ is the charge magnitude of a bead $m$.

where ${E}_{m,n}$ is the electrostatic energy potential expressed as:

${E}_{m,n}=\frac{{q}_{m}{q}_{n}}{8\mathrm{\pi }\mathrm{ϵ}{R}_{m,n}^{2}}$

${V}_{m,n}$ is the van Der Waals force potential expressed as:

${V}_{m,n}=\frac{-A}{6}\left(\frac{2{\mathrm{\rho }}_{m}{\mathrm{\rho }}_{n}}{{R}_{m,n}^{2}-{\left({\mathrm{\rho }}_{m}+{\mathrm{\rho }}_{n}\right)}^{2}}+\frac{2{\mathrm{\rho }}_{m}{\mathrm{\rho }}_{n}}{{R}_{m,n}^{2}-{\left({\mathrm{\rho }}_{m}-{\mathrm{\rho }}_{n}\right)}^{2}}+\mathrm{ln}\left(\frac{{R}_{m,n}^{2}-{\left({\mathrm{\rho }}_{m}+{\mathrm{\rho }}_{n}\right)}^{2}}{{R}_{m,n}^{2}-{\left({\mathrm{\rho }}_{m}-{\mathrm{\rho }}_{n}\right)}^{2}}\right)\right)$

and ${S}_{m,n}$ is the pair solvent energy expressed as:

${S}_{m,n}=\frac{1}{8\mathrm{\pi }}\left(\frac{1}{{\mathrm{\epsilon }}_{0}}-\frac{1}{\mathrm{\epsilon }}\right)\frac{{q}_{m}{q}_{n}}{{f}_{m,n}}$

with

${f}_{m,n}=\sqrt{{R}_{m,n}^{2}+{\mathrm{\rho }}_{m}{\mathrm{\rho }}_{n}{e}^{-{g}_{m,n}}}$

and

${g}_{m,n}=\frac{{R}_{m,n}}{4{\mathrm{\rho }}_{m}{\mathrm{\rho }}_{n}}$

Table 1 provides the remaining variable values.

Table 1 - System parameters
Description Symbol Value
Zinc finger radius ${q}_{\mathit{zinc}}$ $2\cdot 1.60217662\cdot {10}^{-19}C$
Base pair equivalent bead radius ${R}_{\mathit{bead}}$ $1.5\mathit{bp}\left(1\mathit{bp}=350\mathit{pm}\right)$
DNA arm length ${L}_{\mathit{arm}}$ $150\mathit{bp}-1500\mathit{bp}$
DNA arm bead charge magnitude ${N}_{\mathit{zinc}}$ $-2\cdot 1.60217662\cdot {10}^{-19}C$
Real gas constant $R$ $8.3144598$
Temperature $T$ $300K$
Absolute permittivity $\mathrm{\epsilon }$ $8.854187817\cdot {10}^{-12}F/m$
Relative permittivity of water ${\mathrm{\epsilon }}_{r}$ $88$
Hamaker coefficient of nucleotide $A$ $6\cdot {10}^{-10}$
Dielectric constant of water $\mathrm{ϵ}$ $80.4$

#### I.3) Numerical evaluation

We have simulated the effect of arm length on free energy (low free energy implies high binding probability) using the Born salvation free model, pair van der Waals energy, and electrostatic potential, described above. However, there are some parameters that we are not sure about (Equivalent charge and Hamacker coefficient of Zinc fingers, equivalent charge and Hamacker coefficient of base pair) that depend on charge distribution of the DNA strand, which should be determined experimentally as boundary conditions. Alternatively, one could consider each base pair with independent charge.

The systems have been simulated in MATLAB.

Figure 1 shows a rendering of the bacteria - arm compound (E. Coli in blue, zinc fingers in red, arms in orange).

Figure 2 shows relative free energy against arm length. As expected, the longer the arms, the lower is the free energy, and thus the higher is the bonding probability.

Figure 3 shows relative free energy against arm length, where the distribution of zinc fingers is randomized. We notice that the distribution of the zinc fingers plays a very important role in the binding as shown in the figure where the distribution is varied randomly, but the trend is more and less the same as when the distribution is assumed to be deterministic.

#### I.4) Conclusion

The bead approximation and thermodynamic information theoretical framework we proposed effectively captures the challenge of modeling binding probability where different DNA strands compete in binding with a multitude of zinc fingers. The model takes into account the geometry and charge magnitude distribution on the bacteria and the DNA strands.

`NTNU_Trondheim_Warwick_Matlab_code.zip`