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Modeling Approach

Modeling our sensor is a great way to describe the function, operation and design of all of the constructed parts. With our model we are trying to gain a thorough insight in the biosensor’s behavior from input to its signal transduction. It is hard to predict the exact signal a certain event will produce, however a relative approach, by comparing different states of the device, can give valuable information on how to improve and optimize the biosensor. This model compares the signal produced by the intracellular components, both in presence and in absence of the ligand, in order to get a signal-to-noise ratio of the biosensor. (Hereafter “noise” will be used as the measured signal of the sensor in absence of a ligand.) The intracellular components in the model will include a BRET (Bioluminescence Resonance Energy Transfer) or a FRET (Förster Resonance Energy Transfer) pair.

Modeling of Aptamer Binding

To be able to estimate the increase in BRET or FRET when the ligand is added, a simulation of the membrane-bound aptamers and the ligand is required. The software package Smoldyn was used to perform a molecular simulation in order to predict the increased amount of aptamers bound to their ligand. In Smoldyn, a sphere was created to represent an E. coli bacterium. The size of this sphere was set so that the model will simulate one hundredth of the membrane area of an E. coli (approximately the area of a cylinder 2.5 µm in length and 0.5 µm in width). The system makes use of a cubic boundary that bounces off particles; a particle that would leave the cube will be positioned back inside it. The ligand is freely floating in the system and the membrane protein domains are attached to the sphere resembling the E. coli. These domains interact with each other based on the properties, i.e. the association and dissociation constants of the aptamers used in the lab. The simulation was run for 1 second with a time step of 100 nanoseconds, for different concentrations of the ligand. To simulate the noise of this sensor, the same simulation was run, except now in absence of the ligand. This was done with a custom program. The results of these simulations can be viewed in the results page.

Modeling of BRET and FRET

Now that the amount of aptamers bound to the ligand is known, the amount of BRET and FRET can be calculated, which is done in a custom coded program written in Java. The program implements a Monte Carlo method to simulate the behavior of BRET and FRET between two tethered particles. To calculate the amount of BRET and FRET, it is important to know the distance between the intracellular BRET and FRET pairs, as these phenomena’s are heavily dependent on the distance between the interacting domains.

BRET and FRET in Absence of the Ligand
First the simulation is run with a certain amount of membrane proteins (and thus intracellular domains), positioned randomly on a specified 2d membrane area. The program will check automatically if membrane proteins overlap; in that case, the concerned protein will be placed again randomly. The program will repeat the process up to a million times if needed. The program then checks for interacting intracellular domains, by means of the specified maximal Förster distance. It is assumed that each domain forms a BRET or FRET pair with maximal one other domain. Then the distance between these domains is calculated. This distance is not only dependent on the distance between the corresponding membrane proteins, but also on the linker connecting them. The simulation of the position of the intracellular domain in comparison to the attached membrane domain is explained below. Then finally, the amount of BRET and FRET is calculated using this obtained distance and the specified Förster distance.

BRET and FRET in Presence of the Ligand
A second simulation must be run to find out the efficiency of BRET and FRET in presence of the ligand. The calculation in this case follows roughly the same method as the calculation in absence of the ligand.
First, the amount of BRET and FRET is calculated as a result of the binding of the aptamers, simulated using the software packet Smoldyn. In order to calculate the amount of BRET and FRET using the efficiency formula, the distance between the two interacting domains must be known. The distance between the corresponding membrane proteins is approximately the sum of their radii, since the linker between the aptamer and the membrane protein is relatively short. Then the amount of BRET and FRET can be calculated, keeping in mind the influence of the peptide linker on the distance between the intracellular domains.
Secondly, the BRET and FRET resulting from unbound aptamers is calculated using the same method as mentioned above in the section ‘BRET and FRET in absence of the ligand’. However, the amount of membrane proteins involved in this calculation, is the total amount of membrane proteins minus the amount of membrane proteins that are attached to ligand-bound aptamers, calculated by Smoldyn.
The total amount of BRET or FRET is the weighted average (weighted by the amount of particles) of the amount of BRET or FRET calculated in the two section.

Simulating the Peptide Linker

Since the main limitation of the efficiency of BRET and FRET is the distance between the two participating domains, the length of linker in the intracellular compartment plays an important role in the success of our sensor. Simulating the behavior of this peptide linker is essential to understand and predict the amount of BRET or FRET between the two intracellular signaling components. The movement of these components can be seen as a tethered particle motion (TPM), and was simulated using a freely-jointed chain (FJC) model, according to the following formula:



The formula describes the behavior of the linker. The first formula describes the mean perpendicular distance between the membrane and the intracellular domain, the z-radius. The second formula describes the variance of the Gaussian that determines the perpendicular distance to the membrane. The third formula describes the variance of the Gaussian that determines the radius in the xy-plane. Meaning of the symbols: L = linker contour length; ξ = linker persistence length; R_p = radius of membrane particle. Adapted from [1]

When calculating the distance between the two domains, the distance between the corresponding membrane proteins is added in the calculation by translating the intracellular domain in one dimension.

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[1]D. E. Segall, P. C. Nelson, and R. Phillips, “Volume-exclusion effects in tethered-particle experiments: Bead size matters,” Phys. Rev. Lett., vol. 96, no. 8, pp. 1–4, 2006.