The diffusion medium

In diffusion-based MC, the information molecules propagate through the fluidic transmission medium between the transmitter and receiver via diffusion. Generally, the information molecules can be chosen as proteins, protein complexes, peptides, DNA sequences or other molecular structures [38]. The motion of information molecules is inspired by the forces produced by the constant random thermal motion of the molecules within the fluid medium. To be available to communicate through the diffusion channel, the information molecules must be able to be easily fabricated by the transmitter. Also, there must be enough building blocks for the messenger molecules to reside in the environment. In addition, the information molecules must be no harm to the components of the communication system [38].

Molecular diffusion is the thermal motion of all molecules at temperatures above absolute zero. Following this principle, when an uneven distribution of particles exist in a certain environment, they tend to diffuse away in order to reach a uniform concentration throughout all the space. Molecular diffusion could also be considered as a special situation of a random walk or Brownian motion, which is used to model the random motion of particles suspended in a fluid or gas, and also some other phenomena in diverse fields. The emission and propagation of the autoinducers are subject to this physical law. When a certain amount of autoinducers are released by the transmitter node, a peak of concentration appears adjacent to the node. Then the autoinducers would diffuse away as explained before, following the gradient of the concentration, thus going away from the transmitter. The phenomenon of molecular diffusion is typically described mathematically using Fick’s laws of diffusion [187].

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To simplify the communication system, the three dimensional medium is assumed to be extremely large compared to the size of the information molecules. Furthermore, collisions between these molecules are neglected and their motion is inspired by the forces produced by the constant random thermal molecular motion within the fluid medium. The transmitter encodes its information into the concentration of signals. The bacteria inside the transmitter node can produce various concentrations of Type- I molecules to be transmitted through the channel. The emitted signalling molecules then diffuse through the channel to the receiver which is at a distance from the transmitter. At the receiver, each bacterium senses the concentration of Type-I molecules through the corresponding Type-I receptors (LuxR), followed by the production of GFP by the bacteria, which would be used to decode the transmitted information.

Due to the process of QS, the bacteria cells in the receiver can synchronously respond to the molecules as they arrive. In addition, at the receiver, although the molecules can pass through the bacteria cells in the node, the concentration of signalling molecules and the luminescence output will not be affected since the Type-II receptors in the node are not activated, and gene luxI is repressed, which means that no extra Type-I autoinducers could be generated. Thus the channel can be modelled as a diffusion-based MC channel as follows.

The proposed channel is a binary asymmetric channel (BAC) with binary input and binary output and an average probability of error. To effectively represent the transmitted symbols, the propagation time is divided into time slots, also called symbol durations, of equal length, denoted by , in each of which only one symbol propagates. The “intended symbol” and the “received symbol” refer to the symbol sent by the transmitter and received by the receiver, in the current time slot,

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respectively. The information is encoded by concentration with binary representation. Specifically, if the number of information molecules arriving at the receiver in a certain time slot exceeds a threshold τ, the symbol is interpreted as ‘1’; otherwise, it will be taken as ‘0’. Moreover, with OOK modulation employed, the release of molecules in a time slot represents a binary one while their absence for the same duration represents a binary zero. However, errors may be caused by ISI, which is a form of distortion of a signal in which one symbol interferes with subsequent symbols. It is an unavoidable consequence of both wired and wireless communication systems and is known to have adverse effects in communication systems, particularly when the system is stochastic [188]. It should be noticed that the received signals tend to spread to adjacent symbols and smear into each other when a sequence of symbols are transmitted [189]. The ISI effect is related to the properties of the medium used, the distance of the symbol propagation and the selection of the threshold value. In the diffusion communication system here, some information molecules may arrive at the receiver after the current time slot according to the diffusion dynamics, which will lead to the incorrect decoding of the received symbol of the next time slot.

As shown in Figure 4.2, which shows the communication setup of the QS system, the transmitter is at a distance away from the receiver which has a radius of , the value of which is related to the number of bacteria in the receiver node.

In essence, the information molecules propagate through the fluid medium undergoing Brownian motion which is a random procedure and a probabilistic behaviour, which means that the molecules are not ensured to arrive at the receiver. In other words, there is a probability that the molecule will hit the receiver at a time slot. According to Fick’s second law of diffusion [187], the escape probability

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( , ) in a three dimensional environment can be described with the following

backward difference equation at a given time : ( , )

= ∇ ( , ) (4.6)

The diffusion coefficient is 4.9 × 10 , which is settled as a conservative value for AHLs in water at 25℃ [190]. Considering the medium homogeneous, the coefficient will be a constant for all the points in space. It shows that the time rate of change in escape probability is proportional to the curvature of escape probability and to the diffusion coefficient.

However, for this communication model, the capture probability, rather than the escape probability, is more important. Thus by solving equation (4.6), the capture probability ( , ) can be calculated by:

( , ) =

+ 2√

(4.7)

where erfc{∙} is the complementary error function [191]. Equation (4.7) shows the probability that a molecule arrives at the receiver at a time slot from mathematical approach.

To achieve the hit time probability, which refers to the probability that an information molecule arrives at the receiver at a certain time , equation (4.7) is differentiated with respect to time, obtaining the hit time distribution as:

ℎ( , ) =

+ 2√ 1

/ −₄

(4.8)

The capture probability and the hit time probability are both affected primarily by the diffusion coefficient , the radius of the receiver which is affected by the

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number of bacteria in the receiver node , and the distance between the transmitter and receiver, .