Boolean AND and OR logic for cell signalling gateways: a communication perspective

Abstract

Cell signalling plays a vital role in development, sustaining, differentiation, and reproduction of cells. Pathways involved in signalling networks are quite interwoven and complex. Complexity encountered in understanding these pathways is often reduced with the help of Boolean circuit representation. In this study, the authors provide communication aspect of the signalling pathways that have two input Boolean logic AND/OR implemented at the rear effector protein. Communication is assumed to be taking place in extracellular and intracellular environment. The two environments are connected using a receptor protein acting as relay between a molecular source and effector protein. Each relay detects molecules from outside environment and stimulates the production of signals in the intracellular space. These signals/molecules further activate the effector protein which acts as a Boolean switch driven by AND/OR logic. Assuming Poisson reception at the relay as well as at the receiver, the authors provide probability of error of the AND and OR Boolean logic communication systems. Furthermore, reliability and some capacity bounds are deduced for the given Boolean communication system.

1 Introduction

Computational and mathematical analysis of cellular and molecular biology form an essential component of systems as well as synthetic biology [1]. The advent of bioengineering and nanotechnology has led to attempts to mimic various biological phenomena like immune system [2]. Active research is going on using the nanomedicines technology to treat lethal diseases like cancer and AIDS [3]. The use of nanomachines in treating diseases relies on an explicit understanding of cellular signalling pathways. Communication between cells, known as extracellular communication, as well as inside cells, known as intracellular communication, are complex processes. The complexity arises from a lot of signalling pathways, which are quite interwoven and complicated, and use molecules as information carriers. Interaction of signalling pathways is sometimes naturally represented by Boolean logic [4, 5, 6, 7, 8]. Further, the development of genetic circuit [9] and genetic logic gates [10] has enhanced the role of Boolean logic in synthetic biology.

Understanding several signalling processes from communication perspective is an active research area in molecular communication (MC) domain. Modelling of neural signalling is already being explored in MC literature. For instance, upper bounds on the information capacity of bipartite and tripartite synapses have been provided in [11], by giving analogies between the optical and the neuronal communication systems. Study in [11] has been further extended in [12], by incorporating the effect of the number of vesicles available for release, on channel capacity. In [12], authors have employed a realistic pool‐based model for vesicle release and deduced the capacity by modelling channel as binary channel with memory. On the other hand, authors in [13, 14] have established a realistic model of neuron spike trains by modelling them as Cox process, to take into the account the absolute refractory period. A new model which incorporates the effects of three‐dimensional synaptic geometry and re‐uptake of neurotransmitters by the transmitting neuron is provided in [15].

Cellular signalling process generally involves transmitting molecules as signals to the specified location. For the signalling process to work properly, the molecules released from transmitter need to be sensed correctly at the receptor as well as at the effector protein. The molecules move in a fluid media with the help of various forces like diffusion and drift [16, 17, 18]. Movement subjected to these forces is probabilistic in nature. Random movement of molecules leads to a certain probability of error in detecting the stimuli to generate the required response. The probability of error in reception has been aptly dealt in MC literature [19]. Signalling between extracellular and intracellular environment is guided by receptor proteins. Receptor protein, acting as relay, transduces extracellular signal into intracellular signal. Relaying in MC literature is discussed in [20], and the concept has been further envisaged in [21]. In [21], authors used bacteria at source, relay, and destination nodes for emitting, relaying, and detecting the signals, respectively. Relaying under Gaussian noise environment has been studied in [22], employing decode‐and‐forward (DF) protocol. Poisson reception is used in [23] to study the amplify‐and‐forward relay protocol. Generalisation to multi‐hop scenario is proposed in [24], using DF protocol. All these studies are primarily concerned with the various technicalities of MC but none of them explores the Boolean logic gated signalling from communication perspective. This paper presents a simple communication analysis of the AND and OR Boolean logic involved in molecular signalling circuits. The given work provides a basic reliability and capacity analysis of a Boolean signalling pathway, from communication theory perspective. Our study primarily aims at signalling between nanomachines mediated Boolean pathways. Further, the described analysis may be of use in future, if one needs to replace the specific corrupt part of a complex signalling pathway with the nanomachine mediated signalling.

In this paper, following novel contributions are presented:

1. A generalised communication description of the extracellular and intracellular environment, separated by membrane and connected by receptor proteins, is provided.

2. The receptor protein, present at the interface of extracellular and intracellular environments, is modelled as relay. The relay basically connects the molecular source and destination node. An effector protein in the intracellular environment is depicted as a destination node. Effector protein is embedded with AND/OR Boolean logic and has two receivers, acting as the input to the logic gate.

3. Probability of error of the Boolean logic communication circuit, with and without taking relays into consideration is derived.

4. Reliability analysis of the AND and OR Boolean communication circuit is provided.

5. Capacity bounds of the system with AND and OR Boolean logic at the effector protein are deduced using max and min operators. The derived capacity formulates an idea about the switching frequency of the considered communication system.

The rest of the paper is organised as follows. In Section 2, system characterisation describing transmitter, receiver, and relay in terms of source, effector protein, and receptor protein, respectively, is presented. Section 3 deals with relay and effector protein reception and decision threshold calculation. On the other hand, Section 4 presents reliability analysis of the considered Boolean communication circuit. In Section 5, the capacity/maximum switching frequency bounds are calculated. Section 6 gives the numerical results and we end our paper with Section 7 giving conclusions and applications, to which our work may contribute.

2 System characterisation

A simple transmission in the cellular domain from extracellular to intracellular environment is given in Fig. 1. The figure illustrates that the flow of the signal from the outside environment to the inside of a cell mainly consists of three parts [25]. (i) The outside signal in the form of extracellular signalling molecule is received by the receptor protein present on the cellular membrane. (ii) On receiving the intended molecule, receptor protein, acting as relay, gets stimulated. Once stimulated, it starts disseminating intracellular molecules. These molecules act as input to the effector protein, located inside of a cell. (iii) Effector protein is located at the rear end of the communication chain. It may be gene regulatory protein, ion channel, component of a metabolic pathway, or part of the cytoskeleton. Effector protein may be induced by a single signal or may require two different signals in a logical AND/OR fashion to generate its response.

Fig. 1.

Lucid illustration of a simple intracellular signalling pathway getting activated by an extracellular signal molecule

Considering the aforementioned signal flow, we will provide a simple characterisation to the system using a pair of sources, relays, and destination node. Boolean communication model is depicted in Fig. 2 and described as follows:

1. Communication chain starts with a pair of molecular point sources, not shown in Fig. 2, emitting distinct extracellular signals/molecules. Different disseminated molecules are intended for distinct receptor proteins.

2. A pair of receptor proteins is present on the cell membrane. The relays after absorbing molecules for a certain time take a decision, using maximum likelihood (ML) detection. Relays further transmit molecules inside of a cell using DF protocol.

3. Effector protein is described as a destination node, where AND/OR Boolean logic is implemented. To model its two inputs we have two different receivers. These two receivers detect different types of molecules that are coming from different sources and transduced by different receptor proteins as shown in Fig. 2. Note that specific receiver of effector protein is sensitive to only specific type of intracellular signalling molecules, i.e. the receiver absorbs only those molecules that are intended for it.

Fig. 2.

Illustration showing the involvement of Boolean AND or OR in cell signalling pathway

Following assumptions are taken for simplicity:

1. The concentration of the signalling molecule is much smaller than the concentration of the fluid media environment. This assumption is required to prevent the collision of signalling molecules with themselves.

2. We do not assume the presence of any membrane in the considered scenario. Rather we presume that the molecules outside the cell are different from the ones emitted by relay/receptor protein inside the cell. Further, the receptivity of the effector protein is only for those molecules emitted by the relay. This assumption makes the presence or absence of cell boundary quite insignificant. In a natural environment, the molecules which are unabsorbed by the receptor protein on the membrane get degraded [26]. Therefore, they do not affect further communication.

3. The inputs to the gates as well as to the receptor protein are such that one molecule should not act as an inhibitor to another molecule. This assumption is quite implicit in natural environment but taken for incorporating artificial nanomachines/synthetic genetic circuit into the designing scenario [27].

4. The radius of curvature of the cell membrane is quite large with respect to the signalling molecules and hence can be considered flat, for many real cases. This allows both of the receptor proteins to be present at the same distance from respective transmitters and receivers.

5. A timing synchronisation between two inputs of the gates is assumed. The assumption is taken to relax the timing constraints that may arise due to time lag between the MC processes. In practice, issue of synchronisation can be handled using ML estimator [28] or peak based estimation [29].

6. The given work concerns itself with the noise generated because of diffusion, inherently present in the MC scenario. The noise due to chemical binding of molecules to the receptors and the chemical kinetics [30] associated with it, is not taken into consideration for simplicity.

We assume our initial origin of molecules to be a pair of binary point source working on concentration shift keying principle [31]. Each source disseminates a certain number of molecules ${\beta }_m$ for binary $m\in \{0,1\}$. These molecules are mostly proteins or enzymes in the biological scenario [25]. Assuming a diffusive environment with a certain drift, we consider transmitters and relays disseminating in slots of duration T where j th slot is given by $[(j-1)T,jT)$, $j\in \{1,2,\dots \}$. The molecules are disseminated in the form of impulses at the beginning of every time slot. We assume the relays and the receivers on effector protein to absorb the molecules for a certain identical sampling time T. The transferring of molecules from source to relay and relay to destination has a certain time lag. To ease with delay constraints on communication links, we assume time synchronisation as well as channel side information at the relays and receivers in source–relay–destination chain [29].

The probability of molecules, transmitted in the previous i th slot, and reaching the receiver in the current slot, is given by [32]

$$p_i={\int }_{iT}^{(i+1)T}g(t){\mathrm{e}}^{-\alpha t}\mathrm{d}t,$$ (1)

where $\alpha$ is the degradation constant and $g(t)$ is defined as

$$g(t)=\sqrt{\frac{\lambda }{2\pi t^3}}{\mathrm{e}}^{-(\lambda {(t-\mu )}^2/2{\mu }^{2}t)},$$ (2)

where $\mu =d/v$ and $\lambda =d^2/{\sigma }^2$ and ${\sigma }^2=2D$. As can be seen from (1) and (2), $p_i$ is a function of drift velocity v, diffusion coefficient D, and distance between transmitter and receiver d. Basically, $p_i$ is the impulse response of the system. It incorporates the effect of release and the transmission of molecules through the fluid media. The degradation parameter $\alpha$ is incorporated in the probability term to take into consideration average lifetime of molecules, before they denature. Molecular reception can be aptly modelled by presence or absence of the molecules at the receiver. This naturally falls under the domain of binomial distribution. Since we have considered intersymbol interference in the given model, the cumulative effect of molecules at the receiver can be analytically difficult to work with, as suggested in [33]. Therefore, to simplify the analysis, we take the Poisson approximation of binomial distribution.

Considering finite interference from the molecules that are emitted in the previous slot we have an interference term given by ${\sum }_{i=1}^K{p}_i{\beta }_{m_i}$, where K is the total interference considered from previous emissions and ${\beta }_{m_i}$ gives ${\beta }_m$ dispersed in the i th previous slot. The value of K depends on degradation parameter $\alpha$ – the more the $\alpha$, the faster the degradation will be and hence less K contributing to interference. Let $Y_K$ be the number of molecules received at the receiver, distributed as

$$Y_K\sim \mathrm{Poisson}\left({\beta }_{m_0}{p}_0+\sum _{i=1}^K{p}_i{\beta }_{m_i}\right),$$ (3)

where the summation is the result of combining of the independent Poisson processes at the receiver [34].

3 Relay and effector protein reception

The receptor protein on cellular membrane acts as a gateway that allows the molecules coming from external environment to transduce their signals to inner domain. The transduced signals in the form of molecules are then received by effector protein, for further processing. In this section, we deal with the reception at both the receptor protein as well as at the effector protein.

3.1 Relay reception

Receptor protein gateway can be simplistically represented by a relay on the cell membrane. It takes the molecules coming from transmitter and emits another type of molecules, to which the receiver is receptive. As already justified in assumption (ii), we can comfortably ignore the membrane in the reception analysis; conjecturing the little change, it will make on the final analysis. Relay after receiving the signal molecules compares the concentration with a detection threshold and decides in favour of binary ‘1’ or ‘0’. Assuming Poisson reception and a certain decision threshold $(\eta )$, the probability of detection is given as

$$\mathrm{Pr}\{0\to 0\}=\sum _{y=0}^{\eta }{\mathrm{e}}^{-({\sum }_{i=1}^K{p}_i{\beta }_{m_i})}\frac{{({\sum }_{i=1}^K{p}_i{\beta }_{m_i})}^y}{y!},$$ (4a)

$$\mathrm{Pr}\{1\to 1\}=\sum _{y=\eta +1}^{\mathrm{\infty }}{\mathrm{e}}^{-(p_0{\beta }_1+{\sum }_{i=1}^K{p}_i{\beta }_{m_i})}\frac{{(p_0{\beta }_1+{\sum }_{i=1}^K{p}_i{\beta }_{m_i})}^y}{y!},$$ (4b)

$$\mathrm{Pr}\{1\to 0\}=\sum _{y=0}^{\eta }{\mathrm{e}}^{-(p_0{\beta }_1+{\sum }_{i=1}^K{p}_i{\beta }_{m_i})}\frac{{(p_0{\beta }_1+{\sum }_{i=1}^K{p}_i{\beta }_{m_i})}^y}{y!},$$ (4c)

$$\mathrm{Pr}\{0\to 1\}=\sum _{y=\eta +1}^{\mathrm{\infty }}{\mathrm{e}}^{-({\sum }_{i=1}^K{p}_i{\beta }_{m_i})}\frac{{({\sum }_{i=1}^K{p}_i{\beta }_{m_i})}^y}{y!},$$ (4d)

where $\mathrm{Pr}\{\cdot \}$ is the probability operator and $\mathrm{Pr}\{r\to s\}$ indicates the probability of detecting $s\in \{0,1\}$ at the output when $r\in \{0,1\}$ is sent. We consider r and s as binary symbols of the mentioned transmission scheme. We have assumed no emission of molecules for binary ‘0’ and ${\beta }_1$ is the number of molecules disseminated for binary ‘1’. To find the detection threshold, we have taken a continuous evaluation process, i.e. based on the previously estimated input the threshold is concurrently updated. This continuous modification of threshold takes care of finite interference term which changes as per the continuous stream of 1s and 0s. Relay detects the output using ML detection. Employing ML detection for cellular reception can be traced to quorum sensing which is a biological phenomenon involved in calculating decision thresholds [35].

For the proposed analysis, decision threshold can be calculated by reducing the ML detection rule to probability mass function comparison, as given in [32]

$$\begin{array}{rl}& {\mathrm{e}}^{-(p_0{\beta }_1+{\sum }_{i=1}^K{p}_i{\hat{\beta }}_{m_i})}\frac{{(p_0{\beta }_1+{\sum }_{i=1}^K{p}_i{\hat{\beta }}_{m_i})}^{\eta }}{\eta !}\\ & ={\mathrm{e}}^{-({\sum }_{i=1}^K{p}_i{\hat{\beta }}_{m_i})}\frac{{({\sum }_{i=1}^K{p}_i{\hat{\beta }}_{m_i})}^{\eta }}{\eta !},\end{array}$$ (5)

where ${\hat{\beta }}_i$ is the estimated number of molecules that are emitted in i th previous slot. After some mathematical manipulations, (5) reduces to

$$\eta =⌈\frac{p_0{\beta }_1}{\ln ({\sum }_{i=1}^K{p}_i{\hat{\beta }}_{m_i}+p_0{\beta }_1)-\ln ({\sum }_{i=1}^K{p}_i{\hat{\beta }}_{m_i})}⌉,$$ (6)

where $\lceil \cdot \rceil$ is the ceiling operator. The interfering term ${\sum }_{i=1}^K{p}_i{\hat{\beta }}_{m_i}$ in (6) is the one that leads to continuous detection threshold evaluation depending on previously estimated inputs of size K. It is important to note that this detection requires memory of previous estimations. Incorporating memory in the biological circuit design is an active research area [36]. Once the threshold is determined, relay can decide whether to select ‘1’ or ‘0’, as its output for further transmission to the receiver. If the received number of molecules are larger than calculated detection threshold ($Y_K>\eta$), then the receptor protein gets stimulated else remains inert.

3.2 Effector protein reception

Effector proteins are located at the rear end of cellular communication. They can be represented by the pair of receivers situated at the termination of the source–relay–destination chain. The two receivers act as inputs to the Boolean gate. Communication performance analysis of effector protein with embedded AND/OR Boolean logic is the subject of this paper. Molecules absorbed at the receivers generate signals which are then compared with their respective detection thresholds. The decision made is further processed by biological gate to give AND/OR operation. Detection is identical to the one already referred in Section 3.1 for relay reception. Probability of error in the decision‐making process may result from wrong detection at the relays or at the receivers incorporated in the effector protein. We assume that different inputs of the Boolean gate are coming from different receptor proteins (relays) transducing dissimilar molecules to gate inputs. The mentioned binary process can be best represented by binary trees. Each tree depicts source output as its root and probable decisions taken by relay and receiver as branch and leaves, respectively. In Fig. 3, binary trees for source pair output $\mathcal{I}=\{1,1\}$ is shown. The figure illustrates both erroneous as well as right combinations of symbols detected at the receivers of effector protein.

Fig. 3.

Binary trees illustration of erroneous and non‐erroneous combinations when source outputs {1, 1}

3.2.1 AND Boolean gate effector protein reception

Assuming both sources to be on, we now perform probability of error analysis for AND gate communication circuit. Both sources on implies dissemination of ${\beta }_1$ molecules from the source pair. The transmitted molecules are absorbed by the receptor proteins and based on the detection threshold the two relays decide in favour of ‘1’/‘0’. The decision is transmitted to the two receivers at effector protein which further decide between ‘1’/‘0’. The sequence of detections gives us a total of four probable inputs at single receiver of effector protein per source output. The four probable inputs are represented by four leaves of a binary tree, as shown in Fig. 3. Since the effector has two receivers, the eight possible inputs to effector protein are given by: $\{a,b,c,d\}$ and $\{e,f,g,h\}$, as shown in the figure. Error in the processing can occur if the AND gate embedded at the effector protein does not get the combination {1, 1} at its two inputs. As can be seen in the figure, this leads to four non‐erroneous combinations, given by $\{b,f\},\{b,h\},\{d,f\}\mathrm{and}\{d,h\}$. While rest all other combinations will give an erroneous output. The path traversed from root to a single leaf involves two detections: one at the relay or receptor protein and the other at the receiver ends of effector protein. Let the output at the effector protein be given by $\hat{W}$, which is a result of AND/OR operation done on the inputs of Boolean logic gate.

Mathematically, the probability of wrong decision when both sources are excited is given by

$$\begin{array}{rl}& \mathrm{Pr}\{\hat{W}\ne 1|\mathcal{I}=\{1,1\}\}\\ & =\frac{1}{4}(1-[\mathrm{Pr}\{b,f\}+\mathrm{Pr}\{b,h\}+\mathrm{Pr}\{d,f\}+\mathrm{Pr}\{d,h\}]).\end{array}$$ (7)

The inclusion of factor 1/4 in (7) is because of the assumption of equiprobable binary outputs disseminated by the source pair. The probability of getting $\{b,h\}$ combination is given by

$$\mathrm{Pr}\{b,h\}=\mathrm{Pr}\{b\}\times \mathrm{Pr}\{h\},$$ (8)

as both of these events are mutually exclusive. Further, the probability of event $\{b\}$ can be written as

$$\mathrm{Pr}\{b\}=\mathrm{Pr}\{1\to 0\}\times \widetilde{\mathrm{Pr}}\{0\to 1\},$$ (9)

where the definitions of $\mathrm{Pr}\{1\to 0\}$ and $\widetilde{\mathrm{Pr}}\{0\to 1\}$ are already given in (4). Further, $\mathrm{Pr}\{\cdot \}$ and $\widetilde{\mathrm{Pr}}\{\cdot \}$ correspond to link between the source and relay and link between the relay and the receiver of effector protein, respectively. The probability of getting event $\{h\}$ can be calculated in the same manner, given as

$$\mathrm{Pr}\{h\}=\mathrm{Pr}\{1\to 1\}\times \widetilde{\mathrm{Pr}}\{1\to 1\}.$$ (10)

Similarly, probability of error for all other combinations can be calculated for AND gate Boolean communication circuit.

3.2.2 OR Boolean gate effector protein reception

We will now calculate the probability of error for OR Boolean logic, given both sources transmit ‘1’, i.e. $\mathcal{I}=\{1,1\}$. As can be seen in Fig. 3, the erroneous combinations, received at receivers of Boolean gate, are given by $\{a,e\}$, $\{a,g\}$, $\{c,e\}$, and $\{c,g\}$. The probability of error is given by

$$\begin{array}{rl}& \mathrm{Pr}\{\hat{W}\ne 1|\mathcal{I}=\{1,1\}\}\\ & =\frac{1}{4}(\mathrm{Pr}\{a,e\}+\mathrm{Pr}\{a,g\}+\mathrm{Pr}\{c,e\}+\mathrm{Pr}\{c,g\}).\end{array}$$ (11)

To calculate probability of occurrence $\{a,g\}$ we need to find $\mathrm{Pr}\{a\}$ and $\mathrm{Pr}\{g\}$. The probability of event $\{a\}$ is given by

$$\mathrm{Pr}\{a\}=\mathrm{Pr}\{1\to 0\}\times \widetilde{\mathrm{Pr}}\{0\to 0\},$$ (12)

and of event $\{g\}$ by

$$\mathrm{Pr}\{g\}=\mathrm{Pr}\{1\to 1\}\times \widetilde{\mathrm{Pr}}\{1\to 0\}.$$ (13)

Similarly, probability of error for all other source input combinations can be calculated for both AND and OR Boolean communication circuits. Effector protein after having a ‘1’ at its output can proceed with the next step of signalling pathway, of which it is a part.

Note that we have presented the generalised analysis of the Boolean communication circuit with relay. The error probability for AND and OR Boolean communication circuit without relay can be easily obtained using same analysis. However, in the aforementioned case, we just need to consider binary trees with four leaves. This is because the first level of decision taken at receptor protein/relay is omitted now.

4 Reliability of Boolean communication circuit

Any disturbance in cell signalling pathways can lead to perturbation of the whole signalling process. Corruption of a signalling pathway may result in a certain disease. For instance, authors in [5] have shown schizophrenia as the implication of deviation from normal neurotransmitter signalling pathway. Reliability of a given communication system can be described as the precision with which the system works, given some disruptions. In this section, we provide reliability analysis for AND and OR Boolean communication systems, when one of the links in the cellular signalling pathway breaks, i.e. stops functioning. Diseases may be caused either by hyper‐ or hypo‐functioning of the involved signalling protein. However, in the present work, we just consider complete unavailability of the signalling molecules at one of the receivers of the effector protein. Also for simplicity, we just consider two sources at the transmitting end and two receivers at the effector protein, without taking receptor proteins into consideration.

4.1 Reliability analysis of AND Boolean communication circuit

Let us consider AND gate Boolean communication system with two sources transmitting molecules. The receivers present at the effector protein receive and detect the molecules. Based on the detected input the effector protein, incorporated with AND gate, either gets stimulated or remains inert. Functionally, AND gate gets stimulated just for input combination {1, 1}, and remains inert for all the other combinations. Breaking a link corresponding to certain receiver end of effector protein implies that corresponding input to the receiver is always ‘0’. The only permissible detected combinations at the receivers’ inputs are {0, 0} and {1, 0}, implying that effector protein will never get stimulated. To elucidate further, apart from combination {1, 1} all other combinations will never be in error. This is because broken link always leads to ‘0’ at the output of AND gate effector protein, i.e. effector protein remains inert. Assuming equiprobable inputs, the error probability of AND gate communication circuit is then given by

$$P_e^{\mathrm{AND}}=\mathrm{0.25.}$$ (14)

Further, reliability of AND gate Boolean communication circuit can be calculated from (14) as

$$\begin{array}{rl}{\mathcal{R}}_{\mathrm{AND}}& =1-P_e^{\mathrm{AND}}\\ & =\mathrm{0.75.}\end{array}$$ (15)

4.2 Reliability analysis of OR Boolean communication circuit

In this section, we present reliability analysis of OR gate communication circuit with a broken link at one of the receivers of the effector protein. Let us assume the receiver input corresponding to the broken link be n. The probability that $n=0$ is given by

$$\mathrm{Pr}(n=0)=1.$$ (16)

To comprehend the reliability calculation, we break the analysis into four scenarios, as follows:

1. When both the sources are off, i.e. input is {0, 0}: As can be seen in Fig. 4, the error here lies in detecting ‘1’ at the OR effector protein output. This can only happen when the source output at unbroken link, because of channel memory, is sensed as ‘1’ at the receiver's input. The probability of error $\mathrm{Pr}\{b,n\}$ is given by

   $$\begin{array}{rl}\mathrm{Pr}\{b,n\}\stackrel{(\mathrm{i})}{=}& \mathrm{Pr}\{b\}\times \mathrm{Pr}\{n\}\\ \stackrel{(\mathrm{ii})}{=}& \mathrm{Pr}\{0\to 1\}\times 1,\end{array}$$ (17)

   where the equality $(\mathrm{i})$ implies mutually exclusive events and equality $(\mathrm{ii})$ is given by (16).
2. When the source corresponding to unbroken link is off while the one corresponding to broken link is on, i.e. input is {0, 1}: As can be illustrated from Fig. 4, the error will be there when the output of the effector protein is ‘0’. Error probability corresponds to $\mathrm{Pr}\{a,n\}$ given by

   $$\begin{array}{rl}\mathrm{Pr}\{a,n\}& =\mathrm{Pr}\{a\}\times \mathrm{Pr}\{n\}\\ & =\mathrm{Pr}\{0\to 0\}\times 1.\end{array}$$ (18)
3. When the source corresponding to unbroken link is on, while the one corresponding to broken link is off, i.e. input is {1, 0}: For the given scenario, error corresponds to detecting ‘0’ at the output of effector protein, and is given by $\mathrm{Pr}\{a^{\mathrm{\prime }},n\}$

   $$\begin{array}{rl}\mathrm{Pr}\{a^{\mathrm{\prime }},n\}& =\mathrm{Pr}\{a^{\mathrm{\prime }}\}\times \mathrm{Pr}\{n\}\\ & =\mathrm{Pr}\{1\to 0\}\times 1.\end{array}$$ (19)
4. When both the sources are on, i.e. input is {1, 1}: It can be illustrated from Fig. 4 that the error in decision corresponds to detecting ‘0’ at both of the receivers of OR gate, and the probability of error ($\mathrm{Pr}\{a^{\mathrm{\prime }},n\}$) is given by

   $$\begin{array}{rl}\mathrm{Pr}\{a^{\mathrm{\prime }},n\}& =\mathrm{Pr}\{a^{\mathrm{\prime }}\}\times \mathrm{Pr}\{n\}\\ & =\mathrm{Pr}\{1\to 0\}\times 1.\end{array}$$ (20)

Fig. 4.

Detection at the effector protein for OR gate Boolean communication system with a broken link

The total probability of error of OR Boolean communication circuit, given a broken link, is obtained by adding (17)–(20)

$$\begin{array}{rl}{P}_e^{\mathrm{OR}}& =\frac{1}{4}[\mathrm{Pr}\{b,n\}+\mathrm{Pr}\{a,n\}\\ & +\mathrm{Pr}\{a^{\mathrm{\prime }},n\}+\mathrm{Pr}\{a^{\mathrm{\prime }},n\}],\end{array}$$ (21)

where we have assumed each combination at the source to have equiprobable distribution. Reliability of OR gate Boolean communication circuit can now be calculated from (21) as

$${\mathcal{R}}_{\mathrm{OR}}=1-P_e^{\mathrm{OR}}.$$ (22)

Through the proposed analysis a relationship between various diseases, affecting cell signalling, and reliability of communication circuit may be established. Analysing diseases in terms of reliability of Boolean communication circuit may provide a different outlook in quantitative life sciences like systems biology.

5 Maximum switching frequency or the capacity of Boolean communication circuit

The capacity is basically the upper bound on the rate at which information can be sent over a communication channel. This in turn corresponds to the maximum switching frequency of the Boolean communication circuit. The switching frequency, described here, is different than that of the electronic circuits. While switching frequency in electronic circuits concerns itself with the processing speed of the gate. However, here the switching frequency is primarily associated with the rate at which the effector protein switch can turn on and off, given communication constraints of a diffusive‐drift environment. Since we are dealing with the capacity of the wireless Boolean communication circuit, the processing speed of the relay and the gate is not considered in the given analysis.

The capacity of the Poisson channel is dealt in [37, 38] to name a few. In the considered channel model, we assume finite channel interference, which renders memory to the channel. The Poisson channel with memory is examined in [39], where it is assumed that molecules injected in the environment disappear after a fixed number of time slots. Disappearance of molecules makes the channel indecomposable. Further, the vanishing of the molecules has an analogy with the finite interference scenario characterised by degradation parameter, in the assumed channel model.

As mentioned in [39], we can define a channel as a memory limited channel of order K if it satisfies the given relation between its input and output pair

$$p(y_{K+1:\tilde{n}}|x_{1:\tilde{n}})\triangleq \prod _{l=K+1}^{\tilde{n}}p(y_l|x_{l:l-K}),$$ (23)

where ${\mathcal{S}}_{L+1:M}$ gives the symbols transmitted/received over intervals $L+1$ to M. Equation (23) renders $y_l$ to be dependent just on the previous $x_{l:l-K}$ values, and independent of all other inputs. The presented Boolean gate channel model can also be considered as memory limited, owing to the effect of degradation constant. Degradation constant gives a finite lifetime to the molecules and renders the output to depend just on the certain number of previous inputs. As will also be shown in numerical results, the hitting probability goes on decreasing with increasing delay response. Using the aforementioned fact, we can apply (23) to the mentioned channel model, rendering it block memoryless. The upper and lower bounds of the capacity of such a point‐to‐point memory‐limited channel of order K are then bounded by [39]

$$C\ge \underset{p(x_0,\dots ,x_K)\in \mathcal{P}}{max}I(X_0;Y_0|X_{1:K}),$$ (24a)

$$C\le \underset{p(x_0,\dots ,x_K)\in \mathcal{P}}{max}I(X_{0:K};Y_0),$$ (24b)

where C stands for the channel capacity and $p(x_0,\dots ,x_K)$ gives the probability distribution over which the mutual information $I(\cdot ;\cdot )$ needs to be maximised; K is the number of interfering inputs from the previous molecular emissions, and the present input and output are represented by subscript 0. Since the effector protein consists of two receivers, the capacity of a Boolean AND/OR communication circuit cannot be treated as a single communication line. Switching frequency depends on the capacity of the links between sources and relays as well as the links between relays and destination.

Let the capacity of the $\mathrm{lin}{\mathrm{k}}_w$ be $C_w$ where $w\in \{1,2,3,4\}$, then the capacity of the OR Boolean communication system is given by

$$C_{\mathrm{OR}}=max\{min\{C_1,C_2\},min\{C_3,C_4\}\},$$ (25)

and the capacity of the AND Boolean communication system is given by

$$C_{\mathrm{AND}}=min\{C_1,C_2,C_3,C_4\},$$ (26)

where $\mathrm{lin}{\mathrm{k}}_1$, $\mathrm{lin}{\mathrm{k}}_2$, $\mathrm{lin}{\mathrm{k}}_3$, and $\mathrm{lin}{\mathrm{k}}_4$ can be identified as given in Fig. 5.

Fig. 5.

Model showing sources, receptor proteins (relays) and effector protein (destination) for the capacity calculation; molecules disseminated from same source have been depicted using identical colour

Let us first consider the OR gate Boolean communication circuit. To prove the theorem, we define an analogous scenario with four links and OR gate given at the termination of the link chain, as illustrated in Fig. 6. The Boolean gate acts as a gateway to further signalling process. We need to find the maximum rate at which information can be transmitted from the start of the link chains to the output of the OR Boolean gate. To simplify the model we take help of ‘flow of water in pipes’ model as referred in [40], by modelling links as pipe and rate through that link as the flow of water in the given pipe. For the considered system, we have two link pairs; $\mathrm{lin}{\mathrm{k}}_1$ and $\mathrm{lin}{\mathrm{k}}_2$ given by $\mathrm{chai}{\mathrm{n}}_1$ as well as $\mathrm{lin}{\mathrm{k}}_3$ and $\mathrm{lin}{\mathrm{k}}_4$ given by $\mathrm{chai}{\mathrm{n}}_2$. The links in a chain are in series with one another. The series structure of a chain imparts to it the minimum capacity that one of the two links can support. This is similar to the rate of flow of water in chain that will be the minimum of the flow of two cascaded pipes denoted by links. The capacities of the chains are given by

$$C_{\mathrm{chai}{\mathrm{n}}_1}=min\{C_1,C_2\},$$ (27a)

$$C_{\mathrm{chai}{\mathrm{n}}_2}=min\{C_3,C_4\},$$ (27b)

where $C_{{\mathrm{chain}}_u}$ is the capacity of ${\mathrm{chain}}_u$ and $u\in \{1,2\}$. Further, OR gate at the rear end implies that these two chains are in parallel structure. A parallel structure allows the flow of signalling if any of one of the two inputs is stimulated, leading to

$$C_{\mathrm{OR}}=max\{C_{\mathrm{chai}{\mathrm{n}}_1},C_{\mathrm{chai}{\mathrm{n}}_2}\},$$ (28)

where $C_{\mathrm{OR}}$ is the total capacity of the OR Boolean gate communication system and hence proves (25).

We now prove (26) which gives the capacity of AND Boolean communication circuit. The capacities of $\mathrm{chai}{\mathrm{n}}_1$ and $\mathrm{chai}{\mathrm{n}}_2$, as given in Fig. 6, are identical to the one calculated for OR Boolean communication circuit, owing to their series structure. Further, the AND gate at the termination of communication circuit will be stimulated only when both inputs are activated, leading to a series structure. As has been mentioned that for pipes in series, total flow is the minimum of flows over all connected pipes, giving us

$$C_{\mathrm{AND}}=min\{C_{\mathrm{chai}{\mathrm{n}}_1},C_{\mathrm{chai}{\mathrm{n}}_2}\},$$ (29)

which further simplifies to (26).□

Fig. 6.

Model illustrating the chains and links for the given Boolean gate communication system

6 Numerical results

In this section, we present numerical results for error performance of AND and OR Boolean logic communication circuits with and without taking relays into consideration, using (7) and (11). We present the reliability of Boolean communication circuit assuming a broken communication link, given by (15) and (22); and calculate the capacity bounds for AND and OR Boolean logic MC circuits, using (25) and (26). The parameter values taken for plotting the analytical results are typical of short‐range MC [32]. Sampling time is taken as $T=2.5\mathrm{s}$. The specific value of the sampling time corresponds to time difference that can render measures of counting noise statistically independent, as suggested in [32]. Channel memory is decided by the relative loss in the probability of molecules, transmitted in the i th previous slot and reaching the receiver in the current slot. The channel memory depends on the average lifetime of the molecules and the probability of molecules hitting the receiver in the present slot. We have plotted the probability of molecules hitting the receiver as a function of distance between transmitter and receiver in Figs. 7 and 8, for various delay responses. The plots in Figs. 7 and 8 are for degradation constants $\alpha =0.75$ and $\alpha =0.15$, respectively. As can be seen in both the figures, the hitting probability goes on decreasing with the increase in the delay response. Further, it reduces up to ${10}^{-1}$ for the fourth delay response when compared with the no delay response, for both figures. This justifies our assumption of $K=4$, as the memory length. The dissemination of different molecules by source pair and receptor proteins leads to different values of diffusion coefficient. For the given model, we have taken diffusion coefficients to be $20\times {10}^{-11}$ and $25\times {10}^{-11}{\mathrm{m}}^2/s$ in the extracellular space and $35\times {10}^{-11}$ and $40\times {10}^{-11}{\mathrm{m}}^2/s$ in the intracellular space as can be seen in Fig. 5. The diffusion coefficients used for numerical analysis are basically in the range of protein/dye molecules diffusing in extracellular and intracellular environments [41, 42]. Extracellular drift is taken to be $14\mu \mathrm{m}/\mathrm{s}$ and intracellular to be $20\mu \mathrm{m}/\mathrm{s}$, which is the typical speed of extracellular and intracellular currents and molecular drifts [32, 43, 44]. For instance, the intercellular calcium wave can spread up to 200–350 mm in one dimension at a speed of 16–27 μm/s [43], and calcium has a measured diffusion coefficient of $23\times 10^{-11}{\mathrm{m}}^2/s$ in dialysed axoplasm [45]. The degradation parameter $\alpha$ is taken to be 0.5. The number of molecules emitted for binary ‘1’, i.e. ${\beta }_1=10$. Unless stated otherwise, these parameters are considered same for both the sources as well as the relays. The length of $\mathrm{lin}{\mathrm{k}}_1$ is same as of $\mathrm{lin}{\mathrm{k}}_3$, and that of $\mathrm{lin}{\mathrm{k}}_2$ is same as of $\mathrm{lin}{\mathrm{k}}_4$, as shown in Fig. 5. The parameters values are also tabulated in Table 1.

Fig. 7.

Hitting probability versus distance curve for $\alpha =0.75$, $D=35\times {10}^{-11}{\mathrm{m}}^2/\mathrm{s}$, and $v=20\times {10}^{-6}\mathrm{m}/\mathrm{s}$

Fig. 8.

Hitting probability versus distance curve for $\alpha =0.15$, $D=35\times {10}^{-11}{\mathrm{m}}^2/\mathrm{s}$, and $v=20\times {10}^{-6}\mathrm{m}/\mathrm{s}$

Table 1.

Various parameter values used in the paper

Parameter	Symbol	Value
degradation constant	$\alpha$	0.5
sampling time	T	2.5 s
diffusion coefficient for link1	$D_{\mathrm{lin}{\mathrm{k}}_1}$	$20\times {10}^{-11}{\mathrm{m}}^2/s$
diffusion coefficient for link2	$D_{\mathrm{lin}{\mathrm{k}}_2}$	$35\times {10}^{-11}{\mathrm{m}}^2/s$
diffusion coefficient for link3	$D_{\mathrm{lin}{\mathrm{k}}_3}$	$25\times {10}^{-11}{\mathrm{m}}^2/s$
diffusion coefficient for link4	$D_{\mathrm{lin}{\mathrm{k}}_4}$	$40\times {10}^{-11}{\mathrm{m}}^2/s$
velocity in extracellular environment	$v_1$	$14\times {10}^{-6}{\mathrm{m}}^2/s$
velocity in intracellular environment	$v_2$	$20\times {10}^{-6}\mathrm{m}/\mathrm{s}$
number of molecules released for binary 0	${\beta }_0$	0
number of molecules released for binary 1	${\beta }_1$	10

In Figs. 9 and 10, probability of error versus distance plots for AND and OR Boolean logic communication circuits are illustrated, respectively. The figures are drawn without taking membrane and hence receptor proteins into consideration. The source pair and effector protein are assumed to be in the same fluid medium, with two communication links having diffusion coefficients as $40\times {10}^{-11}$ and $35\times {10}^{-11}{\mathrm{m}}^2/s$, respectively. The considered fluid medium has a drift of 20 μm/s. Both Figs. 9 and 10 consist of various plots as a function of degradation parameter. It can be seen from both the figures that the probability of error for AND and OR Boolean communication circuit saturates to 0.25 and 0.75, respectively, after a certain separating distance between transmitters and receivers.

Fig. 9.

Probability of error versus distance curve for AND Boolean logic without relay

Fig. 10.

Probability of error versus distance curve for OR Boolean logic without relay

Some inferences drawn from Figs. 9 and 10 are listed below:

• The maximum probability of error for AND and OR Boolean communication circuit is 0.25 and 0.75, respectively. The disparity in maximum probability of error is because of the inherent logic of both gates as well as the hitting probability model. To elucidate the argument we take help of Fig. 7. As can be seen from the figure, hitting probability decreases with increase in distance. The given behaviour of hitting probability favours AND gate over OR gate communication. For AND gate, ‘1’ at output is rendered just for one combination, i.e. {1, 1} while for OR it is for three combinations, i.e. {0, 1}, {1, 0}, {1, 1}. At larger distances between transmitter and receiver the hitting probability of disseminating molecules decreases, leading to higher $\mathrm{Pr}\{1\to 0\}$. Higher $\mathrm{Pr}\{1\to 0\}$, in turn leads to error for just one input combination of AND. However, it affects three input combinations of OR gate communication circuit. Therefore, the maximum probability of error is 1/4 and 3/4 for AND and OR Boolean communication systems, respectively.

• Error probability favours OR gate Boolean communication circuit for lower degradation constant like $\alpha =0.15$, near error saturating distances. However, the error probability plot for the same degradation constant and near the error saturating distances does not show the similar favourable behaviour for AND gate Boolean communication circuit. For AND gate Boolean communication circuit, the given plot at once crosses saturating error probability of 0.25 and eventually settles for the same error probability on increasing distance. The reason for the aforementioned behaviour can be attributed to the channel memory. As degradation constant decreases, the probability of molecules getting denatured reduces. Aforementioned fact can also be verified by comparing Fig. 7 with Fig. 8. Less denaturing means that molecules from previous slots effectively interfere with the molecules disseminated in the current slot. Thus, decreasing $\alpha$ increases correct detection of ‘1’ and also leads to higher $\mathrm{Pr}\{0\to 1\}$ at the receiver; leading to better detection for OR communication circuit. However, increase in $\mathrm{Pr}\{0\to 1\}$ elevates error probability for AND gate communication circuit. The reason for the given behaviour can be attributed to the functionality of Boolean AND and OR logic gates. Considering equiprobable inputs from source pair, the probability of disseminating molecules in the combinations of {0, 0}, {0, 1}, {1, 0}, {1, 1} is 1/4 each. Increasing $\mathrm{Pr}\{0\to 1\}$ will lead to erroneous detection 3/4 times for AND gate communication circuit, as now the probability of wrongly detecting ‘1’ increases. This will proliferate erroneous detection for three input combinations, i.e. {0, 0}, {0, 1}, {1, 0}, out of four. On the other hand, for OR gate communication circuit the increase in $\mathrm{Pr}\{0\to 1\}$ just leads to improper detection of a single output combination, i.e. {0, 0}. Therefore, total increment in erroneous detection for OR Boolean logic is just 1/4.

• At smaller distances, molecules with relatively higher degradation constant perform better than molecules with lower degradation constant. The given behaviour may be attributed to the channel memory. Higher degradation constant allows molecules to disintegrate relatively fast. This leads to less interference from previous slots, at smaller distances. Further, for lower degradation constant interference at smaller distances dominates leading to wrong decision. On the other hand, for larger distances lower degradation constant favours communication by allowing molecules to survive up to a greater separation distance between the transmitter and the receiver as compared to molecules with higher degradation constant. The presented phenomenon can be illustrated by the magnified portion of the plots, given in Figs. 9 and 10.

In Fig. 11, probability of error versus distance plots are drawn for $\alpha =0.5$. The plots are illustrated considering relays in between molecular source pair and effector protein. The graph is plotted by varying the relays’ position between source pair and receivers for both AND and OR Boolean gate communication circuits. Total distance between the transmitters and the receivers is $100\mu \mathrm{m}$. The position of membrane, containing receptor proteins, is varied from $10$ to $90\mu \mathrm{m}$ from the transmitters. It can be seen from the figure that as relays move away from the transmitters, probability of error decreases. The error probability reduces to a minimum of 0.17 and 0.26 at a separation distance close to $40\mu \mathrm{m}$ for AND and OR Boolean gate communication systems, respectively. On further increasing the distance, error probability again starts elevating as relays move towards the effector protein's receivers. The behaviour may be attributed to hitting probability which decreases with increase in distance, an example of which is shown in Fig. 7. When the relays get closer to the transmitters, the distance between receptor proteins and receivers increases leading to loss of disseminated molecules, and hence less absorption at receivers. On the other hand, making relays closer to the receivers decreases the number of molecules absorbed at the receptor protein, providing near mid‐point as the location for minimum probability of error.

Fig. 11.

Probability of error versus distance curve for AND and OR Boolean logic when relays are moved away from transmitters

In Fig. 12, we provide the reliability versus distance plots, for various values of degradation constants. Reliability curves are for a single broken link, as shown in Fig. 4. The link impeded is the one with $D=40\times {10}^{-11}{\mathrm{m}}^2/s$, and the unbroken link is with $D=35\times {10}^{-11}{\mathrm{m}}^2/s$. The considered fluid medium has a drift of $20\mu \mathrm{m}/\mathrm{s}$. It can be seen from Fig. 12 that for OR Boolean gate communication circuit, disrupting communication link can lead to maximum reliability of 0.75 which decreases to a minimum value of 0.25 after a certain distance. On the other hand, for AND Boolean gate MC circuit reliability is at a constant value of 0.75. The given reliability analysis implies that physically AND gate communication circuit is less susceptible to the breaking of single link than the OR gate communication circuit. It is important to note that higher reliability of AND gate Boolean communication circuit is because of the inherent structure of the AND Boolean logic. Basically, the reliability analysis implies that effector protein corresponding to AND gate communication circuit will never be stimulated. However, given equiprobable source inputs, three out of four times AND gate incorporated effector protein does not need to be activated.

Fig. 12.

Reliability versus distance curve for AND and OR Boolean logic communication circuits

In Fig. 13, the upper and lower bounds of the capacity of AND and OR Boolean logic communication systems are illustrated. These bounds are numerically derived from (24), using the Blahut–Arimoto algorithm [46]. Total distance between the transmitters and the receivers is assumed to be $100\mu \mathrm{m}$. We vary the position of membrane, containing receptor proteins, from $10$ to $90\mu \mathrm{m}$ from the transmitters. The figure illustrates that maximum switching frequency is obtained when the separation of membrane is approximately equidistant from the transmitters as well as the receivers. The peculiar behaviour of the capacity can be attributed to the hitting probability. As the membrane distance from transmitter increases, hitting probability of disseminated molecules to the receptor protein decreases. On the other hand, increasing membrane distance from the effector proteins leads to low reception by the receivers present at the effector protein, owing to decrease in hitting probability. This renders near mid‐point as the best location, supporting communication with maximum switching frequency. Both the probability of error and the capacity appear to be at their best near $40\mu \mathrm{m}$ from the transmitters.

Fig. 13.

Upper and lower bounds for the capacity versus distance curve for AND and OR Boolean logic when relays are moved away from transmitters

7 Conclusions and applications

In this paper, we have provided the communication analysis of Boolean logic AND and OR for biochemical pathways. Probability of error for the Boolean communication circuit, with and without taking relays into consideration, has been analysed. We have derived the reliability of simple cellular signalling pathway, given a broken link. Further, upper and lower bounds of the capacity with the variation in relays’ position from the source pair have been calculated.

We believe our perusal to be of some use in examining biological signalling processes from communication perspective. Also our study may contribute to performance analysis of nanomachines mediated Boolean pathways, which in future may replace the corrupt parts of the cellular signalling. A few of the conjectured future directions are presented here. Improving safety/efficacy ratio of existing drugs is of extreme importance as it reduces monetary cost in developing efficient drugs and also reduces side‐effects of a given medicine. Development of target drug delivery which incorporates controlled release technology (CRT), i.e. emission of drug at a certain rate, forms an indispensable tool in meeting this end. The proposed capacity analysis can be of use if CRT is equipped with Boolean logic MC. For instance, in [47], DNA nanorobot for targeted drug delivery is proposed where the drug is locked in the nanorobot by two aptameer strands. When these aptameer strands come in contact with certain antigens, forming an AND gate, the robot releases drug inside the cell. Similar possibility to kill target cancer cell using AND logic gated‐drug delivery system with the miRNA‐based synthetic cell death device has been proposed in [48] and an OR logic gate for targeted drug delivery has been suggested in [49]. The proposed analysis of probability of error and the capacity/maximum rate achievable of Boolean logic embedded effector proteins can be used in such systems to study the probabilistic effect of fluid media in the mentioned drug delivery, using nanomachines. Another use of our investigation can be conjectured in signalling study of cascaded biological process, i.e. processes where an external signal turns on gene A, which then turns on gene B, and the sequence goes on [50]. There is some evidence that design principles of biological processes inherently lead to cascaded structure. This Boolean property of biological behaviour in general and cell signalling in particular can also be studied from communication point of view using our analysis. Further, providing a connection between reliability of Boolean communication link and diseases may lead to certain insights into systems and synthetic molecular biology.