Optimal Demodulation of Reaction Shift KeyingSignals in Diffusion-based Molecular Communication

Networks

Chun Tung ChouSchool of Computer Science and Engineering,

University of New South Wales,Sydney, NSW, Australia 2052

E-mail: ctchou@cse.unsw.edu.au

March 6, 2015

Abstract

In a diffusion-based molecular communication network, transmitters and receiverscommunicate by using signalling molecules (or ligands) in a fluid medium. This paperproposes a novel modulation mechanism for molecular communication called ReactionShift Keying (RSK). In RSK, the transmitter uses different chemical reactions togenerate different time-varying functions of concentration of signalling molecules torepresent different transmission symbols. We consider the problem of demodulatingthe RSK symbols assuming that the transmitter and receiver are synchronised. Weassume the receiver consists of receptors and signalling molecules may react with thesereceptors to form ligand-receptor complexes. We derive an optimal RSK demodulatorusing the continuous history of the number of complexes at the receiver as the input tothe demodulator. We do that by first deriving a communication model which includesthe chemical reactions in the transmitter, diffusion in the transmission medium andthe ligand-receptor process in the receiver. This model, which takes the form of acontinuous-time Markov process, captures the noise in the receiver signal due to thestochastic nature of chemical reactions and diffusion. We then adopt a maximumposterior framework and use Bayesian filtering to derive the optimal demodulatorfor RSK signals. We use numerical examples to illustrate the properties of the RSKdemodulator.

Keywords: Molecular communication networks; modulation; demodulation; maximum aposteriori; optimal detection; stochastic models; Bayesian filtering; molecular receivers.

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1 Introduction

Molecular communication is a promising approach to realise communications among nano-scale devices [1, 2, 3, 4]. There are many possible applications with these networks ofnano-devices, for example, in-body sensor networks for health monitoring and therapy[5, 3]. This paper considers diffusion-based molecular communication networks.

In a diffusion-based molecular communication network, transmitters and receivers com-municate by using signalling molecules or ligands. The transmitter uses different time-varying functions of concentration of signalling molecules (or emission patterns) to repre-sent different transmission symbols. The signalling molecules diffuse freely in the medium.When signalling molecules reach the receiver, they react with chemical species in the re-ceiver to produce output molecules. The counts of output molecules over time is thereceiver output signal which the receiver uses to decode the transmitted symbols.

Two components in diffusion-based molecular communication system are modulationand demodulation. A number of different modulation schemes have been considered in theliterature. For example, [6, 7] consider Concentration Shift Keying (CSK) where differ-ent concentrations of signalling molecules are used by the transmitter to represent differ-ent transmission symbols. Other modulation techniques that have been proposed includeMolecule Shift Keying (MSK) [8, 9], Pulse Position Modulation (PPM) [10] and FrequencyShift Keying (FSK) [11]. This paper considers a novel modulation scheme, called ReactionShift Keying (RSK), where different chemical reactions are used to generate the emissionpatterns of different symbols. The motivation to study RSK is that chemical reactionsare a natural way to produce signalling molecules. Note that the modulation schemesstudied in the earlier literature are deterministic in the sense that one symbol correspondsto exactly one emission pattern. However, due to stochastic nature of chemical reactions[12, 13], for RSK, it is possible for two chemical reactions to produce the same emissionpattern, though with different probabilities.

We assume the receiver consists of receptors. When the signalling molecules (ligands)reach the receiver, they can react with the receptors to form ligand-receptor complexes(which are the output molecules in this paper). We consider the problem of using thecontinuous-time history of the number of complexes for the demodulation of RSK symbolsassuming that the transmitter and receiver are synchronised. The ligand-receptor complexsignal in RSK is a stochastic process with three sources of noise because the chemicalreactions at the transmitter, the diffusion of signalling molecules and the ligand-receptorbinding process are all stochastic. We derive a continuous-time Markov process (CTMP)which models the chemical reactions at the transmitter, the diffusion in the medium and theligand-receptor binding process. By using this model and the theory of Bayesian filtering,we derive the maximum a posteriori (MAP) demodulator using the history of the numberof complexes as the input.

This paper makes two key contributions: (1) We propose a new modulation schemeRSK. (2) We derive a closed-form expression for the MAP demodulation filter for RSKsignals. The closed-form expression gives insight into the important elements needed foroptimal demodulation, these are the timings at which the receptor bindings occur, the

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number of unbound receptors and the mean concentration of signalling molecules aroundthe receptors.

The rest of the paper is organised as follows. Section 2 presents the system assumptions,as well as a mathematical model from the transmitter to the ligand-receptor complex signalbased on CTMP. We derive the MAP demodulator in Section 3 and illustrate its numericalproperties in Section 4. Section 5 discusses related work. Finally, Section 6 concludes thepaper.

2 End-to-end communication models

This paper considers diffusion-based molecular communication with one transmitter andone receiver in a fluid medium. Figure 1 gives an overview of the setup considered in thispaper. The transmitter uses RSK for modulation. The transmitter acts as the source andemitter of signalling molecules. The signalling molecules diffuses in the fluid medium. Thefront end of the receiver consists of a ligand-receptor binding process and the back-endconsists of the demodulator with the number of complexes as its input.

In this section, we first describe the system assumptions in Section 2.1. We thenpresent an end-to-end model which includes the transmitter, the transmission medium andthe ligand-receptor binding process in the receiver, see the dashed box in Figure 1. Theend-to-end model is a CTMP which includes chemical reactions in the transmitter, diffusionin the medium and the ligand-receptor binding process in the receiver. The presentationof the end-to-end model is divided into two parts. We first present a small example toillustrate the derivation of the CTMP in Section 2.2 and then present the general modelin Section 2.3.

2.1 Model assumptions

We assume that the medium (or space) is discretised into voxels while time is continuous.This modelling framework results in a reaction-diffusion master equation (RDME) [13, 14,15], which is a CTMP commonly used to model systems with both diffusion and reactions.In addition, we assume the communication uses only one type of signalling molecule (orligand) denoted by S. We divide the description of our model into three parts: transmissionmedium, transmitter and receiver. We begin with the transmission medium.

2.1.1 Transmission medium

We model the transmission medium as a three dimensional (3-D) space with dimensionsX ˆ Y ˆ Z, where X, Y and Z are integral multiples of length W . That is, there existpositive integers Nx, Ny and Nz such that X “ NxW and Y “ NyW , Z “ NzW . The 3-Dvolume can be partitioned into Nx ˆNy ˆNy cubic voxels of volume W 3. Figure 2 showsan arrangement with Nx “ Ny “ 4 and Nz “ 1.

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We refer to a voxel by a triple px, y, zq where x, y and z are integers or by a singleindex ξ P r1, NxNyNzs where ξpx, y, zq “ x`Nxpy´ 1q `NxNypz´ 1q. The indices for thevoxels are shown in Figure 2.

Diffusion is modelled by molecules moving from one voxel to a neighbouring voxel. Forexamples, in Figure 2, molecules can diffuse from Voxel 1 to Voxels 2 or 5, from Voxel 2to Voxels 1, 3 and 6, and so on. The diffusion of molecules between neighbouring voxels isindicated by the two-way arrows in Figure 2.

We assume that the signalling molecule S is the only diffusible chemical species in ourmodel and the diffusion coefficient for S is D. This means the signalling molecules diffusefrom one voxel to a neighbouring voxel at a mean rate of d where d “ D

W 2 . In other words,within an infinitesimal time ∆t, the probability that a signalling molecule diffuses to aneighbouring voxel is d ∆t.

Our model can capture standard boundary conditions such as reflecting and absorbingboundaries. For example, in Figure 2, we allow molecules to leave the medium via onesurface of Voxel 4 as indicated by the one-way arrow. Mathematically, this is modelled bya rate of leaving the medium, similar to that of modelling the diffusion between the voxels.

It has been shown in [16, 13] that in order for RDME to produce physically meaningfulresults, the voxel dimension W must be within a certain range. In this paper, we assumethat W comes from a valid range. The choice of W is beyond the scope of the paper andthe reader can refer to [16, 13] for further discussion.

For simplicity, we assume that the medium is homogeneous with a constant diffusion co-efficient D. It is straightforward to extend the framework to cover inhomogeneous medium[17]. It is also possible to use non-cubic voxels, see [18, 19].

2.1.2 Transmitter

We assume the transmitter occupies one voxel. However, it is straightforward to generaliseto the case where a transmitter occupies multiple voxels. In this paper, we limit ourconsideration to one symbol interval and assume that there is no inter-symbol interference,see Remark 1 at the end of Section 3.

We assume that the transmitter uses RSK, see Figure 1. It can send K different symbolss “ 0, 1, .., K´1 where each symbol s is characterised by an emission pattern usptq. The roleof emission pattern in molecular communication is the same as that of transmitted signalin electromagnetic communication. If a transmitter uses a deterministic emission patternusptq to represent symbol s, it means the transmitter emits usptq signalling molecules intothe transmitter voxel at time t. We use an example to illustrate the meaning of emissionpattern. Consider an emission pattern u1ptq for Symbol 1 where u1p1.2q “ u1p5.6q “ 1,u1p8.1q “ 2 and u1ptq is zero at all other times; this means, for Symbol 1, the transmitteremits one signalling molecule at times 1.2 and 5.6, two signalling molecules at time 8.1 anddoes not emit any molecules at any other times.

In this paper, we assume that the emission pattern for each symbol is produced by a setof chemical reactions located in the transmitter voxel. As an example, a class of chemical

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reactions inside living cells [20] is

RNAκÝÑ RNA` A (1)

where RNA (a molecule commonly found in living cells) produces the chemical species A.This class of chemical reactions can be modelled by a Poisson process where molecules ofA are produced at a mean rate of κ [21]. The important point to note is that the emissionpatterns produced by chemical reactions are not deterministic, but stochastic. The meanemission pattern of this chemical reaction is Eruptqs “ κ.

Following on from the above example, one can realise Amplitude Shift Keying (ASK)or CSK in molecular communication by using different chemical reactions that can pro-duce signalling molecules at different mean rates. For example, if there are four differentreactions that can produce signalling molecules at four different mean rates of κ0, κ1, κ2

and κ3, then one can use these four different reactions to produce 4 different symbols. Notethat it is possible for the four chemical reactions to produce the same emission pattern (orrealisation), though with different probabilities.

A standard result in physical chemistry shows that the dynamics of a set of chemicalreactions can be modelled by a CTMP [12]. Therefore, we will model the transmitter by aCTMP. Note that, in this paper, we will not specify the sets of chemical reactions used bythe transmitter except for simulation because the MAP demodulator does not explicitlydepend on the sets of chemical reactions that the transmitter uses.

2.1.3 Receiver

We assume the receiver occupies one voxel and we use R to denote the index of the voxelat which the receiver is located. In Figure 2, we assume the receiver is at Voxel 11 (lightgrey) and hence R “ 11 for this example. In addition, we assume that the transmitter andreceiver voxels are distinct.

We assume that the receiver has M receptors and we use E as the chemical name fora unbound receptor. These receptors are fixed in space and do not diffuse, and they areonly found in the receiver voxel. Furthermore, these receptors are assumed to be uniformlydistributed in the receiver voxel.

The receptor E can bind to a signalling molecule S to form a ligand-receptor complex(or complex for short) C, which is a molecule formed by combining E and S. This is knownas ligand-receptor binding in molecular biology literature [22]. The binding reaction canbe written as the chemical equation:

S` EλÝÑ C (2)

where λ is the kinetic constant of this reaction. The reaction rate of the binding reactionis the product of λ, the number of signalling molecules in the receiver voxel and the thenumber of unbound receptors.

The ligand-receptor complex C can dissociate into unbound receptor E and signallingmolecule. This can be represented by the chemical equation

CµÝÑ E` S (3)

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where µ is the dissociation rate constant. The reaction rate of the unbinding reaction isthe product of µ and the number of complexes.

Since a receptor can either be in a unbound state E or in a complex C, we havethe following conservation relation: the number of unbound receptors plus the number ofcomplexes is equal to the total number of receptors M .

2.2 Example end-to-end model

In order to derive the MAP demodulator for RSK, we need an end-to-end model whichincludes the transmitter, the medium and the ligand-binding process, see Figure 1. Sincechemical reactions (which includes the chemical reactions in the transmitter as well as theligand-receptor binding process in the receiver) and diffusion can be modelled by CTMP,it is possible to use a CTMP as an end-to-end model. In this section, we will present anexample CTMP which models a set of chemical reactions at the transmitter, the diffusion ofsignalling molecules and the ligand-receptor binding process in the receiver. This examplewill help us to explain the general end-to-end model to be presented in Section 2.3. Anexcellent tutorial introduction to the modelling of chemical reactions and diffusion by usingCTMP can be found in [23].

The aim of the end-to-end model is to determine the properties of the receiver signalfrom the transmitter signal. The receiver signal in our case is the number of complexes overtime. Since the transmitter uses RSK with K symbols, the transmitter signal is generatedby one of the K sets of chemical reactions. This means that we need K end-to-end modelswith a model for each of the K symbols or sets of chemical reactions. The principle behindbuilding these K models are identical so without loss of generality, we will assume thatthe example here is for Symbol 0.

For this example, we assume the transmission medium consists of 3 voxels as illustratedin Figure 3. The transmitter and receiver are assumed to be located in, respectively, Voxels1 and 3. We assume reflecting boundary condition which means the signalling moleculescannot leave the medium.

For this end-to-end model, the transmitter is assumed to send Symbol 0 which meansit uses the set of chemical reactions corresponding to this symbol. We therefore view atransmitter as a set of chemical reactions located within the transmitter voxel. It is stillan open problem what chemical reactions are good for communication performance. Theexample being used here is not meant to promote the use of a particular set of chemicalreactions but our purpose is to show how a set of chemical reactions can be modelled by aCTMP.

For this example, we assume that the production of the signalling molecules S requirestwo intermediate chemical species F and G, which are produced by RNA1 and RNA2.There are four reactions and they are assumed to take place within the transmitter voxel

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only. The four chemical reactions are:

RNA1k1ÝÑ RNA1 ` F (4)

RNA2k2ÝÑ RNA2 `G (5)

Fk3ÝÑ S (6)

S`Gk4ÝÑ φ (7)

Reaction (4) says that the molecules of F are produced at a mean rate of k1. Similarly,according to Reaction (5), G is produced at a mean rate of k2. Reaction (6) says thatF is converted to S at a mean rate equals to k3 times the number of F molecules inthe transmitter voxel. Reaction (7) say that S and G can react to produce a moleculeφ that we are not interested to keep track of in the mathematical model. If an S (or aG) molecule takes part in Reaction (7), we can consider this S (G) molecule has left thesystem permanently after the reaction. The rate of Reaction (7) is k4 times the number ofG molecules and the number of signalling molecules S in the transmitter voxel.

We assume that the chemical species RNA1, RNA2, F and G are found in the trans-mitter voxel only, and they cannot leave the transmitter voxel. This means that we donot need to consider the diffusion of these chemical species. The only diffusible chemicalspecies in the entire system is the signalling molecule S. We also assume that there is onlyone of each RNA1 and RNA2 and their counts remain constant.

In order to define the state of the system, we make the following definitions: niptq is thenumber of signalling molecules in Voxel i at time t, nF ptq and nGptq are respectively thenumber of F and G molecules at time t, nAptq is the cumulative number of molecules thathave left the system at time t and bptq is the number of complexes (or bound receptors)at time t. Since a receptor can either be unbound or in a complex, the number of unbounreceptors at time t is M´bptq; therefore, the mathematical model only has to keep track ofeither the number of unbound receptors or the number of complexes, and we have chosento keep track of the latter. The state of the system is completely specified by these sevenmolecular counts: n1ptq, n2ptq, n3ptq, nF ptq, nGptq, nAptq and bptq. All the molecular countsshould be non-negative integers (i.e. belonging to the set Zě0) and a further restriction isthat 0 ď bptq ďM or we write bptq P Zr0,Ms.

We define the vector Nptq as

Nptq ““

n1ptq n2ptq n3ptq nF ptq nGptq nAptq‰T

(8)

where the superscript T is used to denote matrix transpose.Based on the definition of Nptq, the state of the system is the tuple pNptq, bptqq and

a valid state must be an element of the set S “ Z6ě0 ˆ Zr0,Ms. The state of the system

changes when a reaction or diffusion event occurs. Our modelling assumptions mean thatreactions can only take place in the transmitter or the receiver voxels. The reactions inthe transmitter voxel are (4)´(7). The reactions taking place in the receiver voxel are(2) and (3). The only diffusible chemical species in this system is the signalling molecule

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S. Within an infinitesimal time ∆t, at most one diffusion or reaction event can occur.Therefore, the dynamics of the system can be specified by the transition probability fromstate pNptq, bptqq to pNpt`∆tq, bpt`∆tqq. We will now specify these transition probabilitiesand we begin with the transmitter.

Four possible reaction events (4)–(7) can take place in the transmitter voxel. An oc-currence of Reaction (4) increases the number of F molecules in the transmitter voxel by1 and this occurs at a mean rate of k1. By defining 1i to be the standard basis vector witha ‘1’ at the i-th position, we can write the state transition probability due to Reaction (4)as:

PrNpt`∆tq “ Nptq ` 14, bpt`∆tq “ bptq|Nptq, bptqs “ k1 ∆t (9)

Note that we have used 14 because nF ptq is increased by 1 if Reaction (4) occurs and nF ptqis the fourth element of Nptq in the definition of Nptq in (8). The right-hand side (RHS)of Equation (9) is the transition probability that Reaction (4) occurs in pt, t`∆tq, whichis given by the reaction rate k1 times ∆t.

We can write the transition probabilities due to Reactions (5)´(7) as:

PrNpt`∆tq “ Nptq ` 15, bpt`∆tq “ bptq|Nptq, bptqs “ k2 ∆t (10)

PrNpt`∆tq “ Nptq ´ 14 ` 11, bpt`∆tq “ bptq|Nptq, bptqs “ k3 nF ptq ∆t (11)

PrNpt`∆tq “ Nptq ´ 15 ´ 11 ` 216, bpt`∆tq “ bptq|Nptq, bptqs “ k4 nGptq n1ptq∆t(12)

The rationale behind Equation (10) is similar to that of (9). Equation (11) models Reaction(6). If Reaction (6) occurs, an F molecule is converted to an S molecule, so the number ofF molecules nF ptq (which is the fourth element of Nptq) is decreased by 1 and the numberof signalling molecule in the transmitter voxel n1ptq (which is the first element of Nptq)is increased by 1; this change in the number of molecules as a result of Reaction (6) canbe written as Npt ` ∆tq “ Nptq ´ 14 ` 11 in (11). Equation (12) models Reaction (7).When Reaction (7) occurs, a G and an S molecule in the transmitter are consumed, hence´15 ´ 11 in (12). We are not interested to keep track of the molecules as a result of thisreaction, we consider these two molecules have left the system permanently and add ‘2’ tonAptq which is at the sixth position of Nptq. The letter ‘A’ here comes from ’absorbing’because once a molecule is added to nAptq, it will not leave. Note that the RHSs of (9)–(12)show the transition probabilities and they are of the form of the transition rate times ∆t.

The state of the system can also be changed by signalling molecules diffusing from onevoxel to another. For this example, there are four possible diffusion events, which takeplace when a signalling molecule diffuses from a voxel to its neighbouring voxel. The fourdiffusion events are: from Voxel 1 to Voxel 2, from Voxel 2 to Voxel 1, from Voxel 2 to

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Voxel 3, and from Voxel 3 to Voxel 2. The transition probabilities of these four events are:

PrNpt`∆tq “ Nptq ´ 11 ` 12, bpt`∆tq “ bptq|Nptq, bptqs “ d n1ptq ∆t (13)

PrNpt`∆tq “ Nptq ` 11 ´ 12, bpt`∆tq “ bptq|Nptq, bptqs “ d n2ptq ∆t (14)

PrNpt`∆tq “ Nptq ´ 12 ` 13, bpt`∆tq “ bptq|Nptq, bptqs “ d n2ptq ∆t (15)

PrNpt`∆tq “ Nptq ` 12 ´ 13, bpt`∆tq “ bptq|Nptq, bptqs “ d n3ptq ∆t (16)

Equation (13) is the probability that a signalling molecules diffuses from Voxel 1 to Voxel 2.The occurrence of this event means the number of signalling molecules in Voxel 1 (“ n1ptq,which is the first element of Nptq) is decreased by 1 while the number of signalling moleculesin Voxel 2 (“ n2ptq, which is the second element of Nptq) is increased by 1. The probabilityof this occurring is d ∆t. The explanation for the other three transition probabilities aresimilar.

The last category of state transitions occurs when a receptor is bound or unboundaccording to chemical reactions (2) and (3). The state transition probabilities are:

PrNpt`∆tq “ Nptq ´ 13, bpt`∆tq “ bptq ` 1|Nptq, bptqs “ λ n3ptq pM ´ bptqq ∆t (17)

PrNpt`∆tq “ Nptq ` 13, bpt`∆tq “ bptq ´ 1|Nptq, bptqs “ µ bptq ∆t (18)

Equation (17) is the transition probability for receptor binding or the formation of newcomplex. This event occurs when a signalling molecule in the receiver voxel reacts with aunbound receptor to form a complex. As a result of this reaction, the number of signallingmolecules in the receiver voxel (which is Voxel 3 in this example) is decreased by 1 and thenumber of complexes bptq is increased by 1. The rate of this event is proportional to theproduct of the number of signalling molecules in the receiver voxel n3ptq and the number ofunbound receptors pM ´ bptqq. Equation (18) is the transition probability for a receptor tounbind. The unbinding reaction causes the number of signalling molecules in the receivervoxel n3ptq to increase by 1 while the number of complexes bptq to decrease by 1. The rateof this reaction is proportional to number of complexes bptq.

Equations (9) to (18) give the transition probabilities of the possible events that canoccur when the state of the system is pNptq, bptqq. It is possible that no transitions occurs inthe time interval pt, t`∆tq, the probability of this occurring is given by the complementaryto that of an event occurring, that is, one minus the sum of the RHSs of Equations (9) to(18).

Equations (9) to (18) hold for any valid state pNptq, bptqq P S. If we collect all thetransition probability equations for all valid states, then we can form the infinitesimalgenerator of the CTMP. For a given initial probability distribution of the initial statepNp0q, bp0qq, one can in principle solve the first order ordinary differential equation (ODE)associated with the infinitesimal generator to compute the probability of the number ofcomplexes bptq, or the property of the receiver signal. However, in practice, this ODE isof a very high dimension and it is an active area of research to derive algorithms to solvethis ODE efficiently and accurately [24]. We remark that this ODE is commonly knownas the reaction-diffusion Master equation [23, 15, 14] because it describes the dynamics of

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chemical reactions and diffusion. For this paper, it suffices for us to use the equations ofthe form in (9) to (18), therefore, we will not present the Master equation.

2.3 General end-to-end model

In this section we present a general end-to-end model that includes the transmitter, dif-fusion and the ligand-receptor process in the receiver. The model presented here is ageneralisation of the example presented earlier. We assume for the time being that theend-to-end model is for Symbol 0. We begin with a few definitions.

Let Nv “ NxNyNz be the number of voxels and niptq (where 1 ď i ď Nv) be the numberof signalling molecules S in Voxel i at time t. The transmitter is a set of chemical reactionswhich uses H intermediate chemical species Q1, Q2, ... and QH . In the earlier example,H “ 2 and the intermediate species are F and G. Let nQi

ptq be the number of chemicalspecies Qi in the transmitter voxel at time t. As in the example earlier, we use nAptq todenote the number of molecules that have left the system.

We define the vector Nptq P ZNv`H`1ě0 to be:

Nptq ““

n1ptq ... nNvptq nQ1ptq ... nQHptq nAptq

‰T(19)

The state of the general model is the tuple pNptq, bptqq where bptq is the number ofcomplexes (or bound receptors) and Nptq contains all the other molecular counts. We willnow specify the transition probabilities of the general model. From the earlier example,we can divide the transition probabilities from pNptq, bptqq to pNpt ` ∆tq, bpt ` ∆tqq into2 groups depending on whether the number of complexes has changed or not in the timeinterval pt, t ` ∆tq. If the number of complexes has changed from time t to t ` ∆t, i.e.bpt `∆tq ‰ bptq, this means either a binding reaction (2) or a unbinding reaction (3) hasoccurred. Analogous to Equations (17) and (18), we have the state transition probabilitiesare:

PrNpt`∆tq “ Nptq ´ 1R, bpt`∆tq “ bptq ` 1|Nptq, bptqs “ λ nRptq pM ´ bptqq ∆t (20)

PrNpt`∆tq “ Nptq ` 1R, bpt`∆tq “ bptq ´ 1|Nptq, bptqs “ µ bptq ∆t (21)

where nRptq is the number of signalling molecules in the voxel with index R or the receivervoxel. (Recalling that we use R to denote the index for the receiver voxel, see Section2.1.3.) In the earlier example, the receiver is at Voxel 3, note that Equations (20) and (21)become Equations (17) and (18) if we put R “ 3.

We now specify the second group of transition probabilities with bpt ` ∆tq “ bptq.These transitions are caused by either a reaction in the transmitter or diffusion of signallingmolecules between neighbouring voxels. Let ηi, ηj P ZNv`H`1

ě0 be two valid Nptq vectors; letalso β P Zr0,Ms. For ηi ‰ ηj, we write

PrNpt`∆tq “ ηi, bpt`∆tq “ β|Nptq “ ηj, bptq “ βs “ dij ∆t (22)

where dij is the transition rate from state pηj, βq to state pηi, βq. Since this transition isdue to either a reaction in the transmitter or diffusion, dij is independent of the number

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of complexes β. Depending on the type of transition, the value of dij can depend on thereaction constants in the transmitter, diffusion rate and some states of ηj. For example, ifthe transition from ηj to ηi is caused by the diffusion of a signalling molecule from Voxel 1to Voxel 2, we have ηi “ ηj´11`12 at a rate of dηj,1 where ηj,1 is the first element in ηj orequivalently the number of signalling molecules in Voxel 1 in state ηj; so, for this example,dij “ dηj,1. This example can be compared to Equation (13) in Section 2.2. The readercan verify that we can use Equation (22) to cover Equations (9) to (16) in the examplein Section 2.2. The main advantage of using Equation (22) is that it allows us a cleanerabstraction to solve the Bayesian filtering problem when deriving the MAP demodulator.We also remark that we will not specify the exact expression of dij because dij’s do notappear explicitly in the demodulator.

Equations (20), (21) and (22) specify all the possible state transitions. The probabilityof no state transition is therefore:

PrNpt`∆tq “ ηj, bpt`∆tq “ bptq|Nptq “ ηj, bptqs “1´ djj ∆t´ λ nRptq pM ´ bptqq ∆t´ µ bptq ∆t(23)

where

djj “ÿ

j‰i

dij (24)

We have now specified all the state transition probabilities for Symbol 0. If a differentsymbol is used, the value of H, the dimension of Nptq and the dij parameters can change.However, the state transition probabilities still can be summarised by Equations of the form(20), (21) and (22). In any case, the derivation of the MAP demodulator only requiresus to work with one symbol at a time. Hence, we will use Equations (20)-(24) for anytransmission symbol.

Finally, note that the CTMP includes all the three sources of noise in our system, dueto chemical reactions in the transmitter, random diffusive movements in the medium andthe ligand-receptor binding process at the receiver.

3 The MAP demodulator

This section aims to derive the optimal demodulator for RSK signals. We assume thatthe demodulator uses the continuous-time signal bptq, which is the number of complexesat time t, as its input. Let Bptq “ tbpτq : 0 ď τ ď tu denote the continuous history of thenumber of complexes up till time t. The demodulation problem is to use the history Bptqto determine which symbol the transmitter has sent.

There are a number of reasons why we choose to work with the continuous-time signalbptq, rather than its sampled version. First, the signal bptq may not be strictly bandlimited in the frequency domain. Second, our results show that the optimal demodulatorneeds to know the time instances at which a receptor is switching from the unbound to

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bound state. This timing information, which is essentially an impulse, is unfortunatelylost by sampling bptq. Third, the solution of the proposed decoding problem can be usedto benchmark molecular circuit [25] based decoders. Since molecular circuits use chemicalreactions for computation, they are fundamentally analogue circuits. Fourth, there is anincreasing interest in the circuit design community to design low-power analogue signalprocessing circuits [26].

3.1 The MAP framework

We adopt a MAP framework for detection. Let Prs|Bptqs denote the posteriori probabilitythat symbol s has been sent given the history Bptq. If the demodulation decision is to bedone at time t, then the demodulator decides that symbol s has been sent if

s “ arg maxs“0,...,K´1Prs|Bptqs (25)

Instead of working with Prs|Bptqs, we will work with its logarithm. Let

Lsptq “ logpPrs|Bptqsq (26)

The first step is to determine Lspt`∆tq from Lsptq. By using Bayes’ rule and Bpt`∆tq “Bptq Y tbpt`∆tqu, it can be shown that

Lspt`∆tq “ Lsptq ` logpPrbpt`∆tq|s,Bptqsq ´ logpPrbpt`∆tq|Bptqsq (27)

where Prbpt`∆tq|s,Bptqs is the probability that there are bpt`∆tq complexes given thatthe transmitter has sent the symbol s and the previous history Bptq. The last term on theRHS of (27), i.e. Prbpt ` ∆tq|Bptqs, is independent of the transmission symbol so we donot need it for the purpose of detection. We will focus on determining Prbpt`∆tq|s,Bptqs.

3.2 Computing Prbpt`∆tq|s,BptqsThe problem of determining the probability Prbpt ` ∆tq|s,Bptqs is essentially a Bayesianfiltering or hidden Markov model problem. Recall that the complete state of the system ispNptq, bptqq and the receiver can only observe bptq, therefore the task of the receiver is touse the history Bptq and the system model to do prediction. Standard method can be usedto derive Prbpt`∆tq|s,Bptqs but the derivation is long, especially because of the diffusionterms; the derivation can be found in Appendix A. The result is

Prbpt`∆tq|s,Bptqs “δpbpt`∆tq “ bptq ` 1qλpM ´ bptqq ∆t ErnRptq|s,Bptqs`δpbpt`∆tq “ bptq ´ 1qµbptq ∆t `

δpbpt`∆tq “ bptqqp1´ λpM ´ bptqqErnRptq|s,Bptqs ∆t´ µbptq ∆tq(28)

where δp.q is the Kronecker delta function. Note that only one of the three terms on theRHS of Equation (28) is non-zero depending on whether the observed bpt`∆tq is one more,

12

one less or equal to that of bptq; or, in other words, whether the number of complexes hasincreased by one, decreased by one or stayed the same. The term ErnRptq|s,Bptqs is theexpected number of signalling molecules in the receiver voxel given the history and thesymbol s. The meaning of this term is that the receiver uses the history to predict whatthe expected number of signalling molecules in the receiver voxel is. Note that only thechemical kinetic parameters λ and µ of the receptor appear explicitly in Equation (28).Other parameters, such as the set of chemical reaction that generate Symbol s and thediffusion coefficient, do not appear explicitly in Equation (28) but influence the systembehaviour via the term ErnRptq|s,Bptqs.

3.3 The demodulation filter

By substituting Equation (28) into Equation (27) and let ∆t go to zero, we show inAppendix B that

dLsptq

dt“dUptq

dtlogpErnRptq|s,Bptqsq ´ λpM ´ bptqqErnRptq|s,Bptqs ` Lptq (29)

wtih Lsp0q is initialised to the logarithm of the prior probability that Symbol s is sent. Theterm Uptq is the cumulative number of times that the receptors have turned from unboundto bound state. The meaning of Uptq is illustrated in Figure 4 assuming that there aretwo receptors. The top two pictures in Figure 4 show the state transitions for the tworeceptors. The third picture shows the function Uptq which is increased by one every time

a receptor switches from unbound to bound state. The bottom picture shows dUptqdt

which

is the derivative of Uptq. Note that dUptqdt

consists of a train of impulses (or Dirac deltas)where the timings of the impulses are the times at which a receptor binding event occurs.Loosely speaking, one may also view dUptq

dtas maxpdbptq

dt, 0q.

The function Lptq, which is the last term on the RHS of (29), contains all the termsthat are independent of Symbol s. Since Lsptq does not appear on the RHS of (29), thismeans that Lptq adds the same contribution to all Lsptq for all s “ 0, ..., K ´ 1. We cantherefore ignore Lptq for the purpose of demodulation.

The term ErnRptq|s,Bptqs in Equation (29) is the prediction of the mean number ofsignalling molecules in the receiver voxel using the history of receptor state. This is a fil-tering problem which requires extensive computation. Instead, we assume that the receiverhas prior knowledge that if Symbol s is transmitted, then the mean number of signallingmolecules in the receiver voxel is σsptq and the receiver uses σsptq for demodulation. Wecan view σsptq as internal models that the demodulator uses. The use of internal modelsis fairly common in signal processing and communication, e.g. a matched filter correlatesthe measured data with an expected response.

After making the modifications described in the last two paragraphs, we are now readyto describe the demodulator. Using bptq as the input, the demodulator runs the followingK continuous-time filters in parallel:

dZsptq

dt“dUptq

dtlogpσsptqq ´ λpM ´ bptqqσsptq (30)

13

where Zsp0q is initialised to the logarithm of the prior probability that the transmittersends Symbol s. If the demodulator makes the decision at time t, then the demodulatordecides that Symbol s has been transmitted if

s “ arg maxs“0,...,K´1Zsptq (31)

The demodulator structure is illustrated in Figure 5. By comparing Equations (29) and(30), it can be shown that Ls1ptq ´ Ls2ptq “ Zs1ptq ´ Zs2ptq for any two symbols s1 ands2. An interpretation of modulation filter output Zsptq is that exppZsptqq is proportionalto the posteriori probability Prs|Bptqs.

We see from Equation (30) that the calculation of the demodulator output requiresa number of pieces of information. For the calculation of the first term on the RHS ofEquation (30), it needs to know the time instances at which the receptor bindings occurand this timing is used to determine a contribution from the weighting function logpσsptqq.The second term on the RHS of Equation (30) requires the number of unbound receptorsat time t as well as the weighting function σsptq.

In order to understand Equation (30), we consider the situation where Symbol 1 gen-erates a lot more signalling molecules than Symbol 0 such that it results in more signallingmolecules in the receiver voxel, or σ1ptq ą σ0ptq for all t. If the transmitter sends Symbol 1,then more signalling molecules are expected to reach the receiver voxel. The consequence isthat there are more receptor binding events and the number of unbound receptors pM´bptqqis smaller. Therefore, in Equation (30), we expect a big positive contribution from the firstterm on the RHS and a small negative contribution from the second term. The net effect isa big Z1ptq. On the other hand, if the transmitter sends Symbol 0, the number of receptorbinding events is smaller and pM ´ bptqq is big. This results in a smaller Z0ptq. Therefore,Z1ptq is likely to be bigger than Z0ptq, which means correct detection.

Remark 1 It is in principle possible to use the demodulation filters (30) to deal with thecase with inter-symbol interference. Let us assume the transmitter uses K “ 2 symbolss “ 0, 1 and a symbol interval is Tx. Let σsptq (s “ 0, 1) be the mean number of signallingmolecules at the receiver voxel in the absence of inter-symbol interference. For simplicity,we consider decoding over two symbol intervals and the aim is to determine whether thesymbols sent are 00, 01, 10 and 11. This can be done by using four modulation filterswith internal models of the form σs1ptq ` σs2pt ´ Txq for s1, s2 “ 0, 1. However, this is aninefficient method due to exponential increase in the number of internal models. We willleave this problem as future work.

4 Properties of the demodulator

The aim of this section is to study the properties of the MAP demodulator numerically.We begin with the methodology.

14

4.1 Methodology

We consider a medium of 2µm ˆ 2µm ˆ 1 µm. We assume a voxel size of (13µm)3 (i.e.

W “ 13µm), creating an array of 6ˆ6ˆ3 voxels. The transmitter and receiver are located

at (0.5,0.8,0.5) and (1.5,0.8,0.5) (in µm) in the medium. The voxel co-ordinates are (2,3,2)and (5,3,2) respectively.

We assume the diffusion coefficient D of the medium is 1 µm2s´1. The receptor param-eters are λ “ 0.005

W 3 µm s´1 and µ “ 1 s´1. These values are similar to those used in [27]and are realistic for biological systems. We assume an absorbing boundary for the mediumand the signalling molecules escape from a boundary voxel surface at a rate of d

50. The

above parameter values will be used for all the numerical experiments.For each experiment, the transmitter uses either K “ 2 or K “ 3 symbols. Each

symbol is generated by a different sets of chemical reactions. Different experiments mayuse different sets of chemical reactions and will be described later. The number of receptorsalso varies between the experiments.

We use the Stochastic Simulation Algorithm (SSA) [28] to obtain realisations of bptqwhich is the number of complexes over time. SSA is a standard algorithm in chemistry tosimulate diffusion and reactions; it is essentially an algorithm to simulate a CTMP.

In order to use Equation (30), we require the mean number of signalling molecules σsptqin the receiver voxel when Symbol s is sent. Unfortunately, it is not possible to analyticallycompute σsptq from the CTMP because of moment closure problem which arises when thetransition rate is a non-linear function of the state [29]. We therefore resort to simulationto estimate σsptq. Each time when we need an σsptq, we run SSA simulation 500 times andaverage the results to obtain σsptq. Note that these simulations are different from thosethat we use to generate bptq for the performance study. In other words, the simulations forestimating σsptq and for performance study are completely independent.

Once bptq and σsptq are obtained, we use numerical integration to calculate Zsptq usingEquation (30). We assume that all symbols appear with equal probability, so we initialiseZsp0q “ 0 for all s.

4.2 Properties of the demodulator output

For this experiment, we use K “ 2 symbols and M “ 50 receptors. Both Symbols 0 and 1use a reaction of the form:

RNAκÝÑ RNA` S (32)

where the κ’s for Symbols 0 and 1 causes, respectively, 40 and 80 molecules to be generatedper second on average by the transmitter. The simulation time is about 3 seconds.

Figure 6 shows the demodulation filter outputs Z0ptq and Z1ptq if the transmitter sendsa Symbol 0. It can be seen that Z0ptq ą Z1ptq most of the time after t “ 1.2, which meansthe detection is likely to be correct after this time. The sawtooth like appearance of Z0ptqand Z1ptq is due to the fact that every time when a receptor is bound, there is a jump in

15

the filter output according to Equation (30). Figure 7 shows the filter outputs Z0ptq andZ1ptq if the transmitter sends a Symbol 1; the behaviour is similar.

Figure 8 shows the mean filter outputs Z0ptq and Z1ptq if the transmitter sends a Symbol0. The mean is computed over 200 realisations of bptq. It can be seen that the mean filteroutput of Z0ptq is greater than that of Z1ptq. Similarly, if Symbol 1 is sent, then we expectof the mean of Z1ptq to be bigger. The figure is not shown for brevity.

Figure 9 shows the mean symbol error rates (SERs) for Symbols 0 and 1 if the detectionis done at time t. The SER for Symbol 1 is high initially but as more information isprocessed over time, the SER drops to a low value. This experiment shows that it ispossible to use the analogue demodulation filter (30) to compute a quantity that allows usto distinguish between two emission patterns at the receiver.

4.3 Impact of number of receptors

We continue with the setting of 4.2 but we vary the number of receptors between 1 and 20.We assume the demodulator makes the decision at t “ 2.5 and calculate the mean SERfor both symbols at t “ 2.5. Figure 10 plots the SERs versus the number of receptors. Itcan be seen that the SER drops with increasing number of receptors.

We have used K “ 2 symbols so far. We retain the current Symbols 0 and 1, and adda Symbol 2 which is also of the form of Reaction (32) but its mean rate of production ofsignalling molecules is 3 times that of Symbol 0. We vary the number of receptors between1 and 50. We consider SER at t “ 2.5. Figure 11 plots the SERs of the three symbols fordifferent number of receptors. It can be seen that the SER drops with increasing numberof receptors.

4.4 Distinguishability of different chemical reactions

Equation (30) suggests that if the transmitter uses two sets of reactions which have almostthe same mean number of signalling molecules in the receiver voxel, then it may be difficultto distinguish between these two symbols. In this study, Symbol 0 is generated by Reaction(32) with a rate of κ while Symbol 1 is generated by:

rRNAsON ÐÑ rRNAsOFF (33)

rRNAsON

2κÝÑ rRNAsON ` S (34)

where we assume that RNA can be in an ON or OFF state, and signalling molecules S areonly produced when the RNA is in the ON-state. We assume that the there is an equalprobability for the RNA to be in the two states and the reaction rate constant for theproduction of signalling molecule S from rRNAsON is 2κ. This means that the mean rateof production of signalling molecules S by Symbols 0 and 1 are the same. This gives riseto very similar σ0ptq and σ1ptq. Figure 12 shows the demodulation filter outputs Z0ptq andZ1ptq for one simulation. It can be seen that the two outputs are almost indistinguishable.Consequently, the SER is pretty high. This shows that symbols generated by reactions

16

which have similar mean number of signalling molecules at the receiver voxel can be hardto distinguish.

5 Related work

There is a growing interest to understand molecular communication from the communica-tion engineering point of view. For recent surveys of the field, see [1, 2, 3, 4].

This paper differs from earlier work on diffusion based molecular communications in twomain aspects: modulation and demodulation methods. A number of different modulationschemes have been proposed in the literature. The novelty of RSK has already beendiscussed in Section 1 and will not be repeated here.

Demodulation methods for diffusion based molecular communication have been stud-ied in [30, 31]. Both papers also use the MAP framework with discrete-time samples ofthe number of output molecules as the input to the demodulator. Instead, in this paper,we consider demodulation using continuous-time history of the number complexes. Thedemodulation from ligand-receptor signal has also been considered in [32]. The key differ-ence is that [32] uses a linear approximation of the ligand-receptor process while we use anon-linear reaction rate.

The capacity of molecular communications based on ligand-receptor binding has beenstudied in [33, 34] assuming discrete samples of the number of complexes are available.A recent work [35] considers the capacity of such systems in the continuous-time limit.Instead of focusing on the capacity, our work focuses on demodulation.

Receiver design is an important topic in molecular communication and has been studiedin many papers, some examples are [36, 30, 37, 31, 38]. These papers either use onesample or a number of discrete samples on the count of a specific molecule to compute thelikelihood of observing a certain input symbols. This paper takes a different approach anduses continuous-time signals.

Another approach of receiver design for molecular communication is to derive molecularcircuits that can be used for decoding. An attempt was made in [11] to design a molecularcircuit that can decode frequency-modulated signals. However, the work does not takediffusion and reaction noise into consideration. A recent work in [39] analysed end-to-end molecular communication biological circuits from linear time-invariant system pointof view. The work in [25] compares the information theoretic capacity of a number ofdifferent types of linear molecular circuits. This paper differs from the previous work inthat it uses a non-linear ligand-receptor binding model.

The end-to-end model used in this paper is based on the RDME framework. RDMEtype of models have also been used to model molecular communication in [40, 41]. Analternative end-to-end model is based on tracking the particle dynamics of the molecules,see [42, 43].

The noise property of ligand-receptor for molecular communication has been charac-terised in [43]. The case for non-linear ligand-receptor binding does not appear to have ananalytical solution and [43] derives an approximate characterisation using a linear reaction

17

rate assuming that the number of signalling molecules around the receptor is large. Thispaper uses a non-linear ligand-receptor binding model and no approximation is used insolving the filtering problem.

The results of this paper may also be of interest to biologists who are interested tounderstand how living cells can distinguish between different concentration levels. Theresult of this paper can be viewed as a generalisation of [44] which studies how cells candistinguish between two constant levels of ligand concentration.

6 Conclusions and future work

This paper studies a diffusion based molecular communication network that uses differentsets of chemical reactions to represent different transmission symbols. We focus on thedemodulation problem. We assume the receiver uses a ligand-receptor binding process anduses the continuous history of the number of ligand-receptor complexes over time as theinput signal to the demodulator. We derive the maximum a posteriori demodulator bysolving a Bayesian filtering problem.

A Proof of Equation (28)

Let s denote the transmitted symbol, our aim is to determine Prbpt`∆tq|s,Bptqs in termsof the quantity at time t. Recalling that pNptq, bptqq is the state of the CTMP and sinceonly Bptq is observed, the problem of predicting bpt`∆tq from Bptq is a Bayesian filteringor hidden Markov model problem. The first step is to condition on the state of the system,as follows:

Prbpt`∆tq|s,Bptqs (35)

“ÿ

i

PrNpt`∆tq “ ηi, bpt`∆tq|s,Bptqs (36)

“ÿ

i

ÿ

j

PrNpt`∆tq “ ηi, bpt`∆tq|s,Nptq “ ηj,BptqsPrNptq “ ηj|s,Bptqs (37)

“ÿ

i

ÿ

j

PrNpt`∆tq “ ηi, bpt`∆tq|s,Nptq “ ηj, bptqsPrNptq “ ηj|s,Bptqs (38)

where we have used the Markov property PrNpt`∆tq “ ηi, bpt`∆tq|s,Nptq “ ηj,Bptqs “PrNpt`∆tq “ ηi, bpt`∆tq|s,Nptq “ ηj, bptqs to arrive at Equation (38).

We now focus on the term PrNpt `∆tq “ ηi, bpt `∆tq|s,Nptq “ ηj, bptqs in Equation(38). This term is the state transition probability. Using the CTMP in Section 2, we have

PrNpt`∆tq “ ηi, bpt`∆tq|s,Nptq “ ηj, bptqs (39)

“δpbpt`∆tq “ bptq ` 1qP1 ` δpbpt`∆tq “ bptq ´ 1qP2 ` δpbpt`∆tq “ bptqqP3

18

where

P1 “ δpηi “ ηj ´ 1Rqληj,RpM ´ bptqq ∆t (40)

P2 “ δpηi “ ηj ` 1Rqµbptq ∆t (41)

P3 “ ηiÑj (42)

where ηj,R is the R-th element of ηj, i.e. there are ηj,R signalling molecules in the receivervoxel, and

ηiÑj “

"

dij ∆t if i ‰ j1´ pληj,RpM ´ bptqq ´ µbptq ´ djjq ∆t if i “ j

(43)

where

djj “ÿ

i‰j

dij (44)

By substituting Equation (40) into Equation (38), we have

Prbpt`∆tq|s,Bptqs “ δpbpt`∆tq “ bptq ` 1qQ1 ` δpbpt`∆tq “ bptq ´ 1qQ2 ` δpbpt`∆tq “ bptqqQ3

(45)

where

Q` “ÿ

i

ÿ

j

P`PrNptq “ ηj|s,Bptqs (46)

We will now determine Q1, Q2 and Q3.For Q1, we have

Q1 “ÿ

i

ÿ

j

δpηi “ ηj ´ 1Rqληj,RpM ´ bptqq ∆t PrNptq “ ηj|s,Bptqs

“ λpM ´ bptqq ∆tÿ

i

ÿ

j

δpηi “ ηj ´ 1Rqηj,RPrNptq “ ηj|s,Bptqs

“ λpM ´ bptqq ∆tÿ

j s.t. ηi,Rě1

ηj,RPrNptq “ ηj|s,Bptqs (47)

“ λpM ´ bptqq ∆tÿ

j

ηj,RPrNptq “ ηj|s,Bptqs (48)

“ λpM ´ bptqq ∆t ErnRptq|s,Bptqs (49)

Note that in Equation (47), the sum is over all states ηi with at least one signallingmolecule in the receiver voxel, i.e. ηj,R ě 1. Since the summand in Equation (47) is zeroif ηj,R “ 0, we get the same result if we are to sum over all possible states, that is whyEquation (48) holds.

19

For Q2, we have

Q2 “ÿ

i

ÿ

j

δpηi “ ηj ` 1Rqµbptq ∆t PrNptq “ ηj|s,Bptqs

“ µbptq ∆tÿ

i

ÿ

j

δpηi “ ηj ` 1Rq PrNptq “ ηj|s,Bptqs (50)

“ µbptq ∆tÿ

j

PrNptq “ ηj|s,Bptqs (51)

“ µbptq ∆t (52)

Note that Equation (51) follows from Equation (50) because for every ηj, there is aunique ηi such that ηi “ ηj ` 1R holds.

For Q3, we have

Q3 “ÿ

i

ÿ

j

ηiÑjPrNptq “ ηj|s,Bptqs

“ÿ

i

ÿ

j‰i

pdij ∆tqPrNptq “ ηj|s,Bptqs`

ÿ

j

p1´ ληj,RpM ´ bptqq ∆t´ µbptq ∆t´ djj ∆tqPrNptq “ ηj|s,Bptqs

“ÿ

j

p1´ ληj,RpM ´ bptqq ∆t´ µbptq ∆tqPrNptq “ ηj|s,Bptqs`

pÿ

i

ÿ

j‰i

dijPrNptq “ ηj|s,Bptqs ´ÿ

j

djjPrNptq “ ηj|s,Bptqsq ∆t

“p1´ λpM ´ bptqqErnRptq|s,Bptqs ∆t´ µbptq ∆tq`

pÿ

i

ÿ

j‰i

dijPrNptq “ ηj|s,Bptqs ´ÿ

j

ÿ

i‰j

dijPrNptq “ ηj|s,Bptqsqlooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooon

“0

∆t

“p1´ λpM ´ bptqqErnRptq|s,Bptqs ∆t´ µbptq ∆tq (53)

Having obtained Q1, Q2 and Q3, we arrive at:

Prbpt`∆tq|s,Bptqs “δpbpt`∆tq “ bptq ` 1qλpM ´ bptqq ∆t ErnRptq|s,Bptqs`δpbpt`∆tq “ bptq ´ 1qµbptq ∆t `

δpbpt`∆tq “ bptqqp1´ λpM ´ bptqqErnRptq|s,Bptqs ∆t´ µbptq ∆tq(54)

Note that Equation (54) is the same as Equation (28) in the main text.

20

B Proof of Equation (29)

From Equation (27), we have:

dLsptq

dt“ lim

∆tÑ0

logpPrbpt`∆tq|s,Bptqsq∆t

´ lim∆tÑ0

logpPrbpt`∆tq|Bptqsq∆t

(55)

Note that the second term on the RHS is independent of transmission symbol s, we willfocus on the first term.

Note that Prbpt ` ∆tq|s,Bptqs, which is given in Equation (54), is a sum three termswith multipliers δpbpt ` ∆tq “ bptq ` 1q, δpbpt ` ∆tq “ bptq ´ 1q and δpbpt ` ∆tq “ bptqq.Since these multipliers are mutually exclusive, we have:

log pPrbpt`∆tq|s,Bptqsq “δpbpt`∆tq “ bptq ` 1q log pλpM ´ bptqq ∆t ErnRptq|s,Bptqsq`δpbpt`∆tq “ bptq ´ 1q log pµbptq ∆tq`

δpbpt`∆tq “ bptqq log pp1´ λpM ´ bptqqErnRptq|1,Bptqs ∆t´ µbptq ∆tqq

«δpbpt`∆tq “ bptq ` 1q log pErnRptq|s,Bptqsq´δpbpt`∆tq “ bptqqλpM ´ bptqqErnRptq|s,Bptqs ∆t`

P ptq (56)

where we have used the approximation logp1`αxq « αx and have collected all terms thatdo not depend on s in P ptq.

By substituting Equation (56) into Equation (55), and taking limit ∆tÑ 0, we have

dLsptq

dt“ lim

∆tÑ0

δpbpt`∆tq “ bptq ` 1q

∆tlog pErnRptq|s,Bptqsq´

δpbpt`∆tq “ bptqqλpM ´ bptqq pErnRptq|s,Bptqssq ` Lptq (57)

“dUptq

dtlog pErnRptq|s,Bptqsq ´ λpM ´ bptqq pErnRptq|s,Bptqsq ` Lptq (58)

where all terms that are independent of s have been collected in Lptq. Note that Lptqcontains some terms that diverges but this is not an issue because for demodulation it istheir relative difference Ls1ptq ´ Ls2ptq (for any two symbols s1 and s2) that matters.

Note also that we have used the following reasonings to arrive at Equation (58) fromEquation (57):

1. The term lim∆tÑ0δpbpt`∆tq“bptq`1q

∆tis an impulse whenever a receptor changes from the

unbound to the bound state. This is precisely dUptqdt

.

2. The term δpbpt`∆tq “ bptqq is only zero when the number of bound receptors changesand the number of such changes is finite. In other words, δpbpt ` ∆tq “ bptqq “ 1with probability one. This allows us to drop δpbpt`∆tq “ bptqq.

Finally, note that Equation (58) is the same as Equation (29).

21

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Symbol  0  =    Chemical  Reac:on  0  

Symbol  1  =    Chemical  Reac:on  1  

Symbol  K-­‐1  =    Chemical  Reac:on  K-­‐1  

TransmiAer  

Ligand-­‐  Receptor  Binding  

Demodulator  

Receiver  

b(t)  =  number  of  complexes    

Fluid  medium  

Con:nuous-­‐:me  Markov  process    

Figure 1: An overview of the system considered in this paper.

1   2   3   4  

5   6   7   8  

9   10   11   12  

13   14   15   16  

Figure 2: A model of molecular communication network. The volume is divided intovoxels. The indices of the voxels are given in the top right hand corner. Unfilled circlesare signalling molecules. Filled circles are receptors.

26

d  

d  

Voxel  1  TransmiAer  

Voxel  3  Receiver  

d  

d  

Voxel  2  

Figure 3: An example system consisting of 3 voxels. The indices of the voxels are asindicated. Unfilled circles are signalling molecules. Filled circles are receptors.

State  of    Receptor  1  

State  of    Receptor  2  

U(t)  

Deriva:ve  of    U(t)  

unbound  

unbound  

bound  

bound  

Time  

Time  

Time  

Time  

1  2  

3  

Figure 4: This figure explains the meaning of the function Uptq, which is the total numberof unbound-to-bound transitions for all receptors.

27

Demodula:on  filter  for  Symbol  0  

Demodula:on  filter  for  Symbol  1  

Demodula:on  filter  for  Symbol  K-­‐1  

Maximum  b(t)  =  number  of  complexes    

Z0(t)  

Z1(t)  

ZK-­‐1(t)  

Figure 5: The demodulator structure.

0 0.5 1 1.5 2 2.5 3−15

−10

−5

0

5

10

15

20

25

time

De

mo

du

lato

r o

utp

ut

Symbol 0 is sent

Z0(t)

Z1(t)

Figure 6: The output of the modulators Z0ptq (thin line) and Z1ptq (thick line) for Symbol0.

28

0 0.5 1 1.5 2 2.5 3−10

0

10

20

30

40

50

60

70

time

De

mo

du

lato

r o

utp

ut

Symbol 1 is sent

Z0(t)

Z1(t)

Figure 7: The output of the modulators Z0ptq (thin line) and Z1ptq (thick line) for Symbol1.

0 0.5 1 1.5 2 2.5 3−10

−8

−6

−4

−2

0

2

4

6

8

time

Me

an

mm

od

ula

tor

ou

tpu

t

Symbol 0 is sent

Z0(t)

Z1(t)

Figure 8: The mean output of the modulators Z0ptq (thin line) and Z1ptq (thick line) forSymbol 0.

29

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

Sym

bo

l e

rro

r ra

te

Symbol 0

Symbol 1

Figure 9: The SER for Symbols 0 and 1.

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Number of recetpors

Sym

bol err

or

rate

Symbol 0

Symbol 1

Figure 10: The SER for Symbols 0 and 1 for varying number of receptors.

30

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Number of recetpors

Sym

bol err

or

rate

Symbol 0

Symbol 1

Symbol 2

Figure 11: The SER for Symbols 0, 1 and 2 for varying number of receptors.

0 0.5 1 1.5 2 2.5 3−10

0

10

20

30

40

50

60

70

time

De

mo

du

lato

r o

utp

ut

Symbol 0 is sent

Z0(t)

Z1(t)

Figure 12: The output of the modulators Z0ptq (thin line) and Z1ptq (thick line) for Symbol0. The mean number of signalling molecules at the receiver voxel for both symbols is similar.

31