Modeling and Simulation of Molecular Communication Systems with a Reversible Adsorption Receiver

Transcription

1 Modeling and Simulation of Molecular Communication Sytem with a Reverible Adorption Receiver Yanha eng, Member, IEEE, Adam Noel, Member, IEEE, Maged Elkahlan, Member, IEEE, Arumugam Nallanathan, Senior Member, IEEE, and Karen C. Cheung. arxiv:6.68v3 [c.et] 22 Jun 26 Abtract In thi paper, we preent an analytical model for the diffuive molecular communication (MC ytem with a reverible adorption receiver in a fluid environment. The widely ued concentration hift keying (CSK i conidered for modulation. The time-varying patial ditribution of the information molecule under the reverible adorption and deorption reaction at the urface of a receiver i analytically characterized. Baed on the patial ditribution, we derive the net number of adorbed information molecule expected in any time duration. We further derive the net number of adorbed molecule expected at the teady tate to demontrate the equilibrium concentration. Given the net number of adorbed information molecule, the bit error probability of the propoed MC ytem i analytically approximated. Importantly, we preent a imulation framework for the propoed model that account for the diffuion and reverible reaction. Simulation reult how the accuracy of our derived expreion, and demontrate the poitive effect of the adorption rate and the negative effect of the deorption rate on the error probability of reverible adorption receiver with lat tranmit bit-. Moreover, our analytical reult implify to the pecial cae of a full adorption receiver and a partial adorption receiver, both of which do not include deorption. Index Term Molecular communication, reverible adorption receiver, time varying patial ditribution, error probability. I. INTROUCTION Conveying information over a ditance ha been a problem for decade, and i urgently demanded for multiple ditance cale and variou environment. The conventional olution i to utilize electrical- or electromagnetic-enabled communication, which i unfortunately inapplicable or inappropriate in very mall dimenion or in pecific environment, uch a in alt water, tunnel, or human bodie. Recent breakthrough in bio-nano technology have motivated molecular communication [, 2] to be a biologically-inpired technique for nanonetwork, where device with functional component on the cale of nanometer (i.e., nanomachine hare information over ditance via chemical ignal in nanometer to micrometer cale environment. Thee mall cale bio-nanomachine are capable Y. eng and A. Nallanathan are with epartment of Informatic, King College London, London, WC2R 2LS, UK ( {yanha.deng, arumugam.nallanathan}@kcl.ac.uk. A. Noel i with the School of Electrical Engineering and Computer Science, Univerity of Ottawa, Ottawa, ON, KN 6N5, Canada ( anoel2@uottawa.ca. K. C. Cheung i with epartment of Electrical and Computer Engineering, Univerity of Britih Columbia, Vancouver, BC, V6T Z4, Canada ( kcheung@ece.ubc.ca. M. Elkahlan i with Queen Mary Univerity of London, London E 4NS, UK ( maged.elkahlan@qmul.ac.uk. of encoding information onto phyical molecule, ening, and decoding the received information molecule, which could enable application in drug delivery, pollution control, health, and environmental monitoring [3]. Baed on the propagation channel, molecular communication (MC can be claified into one of three categorie: Walkway-baed MC, where molecule move directionally along moleculaail uing carrier ubtance, uch a molecular motor [4]; 2 Flow-baed paradigm, where molecule propagate primarily via fluid flow. An example of thi kind i the hormonal communication through the bloodtream in the human body []; 3 iffuion-baed MC, where molecule propagate via the random motion, namely Brownian motion, caued by colliion with the fluid molecule. In thi cae, molecule motion i le predictable, and the propagation i often aumed to follow the law of a Wiener proce. Example include deoxyribonucleic acid (NA ignaling among NA egment [5], calcium ignaling among cell [6], and pheromonal communication among animal [7]. Among the aforementioned three MC paradigm, diffuionbaed MC i the mot imple, general and energy efficient tranportation paradigm without the need for external energy or infratructure. Thu, reearch ha focued on the mathematical modeling and theoretical analyi [8 2], reception deign [3], receiver modeling [4], and modulation and demodulation technique [5 7], of diffuion-baed MC ytem. In diffuion-baed MC, the tranmit ignal i encoded on the phyical characteritic of information molecule (uch a hormone, pheromone, NA, which propagate through the fluid medium via diffuion with the help of thermal energy in the environment. The information can be encoded onto the the quantity, identity, oeleaed timing of the molecule. In the domain of timing channel, the firt work on diffuion baed MC wa pioneered by Eckford [8], in which the propagation timing channel i ideally characterized a an additive noie channel. In the domain of concentration-baed encoding, the concentration level of information molecule repreent different tranmit ignal. Since the average diplacement of an information molecule i directly proportional to the quare root of diffuion time [5], long ditance tranmiion require much longer propagation time. Moreover, the randomne of the arriving time for each molecule make it difficult for the receiver to ditinguih between the ignal tranmitted in different bit interval, becaue the number of received molecule in the current ymbol depend on the molecule emitted in previou and current ymbol. Thi i known a

2 2 interymbol interference (ISI. In mot exiting literature, ome aumption are made in order to focu on the propagation channel. One aumption i that each molecule i removed from the environment when it contribute once to the received ignal. A uch, the information molecule concentration near the receiver i intentionally changed [8]. Another widely-ued idealitic aumption i to conider a paive receiver, which i permeable to the information molecule paing by, and i capable of counting the number molecule inide the receiver volume [3, 9]. The paive receiver model eaily encounter high ISI, ince the ame molecule may unavoidably contribute to the received ignal many time in different ymbol interval. In a practical bio-inpired ytem, the urface of a receiver i covered with elective receptor, which are enitive to a pecific type of information molecule (e.g., pecific peptide or calcium ion. The urface of the receiver may adorb or bind with thi pecific information molecule [2]. One example i that the influx of calcium toward the center of a receiver (e.g. cell i induced by the reception of a calcium ignal [2, 22]. epite growing reearch effort in MC, the chemical reaction receiver ha not been accurately characterized in mot of the literature except by Yilmaz [4, 5, 7] and Chou [23]. The primary challenge i accommodating the local reaction in the reaction-diffuion equation. In [4] and [24], the channel impule repone for MC with an aborbing receiver wa derived. The MolecUlar CommunicatIoN (MUCIN imulator wa preented in [5] to verify the fully-aborbing receiver. The reult in [4, 5] were then extended to the ISI mitigation problem for the fully-aborbing receiver [7]. In [23], the mean and variance of the receiver output wa derived for MC with a reverible reaction receiver baed on the reaction-diffuion mater equation (RME. The analyi and imulation were performed uing the ubvolume-baed method, where the tranmitter and receiver were cube, and the exact location or placement of individual molecule were not captured. They conidered the reverible reaction only happen inide the receiver (cube rather than at the urface of receiver. Unlike exiting work on MC, we conider the reverible adorption and deorption (A& receiver, which i capable of adorbing a certain type of information molecule near it urface, and deorbing the information molecule previouly adorbed at it urface. A& i a widely-oberved proce for colloid [25], protein [26], and polymer [27]. Within the Internet of Bio-NanoThing (IoBNT, biological cell are uually regarded a the ubtrate of the Bio-NanoThing. Thee biological cell will be capable of interacting with each other by exchanging information, uch a ened chemical or phyical parameter and et of intruction or command [28]. Analyzing the performance characteritic of MC ytem uing biological cell equipped with adorption and deorption receptor allow for the comparion, claification, optimization and realization of different technique to realize the IoBNT. The A& proce alo implifie to the pecial cae of an aborbing receiver (i.e., with no deorption. For conitency in thi paper, we refer to receiver that do not deorb, but have infinite or finite aborption rate, a fullyadorbing and partially-adorbing receiver, repectively. From a theoretical perpective, reearcher have derived the equilibrium concentration of A& [29], which i inufficient to model the time-varying channel impule repone (and ultimately the communication performance of an A& receiver. Furthermore, the imulation deign for the A& proce of molecule at the urface of a planaeceiver wa alo propoed in [29]. However, the imulation procedure for a communication model with a pherical A& receiver in a fluid environment ha never been olved and reported. In thi model, information molecule are releaed by the tranmiion of pule, propagate via free-diffuion through the channel, and contribute to the received ignal via A& at the receiver urface. The challenge are the complexity in modeling the coupling effect of adorption and deorption under diffuion, a well a accurately and dynamically tracking the location and the number of diffued molecule, adorbed molecule and deorbed molecule (which are free to diffue again. epite the aforementioned challenge, we conider in thi paper the diffuion-baed MC ytem with a point tranmitter and an A& receiver. The tranmitter emit a certain number of information molecule at the tart of each ymbol interval to repreent the tranmitted ignal. Thee information molecule can adorb to or deorb from the urface of the receiver. The number of information molecule adorbed at the urface of the receiver i counted for information decoding. The goal of thi paper i to characterize the communication performance of an A&. Our major contribution are a follow: We preent an analytical model for the diffuion-baed MC ytem with an A& receiver. We derive the exact expreion for the channel impule repone at a pherical A& receiver in a three dimenional (3 fluid environment due to one intantaneou releae of multiple molecule (i.e., ingle tranmiion. 2 We derive the net number of adorbed molecule expected at the urface of the A& receiver in any time duration. To meaure the equilibrium concentration for a ingle tranmiion, we alo derive the aymptotic number of cumulative adorbed molecule expected at the urface of A& receiver a time goe to infinity. 3 Unlike mot literature in [9], where the received ignal i demodulated baed on the total number of molecule expected at the paive receiver, we conider a imple demodulator baed on the net number of adorbed molecule expected. When multiple bit are tranmitted, the net number i more conitent than the total number. 4 We apply the Skellam ditribution to approximate the net number of adorbed molecule expected at the urface of the A& receiver due to a ingle tranmiion of molecule. We formulate the bit error probability of the A& receiver uing the Skellam ditribution. Oueult how the poitive effect of adorption rate and negative effect of deorption rate on the error probability of A& receiver with lat tranmit bit-. 5 We propoe a imulation algorithm to imulate the diffuion, adorption and deorption behavior of information molecule baed on a particle-baed imulation framework. Unlike exiting imulation platform (e.g.,

3 3 Smoldyn [3], N3im [3], our imulation algorithm capture the dynamic procee of the MC ytem, which include the ignal modulation, molecule free diffuion, molecule A& at the urface of the receiver, and ignal demodulation. Our imulation reult are in cloe agreement with the derived number of adorbed molecule expected. Interetingly, we demontrate that the error probability of the A& receiver for the lat tranmitted bit i wore at higher detection threhold but better at low detection threhold than both the full adorption and partial adorption receiver. Thi i becaue the A& receiver oberve a lower peak number of adorbed molecule but then a fater decay. The ret of thi paper i organized a follow. In Section II, we introduce the ytem model with a ingle tranmiion at the tranmitter and the A& receiver. In Section III, we preent the channel impule repone of information molecule, i.e., the exact and aymptotic number of adorbed molecule expected at the urface of the receiver. In Section IV, we derive the bit error probability of the propoed MC model due to multiple ymbol interval. In Section V, we preent the imulation framework. In Section VI, we dicu the numerical and imulation reult. In Section VII, we conclude the contribution of thi paper. II. SYSTEM MOEL We conider a 3-dimenional (3 diffuion-baed MC ytem in a fluid environment with a point tranmitter and a pherical A& receiver. We aume pherical ymmetry where the tranmitter i effectively a pherical hell and the molecule are releaed from random point over the hell; the actual angle to the tranmitter when a molecule hit the receiver i ignored, o thi aumption cannot accommodate a flowing environment. The point tranmitter i located at a ditance r from the center of the receiver and i at a ditance d = r from the nearet point on the urface of the receiver with radiu. The extenion to an aymmetric pherical model that account for the actual angle to the tranmitter when a molecule hit the receiver complicate the derivation of the channel impule repone, and might be olved following [32]. We aume all receptor are equivalent and can accommodate at mot one adorbed molecule. The ability of a molecule to adorb at a given ite i independent of the occupation of neighboring receptor. The pherical receiver i aumed to have no phyical limitation on the number or placement of receptor on the receiver. Thu, there i no limit on the number of molecule adorbed to the receiver urface (i.e., we ignore aturation. Thi i an appropriate aumption for a ufficiently low number of adorbed molecule, or for a ufficiently high concentration of receptor. Once an information molecule bind to a receptor ite, a phyical repone i activated to facilitate the counting of the molecule. Generally, due to the non-covalent nature of binding, in the diociation proce, the receptor may releae the adorbed molecule to the fluid environment without changing it phyical characteritic, e.g., a ligand-binding receptor [33]. We alo aume perfect ynchronization between the tranmitter and the receiver a in mot literature [9, 3 7, 9]. The ytem include five procee: emiion, propagation, reception, modulation and demodulation, which are detailed in the following. A. Emiion The point tranmitteeleae one type of information molecule (e.g., hormone, pheromone to the receiver for information tranmiion. The tranmitter emit the information molecule at t =, where we define the initial condition a [24, Eq. (3.6] C(r, t r = 4πr 2 δ(r r, ( where C(r, t r i the molecule ditribution function at time t and ditance r with initial ditance r. We alo define the firt boundary condition a lim C(r, t r =, (2 r uch that at arbitrary time, the molecule ditribution function equal zero when r goe to infinity. B. iffuion Once the information molecule are emitted, they diffue by randomly colliding with other molecule in the environment. Thi random motion i called Brownian motion [5]. The concentration of information molecule i aumed to be ufficiently low that the colliion between thoe information molecule are ignored [5], uch that each information molecule diffue independently with contant diffuion coefficient. The propagation model in a 3 environment i decribed by Fick econd law [5, 4]: (r C(r, t r t = 2 (r C(r, t r r 2, (3 where the diffuion coefficient i uually obtained via experiment [34]. C. Reception We conider a reverible A& receiver that i capable of counting the net number of adorbed molecule at the urface of the receiver. Any molecule that hit the receiver urface i either adorbed to the receiver urface oeflected back into the fluid environment, baed on the adorption rate k (length time. The adorbed molecule either deorb or remain tationary at the urface of receiver, baed on the deorption rate k (time. At t =, there are no information molecule at the receiver urface, o the econd initial condition i C(, r =, and C a ( r =, (4 wherec a (t r i the average concentration of molecule that are adorbed to the receiver urface at time t.

4 4 For the olid-fluid interface located at, the econd boundary condition of the information molecule i [29, Eq. (4] (C(r, t r r = k C(, t r k C a (t r, r=r r (5 which account for the adorption and deorption reaction that can occur at the urface of the receiver. Mot generally, when both k and k are non-zero finite contant, (5 i the boundary condition for the A& receiver. When k and k =, (5 i the boundary condition for the full adorption (or fully-adorbing receiver, wherea when k i a non-zero finite contant and k =, (5 i the boundary condition for the partial adorption (or partially-adorbing receiver. In thee two pecial cae with k =, the lack of deorption reult in more effective adorption. Here, the adorption rate k i approximately limited to the thermal velocity of potential adorbent (e.g., k < 7 6 µm/ for a 5 ka protein at 37 C [29]; the deorption rate k i typically between 4 and 4 [35]. The urface concentration C a (t r change over time a follow: C a (t r t = (C(r, t r r, (6 r=r r which how that the change in the adorbed concentration over time i equal to the flux of diffuion molecule toward the urface. Combining (5 and (6, we write C a (t r = k C(, t r k C a (t r, (7 t which i known a the Robin oadiation boundary condition [36, 37] and how that the equivalent adorption rate i proportional to the molecule concentration at the urface.. Modulation and emodulation In thi model, we conider the widely applied amplitudebaed modulation concentration hift keying (CSK [3, 5, 7, 38, 39], where the concentration of information molecule i interpreted a the amplitude of the ignal. Specifically, we utilize Binary CSK, where the tranmitter emit N molecule at the tart of the bit interval to repreent the tranmit bit-, and emit N 2 molecule at the tart of the bit interval to repreent the tranmit bit-. To reduce the energy conumption and make the received ignal more ditinguihable, we aume that N = N tx and N 2 =. We aume that the receiver i able to count the net number of information molecule that are adorbed to the urface of the receiver in any ampling period by ubtracting the number of molecule bound to the urface of the receiver at the end of previou ampling time from that at the end of current ampling time. The net number of adorbed molecule over a bit interval i then demodulated a the received ignal for that bit interval. Thi approach i in contrat to [7], where the cumulative number of molecule arrival in each ymbol duration wa demodulated a the received ignal (i.e., cumulative counter i reet to zero at each ymbol duration. We claim (and oueult will demontrate that our approach i more appropriate for a imple demodulator. Here, we write the net number of adorbed molecule meaured by the receiver in the jth bit interval a Nnew Rx [j], and the deciion threhold for the number of received molecule i N th. Uing threhold-baed demodulation, the receiver demodulate the received ignal a bit- if Nnew Rx [j] N th, and demodulate the received ignal a bit- if Nnew[j] Rx < N th. III. RECEIVER OBSERVATIONS In thi ection, we firt derive the pherically-ymmetric patial ditribution C(r, t r, which i the probability of finding a molecule at ditance r and time t. We then derive the flux at the urface of the A& receiver, from which we derive the exact and aymptotic number of adorbed molecule expected at the urface of the receiver. A. Exact Reult The time-varying patial ditribution of information molecule at the urface of the receiver i an important tatitic for capturing the molecule concentration in the diffuion-baed MC ytem. We olve it in the following theorem. Theorem. The expected time-varying patial ditribution of an information molecule releaed into a 3 fluid environment with a reverible adorbing receiver i given by { } C(r, t r = 8πr r πt exp (r r 2 4t { 8πr r πt exp where 2πr ϕ Z (w = Z( = ( ( 2 2 4t } ( e t ϕ Z (we t ϕ Z (w dw, ( 2 8πr exp k (k k (k { (rr 2 and ϕ Z (w i the complex conjugate of ϕ Z (w. (8 }, (9 Proof: See Appendix A. Oueult in Theorem can be eaily computed uing Matlab. We oberve that (8 reduce to an aborbing receiver [24, Eq. (3.99] when there i no deorption (i.e., k =. To characterize the number of information molecule adorbed to the urface of the receiver uing C(r,t r, we define the rate of the coupled reaction (i.e., adorption and deorption at the urface of the A& receiver a [24, Eq. (3.6] K(t r = 4πrr 2 C(r,t r r. ( r=rr

5 5 Corollary. The rate of the coupling reaction at the urface of a reverible adorbing receiver i given by K(t r = 2 2 where ϕ Z (w i a given in (9. e t [ ϕ Z (w] dw e t [ ϕ Z (w ] dw, ( Proof: By ubtituting (8 into (, we derive the coupling reaction rate at the urface of an A& receiver a (. From Corollary, we can derive the net change in the number of adorbed molecule expected for any time interval in the following theorem. Theorem 2. With a ingle emiion at t =, the net change in the number of adorbed molecule expected at the urface of the A& receiver during the interval [T, T T ] i derived a E[N A& (Ω rr,t,t T r ] = 2 N tx [ e T e (TT [ ] dw ϕ Z (w e (TT e T [ ] ] ϕ Z (w dw, (2 where ϕ Z (w i given in (9, T i the ampling time, and Ω repreent the pherical receiver with radiu. Proof: The cumulative fraction of particle that are adorbed to the receiver urface at time T i expreed a R A& (Ω rr,t r = [ = 2 T e T e T K(t r dt [ ] ϕ Z (w dw [ ] ] ϕ Z (w dw. (3 Baed on (3, the net change in adorbed molecule expected at the receiver urface during the interval [T, T T ] i defined a E[N A& (Ω rr,t,t T r ] = N tx R A& (Ω rr,t T r N tx R A& (Ω rr,t r. (4 Subtituting (3 into (4, we derive the expected net change of adorbed molecule during any obervation interval a (2. Note that the net change in the number of adorbed molecule in each bit interval will be recorded at the receiver, which will be converted to the recorded net change of adorbed molecule in each bit interval, and compared with the deciion threhold N th to demodulate the received ignal (the ampling interval i maller than one bit interval. B. Aymptotic Behavior: Equilibrium Concentration In thi ection, we are intereted in the aymptotic number of adorbed molecule due to a ingle emiion a T b goe to infinity, i.e., the concentration of adorbed molecule at the teady tate. Note that thi aymptotic concentration of adorbed molecule i an important quantity that influence the number of adorbed molecule expected in ubequent bit interval, and we have aumed that the receiver urface ha infinite receptor. Thu, in the remainder of thi ection, we derive the cumulative number of adorbed molecule expected at the urface of the A& receiver, the partial adorption receiver, and the full adorption receiver, a T b. Reverible A& Receiver: Lemma. A T b, the cumulative number of adorbed molecule expected at the A& receiver implifie to E[N A& (Ω rr,t b r ] = N tx 2r [ ] 4N tx w Im ϕ Z (w dw. (5 Proof: We expre the cumulative fraction of particle adorbed to the urface of the A& receiver at time T b in (3 a R A& (Ω rr,t b r [ ( ] e T b = Re 4 ϕ Z (w dw [ ] inwt b = 4 w Re ϕ Z (w dw [ ] cowt b 4 Im w ϕ Z (w dw [ ( ] inz z = 4 z Re q dz 4 T b [ ( ] z Im q dz 4 where T b q(w = ( ( exp k (k k (k { (r coz z Im[q(w]dw, (6 w 4πr }. (7

6 6 A T b, we have the following: E[N A& (Ω rr,t b r ] = 4 N tx [ inz z Re[q(] coz dz z Im[q(]dz ] w Im[q(w]dw [ (b inz = 4 N tx z Re[q(]dz (c = N tx [ rr πr = N tx 2r 4N tx ] w Im[q(w]dw inz z dz 4r ] r w Im[q(w]dw [ ] w Im ϕ Z (w dw, (8 where (b i due to the fact that Im[q(] =, and (c i due to q( = 4πr. 2 Partial Adorption Receiver: The partial adorption receiver only adorb ome of the molecule that collide with it urface, correponding to k a a finite contant and k = in (5. Propoition. The number of molecule expected to be adorbed to the partial adorption receiver by time T b, a T b, i derived a E[N PA (Ω rr,t b r ] = N txk r 2 r r (k. (9 Proof: We note that the exact expreion for the net number of adorbed molecule expected at the partial adorption receiver during [T, T T ] can be derived from [24, Eq. (3.4] a α E[N PA (Ω rr,t,t T r ] = N tx [ { } r α r erf exp{(r α 4(T T { } (T T α 2} r 2α(T T erfc 4(T T { } rr r erf exp { (r αtα 2} 4T { } ] r 2αT erfc, (2 4T where α = k The cumulative fraction of molecule adorbed at the partial adorption receiver by time T b wa derived in [24, Eq. (3.4] a R PA (Ω r,t b r = r ( { } rα rr r erf r α 4Tb. exp { (r αt b α 2} erfc { } r 2αT b. 4Tb (2 By etting T b and taking the expectation of (2, we arrive at (9. The aymptotic reult in (9 for the partial adorption receiver reveal that the number of adorbed molecule expected at infinite time T b increae with increaing adorption rate k, and decreae with increaing diffuion coefficient and increaing ditance between the tranmitter and the center of the receive. 3 Full Adorption Receiver: In the full adorption receiver, all molecule adorb when they collide with it urface, which correpond to the cae of k and k = in (5. Propoition 2. The cumulative number of adorbed molecule expected at the full adorption receiver by time T b, a T b, i derived a E[N FA (Ω rr,t b r ] = N tx r. (22 Proof: We note that the exact expreion for the net number of adorbed molecule expected at the full adorption receiver during [T, T T ] ha been derived in [4, 24] a E[N FA (Ω rr,t,t T r ] = [ { } { } ] r r N tx erfc erfc. (23 r 4(T T 4T The fraction of molecule adorbed to the full adorption receiver by time T b wa derived in [24, Eq. (3.6] and [4, Eq. (32] a R FA (Ω r,t b r = r erfc { } r. (24 4Tb By etting T b and taking the expectation of (24, we arrive at (22. Alternatively, with the help of integration by part, the reult in (5 reduce to the aymptotic reult in (22 for the full adorption receiver by etting k = and k =. The aymptotic reult for the full adorption receiver in (22 reveal that the cumulative number of adorbed molecule expected by infinite time T b i independent of the diffuion coefficient, and directly proportional to the ratio between the radiu of receiver and the ditance between the tranmitter and the center of receiver. IV. ERROR PROBABILITY In thi ection, we propoe that the net number of adorbed molecule in a bit interval be ued foeceiver demodulation. We alo derive the error probability of the MC ytem uing the Poion approximation and the Skellam ditribution. To calculate the error probability at the receiver, we firt need to model the tatitic of molecule adorption. For a ingle emiion att =, the net number of molecule adorbed during [T,T T b ] i approximately modeled a the difference between two binomial ditribution a N Rx new B(N tx,r A& (Ω rr,t T b r B(N tx,r A& (Ω rr,t r, (25 where the cumulative fraction of particle that are adorbed to the A& receiver R A& (Ω rr,t r i given in (6. Note that the number of molecule adorbed at T T b depend on that at T, however thi dependence can be ignored for a ufficiently large bit interval, and make (25 accurate. The number of adorbed molecule repreented by Binomial

7 7 ditribution can alo be approximated uing either the Poion ditribution or the Normal ditribution. The net number of adorbed molecule depend on the emiion in the current bit interval and thoe in previou bit interval. Unlike the full adorption receiver in [7, 4, 4] and partial adorption receiver where the net number of adorbed molecule i alway poitive, the net number of adorbed molecule of the A& receiver can be negative. Thu, we cannot model the net number of adorbed molecule of the reverible adorption receiver during one bit interval a Nnew Rx B(N tx,r(ω rr,t,t T b r with R(Ω rr,t,t T b r = TT b K(t r T dt, which wa ued to model that of full adorption receiver and partial adorption receiver [4, 4]. For multiple emiion, the cumulative number of adorbed molecule i modeled a the um of multiple binomial random variable. Thi um doe not lend itelf to a convenient expreion. Approximation for the um were ued in [42]. Here, the binomial ditribution can be approximated with the Poion ditribution, when we have ufficiently large N tx and ufficiently mall R A& (Ω rr,t r [43]. Thu, we approximate the net number of adorbed molecule received in the jth bit interval a N Rx new [j] P ( j i=n tx i R A& (Ω rr,(j it b r ( j P N tx i R A& (Ω rr,(j it b r, i= (26 where i i the ith tranmitted bit. The difference between two Poion random variable follow the Skellam ditribution [44]. For threhold-baed demodulation, the error probability of the tranmit bit- ignal in the jth bit i then P e [ŝ j = j =, :j ] = Pr ( Nnew[j] Rx < N j th =, :j where N th n= Ψ = exp{ (Ψ Ψ 2 }(Ψ /Ψ 2 n/2 I n ( 2 Ψ Ψ 2, (27 j N tx i R A& (Ω rr,(j it b r, (28 i= j Ψ 2 = N tx i R A& (Ω rr,(j it b r, (29 i= ŝ j i the detected jth bit, and I n ( i the modified Beel function of the firt kind. Similarly, the error probability of the tranmit bit- ignal in the jth bit i given a P e [ŝ j = j =, :j ] = Pr ( Nnew[j] Rx N j th =, :j exp{ (Ψ Ψ 2 }(Ψ /Ψ 2 n/2 I n n=n th (2 Ψ Ψ 2, (3 where Ψ and Ψ are given in (28 and (29, repectively. Thu, the error probability of the random tranmit bit in the jth interval i expreed by P e [j] =P P e [ŝ j = j =, :j ] P P e [ŝ j = j =, :j ], (3 where P and P denote the probability of ending bit- and bit-, repectively. For comparion, we alo preent the error probability of the tranmit bit- ignal in the jth bit and error probability of the tranmit bit- ignal in the jth bit for the full adorption receiver and the partial adorption receiver uing the Poion approximation a and N th P e [ŝ j = j =, :j ] exp{n tx Γ} n= N th P e [ŝ j = j =, :j ] exp{n tx Γ} In (32 and (33, we have Γ = n= [N tx Γ] n, n! (32 [N tx Γ] n. n! (33 j i R FA (Ω rr,(j it b (j it b r (34 i= for the full adorption receiver, and Γ = j i R PA (Ω rr,(j it b (j it b r (35 i= for the partial adorption receiver. V. SIMULATION FRAMEWORK Thi ection decribe the tochatic imulation framework for the point-to-point MC ytem with the A& receiver decribed by (5, which can be implified to the MC ytem with the partial adorption receiver and full adorption receiver by etting k = and k =, repectively. Thi imulation framework take into account the ignal modulation, molecule free diffuion, molecule A& at the urface of the receiver, and ignal demodulation. To model the tochatic reaction of molecule in the fluid, two option are a ubvolume-baed imulation framework or a particle-baed imulation framework. In a ubvolumebaed imulation framework, the environment i divided into many ubvolume, where the number of molecule in each ubvolume i recorded [23]. In a particle-baed imulation

8 8 framework [45], the exact poition of each molecule and the number of molecule in the fluid environment i recorded. To accurately capture the location of individual information molecule, we adopt a particle-baed imulation framework with a patial reolution on the order of everal nanometer [45]. A. Algorithm We preent the algorithm for imulating the MC ytem with an A& receiver in Algorithm. In the following ubection, we decribe the detail of Algorithm. Algorithm The Simulation of a MC Sytem with an A& Receiver Require: N tx, r,, Ω rr,, t, T, T b, N th : procedure INITIALIZATION 2: Generate Random Bit Sequence {b,b 2,,b j, } 3: etermine Simulation End Time 4: For all Simulation Time Step do 5: If at tart of jth bit interval and b j = 6: Add N tx emitted molecule 7: For all free molecule in environment do 8: Propagate free molecule following N (, 2 t 9: Evaluate ditance d m of molecule to receiver : if d m < then : Update tate & location of collided molecule 2: Update # of collided molecule N C 3: For all N C collided molecule do 4: if Adorption Occur then 5: Update # of newly-adorbed molecule N A 6: Calculate ( adorbed molecule location 7: x A m,ym,zm A A 8: ele 9: Reflect ( the molecule off receiver urface to 2: x Bo m,ym Bo,zm Bo 2: For all previouly-adorbed molecule do 22: if eorption Occur then 23: Update tate & location of deorbed molecule 24: Update # of newly-deorbed molecule N 25: iplace ( newly-deorbed molecule to 26: x m,ym,zm 27: Calculate net number of adorbed molecule, 28: which i N A N 29: Add net number of adorbed molecule in each imulation interval of jth bit interval to determine Nnew Rx [j] 3: emodulate by comparing Nnew[j] Rx with N th B. Modulation, Emiion, and iffuion In our model, we conider BCSK, where two different number of molecule repreent the binary ignal and. At the tart of each bit interval, if the current bit i, then N tx molecule are emitted from the point tranmitter at a ditance r from the center of the receiver. Otherwie, the point tranmitter emit no molecule to tranmit bit-. The time i divided into mall imulation interval of ize t, and each time intant i t m = m t, where m i the current imulation index. According to Brownian motion, the diplacement of a molecule in each dimenion in one imulation tep t can be modeled by an independent Gauian ditribution with variance 2 t and zero mean N (, 2 t. The diplacement S of a molecule in a 3 fluid environment in one imulation tep t i therefore S = {N (,2 t, N (,2 t, N (,2 t}. (36 In each imulation tep, the number of molecule and their location are tored. C. Adorption or Reflection According to the econd boundary condition in (6, molecule that collide with the receiver urface are either adorbed oeflected back. The N C collided molecule are identified by calculating the ditance between each molecule and the center of the receiver. Among the collided molecule, the probability of a molecule being adorbed to the receiver urface, i.e., the adorption probability, i a function of the diffuion coefficient, which i given a [46, Eq. (] P A = k π t. (37 The probability that a collided molecule bounce off of the receiver i P A. It i known that adorption may occur during the imulation tep t, and determining exactly where a molecule adorbed to the urface of the receiver during t i a nontrivial problem. Unlike [29] (which conidered a flat adorbing urface, we aume that the molecule adorption ite during [t m,t m ] i the location where the line, formed by thi molecule location at the tart of the current imulation tep (x m,y m,z m and thi molecule location at the end of the current imulation tep after diffuion(x m,y m,z m, interect the urface of the receiver. Auming that the location of the center of receiver i (x r,y r,z r, then the location of the interection point between thi 3 line egment, and a phere with center at (x r,y r,z r in the mth imulation tep, can be hown to be where Λ = x A m =x m x m x m g, Λ (38 ym A =y m y m y m g, Λ (39 zm A =z m z m z m g, (4 Λ (x m x m 2 (y m y m 2 (z m z m 2, (4 g = b b 2 4ac. (42 2a

9 9 In (42, we have ( 2 ( 2 ( 2 xm x m ym y m zm z m a =, Λ Λ Λ b =2 (x m x m (x m x r Λ 2 (y m y m (y m y r Λ 2 (z m z m (z m z r, (43 Λ c =(x m x r 2 (y m y r 2 (z m z r 2 2, (44 where Λ i given in (4. Of coure, due to ymmetry, the location of the adorption ite doe not impact the overall accuracy of the imulation. If a molecule fail to adorb to the receiver, then in the reflection proce we make the approximation that the molecule bounce back to it poition at the tart of the current imulation tep. Thu, the location of the molecule afteeflection by the receiver in the mth imulation tep i approximated a ( x Bo m,ym Bo,zm Bo = (xm,y m,z m. (45 Note that the approximation for molecule location in the adorption proce and the reflection proce can be accurate for ufficiently mall imulation tep (e.g., t < 7 for the ytem that we imulate in Section V, but mall imulation tep reult in poor computational efficiency. The tradeoff between the accuracy and the efficiency can be deliberately balanced by the choice of imulation tep.. eorption In the deorption proce, the molecule adorbed at the receiver boundary either deorb oemain adorbed. The deorption proce can be modeled a a firt-order chemical reaction. Thu, the deorption probability of a molecule at the receiver urface during t i given by [29, Eq. (22] P = e k t. (46 The diplacement of a molecule after deorption i an important factor for accurate modeling of molecule behaviour. If the imulation tep were mall, then we might place the deorbed molecule near the receiver urface; otherwie, doing o may reult in an artificially higher chance of re-adorption in the following time tep, reulting in an inexact concentration profile. To avoid thi, we take into account the diffuion after deorption, and place the deorbed molecule away from the urface with diplacement ( x, y, z ( x, y, z = (f (P,f (P 2,f (P 3, (47 where each component wa empirically found to be [29, Eq. (27] f (P =.57825P P 2 2 t.5398p p2. (48 In (47, P, P 2 and P 3 are uniform random number between and. Placing the deorbed molecule at a random ditance away from where the molecule wa adorbed may not be ufficiently accurate due to the lack of conideration for the coupling effect of A& and the diffuion coefficient in (48. Unlike [29], we have a pherical receiver, uch that a molecule after deorption in our model mut be diplaced differently. We aume that the location of a molecule after deorption ( x m,y m,z m, baed on it location at the tart of the current imulation tep and the location of the center of the receiver (x r,y r,z r, can be approximated a x m =x A m gn ( x A m x r x, y m =ya m gn( y A m y r y, z m =za m gn( z A m z r z. (49 In (49, x, y, and z are given in (47, and gn( i the Sign function. E. emodulation The receiver i capable of counting the net change in the number of adorbed molecule in each bit interval. The net number of adorbed molecule for an entire bit interval i compared with the threhold N th and demodulated a the received ignal. VI. NUMERICAL RESULTS In thi ection, we examine the channel repone and the aymptotic channel repone due to a ingle bit tranmiion. We alo examine the channel repone and the error probability due to multiple bit tranmiion. In all figure of thi ection, we ue FA, PA, Anal. and Sim. to abbreviate Full adorption receiver, Partial adorption receiver, Analytical and Simulation, repectively. Alo, the unit for the adorption rate k and deorption rate k are µm/ and in all figure, repectively. In Fig. to 4, we et the parameter according to micro-cale cell-to-cell communication, 2 : N tx =, = µm, r = µm, = 8 µm 2 /, and the ampling interval T =.2. A. Channel Repone Fig. and 2 plot the net change of adorbed molecule at the urface of the A& receiver during each ampling time T due to a ingle bit tranmiion. The expected analytical curve are plotted uing the exact reult in (2. The imulation point are plotted by meauring the net change of adorbed molecule during [t,tt ] uing Algorithm decribed in Section IV, where t = nt, and n {,2,3,...}. In both figure, we average the net number of adorbed molecule expected over independent emiion of N tx = information molecule at time t =. We ee that the expected net number of adorbed molecule meaured uing imulation i cloe to the exact analytical curve. The mall gap between the The mall eparation ditance between the tranmitter and receiver compared to the receiveadiu follow from the example of the pancreatic ilet, where the average cell ize i around 5 micrometer and the communication range i around 5 micrometer [4, 5]. 2 Thi diffuion coefficient value correpond to that of a large molecule, however, our analytical reult and imulation algorithm apply to any pecific value.

10 Net Number of Adorbed Molecule during Each Sampling Time k = Anal. Sim Time (.2 Net Number of Adorbed Molecule during Each Sampling Time Anal. Sim. FA Sim. PA k = 3 Sim. PA k = 2 Sim. A& k = 2, k - = Sim. A& k = 2, k - = 2 2 Time ( Fig.. The net number of adorbed molecule for variou adorption rate with k = 5 and the imulation tep t = 5. Fig. 3. The net number of adorbed molecule with the imulation tep t = 5. Net Number of Adorbed Molecule during Each Sampling Time Anal. Sim. 3 2 k - = Time ( Fig. 2. The net number of adorbed molecule for variou deorption rate with k = 2 µm/ and the imulation tep t = 5. curve reult from the local approximation in the adorption, reflection, and deorption procee in (37, (45, and (49, which can be reduced by etting a maller imulation tep. Fig. examine the impact of the adorption rate on the net number of adorbed molecule expected at the urface of the receiver. We fix the deorption rate to be k = 5. The expected net number of adorbed molecule increae with increaing adorption rate k, a predicted by (5. Fig. 2 how the impact of the deorption rate on the expected net number of adorbed molecule at the urface of the receiver. We et k = 2µm/. The net number of adorbed molecule expected decreae with increaing deorption rate k, which i a predicted by (5. From a communication perpective, Fig. how that a higher adorption rate make the bit- ignal more ditinguihable, wherea Fig. 2 how that a lower deorption rate make the bit- ignal more ditinguihable for the decoding proce. In Fig. and 2, the horter tail due to the lower adorption rate and the higher deorption rate correpond to le interymbol interference..2 Fig. 3 plot the net number of adorbed molecule from bit tranmiion over a longer time cale. We compare the A& receiver with otheeceiver deign in order to compare their interymbol interference (ISI. The analytical curve for the A& receiver, the partial adorption receiver, and the full adorption receiver are plotted uing the expreion in (2, (2, and (23, repectively. The marker are plotted by meauring the net number of adorbed molecule during [t,tt ] for one bit interval uing Algorithm decribed in Section IV. We ee a cloe match between the analytical curve and the imulation curve, which confirm the correctne of our derived reult. It i clear from Fig. 3 that the full adorption receiver and the partial adorption receiver with high adorption rate have longer tail. Interetingly, the A& receiver in our model ha the horter tail, even though it ha the ame adorption rate k a one of the partial adorption receiver. Thi might be urpriing ince the A& receiver would have more total adorption event than the partial adorption receiver with the ame k. The reaon for thi difference i that the deorption behaviour at the urface of the receiveeult in more adorption event, but not more net adorbed molecule; molecule that deorb are not counted unle they adorb again. A expected, we ee the highet peak E[N (Ω rr,t,t T r ] in Fig. 3 for the full adorption receiver, which i becaue all molecule colliding with the urface of the receiver are adorbed. For the partial adorption receiver, the peak value of E[N (Ω rr,t,t T r ] increae with increaing adorption rate k a hown in (5. The net number of adorbed molecule expected at the partial adorption receiver i higher than that at the A& receiver with the ame k. Thi mean the full adorption receiver and the partial adorption receiver have more ditinguihable received ignal between bit- and bit-, compared with the A& receiver. B. Equilibrium Concentration

11 Number of Accumulatively-Adorbed Molecule A& k = 3, k - = FA. PA k = 2 PA k = 5 A& k = 3, k - = 2 Anal. Aymptotic 2 3 Time ( Number of Accumulatively-Adorbed Molecule [ Sim. Cumulative adorbed molecule Tranmit Sequence 2 3 Time ( 4 5 Fig. 4. The cumulative number of adorbed molecule. Fig. 5. The cumulative number of adorbed molecule. Fig. 4 plot the number of cumulatively-adorbed molecule expected at the urface of the different type of receiver with a ingle emiion N tx and a T b. The olid curve are plotted by accumulating the net number of adorbed molecule expected in each ampling time E[N (Ω rr,t,t T r ] in (4, (2, and (23. The dahed line are plotted uing the derived aymptotic expreion in (5, (9, and (22. The aymptotic analytical line are in precie agreement with the exact analytical curve a T b. The exact analytical curve of the full adorption receiver and the partial adorption receiver converge to their own aymptotic analytical line fater than the convergence of the A& receiver. Interetingly, we find that the analytical curve of the A& receiver decreae after increaing over a few bit interval, and then increae again, while that of the partial adorption receiver ha an increaing trend a time goe large and how a udden jump at a pecific time. The dicontinuitie in the PA curve are caued by the underflow during the evaluation of erfc(x, which reult from the limitation of Matlab mallet poible double. A expected, the aymptotic curve of the partial adorption receiver degrade with decreaing k, a hown in (9. More importantly, the full adorption receiver ha a higher initial accumulation rate but the ame aymptotic number of bound molecule a that of the A& receiver with k = 3 µm/ and k = 2. C. emodulation Criterion In Fig. 5 and 6, we compare our propoed demodulation criterion uing the net number of adorbed molecule with the widely ued demodulation criterion uing the number of cumulatively-adorbed molecule in [3, 9]. In thee two figure, we et the parameter: k = µm/, k = 5, N tx = 3, = µm, r = µm, = 8 µm 2 /, t = 5, T =.2, the bit interval T b =.2, and the number of bit N b = 25. Fig. 5 plot the number of cumulatively-adorbed molecule expected at the urface of the A& receiver in each ampling time due to the tranmiion of multiple bit, wherea Fig. 6 plot the net Fig. 6. Net Number of Adorbed Molecule during Each Sampling Time Sim. Newly adorbed molecule Tranmit Sequence Time ( The net number of adorbed molecule. 4 5 number of adorbed molecule expected at the urface of the A& receiver at each ampling time due to the tranmiion of multiple bit. In both figure, the olid line plot the tranmit equence, where each bit can be bit- or bit-. Note that in both figure, the y-axi value of the tranmit ignal for bit- are zero, and thoe for bit- are caled in order to clearly how the relationhip between the tranmit equence and the number of adorbed molecule. The dahed line are plotted by averaging the number of adorbed molecule over independent emiion for the ame generated tranmit equence in the imulation. In Fig. 5, it i hown that the number of cumulativelyadorbed molecule expected at the urface of the A& receiver increae in bit- bit interval, but can decreae in bit- bit interval. Thi i becaue the new information molecule injected into the environment due to bit- increae the number of cumulatively-adorbed molecule, wherea, without new molecule due to bit-, the deorption reaction can eventually decreae the cumulative number of adorbed molecule. In Fig. 6, we oberve a ingle peak net number of adorbed

12 2 Error Probability - -2 Anal. k= 2, k-= Sim. k = 2, k - = Anal. k= 2, k-= 2 Sim. k= 2, k-= 2 Anal. k = 8, k - = Sim. k = 8, k - = N th Error Probability Anal. A& k = 2, k-= Sim. A& k = 2, k-= Anal. PA k = 2 Sim. PA k = 2 Anal. FA Sim. FA N th Fig. 7. The error probability for the lat tranmit bit-. Fig. 8. The error probability for the lat tranmit bit-. molecule for each bit- tranmitted, imilar to the channel repone for a ingle bit- tranmiion in Fig.. We alo ee a noiier ignal in each bit- interval due to the ISI effect brought by the previou tranmit ignal. To motivate our propoed demodulation criterion, we compare the behaviour of the accumulatively and net change of adorbed molecule at the receiver in Fig. 5 and Fig. 6. We ee that the number of cumulatively-adorbed molecule increae with increaing time, wherea the met number of adorbed molecule have comparable value (between and 5 for all bit- ignal. A uch, the threhold for demodulating the number of cumulatively-adorbed molecule hould be increaed a time increae, while the ame threhold can be ued to demodulate the net number of adorbed molecule in different bit interval. We claim that the received ignal hould be demodulated uing the net number of adorbed molecule. Note that the net number of adorbed molecule refer to the net change, ince the receiver cannot ditinguih between the molecule that jut adorbed and thoe that were already adorbed.. Error Probability Fig. 7 and 8 plot the error probability a a function of deciion threhold for the third bit in a 3-bit equence where the lat bit i bit- and bit-, repectively. The firt 2 bit are. In thee two figure, we et the parameter: N tx = 5, = 5 µm, r = 2 µm, = 5 µm 2 /, t = 6, T =.2, and the bit interval T b =.2. Note that with lower diffuion coefficient and larger ditance between the tranmitter and the receiver, a weaker ignal i oberved. The imulation reult are compared with the evaluation of (27 for bit- and (3 for bit-, where the net number of adorbed molecule expected at the urface of the receiver are approximated by the Skellam ditribution. There are negative threhold with meaningful error probabilitie, thu confirming the need for the Skellam ditribution. The imulation point are plotted by averaging the total error over 5 independent emiion of tranmit equence with lat bit- and bit-. In Fig. 9. Error Probability - Anal. FA Anal. PA k = 3 4 Anal. A& k =, 4 Anal. A& k =, 3 Anal. A& k =, 3 k - = 2 k - = 2 k - = N The error probability for the lat random tranmit bit. both figure, we ee a cloe match between the imulation point and the analytical line. Fig. 7 plot the error probability of the lat tranmit bit- at the A& receiver with N b = 3 bit tranmitted for variou adorption rate k and deorption rate k. We ee that the error probability of the lat tranmit bit- increae monotonically with increaing threholdn th. Interetingly, we find that for the ame k, the error probability improve with increaingk. Thi can be explained by the fact that increaing k increae the amplitude of the net number of adorbed molecule expected (a hown in Fig., which make the received ignal for bit- more ditinguihable than that for bit-. For the ame k, the error probability degrade with increaing k, which i becaue the received ignal for bit- i le ditinguihable than that for bit- with increaing k, a hown in Fig. 2. Fig. 8 plot the error probability of the lat tranmit bit- for different type of receiver with N b = 3 bit tranmitted. The error probability of the full adorption receiver and the partial adorption receiver are plotted uing (32 and (33. We ee th