Nano Communication Networks 1 (2010) 86–95

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Nano Communication Networks

journal homepage: www.elsevier.com/locate/nanocomnet

Energy model for communication via diffusion in nanonetworksMehmet Şükrü Kuran a,∗, H. Birkan Yilmaz a, Tuna Tugcu a, Bilge Özerman ba Department of Computer Engineering, Bogazici University, 34342, Bebek, Istanbul, Turkeyb Department of Biophysics, Istanbul Faculty of Medicine, Istanbul University, 34080, Çapa, Istanbul, Turkey

a r t i c l e i n f o

Article history:Received 6 July 2010Accepted 9 July 2010Available online 16 July 2010

Keywords:Molecular communicationNanonetworksCommunication via diffusionEnergy modelChannel model

a b s t r a c t

Abstract Molecular communication is a new communication paradigm that usesmolecules for information transmission between nanomachines. Similar to traditionalcommunication systems, several factors constitute limits over the performance of thiscommunication system. One of these factors is the energy budget of the transmitter. Itlimits the rate at which the transmitter can emit symbols, i.e., produce the messengermolecules. In this paper, an energy model for the communication via diffusion systemis proposed. To evaluate the performance of this communication system, first a channelmodel is developed, and also the probability of correct decoding of the information isevaluated. Two optimization problems are set up for system analysis that focus on channelcapacity and data rate. Evaluations are carried out using the human insulin hormone as themessengermolecule and a transmitter device whose capabilities are similar to a pancreaticβ-cell. Results show that distance between the transmitter and receiver has a minor effecton the achievable data ratewhereas the energy budget’s effect is significant. It is also shownthat selecting appropriate threshold and symbol duration parameters are crucial to theperformance of the system.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Nanotechnology is a new, emerging field dealing withthe development and manufacturing of nanoscale materi-als and machines. These machines, called nanomachines,are expected to have the capabilities of their higher-scalecounterparts in the nanoscale. In order to perform morecomplex tasks, the nanomachines should be able to com-municate with each other. Communication systems thatenable information exchange between nanomachines arecalled nanonetworks [1]. Molecular communication is oneof these communication systems that focuses on commu-nication via molecules as information carriers. Inspired bythe communication methods used by biological systems

∗ Corresponding author.E-mail addresses: sukru.kuran@boun.edu.tr (M.Ş. Kuran),

yilmhuse@boun.edu.tr (H.B. Yilmaz), tugcu@boun.edu.tr (T. Tugcu),bilgeozerman@hotmail.com (B. Özerman).

1878-7789/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.nancom.2010.07.002

(eukaryotic and prokaryotic cells), a variety of commu-nication systems are proposed in the literature. Amongthese systems, microtubule communication is envisionedfor short range communication [9], ion signaling and com-munication via diffusion are proposed for short tomediumrange communication [14,19] and lastly pheromone, spore,and pollen-based communication for long range commu-nication [11]. In this paper, we focus on the communi-cation via diffusion method in which the information istransmitted via the propagation of messenger moleculesthrough the environment where the transmitter and re-ceiver reside. In the nano- and micro-scales, objectspropagate through the environment following Diffusiondynamics.Similar to higher-scale machines, micro- and nano-

scale machines are also constrained by energy limitations.Unlike battery-powered devices in conventional wirelessnetworking devices, micro- and nano-scale machines areexpected to be self-sufficient regarding energy regula-tion. They are expected to have some means of producing

M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95 87

energy to use in their tasks. In these machines, communi-cation capabilities (e.g., capacity, maximum distance) arealso affected by this energy constraints. However, cur-rent energy models are generally developed for battery-powered machines. Hence, they cannot be directly appliedto micro- and nano-scale machines that use sustainable-energy.In this paper, we propose an energy model for the com-

munication via diffusion system including energy expen-diture for molecule synthesis, production of the secretoryvesicle, and carrying vesicle close to the cell membrane.The energy budget is necessary to take into account for an-alyzing capabilities and achievable performance. To eval-uate the performance of this communication system, wedevelop a channel model, and derived formulas for theprobability of correct decoding of the information in a timeslotted communication system.We consider two consecu-tive time slots where the previous affects the current onefor analyzing interference, misdecoding, and degradationover performance. Two optimization problems are set upfor system analysis that focus on channel capacity and datarate. Performance evaluation is carried out using the hu-man insulin hormone as themessengermolecule and a de-vice whose capabilities are similar to a pancreatic β-cell asthe transmitter.Wemodel a communication system composed of a pair

of devices called ‘‘unit’’s; one as the transmitter and oneas the receiver (Fig. 1). These units have micrometer scalesizes and can be designed by a variety of methods. Inthe nanomachine collection method, units are a collectionof nanomachines working cooperatively to form a device,which is similar to an eukaryotic cell function-wise. Inthe engineered cell method, they are eukaryotic cells withaltered DNAs that perform functions as programmed intheir DNAs. They can also be eukaryotic cells that arecontrolled by nanomachines injected into the cell. Thedesign methodology of the units is a separate researchissue on its own and is out of scope of this paper. Regardlessof its design method, the unit is assumed to have thefollowing properties:

• It can generate energy from raw materials available inthe environment.• It can synthesize proteins and vesicles.• It can synchronize with another unit and communicatewith it using the communication via diffusion method.• It incorporates an internal communication systemsimilar to the microtubule in cells.• It is wrapped inside a protective shielding layer(e.g., cell membrane) to prevent it from dissolving intothe environment.

The unit is also assumed to have several subunits (ornanomachines in the case of the nanomachine collectionmethod) to perform these functions. These subunitsare generally analogous to well known organelles ofeukaryotic cells and are as follows:

• Power Plant: Converts raw materials to provide energyto the unit in a way analogous to the mitochondria.• Factory: Synthesizes proteins from raw amino acids,analogous to the endoplasmic reticulum.

• Packager: Able to wrap several proteins or othermolecules together inside a protective shell and sendthem to their appropriate destinations either inside oroutside the unit, analogous to the golgi apparatus.• Protective Shielding: Protects the unit from dissolvinginto the environment, analogous to the cell membrane.The main contribution of this paper is twofold: first an

energy model that constitutes a constraint on the com-munication capabilities of the unit is developed, second achannelmodel for the communication via diffusion systemis developed.We also derive the successful decoding prob-abilities in the proposed channel model. While analyzingthe performance of the communication via diffusion sys-tem we take into account these proposed models.The rest of the paper is as follows. In Section 2, we

explain communication via diffusion approaches to realizethis communication paradigm and suitable propagationand communication channel models. In Section 3, theproposed energy model for communication via diffusionis given. Section 4 defines several problems that are vitalto communication via diffusion, and the evaluation of theperformances of this communication method is presentedin Section 5. Finally, Section 6 concludes the paper.

2. Communication via diffusion

In communication via diffusion, the information istransmitted between the transmitter and receiver throughthe propagation of certain molecules via diffusion [15].These molecules are called messenger molecules and theycan be chosen as a specific type of protein, peptide, DNAsequence, or other molecular structure. At the shieldinglayer around the receiver, there are receptors that formchemical bonds with messenger molecules when theyare in close proximity. The formation of such a chemicalbond triggers an event inside the unit and thus thetransmission is received. A messenger molecule musthave several properties to be suitable for this kind ofcommunication. First, itmust be intoxic to the componentsof the communication system (i.e. transmitter, receiver,and communication medium). It should also be easilymanufactured by the transmitter, and its building blocksshould be abundantly available in the environment wherethe transmitter resides.In this communication system, information is sent

using a sequence of symbols which are spread oversequential time slots with one symbol in each slot. Thesymbol sent by the transmitter is called the ‘‘intendedsymbol’’, and the symbol received at the receiver is calledthe ‘‘received symbol’’. A received symbol represents one-bit information: ‘‘1’’ if the number of messenger moleculesarriving at the receiver during a time slot exceeds athreshold, ‘‘0’’ otherwise. This communication system canbe affected adversely from Inter Symbol Interference (ISI).Due to the diffusion dynamics, somemessenger moleculesmay arrive after the current time slot. Thus, the receiverincorrectly decodes the received symbol of the next timeslot. This error, caused by ISI, is heavily affected by theselection of the threshold value.

88 M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95

Fig. 1. Communication model.

2.1. Propagation model

The messenger molecules are particles in molecularscale. In this scale, the movement of particles inside a fluidis modeled by Brownian motion. The motion is governedby the combined forces applied to themessengermoleculeby the molecules of the liquid, which are in constantmotion due to random thermal motion. Our model ignorescollisions between the messenger molecules, as done inthe literature for the sake of simplicity [13]. In a onedimensional space, the displacement of a singlemessengermolecule in unit time is a random variable (1X) whichfollows a normal distribution

1X ∼ N (µ, σ 2) (1)

whereµ is taken as 0, andσ is chosen based on the selectedmotion type. The value ofµ reflects the particle’s tendencyto stay still at its current position. Among the variousmotion types in the Brownian motion [17], we choose thegeneral diffusionmodel. The corresponding σ value can becalculated as

σ =√2D1t (2)

where D is the diffusion coefficient (or diffusivity) and 1tis the step time. The diffusion coefficient describes thetendency of the propagating molecules’ to diffuse throughthe fluid. It can be calculated as

D =Kb · Tb

(3)

where Kb is the Boltzmann constant, T is the temperatureof the environment, and b is the drag constant (or thefriction coefficient) of the propagating molecule insidethe given fluid, which depends on the characteristics ofboth the propagating molecule and the type of the fluid.The comparative sizes between the propagating molecule(spm) and the molecules of the fluid (sfluid) also affect thisconstant [20]. If the propagating molecule’s size is similarto the size of the molecules of the fluid, it is consideredas part of the fluid whereas if it is much larger than themolecules of the fluid, the fluid can be considered as a

continuum. Based on these two different conditions, thisconstant is calculated as

b ={4πηrs, if spm ≈ sfluid6πηrs, if spm � sfluid

(4)

where η defines the viscosity of the fluid and rs defines theStokes’ radius of the propagating molecule. Stokes’ radiusof a molecule is different from the radius of the molecule;it is defined as the radius of a sphere whose diffusiondynamics are the same as the molecule in question in thesame environment (such as fluid type, temperature).Following the discussion above, the formula of the

standard deviation of this displacement becomes

σ =

√Kb · T2πηrs

1t, if spm ≈ sfluid√Kb · T3πηrs

1t, if spm � sfluid.

(5)

In our model, the particles propagate through athree dimensional environment. This movement can bemodeled as three independent displacements (one for eachdimension) [17] and the total displacement,−→r , in one timestep can be found as−→r = (1x,1y,1z). (6)

Each messenger molecule and the receiver is as-sumed to have spherical bodies. Whenever a messengermolecule’s body coincides with the body of the receiver,the molecule is received and removed from the environ-ment. A single messenger molecule reaches the receiverbefore a given deadline with a certain probability. Thisprobability is affected mainly by the diffusion coefficientand the distance between transmitter and receiver.

2.2. Communication channel model

In the previous subsections, the basic workings ofcommunication via diffusion and the behavior of themolecule in the medium are explained. To analyze theperformance of this communication method, its channelcapacity should be evaluated; thus, the channel should

M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95 89

be modeled. We assume that, time is divided into equalsized slots in which a single symbol can be sent. Thesetime slots are called symbol durations and denoted by ts.The threshold value for choosing between ‘‘0’’ and ‘‘1’’ isdenoted by τ .Due to the probabilistic behavior of the molecules

exhibiting Brownianmotion,molecules are not guaranteedto reach the receiver; instead they have a probability ofhitting the receiver. This probability depends on variousfactors: the distance between the transmitter and thereceiver (d), ts, environment (Env), and the type of themolecule (Tm). The probability of hitting the receiveris denoted by Phit(d, ts, Env, Tm) or Phit(d, ts) for theshorthand notation where the Env and Tm is invariant.Under the assumption that each molecule propagatesindependently, if n molecules are sent at the start of asymbol duration, the number of molecules receivedwithinthe current symbol duration among these nmolecules (Nc)is a random variable and follows a binomial distribution[4,13],

Nc ∼ Binomial(n, Phit(d, ts)). (7)

The number molecules received in a symbol duration(Nhit) is composed of molecules sent at the start of thecurrent symbol duration (Nc) and at the start of all pastsymbol durations. As we later show in Section 5.2, onlythe previous symbol has a significant effect on the currentsymbol. Np, the number of left over molecules belongingto the previous symbol to the current symbol duration, isalso a random variable that follows the subtraction of twobinomial distributions: the number of molecules receivedwithin two symbol durations and the number ofmoleculesreceived within one symbol duration.

Np ∼ Binomial(n, Phit(d, 2ts))− Binomial(n, Phit(d, ts)).(8)

ABinomial distribution (Binomial(n, p)) can be approx-imated with a normal distribution (N (np, np(1 − p))),when p is not close to one or zero and np is large enough.Therefore the Eq. (8), can be approximated as

Np ∼ N (nPhit(d, 2ts), nPhit(d, 2ts)[1− Phit(d, 2ts)])

−N (nPhit(d, ts), nPhit(d, ts)[1− Phit(d, ts)]). (9)

As the current received symbol is only dependent onthe one-bit information of the current and the previousintended symbols (sc and sp, respectively), in this channelmodel there are four different cases for received symboldecoding. The probability of successfully receiving thecurrent intended symbol in this symbol duration isdependent on these four cases and can be evaluated asbelow. This probability is denoted as PR(p,c) where p andc are the one-bit information represented by the previousand current intended symbols, respectively.Case (sp = 1, sc = 1): The current received symbol

is both affected by the current and previous intendedsymbols. Since some of the molecules sent at the startof the previous symbol duration arrive in this symbolduration, the probability of successfully receiving thecurrent symbol as ‘‘1’’ increases. Thus, regarding signal

Fig. 2. Binary channel model.

reception this is a favorable case.Nhit ∼ N (nP2, n[P2(1− P2)+ 2P1(1− P1)]) (10)

PR(1,1) = P(Nhit ≥ τ)

≈ Q(

τ − nP2√n[P2(1− P2)+ 2P1(1− P1)]

)(11)

where P1 = Phit(d, ts), P2 = Phit(d, 2ts), and Q (.) denotestail probability of standard normal distribution.Case (sp = 1, sc = 0): The molecules overflowing

from the previous symbol duration negatively affectthe successful decoding of the current intended symbol.The new symbol can be detected correctly only if theoverflowing molecule count does not exceed τ . Contraryto the first case, regarding signal reception this is aunfavorable case.Nhit ∼ N (n(P2 − P1), n[P2(1− P2)+ P1(1− P1)]) (12)

PR(1,0) = P(Nhit < τ)= P(Np < τ)

≈ 1− Q(

τ − n(P2 − P1)√n[P2(1− P2)+ P1(1− P1)]

)(13)

where P1 = Phit(d, ts), P2 = Phit(d, 2ts), and Q (.) denotesthe tail probability of standard normal distribution.Case (sp = 0, sc = 1): Since the previous intended

symbol contains a one-bit information of ‘‘0’’, there are nooverflowingmolecules from the previous symbol duration.PR(0,1) = P(Nhit ≥ τ)

= P(Nc ≥ τ)

=

n∑k=τ

(nk

)Pk

1(1− P1)n−k

= IP1(τ , n− τ + 1) (14)where P1 = Phit(d, ts) and Ip(., .) denotes regularizedincomplete beta function.Case (sp = 0, sc = 0): Because τ is always greater than

zero and the current received symbol is not affected by theprevious symbol, the received symbol is always equal tozero in this case.PR(0,0) = 1. (15)

Considering these cases, we model the channel as abinary channel (Fig. 2). Using this model the channelcapacity (C) can be calculated using the PR(p,c) values andthe Bayesian rule for the marginal probabilities.C = max

τI(X, Y )

= maxτ

∑y∈0,1

∑x∈0,1

PX,Y (x, y) log2PX,Y (x, y)PX (x)PY (y)

(16)

where I(X, Y ) stands for mutual information [6,3].

90 M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95

This channel capacity analysis refers to using a sin-gle type of molecule for sending the information. Thethroughput can be increased via the use different types ofmolecules, each representing a single bit of informationas independent channels (molecular diversity). However,a single transmitting unit has a limited energy budget forcommunication. Hence, the number of independent chan-nels that can be used by a transmitting unit is constrainedby this budget.

3. Energy model for communication via diffusion

In a wireless communication system, the energybudgets of the transmitting and receiving units introducea serious limitation over the performance of the system.In classical wireless communication, the transmitter andthe receiver units have a limited battery power, whichshould be used efficiently by the communication systemto maximize the system performance while attainingacceptable operational device lifetime values. However,in a micro-scale unit, due to the device’s scale, thebattery of the unit can store very little energy whichin turn cannot provide the unit with long operationallifetimes. In the literature it is explained that insteadof using batteries, these devices are expected to havesome energy production capabilities [1,7,10]. Thus, energymodels designed for conventional wireless systems are notsuitable for this kind of communication and new modelsare necessary.In this paper, we consider systems in which each unit

is able to produce and store energy. Some of the producedenergy is used by routine activities of the unit while therest is available for communication purposes. In orderto remain operational, the unit should not spend energymore than what it produces and has in its energy reserves.Also, it should not consume energy for communicationpurposes more than its allowance. Otherwise, it cannotprovide energy to some vital system and is exhausted. Ifthere is abundant energy production, the unit can storethat energy for later provided that it has available energystorage capacity. We define the total power produced by aunit as PwT , power used for routine background activities(i.e., activities for maintaining metabolic operations ofthe unit) as PwB, and power available for communicationpurposes as PwC . The total power is divided into two partsas

PwT = PwB + PwC . (17)

In a communication via diffusion systemenergy is spentfor the production of the messenger molecules and theirrelease to the environment. As explained in Section 1, thiscommunication system is already being used by eukaryoticand prokaryotic cells in nature. We model the steps ofthe messenger molecule production and release followingthe techniques used by eukaryotic cells. In molecular cellbiology, this process is known as exocytosis and is mainlycomposed of four steps [2] (Fig. 3):

1. Synthesis of the messenger molecules from theirbuilding blocks.

2. Production of a secretory vesicle.

Fig. 3. Steps of exocytosis.

3. Carrying the secretory vesicles to the cell membrane.4. Releasing themolecules via the fusion of the vesicle andthe cell membrane.

A transmitting unit consumes energy in all of thesesteps. We define the energy cost of synthesis of a singlemessenger molecule as ES , production cost of a vesicle thatwill carry messenger molecules as EV , carrying cost of avesicle as EC , and the extraction cost of a vesicle to theenvironment as EE . Vesicles are much larger in size thanthe messenger molecules, so each vesicle has a capacityof cv messenger molecules.1 Thus, if nmolecules are to besent by the transmitter, the total energy to be spent can befound as

ET = nES +⌈ncv

⌉(EV + EC + EE) (18)

assuming the transmitter fills each vesicle completely.We use a protein based messenger molecule in our

communication model. Proteins are synthesized at theFactory subunit by combining amino acids to form aspecific amino acid chain. The number of amino acidsrequired for a protein depends on the size of the protein.Adding a single amino acid to an amino acid chain has afixed energy cost of 202.88 zJ [5], independent of the typeof the amino acid used. Thus, the total cost to synthesize aprotein that is composed of naa number of amino acids is

ES = 202.88 (naa − 1) zJ. (19)

After the synthesis step is complete, the messengermolecules are passed to the Packager to be packed insidesecretory vesicles. These vesicles are similar to the cellmembrane in structure and can roughly be consideredas hollow spheres whose walls are constructed out of avariety of simple structures, most notably phospolipids.The cost of synthesizing one phospolipids is 1 unit ofATP [12], which in turn equals 83 zJ [10]. In a secretoryvesicle, there are 5 phospolipids in 1 nm2 area [10]. Thus,the total cost of synthesizing a vesicle with a radius of rv is

EV = 83× 5(4πr2v ) zJ. (20)

1 For the sake of simplicity of the model, we assume fixed size vesicles.

M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95 91

Also, the capacity of the vesicle can be approximated by

cv =(

rvrmm√3

)3(21)

where rmm is the radius of the messenger molecule.The vesicle moves along the microtubule from the

Packager to the Protective Shielding with the help of themotor proteins. Motor proteins have two parts; one thatforms a bondwith the vesicle, and the other that forms twobonds with the microtubule. The motor protein releasesone of its bonds with the microtubule with a processcalled phosphorylation, then the released part moves asingle step along the microtubule and again forms thesame bondwith themicrotubule. At each phosphorylation,the motor protein (and its load, the vesicle) travels 8 nmand spends 1 ATP of energy [2]. In an eukaryotic cell, theorganelle analogous to the Packager subunit is roughlyconsidered to be at a distance of half the cell radius tothe cell membrane [2]. Assuming the same organizationalstructure in the unit, the overall cost of this intracelltransportation is

EC = 83⌈runit/28

⌉zJ (22)

where runit denotes the radius of the unit in nanometers.At the last step, the vesicle merges with the cell

membrane with a process called Membrane Fusion andthe messenger molecules inside are released to theenvironment. After being carried to the cell membranevia molecular motors, first the vesicle docks on the cellmembrane, then it is primed, and lastly the merge ofvesicle and the cell membrane occurs. For a single vesicle,roughly 10 ATP of energy is consumed in this step [8].

EE = 83 · 10 zJ. (23)

Using all of the costs explained above, the total cost torelease nmolecules can be written as

ET = 202.88 (naa − 1) n+⌈ncv

⌉(83× 5(4πr2v )

+ 83⌈runit/28

⌉+ 83 · 10

). (24)

The number of molecules that can be produced andreleased in a time slot (ns) is limited by the energyproduced. In a long transmission, the stored energy will bedepleted in a short time. Therefore, we consider the long-term behavior where only the energy produced duringcommunications is used.

PwC ts ≥ 202.88 (naa − 1) ns +⌈nscv

⌉(83× 5(4πr2v )

+83⌈runit/28

⌉+ 83 · 10

)(25)

where the units of Pwc and ts are zeptowatt (zW) andsecond respectively; the units of rv and runit are nm.If nt independent channels are used to increase the

data rate, each with the same number of molecules andthe same threshold value of τ , the maximum number of

independent channels that can be used for a given energybudget can be calculated by using the inequality,

PwC ts ≥ nt

(202.88 (naa − 1) ns +

⌈nscv

⌉(83(4πr2v )

+ 83⌈runit/28

⌉+ 83 · 10

)). (26)

This nt value, combined with the selected ts andthe corresponding channel capacity (C) can be used tocalculate the maximum data rate of the system.

4. Optimization model formulation

So far we have explained the propagation, channel,and the energy model for the communication via diffusionsystem. The behaviors of these models depend on severalparameters. The propagation model is generally based onthe environment and the type of messenger molecule;the channel model is affected by ts, τ , d, and PX (x) valuesand the energy model depends on Pwc, ts, naa, cv, rv , andrunit. While some of these parameters are determined bythe application type (e.g. environment, d), others can beoptimized to achieve better communication performance.For a communication system the maximum value of theI(X, Y ), which is known as C , is a definitive performancemetric. We define an optimization problem that finds thisC value for a single communication via diffusion channelconstrained by the communication energy budget (PwC ) ofthe unit and based on the given values of d, ts, and PX (x).Also, using C , PwC , and the ts values the maximum datarate of the system can be found by using the appropriatent value. We define yet another optimization problem thatcalculates themaximumdata rate

(C ·ntts

)value given a PwC

value.

4.1. Maximizing mutual information

The C value can be evaluated by an optimizationproblem. Since we are not considering the coding aspectsof the communication channel we optimize C under givenPX (x) values. Therefore, the objective function, I(X, Y ),depends on τ and ns:

maximize I(X, Y )τ ,ns

s.t. ns ≤ nmax(Pwc, Tm, ts, 1), ns ∈ Z+ (27)ts = α1d = α2PX (x = 0) = α3 (28)

where ts, d, and PX (x = 0) values are selected as α1, α2,and α3 respectively, and ns is a positive integer be-tween one and the maximum number of moleculesthat can be produced with a given energy budget(nmax(Pwc, Tm, ts, 1)). This value is a function of poweravailable for communicationpurposes, type of themolecule,symbol duration, and the number of independent chan-nels. Since the number of values that can be taken by nsand τ is finite, this problem is solvable.

92 M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95

4.2. Maximizing data rate

Similarly, the maximum data rate(C ·ntts

)of the system

can be found by an optimization problem. The problemis similar to the one given for mutual informationmaximization. However, τ and ns have no degree offreedom. For a given nt value, ns ≤ nmax

nt, and C are

evaluated by finding the optimal τ value (τ ∗) as inSection 4.1. Using this nt , its corresponding C , and theselected ts values,we find the data rate bymultiplyingC ,nt ,and 1

ts. Therefore, the objective function,

(C ·ntts

), depends

on nt :

maximize(C · ntts

)nt

s.t. ns · nt ≤ nmax(Pwc, Tm, ts, nt), ns, nt ∈ Z+ (29)ts = α1d = α2PX (x = 0) = α3 (30)

where ts, d, and PX (x = 0) values are selected as α1, α2,and, α3 respectively, and ns · nt is a positive integerbetween one and the maximum number of moleculesthat can be produced with a given energy budget(nmax(Pwc, Tm, ts, nt)).

5. Performance evaluation

We have developed a simulator for the propagationmodel explained earlier in this paper. Using this simulatorwe evaluate the Phit(d, ts) values at various distances anddetermine appropriate ts values for these distances. Next,we show the effect of τ on the mutual information fordifferent values of d and nmax. Lastly, we calculate themaximum attainable data rate for a given energy budgetvia using different communication channels as explainedin Section 3. In this analysis, we optimize C by adjusting τfor the case PX (x = 0) = PX (x = 1) and are equal to 0.5.

5.1. Simulation parameters and performance metrics

We evaluate the performance of communication viadiffusion system using the human insulin hormone as themessenger molecule and the molecule type specific valuesare selected correspondingly (e.g., Stokes’ radius (rs)). Thepropagation environment is chosen as water in the humanbody temperature. The size of the receiver unit is chosen asan average eukaryotic cell. The energy budget of the unit ischosen following the energy production capabilities of thepancreating β-cell that is responsible for insulin secretion.We assume that the unit uses half of its energy budgetfor communication purposes. These values are given inTable 1.We analyze the channel capacity of the simulated com-

munication systemwith different τ values. As performancemetrics, we use the channel capacity (C), and the maxi-mum achievable data rate via employing molecular diver-sity

(C ·ntts

).

Table 1Simulation parameters.

Parameter Value

# of amino-acids (naa) 51Stokes’ radius of insulin (rs) 2.86 nm [18]Radius of insulin molecule (rmm) 2.5 nm [10]Viscosity of the fluid (η) 0.001 kg

s .mTemperature (T ) 310 KDrag constant (b) 5.391 10−11 kgsDiffusion coefficient (D) 79.4 µm2

sRadius of the receiver 10 µm [2]Radius of the vesicle (rv) 0.05 µm [10,16]Capacity of the vesicle (cv) 1540 InsulinRadius of the transmitter (runit) 10 µm [2]Energy budget for communication (PwC ) 4.5 pW [10]

Fig. 4. Effect of distance on Phit .

5.2. Hitting probability analysis

We evaluate the Phit values for various distancesby taking a very large ts value (30,000 s). The Phitvalues are calculated using the number of moleculesreaching at the receiver in 9000 trials. As seen inFig. 4, the hitting probabilities decrease exponentially withincreasing distance. This behavior is due to the Browniandynamics prevalent in the propagation of the messengermolecules.In order to find appropriate ts values for communication

over different distances, we record the average hittingtimes of the molecules in the simulation. However, asseen in Table 2, the average hitting times for distancesover 8 µm are too large and not suitable to be selectedas the symbol duration of a communication system. Fromthe hitting time histogram in Fig. 5, it is observed thatmost of the molecules arrive in a short time whereas afew molecules arrive after a very long period of time.Therefore, the average hitting time values are increasedto inappropriate values. Based on these histograms, wechoose the ts values as the time before which most ofthe molecules arrive at the receiver. For the rest of theanalysis, we choose ts values as the time before which60% of the molecules arrive at the receiver (Table 3).Since for different distance values the ts values are also

M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95 93

Table 2Average hitting times.

Distance (µm) Average hit time

1 0.5612 1.8644 7.3468 31.29616 107.28332 313.939

Fig. 5. Hitting time histogram cutoff at 80% (d = 16 µm). The lines referto the time before which the mentioned percentage of hitting moleculesarrived at the receiver.

Table 3ts values for different distances.

Distance (µm) ts

1 0.032 0.114 0.408 1.5416 5.932 22.01

chosen differently, we assume that before the start ofthe communication, the communicating pair negotiatesthe ts value according to the distance between theunits.After choosing suitable ts values for different distances,

we recalculate the Phit values using the chosen symboldurations using 150,000 trials. In order to see the numberof molecules surplus to the next couple of symboldurations, we also calculate the Phit values if the symbolduration is selected as 2ts, 3ts, and 4ts. Fig. 6 shows thatafter the second symbol duration, the rest of themoleculesare spread over a wide duration.

5.3. Threshold and channel capacity analysis

Using the ts values selected, we calculate the channelcapacities using different nmax and τ values (where nt =1). Fig. 7 shows the effect of τ over I(X, Y ) where nmaxis fixed as 100 and 500 molecules, respectively. For each

Fig. 6. Hitting probabilities using the chosen ts values.

Table 4Channel capacities for different distances and nmax values.

nmax 2 µm 8 µm 32 µmC τ ∗ C τ ∗ C τ ∗

10 0.642 2 0.542 1 0.322 150 0.935 15 0.8 9 0.603 2100 0.993 31 0.934 20 0.713 7500 0.999 153 0.999 103 0.985 431000 0.999 352 0.999 206 0.999 86

distance and nmax couple, there is an ideal threshold value(τ ∗) which maximizes the mutual information rate. Asseen in the figures τ ∗ and C decrease as the distanceincreases.At low τ values, after the arrival of a few molecules

the receiver decides that the current symbol represents‘‘1’’ whereas at high τ values, a large number of moleculesshould arrive at the receiver for the receiver to decode thecurrent symbol as ‘‘1’’. If τ is selected as τ < τ ∗, PR(1,0)attains a very low value and results in a low I(X, Y ) valuesince the surplus molecules from the previous symbolduration cause passing the threshold. On the other hand,if τ is selected as τ > τ ∗, as the threshold value increases,first both PR(0,1) and PR(1,1) drop. After a certain thresholdvalue these probabilities sharply drop down to zero sincethe probability of an adequate number of moleculespassing the threshold decreases down to zero even withthe help of the molecules surplus from the previoussymbol in the case of PR(1,1). This behavior resultsin the sharp decrease of the I(X, Y ) values in thefigures.Table 4 gives the solution of the optimization problem

defined in Section 4.1 for different distances and usingvarious nmax values. As seen from the table, the capacity Ccan attain very high values at each distance after a certainnmax value. However, with the increase of the distance,the smallest nmax value that achieves high C values alsoincrease. Since Phit values for longer distances are lowcompared to that for shorter distances, the τ ∗ valuesdecrease with distance.

94 M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95

(a) nmax = 100. (b) nmax = 500.

Fig. 7. Effect of τ on the mutual information.

Fig. 8. Effect of distance on data rate.

5.4. Data rate analysis

The energy budget of the unit allocated for communica-tionpurposes is far greater than the energyused for a singlechannel even when channel capacities are high (e.g., C ≥0.999). This surplus energy budget can be utilized forincreasing data rate by using multiple communicationchannels via molecular diversity. Using the optimizationproblem given in Section 4.2, we calculate the maximumdata rate in terms bps over different distances and withthree PwC values (Fig. 8). The reader should note that foreach distance value we use the corresponding ts value asgiven in Table 3.We observe that as the energy budget increases, the

achievable data rate increases as expected. At a distanceof 1 µm, the corresponding ts value is too low (0.03 s).Therefore, in this case PwC ts is very low and adequateonly for small nmax values and few vesicle production. Thisvesicle production cost can be seen as a communicationoverhead and severely limits the usage of molecular

diversity for this distance and results in a low data rate.However, when the energy budget is high enough, as inthe case when PwC = 9.0 pW, this overhead problemis eliminated. At longer distances (d ≥ 2 µm), ts valuesdo not hinder the effective use of molecular diversity. Thedecrease in the Phit due to distance is compensated by theincrease in PwC ts due to the increase in ts. Therefore, thedata rate is only slightly affected by the distance.

6. Conclusion

In this paper, we explain how the communication viadiffusion systemworks and propose a communication andan energy model for this system. Following the literature,we use a time slotted structure in our communicationsystem model. In order to analyze the performance of thissystem, we define two optimization problems for channelcapacity and data rate. We solve these problems usinghuman insulin hormone as the messenger molecule anda transmitting unit whose capabilities are similar to thepancreatic β-cells.Based on our evaluations, we have seen that the symbol

duration (ts) of the system is greatly affected by the dis-tance between the transmitter and the receiver. We alsosee that only the previous symbol has a significant ISI ef-fect on the current symbol. Hence, we model the channelaccordingly and derive the successful reception probabili-ties of the current symbol. Next, we show that the selectionof the appropriate threshold value is crucial for the chan-nel capacity. Finally, we show howmuch molecular diver-sity can be utilized subject to energy budget in order toimprove the data rate. After a certain distance and energybudget for communication purposes, the achievable datarate is only slightly affected by the distance and mostlyconstrained by the energy budget of the unit.

Acknowledgements

This work has been supported by the State Planning Or-ganization (DPT) of Republic of Turkey under the project

M.. Kuran et al. / Nano Communication Networks 1 (2010) 86–95 95

TAM,with the project number 2007K120610 and by Scien-tific and Technical Research Council of Turkey (TUBITAK).We also thank Yigit Sevinc for his invaluable commentsand contributions during the technical discussions for thispaper, and Emrecan Cakir for his assistance in the illustra-tions of this paper.

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Mehmet Şükrü Kuran received his B.S. degreein Computer Engineering from Yildiz Techni-cal University, Turkey, in 2004 and his M.S.degree in Systems and Control Engineeringfrom Bogazici University, Turkey, in 2007. Heis currently studying as a Ph.D. student in theComputer Engineering Department of BogaziciUniversity. He also works as a research assistantin the same department. His research interestsare molecular communications, nanonetworks,WiMAX networks, MAC layer mechanisms, and

performance analysis of wireless LAN and wireless MAN.

H. Birkan Yilmaz received his B.S. degree inMathematics in 2002 and received M.Sc degreein Computer Engineering in 2006 from BogaziciUniversity. He is pursuing his Ph.D. study inComputer Engineering at Bogazici Universitysince 2006. He also worked as a teaching as-sistant in Mathematics Department. He holdsTUBITAK (The Scientific and Technological Re-search Council of Turkey) National Ph.D. Schol-arship and is amember of IEEE and TMD (TurkishMathematical Society). His research interests in-

clude molecular communication, cognitive radio, spectrum sensing, nextgeneration wireless systems, mathematical modeling, Markov models,routing in ad-hoc networks, cryptography, algebraic structures.

Tuna Tugcu received his B.S. and Ph.D. degreesin Computer Engineering from Bogazici Univer-sity in 1993 and 2001, respectively, his M.S. de-gree in Computer and Information Science fromthe New Jersey Institute of Technology in 1994.He worked as a post-doctoral fellow and visitingprofessor at Georgia Institute of Technology. Heis currently an associate professor in the Com-puter Engineering Department of Bogazici Uni-versity. His research interests include WiMAX,cognitive radio networks, wireless sensor net-

works, and molecular communications.

Bilge Özerman received her B.S. degree in Bi-ology from Istanbul University, Turkey, in 1999,her M. S. Degree in Neuroscience from IstanbulUniversity, in 2002, and her Ph.D. degree in Bio-physics in Istanbul University, in 2009. Her re-search interests include protein-protein interac-tions; intracellular actions of diphtheria toxin,actin based intracellular communication,molec-ular mechanisms of ischemia and receptor-ligand interactions in experimental models ofepilepsy.