A Molecular Communication System Model

Based on Biological Circuits

Massimiliano Pierobon

Department of Computer Science and Engineering

University of Nebraska-Lincoln

Lincoln, NE, USA

maxp@unl.edu

ABSTRACTMolecular Communication (MC), i.e., the exchange of in-formation through the emission, propagation, and receptionof molecules, is a promising paradigm for the interconnec-tion of autonomous nanoscale devices, known as nanoma-chines. Synthetic biology techniques, and in particular theengineering of biological circuits, are enabling research to-wards the programming of functions within biological cells,thus paving the way for the realization of biological nanoma-chines. The design of MC systems built upon biologicalcircuits is particularly interesting since cells naturally em-ploy the MC paradigm in their interactions, and possessmany of the elements required to realize this type of com-munication. This paper focuses on the identification andsystems-theoretic modeling of a minimal subset of biologicalcircuit elements necessary to be included in an MC systemdesign where the message-bearing molecules are propagatedvia free di↵usion between two cells. The system-theoreticmodels are here detailed in terms of transfer functions, fromwhich analytical expressions are derived for the attenuationand the delay experienced by an information signal throughthe MC system. Numerical results are presented to eval-uate the attenuation and delay expressions as functions ofrealistic biological parameters.

Categories and Subject DescriptorsC.2.1 [Computer-Communication Networks]: NetworkArchitecture and Design—Wireless Communication; H.1.1[Models and Principles]: Systems and Information The-ory—Information Theory

General TermsTheory

KeywordsMolecular Communication; Nanonetworks; Synthetic Biol-ogy

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1. INTRODUCTIONThe realization of communication systems and networks

based on the exchange of message-bearing molecules, oftenreferred to as molecular communication systems, promise tobe a key enabling technology to realize the interconnectionof autonomous nanotechnology-enabled devices, also knownas nanomachines [2]. The potential applications of this typeof systems range from the biomedical [12], to the indus-trial and surveillance fields [23]. The design of a molecularcommunication system has to necessarily take into accountthe underlying type of nanomachines for which the systemis intended. In this direction, we envision a particularlypromising type of nanomachines based on key synthetic bi-ology techniques [21], which allow to genetically engineercells, such as bacteria, and program functionalities in thebiological environment. In particular, a synthetic biologi-cal circuit [19], or simply biological circuit, allows to pro-gram logical functions from simple controlled production ofspecific types of protein molecules, to complete engineeredcell-to-cell interactions [18], such as a di↵usion-based molec-ular communication system, in a similar way as it is donewith electrical circuits. The focus of this paper is on thestudy from the communication engineering point of view ofa molecular communication system based on biological cir-cuits where the message-bearing molecules are propagatedvia free di↵usion between two genetically-engineered cells, atransmitter and a receiver.

A biological circuit is normally defined as a genetic reg-ulatory network [19] embedded in a biological cell, whereDNA genes are linked together by activation and repres-sion mechanisms that regulate their expression into proteins,which are biological macromolecules. Each DNA gene con-tains coding sequences, which are chemical information forbuilding proteins, and regulatory sequences, which are siteswere proteins can bind and control the rate of the gene ex-pression, either by increasing (activation) or decreasing (re-pression) the protein building rate. In biological circuits,genes are interconnected such as the proteins produced byone or more genes regulate the expression of one or moregenes. In recent years, a great e↵ort is being devoted to thestandardization and the establishment of catalogues of bio-logical circuit parts [7]. By following the BioBrickTM stan-dard [5], the units to measure the input and the output ofa biological circuit are defined as Polymerases Per Second(PoPS), which correspond to the rates of the transcriptionprocess of the first and last biological circuit genes, respec-tively, proportional to their rate of expression. A biologicalcircuit can process a PoPS signal as a function of the time in

http://dx.doi.org/10.1145/2619955.2619958

input by returning in output another PoPS signal as a func-tion of the time through the aforementioned interconnectionof gene regulations.

Some recent literature can be found on the analytical mod-eling of biological circuits, but with no specific mention todi↵usion-based cell-to-cell communication through moleculeexchange, for which only a biological description is providedin some specific works. Notable examples from this litera-ture are given as follows. In [19] the genetic circuit design isintroduced as an engineering discipline and the main math-ematical framework for the modeling of biological circuitfunctions is introduced. The models of some important bio-logical circuit patterns, called network motifs, are presentedin a very complete theoretical framework in [3]. The stan-dardization e↵orts of biological circuit parts are reviewedin [5], while the modeling techniques for biological circuitsare discussed in [4]. The frequency domain analysis of bio-logical circuits is presented in [8] both from a deterministicand a stochastic point of view, while the noise in biologicalcircuit is discussed in [22]. In [16], the specific noise sourcesa↵ecting cellular signaling pathways are described. Finally,the work in [21] treats engineering techniques to implementsignals and sensors in bacteria through biological circuits.

In this paper, a systems-theoretic communication engi-neering model is presented for a biological circuit where asignal is transmitted from a PoPS input in a biological cell(transmitter cell) to a PoPS output in another biologicalcell (receiver cell), located at a predefined distance fromthe transmitter cell. This biological circuit, inspired by thecell-to-cell communication circuit sketched in [19], realizes adi↵usion-based molecular communication system by encod-ing the signal to be transmitted into signaling molecules,which propagate between the transmitter cell and the re-ceiver cell through their di↵usion in the intercellular space.In addition, the biological circuit detailed in this paper iscomposed by the minimal subset of elements necessary torealize di↵usion-based molecular communication between bi-ological cells, and the resulting models are expected to havea general validity over other more complex implementations.

This paper is organized as follows. In Section 2 a biologicalcircuit for molecular communication is identified through aminimal subset of elements. In Section 3 a systems-theoreticmodel is detailed in terms of transfer functions, from whichanalytical expressions are derived for the attenuation andthe delay experienced by an information signal through thebiological circuit. In Section 4 we present some numericalresults obtained by applying to the developed models somerealistic biological parameters from the literature. Finally,in Section 5 we conclude the paper.

2. A BIOLOGICAL CIRCUITFOR MOLECULAR COMMUNICATION

2.1 Functional Blocks DescriptionThe main functional blocks of this biological circuit are

shown in Figure 1, where a space is divided into the intracel-lular environments of a transmitter cell and a receiver cell,respectively, which are assumed chemically homogeneous,or well-stirred, and they are divided by an intercellular en-vironment. As a consequence, in the intracellular environ-ment the molecule concentrations are assumed homogeneousin the space, while in the intercellular environment there

is in general a non-homogeneous concentration of signalingmolecules, which is subject to propagation via di↵usion. Weassume that the intracellular space of the transmitter cellis a volume with size ⌦Tx, while the intracellular space ofthe receiver cell is a volume with size ⌦Rx. The main func-tional blocks of this biological circuit, shown in Figure 1, aredetailed as follows:

Figure 1: Main functional blocks of a biological cir-cuit for di↵usion-based molecular communication.

• The Signaling Enzyme Expression takes place in-side the transmitter cell, and it is initiated by a PoPSsignal in input, PoPSin, which promotes the transcrip-tion of an enzyme coding sequence and the translationof the contained information into a protein, denoted byE and called enzyme because of its specific chemicalfunction, as explained in the following. The output ofthe signaling enzyme expression is the concentrationof the produced enzymes, denoted by [E].

• The Signaling Molecule Production is an enzy-matic chemical reaction that occurs inside the trans-mitter cell, where the enzymes E catalyze the con-version of molecules present in the intracellular envi-

ronment, called substrates, into other molecules, calledproducts, by forming enzyme-substrate complexes. A-mong these products, the signaling molecules, denotedby S, are small organic molecules whose size allowsthem to cross the cell membrane and propagate throughdi↵usion in the intercellular environment. The otherproducts of the enzymatic reaction, denoted here assubproducts, remain in the intracellular environmentand do not take part in the di↵usion-based molecularcommunication. As a consequence, the input of thesignaling molecule production is the concentration ofenzymes [E], while the output is the concentration ofproduced signaling molecules at the transmitter, de-noted by [S]Tx.

• The Di↵usion Process realizes the propagation ofthe signaling molecules S in the intercellular environ-ment, and it is the macroscopic e↵ect of the randomBrownian motion of the signaling molecules in the spa-ce. The di↵usion process has the e↵ect to propagatedi↵erences in the signaling molecule concentration from

the transmitter cell to the receiver cell, where theycross the membrane and have access to the receiverintracellular environment. The input of the di↵usionprocess is the concentration [S]Tx of signaling moleculesat the transmitter cell, while the output is the concen-tration [S]Rx of signaling molecules at the receiver cell.

• The Receptor Activator Expression takes placeinside the receiver cell, and it is initiated by an in-put PoPS auxiliary signal, PoPSaux, which promotesthe transcription of a receptor coding sequence and thetranslation of the contained information into proteins,called receptors, and denoted by R. The output ofthe receptor activator expression is the concentrationof the produced receptors, denoted by [R].

• The Ligand-Receptor Binding is a reaction thatoccurs inside the receiver cell, where the incoming sig-naling molecules S bind to the receptors R and formactivator complexes, denoted by RS. The inputs ofthe ligand-receptor binding are the concentration ofproduced receptors [R] and the concentration [S]Rx ofsignaling molecules at the receiver cell, and the outputis the concentration [RS] of activator complexes.

• The Output Transcription Activation is initiatedby the activator complexes RS upon binding to theactivator site, where a PoPS output signal is producedaccording to the binding of RNA polymerase proteins,denoted as RNAP , to the promoter sequence. Theinputs of the transcription activation are the concen-tration [RS] of activator complexes, the concentration[PRx] of promoter sequences, and the concentration[RNAP ] of the RNA polymerase protein, respectively,while the output PoPS signal is denoted as PoPSout.

2.2 Reaction-based DescriptionIn the following, we provide a description of the biological

circuit in terms of the chemical reactions undergoing in theaforementioned elements. This description serves to defineall the chemical parameters of the biological circuit understudy, and it sets the basis to build the systems-theoreticmodel detailed in Section 3.

The Signaling Enzyme Expression is based on a tran-

scription and translation reaction, which models the pro-duction of the npE enzymes stimulated by the input signalPoPSin with a rate kE , expressed as follows:

PoPSinkE��! npEE + PoPSin . (1)

The enzymes are also subject to degradation, with a degra-dation rate kd

E

, expressed as:

Ekd

E���! () . (2)

The Signaling Molecule Production is based on anenzymatic reaction where the enzyme E and the substrates(one or more), here denoted as S0, according to the ratekS1 form a complex Cs, which can then either dissociateback into the enzyme E and the substrates S0, with a ratekS�1 , or evolve into the sum of the enzyme E and the sig-naling molecule S according to a rate kS2 . This reaction isexpressed as follows:

E + S0

kS1 ��!

kS�1

CS

kS2��! E + S . (3)

The Di↵usion Process is based on the assumption tohave a 3-dimensional intercellular space, which contains afluidic medium and has infinite extent in all the three di-mensions. The di↵usion process is based on the followingDi↵usion Equation [20, 9] in the variable [S](r, t), whichis the concentration [S] of signaling molecules present at dis-tance r from the transmitter and time instant t:

@[S](r, t)@t

= Dr2[S](r, t) , (4)

where @(.)/@t and r2(.) are the time first derivative andthe Laplacian operator, respectively. D is the di↵usion co-e�cient and it is considered a constant parameter withinthe scope of this paper. This is in agreement with the as-sumption of having independent Brownian motion for everymolecule in the space.

The Receptor Activator Expression is based on thetranscription and translation reaction for the production ofnpR receptors R stimulated by the signal PoPSaux with arate kR, expressed as follows:

PoPSauxkR��! npRR+ PoPSaux . (5)

The degradation reaction of the receptors R is expressed asfollows according to a degradation rate kd

R

:

Rkd

R���! () . (6)

The Ligand-Receptor Binding is based on the bind-ing and release reactions between receptors R and signalingmolecules S. Upon binding, which occurs with a rate kRS ,a receptor R and a signaling molecule S form an activatorcomplex RS, which will be the input of the next transcrip-tion activation reaction. A complex RS unbinds and releasesa receptor R and a signaling molecule S according to a ratek�RS . This is expressed as follows:

R+ SkRS ��!

k�RS

RS . (7)

The Output Transcription Activation is based on anopen complex formation reaction, where an activator com-plex RS, a promoter sequence PRx, and an RNA poly-merase RNAP trigger the open complex formation, quanti-fied through the output signal PoPSout, according to a ratekRx. The open complex can dissociate back into an acti-vator complex RS, a promoter sequence PRx, and an RNApolymerase RNAP according to a rate k�Rx. This has thefollowing expression:

RS + PRx +RNAPkRx ��!

k�Rx

PoPSout . (8)

In the following, with reference to the aforementionedchemical reactions, we detail the systems-theoretic modelof this biological circuit, which allows to derive the trans-fer function and, consequently, the attenuation and delayparameters for each functional block and for their overallend-to-end cascade.

3. SYSTEMS-THEORETIC MODELThe objective of the systems-theoretic model is to derive

the mathematical relation between the input signal PoPSin

(t) and the output signal PoPSout(t) of the aforementionedbiological circuit for di↵usion-based molecular communica-tion, where the input and output signals are function of the

time t. As detailed in the following, we express this mathe-matical relation in terms of transfer function H(!), where !corresponds to the frequency of the Fourier transforms [10]of the signals, namely, PoPS

in

(!) and PoPSout

(!), ex-pressed as follows:

H(!) =PoPS

out

(!)PoPS

in

(!), PoPS

i

(!) =

ZPoPS

i

(t)e�j!tdt ,

(9)where i 2 {in, out}, and H(!) depends from all the chemi-cal parameters defined in Section 2, namely, the transmittercell volume ⌦Tx and the receiver cell volume ⌦Rx, the reac-tion rates kE ,kd

E

,kS1 ,kS�1 ,kS2 ,kR,kdR

,kRS ,k�RS ,kRx,k�Rx

and numbers of produced molecules npE and npR, the di↵u-sion coe�cient D, the auxiliary signal PoPSaux, assumedconstant in time, the concentration of substrates [S0] atthe transmitter cell, and the concentrations of promoter se-quence [PRx] and RNA polymerase [RNAP ] at the receivercell.

Figure 2: Decomposition of the transfer functionof a biological circuit for di↵usion-based molecularcommunication into the transfer functions of eachfunctional block.

As explained in the following and graphically shown inFigure 2, the linearity property of the mathematical expres-sions of the chemical reactions described in Section 2.2 al-lows the decomposition of the transfer function H(!) intothe cascade of the transfer functions of each functional block,as shown in Figure 1. This decomposition has the followingexpression:

H(!) =HA(!)HB(!)HC(!)HD(!)HE(!)HF (!)[S0]··PoPSaux[PRx][RNAP ] , (10)

whereHX(!), X 2 {A,B,C,D,E, F}, are the transfer func-tions of each functional block, as function of the frequency!, detailed in the following. The parameters [S0], PoPSaux,[PRx] and [RNAP ] are the concentration of substrates atthe transmitter cell, and the auxiliary input signal, the con-centration of promoter sequences and the concentration ofRNA polymerase at the receiver cell, respectively, assumedconstant in time for the scope of this paper.

In the following, we analytically derive the transfer func-tion of each functional block shown in Figure 2 from thereaction-based description provided in Section 2.2. Subse-quently, we provide an approximation H(!) of the transferfunction of the biological circuit through considerations onthe di↵erences in the time scales of the chemical reactionsof di↵erent functional blocks. Finally, starting from the ex-pression of the approximated transfer function H(!) of thebiological circuit, we provide analytical expressions for theattenuation and delay experienced by an information signalthrough the biological circuit.

3.1 Functional Block Transfer Functions

The transfer function of each functional block, with refer-ence to Figure 2, is analytically derived by applying the Clas-sical Chemical Kinetic (CCK) modeling [6] to the reaction-based description provided for each block in Section 2.2.

The CCK model of the Signaling Enzyme Expressionis expressed through the following Reaction-Rate Equation(RRE), which analytically models the chemical reactionsin (1) and (2):

d[E](t)dt

= npEkEPoPSin(t)� kdE

[E](t) , (11)

where [E](t) and PoPSin(t) are the concentration of pro-duced enzymes inside the transmitter cell and the input sig-nal, respectively, as functions of the time t. By applying theFourier transform [10] to (11), we obtain the following:

j = npEkEPoPSin

(!) + kdE

[E](!) , (12)

As a consequence, the transfer function HA(!) of the signal-ing enzyme expression functional block is derived by solv-ing (12) with respect to the concentration of produced en-zymes [E](!) as function of the PoPS

in

(!), expressed as

HA(!) =npEkE

j! + kdE

, (13)

where npE , kE , and kdE

are the number of enzymes pro-duced per reaction, the enzyme expression rate, and the en-zyme degradation rate, respectively.

The Signaling Molecule Production is expressed thro-ugh the following two RREs, which analytically model thechemical reactions in (3):

d[CS ](t)dt

= kS1 [E](t)[S0]� kS�1 [CS ](t)� kS2 [CS ](t)

d[S]Tx(t)dt

= kS2 [CS ](t) , (14)

where [CS ](t), [E](t), and [S]Tx(t) are the concentration offormed complexes, produced enzymes and produced signal-ing molecules inside the transmitter cell, respectively, asfunctions of the time t, and [S0] is the concentration ofthe substrates, assumed constant in time. By applying theFourier transform [10] to (14) and by substituting the firstexpression in the second expression, we obtain

j![S]Tx

(!) =kS1

kS�1 + kS2 + j![S0][E](!) . (15)

Starting from (15), by expressing the concentration of pro-duced signaling molecules [S]

Tx

(!) as function of the pro-duced enzymes [E](!) and the frequency !, we derive theexpression of the transfer function HB(!) of the signalingmolecule production functional block as follows:

HB(!) =kS1

!{j(kS�1 + kS2)� !} , (16)

where kS1 , kS�1 , and kS2 are the complex formation rate,the complex dissociation rate, and the signaling moleculeproduction rate, respectively.

TheDi↵usion Process functional block is expressed thro-ugh the Inhomogeneous Di↵usion Equation, which is basedon the di↵usion equation expression in (4), as follows:

@[S](r, t)@t

= Dr2[S](r, t) +d[S]Tx(t)

dt�(r) , (17)

where [S](r, t) and d[S]Tx

(t)dt

are the concentration of signal-ing molecules present at distance r from the transmitter andthe first time derivative of the concentration of signalingmolecules at the transmitter, respectively, as function of thetime t. �(r) is a Dirac delta centered at the transmitter loca-tion and D is the di↵usion coe�cient. The solution of (17)in terms of Fourier transform [10] of the concentration ofsignaling molecules [S]

Rx

(!) at the receiver, located at adistance rRx from the transmitter, as function of the pro-duced signaling molecules [S]

Tx

(!), is as follows [17]:

[S]Rx

(!) =e�(1+j)

p!

2D rRx

⇡DrRxj![S]

Tx

(!) . (18)

As a consequence, the expression of the transfer functionHC(!) of the di↵usion process functional block is as follows:

HC(!) = j!e�(1+j)

p!

2D rRx

⇡DrRx, (19)

where D and rRx are the di↵usion coe�cient and the dis-tance of the receiver from the transmitter, respectively.

The CCK model of the Receptor Activator Expres-sion is expressed through the following RRE, which analyt-ically models the chemical reactions in (5) and (2):

d[R](t)dt

= npRkRPoPSaux � kdR

[R](t) , (20)

where [R](t) and PoPSaux are the concentration of recep-tors inside the receiver cell as functions of the time t andthe PoPS signal that controls the receptor expression, re-spectively. Since we assume that the auxiliary input signalPoPSaux is constant in time, the resulting concentration ofreceptors inside the receiver cell is also constant in time. Bysolving (20), the expression of the transfer function HD(!)of the di↵usion process functional block is as follows:

HD(!) =npRkRkd

R

, (21)

where npR, kR, and kdR

are the number of receptors pro-duced per reaction, the receptor expression rate, and thereceptor degradation rate, respectively.

The Ligand-Receptor Binding has a RRE CCK modelwhich derives from the chemical reaction expression in (7),and it is as follows:

d[RS](t)dt

= kRS [R](t)[S]Rx(t)� k�RS [RS](t) , (22)

where [RS](t), [R](t) and [S]Rx(t) are the concentration ofactivator complexes, receptors and signaling molecules in-side the receiver cell, respectively, as functions of the timet. By applying the Fourier transform [10] to (22), we ex-press the concentration of activator complexes [RS](!) asfunction of [R](!), [S](!), and the frequency ! as follows:

j = ([R](!) ⇤ [S](!)) kRS � k�RS [RS](!) , (23)

where . ⇤ . is the convolution operator [10]. As explainedabove, we assume a constant auxiliary input signal PoPSaux

is constant in time, which results in a constant concentra-tion of receptors inside the receiver cell. As a consequence,the expression of the transfer function HE(!) of the ligand-receptor binding functional block is as follows:

HE(!) =kRS

j! + k�RS, (24)

where kRS and k�RS are the ligand-receptor binding andrelease rates, respectively.

The CCK model of the Output Transcription Activa-tion functional block is expressed through the RRE, which isderived from the description of the chemical reaction in (8).This RRE has the following expression:

dPoPSout(t)dt

= kRx[PRx][RNAP ][RS](t)� k�Rx[RS](t) ,

(25)where PoPSout(t) and [RS](t) are the biological circuit out-put PoPS signal and the concentration of activator com-plexes inside the receiver cell, respectively, as functions ofthe time t, and [PRx] and [RNAP ] are the concentrationsof promoter sequences and RNA polymerase at the receivercell, respectively, assumed constant in time. The expres-sion in (22) is solved in the same way as done for the sig-naling enzyme expression functional block in (12). Finally,the expression of the transfer function HF (!) of the outputtranscription activation functional block is as follows:

HF (!) =kRx

j! + k�Rx, (26)

where kRx, and k�Rx are the open complex formation anddissociation rates, respectively.

Since the RRE expression of the functional blocks in (11),(14), (17), (20), (22), and (22) are Ordinary Di↵erentialEquations (ODE), they represent Linear Time-Invariant sys-tems, whose transfer function solutions can be combinedthrough the formula in (10) to derive the transfer functionH(!) of a biological circuit for di↵usion-based molecularcommunication, expressed as

H(!) =npEkE

j! + kdE

kS1 [S0]!{j(kS�1 + kS2)� !} j!

e�(1+j)p

!

2D rRx

⇡DrRx·

· npRkRkd

R

PoPSauxkRS

j! + k�RS·

· kRx

j! + k�Rx[PRx][RNAP ] , (27)

where npE , kE , and kdE

are the number of enzymes pro-duced per reaction, the enzyme expression rate, and the en-zyme degradation rate, respectively, kS1 , kS�1 , and kS2 arethe complex formation rate, the complex dissociation rate,and the signaling molecule production rate, respectively, Dand rRx are the di↵usion coe�cient and the distance of thereceiver from the transmitter, respectively, npR, kR, andkd

R

are the number of receptors produced per reaction, thereceptor expression rate, and the receptor degradation rate,respectively, PoPSaux is the auxiliary input signal, assumedconstant in time, kRS and k�RS are the ligand-receptorbinding and release rates, respectively, and kRx, and k�Rx

are the open complex formation and dissociation rates, re-spectively. [S0], [PRx] and [RNAP ] are the concentrationsof substrates at the transmitter cell, promoter sequences andRNA polymerase at the receiver cell, respectively, assumedconstant in time.

3.2 Time Scale ApproximationAccording to [3], the chemical reactions involved in bi-

ological circuits have di↵erent time scales. In particular,the chemical reactions described in Section 2.2, and mod-eled through the transfer function expressions in Section 3.1,occur at significantly di↵erent speeds. As experimentally

demonstrated in [3], the chemical reactions where a proteinis expressed from the DNA coding sequence and accumu-lates/propagates in the space, such as the signaling enzymeexpression, the receptor activator expression, and the dif-fusion process, are significantly slower than the reactionsbetween two or more molecules for the formation of com-plexes, such as in the signaling molecule production, theligand-receptor binding and the output transcription activa-tion. Therefore, the former reactions dominate the dynamicbehavior of the circuit, and the transfer functions of the lat-ter reactions can be approximated with their steady stateversions, as shown in Figure 3 and analytically derived inthe following.

Figure 3: Approximation of the decomposition ofthe transfer function of a biological circuit fordi↵usion-based molecular communication.

As a result, we define H(!) as the approximate transferfunction of the biological circuit, derived through consider-ations on the chemical reaction time scales. The transferfunction H(!) is expressed as follows:

H(!) =HA(!)KB(!)[S0]HC(!)HD(!)·· PoPSauxKEKF [PRx][RNAP ] , (28)

where HA(!), HC(!), and HD(!) are the transfer func-tions of the signaling enzyme expression, the receptor ac-tivator expression, and the di↵usion process, respectively,[S0], PoPSaux, [PRx] and [RNAP ] are the concentrationsof substrates at the transmitter cell, the auxiliary input,the concentration promoter sequences and the concentrationRNA polymerase at the receiver cell, respectively, assumedconstant in time, KB(!) is the steady state transfer functionof the signaling molecule production, KE , and KF are thesteady state approximations to constant values of the trans-fer functions of the ligand-receptor binding and the outputtranscription activation, respectively.The steady state approximation KB(!) of the Signal-

ing Molecule Production functional block is computedby setting in (14) the first time derivative d[CS ](t)/dt in theconcentration of formed complexes to 0. The solution to (14)becomes as follows:

d[CS ](t)dt

= 0! [S]Tx

(!) =kS1

j!(kS�1 + kS2)[S0][E](!) .

(29)The steady state transfer function KB(!) of the signalingmolecule production is therefore given by

KB(!) =kS1

j!(kS�1 + kS2), (30)

where kS1 , kS�1 , and kS2 are the complex formation rate,the complex dissociation rate, and the signaling moleculeproduction rate, respectively.

The steady state approximations of the Ligand-ReceptorBinding and theOutput Transcription Activation func-tional blocks to the constant values KE and KF result fromcomputing the transfer functions HE(!) and HF (!) for avalue of the frequency ! = 0, expressed as

KE =kRS

k�RS, KF =

kRX

k�RX, (31)

which correspond to the solution of (22) and (25) when weset to 0 the time first derivative d[RS](t)/dt in the concen-tration of activator complexes and the time first derivativedPoPSout(t)/dt in the biological circuit output PoPS signal.

The approximate transfer function H(!) of the biologicalcircuit, derived through the steady state approximations,has the following expression:

H(!) = KH

e�(1+j)p

!

2D rRx

j!(j! + kdE

), (32)

where the constant KH is as follows:

KH =npEkEkS1 [S0]npRkRPoPSauxkRSkRx[PRx][RNAP ]

(kS�1 + kS2)⇡DrRxkdR

k�RSk�Rx,

(33)where all the parameters are the same as in (27).

3.3 Attenuation and Delay ExpressionsThe attenuation and delay experienced by a signal through

the biological circuit are analytically derived from the ap-proximate transfer function H(!) expressed in (32),

The attenuation ↵(!), as function of the frequency !, iscomputed through the reciprocal of the absolute value of theapproximate transfer function H(!) in (32), which has thefollowing expression:

↵(!) =1���H(!)

���=

!q

!2 + k2dE

KHe�p

!

2D rRx

, (34)

where kdE

is the enzyme degradation rate, D is the di↵u-sion coe�cient, rRx is the distance of the receiver from thetransmitter, and KH is given in (33).

The delay �(!), as function of the frequency !, is com-puted as the frequency first derivative of the phase �H(!) of

the approximate transfer function H(!) in (32). The phase�H(!) has the following expression

�H(!) = arctan

✓kd

E

!

◆�

r!

2DrRx , (35)

As a consequence, the delay �(!) is expressed as

�(!) = �d�H(!)

d!=

rRx!2 + 2kd

E

p2D! + rRxk

2dE

2p2D!

⇣!2 + k2

dE

⌘ ,

(36)where rRx, kd

E

, and D are the distance of the receiver fromthe transmitter, the enzyme degradation rate, and the dif-fusion coe�cient, respectively.

4. NUMERICAL RESULTSIn this section, we present some preliminary numerical

results obtained through the evaluation of the expressionsof the systems-theoretic model of the biological circuit fordi↵usion-based MC analyzed in this paper.

In the synthetic biology and biological circuit engineer-ing literature, there are currently very few works that focuson the joint experimental determination of the biochemicalparameters of a complete biological circuit implementation,such as the parameters we introduced in Section 2.2. Mostof the results presented in the literature focus on the studyof one specific element or biochemical reaction rather than acomplete architecture. As a consequence, the values of thebiochemical parameters used for our numerical results aretaken from a diverse pool of papers and, although they sat-isfy the goal of having a realistic order of magnitude, theydo not necessarily capture the values that they would havein a real implementation of the biological circuit we analyzehere.

For the numerical results obtained from the system-theore-tic model, presented in the following, we applied the fol-lowing parameter values from the LuxR-LuxI quorum sens-ing system [13] in E. coli bacteria, which has been alreadyused for the engineering of a biological circuit for di↵usion-based molecular communication, as in [1]. The rates ofsignaling enzyme and receptor activator translation equalto the rate of Lux protein translation from [15], namely,kE = kR = 9.6x10�1 min�1, the rate of signaling enzymedegradation equal to the degradation rate of LuxI proteinin [15], namely, kd

E

= 1.67x10�2 min�1, the rate of recep-tor degradation equal to the degradation rate of LuxR pro-tein in [15], namely, kd

R

= 2.31x10�2 min�1, the complexformation rate from the binding of the signaling enzymesand the substrates equal to the forward LuxI-substrates re-action for autoinducer molecule production in [14], namely,kS1 = 0.6 molecules�1 min�1, the binding and unbindingrates between receptors and signaling molecules equal to thevalues in [15], namely, kRS = 6x10�4 molecules�1 min�1

and k�RS = 2x10�2 min�1, respectively, and the rate ofopen complex formation upon output transcription activa-tion and open complex dissociation at the receiver as in [15],namely, kRx = 10�2 molecules�1 min�1 and k�Rx = 4x10�2

min�1. The di↵usion coe�cient D ⇠ 60x10�9 m2min�1 isset to the di↵usion coe�cient of molecules di↵using in a bi-ological environment (cellular cytoplasm, [11]). We set thevolume of the transmitter and receiver volume ⌦Tx = ⌦Rx

to 1 µm2, the number of substrates at the transmitter cellS0 = 100, the auxiliary input signal PoPSaux = 1, thenumber of promoter sequences and RNA polymerases at thereceiver cell PRx = 1 and RNAP = 100, respectively.In Figure 4 and Figure 5 we show the numerical results

for the attenuation ↵ and delay � experienced by a signalthrough the biological circuit, computed by using the ex-pressions in (34) and (36), respectively. The value of thereceiver distance from the transmitter rRx ranges from 5 to500 µm, while the range of observed frequency ! values isbetween 0 and 1 Hz. The values for the attenuation ↵ of thebiological circuit, shown in dB, range from a minimum of 0at the minimum distance rRx and frequency ! equal to 0,to a maximum of 117.8 dB for a distance rRx of 50 µm andfrequency ! equal to 1 Hz with a monotonic increasing trendboth when increasing the distance rRx and the frequency !.The values for the delay � range from a minimum of 0 minat the minimum distance rRx and maximum frequency !equal to 1 Hz, to a maximum of 1.4 min at a distance rRx

of 50 µm and for a frequency ! equal to 0. The curves ofthe delay � always show a monotonically decreasing trendas function of the frequency !, more pronounced for higher

0.5

1

1.5

2

1

2

3

4

5

x 10−4

0

20

40

60

80

100

ω [Hz]

Biological Circuit − Attenuation

rRx

[m]

α [

dB

]

10

20

30

40

50

60

70

80

90

100

110

Figure 4: Attenuation of the biological circuit fordi↵usion-based MC as function of the receiver dis-tance from the transmitter rRx and the frequency!.

0.5

1

1.51

2

3

4

5

x 10−4

0.2

0.4

0.6

0.8

1

1.2

rRx

[m]

Biological Circuit − Group Delay

ω [Hz]

∆ [

min

]

0.2

0.4

0.6

0.8

1

1.2

Figure 5: Delay of the biological circuit for di↵usion-based MC as function of the receiver distance fromthe transmitter rRx and the frequency !.

values of the distance rRx.

5. CONCLUSIONIn this paper, systems-theoretic models are derived for

a di↵usion-based molecular communication system designbased on biological circuits. Biological circuits are definedas genetic regulatory networks embedded in a biological cell,and they are envisioned to allow the future engineering ofcomplete biological nanomachines. Some recent literaturecan be found on the analytical modeling of biological cir-cuits, but with no specific mention to di↵usion-based cell-to-cell communication through molecule exchange, for whichonly a biological description is provided in some specificworks.

In our work, first, a biological circuit for di↵usion-basedmolecular communication is described through a minimalsubset of elements. Then, a mathematical model is detailed

in terms of transfer functions, from which analytical expres-sions are derived for the attenuation and the delay experi-enced by an information signal through the biological cir-cuits. Finally, we present some numerical results obtainedby applying to the developed models some realistic biologicalparameters from the literature.

The work presented in this paper is a preliminary analy-sis of the application of communication engineering to thedesign of communication systems through biological circuitsand their biochemical reactions within genetically-engineeredcells. We believe this is the first step towards a potentiallytransformative interdisciplinary research area that could cha-nge the way we design devices and interact with the nature.

AcknowledgmentThis material is based upon work in part supported by theUS National Science Foundation under Grant no. CNS-1110947, and in part supported by the Jane Robertson Lay-man Fund under project no. WBS #26-0511-9001-015.

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