A Biochemical Oscillator Using Excitatory Molecules for Nanonetworks

10 Pages·2016·1.648 MB·english

by  Shitiri, Ethungshan and Cho, Ho-Shin

IEEETRANSACTIONSONNANOBIOSCIENCE,VOL.15,NO.7,OCTOBER2016 765
A Biochemical Oscillator Using Excitatory
Molecules for Nanonetworks
∗
Ethungshan Shitiri, Student Member, IEEE, and Ho-Shin Cho, Member, IEEE
Abstract—For nanonetworks to be able to achieve large-scale With the advantage of being minuscule, the applications
functionality, such as to respond collectively to a trigger, syn- of nanonetworks are envisioned over a wide range of fields:
chrony between nanomachines is essential. However, to facili-
for quality control and to develop new materials in industrial
tate synchronization, some sort of physical clocking mechanism
applications, to control pollution in environment monitoring
is required, such as the oscillators driven by auto-inhibitory
molecules or by auto-inducing molecules. In this study, tak- applications [1], to deliver drugs in nanomedicine applica-
ing inspiration from the widely studied biological oscillatory tions[2],andtodetectnuclear,biological,andchemicalagents
phenomena called Calcium (Ca2+) oscillations, we undertake a in military applications [5]. However, being minuscule limits
different approach to design an oscillator. Our model employs the capability of the nanomachines and would require them
threedifferenttypesofexcitatorymoleculesthatworkintandem
to be coordinated or synchronized in order to achieve large-
to generate oscillatory phenomenon in the concentration levels
scale functionality, such as releasing drugs in coherence to
of the molecule of interest. The main objective of the study is
to model a high frequency biochemical oscillator, along with a targeted site. In this regard, the pertinent issue here is,
the investigations to identify and determine the parameters that however, not about how synchronization can be achieved, but
affect theperiodoftheoscillations.Theinvestigationsentailand most importantly the issue of how to implement a physical
highlightthedesignofthereserveunit,areservoirofthemolecule
timing unit [6], such as a clocking device or an oscillator
of interest, as a key factor in realizing a high frequency stable
biochemical oscillator. within a nanomachine, so that the nanomachines can simply
synchronize their oscillators in order to be synchronizedwith
Index Terms—Biochemical oscillator, calcium oscillations,
each other. In that way, the oscillator(s) can facilitate the
molecular communication, nanonetworks, nanomachines.
process of synchronization.
Until now, only a couple of studies have addressed this
I. INTRODUCTION
issue. In [7], auto-inhibitorymoleculesare employedto make
MOLECULAR nanonetworks or simply nanonetworks
nanomachines oscillate. Auto-inhibitory molecules have a
are defined as the internetworking of nanomachines
peculiar behavior of inhibiting the chemical process that is
based on, but not limited to, molecular communication [1].
responsible for their generation. This inhibiting behavior is
Molecular communication employs molecules as information
triggered when the auto-inhibitor’s concentration increases to
carriers [2] and, over the years, it has been garnering a
an upper threshold, and, it subsequently causes a decrease in
lot of interest over other forms of nanoscale communica-
their concentration. As the concentration decreases and falls
tion techniques such as the traditional electromagnetic and
below a lower threshold, the inhibition is stopped and the
acoustic communication, and a not so traditional contact-
generationprocessisreinitiated,allowingthe concentrationto
based communication called nano-mechanical communica-
increase. These processes keep repeating, forming the oscil-
tion [3, p. 177]. What makes molecular communication so
lations. In a more recent study [8], auto-inducing molecules
attractive for nanonetworks are for the facts that it is natu-
are employed to make a group of nanomachines oscillate.
rally designed for the nanoscale, compatible with biological
Unlike the auto-inhibitors, auto-inducers facilitate their own
systems, and energy-efficient [4]. Nanomachines, which can
generation.Whenthenanomachinessensetheauto-inducersin
be biological or electromechanical, are the functional units
the environment, they generate and release the auto-inducers
of a nanonetwork that are capable of performing basic func-
until the concentration of the auto-inducers reaches an upper
tions of sensing, actuating, computing, and communicating
threshold.Oncetheconcentrationreachestheupperthreshold,
(information transfer) with one another [1] [4].
the nanomachines stop the release of the auto-inducers and
Manuscript received August11,2016;revised October 14,2016;accepted this action causes a decrease in the concentrationof the auto-
November 1, 2016. Date of publication October 18, 2016; date of current inducers. As the concentration falls to a lower threshold, the
version December 20, 2016. This study was supported by the BK21 Plus
nanomachinesbegintoreleasetheauto-inducerscausingarise
project funded by the Ministry of Education, Korea (21A20131600011)
Asteriskindicates corresponding author. intheconcentrationagain.Theresultantperiodicincreaseand
E.ShitiriiswiththeMobileCommunicationLaboratory,SchoolofElectron- decrease in the concentration of the auto-inducers constitutes
icsEngineering,KyungpookNationalUniversity,Daegu41556,SouthKorea.
∗ the oscillations.
H.-S. Cho is with the Mobile Communication Laboratory, School of
ElectronicsEngineering,KyungpookNationalUniversity,Daegu41556,South The basis of the first oscillator [7] is from a naturally
Korea(e-mail: [email protected]). occurringphenomenon,wherein,thecouplingoftwofeedback
Color versions of one or more of the figures in this paper are available
signals, excitatory and delayed inhibitory, results in oscilla-
onlineathttp://ieeexplore.ieee.org.
Digital ObjectIdentifier 10.1109/TNB.2016.2616539 tions [9]. Similarly, the basis of the second oscillator [8] is
1536-1241©2016IEEE.Personaluseispermitted, butrepublication/redistribution requires IEEEpermission.
Seehttp://www.ieee.org/publications_standards/publications/rights/index.html formoreinformation.
766 IEEETRANSACTIONSONNANOBIOSCIENCE,VOL.15,NO.7,OCTOBER2016
also from a naturally occurring phenomenon, called quorum
sensing[10].Infact,severalbiologicalactivitiesexhibitoscil-
latorybehaviorandarewidelystudied.Theperiodicsignaling
of cAMP (cyclic adenosine monophosphate) between cells,
periodicincreaseanddecreaseincalciumconcentrationinside
a cell, and circadian rhythms are few examples of biological
oscillations [11]. Such systems are commonly referred to as
biological oscillators. Biological oscillators, in general, have
been studied for decades and in recent years there have been
several attempts and breakthroughs in designing biological
oscillators, such as genetic oscillators [12] [13] and chemical
micro-oscillators [14]. These oscillators are designed from a
biologicalsystemengineerperspectiveandtendtohaveacom- Fig. 1. Working dynamics of Ca2+ oscillations found in astrocytes. Solid
mon feature—large period of oscillations. Such feature may lines indicate the transport/feedback mechanisms and dashed lines indicate
not be desirable for nanonetworks that are deployed in time- the fluxes. Plus sign indicates positive feedback and minus sign indicates
critical applications. The oscillators proposed in [7] and [8] negativefeedback. (ThefiguredepictsonlythemajorsignalingpathsofIP3,
Ca2+,andPLCδ.)
are probably the first studies on biological oscillators (with
comparativelysmaller periodsof oscillations) from a commu-
nication engineering perspective. However, in [7] there is no signals is reduced. To validate this statement, we provide an
clarity on developing the oscillator as a separate functional elaborate discussion in Section III-C.
unit within the nanomachine, while in [8] there is no such The remainder of this paper is organized as follows.
attemptatallas a groupof nanomachinesis consideredto act Section II provides an overview of the signaling principles
as an oscillator. of Ca2+ oscillations. Section III presents the working of the
Giventhestochasticpropagationnatureofthebiomolecules, proposed biochemical oscillator along with the mathematical
viz. Brownian motion, it is imperative to design the oscillator framework,andtoshowthatahighfrequencyoscillatorcanbe
as a separate functional unit such that the oscillations, whose obtained,theconceptofanoperatorisintroducedhereaswell.
molecules are already impaired by Brownian motion, are SectionIVdiscussesthesimulationresultsandtheparameters
not interfered by unwanted noises. Noises, here, refers to thataffectthe meanandvariancein the periodof oscillations.
the residual or lingering molecules from other processes that Finally, Section V concludes the paper. Beyond this point,
can interfere with movements of the oscillatory molecules. the proposed biochemical oscillator will be interchangeably
Moreover,thefunctionofabiochemicaloscillatorneednotbe referred to as the oscillator.
limited only to facilitate synchronization but can be extended
to perform other functions such as providing timing infor-
II. CALCIUMOSCILLATIONS
mation for coordination among communication modules in
Ca2+ oscillations are the fluctuations in [Ca2+]1 that arise
a nanomachine or for scheduling channel access [6]. Hence,
either from the entry of external calcium into a cell or from
it would be a fair requirement of the biochemical oscillators
the periodic release of intracellular calcium to the cytosol
to operate and function as a separate unit within a nanoma-
from internal stores [15]. The former is commonly found in
chine but with the ability to communicate or pass timing
excitablecellswhilethelatterinnon-excitablecells[16].Over
information to other communication modules. The primary
the years, several models have been developed to describe
focus of our study is to address the former requirement,
the mechanisms and dynamics of the Ca2+ oscillations.
while the latter requirement is left as an open research
For example, while some models consider Ca2+ oscillations
challenge.
with a constant Inositol 1,4,5-triphosphate (IP ) concentra-
Based on the aforementioned issues and requirements, 3
tion [17]–[20], other modelsconsiderCa2+ oscillationswith a
we propose a biochemical oscillator model by taking inspi-
varying [IP ] [21]–[24].
ration from the widely investigated and studied biological 3
oscillatory phenomena called calcium (Ca2+) oscillations. The major mechanism and signaling principles of Ca2+
oscillationsfoundinastrocytes[23]isreviewedhereinabrief
A bottom up approach is applied to model the biochemical
manner. As shown in Fig. 1, in response to an external or
oscillator as a separate functional unit within a nanomachine.
agoniststimulation, the plasma membranetriggers the release
Itemploysthreetypesofexcitatorymoleculesthatworkintan-
of IP , which is a secondary messenger2, into the cytosol
demtogenerateoscillatorypatternsintheconcentrationofthe 3
(denoted by ❶ ). The IP molecules propagate via diffusion
molecule of interest — the molecule whose periodic increase 3
and bind to the bi-phasic IP receptors(IP Rs) located on the
and decrease in concentration is considered to constitute the 3 3
membranes of Endoplasmic Reticulum (ER) (denoted by ❷ ).
oscillator’s oscillations. The main contributions of this study
The ER serves as an internal store of Ca2+ molecules. When
are the design of a high frequency biochemical oscillator, the
the IP Rs are bounded to by the IP molecules, they open
investigationsto identify the parameters that affect the period 3 3
of the oscillations, and the identification of a key design unit.
1Squarebrackets denote theconcentration ofthechemical ormolecule.
We state that a high frequency oscillator will allow faster 2Secondary messengers are molecules that relay information from a cell’s
synchronization as the interval between the synchronization surfacetoothermolecules ororganelles thatresidewithinthecell.
SHITIRIANDCHO:BIOCHEMICALOSCILLATORUSINGEXCITATORYMOLECULESFORNANONETWORKS 767
Fig.2. Ca2+ oscillations.
up and allow the stored Ca2+ molecules to diffuse into the
Fig.3. Internal signalingpathways oftheproposedoscillator.
cytosol (denoted by ❸ ). This results in an increase in the
cytosolic [Ca2+]. The increase in cytosolic [Ca2+] causes two
actions.First,athigherconcentrationlevelsitbeginstoinhibit
First, the oscillator is assumed to be equipped with an
the IP Rs, thereby stopping the flow of Ca2+ from the ER to
3 internal store that holds B molecules; we call it the reserve
the cytosol (denotedby ❹ ). Second, it stimulates the regener-
unit(RU).ThesurfaceoftheRUisassumedtobecoveredwith
ation of IP3 via PLCδ, also a secondary messenger (denoted receptorsthat are sensitive to both A and B molecules, called
by ❺ ). Meanwhile, special structures called Ca2+ ATPase and
the type AB sensitive receptors (tABRs), and with special
Na+-Ca2+ exchanger regulate the cytosolic [Ca2+] to avoid
suction-like receptors that are sensitive and activated only
cell death caused by high levels of cytosolic [Ca2+]. The
by B molecules when B’s concentration increases, called the
Ca2+ATPase structures are found on the ER’s surface and the
reserve unit pumps (RUPs). The aforementioned assumptions
Na+-Ca2+ exchangerstructuresare foundon the cell’s surface
are based on the internal Ca2+ stores and their receptors
or membrane. Both, Ca2+ATPase and Na+-Ca2+ exchanger,
(and pumps) [15] [16]. Then, we assumed the oscillator to be
structures regulate the cytosolic [Ca2+] by pumping the Ca2+
a well-mixed3 closed unit [25], that is, molecules can neither
intotheERandoutofthecell,respectivelyandarecommonly
enter nor exit the oscillator unit. This assumption ensures
referred to as pumps attributed to the pump-like action they
two things. First, it ensures that unwanted noises, such as
perform. A majority of the cytosolic Ca2+ is pumped into the
the residual or lingering molecules from other processes that
ER (denoted by ❻ ) and only a fraction of it is pumped out
occurwithinthenanomachines,donotentertheoscillatorunit
of the cell (not shown in the figure). The second action (❺ is
and affect the oscillations. Second, it ensures the oscillator to
responsible for increasing the cytosolic [Ca2+], while the first
be a separate functional unit that can be embedded within
action(❹ )combinedwiththepumpingactionsofCa2+(❻ )are
a nanomachine. Next, the excitatory molecules are assumed
responsible for reducing the cytosolic [Ca2+]. This periodic
to have a lifetime and undergo biodegradation once they
increaseand decreasein cytosolic[Ca2+] constitutein whatis
accomplishtheirtask,therebyeliminatingthemselvesfromthe
known as the Ca2+ oscillations shown in Fig. 2.
oscillatorandminimizinganychancesofintra-oscillatorinter-
Atthecellularlevel,theoscillationsof[Ca2+]ortheculmi-
ferences. However, we consider the lifetime of B molecules
nationof the oscillationsinto a wave is a formof information
to be long enough so that the B oscillations can be sustained
that can triggera variety of biologicalactivities dependingon
overthelifetimeofthenanonetworkortheapplication.Finally,
the type of cell in which they are produced. However, from
while we show that a chemical process can regenerate A and
a communication system point of view, the oscillations could
theRU canstoreB, wesimplyassumeCtobesynthesizedby
very well be interpreted as a clocking mechanism. As stated
a chemical process [17]. For clarity and ease of readability,
earlier, nanonetworks and its nanomachines are envisioned to
we denote the concentration of B molecules in the RU and
be largely based on molecules for their functionality,and this intra-oscillator space4 by [B] and [B] , respectively.
RU INTRA
indicates the importance and relevance of an oscillator that
Fig. 3 depicts the proposed oscillator model along with
employs molecules as well. Therefore, in the next section we
the internal signaling pathways. The oscillator’s surface is
introduce the oscillator model, describing the working mech-
assumed to be covered with special receptors [1] that can
anism along with elaborate discussions on the mathematical
sense a beaconsignal(Bcon). Bcon isassumedto bespecific
framework.
moleculethatmayactasatriggertoinitializetheoscillator(s)
ormaycarrythe informationrelatedtothesynchronizationof
III. MODELOF THEBIOCHEMICAL OSCILLATOR oscillators. We will not elaborate on Bcon as it not the focus
A. Working Principle of this paper and leave it aside for future work. When Bcon
is sensed, the oscillator triggers A molecules. A molecules
Theproposedoscillatormodelemploysthreedifferenttypes
of excitatory molecules: A, B, and C to generate oscillatory
patternsintheconcentrationofthemoleculeofinterest,which 3Theconcentrationsofthebiomoleculesarehomogeneousthroughout.This
assumptionallowsustoobtainthesamemeasurementof[B]atanypointin
is B. Before we describe the oscillator in detail, we provide
theoscillator unit.
the assumptions that are considered in this study. 4Volumeoftheoscillator excluding thereserveunit’svolume.
768 IEEETRANSACTIONSONNANOBIOSCIENCE,VOL.15,NO.7,OCTOBER2016
propagate via diffusion and bind to the tABRs present on the For our model, we require the flux balance equations pertain-
surface of the RU. ing to two molecules, that of B and A.
Now for the tABRs to able to open and close, we consider The changes in [B] are mainly governed by the
INTRA
each tABR to consist of three identical gate-like structures outward B flux through the tABRs and the inward B flux
called sites. These gate-like structures have been observed in through the RUPs as shown in (1). J is the flux of B
tABR
biologicalreceptorssuchastheIPRs[17],[18].Thethreesites from the RU to the intra-oscillator space and J is the
RUP
aresiteA ,siteB ,andsiteB .Thesubscripts“actv” flux of B pumped from the intra-oscillator space into the RU.
actv actv deactv
and “deactv” denote the activation and deactivation functions Similarly, the balance equation for [A] can be formulated as
performedbythesites.SiteA andsiteB areresponsible the difference between the maximum rate of A’s production
actv actv
fortheactivationofthetABRs,whilesiteBdeactvisresponsible stimulatedby B via C, JB+C→A, andthe loss rate of A, Jloss,
forthe deactivationof the tABRs. Onlya single moleculecan as shown in (2).
occupy each site at a time [26], e.g., a single A molecule
d[B ]
can bind to the Aactv site at a given time. When A molecules INTRA = JtABR − JRUP (1)
dt
bindto theA sites andBmoleculesbindto the B sites,
actv actv d[A]
only then the tABRs are activated and are said to be in the = JB+C→A− Jloss (2)
dt
open state. However, when B molecules occupy the B
deactv
site (occurs only when B concentration increases), then the Equations(1) and (2) are an over simplification and do not
tABRs are deactivated and are said to be in the closed state. provide adequate details of the physiological processes that
When the tABRs are in the open state, B molecules begin are involved in obtaining the changes in [B] and [A].
INTRA
to diffuse out due to the concentration gradient ([B] is Hence, to add details of the physiological processes, the
RU
assumed to be higherthan [B] [17]).This results in an definitions and frameworks, hereon, provide the necessary
INTRA
increase in [B] . As [B] increases, B molecules mathematical formulations of the physiological processes.
INTRA INTRA
begin to occupy the third site, B , and the tABRs are We begin by modelling the dynamics of tABRs and as stated
deactv
forcedtogointotheclosedstate,therebystoppinganyfurther earlier, the tABRs can exist either in the open or closed state.
outflow of B to the intra-oscillator space. Additionally, the Thus, we obtain the fraction of tABRs that are in the open
RUPs begin to absorb B molecules back into the RU, aiding state (F ) [27] as shown by
O
in the reduction of [B] . By now, the concentration of
INTRA N
A and B molecules are decreasing, but in order to sustain F = O (3)
O
N
the oscillations, B molecules have to be released again from T
the RU. However, to release B molecules, the tABRs have where, N and N are the number of tABRs that are in
O T
to be opened and this means A molecules have to be regen- the open state and the total number of tABRs, respectively.
erated as they are partly responsible for opening the tABRs. In addition, we stated that an open tABR allows B molecules
Therefore,achemicalprocessinvolvingBandCmoleculesis to diffuse into the intra-oscillator space aided by the concen-
considered for the regeneration or synthesis of A molecules. tration gradient ([B] −[B] ). However, for a tABR
RU INTRA
This chemical process is assumed to be initialized around the tobeintheopenstate,A andB sitesshouldbeactivated
actv actv
time when [B]INTRA is approaching its peak such that the by A and B molecules, respectively. To model this behavior,
total time taken for A molecules to regenerate and propagate we incorporate the ion-channel formulation as described by
to the tABRs is proportional to the time at which [B]INTRA Li and Rinzel [18]. Then the probability that Aactv and Bactv
decreasesto a lower threshold.With the bindingof the regen- sites are in the activated state, such that an arbitrary tABR is
erated A molecules (along with B the molecules of the intra- in the open state, is given by
oscillator space) to the tABRs, B molecules are once again (cid:2) (cid:3)(cid:2) (cid:3)
rTehleeaesendtirientporotcheessienstrac-uolmsciinllaatteorinstpoatcheeipnecrrieoadsiicngin[cBre]aIsNeTaRnAd. PB = [A][+A]K1 [B][IBN]TINRATR+AK2 (4)
decreaseof[B] andarewhatconstitutestheoscillations
INTRA where K and K are the dissociation constants5 of A and B
of the proposed oscillator. 1 2
molecules from their respective sites. The first term signifies
the activation of A sites and the second term signifies
actv
B. Mathematical Formulation the activation of the Bactv sites. Then, the product of (3)
and (4) gives the overall probability of the tABRs that are
In this section, we present the mathematical formulation
in the open state. From these formulations, we can calculate
of the oscillator model. For anydynamicsystems, such as the
the flux of B from the RU, J , by taking into account
oscillatorproposedinthispaper,thechangesinthesystemare tABR
the concentration gradient ([B] −[B] ) that allows
modelledusingordinarydifferentialequations(ODEs)andare RU INTRA
B molecules to diffuse out with a rate r , the fraction of
generally termed as the general flux balance equations. Flux, 1
tABRs that are in the open state, F , and the probability
here,refersto the rate atwhich the moleculesflow,where the O
of A and B sites that are in the activated state, P ,
motion of the flow is assisted by diffusion. In simple terms, actv actv B
we need to identify the process involved in the generation
5Dissociationconstantdescribestheaffinitybetweenaprotein(e.g.,tABRs)
and degradation of a molecule, and calculate the change in
and a ligand (e.g., A). A small dissociation constant indicates a stronger
the concentration of the molecule for every time instant. affinity.
SHITIRIANDCHO:BIOCHEMICALOSCILLATORUSINGEXCITATORYMOLECULESFORNANONETWORKS 769
as shown by (5). The power, which is three, represents the Since A is dependent on B for its regeneration, we can
number of sites per tABR. incorporate a factor α [17] to denote by how much the
production of A depends on B. The value of α can range
J =r (F )3(P )3([B] −[B] ) (5)
tABR 1 O B RU INTRA between 0 and 1, with 0 denoting no dependenceat all and 1
denoting full dependence. Hence, we can rewrite (7) as
Next, to quantify the flux of B molecules into the RU,
aJsRU2P,[2w5e, ups.e2t9h3e]H. iHllilelqueaqtuioatniownitdhetshceriHbeilslcthoeeffibceiheanvtiovraluoef JB+C→A =r3[[BB]]INTRA++αKK4 (10)
INTRA 4
molecules that have more than one binding site and a Hill’s
Substituting(9),(6),(7),and(10)in(1)and(2),weobtainthe
coefficientvaluegreaterthan1indicatespositivecooperativity
complete flux balance equation for the oscillator as follows,
among these binding sites. Positive cooperativity refers to
the increase in the affinity of a biological substrate to other d[B]INTRA =η(r (F )3(P )3+r )([B] −[B] )
1 O B 5 RU INTRA
moleculesonceamoleculebindstoit.Equation(6)showsthe dt
formulation of J using the Hill’s equation, [B]2
RUP − r INTRA (11)
[B]2 2K32+[B]2INTRA
JRUP =r2K2+[IBN]T2RA (6) d[A] =r [B]INTRA+αK4 −r [A] (12)
3 INTRA dt 3 [B] +K 4
INTRA 4
where,r isthemaximumrateatwhichBisabsorbedintothe
2 Now, given that the oscillator is assumed to be a closed
RU and K is the dissociation constant of B molecules from
3 unit, we can apply the law of conservation of mass to the
the RUPs. totalconcentrationof Bin theoscillator ([B] ). Therefore,
TOT
Thetermsonthe righthandside of(2)are merelytherates [B] is the sum of [B] and [B] , which implies
TOT INTRA RU
at which A is regenerated and degraded. As the regeneration that [B] can be calculated by
RU
of A is achieved by a chemical process involving B and C
[B] −[B]
molecules, this process can be modeled by [B] = TOT INTRA (13)
RU η
[B]
JB+C→A =r3[B] INTR+AK (7) One final consideration to be made is to formulate the
INTRA 4
rates at which the tABRs switch between the open states
where, r3 and K4 are the maximum rate at which A can and the closed states. Such formulationis necessary to model
be regenerated and the dissociation constant of B from C, the tABRs behavior as close as possible to the behavior
respectively. Equation (7) is the Michaelis-Menten equation of the actual biological receptors and also to obtain N .
O
and is a special case of the Hill equation where the molecule Therefore,wemodelthestatetransitionsasatwostateMarkov
has only one binding site. Meanwhile the degradationof A is process [27] [29, p. 286]
proportional to a loss rate and is given by [17]
J =r [A] (8)
loss 4
where, r is the rate at which A is degraded.
4
Now,toprovideformaldetailstotheformulationsdiscussed where, Closed and Open refer to the closed state of tABRs
so far (Equations (3) – (8)), first, we define and assign the when the B sites are bounded by B molecules and the
deactv
following biochemical molecules: type A, IP3; type B, Ca2+; open state of tABRs when B molecules are not bounded to
andtypeC, PLCδ basedontheirresemblanceinfunctionality. them, respectively. The state transition parameters β and β
o c
Additionally, tABRs are defined by IP3Rs. Based on these are the rates at which the tABRs switch from the open states
definitions, certain considerations have to be addressed in (5) to the closed states and vice versa, respectively. To determine
and (8), and are discussed in the following paragraphs. the parameterswe apply the formulationdevelopedfor IP Rs
3
In(5),thedifferencesintheunitsofthefluxratesofBfrom by Jung and Shuai [26].
the RU and into the RU have not been taken into account. (cid:2) (cid:3)
[A]+K
The fact is the flux rate of B from the RU is measured in β =r K 1 (15)
micromoles (μM) per liter RU per second, while the flux o 6 5 [A]+K6
rate of B into the RU is measured in micromoles per liter βc =r6[B]INTRA (16)
intra-oscillatorper second.Thisis becausethe volumesof the
where,r istherateatwhichBbindstothetABRs, K andK
intra-oscillator space and the RU are not the same; volume 6 5 6
are the dissociation constants of B (inactivation) and A,
of RU being smaller. To compensate for the differences in
respectively.
the volumes, a scaling factor η is introduced and is defined
as the ratio of the intra-oscillator’s volume to the RU’s C. Operator, (cid:6)
k
volume [28, p. 23]. Moreover, as the RU is not considered
To realize the characteristics befitting an oscillator, specifi-
to be a closed unit, there will be a tiny amount of B leakage
callytoattain highfrequencyofoscillations,we introducethe
from the RU at a rate r . Therefore, (5) can be rewritten as,
5 concept of an operator ((cid:6) ). The operator is defined as the
k
J =η(r (F )3(P )3+r )([B] −[B] ) (9) factorbywhichavalueofcertainparameter,k,oftheoscillator
tABR 1 O B 5 RU INTRA
770 IEEETRANSACTIONSONNANOBIOSCIENCE,VOL.15,NO.7,OCTOBER2016
can be varied to satisfy a criterion: attain high frequency of TABLEI
oscillations while maintaining stable period of oscillations. PARAMETERVALUES
We use the term “ variant ” to denote the parameter whose
value is changed to the amount of modification made to it by
the operator, all the while satisfying the criterion. Let Zk be
the value of the kth parameter, then the variant value of the
kth parameter, Zk , can be obtained by
Var
Zk = Zk(cid:6) , k ∈{(cid:7)} (17)
Var k
where, (cid:7) is the set of parameters that satisfy the criterion
on being modified by the operator. Using Equation 17, we
areabletoidentifytheparametersthatwillplayavitalrolein
designingahighfrequencyoscillatorandthediscussionstothe
same are providedin Section IV. To highlightthe importance
of faster oscillations for faster synchronization, let us outline
two case examples.
In both cases, we consider synchronization using a pulse-
coupling like scheme [30], where the oscillators sense the
oscillation peaks of the other nodes through the synchro-
nization signals such as Bcon, adjust their peaks accord-
ingly, and over a period of time converge towards peaking
at the same times. In the first case, a reference oscillator
sends out Bcon signals at the beginning of every period, a
period being made up of the time between two consecutive
peaks. Now, if the reference oscillator has a period of, say,
2 seconds, it means that the Bcon signal can be sent after
every2seconds.Butiftheoscillatorhasalargerperiodof,say,
5 seconds, then the Bcon signal can be sent out after
every 5 seconds. Intuitively, notwithstanding the stochastic
nature of the molecular channel, the Bcon sent out from the
faster oscillator (period = 2 seconds) will arrive at a specific
destination much earlier or will have a higher probability of
reaching earlier than that sent out from the slower oscillator
(period= 5 seconds).In the secondcase, pulse-couplingwith Fig. 4. Illustration of two consecutive periods having certain
a two-way exchange mechanism, where a reference oscillator periods Xi and Xi+1.
sends out the Bcon signal only when it receives a response
signal from the other oscillators is considered. Such mech-
anism avoids redundant signaling as the reference oscillator thevaluesoftheparametersusedinthesimulations.Aparame-
needs to send out the Bcon signal only if the response signal ter’svalueissaidtobetheoptimumvalueifthatvalueresults
arrives before the beginning of an immediate period; if not, inthesmallestmeanandvarianceintheperiodofoscillations.
Bconisnottransmitted.Again,intuitively,iftheoscillatorhas Period of oscillation (Xi,i =1,2,3...n−1) refers to the
a smaller period (e.g., 2 seconds), then the Bcon signal can time between consecutive peaks, where “n” is the number of
be sent out at the immediate peak, which will be at a shorter peaks observedfor t seconds. If X is a random variable, then
time, allowing Bcon to reach at a destination much earlier. the varianceof the peri(cid:4)odcan be obtainedas σ =(Xi − X¯)2,
Besidesfacilitatingfastersynchronization,ahighfrequency withmeanperiod X¯ = X /n−1asshowninFig.4.Unless
i
oscillator has another benefit and that is it will allow higher specified or denoted by ∗, the values of the parameters listed
data rates. For instance, when a sender has data to send, it are the values used in [17], which we refer to as the standard
has to encode the message through, say, the concentration values (Zk ).
Std
or frequency of the message molecule(s). The rate, at which We define (cid:7) as r , r , r ,K , and α, thus, k ∈ {r , r , r ,
1 2 3 3 1 2 3
the data are encoded, regardless of the encoding scheme, is K ,andα}and Zk isreplacedby Zk .Theaforementionedk
3 Std
proportionalto the oscillator’sfrequency— a faster oscillator parameterswere identifiedthroughextensivesimulationssuch
meanshigherdatarate.Andtherefore,havingahighfrequency that the criterion is satisfied. In addition, the following bases
oscillator can benefit a nanonetwork in two significant ways. werealso usedindeterminingthe parameters.Theparameters
r , r , and K were determined so that the resultant B oscil-
1 2 3
IV. SIMULATIONRESULTSAND lations are bounded within reasonable [A]; while r and α
3
ANALYSIS OF PARAMETERS were determined to maintain the balance in (9) and (10).
The simulations were carried out on a time scale Thus, from these bases, extensive simulations were carried
of 100 seconds with a time step of 0.01 seconds. Table I lists out to determine the optimum values of the k parameters
SHITIRIANDCHO:BIOCHEMICALOSCILLATORUSINGEXCITATORYMOLECULESFORNANONETWORKS 771
TABLEII
PARAMETERSWITHTHEIRSTANDARD,OPERATOR,
ANDVARIANTVALUES
theiroptimumvaluescauseanincrease,whichisnotdesirable,
in the mean and variance. Therefore, one must be careful in
choosing these values. In Fig. 5(c) and Fig. 5(d), the graphs
followadecreasingslopeforthemeanandanincreasingslope
for the variances. For both of the figures, a trade-offbehavior
isobserved,thatis, forlowervaluesofmean,largervariances
are obtained and vice versa, suggesting that the optimum
values of the mean and variance for r and k lie in and
3 3
around the point of intersection. In Fig. 5(e), increase in α
cause the graphs to decrease. Note that the graphs after 0.6
is actually a slope but with very small differences (≈4 μs),
and it gives the impressions of a straight line. Therefore, a
minimum of ≈60% dependencerate of A must be allowed to
achieve negligible variance and uniform periods. It should be
notedthatinFig.5(a)–(e),therangesonthex-axesarelimited
only to those values beyond which the oscillations die out or
the mean and variance are too large to consider the oscillator
suitable for a communication nanonetwork. Table II lists the
operator value associated to each standard value that results
in the corresponding variant values.
To show how much of a difference was obtained in the
frequencies of oscillations using the variant values of the
proposed oscillator and the standard values of the standard
oscillator [17], we plot the oscillations for each oscillator
in Fig. 6. As expected, in Fig. 6(a), a higher frequency of
oscillationwithameanperiodof2.0148secondswasobtained
for the variant values, while in Fig. 6(b) a lower frequency
of oscillations with a mean period of 17.71 seconds was
obtained for the standard values. The oscillations in Fig. 6(a)
were achieved by using the concept of the operator and the
associated criterion.
One may notice the difference in the “optimum” values
of the variant parameters used (Table I) to those obtained
in Fig. 4 (a)–(e). That is because we numerically evaluated
and chose the variant values that were more suitable in gen-
erating the resultant oscillations shown in Fig. 6(a). However,
the variant values used satisfy the criterion and reside well
within the range of the optimum values, and therefore, has
Fig. 5. Mean and variance of B oscillations for different values of (a) r1 insignificantornoeffectontheoutcome.Infact,asanexample
(b)r2 (c)r3 (d) K3 and(e)α.
to justify that difference, we plot the mean and variance
as shown(cid:4)in Fig. 5 (a)–(e). Hereon, unless(cid:4)specified, mean versus two parameters, r1 and r2, simultaneously. r1 and r2
refers to X¯ /N and variance refers to σ /N; where were specifically chosen due to their association to the RU.
j j
j =1,2,3...N and N is the number of simulation runs. Atrade-offcanbeobservedbetweenthemeanandthevariance
In Fig. 5(a) and Fig. 5(b) the graphs of the mean and (Fig. 7 (a), (b)) but the key observation to note here is that
variance follow a valley-like shape, suggesting the optimum the values of r and r (ref. Table 1) results in a very low
1 2
values of r and r lie in and around the valleys. Both these variance and a relatively decent mean (denoted by the white
1 2
parameters are associated with the RU and deviations from cross). Another observation to note is that a large number of
772 IEEETRANSACTIONSONNANOBIOSCIENCE,VOL.15,NO.7,OCTOBER2016
Fig.8. Meanandvariance fordifferent initial concentrations ofA.
Fig.9. Variancefordifferent numberofreceptors, NTOT.
Fig.6. Boscillations with(a)variant values (b)standardvalues.
whichmolecularcommunicationoccurs,itisappropriatetobe
termedashigh.TheinitialconcentrationofAfortheproposed
oscillatorissetto2μMasthisvalueaccountsforthesmallest
mean as shown in Fig. 8, while that of the standard oscillator
is set to 0.5 μM [17].
On careful examination of the values listed in Table II,
the operator’s value associated to parameters r and r are
1 2
quite large, indicating that the tABRs and RUPs should be
designed to release and absorb B molecules, respectively,
in larger quantities or be able to perform their functions at
a faster rate. Either way their design will play a key role
in satisfying the criterion mentioned in Section III. Another
parameterthat couldplaya vitalrole in maintainingthe crite-
rion is the number of tABRs, N . A higher N means
TOT TOT
that more tABRs will be in the open state, allowing more B
molecules to be released into the intra-oscillator space at a
given instant of time. This, in a way, also aides in generating
smoothoscillationcurves,whichotherwisewouldbejaggedor
spiky caused by the random nature in which the B molecules
are released. Therefore, with higher N , [B] will
TOT INTRA
increase faster and it results in a proportionate change in the
regeneration of A as well as the inhibition of tABRs. As a
result, shorter periods can be obtained. Furthermore, a high
N is desirable to ensure that the variances are very small
TOT
Fig. 7. (a) Mean and (b) variance obtained by varying the ornegligible.AsshowninFig.9,thevariancetendstobecome
parameters r1 andr2 simultaneously. smaller as NTOT is increased incrementally from 1×105 to
10×105 and a gentle slope can be observed when N is
TOT
equal to or greater than 5×105.
the values of variance reside in the lows, and therefore most From these observations, we can infer that the parameters
of the combinations of r and r will lead to a minimum in N , r , and r have great influences on the oscillator’s
1 2 TOT 1 2
the variance. This may allow flexibility in choosing any of behaviorandinterestinglyallthreeareassociatedwiththeRU.
the combinations of r and r within the lows as long as the This suggests the design of RU to be a key role in realizing a
1 2
criterion is not violated. stable high frequency biochemical oscillator. That being said,
In terms of magnitude, the high frequency of oscillations it remains an open research challenge to identify methods on
for biochemical oscillators will be nowhere close to those how such reserve unit(s) can be physically designed or to
of traditional oscillators. However, considering the speed at be more specific how the operator’s value can be physically
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[29] C.P.Fall,Comput.CellBiol.,vol.20,2004. Ho-ShinCho(S’92–M’99)receivedtheB.S.,M.S.,
[30] R. E. Mirollo and S. H. Strogatz, “Synchronization of pulse- andPh.D.degreesinelectricalengineeringfromthe
coupled biological oscillators,” SIAM J. Appl. Math., vol. 50, no. 6, KoreanAdvancedInstitute ofScienceandTechnol-
pp.1645–1662, 1990. ogy(KAIST)in1992,1994,and1999,respectively.
From 1999 to 2000, he was a Senior Member of
theResearchStaffwiththeElectronicsandTelecom-
munications Research Institute (ETRI), where he
EthungshanShitiri(S’16)receivedtheB.E.degree was involved in developing a base station system
inelectronics andcommunication engineering from forIMT-2000.From2001to2002,hewasafaculty
Thanthai Periyar Government Institute of Tech- member at the School of Electronics, Telecommu-
nology, India, in 2010 and the M. Tech. degree nications, andComputer Engineering, Hankuk Avi-
in communication systems from Christ University, ation University. In2003, hejoined the faculty ofthe School ofElectronics
India, in 2013. He is currently working toward the Engineering, Kyungpook National University, South Korea, and now is a
Ph.D.degreeatMobileCommunicationLaboratory, Professor. His current research interests include mobile communication—
Kyungpook National University, South Korea. His IoT, 5G cellular; underwater communication—CDMA, SON, NOMA; and
currentresearchinterestsincludemolecularcommu- molecular communication—synchronization andmultiplexing.
nication,synchronizationandmediumaccesscontrol Prof. Cho was awarded the rising researcher fellowship of NRF in 1998,
forwireless networks,suchasnanonetworks. andisamemberofIEEE,IEICE,IEEK,KICS,andASK.