Queuing Models for Abstracting Interactions in …584 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 34, NO. 3, MARCH 2016 Queuing Models for Abstracting Interactions in Bacterial

584 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 34, NO. 3, MARCH 2016

Queuing Models for Abstracting Interactionsin Bacterial Communities

Nicolò Michelusi, Member, IEEE, James Boedicker, Mohamed Y. El-Naggar, and Urbashi Mitra, Fellow, IEEE

Abstract—Microbial communities play a significant role inbioremediation, plant growth, human and animal digestion, globalelemental cycles including the carbon-cycle, and water treat-ment. They are also posed to be the engines of renewable energyvia microbial fuel cells, which can reverse the process of elec-trosynthesis. Microbial communication regulates many virulencemechanisms used by bacteria. Thus, it is of fundamental impor-tance to understand interactions in microbial communities and todevelop predictive tools that help control them, in order to aid thedesign of systems exploiting bacterial capabilities. This positionpaper explores how abstractions from communications, network-ing and information theory can play a role in understanding andmodeling bacterial interactions. In particular, two forms of inter-actions in bacterial systems will be examined: electron transfer andquorum sensing. While the diffusion of chemical signals has beenheavily studied, electron transfer occurring in living cells and itsrole in cell–cell interaction is less understood. Recent experimen-tal observations open up new frontiers in the design of microbialsystems based on electron transfer, which may coexist with themore well-known interaction strategies based on molecular dif-fusion. In quorum sensing, the concentration of certain signaturechemical compounds emitted by the bacteria is used to estimatethe bacterial population size, so as to activate collective behav-iors. In this position paper, queuing models for electron transferare summarized and adapted to provide new models for quorumsensing. These models are stochastic, and thus capture the inher-ent randomness exhibited by cell colonies in nature. It is shownthat queuing models allow the characterization of the state of asingle cell as a function of interactions with other cells and theenvironment, thus enabling the construction of an information the-oretic framework, while being amenable to complexity reductionusing methods based on statistical physics and wireless networkdesign.

Index Terms—Quorum sensing, electron transfer, bacterialinteractions, stochastic modeling, queuing models.

Manuscript received August 6, 2015; revised December 19, 2015; acceptedJanuary 5, 2016. Date of publication February 3, 2016; date of current versionMarch 15, 2016. The work of N. Michelusi and U. Mitra was supported by oneor all of these grants: ONR N00014-09-1-0700, CCF-0917343, CCF-1117896,CNS-1213128, AFOSR FA9550-12-1-0215, and DOT CA-26-7084-00. Thework of J. Boedicker was supported by the ONR award N00014-15-1-2573. Thework of M. Y. El-Naggar was supported by the NASA Cooperative AgreementNNA13AA92A and PECASE award FA955014-1-0294 from the Air ForceOffice of Scientific Research.

N. Michelusi is with the Department of Electrical and ComputerEngineering, Purdue University, West Lafayette, IN 47907 USA (e-mail:[email protected]).

U. Mitra is with the Ming Hsieh Department of Electrical Engineering,University of Southern California, Los Angeles, CA USA.

M. Y. El-Naggar and J. Boedicker are with the Departments of Physics,Biological Sciences, and Chemistry, University of Southern California, LosAngeles, CA USA (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2016.2525558

I. INTRODUCTION

B ACTERIA constitute some of the earliest life forms onEarth, existing anywhere from four to three billion years

ago [1]. Bacteria are unicellular organisms that lack a nucleusand rarely harbor membrane-bound organelles. While bacteriahave certain elements in common (such as being unicellular),their numbers and variety are vast, with an aggregate biomasslarger than that of all animals and plants combined [2]. Bacteriaexist all over the planet, in and on living creatures, underground,and underwater. They can survive in Antarctic lakes and in hotsprings, showing tremendous robustness and tenacity. It is the-orized that bacteria enabled changes in the Earth’s environmentleading to atmospheric oxygen as well as being the precursorsto more complex organisms. Microbial communities play a sig-nificant role in bioremediation, plant growth, human and animaldigestion, global elemental cycles including the carbon-cycle,and water treatment [3]. Despite their simplicity as unicellularorganisms, the operations and interactions of bacterial com-munities are not fully understood. Elucidating how bacteriainteract with each other and with their environment is of fun-damental importance in order to fully exploit their potential. Tothis end, realistic tools for the modeling and prediction of thesesystems are needed.

In this position paper, we explore how queuing models canplay a role in understanding and modeling bacterial interac-tions. Indeed, molecules, electrons and cells are countable units.It is thus natural to represent them as "quanta", and their amountas the state of a "queue" in which these quanta are collected andfrom which they are dropped. Queue evolution then depends onthe interaction of these elements with each other and with thesurrounding environment. The advantage of a queuing modelcompared, for instance, to continuous models based on ordinarydifferential equations, is their amenability to capture random-ness and interactions in small bacterial systems, such as thequorum sensing system studied in [4], where the discrete natureof these interactions is predominant. In particular, we shallexamine two forms of interactions:

a) Electron Transfer [5], [6]: While chemical and molecu-lar diffusion are widely investigated [7], [8], electron transfer isemerging as an exciting strategy by which bacteria exchangenutrients and potentially information. Each cell relies on acontinuous flow of electrons from an electron donor to anelectron acceptor through the cell’s electron transport chain toproduce energy in the form of the molecule adenosine triphos-phate (ATP), and to sustain its vital operations and functions.This strategy, known as oxidative phosphorylation, is employedby all respiratory microorganisms. While the importance of

0733-8716 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

MICHELUSI et al.: QUEUING MODELS FOR ABSTRACTING INTERACTIONS 585

biological electron transport is well-known for individual cells,the past decade has also brought about remarkable discoveriesof multicellular microbial communities that transfer electronsbetween cells and across much larger length scales than previ-ously thought [9], spanning from molecular assemblies knownas bacterial nanowires, to entire macroscopic architectures,such as biofilms and multicellular bacterial cables [5], [6].Electron transfer has been observed in nature [6] and in coloniescultured in the laboratory [10]. This multicellular interaction istypically initiated by a lack of either electron donor or acceptor,which in turn triggers gene expression. The overall goal in theseextreme conditions is for the colony to survive despite depri-vation, by relaying electrons from the donor to the acceptor tosupport the electron transport chain in each cell. The survival ofthe whole system relies on this division of labor, with the inter-mediate cells operating as relays of electrons to coordinate thiscollective response to the spatial separation of electron donorand acceptor.

These observations open up new frontiers in the modelingand design of microbial systems based on electron transfer,which may coexist with the more well-known interaction strate-gies based on molecular diffusion. A logical question to askis whether classical approaches to cascaded channels [11],[12] are suitable for the study of bacterial cables, which areindeed multi-hopped networks [13]. A straightforward cut-setbound analysis [14] shows that the capacity of such a sys-tem is achieved by a decode-and-forward strategy. However,in microbial systems, there is a stronger interaction betweenthe bacteria, due to the coupling of the electron signal withthe energetic state of the cells via the electron transport chain.Thus, alternative strategies for analysis and optimization mustbe undertaken,1 which take into account the specificity ofthe interactions between cells in a bacterial cable. In turn, itis important to understand how to design capacity-achievingsignaling schemes over bacterial cables, which exploit thesespecific interactions, as a preliminary step towards optimiz-ing fuel cell design (see Sec. I-B). In Sec. II, we summarizea stochastic queuing model of electron transfer for a singlecell, which links electron transfer to the energetic state of thecell (e.g., ATP concentration). This model abstracts the detailedworking of a single cell by using interconnected queues tomodel signal interactions. In Sec. III, we show how the pro-posed single-cell model can be extended to larger communities(e.g., cables, biofilms), by allowing electron transfer betweenneighboring cells. In Sec. III-A, based on such a model, weperform a capacity analysis for a bacterial cable, and discussthe design of signaling schemes.

b) Quorum Sensing [16]–[18]: As previously noted, biolog-ical systems are known to communicate by diffusing chemicalsignals in the surrounding medium [7], [8]. In quorum sens-ing, the concentration of certain signature chemical compoundsemitted by the bacteria is used to estimate the bacterial popula-tion size, so as to regulate collective gene expression. However,this simplistic explanation does not fully capture the complexity

1Relay systems employing decode and forward for molecular diffusion basedcommunication systems are considered in [15].

and variety of quorum sensing related processes. In this paper,we provide a model for quorum sensing, informed by our workon electron transfer, that will enable more sophisticated studyand analyses.

While quorum sensing is based on the emission, diffusionand detection of these chemical signatures across the envi-ronment in which the cell colony lives, it differs from recentwork endeavoring to model and design transceiver algorithmsbased on molecular diffusion channels (see, e.g., [19]–[22]) andto analyze their capacity (see, e.g., [23]–[26]). The focus ofthese works is the design of models and algorithms for engi-neered systems for future nanomachines to act as transmittersand receivers. Herein, instead, we attempt to understand naturalsystems and their optimization via the control of environmen-tal conditions. Rather than looking at the characteristics of thediffusion channel, we consider the system (the cell colony andthe surrounding environment) as a whole, and attempt to modelthe dynamic interactions between cells and of the cells with theenvironment. We abstract the diffusion channel by consideringa queueing model to represent it in terms of concentration ofsignals. As will be seen in our model depicted in Figs. 6 and 7,quorum sensing does not fit well into traditional multi-terminalframeworks such as the multiple access channel [27]–[30], thebroadcast channel [31], [32] or two-way communication [27].In fact, recent work examining capacity questions relating toquorum sensing have considered the binding of molecules andmulticellular processes with molecular diffusion [25], [33]. Themodels adopted therein do not take the dynamics of the bindingprocesses into consideration as we do here.

A. Prior Art on Mathematical Modeling of BacterialPopulations

The modeling of natural populations such as swarms (motilebacteria, swimming fish, flying birds, or migratory herds of ani-mals) has been consistently studied over the years. Microscopicmodels model individuals as point particles via dynamical equa-tions [34]–[36]. These models have precision, but become muchmore complex as the interactions are finely modeled and donot scale well as the population size increases. To combatsuch issues, the computational approach of simulating a largenumber of agents with prescribed interaction rules has alsobeen considered [34], [37]. These studies duplicate experimen-tal results well, but do not lead to design methodologies oroptimization strategies as the underlying operations are notconsidered. In contrast, macroscopic models model biologi-cal swarm dynamics via advection-diffusion-reaction partialdifferential equations [36], [38]–[41]. The partial differentialequations are used to describe the evolution of the probabil-ity distribution of group sizes or fraction of individuals withspecific characteristics (e.g., motion orientation [42]). Theseapproaches also suffer from computational complexity, oftenrelying on numerical solvers, and do not have an eye towardsoptimization and control.

From an information theoretic perspective, there havebeen ongoing efforts to examine how entropy and mutual


information2 can provide insights into experimental data [44]–[46]; however these works do not endeavor to model theunderlying processes, but rather compute empirical informa-tion theoretic measures based on experimental data. Anotherkey distinction is a lack of an explicit modeling of the dynamicsof the constituent systems.

Our modeling approach differs from these mathematicalmodels on the following aspects: 1) We propose an accuratequeueing approach to characterize the state of a single cellas a function of interactions with other cells and the environ-ment. Thus, not only this approach can be used to describe themetabolic state of a cell, but it also enables the construction ofan information theoretic framework; 2) our proposed modelsare inherently stochastic (rather than deterministic), and thuscapture the inherent randomness exhibited by cell colonies innature; 3) finally, they are amenable to complexity reduction,e.g, using methods based on statistical physics [47] or wirelessnetwork design [48]. We show an example of application ofthese principles to the capacity analysis of bacterial cables inSec. III-A.

B. Applications

The proposed queueing models serve as powerful predictivetools, which will aid the design of systems exploiting bacterialcapabilities. We highlight two relevant applications.

1) Microbial Fuel Cells: Renewable energy technologiesbased on bioelectrochemical systems are now attracting tensof millions of dollars in government and industry funding[9]. Microbial fuel cells and the essentially reverse processof microbial electrosynthesis are well known bioelectrochem-ical technologies; both use microorganisms to either generateelectricity or biofuels in a sustainable way [49]. Microbialfuel cells are constructed with microbes that oxidize diverseorganic fuels, including waste products and raw sewage, whilerouting the resulting electrons to large-area electrodes wherethey are harvested as electricity [50]. Certain bacterial strains(e.g., Shewanella oneidensis [50], [51]) are of particular inter-est for microbial fuel cells as they can attach to electrodesand transfer electrons without mediators which are often toxic,especially in the high concentrations needed to overcome theirdiffusion.

The challenge to realizing microbial fuel cells is under-standing the cooperative and anti-cooperative behaviors ofcollections of bacteria. Ideally, as the number of bacteriaincreases, the amount of current generated should increase aswell. However, this is not always observed in many experi-mental scenarios [52]. We posit that the cell-cell interactionsnecessitated by a multicellular (i.e. biofilm) lifestyle, includingelectron transfer across and through the biofilms themselves,can limit device performance. In other words, the biofilm’senergy output cannot be casually gleaned from what we knowabout the mechanisms relevant to individual or “disconnected”microbes. Optimization of biofilms, and multicellular perfor-mance in general, requires sophisticated models and predictivetools that account for the electronic and nutrient fluxes within

2See [43] for a review of the application of information theory to biology.

whole communities. In this paper, we propose queuing modelsto serve this purpose.

2) Suppression of Infections: Quorum sensing processeshave been of significant interest since first observed in Vibrioharveyi [18]. The discovery of quorum sensing in Pseudomonasaeruginosa, a human pathogen associated with cystic fibrosisand burn wound infections, demonstrates the medical implica-tions of quorum sensing regulation [53], [54]. In Pseudomonasaeruginosa infections, quorum sensing plays an important rolein activating many critical virulence pathways [55], [56]. Thegenes regulated by quorum sensing enable the bacteria to mounta successful attack or evade the body’s defenses. In these cases,blocking quorum sensing activation may help to prevent or treatmicrobial-associated diseases [57]. As bacteria build resistanceto antibiotics, reducing virulence through inhibition of quo-rum sensing may be more effective in mitigating disease versuseradicating the bacteria [58]. Naturally occurring compoundsand enzymes exist which can inhibit quorum sensing [57], [59].However, the complexity of signaling interactions within com-plex communities of microbes points to the need for a morequantitative understanding of quorum sensing regulation. Sincemany different types of bacteria use a similar set of signals,and many signals work in conjunction with other autoinduc-ers within the same cell, the potential outcomes of quorumsensing manipulation are not obvious. The combination of pre-dictive model development with quantitative experiments toparametrize and test models are essential to develop successfulstrategies for virulence reduction.

This paper is organized as follows. In Secs. II and III, wepropose a model of electron transfer for a single cell and a bac-terial cable, respectively, and present some capacity results. InSec. IV, we present a model of quorum sensing in a homo-geneous bacterial community. In Sec. V, we present somesimulation results and experimental data and, in Sec. VI, we dis-cuss extensions of the proposed quorum sensing model. Finally,in Sec. VII, we conclude the paper.

II. SINGLE CELL MODEL OF ELECTRON TRANSFER

In this section, we propose a queuing model of electrontransfer in a single cell. We will extend this model to inter-connections of cells in a bacterial cable in Sec. III. The cellis modeled as a system with an input electron flow comingeither from the electron donor via molecular diffusion, or froma neighboring cell via electron transfer, and an output flow ofelectrons leaving either toward the electron acceptor via molec-ular diffusion, or toward the next cell in the cable via electrontransfer (Fig. 1). Inside the cell, the conventional pathway ofelectron flow, enabled by the presence of the electron donorand the acceptor, is as follows (see the numbers in Fig. 1): elec-tron donors permeate inside the cell via molecular diffusion (1),resulting in reactions that produce electron-containing carriers(e.g., NADH) (2), which are collected in the internal electroncarrier pool. The electron carriers diffusively transfer electronsto the electron transport chain, which are then discarded byeither a soluble and internalized electron acceptor (e.g., molecu-lar Oxygen) or are transferred through the periplasm to the outer


Fig. 1. Stochastic model of electron transfer within a bacterial cable.

membrane and deposited on an extracellular electron accep-tor (3). The electron flow through the electron transport chainresults in the production of a proton concentration gradient(proton motive force, [60]) across the inner membrane of thecell (4), which is utilized by the inner membrane protein ATPsynthase to produce ATP, collected in the ATP pool and laterused by the cell as an energy source (5).

A. Stochastic Cell Model

In this section, we present our proposed cell model. Each cellincorporates four pools (see Fig. 1), each with an associatedstate, as a function of time t :

1) The internal electron carrier pool, which contains theelectron carrier molecules (e.g., NADH) produced as aresult of electron donor diffusion throughout the cellmembrane and chemical processes occurring inside thecell, with state mC H (t) ∈ MC H ≡ {0, 1, . . . , MC H } rep-resenting the number of electrons3 bonded to electroncarriers in the internal electron carrier pool, where MC H

is the electro-chemical storage capacity;2) The ATP pool, containing the ATP molecules pro-

duced via electron transport chain, with state n AT P (t) ∈NAX P ≡ {0, 1, . . . , NAX P }, representing the number ofATP molecules within the cell, where NAX P is the ATPcapacity of the cell;

3) The external membrane pool, which involves the extra-cellular respiratory pathway of the cell in the outermembrane. This is further divided into two parts: a) ahigh energy4 external membrane, which contains highenergy electrons coming from previous cells in thecable; and b) a low energy external membrane, whichcollects low energy electrons from the electron trans-port chain, before they are transferred to a neighbor-ing cell. We denote the number of electrons in the

3While in the following analysis we assume that one queuing "unit" corre-sponds to one electron, this can be generalized to the case where one "unit"corresponds to NE electrons.

4Note that the terms high and low referred to the energy of electrons are usedhere only in relative terms, i.e., relative to the redox potential at the cell surface.In bacterial cables, the redox potential slowly decreases along the cable, thusinducing a net electron flow.

TABLE IDESCRIPTION OF THE QUEUING ABSTRACTION FOR THE ELECTRON

TRANSFER MODEL

high energy and low energy external membranes asq(H)

E XT (t) ∈ Q(H)E XT ≡ {0, 1, . . . , Q(H)

E XT }, and q(L)E XT (t) ∈

Q(L)E XT ≡ {0, 1, . . . , Q(L)

E XT }, respectively, where Q(H)E XT

and Q(L)E XT are the respective electron “storage capaci-

ties”. The internal cell state at time t is thus

sI (t) =(

mC H (t), n AT P (t), q(L)E XT (t), q(H)

E XT (t))

. (1)

Cell behavior is also influenced by the concentrationsσD(t) and σE (t) of the electron donor and acceptor in thesurrounding medium, respectively. Therefore, we definethe external state as sE (t) = (σD(t), σA(t)).

Each pool in this model has a corresponding inflow and out-flow of electrons, each modeled as a Poisson process with ratefunction of the internal and external state (sI (t), sE (t)). Theintensities of these Poisson processes are labeled as in Fig. 1.For notational simplicity, the input and output flows are denotedas λ and μ, respectively, which commonly correspond to arrivaland service rates in the queueing literature. A description of thequeuing model abstraction is given in Table I.

B. Model Validation for an Isolated Cell

The experimental investigation of a multi-cellular network ofbacteria is very challenging, due to the technical difficulties ofplacing multiple cells next to each other in a controlled way,and of performing measurements of electron transfer, ATP andNADH concentrations along the cable [61]. Therefore, we startby investigating the properties of single, isolated cells, whichconstitute the building blocks of more complex multicellularsystems.

In the case of an isolated cell, multicellular electron trans-fer does not occur, hence μ

(L)E XT = λ

(H)E XT = 0 (see Fig. 1 for an

explanation of the intensities μ(L)E XT and λ

(H)E XT ). Therefore, after

a transient phase during which the low and high energy externalmembranes get emptied and filled, respectively, the cell’s stateand the corresponding Markov chain and state transitions are asdepicted in Fig. 2. In [61], we have validated this model basedon experimental data available in [62], using the following para-metric model for the rates, inspired by biological constraints(e.g., Fick’s law of diffusion [63]):

λC H (sI (t); sE (t)) = ρ(

1 − mC H (t)MC H

)σD(t),

μOU T (sI (t); sE (t)) = μC H (sI (t); sE (t))

= ζ(

1 − n AT P (t)NAX P

)σA(t),

μAT P (sI (t); sE (t)) = βσD(t),

(2)

where ρ, ζ, β ∈ R+ are parameters, estimated via curve fitting,and MC H = NAX P = 20.


Fig. 2. Stochastic model for an isolated cell after the transient phase, andMarkov chain with the corresponding transitions, for the case where MC H =NAX P = 4. The transition rates from state (2, 2) are also depicted.

Initially, cells are starved and the electron donor concentra-tion is 0. At time t = 80s, glucose is added to the cell culture.Afterwards, the electron donor concentration profile decreasesin a staircase fashion from σD(t) = 30[mM] glucose at timet = 80s, to σD(t) = 0[mM] at time t = 1300s, and σD(t) =0[mM] for t < 80s, whereas the electron acceptor concentra-tion (molecular Oxygen) is constant throughout the experiment,and sufficient to sustain reduction (σA(t) = 1, ∀t) [62]. Wehave designed a parameter estimation algorithm based on maxi-mum likelihood principles, outlined in [61], to fit the parametervector x = [ρ, ζ, β].

Fig. 3 plots the ATP time-series, related to the cell cul-ture, and the expected predicted values based on our proposedstochastic model. We observe a good fit of the prediction curvesto the experimental ones. In accordance with [62], the additionof the electron donor to the suspension of starved cells trig-gers an increase in ATP and NADH production as well as ATPconsumption, as observed experimentally and predicted by ourmodel. Further discussion can be found in [61].

III. TOWARDS A MODEL OF BACTERIAL CABLES

In multicellular structures, such as bacterial cables, an addi-tional pathway of electron flow may co-exist, termed intercel-lular electron transfer, which involves a transfer of electronsbetween neighboring cells, as opposed to molecules (electrondonor and acceptor) diffusing through the cell membrane. Inthis case, one or both of the electron donor and the acceptor arereplaced by neighboring cells in a network of inter-connectedcells. This cooperative strategy creates a multicellular elec-tron transport chain that utilizes intercellular electron transferto distribute electrons throughout an entire bacterial network.Electrons originate from the electron donor localized on oneend of the network and terminate to the electron acceptor onthe other end. The collective electron transport through this net-work provides energy for all cells involved to sustain their vitaloperations.

Since typical values for transfer rates between electron carri-ers (e.g. outer-membrane cytochromes) on the cell exterior arerelatively high [13], we assume that μ

(L)E XT = λ

(H)E XT = ∞ when

two cells are connected, so that any electron collected in thelow energy external membrane is instantaneously transferred tothe high energy external membrane of the neighboring cell inthe cable. Therefore, the low energy and high energy external

Fig. 3. Prediction of expected ATP level over time (in mM) and experimentaltime-series [61].

membrane pools of two neighboring cells can be joined intoa common pool for intercellular electron transfer. The internalstate of cell i at time t is thus

s(i)I (t) =

(m(i)

C H (t), n(i)AT P (t), q(L)(i)

E XT (t), q(H)(i)E XT (t)

), (3)

and its external state is s(i)E (t)=(σ

(i)D (t), σ (i)

A (t)). The state ofthe cable of length N is denoted as s(t) = (s(1)(t), . . . , s(N )(t)),where s(i)(t)=(s(i)

I (t), s(i)E (t)).

The internal state s(i)I (t) is time-varying and stochastic,

and evolves as a consequence of electro/chemical reactionsoccurring within the cell, chemical diffusion through thecell membrane, and intercellular electron transfer from theneighboring cell i − 1 to cell i , and then to the neighbor-ing cell i + 1. The evolution of s(i)

I (t) is also influenced

by the external state s(i)E (t) experienced by the cell. Let

s(i)I (t) =

(mC H , n AT P , q(L)

E XT , q(H)E XT

)be the state of the i th

cell at time t , and t+ be the time instant immediately followingt . We define the following processes which affect the evolutionof s(i)

I (t) (see Fig. 1):• Electron donor diffusion through the mem-

brane, joining the internal electron carrier poolwith rate λC H (s(i)

I (t); s(i)E (t)). The state becomes

s(i)I (t+) =

(mC H + 1, n AT P , q(L)

E XT , q(H)E XT

);

• Intercellular electron transfer from the neighboringcell i − 1, joining the high energy external mem-brane with rate λ

(L)E XT (s(i−1)

I (t); s(i−1)E (t)) (in fact, owing

to the high transfer rate approximation, the elec-tron joining the low energy external membrane ofcell i − 1 is immediately transferred to the highenergy external membrane of cell i), resulting in

s(i)I (t+) =

(mC H , n AT P , q(L)

E XT , q(H)E XT + 1

);

• Conventional ATP synthesis: this process involvesthe transfer of one electron from the internal elec-tron carrier pool to the internal membrane withrate μC H (s(i)

I (t), s(i)E (t)), resulting in the synthesis

of one unit of ATP. The electron then leaves theinternal membrane and either follows the aerobic


pathway (i.e., it is captured by an electron accep-tor), with rate μOU T (s(i)

I (t); s(i)E (t)), resulting in

s(i)I (t+) = (mC H − 1, n AT P + 1, q(L)

E XT , q(H)E XT ), or the

anaerobic one (i.e., it is collected in the low energy exter-nal membrane, also the high energy external membraneof cell i + 1), with rate λE XT (s(i)

I (t); s(i)E (t)), so that

s(i)I (t+) = (mC H − 1, n AT P + 1, q(L)

E XT + 1, q(H)E XT );

• Unconventional ATP synthesis: this process involves thetransfer of one electron from the high energy exter-nal membrane to the internal membrane to synthe-size one unit of ATP, with rate μ

(H)E XT (s(i)

I (t), s(i)E (t)).

Afterwards, the electron follows either the aerobicpathway with rate μOU T (s(i)

I (t); s(i)E (t)), resulting in

s(i)I (t+) = (mC H , n AT P + 1, q(L)

E XT , q(H)E XT − 1); or the

anaerobic one with rate λ(L)E XT (s(i)

I (t); s(i)E (t)), so that

s(i)I (t+) = (mC H , n AT P + 1, q(L)

E XT + 1, q(H)E XT − 1);

• ATP consumption, with rate μAT P (s(i)I (t); s(i)

E (t)), result-

ing in s(i)I (t+) = (mC H , n AT P − 1, q(L)

E XT , q(H)E XT ).

A. Information Capacity of Bacterial Cables

In [64], we have studied the information capacity of bacte-rial cables. By treating the bacterial cable as a communicationmedium, its capacity represents the maximum amount of infor-mation that can be transferred through the cable, and thus setsthe limit on the rate at which cells can communicate via electrontransfer.

Importantly, as shown in [64] and in the following anal-ysis, communication over a bacterial cable entails two con-flicting factors: 1) achieving high instantaneous informationrate; 2) inducing the bacterial cable to operate in informationefficient states. In fact, the input electron signal affects the ener-getic state of the cells along the cable via the electron transportchain, and thus affects their ability to relay electrons and totransfer information.

Indeed, our analysis reveals that the size of the ATP queuesalong the cable is extremely important to the health of thecable and to its capacity to transfer information (or electrons).Thus, with a greedy approach (denoted as myopic signalingand analyzed in Sec. III-A3), which ignores these biologicalconstraints, the overall achievable rate is measurably reducedcompared to the optimal capacity achieving scheme, analyzedin Sec. III-A2. Thus, the importance and impact of includ-ing the unique constraints and dynamics of bacterial cables isunderscored.

1) Queuing Model: The model [61] presumes that the chan-nel state is given by the interconnection of the state of each cellin the cable, leading to high dimensionality. As shown in Fig. 4,the bacterial cable contains N cells. Letting Scell be the statespace of the internal state of each cell, then the overall statespace of the cable is S ≡ SN

cell , which grows exponentially withthe bacterial cable length N . Indeed, one of the advantages ofour proposed queuing model is its amenability to complexityreduction. Herein, we propose an abstraction of [61] by treat-ing the bacterial cable as a black box, which captures only theglobal effects on the electron transfer efficiency of the cable,

Fig. 4. Communication system over a bacterial cable. "ENC"=encoder,"DEC"=decoder [64].

resulting from the local interactions and cells’ dynamics pre-sumed by [61]. Specifically, we let E(t) be the number ofelectrons carried in the cable, i.e., the sum of the number ofelectrons carried in the external membrane of each cell, whichparticipate in the electron transport chain to produce ATP forthe cell.5 The state space for this approximate model is E ≡{0, 1, . . . , Emax}, where Emax is the electron carrying capacityof the bacterial cable. Letting E (cell)

max be the electron carryingcapacity of a single cell, we have that Emax = N · E (cell)

max , whichscales linearly with the cable length, rather than exponentially.

The rationale behind this approximate model is as follows:when E(t) is large, the large number of electrons in the cablecan sustain a large ATP production rate, so that the ATP poolsof the cells are full and the cable is clogged. When this happens,the electron relaying capability of the bacterial cable is reduced,so that only a portion of the input electrons can be relayed.On the other hand, when E(t) is small, a weak electron flowoccurs along the cable, so that the ATP pools are almost empty,the cells are energy-deprived, and the cable can thus sustain alarge input electron flow to recharge the ATP pools. We thusdefine the clogging function α(E(t)) ∈ [0, 1], which representsthe fraction of electrons which can be relayed at the input of thecable. This model is in contrast to classical relay channels overcascades, where the performance is dictated by the worst linkrather than by a clogging function [14].

The communication system includes an encoder, which mapsthe message m ∈ {1, 2, . . . , M} to a desired electron inten-sity λ(t), taking binary values λ(t) ∈ {λmin, λmax}, t ∈ [0, T ],6

where T is the codeword duration, and λmin > 0 and λmax >

λmin are, respectively, the minimum and maximum electronintensities allowed into the cable.

The channel state E(t) evolves in a stochastic fashion asa result of electrons randomly entering and exiting the cable.Electrons enter the bacterial cable following a Poisson pro-cess with rate λin(t)=α(E(t))λ(t), resulting in the state E(t) toincrease by one unit, where α(i) ∈ [0, 1] is the clogging func-tion and λ(t) is the desired electron intensity, to be designed.

5Our model in [64] studies a more general model with leakage andinterference.

6Herein, we restrict the input signal to take binary values, since this choiceis optimal [64].


Electrons exit the cable following a Poisson process withrate μ(E(t)), resulting in the state E(t) to decrease by oneunit. In general, α(i) is a non-increasing function of i , withα(Emax)=0, α(0)=1, α(i)>0, ∀i<Emax. In fact, the largerthe amount of electrons carried by the cable E(t), the moresevere ATP saturation and electron clogging, and thus thesmaller α(E(t)).

2) Capacity Analysis: Using results on the capacity offinite-state Markov channels [65], we have proved that thecapacity of bacterial cables is given by

J = maxλ:E �→[λmin,λmax]

Emax∑i=0

πλ(i)I (λ(i); i), (4)

where• λ(i) is the average desired input electron intensity in state

E(t) = i , defined as

λ(i) � E [λ(t)|E(t) = i] ; (5)

• I (λ(i); i) is the instantaneous mutual information ratein state E(t) = i with expected desired input electronintensity λ(i), given by

I (x; i) � α(i)x log2

(λmax

x

)

+ α(i)λmax − x

λmax − λminλmin log2

(λmin

λmax

); (6)

• πλ(i) is the asymptotic steady-state distribution of thebacterial cable, induced by the desired input electronintensity λ(·), given by

πλ(i) =i−1∏k=0

α(k)λ(k)

μ(k + 1)πλ(0), (7)

and πλ(0) is obtained via normalization (∑

i πλ(i) = 1).The capacity optimization problem in Eq. (4) is a Markov

decision process [66], with state E(t), action λ(i) in stateE(t) = i , which generates the binary intensity signal, andreward function I (λ(i); i) in state E(t) = i , and can thusbe solved efficiently using standard optimization algorithms,e.g., policy iteration (see [66]). The following trade-off arises:1) the optimal input signal should, on the one hand, achievehigh instantaneous information rate, i.e., it should maximizeI (λ(E(t)); E(t)) at each time instant t ; 2) on the other hand,it should induce an "optimal" steady-state distribution of thecable, such that those states characterized by less severe clog-ging and where the transmission of information is more favor-able are visited more frequently. These two goals are in tension.In fact, the instantaneous information rate is maximum in stateswith large clogging state α(i) � 1, i.e., when E(t) is smalland the bacterial cable is deprived of electrons. Visits to thesestates are achieved more frequently by choosing λ(t) = λminwith probability one. However, under a deterministic inputdistribution the information rate is zero.

3) Myopic Signaling: While the optimal signaling λ∗(i)balances this tension by giving up part of the instantaneous

Fig. 5. Achievable information rate as a function of the minimum cloggingstate value, αmin, for the MP and OPT input distributions [64].

information rate in favor of better states in the future, themyopic signal greedily maximizes the instantaneous informa-tion rate, without considering its impact on the steady-statedistribution of the cable. This is defined, ∀i ∈ E, as

λM P (i) � arg maxx∈[λmin,λmax]

I (x; i)=λmax

e

(λmin

λmax

)− λminλmax−λmin

, (8)

and is constant with respect to the cable state E(t) = i , so thatit does not require channel state information at the encoder.Importantly, the myopic signal neglects the steady-state behav-ior of the cable, and thus it tends to quickly recharge the ATPreserves of the cells, resulting in severe clogging of the cable.Thus, the myopic signaling may induce frequent visits to statescharacterized by small clogging state α(E(t)) � 0, i.e., whenE(t) approaches Emax.

Indeed, in [64] we have shown that any input distribution λ(i)larger than the myopic one λM P (i) is deleterious to the capacityfor the following two reasons: 1) a lower instantaneous mutualinformation rate is achieved, compared to λM P (i) (by definitionof the myopic distribution, which maximizes I (x; i)); 2) fasterrecharges of electrons within the cable are induced, result-ing in frequent clogging of the cable, where the instantaneousinformation rate is small.

4) Numerical Results: We consider a cable with electroncapacity Emax = 1000. The clogging state α(i) and output rateμ(i) are given by7

α(i) = χ(i < Emax)

[1 − (1 − αmin)

i

Emax

], (9)

μ(i) = 0.6 + 0.8i

Emax.

In Fig. 5, we note that the achievable information rateincreases with αmin under both the optimal signaling (OPT) andthe myopic signaling (MP). This is because, as αmin increases,both α(i) and the instantaneous information rate I (x; i)increase as well (see Eqs. (9) and (6)). Intuitively, the larger

7The specific choices of α(i) and μ(i) have been discussed with Prof. M. Y.El-Naggar and S. Pirbadian, Department of Physics and Astronomy, Universityof Southern California, Los Angeles, USA.


α(i), the better the ability of the bacterial cable to transportelectrons.

OPT outperforms MP by ∼ 9% for small values of αmin.In fact, clogging is severe when the cable state approachesthe maximum value Emax. Therefore, in order to achieve highinstantaneous information rate, the state of the cable E(t)should be kept small. MP greedily maximizes the instanta-neous information rate, but this action results in an unfavorablesteady-state distribution, such that the cable is often in largequeue states E(t) � Emax, where clogging is severe and mostelectrons are dropped at the cable input. On the other hand,OPT gives up some instantaneous information rate in order tofavor the occupancy of low queue states, where α(i) approachesone and the transfer of information is maximum. The perfor-mance degradation of MP with respect to OPT is less severewhen αmin→1. In fact, in this case the instantaneous informa-tion rate I (x; i) is the same in all states (except Emax, whereα(Emax)=0 and I (x; Emax)=0), hence optimizing the steady-state distribution of the cable is unimportant. Further discussioncan be found in [64].

IV. TOWARDS A MODEL OF QUORUM SENSING

IN A HOMOGENEOUS CELL COLONY

In this paper, we propose stochastic queuing models as ageneral framework to capture signal interactions in microbialcommunities. We apply this general framework to two differ-ent microbial systems: electron transfer over bacterial cablesand quorum sensing. In the previous section, we summarized aqueuing model of electron transfer over bacterial cables [61],and we have shown how complexity reduction can be achievedby a more compact queuing model, which represents the num-ber of electrons carried in the cable by the state variable E(t)and defines a clogging function α(E(t)) over its state space[64]. Similarly, in this section, we propose a queuing model forthe simplest quorum sensing signaling, for the case of a homo-geneous cell colony which uses a single autoinducer-receptorpair, depicted in Figs. 6 and 7. This is the case, for instance,for Chromobacterium violaceum [67]. Similarly to the queuingmodel of electron transfer, this model can then be used as abaseline for further model reduction techniques.

We consider a colony composed of N (t) identical cellsat time t . We assume that these cells can duplicate, but donot die. In fact, only a small fraction of cells under favor-able growth conditions are dead (< 0.01%) [68], therefore celldeath has a negligible influence on quorum sensing dynamics.We model cell duplication as a Poisson process with intensityρ(n) per cell, function of the cell population size N (t) = n,i.e., each cell duplicates once every 1/ρ(n) hours, on average.Therefore, N (t) is a counting process with time-varying inten-sity N (t)ρ(N (t)). One model of ρ(n) is the logistic growthmodel ρ(n) = ρmax (1 − n/Nmax) [69], [70], where ρmax is themaximum growth rate and Nmax is the maximum populationsize. This model presumes that growth slows down as the pop-ulation increases, until N (t) = Nmax when growth stops. Wedenote the volume of a single cell as φcell , so that the total cellvolume of a colony of N (t) = n cells is Vcell(n) = nφcell , and

Fig. 6. Quorum sensing signaling system.

the total volume occupied by the colony (cell volume and extra-cellular environment) as Vtot (n). Here we consider both closedsystems, with a fixed total volume Vtot (n) = constant, and opensystems, in which the volume changes with the number of cells,Vtot (n) = nφex , where φex is the volume occupied by each cell(cell volume and extracellular environment).

The quorum sensing signaling system is characterized by dif-ferent signals, represented in Fig. 6: autoinducer molecules,produced by the synthases within each cell and released in theenvironment; receptors, located within each cell, which bindwith the autoinducer molecules to form autoinducer-receptorcomplexes; these complexes then bind to the active DNA sitesto produce further synthases, receptors and virulence factors.The dynamics of each of these signals and their interaction arecharacterized herein.

A. Environment State

Let A(t) be the number of autoinducer molecules in the sys-tem. Thus, the concentration of autoinducers in the volumeoccupied by the colony is

ηA(t) � A(t)

Vtot (N (t)). (10)

The dynamics of A(t) over time is affected by autoin-ducer production, degradation, receptor binding, and leakage.Leakage accounts for the autoinducer molecules that leak outof the volume before being captured by the cells. We assumethat the larger the colony, the smaller the leakage. In fact, thelarger the colony of cells, the higher the likelihood that thesemolecules are detected before they diffuse out of the system.Additionally, autoinducer molecules may chemically degradethus becoming inactive. We thus let δA(n) be the rate of leakageand degradation for each autoinducer molecule as a function ofthe number of cells N (t)=n, where δA(n)≥δA(n+1), ∀n≥1.An example is

δA(n) = ξD + ξL ,1

1 + ξL ,2(n − 1), (11)

where ξD is the degradation rate of each autoinducer molecule,ξL ,1 is the leakage rate when only one cell is present, and ξL ,2 isthe rate of leakage decay as the colony size increases. Given thatthere are A(t) autoinducer molecules in the system, the overallleakage rate is A(t)δA(N (t)).


Fig. 7. Queuing model of quorum sensing signaling system.

Remark 1: Both ξL ,1 and ξL ,2 depend on the diffusionproperties of the medium and the spatial distribution of thecells [71]. For a closed system without leakage, we have thatξL ,1 = 0, hence δA(n) = ξD, ∀n. However, (11) may not cap-ture all possible scenarios. Other scenarios can be accommo-dated by an appropriate selection of δA(n) (possibly, differentfrom (11)).

B. Cell State

Each cell is described by three state variables, as repre-sented in Fig. 7. For cell i , we let Ri (t) be the number ofunbound receptors at time t , and Ci (t) be the number ofautoinducer-receptor complexes within the cell. Each cell usesa protein called synthase to synthesize autoinducers. In turn,these autoinducers are released outside the cell through itsmembrane. We let Si (t) be the number of synthases within cell iat time t . The state of cell i is thus given by (Ri (t), Ci (t), Si (t))at time t . No leakage of receptors, complexes or synthasesoutside of the cell occurs.

Each of the Ci (t) autoinducer-receptor complexes randomlybinds to active sites in the DNA sequence, resulting in specificgene expression (see Fig. 6). We assume three active bindingsites: 1) the first site contains the code to produce synthases,resulting in the increase of Si (t) by one unit; 2) the secondsite contains the code to produce receptors, resulting in theincrease of Ri (t) by one unit; 3) the third site contains the codeto produce the virulence factor.

Remark 2: It is worth noting that, while in the case ofChromobacterium violaceum the receptor and the synthase aremade from distinct DNA binding sites, in some cases both com-pounds are made from the same DNA binding site [72]. Thisscenario can be incorporated by letting the first site contain thecode to synthesize both synthases and receptors, resulting in theincrease of both Si (t) and Ri (t) by one unit. Additionally, indi-vidual binding events may lead to the production of multiplesynthases or receptors [73]. This scenario can be included bydefining a random number of synthases or receptors producedat each binding event, with a given probability distribution.

After the gene is expressed, the autoinducer-receptor com-plex unbinds from the DNA site. We assume that the

binding-unbinding process is instantaneous, i.e., we neglectthe amount of time that the complex occupies the DNA site.We model the process of autoinducer-receptor complexes bind-ing to the DNA sites as Poisson processes, whose intensitiesdepend on the affinity of the complex to a specific site. Thus,we let εC, j , j ∈ {1, 2, 3} be the binding rate of each complexto site j (see Fig. 6). Additionally, gene expression occurs evenin the absence of complexes, at basal rate ε0, j , j ∈ {1, 2, 3}.Therefore, since there are Ci (t) autoinducer-receptor com-plexes within cell i , the gene located in site j is expressed withoverall rate ε0, j + Ci (t)εC, j .

C. State Dynamics

N (t) increases over time due to cell growth; A(t) increasesover time as the synthases located within each cell produceautoinducers, and decreases over time as a result of autoinduc-ers binding to receptors, leakage and degradation (Sec. IV-A);Ri (t) increases over time as the cell synthesizes receptors, anddecreases over time as a result of autoinducers binding to recep-tors and receptor degradation; similarly, Ci (t) increases overtime as the unbound receptors bind to the autoinducer signal inthe environment, and decreases over time as a result of degrada-tion and complex unbinding; finally, Si (t) increases over timeas the cell produces synthases, and decreases over time as aresult of degradation. We let⎧⎪⎨

⎪⎩RTOT(t) �

∑N (t)i=1 Ri (t),

CTOT(t) �∑N (t)

i=1 Ci (t),

STOT(t) �∑N (t)

i=1 Si (t),

(12)

be the total amount of receptors, complexes and synthases ofthe cell colony, and ⎧⎪⎨

⎪⎩ηR(t) � RTOT(t)

Vcell (N (t)) ,

ηC (t) � CTOT(t)Vcell (N (t)) ,

ηS(t) � STOT(t)Vcell (N (t)) ,

(13)

be their concentrations, respectively.8 There are several pro-cesses that affect the dynamics of (N (t), A(t), RTOT(t),CTOT(t), STOT(t)), as detailed below.

a) Cell duplication: Cell duplication occurs with rateρ(N (t)) for each cell, so that the cell population N (t)increases by one unit with rate ρ(N (t))N (t). When onecell duplicates, its content is split randomly among the twocells. Therefore, if (Ri (t), Ci (t), Si (t)) is the state of cell ibefore its duplication, after the duplication (at time instant t+)there are two cells with state (R(1)

i (t+), C (1)i (t+), S(1)

i (t+))

and (R(2)i (t+), C (2)

i (t+), S(2)i (t+)), respectively, where

R(1)i (t+) + R(2)

i (t+) = Ri (t), C (1)i (t+) + C (2)

i (t+) = Ci (t),

and S(1)i (t+) + S(2)

i (t+) = Si (t), with probability distribution

P

((R(1)

i (t+), C (1)i (t+), S(1)

i (t+))=(r, c, s)|Ri (t), Ci (t), Si (t))

=(

Ri (t)r

) (Ci (t)

c

)(Si (t)

s

)2−Ri (t)−Ci (t)−Si (t). (14)

8Note that receptors, complexes and synthases are located inside the cellrather then in the extracellular environment, therefore their concentrations arecalculated with respect to the total cell volume Vcell (N (t)).


TABLE IIDESCRIPTION OF THE QUEUING ABSTRACTION FOR THE QUORUM SENSING MODEL

b) One autoinducer is created: Each synthase producesautoinducers with rate β. Since there are STOT(t) synthases, thecell colony produces autoinducers with rate βSTOT(t), resultingin the increase of A(t) by one unit.

c) One autoinducer leaks or degrades: Each autoinducerleaks or degrades with rate δA(N (t)). Therefore, autoinducerleakage and degradation occurs with rate δA(N (t))A(t), result-ing in the decrease of A(t) by one unit. Degradation rates aredependent on auto-inducer type and environmental factors, asdiscussed in [70], [74].

d) One receptor is created: One receptor is created wheneverone complex binds to the second active DNA site. This occurswith rate N (t)ε0,2 + CTOT(t)εC,2 across the whole cell colony,resulting in the increase of RTOT(t) by one unit.

e) One receptor degrades: Each receptor degrades withrate δR . Therefore, receptor degradation occurs with rateδR RTOT(t), resulting in the decrease of RTOT(t) by one unit.

f) One complex is created: Typically, binding of autoinduc-ers to receptors to form autoinducer-receptor complexes doesnot occur until a threshold concentration of autoinducers isachieved [75]. We let this threshold be ηA,th. Thus, if ηA(t) <

ηA,th, then no binding of autoinducers to receptors occurs.When the threshold is exceeded, i.e., ηA(t) ≥ ηA,th, then thebinding of autoinducers to receptors is a Poisson process withintensity γ [per unit of autoinducer and receptor concentra-tions, per unit volume, per hour]. Since the concentrations ofautoinducers and free receptors is ηA(t) and ηR(t), respectively,and the total cellular volume where these reactions occur isVcell(N (t)) = N (t)φcell , complexes form with rate

λC (A(t), RTOT(t), N (t))

� γ ηA(t)ηR(t)χ(ηA(t) ≥ ηA,th)Vcell(N (t))

= γA(t)RTOT(t)

Vtot (N (t))χ(A(t) ≥ ηA,thVtot (N (t))), (15)

so that A(t) and RTOT(t) decrease by one unit,9 and CTOT(t)increases by one unit.

g) One complex degrades: Each autoinducer-receptor com-plex degrades with rate δC . Therefore, autoinducer-receptorcomplex degradation occurs with rate δC CTOT(t), resulting inthe decrease of CTOT(t) by one unit.

h) One complex unbinds: Each complex may randomlyunbind, and the constituent autoinducer and receptor moleculesbecome active again. This event occurs with rate υC for eachcomplex. Therefore, unbinding of complexes occurs with rate

9In this paper, for simplicity, we assume that one receptor binds to oneautoinducer to form one complex. However, in most systems, two receptorsbind an autoinducer.

υC CTOT(t), resulting in the decrease of CTOT(t) by one unitand in the increase of A(t) and RTOT(t) by one unit.

i) One synthase is created: One synthase is created when-ever one complex binds to the first active DNA site. This occurswith rate N (t)ε0,1 + CTOT(t)εC,1 across the whole cell colony,resulting in the increase of STOT(t) by one unit. Since synthasescreate autoinducers and autoinducer-receptor complexes createnew synthases, the production of autoinducer can be consideredto be autocatalytic.

j) One synthase degrades: Each synthase degrades with rateδS . Therefore, synthase degradation occurs with rate δS STOT(t),resulting in the decrease of STOT(t) by one unit.

k) Virulence factor expression: Finally, the gene located insite 3 is expressed with intensity ε0,3 + Ci (t)εC,3 in cell i ,resulting in the overall expression with intensity N (t)ε0,3 +CTOT(t)εC,3 over the whole cell colony.

D. State Representation and State Space Reduction

Let R(t) = [R1(t), R2(t), . . . , RN (t)(t)] be the vector ofreceptor concentrations, C(t) = [C1(t), C2(t), . . . , CN (t)(t)]be the vector of autoinducer-receptor complexes concen-trations, and S(t) = [S1(t), S2(t), . . . , SN (t)(t)] be the vec-tor of synthase concentrations. These vectors have time-varying length N (t), due to cell duplication. Then, thestate of the system at time t is described by the tuple(N (t), A(t), R(t), C(t), S(t)). We notice that the state spacegrows exponentially with the cell population size, sim-ilarly to the model of electron transfer in Sec. III.However, the analysis of state dynamics above highlightsthat (N (t), A(t), RTOT(t), CTOT(t), STOT(t)) is a Markov chainwith dynamics described in Table II, hence complexity reduc-tion can be achieved. Indeed, this stochastic model and suchMarkov property can be exploited to infer the overall quorumsensing dynamics, rather than tracking the dynamics of eachspecific cell, i.e.,

pt (n, a, r, c, s) (16)

� P (N (t)=n, A(t)=a, RTOT(t)=r, CTOT(t)=c, STOT(t)=s) .

V. SIMULATIONS RESULTS AND EXPERIMENTAL DATA

In this section, we present simulation results for the homo-geneous cell colony considered in Sec. IV. The param-eters are listed in Table III. We consider two differentsetups: open system and closed system, both with initial state(N (0), A(0), RTOT(0), CTOT(0), STOT(0)) = (1, 0, 0, 0, 0) andthe following parameters.

a) Open system: Here, cells in an open system exist as adensely packed biofilm attached tho a surface. Autoinducers


TABLE IIISIMULATION PARAMETERS (NM DENOTES NANOMOLAR CONCENTRATION)

may leak, with parameters ξL ,1 = 5000 and ξL ,2 = 0.1, and thecell colony grows in an open space. Cells grow tightly packedtogether, so that the extracellular environment takes only 10%of the cell volume (φex = 1.1φcell ). The maximum populationsize is Nmax = ∞, so that cell growth does not slow down. Theconcentration of autoinducers is computed as in (10), over thetotal volume Vtot (N (t)) = N (t)φex , whereas the concentrationof receptors, complexes and synthases is computed as in (13),over the cell volume Vcell(N (t)) = N (t)φcell .

b) Closed system: Autoinducers do not leak (ξL ,1 = ξL ,2 =0). The cell colony grows in a closed space of volume Vtot (n) =0.1nL. The maximum population size is Nmax = 1000, so that,when growth is complete, the cell volume takes 1% of the totalvolume. The concentration of autoinducers is computed as in(10), over a constant total volume Vtot (N (t)) = 0.1nL, whereasthe concentration of receptors, complexes and synthases is com-puted as in (13), over the cell volume Vcell(N (t)) = N (t)φcell ,which grows with the population size.

Open and closed systems occur both in nature and in exper-imental testbeds: microdroplets [4], [76] and well-plates [77]are examples of closed systems, whereas cells on agar plates[69], [78] or in the presence of flow [79] are examples of opensystems.

We use Gillespie’s exact stochastic simulation algorithm [83]to simulate the dynamics of the cell colony. In Figs. 8.a-e,we plot the time series of cell concentration, the concentrationof autoinducers, receptors, complexes and synthases, respec-tively. From Figs. 8.d-e, we notice that the concentrations ofautoinducer-receptor complexes and of synthases are small andapproximately constant before 6 hours and 8 hours for theopen and closed systems, respectively, and then grow quicklyand steadily afterwards. We deduce that quorum sensing acti-vation occurs after approximately 6 hours in the open systemand after 8 hours in the closed system. Indeed, as can beobserved in Fig. 8.b, the concentration of autoinducers growsslowly before quorum sensing activation, until reaching the crit-ical concentration ηA,th = 21.4 [nM]. Once this threshold isreached, complexes start to get formed with rate given by Eq.(15), thus explaining the steep increase observed in Fig. 8.d.In turn, the high concentration of complexes, which frequentlybind to the DNA sites, contributes to the buildup of synthasesin the system, thus explaining the steep increase observed in

Fig. 8.e. Finally, each synthase produces autoinducers withrate β, thus contributing to the buildup of autoinducers andsustaining the quorum sensing positive feedback loop.

Quorum sensing activation is faster in the open system, sincea much higher cell density is achieved (Fig. 8.a), resulting ina steeper growth of autoinducer concentration, and thus earlierquorum sensing activation. In the open system the parameterthat accounts for the loss of autoinducers to the surround-ings will depend on the system, and further work is needed todetermined if closed systems in general have slower activationkinetics.

In Fig. 8.a, we note that, when quorum sensing activationoccurs, cell density is much smaller in the closed system than inthe open system. This is because, in the closed system, autoin-ducers do not leak, and thus a lower concentration of cells issufficient to reach the critical concentration of autoinducers foractivation. In closed systems autoinducers are produced by dis-persed cells and get diluted and retained in the entire volume ofthe system, whereas in open systems autoinducers are producedin concentrated regions of cells but escape in the surroundings.The local concentration of producers and autoinducer escapeare key factors that dictate quorum sensing activation. Similardependencies have been observed in other nonlinear systems[84].

In Fig. 8.c, we note that, in the open system, the concentra-tion of receptors drops after quorum sensing activation. This isdue to the high rate of autoinducers binding to receptors to formcomplexes, thus reducing the availability of free receptors. Asimilar behavior can be noticed in the closed system. However,the drop in this case is followed by a build up of receptors.In fact, the lower concentration of autoinducers in this caselimits the production of complexes, resulting in excess freereceptors.

Finally, we provide experimental data for a closed sys-tem. The genes required for quorum sensing were placed intoEscherichia coli using a plasmid constructed in [85]. Thegenetic constructs place a green fluorescent protein under con-trol of quorum sensing. Cells were grown in LB media at 37 ◦C.Growing cells are diluted such that cells remain in the earlyto mid exponential phase of growth, before quorum sensing isactivated, for several generations before measurement. Sincemost proteins are stable for long periods of time, maintaining


Fig. 8. Simulation of quorum sensing system and experimental data. Comparison of open and closed systems. Sampling period 10m.

cultures at low density for several generations is needed to ini-tialize the population of cells to a quorum sensing off state.Fluorescence was monitored using a 96 well-plate reader [77]and cell concentration was measured by absorbance of light at600 nm.

As seen in Fig. 8.e, the dynamics of the green fluorescentprotein regulated by quorum sensing in experiments is qual-itatively similar to the predicted expression of synthase: theyboth exhibit the characteristic switch from a low to high pro-duction rate after quorum sensing is activated. However, green


fluorescent protein production is immeasurable or zero prior toquorum sensing activation, which suggests that it has a lowerbasal level of gene expression than that assumed for synthase inour model. The basal level of expression can be independentlytuned for each quorum sensing regulated gene. Genes that arepart of the quorum sensing machinery, such as synthases andreceptors, likely need a higher basal expression rate to primequorum sensing activation, The consequences of differences inthe basal and activated rates of expression for quorum sensingcontrolled genes warrants further investigation.

Fig. 8.f plots the cell concentration over time and quorumsensing activation time. Note that the dynamics of cell concen-tration over time for the closed system (Fig. 8.a) are similarto experimental data (Fig. 8.f), i.e., the cell population growsexponentially fast until reaching a certain maximum concen-tration, thus validating the logistic growth model employed.However, both the cell duplication rate and the maximum con-centration exhibit different values in our model (ρmax = 1 [percell per hour] and Nmax/Vtot (n) = 1010 [cells per mL], respec-tively) and in the experimental data (ρmax = 0.65 [per cellper hour] and Nmax/Vtot (n) = 8 × 108 [cells per mL], respec-tively). Indeed, these parameters are variable and depend onspecifics of each experiment, such as the cell type, compositionof the media, and growth conditions.

VI. EXTENSIONS

Our framework can accommodate several aspects of quorumsensing observed in nature.

A. Signal Integration

Indeed, most quorum sensing signaling systems are morecomplex than the scenario considered in Sec. IV, whichincludes one receptor-autoinducer pair only. For instance,Pseudomonas aeruginosa contains multiple quorum sens-ing networks that work together to monitor changes in thelocal population of cells. Since the initial discovery of theLasI/LasR system, two other quorum sensing circuits inPseudomonas aeruginosa have been discovered and character-ized, the RhlI/RhlR and PQS [86]. Recently a fourth quorumsensing system has been identified, IQS [87]. Although each ofthese networks has a unique signal and receptor, these four sys-tems are coupled in a feedback network in which each signalingpair regulates the production of both the synthase and receptorof at least one other network. Vibrio species have similar com-plexity in their quorum sensing networks [88], integrating infor-mation from multiple signals to direct regulatory decisions.

B. Interference

In addition to individual microbes producing multiple sig-nals, quorum sensing systems also mediate communicationbetween different species. Many bacteria find themselvesin diverse and crowded environments, and many of theseneighboring species will also produce autoinducers. The typeof autoinducer produced by many Vibrio and Pseudomonasspecies has been identified in more than 70 different species

[89], and there are examples of signals or enzymes that inhibitquorum sensing activation [90]. Within these multispecies com-munities, there is significant potential for interference with theprocess of quorum sensing activation, i.e., any process whichalters the ability of cells to produce, exchange, recognize, orrespond to a quorum sensing autoinducer. Several mechanismsof interference that occur naturally have been identified.

1) Signal synthesis inhibition: Although no examples ofthis interaction have been found in nature, a syn-thetic compound that prevented acyl-homoserine lactone(HSL) production by the TofI synthase in the soil microbeBurkholderia glumae has been identified [91]. This com-pound occupies the active site of the synthase, preventingautoinducer production.

2) Destruction or chemical modification of the autoinducer:The common example of signal destruction is the AiiAenzyme isolated from Bacillus [90]. AiiA is an enzymethat degrades a common type of autoinducer, thereby pre-venting the autoinducer from binding to the receptor. Inanother example, the oxidoreductase from Burkholderiais also capable of inactivating some autoinducers throughchemical modification, thereby altering the specificity forreceptor molecules.

3) Competitive binding to the receptor: Some species havebeen shown to produce small molecules that interferewith autoinducer binding to the receptor, for examplea furanone compound produced by the alga Deliseapulchra [92]. In other cases the competition is fromanalogous autoinducers. There are many versions of theautoinducer HSL, with differing carbon tails, and eachbacteria produces one or more types of HSL. For exam-ple, Pseudomonas aeruginosa produces the autoinducersC4-HSL and 3oxo-C12-HSL, whereas Chromobacteriumviolaceum produces C6-HSL [93]. The receptor ofChromobacterium violaceum has been found to be acti-vated by C6-HSL, but deactivated or inhibited by HSLwith longer carbon tails, such as 3oxo-C12-HSL [94].In this way, crosstalk between autoinducers produced byneighboring bacteria can influence on signal transductionand quorum sensing activation.

Our model in Sec. IV can be extended to include differ-ent types of interference. Consider an interfering signal withconcentration ηI (t) � I (t)/Vtot (N (t)), where I (t) is the totalamount of interfering molecules in the volume of interest. Theinterfering signal is injected in the system with rate μI , andleaks with rate δI (N (t)), which is a function of the colonysize, similarly to autoinducers. We may consider the followingmodels of interference:

• Receptor inhibition: the interfering signal binds withthe receptors, thus competing with the autoinducer sig-nal; unlike the autoinducer-receptor complex, whichinduces specific gene expression by binding to the DNAsites, the complex formed by the interfering signal andthe receptor becomes inactive; the binding of interfer-ing molecules to receptors is a Poisson process withintensity γI R [per unit of interfering signal and recep-tor concentrations, per unit volume, per hour]. Sincethe concentrations of the interfering signal and of free


receptors is ηI (t) and ηR(t), respectively, the overallbinding rate is γI RηI (t)ηR(t)Vcell(N (t)), since thesereactions occur inside the cells and the total cell volumeis Vcell(N (t)); correspondingly, both RTOT(t) and I (t)decrease by one unit;

• Synthase blocking: the interfering signal binds withthe active site of the synthase, thus preventing autoin-ducer production; the binding of interfering moleculesto synthases is a Poisson process with intensity γI S

[per unit of interfering signal and synthase concentra-tions, per unit volume, per hour]. Since the concen-trations of the interfering signal and of synthases isηI (t) and ηS(t), respectively, the overall binding rate isγI SηI (t)ηS(t)Vcell(N (t)); correspondingly, both STOT(t)and I (t) decrease by one unit;

• Autoinducer degradation: the interfering signal destroysor chemically modifies the autoinducer signal, thus pre-venting it from binding to the receptor; the binding ofinterfering molecules to autoinducers is a Poisson pro-cess with intensity δI A [per unit of interfering signaland autoinducer concentrations, per unit volume, perhour]. Since the concentrations of the interfering sig-nal and of autoinducers is ηI (t) and ηA(t), respectively,and these reactions occur over the volume Vtot (N (t))occupied by the cell colony, the overall binding rate isδI AηI (t)ηA(t)Vtot (N (t)); correspondingly, both A(t) andI (t) decrease by one unit; this type of interference canbe also represented as an additional source of leakagefor the autoinducers, so that their overall leakage rate is[δA(N (t)) + δI AηI (t)]A(t).The model can be extended to include multiple interferingsignals in a similar fashion.

VII. CONCLUSIONS

In this position paper, we have explored how abstractionsfrom communications, networking and information theory canplay a role in understanding and modeling bacterial interac-tions. In particular, we have examined two forms of interactionsin bacterial systems: electron transfer and quorum sensing. Wehave presented a stochastic queuing model of electron transferfor a single cell, and we have showed how the proposed single-cell model can be extended to bacterial cables, by allowing forelectron transfer between neighboring cells. We have performeda capacity analysis for a bacterial cable, demonstrating thatsuch model is amenable to complexity reduction. Moreover, wehave provided a new queuing model for quorum sensing, whichcaptures the dynamics of the quorum sensing signaling sys-tem, signal interactions, and cell duplication within a stochasticframework. We have shown that sufficient statistics can be iden-tified, which allow a compact state space representation, and wehave provided preliminary simulation results. Our investigationreveals that queuing models effectively capture interactions inmicrobial communities and the inherent randomness exhibitedby cell colonies in nature, and they are amenable to complexityreduction using methods based on statistical physics and wire-less network design. Thus, they represent powerful predictivetools, which will aid the design of systems exploiting bacterialcapabilities.

ACKNOWLEDGMENTS

The authors would like to thank P. Silva for providing theexperimental quorum sensing data.

REFERENCES

[1] A. J. Kaufman, “Early earth: Cyanobacteria at work,” Nat. Geosci., vol. 7,no. 4, pp. 253–254, Apr. 2014.

[2] C. M. Hogan, “Bacteria,” in Encyclopedia of Earth. Washington, DC,USA: National Council for Science and the Environment, 2010.

[3] G. A. Kowalchuk, S. E. Jones, and L. L. Blackall, “Commentary:Microbes orchestrate life on earth,” ISME J., vol. 2, pp. 795–796, 2008.

[4] J. Boedicker, M. Vincent, and R. Ismagilov, “Microfluidic confinement ofsingle cells of bacteria in small volumes initiates high-density behavior ofquorum sensing and growth and reveals its variability,” Angew. Chem. Int.Ed., vol. 48, pp. 5908–5911, 2009.

[5] G. Reguera, K. D. McCarthy, T. Mehta, J. S. Nicoll, M. T. Tuominen, andD. R. Lovley, “Extracellular electron transfer via microbial nanowires,”Nature, vol. 435, no. 7045, pp. 1098–1101, 2005.

[6] C. Pfeffer et al., “Filamentous bacteria transport electrons over centimetredistances,” Nature, vol. 491, no. 7423, pp. 218–221, 2012.

[7] T. Nakano, A. W. Eckford, and T. Haraguchi, Molecular Communication.Cambridge, U.K.: Cambridge Univ. Press, 2013.

[8] I. Mian and C. Rose, “Communication theory and multicellular biology,”Integr. Biol., vol. 3, no. 4, pp. 350–367, 2011.

[9] M. Y. El-Naggar and S. E. Finkel, “Live wires: Electrical signalingbetween bacteria,” Scientist, vol. 27, no. 5, pp. 38–43, 2013.

[10] S. Kato, K. Hashimoto, and K. Watanabe, “Microbial interspecies elec-tron transfer via electric currents through conductive minerals,” Proc.Nat. Acad. Sci., vol. 109, no. 25, pp. 10042–10046, Jun. 2012.

[11] R. Silverman et al., “On binary channels and their cascades,” IRE Trans.Inf. Theory, vol. 1, no. 3, pp. 19–27, Dec. 1955.

[12] M. Simon, “On the capacity of a cascade of identical discrete memory-less nonsingular channels,” IEEE Trans. Inf. Theory, vol. IT-16, no. 1,pp. 100–102, Jan. 1970.

[13] S. Pirbadian and M. Y. El-Naggar, “Multistep hopping and extracellularcharge transfer in microbial redox chains,” Phys. Chem. Chem. Phys.,vol. 14, pp. 13802–13808, 2012.

[14] T. M. Cover and J. A. Thomas, Elements of Information Theory. NewYork, NY, USA: Wiley-Interscience, 1991.

[15] A. Einolghozati, M. Sardari, and F. Fekri, “Relaying in diffusion-basedmolecular communication,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT),2013, pp. 1844–1848.

[16] M. B. Miller and B. L. Bassler, “Quorum sensing in bacteria,” Annu. Rev.Microbiol., vol. 55, no. 1, pp. 165–199, 2001.

[17] K. L. Visick and C. Fuqua, “Decoding microbial chatter: Cell-cell com-munication in bacteria,” J. Bacteriol., vol. 187, no. 16, pp. 5507–5519,2005.

[18] K. Nealson, T. Platt, and J. Hastings, “Cellular control of synthesis andactivity of bacterial luminescent system,” J. Bacteriol., vol. 104, no. 1,pp. 313–322, 1970.

[19] M. S. Kuran, H. B. Yilmaz, T. Tugcu, and I. F. Akyildiz, “Interferenceeffects on modulation techniques in diffusion based nanonetworks,” NanoCommun. Netw., vol. 3, no. 1, pp. 65–73, 2012.

[20] H. Arjmandi, A. Gohari, M. N. Kenari, and F. Bateni, “Diffusion-based nanonetworking: A new modulation technique and perfor-mance analysis,” IEEE Commun. Lett., vol. 17, no. 4, pp. 645–648,Apr. 2013.

[21] R. Mosayebi, H. Arjmandi, A. Gohari, M. Nasiri-Kenari, and U. Mitra,“Receivers for diffusion-based molecular communication: Exploitingmemory and sampling rate,” IEEE J. Sel. Areas Commun., vol. 32, no. 12,pp. 2368–2380, Dec. 2014.

[22] A. Noel, K. C. Cheung, and R. Schober, “Optimal receiver design fordiffusive molecular communication with flow and additive noise,” IEEETrans. NanoBiosci., vol. 13, no. 3, pp. 350–362, Sep. 2014.

[23] K. Srinivas, A. W. Eckford, and R. S. Adve, “Molecular communicationin fluid media: The additive inverse Gaussian noise channel,” IEEE Trans.Inf. Theory, vol. 58, no. 7, pp. 4678–4692, Jul. 2012.

[24] H. Li, S. M. Moser, and D. Guo, “Capacity of the memoryless additiveinverse Gaussian noise channel,” IEEE J. Sel. Areas Commun., vol. 32,no. 12, pp. 2315–2329, Dec. 2014.

[25] A. Einolghozati, M. Sardari, and F. Fekri, “Design and analysis ofwireless communication systems using diffusion-based molecular com-munication among bacteria,” IEEE Trans. Wireless Commun., vol. 12,no. 12, pp. 6096–6105, Dec. 2013.


[26] M. Pierobon and I. F. Akyildiz, “Capacity of a diffusion-based molecularcommunication system with channel memory and molecular noise,” IEEETrans. Inf. Theory, vol. 59, no. 2, pp. 942–954, Feb. 2013.

[27] C. E. Shannon et al., “Two-way communication channels,” in Proc. 4thBerkeley Symp. Math. Statist. Probab., 1961, vol. 1, pp. 611–644.

[28] E. C. Van Der Meulen, “The discrete memoryless channel with twosenders and one receiver,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT),1971, p. 78.

[29] H. H.-J. Liao, “Multiple access channels,” DTIC Document, DefenseTech. Inform. Center, Fort Belvoir, VA, Tech. Rep., 1972 [Online].Available: http://www.dtic.mil/docs/citations/AD0753127

[30] R. Ahlswede, “The capacity region of a channel with two senders and tworeceivers,” Ann. Probab., vol. 2, no. 5, pp. 805–814, 1974.

[31] T. M. Cover, “Broadcast channels,” IEEE Trans. Inf. Theory, vol. IT-18,no. 1, pp. 2–14, Jan. 1972.

[32] T. M. Cover, “Comments on broadcast channels,” IEEE Trans. Inf.Theory, vol. 44, no. 6, pp. 2524–2530, Oct. 1998.

[33] A. Einolghozati, M. Sardari, A. Beirami, and F. Fekri, “Consensus prob-lem under diffusion-based molecular communication,” in Proc. 45thAnnu. Conf. Inf. Sci. Syst. (CISS), 2011, pp. 1–6.

[34] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, “Noveltype of phase transition in a system of self-driven particles,” Phys. Rev.Lett., vol. 75, pp. 1226–1229, Aug. 1995.

[35] T. Vicsek and A. Zafeiris, “Collective motion,” Phys. Rep., vol. 517,pp. 71–140, Aug. 2012.

[36] H.-S. Song, W. R. Cannon, A. S. Beliaev, and A. Konopka,“Mathematical modeling of microbial community dynamics: A method-ological review,” Processes, vol. 2, no. 4, pp. 711–752, 2014.

[37] P. Mina, M. di Bernardo, N. J. Savery, and K. Tsaneva-Atanasova,“Modelling emergence of oscillations in communicating bacteria: Astructured approach from one to many cells,” J. Roy. Soc. Interface,vol. 10, no. 78, 2012.

[38] R. E. Baker, C. A. Yates, and R. Erban, “From microscopic to macro-scopic descriptions of cell migration on growing domains,” Bull. Math.Biol., vol. 72, no. 3, pp. 719–762, 2010.

[39] I. Klapper and J. Dockery, “Mathematical description of microbialbiofilms,” SIAM Rev., vol. 52, no. 2, pp. 221–265, 2010.

[40] J. F. Hammond, E. J. Stewart, J. G. Younger, M. J. Solomon, andD. M. Bortz, “Spatially heterogeneous biofilm simulations using animmersed boundary method with Lagrangian nodes defined by bacteriallocations,” CoRR, vol. abs/1302.3663, 2013.

[41] A. Friedman, B. Hu, and C. Xue, “On a multiphase multicomponentmodel of biofilm growth,” Arch. Ration. Mech. Anal., vol. 211, no. 1,pp. 257–300, 2014.

[42] C. A. Yates, R. Baker, R. Erban, and P. Maini, “Refining self-propelledparticle models for collective behaviour,” Can. Appl. Maths Quart.,vol. 18, no. 3, 299 p., 2011.

[43] C. Waltermann and E. Klipp, “Information theory based approaches tocellular signaling,” Biochim. Biophys. Acta Gen. Subj., vol. 1810, no. 10,pp. 924–932, 2011.

[44] P. Mehta, S. Goyal, T. Long, B. L. Bassler, and N. S. Wingreen,“Information processing and signal integration in bacterial quorum sens-ing,” Mol. Syst. Biol., vol. 5, no. 1, p. 325, 2009.

[45] R. Cheong, A. Rhee, C. J. Wang, I. Nemenman, and A. Levchenko,“Information transduction capacity of noisy biochemical signaling net-works,” Science, vol. 334, no. 6054, pp. 354–358, 2011.

[46] P. D. Pérez, J. T. Weiss, and S. J. Hagen, “Noise and crosstalk in twoquorum-sensing inputs of Vibrio fischeri,” BMC Syst. Biol., vol. 5, no. 1,p. 153, 2011.

[47] U. Mitra, N. Michelusi, S. Pirbadian, H. Koorehdavoudi, M. El-Naggar,and P. Bogdan, “Queueing theory as a modeling tool for bacterial inter-action: Implications for microbial fuel cells,” in Proc. Int. Conf. Comput.Netw. Commun. (ICNC), Feb. 2015, pp. 658–662.

[48] M. Levorato, S. Narang, U. Mitra, and A. Ortega, “Reduced dimensionpolicy iteration for wireless network control via multiscale analysis,” inProc. IEEE Global Commun. Conf. (GLOBECOM), Dec. 2012, pp. 3886–3892.

[49] K. Rabaey and R. A. Rozendal, “Microbial electrosynthesis—Revisitingthe electrical route for microbial production,” Nat. Rev. Microbiol., vol. 8,no. 10, pp. 706–716, 2010.

[50] B. E. Logan, “Exoelectrogenic bacteria that power microbial fuel cells,”Nat. Rev. Microbiol., vol. 7, no. 5, pp. 375–381, 2009.

[51] H. J. Kim, H. S. Park, M. Hyun, I. S. Chang, M. Kim, and B. Kim,“A mediator-less microbial fuel cell using a metal reducing bacterium,Shewanella putrefaciens,” Enzyme Microb. Technol., vol. 30, no. 2,pp. 145–152, 2002.

[52] J. S. McLean et al., “Quantification of electron transfer rates to a solidphase electron acceptor through the stages of biofilm formation from sin-gle cells to multicellular communities,” Environ. Sci. Technol., vol. 44,no. 7, pp. 2721–2727, 2010.

[53] M. Gambello and B. Iglewski, “Cloning and characterization of thePseudomonas aeruginosa lasR gene, a transcriptional activator of elastaseexpression,” J. Bacteriol., vol. 173, pp. 3000–3009, 1991.

[54] P. Singh, A. Schaefer, M. Parsek, T. Moninger, M. Welsh, andE. Greenberg, “Quorum-sensing signals indicate that cystic fibrosis lungsare infected with bacterial biofilms,” Nature, vol. 407, pp. 762–764, Oct.2000.

[55] D. Erickson et al., “Pseudomonas aeruginosa quorum-sensing sys-tems may control virulence factor expression in the lungs ofpatients with cystic fibrosis,” Infect. Immun., vol. 70, pp. 1783–1790,2002.

[56] K. Rumbaugh, J. Griswold, B. Iglewski, and A. Hamood, “Contributionof quorum sensing to the virulence of Pseudomonas aeruginosa in burnwound infections,” Infect. Immun., vol. 67, pp. 5854–5862, 1999.

[57] T. Rasmussen and M. Givskov, “Quorum-sensing inhibitors as anti-pathogenic drugs,” Int. J. Med. Microbiol., vol. 296, pp. 149–161,2006.

[58] R. Allen, R. Popat, S. Diggle, and S. Brown, “Targeting virulence: Can wemake evolution-proof drugs?” Nat. Rev. Microbiol., vol. 12, pp. 300–308,2014.

[59] V. Kalia, “Quorum sensing inhibitors: An overview,” Biotechnol. Adv.,vol. 31, pp. 224–245, 2013.

[60] N. Lane, “Why are cells powered by proton gradients?” Nat. Educ., vol. 3,no. 9, p. 18, 2010.

[61] N. Michelusi, S. Pirbadian, M. El-Naggar, and U. Mitra, “A stochas-tic model for electron transfer in bacterial cables,” IEEE J. Sel. AreasCommun., vol. 32, no. 12, pp. 2402–2416, Dec. 2014.

[62] V. Ozalp, T. R. Pedersen, L. J. Nielsen, and L. F. Olsen, “Time-resolvedmeasurements of intracellular ATP in the yeast Saccharomyces cerevisiaeusing a new type of nanobiosensor,” J. Biol. Chem., vol. 285, no. 48,pp. 37579–37588, Nov. 2010.

[63] W. Smith and J. Hashemi, Foundations of Materials Science andEngineering. New York, NY, USA: McGraw-Hill, 2003.

[64] N. Michelusi and U. Mitra, “Capacity of electron-based communicationover bacterial cables: The full-CSI case,” IEEE Trans. Mol. Biol. Multi-Scale Commun., vol. 1, no. 1, pp. 62–75, Mar. 2015.

[65] J. Chen and T. Berger, “The capacity of finite-state Markov channels withfeedback,” IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 780–798, Mar.2005.

[66] D. Bertsekas, Dynamic Programming and Optimal Control. Belmont,MA, USA: Athena Scientific, 2005.

[67] D. L. Stauff and B. L. Bassler, “Quorum sensing in Chromobacteriumviolaceum: DNA recognition and gene regulation by the CviR receptor,”J. Bacteriol., vol. 193, no. 15, pp. 3871–3878, Aug. 2011.

[68] E. Stewart, R. Madden, G. Paul, and F. Taddei, “Aging and death in anorganism that reproduces by morphologically symmetric division,” PLoSBiol., vol. 3, no. 2, Feb. 2005.

[69] G. Dilanji, J. Langebrake, P. De Leenheer, and S. Hagen, “Quorum acti-vation at a distance: Spatiotemporal patterns of gene regulation fromdiffusion of an autoinducer signal,” J. Amer. Chem. Soc., vol. 134,pp. 5618–5626, 2012.

[70] A. Pai, J. Srimani, Y. Tanouchi, and L. You, “Generic metric to quantifyquorum sensing activation dynamics,” ACS Synth. Biol., vol. 3, pp. 220–227, 2014.

[71] C. J. Kastrup, F. Shen, and R. F. Ismagilov, “Response to shape emergesin a complex biochemical network and its simple chemical analogue,”Angew. Chem. Int. Ed., vol. 46, no. 20, pp. 3660–3662, 2007.

[72] K. Gray and J. R. Garey, “The evolution of bacterial LuxI and LuxR quo-rum sensing regulators,” Microbiology, vol. 147, no. 8, pp. 2379–2387,Aug. 2001.

[73] S.-W. Teng et al., “Measurement of the copy number of the masterquorum-sensing regulator of a bacterial cell,” Biophys. J., vol. 98, no. 9,pp. 2024–2031, May 2010.

[74] M. G. Surette, M. B. Miller, and B. L. Bassler, “Quorum sensing inEscherichia coli, Salmonella typhimurium, and Vibrio harveyi: A newfamily of genes responsible for autoinducer production,” Proc. Nat. Acad.Sci. USA, vol. 96, no. 4, pp. 1639–1644, 1999.

[75] J. Perez-Velazquez, B. Quinones, B. A. Hense, and C. Kuttler, “Amathematical model to investigate quorum sensing regulation and itsheterogeneity in Pseudomonas syringae on leaves,” Ecol. Complexity,vol. 21, pp. 128–141, 2015.


[76] E. Carnes et al., “Confinement-induced quorum sensing of individualStaphylococcus aureus bacteria,” Nat. Chem. Biol., vol. 6, pp. 41–45,2010.

[77] J. Boedicker, H. Gargia, and R. Phillips, “Theoretical and experimentaldissection of DNA loop-mediated repression,” Phys. Rev. Lett., vol. 110,no. 1, 2013.

[78] R. McLean, L. Pierson, III, and C. Fuqua, “A simple screening proto-col for the identification of quorum signal antagonists,” J. Microbiol.Methods, vol. 58, pp. 351–360, 2004.

[79] M. Kirisits et al., “Influence of the hydrodynamic environment on quo-rum sensing in Pseudomonas aeruginosa biofilms,” J. Bacteriol., vol. 189,pp. 8357–8360, 2007.

[80] J. R. Chandler, S. Heilmann, J. E. Mittler, and E. P. Greenberg, “Acyl-homoserine lactone-dependent eavesdropping promotes competition in alaboratory co-culture model,” ISME J., vol. 6, no. 12, pp. 2219–2228,Dec. 2012.

[81] C. Smith, H. Song, and L. You, “Signal discrimination by differentialregulation of protein stability in quorum sensing,” J. Mol. Biol., no. 382,pp. 1290–1297, Oct. 2008.

[82] Milo et al., Nucl. Acids Res., 2010, 38 (suppl 1), pp. D750–D753.[83] D. T. Gillespie, “Exact stochastic simulation of coupled chemical reac-

tions,” J. Phys. Chem., vol. 81, no. 25, pp. 2340–2361, Dec. 1977.[84] C. J. Kastrup et al., “Spatial localization of bacteria controls coagulation

of human blood by ‘quorum acting’,” Nat. Chem. Biol., vol. 4, no. 12,pp. 742–750, 2008.

[85] A. Prindle, P. Samayoa, I. Razinkov, T. Danino, L. Tsimring, and J. Hasty,“A sensing array of radically coupled genetic “biopixels”,” Nature,vol. 481, no. 7379, pp. 39–44, Dec. 2011.

[86] P. Jimenez, G. Koch, J. A. Thompson, K. B. Xavier, R. H. Cool, andW. J. Quax, “The multiple signaling systems regulating virulence inPseudomonas aeruginosa,” Microbiol. Mol. Biol. Rev., vol. 76, no. 1,pp. 46–65, Mar. 2012.

[87] J. Lee et al., “A cell-cell communication signal integrates quorum sensingand stress response,” Nat. Chem. Biol., vol. 9, no. 5, pp. 339–343, Mar.2013.

[88] S. Teng et al., “Active regulation of receptor ratios controls integrationof quorum-sensing signals in Vibrio harveyi,” Mol. Syst. Biol., vol. 7,no. 491, May 2011.

[89] N. Kimura, “Metagenomic approaches to understanding phylogeneticdiversity in quorum sensing,” Virulence, vol. 5, no. 3, pp. 433–442, Apr.2014.

[90] G. Rampioni, L. Leoni, and P. Williams, “The art of antibacterialwarfare: Deception through interference with quorum sensing-mediatedcommunication,” Bioorg. Chem., vol. 55, pp. 60–68, 2014.

[91] K.-G. Chan et al., “Characterization of N-acylhomoserine lactone-degrading bacteria associated with the Zingiber officinale (ginger) rhi-zosphere: Co-existence of quorum quenching and quorum sensing inAcinetobacter and Burkholderia,” BMC Microbiol., vol. 11, no. 51, 2011.

[92] J. E. Gonzalez and N. D. Keshavan, “Messing with bacterial quorumsensing,” Microbiol. Mol. Biol. Rev., vol. 70, no. 4, p. 859–875, Dec.2006.

[93] P. Williams, K. Winzer, W. Chan, and M. Camara, “Look who’s talk-ing: Communication and quorum sensing in the bacterial world,” Philos.Trans. Roy. Soc., Biol. Sci., vol. 362, no. 1483, pp. 1119–1134, Jul. 2007.

[94] K. McClean et al., “Quorum sensing and Chromobacterium violaceum:Exploitation of violacein production and inhibition for the detection ofN-acylhomoserine lactones,” Microbiology, vol. 143, no. 12, pp. 3703–3711, Dec. 1997.

Nicolò Michelusi (S’09–M’13) received the B.Sc.(with Hons.), M.Sc. (with Hons.), and Ph.D. degreesfrom the University of Padova, Padova, Italy, in2006, 2009, and 2013, respectively, and the M.Sc.degree in telecommunications engineering fromthe Technical University of Denmark, Lyngby,Denmark, in 2009, as part of the T.I.M.E. doubledegree program. In 2011, he was with the Universityof Southern California, Los Angeles, CA, USA,and, in Fall 2012, at Aalborg University, Aalborg,Denmark, as a Visiting Research Scholar. He

was a Postdoctoral Research Fellow with the Ming Hsieh Department ofElectrical Engineering, University of Southern California, Los Angeles,CA, USA, from 2013 to 2015. He is currently an Assistant Professorwith the Department of Electrical and Computer Engineering, PurdueUniversity, West Lafayette, IN, USA. His research interests include wirelessnetworks, stochastic optimization, distributed estimation and modeling ofbacterial networks. He serves as a Reviewer for the IEEE TRANSACTIONS

ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS, the IEEE TRANSACTIONS ON INFORMATION THEORY,the IEEE TRANSACTIONS ON SIGNAL PROCESSING, the IEEE JOURNAL

ON SELECTED AREAS IN COMMUNICATIONS, and the IEEE/ACMTRANSACTIONS ON NETWORKING.

James Boedicker received the S.B. degree from theMassachusetts Institute of Technology, Cambridge,MA, USA, and the M.A. and Ph.D. degrees from theUniversity of Chicago, Chicago, IL, USA. He was aPostdoctoral Scholar with the Department of AppliedPhysics California Institute of Technology, Pasadena,CA, USA, from 2010 to 2013. He is currentlyan Assistant Professor of Physics and Astronomyand the Biological Sciences with the University ofSouthern California, Los Angeles, CA, USA. Hisresearch interests include experimental and theoret-

ical tools to understand the emergent properties of cellular networks, signalexchange in microbial ecosystems, and the consequences of stochasticity inbiological systems.

Mohamed Y. El-Naggar received the B.S. degreein mechanical engineering from Lehigh University,Bethlehem, PA, USA, in 2001, and the Ph.D.degree in engineering and applied science from theCalifornia Institute of Technology, Pasadena, CA,USA, in 2007. He is an Assistant Professor of Physicswith the University of Southern California: DornsifeCollege of Letters, Arts, and Sciences, Los Angeles,CA, USA. He studies energy conversion and chargetransmission at the interface between living cells andnatural or synthetic surfaces. In 2012, he was named

one of Popular Science’s “Brilliant 10.” His work has important implicationsfor cell physiology and astrobiology: it may lead to the development of newhybrid materials and renewable energy technologies that combine the exquisitebiochemical control of nature with the synthetic building blocks of nanotech-nology. In 2014, he was the recipient of the Presidential Early Career Award forScientists and Engineers (PECASE).

Urbashi Mitra (F’07) received the B.S. and M.S. degrees from the Universityof California at Berkeley, Berkeley, CA, USA, and the Ph.D. degree fromPrinceton University, Princeton, NJ, USA. After a six-year stint at the OhioState University, Columbus, OH, USA, she joined the Department of ElectricalEngineering, University of Southern California, Los Angeles, CA, USA, whereshe is currently a Dean’s Professor. Her research interests include wirelesscommunications, biological communication, underwater acoustic communica-tions, communication and sensor networks, detection and estimation and theinterface of communication, sensing and control. She is the inaugural Editor-in-Chief of the IEEE TRANSACTIONS ON MOLECULAR, BIOLOGICAL, AND

MULTI-SCALE COMMUNICATIONS. She currently serves on the IEEE FourierAward for Signal Processing, the IEEE James H. Mulligan, Jr. EducationMedal and the IEEE Paper Prize committees. She has been an AssociateEditor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING (2012–2015),the IEEE TRANSACTIONS ON INFORMATION THEORY (2007–2011), theIEEE JOURNAL OF OCEANIC ENGINEERING (2006–2011), and the IEEETRANSACTIONS ON COMMUNICATIONS (1996–2001). She has cochaired:(technical program) 2014 IEEE International Symposium on InformationTheory in Honolulu, HI, 2014 IEEE Information Theory Workshop in Hobart,Tasmania, the IEEE 2012 International Conference on Signal Processing andCommunications, Bangalore India, and the IEEE Communication TheorySymposium at ICC 2003 in Anchorage, AK, USA; and was the General Co-Chair for the first ACM Workshop on Underwater Networks at Mobicom 2006,Los Angeles, CA. She was the Tutorials Chair for IEEE ISIT 2007 in Nice,France and the Finance Chair for the IEEE ICASSP 2008 in Las Vegas, NV,USA. She served as the Co-Director of the Communication Sciences Institute,University of Southern California, from 2004 to 2007. She was the recipi-ent of: INSIGHT Into Diversity 2015 Inspiring Women in STEM Award, the2014–2015 IEEE Communications Society Distinguished Lecturer, a TutorialLecturer for the IEEE North American School on Information Theory, UCSan Diego, San Diego, CA, USA, August 2015, the 2012 Globecom SignalProcessing for Communications Symposium Best Paper Award, the 2012 NAELillian Gilbreth Lectureship, the USC Center for Excellence in ResearchFellowship (2010–2013), the 2009 USC VSoE Dean’s Service Award, the2009 DCOSS Applications and Systems Best Paper Award, the 2009 USCRemarkable Woman Award (Faculty), the 2008 USC Mellon Mentoring Award(Faculty-to-Faculty), the Texas Instruments Visiting Professorship (Fall 2002,Rice University), 2001 Okawa Foundation Award, 2000 OSU College ofEngineering Lumley Award for Research, 1997 OSU College of EngineeringMacQuigg Award for Teaching, and the 1996 National Science Foundation(NSF) CAREER Award.

Documents

Queuing Models for Abstracting Interactions in …584 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 34, NO. 3, MARCH 2016 Queuing Models for Abstracting Interactions in Bacterial