Queues & Bacteria: a Tale of Modeling and Experiment
Urbashi Mitra Ming Hsieh Department of Electrical Engineering Department of Computer Science Viterbi School of Engineering University of Southern California Thanks to ONR N00014-15-1-2550, NSF CPS-1446901, NSF CCF-1410009, [email protected] NSF CNS-1213128 , AFOSR FA9550-12-1-0215, NSF CCF-1117896 Bacteria and RF analogies?
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for their discoveries concerning gene c control of enzyme and virus synthesis Bacterial Communi es
Analyze electron transport in bacterial cables Develop models for quorum sensing Why bacteria?
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¨ Unicellular organisms
¨ One of the earliest forms of life (3-4 billion years ago)
¨ Aggregate biomass larger than all other plants/animals combined
¨ Relevant for bioremedia on, plant growth, human and animal diges on, global elemental cycles including the carbon-cycle, and water treatment
Escherichia coli cell Not as simple as they seem..
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E. Coli genome
h p://ecocyc.org http://ecocyc.org What is communica on?
The fundamental problem of communica on is that of reproducing at one point either exactly or approximately a message selected at another point.
Claude Shannon, A Mathema cal Theory of Communica on, 1948
TX RX What is capacity for bacteria?
13 MICROBIAL FUEL CELL (MFC) q Bacteria as “devices” that sense the environment, es mate, react q Idea: tools from informa on theory and communica ons to model & predict bacterial networks: § Interac ons, dynamics, forma on § Do bacteria realize op mal control strategies? § E.g.: op mize MFC growth
) W-COMM INSPIRED ELECTRODE MODEL How to achieve the sweet spot?
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¨ Microbes on a 2D electrode ¨ Popula on grows as a func on of me q What happens at 15h? q How are bacteria interac ng?
q What is electron transport?
From Ken Nealson group, USC, Earth Sciences e- transport chain (ETC)
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q Fundamental (and well known) process in cellular respira on
q e- flow creates proton gradient to pump synthesis of Adenosine triphosphate (ATP, cell’s energy currency) q Cells need a constant electron flow to produce energy (ATP) e- transport
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q e- transport over cm-lengths Ø bacterial nanowires, biofilms, bacterial cables
mul -hop communica on ``network’’ from e- donor (ED) to e- acceptor (EA)
Bacterial cable (1D), Source: [Reguera,Nature’12] e- transport
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q e- transport over cm-lengths Ø bacterial nanowires, biofilms, bacterial cables
q Use communica on strategies to op mize e- transfer
q Capacity as proxy for maximal e- transfer
q New dimension (& new models) informa on-energy coupling § Capacity bo leneck, = vs wireless communica ons 6 § Instead, local interac ons & Bacterial cable (1D), Source: [Reguera,Nature’12] energy coupling, cells may die Stochas c queuing model
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q BEFORE: modeling via determinis c ODE/PDE • E.g., [Klapper&Dockery,’10], [Picioreanua et al.,’07], [Marcus et al.,’07] • Detailed master equa ons & chemical reac ons à huge complexity
q NOW: abstrac on à stochas c queuing model inspired by W-COM • Chemical/electron concentra ons à queues • Small-scale, local interac ons as nuisances à stochas c model • S ll high complexity, but tools for complexity reduc on from W-COM can be exploited Stochas c queuing model
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q External high energy membrane: e- from previous cell
q e- par cipate in electron transport chain to produce ATP
e- from previous cell Stochas c queuing model
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q Internal e- carrier pool: e- from external ED q e- par cipate in electron transport chain to produce ATP
Nico namide adenine dinucleo de (NADH)
electro-chemical energy from environment (NADH pool) Stochas c queuing model
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q ATP pool: ATP molecules produced via e- transport
q ATP consumed to sustain cell
ATP molecules generated via ET Stochas c queuing model
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q External low energy membrane carries e- a er ETC. 2 paths: • Reduc on to EA • e- transfer to next cell or electrode
e- then transferred to next cell Stochas c queuing model
24 q Molecular diffusion & e- transfer coexist q ATP level = energe c state of the cell Ø Energy dimension: ATP low à cell may die Ø Informa on dimension: ATP level too high à clogging
BACTERIAL CABLE 23
ADP molecules in the cell is conserved, a higher ATP level corresponds to a lower ADP level. Therefore, as there is more ATP available in the cell, there are less ADP molecules available for ATP synthase to produce additional ATP molecules, which, in turn, results23 in reduction of the ATP production rate. Finally, the larger the ED concentration, the larger the ATP consumption ADP molecules in the cell is conserved, a higher ATP level corresponds to a lower ADP level. rate µ (s (t); s (t)). This is shown to be true experimentally, for instance in [25]. The reason Therefore, as there is more ATPAT PavailableI E in the cell, there are less ADP molecules available for behind this correlation is that cellular operations that consume ATP (e.g., ATP-ases) are directly ATP synthase to produce additional ATP molecules, which, in turn, results in reduction of the µ µ ATP production rate. Finally,regulated the larger by the the ED ED concentration. concentration, Note the that largerCH the, ATPCH and consumptionAT P further need to satisfy the constraints listed in Sec. II-A. We assume that all the cells are alive throughout the experiment, rate µAT P (sI (t); sE(t)). This is shown to be true experimentally, for instance in [25]. The reason behind this correlation isand that set cellular the death operations rate (s thatI (t); consumesE(t)) = ATP 0. (e.g., ATP-ases) are directly We define the parameter vector x =[ , ⇢, ⇣, ], which is estimated via Maximum Likelihood regulated by the ED concentration. Note that CH, µCH and µAT P further need to satisfy the constraints listed in Sec.(ML) II-A. in We the assume next thatsection. all the Therefore, cells are the alive flow throughout matrix A the, defined experiment, in (15), is a linear function
of the entries of x. We write such a dependence as A(x, sE). Similarly, from (18), we write and set the death rate (sI (t); sE(t)) = 0. We define the parameterP ( vectorx, sE)=x =[I + , ⇢,A⇣(,x ,],sE which). is estimated via Maximum Likelihood (ML) in the next section. Therefore, the flow matrix A, defined in (15), is a linear function B. Maximum Likelihood estimate of x of the entries of x. We write such a dependence as A(x, sE). Similarly, from (18), we write
P (x, s )=I + A(x, s For) a given time series (yk,tk),k=0, 1,...,N , and the piecewise constant profile of the E E . { } external state sE(t), in this section we design a ML estimator of x. Since the initial distribution ⇡0 B. Maximum Likelihoodis estimate unknown, of wex also estimate it jointly with x. Note that, since the death rate is zero, the entries of Isolated cell: T T For a given time series⇡0 (needy ,t to),k sum=0 to, one,1,...,Ni.e., 1, and⇡0 =1 the. piecewise Moreover, constant we further profile enforce of the the constraints ⇡0 Z = y0, { k k } t y external state sE(t), in thisi.e. section, theexperimental valida on expected we design values a ML of estimator the NADH of x. and Since ATP the poolsinitial atdistribution time 0 equal⇡0 the measurement 0. 25 T is unknown, we also estimateTherefore, it jointly we with havex the. Note linear that, equality since the constraint death rate⇡0 is[Z zero,, 1]=[ they entries0, 1], and of the inequality constraint Tq Setup [Ozalp&al.,Journal of Biological Chemistry,’10] T ⇡ need to sum to one, i.e.⇡0, 1 0⇡(component-wise).=1. Moreover, we We further denote enforce the constraint the constraints set as , so⇡ thatZ =⇡y0 , . Due to the Gaussian 0 0 P 0 02 P • Suspension of yeast Saccharomyces cerevisiae (isolated cells) i.e., the expected valuesobservation of the NADH model and (38), ATP the pools ML at estimate time t0 ofequal(x, ⇡0 the) is measurement given by y0. • ED added abruptly to starved cells T Therefore, we have the linear equality constraint ⇡0 [Z, 1]=[y0, 1], and the inequality constraint • NADH (electron-carrier pool) and ATP levels measured (xˆ, ⇡ˆ0)=arg min f(x, ⇡0), (40) x 0,⇡0 ⇡ 0 (component-wise). We denote the constraint set as , so that ⇡ .2 DueP to the Gaussian 0 P 0 2 P whereq we Use experimental data to fit parameters have defined the negative log-likelihood cost function observation model (38), the ML estimate of (x, ⇡0) is given by N k 2 1 T nj nj 1 (xˆ, ⇡ˆ0)=argf(x, ⇡ min0) , f(x, ⇡0y),k ⇡0 P (x, sE,j 1) (40) Z . (41) x 0,⇡0 2 2P j=1 Xk=0 Y F where we have defined the negative log-likelihood cost function For a fixed x, the optimization over ⇡0 is aState distribu on quadratic programming problem, which can be 2 solved1 efficientlyN using,k e.g., interior-point methods [27], [28]. On the other hand, for fixed T Measured ATP/NADH nj nj 1 f(x, ⇡0) , yk ⇡0 P (x, sE,j 1) Z . Fixed matrix: (41) 2 ⇡ , the optimizationk=0 overj=1 x is a non-convex optimization problem. Therefore, we resort to a 0 (average over popula on) F State indicator X Y For a fixed x, the optimization over ⇡0 is a quadratic programming problem, which can be November 18, 2013 DRAFT solved efficiently using, e.g., interior-point methods [27], [28]. On the other hand, for fixed
⇡0, the optimization over x is a non-convex optimization problem. Therefore, we resort to a
November 18, 2013 DRAFT Isolated cell: experimental valida on 26
3 Model Experimental [Ozalp&al.’10] 2.5 Glucose added (30mM)
EXPERIMENTAL DATA ) 2 [Ozalp et al.,’10] mM 1.5 ATP (mM) ATP ( 1
0.5 MODEL
0 0 500 1000 TimeTIME (s) (s) 30mM GLUCOSE ADDED q Good fi ng of experimental curve and model
¨ Stochas c queuing model simpler than detailed “physical” models, but captures salient features Informa on theore c aspects
M Mˆ
ENC DEC Mul hop Capacity
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¨ Cascade of links • Persistently studied
¨ Via cut-set bounds can show that capacity is that of worst-case link
• Achieved by decode-and-forward
¨ What happens in biological systems? bo leneck • Communica on over channels with state! • There is coupling between the links Bacterial cable as communica on system 31 M Mˆ CHANNEL STATE INFORMATION (CSI) ENC DEC