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Model and analysis of cell population density estimation via Quorum Sensing

Nicolò Michelusi Purdue University https://engineering.purdue.edu/~michelus/ michelus@purdue.edu Nafplio, Communications Theory Workshop May 18, 2016

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q Advances in bio-nano

technology and biology

Ø bio-inspired nano-devices,

biological nanosensors, prosthetic devices

q Biomedical, industrial,

environmental applications

q Comm. networks & protocols

at nanometer length scales

Ø Actuation, coordination,

chemotaxis, etc.

Molecular bio-nano communication networks

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Molecular nanonetwork with 2 nodes Source: http://www.ece.gatech.edu/research/labs/bwn/monaco/

q Electromagnetic comm.

unfeasible

q Communication via

molecular diffusion

q Most recent literature:

[Nakano&al.’13],[Akyildiz&al.’08], [Mian&Rose,’11],[Eckford,’07], [Einolghozati&al.’13],[Kadloor&al.’12], [Arjmandi&al.’13]…

q Capacity achieving schemes? Ø Any simpler scheme?

Diffusion Channel Transmitter Receiver

Molecular nanonetwork with 2 nodes Source: http://www.ece.gatech.edu/research/labs/bwn/monaco/

Molecular bio-nano communication networks

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What is Quorum Sensing?

q Quorum sensing in bacteria: Ø Gene expression

= f(cell density)

Ø How does it work? 1.

AutoInducers (AI) produced

2.

AI concentration= f(cell density)

3.

AI reception activates gene expression

Ø Regulates biofilm formation,

virulence, diseases, antibiotic resistance, etc.

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What is Quorum Sensing?

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Hawaiian bobtail squid

q Light emission very

demanding, should be done

• nly when population size is

large enough

q Symbiotic relationship Ø The squid provides nutrients to

V.Fischeri to grow

Ø V.Fischeri provides light to hide

from predators

Ø Every morning, the squid gets

rid of 95% of bacteria, & the process repeats

Motivation

q Quorum Sensing enables coordination among large

populations of cells

q Coordinate mechanism in future nanonetworks q Goals: Ø Develop a model of Quorum Sensing Ø Necessary conditions for QS to function Ø Analysis of QS dynamics & cell population density estimation

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Towards a model of QS

q “Ingredients” of QS 1.

Microbial community (e.g., Vibrio Fischeri)

2.

Autoinducers (AI)

v Produced by each cell and released in the extracellular environment

3.

Receptors (R)

v They bind to AIs within each cell to form complexes

4.

Complexes (C)

v 1 AI bound to 1 R

5.

Synthases (S)

v “machines” that produce AIs

6.

DNA binding sites

v C activates DNA transcription (produce more S,R) v Costly gene expression

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Transmission Reception Actuation

Towards a model of QS

q “Ingredients” of QS 1.

Microbial community (e.g., Vibrio Fischeri)

2.

Autoinducers (AI)

v Produced by each cell and released in the extracellular environment

3.

Receptors (R)

v They bind to AIs within each cell to form complexes

4.

Complexes (C)

v 1 AI bound to 1 R

5.

Synthases (S)

v “machines” that produce AIs

6.

DNA binding sites

v C activates DNA transcription (produce more S,R) v Costly gene expression

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Reception Actuation

Towards a model of QS

q “Ingredients” of QS 1.

Microbial community (e.g., Vibrio Fischeri)

2.

Autoinducers (AI)

v Produced by each cell and released in the extracellular environment

3.

Receptors (R)

v They bind to AIs within each cell to form complexes

4.

Complexes (C)

v 1 AI bound to 1 R

5.

Synthases (S)

v “machines” that produce AIs

6.

DNA binding sites

v C activates DNA transcription (produce more S,R) v Costly gene expression

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Actuation

Towards a model of QS

q “Ingredients” of QS 1.

Microbial community (e.g., Vibrio Fischeri)

2.

Autoinducers (AI)

v Produced by each cell and released in the extracellular environment

3.

Receptors (R)

v They bind to AIs within each cell to form complexes

4.

Complexes (C)

v 1 AI bound to 1 R

5.

Synthases (S)

v “machines” that produce AIs

6.

DNA binding sites

v C activates DNA transcription (produce more S,R) v Costly gene expression

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Model of each cell

11 Sites 1 2 3

Ri(t) receptors ✏C,1 ✏C,2 ✏C,3 Si(t) synthases

q Synthases & Receptors produced at low basal rate

Model of each cell

Sites 1 2 3

β

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Ai(t) AIs (inside)

q Synthases produce AIs inside cell

Si(t) synthases Ri(t) receptors

Model of each cell

Sites 1 2 3

Aext(t) Ais (outside) Ci(t) complexes β

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q Transmission: AI diffusion across cell membrane (in/out)

Si(t) synthases Ri(t) receptors Ai(t) AIs (inside)

Model of each cell

Sites 1 2 3

Ci(t) complexes γ β

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q Reception: Complex formation

Si(t) synthases Ri(t) receptors Ai(t) AIs (inside) Aext(t) Ais (outside)

Model of each cell

Sites 1 2 3

γ Gene expression β ✏C,1 ✏C,2 ✏C,3

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q Actuation: DNA binding à gene expression

Si(t) synthases Ri(t) receptors Ai(t) AIs (inside) Aext(t) Ais (outside) Ci(t) complexes

Queuing model of QS

S1(t)

S2(t) S3(t) S4(t)

R1(t)

R2(t) R4(t) R3(t)

C1(t)

C2(t) C4(t) C3(t)

Aext(t)

R1(t)A(t)γ S1(t)β A(t)δ(N(t)) C1(t)✏1 C1(t)✏2

AUTOINDUCER QUEUE SYNTHASE QUEUE RECEPTOR QUEUE COMPLEXES QUEUE

AI leakage

q Each cell modeled by

queues

q Cell population N(t)

Ø Increases with cell

growth

q Huge complexity! q Cells coupled via AI

queue

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A1(t)

AI QUEUE

A2(t) A4(t) A3(t)

Simulations

q Simulation tools based on

queuing model

[Michelusi&al.2015]

2 4 6 8 10 10 20 30 40 50 60 70 80 90 100 TIME [hours] Autoinducers concentration [nM] Open system, time series Open system, QS activation time Closed system, time series Closed system, QS activation time

OPEN SYSTEM: cells grow in

• pen space & closely packed

(no boundariesàleakage of AI) CLOSED SYSTEM: cells grow in finite box & sparse (boundariesàno leakage)

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Experimental activation time

2 4 6 8 10 10

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10

8

10

9

10

10

10

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TIME [hours] Cell density [cells per mL] Open system, time series Open system, QS activation time Closed system, time series Closed system, QS activation time

Simulations

q Higher density in open

system

Ø > AI concentration Ø Faster activation time

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OPEN SYSTEM: cells grow in

• pen space & closely packed

(no boundariesàleakage of AI) CLOSED SYSTEM: cells grow in finite box & sparse (boundariesàno leakage)

Simplified model

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+ + DNA

More comprehensive model Simplified model

AIs generate more AIs

Simplified model

q Internal AIs for each cell (local state) & external AIs Ø Ai(t): local AI Ø AI diffusion in-out

proportional to AI concentration,

Ø AI synthesis

proportional to local AI availability

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Aext(t)

EXTERNAL AUTOINDUCER QUEUE

A2(t) A4(t) A3(t) A1(t) αext(t) = Aext(t) V − NVc αi(t) = Ai(t) Vc ρS,0 + ρSAi(t) λα1(t) λα2(t) λα3(t) λα4(t) λαext(t) λαext(t) λαext(t) λαext(t) λα Local: Ext:

Simplified model

q Local AI Ai(t): Ø Diffusion out-in:

Augments w.r.

Ø Diffusion in-out:

Diminishes w.r.

Ø Synthesis:

Augments w.r.

Ø AI & complex deg.: Diminishes w.r. q External AI Aext(t): Ø Diffusion out-in:

Diminishes w.r.

Ø Diffusion in-out:

Augments w.r.

Ø AI degradation:

Diminishes w.r.

q Cells coupled through external AI q Local AI captures local state and cell fluctuations

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ρS,0 + ρSAi(t) λαext(t) λαi(t) µDAi(t) Nλαext(t) λ αi(t)

N

X

i=1

µD,extAext(t)

A1(t)

Analysis of expected AI evolution

q State is q We want to compute &

(note: is the same for all cells)

q

characterizes cell sensitivity to population density

q Asymptotic analysis with fixed q Study eigenvalues of W: 0s of

,

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¯ αext(t) = E[αext(t)] ¯ αcell(t) = E[αi(t)] (αext(t), α1(t), α2(t), . . . , αN(t)) ¯ αcell(t) d dt  ¯ αext(t) ¯ αi(t)

• = W

 ¯ αext(t) ¯ αi(t)

• +



ρS,0 Vc

• V → ∞

¯ αcell(t)

det(W − sI) = 0 s(+), s(−)

β = N/V

Analysis of expected AI evolution

q Average response: q Response driven by largest eigenvalue, Ø

: exponentialincrease, positive feedback loop

Ø

: linear increase, positive feedback loop

Ø

: exponential decay, steady state regime

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s(+)(ρ)

s(+)(ρ) > 0 s(+)(ρ) = 0 s(+)(ρ) < 0 8 > < > : ¯ αext(t) ' X(+)

ext exp

• s(+)t

+ Cext ¯ αcell(t) ' X(+)

cell exp

• s(+)t

+ Ccell

Desirable properties of QS

q In the limit , response should decay, Ø Why? Positive feedback loop only for Ø Intuition: if synthesis > diffusion + degradation, local AI grows

unbounded, not informative!

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s(+)(0) < 0 ρS < λ Vc + µD

AI synthesis rate AI diffusion AI degradation

β → 0

β ≥ βth

q Response should be stronger for larger , Ø Why? QS activity should increase for larger population size Ø Intuition: if internal degradation too intense, diffusion to the

external environment is negligible

Desirable properties of QS

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AI internal degradation AI external degradation AI synthesis rate

µD − µD,ext < ρS < λ Vc + µD β ds(+)(β) dβ > 0

QS response

q Convergence speed

is driven by largest eigenvalue

q Steady-state

concentration difference between in & out

Ø net flow from inside

cells to external environment

Ø Difference due to

external degradation

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2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time AI concentration

s(+)

• Conv. time ∝

1 |s(+)| Internal External λ = 1 ρS,0 = 1 ρS = 1 µD = 1 µD,ext = 1 V c = 1 β = 0.5

0.2 0.4 0.6 0.8 1 1.2 10-2 10-1 100 101 102

Asymptotic analysis

q External signal is

deterministic!

q Internal signal

exhibits fluctuations

• ver time and across

cells

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Population concentration AI concentration

Internal External

Exponential growth

β

Asymptotic analysis

q Fluctuations in local AI

concentration à noisy cell concentration estimate

q Different cells have

different estimates

q However, external

signal is deterministic!

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0.2 0.4 0.6 0.8 1 1.2 100 101 102 Population concentration AI concentration

Internal + stdev/2

• stdev/2

Exponential growth Fluctuations

β

0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5

Cell concentration estimation

q Error can be reduced by

filtering local AI concentration sequence

q Conv. time

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Population concentration Exponential growth

Square estimation error ˆ αk+1,i =(1 − γ∆)ˆ αk,i + γ∆αk+1,i

⇒

var(ˆ α) = γ γ + constvar(α)

∝ 1 |s(+)| + 1 γ

β

Conclusions

q Quorum Sensing has evolved over millions of years as a

coordination mechanism among large populations of cells

q It can serve to coordinate nanonetworks as well q We have Ø Developed a model of Quorum Sensing Ø Found necessary conditions for QS to function Ø Analysis of QS dynamics & cell population density estimation

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