5D Bacterial quorum sensing
Quorum sensing is a form of system stimulus and response that is correlated to
population density. Many species of bacteria use quorum sensing to coordinate various types of behavior including bioluminescence, biofilm formation, virulence,
and antibiotic resistance, based on the local density of the bacterial population
[9, 6, 8, 21, 34, 31, 36, 23]. In an analogous fashion, some social insects use quorum sensing to determine where to nest [5]. Quorum sensing also has several useful
applications outside the biological realm, for example, in computing and robotics.
Roughly speaking, quorum sensing can function as a decision-making process in
any decentralized system, provided that individual components have (i) some mechanism for determining the number or density of other components they interact with
and (ii) a stereotypical response once some threshold has been reached.
In the case of bacteria, quorum sensing involves the production and extracellular secretion of certain signaling molecules called autoinducers. Each cell also has
receptors that can specifically detect the signaling molecule (inducer) via ligandreceptor binding, which then activates transcription of certain genes, including those
for inducer synthesis. However, since there is a low likelihood of an individual
bacterium detecting its own secreted inducer, the cell must encounter signaling
molecules secreted by other cells in its environment in order for gene transcription to be activated. When only a few other bacteria of the same kind are in the
vicinity (low bacterial population density), diffusion reduces the concentration of
the inducer in the surrounding medium to almost zero, resulting in small amounts of
inducer being produced. On the other hand, as the population grows, the concentration of the inducer passes a threshold, causing more inducer to be synthesized. This
generates a positive feedback loop that fully activates the receptor, and induces the
up-regulation of other specific genes. Hence, all of the cells initiate transcription at
approximately the same time, resulting in some form of coordinated behavior. The
basic process at the single-cell level is shown in Fig. 5D.1.
Most models of bacterial quorum sensing are based on deterministic ordinary
differential equations (ODEs), in which both the individual cells and the extracellular medium are treated as well-mixed compartments (fast diffusion limit)
[19, 32, 7, 24, 10, 2, 15, 4, 6]. (For a discussion of spatial models that take into
account bulk diffusion of the autoinducer in the extracellular domain see Refs.
[7, 25, 20, 26, 17].) Suppose that there are N cells labeled i = 1, . . . , N. Let U(t)
denote the concentration of signaling molecules in the extracellular space and let ui
be the corresponding intracellular concentration within the i-th cell. Suppose that
there are K other chemical species within each cell, which together with the signaling molecule comprise a regulatory network. Let vi = (vi,1 , . . . , vi,K ) with vi,k the
concentration of species k within the i-th cell. A deterministic model of quorum
sensing can then be written in the general form [24]
1
synthesis
autoinducer
receptor
coordinated behavior
low population density
high population density
Fig. 5D.1: A schematic illustration of quorum sensing at the single-cell level.
dui
= F(ui , vi ) − κ(ui −U),
dt
dvi,k
= Gk (ui , vi ),
dt
dU
α N
= ∑ κ(u j −U) − γU,
dt
N j=1
(5D.1a)
(5D.1b)
(5D.1c)
Here F(u, v) and Gk (u, v) are the reaction rates of the regulatory network based
on mass action kinetics, the term κ(u j − U) represents the diffusive exchange of
signaling molecules across the membrane of the i-th cell, and γ is the rate of degradation of extracellular signaling molecules. Finally, α = Vcyt /Vext is a cell density
parameter equal to the ratio of the total cytosolic and extracellular volume. Note that
Vcyt = vcyt N, where vcyt is the single-cell volume. We can also write
α=
ρ
,
1−ρ
κ=
δ
,
ρ
(5D.2)
where ρ is the volume fraction of cells and δ is the effective conductance, which is
independent of ρ.
From a dynamical systems perspective, two basic forms of collective behavior
are typically exhibited by equations (5D.1): either the population acts as a biochemical switch [19, 7, 32, 18] or as a synchronized biochemical oscillator [24, 4, 6].
In the former case, two distinct mechanisms for a biochemical switch have been
identified. The first mechanism involves the occurrence of bistability in a gene regulatory network, as exemplified by the mathematical model of quorum sensing in
the bacterium Pseudomonas aeruginosa developed by Dockery and Keener [7]. P.
aeruginosa is a human pathogen that monitors its cell density in order to control
2
the release of various virulence factors [6, 8]. That is, if a small number of bacteria
released toxins then this could easily be neutralized by an efficient host response,
whereas the effectiveness of the response would be considerably diminished if toxins were only released after the bacterial colony has reached a critical size via quorum sensing. Multiple steady-steady states have also been found in a related ODE
model of quorum sensing in the bioluminescent bacteria V. fisheri [19]. In this system, quorum sensing limits the production of bioluminescent luciferase to situations
where cell populations are large; this saves energy since the signal from a small
number of cells would be invisible and thus useless. Recent experimental studies
of quorum sensing in the bacterial species V. harveyi and V. cholerae [21, 31, 36]
provide evidence for an alternative switching mechanism, which can provide robust switch-like behavior without bistability. In these quorum sensing systems two
or more parallel signaling pathways control a gene regulatory network via a cascade of phosphorylation-dephosphorylation cycles (PdPCs), see Sect. 5C. Within
the context of quorum sensing, the binding of an autodinducer to its cognate receptor switches the receptor from acting like a kinase to one acting like a phosphotase.
Thus the PdPCs are driven by the level of autoinducer, which itself depends on the
cell density. One source of the switch-like behavior is thus ultrasenstivity of the
PdPCs [13, 2, 27, 11, 28].
In the following we derive conditions for the global convergence of the general
quorum sensing systems (5D.1, and then consider models of switching in P. aeruginosa and V. harveyi under the assumption that they operate in the regime of global
convergence
5D.1 Global convergence of a quorum sensing network
The global convergence properties of quorum sensing networks, where coupling between nodes in the network is mediated by a common environmental variable, has
been analyzed within the context of nonlinear dynamical systems in Ref. [29]. These
authors consider a more general class of model than given by equations (5D.1), including non-diffusive coupling and non-identical cells. In order to apply their analysis , to the specific system (5D.1), it is necessary to review some basic results of
nonlinear contraction theory [30]. Consider the m-dimensional dynamical system
dx
= f(x,t),
dt
x ∈ Rn ,
(5D.3)
with f : Rn → Rn a smooth nonlinear vector field. Introduce the vector norm |x| for
x ∈ Rn and let kAk be the induced matrix norm for an arbitrary square matrix A, that
is,
kAk = sup{|Ax| : x ∈ Rn with |x| = 1}.
Some common examples are as follows:
3
n
|x|1 =
n
∑ |x j |,
kAk1 = max
∑ |ai j |,
1≤ j≤n
j=1
n
|x|2 =
∑ |x j |2
j=1
!1/2
,
i=1
kAk2 =
p
λmax (A∗ A)
n
|x|∞ = max |x j |,
1≤ j≤n
kAk∞ = max
∑ |ai j |,
1≤i≤n
j=1
where A∗ is the transpose of A and λmax (A∗ A) is the largest eigenvalue of the positive
semi-definite matrix A∗ A. Define the associated matrix measure µ as
µ(A) = lim
h→0+
1
(kI + hAk − 1),
h
where I is the identity matrix. For the three above norms on Rn , the associated
matrix measures are
µ1 (A) = max {a j j + ∑ |ai j |}
1≤ j≤n
i6= j
µ2 (A) = max {λi ([A + A∗ ]/2)}
1≤i≤n
µ∞ (A) = max {aii + ∑ |ai j |}.
1≤i≤n
j6=i
Given these definitions, the basic contraction theorem is as follows [29]:
Theorem 5D.6. The n-dimensional dynamical system (5D.3) is said to be contracting if any two trajectories, starting from different initial conditions, converge exponentially to each other. A sufficient condition for a system to be contracting is the
existence of some matrix measure µ for which there exists a constant λ > 0 such
that
∂ fi
µ(J(x,t)) ≤ −λ , Ji j =
(5D.4)
∂xj
for all x,t. The scalar λ defines the rate of contraction.
A related concept is partial contraction [29]. Consider a smooth nonlinear dynamical system of the form ẋ = f (x, x,t) with x ∈ Rn . Suppose that the so-called virtual
non-autonomous system ẏ = f (y, x,t) with x(t) evolving as specified, is contracting
with respect to y. If a particular solution of the virtual system has some smooth specific property, then all trajectories of the original x system exhibit the same property
in the large t limit. This follows from the fact that y(t) = x(t), t ≥ 0, is another particular solution of the virtual system, and all trajectories of the y system converge
exponentially to a single trajectory.
In order to apply the above results to Eqs. (5D.1), we rewrite the latter in the form
4
dxi
= f(xi ) − κ((xi )1 −U)e1 , i = 1, . . . , N
dt
dU
ακ N
=
∑ ((x j )1 −U) − γU,
dt
N j=1
(5D.5a)
(5D.5b)
with xi = (ui , vi ) ∈ R1+K , (xi )1 = ui , f = (F, G1 , . . . , GK ) and e1 = (1, 0, . . . , 0). From
the contraction theorem and the notion of partial contraction, one can show that the
global convergence condition
|xi (t) − x j (t)| → 0 as t → ∞
holds provided that f(x) − κ(x)1 e1 is contracting. The proof follows from considering the reduced order virtual system
ẏ = f(y) − κ(y)1 e1 + κU(t)e1 ,
where U(t) is treated as an external input. Setting y(y) = xi (t) in the virtual system
recovers the dynamics of the ith cell. Hence, xi (t) for i = 1, . . . , N are particular
solutions of the virtual system so that if the virtual system is contracting in y, then
all of its solutions converge exponentially toward each other, including the solutions
xi (t). In this asymptotic limit, we effectively have a single cell diffusively coupled
to the extracellular medium, that is ui (t) → u(t) and vi (t) → v(t) with:
du
= F(u, v) − κ(u −U),
dt
dvk
= Gk (u, v),
dt
dU
= ακ(u −U) − γU.
dt
(5D.6a)
(5D.6b)
(5D.6c)
5D.2 Bistability in a model of Pseudomonas aeruginosa quorum
sensing
In P. aeruginosa there are two quorum-sensing systems working in series, known as
the las and rhl system, respectively. Following Ref. [7], we consider the upstream
las system, which is composed of lasI, the autoinducer synthase gene responsible
for synthesis of the autoinducer 3-oxo-C12-HSL via the enzymatic action of the
protein LasI, and the lasR gene that codes for transcriptional activator protein LasR
(R), see Fig. 5D.2. Positive feedback occurs due to the fact that LasR and 3-oxoC12-HSL can form a dimer, which promotes both lasR and lasI activity. (We ignore
an additional negative feedback loop that is thought to play a relatively minor role in
quorum sensing.) Exploiting the fact that the lifetime of each type of mRNA is much
shorter than its corresponding protein, we can eliminate the mRNA dynamics, and
write down a system of ODES for the concentrations of LasR, the dimer LasR/35
oxo-C12-HSL and the autoinducer 3-oxo-C12-HSL, which we denote by R, P and
A, respectively. The resulting system of ODES for mass action kinetics thus take the
form [7]
dP
= kRA RA − kP P,
dt
dR
P
= −kRA RA + kP P − kR R +VR
+ R0 ,
dt
KR + P
dA
P
= −kRA RA + kP P +VA
+ A0 − kA A.
dt
KA + P
(5D.7a)
(5D.7b)
(5D.7c)
Here kA , kR are the rates of degradation of A, R, kRA , kP are the rates of production
and degradation of the dimer P, and A0 , R0 are baseline rates of production of A, R.
Finally, the positive feedback arising from the role of the dimer P as an activator
protein that enhances the production of LasR and arising from the activation of R
and A (with the latter mediated by lasI) is taken to have Michaelis-Menten form.
Dockery and Keener [7] carry out a further reduction by noting that formation and
degradation of dimer is much faster than transcription and translation so that we can
assume P is in quasi steady-state so that
P=
kRA
RA = kRA.
kP
Then we have
dR
kRA
= −kR R +VR
+ R0 ,
dt
KR + kRA
dA
kRA
= VA
+ A0 − kA A.
dt
KA + kRA
(5D.8a)
(5D.8b)
Comparison of equations (5D.8) with the general kinetic system in equations
(5D.6a,b), shows that we have one auxiliary species and we can make the following
identifications (after dropping the k = 1 index): u = A, v = R with
3-oxo-C12-HSL
lasR
+
lasI
lasR
lasI
+
lasR
dimer
Fig. 5D.2: Simplified regulatory network for the las system in P. aerginosa.
6
dR/dt = 0
4
ρ = 0.15
3
dA/dt = 0
ρ = 0.10
A 2
1
ρ = 0.05
0
0
1
2
3
4
R
Fig. 5D.3: Bistability in planar model of las system in P. aerginosa.. Nullcline Ṙ = 0 is shown
by the gray curve and the ρ-dependent nullclines Ȧ = 0 are shown by the black curves for three
different values of ρ. It can be seen that for intermediate ρ values there are two stable fixed points
separated by an unstable fixed point. Parameter values are VR = 2.0, VA = 2.0, KR = 1.0, KA = 1.0,
R0 = 0.05, A0 = 0.05, δ = 0.2, γ = 0.1, kR = 0.7, and kA = 0.02. [Redrawn from Dockery and
Keener [7].]
F = VA
uv
+ A0 − kA u.
KA + kuv
G = −kR v +VR
kuv
+ R0 .
KR + kuv
Now suppose equation (5D.8b) is diffusively coupled to an extracellular concentration along the lines of (5D.6), and as a further simplification take U to be in
quasi-equilibrium. The latter conditions allows us to carry out a phase-plane analysis without changing the essential behavior of the system. Set α = ρ/(1 − ρ) and
κ = δ /ρ, where ρ is the volume fraction of cells and the conductance δ is independent of ρ. The only modification is that the last term on the right-hand side of
(5D.8b) which is transformed according to the scheme
δ
γ
kA A → d(ρ) ≡ kA +
A.
ρ γ + δ /(1 − ρ)
Quorum sensing thus arises due to the dependence of the effective degradation rate
d(ρ) on ρ. More specifically, when ρ is small the rate d(ρ) is large, whereas d(ρ)
is small when ρ → 1. In Fig. 5D.3, we plot nullclines of the system at various values
of ρ, illustrating that bistability occurs at intermediate values of ρ. In this parameter
7
regime the system acts like a bistable switch, where one stable state has low levels of
autoinducer and the other high levels of autoinducer [7]. Note that multiple steadysteady states have also been found in a related ODE model of quorum sensing in the
bioluminescent bacteria V. fisheri [19, 32].
5D.3 Ultrasensitivity in a model of V. harveyi quorum sensing
In the bacterium V. harveyi, there are three parallel quorum sensing systems, each
consisting of a distinct autoinducer (HAI-1, AI-2, CAI-1), cognate receptor (LuxN,
LuxP/Q, CqS), and associated enzyme (LuxM, LuxS, CqsA) that helps produce the
autoinducer, see Fig. 5D.4. (The human pathogen V. cholerae has a similar quorum
sensing network, except there are now only two parallel pathways.) Each autoinducer moves freely between the intracellular and extracellular domains. At low cell
densities there are relatively low levels of autinducer due to diffusion, so that there is
a low probability that the autoinducer can bind to its cognate receptor. Consequently,
the receptor acts as a kinase that autophosphorylates, and subsequently transfers its
phosphate to the cytoplasmic protein LuxU. LuxU-P then passes its phosphate to the
DNA-binding regulatory protein LuxO to yield LuxO-P. The upshot is that at low
cell densities, the ratio of [LuxO-P] to [LuxO] is high and this activates transcrip-
AI-2
extracelular domain
HAI-1
HAI-1
LuxP
CAI-1
AI-2
CAI-1
HAI-1
CAI-1
outer
membrane
AI-2
LuxQ
HAI-1
CqS
CAI-1
LuxS
LuxN
inner
membrane
LuxM
P+
CqsA
P+
P+
LuxU
P+
LuxO
+
sRNA
LuxR
regulation of
gene transcription
Fig. 5D.4: Summary of the V. harveyi quorum sensing circuit. Three phosphorylation cascades
work in parallel to control the ratio of LuxO to LuxO-P based on local cell-population density.
Five sRNA, qrr1-5, then regulate expression of quorum sensing target genes including the master
transcriptional regulator LuxR, which upregulates downstream factors. [Redrawn from Hunter et
al. [18].]
8
tion of the genes encoding five regulatory small RNAs (sRNAs) termed Qrr1-Qrr5
(Quorum Regulatory RNA). Bacterial sRNAs are small (50-250 nucleotide) noncoding RNA molecules that can either bind to a protein and alter its function or
bind to mRNA and regulate gene expression. In the case of quorum sensing in V.
harveyi, the small sRNAs Qrr–Qrr5 destabilize the transcriptional activator protein
LuxR, thus preventing the activation of target genes responsible for the production
of various proteins, including luciferase. Hence, at low cell density the bacteria do
not bioluminesce. On the other hand, at high cell density, the concentration of intracellular autodinducers is increased so that they have a higher probability of binding
to their receptors, which then switch from being kinases to being phosphotases, significantly reducing the ratio of [LuxO-P] to [LuxO]. The sRNAs are thus no longer
expressed, allowing the synthesis of LuxR and the expression of bioluminescence,
for example. Both the phosphorylation-dephosphorylation cascades and the sRNA
regulatory network provide a basis for an ultrasensitive response of the concentration of LuxR to smooth changes in cell density.
We will illustrate the occurrence of ultrasensitivity in the above quorum sensing system by focusing on a single phosphorylation pathway and adapting the
Goldbetter-Koshland of PdPCs in Sect. 5C. (For a corresponding model of switching due to the action of sRNAs see Hunter et al. [18]. In their model, the fraction
of phosphorylated LuxO is taken to be the external input to the sRNA network. The
latter itself depends on the level of phosphorylated LuxU, which is the output of our
model. Note that we could also consider ultrasensitivity in a bicyclic PdPC cascade
involving both LuxU and LuxO.) In particular, we consider the phosphorylationdephosphorylation of LuxU by the enzymatic action of a particular quorum sensing
receptor, which is denoted by R when acting as a kinase and by Rb when it is it is
bound by an autoinducer (A) and acts like a phosphotase. Denoting the protein LuxU
by W, we have the following reaction schemes:
a1
k
1
W +R WR →
W ∗ + R,
d1
a2
k
2
b
W ∗ + Rb W ∗ Rb →
W + R,
d2
k+
b
R + A R.
k−
(5D.9a)
(5D.9b)
(5D.9c)
b v=
Introducing the concentrations u = [A], w = [W ], w∗ = [W ∗ ], r = [R], b
r = [R],
∗
∗
b
[W R] and v = [W R], the corresponding kinetic equations are
dw
= −a1 w(r − v) + d1 v + k2 v∗
dt
dv
= a1 w(r − v) − (d1 + k1 )v
dt
9
(5D.10a)
(5D.10b)
dw∗
= −a2 w∗ (b
r − v∗ ) + d2 v∗ + k1 v
dt
dv∗
= a2 w∗ (b
r − v∗ ) − (d2 + k2 )v∗
dt
dr
= k−b
r − k+ ur.
dt
(5D.10c)
(5D.10d)
(5D.10e)
These equations are supplemented by the conservation equations
WT = w + w∗ + v + v∗ ,
RT = r + b
r,
(5D.11a)
(5D.11b)
where RT is the total concentration of receptors and WT is the total concentration
of LuxU. For the moment we are assuming that the intracellular concentration u of
autoinducer is fixed. Suppose that rates of binding/unbinding of A are sufficiently
fast so that we can treat the concentrations of kinases and phosphotases as in quasiequilibrium:
r=
k−
RT ≡ E1T ,
k+ u + k−
b
r=
k+ u
RT ≡ E2T .
k+ u + k−
(5D.12)
The system of equations (5D.10) then reduces to the classical Goldbetter-Koshland
model given by equations (5C.2) with w1 → v and w∗2 → v∗ . It follows that the
relative amount of LuxU-P is given by
[LuxU-P]
= φ (σ ),
[LuxU-P]+[LuxU]
σ=
k1 E1T
k− k1
=
,
k2 E2T
k+ u k2
(5D.13)
with φ given by equation (5C.7).
We can now couple the above kinetic equations to the extracellular space by noting that the autoinducer A can transfer across the cell membrane to the extracellular
domain. Equation (5D.10e) is then replaced by a system of equations of the form
(5D.6),
du
= Γ + k−b
r − k+ ur − κ(u −U)
dt
dr
= k− b
r − k+ ur
dt
dU
= ακ(u −U) − γU,
dt
(5D.14a)
(5D.14b)
(5D.14c)
with U the extracellular concentration of autoinducer, and Γ the rate of production
of the autoinducer. If we now take these equations to be in quasi-equilibrium relative
to the phosphorylation-dephosphorylation cycle, we have u = u∗ with
Γ (ακ + γ)
δ + (1 − ρ)γ
∗
=Γρ
≡ ψ(ρ),
(5D.15)
u =
κγ
(1 − ρ)γ
10
molar fraction Ψ
1
K = 0.01
0.8
K =1
K = 0.1
0.6
0.4
0.2
0
0
0. 2
0. 4
0. 6
cell density ρ
0. 8
1
Fig. 5D.5: Molar fraction of modified protein W ∗ at steady-state as a function of the cell density ρ
for different values of K, K = K1 = K2 . Parameter values are chosen so that Ψ (ρ) = φ (1/ψ(ρ))
and ψ(ρ) = ρ(2 − ρ)/(1 − ρ).
after setting α = ρ/(1 − ρ) and κ = δ /ρ. Setting u = u∗ in equation (5D.16) finally
shows that
[LuxU-P]
k− k1
=φ
≡ Ψ (ρ),
(5D.16)
[LuxU-P]+[LuxU]
k+ ψ(ρ) k2
For low density cells we have ρ → 0 so that ψ(ρ) → 0 and Ψ (ρ) → 1. The last
limit follows from the functional form of φ , see Fig. 5C.1. Hence the fraction of
phosphorylated LuxU-P is high, which ultimately means that the expression of the
gene regulator protein LuxR is suppressed. On the other hand, for large cell densities
we have ρ → 1, ψ(ρ) → ∞ and Ψ (ρ) → 0. Now the fraction of phosphorylated
LuxU-P is small, allowing the expression of LuxR and downstream gene regulatory
networks. The ρ-dependence is illustrated in Fig. 5D.5
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