5D Bacterial quorum sensing

Transcription

1 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 [10, 7, 9, 24, 40, 36, 42, 27]. In an analogous fashion, some social insects use quorum sensing to determine where to nest [6]. 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) [21, 38, 8, 28, 11, 2, 17, 5, 6]. (For a discussion of spatial models that take into account bulk diffusion of the autoinducer in the extracellular domain see Refs. [8, 29, 23, 30, 19].) 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 u i 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 v i = (v i,1,...,v i,k ) with v i,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 [28] 1

2 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. du i = F(u i,v i ) κ(u i U), (5D.1a) dv i,k = G k (u i,v i ), (5D.1b) du = α N κ(u j U) γu, (5D.1c) N j=1 Here F(u,v) and G k (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 = V cyt /V ext is a cell density parameter equal to the ratio of the total cytosolic and extracellular volume. Note that V cyt = v cyt N, where v cyt 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 [21, 8, 38, 20] or as a synchronized biochemical oscillator [28, 5, 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 [8]. P. aeruginosa is a human pathogen that monitors its cell density in order to control 2

3 the release of various virulence factors [7, 9]. 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 [21]. 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 [24, 36, 42] 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 supplementary 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 [15, 2, 32, 12, 33]. 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. [34]. 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 [25]. Consider the m-dimensional dynamical system dx = f(x,t), x Rn, (5D.3) with f : R n R n a smooth nonlinear vector field. Introduce the vector norm x for x R n and let A be the induced matrix norm for an arbitrary square matrix A, that is, A = sup{ Ax : x R n with x = 1}. Some common examples are as follows: 3

4 x 1 = x 2 = n j=1 x j, A 1 = max n 1 j n i=1 a i j, ( n ) 1/2 x j 2, A 2 = λ max (A A) j=1 x = max 1 j n x j, A = max n 1 i n j=1 a i j, where A is the transpose of A and λ max (A A) is the largest eigenvalue of the positive semi-definite matrix A A. Define the associated matrix measure µ as µ(a) = lim ( I + ha 1), h h where I is the identity matrix. For the three above norms on R n, the associated matrix measures are µ 1 (A) = max {a j j + a i j } 1 j n i j µ 2 (A) = max 1 i n {λ i([a + A ]/2)} µ (A) = max {a ii + a i j }. 1 i n Given these definitions, the basic contraction theorem is as follows [34]: Theorem 5D.5. 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 µ(j(x,t)) λ, J i j = f i (5D.4) x j for all x,t. The scalar λ defines the rate of contraction. A related concept is partial contraction [34]. Consider a smooth nonlinear dynamical system of the form ẋ = f (x,x,t) with x R n. 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 j i 4

5 dx i = f(x i ) κ((x i ) 1 U)e 1, i = 1,...,N (5D.5a) du = ακ N ((x j ) 1 U) γu, N (5D.5b) j=1 with x i = (u i,v i ) R 1+K, (x i ) 1 = u i, f = (F,G 1,...,G K ) and e 1 = (1,0,...,0). From the contraction theorem and the notion of partial contraction, one can show that the global convergence condition x i (t) x j (t) 0 as t holds provided that f(x) κ(x) 1 e 1 is contracting. The proof follows from considering the reduced order virtual system ẏ = f(y) κ(y) 1 e 1 + κu(t)e 1, where U(t) is treated as an external input. Setting y(y) = x i (t) in the virtual system recovers the dynamics of the ith cell. Hence, x i (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 x i (t). In this asymptotic limit, we effectively have a single cell diffusively coupled to the extracellular medium, that is u i (t) u(t) and v i (t) v(t) with: (5D.6a) du = F(u,v) κ(u U), dv k = G k (u,v), (5D.6b) du = ακ(u U) γu. (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. [8], 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-oxo- C12-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/3-5

6 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 [8] dp = k RARA k P P, dr = k P RARA + k P P k R R +V R K R + P + R 0, da = k P RARA + k P P +V A K A + P + A 0 k A A. (5D.7a) (5D.7b) (5D.7c) Here k A,k R are the rates of degradation of A,R, k RA,k P are the rates of production and degradation of the dimer P, and A 0,R 0 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 (see supplementary Sect. 5C). Dockery and Keener [8] 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 = k RA k P RA = kra. Then we have dr = k kra RR +V R K R + kra + R 0, da = V kra A K A + kra + A 0 k A A. (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 lasr + lasr 3-oxo-C12-HSL lasi lasr dimer lasi + Fig. 5D.2: Simplified regulatory network for the las system in P. aerginosa. 6

7 (a) 4 dr/ = 0 ρ = 0.15 (b) 3 A 2 da/ = 0 ρ = 0.10 autoinducer A SN switch 1 ρ = 0.05 SN R cell density ρ Fig. 5D.3: Bistability in planar model of las system in P. aerginosa.. (a) 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. (b) The appearance and subsequent disappearance of a stable/unstable pair of fixed points via saddle-node bifurcations can switch the system from a low activity state to a high activity state. Decreasing the density again leads to hysteresis. Parameter values are V R = 2.0, V A = 2.0, K R = 1.0, K A = 1.0, R 0 = 0.05, A 0 = 0.05, δ = 0.2, γ = 0.1, k R = 0.7, and k A = uv F = V A K A + kuv + A 0 k A u. kuv G = k R v +V R K R + kuv + R 0. Now suppose equation (5D.8b) is diffusively coupled to an extracellular concentration along the lines of (5D.6a-5D.6c), 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 k A A d(ρ) [ k A + δ ρ γ γ + δ/(1 ρ) ] A. 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(a), we plot nullclines of the system at various values of ρ, illustrating that bistability occurs at intermediate values of ρ. In this parameter regime the system acts like a bistable switch, where one stable state has low levels of autoinducer and the other high levels of autoinducer [8]. Switching occurs due to a saddle-node bifurcation as illustrated in Fig. 5D.3(b). When noise is included, the resulting steady-state distribution exhibits peaks around the stable 7

8 fixed points of the deterministic system, and the transition between a unimodal and a bimodal distribution is used as one indictator of a stochastic bifurcation [35, 14]. 5D.3 Ultrasensitivity in a model of V. harveyi quorum sensing The bioluminescent bacterium V. harveyi has three parallel quorum sensing systems, each consisting of a distinct autoinducer (HAI-1, AI-2, CAI-1), cognate receptor (LuxN, LuxPQ, CqsS), and associated enzyme that helps produce the autoinducer, see Fig. 5D.4. (The human pathogen V. cholerae has a similar quorum sensing network, except that there could be up to four parallel pathways [22].) Each autoinducer is freely exchanged between the intracellular and extracellular domains. Extracellular diffusion at low (high) cell densities leads to small (large) intracellular autoinducer concentrations, resulting in a low (high) probability that the autoinducer can bind to its cognate receptor. The receptors target downstream DNA-binding regulatory proteins LuxU and LuxO. At low cell densities the receptors act as kinases and the resulting phosphorylation of LuxU and LuxO activates transcription of the genes encoding five regulatory small non-coding RNAs (srnas) termed Qrr1-Qrr5, which destabilize the transcriptional activator protein LuxR. This prevents the activation of target genes responsible for the production of various proteins, including bioluminescent luciferase. Hence, at low cell densities the bacteria do not bioluminesce. On the other hand, at high cell densities the receptors switch from kinases to phosphotases, significantly reducing the levels of LuxU-P and LuxO-P. 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 a sharp, sigmoidal response of the concentration of LuxR to smooth changes in cell density. Following Ref. [4], we analyze the occurrence of ultrasensitivity at the single cell level by focusing on a single phosphorylation pathway and adapting the Goldbeter- Koshland of phosphorylation-dephosphorylation cycles (PdPCs) [15]. (For a corresponding model of switching due to the action of srnas see Hunter et al. [20].) In particular, we consider the phosphorylation-dephosphorylation of LuxU by the enzymatic action of a particular quorum sensing receptor, which is denoted by R when acting as a kinase and by R when it is it is bound by an autoinducer (A) and acts like a phosphotase. Denoting the protein LuxU by W, we define the following reaction schemes: W + R a 1 d1 WR k 1 W + R W + R a 2 d2 W R k 2 W + R, (5D.9a) (5D.9b) R + A k + k R. (5D.9c) 8

9 HAI-1 HAI-1 LuxN HAI-1 extracelular domain LuxPQ AI-2 AI-2 CAI-1 CAI-1 CqsS CAI-1 outer membrane inner membrane P + P + P + LuxU LuxO P + + srna LuxR regulation of gene transcription Fig. 5D.4: Summary diagram 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. For simplicity, we assume that both the phosphorylation and the dephosphorylation steps are irreversible. (See the work of Qian and collaborators for an analysis of more detailed, reversible models of PdPCs [32, 12, 33].) Introducing the concentrations u = [A], w = [W], w = [W ], r = [R], r = [ R], v = [WR] and v = [W R], the corresponding kinetic equations for a single cell with a fixed intracellular concentration of autoinducer, are dw dv dw = a 1 w(r v) + d 1 v + k 2 v (5D.10a) = a 1 w(r v) (d 1 + k 1 )v (5D.10b) = a 2 w ( r v ) + d 2 v + k 1 v (5D.10c) dv = a 2 w ( r v ) (d 2 + k 2 )v (5D.10d) dr = k r k + ur. (5D.10e) These are supplemented by the conservation equations W T = w + w + v + v, R T = r + r, (5D.11) 9

10 where R T is the total concentration of receptors and W T is the total concentration of LuxU. For simplicity, we will assume that the conversion of the receptors from kinase to phosphotase activity is independent of the PdPC. That is, we ignore any positive feedback pathways, in which the regulation of gene expression by the phosphorylation/dephsophorylation of LuxU alters the production of the autoinducer [31]. This allows us to treat the receptor-ligand dynamics given by Eq. (5D.10e), or subsequent extensions, independently of the PdPC dynamics given by Eqs. (5D.10a-5D.10d). In particular, for fixed concentration u, we can take the concentration of kinases and phosphotases to be at equilibrium: r eq = k k + u + k R T R(u), r eq = R T R(u). (5D.12) The system of Eqs. (5D.10a-5D.10e) then reduces to the classical Goldbeter- Koshland model of PdPCs [15], and we can apply their analysis based on a generalization of Michaelis-Menten kinetics. The first step is to assume that the concentration of W and W is much larger than that of the receptor, that is, W T R T or equivalently W T = w + w. This implies that the time scale for the dynamics of the complexes WR and W R is much faster than that for the dynamics of W and W. Performing a separation of time-scales, we can treat the concentrations w and w as constants when analyzing equations (5D.10b) and (5D.10b), while we can take the steady-state values of the concentrations v,v when solving equations. (5D.10a) and (5D.10c). Hence, setting dv/ = 0 and r = R(u) in (5D.10b) we can solve for v in terms of w. Similarly, setting dv / = 0 and r = R T R(u) in (5D.10d) we can solve for v in terms of w. We thus obtain the reduced kinetic scheme W f 1(w) W, f 2 (w ) (5D.13) with and f 1 (w) = k 1R(u)w K 1 + w, f 2(w ) = k 2[R T R(u)]w K 2 + w, (5D.14) K 1 = d 1 + k 1 a 1, K 2 = d 2 + k 2 a 2. Imposing the conservation condition W T = w+w thus yields the single independent kinetic equation dw = f 1 (W T w ) f 2 (w ). (5D.15) The steady-state concentration w eq of LuxU-P is thus obtained by solving f 1 (W T w ) = f 2 (w ), which yields a quadratic equation for w. Taking the positive root, and expressing it as a function of the fixed autoinducer concentration u, we find that [LuxU-P] = φ(v (u)), [LuxU-P]+[LuxU] (5D.16) 10

11 molar fraction φ K =1 K = 0.01 K = 0.1 (a) molar fraction Ψ K =1 K = 0.01 K = 0.1 (b) inducer concentration u cell density α Fig. 5D.5: Molar fraction of modified protein W at steady-state as a function of (a) the autoinducer concentration u and (b) cell density α for different values of K, with K = K 1 = K 2, k k 1 = k + k 2, Γ = 1, κ = 10, and α c = with and φ given by for V 1, where V (u) = k 1 R(u) k 2 [R T R(u)] = k k + u k 1 k 2 (5D.17) φ = B + B 2 + 4AC 2AW T, (5D.18) A = V (u) 1, B = 1 W T (K 1 + K 2 V (u)) (V (u) 1), C = K 2 W T V (u). A plot of φ, as a function of the autoinducer concentration u is shown in Fig. 5D.5(a) for K 1 = K 2 = K. At low values of K, there is a sharp change from high to low levels of modified protein over a very small change in u (ultrasensitivity); this corresponds to a regime in which the two enzymes are saturated. On the other hand, for large values of K, the curve is relatively shallow, and one obtains a response similar to first-order kinetics. Now consider a population of N identical cells that are coupled via a common extracellular domain due to the transfer of the autoinducer A across the cell membrane. Eq. (5D.10e) for a single cell is then replaced by a system of equations of the form du i dr i du = Γ + k (R T r i ) k + u i r i κ(u i U), (5D.19a) = k (R T r i ) k + u i r i i = 1,...,N (5D.19b) = ακ(u av U) γu, (5D.19c) 11

12 where u av = 1 N N j=1 u j, (5D.20) is the population-averaged intracellular concentration of A, U is the extracellular concentration of A, and Γ is the rate of production of A due to the action of enzymes. Suppose that the above system is globally convergent with u i (t) u(t) and r i (t) r(t) in the limit t, with (u,r) evolving according to the effective singlecell model du = Γ + k [R T r] k + ur κ(u U) (5D.21a) dr = k [R T r] k + ur (5D.21b) du = ακ(u U) γu. (5D.21c) (Conditions for global convergence based on Sect. 5D.1 have been derived elsewhere [4].) Equations (5D.21a)-(5D.21c) have a unique stable fixed point u eq with u eq = Γ (ακ + γ) κγ ψ(α). Putting u = u eq in equation (5D.16) finally shows that ( [LuxU-P] [LuxU-P]+[LuxU] = φ k k + ψ(α) k 1 k 2 ) Ψ(α), (5D.22) (5D.23) In order that the system exhibit switch-like behavior as a function of cell density α, we require a critical value α c, 0 < α c < such that (for Γ = 1) χ k k + k 1 k 2 = α c γ + 1 κ, that is α c = γ(χ κ 1 ). Assuming that χ = 1 and taking α c = 0.05 [28], we require κ > 1 and γ = γ c = 0.05(1 κ 1 ). For low cell densities (α α c ) we have Ψ(α) 1, which follows from the functional form of φ, see Fig. 5D.5(a). 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 (α α c ) we find that Ψ(α) 0. Now the fraction of phosphorylated LuxU-P is small, allowing the expression of LuxR and downstream gene regulatory networks. The α-dependence is illustrated in Fig. 5D.5(b) with α and u linearly related. 12

13 5D.4 Noise amplification of intrinsic fluctuations in an ultrasensitive switch One of the characteristic features of ultrasensitive biochemical signaling networks is that they tend to amplify noise [2, 14, 12, 9]. On the other hand, collective behavior at the population level, as exhibited by quorum sensing networks, can mitigate the effects of noise [37, 34]. The amplification of intrinsic fluctuations in the V. harveyi quorum sensing model has recently been investigated within the framework of linear response theory [4]. Linear response of single-cell model First, consider the single-cell model with fixed concentration u of autoinducer. In the deterministic case, with r given by the equilibrium solution (5D.12), the dynamics of LuxU is given by equation (5D.15), which we write explicitly in the form dw = F (w,r) k 1r(W T w ) K 1 + [W T w ] k 2[R T r]w K 2 + w. (5D.24) Several studies have analyzed noise signal amplification in ultrasensitive signal transduction networks by considering stochastic versions of equation (5D.24) [2, 14, 12, 9]. (A more detailed analysis would need to start from a stochastic version of the full system of equations (5D.10a-5D.10e), since r(t) is now time-dependent.) Suppose that r(t) undergoes small fluctuations about the equilibrium state r eq of Eq. (5D.12). This can be incorporated by replacing equation (5D.10e) for fixed u by the Langevin equation dr = k (R T r) k + ur + σ 0 ξ (t), (5D.25) with ξ (t) a white noise process, ξ (t) = 0, ξ (t)ξ (t ) = δ(t t ), and σ 0 a fixed noise intensity. One possible source of noise is fluctuations in the autoinducer concentration u (see below). Under the linear noise approximation, we set r(t) = r eq + R(t) such that dr = (k + k + u)r + σ 0 ξ (t). (5D.26) Similarly, linearizing equation (5D.24) about the equilibrium solution by setting w = w eq +W, w eq = φ(v (u)), gives dw = β 1 W + β 2 R, (5D.27) 13

14 with and β 1 = F k 1 K 1 r eq w = eq [K 1 + [W T w eq]] 2 + k 2K 2 [R T r eq ] [K 2 + w eq] 2 (5D.28) β 2 = F r = eq k 1(W T w eq) K 1 + [W T w eq] + k 2w eq K 2 + w eq (5D.29) Note that the gain of the underlying deterministic system (5D.24) at equilibrium is given by g = W/w eq = β 2 r eq R/r eq β 1 w. (5D.30) eq Incorporating the u-dependence of r eq and β 1,2 one can determine g = g(u) and show that there is a sharp gain around the critical concentration u = 1. (Note that the units of autoinducer concentration are fixed by taking the PdPC switch to occur at u = 1. Typical intracellular molar concentrations are of order nm.) In order to characterize noise amplification in the system, one can Fourier transform the linear equations (5D.26) and (5D.27) with R(ω) = e iωt R(t) etc.. This gives W(ω) = β 2 β 1 + iω R(ω), R(ω) = σ 0 k + k + u + iω ξ (ω), where ξ (ω) is the Fourier transform of a white noise process with relative gain G/g K=1 K=0.01 (a) K= inducer concentration u stochastic gain G K=1 (b) K=0.01 K= inducer concentration u Fig. 5D.6: Plot of (a) relative gain G/g and (b) stochastic gain G as a function of autoinducer concentration u for different values of K, K = K 1 = K 2. Other parameter values are k = k + = k 1 = k 2 = 1, and R T = W T = 1. 14

15 ξ (ω) = 0, ξ (ω)ξ (ω ) = 2πδ(ω ω ). Using the Wiener-Khinchine theorem, the variance of the receptor concentration is given by the integral of the power spectrum defined by That is, 2πS R (ω)δ(ω ω ) = R(ω)R(ω ) σr 2 = S R (ω) dω 2π = σ0 2 dω [k + k + u] 2 + ω 2 2π = σ0 2 2[k + k + u]. (5D.31) Similarly, the variance of the Lux-P concentration is σ 2 W = S W (ω) dω 2π = β2 2 = σ β 1 [k + k + u] k + k + u + β 1 β2 2 σ0 2 dω β1 2 + ω2 [k + k + u] 2 + ω 2 2π (5D.32) If we interpret σ W /w eq as the relative noise intensity of the output and σ R /r eq as the relative noise intensity of the output, then the noise amplification of the PdPC in response to receptor fluctuations is defined by the stochastic gain [14] G = σ W /w eq σ R /r eq = r eq w eq β2 2 1 = g β 1 k + k + u + β 1 β 1 k + k + u + β 1, (5D.33) where g is the deterministic gain (5D.30). In Fig. 5D.6 we plot the relative gain G/g and the stochastic gain G as a function of u. It can be seen that the sharp amplification around the critical density u = 1 in the ultrasensitive regime (K = 0.01) is suppressed relative to the deterministic gain. Linear response of population model In order to extend the analysis of noise amplification to the population model, it is necessary to explicitly model the stochastic dynamics of the autodinducer concentration. In a stochastic model of quorum sensing, it is no longer possible to identify the state of all the cells, even if the deterministic system (5D.1a-5D.1c) is globally convergent. Therefore, in the case of V. harveyi, we have to consider a stochastic version of equations (5D.19a) and (5D.19b) 15

16 for i = 1,...,N, with du i = Γ + k (R T r i ) k + u i r i κ(u i U) + θ 0 ξ i (t), (5D.34a) dr i = k (R T r i ) k + u i r i, (5D.34b) du = ακ(u av U) γu (5D.34c) ξ i (t) = 0, ξ i (t)ξ j (t ) = δ(t t )δ i j. We assume that each cell is driven by an independent white noise term with constant noise intensity θ 0 ; one source of the noise could be fluctuations in the production of the autoinducer. Linearizing about the global steady-state by setting yields u i (t) = u eq + U i (t), r i (t) = r eq + R i (t), U(t) = U eq + V (t), du i dr i dv = k R i k + (U i r eq + u eq R i ) κ(u i V ) + θ 0 ξ i (t), (5D.35a) = k R i k + (U i r eq + u eq R i ), (5D.35b) = ακ(u av V ) γv, U av = 1 N N i=1 U i. (5D.35c) (σr/σ0)2 0.5 (a) κ=1 0.1 κ= 2 κ= inducer concentration ueq gain G 8 κ=2 κ=4 6 κ=8 4 2 (b) inducer concentration ueq Fig. 5D.7: (a) Plot of variance σr 2 in receptor fluctuations as a function of autoinducer concentration u eq for different values of κ. Also shown is the corresponding quantity in the single-cell model (dashed curve). (b) Plot of stochastic gain G of population model as a function of autoinducer concentration u eq for different values of κ and K = Other parameter values are k = k + = k 1 = k 2 = 1, and R T = W T = 1. Dashed curve is gain of single-cell model from Fig. 5D.6(b). 16

17 These linear equation can be analyzed along similar lines to the single-cell case using Fourier transforms and contour integration [4]. Here, we simply state the final results (for large N). First, the variance in the receptor concentration r i at the single cell level is σ 2 R = σ 2 0 2z (z 2 + z 2 ) σ 2 0 2z + (z 2 + z 2 ) = σ 2 0 2z + z (z + + z ), (5D.36) where σ 0 = k + r eq θ 0 /Γ with Γ = 1, and z ± = 1 [ ] (k + k + (u eq + r eq ) + κ) ± (k + k + (u eq + r eq ) + κ) 2 2 4κ(k + k + u eq ). The corresponding stochastic gain of individual cell PdPCs driven by the receptor fluctuations of the population model is G = g β 1 [β1 2 (z2 + z z + + z 2 +)] + (z + + z )z + z, β 1 (β1 2 z2 + )(β 1 2 z2 ) One finds that σr 2 0 as κ, see Fig.5D.7(a). On the other hand, expressing r eq and β 1,2 in terms of u eq, one finds that the stochastic gain G is only weakly dependent on κ, and approaches the gain of the single-cell model as κ increases, see Fig. 5D.7(b). Given that the receptor fluctuations σr 2 are suppressed in the quorum sensing model for large κ, while the gain is hardly changed, we conclude that the fluctuations in the LuxU-P concentration w are greatly suppressed compared to an isolated cell. Supplementary references 1. Anetzberger, C., Pirch, T., Jung, K.: Heterogeneity in quorum sensing- regulated bioluminescence in Vibrio harveyi. Molecular Microbiology 73, (2009) 2. Anguige, K., King, J. R., Ward, J. P.: A multi-phase mathematical model of quorum-sensing in a maturing Pseudomonas aeruginosa biofilm. Math.Biosci. 203, (2006). 3. Berg, O. G., Paulsson, J., Ehrenberg, M: Fluctuations and quality of control in biological cells: zero-order ultrasensitivity reinvestigated. Biophys. J. 79, (2000) 4. Bressloff, P. C.: Ultrasensitivity and noise amplification in a model of V. harveyi quorum sensing Phys Rev E (2016) 5. Chiang, W. Y., Li, Y. X., Lai, P. Y.: Simple models for quorum sensing: Nonlinear dynamical analysis. Phys. Rev. E 84, (20111) 6. Couzin, I. D.: Collective cognition in animal groups. Trends. Cogn. Sci.13, (2009). 7. Davies, D. G., Parsek, M. R., Pearson, J. P., Iglewski, B. H., Costerton, J. W., Greenberg, E. P.: The involvement of cell-to-cell signals in the development of bacterial biofilm. Science (1998). 8. Dockery, J. D., Keener, J. P.: A mathematical model for quorum-sensing in Pseudomonas aeruginosa. Bull. Math. Biol. 63, (2001). 9. Dunlap, P. V. Quorum regulation of luminescence in Vibrio fischeri. J. Mol. Microbiol. Biotechnol (1999). 17

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