Dynamics and Mechanism of A Quorum Sensing Network Regulated by Small RNAs in Vibrio Harveyi

Commun. Theor. Phys. 55 (2011) 465 472 Vol. 55, No. 3, March 15, 2011 Dynamics and Mechanism of A Quorum Sensing Network Regulated by Small RNAs in Vibrio Harveyi SHEN Jian-Wei ( å) Institute of Applied Mathematics, Xuchang University, Xuchang 464000, China Institute of System Biology, Shanghai University, Shanghai 200444, China (Received August 16, 2010; revised manuscript received November 17, 2010) Abstract Bacterial quorum sensing (QS) has attracted much interests and it is an important process of cell communication. Recently, Bassler et al. studied the phenomena of QS regulated by small RNAs and the experimental data showed that small RNAs played important role in the QS of Vibrio harveyi and it can permit the fine-tuning of gene regulation and maintenance of homeostasis. According to Michaelis Menten kinetics and mass action law in this paper, we construct a mathematical model to investigate the mechanism induced QS by coexist of small RNA and signal molecular (AI) and show that there are periodic oscillation when the time delay and Hill coefficient exceed a critical value and the periodic oscillation produces the change of concentration and induces QS. These results are fit to the experimental results. In the meanwhile, we also get some theoretical value of Hopf Bifurcation on time deday. In addition, we also find this network is robust against noise. PACS numbers: 05.45.-a Key words: quorum sensing, genetic network, oscillation, small RNA, bifurcation, negative feedback loop 1 Introduction The survival of bacteria relies mainly on regulatory networks, which detect and integrate multiple environmental input and respond with appropriate behavioral output. These regulatory networks are involved in quorum sensing (QS), which is a mechanism of chemical communication that enable bacteria to track population density by secreting and detecting extracellular signaling molecules called autoinducers (AIs). [1 3] QS can regulate critical bacterial process by monitoring the concentration of AIs, and alter the expression of genes to carry out task. Quantitative modeling of QS pathway can be useful to understand interaction between different genes. In 1970, Nealson et al. [4] had studied the model of the bioluminescent marine bacterium V ibrio harveyi for QSbased regulation, experimental studies indicate that there are multiple autoinducers and corresponding sensor proteins, which can regulate the expression of bioluminescent. The phosphorylation of regulator protein LuxO can be controlled by the interaction between AIs and sensors. At negligible concentrations of AIs, i.e. at low cell density (LCD), these sensors act as kinases that transfer phosphate through LuxU to LuxO. [5 6] LuxO-P activates the expression of genes encoding small RNAs which in turn post-transcriptionally repress the QS master regulatory protein LuxR. At the high cell density (HCD), AIs accumulate and bind to their cognate sensors and the sensors act as phosphatases, reversing the phosphate flow through the QS circuit. This results in dephosphorylation and inactivation of LuxO, so that the expression of genes encoding the small RNAs is terminated. Recent experiments [7 8] indicate that there is feedback loop between small RNAs and the QS master regulatory protein LuxR, LuxR directly activates transcription of genes encoding the small RNAs and inhibits the expression of genes encoding itself. The feedback loop involving in small RNA is essential for optimizing the dynamics of transition between individual and group behaviors. Due to the complexity of this regulatory network, we would desire to develop a quantitative framework for explaining the mechanism how to optimize the dynamics of transition between individual and group behaviors. The corresponding quantitative model can then be used to make testable predictions for future experiments as well as to further analyze existing experimental data. The rest of the paper is organized as follows. In Sec. 2, we give an overview of the QS network in Vibrio harveyi and develop a mathematical model of QS network mediated by small RNAs and define some dimensionless parameters. In Sec. 3, we investigate the oscillatory dynamics of the quorum sensing model. In Sec. 4, we explain the mechanism on this quorum sensing. 2 Formulation of Model The quorum-sensing network in Vibrio harveyi is shown in Fig. 1. Supported by National Natural Science Foundation of China under Grant Nos. 10802043 and 10832006, Program for Science & Technology Innovation Talents in Universities of Henan under Grant No. 2009HASTIT033 and Key Disciplines of Shanghai municipality (S30104) Corresponding author, E-mail: xcjwshen@gmail.com c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

466 Communications in Theoretical Physics Vol. 55 Fig. 1 Schematic diagram showing the gene regulation mediated by srna with a delayed negative feedback loop. These small RNAs are small noncoding RNAs, 18-24 nt in length, that are predicted to regulate the expression of approximately one-third of all human genes. This regulation occurs posttranscriptionally through small RNA binding to mrna targets which can lead to degradation of target genes or inhibition of translation. Experimental data [7 8] indicates that phosphorylated LuxO can activate the expression of small RNAs, and the master regulatory protein LuxR is the target protein of srna (i.e. Qrr2-4) and can activate the qrr (genes of srnas) promoters. At the same time, LuxR can autorepress the expression of itself (see Fig. 1). To understand the mechanism of the quorum sensing mediated by small RNAs, we consruct a mathematical model abstracted from the complex network (Figs. 1 and 2). In Fig. 2, we simplify the process of QS and abstract the model from Fig. 1, the three sensors (LuxN, LuxPQ, CqsS), Synthases (LuxM, LuxS, CqsA) autoinducers (HAI-1, AI-2, CAI-1) and srnas (Qrr2-4) can be viewed as one component. In this QS pathway, the sensor protein can be modeled as two-state systems, [9 10] i.e., kinase sate and phosphatase state. In the kinase state, the sensor (S k ) can be autophosphorylated and then transfer the phosphate group to the downstream protein LuxU, whereas in the phosphatase state, the phosphate flow is reversed. From the experiments we can know that the sensors are in the kinase state at LCD, whereas at HCD, the sensors are primarily in phosphatase state. So we can consider a network wherein the free sensor corresponds to the kinase state (S k ), whereas sensors binded by autoinducers are in phosphatase state (S p ). According to the chemical reaction, we can obtain the chemical reaction equation as follows. M k1 AI, (1) nai k2 k 2 A n, (2) k 3 A n + S k S p, (3) k 3 Fig. 2 Reduction of Synthase and Qrr1-4 (srnas) to an abstract model. There are three pathways to control LuxU. The principal components of the pathway are three sensor (LuxN, LuxPQ, CqsS) and the corresponding autoinducer synthase (LuxM, LuxS, CqsA), which can produce the three autoinducers: HAI1, AI-2, and CAI-1, respectively. The binding of the single autoinducer to a sensor is highly specific. The interaction between autoinducers (AI) and sensor proteins can determine the overall phospherylation state of LuxO through the phosphorelay mechanism. the phosphorylation state of LuxO can determine the activation state of corresponding small RNAs (srnas), Qrr2-4. S k + U k4 S k + U p, (4) U p + S p k 5 S p + U, (5) U p + O k6 k 6 U + O p. (6) In the above reaction, M represents synthase which produces the corresponding autoinducer, AI represents autoinducer, S k represents sensor corresponding to the kinase state, S p represents the sensor corresponding to the phosphatase state. U represents LuxU, U p represents phosphorylated LuxU. O represents LuxO and O p represents phosphorylated LuxO, k 1 k 6 represent the reaction rate and the k 1 k 3, k 6 represent the dissociation constant.

No. 3 Communications in Theoretical Physics 467 By using the Mass Action Law and Michaelis-Mentens Kinetics, we can obtain the mathematical model as follows d[luxm] d[ai] = a 1 k 1 [LuxM], = k 1 [LuxM] d 1 [AI], d[luxu p ] = k 4 [S k ](U 0 [LuxU p ]) k 5 [S p ][LuxU p ] d 3 [LuxU p ], d[srna] d[mrna] d[protein] = 1 + k 7[protein] m (t τ 1 ) + δ 1 [LuxO p ] 1 + [protein] m r[srna][mrna] d 4 [srna], (t τ 1 ) + δ 2 [LuxO p ] k 8 = 1 + [protein] m (t τ 1 ) r[srna][mrna] d 5[mRNA], = k 9 [mrna](t τ 2 ) d 6 [protein], d[lux] = k 10[protein] m (t τ 1 ) 1 + [protein] m (t τ 1 ) d 7[Lux], (7) where [LuxO p ] = [LuxU p ]O 0 k[u 0 ] + (1 k)[luxu p ], k = k 5 k 5, [O 0 ] = [LuxO] + [LuxO p ], U 0 = [LuxU] + [LuxU p ], S 0 = [S k ] + [S p ], S p = [AI]n [S 0 ] κ n + [AI] n, S k = κn [S 0 ] κ n + [AI] n, κ = (k 2 k 3 /k 2 k 3 ) 1/n, [mrna] and [protein] represent the concentration of the master regulator HaR mrna and its protein, respectively. [Lux] represents the concentration of luciferase, which is required for bioluminescence. d 1 d 7 represent the degradation rate. k 7 k 10 and δ 1 are the basal rate of transcription in the absence of transcription. The small RNA base pairs with the target mrna at a rate r and the m represents the Hill coefficient. 3 Oscillation Dynamics of Quorum Sensing Model 3.1 Dynamics with Time Delay Delay often appears in the process of gene regulation, and affects the dynamics of gene network. From the theory of bifurcation, we can know that oscillatory dynamics is related to the total values of all the delay and obtain the oscillatory threshold of time delay. In addition, we also derive the other conditions when the system (7) is stable (see Theorem 1 in Appendix). Fig. 3 Bifurcation diagram with total time delay τ as a parameter.

468 Communications in Theoretical Physics Vol. 55 As an example, assume that a1 = 1, k1 = 0.2, d1 = 0.5 k5 = 0.3, s0 = 2, u0 = 2, d2 = 0.6, k4 = 0.3, κ = 0.8, n = 2, k6 = 1, r = 1, d3 = 0.5, δ1 = 1, δ2 = 1.1, d3 = 0.5, d4 = 0.3, k8 = 2, d5 = 0.2, k9 = 1, d6 = 0.1, k10 = 1, d7 = 0.8, O0 = 2, k = 0.6. As the time delay τ changes, the dynamics of the system is changed. There is a critical value τ0, when τ > τ0, the steady state becomes unstable and the sustained oscillations occur, in the meantime, we also find the robustness of amplitudes against variation in delays. So we can control the dynamics of the system by tuning the time delay (see Fig. 3). In the system, delay τ1 and τ2 represent transportation or diffusion process from nucleus to cytoplasm of mrna and from cytoplasm to nucleus of protein, respectively. From the analysis, we can know that τ1 and τ2 affect the dynamical behaviors in the form of τ1 + τ2 due to the cyclic structure of system. In addition, with the increase of τ, the period will also increase (see Fig. 4 or the expression of period in Appendix). Fig. 4 Time history diagram with different time delay. 3.2 Dynamics with Hill Coefficient m In the process of gene regulation, protein as TF (Transcription Factor) is often polymerized and forms polymer, the degrees of polymerization (i.e. Hill coefficient) often affect the dynamical behaviors. As an example, we make use of parameters in Subsec. 3.1. In the steady state, the concentration of small RNA, mrna, HapR protein, and Lux is changed with the variance of Hill coefficient m. With the increase of m, the bifurcation occurs at a critical value m0 7.5 (see Fig. 5) and there will be periodic oscillation. In this case, the amplitude of oscillation will be also changed, and the maximum of amplitude also increases. At the same time, the period will also increase due to increase of hill coefficient m (See Fig. 6). Fig. 5 Bifurcation induced by the hill coefficient m.

Communications in Theoretical Physics No. 3 469 Fig. 6 Time history diagram with different m. 3.3 Dynamics with Others In this paper, we also study the changes of concentration with the base pairing rate r. Obviously, the concentration of small RNA, mrna, HapR protein, and Lux will decrease with the increase of r (See Fig. 7). So the small RNAs play an important role in repressing the expression of target gene. And we can control the expression of target mrna by regulating the base pairing rates. Fig. 7 Changes of srna, mrna, protein and Lux with the base pair rate r. In addition, gene expression is often accompanied by the noise, the noises often affect the dynamics of gene regulation, so we should consider these noise. In this paper, we take the noises as Gaussian white noise, and find that this network can not be almost affected by the noises and the small RNA can filter high frequency noise without compromising the ability to rapidly respond to large changes in input signals. In this case, we can explain this phenomenon due to a large pool of srnas shortens the effective mrna lifetime and buffers against target mrna fluctuations.[11] 4 Mechanism of Quorum Sensing 4.1 Relation Between hapr mrna and Small RNA Levels More and more experimental data show that small RNA, hapr mrna, and master regulator HapR protein levels vary reciprocally,[7] HapR protein activates the expression of small RNAs. As τ increases and goes beyond the critical value τ0, a limit cycle occurs, so small RNA, hapr mrna, and HapR protein levels vary reciprocally (see Fig. 9(d)). The LapR proteins directly activate the transcription of small RNA (see Fig. 9), which makes up the deficiency at HCD mode, at the same time, they also repress the expression of mrna, which lead to the downregulation of mrna, which verify the results of experimental data.[7]

470 Communications in Theoretical Physics Vol. 55 Fig. 8 Comparsion with noise and without noise. Fig. 9 hapr mrna and small RNA levels vary reciprocally and hapr mrna levels influence small RNA levels. At first, the system (7) is in LCD mode and the concentration of small RNAs increases which induce the decrease of the concentration of mrna, in the meanwhile, the HapR protein actives the expression of small RNA and represses the expression of mrna (see Fig. 9). In the steady state, as time delay τ exceeds τ0, there is attractor among small RNA, mrna, HapR protein, and Lux, which attracts to a limit cycle. Thus it creates a periodic cycle from LCD to HCD mode. In this mechanism of quorum sensing, as the signal molecules (AI) first increase and then decrease, the srna-luxr feedback loop activates the transition from LCD to HCD and accelerates the transition from HCD to LCD. From the analysis and previous results,[12] we can know that the small RNAs mainly provide fine-tuning of HapR expression and defer the effect on changing the concentration of AI and make up the deficiency of decreasing concentration. In addition, the recovery time in the case of regulation by small RNAs is faster than that in the absence of small RNAs, therefore, it can improve the deficiency when AI becomes rare[12] (See Fig. 11). According to analysis of Sec. 3, we can know that this feedback loop can filter the noise, so this is very good mechanism and it may be the result of natural selection. 4.2 Transition from Coexistence of LuxOp and LuxR Protien During the transition from the low to high cell density

No. 3 Communications in Theoretical Physics 471 mode, AI levels first increase and the cell switches from LCD dominated by LuxO p mode to HCD mode dominated by LuxR protein. During this switch, LuxO p and LuxR protein could transiently coexist, which allow LuxR protein feedback-activates small RNAs (see Figs. 10(c) and 10(d)). When the time delay τ of transportation or diffusion from nucleus to cytoplasm of mrna or from cytoplasm to nucleus of protein, the Hill coefficient m in Eq. (7) exceed the critical value, the concentration of small RNA, mrna, protein, Lux changes periodically, which make up the deficiency of increase of AI. and these defer the transition from LCD mode to HCD mode. When AIs begin decreasing, these periodical oscillation accelerates the transition from HCD mode to LCD mode. These periodical oscillations result in the Hopf bifurcation (see Appendix) induced by time delay τ and Hill coefficient m. Fig. 10 Protein feedback on the small RNA in the presence of protein and LuxO p. gene network, which can induce the QS phenomenon. In this gene network, changes in the concentration of small RNA, LuxR protein will lead to the change in the concentration of bioluminescent protein Lux by time delay and the hill coefficient. From the theoretical analysis and simulation, we can know that the coexistence of AI and small RNA produce this kind of QS. In this mechanism, the system is robust against the noise. Appendix Fig. 11 Comparsion with small RNA and without small RNA. 5 Conclusions The bioluminescent marine bacterium often uses a cellcell communication process called quorum sensing (QS) to co-ordinate behaviors in response to changes in population density. In this paper, we investigate the dynamics of a dx 1 dx 2 = a 1 k 1 x 1, = k 1 x 1 d 1 x 2, dx 3 = f 1 (x 2 ) f 2 (x 2 )x 3 d 3 x 3, dx 4 = g(x 6, x 3 ) rx 4 x 5 d 3 x 4, dx 5 = f 3 (x 6 ) rx 4 x 5 d 4 x 5, dx 6 = k 9 x 5 (t τ 2 ) d 5 x 6,

472 Communications in Theoretical Physics Vol. 55 dx 7 = f 4 (x 6 ) d 5 x 7, (A1) where x 1, x 2, x 3, x 4, x 5, x 6, x 7 represent the concentration of LuxM, AI, LuxU p, srna, mrna, protein, and Lux, respectively. k 6 O 0 x 3 [LuxO p ] =, a 2 = k 4 κ n s 0 u 0, κu 0 + (1 κ)x 3 a 3 = k 4 κ n s 0 d 3 κ n, a 4 = k 5 s 0 d 3, f 1 (x) = f 3 (x) = a 2 κ n + x n, f 2(x) = a 3 + a 4 x n κ n + x n, k 8 1 + x m (t τ 1 ), f 4(x) = k 10x m (t τ 1 ) 1 + x m (t τ 1 ), g(x 6, x 3 ) = 1 + k 7x n 6(t τ 1 ) + δ 1 [LuxO p ] 1 + x n 6 (t τ. 1) + δ 2 [LuxO p ] Clearly, a time delay cannot change the number and location of equilibria of system (A1). Let (x 10, x 20, x 30, x 40, x 50, x 60, x 70 ) be an equilibrium of Eq. (A1), then we can obtain the linearized system of Eq. (A1). The linearized system of Eq. (A1) at above equilibrium is as follows. Defining the A(λ) which is involved in the coeffecient matrix of the linear system for Eq. (A1) as follows k 1 0 0 0 0 0 0 k 1 d 2 0 0 0 0 0 0 b 1 b 2 0 0 0 0 A(λ)= 0 0 b 3 b 4 b 5 b 6 e λτ1 0, 0 0 0 b 7 b 8 b 9 e λτ1 0 0 0 0 0 k 9 e λτ2 d 6 0 0 0 0 0 0 b 10 e λτ1 d 7 where b 1 = df 1(x 20 ) dx 2 df 2(x 20 ) dx 2 x 30, b 2 = (f 2 (x 20 ) + d 3 ), b 3 = dg(x 30, x 60 ) dx 3, b 4 = d 4 + rx 50, b 5 = rx 40, b 6 = dg(x 30, x 60 ) dx 6, b 7 = rx 50, b 8 = (rx 40 + d 5 ), b 9 = df 3(x 60 ), b 10 = df 4(x 60 ). dx 6 dx 6 Assume that ǫ 0 = b 4 b 8 d 6 b 5 b 7 d 6, ǫ 1 = b 4 b 8 b 5 b 7 b 4 b 6 b 6 b 8, ǫ 2 = d 6 b 4 b 8, ǫ 3 = b 9 k 9, ǫ 4 = b 4 b 9 k 9 b 6 b 7 k 9, the characteristic equation of Eq. (A1) can be deduced as follows λ 3 + ǫ 2 λ 2 + ǫ 1 λ + ǫ 0 + ǫ 3 λe λτ + ǫ 4 e λτ = 0, (A2) where τ = τ 1 + τ 2. We assume that iω (ω > 0) is the root of the above transcendental equation, then we can obtain the following equation h(z) = z 3 + p 2 z 2 + p 1 z + p 0 = 0, (A3) where ω 2 = z, p 0 = ǫ 2 0 ǫ2 4, p 1 = ǫ 2 1 ǫ2 3 2ǫ 0ǫ 2, p 2 = ǫ 2 2 2ǫ 1. τ j k = 1 arcsin (ǫ 2ǫ 3 ǫ 4 )ωk 3 + (ǫ 1ǫ 4 ǫ 0 ǫ 3 )ω k ω k ǫ 2 4 + ǫ2 3 ω, (A4) k where k = 1, 2, 3, j = 0, 1,... Define τ 0 = τ j0 k 0 = min 1 k 3,j 1 τj k, ω 0 = ω k0. (A5) From the above analysis, we can obtain the following theorem Theorem Suppose that ǫ 0 > 0, ǫ 2 > 0, ǫ 1 ǫ 2 ǫ 0 > 0, and the ω 0, τ 0, λ(τ) be defined as above, then (i) If p 0 0 and = p 2 2 3p 1 < 0, then all roots of Eq. (A2) have negative real parts for all τ > 0, thus the steady state of origin system is absolutely stable. (ii) If p 0 < 0 or p 0 0, h(z 1 ) 0 (z 1 = ( p 2 + )/3), then all roots of Eq. (A2) have negative real parts when τ [0, τ 0 ], thus the steady state of origin system is asymptotically stable. (iii) If the condition (2) is satisfied, τ = τ 0 and h (z 0 ) 0, z 0 = ω 2 0, then ±ω 0 is a pair of simple purely imaginary roots of Eq. (A2) and all other roots have negative real parts. Moreover, d Re λ(τ 0 )/dτ > 0. Thus the original system exhibits the Hopf bifurcation at stable state. References [1] M.B. Miller and B.L. Bassler, Annu. Rev. Microbiol. 55 (2001) 165. [2] C.M. Waters and B.L. Bassler, Annu. Rev. Cell. Dev. Biol. 21 (2005) 319. [3] B.L. Bassler and R. Losick, Cell 125 (2006) 237. [4] K.H. Nealson, T. Platt, and J.W. Hastings, J. Bacteriol 104 (1970) 313. [5] J.A. Freeman and B.L. Bassler, Mol. Microbiol. 31 (1999) 665. [6] B.N. Lilley and B.L. Bassler, Mol. Microbiol. 36 (2000) 940. [7] K.C. Tu, C.M. Waters, S.L. Svenningsen, and B.L. Bassler, Mol. Microbiol. 70 (2008) 896. [8] S.L. Svenningsen, C.M. Waters, and B.L. Bassler, Gene & Development 22 (2008) 226. [9] M.B. Neiditch, M.J. Federle, A.J. Pompeani, et al., Cell 126 (2006) 1095. [10] L.R. Swem, D.L. Swem, N.S. Wingreen, and B.L. Bassler, Cell 134 (2008) 461. [11] P. Mehta, S. Gojal, and N.S. Wingreen, Molecular Systems Biology 4 (2008) 221. [12] Y. Shimoni, G. Friedlander, G. Hetzroni, et al., Molecular Systems Biology 3 (2007) 1.