Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 1 / 27Co. Quorum Sensing

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1 Colored Noise Induced Synchronized Switching in the Genetic Toggle Switch Systems Coupled by Quorum Sensing 王沛, 吕金虎 School of Mathematics and Statistics, Wuhan University 第 Ô3 国 ä 科学 Ø 坛, 北 2011 c 4 27 F Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 1 / 27Co

2 Acknowledgement Many thanks to Prof. Lü Jinhu, Prof. Lu Jun-an for their guidance and supporting; Many thanks to the teachers and fellow apprentices of the workshop on complex networks. NSF: , , , ; 973 Program of China: 2007CB310805; Fundamental Research Funds for the Central Universities of China( ). Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 2 / 27Co

3 Contents 1 Brief Overview 2 Mathematical Model 3.1 Mathematical Model 3.2 Measurements 3 Main Results 3.1 Feature Comparison Between Colored Noise And White Noise On Synchronized Switching 3.2 Colored Noise Can Promote Protein Production 3.3 Robustness of The Synchronized Switching Behavior 3.4 Autocorrelation Time And Synchronized Switching 4 4. Summary Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 3 / 27Co

4 1 Brief Overview 2 Mathematical Model 3 Main Results 4 4. Summary Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 4 / 27Co

5 2.1 Terminology Noise: a stochastic process ζ(t), Gaussian white noise: < ζ(t) >= 0, < ζ(t)ζ(t ) >= Dδ(t t ), D denotes noise strength; Colored noise: < ζ(t) >= 0, < ζ(t)ζ(t ) >= κ(t t ), the autocorrelation function κ is not a δ function. Ornstein-Uhlenbeck process is often used. Bistable system: for system dx/dt = f(x), X R n, there are two stable equilibria, system can reach its high steady states under some initial values, while low steady state under other initial values. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 4 / 27Co

6 2.1 Terminology Synchronized Switching: under some perturbation to the coupled bistable systems, every unit of the systems can switch between its high and low steady states synchronously; The synchronized switching behavior is quantitatively described by the average synchronization error (ASE) and spectral amplification factor. Quorum Sensing: A type of decision-making process used by decentralized groups to coordinate behavior. Many species of bacteria use quorum sensing to coordinate their gene expression according to the local density of their population 1. 1 M.B. Miller, B.L. Bassler, Quorum sensing in bacteria, Annu. Rev. microbiol., 2001,55, Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 5 / 27Co

7 2.2 Biological Implications of synchronized switching Synchronized Switching of cell ensembles is an interesting phenomena, which can be used to explain some biological behaviors 2,3,4 : Cell communications Chemotaxis Cell differentiation... 2 T. Zhou, L. Chen, K. Aihara, Molecular communication through stochastic synchronization induced by extracellular fluctuations. Phys. Rev. Lett., 2005, 95(17): art. no Laurent M., Kellershohn N., Multistability: a major means of differentiation and evolution in biological systems. Trends Biochem. Sci., 1999, 24: Smolen P., Douglas A. Baxter, John H. Byrne, Mathematical Modeling of gene networks. Neuron, 2000, 26: Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 6 / 27Co

8 2.2 Brief Overview: The toggle switch system The genetic toggle switch system 5 is firstly constructed in Escherichia coli by Gardner T. S. et al in the year T. S. Gardner, C. R. Cantor, J. J. Collins, Construction of a genetic toggle switch in Escherichia coli. Nature, 2000, 403(6767): Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 7 / 27Co

9 2.2 Brief Overview: the toggle switch system The toggle switch system is a typical bistable system. Hasty et al. 6 investigated multiplicative as well as additive noise induced switch in a single gene bistable system; and Yuan et al. 7 investigated noise induced switch behaviors of the single toggle switch system, and they find: Both multiplicative and additive noise can induce continuous switch, moreover, Yuan et al. found that there exists an optimal noise strength that can induce this switch effectively. 6 J. Hasty, J. Pradines, M. Dolnik, J. J. Collins, Noise-based switches and amplifiers for gene expression. Proc. Natl. Acad. Sci. USA, 2000, 97(5): Yuan Zh.,Zhang J., Zhou T., Sci. in China, Ser.B, 2007, 37: (in chinese). Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 8 / 27Co

10 2.2 Brief Overview: coupled toggle switch systems Wang J. et al. 8 investigated the multicellular toggle switch systems perturbed by Gaussian white noise and found that: Extrinsic noise can induce synchronized switching; There exists an optimal extrinsic noise strength, which induces the best synchronized switching behavior. Extracellular white noise can enhance ordering behavior. 8 J.Wang, J. Zhang, Z. Yuan, T. Zhou, Noise-induced switches in network systems of the genetic toggle switch. BMC Syst. Biol., 2007, 1: art. no. 50. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 9 / 27Co

11 2.2 Brief Overview: coupled toggle switch systems Wang J. et al. 8 investigated the multicellular toggle switch systems perturbed by Gaussian white noise and found that: Extrinsic noise can induce synchronized switching; There exists an optimal extrinsic noise strength, which induces the best synchronized switching behavior. Extracellular white noise can enhance ordering behavior. 8 J.Wang, J. Zhang, Z. Yuan, T. Zhou, Noise-induced switches in network systems of the genetic toggle switch. BMC Syst. Biol., 2007, 1: art. no. 50. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 9 / 27Co

12 2.2 Brief Overview: coupled toggle switch systems Wang J. et al. 8 investigated the multicellular toggle switch systems perturbed by Gaussian white noise and found that: Extrinsic noise can induce synchronized switching; There exists an optimal extrinsic noise strength, which induces the best synchronized switching behavior. Extracellular white noise can enhance ordering behavior. 8 J.Wang, J. Zhang, Z. Yuan, T. Zhou, Noise-induced switches in network systems of the genetic toggle switch. BMC Syst. Biol., 2007, 1: art. no. 50. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 9 / 27Co

13 switch 2.2 Brief Overview: Other investigations on bistable Other researches on the toggle switch systems includes: T. H. Tian, K. Burrage, Stochastic models for regulatory networks of the genetic toggle switch. Proc. Natl. Acad. Sci. USA, 2006, 103(22): A. Lipshtat, A. Loinger, N. Q. Balaban, O. Biham, Genetic toggle switch without cooperative binding. Phys. Rev. Lett., 2006, 96(18): art. no A. Lipshtat, A. Loinger, N. Q. Balaban, O. Biham, Stochastic simulations of genetic toggle switch system. Phys. Rev. E, 2007, 75(2): art. no Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 10 / 27Co

14 switch 2.2 Brief Overview: Other investigations on bistable Other researches on the toggle switch systems includes: T. H. Tian, K. Burrage, Stochastic models for regulatory networks of the genetic toggle switch. Proc. Natl. Acad. Sci. USA, 2006, 103(22): A. Lipshtat, A. Loinger, N. Q. Balaban, O. Biham, Genetic toggle switch without cooperative binding. Phys. Rev. Lett., 2006, 96(18): art. no A. Lipshtat, A. Loinger, N. Q. Balaban, O. Biham, Stochastic simulations of genetic toggle switch system. Phys. Rev. E, 2007, 75(2): art. no Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 10 / 27Co

15 switch 2.2 Brief Overview: Other investigations on bistable Other researches on the toggle switch systems includes: T. H. Tian, K. Burrage, Stochastic models for regulatory networks of the genetic toggle switch. Proc. Natl. Acad. Sci. USA, 2006, 103(22): A. Lipshtat, A. Loinger, N. Q. Balaban, O. Biham, Genetic toggle switch without cooperative binding. Phys. Rev. Lett., 2006, 96(18): art. no A. Lipshtat, A. Loinger, N. Q. Balaban, O. Biham, Stochastic simulations of genetic toggle switch system. Phys. Rev. E, 2007, 75(2): art. no Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 10 / 27Co

16 switch 2.2 Brief Overview: Other investigations on bistable Other researches on the toggle switch systems includes: T. H. Tian, K. Burrage, Stochastic models for regulatory networks of the genetic toggle switch. Proc. Natl. Acad. Sci. USA, 2006, 103(22): A. Lipshtat, A. Loinger, N. Q. Balaban, O. Biham, Genetic toggle switch without cooperative binding. Phys. Rev. Lett., 2006, 96(18): art. no A. Lipshtat, A. Loinger, N. Q. Balaban, O. Biham, Stochastic simulations of genetic toggle switch system. Phys. Rev. E, 2007, 75(2): art. no Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 10 / 27Co

17 2.3 Limitations of existing researches Limitations of the above researches: Only Gaussian white extrinsic noise is consider, while experimentally observed extrinsic noise is colored one. N. Rosenfeld, J. W. Young, U. Alon, P. S. Swain, M. B. Elowitz, Gene regulation at the single cell level. Science, 2005, 307(5717): V. Shahrezaei, J.F. Ollivier, P. S. Swain, Colored extrinsic fluctuations and stochastic gene expression. Mol. Syst. Bio., 2008, 4: art. no J. Lei, Stochasticity in single gene expression with both intrinsic noise and fluctuation in kinetic parameters. J. Theor. Bio., 2009, 256(4): Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 11 / 27Co

18 1 Brief Overview 2 Mathematical Model 3.1 Mathematical Model 3.2 Measurements 3 Main Results 4 4. Summary Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 12 / 27Co

19 3.1 Mathematical Model The following system coupled by Quorum Sensing (QS) 9 will be investigated: 9 J.Wang, J. Zhang, Z. Yuan, T. Zhou, Noise-induced switches in network systems of the genetic toggle switch. BMC Syst. Biol., 2007, 1: art. no. 50. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 12 / 27Co

20 3.1 Mathematical Model The N coupled systems are described as: dx i dt = α1 1+y n 1 d 1 x i + γ 1 + ψ 1i (t) + βai 1+A i i, d 2 y i + γ 2 + ψ 2i (t), dy i dt = α2 1+x n 2 i da i dt = εy i µa i + k(a e A i ), da e dt = Q N N i=1 (A i A e ) d e A e + ζ(t) + Asin(Ωt), (1) α j, d j, γ j (j = 1, 2) : the dimensionless maximal transcriptional rate, degradation rate and basal synthesis rate for jth species; n 1 and n 2 : Hill coefficients; ε and µ are the synthesis and degradation rates of the intracellular AI, respectively; d e is degradation rate of the extracellular AIs; Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 13 / 27Co

21 3.1 Mathematical Model ψ ji (t) indicates the intrinsic additive noise, such that < ψ ji (t) >= 0, < ψ ji (t)ψ ji (t ) >= D int δ ij (t t ), ζ(t) is the environmental perturbation, which can be white or colored. For white noise < ζ(t) > = 0; < ζ(t)ζ(t ) > = D ext δ(t t ); For colored one; Ornstein-Uhlenbeck process is used: dζ dt = ζ τ + 2/τξ(t), (2) where τ is the autocorrelation time, ξ(t) is Gaussian white noise, ζ(t) is the colored noise satisfying: < ζ(t) > = 0, < ζ(t)ζ(t ) > = D ext exp( t t /τ). Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 14 / 27Co

22 3.1 Mathematical Model ψ ji (t) indicates the intrinsic additive noise, such that < ψ ji (t) >= 0, < ψ ji (t)ψ ji (t ) >= D int δ ij (t t ), ζ(t) is the environmental perturbation, which can be white or colored. For white noise < ζ(t) > = 0; < ζ(t)ζ(t ) > = D ext δ(t t ); For colored one; Ornstein-Uhlenbeck process is used: dζ dt = ζ τ + 2/τξ(t), (2) where τ is the autocorrelation time, ξ(t) is Gaussian white noise, ζ(t) is the colored noise satisfying: < ζ(t) > = 0, < ζ(t)ζ(t ) > = D ext exp( t t /τ). Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 14 / 27Co

23 3.1 Mathematical Model Asin (Ωt) denotes the extracellular stimulus 10, H. Kori, A. S. Mikhailov, Entrainment of randomly coupled oscillator networks by a pacemaker. Phys. Rev. Lett., 2004, 93(25): art. no J. Hasty, F. Isaacs, M. Dolnik, D. McMillen, J. J. Collins, Designer gene network: towards fundamental cellular control. Chaos, 2001, 11(1): Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 15 / 27Co

24 3.2 Measurements Of Synchronized Switching Average synchronization error (ASE) 12 can be used to measure synchronization behavior, which is defined as: ASE: ASE =< 1 C 2 N [x i (t) x j (t)] 2 >. (3) i>j Here C 2 N = N(N 1) 2 ; N: cell numbers; <. >: time averaging. When all the cells are synchronized, ASE 0, otherwise, ASE will be above zero. 12 J.Wang, J. Zhang, Z. Yuan, T. Zhou, Noise-induced switches in network systems of the genetic toggle switch. BMC Syst. Biol., 2007, 1: art. no. 50. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 16 / 27Co

25 3.2 Measurements of Synchronized Switching The spectral amplification factor 13 can be used to measure switching behavior: Spectral amplification factor: η = 4A 2 < e iωt M(t) > 2. (4) Here M(t) = 1 N N x i (t), <. >: time averaging. i=1 The bigger η, the better the switching behavior. 13 C. J. Tessone, C. R. Mirasso, R. Toral, J. D. Gunton, Diversity-induced resonance. Phys. Rev. Lett., 2006, 97(19): art. no Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 17 / 27Co

26 3.3 Parameters Chosen And Simulation Of SDE Parameters are chosen as follows, unless to be otherwise noted: α 1 = 2.5, α 2 = 5; d 1 = 1; d 2 = 1; γ 1 = 0.5; γ 2 = 0.5; n 1 = 4; n 2 = 4; d e = 3.0; β = 15; ε = 0.07, µ = 1, k = 10; A = 0.08; Ω = 2π/400.N = 100. The SDE system will be numerically integrated by Euler-Maruyama method 14 in Matlab. 14 D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 2001, 43(3): Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 18 / 27Co

27 1 Brief Overview 2 Mathematical Model 3 Main Results 3.1 Feature Comparison Between Colored Noise And White Noise On Synchronized Switching 3.2 Colored Noise Can Promote Protein Production 3.3 Robustness of The Synchronized Switching Behavior 3.4 Autocorrelation Time And Synchronized Switching 4 4. Summary Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 19 / 27Co

28 3.1 Feature comparison: colored noise versus white noise To see and compare the performance of Gaussian extrinsic white noise and colored noise on synchronized switching, we vary the noise strength D ext in the interval [0, 0.1], and intrinsic noise strength D int in the interval [0, 0.05]. ASE versus D ext and D int for white extrinsic noise (left) and colored extrinsic noise (right) QS: ASE under Gaussian white noise QS: ASE under colored extrinsic noise 1.4 White intrinsic noise strength D int White intrinsic noise strength D int White extrinsic noise strength D ext Colored extrinsic noise strength D ext Colored extrinsic noise is favorable for synchronization 年 4 月 27 日 19 / 27 Colored Noise Induced Synchronized Switching in the Genetic Toggle Switch Systems Co

29 noise 3.1 Feature comparison: colored noise versus white η versus D ext and D int. Left panel: white extrinsic noise; Right: Colored extrinsic noise. White intrinsic noise strength D int QS: η under Gaussian white noise White extrinsic noise strength D ext White intrinsic noise strength D int QS: η under colored noise Colored extrinsic noise strength D ext White extrinsic noise is favorable for switch. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 20 / 27Co

30 3.2 Colored Noise Can Promote Protein Production Bimodal distributions of molecular numbers 15 under these two cases and average molecular numbers <X>= <X>=928 Proportion Proportion Molecule numbers of protein LacI Molecule numbers of protein LacI Colored noise can promote protein production. 15 There are about 500 molecules in each 1µmol.L 1 for each E.coli cell.(j. Hasty, F. Isaacs, M. Dolnik, D. McMillen, J. J. Collins, Designer gene network: towards fundamental cellular control. Chaos, 2001, 11(1): ) Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 21 / 27Co

31 3.2 Colored Noise Can Promote Protein Production From [V. Shahrezaei, J. F. Ollivier, P. S. Swain, Colored extrinsic fluctuations and stochastic gene expression. Mol. Syst. Bio., 2008, 4: art. no. 196.], colored noise can: Alter mean protein numbers; Speed up typical network response times; Explain trends in high-throughput measurements of variation; If extrinsic fluctuations in two components of the network are correlated, they may combine constructively (amplifying each other) or destructively (attenuating each other). Our conclusion:colored noise can promote protein production Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 22 / 27Co

32 η η 3.3 Robustness Synchronized Switching To further investigate the robustness of the synchronized switching behavior with respect to parameters, we fix intrinsic noise strength D int at 0.005, and then observe the curves η versus D ext under different extracellular periodical stimuli strengths A, extracellular diffusion rates Q, and intracellular diffusion rates k. k=10,q=0.5,d int =0.005,τ=10 k=10,q=0.5,d int =0.005,τ= A=0.03 A=0.05 A=0.08 A=0.1 A= A=0.04 A=0.05 A=0.08 A=0.1 A= White extrinsic noise strength D ext Colored extrinsic noise strength D ext Robustness with respect to A Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 23 / 27Co

33 3.3 Robustness Synchronized Switching Robustness with respect to Q and k. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 24 / 27Co

34 η 3.4 About The Autocorrelation Time White noise has zero autocorrelation time and colored noise has a nonzero autocorrelation time. k=10,a=0.08,q=0.5,d int =0.005 k=10,a=0.08,q=0.5,d int = τ=2 τ=5 τ=10 τ=15 τ= τ=2 τ=5 τ=10 τ=15 τ= ASE Colored extrinsic noise strength D ext Colored extrinsic noise strength D ext Curves of η, ASE versus D ext for different noise autocorrelation time τ, where D int = 0.005, k = 10, Q = 0.5, A = The longer τ is, the worse the switch behavior. Autocorrelation time indeed contributes to the performance differences between white noise and colored noise. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 25 / 27Co

35 1 Brief Overview 2 Mathematical Model 3 Main Results 4 4. Summary Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 26 / 27Co

36 4. Summary Colored extrinsic noise can also induce synchronized switching; And there also exists an optimal strength to induce best switch behavior; Colored extrinsic noise is favorable for synchronization, while white noise is favorable for switch. Colored noise can promote protein production. Synchronizing switching of cell population is robust to parameter perturbations. Autocorrelation time contributes to the performance differences between white noise and colored noise. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 26 / 27Co

37 4. Summary Colored extrinsic noise can also induce synchronized switching; And there also exists an optimal strength to induce best switch behavior; Colored extrinsic noise is favorable for synchronization, while white noise is favorable for switch. Colored noise can promote protein production. Synchronizing switching of cell population is robust to parameter perturbations. Autocorrelation time contributes to the performance differences between white noise and colored noise. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 26 / 27Co

38 4. Summary Colored extrinsic noise can also induce synchronized switching; And there also exists an optimal strength to induce best switch behavior; Colored extrinsic noise is favorable for synchronization, while white noise is favorable for switch. Colored noise can promote protein production. Synchronizing switching of cell population is robust to parameter perturbations. Autocorrelation time contributes to the performance differences between white noise and colored noise. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 26 / 27Co

39 4. Summary Colored extrinsic noise can also induce synchronized switching; And there also exists an optimal strength to induce best switch behavior; Colored extrinsic noise is favorable for synchronization, while white noise is favorable for switch. Colored noise can promote protein production. Synchronizing switching of cell population is robust to parameter perturbations. Autocorrelation time contributes to the performance differences between white noise and colored noise. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 26 / 27Co

40 4. Summary Colored extrinsic noise can also induce synchronized switching; And there also exists an optimal strength to induce best switch behavior; Colored extrinsic noise is favorable for synchronization, while white noise is favorable for switch. Colored noise can promote protein production. Synchronizing switching of cell population is robust to parameter perturbations. Autocorrelation time contributes to the performance differences between white noise and colored noise. Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 26 / 27Co

41 Thank you! Author: Wang Pei Address: School of Mathematics & Statistics Wuhan University Wuhan, , China Colored Noise Induced Synchronized Switching in the Genetic 2011 年 Toggle 4 月 27 Switch 日 Systems 27 / 27Co

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