Isolating intrinsic noise sources in a stochastic genetic switch

The stochastic mutual repressor model is analyzed using perturbation methods. This simple model of a gene circuit consists of two genes and three promotor states. Either of the two protein products can dimerize, forming a repressor molecule that binds to the promotor of the other gene. When the repressor is bound to a promotor, the corresponding gene is not transcribed and no protein is produced. Either one of the promotors can be repressed at any given time or both can be unrepressed, leaving three possible promotor states. This model is analyzed in its bistable regime in which the deterministic limit exhibits two stable fixed points and an unstable saddle, and the case of small noise is considered. On small timescales, the stochastic process fluctuates near one of the stable fixed points, and on large timescales, a metastable transition can occur, where fluctuations drive the system past the unstable saddle to the other stable fixed point. To explore how different intrinsic noise sources affect these transitions, fluctuations in protein production and degradation are eliminated, leaving fluctuations in the promotor state as the only source of noise in the system. The process without protein noise is then compared to the process with weak protein noise using perturbation methods and Monte Carlo simulations. It is found that some significant differences in the random process emerge when the intrinsic noise source is removed.

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1. Isolating Intrinsic Noise Sources in a Stochastic Genetic Switch Jay Newby Oxford Center for Collaborative Applied Mathematics (OCCAM) University of Oxford
2. Stochastic gene expression a b c d e mRNA ‘On’ ‘Off’ Protein Target mRNA 0 20 40 0 50 100 150 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 10 20 Bursting Time-averaging Propagation Time (cell cycles) X X Translation Target gene activation Degradation Promoter state: ‘on’ ‘off’ Figure 1 | Gene expression noise is ubiquitous, and affects diverse systems at several levels. a, E. coli expressing two identical promoters driving two Gene expression Crz1P Crz1 Calcineurin Ca2+ A a b Target gene respo Nuclear Crz1 Figure 2 | Frequen enables coordinat calcineurin, which Crz1 (Crz1P) tran where it activates b, Response curve as a function of n vary in the effecti c, d, Regulation o frequency-modula levels of calcium l true at high levels genes A and B, no expression profile between genes. d, average) the same Eldar and Elowitz. Nature, 2010. Jay Newby, Oxford
3. 988 R. ERBAN, S. J. CHAPMAN, I. G. KEVREK (a) 0 100 200 300 400 500 0 50 100 150 200 250 300 350 400 time number of X molecules k 1d =12 stochastic deterministic (b) 200 400 600 800 1000 1200 1400 number of Y molecules Fig. 2.2. (a) The time evolution of X, given by the s line) models of the chemical system (2.1)–(2.2). The rat (b) The same trajectory plotted in the X-Y plane. (a) 400 (b) Metastable behavior Can’t be described by deterministic model focus on bistable switching Erban et. al., 2009 Jay Newby, Oxford
4. 990 R. ERBAN, S. J. CHAPMAN, I. G. KEVREKI (a) 0 20 40 60 80 100 0 100 200 300 400 500 time number of molecules (b) 0 0 1 2 3 4 5 6 7 8 x stationary distribution Fig. 3.1. (a) Time evolution of X obtained by the Gil The values of the rate constants are given by (3.5). (b) by the Gillespie SSA (yellow histogram) for the parameter solution (3.11) of the chemical Fokker–Planck equation. the concentration x = X/V : Metastable behavior Can’t be described by deterministic model focus on bistable switching Erban et. al., 2009 Jay Newby, Oxford
5. Mutual repressors Two genes, X and Y Jay Newby, Oxford
6. Mutual repressors Either gene can be repressed (turned off) Jay Newby, Oxford
7. Mutual repressors ... by a dimer of the other gene’s protein product Jay Newby, Oxford
8. Mutual repressors ... by a dimer of the other gene’s protein product Jay Newby, Oxford = 1 x, y = # of X,Y proteins ↵ Let
9. Mutual repressors Genes switch on and off randomly Jay Newby, Oxford XY b/✏ ! x 2 /✏ XY y 2 /✏ ! b/✏ XY XY Y X XY
10. Deterministic limit Jay Newby, Oxford x y x + y ˙ x = f ( x, y ) ˙ y = f ( y, x ) f ( x, y ) = b + x 2 b + x 2 + y 2 x
11. Deterministic limit stable fixed point Jay Newby, Oxford x y x + y ˙ x = f ( x, y ) ˙ y = f ( y, x ) f ( x, y ) = b + x 2 b + x 2 + y 2 x
12. Deterministic limit stable fixed point saddle Jay Newby, Oxford x y x + y ˙ x = f ( x, y ) ˙ y = f ( y, x ) f ( x, y ) = b + x 2 b + x 2 + y 2 x
13. Deterministic limit absorbing BC stable fixed point saddle x y x + y transition rate is the mean exit time to the absorbing boundary
14. Metastable behavior Goals: stationary density transition times Jay Newby, Oxford
15. Metastable behavior Goals: stationary density transition times Jay Newby, Oxford double-well potential
16. Metastable behavior Goals: stationary density transition times Jay Newby, Oxford double-well potential
17. Perturbation theory can answer these questions... for diffusion process or certain birth/death processes see Ludwig, Schuss & Matkowsky, Hanggi, Talkner, Dykman, Chapman, Ward and many others Jay Newby, Oxford Ps ⇠ N exp[ 1 ✏ (x)] T / exp[ 1 ✏ ( (xmax) (xmin))]
18. Perturbation theory can answer these questions... for diffusion process or certain birth/death processes see Ludwig, Schuss & Matkowsky, Hanggi, Talkner, Dykman, Chapman, Ward and many others Jay Newby, Oxford Ps ⇠ N exp[ 1 ✏ (x)] potential function T / exp[ 1 ✏ ( (xmax) (xmin))]
19. Perturbation theory can answer these questions... for diffusion process or certain birth/death processes see Ludwig, Schuss & Matkowsky, Hanggi, Talkner, Dykman, Chapman, Ward and many others Jay Newby, Oxford well height T / exp[ 1 ✏ ( (xmax) (xmin))] Ps(x) ⇠ N exp[ 1 ✏ (x)]
20. What about gene expression? Model has an “internal state” representing promotor switching QSA doesn’t apply quasi-steady state (adiabatic) approx. is common solution Jay Newby, Oxford Ps(n, x) ⇠ p(n | x) ✓ N exp[ 1 ✏ (x)] ◆
21. What about gene expression? Model has an “internal state” representing promotor switching QSA doesn’t apply quasi-steady state (adiabatic) approx. is common solution Jay Newby, Oxford Ps(n, x) ⇠ p(n | x) ✓ N exp[ 1 ✏ (x)] ◆
22. Two limits yield model reduction Adiabatic limit (remove promotor noise) Quasi-deterministic (QD) limit (remove protein noise) Jay Newby, Oxford ✏ ! 0 ↵ ! 1
23. Two limits yield model reduction Adiabatic limit (remove promotor noise) Quasi-deterministic (QD) limit (remove protein noise) Jay Newby, Oxford ✏ ! 0 ↵ ! 1
24. Two limits yield model reduction Adiabatic limit (remove promotor noise) Quasi-deterministic (QD) limit (remove protein noise) define ratio of noise sources Jay Newby, Oxford ✏ ! 0 ↵ ! 1 ' = (1/↵) ✏
25. Two limits yield model reduction QD limit large protein population (system-size limit) Jay Newby, Oxford
26. Two limits yield model reduction QD limit large protein population (system-size limit) Jay Newby, Oxford
27. Two limits yield model reduction QD limit large protein population (system-size limit) Jay Newby, Oxford
28. No protein noise Discrete Markov process for the promotor state Protein concentration is deterministic between promotor switch Jay Newby, Oxford XY b/✏ ! x 2 /✏ XY y 2 /✏ ! b/✏ XY XY Y X XY
29. No protein noise SDE representation Jay Newby, Oxford X(t) = X(0) + Z t 0 (S x (t0) X(t0))dt0 Y (t) = Y (0) + Z t 0 (S y (t0) Y (t0))dt0 S x (t) = S x (0) + 1 ✏  0(b Z t 0 (1 S x (t0))dt0 1( Z t 0 S x (t0)Y (t0)2dt0) S y (t) = S y (0) + 1 ✏  0(b Z t 0 (1 S y (t0))dt0 1( Z t 0 S y (t0)X(t0)2dt0) j, j unit Poisson processes
30. Analysis @ p @t = @ @x [⌃ x ( x, y )p] @ @y [⌃ y ( x, y )p] + 1 ✏ A ( x, y )p p( x, y, t ) = 2 4p0( x, y, t ) p1( x, y, t ) p2( x, y, t ) 3 5 Probability density ...satisfies differential Kolmogorov equation Jay Newby, Oxford
31. Analysis L✏ ⌘ @ @x [⌃ x ( x, y ) ] + @ @y [⌃ y ( x, y ) ] 1 ✏ A ( x, y ) = p( x, y, t ) = 1 X j=0 cj j( x, y ) e j t Jay Newby, Oxford represent solution with eigenfunction expansion for eigenfunctions/eigenvalues of operator
32. Quasi-stationary approximation If BC is reflecting Approximate solution If BC is absorbing x y p( x, y, t ) ⇠ 0( x, y ) e 0t 0 = O(e C/✏), C > 0 0 ⌧ j, j = 1, 2, · · · 0 = 0 boundary x y
33. Quasi-stationary approximation WKB anzatz: assume the solution has the form Eigenvalue is Escape time is exponential RV with mean 0(x, y) = r (x, y) exp  1 ✏ (x, y) 0 / exp  1 ✏ (x⇤, y⇤) T ⇠ 1 0
34. Results Orange = with protein noise Blue = no protein noise Potential ' ! 0, no protein noise ( x, y )
35. Results Orange = with protein noise Blue = no protein noise ' ! 0, no protein noise
36. Results Orange = with protein noise Blue = no protein noise Potential ' ! 0, no protein noise ( x, y )
37. Results Orange = with protein noise Blue = no protein noise Potential ' ! 0, no protein noise ( x, y )
38. Results T / exp  1 ✏ (x⇤, y⇤) currently working on exit time approximation for general case Jay Newby, Oxford
39. Thanks Jay Newby, Oxford Jim Keener (U. of Utah) Jon Chapman (Oxford)
40. References JMN. Isolating intrinsic noise sources in a stochastic genetic switch. Physical Biol., 9(2):026002, 2012 JMN and J. P. Keener. An asymptotic analysis of the spatially inhomogeneous velocity-jump process. Multiscale Model. Simul., 9(2): 735–765, 2011. J. P. Keener and JMN. Perturbation analysis of spontaneous action potential initiation by stochastic ion channels. Phys. Rev. E, 84(1): 011918, 2011. Jay Newby, Oxford