BioControl - Week 2, Lecture 1

Transcription

1 BioControl - Week 2, Lecture Goals of this lecture: CDS tools for bio-molecular systems Study of equilibria Dynamic performance Periodic behaviors Main eample: Negative autoregulation Suggested readings Angeli D, Ferrell JE, Sontag ED (2004) Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. Proc Natl Acad Sci USA 0: Rosenfeld, N., M.B. Elowitz, and U. Alon, Negative autoregulation speeds the response times of transcription networks. J Mol Biol, (5): p Mathematical techniques for the study of oscillations in biochemical control loops. Rapp PE - Bull. Inst. Math. Appl, 976 Elisa Franco, Caltech

2 Macroscopic models: ordinary differential equations Summary of last lecture: Modeling: Tools chosen depending on the time/space scale ODE models: Molecules in high copy numbers, negligible stochasticity Hill functions: Ligand/receptor dynamics Michaelis-Menten kinetics: Enzyme catalysis kinetics Modeling transcriptional regulation: RNA Transcription + Translation, low pass filters with Hill-type nonlinearity The trp operon Elisa Franco, Caltech 2

3 Overview of CDS tools for analysis of ODEs: Equilibria Given a system: ẋ = f(, u) R n,u R m Note: biological systems are, in general, nonlinear Equilibria: Find all such that f(, ū) =0, ū = const In a neighborhood of each equilibrium, we can approimate the nonlinear system with its linearized dynamics. f f f Jacobian: u n f f δẋ = Df (, ū)δ + Df u (, ū)δu 2 f 2 Df u (, ū) = u... Df (, ū) =... 2 n.....,.... f n f n... u... The eigenvalues of the Jacobian tell us what is the local nature of the equilibrium. Phase portrait for typical situations in 2D systems: sink y source y hyperbolic y f n n center y f u m f 2 u m. f n u m Real e-values: Re<0 Real e-values: Re>0 Real e-values: Re>0, Re2<0 Pure Imaginary e-values Elisa Franco, Caltech 3

4 Eamples, transcriptional regulation 2 NEGATIVE AUTOREGULATION 40% of E coli genes ẋ = ẋ 2 =2 2 One real equilibrium point: =.5, 2 = 2 2 Jacobian: Df ( ) = (+ 2 2 )2 2 Eigenvalues: λ,2 =± i Solutions spiral towards the equilibrium point. Spiraling direction: sign of Jacobian elements 2 Figure obtained with Matlab pplane8 Toggle Switch 2 Ignoring RNA dynamics ẋ = ẋ 2 = Find: Equilibria Eigenvalues 2 Elisa Franco, Caltech 4

5 Equilibria: bifurcations Equilibria can change their nature as a function of a specific parameter Classical eample: Hopf bifurcation ẋ = 2 + (µ 2 2 2) ẋ 2 = + 2 (µ 2 2 2) r 2 = 2 ṙ = rµ r r =0 θ = atan( y θ = ) r =0 r = ± µ Cases: µ<0 µ =0 µ>0 y y y Elisa Franco, Caltech 5

6 CDC 2 / Wee eample Angeli D, Ferrell JE, Sontag ED (2004) Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. Proc Natl Acad Sci USA 0: CDC 2 Wee Active Inactive 2 y y 2 Full model: active proteins inhibit each other with a kinase action. Mass conservation => model reduction 2 = y 2 = y Changing the feedback parameter v results in a bifurcation. On the choice of v: this is a coefficient that reflects the strength of the influence of Wee on Cdc2. Alternatively one could have picked / y NUMERICAL ANALYSIS: The system is bistable or monostable 2 3 The numerical analysis can be further investigated with the theory of monotone systems: Week 5 Elisa Franco, Caltech 6

7 Dynamic performance Linear systems, Laplace transforms and and transfer functions t ẋ = A + Bu General solution in the time domain? (t) =e At (0) + Be A(t τ) u(τ)dτ t 0 Laplace transforms: :O L d dt = sl{} ẋ = A + Bu sx = AX + BU 2. Convolution integrals become products in the Laplace domain :) Open loop: Closed loop: u r + G e y G u =(si A) B u Controller Actuators G Sensors C=C(s), A=, S= Open loop transfer function: L=C(s)G(s) H er = +L If actuators and sensors have linear dynamics, we can just factor them in! Performance analysis... H yr = L +L Elisa Franco, Caltech 7

8 Dynamic performance Linear systems, Laplace transforms and and transfer functions Eample: G u = 0 (s + )(s + 2) Comple function... Characterization: Bode plot Im Nyquist diagram real & imaginary parts magnitude & phase Mag (db) Re Phase Stable OLTF => Stable CLTF iff Nyquist does not encircle - How much we can push in gain and phase before we destabilize the system? Design specifications: Steady state error Maimum Overshoot Settling time Gain and phase margins... L = C(s) 0 (s + )(s + 2) Based on the Bode Plot of the Open Loop TF we can iteratively do loop shaping by designing the poles of the controller. H er = +L log 0 ω In some cases we can etend these notions to biological networks... H yr = L +L Elisa Franco, Caltech 8

9 Dynamic performance: negative autoregulation ẋ = ẋ 2 =2 2 =.5, 2 = Open Loop: ẋ = α 0 α ẋ = + u ẋ 2 = y = 0 OPEN LOOP TF: L = C (si A) B Let us close the loop: f(y) ẋ = 2 u = f(y) y = 0 For this illustrative eample we pick: u = ky Closed Loop TF: H yr = L +L *Austin, DW et al. Gene network shaping of inherent noise spectra. Nature 439, (2006) Mag (db) Slight high-frequency performance deterioration* Step response Open Loop, α=0.5 Closed Loop, k= Closed Loop, k=0 Time [s] log 0 ω Open Loop, α=0.5 Closed Loop, k= Closed Loop, k=0 Increasing k: Settling time decreases Steady state error decreases Elisa Franco, Caltech 9

10 Dynamic performance: eperiments on negative autoregulation (NAR) Rosenfeld, N., M.B. Elowitz, and U. Alon, Negative autoregulation speeds the response times of transcription networks. J Mol Biol, (5): p Hill coefficient = Analysis is done finding eplicit solution to the ODEs Rise time in absence of NAR is the cell cycle timescale Steady state of NAR is comparable with unregulated gene because of STRONG PROMOTER. Elisa Franco, Caltech 0

11 Periodic behavior: tools for nonlinear systems Poincare -Bendison theorem Planar systems cannot ehibit chaos! See: Perko L, Differential equations and dynamical systems, Springer, page 250. Given a 2-D invariant set for a vector field, for t the trajectories converge to an equilibrium, to a periodic orbit or to a finite number of steady states and periodic orbits. Mallet-Paret theorem (990) Etension of PB theory to systems of higher order Small gain theorem: If we can prove that a system converges to a fied point (contraction mapping), we can eclude the presence of periodic orbits. See Week 5, monotone systems Describing Functions: Theory developed in the (before Mallet-Paret was proved). Approimate technique, useful in plant engineering to predict limit cycles in the presence of saturating actuators, deadband or other piecewise discontinuous nonlinearities in the feedback loop. Elisa Franco, Caltech

12 Periodic behavior: Describing Functions Closed loop linear system + nonlinearity in the loop r + N. Approimate the I/O characteristic of the nonlinearity using Fourier series (t) =A cos(ωt) w(t) = N a n cos(nωt)+b n sin(nωt) 2. If G(iω) is a low pass filter, we can ignore high order harmonics w G(iω) n=0 3. Apply the harmonic balance technique: X(iω) = G(iω)W (iω) = G(iω)N(A, ω)x(iω) N(A, ω)g(iω) = Book Reference: Gelb, A., and W. E. Vander Velde. Multiple-Input Describing Functions and Nonlinear System Design. McGraw Hill, 968 Fully downloadable at: - N(A) G(iω) I R Elisa Franco, Caltech 2

13 Application: generalized NAR Goodwin oscillator: periodic behavior depends on n (Hill coeff) and m (num of integration steps) w ẋ = + n m δ N G(iω) ẋ 2 = α 2 δ ẋ m = α m m δ m m N = + n m G(iω) = Π m i=2 α i Π m i= (iω + δ i). Approimate the I/O characteristic of the nonlinearity using Fourier series a n = π b n = π 2π 0 2π 0 w(σ) cos(nσ)dσ w(σ)sin(nσ)dσ Static Memoryless a 0 = 2π a = π 2π 0 2π 0 +(A 0 + A cos(σ)) m dσ cos(σ) +(A 0 + A cos(σ)) m dσ 2. G(iω) is a low pass filter=> we can ignore high order harmonics Elisa Franco, Caltech 3

14 Application: generalized NAR Goodwin oscillator w ẋ = + n δ m N G(iω) ẋ 2 = α 2 δ N = + n m G(iω) = Π m i=2 α i Π m i= (iω + δ i) ẋ m = α m m δ m m 3. Apply the harmonic balance technique: Offset: Fundamental: We get approimated: Period Amplitude N 0 (A 0,A ) G(0) = N(A 0,A ) G(iω) = Stability? Analytical Graphical - N(A) S U S G(iω) I R Elisa Franco, Caltech 4

15 Application: generalized NAR For the NAR case, the DF is localized on the negative real semi-ais. To get intersections in Nyquist: Hill coefficient n> The more integration steps, the easier to find an intersection Effect of increasing m m= m=2 m=3 m=4 m=5 Eample: G(s) = 0 (s + 2) m Mathematical techniques for the study of oscillations in biochemical control loops. Rapp PE - Bull. Inst. Math. Appl, 976 Elisa Franco, Caltech 5

16 Application: generalized NAR NOTE: Delays are comparable to increasing m, increase number of intersections with negative semi-ais 0 G(s) = (s + 2) 2 e sτ Delays => rotations COMPARE: Improved cellular oscillator that relies on delays τ=.25 τ=.5 τ= τ=2 Stricker et al Add delay -Eliminate positive feedback loop Stochastic analysis of delay-induced oscillations: Delay-Induced Degrade-and-Fire Oscillations in Small Genetic Circuits. William Mather, Matthew Bennett, Jeff Hasty, Lev Tsimring Physical Review Letters, 02(6):06805(4) (2009) Elisa Franco, Caltech 6