Modeling and Systems Analysis of Gene Regulatory Networks

Transcription

1 Modeling and Systems Analysis of Gene Regulatory Networks Mustafa Khammash Center for Control Dynamical-Systems and Computations University of California, Santa Barbara

2 Outline Deterministic A case study: the bacterial heat shock response A systems analysis of the regulation strategy What s behind the complexity? Robustness, performance, and optimality Reconciling known components with experimental data Stochastic Why are stochastic models needed? A case study: the Pap pili stochastic switch The stochastic formulation of chemical kinetics Gillespie s SSA A New Approach: the Finite State Projection

3 Loss of Protein Function Network failure Cell Unfolded Proteins Aggregates Death Folded Proteins Temp cell Temp environ

4 Function of Heat-Shock Proteins I. Protein Folding DnaK GroEL/ GroES Unfolded/partially folded Proteins II. Protein Degradation Proteases Proteins Aggregates Amino Acids Folded Proteins

5 Synthesis of Heat Shock Proteins RNA Polymerase + 32 factor start DNA end hsp1 hsp2 promoter terminator Heat-Shock Proteins mrna ribosomes

6 Regulation of HSP Tight regulation of 32 at 3 levels Synthesis Activity Stability Feedforward Feedback

7 I. Regulation of HSP Synthesis At low temperature, mrna has a secondary structure 32 mrna Translation Heat mrna

8 II. Regulation of 32 Activity: A Feedback Mechanism RNAP hsp1 hsp2 Transcription & Translation Chaperones Heat 32

9 III. Regulation of 32 Degradation FtsH degrades sigma-32 only when bound to cheperones RNAP hsp1 hsp2 Proteases FtsH Chaperones Transcription & Translation Heat FtsH FtsH FtsH 32

10 Regulation of E. coli s Heat-Shock Response Heat rpoh gene Transcription Degradation - Activity - 32 Translation Feedforward hsp1 hsp2 Heat Feedback DnaK GroEL GroES HslVU FtsH Transcription & Translation Chaperones Proteases

11 Protein Synthesis Mathematical Model

12 Binding Equations Mass Balance Equations

13 Full Model Simulations SIGMA32 TOTAL TIME 30 o 42 o DNAK TOTAL TIME 30 o 42 o E SIGMA32:RNAP FOLDED PROTEINS 1.8E E E TIME 1.2E TIME

14 Analysis of the HS System What are the advantages of the different control strategies? Disturbance feedforward Sequestration feedback loop Degradation feedback loop High synthesis/degradation rates of sigma-32 Fast response Robustness, efficiency Fast response, noise suppression Fast response Tools Dynamic analysis and simulations Sensitivity/robustness analysis Sum-of-Squares tools Optimal control

15 Effect of Sequestration Feedback Loop 10 1 Sensitivity of DnaK to model parameters 10 0 Sensitivity of DnaK K-s32/dnaK K-s32:dnaK/D transcription rate No feedback loop Wild type Time (min) 30 o 42 o

16 Role of the degradation feedback loop (cont.) Wild type Constitutive degradation DNAK Time

17 Total DNAK E+ WildType No Feedforward Low S32 Flux Constitutive 32 Deg No DnaK interaction 6 4 Free 32 Unfolded Time Time

18 Is the WT Response optimal? Observation: A hypothetical heat-shock response system maybe devised to achieve a very small number of unfolded proteins minimal complexity (no feedback necessary) E.g. over-expressing chaperones & eliminating feedback However chaperone over-expression involves a high metabolic cost is toxic to the cell The existing design appears to achieve a tradeoff: good folding achieved with minimal number of chaperones How optimal is the WT design?

19 Cost function: For a given >0, solve: J The optimal solution yields a point in R 2 : p opt t 1 2 ( ) = [ chaperones] dt + t 0 t 1 t1 opt = 2 ( ) : [ chaperones ] dt, [ P t0 t0 { p ( ) : > 0} min J ( ) st.. x& = F( x, y, ) 0 = Gxy (,, ) opt The curve gives the set of Pareto optimal designs t 1 t 0 opt un [ P ] 2 un ] 2 dt dt opt { p ( ) : > 0} t 1 t0 [ 2 P un ] dt Non-optimal unachievable t 2 [ chaperones] dt 1 t 0

20 Pareto Optimal Design of the Heat Shock System 100 Pareto Optimal curve t 1 t 0 [ 2 P un ] dt Cost of unfolded proteins various nonoptimal values of parameters 20 0 Wild type heat shock Cost of chaperones t 1 t 0 [ chaperones] 2 dt

21 9600 Constitutive Degradation of S32 Deterministic Wild Type S32 Chaperone level Wild Type No Feedforward Low Flux Constitutive degradation of S32 Time The architecture of Sensitivity the heat of DnaK shock to model system parameters is a necessary Sensitivity of DnaK No DnaK 10-1 interaction Time 10 1 outcome of robustness and performance requirements Level of unfolded proteins (Uf) 1E Time (min) Wild type No DnaK interaction No feedforward Low S32 flux Constitutive degradation of S Time

22 Molecular Biology and Phenotype A dynamic model is to be viewed as a tool for capturing biological knowledge about known components and their frequently complicated interactions. Question: Are known components and their interactions consistent with experimental observation? Two scenarios: Yes. A dynamic model capturing known players and their interactions can robustly reproduce the data. No. A proof is generated showing that with the existing components it is impossible to reproduce the data.

23 Level of sigma Induction phase Case Study: Origin of Adaptation Phase Adaptation phase Time i A hypothesis If induction phase is caused by increased sigma32 translation, then surely the adptation phase is caused by a shut off in translation. Several experiments to search for hypothesized shut-off mechanism But Dynamic simulations of the model based on known components always exhibit an adaptation phase. No need for a shut off mechanism! Feedback through the FtsH loop is responsible for such adaptation Translation of sigma32 does not shut off at higher temperatures. Dynamic interplay between antagonistic pathways controlling the s32 level in E. coli, Morita et. al, PNAS, 2000.

24 Missing Players? Question: Is a model without the FtsH feedback loop consistent with the data?

25 Model Invalidation Given: Proposed state-space model with constraints on parameters Measurement data Provide (if possible) a proof that the measurements are inconsistent with the model.

26 Model Invalidation using SOStools model measurements

27 Deficient model (no FtsH feedback) Measured data with uncertainty A barrier function B(x,p,t) was constructed using SOStools A model w/o FtsH cannot possibly produce measured data

28 Stochastic Modeling In the HS system, the total number of sigma-32 molecules per cell is very small (~30 per cell) Does it make sense to treat these quantities as concentrations? Deterministic models describe macroscopic behavior, but there are many examples when deterministic models are not adequate

29 Outline Why are stochastic models needed? Randomness and variability Motivating examples Stochastic Chemical Kinetics The Chemical Master Equation The Linear Noise Approximation Monte Carlo methods: Gillespie s Stochastic Simulation Algorithm A New Approach: The Finite State Projection Method (FSP) A case study: the Pap pili stochastic switch

30 Why Are Stochastic Models Needed? Much of the mathematical modeling of gene networks represents gene expression deterministically Considerable experimental evidence indicates that significant stochastic fluctuations are present

31 Deterministic vs. Stochastic Deterministic model Protein mrna Stochastic model DNA...

32 Fluctuations at Small Copy Numbers Protein (protein) mrna (mrna) DNA Mean equals deterministic steady-state Coefficient of variation of grows as 1/sqrt(mean) For large mean, fluctuations are small (relative to the mean)

33 Intrinsic Variability in Gene Expression Protein mrna Source of variability at cellular level. Small # of molecules Random events DNA DNA DNA Cell 1 Cell 2 Cell 3 Or T=1 T=2 T=3 Intrinsic noise Noise propagate through the network Cannot be captured by deterministic models

34 Deterministic Models May Not Capture the Stochastic Mean Number of Molecues of P stochastic deterministic Johan Paulsson, Otto G. Berg, and Måns Ehrenberg, PNAS 2000 Time Stochastic mean value different from deterministic steady state Noise enhances signal!

35 Noise induced oscillations Circadian rhythm 2500 deterministic stochastic A R 2000 C 1500 A + R R 1000 A a A A ' a A A a R A ' a R D A D R 500 Vilar, Kueh, Barkai, Leibler, PNAS Time (hours) Oscillations disappear from deterministic model after a small reduction in deg. of repressor (Coherence resonance) Regularity of noise induced oscillations can be manipulated by tuning the level of noise [El-Samad, Khammash]

36 Stochastic Switches The Lysis-lysogony switch CI 60 Equilibrium Points Time

37 Stochastic Model: Pap Pili Phase Variation System Pili are hair-like structures on a bacterium Pili enable uropathogenic E. coli to attach to the upper urinary tract; plays an essential role in the pathogenesis of UTIs Pili expression switches ON or OFF such that bacterial cells are either piliated or unpiliated Piliation is controlled by a stochastic switch that involves molecular events

38 A Simplified Pap Switch Model local regulatory protein Lrp Site 1 Site 2 One gene g Lrp can (un)bind either or both of two binding sites A (un)binding reaction is a random event

39 Lrp State g 1 R 1 R 2 State g 2

40 Lrp State g 1 R 1 R 2 R 3 R 4 State g 2 State g 3

41 Lrp State g 1 R 1 R 2 R 3 R 4 State g 2 State g 3 R 5 R 6 R 7 R 8 State g 4

42 Lrp State g 1 PapI R 1 R 2 R 3 R 4 State g 2 State g 3 R 5 R 6 R 7 R 8 State g 4

43 Lrp OFF State g 1 PapI R 1 R 2 R 3 R 4 State g 2 ON R 5 R 6 R 7 R 8 State g 3 OFF State g 4 OFF

44 Lrp OFF State g 1 PapI R 1 R 2 R 3 R 4 State g 2 ON R 5 R 6 R 7 R 8 State g 3 OFF Piliation takes place if gene is ON at a specific time: T State g 4 OFF

45 Identical genotype results in different phenotype gene gene. gene gene Piliated Unpiliated

46 Lrp PapI R 1 R 2 R 3 R 4 OFF What is the probability of being in State g 2 at time T? State g 1 State g 2 ON Piliation takes place if gene is ON at a specific time: T R 5 R 6 R 7 R 8 State g 4 OFF State g 3 OFF

47 The Stochastic Formulation of Chemical Kinetics

48 The Stochastic Formulation of Chemical Kinetics

49 The Chemical Master Equation The Chemical Master Equation (CME):

50 The Stochastic Simulation Algorithm (SSA)

51 The Stochastic Simulation Algorithm (SSA) P x (,μ) μ M d

52 # of specie 2 An SSA run # of specie 1

53 A New Approach: The Finite State Projection (FSP) Algorithm

54 The Chemical Master Equation (CME): The states of the chemical system can be enumerated: The probability density state vector is defined by: The evolution of the probability density state vector is governed by

55 The probability density vector evolves on an infinite integer lattice

56 A finite subset is appropriately chosen

57 A finite subset is appropriately chosen The remaining (infinite) states are projected onto a single state (red).

58 The projected probability density state vector evolves on a finite set. The projected system can be solved exactly!

59 Finite Projection Bounds

60 The Finite State Projection (FSP) Algorithm

61 FSP Applied to the Pap System Lrp PapI State g 2 R 1 R2 R 3 R 4 State g 1 State g 3 R 5 R 6 R 7 R 8 State g 4 10 reactions An infinite number of possible states:

62 Parameters and initial conditions

63 Results and Comparisons Unlike Monte-Carlo methods (such as the SSA), the FSP directly approximates the solution of the CME.

64 FSP Results (cont.) PapI Probability Density Probability Density Guaranteed error bound: 10-6 PapI Concentration

65 FSP Results (cont.) Joint Probability Density g1 g2 Probability Density g3 g4 tf=10s PapI concentration Guaranteed error bound: 10-6

66 The Systems Paradigm Stochastic Deterministic Master Equation Langevin Fokker- Planck Continuous Boolean Hybrid Simplified & Toy Models Structure Model Reduction Detailed Mechanistic Models Experiments Validation Sensitivity Analysis Monte Carlo Simulation (SSA, ) Simulation/ Analysis Tools Linear Noise Approximation Stochastic Stability Moment Closure method New! Finite State Projection (FSP) Algorithm Reverse Engineering / Design Principles Noise Exploitation Noise Attenuation Performance Efficiency & Optimality Robustness

67 Stochastic Stiffness in the Heat Shock Response Fast and Frequent reactions 70 RNAP rpoh free: 0-1 mol./cell mrna total: mol./cell DnaK total: mol./cell folded Heat unfolded DNAK Fast, rare and important reaction RNAP RNAP ftsh DNAK Lon aggregate degradation Slow reactions DnaKFtsH Lon

68 Conclusions Systems Biology is concerned with the quantitative analysis of networks of dynamically interacting biological components, with the goal of reverse engineering these networks to understand how they robustly achieve biological function. Control strategies abound at all levels of organization Systems notions are required for a deeper understanding of biological function. Many similarities with engineering systems.

69 Conclusions Low copy numbers of important cellular components give rise to stochasticity in gene expression. This in turn may result in cell to cell variations. Organisms use stochasticity to their advantage. Stochastic modeling and computation is an emerging area of research in Systems Biology. New tools are being developed, e.g. FSP, SSA. More are needed. Many challenges and opportunities for control and system theorists!

70 Acknowledgments Heat Shock: Hana El-Samad (UCSF), Hiro Kurata (Kyushu), John Doyle (Caltech), Carol Gross (UCSF) Finite State Projection: Brian Munsky (UCSB) Pap Switch: Brian Munsky (UCSB), David Low (UCSB), Aaron Hernday(UCSB)