Transcript Computing probabilistic landscape of stochastic network for

Computing probabilistic landscape
of stochastic network for
maintenance of epigenetic states of
cell and for regulation of
cellular fate
Youfang Cao, Hsiao-Mei Lu and Jie Liang
Department of Bioengineering
University of Illinois at Chicago
Oct. 21, 2011 Ninth Annual CBC Symposium
Phage lambda: How is cell fate determined?
Infected
by phage 
http://en.wikipedia.org/wiki/
Lambda_phage
Different life styles upon infection
St-Pierre and Endy, PNAS 2008
How does it work?
• Systems stability against perturbation
• Robustness against genetic mutation
• Regulation mechanism of the switch
• Heritable epigenetic state
• DNA damage due to UV
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
Discrete Chemical Master Equation (dCME):
A general framework for studying stochastic networks.
Stochastic nature of molecular
network and formulation
• Stochasticity:
– networks with small copy
numbers (eg. mM – nM) are
stochastic:
• transcription regulation, protein
synthesis, signal transduction.
– Slow reactions: multistable
systems
– Burst in transcription and
translation
– Cell-cell variaton
– External environment
Definitions:
1. Molecular species: X={X1, …, Xn}.
2. Reactions: R1, …, Rm.
rk
'
c1k X 1  c2 k X 2    cnk X n 
c1' k X 1  c2' k X 2    cnk
Xn
3. Microstate: x(t)={x1(t), …, xn(t )}  ℕn
4. The reaction rate of reaction Rk:
 xl 
Ak ( xi , x j )  rk   
l 1  clk 
n
• Stochasticity important:
– Signal discrimination and
amplification
– Cell fate, cell-differentiation, stem
cells
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
•
Discrete Chemical master equation (dCME)
dp( x, t )
CME of biological networks:
  A( x, x' ) p( x' , t )  A( x' , x) p( x, t )
dt
– Matrix form of CME:
• in which
dP ( x, t )
 P( x, t )
dt
x'
  { A( xi ,x j )} and A( x j ,x j )  
• Full account of probabilities of jumps:
 A( x , x ),
x j , x j  xi
i
j
xi , x j  X
– Regardless whether copy numbers and jumps are small or large
– Full stochasticity
• Can generate trajectories with correct probabilities
– based on the network architecture and reaction rates with enumerated states
• Little applications
– Not feasible beyond very simple systems
• No analytical solution for non-trivial systems
• Direct numerical solution infeasible due to enormously large state space
• Approximations
–
–
–
–
Stochastic simulation (SSA)
Chemical Langevin Equation (CLE, SDE)
Fokker-Planck Equation (FPE)
Finite State Projection (FSP)
Challenge: Requires full description of the discrete state space! 4
Optimal method for enumeration of accessible states
• Optimal enumeration of state
space
– Assuming finite number of net
molecules synthesized
– Starting from a given initial
condition
• Works for networks of
reactions with arbitrary
stoichiometry
• Optimal in memory
requirement and time
complexity
– All states reachable from
an initial condition will be
accounted for
– No irrelevant states are
included
– All possible transitions
will be recorded
– No infeasible transitions
will be attempted
(Cao and Liang, BMC Systems Biology, 2008, 2:30)
Our model for control of epigenetic state of phage lambda
• 13 molecular species
• 54 reactions
• ~1.7M microstates
• Cooperativities
– CI2, Cro2
– Neighboring sites
• Implicit OR-OL looping
effect
− Stabilized CI2 binding
to OR2, with 10 times
higher CI synthesis
rate with CI2 bound
OR2
‒ PRM suppressed
when CI2 bound to
OR3
•
Only possible with
looping
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
6
Steady State Probability landscapes
• Lysogenic induction ( phage induction):
-- Switching from lysogeny to lytic development
• Different UV dosage: varying CI degradation rate
– Projection of 13D landscape to CI2-Cro2 subspace
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
7
Titration Curve: A Mechanistic Picture
• CI2 and Cro2 level:
– Integrate CI2 and Cro2 over
landscape
• Wild type: Deep threshold
– Stable to UV irradiation
• CI degradation rate can
fluctuate over a wide region
• CI level changes little and Cro
suppressed
– Efficient switch over a narrow
region:
• Ultransensitivity for true signal
after set point
• Maintenance of epigenetic
state
– Same lysogeny upon cell growth
and cell division
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
8
Mechanism: Recruitment and Cooperativity
• Cooperativity between OR1 and OR2
– Key to maintenance of epigenetics and phenotype
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
Effects of Altered Operators
• Wild Type
• Mutants:
–
–
–
–
1-2-1
3’-2-3’
3-2-3
1-2-3
(Little et al, 2003, EMBO J)
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
Titration Curves: A Mechanistic Picture of Little Study
• All mutants have lower threshold in lysogenic-lytic state
transition
– And also leaky: switching not efficient
• Mutants can be induced to lytic state with lower UV dosage
– Consistent with Little’s “hair trigger” mechanism
• Some cannot lysogenize
– 1-2-3
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
11
• Gillespie algorithm
direct method: StochKit
Comparisons with SSA
– Same model
parameters, with CI kd
= 0.002/s
– Three different initial
conditions
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
• Time:
– dCME: 8 hours, 2GHz
quad core, AMD
– SSA: not yet
converged after 48
hours
• Wrong conclusion
dCME solution
12
Comparisons with Langevin SDE Method
• Langevin SDE
– Same model parameters, with
CI kd = 0.0021/s
– Three different initial
conditions
• Time:
– dCME: 8 hours,
– Langevin SDE: not yet
converged after 48 hours
• Wrong conclusion
– Transition phase vs lytic state
• It is unclear how much
improvement can be achieved
if different formulation of SDE
and different stochastic
parameters are chosen.
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
13
Comparison with ODE
•
•
•
ODE model
Cao, Lu and Liang, PNAS 2010, 107(43):18445-18450
– Based on Santillan and Mackey (Biophy J. 2004, 86, 75–84) but without OL for direct
comparison
No probability landscape description possible
Major differences
– WT cannot maintain stable CI level when UV increases
– Mutant 1-2-1 would behave similarly to WT
– Mutants 3’-2-3’ and 1-2-1 can tolerate higher UV then wt
• Transition point from lysogeny to lysis is higher for 3’-2-3’ and 1-2-1
14
Conclusions
• dCME method can compute exact probability landscape
solution
– Can be used as a general tool to study complex biological systems:
• Bacteria, eukaryotic epigenetic state
• Cell differentiation, stem cell
– Can be used as an exact tools to test theoretical constructs, model
development, and new methodology
• Mechanistic understanding of maintenance of epigenetic state
in phage lambda
– Reproduce Little mutant studies
– Generate testable predictions
– Help to understand general mechanism for maintaining epigenetic
state and for acquiring desirable phenotypes
– Evolutionary studies of emergency of modern decision network
15
Acknowledgements
•
•
•
NSF
NIH
SCSB, Shanghai Jiao Tong University
Selected References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
Cao Y, Liang J (2008) Optimal enumeration of state space of finitely buffered stochastic molecular networks and exact computation of
steady state landscape probability. BMC Syst Biol 2:30.
Shea MA, Ackers GK (1985) The OR control system of bacteriophage lambda a physical-chemical model for gene regulation. J Mol Biol
181:211–230.
Arkin A, Ross J, McAdams HH (1998) Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected
Escherichia coli cells. Genetics 149:1633–1648.
Aurell E, Brown S, Johanson J, Sneppen K (2002) Stability puzzles in phage λ. Phys Rev E 65:051914.
Ptashne M (2004) A Genetic Switch: Phage Lambda Revisited (Cold Spring Harbor Laboratory Press, Plainview, NY), 3rd ed, pp 1–66.
Ptashne M, et al. (1976) Autoregulation and function of a repressor in bacteriophage lambda. Science 194:156–161.
Little JW, Shepley DP, Wert DW (1999) Robustness of a gene regulatory circuit. EMBO J 18:4299–4307.
Anderson LM, Yang H (2008) DNA looping can enhance lysogenic CI transcription in phage lambda. Proc Natl Acad Sci USA 105:5827–
5832.
Zhu XM, Yin L, Hood L, Ao P (2004) Robustness, stability and efficiency of phage lambda genetic switch: Dynamical structure analysis. J
Bioinform Comput Biol 2:785–817.
Van Kampen NG (2007) Stochastic Processes in Physics and Chemistry (North–Holland, Amsterdam), 3rd Ed, pp 96–133, pp 193–200.
Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361.
Li H, Cao Y, Petzold LR, Gillespie DT (2008) Algorithms and software for stochastic simulation of biochemical reacting systems.
Biotechnol Progr 24:56–61.
Koblan KS, Ackers GK (1992) Site-specific enthalpic regulation of DNA transcription at bacteriphage λ OR. Biochemistry 31:57–65.
Darling PJ, Holt JM, Ackers GK (2000) Coupled energetics of λ cro repressor self-assembly and site-specific DNA operator binding II:
Cooperative interactions of cro dimers. J Mol Biol 302:625–638.
Ackers GK, Johnson AD, Shea MA (1982) Quantitative model for gene regulation by lambda phage repressor. Proc Natl Acad Sci USA
79:1129–1133.
Santillán M, Mackey MC (2004) Why the lysogenic state of phage lambda is so stable: A mathematical modeling approach. Biophys J
86:75–84.
Gillespie DT (2000) The chemical Langevin equation. J Chem Phys 113:297–306.
Dodd IB, Micheelsen MA, Sneppen K, Thon G (2007) Theoretical analysis of epigenetic cell memory by nucleosome modification. Cell
129:813–822.
Meyer BJ, Ptashne M (1980) Gene regulation at the right operator (OR) of bacteriophage lambda. III lambda Repressor directly
activates gene transcription. J Mol Biol 139:195–205.
17