Noisebased switches and amplifiers for gene expression

1
Noise-based switches and amplifiers for gene
expression

• Hasty et al, 2000
• Discussion led by Morgan Price
• MorganNPrice_at_yahoo.com

2
Context of this Paper

• Gene expression-based switches are known
• Signal to switch?
• External input causes a state change
• Model system Gardner et al 2000
• Random fluctuations
• New Change in noise level ? switching

3
Overview of paper

• Analyze effects of noise levels on a simple gene
  expression-based switch
• Average state changes by orders of magnitude
• Simple, theoretical noise model
• No simulation
• Relevance gene therapy?

4
Simplified lambda switch


OR1
OR2
OR3
CI

• Cooperative binding
• OR2 OR3 in binding strength both is 5 weaker
• Two states CI low and CI high
• determined by DNA levels and noise
• Mutant OR1 knocked out

5
Actual lambda circuit for comparison
from Arkin et al 1998
6
Chemical reactions for mutant

• CI CI ? CI2
• Promoter CI2 ? Promoter(2)
• Promoter CI2 ? Promoter(3)
• Promoter(2) CI2 ? Promoter(2,3)
• Promoter(3) CI2 ? Promoter(2,3)
• (left this one out, irrelevant)
• Promoter(2) RNAP ? Promoter(2) RNAP nCI
• CI ? amino acids
• X CI, D Promoter, Promoter(2)DX2,Promoter(3)
  DX2, Promoter(2,3) DX2X2

bindingreactions
7
Kinetic equations for the mutant

• Equilibrium binding assumptions
• Fixed total promoter concentration
• Promoter(2) Total Promoterx2 / (1 C1x2
  C2x4)
• dx/dt ?Promoter(2) ?x 1
• Simplified basal rate assumption rescaling
• DNA level implicit in ?
• x CI

8
Bifurcation plots of steady states

• Steady states ?Promoter(2) ?x 1
• Count crossings of ?x 1 versusf(x)
  ?Promoter(2)

Figure 1A
9
OR1 increases range, stability

• CI vs. gamma hysteresis
• Intuition at moderate CI, OR1 causes a
  preference for OR2 over OR3

Figure 1B
10
Analysis of Additive Noise

• Turn dx/dt into a potential function (Fig 2A)
• Steady state distribution analogous to the
  Boltzmann distribution (math)
• 10x more noise shifts average by 10x

Figure 2A
Figure 2C
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Multiplicative Noise

• Is like varying ?
• rate of CI synthesis from activated promoters
• high ? is like high copy number
• High steady state CI sensitive to ?

Figure 3A
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Multiplicative Noise

• Solve for a new potential function
• Note log scale!
• Noise-based amplification
• CI not gt ratio of noise?
• High-copy plasmids not really so inducible?
• High-copy plasmids and gene therapy?

Figure 3C
13
Theoretical issues with noise (Gillespie 2000)

• Adding noise terms to the deterministic solution
  is not theoretically justifiable, inaccurate
• But equilibrium binding assumption is OK
• Adding noise terms to deterministic rates is
  justifiable if a time scale exists where
  molecules changes slowly while reaction rates
  are gtgt 1
• Probably false for both transcription and
  translation of regulatory proteins
• Relative noise in reaction rate
  1/sqrt(molecules)
• Constant and multiplicative noise bracket this?
• But Promoter(2 not 3) is just one reactant

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Un-correlated Gaussian noise is not realistic

• Transcription is bursty (Ko 1992)
• Auto-correlation on time scale of hours in
  eukaryotes
• Transcripts/RNA has a skewed distribution
  (McAdams and Arkin 1997)
• Constant rate in deterministic model
• Multiplicative noise a substitute for handling
  this?
• Stochastic kinetic simulation would resolve these
  issues

15
Biological Interpretation of the External Noise

• Stochastic fluctations in chemical kinetics
• Controlled by temperature (limited range)
• Determined by circuit architecture
• Low copy number of high-affinity factor ? high
  noise
• High transcription, low translation rates ? low
  noise
• Electric noise or sinusoids can drive an ion pump
  (Xie et al 1994)
• Why use noise instead of a short-term pulse as in
  Gardner et al 2000?
• Chemical source of noise?

16
Summary

• Analyzed effects of noise levels on a simple gene
  expression-based switch
• Average state changes by orders of magnitude
• Simple, theoretical noise model
• Elegant analysis with potential functions
• Accuracy issues
• Relevance of noise-based switching unclear

17
References

• (Arkin et al 1998) Stochastic Kinetic Analysis of
  Developmental Pathway Bifurcation in Phage
  ?-Infected Escherichia coli Cells. Genetics
  1491633-1648.
• (Becksei and Serrano 2000) Engineering stability
  in gene networks by autoregulation. Nature
  405(6786)590-3.
• (Bialek 2000) Stability and Noise in Biochemical
  Switches. Neural Information Processing Systems
  13103-109
• (Gardner et al 2000) Construction of a genetic
  toggle switch in Escherichia coli. Nature
  403339-42.
• (Gillespie 2000) The chemical Langevin equation.
  J. Chem. Phys. 113(1)297-306
• (Ko 1992) Induction mechanism of a single gene
  molecule stochastic or deterministic? Bioessays
  14(5)341-6
• (McAdams and Arkin 1997) Stochastic mechanisms in
  gene expression. PNAS 94814-9.
• (Xie et al 1994) Recognition and processing of
  randomly fluctuating electric signals by
  Na,K-ATPase. Biophys J 67(3)1247-51

18
Non-rigorous derivation of Fokker-Planck

• If f(x)D and -f(x)D are probabilities of
  transitions up and down by dx/2 (respectively)
  during dt, then
• P(x,tdt)-P(x,t) P(up from x dx/2) P(down
  from x dx/2) - D change from x
• f(x-dx/2)D)P(x-dx/2,t) (-f(xdx/2)D)P(xd
  x/2,t) DP(x,t)
• -f(xdx/2)P(xdx/2,t)-f(x-dx/2)P(x-dx/2,t)
  D(P(xdx/2,t)P(x-dx/2,t)-2P(x,t)
• dP(x,t)/dt - (d/dx)f(x)P(x,t) D/2
  (d2/dx2)P(x,t)

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Boltzmann-like distribution

• P(x) Aexp-?(x)2/D, d?/dx -f(x)
• dP/dx -2/DP(x)-f(x)
• dP/dt -d/dx(f(x)P(x)) D/2d2P/dx2
  -d/dx(f(x)P(x)) d/dx(P(x)f(x)) 0

20
Modifying the potential for varying temperature

• P(y) exp- Integral(dxd/dx(potential) / Teff)
  
• From Bialek 2000