1 stochasticity and robustness steve andrews brent lab, basic sciences division, fhcrc lecture 5 of...
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Stochasticity and robustness
Steve Andrews
Brent lab, Basic Sciences Division, FHCRC
Lecture 5 of Introduction to Biological ModelingOct. 20, 2010
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Last week• gene regulatory networks• graphs• Boolean networks• transcription dynamics• motifs
ReadingRao, Wolf, and Arkin, “Control, exploitation and tolerance of intracellular noise” Nature 420:231-237, 2002.
(Arkin, Ross, and McAdams, “Stochastic kinetic analysis of developmental pathway bifurcation in phage -infected Eshcherichia coli cells” Genetics 149:1633-1648.)
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Sources and amount of stochasticity
Amplifying stochasticity
Reducing stochasticity
Modeling stochasticity
Summary
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Deterministic vs. stochastic
d A[ ]dt
=−k A[ ]
Deterministic
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A[ ] = A[ ]0 e−kt
A Bk
A
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Deterministic vs. stochastic
d A[ ]dt
=−k A[ ]
Deterministic
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A[ ] = A[ ]0 e−kt
Stochastic
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A Bk
many possible trajectories
A A
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Stochasticity origins
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Reaction timing is randomA B
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probability density of reaction time
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Stochasticity origins
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Reaction timing is randomnow, determined by diffusionand reaction rate upon collision
A + B C
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probability density of reaction time
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Probability vs. probability density
probability for discrete events
probability density for continuous events
1/6
0
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1 2 3 4 5 6
each bar is probability of a specific eventpartial sum is cumulative probabilitytotal sum of bars = 1
rectangle area is probability of event being in that rangeintegral is cumulative probabilitytotal area under curve = 1
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How much variation?
A B
At equilibrium,
average =
std. dev. ~ n
n
Absolute noise increases with more molecules:
Relative noise decreases with more molecules:
n
n
n=
1n
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How much variation?
A B
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probability density for fraction in state A
the relative noise decreaseswith more molecules ~ 1/ n
Credit: Rao et al. Nature 420:231, 2002.
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How much variation?
At, or near, steady state
variation is still ~ 1
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molecules variation stochasticity10,000 1% not important1000 3% probably not important100 10% probably important10 30% very important1 100% essential
Credit: Klamt and Stelling in System Modeling in Cellular Biology ed. Szallasi et al. p. 73, 2006.
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where does stochasticity matter?
species copies stochastic?DNA 1 or 2 (or 4) no, tightly controlled
mRNA 0 to 100s yes
proteins 1 nM = 1 molec. in E. coli (2 fl) yes = 300 molec. in yeast (500 fl) probably1 M = 1000 molec. in E. coli maybe = 300,000 molec. in yeast no
metabolites M to mM usually no
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Sources and amount of stochasticity
Amplifying stochasticity
Reducing stochasticity
Modeling stochasticity
Summary
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Benefits of stochasticity
Population heterogeneity(phase variation)• E. coli pili variation• Salmonella flagella• phage lysis-lysogeny
Waiting mechanism• phage lysis-lysogeny
Exploration strategies• E. coli chemotaxis
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QuickTime™ and a decompressor
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Credits: http://www.biologyjunction.com/fimbriae_article.htm; Alberts, et al., Molecular Biology of the Cell, 3rd ed. Garland Publishing, 1993.
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Phage lysis-lysogeny
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Credit: http://textbookofbacteriology.net/themicrobialworld/Phage.html
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Phage lysis-lysogeny
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Lysis-lysogeny model of Arkin, Ross, and McAdams, 1998.
Stochastic decision upon initial infection.
Stochastic departure from lysogeny to lysis.
Credit: Arkin et al. Genetics 149:1633, 1998
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Amplification with transcription/translation
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DNA RNA protein
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DNA is fixed at 1 (or 2 or 4)
RNA follows rule
noise is amplifiedin translation, soprotein noise >>
n
n
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Amplification with positive feedback
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Co
nce
ntr
atio
n A
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small variation of xcauses switchingbetween A and B C
on
cen
tra
tion
A
Credit: Rao et al. Nature 420:231, 2002.
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More amplification with positive feedback
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Positive feedback in E. coli motorrandomly switches it between cwand ccw.
ccw cw
Credits: Duke, Le Novere, and Bray, J. Mol. Biol. 308:541, 2001.
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intrinsic vs. extrinsic noise
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Question: is noise arising from factors intrinsic to geneexpression, or upstream extrinsic factors?
IPTG LacI DNA RNA protein
intrinsic noiseextrinsic noise
Solution: 2 GFP genes, same chromosome, same promotors, same extrinsic noise. Different intrinsic noise.
only extrinsic noise
only intrinsic noise
Credits: Elowitz et al. Science 297:1183, 2002.
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intrinsic vs. extrinsic noise
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IPTG LacI DNA RNA protein
intrinsic noiseextrinsic noise
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quantify using a scatter plot
Credits: Elowitz et al. Science 297:1183, 2002.
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intrinsic vs. extrinsic noise
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IPTG LacI DNA RNA protein
intrinsic noiseextrinsic noise
low expression levelfew mRNAhigh intrinsic noise
high expression levelmany mRNAlow intrinsic noise
intrinsic noise decreases withhigher expression levels
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Relative fluorescence
Credits: Elowitz et al. Science 297:1183, 2002.
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Sources and amount of stochasticity
Amplifying stochasticity
Reducing stochasticity
Modeling stochasticity
Summary
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Problems of stochasticity
• development• signaling
Noisy inputs, and want to make bestdecision possible.
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Noise reduction with negative feedback
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Credit: Sotomayor et al. Computational and Applied Mathematics, 26:19, 2007.
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Noise reduction with negative feedback
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Becksei and Serrano, Nature 405:590, 2000.
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Noise reduction with integral negative feedback
Barkai and Leibler, 1997 showed bacterical chemotaxis adaptation is robustto protein number variation.
Postulated: CheBonly demethylatesactive receptors
Resultadaptation robust to variableprotein concentrations
Credit: Barkai and Leibler, Nature, 387:913, 1997.
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Yi, Huang, Simon, and Doyle, 2000 showed that this arises from integralnegative feedback.• adaptation is from integral• robustness is from negative feedback
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Noise reduction with feed-forward motif
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Credit: Shen-Orr et al., Nat. Genetics 31:64, 2002.
filters out brief inputs – noise reduction
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Sources and amount of stochasticity
Amplifying stochasticity
Reducing stochasticity
Modeling stochasticity
Summary
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Stochastic modeling
2 approaches
• compute every possible outcome at once, with probabilities
• simulate individual trajectories, and then analyze results
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Chemical master equation approach
A Bk
State is number of A moleculesPa is probability system is in state a.
dPadt
=kPa+1 −kPa
rate ofenteringstate
rate ofleavingstate
states
state 9entering state
leaving state
Calculate the probability of every possible system state,as a function of time
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Chemical master equation approach
Exact solution for all trajectories, but• doesn’t give sense of trajectories• is hopelessly complicated for realistic system
A molecules
B m
olec
ules
More generally, a trajectory is a random walk in state space. The master equation computes the probability of being in each state as a function of time.
A molecules
B m
olec
ules
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Langevin approach
d A[ ]dt
=−k A[ ]
A Bk
Δ A[ ] = −k A[ ]Δt
d A[ ]dt
=−k A[ ] + x(t)
deterministic stochastic
Δ A[ ] = −k A[ ]Δt + X k A[ ]Δt
noise term
Gaussian distributed random variable with mean 0, std. dev. 1
Result has correct level of noise, but• number of molecules is not discrete• A can increase as well as decrease
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Gillespie algorithm
A Bk
1. For each reaction path• decide when the next reaction will be:
is drawn from an exponential distribution• decide what the product will be:
Here, B is the only option
2. Step the system forward to the next reaction
3. Perform the reaction
4. Repeat
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probability distribution of reaction time
Method is exact, but simulates slowly.
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Gillespie algorithm
A Bk
1. For each reaction path• decide when the next reaction will be:
is drawn from an exponential distribution• decide what the product will be:
Here, B is the only option
2. Step the system forward to the next reaction
3. Perform the reaction
4. Repeat
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probability distribution of reaction time
variants: Gibson-Bruck, direct method, first-reaction method, optimized direct method.
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Sources and amount of stochasticity
Amplifying stochasticity
Reducing stochasticity
Modeling stochasticity
Summary
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Summary
Source of stochasticity• reaction timing
Amount of stochasticity• √n is rule-of-thumb
Amplification• good for population heterogeneity, etc.• transcription/translation• positive feedback• intrinsic/extrinsic noise
Reduction• negative feedback• feed-forward motif
Modeling• chemical master equation• Langevin equation• Gillespie algorithm
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Homework
No class next week. Instead, a talk by Herbert Sauro.
In two weeks, development and pattern formation.
Read?