An Intro To Systems An Intro To Systems Biology: Design Principles Biology: Design Principles
of Biological Circuitsof Biological CircuitsUri AlonUri Alon
Presented by: Sharon HarelPresented by: Sharon Harel
AgendaAgenda
IntroductionIntroduction
Auto-regulationAuto-regulation
Feed-forward loopFeed-forward loop
Life of a cellLife of a cell
Cells live in complex environments and Cells live in complex environments and can sense many different signals:can sense many different signals: Physical parametersPhysical parameters Biological signaling moleculesBiological signaling molecules Nutrients or harmful chemicalsNutrients or harmful chemicals Internal state of the cellInternal state of the cell
Cell response is producing appropriate Cell response is producing appropriate proteins that act on the internal or external proteins that act on the internal or external environmentenvironment
Transcription factorsTranscription factors
Cells use transcription factors to represent Cells use transcription factors to represent environmental states.environmental states.
Designed to switch rapidly between active Designed to switch rapidly between active & inactive.& inactive.
Regulate the rate of transcription of genes:Regulate the rate of transcription of genes: Change the probability per unit time that Change the probability per unit time that
RNAp binds to the promoter and creates an RNAp binds to the promoter and creates an mRNA molecule.mRNA molecule.
Can be activators or repressors.Can be activators or repressors.
Transcription networkTranscription network
Transcription factors are encoded by Transcription factors are encoded by genes, which are regulated by genes, which are regulated by transcription factors, which are regulated transcription factors, which are regulated by transcription factors …by transcription factors …
Transcription networks describe all the Transcription networks describe all the regulatory transcription interactions in a regulatory transcription interactions in a cellcell
Nodes: genesNodes: genes
Directed edges: transcriptional regulationDirected edges: transcriptional regulation
Sign on edged: activation or repressionSign on edged: activation or repression
Network input: environmental signalsNetwork input: environmental signals
Input function - activatorInput function - activator
Input function – strength of the effect of a Input function – strength of the effect of a t.f on the transcription rate of target gene.t.f on the transcription rate of target gene.
Hill function:Hill function:
Logical function:Logical function:
*
**
rate production of Y=n
n n
Xf X
K X
* *f X X K
Input function - repressorInput function - repressor
Hill function:Hill function:
Logical function:Logical function:
*
*rate production of Y=
1nf X
XK
* *f X X K
Multi dimensional input functionsMulti dimensional input functions
All activators present:All activators present:
At least one activator present:At least one activator present:
Non Boolean:Non Boolean:
* * * * * *, and X Yf X Y X K Y K X Y
* * * *, X Yf X Y X Y
* * * * * *, OR or X Yf X Y X K Y K X Y
Dynamics and response timeDynamics and response time
Single edge in a network:Single edge in a network:
Production of Y is balanced by protein Production of Y is balanced by protein degradation and dilution:degradation and dilution:
Change in concentration of Y:Change in concentration of Y:
Steady state:Steady state:
X Y
dil dega a a
dYaY
dt
stY a
Unstimulated Unstimulated StimulatedStimulated
Stimulated Stimulated UnstimulatedUnstimulated
atstY t Y e 1 at
stY t Y e
1/ 2 log 2 /T a 1/ 2 log 2 /T a
Detecting network motifsDetecting network motifs
Looking for meaningful network patterns Looking for meaningful network patterns with statistical significance.with statistical significance.
Network Motif – Patterns that occur in the Network Motif – Patterns that occur in the real network significantly more often than real network significantly more often than in randomized network.in randomized network.
Idea: these patterns have been preserved Idea: these patterns have been preserved over evolutionary timescale against over evolutionary timescale against mutations that randomly change edges.mutations that randomly change edges.
Erdos-Renyi random networksErdos-Renyi random networks
Same number of nodes and edges.Same number of nodes and edges.
Directed edges assigned at random.Directed edges assigned at random.
N nodes N nodes N N22 possible edges. possible edges.
Probability edge position is occupied:Probability edge position is occupied:
2
EP
N
AutoregulationAutoregulation
Autoregulation – A network motifAutoregulation – A network motif
Autoregulation – regulation of a gene by Autoregulation – regulation of a gene by its own product.its own product.
Graph: a self edge.Graph: a self edge.
Example E.coli graph has 40 self edges, Example E.coli graph has 40 self edges, 34 of them are repressors (negative 34 of them are repressors (negative autoregulation).autoregulation).
Is that significant?Is that significant?
Autoregulation – the statisticsAutoregulation – the statistics
What is the probability of having k self What is the probability of having k self edges in an ER network?edges in an ER network?
One self edge:One self edge: PPselfself=1/N=1/N
k self edges:k self edges:
1E kk
self self
EP k P P
k
/self selfrandE N EP E N
/rand E N
Statistics – contStatistics – cont..
In our E. coli network: N=424, E=519In our E. coli network: N=424, E=519
Difference in STD units:Difference in STD units:
/ 1.2self randE N E N / 1.1rand E N
32
self selfreal rand
rand
E N E NZ
Why negative autoregulation?Why negative autoregulation?
Dynamics of X:Dynamics of X:
At early times:At early times:
Steady state:Steady state:
*dXf X aX
dt * *f X X K
dXaX X K
dt
stX K
Negative AutoregulationNegative Autoregulation
Response time:Response time:
Evolutionary selection Evolutionary selection on on ββ and K and K
1/ 2 2
KT
Negative auto vs. simpleNegative auto vs. simple
Mathematically controlled comparisonMathematically controlled comparison
Best of both worlds: rapid production and Best of both worlds: rapid production and desired steady statedesired steady state
. .1/ 2
1/ 2
/
2 log 2
simplesimple st
simple
n a rsimple
simple
a a X Ka
T
T
Robustness to production Robustness to production fluctuationsfluctuations
Production rate Production rate ββ fluctuates over time. fluctuates over time.
Twin cells differ in production rate of all Twin cells differ in production rate of all proteins in O(1) up to O(10).proteins in O(1) up to O(10).
Repression threshold K is more fixed.Repression threshold K is more fixed.
Simple regulation is affected strongly by Simple regulation is affected strongly by ββ::
Negative autoregulation is not:Negative autoregulation is not:
/stX a
stX K
Feed-forward loopFeed-forward loop
Sub graphs in ER networksSub graphs in ER networks
Probability edge position is occupied: Probability edge position is occupied: P=E/NP=E/N22
Occurrences of sub graph G(n,g) in an ER Occurrences of sub graph G(n,g) in an ER network:network:
Mean connectivity:Mean connectivity: λλ=E/N=E/N
1 1n g g n gGE N a N P a N
Three-node patternsThree-node patterns
There are 13 possible sub-graphs with 3 nodesThere are 13 possible sub-graphs with 3 nodes
Feed forward loopFeed forward loop
X Y
Z
Feedback loopFeedback loop
X Y
Z
Feed-Forward is a network motifFeed-Forward is a network motif
The feed-forward loop (FFL) is a strong motif.The feed-forward loop (FFL) is a strong motif.
The only motif of the 13 possible 3-node The only motif of the 13 possible 3-node patternspatterns
Feed forward loopFeed forward loop33 node feedback node feedback
E. ColiE. Coli424200
ER networksER networks1.7±1.3 (Z=31)1.7±1.3 (Z=31)0.6±0.80.6±0.8
Degree preserving Degree preserving random netsrandom nets
7±5 (Z=7)7±5 (Z=7)0.2±0.60.2±0.6
Feed-forward typesFeed-forward types
C1-FFL with AND logicC1-FFL with AND logic
C1-FFL equationsC1-FFL equations
For transcription factor Y:For transcription factor Y:
*y XYproduction Y X K
*/ y XY YdY dt X K a Y
For gene Z:For gene Z:
* *Z XZ YZproduction Z X K Y K
* */ Z XZ YZ ZdZ dt X K Y K a Z
C1-FFL as a delay elementC1-FFL as a delay element
Consider the response to 2 steps of signal Consider the response to 2 steps of signal SSxx : : ON step – SON step – Sxx is absent and then appears. is absent and then appears.
OFF step – SOFF step – Sxx is present and then disappears. is present and then disappears.
Assumption: SAssumption: SY Y is always present.is always present.
Delay following ON stepDelay following ON step
ON step ON step Production of Y* Production of Y*
* 1 Ya tstY t Y e accumulation of Y* accumulation of Y*
Y*Y*
threshold threshold
1
1/ log 1/ 1 /
Y delaya T
delay st YZ
delay Y YZ st
Y T Y e K
T a K Y
Production of Z Production of Z
C1-FFL + AND graphsC1-FFL + AND graphs
C1-FFL + OR logic - ExampleC1-FFL + OR logic - Example
Sign-sensitive delay in the OFF step: X* can Sign-sensitive delay in the OFF step: X* can activate gene Z by itself, but both X* and Y* activate gene Z by itself, but both X* and Y* have to fall below their Khave to fall below their KZZ levels for the levels for the activation to stop.activation to stop.Allows maintaining expression even if signal Allows maintaining expression even if signal momentarily lost.momentarily lost.
I1-FFLI1-FFL
Two parallel but opposing paths: Two parallel but opposing paths: the direct path activates Z and the direct path activates Z and the other represses Z.the other represses Z.
Z shows high expression when Z shows high expression when X* is bound and low expression X* is bound and low expression when Y* is bound.when Y* is bound.
Use: pulse generator & fast Use: pulse generator & fast response time.response time.
I1-FFL equationsI1-FFL equations
Accumulation of Y:Accumulation of Y:
For gene Z:For gene Z:
* 1 Ya tstY t Y e
1 Za tmZ t Z e
X*, Y*<KYZ Z production at βz
Y* accumulates until Y*=KYZ
I1-FFL equations – contI1-FFL equations – cont..
1/ log 1/ 1 /rep Y YZ STT a K Y
Y* represses Z
0
0 1
Z rep
Z rep
a t T
st st
a T
m
Z t Z Z Z e
Z Z e
Z production at β’z
' /st Z ZZ a
I1-FFL graphsI1-FFL graphs
I1-FFL response timeI1-FFL response time
Half of steady state is reached during the Half of steady state is reached during the fast stage:fast stage:
F – repression coefficient. The larger the F – repression coefficient. The larger the coefficient (the stronger the repression) coefficient (the stronger the repression) the shorter the response time. the shorter the response time.
1/ 21/ 2
1/ 2
/ 2 1
1/ log 2 / 2 1
Za Tst m
Z
Z Z Z e
T a F F
/m stF Z Z
I1-FFL - exampleI1-FFL - example
Galactose system in E. coliGalactose system in E. coli Low expression of Gal genes when Glu present.Low expression of Gal genes when Glu present. When both are absent Gal genes have low but When both are absent Gal genes have low but
significant expression (“getting ready”).significant expression (“getting ready”). When Gal appears – full expression of Gal genesWhen Gal appears – full expression of Gal genes
Other FFL typesOther FFL types
The other 6 types of FFL are rare in transcription The other 6 types of FFL are rare in transcription networks.networks.Some of the lack responsiveness to one of the Some of the lack responsiveness to one of the signals.signals.Example: I4-FFLExample: I4-FFL
I4-FFL vs. I1-FFLI4-FFL vs. I1-FFL
SSxxSSYYZZstst – I1 – I1ZZstst – I4 – I4
00000000
00110000
1100high high ββzz/a/azzlow low ββ’’zz/a/azz
1111low low ββ’’zz/a/azzlow low ββ’’zz/a/azz
QuestionsQuestions??
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