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??