An Intro To Systems
Biology: Design Principles
of Biological Circuits
Uri Alon
Presented by: Sharon Harel
Agenda
Introduction
Auto-regulation
Feed-forward loop
Life of a cell
Cells live in complex environments and
can sense many different signals:




Physical parameters
Biological signaling molecules
Nutrients or harmful chemicals
Internal state of the cell
Cell response is producing appropriate
proteins that act on the internal or external
environment
Transcription factors
Cells use transcription factors to represent
environmental states.
Designed to switch rapidly between active
& inactive.
Regulate the rate of transcription of genes:

Change the probability per unit time that
RNAp binds to the promoter and creates an
mRNA molecule.
Can be activators or repressors.
Transcription network
Transcription factors are encoded by
genes, which are regulated by
transcription factors, which are regulated
by transcription factors …
Transcription networks describe all the
regulatory transcription interactions in a
cell
Nodes: genes
Directed edges: transcriptional regulation
Sign on edged: activation or repression
Network input: environmental signals
Input function - activator
Input function – strength of the effect of a
t.f on the transcription rate of target gene.
Hill function:
rate production of Y=f  X *  
Logical function:
f X
*
    X
*
 K
 X *n
K n  X *n
Input function - repressor
Hill function:
rate production of Y=f  X
Logical function:
f X
*
    X
*
 K
*


X 
1 

K


*
n
Multi dimensional input functions
All activators present:
f  X * , Y *     X *  K X  Y *  KY 
X * and Y *
At least one activator present:
f  X * , Y *     X *  K X OR Y *  KY 
Non Boolean:
f  X * , Y *    X X *  Y Y *
X * or Y *
Dynamics and response time
Single edge in a network: X  Y
Production of Y is balanced by protein
degradation and dilution:
a  adil  adeg
Change in concentration of Y:
dY
   aY
dt
Steady state:
Yst 

a
Unstimulated 
Stimulated
Y  t   Yst 1  e
T1/ 2  log  2  / a
 at
Stimulated 
Unstimulated

Y  t   Yst e
 at
T1/ 2  log  2  / a
Detecting network motifs
Looking for meaningful network patterns
with statistical significance.
Network Motif – Patterns that occur in the
real network significantly more often than
in randomized network.
Idea: these patterns have been preserved
over evolutionary timescale against
mutations that randomly change edges.
Erdos-Renyi random networks
Same number of nodes and edges.
Directed edges assigned at random.
N nodes  N2 possible edges.
Probability edge position is occupied:
E
P 2
N
Autoregulation
Autoregulation – A network motif
Autoregulation – regulation of a gene by
its own product.
Graph: a self edge.
Example E.coli graph has 40 self edges,
34 of them are repressors (negative
autoregulation).
Is that significant?
Autoregulation – the statistics
What is the probability of having k self
edges in an ER network?
One self edge:
Pself=1/N
k self edges:
E k
E
k
P  k     Pself 1  Pself 
k
E  N self 
rand
EPself
E/N
 rand
E/N
Statistics – cont.
In our E. coli network: N=424, E=519
E  N self 
rand
E / N  1.2
 rand
E / N  1.1
Difference in STD units:
Z
E  N self 
real
 E  N self 
 rand
rand
 32
Why negative autoregulation?
Dynamics of X:
dX
*
 f  X   aX
dt
f X
*
    X
At early times:
dX
   aX
dt
Steady state:
X K
X st  K
*
 K
Negative Autoregulation
Response time:
T1/ 2
K

2
Evolutionary selection
on β and K
Negative auto vs. simple
Mathematically controlled comparison
asimple  a
X st 
 simple
asimple
K
 simple /  

T1/ 2

simple
T1/ 2
2 log  2 
n.a .r
Best of both worlds: rapid production and
desired steady state
Robustness to production
fluctuations
Production rate β fluctuates over time.
Twin cells differ in production rate of all
proteins in O(1) up to O(10).
Repression threshold K is more fixed.
Simple regulation is affected strongly by β:
X st   / a
Negative autoregulation is not:
X st  K
Feed-forward loop
Sub graphs in ER networks
Probability edge position is occupied:
P=E/N2
Occurrences of sub graph G(n,g) in an ER
network:
E  NG   a N P  a  N
1
n
Mean connectivity:
g
1
λ=E/N
g
n g
Three-node patterns
There are 13 possible sub-graphs with 3 nodes
Feed forward loop
X
Y
Z
Feedback loop
X
Y
Z
Feed-Forward is a network motif
Feed forward loop 3 node feedback
E. Coli
ER networks
Degree preserving
random nets
42
0
1.7±1.3 (Z=31)
0.6±0.8
7±5 (Z=7)
0.2±0.6
The feed-forward loop (FFL) is a strong motif.
The only motif of the 13 possible 3-node
patterns
Feed-forward types
C1-FFL with AND logic
C1-FFL equations
For transcription factor Y:
production Y    y  X *  K XY 
dY / dt   y  X *  K XY   aY Y
For gene Z:
production  Z    Z  X  K XZ  Y  KYZ 
*
*
dZ / dt   Z  X *  K XZ  Y *  KYZ   aZ Z
C1-FFL as a delay element
Consider the response to 2 steps of signal
Sx :


ON step – Sx is absent and then appears.
OFF step – Sx is present and then
disappears.
Assumption: SY is always present.
Delay following ON step
ON step
Production of Y*
accumulation of Y*
Y*
threshold
Y  t   Yst 1  e
*

Y Tdelay   Yst 1  e
 aY Tdelay
 aY t
K

YZ
Tdelay  1/ aY log 1/ 1  KYZ / Yst  
Production of Z
C1-FFL + AND graphs
C1-FFL + OR logic - Example
Sign-sensitive delay in the OFF step: X* can
activate gene Z by itself, but both X* and Y*
have to fall below their KZ levels for the
activation to stop.
Allows maintaining expression even if signal
momentarily lost.
I1-FFL
Two parallel but opposing paths:
the direct path activates Z and
the other represses Z.
Z shows high expression when
X* is bound and low expression
when Y* is bound.
Use: pulse generator & fast
response time.
I1-FFL equations
Accumulation of Y:
Y  t   Yst 1  e
*
 aY t

For gene Z:
Z production at βz
X*, Y*<KYZ
Z  t   Z m 1  e
 aZ t

Y* accumulates until Y*=KYZ
I1-FFL equations – cont.
Trep  1/ aY log 1/ 1  KYZ / YST  
Z production at β’z
Z  t   Z st   Z 0  Z st  e

Z0  Zm 1  e
 aZ Trep

Y* represses Z

 aZ t Trep

Z st   'Z / aZ
I1-FFL graphs
I1-FFL response time
Half of steady state is reached during the
fast stage:

Z1/ 2  Z st / 2  Z m 1  e
 aZ T1/ 2

T1/ 2  1/ aZ log  2 F /  2 F  1 
F – repression coefficient. The larger the
coefficient (the stronger the repression)
the shorter the response time.
F  Z m / Z st
I1-FFL - example
Galactose system in E. coli



Low expression of Gal genes when Glu present.
When both are absent Gal genes have low but
significant expression (“getting ready”).
When Gal appears – full expression of Gal
genes
Other FFL types
The other 6 types of FFL are rare in transcription
networks.
Some of the lack responsiveness to one of the
signals.
Example: I4-FFL
I4-FFL vs. I1-FFL
Sx
SY
Zst – I1
Zst – I4
0
0
0
0
0
1
0
0
1
0
high βz/az
low β’z/az
1
1
low β’z/az
low β’z/az
Questions?