Network Design I Random networks Degree distribution Random networks …home.thep.lu.se/~henrik/bnf079/2006_network_design.pdf · 2006. 12. 7. · Boolean networks Assume the biological

Complex Systems Lund University

Network Design I

Patrik EdénComplex Systems

Theoretical Physics;Lund Stem Cell Center

[email protected]

BNF 079 Fall 2005

● Random networks

● Degree distribution

● Random networks revisited

● Motifs


Are biological networks special?

Well, what is “special”?

● Subjectiveexample: humans have 56 times as many genes

as the bacteria E.Coli.Interestingly few! Complexity does not grow with network size as we expect, or humans are simply not as complex as we think.

● Compare with other networksOther fields (e.g., social networks)Other biological networks

protein interaction versus protein regulation different species

● Compare with random artificial networksToday's topic!


Random networks

Compare:Biological network: N nodes, L linksRandom network: N nodes, probability for a link p

What is p?

Undirected: node pairs, p =

Directed: N starting points, N destinations, p =

N(N1)2

2LN(N1)

LN 2


Random network generation 1

Start with N nodes. In every possible place, insert a link with probability p.Compare with your real network.

Degree distribution

Probability that a node has k links: binomial distribution.

p k(1p) N1k N1

k( )


Degree distribution

Random network, N=5800, L=28110

The network is not single connected, but almost.


Degree distributioncomparison, N=5800, L=28110

The pvalue of getting the yeast distribution by chance is0.00000000000000000000...


Degree distribution

Roughly a straight line in loglog plot“Scalefree network”

pk~kγ

True for almost all social and biological networks.(kdepedence differs. γ=13 common.)

Not true for the random network discussed so far.

Biological networks contain more nodes with very many links than you expect by random.

● Transcription factors controlling really many genes.● Proteins interacting with really many other proteins.


Can we study more than degree distribution?

For example, how many connected 3node groups have all 3 links?

p2(1p) p2(1p) p2(1p)

p3

Expected fraction complete triangles in random networks:p3/[3p2(1p)+p3]=p/(32p)


Is this legitimate?


No!!

● Probability for a triangle depends on the probability for a link to be present

● The probability for a link to be present depends on the degree of the node in question

● Have to ensure our random networks have the right degree distribution.


Alternative: growing random networks

● Start with 2 nodes.● Insert a link with probability p.● Add a node.● Insert a link to every other node, with a probability that depends on the number of links the node already has (“Many gets more”).● Stop when you have N nodes.

Finetune p and “many gets more” probabilitiesto get roughly L links and correct degree distribution.

Only samples a subset of possible random networks(e.g., the higher order connectivity discussed

in “network basics” will not really be random).


Can be tuned to give scale free networks,but difficult to control what subset of random networks

that are generated

Alternative: mimic evolution

Copying + diversing


Heuristic method: randomize given network

Flexible in preserving other known features of the network.

Believed to sample a large subset of networks (not proven to my knowledge)

De facto standard

Conserves the degree of all nodes, i.e.,keeps degree distribution of every kind of node

(transcription factors, other genes...)(kinases, receptors, other protein subgroups...)


Randomization

1a. Starting network 1b. Pick two links

1c. If undirected, pick directions 1d. Swap arrow heads


Randomization

2a. Modified network 2b. Pick two new links

2c. Pick directions 2d. Swap arrow heads


Randomization

3a. Modified network 3b. Go back to previously Is it acceptable? accepted network (2a). No, double undirected link Pick two new links...

Check for what ever you want to avoid:Double undirected linksDouble directed links in same directionSelfself connectionSingle connected network / many components.


“Other features”

● Functional classification of the gene (Are genes with high degree lethal? Part of known gene family?) This can be studied without random networks.

Using random networks with correct degree distribution,we can ask:

Is our biological network different in...

● Motifs (small often occurring subgroups in the network)● Degree distribution of neighbours to nodes of a fixed degree (the “higher order” connectivity from network basics)● ...


Motifs

● Big networks are never 100% identical● Small subparts of networks might be, called “motifs”● An example of a simple motif is the triangle

Triangles

In directed networksTriangles are feed forward loops or feed backward loops


Transcriptional regulation in E. coli(nature genetics 2002, ShenOrr)

Only feed forward loops are overrepresented

40 in real network Zscore (407)/4 = 87 ± 4 in randomized networks pvalue ~ 1017

Regulation comes with a sign

● Activate (336 links). Turns on the gene.● Repress (214 links). Turns off the gene.● Dual (29 links). Sometimes activate, sometimes repress.

(This result for E. coli may change with experimental development,

and might be different for eukariots.)


Transcriptional regulation

Two kinds of feed forward loops● Coherent

E. coli: 34 in real, 4.4 ± 3 in random, Z=(344.4)/3=9, p=1022 ● Incoherent

E. coli: 6 in real, 2.5 ± 2 in random, Z=(62.5)/2=1.7, p=0.04



Function of triangles (dynamics preview).

Coherent

feed forward:

x y z

Noise reduction (z independent of sudden burst in x)Cannot be achieved with less than 3 nodes (to my knowledge)




Incoherent

feed forward:

x y z

Transient response (z only responds while x increases)Cannot be achieved with less than 3 nodes (to my knowledge)




Positive feed backward: Toggle switch.

x y z

Temporary external signal activates x (or y or z)=> all three remain activeTemporary external signal inactivates x (or y or z)=> all three remain inactive

Simpler alternatives:




Negative feed backward: Stable state

x y z

External signal activates/represses x (or y or z)=> Negative feedback restores all three to approximately the previous concentrations.

Simpler alternatives:



Function of triangles (dynamics preview). Summary

Coherent feed forward: Noise reductionIncoherent feed forward: Transient responsePositive feed backward: Toggle switchNegative feed backward: Stable state

Note: These functions are just examples.They depend on time scales

(is x>y>z much slower than x>z?)and logical combinations of signals

(is z activated by [x AND y] or [x OR y]?)

In particular, negative feedback with large time delay becomesan oscillator (dynamics lectures, computer exercise).



Function of triangles (dynamics preview). Comment

The feed forward triangles solve tasks that cannot be solved bysimpler means.

The feed backward triangles can be replaced by simpler(smaller) motifs.

Our systems biology finding (overrepresentation in E. coli transcriptional regulation

of feed forward triangles, but not of feed backward) makes biological sense!


Transcriptional regulation, other motifs (Science 2002, Lee)


Autoregulation

Overrepresented in the E.coli transcriptional network.What is its function?

An example showing that the answer we find depends on the question we ask!

Protein concentration [P ]Promoter strength V

Degradation constant 1/τDissociation constant K


Autoregulation

Neglect timedelay from transcription to protein (~ 3 minutes for E.coli )

Assume MichaelisMenten and mass action

d[P ] VK [P ]

d[P ] [P ]

d[P ] VK [P ]

dt τ= V - , no inhibition (large K )

dt [P ] τ= - , strong inhibition (small K )

dt K + [P ] τ= -


Autoregulation

Autoregulation reduces the stable protein concentration Pst

Same effect with lower V or larger 1/τ. Why autoregulation?


Autoregulation

Assume the cell prefers one Pst. Vary both K and V .

Quicker responsewith autoregulation

Same effect withactive degradation

(larger 1/τ ), but expensive to produce and degradeproteins.


Autoregulation

Assume downstreameffects startat half the stableconcentration

No degradation (only dilution): 0.7 τ = 1 cellcycle

0.15 0.7


Network Design II

None of these questions will be definitely answered

Patrik EdénComplex Systems

Theoretical Physics;Lund Stem Cell Center

[email protected]

BNF 079 Fall 2005

● Digital or analogue networks?

● Boolean networks: How does nature get stable solutions?

● Modules or mess?


Are biological networks analogue or digital?

● Analogue: “What is the concentration of protein P?” Infinitely many answers

● Digital: “Is the concentration of protein P above a threshold?” Yes/no answer

We get much more information from protein P inthe analogue case, but we also get very sensitive to noise.

Digital networks are safer, but more expensive

Nature probably uses both kinds...


The protein activation/deactivation cycle

Consider a protein that can be modified somehow(Phosphorylation, methylation, ...)

In the modified state it interacts with other proteins

The process can be reversed

Inactive Active

state state

(the arrow means “turns into”, not activate or repress)



1X X

The fraction X of active protein is often determined by

(MichaelisMenten dynamics, coming later)

For simplicity assume K1=K

2=K and set V

2 / V

1 = f

K+(1X) = f

K+X

X (1X)

K2+(1X)

=K

1+X

V1X V

2(1X)



input f

1X X output

External signal can help activate the protein(increasing f)

How does our output (X) depend on our input (f) ?



Large K : Analogue amplifier

Small K : Digital switch

K depends on enzyme affinities.Designing enzymes = choosing between analogue and digital

Both are possible in nature.


0 2 4 6 8 100

0.5

1

0 2 4 6 8 100

0.5

1

0 0.5 1 1.5 20

0.5

1

-0.5 0 0.5 1 1.5 20

20

40

SN

SN

SN

SN

Unphysical Region

input

outputSwitches in general

Ultrasensitive

Bistable

Irreversible


Boolean networks

Assume the biological network is digitalEach node is a boolean variable (“on” or “off”)

A node value depends on its inputs, e.g.,

yeast transcriptional network

In1 in2 in3 out0 0 0 10 0 1 10 1 0 00 1 1 01 0 0 11 0 1 01 1 0 01 1 1 0


Boolean networks

Simplify rule table

One input has a value that This can be iteratedimmediately determines the nested canalizing ruleoutput: canalizing rule

in1 in2 in3 outx 1 x 00 0 x 11 0 0 11 0 1 0

in1 in2 in3 out0 0 0 10 0 1 10 1 0 00 1 1 01 0 0 11 0 1 01 1 0 01 1 1 0

in1 in2 in3 outx 1 x 00 0 0 10 0 1 11 0 0 11 0 1 0


Boolean networks

nested canalizing rule, example

in2: strong suppressorin1 & in3: suppressors when together

Almost all documented biological rules are nested canalizing!

Is it because they are most common, or because theyare easiest to find? ...

in1 in2 in3 outx 1 x 00 0 x 11 0 0 11 0 1 0


Boolean networks

Dynamics.

Start with random variables 0 or 1Update all nodes according to their rule

Iterate, and you eventually reach:

● Fixpoint (nothing changes in an update)● Limit cycle (return to the same state after a few updates)● Chaos (A “limit cycle” almost as long as the number of possible states)


Boolean networks

How does nature avoid chaos?

Assign random rulesto a known network=>You often get chaos

Assign randomnested canalizingrules=>You do not get chaos!

Nested canalizing rules may dominate in biologysince they give stable networks.

yeast transcriptional network


Modules or mess?

A motif is a small pattern occurring often.A module can be large, perhaps only occurring once,

but we still expect it to have a special function

Skeleton analogy:10 fingers, 10 toes, 3 similar bones in each

=>A fingerbone “motif” occurring 60 times

The skull only occurs once, but is placed on topand forms a unique structure

(the almost spherical shell with well defined openings)=>

Aliens would pretty soon consider it a “module”(especially when they find similar modules on different species)


Modules or mess?

Analogy between biological networksand electric circuits

transistor

cascade cycle

Very similar amplifying inputoutput relations!


Modules or mess?

Analogy between biological networksand electric circuits

Today's complex electric circuits (logical circuits)are build up by modules.

Safe: Stay in control of what your circuit doesCheap development: Use what you already have in a new way.

Costly running: Missing fast solution with fewer components?

Some try “genetic programming”.


Modules or mess?

Genetic programming

Seek wanted inputoutput relation.Use a set of simple motifs (capacitor, contact, ...)

Start with randomly designed circuit with few elementsOutput very wrong!

Modify: duplicate in parallell, in seriesdelete item redirect contact ...

If output worse, discard modification with some probabilityIf output better, keep modification and modify again

Continue until output good enough


Modules or mess?

Analogue squareroot operator, conventional solution:

One amplifier motif and one multiplication moduleThe mulitplication module consists of many amplifier motifs90 % of the time, genetic programming gives similar circuit

(a good electric engineer recognizes the evolved circuit)


Modules or mess?

Sometimes, genetic programming givescompletely different solution:

no modules

impossible to reconstruct function

fewer components

Solution without modules rare, but possible, and seemsless expensive!

Relevance for biological evolution of “circuits” unclear.

Documents

Network Design I Random networks Degree distribution Random networks …home.thep.lu.se/~henrik/bnf079/2006_network_design.pdf · 2006. 12. 7. · Boolean networks Assume the biological