Statistical Physics of Complex Networks

Shai CarmiThesis defense

June 2006

Protein Interaction Networks

The Thesis

Relating the topological structure of protein networks to the properties of the proteins.

Showing that interacting proteins tend to be expressed uniformly in the cell.

Presenting a simple model that has this feature.

The People

Together with :

Shlomo Havlin, Bar-Ilan University, my supervisor.

Eli Eisenberg, Tel-Aviv University.

Erez Levanon, Compugen Ltd.

S. Carmi, E. Y. Levanon, S. Havlin, and E. Eisenberg, “Connectivity and expression in protein networks: Proteins in a complex are uniformly expressed”, Phys. Rev. E. 73, 031909 (2006).

Outline

• Introduction to complex networks• Biological networks• In-vivo similarity of concentrations in

interacting proteins• Presentation of a model, its properties

and their explanation.• Summary

Complex Networks

Every system with interactions between its elements can be described as a network.

The elements are called nodes (vertices, sites); the interactions are called edges (links, bonds).

Interaction can be binary/weighted, symmetric/asymmetric.

Complex Networks

Describe systems from many fields. For example :

Communication and computer networks.

Social networks.

Transportation and infrastructure networks.

Biological networks, which is the focus of our work.

Complex Networks

Massive data collection in recent years. New discoveries since 1999, including the

group of Prof. Havlin in Bar-Ilan. Most striking discovery – The distribution of

degrees (number of links) almost always follows a power-law.

Contradicting the common belief that large enough networks are random, with exponentially decaying degree distribution.

The new networks are called ‘scale-free’.

Complex Networks

Illustrating the difference between (L) pure random and (R) scale-free networks -

Biological Networks - Importance

In the past, scientists made efforts to decode the genomic sequence.

In the ‘post-genomic era’, we try to understand the more complicated question of how proteins function.

It is thus of great significance to understand the protein interaction network.

Understand the way proteins work may help in the development of therapeutic drugs.

Experiments performed

Most investigated organism is the baker's yeast Saccharomyces cerevisiae.

Known are -

The complete set of genes and proteins.

Large datasets of protein-protein interactions based on a wide range of experimental methods.

Intracellular location and the protein levels of most proteins.

The Interactions Network

Every node is a protein, two proteins are linked if they interact.

Various levels of confidence.

80,000 interactions between 5,300 proteins when taking all confidence levels.

Interactions were deduced by many different experimental methods, and they describe different biological relations between the involved proteins.

The Interactions Network

The Interactions network

Some early findings –

Power law degree distribution; high clustering; small distances; degree correlations.

Models for growth.

Topological and functional modules.

Resilience to random nodes removal, but cell would die following the removal of high degree proteins.

Concentrations

Concentrations (number of molecules per cell) of the baker's yeast proteins are distributed log-normally.

For our analysis, we will look at the concentration’s natural logarithm.

Correlations

To begin with our analysis of the network, we study the correlations between concentrations of interacting proteins.

Results in significant correlations (comparing to randomly shuffled protein concentrations).

Correlations

The complete table of correlations -

Correlations

Strongest correlation is in synexpression interaction, which is inferred from correlated mRNA expression, thus confirming the expectation that genes with correlated mRNA expression would yield correlated protein levels.

Strong effects also for HMS and TAP which correspond to physical interactions.

Complexes

As a result, we suggest the conjecture that proteins in physical complexes have uniform concentrations.

To verify our conjecture, we study a dataset of directly observed protein complexes.

We also study small complexes of size 5 (‘pentagons’) extracted from the network.

Complexes

As a measure of uniformity, we study the variance in the concentrations of the proteins forming the complex.

We find this measure to be significantly lower than in randomly generated complexes.

Robust for two different ways of randomization.

Complexes

The Model

We suggest a simple model of complex formation in order to understand our findings.

We show that within this model, complex production is most effective when its constituents are uniformly concentrated.

Thus, the experimental observation can be explained as a selection for efficiency.

The Model

We start by investigating a complex made up from 3 different particles (A,B, and C).

[A],[B],[C] – Concentrations of A,B,C.

[AB],[AC],[BC] – Concentrations of the sub-complexes.

[ABC] – Concentration of the full complex, which is the desired outcome of the process.

[A0],[B0],[C0] – The total number of available particles (per unit volume) of each type.

The Model

The Model

One can easily write the kinetic reaction equations + conservation of material equations.

Equations depend on the association and dissociation rate constants.

One can usually ignore 3-body processes, but adding them do not impose any further complications.

Sample equations

This is the kinetic equation for A-

and this is the conservation of material equation for particles of type A-

Properties of the model

We start by exploring the totally symmetric case. We look at the absolute quantity of the complex product ABC (for fixed C0=100).

Fixing B0 and C0, we find that ABC is maximized for finite optimal A0.

00max0 ,CBmaxA

Properties of the model

We also solve the more general case where the ratio of the dissociation to association rate constants can take values other than one. The picture remains the same.

Also valid for components with 4 particles.

Explanation of the results

•Why is it that adding more particles of one type deteriorates the complex production ?

•Assume a complex is made up from 3 components. One of them (A) is in excess of the others.

•Almost all B particles bind to A to form AB complexes.

•Almost all C particles bind to A to form AC complexes.

Explanation of the results

Explanation of the results

To produce ABC, we need free B’s to stick to AC, or free C’s to stick to AB, but ...

Very few free B's and C's are available, as opposed to many half-done AB's and AC's.

Lowering the concentration of A, more B's and C's will remain unbound, thus the total production of ABC will increase.

Thus we conclude, complex production is most efficient when all members are expressed uniformly, as found in-vivo.

Explanation of the results

Summary

We present and solve a simple model of complex formation.

We find that the efficiency is maximized when the concentrations of the different complex constituents are roughly equal.

Adding more particles beyond the optimal point results in less product yield.

Explained by simple arguments.

Summary

Enables us to understand the tendency of members of cellular protein complexes to have uniform concentrations, as a selection towards efficiency.

Important for the understanding of the cell’s regulation pathways.

Can be extended to study the behaviour of protein levels under stress conditions.

Thank you for your attention!

More on Complex Networks

Several models have been suggested to explain this phenomenon.

Most of them require growing the network while connecting the new nodes preferentially to high degree nodes.

It was also discovered that most networks are ‘small-worlds’ – average distance on the net scales logarithmically with the network size.

More on Complex Networks

Many other discoveries and models.

Some networks show hierarchical structure.

It was shown how to measure the network’s fractal dimension and how to observe self-similarity.

The resilience of networks to random and targeted attack was explored.

Extensive work on networks describing cellular processes.

Molecular Biology in a nutshell

Living creatures body is made of cells.

Proteins are the building blocks of the cell and they participate in almost every biological activity.

Proteins are macro-molecules (huge polymers) – long chains of ( ) small organic molecules called amino-acids.

There are only 20 possible amino-acids.

42 1010

Molecular Biology in a nutshell

The order of the amino-acids assembling a protein is coded as a gene.

The DNA is a list of genes, coded using a sequence of only 4 different nucleotides.

To produce a protein, the relevant DNA segment is copied into mRNA (transcription), then the protein is built from amino-acids according to the code (translation).

Types of Biological Interactions

Two main classes.

First is transmission of information within the cell -

Protein A interacts with protein B and changes it, by a conformational or chemical transformation.

Usually the two proteins dissociate shortly after the completion of the transformation.

Types of Biological Interactions

Second is a formation of a protein complex-

In this mode of operation the physical attachment of two or more proteins is needed in order to allow for the biological activity of the combined complex.

Typically stable over relatively long time scales.

Interactions finding experiments

HMS and TAP – One protein is being tagged and used as a bait, to fish other proteins that are physically attached to it in the cell.

Synthetic Lethality – Two proteins that are not-essential interact if mutation in both kills the cell.

Interactions finding experiments

Synexpression – The expression levels of the mRNA was measured in 300 different states of the cell cycle. Interaction between proteins happens when there is linear correlation between the series of expression levels.

Yeast 2-hybrid – Systematic identification of pairs of physically interacting proteins, by fusing them into parts of the DNA and watching when they interact.

Interactions finding experiments

Gene Fusion and 2-Neighborhood – those methods predict protein interactions by looking (in-silico) in their genomic sequence. In gene-fusion method, two proteins that are fused in a different species are predicted to interact, in 2-neighborhood method, proteins are predicted to interact if their code is adjacent in the DNA sequence.

Measuring Concentrations

Two methods of measuring the amount of a protein in the cell –

1. Measuring the expression level of the mRNA segment that codes a certain protein – this is only an indirect evidence for the existence of the protein due to post-transcriptional regulation.

2. Measuring the concentration of the protein directly – experiments were performed only recently – our main data-set.

More on Correlations

It can be shown that proteins interact significantly more with other proteins that has the same order of magnitude of concentration.

Pentagons

We look at another yeast netowrk.

We study the uniformity of concentrations in pentagons (groups of 5 proteins which form a clique in the network, which we consider as good candidates for protein complexes).

Again, we see significantly lower deviation in the concentrations of the pentagon members.

Pentagons and mRNA

We further study the mRNA expression levels.

For each protein, we have a list of 300 expression values obtained under different cellular conditions.

We notice, that for each pair of proteins in a pentagon, the mRNA expression levels are significantly more correlated than for a random pair.

Symmetric case

We start with the simplest case – for each possible reaction – the ratio between the dissociation and association rate constants is equal to some constant X0 (which takes the concentrations units).

Effectiveness

•We define the effectiveness of the production as

•This takes into account the obvious waste due to excess in one constituent.

][][][min

][

000 C+B+A

ABC=eff

Properties of the effectiveness

•For fixed C0 (=10^2), we plot eff vs. A0 and B0.

•The production is most effective when the two more abundant components have approximately the same concentration.

•For example, if A0,B0 > C0, then we’re efficient if

00 BA

4-components

•We have validated that the picture holds also for 4-component complex.

•For example, assume that we have A,B,C,D.

•A and D do not interact.

•The product complexes are ABC, BCD.

•Consider the totally symmetric case.

4-components

Can write again the set of kinetic and conservation equations.

Solution shows that the production of ABC and BCD is maximized when (for a fixed ratio of A0 and D0)

But only the few proteins that participate in many complexes with extremely different concentrations will show deviations from our previous conclusion.

0000 CBDA