Statistical Physics of Complex Networks

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Statistical Physics of Complex Networks
Protein Interaction Networks

• Shai Carmi
• Thesis defense
• June 2006

2
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.

3
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).
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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

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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.

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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.

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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.

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Complex Networks

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

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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.

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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.

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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.

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The Interactions Network
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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.

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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
concentrations natural logarithm.
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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).

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Correlations

• The complete table of correlations -

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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.

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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.

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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.

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Complexes
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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.

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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.

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The Model
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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.

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Sample equations

• This is the kinetic equation for A-

and this is the conservation of material equation
for particles of type A-
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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 C0100).

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

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Explanation of the results
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Explanation of the results

• To produce ABC, we need free Bs to stick to AC,
  or free Cs 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.

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Explanation of the results
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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.

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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 cells
  regulation pathways.
• Can be extended to study the behaviour of protein
  levels under stress conditions.

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Thank you for your attention!
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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.

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More on Complex Networks

• Many other discoveries and models.
• Some networks show hierarchical structure.
• It was shown how to measure the networks 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.

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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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.

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More on Correlations

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

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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.

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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.

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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).

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Effectiveness

• We define the effectiveness of the production as
• This takes into account the obvious waste due to
  excess in one constituent.

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Properties of the effectiveness

• For fixed C0 (102), 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 gt C0, then were efficient
  if

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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.

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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.